Algebraic Geometry II: Cohomology of Schemes [1 ed.] 9783658430306, 9783658430313

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Springer Studium Mathematik – Master

Ulrich Görtz Torsten Wedhorn

Algebraic Geometry II: Cohomology of Schemes With Examples and Exercises

Springer Studium Mathematik – Master Reihe herausgegeben von Martin Aigner, Freie Universität Berlin, Berlin, Germany Heike Faßbender, Technische Universität Braunschweig, Braunschweig, Germany Barbara Gentz, Bielefeld, Germany Daniel Grieser, Institut für Mathematik, Carl von Ossietzky Universität, Oldenburg, Germany Peter Gritzmann, Zentrum Mathematik, Technische Universität München, Garching, Germany Jürg Kramer, Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany Volker Mehrmann, Institut für Mathematik, TU Berlin, Berlin, Germany Gisbert Wüstholz, ETH Zürich, Wermatswil, Switzerland

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Ulrich Görtz · Torsten Wedhorn

Algebraic Geometry II: Cohomology of Schemes With Examples and Exercises

Ulrich Görtz Fakultät für Mathematik Universität Duisburg-Essen Essen, Germany

Torsten Wedhorn Fachbereich Mathematik TU Darmstadt Darmstadt, Germany

ISSN 2509-9310 ISSN 2509-9329 (electronic) Springer Studium Mathematik – Master ISBN 978-3-658-43030-6 ISBN 978-3-658-43031-3 (eBook) https://doi.org/10.1007/978-3-658-43031-3 © Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Spektrum imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany Paper in this product is recyclable.

V

Contents 1

Introduction . . . .

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5 5 10 24 29

´etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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31 31 39 48 63

19 Local complete intersections The Koszul complex and completely intersecting immersions . . . . . . . . . Local complete intersection and syntomic morphisms . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 68 81 87

17 Differentials Differentials for rings and extensions of algebras . Differentials for sheaves on schemes . . . . . . . . The de Rham complex . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . ´ 18 Etale and smooth morphisms Formally unramified, formally smooth and Unramified and ´etale morphisms . . . . . Smooth morphisms . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

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formally . . . . . . . . . . . . . . .

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91 92 101 115 142

21 Cohomology of OX -modules Categories of abelian sheaves and of OX -modules . . . . . Cohomology and derived direct image . . . . . . . . . . . ˇ Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . Derived inverse image, Hom sheaves, and tensor products Relations between derived functors . . . . . . . . . . . . . Perfect and pseudo-coherent complexes . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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151 153 158 178 187 199 210 225

22 Cohomology of quasi-coherent modules ˇ Cohomology of quasi-coherent modules and Cech cohomology Derived categories of quasi-coherent modules . . . . . . . . . Finiteness properties of complexes on schemes . . . . . . . . . Projection formula, base change and the K¨ unneth formula . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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232 233 243 255 269 288

´tale topology 20 The e Henselian rings . . . . . . . . . The ´etale topology . . . . . . . The ´etale fundamental group of Exercises . . . . . . . . . . . .

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VI

Contents

23 Cohomology of projective and proper schemes Cohomology of projective schemes . . . . . . . . . . . . . . . . . . . . . . . Coherence of higher direct images for proper morphisms . . . . . . . . . . . Numerical intersection theory, Euler characteristic, and Hilbert polynomial The Grothendieck-Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . Cohomology and base change . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert polynomials and flattening stratification . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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297 298 306 314 332 342 361 367

24 Theorem on formal functions Derived Completion . . . . . . . . The theorem of formal functions . Stein factorization . . . . . . . . . Algebraization . . . . . . . . . . . Exercises . . . . . . . . . . . . . .

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376 376 390 398 411 431

25 Duality The right adjoint f × of Rf∗ . . . . Computation of f × in special cases The functor f ! . . . . . . . . . . . Dualizing complexes . . . . . . . . Dualizing sheaves . . . . . . . . . . Duality for schemes over fields . . Applications to algebraic surfaces . Exercises . . . . . . . . . . . . . .

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435 436 456 463 469 483 493 500 509

26 Curves Basic notions . . . . . . . . . . The Theorem of Riemann-Roch Special classes of curves . . . . Vector bundles on curves . . . Further topics . . . . . . . . . . Exercises . . . . . . . . . . . .

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513 513 534 547 571 584 598

27 Abelian schemes Preliminaries and general results about group schemes Definition and basic properties of abelian schemes . . The Picard functor . . . . . . . . . . . . . . . . . . . . Duality of abelian schemes . . . . . . . . . . . . . . . . Cohomology of line bundles on abelian schemes . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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603 604 635 643 658 692 734

F Homological Algebra Addenda to the language of categories . . . . Additive and abelian categories . . . . . . . . Complexes in additive and abelian categories Spectral sequences . . . . . . . . . . . . . . . Triangulated categories . . . . . . . . . . . . Sign conventions . . . . . . . . . . . . . . . .

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VII Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

781 791 818

G Commutative Algebra II On regular and Cohen-Macaulay rings . . . . . . . . . . . . . . . . . . . . . . On injective modules and Gorenstein rings . . . . . . . . . . . . . . . . . . . .

826 826 829

Bibliography

833

Detailed List of Contents

846

Index of Symbols

857

Index

860

Introduction Algebraic geometry is at the same time a very classical subject, and one that remains extraordinarily active to this day. There has been fascinating and spectacular progress in the last century, especially after Grothendieck had introduced his language of schemes which for instance allowed for a much easier treatment of families of varieties, and also immensely strengthened the connection between algebraic geometry and algebraic number theory. On the other hand, many intriguing questions remain open. Not surprisingly, the study of solutions of systems of polynomial equations, which is at the heart of algebraic geometry, is important in many mathematical subjects. In addition to the numerous applications of algebraic geometry in number theory and representation theory, say, there is also a close connection to mathematical physics, and over the last ten or twenty years advanced methods of algebraic geometry play an increasing role in solving problems arising in biology, chemistry, etc. Many of the results in modern algebraic geometry are deep in the sense that they rely on several layers of theory building upon each other. While it is particularly exciting when this large body of abstract machinery can be used to prove innocuous looking elementary statements, this also means that the subject is not easily accessible. The aim of this book, which is a sequel to our introduction [GWI] O to the theory of schemes, is to provide a systematic account of several key results and methods in algebraic geometry that were not discussed in [GWI] O , but are required knowledge for (almost) every advanced student and researcher in the field, at a level of generality that is suitable for using the theory in a more classical, geometric context, as well as in a more arithmetically oriented setting. The content of the book can be grouped into three parts: • Smooth morphisms (including the closely related notions of ´etale and unramified morphisms, and also locally complete intersections, and the beginnings of the theory of the ´etale topology and the ´etale fundamental group of a scheme), Chapters 17 – 20, • cohomology of OX -modules and of quasi-coherent OX -modules (including finiteness results for cohomology of projective and proper schemes, the theorem of formal functions and some of its applications such as the Stein factorization, Grothendieck duality, and a rather long appendix on homological algebra), Chapters 21 – 25 and Appendix F, and • as examples to illustrate how to put the technical machinery to use, and as an outlook on two more specific, fascinating topics in algebraic geometry, Chapter 26 on algebraic curves and Chapter 27 on abelian schemes. The material in the first chapters, which center around the notion of smoothness, is of great importance in basically all areas of algebraic geometry, and is to a large extent “classical” in the sense that it can be found in [EGAIV] O and [SGA1] O X . Some of the key results here are the infinitesimal lifting criterion for smoothness (and similarly for ´etale and for unramified morphisms), and the characterization of ´etale morphisms as flat and unramified morphisms. In Chapter 20 we explain the construction and basic properties of the ´etale fundamental group of a connected scheme. © Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_1

2

Introduction

The bulk of the book is concerned with the theory of cohomology of (quasi-)coherent sheaves. It was our intention to present the topic in a way that makes full use of the powerful machinery of derived categories, including results such as (a suitable version of) the Brown representability theorem, which is used in Chapter 25 to construct a right adjoint of the derived direct image functor. Beyond the classical results by Serre, Grothendieck, Verdier and others, this approach relies on work of Neeman, Lipman and others. To set up the notation and to collect references for the results from homological algebra we use, we have included Appendix F. Of course, other approaches, either more elementary or more advanced, are possible and have their merit. For instance using derived functors, but no derived categories, as ˇ in [Har3] O or in [Vak] X , or focusing on Cech cohomology as in [Liu] O . In comparison O to [Har1] , we work with the unbounded derived category wherever possible. We neither use ∞-categories, nor go into the more recent approach to Grothendieck duality using Clausen’s and Scholze’s theory of condensed mathematics. The chapters on cohomology are roughly organized as follows. Chapter 21 contains general results on OX -modules on a ringed space X, their derived category, derived functors and relations between them. Starting from Chapter 22, we specialize to the case of schemes, and (mostly) to categories of quasi-coherent sheaves. We prove vanishing of higher cohomology of quasi-coherent modules on affine schemes and compare the derived category of quasi-coherent sheaves on a scheme X and the subcategory Dqcoh (X) of the derived category of OX -modules consisting of complexes with quasi-coherent cohomology. In Chapter 22 we also prove some important general results such as the (derived) projection formula and the K¨ unneth isomorphism. The next three chapters contain some of the cornerstones of the theory. We start with finiteness theorems for cohomology of coherent sheaves on proper schemes in Chapter 23. Building on this, we discuss intersection numbers and give an outlook on the famous theorem of Grothendieck-Riemann-Roch; however, we do not introduce Chow groups but rather choose an axiomatic approach and use the theorem that K-theory is the “initial multiplicative cohomology theory” (which we state without proof). In the final parts of the chapter, we develop the theory of cohomology and base change and apply it to Hilbert polynomials and the flattening stratification. The topic of Chapter 24 is the theorem on formal functions which relates, given a closed subscheme Z of a proper scheme X, the completion of the cohomology of X (with respect to the ideal corresponding to Z) to the cohomology of the “completion of X along Z”, i.e., the limit of the cohomology groups of infinitesimal thickenings of Z. We base the proof of this fact on a detailed discussion of derived completion. The notion of formal scheme is not discussed in detail, however. As two important applications, we discuss the Stein factorization and Grothendieck’s algebraization theorem. Next, in Chapter 25, we study Grothendieck duality for (complexes of) quasi-coherent sheaves. We first discuss that the derived direct image functor has a right adjoint (under mild conditions) and compute it for smooth and proper morphisms. We go on to construct the twisted inverse image functor, a variant which is better behaved for non-proper morphisms. At the end of the chapter we apply results on cohomology to the theory of algebraic surfaces. The final two chapters serve to highlight some of the beautiful consequences of the theory that was developed before. In particular, we freely use all previous results. Nevertheless we have aimed at giving a somewhat coherent exposition of basics of the theory of algebraic curves over a field in Chapter 26, and of the foundations of the theory of abelian schemes in Chapter 27. Clearly, each of these topics could easily fill a whole (or several . . . ) books

3 of its own, so it was impossible to give a comprehensive account. Thus we just hope that the material presented here will help the reader to get started. Probably the single most important theorem in the theory of algebraic curves, and in Chapter 26, is the theorem of Riemann-Roch which relates the degree of a line bundle to its cohomology. At this point of the book, it is an easy corollary of facts on the Euler characteristic and of Serre duality. We prove some consequences such as the formula of Riemann and Hurwitz, discuss curves of genus 0, of genus 1, and hyperelliptic curves, show the existence of the Harder-Narasimhan filtration of vector bundles on curves, and prove the Weil conjectures for curves over finite fields. In Chapter 27 on abelian schemes we focused on studying abelian schemes over arbitrary base schemes. For instance we sketch Deligne’s proof of the existence of the dual abelian scheme At of an abelian scheme A over an arbitrary base scheme. To this end we briefly introduce (but do not discuss in full detail) the notion of algebraic space, and use the representability of the relative Picard functor by an algebraic space without proof. We also prove a few other foundational results of the theory, for instance the Fourier-Mukai equivalence for abelian schemes or that every abelian scheme over a normal noetherian base scheme is projective. Let us conclude this introduction by emphasizing that we had to make a choice what topics to present and how to present them. Our approach is certainly influenced by our mathematical background. There are many other excellent introductions to this beautiful and vast theory ([EGAInew] – [EGAIV] O , [Har3] O , [Vak] X , [Sta], [Liu] O , [Lu-DAG] – just to list a few), different in style and in choice of material. We engourage the reader to take advantage of this diversity of literature. Leitfaden The following diagram displays dependencies between the chapters. A double arrow indicates that the target chapter relies in an essential way on (some of) the results developed in the source chapter. A single arrow stands for a weaker dependency, in the sense that it might be possible to use some results of the source chapter as a black box in order to read the target chapter. The diagram is not meant to be complete in a formal sense; in particular, some references to “standard definitions outside Algebraic Geometry” given in previous chapters are not visible here. Ch. 17

Ch. 21 ks

 Ch. 18

 80 Ch. 22 !) Ch. 19

 y Ch. 20

 Ch. 23 u} +3 Ch. 24

% r Ch. 26

App. F

!) / Ch. 25 !) y Ch. 27

4

Introduction

Corrigenda and addenda Additions and corrections of the text will be posted on the web page https://www. algebraic-geometry.de/ of this book. We encourage all readers to send us remarks and to give us feedback. Acknowledgements Almost none of the results presented here are new. The EGA volumes [EGAInew], [EGAII] O , [EGAIII] O , [EGAIV] O written by Dieudonn´e and Grothendieck, and their modern successor, the Stacks project [Sta] launched and largely written by de Jong, were of great importance for us when learning the topics discussed in this book. In addition to individual references in many places, some of the chapters have a short section at the end with pointers to the literature. While we have devised many of the exercises ourselves, others are standard or folklore problems or exercises whose source is hard to track. For some of the more elaborate exercises, we gave references, but please notify us if there are places, in the exercises or elsewhere, where a reference would be appropriate but is currently missing. We thank everybody who helped us writing this book, in discussions or via concrete feedback or comments, in particular: Johannes Ansch¨ utz, Elmar Große-Kl¨onne, Hasan Hasan, Tim Holzschuh, Moritz Kerz, Alex K¨ uronya, Christopher Lang, Eike Lau, Jonas Lenz, Kin-Lok Li, Joseph Lipman, Martin L¨ udtke, Catrin Mair, Lucas Mann, Timo Richarz, Benjamin Rosswinkel, Alexander Schmidt, Jakob Stix, Georg Tamme, Burt Totaro, Can Yaylali, Heer Zhao. We would also like to use the occasion to make explicit here our thanks to Peng Du for his extensive list of remarks on the first volume, because his name is unfortunately missing in the list of acknowledgments in the preface to the second edition of that volume. Notation References to volume 1 are given by the number of the result in the second edition of that book, without the prefix [GWI] O . They are “clickable” if you put the pdf file of volume 1 in the same folder as this file, under the name GW1.pdf. We collect some general notation used throughout the book. By ⊆ we denote an inclusion with equality allowed, and by ⊊ we denote a proper inclusion; by ⊂ we denote an inclusion where we do not emphasize that equality must not hold, but where equality never occurs or would not make sense (e.g., m ⊂ A a maximal ideal in a ring). By Y c we denote the complement of a subset Y of some bigger set. By Y we denote the closure of some subspace Y of a topological space. By convention, the empty topological space is not connected. If R is a ring, then we denote by Mm×n (R) the additive group of (m × n)-matrices over R, and by GLn (R) the group of invertible (n × n)-matrices over R. The letters Z, Q, R, C denote the ring of integers and the fields of rational, real and complex numbers, respectively. By N we denote the set of natural numbers (including 0). Given a prime number p, we denote by Zp the ring of p-adic integers and by Qp its field of fractions. We denote (projective) limits by lim, and colimits, i.e., inductive limits, by colim, see also Section (F.3).

17

Differentials

Content – Differentials for rings and extensions of algebras – Differentials for schemes – The de Rham complex In differential geometry the tangent bundle TM of a manifold M and its dual T ∨ M play a central role. As both are vector bundles, one determines the other. Moreover, vector bundles over M are essentially the same as finite locally free modules over the structure sheaf of M , and the sections of the cotangent bundle over an open subset U form the module of differential forms Ω1 (U ). In the presence of singularities, such as for nonsmooth schemes over a field, it turns out that the module of differential forms is the more fundamental object. For a morphism X → S of schemes there is a suitable algebraic notion of differential forms, giving rise to a quasi-coherent OX -module Ω1X/S of differential 1-forms of X over S, called the sheaf of K¨ ahler differentials. Its dual will be called the tangent sheaf TX/S whose sections are the sections of the tangent bundle TX/S . If X → S is smooth, we will see in the next chapter that Ω1X/S is finite locally free (Corollary 18.58) and hence Ω1X/S and TX/S determine each other. But in general the biduality homomorphism Ω1X/S → (Ω1X/S )∨∨ is not an isomorphism and it is not possible to recover Ω1X/S from TX/S . Taking exterior powers of Ω1X/S , we obtain the sheaves of differential forms of higher order. Together they form the de Rham complex of X over S which we will introduce at the end of the chapter.

Differentials for rings and extensions of algebras We start by defining the module of differentials Ω1A/R for a homomorphism of rings R → A. It is an A-module which is characterized by the property that for every A-module M there is a functorial identification of HomA (Ω1A/R , M ) with the A-module of R-derivations of A with values in M . (17.1) Derivations and K¨ ahler differentials for rings. Let R be a ring, let A be an R-algebra, and let M be an A-module. Definition 17.1. An R-derivation of A with values in M is a map D : A → M such that (a) D is R-linear, (b) D(xy) = xD(y) + yD(x) for all x, y ∈ A (“Leibniz rule”). We denote by DerR (A, M ) the set of R-derivations from A to M .

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_2

6

17 Differentials

The sum of two R-derivations is again an R-derivation, and for D ∈ DerR (A, M ) and a ∈ A the map aD : x 7→ aD(x) is again an R-derivation. This endows DerR (A, M ) with the structure of an A-module. For M = A we write DerR (A) instead of DerR (A, A). Remark 17.2. (1) For D ∈ DerR (A, M ) the Leibniz rule implies D(1) = D(1 · 1) = D(1) + D(1) which shows D(λ · 1) = 0,

(17.1.1)

for all λ ∈ R.

(2) By induction the Leibniz rule implies D(xn ) = nxn−1 D(x),

(17.1.2)

for all x ∈ A, n ≥ 1.

(3) Finally, if D ∈ DerR (A), another easy induction argument shows that (17.1.3)

Dn (xy) =

n   X n i=0

i

Di (x)D n−i (y),

for all x, y ∈ A, n ≥ 1.

Example 17.3. Let R be a ring and let A = R[T1 , . . . , Tn ]. Then the partial derivative ∂/∂Ti : A → A is a derivation for all i. The next two remarks establish a connection between derivations and liftings of Ralgebra homomorphisms. Remark 17.4. Let A ψ

P

π

 /C

be homomorphisms of R-algebras such that π is surjective and I := Ker(π) is an ideal of square zero. Thus I = I/I 2 is also a C-module and hence, via ψ, an A-module. Let Lψ be the set of lifts of ψ, i.e., the set of R-algebra homomorphisms ψ˜ : A → P with π ◦ ψ˜ = ψ. We claim that (17.1.4)

DerR (A, I) × Lψ → Lψ ,

˜ 7→ D + ψ˜ (D, ψ)

defines a simply transitive action of the additive group DerR (A, I) on the set Lψ . Note that Lψ might be empty. We first check that D + ψ˜ ∈ Lψ : For m ∈ I and p ∈ P one has pm = π(p)m, depending whether we consider I as ideal in P or as a C-module. For D ∈ DerR (A, I) and ψ˜ ∈ Lψ one has ˜ ˜ ˜ ψ(b) ˜ (D + ψ)(a)(D + ψ)(b) = ψ(b)D(a) + ψ(a)D(b) + ψ(a) ˜ = ψ. for all a, b ∈ A. Here we use that I 2 = 0. Moreover, π ◦ (D + ψ) ˜ ˜ The action (17.1.4) is simply transitive: Indeed if ψ1 , ψ2 ∈ Lψ , then D := ψ˜1 − ψ˜2 is an R-linear map D : A → I such that for a, b ∈ A ψ(a)D(b) + ψ(b)D(a) = ψ˜1 (a)(ψ˜1 (b) − ψ˜2 (b)) + ψ˜2 (b)(ψ˜1 (a) − ψ˜2 (a)) = D(ab), hence D ∈ DerR (A, I).

7 Remark 17.5. We consider the following special case of Remark 17.4. Let ψ : A → C be an R-algebra and let N be a C-module. Define a C-algebra by (17.1.5)

DC (N ) := C ⊕ N,

(c, n) · (c′ , n′ ) := cc′ + (cn′ + c′ n).

Then π : DC (N ) → C, (c, n) 7→ c is a C-algebra homomorphism with kernel N of square zero. For N = C we obtain DC (C) ∼ = C[T ]/(T 2 ), the ring of dual numbers over C. The composition of ψ with the homomorphism of C-algebras C → DC (N ), c 7→ (c, 0) is an element of Lψ . Applying Remark 17.4 we obtain a bijection (17.1.6)

∼ DerR (A, N ) → { ψ˜ : A → DC (N ) R-algebra homomorphism ; π ◦ ψ˜ = ψ }, D 7→ (a 7→ ψ(a) + D(a)).

Applied to the special case C = A and ψ = idA we obtain ∼

(17.1.7)

DerR (A, N ) → { ψ : A → DA (N ) R-algebra homomorphism ; π ◦ ψ = idA }, D 7→ (a 7→ a + D(a)).

We will now show that there is an A-module Ω1A/R and an R-derivation d : A → ΩA/R such that for all A-modules the map HomA (Ω1A/R , M ) → DerR (A, M ),

(17.1.8)

u 7→ u ◦ d

is an isomorphism of A-modules. The bijection (17.1.8) is clearly functorial in M , thus Ω1A/R represents the covariant functor M 7→ DerR (A, M ). To construct Ω1A/R , let I be the kernel of ∆∗ : A ⊗R A → A, b1 ⊗ b2 7→ b1 b2 , and set 1 PA/R := (A ⊗R A)/I 2 ,

(17.1.9)

Ω1A/R := I/I 2 .

1 → A induced by ∆∗ . It Then Ω1R/A is the kernel of the R-algebra homomorphism π : PA/R 1 is an ideal of square zero in PA/R and so it is an A-module. With respect to this A-module structure an element a ∈ A acts by multiplication by a ⊗ 1 ∈ A ⊗R A (or equivalently by 1 ⊗ a or by any element α ∈ A ⊗R A such that ∆∗ (α) = a). The R-algebra homomorphisms 1 i1 , i2 : A → PA/R ,

i1 (a) := a ⊗ 1 mod I 2 ,

i2 (a) := 1 ⊗ a mod I 2

satisfy π ◦ i1 = π ◦ i2 = idA . Thus Remark 17.4 shows that (17.1.10)

d := dA/R : A → Ω1A/R ,

b 7→ i2 (a) − i1 (a)

is an R-derivation. Proposition 17.6. Let X ⊆ A be a subset that generates A as an R-algebra. Then the A-module Ω1A/R is generated by { d(x) ; x ∈ X }. Proof. It suffices to show that I is generated as A⊗R A-module by the elements of the form ′ ′ ′ ′ ′ for x ∈ X. But for 1⊗x−x⊗1 P P all a,′ a ∈ A one has a⊗a = aa ⊗1+(a⊗1)(1⊗a −a ⊗1). ′ For i ai ⊗ ai ∈ I one has i ai ai = 0 by definition and therefore

8

17 Differentials X

(ai ⊗ a′i ) =

i

X

(ai ⊗ 1)(1 ⊗ a′i − a′i ⊗ 1).

i

This proves that I is generated by 1 ⊗ a − a ⊗ 1 for a ∈ A. Moreover, if a = xy, then 1 ⊗ a − a ⊗ 1 = (1 ⊗ y)(1 ⊗ x − x ⊗ 1) + (x ⊗ 1)(1 ⊗ y − y ⊗ 1) and the claim follows by induction. Proposition 17.7. The map α : HomA (Ω1A/R , M ) → DerR (A, M ),

(17.1.11)

u 7→ u ◦ d

is an isomorphism of A-modules, functorial in M . Proof. It is clear that α is A-linear and functorial in M . The injectivity of α follows from Proposition 17.6. For the surjectivity let D ∈ DerR (A, M ). By (17.1.7), D corresponds to a homomorphism ψD : A → DA (N ) of R-algebras. We obtain a commutative diagram of R-linear maps with exact rows 0

/ P1

/ Ω1 A/R

A/R

/0

/A

/ 0.

a⊗a′ 7→aψD (a′ )

uD

0

/A

 / DA (M )

 /M

Then (uD ◦ dA/R )(a) = u(1 ⊗ a − a ⊗ 1) = ψD (a) − a = D(a) shows α(uD ) = D. Example 17.8. Let A = R[(Ti )i∈I ] be a polynomial algebra. Then Ω1A/R is a free A-module with basis (dTi )i∈I . Indeed, Ω1A/R is generated by (dTi )i∈I (Proposition 17.6). From the definition of derivations, it follows that for every polynomial f ∈ A one has df =

X ∂f dTi . ∂Ti i∈I

∂ ∈ DerR (A) corresponds via the bijection (17.1.11) to an A-linear map Moreover ∂T j 1 uj : ΩA/R → A such that uj (dTi ) = δij . This shows that (dTi )i∈I is also linearly independent.

(17.2) Extensions of algebras by modules. In this section, we introduce certain “extension groups”, which will later be useful when we investigate functoriality properties of derivations and differential forms. Definition 17.9. Let R be a ring, A an R-algebra, and M an A-module. (1) An R-extension of A by M is an exact sequence of R-modules j

π

0 → M −→ E −→ A → 0, where E is an R-algebra, π is a homomorphism of R-algebras, and one has (17.2.1)

j(π(e)m) = ej(m)

for all m ∈ M , e ∈ E.

9 (2) Two such extensions E and E ′ are called equivalent, if there exists an R-algebra homomorphism u : E → E ′ making the following diagram commutative =E 0

/M

u

>A

/0

 E′ The Five Lemma shows that u as in (2) is automatically an isomorphism of R-algebras. The set of equivalence classes of R-extensions of A by M is denoted by ExR (A, M ). j π We think of an extension 0 → M −→ E −→ A → 0 as a thickening E of A of order 1 with kernel M : Condition (17.2.1) implies that j(M ) is an ideal of E and j(m)j(m′ ) = j(π(j(m))m′ ) = 0 shows that j(M ) is an ideal of square zero in E. Remark 17.10. Let R be a ring, A an R-algebra, and M an A-module. (1) The algebra DA (M ) (17.1.5) is an extension of A by M when we define π(a, n) = a and j(m) = (0, m) for m ∈ M and a ∈ A. It is called the trivial extension. j π An extension 0 → M −→ E −→ A → 0 is equivalent to the trivial extension if and only if there exists an R-algebra homomorphism ι : A → E which is a section of π, i.e., such that π ◦ ι = idA . j π (2) Let 0 → M −→ E −→ A → 0 be an extension and let ψ : C → A be a homomorphism of R-algebras. Remark 17.4 shows that DerR (C, M ) acts simply transitively on the set of R-algebra homomorphisms ψ˜ : C → E with π ◦ ψ˜ = ψ. Remark 17.11. Let R be a ring, A an R-algebra, and M an A-module. (1) ExR (A, M ) is functorial in M : Let u : M → M ′ be a homomorphism of A-modules j π and let 0 → M −→ E −→ A → 0 be in ExR (A, M ). Then the pushout with u is an extension j′ π′ 0 → M ′ −→ E ⊕M M ′ −→ A → 0 in ExR (A, M ′ ). Here E ⊕M M ′ := (E ⊕ M ′ )/{ (j(m), −u(m)) ; m ∈ M } is the pushout in the category of R-modules, j ′ is given by m′ 7→ (0, m′ ) and π ′ is given by (e, m′ ) 7→ π(e). Q (2) Let M = i∈I Mi be a product of A-modules. Then the projections pi : M → Mi induce via functoriality a map Y ExR (A, M ) → ExR (A, Mi ). i∈I ji

π

i This mapQis bijective: An inverse map is given by sending (0 → Mi −→ Ei −→ A→ j π 0)i∈I in i∈I ExR (A, Mi ) to 0 → M −→ E −→ A → 0, where Y E := { (ei )i∈I ∈ Ei ; ∀ i, j ∈ I : πi (ei ) = πj (ej ) }

i∈I

and where j is given by

Q

ji and π is given by (ei ) 7→ πi (ei ).

10

17 Differentials

(3) Assertions (1) and (2) show that the addition and scalar multiplication on M yield the structure of an A-module on ExR (A, M ) and that M 7→ ExR (A, M ) is an A-linear functor from the category of A-modules to itself. The zero element in ExR (A, M ) is given by the trivial extension DA (M ). Proposition 17.12. Let φ : R → A, ψ : A → B be ring homomorphisms, and let M be a B-module. Then D7→D◦ψ

D7→D

(17.2.2)

0 → DerA (B, M ) −→ DerR (B, M ) −→ DerR (A, M ) E7→E× A ∂ E7→E −→ ExA (B, M ) −→ ExR (B, M ) −→B ExR (A, M ),

is an exact sequence of R-modules, where ∂ is given by (D : A → M ) 7→ (0 → M → E → B → 0) with E = DB (M ) with A-algebra structure A → B ⊕ M , a 7→ (ψ(a), D(a)). Proof. This is a straightforward calculation. If A is a quotient of R we have the following description of ExR (A, −). Proposition 17.13. Let R be a ring, I ⊆ R an ideal, A := R/I. Then for every A-module M there exists an isomorphism, functorial in M ∼

Φ : HomA (I/I 2 , M ) → ExR (A, M ). Proof. Consider the canonical exact sequence 0 → I/I 2 −→ R/I 2 −→ A → 0 which yields an element in ExR (A, I/I 2 ). For a homomorphism of A-modules w : I/I 2 → M we obtain by functoriality of ExR (A, −) an element Φ(w) ∈ ExR (A, M ). π Conversely, to 0 → M → E −→ A → 0 in ExR (A, M ) we attach the R-linear map u : I ,→ R → E. Then π ◦ u = 0 and we may consider u as an R-linear map I → M , which induces an A-linear map I/I 2 → M because M is an ideal of square zero in E. This defines an inverse to Φ.

Differentials for sheaves on schemes We now globalize the construction of the module of differentials to a morphism of schemes X → S. In fact, the construction in the affine case (17.1.9) suggests that instead of gluing local Ω1A/R ’s for open affine Spec A ⊆ X and Spec R ⊆ S (which is one possible definition) we may define Ω1X/S as the “conormal sheaf” of the diagonal X → X ×S X. Hence we start with a short section on the conormal sheaf of an arbitrary immersion.

11 (17.3) Conormal sheaf of an immersion. Let i : Y → X be an immersion of schemes. By definition (Definition 3.43) there exists an open subscheme U of X and a closed immersion i1 : Y → U such that i is the composition of i1 followed by the inclusion U ,→ X. Let I ⊆ OU be the quasi-coherent ideal defining i1 . For any integer n ≥ 0 the closed subscheme (n)

Yi

:= V (I n+1 )

of U is called the n-th infinitesimal neighborhood of Y in X. Moreover we call (17.3.1)

CY /X := Ci := I /I 2

the conormal sheaf of i1 . It is a quasi-coherent OU -module that is annihilated by I . Thus we will consider it as a quasi-coherent OY -module. One obtains a sequence of closed immersions of subschemes of X, (0)

Y = Yi

(1)

,→ Yi

(2)

,→ Yi

,→ · · ·

each closed immersion defined by a quasi-coherent ideal of square 0. The immersion (n−1) (n) Yi ,→ Yi is defined by the quasi-coherent ideal I n /I n+1 which we consider as a (1) quasi-coherent OY -module. In particular, Ci defines the closed immersion Y → Yi . (n) We have the following alternative description of Yi and of Ci which shows in particular that they do not depend on the choice of U . The homomorphism of sheaves of rings i# : i−1 OX → OY on Y is surjective. If J denotes its kernel, then J = i−1 1 I . Hence (n) −1 n+1 2 Yi is the ringed space (Y, i OX /J ) and Ci = J /J as OY -modules. (n)

Remark 17.14. The formation of Yi

and of Ci is functorial in the following sense. Let

Y′ (17.3.2)

g

i′

 X′

/Y i

f

 /X

be a commutative diagram of scheme morphisms such that i and i′ are immersions. Let i = j ◦ i1 , where i1 : Y → U is a closed immersion, U is an open subscheme of X, and j is the inclusion U ,→ X. Let j ′ : U ′ ,→ X ′ be an open subscheme such that U ′ ⊆ f −1 (U ) such that i′ = j ′ ◦ i′1 , where i′1 : Y ′ → U ′ is a closed immersion. Replacing X by U and X ′ by U ′ we may assume that i and i′ are closed immersions. Let I (resp. I ′ ) be the quasi-coherent ideal defining i (resp. i′ ). The commutativity of (17.3.2) implies that f ∗ (I n ) → OX ′ factors through (I ′ )n and thus yields for all n ≥ 1 a homomorphism of OX ′ -modules u(n) : f ∗ (I n ) → (I ′ )n . Hence f induces scheme morphisms (17.3.3) 1

(n)

(n)

g (n) : Y ′ i′ → Yi

It is denoted by NY /X in [EGAIV] O (16.1.1). But we will denote its dual, the normal bundle, by Ni , at least for quasi-regular immersions, see Definition 19.21.

12

17 Differentials

and u(1) induces a homomorphism of OY ′ -modules (17.3.4)

w := wi′ ,i : g ∗ (Ci ) → Ci′ .

Remark 17.15. Assume that the diagram (17.3.2) is cartesian. (1) Then u(n) is surjective. Indeed, as above one can assume that i and i′ are closed immersions. As the diagram is cartesian, I ′ = f −1 (I )OX ′ and u(n) is the surjective homomorphism (17.3.5)

f −1 (I n ) ⊗f −1 OX OX ′ → (I ′ )n .

In particular w : g ∗ (Ci ) → Ci′ is surjective. (2) If f is flat, then the description of u(n) in (17.3.5) shows that u(n) is an isomorphism for all n ≥ 1. In particular, w is an isomorphism. (3) Assume that there exist morphisms p : X → Y and p′ : X ′ → Y ′ such that p ◦ i = idY , p′ ◦ i′ = idY ′ and such that the following diagram is cartesian YO ′

/Y O

g

p′

X′

p

/ X.

f

Then p (resp. p′ ) makes OY (n) (resp. OY ′ (n) ) into a quasi-coherent OY -algebra (resp. OY ′ -algebra) endowed with augmentations defined by i (resp. i′ ). Thus we obtain a direct sum decomposition of OY -modules OY (n) = OY ⊕ I /I n+1 ; similarly for OY ′ (n) . Moreover, if i(n) (resp. (i′ )(n) ) denotes the immersion Y (n) → X (n) (resp. Y ′ → X), then the diagram Y′

(n)

g (n)

p′ ◦(i′ )(n)

 Y′

/ Y (n)  /Y

g

p◦i(n)

is cartesian. Hence g ∗ OY (n) → OY ′(n) is an isomorphism of OY ′ -algebras, restricting n+1 to an isomorphism g ∗ (I /I n+1 ) → I ′ /I ′ of OY ′ -modules. For n = 1 this shows that (17.3.4) is an isomorphism ∼

w : g ∗ (Ci ) → Ci′ . Remark 17.16. If i is an immersion locally of finite presentation (i.e., I is an OU -module of finite type (Proposition 10.35)), then Ci is a quasi-coherent OY -module of finite type. Proposition 17.17. Let i : Y → X be an immersion locally of finite presentation, and let y ∈ Y . Then the following assertions are equivalent. (i) There exists an open neighborhood V of y in Y such that i|V is an open immersion. (ii) One has (Ci )y = 0.

13 Proof. Clearly, (i) implies (ii). The question is local on X and we may assume that i is a closed immersion, X = Spec A is affine, Y = V (I), where I is a finitely generated ideal of g2 . As Ci is of finite type, its support is closed (Corollary 7.32). Hence A. Thus Ci = I/I (ii) implies that there exists an open neighborhood V of y in Y such that Ci|V = 0. So we may assume that I/I 2 = 0. Nakayama’s lemma then implies I = 0 because Iy ⊆ my for all y ∈ Y . (17.4) Derivations and K¨ ahler differentials for schemes. The following is the central definition of this chapter; the sheaf of K¨ahler differentials will also be of crucial importance in Chapter 18 where we study smooth morphisms in detail. Definition 17.18. Let f : X → S be a morphism of schemes and let ∆f : X → X ×S X be the diagonal. Its conormal sheaf Ω1f := Ω1X/S := C∆f is called the OX -module of K¨ ahler differentials (of X over S). Remark 17.19. Let f : X → S be a morphism of schemes. (1) If S = Spec R and X = Spec A are affine, then the construction of Ω1A/R (17.1.9) shows that Ω1X/S is the quasi-coherent OX -module attached to Ω1A/R . (2) Let j : U → X be a monomorphism of S-schemes. Then U

j

∆U/S

 U ×S U

/X ∆X/S

j×j

 / X ×S X

is a cartesian diagram. If j is in addition flat, then j ∗ (Ω1X/S ) = Ω1U/S by Remark 17.15. In particular, if U is an open subscheme of X, we obtain Ω1X/S |U = Ω1U/S . (3) It follows from (1) and (2) that Ω1X/S is a quasi-coherent OX -module for an arbitrary morphism X → S of schemes. (4) Assume that S = Spec R and X = Spec A are affine and let φ : R → A be the ring homomorphism corresponding to f . Let T ⊆ A be a multiplicative set. Then Spec(T −1 A) → Spec A is a flat monomorphism and hence (2) shows (17.4.1)

T −1 Ω1A/R = Ω1T −1 A/R .

If U ⊆ R is a multiplicative set such that f (U ) ⊆ T , then T −1 A ⊗R T −1 A = T −1 A ⊗U −1 R T −1 A which shows Ω1T −1 A/R = Ω1T −1 A/U −1 R and thus (17.4.2)

T −1 Ω1A/R = Ω1T −1 A/U −1 R .

In particular one obtains for an arbitrary scheme morphism f : X → S and for x ∈ X an isomorphism of OX,x -modules (17.4.3)

(Ω1X/S )x = Ω1OX,x /OS,f (x) .

14

17 Differentials

Remark 17.20. Let f : X → S be a monomorphism (e.g., if f is an immersion), then ∆f is an isomorphism (Exercise 9.1) and therefore Ω1X/S = 0. We have the following finiteness properties of the quasi-coherent OX -module Ω1X/S . Remark 17.21. If f : X → S is a morphism locally of finite type, then the first projection X ×S X → X is locally of finite type. Thus ∆f is locally of finite presentation (Proposition 10.35 (3)) and Ω1X/S is of finite type (Remark 17.16). Alternatively, this follows from Proposition 17.6. If S is in addition locally noetherian, then X is locally noetherian (Proposition 10.9) and hence Ω1X/S is coherent (Proposition 7.46). In Corollary 17.34 below we will see that Ω1X/S is of finite presentation, if f is locally of finite presentation. Remark 17.22. Let f : X → S be a morphism of schemes and let e : S → X be a section of f . Then e is an immersion by Example 9.12. Applying Remark 17.15 (3) to the cartesian diagram e /X S H V f

∆X/S

e

 X

pr1

 / X ×S X

(ef,idX )S

one obtains an isomorphism of OS -modules (17.4.4)

∼

e∗ (Ω1X/S ) → Ce .

Example 17.23. Let k be a field, X be a k-scheme and let x ∈ X(k) be a k-rational point, considered as a closed point of X. Let mx ⊂ OX,x be the maximal ideal such that OX,x /mx = k. Then (17.4.4) shows that x∗ (Ω1X/k ) = Cx = mx /m2x , in other words, the absolute tangent space as defined in Section (6.2) is the dual space of x∗ (Ω1X/k ) = Cx . If X is locally noetherian (e.g., if X is locally of finite type over k), then Cx is finite-dimensional. For affine schemes we saw in Proposition 17.7 that the module of differentials represents the functor of derivations. Globally, we have the results given by Proposition 17.27 and Corollary 17.28. Definition and Remark 17.24. Let f : X → S be a morphism of schemes, and let F be an OX -module. An S-derivation of X with values in F is an f −1 (OS )-linear homomorphism D : OX → F satisfying the Leibniz rule DU (ab) = aDU (b) + bDU (a),

U ⊆ X open,

a, b ∈ OX (U ).

The sum of two S-derivations and the product of an element in Γ(X, OX ) with a derivation are again derivations. This makes the set DerS (OX , F ) of all S-derivations of X with values in F a Γ(X, OX )-module. For D ∈ DerS (OX , F ) and U ⊆ X open, the restriction D|U is an S-derivation of U with values in F |U . We obtain an OX -module Der S (OX , F ) by setting

15 Γ(U, Der S (OX , F )) = DerS (OU , F |U ),

U ⊆ X open.

Remark 17.25. Let f : X → S be a morphism of schemes. Remark 17.19 shows that there exists a unique S-derivation d := dX/S : OX → Ω1X/S

(17.4.5)

such that for all open affine subschemes Spec R ⊆ S and U = Spec A ⊆ f −1 (Spec R) the derivation dU : A = Γ(U, OX ) → Ω1A/R = Γ(U, Ω1X/S ) is the derivation dA/R defined in (17.1.10). Remark 17.26. Let f : X → S be a morphism of schemes, and let F be an OX -module. Globalizing (17.1.5) we set DOX (F ) := OX ⊕ F , endowed with the structure of an OX -algebra as in (17.1.5). Assume now that F is quasi-coherent, then DOX (F ) is a quasi-coherent OX -algebra and we set (17.4.6)

DX (F ) := Spec(DOX (F ))

(see Section (11.2) for the definition of spectra of quasi-coherent OX -algebras). We obtain a contravariant functor F 7→ DX (F ) from the category of quasi-coherent OX -modules to the category of X-schemes that are affine over X. Let ε : X → DX (F ) be the morphism corresponding to the OX -algebra homomorphism DOX (F ) → OX , (a, n) 7→ a for a ∈ OX (U ), n ∈ F (U ) and U ⊆ X open. By restriction to the affine case, (17.1.7) yields a bijection, functorial in F , (17.4.7)

∼

DerS (OX , F ) → { f ∈ HomS (DX (F ), X) ; f ◦ ε = idX }

˜ for an A-module M , then the If S = Spec R and X = Spec A are affine and F = M description of derivations in terms of algebra homomorphisms in (17.4.7) shows that evaluation on global sections yields an isomorphism of A-modules (17.4.8)

∼

DerS (OX , F ) → DerR (A, M ).

Restricting to the affine case and using Remark 17.19 (1) and (17.4.8) shows that there is the following global version of Proposition 17.7. Proposition 17.27. Let f : X → S be a morphism of schemes, and let F be a quasicoherent OX -module. The map (17.4.9)

HomOX (Ω1X/S , F ) → DerS (OX , F ),

u 7→ u ◦ dX/S ,

is an isomorphism of Γ(X, OX )-modules, functorial in F . Corollary 17.28. The morphism (17.4.10)

Hom OX (Ω1X/S , F ) → Der S (OX , F ),

u 7→ u ◦ dX/S ,

is a functorial isomorphism of OX -modules for every quasi-coherent OX -module F . In fact, one can show that (17.4.9) and hence (17.4.10) are isomorphisms for an arbitrary OX -module F . We leave this as an exercise (Exercise 17.12).

16

17 Differentials

Example 17.29. Let S be a scheme and let E be a quasi-coherent OS -module. Let X := V(E ) = Spec(Sym(E )) be the corresponding quasi-coherent bundle (Definition 11.2) and let f : X → S be the structure homomorphism. Then there is an OX -linear isomorphism (17.4.11)

∼

f ∗ (a) 7→ dX/S (a),

f ∗ (E ) → Ω1X/S ,

where a is a local section of E ⊆ Sym(E ). Indeed, for every quasi-coherent OX -module F there are bijections, functorial in F , HomOX (Ω1X/S , F )

(17.4.9)

=

(17.4.7)

= = =

(11.3.3)

DerS (OX , F ) { f ∈ HomS (DX (F ), X) ; f ◦ ε = idX } { u ∈ HomOX (f ∗ E , DOX (F ) ; ε∗ ◦ u = idOX } HomOX (f ∗ E , F ),

n and this bijection is induced by (17.4.11). In particular Ω1An /S ∼ . This also follows by = OX S globalizing Example 17.8.

(17.5) Fundamental exact sequences for K¨ ahler differentials. Consider a commutative diagram of scheme morphisms X′ (17.5.1)

h

/X

g

 / S.

f′

 S′

f

We obtain a commutative diagram X′ (17.5.2)

∆X ′ /S ′



X ′ ×S ′ X ′

h

/X ∆X/S

 / X ×S X

and thus by Remark 17.14 a homomorphism of OX ′ -modules (17.5.3)

u := uX ′ /S ′ ,X/S : h∗ Ω1X/S → Ω1X ′ /S ′

In the affine case (S = Spec R, S ′ = Spec R′ , X = Spec A, X ′ = Spec A′ and ψ : A → A′ corresponding to h) (17.5.3) is given by (17.5.4)

A′ ⊗A Ω1A/R → Ω1A′ /R′ ,

a′ ⊗ d(a) 7→ a′ d(ψ(a)).

The homomorphism (17.5.3) satisfies an obvious transitivity property for a composition of commutative squares. Proposition 17.30. If the diagram (17.5.1) is cartesian, then the homomorphism h∗ Ω1X/S → Ω1X ′ /S ′ in (17.5.3) is an isomorphism.

17 Proof. This is Remark 17.15 (3) applied to i = ∆X/S , i′ = ∆X ′ /S ′ , and the cartesian diagram h /X XO ′ O p′

p

/ X ×S X,

X ′ ×S ′ X ′ where p and p′ are the first projections.

Now consider a commutative diagram of scheme morphisms /Y

f

X (17.5.5) g

S.

~

h

As special cases of (17.5.3) we obtain homomorphisms of OX -modules (17.5.6)

v := vf /S := uX/S,Y /S : f ∗ Ω1Y /S → Ω1X/S , w := wX/h := uX/Y,X/S : Ω1X/S → Ω1X/Y .

In the affine case, S = Spec R, X = Spec B, Y = Spec A, and f corresponding to φ : A → B, these maps are given by (17.5.7)

v : Ω1A/R ⊗A B → Ω1B/R , w:

Ω1B/R

→

d(a) ⊗ b 7→ bd(φ(a)),

Ω1B/A ,

d(b) 7→ d(b).

Proposition 17.31. The following sequence of OX -modules is exact. (17.5.8)

v

w

f ∗ Ω1Y /S −→ Ω1X/S −→ Ω1X/Y → 0.

Proof. This is a local question, hence we may assume that S = Spec R, Y = Spec A and X = Spec B are affine. We have to show that B ⊗A Ω1A/R → Ω1B/R → Ω1B/A → 0 is exact. But this sequence of B-modules is exact, if and only if it is exact after applying HomB (·, N ) for all B-modules N (this follows from the Yoneda Lemma, but can also easily be checked directly). Thus by Proposition 17.7 we have to show the exactness of the sequence D7→D◦ψ

D7→D

0 → DerA (B, M ) −−−−−−→ DerR (B, M ) −−−−−−→ DerR (A, M ), where ψ : A → B is the homomorphism corresponding to f . This follows from Proposition 17.12. Corollary 17.32. Let f : X → S, g : Y → S be morphisms of schemes, set Z := X ×S Y and let p : Z → X and q : Z → Y be the projections. Then vp/S and vq/S (17.5.6) yield an isomorphism of OZ -modules ∼

p∗ Ω1X/S ⊕ q ∗ Ω1Y /S → Ω1X×S Y /S .

18

17 Differentials

Proof. Proposition 17.31 yields an exact sequence vp/S

wZ/f

p∗ Ω1X/S −→ Ω1Z/S −→ Ω1Z/X → 0.

(*)

The composition of wZ/g : Ω1Z/S → Ω1Z/Y followed by the isomorphism Ω1Z/Y ∼ = p∗ Ω1X/S (Proposition 17.30) is a left inverse of vp/S . Hence vp/S is injective, and the sequence (*) splits. Using Ω1Z/X = q ∗ Ω1Y /S (again by Proposition 17.30) we obtain Ω1Z/S ∼ = p∗ Ω1X/S ⊕ ∗ 1 q ΩY /S . Let S be a scheme and let i : Y → X be a closed immersion of S-schemes defined by a quasi-coherent ideal I of OX . Then the restriction of dX/S : OX → Ω1X/S to I induces an OY -linear map d : Ci = I /I 2 → i∗ Ω1X/S .

(17.5.9)

Indeed, we may assume that S = Spec R, X = Spec A, Y = Spec A/I are affine. Then δ := dA/R|I : I → Ω1A/R /IΩ1A/R maps I 2 to 0: For a, b ∈ I one has δ(ab) = aδ(b) + bδ(a) = 0 (mod IΩ1A/R ). Moreover, δ is A-linear because for a ∈ A, b ∈ I one has δ(ab) = aδ(b) + bδ(a) = aδ(b). Proposition 17.33. The following sequence of OY -modules is exact. d

vi/S

Ci −→ i∗ Ω1X/S −→ Ω1Y /S → 0.

(17.5.10)

Proof. Again we may assume that S = Spec R, X = Spec A, Y = Spec B, B = A/I are affine as above. Let π : A → B the canonical projection. It suffices to show that (17.5.10) is exact after applying HomB ( · , N ) for all B-modules, i.e., we have to show that 0

/ HomB (Ω1 , N ) B/R

/ HomB (B ⊗A Ω1 , N ) A/R

/ HomB (B ⊗A I, N )

DerR (B, N )

DerR (A, N )

HomA (I, N )

ist exact. Here DerR (B, N ) → DerR (A, N ) is given by D 7→ D ◦ π, and DerR (A, N ) → HomA (I, N ) is given by D′ 7→ D′ |I . This follows from Proposition 17.12 using Proposition 17.13 and DerA (B, N ) = 0. Corollary 17.34. Let f : X → S be locally of finite presentation. Then Ω1X/S is an OX -module of finite presentation. Proof. The question is local, thus we may assume that S = Spec R and X = Spec B with B = R[T1 , . . . , Tn ]/I, where I is a finitely generated ideal. Then (17.5.10) yields an exact sequence I/I 2 → B ⊗R[T1 ,...,Tn ] Ω1R[T1 ,...,Tn ]/R → Ω1B/R → 0. By Example 17.8 the middle term is a free B-module of rank n and I/I 2 is finitely generated by hypothesis. Thus Ω1B/R is of finite presentation.

19 Remark 17.35. Let R be a ring, and let A be an R-algebra. By choosing generators of A as R-algebra we may write A = R[(Ti )i∈I ]/(fj )j∈J . As Ω1R[(Ti )i∈I ]/R is freely generated by (dTi )i∈I , Proposition 17.33 shows that ! M 1 ∼ Ω A dTi /N, = A/R

i∈I

where N is generated by ( dfj =

X ∂fj i∈I

∂Ti

) dTi ; j ∈ J

Example 17.36. Let R be a ring. (1) Let A = R[T ]/(f ) for some f ∈ R[T ]. Then Ω1A/R ∼ = R[T ]/(f, f ′ ), where f ′ is the formal derivative of f . (2) Consider A = R[T1 , T2 ]/(T1 T2 ) (so Spec A is the “union of the coordinate axes in A2R ”). Then dT1 and dT2 are generators of Ω1A/R as an A-module with the single relation T1 dT2 + T2 dT1 = 0. Let N ⊆ Ω1A/R be the submodule generated by T1 dT2 = −T2 dT1 . Then T1 N = T2 N = 0, and N is a free R-module of rank 1. In Ω1A/R /N one has T1 dT2 = T2 dT1 = 0 and we obtain an exact sequence 0 → N → Ω1A/R → Ω1R[T1 ]/R ⊕ Ω1R[T2 ]/R → 0. Thus Ω1A/R is an extension of Ω1R[T1 ]/R ⊕ Ω1R[T2 ]/R by a torsion module. Remark 17.37. Let R be a ring, let A be an R-algebra, let B = A/I for an ideal I ⊆ A and let π : A → B be the canonical projection. Combining the map d (17.5.9), the boundary map in the exact sequence (17.2.2), and Proposition 17.13 one obtains for all B-modules N a diagram DerR (A, N ) ∼ =

 HomA (Ω1A/R , N )

∂

/ ExA (B, N ) ∼ =

 (−)◦d / HomB (I/I 2 , N ),

where the vertical maps are isomorphisms. This diagram is commutative. In fact the definitions show that both compositions send an R-derivation D : A → N to the B-linear map I/I 2 → N induced by the A-linear map I → N that sends a ∈ I to D(a). (17.6) Tangent bundles. As mentioned in the introduction, we define the tangent bundle of an S-scheme X as the dual of the sheaf of differentials of X over S. See Remark 17.44 below for the connection with the notion of (relative) tangent space defined in Section (6.6). Definition 17.38. Let g : X → S be a morphism of schemes. We call the OX -module (17.6.1)

Tg := TX/S := Hom OX (Ω1X/S , OX )

(17.4.10)

=

Der S (OX , OX )

20

17 Differentials

the tangent sheaf of g or of X over S. Remark 17.39. If g is locally of finite presentation, Ω1X/S is of finite presentation and hence TX/S is a quasi-coherent OX -module by Proposition 7.29. Remark 17.40. Let g : X → S be a morphism of schemes. For D1 , D2 ∈ DerS (OX , OX ) the bracket [D1 , D2 ] := D1 ◦ D2 − D2 ◦ D1

(17.6.2)

is again an S-derivation of OX . It is easy to check that [·, ·] is Γ(S, OS )-bilinear (but not Γ(X, OX )-bilinear in general) and that it satisfies the Jacobi identity [D1 , [D2 , D3 ]] + [D2 , [D3 , D1 ]] + [D3 , [D1 , D2 ]] = 0. Therefore DerS (OX , OX ) is a Γ(S, OS )-Lie algebra (see also Exercise 17.1). This structure is compatible with restrictions to open subsets. Thus TX/S obtains the structure of an g −1 (OS )-Lie algebra. Note that the tangent sheaf is in general not compatible with base change: For a cartesian diagram p1 /X X ×S T p2

 T

 /S

one obtains a canonical map p∗1 TX/S −→ TX×S T /T

(17.6.3) as the composition

∼

p∗1 TX/S = p∗1 ((Ω1X/S )∨ ) −→ (p∗1 Ω1X/S )∨ −→ (Ω1X×S T /T )∨ = TX×S T /T , where the second isomorphism is induced by the inverse of the isomorphism in Proposition 17.30. It is in general not an isomorphism because forming the dual is not compatible with pullback. If Ω1X/S is locally free of finite type (for instance if X → S is smooth, see Corollary 18.58 below), then (17.6.3) is an isomorphism (Proposition 7.7). The notion of the tangent bundle is better behaved. To define it, recall that we attached in Section (11.3) to every quasi-coherent OX -module E an X-scheme V(E ) = Spec(Sym(E )) such that the sections of V(E ) over some open U ⊆ X are identified with Γ(U, E ∨ ). If π : V(E ) → X is the structure morphism, π is affine and π∗ OV(E ) ∼ = Sym(E ) is a graded quasi-coherent OX -algebra. Definition 17.41. Let g : X → S be a morphism of schemes. We call Tg := TX/S := V(Ω1X/S ) the tangent bundle of g or of X over S. The tangent bundle of a morphism g is of finite type (resp. of finite presentation), if g is locally of finite type (resp. locally of finite presentation). As recalled above, for every open subscheme U of X there is an identification

21 HomX (U, TX/S ) = Γ(U, TX/S ). Remark 17.42. The tangent bundle is functorial in the following sense. Let f : X → Y be a morphism of S-schemes. Then the homomorphism f ∗ Ω1Y /S → Ω1X/S (17.5.6) yields by functoriality of V( ) a morphism of X-schemes TX/S → V(f ∗ Ω1Y /S ) = TY /S ×Y X, or equivalently a morphism Tf : TX/S → TY /S making the diagram TX/S

Tf

/ TY /S

(17.6.4)  X

 /Y

f

commutative. The tangent bundle has also the following description. For every scheme Y we set Y [ε] := DY (OY ) = Spec OY [T ]/(T 2 ). We consider Y [ε] as Y -scheme and denote by ιY : Y → Y [ε] the section corresponding to T 7→ 0. Proposition 17.43. Let f : X → S be a morphism of schemes. Then for all S-schemes Y there is a commutative diagram, functorial in Y , HomS (Y [ε], X) o

/ HomS (Y, TX/S )

τY

XS (ιY )

( v HomS (Y, X),

such that the horizontal map is bijective. Proof. We have to construct an isomorphism τ of functors over hX : Y 7→ HomS (Y, X) which is the same as to construct an isomorphism of functors on the category of X-schemes. Thus we have to show that for all X-schemes g : Y → X there is a bijection, functorial in Y, (17.6.5)

{ g˜ ∈ HomS (Y [ε], X) ; g˜ ◦ ιY = g } ↔ HomX (Y, TX/S ).

But by restriction to the affine case (17.1.6) (for N = C) implies that the left hand side can be identified functorially with DerS (OX , g∗ OY ) = HomOX (Ω1X/S , g∗ OY ) = HomOY (g ∗ Ω1X/S , OY ) = HomX (Y, TX/S ). The proposition shows in particular that TX/S represents the functor (Sch/S)

opp

→ (Sets),

Y 7→ HomS (Y [ε], X).

Remark 17.44. In Section (6.6) we defined the relative tangent space Tξ (X/S) of X → S at a K-valued point ξ : Spec K → X, where K is field. It was defined as the left hand side of (17.6.5) with Y = Spec K, and we obtain Tξ (X/S) = HomX (Spec K, TX/S ) = HomK (ξ ∗ Ω1X/S , K).

22

17 Differentials

In other words, the relative tangent spaces are the K-valued points of the tangent bundle. If X → S is locally of finite type, this is a finite-dimensional K-vector space (because Ω1X/S is then of finite type). More generally, we can define for every morphism of schemes x : Z → X the tangent space of X over S at x as the OZ -module Tx (X/S) := Hom OZ (x∗ Ω1X/S , OZ ).

(17.6.6) Then we have

TX/S = TidX (X/S)

(17.6.7) and (17.6.8)

Γ(Z, Tx (X/S)) = HomOZ (x∗ Ω1X/S , OZ ) = HomX (Z, TX/S ).

Let Y → S be another S-scheme and let f : X → Y be a morphism of S-schemes. Then the homomorphism f ∗ Ω1Y /S → Ω1X/S (17.5.6) induces by functoriality an OZ -linear map (17.6.9)

Tx (X/S) −→ Tf ◦x (Y /S).

Given S schemes X and Y with Z-valued points x : Z → X and y : Z → Y , Corollary 17.32 shows that one has an isomorphism (17.6.10)

∼

T(x,y) (X ×S Y /S) −→ Tx (X/S) × Ty (Y /S)

of OZ -modules. Remark 17.45. Let k be a field, let X and Y be k-schemes locally of finite type, and let f : X → Y be a proper k-morphism. By the functorial description of TX/k , Proposition 12.94 may be reformulated as follows. The morphism f is a closed immersion if and only if there exists an algebraically closed extension K of k such that Tf (K) : TX/k (K) → TY /k (K) is injective. (17.7) Differentials of Grassmannians and of projective bundles. We now determine the sheaf of differentials of Grassmannians and thus in particular of projective bundles. Let S be a scheme, let E be a quasi-coherent OS -module, and let e ≥ 1 be an integer. Recall that X := Grasse (E ) is the S-scheme parametrizing submodules U of E such that E /U is locally free of rank e, see Section (8.6). Let π : X → S be the structure morphism. For e = 1, X is the projective bundle P(E ). In particular, if E = (OSn+1 )∨ , then P(E ) = PnS . Recall that by definition of X = Grasse (E ) there is a universal surjection of OX -modules (17.7.1)

u : π ∗ (E ) → Q → 0,

where Q is a locally free OX -module of rank e, such that for every S-scheme f : T → S the map (17.7.2) is a bijection.

HomS (T, X) → { U ⊆ f ∗ E ; f ∗ E /U locally free of rank e}, h 7→ ker(h∗ (u) : h∗ (E ) → h∗ (Q))

23 Theorem 17.46. Let S be a scheme, let E be a finite locally free OS -module and set X = Grasse (E ). Define K := ker(u) so that there is an exact sequence 0 → K → π ∗ E → Q → 0 of finite locally free OX -modules. Then Ω1X/S ∼ = Hom OX (Q, K ) Proof. We will show that the duals of these two OS -modules are isomorphic. This suffices because both OX -modules are finite locally free: This is clear for Hom OX (Q, K ) because Q is locally free of rank e by definition and K is finite locally free because E and Q are. Its dual is isomorphic to (Q ∨ ⊗ K )∨ ∼ =K ∨⊗Q ∼ = Hom OX (K , Q). To check that Ω1X/S is finite locally free2 , we can work locally on X and in particular on S. Hence we ∼ Ae(n−e) may assume that E = OSn . Then X has an open covering (Ui )i such that Ui = S 1 for all i (Corollary 8.15). Therefore (ΩX/S )|Ui is a free OUi -module of rank e(n − e) (Example 17.29). Let f : Spec R → S be a morphism with affine domain and let h : Spec R → X be an Smorphism. Then h corresponds via (17.7.2) to an R-submodule N of M := Γ(Spec R, f ∗ E ) such that M/N is projective of rank e. Verbatim the same argument as in the calculation of the tangent space of the Grassmannian (Section (8.9)) shows that HomX (Spec R[ε], X) = HomR (N, M/N ), functorially in R. Therefore we obtain for Spec R = U ⊆ X open affine: Γ(U, (Ω1X/S )∨ ) = HomX (U, V(Ω1X/S )) = HomX (Spec R[ε], X) = HomR (N, M/N ) = HomOU (K |U , Q |U ) = Γ(U, Hom OX (K , Q)), where the first equality is (11.3.3) and the second equality holds by Proposition 17.43 For e = 1 one has Grass1 (E ) = P(E ) = Proj(Sym(E )) =: P and Q = OP (1) (Theorem 13.32). Hence we obtain the following description of the sheaf of differentials for projective bundles. Corollary 17.47. Let E be a finite locally free OS -module, P = P(E ), π : P → S its structure morphism. Let K be the kernel of the universal quotient π ∗ E → OP (1). Then Ω1P/S ∼ = Hom OP (OP (1), K ) = K (−1). In particular there is an exact sequence of locally free OP -modules (17.7.3)

0 → Ω1P/S → π ∗ (E )(−1) → OP → 0,

called the Euler sequence. Example 17.48. For P = P1S the universal quotient (17.7.1) is given by OP2 → OP (1) → 0 and its kernel is OP (−1). Therefore Ω1P/S ∼ = OP (−2). 2

This also follows from Corollary 18.58 below since X → S is smooth as the argument here shows.

24

17 Differentials

The de Rham complex Our next goal is the definition of the de Rham complex. It is the exterior algebra of Ω 1X/S together with an extension of dX/S which makes it into an “strictly graded commutative differential graded algebra”. We start by recalling these notions.

(17.8) The exterior algebra. Let R be a ring (commutative as usual). L Definition 17.49. Let L = p∈Z Lp be a Z-graded not necessarily commutative R-algebra. Then L is called graded commutative if for all homogeneous elements x, y ∈ L one has xy = (−1)deg(x) deg(y) yx. It is called strictly graded commutative if in addition x2 = 0 for x ∈ L homogeneous of odd degree. Note that if 2 is not a zero-divisor in L, then every graded commutative graded algebra is automatically strictly graded commutative. is the exteriorValgebra of an R-module M . It is defined as V An importantVexample p p (M ) is generated by elements of the from R M := ⊕p≥0 R (M ). The R-module m1 ∧ · · · ∧ mp for m1 , . . . , mp ∈ M and the multiplication is given by

Vp

Vq

Vp+q

(M ) × (M ) −→ (M ), ′ ′ (m1 ∧ · · · ∧ mp ) ∧ (m1 ∧ · · · ∧ mq ) := m1 ∧ · · · ∧ mp ∧ m′1 ∧ · · · ∧ m′q . This is well defined and defines byVbilinear extension the structure of a strictly graded commutative graded R-algebra V on R M . We obtain a functor M 7→ R M from the category of R-modules to the category of strictly graded commutative graded R-algebras. By [BouAI] O III, §7, 1, Rem. (1) we have the following adjointness property. V Proposition 17.50. The functor M 7→ R M is left adjoint to the functor that sends a strictly graded commutative graded R-algebra E to the R-module E1 . In other words, given any strictly graded commutative graded R-algebra E and any u : M → E1 of R-modules, there is a unique extension of u to a homohomomorphism V morphism R M → E of graded R-algebras. All these notions have obvious globalizations. Let (X, OX ) be a ringed space. It is clear how to define the notion of a (strictly) graded commutative graded OX -algebra. If F is an OX -module we define the exterior algebra M Vp V V F := OX F := OX F . p≥0

V Vp The formation of F is clearly functorial in F . Moreover, the properties of OX (F ) (see Section (7.20)) immediately imply the following properties of the exterior algebra.

25 Remark 17.51. Let X be a scheme and let F be quasi-coherent. V (1) F is a quasi-coherent OX -algebra. V ∼ For every morphism f : X ′ → X of schemes there is an isomorphism f ∗ ( F ) → (2) V ∗ (f F ) of OX ′ -algebras, functorial in F . ˜ for an A-module M , then V F is the quasi-coherent (3) If X = Spec A and F = M V OX -algebra corresponding to the A-algebra A M . (17.9) Differential graded algebras. The notion of differential graded algebra which we will define below combines the notions of graded algebra and of a derivation, giving the algebra the structure of a complex. Definition 17.52. Let R be a ring. (1) Let L be a graded R-algebra (not necessarily commutative). A graded R-derivation of L is an R-linear map d : L → L homogeneous of degree 1 such that for all x ∈ Lp , y ∈ L one has the graded Leibniz rule d(xy) = (dx)y + (−1)p xdy.

(17.9.1)

(2) A differential graded R-algebra is a graded algebra L together with a graded Rderivation d of L such that d ◦ d = 0. (3) A differential graded R-algebra L is called (strictly) graded commutative if the underlying graded algebra is (strictly) graded commutative. Sometimes a graded R-derivation is instead defined to be of degree −1. We will always stick with the degree 1 version as defined above. In the literature, the term differential graded algebra is often abbreviated as dga. Again we have the obvious globalizations. Let f : (X, OX ) → (S, OS ) be a morphism of ringed spaces and let L be a graded OX -algebra. An f −1 (OS )-linear map d : L → L is called a graded OS -derivation of L , if it is homogeneous of degree 1 and if (17.9.1) holds for all local sections x of Lp and y of L . A pair (L , d) consisting of a graded OX -algebra and a graded OS -derivation of L is called differential graded OS -algebra over X if d ◦ d = 0. Such a differential graded OS -algebra over X is called (strictly) graded commutative if xy = (−1)deg(x) deg(y) yx holds for all local homogeneous sections x and y (and if x2 = 0 for all local homogeneous sections x of odd degree). (17.10) The de Rham complex. We now come to the definition of the de Rham complex. Let X → S be a morphism of schemes, p ≥ 0 an integer. The p-th exterior power (Section (7.20)) Ωpf := ΩpX/S :=

Vp

1 OX ΩX/S

is called the sheaf of p-differential forms for f or of X over S. In particular Ω0X/S = OX . Moreover, for an integer p < 0 we set ΩpX/S := 0. We also denote by M p V Ω•X/S := OX Ω1X/S = ΩX/S p∈Z

Ω1X/S .

the exterior algebra of To define the differentials of the de Rham complex we will use the following result which follows from [BouAI] O III, §10.9, Prop. 14.

26

17 Differentials

Lemma 17.53. Let RVbe a ring and let A be a commutative R-algebra, let M be an A-module, and let L := A (M ) be its exterior algebra. Let d0 : A → M be an R-derivation V2 and let d1 : M → (M ) be an R-linear map such that for all a ∈ A and m ∈ M we have (17.10.1)

d1 (am) = ad1 (m) + d0 (a) ∧ m.

Then there exists a unique R-derivation d : L → L of degree 1 such that d|L0 = d0 and d|L1 = d1 . Proposition 17.54. Let f : X → S be a morphism of schemes. There exists a unique graded S-derivation d : Ω•X/S → Ω•X/S of degree 1 such that (a) d ◦ d = 0. (b) For f ∈ Γ(U, OX ), U ⊆ X open, one has d(f ) = dX/S (f ) ∈ Γ(U, Ω1X/S ). For local sections b, a1 , . . . , ap of OX , the differential is given by (17.10.2)

d : ΩpX/S −→ Ωp+1 X/S ,

d(bda1 ∧ · · · ∧ dap ) = db ∧ da1 ∧ · · · ∧ dap .

Proof. The question is local on X, thus we may assume that S = Spec R and X = Spec A are affine. To show the uniqueness of d we remark that d ◦ d = 0 shows that for all b, a1 , . . . , ap ∈ A one has d(bda1 ∧ · · · ∧ dap ) = db ∧ da1 ∧ · · · ∧ dap . As the A-module ΩpA/R is generated by elements of the form da1 ∧ · · · ∧ dap , this proves the uniqueness of d. To show the existence we set d0 := dA/R : A → Ω1A/R . We will construct an R-linear map d1 : Ω1A/R → Ω2A/R such that d1 ◦ d0 = 0 and such that (17.10.3)

d1 (aω) = d0 (a) ∧ ω + ad1 (ω)

for all a ∈ A, ω ∈ Ω1A/R . Then Lemma 17.53 ensures the existence of d and we have d ◦ d = 0 because d1 ◦ d0 = 0 and Ω•A/R is generated as A-algebra by elements of the form d0 (a) for a ∈ A. Let I be the kernel of the multiplication m : A ⊗R A → A. Then Ω1A/R = I/I 2 by definition. Consider the R-linear homomorphism u : A ⊗R A → Ω2A/R ,

u(a ⊗ b) := d0 (b) ∧ d0 (a).

Let a ∈ A and τ ∈ A ⊗R A. Writing τ as sum of elementary tensors, one easily sees that (17.10.4)

u((a ⊗ 1 − 1 ⊗ a)τ ) = d0 (m(τ )) ∧ d0 (a).

As I is generated by elements of the form a⊗1−1⊗a for a ∈ A (proof of Proposition 17.6), this shows that u(I 2 ) = 0. Thus u defines by restriction to I an R-linear map d1 : I/I 2 → Ω2A/R . For τ = b ⊗ 1 with b ∈ A, (17.10.4) shows d1 (bd0 (a)) = d0 (b) ∧ d0 (a). For b = 1 this shows d1 ◦ d0 = 0. For c ∈ A and ω = bd0 (a) it shows d1 (cω) = d0 (cb) ∧ d0 (a) = d0 (c) ∧ ω + cd1 (ω), As Ω1A/R is generated as R-module by elements of the form bd0 (a), this implies (17.10.3).

27 Definition 17.55. Let f : X → S be a morphism of schemes. The complex (Ω•X/S , d) is called the de Rham complex of X over S or of f . Sections of ΩiX/S are called differential forms of degree i. If S = Spec R and X = Spec A are affine, taking global sections of (Ω•X/S , d) one obtains a complex (Ω•A/R , d) of A-modules with R-linear differential of degree 1. The de Rham complex has the following universal property. Proposition 17.56. Let f : X → S be a morphism of schemes. Then the de Rham complex (Ω•X/S , d) of f is an initial object in the category of strictly graded commutative differential graded quasi-coherent OS -algebras over X. In other words, given any strictly graded commutative differential graded quasi-coherent OS -algebra (E • , d) over X, there is a unique homomorphism (Ω•X/S , d) → (E • , d) of differential graded OS -algebras over X. Proof. We may assume that S = Spec R and X = Spec A. Then we have to show that (Ω•A/R , d) is the initial object in the category of strictly graded commutative graded A-algebras with an R-linear differential d of degree 1 with d ◦ d = 0 satisfying the graded Leibniz rule (17.9.1). Let (E, d) such an object. The structure of E as a graded A-algebra yields a map A → E 0 of commutative rings whose composition with d : E 0 → E 1 is an R-linear derivation A → E 1 . By the universal property of Ω1A/R , this derivation corresponds to an A-linear map Ω1A/R → E 1 . By the universal property of the exterior algebra (Proposition 17.50), this linear map corresponds to a map of graded algebras φ : Ω•A/R → E. It remains to show that φ preserves the differentials. By construction we have d(φ(a)) = φ(d(a)) for a ∈ A, where on left hand side we consider a as element of E 0 via the ring map A → E 0 . But as an R-module ΩpA/R is generated by elements of the form x = bda1 ∧ · · · ∧ dap for b, a1 , . . . , ap ∈ A and we have φ(dx) = φ(db ∧ da1 ∧ · · · ∧ dap ) = φ(db)φ(da1 ) · · · φ(dap ) = d(b)d(φ(a1 )) · · · d(φ(ap )), d(φ(x)) = d(bφ(da1 ) · · · φ(dap )) = d(bd(φ(a1 )) · · · d(φ(ap ))) = d(b)d(φ(a1 )) · · · d(φ(ap )). Example 17.57. Let R be a ring and let A = R[T1 , . . . , Tn ]. Then Ω1A/R is a free Amodule with basis (dT1 , . . . , dTn ) (Example 17.8). For every subset I = {i1 , . . . , ip } of {1, . . . , n} with i1 < · · · < ip we set dTI := dTi1 ∧ · · · ∧ dTip . Then if I runs through the subsets of {1, . . . , n} with cardinality p, the dTI form a basis of ΩpA/R . For all f ∈ A one has by (17.10.2) d(f dTI ) = df ∧ dTI =

X i∈I /

(−1)n(I,i)

∂f dTI∪{i} , ∂Ti

where n(I, i) denotes the number of elements j ∈ I with j < i. One can show (Exercise 19.7) that Ω•A/R is exact in degrees > 0 if R is a Q-algebra and deduce that Ω•X/S is exact in degrees > 0 if S is a Q-scheme and X → S is a smooth morphism (Exercise 20.28).

28

17 Differentials

In Corollary 17.47 we have seen that Ω1Pn /S is a locally free module of rank n. Hence S is locally free of rank 1. For its calculation we will use the following general remark (see also Exercise 7.29). ΩnPn /S S

Definition and Remark 17.58. Let X be a ringed space. If E is a finite locally free OX -module, we denote by rk(E ) its rank, which is in general a locally constant function X → Z≥0 . Moreover, we call det(E ) :=

Vrk(E )

(E )

its determinant, which is a line bundle. By this we mean that if Xn ⊆ X is the open and closed subscheme of X where the rank of E V is equal to n ∈ Z≥0 , then det(E ) is the n E |Xn . unique line bundle on X such that det(E )|Xn = For any short exact sequence 0 −→ E ′ −→ E −→ E ′′ −→ 0 of vector bundles on X there exists a unique isomorphism ∼

δ : det(E ′ ) ⊗ det(E ′′ ) −→ det(E )

(17.10.5)

that makes the following diagram commutative

Vr′

E′ ⊗

Vr′′

E η

θ

Vr′

E′ ⊗

Vr′′

w E ′′

δ

/

&

Vr′ +r′′

E,

where r′ := rk(E ′ ), r′′ := rk(E ′′ ), and where θ is induced by functoriality from E → E ′′ ′ and where V η is induced by functoriality E → E and by the multiplication in the exterior algebra E . Indeed, as θ is surjective, the uniqueness is clear and we can work locally on X to see its existence. Therefore we may assume that E ′ and E ′′ are free of constant rank and that we find a basis e′′r′ +1 , . . . , e′′r′ +r′′ ∈ Γ(X, E ′′ ) that can be lifted to global sections er′ +1 , . . . , er′ +r′′ of E . Denote by e1 , . . . , er′ images of a basis e′1 , . . . , e′r′ of E ′ . Then E is a free, and e1 , . . . , er′ +r′′ ∈ Γ(X, E ) is a basis. Therefore, we can define δ by sending the single basis element (e′1 ∧ · · · ∧ e′r′ ) ⊗ (e′′r′ +1 ∧ · · · ∧ e′′r′ +r′′ ) of det(E ′ ) ⊗ det(E ′′ ) to the basis element e1 ∧ · · · ∧ er′ +r′′ . Example 17.59. Let S be a scheme, let E be a locally free OS -module of constant rank n + 1, and let P = P(E ) be the corresponding projective bundle over S. Denote by π : P → S the natural morphism. Let L = π ∗ det(E ) be the pullback of the determinant of E (Remark 17.58). We then have (17.10.6)

ΩnP/S ∼ = L (−n − 1).

This follows by passing to the top exterior powers of the terms in the short exact sequence (17.7.3) using (17.10.5). In particular, for E = OSn+1 , we obtain that (17.10.7)

ΩnPn /S ∼ = OPnS (−n − 1). S

29

Exercises Exercise 17.1. Let R → A be a homomorphism of rings. For D, D ′ ∈ DerR (A) set [D, D ′ ] := D ◦ D′ − D′ ◦ D. (1) Show that [D, D ′ ] ∈ DerR (A) and that [ , ] endows DerR (A) with the structure of an R-Lie algebra. (2) Let g ⊆ DerR (A) be an A-submodule generated by a set X. Show that [ , ] preserves g if and only if [D, D] ∈ g for all D ∈ X. i

f

Exercise 17.2. Let Z −→ X −→ Y be morphisms of schemes such that i and j := f ◦ i are immersions. Show that there is a canonical exact sequence of OZ -modules Cj −→ Ci −→ i∗ Ω1X/Y −→ 0. Exercise 17.3. Let R → A be a ring homomorphism, I ⊆ A an ideal, and let Aˆ = limn A/I n be the I-adic completion of A. Let D ∈ DerR (A). Show that D(I n ) ⊆ I n−1 for ˆ all integers n ≥ 1. Deduce that D induces a derivation on A. Exercise 17.4. Let k be a field of characteristic zero, let K/k be a field extension, and let x1 , . . . , xn ∈ K. Show that dx1 , . . . , dxn ∈ Ω1K/k are K-linearly independent if and only if x1 , . . . , xn are algebraically independent over k. Give a counterexample to this assertion in characteristic > 0. Exercise 17.5. Let R → A be a homomorphism of rings. For D ∈ DerR (A) and for an integer i ≥ 1 write Di = D ◦ · · · ◦ D (i times). Let p be a prime number and assume that pA = 0. (1) Let D ∈ DerR (A). Show that Dp ∈ DerR (A). (2) Assume that A is an integral domain and let 0 ̸= D ∈ DerR (A). Show that a0 + a1 D + a2 D2 +· · ·+ap−1 Dp−1 is a derivation (ai ∈ A) if and only if a0 = a2 = · · · = ap−1 = 0. Hint: Show first that 1, D, D 2 , . . . , D p−1 are linearly independent over Frac A. (3) Show that an analogous assertion as in (2) does not hold for A = Fp [T ]/(T p ). (4) Now let A again be an arbitrary ring with pA = 0. Show the “Hochschild formula”: For a ∈ A and D ∈ DerR (A) one has (aD)p = ap Dp + (aD)p−1 (a) · D. In particular (aD)p is a linear combination of D and Dp . Hint: Reduce to an identity over Fp [T1 , T2 , . . . ]. Then use (2). Exercise 17.6. Let k be a field of characteristic p > 0 and let K be a field extension of k such that ap ∈ k for every a ∈ K. Let V be a k-vector space. For every D ∈ Derk (K) we obtain a map DV := idV ⊗D : V ⊗k K → V ⊗k K. Show that a sub K-vector space U ′ ⊆ V ⊗k K is of the from U ⊗k K for some sub K-vector space U ⊆ V if and only if DV (U ′ ) ⊆ U ′ for all D ∈ Derk (K). Exercise 17.7. Let R be a ring, n ≥ 1, and let X ⊂ PnR be a hypersurface of degree d, i.e. X = V+ (f ) for a non-zero homogeneous polynomial f of degree d. Show that the conormal sheaf of the inclusion i : X → PnR is i∗ OPnR (−d). Exercise 17.8. Let R be a discrete valuation ring and let π ∈ R be a uniformizer. Fix i ∈ N and let A = R[X, Y ]/(XY i − π), a regular domain. For which i is ΩA/R a locally free A-module? For which i is it torsion-free?

30

17 Differentials

Exercise 17.9. Let k be a field of characteristic ̸= 2, 3, let a, b ∈ k such that g(X) := 1 X 3 +aX+b is separable, and let E = V+ (Y 2 Z−g(X)) ⊂ P2k . Show that ω := dX Y ∈ ΩK(E)/k 1 1 actually lies in Γ(E, ΩE/k ) ⊂ ΩK(E)/k , and is a nowhere vanishing global section of Ω1E/k . Conclude that Ω1E/k is a free OE -module of rank 1. Remark : The curve E is an elliptic curve, see Section (16.35) and Section (26.17). Exercise 17.10. Let K = k((Ti )i∈I ) be a purely transcendental extension of a field k. Show that (dTi )i∈I is a basis of Ω1K/k . Exercise 17.11. Let X → S be a morphism of schemes that is locally of finite presentation. Show that Der S (OX , F ) is a quasi-coherent OX -module for every quasi-coherent OX module F . Exercise 17.12. Let X → S be a morphism of schemes. Show that for every OX -module F the morphism Hom OX (Ω1X/S , F ) → Der S (OX , F ), u 7→ u ◦ dX/S , is an isomorphism of OX -modules. Exercise 17.13. Let p be a prime, let S be a scheme of characteristic p, let X → S be an Sscheme, and let FX/S : X → X (p) be the relative Frobenius morphism (Remark 4.24). (1) Show that the canonical homomorphism vF/S : F ∗ Ω1X (p) /S → Ω1X/S is zero. (2) Show that Ω1X/S → Ω1X/X (p) is an isomorphism. Exercise 17.14. Let f : X → S be a morphism of schemes and let F be a quasi-coherent OX -module. An f −1 (OS )-linear homomorphism ∇0 : F → F ⊗OX Ω1X/S is called a connection if ∇0 (am) = a∇0 (m) + m ⊗ da,

a ∈ OX (U ), m ∈ F (U ), U ⊆ X open.

In the sequel we will consider F ⊗Ω•X/S as a right module over the OX -algebra Ω•X/S . (1) Show that there exists a unique f −1 (OS )-linear graded endomorphism ∇ of F ⊗ Ω•X/S of degree 1 which extends ∇0 in degree 0 and which satisfies the equality ∇(xω) = (∇x)ω + (−1)p x(dω) for x a local section of F ⊗ ΩpX/S , ω a local section of Ω•X/S . (2) Show that the composition ∇ ◦ ∇ is Ω•X/S -linear. (3) Set R := ∇1 ◦ ∇0 : F → F ⊗ Ω2 . Deduce that R is OX -linear and that (∇ ◦ ∇)(m ⊗ ω) = R(m)ω for all local sections m of F and ω of Ω•X/S . The homomorphism R is called the curvature of the connection ∇0 . If R = 0, then the connection ∇0 is called flat and the complex (F ⊗Ω•X/S , ∇) is called the de Rham complex of (F , ∇0 ) over S.

18

´ Etale and smooth morphisms

Content – Formally unramified, formally smooth and formally ´etale morphisms – Unramified and ´etale morphisms – Smooth morphisms In Chapter 6 we defined a morphism of schemes f : X → S to be smooth if Zariski locally on S and on X one may write f as an open immersion followed by a morphism of the form Spec R[T1 , . . . , Tn ]/(f1 , . . . , fr ) → Spec R such that the rank of the Jacobian matrix of the fi is equal to r in each point x ∈ X. We called f ´etale if in addition we may choose r = n. As the term suggests, smoothness should be thought of as being “without relative singularities”. This point of view is also supported by the fact that for a smooth surjective morphism X → S the singularities of X “are as bad as” the singularities of S (see Corollary 14.60 and Remark 14.61 for precise statements). Moreover, in (20.12) we will attach to each scheme X locally of finite type over C a complex analytic space X an . Then X is smooth over Spec(C) if and only if X an is a complex manifold. This follows from the theorem on inverse functions in complex analysis and justifies the definition of smoothness via the Jacobian matrix. If S = Spec k for a field k, then we showed that a morphism locally of finite type X → Spec k is smooth if and only if X ⊗k K is a regular for some (or, equivalently, for all) algebraically closed extension K of k (Corollary 6.32). Moreover we saw in Theorem 14.24 that every smooth morphism is flat. In this chapter we will prove that conversely every flat morphism with geometrically regular fibers is smooth (Theorem 18.56). That theorem will also show that if f : X → S is locally of finite presentation, then f is smooth if and only if for every S-morphism Spec R → X (R some ring) and for every nilpotent ideal I ⊂ R the map X(R) → X(R/I) is surjective. This last property is called formal smoothness and can be expressed by properties of K¨ahler differentials (Proposition 18.18). We start by studying these properties.

Formally unramified, formally smooth and formally ´ etale morphisms Let T be a scheme and let T0 be a closed subscheme. Given a morphism X → S of schemes it is an important question whether it is possible to extend (maybe even uniquely) an S-morphism T0 → X to an S-morphism T → X. In other words, we look at a commutative diagram of scheme morphisms

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_3

´ 18 Etale and smooth morphisms

32 T0 (18.0.1)

a0

/X

a

 / S,

f

i

 T

where i is a closed immersion, and we study the question whether there exists a morphism b : T → X that commutes with (18.0.1) by which we mean that b ◦ i = a0 and f ◦ b = a. In general there is probably no hope to answer this question, but if T = Spec R is affine and hence T0 = Spec R/I for an ideal I ⊂ R, and if I is “not too big”, then the situation is more manageable. Here we study this question if I is a nilpotent ideal, i.e., there exists some n ≥ 0 such that I n+1 = 0. Then we think of Spec R as an infinitesimal thickening of Spec R/I. Morphisms f for which there always exists a lift b (resp. a unique lift b) as above whenever I is nilpotent will be called formally smooth (resp. formally ´etale). As the definition suggests, we will see then in the following sections that these notions are linked to the question whether X → S is smooth or ´etale. Later in Sections (20.3) (for local rings) and (20.4) in general, we will also consider the more general case if (R, I) is a henselian pair (see Definition 20.15 below) which is in particular the case if I is nilpotent (Proposition 20.17 below). (18.1) Definition of formally unramified, formally smooth and formally ´ etale morphisms. Definition 18.1. Let i : Y → X be a closed immersion defined by a quasi-coherent ideal I ⊆ OX . Then i is called a nil-immersion if i is a homeomorphism. We call i a thickening if I is locally nilpotent (i.e., there exist an open covering (Ui )i∈I of X and for all i an ni ∈ N such that (I |Ui )ni +1 = 0). For n ∈ N, a thickening is called of order at most n if I n+1 = 0. Remark 18.2. Let i : Y → X be a closed immersion and let I be the quasi-coherent ideal of OX defining i. (1) Then i is a nil-immersion if and only if I ⊆ NX , where NX denotes the nil-radical of OX , cf. Section (3.18). (2) If X is quasi-compact and i is a thickening, then there exists an integer n ≥ 0 such that I n+1 = 0. (3) Consider the situation locally on X, i.e. X = Spec A is affine. Then Y ∼ = Spec A/I for an ideal I ⊆ A and i is given by the canonical ring homomorphism A → A/I. Then i is a nil-immersion (resp. a thickening) if and only if every element of I is nilpotent (resp. there exists an n ≥ 0 such that I n+1 = 0). (4) Every thickening is a nil-immersion. Conversely, if i is a nil-immersion and I is of finite type (e.g., if X is locally noetherian), then i is a thickening. (5) The n-th infinitesimal neighborhood of Y in X (17.3) is a thickening of order at most n of Y . Definition 18.3. A morphism f : X → S of schemes is called formally unramified (resp. formally smooth, resp. formally ´etale) if for every diagram of the form (18.0.1) with T affine and i a thickening of order at most 1 there exists at most one (resp. at least one, resp. a unique) morphism b : T → X commuting with (18.0.1). If R is a ring, an R-algebra A is called formally unramified (resp. formally smooth, resp. formally ´etale) over R, if Spec A → Spec R has this property.

33 Remark 18.4. Let f : X → S be a morphism of schemes. (1) f is formally ´etale if and only if f is formally unramified and formally smooth. (2) Let f be formally unramified (resp. formally smooth, resp. formally ´etale) and let i be an arbitrary thickening of affine schemes given by the projection R → R/I for an ideal I such that I n+1 = 0 for some n ≥ 0. We factorize R → R/I into R = R/I n+1 → R/I n → · · · → R/I 2 → R/I and note that Ker(R/I i+1 → R/I i ) = I i /I i+1 is an ideal of square zero in R/I i+1 . This shows that there exists at most one (resp. at least one, resp. a unique) morphism b : T → X commuting with (18.0.1). (3) The fact that we may glue morphisms and the uniqueness properties shows that if f is formally unramified (resp. formally ´etale), then there exists at most one (resp. a unique) morphism b : T → X commuting with (18.0.1) if i is an arbitrary thickening of arbitrary schemes. For formally smooth morphisms a similar property as in (3) does not hold. We will see more precisely in Section (18.3) below that the obstruction to glue local lifts to a global lift is an element in some cohomology group. See also Exercise 18.24. The proof of the following proposition is easy and mostly formal and is therefore omitted. Proposition 18.5. (1) Every monomorphism of schemes (in particular every immersion) is formally unramified. Every open immersion is formally ´etale. (2) The properties “formally unramified”, “formally smooth”, and “formally ´etale” are stable under composition and stable under base change. We will also see (Corollary 18.8 and Proposition 18.12) that all the above properties are local on source and target. (18.2) Formally unramified morphisms and differentials. Proposition 18.6. A morphism of schemes f : X → S is formally unramified if and only if Ω1X/S = 0. Proof. Let X (1) be the first infinitesimal neighborhood of X in X ×S X such that Ω1X/S is the quasi-coherent ideal defining X ,→ X (1) , considered as a quasi-coherent OX module. Let p1 , p2 : X (1) → X be the restrictions of the projections X ×S X → X. Then p1|X = p2|X . Assume that f is formally unramified. We apply Remark 18.4 (2) to T0 = X and T = X (1) and see that p1 = p2 . By Proposition 17.6, Ω1X/S is generated by local sections of the form p♭1 (a) − p♭2 (a) for local sections a of OX . Hence Ω1X/S = 0. Conversely assume that Ω1X/S = 0 and hence X = X (1) . Let b, b′ : T → X be morphisms commuting with (18.0.1), where T0 → T is a thickening of order at most 1. They yield a morphism (b, b′ ) : T → X ×S X. Then b = b′ if and only if (b, b′ ) factors through the subscheme X of X ×S X. By hypothesis, (b, b′ )|T0 factors through X. As T0 is defined in T by an ideal of square zero, (b, b′ ) factors through X (1) = X. Therefore the exact sequence (17.5.8) shows:

´ 18 Etale and smooth morphisms

34

Corollary 18.7. Let S be a scheme and let f : X → Y be a morphism of S-schemes. Then f is formally unramified if and only if the morphism v : f ∗ Ω1Y /S → Ω1X/S (17.5.6) is surjective. As K¨ ahler differentials are compatible with passage to open subschemes on source (Remark 17.19 (2)) and target (Proposition 17.30) we also deduce from Proposition 18.6 the following result. Corollary 18.8. The property “formally unramified” is local on the source and local on the target (Section (4.9)). In other words, if f : X → Y is a morphism of schemes and (Ui )i and (Vi )i are open coverings of X and Y , respectively, such that f (Ui ) ⊆ Vi for all i, then f is formally unramified if and only if the restriction Ui → Vi of f is formally unramified for all i. (18.3) Gluing local lifts. For thickenings of order at most 1 we have the following tool to glue local liftings b. Suppose given a diagram (18.0.1) where T0 → T is a thickening of order at most 1 defined by a quasi-coherent ideal J of OT with J 2 = 0. Let La be the sheaf of lifts of a, i.e., for U ⊆ T open we define La (U ) as the set of morphisms b : U → X such that f ◦ b = a|U and b ◦ i|i−1 (U ) = a0|i−1 (U ) . As the underlying topological spaces of T and T0 are equal, we can consider La also as a sheaf on T0 . Lemma 18.9. The sheaf of groups G := Hom OT0 (a∗0 Ω1X/S , i∗ J ) on T0 acts simply transitively on the sheaf of sets La on T0 . If all schemes are affine, this is essentially Remark 17.4 and we will reduce to it. Proof. For U0 ⊆ T0 open we have to define an action of G (U0 ) on La (U0 ), compatible with restrictions to smaller open subsets, and to show that this action is simply transitive. It suffices to do this for open subsets U0 that run through a basis of the topology. As the underlying topological spaces of T and T0 are the same, we may do so for U0 that are of the form i−1 (U ) for an open affine subscheme U ⊆ T such that a(U ) is contained in an open affine W = Spec R of S and that a0 (U0 ) is contained in an open affine V = Spec A of f −1 (W ). Let U = Spec C be such an open affine subset. Then U0 = i−1 (U ) = Spec C/I with I 2 = 0 and we are now given a commutative diagram of rings C/I o O

AO

Co

R

and we have defined in Remark 17.4 a simply transitive action of DerR (A, I) = HomA (Ω1A/R , I) = HomC/I (Ω1A/R ⊗A C/I, I) = G (U0 ) on La (U0 ) which is easily seen to be compatible with passing to principal open subsets of U .

35 Remark 18.10. With the previous notations assume that there exists an open covering (Ui )i∈I of T and for each i ∈ I a lift bi : Ui → X (i.e. b ∈ La (Ui )). Then La is a G -torsor and hence defines a class in H 1 (T0 , G ) (Section (11.5)). This class is trivial if and only if La (T ) ̸= ∅, i.e., if and only if there exists a global lift b : T → X. Proposition 18.11. Suppose there exists an open covering (Ui )i∈I of T and for each i ∈ I a lift bi : Ui → X. If (*)

H 1 (T0 , Hom OT0 (a∗0 Ω1X/S , i∗ J )) = 0,

then there exists a global lift b : T → X such that b|Ui = bi for all i. Moreover, (*) holds if T0 is affine and X → S is locally of finite presentation. Proof. If (*) holds, b exists by Remark 18.10. It remains to show the last assertion. As T0 is affine, we have H 1 (T0 , F ) = 0 for any quasi-coherent OT0 -module F (Theorem 12.35). Hence it suffices to show that G := Hom OT0 (a∗0 Ω1X/S , i∗ J ) is quasi-coherent. But if X → S is locally of finite presentation, then Ω1X/S is of finite presentation (Corollary 17.34) and hence a∗0 Ω1X/S is an OT0 -module of finite presentation. Therefore G is quasi-coherent by Proposition 7.29. Proposition 18.12. The properties “formally smooth”, and “formally ´etale” are local on the source and local on the target (Section (4.9)). We will show the result only for morphisms locally of finite presentation. The general result is sketched in Exercise 18.5 (or can be found in [Sta] 0D0F). Proof. As we have already seen that “formally unramified” is local on source and target (Corollary 18.8) it suffices to show the claim for “formally smooth”. Let us show that the property is local on the source. Let f : X → Y be a morphism of schemes and let (Ui )i be an open covering of X. If f is formally smooth, then f |Ui is formally smooth because the inclusion Ui → X is formally ´etale and all properties are stable under composition. Conversely, suppose that for all i the morphism f |Ui is formally smooth. Suppose we are given a thickening square (18.0.1), where T0 is a closed subscheme defined by an ideal of square 0. In particular T and T0 have the same underlying topological space. Set V0,i := a−1 0 (Ui ) and let Vi be the open subscheme of T that has the same underlying topological space as V0,i . Choosing open affine coverings of the Vi we see that Zariski locally on T there exists a lifting of a to a morphism to X. As T is affine, we may apply Proposition 18.11 (by our additional assumption that f is locally of finite presentation) to see that there exists a morphism b that commutes with (18.0.1). It remains to show that “formally smooth” is local on the target. Let f : X → Y be a morphism of schemes and let (Vj )j be an open covering of Y . If f is formally smooth, then so is its restriction f −1 (Vj ) → Vj since “formally smooth” is stable by base change. Conversely, suppose that f −1 (Vj ) → Vj is formally smooth for all j. As open immersions are formally ´etale, the composition f −1 (Vj ) → Vj → Y is formally smooth for all j. Hence f is formally smooth since we already have seen that the property is local on the source.

´ 18 Etale and smooth morphisms

36

(18.4) Formally smooth resp. formally ´ etale morphisms and differentials. As the notions of being formally smooth and formally ´etale are local on the source and target (Proposition 18.12), it often suffices to consider the affine case. Recall that an R-algebra A is by definition formally smooth (resp. formally ´etale) if and only if for every ring C, every ideal I ⊂ C with I 2 = 0 and every commutative diagram C/I o O

AO

Co

R

(18.4.1)

there exists a (resp. there exists a unique) homomorphism A → C commuting with the diagram. Example 18.13. Let R be a ring. Then every polynomial algebra A = R[(Tλ )λ∈Λ ] is formally smooth over R. Indeed, in this case an R-algebra homomorphism φ¯ : A → C/I corresponds via φ¯ 7→ (φ(T ¯ λ ))λ∈Λ simply to a Λ-tuple (¯ cλ )λ of elements in C/I, and we can lift φ¯ to an R-algebra homomorphism A → C by choosing a lift cλ ∈ C of every c¯λ . Remark 18.14. Let R be a ring and let A be an R-algebra of finite presentation. Then A ∼ = R[T1 , . . . , Tn ]/(f1 , . . . , fr ) for polynomials fi ∈ R[T1 , . . . , Tn ]. Consider the diagram (18.4.1). The homomorphism A → C/I corresponds to elements c¯1 , . . . , c¯n ∈ C/I such that fi (¯ c1 , . . . , c¯n ) = 0 for all i, i.e., to a solution of the system of equations f1 = · · · = fr = 0 in C/I. A homomorphism A → C commuting with (18.4.1) corresponds to a lift c1 , . . . , cn ∈ C of this solution. Corollary 18.57 below will show that A is a formally smooth R-algebra, i.e., such a lift of solution always exists, if and only if Spec A → Spec R is a smooth morphism. By definition (Definition 6.14), this is for instance the case if the Jacobian matrix   ∂fi ∈ Mr×n (κ(x)) (x) ∂Tj i,j has rank r for all x ∈ Spec A. The following result relates the notion of extensions (17.2) and the property of being formally smooth. Proposition 18.15. Let R be a ring and let A be an R-algebra. Then the following assertions are equivalent. (i) A is a formally smooth R-algebra. (ii) ExR (A, M ) = 0 for every A-module M . j

π

Proof. (i) ⇒ (ii). Let 0 → M −→ E −→ A → 0 be in ExR (A, M ). As A is formally smooth over R, idA can be lifted to an A-algebra homomorphism A → E. Therefore the extension is trivial. (ii) ⇒ (i). Consider the diagram (18.4.1), where C is an arbitrary ring and I 2 = 0. Then 0 → I → C ×C/I A → A → 0 is an element of ExR (A, I). Hence (ii) implies that there exists a homomorphism s : A → C ×C/I A of R-algebras. Then the composition of s followed by the projection C ×C/I A → C commutes with the diagram. For field extensions, formal smoothness has the following description which we will not use in the sequel except in the exercises (see [Mat2] Theorem 26.9 for a proof).

37 Example 18.16. A field extension K ⊇ k is formally smooth if and only if it is separable (in the sense of Definition B.91). Once we have shown that a formally smooth morphism locally of finite presentation is smooth (Corollary 18.57), this result can be viewed, at least for finitely generated extensions K, as a variant of generic smoothness (Theorem 6.19 and Remark 6.20), see also Section (18.17) below. We will often use the following easy remark in the sequel. Remark 18.17. Let A be a ring (not necessarily commutative) and let 0 → M ′ → M → M ′′ → 0

(*)

be a sequence of left A-modules. Then (*) is exact and split if and only if for every left A-module N the induced sequence 0 → HomA (M ′′ , N ) → HomA (M, N ) → HomA (M ′ , N ) → 0

(**)

is exact. Indeed, the map M → M ′′ is a cokernel of u : M ′ → M if and only if for every left module N a linear map v : M → N factors through M → M ′′ if and only if v ◦ u = 0. Hence (*) is right exact if and only if (**) is left exact for all N . Moreover, it is clear that if (*) is exact and split, so is (**). Conversely, suppose that (**) is exact for all N . To see that M ′ → M is has a left inverse r, take N = M ′ and choose a preimage r of idM ′ in HomA (M, M ′ ). Proposition 18.18. Let f : X → Y be a morphism of S-schemes and consider the sequence (17.5.8) (18.4.2)

0 → f ∗ Ω1Y /S −→ Ω1X/S −→ Ω1X/Y → 0.

(1) If f is formally smooth, then (18.4.2) is exact and Zariski locally split. (2) If (18.4.2) is exact and Zariski locally split and X is formally smooth over S, then f is formally smooth. The proof will show that if S, X, and Y are affine and f is formally smooth, then (18.4.2) is split exact. One can also use Exercise 18.4 which shows that if X = Spec B and Y = Spec A are affine and f is formally smooth, then Ω1B/A is a projective B-module, which also implies that (18.4.2) is split exact. Proof. We may assume that S = Spec R, X = Spec B, and Y = Spec A are affine. For a B-module M consider the exact sequence (17.2.2) 0 → DerA (B, M ) → DerR (B, M ) → || HomB (Ω1B/A ,M )

|| HomB (Ω1B/R ,M )

DerR (A, M )

→ ExA (B, M ) → ExR (B, M ).

|| HomB (Ω1A/R ⊗A B,M )

If f is formally smooth, then ExA (B, M ) = 0 (Proposition 18.15). Therefore (18.4.2) is exact and split by Remark 18.17. This shows (1). To show (2) we may assume that (18.4.2) is exact and split. Then DerR (B, M ) → DerR (A, M ) is surjective. If moreover ExR (B, M ) = 0, then ExA (B, M ) = 0. This shows (2).

´ 18 Etale and smooth morphisms

38

Corollary 18.19. Let S be a scheme and f : X → Y a morphism of S-schemes. (1) If f is formally ´etale, the morphism v : f ∗ Ω1Y /S → Ω1X/S (17.5.6) is an isomorphism. (2) If v : f ∗ Ω1Y /S → Ω1X/S is an isomorphism and X is formally smooth over S, then f is formally ´etale. Proof. The assertions follow from Proposition 18.18 because f is formally ´etale if and only if f is formally smooth and Ω1X/Y = 0 (Proposition 18.6). Proposition 18.20. Let f : X → S and g : Z → S be S-schemes and let i : X → Z be an immersion of S-schemes. Consider the sequence (17.5.10) (18.4.3)

0 −→ Ci −→ i∗ Ω1Z/S −→ Ω1X/S −→ 0.

(1) If f is formally smooth, then (18.4.3) is exact and Zariski locally split. (2) If g is formally smooth and (18.4.3) is exact and Zariski locally split, then f is formally smooth. The proof will show that if f is formally smooth, i is a closed immersion and S and Z are affine, then (18.4.3) is split. Proof. Let U be an open subscheme of Z such that i factors through a closed immersion X → U . By replacing Z by U we may assume that i is a closed immersion. Moreover we may assume that S = Spec R, Z = Spec C and X = Spec A with A = C/I for an ideal I ⊆ C. For every A-module M we have DerC (A, M ) = 0. Hence the exact sequence (17.2.2) takes the form 0 → DerR (A, M ) → || HomA (Ω1A/R ,M )

DerR (C, M ) || HomA (Ω1C/R ⊗C A,M )

→ ExC (A, M ) → ExR (A, M ) → ExR (C, M ), || HomA (I/I 2 ,M )

where the description of ExC (A, M ) comes from Proposition 17.13 and where the map DerR (C, M ) → ExC (A, M ) is induced by the map I/I 2 → Ω1C/R ⊗C A defining the first arrow of (18.4.3) by Remark 17.37. If f is formally smooth, then ExR (A, M ) = 0 by Proposition 18.15. This shows (1) by Remark 18.17. To show (2), we may assume that (18.4.3) is split exact, in particular DerR (C, M ) → ExC (A, M ) is surjective. As g is formally smooth, we have ExR (C, M ) = 0. Therefore ExR (A, M ) = 0 for all A-modules M . This proves that f is formally smooth (again by Proposition 18.15). Corollary 18.21. Let S be a scheme and let i : X → Z be an immersion of S-schemes with conormal sheaf Ci . (1) If X is formally ´etale over S, then the morphism d : Ci → i∗ Ω1Z/S (17.5.9) is an isomorphism. (2) If d : Ci → i∗ Ω1Z/S (17.5.9) is an isomorphism and Z is formally smooth over S, then X is formally ´etale over S. Proof. Assertion (1) follows from Proposition 18.20 because Ω1X/S = 0 if X is formally ´etale over S (Proposition 18.6). Let us show (2). If d is an isomorphism, the exact sequence (17.5.10) shows that Ω1X/S = 0. Hence X is formally unramified over S by Proposition 18.6. Moreover Proposition 18.20 (2) shows that X → S is also formally smooth. Hence it is formally ´etale.

39

Unramified and ´ etale morphisms We will call a morphism unramified (resp. ´etale) if it is formally unramified (resp. formally ´etale) and satisfies certain finiteness conditions. The main result is a theorem describing the local shape of unramified and ´etale morphisms (Theorem 18.42). Its proof relies on Zariski’s main theorem. This result will then in particular show (Theorem 18.44) that our new definition of ´etale is equivalent to the previous definition given in Section (6.8). (18.5) Unramified morphisms. Definition 18.22. Let f : X → S be a morphism of schemes. (1) f is called unramified if and only if it is formally unramified and locally of finite type. (2) Let x ∈ X. Then f is called unramified at x if there exists x ∈ U ⊆ X open such that f |U is unramified. Here we follow Raynaud [Ray2] O and deviate from the definition of [EGAIV] O (17.3.1), where unramified morphisms are defined to be formally unramified and locally of finite presentation, because for many results below (e.g., Proposition 18.29 or Theorem 18.42 and the following remark) this slightly weaker finiteness assumption suffices. Remark 18.23. The permanence properties of “formally unramified” and “locally of finite type” imply: (1) Every immersion is unramified. (2) The property “unramified” is stable under composition, stable under base change, local on the source, and local on the target. We now characterize unramified morphisms, becoming more and more global. We first consider unramified schemes over a field. Proposition 18.24. Let k be a field and let f : X → Spec k be locally of finite type. Then the following assertions are equivalent. (i) f is unramified. ` (ii) The k-scheme X is isomorphic to i∈I Spec ki with I some index set and finite separable field extensions ki of k. closed extension K of k, such that the K-scheme X ⊗k K (iii) There exists an algebraically ` is isomorphic to j∈J Spec K for some index set J. (iv) X is a geometrically reduced k-scheme and dim(X) = 0. Proof. We have “(ii) ⇔ (iii)” by Proposition B.97 and “(ii) ⇔ (iv)” by Proposition 5.49. It remains to prove that (iii) holds if and only if Ω1X/k = 0 (Proposition 18.6). As Ω1X/k ⊗k K ∼ = Ω1X⊗k K/K (Proposition 17.30), we may assume that k = K is algebraically closed. The condition is clearly necessary. Conversely, assume that for every closed point x ∈ X one has 0 = Ω1X/k (x) = mx /m2x (Example 17.23). Then for all x ∈ X closed one has mx = 0 by Nakayama’s lemma and hence OX,x = k. This shows that dim X = 0 (Theorem 5.22) and implies (iii) by Proposition 5.11. Generalizing the case of a field extension, a k-algebra A is called separable if Spec A is geometrically reduced over k. Using this language, Proposition 18.24 implies:

´ 18 Etale and smooth morphisms

40

Remark 18.25. Let k be a field and let A be a k-algebra. Then the following assertions are equivalent. (i) Spec A → Spec k is unramified. (ii) A is a finite-dimensional separable k-algebra. (iii) A is isomorphic to a finite product of finite separable field extensions of k. Example 18.26. Let k be a field and let X → Spec k be a proper geometrically reduced k-scheme. Then Γ(X, OX ) is a finite separable k-algebra. Indeed, Γ(X, OX ) is a finite k-algebra by Theorem 12.65. For every field extension K of k one has Γ(XK , OXK ) = Γ(X, OX ) ⊗k K as the formation of global sections commutes with flat base change (Corollary 12.8). As XK is reduced, it follows that Γ(X, OX ) is geometrically reduced. Proposition 18.27. Let f : X → S be locally of finite type, x ∈ X, s := f (x). Recall that Ω1X/S (x) = Ω1X/S,x ⊗OX,x κ(x) denotes the fiber of Ω1X/S at x. The following assertions are equivalent. (i) f is unramified at x. (ii) Ω1X/S (x) = 0. (iii) For every local ring C, every ideal I ⊂ C with I 2 = 0 and for every commutative diagram of local ring homomorphisms C/I o O

OX,x O fx♯

Co

OS,s

there exists at most one local ring homomorphism OX,x → C commuting with the diagram. (iv) OX,x /ms OX,x is a finite separable field extension of κ(s). Proof. As Ω1X/S is quasi-coherent of finite type (Remark 17.21), we have Ω1X/S (x) = 0 if and only if the stalk (Ω1X/S )x is zero by Nakayama’s lemma. As the support of an OX module of finite type is closed, this means that we find an open neighborhood x ∈ U ⊆ X such that Ω1X/S |U = Ω1U/S = 0. Hence (i) and (ii) are equivalent by Proposition 18.6. Clearly, (i) implies (iii). Next we show that (iii) implies (ii) without assuming that f is locally of finite type. We may assume that X = Spec A with A = OX,x and S = Spec R with R = OS,s (17.4.3). Let J be the kernel of the multiplication A ⊗R A → A. Applying (iii) to C = (A ⊗R A)/J 2 and to the image I of J in C we see that the homomorphisms χ : A → C, χ(a) = a ⊗ 1 and χ′ : A → C, χ(a) = 1 ⊗ a are equal. This implies Ω1A/R = 0 because Ω1A/R is generated by elements of the form χ(a) − χ′ (a) (Proposition 17.6). For the equivalence of (iv) and (ii) we may assume that S = Spec k, where k is a field. Hence the equivalence follows from Proposition 18.24. In particular we see that x is isolated in its fiber if f is unramified at x. Hence Zariski’s main theorem (in the version of Corollary 12.79) shows the following corollary (using the notation of ramification index and of inertia index introduced in Section (12.5)).

41 Corollary 18.28. A morphism of finite type h : X → S is unramified at x ∈ X if and ′ only if there exists x ∈ U ⊆ X open such that h|U is quasi-finite and ex/s = fx/s = 1, where s := h(x) ∈ S. From these local characterizations it is now easy to deduce the following global characterization of a morphism to be unramified. Proposition 18.29. Let f : X → S be locally of finite type. Then the following assertions are equivalent. (i) f is unramified. (ii) Ω1X/S = 0. (iii) The diagonal ∆X/S : X → X ×S X is an open immersion. (iv) For an arbitrary nil-immersion of schemes i : T0 → T there exists at most one morphism b : T → X commuting with (18.0.1). (v) For all s ∈ S the fiber Xs of f at s is unramified over κ(s). Therefore to check that a morphism locally of finite type is unramified can be done fiber by fiber, and for the fibers one has the characterizations from Proposition 18.24 to be unramified. Proof. We have already seen “(i) ⇔ (ii)”, and “(ii) ⇔ (iii)” holds by Proposition 17.17 because Ω1X/S is of finite type (Remark 17.21). The implication “(iv) ⇒ (i)” is clear. We show that (iii) implies (iv). Let i : T0 → T be a nil-immersion and let b, b′ : T → X be two morphisms commuting with (18.0.1). Then (b, b′ )|T0 : T0 → X ×S X factors through ∆X/S . As T0 and T have the same underlying topological space and as ∆X/S is an open immersion, this implies that (b, b′ ) factors through ∆X/S . Hence b = b′ . Finally, if f is unramified, then each fiber is unramified because “unramified” is stable under base change. This shows that (i) implies (v). The converse follows from the implication “(iv) ⇒ (i)” of Proposition 18.27. Remark 18.30. Consider a commutative diagram of schemes /Y

f

X u

S.

~

v

Assume that v is unramified. Then the diagonal Y → Y ×S Y is an open immersion by Proposition 18.29. (1) The cartesian diagram (9.1.4) shows that the graph Γf of f is an open immersion because it can be written as base change of the diagonal ∆Y /S . The same cartesian diagram shows that if g : X → Y is a second S-morphism, then the equalizer Eq(f, g) is an open subscheme of X. (2) This can be used to show “cancellation” for properties P of scheme morphisms (in the sense of Appendix C). Assume that P is stable under composition and under base change. Moreover, we assume that every open immersion possesses P. Examples are the properties “flat”, “(formally) unramified/smooth/´etale”, “open immersion”, or “universally open”. Then the same argument as in Remark 9.11 shows that if u possesses P and v is unramified, f also possesses P (simply write f as the composition of Γf followed by the projection X ×S Y → Y ).

42

´ 18 Etale and smooth morphisms

(3) Applying this to u = idS and to the property “open immersion”, we see that every section of an unramified morphism is an open immersion. Another implication of this remark is the following. Proposition 18.31. Let S be a scheme, let X → S and Y → S be S-schemes, and let g, f : X → Y be morphisms of S-schemes. Suppose that Y → S is unramified and separated and that X is connected. If there exists a non-empty S-scheme T and a morphism h : T → Y such that f ◦ h = g ◦ h, then f = g. One usually applies this proposition to T = Spec K, where K is some field extension of κ(x) for some point x ∈ X, and to T → Spec κ(x) → X the canonical morphism. The proposition says that if f and g are equal in some K-valued point, then they are equal. Proof. Since Y → S is unramified and separated, the diagonal ∆Y /S : Y → Y ×S Y is an open and closed immersion. Consider the equalizer subscheme Eq(f, g) (Definition 9.1). As the immersion Eq(f, g) → X is obtained by base change from ∆Y /S (Proposition 9.3), Eq(f, g) is an open and closed subscheme of X. By hypothesis, T → X factors through Eq(f, g), so Eq(f, g) is non-empty. Hence Eq(f, g) = X because X is connected. This means f = g. The characterization of unramified morphisms in Proposition 18.29 also allows us to prove the following descent property. Corollary 18.32. Let f : X → S be locally of finite type and let g : S ′ → S be a morphism of schemes. Let x′ ∈ X ′ := X ×S S ′ and let x ∈ X be the image of x′ . Then f is unramified at x if and only if its base change f ′ : X ×S S ′ → S ′ is unramified at x′ . Proof. The condition is necessary since “unramified” is stable under base change. To see that it is sufficient by Proposition 18.29 it is enough show that if f ′ is unramified at x′ , then the fiber f −1 (f (x)) of f is unramified at x. Hence we can assume that S = Spec k and S ′ = Spec k ′ for a field extension k → k ′ . In this case we can apply Proposition 18.24 (iii) by choosing an algebraically closed extension K of k ′ . Remark 18.33. Using Corollary 18.32 it is not difficult to see that the property “unramified” is compatible with cofiltered limits of schemes in the following sense. In the situation of Section (10.13) 1.-3.,5., and 6., we assume that f0 is locally of finite type (e.g., if X0 and Y0 are locally of finite type over S0 ). Let x ∈ X be a point and let xλ be the image of x in Xλ . Then f is unramified at x if and only if there exists λ such that fλ is unramified at xλ . Indeed, set y = f (x) and yλ = fλ (xλ ) be the image of y in Yλ . Then f −1 (y) = −1 fλ (yλ ) ⊗κ(yλ ) κ(y). By Proposition 18.27, it suffices to show that f −1 (y) is unramified at x if and only if fλ−1 (yλ ) is unramified at xλ . This follows from Corollary 18.32. ´ (18.6) Etale morphisms. Definition 18.34. A morphism of schemes f : X → S is called ´etale if f is formally ´etale and locally of finite presentation. Let x ∈ X. Then f is called ´etale at x if there exists x ∈ U ⊆ X open such that f |U is ´etale. An R-algebra A is called ´etale, if Spec A → Spec R is ´etale.

43 We already defined in Section (6.8) an ´etale morphism to be a smooth morphism of relative dimension 0. We will see in Theorem 18.44 that both notions are equivalent. Until then we will use the notion ´etale only in the sense of Definition 18.34. Remark 18.35. Every open immersion is ´etale. The property “´etale” is stable under composition, stable under base change, local on the source and local on the target. Indeed, all these permanencies hold for the properties “formally ´etale” and “locally of finite presentation”. In Corollary 18.43 we will see that it is compatible with filtered colimits and in Remark 18.46 we will see that “´etale” is stable under faithfully flat descent. Moreover, Remark 18.30 implies the following cancellation property. Remark 18.36. Let S be a scheme and let f : X → Y be a morphism of S-schemes. If X is ´etale over S and Y is unramified over S, then f is ´etale. (18.7) Local description of ´ etale morphisms. Our next goal is to show that locally all ´etale morphisms have a particularly simple form. We will use the following easy lemma. Lemma 18.37. Let A be a ring, I ⊆ A a nil-ideal (i.e., every element of I is nilpotent). Let u ∈ A, u0 its image in A/I. Then u is a unit in A if and only if u0 is a unit in A0 . × b∈A Proof. The condition is certainly necessary. Conversely, if u0 ∈ (A/I) P∞ there exists such that ab = 1 + i with i ∈ I. But 1 + i is invertible with inverse n=0 (−1)n in , a finite sum because i is nilpotent.

Proposition 18.38. Let R be a ring, f ∈ R[T ] a polynomial and set A := R[T ]/(f ). Let g ∈ R[T ] such that the image of the formal derivative f ′ of f in Ag is invertible. Then Ag is an ´etale R-algebra. We will see below that locally on the source every ´etale morphism has this form with f monic. Hence we coin the following notion. Definition 18.39. An R-algebra of the form (R[T ]/(f ))g , where f, g ∈ R[T ] such that f is monic and such that f ′ is invertible in (R[T ]/(f ))g , is called a standard ´etale algebra. Proof. [of Proposition 18.38] Set A := R[T ]/(f ). Clearly, Ag is an R-algebra of finite presentation. Hence it remains to prove that Ag is a formally ´etale R-algebra. Let C be an R-algebra, I ⊆ C an ideal with I 2 = 0, and let p : C → C0 := C/I be the canonical homomorphism. Let u0 : Ag → C0 be an R-algebra homomorphism. Let x ∈ Ag be the image of T . Let B an R-algebra. Then sending an R-algebra homomorphism v : Ag → B to b := v(x) defines a bijection (*)

∼

HomR-Alg (Ag , B) −→ { b ∈ B ; f (b) = 0, g(b) ∈ B × }.

By hypothesis we have f ′ (b) ∈ B × for any such b. Let c0 ∈ C0 be the element corresponding to u0 and c˜ ∈ C be any element with p(˜ c) = c0 . Then g(˜ c) ∈ C × by Lemma 18.37. It remains to show that there exist a unique i ∈ I such that f (˜ c + i) = 0. Then c˜ + i defines the unique lift u : Ag → C of u0 . We have f (˜ c) ∈ I and for i ∈ I we obtain as Taylor expansion

44

´ 18 Etale and smooth morphisms f (˜ c + i) = f (˜ c) + if ′ (˜ c)

because I 2 = 0. As f ′ (c0 ) ∈ C0× , we have f ′ (˜ c) ∈ C × (again by Lemma 18.37) and hence there exists a unique i ∈ I such that f (˜ c + i) = 0. Remark 18.40. It is also easy to see that standard ´etale algebras are smooth of relative dimension 0. For f, g ∈ R[T ] we have (R[T ]/(f ))g = R[T, U ]/(f, gU − 1) and the Jacobian matrix is given by   ′ f U g′ . 0 g This matrix is invertible over (R[T ]/(f ))g if f ′ is invertible in (R[T ]/(f ))g . To show that every ´etale morphism is locally given by a standard ´etale algebra, we start with the easy remark that being “standard ´etale” is compatible with filtered colimits of rings. Remark 18.41. Let (Ri )i∈I be a filtered diagram of rings, R its colimit, let i0 ∈ I, let Ai0 be an Ri0 -algebra of finite presentation and set Ai := Ai0 ⊗Ri0 Ri for i ≥ i0 and A := Ai0 ⊗Ri0 R. If A is a standard ´etale R-algebra there exists i ≥ i0 such that Ai is a standard ´etale Ri -algebra. Indeed, this is a standard limit argument. Let A = (R[T ]/(f ))g with f monic and f ′ invertible in A. Let i ∈ I such that there exist fi , gi ∈ Ri [T ] whose images in R[T ] are f and g, respectively. After possibly enlarging i we may assume that fi is monic and that fi′ is invertible in (Ri [T ]/(fi ))gi . For all j ≥ i let fj and gj be the image of fi and gi in Rj [T ]. We show that there exists a j ≥ i such that Aj ∼ = (Rj [T ]/(fj ))gj . For large j ≥ i we find tj ∈ Aj whose image in A is the image of T , and that under the map of Rj -algebras Rj [T ] → Aj , T 7→ tj , the polynomial fj is sent to zero and the polynomial gj is sent to a unit. We obtain a map φj of Rj -algebras of the standard ´etale Rj -algebra (Rj [T ]/(fj ))gj to Aj which is an isomorphism after base change to R. As Aj is generated by finitely many elements as an Rj -algebra, we may assume that φj is surjective, again after possibly enlarging j. As Aj is of finite presentation over Rj , the kernel Ker(φj ) is a finitely generated ideal (Proposition B.11). Hence again after enlarging j we may assume that φj is also injective. Theorem 18.42. Let f : X → S be a morphism of schemes locally of finite presentation, let x ∈ X, s := f (x). Then f is ´etale at x (resp. unramified at x) if and only if there exists an open affine neighborhood V = Spec R of s and an open affine neighborhood U = Spec A ⊆ f −1 (V ) of x such that A is isomorphic to a standard ´etale R-algebra (resp. such that A is isomorphic to a quotient of a standard ´etale R algebra). The hypothesis that f is locally of finite presentation is in fact superfluous (in the ´etale case this holds by assumption and in the unramified case one can first reduce to a noetherian situation, see Exercises 18.12, 18.13). The essential ingredient in the proof is Zariski’s main theorem. Proof. The condition is sufficient by Proposition 18.38 and because closed immersions are unramified. We prove the converse. We may assume that X = Spec A and S = Spec R are affine, and A is an R-algebra of finite presentation. Any localization by one element in a (quotient of a) standard ´etale algebra is again a (quotient of a) standard ´etale algebra. Hence if there exists an R-algebra B and elements g ∈ A and h ∈ B such that g(x) ̸= 0 and Ag ∼ = Bh , we may replace A by B.

45 (I). We start by assuming that f is unramified at x. Let m ⊂ R be the prime ideal corresponding to s. Writing Rm as an inductive limit of Rh for h ∈ R \ m we may assume that R is local with maximal ideal m by Remark 18.41. As f is unramified at x, the point x is isolated in its fiber (Proposition 18.24). Hence after passing to a principal open neighborhood of X we may assume that A = A˜g for some finite R-algebra A˜ and some g ∈ A˜ by a suitable version of Zariski’s main theorem (Proposition 12.77). Replacing A by A˜ we may assume that A is a finite R-algebra. Then a prime ideal of A is maximal if and only if it lies over m and there are only finitely many maximal ideals in A. (II). Next we show that we may assume that A is generated by one element. This follows from the fact that finite separable field extensions are generated by one element. Indeed, let k = κ(s) be the residue field of R and let p ⊂ A be the maximal ideal corresponding to x. As f is unramified at x, K := Ap /mAp is a finite separable field extension of k (Proposition 18.27) and hence generated by one element a ¯ ∈ K (Proposition B.98). Let a ∈ A be a lift of a ¯ and define C := R[a] ⊆ A and q := p ∩ C. Then p is the unique prime ideal over q. As A is finite over R, it is finite over C. Therefore Ap = A ⊗C Cq is finite over Cq . By the choice of a, the inclusion Cq ,→ Ap is surjective modulo m. Therefore it is an isomorphism by Nakayama’s lemma. As A and C are of finite presentation over R, ∼ there exists g ∈ C \ q such that Cg → Ag (Proposition 10.52). Replacing A by C we may therefore assume that A is finite and generated by a ∈ A. (III). Let r := [K : k]. Then Nakayama’s lemma implies that 1, a, . . . , ar−1 generate the R-module A and hence h(a) = 0 for some monic polynomial h ∈ R[T ] of degree r. Hence we obtain a surjection A′ := R[T ]/(h) → A which modulo m is an isomorphism ∼ ¯ → k[T ]/(h) K. Let p′ be the inverse image of p in A′ . As A is unramified over R at p and the property of being unramified can be checked on fibers (Proposition 18.27), A′ is unramified at p′ and hence (Ω1A′ /R )p′ = 0. As Ω1A′ /R = A′ /(h′ ), where h′ is the formal derivative of h (Example 17.36 (1)), the image of h′ in A′p′ is invertible. Hence there exists g ∈ A′ \ p′ such that A′g is standard ´etale. This proves the theorem if f is unramified at p. (IV). Now assume that f is ´etale at p. We have already seen in Step (III) that we find a standard ´etale R-algebra B and surjective homomorphism of R-algebras φ : B → A which is modulo m an isomorphism. Let q := φ−1 (p) and let I := Ker(φ). We will show that I is zero in a neighborhood of q. As A is of finite presentation, I is finitely generated (Proposition B.11) and hence it suffices to show that Iq = 0 or even Iq /Iq2 = (I/I 2 )q = 0 by Nakayama’s lemma. As A is ´etale over R, there exists a unique R-algebra homomorphism u : A → B/I 2 ∼ lifting the isomorphism A → B/I = (B/I 2 )/(I/I 2 ). Then u is in particular a splitting of the exact sequence of R-modules φ ¯

0 → I/I 2 → B/I 2 −→ A → 0. Hence this exact sequence stays exact after tensoring with k. As φ⊗ ¯ R idk is an isomorphism, this shows I/I 2 ⊗R k = 0. Hence its base change to residue field of q is also zero, i.e., (I/I 2 )q ⊗Bq κ(q) = 0 and therefore (I/I 2 )q = 0 by Nakayama’s lemma. Corollary 18.43. In the situation of Section (10.13) 1.-3.,5., and 6., we assume that X0 and Y0 are locally of finite presentation over S0 , let x ∈ X and let xλ be its projection in Xλ . Then f is ´etale at x if and only if there exists λ such that fλ is ´etale at xλ .

´ 18 Etale and smooth morphisms

46

Proof. As being ´etale is stable under base change, the condition is sufficient. For the converse we may assume that Y0 = S0 because f0 is locally of finite presentation (Proposition 10.35). We may assume that all schemes are affine and that X is standard ´etale over Y = S by Theorem 18.42. Then the result is easy (Remark 18.41). (18.8) Characterization of ´ etale morphisms. We now can characterize ´etale morphisms (in the sense of Definition 18.34) as smooth morphisms of relative dimension 0 (Definition 6.14, see also Section (18.10) below for further characterizations of smooth morphisms) and also as flat unramified morphisms. Theorem 18.44. Let f : X → S be locally of finite presentation. Then the following assertions are equivalent. (i) f is ´etale. (ii) f is flat and unramified. (iii) f is smooth of relative dimension 0. Proof. (iii) ⇒ (ii). If f is smooth, then f is flat (Theorem 14.24). Moreover, all fibers are geometrically reduced and of dimension 0 (Corollary 6.32). Hence all fibers of f are unramified (Proposition 18.24) and therefore f is unramified (Proposition 18.29). (i) ⇒ (iii). As the question is local on S and X, we may assume that X = Spec A and S = Spec R, where A is a standard ´etale algebra over R. But a standard ´etale algebra is smooth of relative dimension 0 (Remark 18.40). (ii) ⇒ (i). Consider the usual diagram a0

T0 = Spec(C/I)

f

i

 T = Spec C

/X

a

 / S,

where I is an ideal of square zero. As f is unramified, we have to show that there exists b : T → X commuting with the diagram. This we may do locally on T (as we already know uniqueness of b). Hence after possibly shrinking T , we may assume that X and S are affine. Define affine schemes Z and Z0 by the following diagram with cartesian squares

f0′

 T0

/X

/Z

Z0

f′ i

 /T

f

a

 / S.

Then f0′ has a section, namely t0 := (idT0 , a0 ) : T0 → T0 ×S X = Z0 . As f0′ is separated (as morphism between affine schemes), t0 is a closed immersion (Example 9.12). As f0′ is unramified (being the base change of f ), t0 is an open immersion (Remark 18.30). Therefore there exists an open and closed subscheme W0 ⊆ Z0 such that f0′ |W0 : W0 → T0 is an isomorphism. As Z0 and Z have the same underlying topological spaces, there exists a unique open and closed subscheme W of Z such that W ×Z Z0 = W0 .

47 Then f ′ |W : W → T is a surjective closed immersion (because the corresponding homomorphisms of rings is surjective modulo I and hence surjective by Nakayama’s lemma). Moreover it is flat and of finite presentation because f has these properties. Therefore f ′ |W : W → T is an open immersion (Proposition 14.20) and hence an isomorphism. ∼ Therefore f ′ has a section t : T → W ,→ Z and we may take b as the composition of t followed by Z → X. Corollary 18.45. Let f : X → S be a morphism locally of finite presentation. Then f is ´etale if and only if it is flat and all fibers of f satisfy the equivalent conditions in Proposition 18.24. Proof. Combine Theorem 18.44 and Proposition 18.29. Remark 18.46. Theorem 18.44 implies easily that “´etale” is stable under fpqc descent in the following sense. Let f : X → S be a morphism of schemes and let g : S ′ → S be a faithfully flat quasi-compact morphism. Then f is ´etale if and only if its base change f ′ : X ×S S ′ → S ′ is ´etale. Indeed, the condition is clearly necessary as “´etale” is stable under base change. Conversely, “´etale” is stable under faithfully flat descent as this is true for the properties “unramified” (Corollary 18.32), “flat” (Corollary 14.12) and “locally of finite presentation” (Proposition 14.53). Example 18.47. Let R be a ring, f ∈ R[T ] a monic polynomial of degree n ≥ 1. Set A := R[T ]/(f ). Then A is a free R-module of rank n and in particular flat over R. Hence A is ´etale over R if and only if Ω1A/R = R[T ]/(f, f ′ ) = 0 (Proposition 18.29), i.e., if and only if (f ) + (f ′ ) = R[T ]. Moreover (f ) + (f ′ ) = R[T ] if and only if for all maximal ideals m ⊂ R the images of f and of f ′ in (R/m)[T ] are prime to each other (because it suffices to show that Ω1A/R /mΩ1A/R = Ω1(A/mA)/(R/m) = 0 for all m by Nakayama’s lemma). As a specific example, consider µn,R , the group scheme of n-th roots of unity (n an integer ≥ 1) over R, i.e., µn,R (B) = { b ∈ B ; bn = 1 }. The underlying scheme of µn,R is Spec R[T ]/(T n − 1). Then the above discussion shows that µn,R is ´etale over R if and only if n is invertible in R. In some cases, flatness is easy to see. For instance we get the following examples of ´etale morphisms using Theorem 14.128. Corollary 18.48. Let f : X → Y be a morphism of locally noetherian schemes which is locally of finite type, let x ∈ X, y := f (x). Assume that the following conditions are satisfied. (a) dim(OY,y ) = dim(OX,x ). (b) OY,y is regular and OX,x is Cohen-Macaulay. (c) f is unramified at x. Then f is ´etale at x. It is often useful to express the fact that f is unramified in x by the triviality of ′ = 1 (Corollary 18.28). ramification and inseparable inertia index ex/y = fx/y Proof. Replacing X by an open neighborhood of x we may assume that f is quasi-finite (Corollary 18.28). Then Conditions (a) and (b) imply that f is flat at x by Theorem 14.128. Therefore f is ´etale at x by Theorem 18.44.

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48

Smooth morphisms We now examine the notion of smooth morphisms. The main results on smooth morphisms are Theorem 18.56 and its noetherian variant Theorem 18.63 which give several characterizations of smooth morphisms.

(18.9) Geometrically regular schemes. One characterization of smoothness will be via flatness and geometric regularity of the fibers. We already touched upon this topic briefly in Section (6.12). Therefore we start our examination of smooth morphisms by studying geometrically regular schemes. We first recall the definition of “geometrically regular” (see also Exercise 5.20 and Exercise 6.19). Definition 18.49. Let k be a field, let X be a k-scheme locally of finite type, and let x ∈ X. Then X is called geometrically regular at x over k if for every field extension K ⊇ k the local ring OX⊗k K,¯x is regular for all points x ¯ ∈ X ⊗k K whose projection in X is x. The k-scheme X is called geometrically regular over k if it is geometrically regular in all its points. In other words, X is geometrically regular if X ⊗k K is regular for all field extensions K of k. This definition is a special case of the Definition of a geometrically regular scheme given in Exercise 6.19. The exercise also shows that it suffices to check regularity of X ⊗k K for finite purely inseparable extensions K, see also Proposition 18.53 below for a similar result. Remark 18.50. As in the discussion of geometric reducedness/irreducibility/connectedness in Section (5.13), the interesting question is whether a point stays regular after base change. The converse is always true: Let K be a field extension of k and let x ¯ be a point of X ⊗k K lying over x. If OX⊗k K,¯x is regular, then OX,x is regular. This follows from Proposition 14.59 because X ⊗k K → X is faithfully flat. Similarly, Proposition 14.59 also implies that if X → Y is a flat morphism of kschemes locally of finite type, and X is geometrically regular at x ∈ X over k, then Y is geometrically regular at f (x) over k. Lemma 18.51. Let (Aλ )λ be a filtered inductive systems of regular rings. Assume that A := lim Aλ is noetherian. Then A is regular. −→

Proof. We have to show that for all p ∈ Spec A the local ring Ap is regular. As filtered colimits commute with localization and localization preserves flatness, we may replace A by Ap and Aλ by (Aλ )pλ , where pλ is the inverse image of p in Aλ . Hence we can assume that A and all Aλ are local and that the transition homomorphisms φλµ are local. Let mλ (resp. m) be the maximal ideal of Aλ (resp. of A). Then m is the inductive limit of the mλ . RecallLthat A is regular if and only if the canonical homomorphism n n+1 σA : SymA (m/m2 ) → is an isomorphism (Proposition B.76). As the n≥0 m /m formation of symmetric algebra commutes with filtered colimits and as filtered colimits preserve exact sequences, the bijectivity of σAλ implies the bijectivity of σA .

49 Proposition 18.52. Let k be a field, let X be a k-scheme locally of finite type, and let x ∈ X such that OX,x is regular. Let K be a separable extension of k. Then OX⊗k K,¯x is regular for all points x ¯ in X ⊗k K lying over x. Proof. As OX,x → OXK ,¯x is a flat local homomorphism, OXK ,¯x is regular if and only if OXK ,¯x /mx OXK ,¯x = κ(x) ⊗k K is regular (Proposition B.77 (5)). Hence it suffices to show that L ⊗k K is regular, where L is the finitely generated extension κ(x) of k. First note that L ⊗k K is noetherian. Indeed, L is the field of fractions of some finitely generated k-algebra B and hence L ⊗k K is the localization of the ring B ⊗k K which is noetherian because it is of finite type over K. Writing K as filtered union of finitely generated subextensions Kλ we may assume that K is finitely generated by Lemma 18.51. If K is finitely generated, then K is a finite separable extension of a purely transcendental extension k(T) := k(T1 , . . . , Tn ) (Proposition B.97). Set A := L ⊗k k(T1 , . . . , Tn ). Then A is a localization of the regular ring L ⊗k k[T1 , . . . , Tn ] ∼ = L[T1 , . . . , Tn ] and therefore regular. As K is faithfully flat over k(T1 , . . . , Tn ), p : Spec L ⊗k K → S := Spec A is faithfully flat. Moreover, for every s ∈ S the fiber p−1 (s) = Spec(κ(s) ⊗k(T) K) is a product of finite separable field extensions of κ(s), in particular p−1 (s) is regular. Therefore L ⊗k K is regular by Proposition 14.59. Proposition 18.53. Let k be a field, let X be a k-scheme locally of finite type, and let x ∈ X. Then X is geometrically regular in x if and only if there exists a perfect field extension K of k and a point x ¯ of X ⊗k K lying over x such that OX⊗k K,¯x is regular. Proof. By Remark 18.50 it suffices to show that if k is perfect and OX,x is regular, then X is geometrically regular in x. This follows from Proposition 18.52 because every extension of a perfect field is separable. (18.10) Characterization of smooth morphisms. Recall that a morphism of schemes f : X → S is called smooth of relative dimension d at x ∈ X, if there exist affine open neighborhoods U of x and V = Spec R of f (x) such that f (U ) ⊆ V , and an open immersion j : U ,→ Spec R[T1 , . . . , Tn ]/(f1 , . . . , fn−d ) of R-schemes for suitable n ≥ d and f1 , . . . , fn−d , such that the Jacobi criterion holds:   ∂fi ∈ M(n−d)×n (κ(x)) has rank n − d. (x) (18.10.1) Jf1 ,...,fn−d (x) = ∂Tj i,j In fact, using the results of this chapter one can show that one may assume that j is an isomorphism (Exercise 18.21). The morphism f is called smooth at x if it is smooth of some relative dimension d at x. Finally f is called smooth (of relative dimension d) if f is smooth (of relative dimension d) at all x ∈ X. If f is smooth of relative dimension d at x, then we set (18.10.2)

dimx (f ) := d.

For a morphism f : X → S locally of finite presentation, the set { x ∈ X ; f is smooth of relative dimension d at x} is an open subset of X. For a smooth morphism f : X → S the map x 7→ dimx (f ) is locally constant. Smoothness has the usual permanence properties:

´ 18 Etale and smooth morphisms

50

Remark 18.54. Clearly, all open immersions are smooth. We have already seen in Proposition 6.15 that the property of being smooth is local on the source, local on the target, and stable under base change. In Proposition 18.59 below we will see that smoothness is stable under composition, under faithfully flat descent, and under passage to cofiltered limits of schemes. The Jacobi criterion can be expressed via differentials as follows. Proposition 18.55. Let R be a ring, I ⊆ B := R[T1 , . . . , Tn ] a finitely generated ideal, A := B/I. Let p be a prime ideal of A. Then the following assertions are equivalent. (i) A is smooth over R at p. (ii) The localization of the sequence (17.5.10) 0 → I/I 2 −→ Ω1B/R ⊗ A −→ Ω1A/R → 0 at p is exact and split. (iii) The κ(p)-linear map I/I 2 ⊗A κ(p) → Ω1B/R ⊗B κ(p) induced by I → Ω1B/R , b 7→ d(b), is injective. Proof. Recall that Ω1B/R is a free B-module with basis dT1 , . . . , dTn (Example 17.8) and that the sequence in (ii) is always right exact (Proposition 17.33). Therefore the equivalence of (ii) and (iii) follows from Proposition 8.10. ∂ Let q ⊂ B be the unique (prime) ideal of B with q/I = p. Let ( ∂T )1≤i≤n be the i dual basis in DerR (B, B) = HomB (Ω1B/R , B) of (dTi )i . As I is finitely generated, the smoothness of A at p is equivalent to the existence of f1 , . . . , fm ∈ I with 0 ≤ m ≤ n such that their images in Iq generate Iq and such that (after a possible renumbering ∂ / q. But this is equivalent to the , dfj ⟩)1≤i,j≤m ∈ of the indeterminates) we have det(⟨ ∂T i 2 injectivity of the map I/I ⊗A κ(q) = I ⊗B κ(q) → Ω1B/R ⊗B κ(q) induced by d (again by Proposition 8.10). As we have κ(p) = κ(q), this proves the equivalence of (i) and (iii). The same proof shows a similar equivalence if B is an arbitrary R-algebra such that Ω1B/R is a projective B-module of finite type. Then locally on Spec B one can choose T1 , . . . , Tn ∈ B such that (dTi )i yields a basis of Ω1B/R ⊗B κ(p). This holds in particular if B is a smooth R-algebra (see Corollary 18.58 below). In fact one can show that a similar result holds if B is only formally smooth over R (see [BouAC10] O X, §7.9, Th´eor`eme 3). We will now state the main theorem on smooth morphisms and deduce some corollaries. We will give further characterizations of smoothness in the situation where all involved schemes are locally noetherian in Theorem 18.63. Theorem 18.56. Let f : X → S be a morphism locally of finite presentation, let x ∈ X, and s := f (x). Then the following assertions are equivalent. (i) f is smooth at x. (ii) There exist an open neighborhood U of x in X and a morphism g : U → AnS such that g is ´etale in x and such that f |U is the composition g

U −→ AnS −→ S, where the second morphism is the canonical morphism.

51 (iii) f is flat in x and dimκ(x) Ω1X/S (x) ≤ dimx f −1 (f (x)). (iv) f is flat in x and the κ(s)-scheme f −1 (f (x)) is geometrically regular at x. (v) There exists x ∈ U ⊆ X open such that f |U is formally smooth. Moreover, if f is smooth of relative dimension d at x, then n = d in (ii) and (18.10.3)

d = dimκ(x) Ω1X/S (x) = dimx f −1 (f (x)) = dimK Tξ (X/S)

for every field K and every S-morphism ξ : Spec K → X with image x. Proof. Step I: (i) ⇒ (iii)+(iv)+(18.10.3) and (iii) ⇔ (iv). We have already seen in Theorem 14.24 that a smooth morphism is flat. Hence we may assume that S = Spec k is a field. We next claim that we may assume that k = κ(x) by replacing X by X ⊗k κ(x). Indeed being smooth is stable under base change. Moreover, we have Ω1X/S (x) ∼ = Tx (X/k)∨ = Tx (X ⊗k κ(x)/κ(x)) by Remark 17.44 and we have dimx (X) = dimx (X ⊗k κ(x)) (apply Proposition 5.38 to all components of X containing x). Finally, X is geometrically regular in x if and only if X ⊗k κ(x) is geometrically regular in x by Remark 18.50. This shows our claim. But if κ(x) = k, then dimx (X) = dim OX,x (Proposition 5.26) and dimk Ω1X/k (x) = dim Tx (X/k). Hence (i) implies (iv) by Corollary 6.32, and (iv) and (iii) are equivalent by Theorem 6.28. This also shows (18.10.3), where we also use the compatibility of the relative tangent base change with field extensions (Remark 6.12 (2)). Step II: (iii)+(iv)+(18.10.3) ⇒ (ii) with n = dimκ(x) Ω1X/S (x). We may assume that S = Spec R and X = Spec A are affine. Set n := dimκ(x) Ω1X/S (x). After possibly localizing A we find a1 , . . . , an ∈ A such that the images of da1 , . . . , dan ∈ Ω1A/R in Ω1A/R ⊗A κ(x) are a κ(x)-basis. Let ψ : B := R[T1 , . . . , Tn ] → A be the R-algebra homomorphism with ψ(Ti ) = ai . We will show that g := Spec(ψ) : X → AnR = Spec(B) is ´etale in x. The morphism g is of finite presentation by Proposition 10.35. By Theorem 18.44 it suffices to show that g is unramified and flat at x. g is unramified at x. As f is locally of finite presentation, Ω1A/R is of finite presentation by Corollary 17.34. Moreover, we have an exact sequence (17.5.8) u

Ω1B/R ⊗B A −→ Ω1A/R −→ Ω1A/B → 0, dTi ⊗ 1 7−→ dai . As da1 , . . . , dan generate (Ω1A/R )px (Nakayama’s lemma), the homomorphism upx is surjective. Hence (Ω1A/B )px = 0. Therefore g is unramified at x by Proposition 18.27. g is flat at x. As X is flat over S in x, we can apply the fiber criterion for flatness (Theorem 14.25) and hence we may assume that R = k is a field. As flatness can be checked after a faithfully flat base change, the same arguments as in Step I show that we may assume κ(x) = k. By (18.10.3) we then have n = dim(OX,x ) = dim(OAnk ,g(x) ). Therefore Corollary 18.48 shows that g is ´etale at x because both local rings are regular. In particular g is flat at x.

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52

Step III: (ii) ⇒ (v). We may assume that f is the composition of an ´etale morphism g : X → AnS followed by the canonical morphism AnS → S. As being formally smooth is stable under base change, the implication follows because ´etale morphisms are formally smooth and AnS → S is formally smooth by Example 18.13. Step IV: (v) ⇒ (i). Again we may assume that S = Spec R and X = Spec A are affine. As A is of finite presentation over R, we can write A = R[T1 , . . . , Tn ]/I for some finitely generated ideal I. By Proposition 18.55 it suffices to show that the sequence (17.5.10) 0 → I/I 2 −→ Ω1R[T1 ,...,Tn ]/R ⊗ A −→ Ω1A/R → 0 is split and exact (as the middle is a free A-module, this implies that I/I 2 and Ω1A/R are finitely generated projective modules). This follows from Proposition 18.20. We deduce the following global version of Theorem 18.56. The equivalence between (i) and (ii) is called the infinitesimal lifting criterion for smoothness. Corollary 18.57. Let f : X → S be a morphism of schemes. Then the following assertions are equivalent. (i) f is smooth. (ii) f is locally of finite presentation and formally smooth. (iii) There exists an open covering (Ui )i of X and for all i a factorization of f |Ui of the form Ui → AdS → S with Ui → AdS ´etale. (iv) f is flat, locally of finite presentation, and all fibers of f are geometrically regular. Proof. All assertions are local on source and target (for (ii) use Proposition 18.12) and hence the equivalences follow from Theorem 18.56. The following corollary shows in particular that the tangent bundle TX/S is a vector bundle of rank d over X if X is smooth of relative dimension d over S. Corollary 18.58. Let X → S be smooth of relative dimension d. Then Ω1X/S is a locally free OX -module of rank d. g

Proof. We may assume that there exists a factorization of f of the form X −→ AdS → S with g ´etale. Then g ∗ Ω1Ad /S → Ω1X/S is an isomorphism by Corollary 18.19. As Ω1Ad /S is S

free of rank d (Example 17.8), Ω1X/S is a free OX -module of rank d.

S

In particular, for a smooth S-scheme X of relative dimension d the top exterior power ΩdX/S = det(Ω1X/S ) (see Remark 17.58) is a line bundle on X; it is called the canonical bundle of X over S. Any divisor with associated line bundle ΩdX/S is called a canonical divisor . Using Theorem 18.56 the following permanence properties for smoothness are now easy to see. Proposition 18.59. The property of being smooth is stable under composition, stable under base change and stable under faithfully flat descent. More precisely: (1) Let f : X → Y , g : Y → Z be morphisms of schemes, let x ∈ X and y := f (x). If f is smooth at x and g is smooth at y, then g ◦ f is smooth at x with (18.10.4)

dimx (g ◦ f ) = dimx (f ) + dimy (g).

53 In (2) and (3), we consider a cartesian diagram of schemes X′ f′

 S′

g′

□ g

/X f

 / S,

and let x′ ∈ X ′ , x := g ′ (x) ∈ X. (2) If f is smooth of relative dimension d at x, then f ′ is smooth of relative dimension d at x′ . (3) If f ′ is smooth of relative dimension d at x′ and g is flat at f ′ (x′ ), then f is smooth of relative dimension d at x. (4) In the situation of Section (10.13) 1.-3., 5.,6., assume that X0 and Y0 are of finite presentation over S0 . Let x ∈ X and let xλ ∈ Xλ be its image for all λ. Then f : X → Y is smooth at x if and only if it there exist a λ such that fλ is smooth at xλ . The stability under base change was also easy to see with our definition of smoothness given in Volume I (Proposition 6.15). Proof. By Theorem 18.56 we know that a morphism is smooth if and only if it is formally smooth and locally of finite presentation, and these two properties are stable under composition. To show (18.10.4) we may assume, after shrinking X and Y , that f and g are smooth of relative dimension d and e, respectively. By Proposition 18.18 there is a locally split exact sequence 0 → f ∗ Ω1Y /Z → Ω1X/Z → Ω1X/Y → 0. By Corollary 18.58, Ω1X/Y (resp. f ∗ Ω1Y /Z ) is locally free of rank d (resp. e). Therefore Ω1X/Z is locally free of rank d + e. This proves (18.10.4). To show (2) we use characterization (iii) of Theorem 18.56. As the formation of Ω1X/S is stable under base change S ′ → S (Proposition 17.30), we have Ω1X/S (x) ⊗κ(x) κ(x′ ) ∼ = Ω1X ′ /S ′ (x′ ) = (g ′∗ Ω1X/S )(x′ ) ∼ and in particular dimκ(x) Ω1X/S (x) = dimκ(x′ ) Ω1X ′ /S ′ (x′ ). Moreover, Proposition 5.38 shows that dimx f −1 (f (x)) = dimx′ f ′−1 (f ′ (x′ )). Hence (2) follows because flatness is stable under base change. To see (3) we have to show that f is flat at x. To see this we may replace S by Spec OS,f (x) and S ′ by Spec OS ′ ,s′ . But then g is faithfully flat (Example B.18) and the flatness of f ′ in x′ implies that f is flat in x because flatness is stable under faithfully flat descent. Let us show (4). The hypotheses imply that f0 is of finite presentation by Proposition 10.35 3. Hence their base changes fλ and f are all of finite presentation. The condition is clearly sufficient since smoothness is stable under base change. Suppose that f is smooth at x. Choose an open quasi-compact neighborhood U of x such that there g f |U can be factorized into U −→ AdY → Y with g ´etale. By Theorem 10.57 there exists a λ and a quasi-compact open neighborhood Uλ of xλ such that the preimage of Uλ in X is U . After possibly enlarging λ we also find a factorization of fλ|Uλ into a morphism gλ : Uλ → AdYλ followed by AdYλ → Yλ by Theorem 10.63. Again after possibly enlarging λ we may assume gλ is ´etale by Corollary 18.43. Hence fλ is smooth in xλ .

54

´ 18 Etale and smooth morphisms

To show (4) it is also possible to use the fact that morphisms locally of finite presentation are smooth if and only if they are flat with geometrically regular fibers. An argument similar as in the proof of Remark 18.33 then shows that if f is smooth in x then there exists λ such that fλ has geometrically regular fiber in xλ and it remains to prove a statement as in (4) for “smooth” replaced by “flat”. This can be done, see [EGAIV] O (11.2.6) and it gives even a slightly more general statement as one has only to assume that X0 and Y0 are locally of finite presentation over S0 , see [EGAIV] O (17.7.8) for details. Together with Remark 18.33 this line of argument also gives a new proof of Corollary 18.43. Here we decided to circumvent the (difficult) theorem [EGAIV] O (11.2.6). For yet another proof of (4) using the characterization of smoothness given in Proposition 18.55 see [Sta] 0C0C. Proposition 18.60. Let S be a scheme and let f : X → Y be a morphism of S-schemes locally of finite presentation over S. Let x ∈ X, y := f (x). (1) If X is smooth at x over S and Y is unramified at y over S, then f is smooth at x. (2) If X is smooth at x over S and f is flat at x, then Y is smooth at y over S. Proof. To show (1) we may pass from Y and X to open neighborhoods of y and of x, respectively. Thus we may assume that X is smooth over S and that Y is unramified over S. Then Assertion (1) is a special case of Remark 18.30. Let us show (2). As f is flat at x and X is flat at x over S, Y is flat over S by Corollary 14.27. Hence we may assume S = Spec k is a field. As X is smooth over k, X is geometrically regular at all x ∈ X, and as f is flat at x, Y is geometrically regular at y (Remark 18.50). Therefore Y is smooth over k. We may express the fact that a scheme is smooth in terms of the functor it represents. Remark 18.61. Let S be a scheme, and let us say that a ring C is an S-algebra if Spec C is given an S-scheme structure. Let X be an S-scheme, and let hX : (Sch/S) → (Sets) be the corresponding functor. As usual we write hX (C) instead of hX (T ) if T = Spec C is an S-algebra. Then by definition X is formally unramified / formally smooth / formally ´etale over S if and only if hX (C) → hX (C/I) is injective / surjective / bijective for every S-algebra C and every nilpotent (or, equivalently, every square zero) ideal I ⊂ C. Exercise 18.16 shows that if f is locally of finite presentation, then it suffices to consider only local rings C. Below in Theorem 18.63 we will see that if S is in addition locally noetherian, then it even suffices to consider local Artinian rings C. Moreover, in Theorem 20.12 we will see that if X is smooth (resp. ´etale) over S, then hX (C) → hX (C/mC ) is surjective (resp. bijective) for every local henselian (see Definition 20.1 below) S-algebra C with maximal ideal mC (not necessarily nilpotent). In particular, this holds if C is a complete local ring (Example 20.3). Finally, the property of being locally of finite presentation can also be expressed in terms of the functor hX ([EGAIV] O (8.14.2), see also Exercise 10.22): We call a functor opp F : (Sch/S) → (Sets) locally of finite presentation if F is a sheaf for the Zariski topology (Section (8.3)) and if for all filtered system (Cλ )λ of S-algebras the canonical map colim F (Spec Cλ ) → F (Spec(colim Cλ )) is bijective. Then an S-scheme X is locally of finite presentation over S if and only if the functor hX is locally of finite presentation.

55 (18.11) Characterizations of smooth morphisms in the noetherian case. If S is locally noetherian, there are further important characterizations of smooth morphisms X → S. We use the following notion. Definition 18.62. A surjective ring homomorphism π : C → C0 of local Artinian rings is called small if I = ker(π) has length 1 as a C-module. In other words, I = (ε) for some 0 ̸= ε ∈ mC with εmC = 0 (in particular ε2 = 0). Theorem 18.63. Let S be a locally noetherian scheme, and let f : X → S be a morphism locally of finite type. Let x ∈ X and s := f (x). Then the following assertions are equivalent. (i) f is smooth at x. (ii) For all small surjections C → C0 of local Artinian rings and for all commutative diagrams of the form a0 /X Spec C0  Spec C

f

a

 / S,

∼

such that a0 has image {x} and induces an isomorphism κ(x) → C/mC , there exists b : Spec C → X commuting with the diagram. If further κ(s) = κ(x), then (i) and (ii) are equivalent to (iii) The ObS,s -algebra ObX,x is isomorphic to the algebra ObS,s [[T1 , . . . , Td ]] of formal power series. In this case the integer d in (iii) is the relative dimension of f in x. In the proof of the theorem we will use the following lemma. Lemma 18.64. Let R → A and R → B be a local homomorphisms of local rings, and let φ : A → B be a local homomorphism of R-algebras. Suppose that φ induces an isomorphism on residue fields and a surjective map φ¯ : mA /(m2A + mR A) → mB /(m2B + mR B). ˆ induced by φ on completions is surjective. Then the map φˆ : Aˆ → B Proof. Let κR = R/mR be the residue field of R and similarly define κA and κB . The homomorphism φ yields a commutative diagram mR /m2R ⊗κR κA

/ mA /m2

/ mA /(m2 + mR A) A



 / mB /m2 B



mR /m2R ⊗κR κB

A

/0

φ ¯

/ mB /(m2 + mR B) B

/0

with exact rows. By hypothesis, the left vertical map is an isomorphism and φ¯ is surjective. Hence the vertical map in the middle is surjective. Hence φ induces a surjective map grmA (A) → grmB (B) because mB /m2B generates grmB (B) as a κB -algebra. Now Proposition B.49 implies that φ ˆ is surjective.

´ 18 Etale and smooth morphisms

56

Proof. [of Theorem 18.63] All assertions of Theorem 18.63 are local on S and on X. Hence we can assume that S = Spec R for a noetherian ring R and X = Spec A for a finitely generated R-algebra A. If f is smooth at x, then we have already seen that f is formally smooth. Hence (i) implies (ii). Step I: (iii) ⇒ (i) if κ(x) = κ(s). Theorem 18.56 shows that it suffices to show that f is flat at x and that the fiber Xs is smooth at x over κ(s). As ObS,s is noetherian, ObS,s [[T1 , . . . , Td ]] is flat over ObS,s (Example B.47 and Proposition B.41 (1)). Hence (iii) implies that ObX,x is flat over ObS,s . As the completion of a local noetherian ring C is faithfully flat over C, this implies that OX,x is flat over OS,s which means that f is flat at x. Hence we may replace X by the fiber Xs and thus S by Spec κ(s). By hypothesis x is then κ(s)-rational. But in this case we have already shown in Theorem 6.28 that (iii) implies (i). Step II: (ii) ⇒ (iii) if κ(x) = κ(s). Preliminary remark. If C is a local Artinian ring and I ⊆ mC is an ideal, then I is of finite length as a C-module (Proposition B.36). Hence there exists a sequence 0 = I0 ⊆ I1 ⊆ · · · ⊆ Il = I such that Ij /Ij−1 has length 1. In other words, C/Ij−1 → C/Ij is a small surjection of local Artinian rings. Hence we may assume that (ii) is satisfied for an arbitrary ideal I ̸= C of the local Artinian ring C. Construction of ψ : ObX,x → B := ObS,s [[T1 , . . . , Td ]]. Let (t¯1 , . . . , t¯d ) be a basis of the κ(x)-vector space Ω1f −1 (s)/κ(s) (x) = mx /(m2x +ms OX,x ) and let ti ∈ mx be a representative of t¯i . We have B/(m2B + ms B) ∼ = κ(s) ⊕ κ(s)T1 ⊕ · · · ⊕ κ(s)Td , 2 ∼ κ(s) ⊕ κ(s)t¯1 ⊕ · · · ⊕ κ(s)t¯d , OX,x /(mx + ms OX,x ) = where we use κ(s) = κ(x) for the second isomorphism. Hence there exists a homomorphism of OS,s -algebras ψ0 : OX,x → B/(m2B + ms B) sending ti to the images of the Ti . By the preliminary remark we can lift ψ0 first to ψ1 : OX,x → B/m2B and then successively to a projective system (ψn : OX,x → B/mn+1 B )n≥1 . By passing to the limit we obtain a continuous homomorphism OX,x → lim B/mn+1 = B and hence a continuous B homomorphism

←− n

ψ : ObX,x → B. Moreover, ψ is surjective by Lemma 18.64. Construction of an inverse. As ψ is surjective, we find τi in the maximal ideal of ObX,x with ψ(τi ) = Ti for all i. Let φ : B → ObX,x be the unique continuous homomorphism with φ(Ti ) = τi for all i. Then (ψ ◦ φ)(Ti ) = Ti for all i and hence ψ ◦ φ = idB . Hence φ induces an injective map φ¯ : mB /(m2B + ms B) → mx /(m2x + ms OX,x ) of κ(s)-vector spaces of the same dimension. Hence φ¯ is surjective. Now we again apply Lemma 18.64 to obtain that φ is surjective. Hence ψ and φ are mutually inverse isomorphisms. Step III: (ii) ⇒ (i). It is enough to show that the proof of “(ii) ⇒ (i)” may be reduced to the case that κ(s) = κ(x) (then we are done by Steps I and II). By Lemma 18.65 below there exists a faithfully flat local ring homomorphism OS,s → R′ with R′ noetherian such that R′ /mR′ = κ(x). Define X ′ by the cartesian diagram

57 g

X′ f′

 Spec(R′ )

□

/ Spec(OS,s )

/X  /S

f

The points x ∈ X and s′ := mR′ ∈ Spec R′ are both sent to s ∈ S. Hence g −1 (x) ∩ f ′−1 (s′ ) = Spec(κ(x) ⊗κ(s) κ(s′ )) (Lemma 4.28), and the multiplication κ(x) ⊗κ(s) κ(s′ ) → κ(x) corresponds to a point x′ ∈ X ′ with f ′ (x′ ) = s′ , g(x′ ) = x and κ(x′ ) = κ(s′ ). As Spec R′ → S is flat, it suffices to show that f ′ is smooth at x′ (Proposition 18.59). As κ(x′ ) = κ(s′ ) = κ(x), the morphism f ′ still satisfies (ii). This gives the desired reduction to the case κ(s) = κ(x). Lemma 18.65. Let R be a local ring with residue field k and let k ′ be a field extension of k. Then there exists a local ring R′ with residue field k ′ and a faithfully flat ring homomorphism φ : R → R′ such that φ(mR )R′ = mR′ . If k ′ ⊇ k is finite, one can assume that R′ is a finite free R-algebra. If R is noetherian, one can assume that R′ is a complete local noetherian ring. In the proof of Theorem 18.63 we used the lemma only in the case where k ′ is a finitely generated extension of k, and we will prove the lemma only in this case. For the general case see [EGAInew] 0I (6.8.2) and (6.8.3). Proof. [if k ′ is a finitely generated field extension of k] By induction on the number of generators of k ′ over k we may assume that k ′ is generated by one element over k. We first assume in addition that k ′ is a finite extension of k, hence k ′ ∼ = k[T ]/(f ) for some monic polynomial f ∈ k[T ]. Choose a monic polynomial F ∈ A[T ] with image f in k[T ] and set A′ := A[T ]/(F ). Then A′ is a finite free A-module. Let I := mA A[T ] + (F ) ⊂ A[T ]. Then A[T ]/I ∼ = k[T ]/(f ) = k ′ . Hence IA′ = mA A′ is a maximal ideal of A′ and this is the only prime ideal over mA . As A′ is a finite A-algebra, the maximal ideals of A′ are exactly the prime ideals lying over mA (by the going up theorem, see Theorem B.56). Hence A′ is local. For a general monogeneous extension k ′ of k it thus suffices to consider the case of ′ k = k(T ) for a transcendental element T ∈ k ′ . But then we can set A′ := A[T ]p , where p ⊂ A[T ] is the prime ideal of polynomials with coefficients in mA . In both cases we see that A′ is noetherian if A is noetherian. By replacing A′ by its completion we may then even assume that A′ is complete. As ´etale morphisms are smooth morphisms of relative dimension 0 we immediately obtain the following corollary. Corollary 18.66. Let S be a locally noetherian scheme and let f : X → S be a morphism locally of finite type over S, let x ∈ X, s := f (x) such that κ(s) = κ(x). Then f is ´etale at x if and only if the induced homomorphism ObS,s → ObX,x between the complete local rings is an isomorphism. (18.12) Smooth schemes over a field. Using the results above we obtain the following criteria for smoothness over a field (see also Theorem 6.28). For a further characterization of smoothness over a field see Exercise 18.18.

58

´ 18 Etale and smooth morphisms

Proposition 18.67. Let k be a field, let X be a k-scheme locally of finite type, and let x ∈ X be a point. Consider the following assertions. (i) OX,x is regular and κ(x) is a separable extension of k. (ii) X is smooth over k at x. (iii) X is geometrically regular at x. ¯ ∈ X ⊗k K whose projection (iv) There exists a perfect field extension K ⊇ k and a point x in X is x such that OX⊗k K,¯x is regular. (v) Ω1X/k,x is a free OX,x -module of rank dimx (X). (vi) OX,x is regular. Then the implications “(i) ⇒ (ii) ⇔ (iii) ⇔ (iv) ⇔ (v) ⇒ (vi)” hold. If k is perfect, then all assertions are equivalent. Proof. We have already seen “(ii) ⇔ (iii)” (Theorem 18.56), “(v) ⇔ (ii)” (Theorem 18.56 and Corollary 18.58), and “(iii) ⇔ (iv)” (Proposition 18.53). Moreover, the implication “(iii) ⇒ (vi)” is trivial. If k is perfect, then κ(x) is automatically separable over k, hence (vi) implies (i) in this case. If (i) holds, then Proposition 18.52 shows that X ⊗k κ(x) is still regular at the κ(x)¯ lying over x. But then X ⊗k κ(x) is smooth at x ¯ (Theorem 6.28) and rational point x hence X is smooth at x by faithfully flat descent (Proposition 18.59). Corollary 18.68. Let X be a k-scheme locally of finite type over a field k, and let x ∈ X(k) be a k-valued point, considered as a closed point of X. Then the following assertions are equivalent (i) X is smooth over k in x. (ii) OX,x is regular. (iii) One has Tx (X) = dimx (X). Proof. The equivalence of (i) and (ii) follows from Proposition 18.67. As x ∈ X(k), Tx (X) = (mx /m2x )∨ and dimx (X) = dim OX,x . This shows the equivalence of (ii) and (iii). In Assertion (v) of Proposition 18.67 it is in general important that Ω1X/k,x is a free OX,x -module of the correct rank, see Example 18.71 below. If the characteristic of k is 0, then the rank hypothesis is superfluous. Proposition 18.69. If char(k) = 0, then a k-scheme X locally of finite type is smooth at x ∈ X if and only if Ω1X/k,x is a free OX,x -module. Proof. The condition is necessary by Proposition 18.67. Let R := OX,x . Then Ω1X/k,x = Ω1R/k (17.4.3). As formation of K¨ahler differentials commutes with base change and as we can check smoothness after the field extension k → κ(x), we may assume that the residue field of R is k. As k is perfect, it suffices to show that R is regular (Theorem 18.56). This follows from the following lemma. Lemma 18.70. Let k be a field of characteristic 0 and let R be a local noetherian k-algebra with residue field k. Let n ≥ 0 be an integer. If Ω1R/k is a free R-module of rank n, then R is a regular ring of dimension n. Proof. Let m ⊂ R be the maximal ideal. By Example 17.23, the map f 7→ df yields an isomorphism of k-vector spaces

59 (*)

∼

m/m2 → Ω1R/k ⊗R k,

in particular dimk (m/m2 ) = n. We now prove the lemma by induction on n. If n = 0, then m = 0 by (*) and by Nakayama’s lemma. Hence R = k. Now suppose n > 0. Then there exists f ∈ m \ m2 . By (*) multiplication by df induces an injective map k → Ω1R/k ⊗R k. Therefore the submodule ⟨df ⟩ of Ω1R/k generated by df is a direct summand (Proposition 8.10). Hence Ω1(R/(f ))/k = Ω1R/k /⟨df ⟩ is free of rank n − 1. The induction hypothesis therefore shows that R/(f ) is regular of dimension n − 1. To deduce that R is regular it suffices to show that f is a nonzero divisor (Proposition G.19). As ⟨df ⟩ is a direct summand, we find a derivation D : R → R with D(f ) = 1 and hence D(f r ) = rf r−1 for all r ≥ 1. Let g ∈ R with gf = 0. By Proposition B.43 it suffices to show that g ∈ (f r ) for all r ≥ 1. As we have 0 = D(f g) = g + f D(g), we find g ∈ (f ). By induction we may assume that g = hf r−1 for some h ∈ R. Then 0 = D(f g) = D(f r h) = rf r−1 h + f r D(h) = rg + f r D(h). Since we are in characteristic 0, this implies that g ∈ (f r ). The implications “(ii) ⇒ (i)” and “(vi) ⇒ (v)” of Proposition 18.67 do not hold over non-perfect fields. Neither does Proposition 18.69 hold in positive characteristic: Example 18.71. (1) Let k be a field that is not perfect, let p be its characteristic, and let a ∈ k be an element that is not a p-th power of an element in k. Then K := k[a1/p ] ∼ = k[T ]/(T p −a) is a purely inseparable field extension of degree p. Consider X = Spec K. As the derivative of T p − a is zero, Ω1K/k is a 1-dimensional K-vector space (Example 17.36 (1)). Moreover OX,x = K is regular, if x denotes the unique point of X. This example shows that in Proposition 18.67 Assertion (vi) does not imply Assertion (v) in general. Now consider X = A1k and let x be the closed point corresponding to the irreducible polynomial T p − a. Then κ(x) = k[a1/p ] is a non-separable extension of k. This shows that in Proposition 6.28 Assertion (ii) does not imply Assertion (i) in general. (2) Let k be an arbitrary field of characteristic p > 0 and consider X = Spec A with A = k[T ]/(T p ), x ∈ X its unique point. Then again Ω1X/k is free of rank 1, but OX,x = A is not reduced and in particular not regular. Hence the freeness of Ω1X/k does not imply regularity of OX,x . (3) Finally, Exercise 18.19 gives an example of a scheme X over a non-perfect field k and a point x ∈ X such that OX,x is regular and geometrically reduced, but Ω1X/k,x is not a free OX,x -module. (18.13) Smooth morphisms and differentials. If X and Y are schemes locally of finite presentation over a scheme S, then every Smorphism X → Y is locally of finite presentation (Proposition 10.35). Hence the two following results follow from the analogous assertions for formally smooth morphisms (Proposition 18.18 and Proposition 18.20) and the fact that a morphism is smooth if and only if it is locally of finite presentation and formally smooth (Corollary 18.57). Corollary 18.72. Let f : X → Y be a morphism of S-schemes locally of finite presentation.

´ 18 Etale and smooth morphisms

60 (1) Let f be smooth, then the sequence (17.5.8) (18.13.1)

0 −→ f ∗ Ω1Y /S −→ Ω1X/S −→ Ω1X/Y −→ 0

is exact and locally on X split. (2) Conversely, assume that X is smooth over S and that (18.13.1) is exact and locally split. Then f is smooth. Corollary 18.73. Let i : X → Z be an immersion of S-schemes locally of finite presentation. (1) If X is smooth over S, then the sequence (17.5.10) (18.13.2)

0 −→ Ci −→ i∗ Ω1Z/S −→ Ω1X/S −→ 0

is exact and locally split. (2) Conversely, assume that Z is smooth over S and that (18.13.2) is exact and locally split, then X is smooth over S. (18.14) Smooth and ´ etale morphisms between smooth schemes. Theorem 18.74. Let f : X → Y be a morphism of S-schemes locally of finite presentation, let x ∈ X and y := f (x). Then the following assertions are equivalent. (i) Y is smooth over S at y and f is smooth (resp. ´etale) at x. (ii) X is smooth over S at x and f is smooth (resp. ´etale) at x. (iii) X is smooth over S at x and the homomorphism Ω1Y /S (y) ⊗κ(y) κ(x) → Ω1X/S (x), which is induced by f ∗ Ω1Y /S → Ω1X/S , is injective (resp. bijective). If f induces a bijection κ(y) → κ(x), then (i), (ii), and (iii) are equivalent to the following assertion. (iv) X is smooth over S at x and the map on relative tangent spaces Tx (X/S) → Ty (Y /S) induced by f is surjective (resp. bijective). Proof. Let us consider the smooth case first. As smoothness ist stable under composition, (i) implies (ii). The implication “(ii) ⇒ (i)” follows from Proposition 18.60 (2). Corollary 18.72 (1) shows that (ii) implies (iii). Conversely, if X is smooth at x, then (Ω1X/S )x is a free finitely generated OX,x -module (Corollary 18.58). Moreover, the hypothesis means that ux : f ∗ (Ω1Y /S )x → (Ω1X/S )x is injective after passing to fibers in x. Therefore ux has a left inverse (Proposition 8.10). Thus there exists an open neighborhood U of x such that the exact sequence 0 → f ∗ (Ω1Y /S )|U → (Ω1X/S )|U → (Ω1X/Y )|U → 0 splits. Hence Corollary 18.72 (2) shows that f is smooth at x. Finally, if κ(y) = κ(x), then Tx (X/S) → Ty (Y /S) is the dual of Ω1Y /S (y) → Ω1X/S (x). This proves the equivalence of (iii) and (iv) in this case. The equivalences in the ´etale case are proved in the same way using Corollary 18.19 instead of Corollary 18.72 and using that a homomorphism of free finitely generated modules over a local ring is an isomorphism if and only if it is an isomorphism modulo the maximal ideal.

61 (18.15) Open immersions and ´ etale morphisms. Lemma 18.75. Let f : X → S be a morphism locally of finite type. Then f is a monomorphism if and only if f is unramified and universally injective. Proof. The morphism f is a monomorphism, if and only if ∆f : X → X ×S X is an isomorphism. This is equivalent to ∆f being a surjective open immersion. But ∆f is an open immersion if and only if f is unramified (Proposition 18.29) and ∆f is surjective if and only if f is universally injective (Exercise 9.9). Proposition 18.76. Let f : X → S be a morphism of schemes. Then the following assertions are equivalent. (i) f is an open immersion. (ii) f is ´etale and universally injective. (iii) f is a flat monomorphism locally of finite presentation. Proof. Clearly, (i) implies (iii). The equivalence “(iii) ⇔ (ii)” follows from Lemma 18.75 and the fact that a morphism is ´etale if and only if it is flat, unramified and locally of finite presentation (Theorem 18.44). It remains to show that (ii) and (iii) imply (i). The question is local on the target and, as f is injective, local on the source. Hence we may assume that X and S are affine. In particular f is in addition quasi-compact. As flat morphisms locally of finite presentation are open (Theorem 14.35), we may replace S by the open subscheme f (S) and hence may assume that f is also surjective and therefore faithfully flat. The base change of f with f itself is the second projection X ×S X → X which is an isomorphism because the diagonal ∆f is an isomorphism. As the property of being an isomorphism is stable under descent by faithfully flat quasi-compact morphisms, this implies that f is an isomorphism. In the proof of “(iii) ⇒ (i)” the reduction to the case that f is quasi-compact is superfluous because the property of being an isomorphism is also stable under descent by faithfully flat morphisms locally of finite presentation (Exercise 14.9). (18.16) Fibre criterion for smooth and ´ etale morphisms. We deduce for a number of properties that they can be checked on fibers in the presence of flatness. Corollary 18.77. Let f : X → Y be a morphism of S-schemes. For a point s ∈ S let fs : Xs → Ys be the morphism induced by f on the fiber over s. Let X and Y be locally of finite presentation over S and let X be flat over S. Let P be one of the following properties. (a) flat, (b) smooth, (c) ´etale, (d) open immersion, (e) isomorphism. Then f has property P if and only if fs has property P for all s ∈ S. This assertion holds also holds for the property “syntomic” defined below (see Corollary 19.54).

62

´ 18 Etale and smooth morphisms

Proof. If X and Y are locally of finite presentation over S, then f is locally of finite presentation (Proposition 10.35). For the property “flat”, the assertion is the fiber criterion of flatness (Theorem 14.25). If f is flat, then we have seen that all other properties hold if and only if they hold for the fibers f −1 (y) → Spec(κ(y)) for all y ∈ Y : For “smooth” this is Theorem 18.56. For “´etale” this holds because the flat morphism f is ´etale if and only if it is unramified (Theorem 18.44) and being unramified can be checked on fibers (Proposition 18.29). For “open immersion” this follows from Proposition 18.76 and the fact that a morphism is universally injective if and only if the induced extensions of residue fields are purely inseparable (Proposition 4.35). Finally, a morphism is an isomorphism if and only if it is a surjective open immersion, and “surjective” can of course also be checked on fibers. As all properties are stable under base change, they hold for fs for all s ∈ S if they hold for f . Conversely, if they hold for all fs , then they hold for all s ∈ S and y ∈ Ys for the fiber fs−1 (y) = f −1 (y). Therefore they hold for all fibers of f and hence for f by the remarks above. (18.17) Generic Smoothness. For schemes locally of finite type over a field let us recall (and slightly refine) the following result of generic smoothness already obtained in Volume I. Proposition 18.78. Let k be a field and let X be a scheme locally of finite type over k. Then the following assertions are equivalent. (i) There exists an open dense subscheme U0 of X such that U0 is smooth over k. (ii) There exists an open dense subscheme U1 of X such that U is geometrically reduced over k. (iii) There exists an open dense subscheme U2 of X such that U2 is reduced and for every maximal point η of X the field extension k → κ(η) is separable. If k is perfect, then every field extension of k is separable. Hence in this case X has an open dense smooth subscheme if and only if it has an open dense open reduced subscheme. Proof. Any smooth k-scheme is geometrically regular and in particular geometrically reduced. This shows that (i) implies (ii). The converse holds by Remark 6.20. The equivalence of (ii) and (iii) holds by Proposition 5.49. We now generalize this result to morphisms of arbitrary schemes. The following result (see also Exercise 10.40) shows in particular that in characteristic 0 dominant morphisms of finite type between integral noetherian schemes are automatically smooth on an open dense subscheme of the source. Proposition 18.79. Let f : X → Y be a dominant morphisms of integral schemes that is locally of finite presentation. Assume that the extension of functions fields K(Y ) ⊆ K(X) is separable (e.g., if the field K(Y ) is of characteristic 0). (1) Then the smooth locus of f is an open dense subscheme of X. (2) Assume that X is regular and that K(Y ) is perfect. Then there exists a dense open subscheme V ⊆ Y such that the restriction f −1 (V ) → V of f is smooth. Recall that there is a similar result on generic flatness which only assumes that Y is integral (Theorem 10.84).

63 Proof. Let η ∈ Y be the generic point. For U = Spec A ⊆ X open affine, f −1 (η) ∩ U is a localization of A. Hence f −1 (η) is reduced. (1). As K(X) is a separable extension of K(Y ), f −1 (η) is even geometrically reduced (Proposition 5.49). Hence by Remark 6.20 there exists an open dense subscheme U ⊆ X such that U ∩ f −1 (η) is smooth over K(Y ). Therefore by replacing X by U we may assume that f −1 (η) is smooth. Now K(Y ) = OY,η is the filtered colimit of Γ(V, OY ), where V runs through the open affine neighborhoods of η. As smoothness is compatible with colimits, there exists η ∈ V ⊆ Y open affine such that the restriction f −1 (V ) → V is smooth. (2). If X is regular, f −1 (η) is regular because localizations of regular rings are again regular (Proposition B.77). Therefore f −1 (η) is a smooth K(Y )-scheme because K(Y ) is perfect. Now we conclude as in the proof of (1).

Exercises Exercise 18.1. Let A be a ring, S ⊆ A a multiplicative set. Show that S −1 A is a formally ´etale A-algebra. Exercise 18.2. Let f : X → Y and g : Y → Z be morphisms of schemes. (1) Suppose that g is formally unramified. Show that if g ◦ f is formally smooth (resp. formally ´etale), then so is f . (2) Let g be formally ´etale. Show that g ◦ f is formally smooth (resp. formally unramified, resp. formally ´etale) if and only if so is f . Exercise 18.3. Show that the property “formally unramified” is stable under faithfully flat descent. Hint: Use that two morphisms are equal if they are equal after a faithfully flat quasicompact base change. Exercise 18.4. Let R be a ring and let A be a formally smooth R-algebra. Show that Ω1A/R is a projective A-module. Exercise 18.5. Show that the property “formally smooth” is local on the source and local on the target. Hint: If A is a ring, X = Spec A, and M is a projective A-module, then show that ˜ , F )) = 0 for every quasi-coherent OX -module F . Now use ExerH 1 (X, Hom OX (M cise 18.4 and Remark 7.44. Exercise 18.6. This exercise shows that formally ´etale morphisms are not necessarily flat. Let A be a ring and let a ⊆ A be an ideal with a = a2 . (1) Show that the projection A → A/a is formally ´etale. (2) Give an example of a pair (A, a) as above such that A → A/a is not flat. Hint: One can choose as A the integral closure of the ring of p-adic integers Zp in an algebraic closure of Qp and a its maximal ideal. Exercise 18.7. Let k → K be a field extension. (1) Let char(k) = 0. Show that K is formally unramified over k if and only if K is an algebraic extension of k.

´ 18 Etale and smooth morphisms

64

(2) Let char(k) = p and let K p = { ap ; a ∈ K }. Show that K if formally unramified over k if and only if K = k(K p ). Exercise 18.8. Let X be a scheme and let g : Y → X be a morphism of schemes. A universal thickening of order at most 1 of g is a thickening Y → Y ′ of order at most 1 of X-schemes such that for every thickening T → T ′ of order at most 1 of X-schemes and for every morphism f : T → Y of X-schemes there exists a unique morphism f ′ : T ′ → Y ′ of X-schemes such that f /Y T  T′

f′

 / Y′

commutes. (1) Define a category of thickenings of order at most 1 over X and show that a universal thickening of order at most 1 of a morphism Y → X is unique up to unique isomorphism in this category. (1) (2) Suppose that g is an immersion. Show that the first infinitesimal neighborhood Yg of Y on X is the universal thickening of order at most 1 of g. (3) Show that the universal thickening Y → Y ′ of order at most 1 of g exists if g is formally unramified. Hint: Reduce to the case that Y = Spec B and X = Spec A are affine. Let P = A[(Ti )i ] be a polynomial A-algebra such that there exists a surjection of A-algebras P → B. Let I be its kernel. Use that g is formally unramified to see that d : I/I 2 → Ω1P/A ⊗P B has a B-linear section and deduce that there exists an ideal I 2 ⊆ I ′ ⊆ I such that d induces an isomorphism I ′ /I 2 ∼ = Ω1P/A ⊗P B. Define Y ′ := Spec P/I ′ . (4) Suppose that g is formally unramified and that Y → Y ′ is the universal thickening of order at most 1 for g. Show that Y ′ → X is formally unramified. Let g be formally unramified and let Y → Y ′ be the universal thickening of order at most 1 for g. Then the quasi-coherent ideal defining Y → Y ′ , considered as a quasi-coherent OY -module is called the conormal sheaf of g and denoted by Cg . Exercise 18.9. Let R → A → B be ring homomorphism. Then the B-module ΓB/A/R := Ker(Ω1A/R ⊗A B → Ω1B/R ) is called the imperfection module of the A-algebra B over R. (1) Let B → C be a flat homomorphism. Show that there exists a natural exact sequence of C-modules 0 → ΓB/A/R ⊗B C → ΓC/A/R → ΓC/B/R → Ω1B/A ⊗B C → Ω1C/A → Ω1C/B → 0. (2) Show that if B is a formally smooth A-algebra, then ΓB/A/R = 0. (3) Now let k → K → L be field extensions. Assume that k is perfect. Show that L is a separable extension of K if and only if ΓL/K/k = 0 Exercise 18.10. Let k be a perfect field, let K be an extension of k and let L be a finitely generated extension of K. Show the “Cartier equality” dimL (Ω1L/K ) = trdegK L + dimL ΓL/K/k . Hint: Exercise 18.9.

65 Exercise 18.11. Let f : X → S be a morphism locally of finite type. Show that f is unramified if and only if for every S-scheme S ′ → S every section of fS ′ × X ×S S ′ → S ′ is an open immersion. Exercise 18.12. Let (Ri )i∈I be a filtered system of rings and let R be its colimit. Let A be an unramified R-algebra (resp. an unramified R-algebra of finite presentation). Show that there exists i ∈ I and an unramified Ri -algebra Ai such that A is a quotient of R ⊗Ri Ai (resp. such that A ∼ = R ⊗Ri Ai ). Exercise 18.13. Let f : X → S be a morphism of schemes, let x ∈ X, s := f (x). Show that f is unramified at x if and only if there exists an open affine neighborhood V = Spec R of s and an open affine neighborhood U = Spec A ⊆ f −1 (V ) of x such that A is isomorphic to a quotient of a standard ´etale algebra. Hint: Use Exercise 18.12 to reduce to the noetherian case. Exercise 18.14. Let S be a scheme, let L be an invertible OS -module, let n ≥ 1 ∼ be an integer, and let u : L ⊗n → OS be a fixed isomorphism of OS -modules. Using u ⊗(d+n) ∼ → L ⊗d for all d ≥ 0. Via this identification the canonical we may identify L ⊗d ⊗e morphism L ⊗L → L ⊗(d+e) defines the structure of a Z/nZ-graded OS -algebra Ln−1 ⊗d on A := d=0 L . Show that X := Spec(A ) is a finite locally free S-scheme which is ´etale if and only if n is invertible in OS . Exercise 18.15. A morphism f : X → S of schemes is called weakly ´etale if f and the diagonal ∆f are flat. Show that a morphism f of schemes is ´etale if and only if f is weakly ´etale and locally of finite presentation. Remark : See also Section (20.4). Exercise 18.16. Let f : X → S be a morphism locally of finite presentation. Show that f is unramified (resp. smooth, resp. ´etale) if and only if for every square (18.0.1), where T = Spec C is a local ring and where i is a thickening of order at most 1, there exists at most one (resp. there exists a, resp. there exists a unique) morphism b : T → X commuting with (18.0.1). Hint: Exercise 10.22. Exercise 18.17. Let S be a scheme and let E be a quasi-coherent OS -module. Then the attached quasi-coherent bundle V(E ) is smooth over S if and only if E is finite locally free. Exercise 18.18. Let X be a scheme locally of finite type over a field k and let x ∈ X. Show that X is smooth at x if and only if Ω1X/k,x is a free OX,x -module and X is geometrically reduced at x. Hint: Use Cartier’s equality (Exercise 18.10). Exercise 18.19. Let k be a non-perfect field of characteristic p > 2 and let a ∈ k \ { αp ; α ∈ k }. Let A := k[T, S]/(T 2 − S p + a) and let X = Spec A. Let x ∈ X be the point corresponding to the maximal ideal (T, S p − a). (1) Show that X is a regular, geometrically reduced, and integral scheme of dimension 1. (2) Show that the smooth locus of X over k is X \ {x}. (3) Show that Ω1X/k,x ∼ = OX,x ⊕ κ(x). Exercise 18.20. Let f : X → S be a morphism locally of finite presentation, x ∈ X, s := f (x). Show that the following assertions are equivalent.

66

´ 18 Etale and smooth morphisms

(i) f is smooth at x. (ii) OX,x is a formally smooth OS,s -algebra. If S is locally noetherian, show that (i) and (ii) are equivalent to: (iii) ObX,x is a formally smooth ObS,s -algebra. Exercise 18.21. Let R → A be a ring homomorphism such that Spec A → Spec R is smooth. Show that there exists an open covering of Spec A be principal open subsets D(g) such that Ag ∼ = R[T1 , . . . , Tn ]/(f1 , . . . , fr ) where fi ∈ R[T1 , . . . , Tn ] such that the determinant of the Jacobian matrix det(Jf1 ,...,fr ) ∈ R[T1 , . . . , Tn ] maps to an invertible element in Ag . Exercise 18.22. Let S be a scheme, let E be a finite locally free OS -module of rank n ≥ 1, and let 0 ≤ e ≤ n be an integer. Show that the Grassmannian Grasse (E ) of quotients of E of rank e is smooth of relative dimension e(n − e) over S. Deduce that the projective bundle P(E ) is smooth of relative dimension n − 1 over S. Exercise 18.23. Let f : X → S be a separated unramified morphism of schemes. Show that every section of f is an open and closed immersion and that this defines a bijection between the set of sections of f and the set of open and closed subscheme Y ⊆ X such that f |Y : Y → S is an isomorphism. Exercise 18.24. Let f : X → S be a morphism locally of finite presentation. Show that f is ´etale if and only if for every diagram (18.0.1), where i is a nilpotent thickening of order at most 1 of arbitrary (not necessarily affine) schemes, there exists a morphism T → X commuting with the diagram. Hint: To show that the condition implies that f is ´etale, it is sufficient to show that all geometric fibers of f are 0-dimensional. Hence one can assume that S = Spec k for an algebraically closed field and that X is ´etale over Ank for some n ≥ 0. Let Z be a proper integral k-scheme and let F → G be a surjection of coherent OZ schemes that is not surjective on global sections. It induces a nilpotent thickening T0 := DZ (G ) → T := DZ (F ) of order at most 1 (Remark 17.26). Use that every morphism Z → Ank of k-schemes is constant to show that n is necessarily 0. Remark : The result is due to Bj¨orn Poonen, https://mathoverflow.net/questions/ 22015 Exercise 18.25. Let S be a scheme of characteristic p > 0, let f : X → S be a morphism of schemes, and let FX/S : X → X (p) be the relative Frobenius. (1) Show that FX/S is a universal homeomorphism. (2) Show that the homomorphism w : Ω1X/S → Ω1X/X (p) (17.5.6) is an isomorphism. Deduce that f is formally unramified if and only if FX/S is formally unramified. (3) From now on let f be locally of finite presentation. Show that FX/S is locally of finite presentation and quasi-finite. (4) Show that f is ´etale if and only if FX/S is an isomorphism. (5) Show that if f is smooth, then FX/S is flat. (6) Show that if f is smooth of relative dimension r, then FX/S is finite locally free of degree pr . Exercise 18.26. Let k be a field of characteristic p > 0 and let X be an integral k-scheme of finite type. Show that the following assertions are equivalent. (i) The k-scheme X is geometrically reduced. (ii) The k-scheme X (p) is reduced.

67 (iii) The ring K(X) ⊗k k 1/p is a field. (iv) The k-scheme X is generically smooth over k, i.e., there exists a non-empty open subset U ⊆ X such that U is smooth over k. Hint: Use Proposition G.33. Exercise 18.27. Let S be a scheme and let Y′

g

i′

 X′

/Y i

f

 /X

be a cartesian diagram of S-schemes, where i is an immersion. Suppose that X, Y , and X ′ are smooth over S. Show that the following assertions are equivalent. (i) The scheme Y ′ is smooth over S. (ii) The canonical map g ∗ (Ci ) → Ci′ is bijective. If these equivalent conditions are satisfied, f is said to be transversal for Y . Show the following assertions. (1) The condition to be transversal is stable under base change and under fpqc descent. (2) The morphism f is transversal for Y if and only if for every s ∈ S the morphism of the fibers fs : Xs′ → Xs is transversal for Ys . Remark : Combining (2) with the faithfully flat base change to some algebraic closure of κ(s) for all s allows to check transversality in the case that the base scheme is the spectrum of an algebraically closed field. See Exercise 18.28. Exercise 18.28. With the notation of Exercise 18.27 suppose that in addition S = Spec k for an algebraically closed field k. Show that f is transversal for Y if and only if for every y ′ ∈ Y ′ (k) the sum of the tangent map Ty′ (X ′ ) ⊕ Tf (y′ ) (Y ) → Tf (y′ ) (X) is surjective. Exercise 18.29. Let f : X → Y be a morphism locally of finite type. Show that f is a monomorphism if and only if for all y ∈ Y the fiber f −1 (y) is empty or f −1 (y) → Spec κ(y) is an isomorphism.

19

Local complete intersections

Content – The Koszul complex and regular immersions – Local complete intersection morphisms If k is a field and X = Spec A is a closed subscheme of Ank , then in general it is not possible to write X as the vanishing scheme of r = codimAnk (X) polynomials fi ∈ k[T1 , . . . , Tn ] (cf. Exercise 19.17). If this is possible at least locally on X, then we call X a local complete intersection over k. By Krull’s principal ideal theorem (Corollary B.61) we know that for a local noetherian ring A and for f ∈ A one has dim A/(f ) = dim A − 1 if f is a regular element. Hence it is not surprising that those subschemes that are locally the vanishing set of a regular sequence (Definition B.58) of polynomials are a local complete intersection. In fact the converse is also true (Proposition 19.50). More generally, a morphism f : X → S of locally noetherian schemes will be defined a local complete intersection morphism if f is locally on X the composition of an immersion X ,→ P which is locally defined by a regular sequence (a so-called regular immersion) followed by a smooth morphism P → S (Definition 19.37). This property is independent of the factorization. But it turns out that the same definition for arbitrary schemes yields a notion which is not known to be independent of the factorization into an immersion followed by a smooth morphism. Hence we replace the notion of a regular sequence by the slightly weaker notion of a completely intersecting sequence (Definition 19.10) and call a morphism of general schemes a local complete intersection if locally on the source it can be factorized as a completely intersecting immersion (an immersion defined locally by a completely intersecting sequence) followed by a smooth morphism. For locally noetherian schemes both notions of local complete intersection morphism are equivalent (Remark 19.22).

The Koszul complex and completely intersecting immersions We start by defining the notion of a completely intersecting sequence. It is defined via its Koszul complex whose definition we explain first. The Koszul complex will also play an important role later elsewhere, for instance for computing the cohomology of twisted line bundles on projective spectra (Section (22.6)), for the notion of derived completion (Section (24.3)) and in Chapter 25. (19.1) Koszul complex. Let A beVa ring, let L be an A-module, and let u : L → A be an A-linear map. For p ≥ 0 Vp−1 p (L) be the contraction with u, i.e., (L) → let du :

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_4

69 du (e1 ∧ · · · ∧ ep ) =

p X

(−1)i+1 u(ei )e1 ∧ · · · ∧ ebi ∧ · · · ∧ ep

i=1

for e1 , . . . , ep ∈ L, where as usual ebi means that ei is omitted. Then it is easy to check that d

· · · −−u→

Vp

d

(L) −−u→

Vp−1

d

d

(L) −−u→ · · · −−u→

V1

d =u

(L) −−u−−→

V0

(L) → 0

is a complex and that

Vp

V

(L), b ∈ (L). du (a ∧ b) = du (a) ∧ b + (−1)p adu (b), a∈ V In other words, ( (L), du ) is a graded commutative differential graded algebra with differential of degree −1 (Section (17.8)). V Definition 19.1. The differential graded algebra ( (L), du ) is called the Koszul complex. It is denoted by K• (u). If f = (f1 , . . . , fn ) P is a tuple of elements fi ∈ A and u : An → A is the associated linear map (a1 , . . . , an ) 7→ i fi ai , we also write K• (f ) instead of K• (u). If M is an A-module, considered as complex of A-modules concentrated in degree 0, we define K• (u, M ) := K• (u) ⊗A M and K• (f , M ) := K• (f ) ⊗A M . The i-th homology of the Koszul complex is denoted by Hi (u) (or Hi (f ), Hi (u, M ), or Hi (f , M ), respectively).

(19.1.1)

Remark 19.2. (1) For f ∈ A and an A-module M , considered as the endomorphism A → A given by multiplication by f , or as the one-element tuple consisting of f , the Koszul complex K• (f, M ) is the chain complex f

0 → M −→ M → 0 concentrated in degrees 1 and 0. (2) If f = (f1 , . . . , fn ) is a sequence of elements of A of length n, then Hi (f , M ) = 0 for Vi n (A ) = 0 for i > n. all i > n because In particular, the Koszul complex K• (f ) is simply the complex f

0 −→ A −→ A −→ 0. For an arbitrary finite sequence f = (f1 , . . . , fr ) of elements of A we will see that K• (f ) ∼ = K• (f1 ) ⊗g · · · ⊗g K• (fr ), where the tensor product is the graded tensor product of differential graded algebras that we define now. L n Definition L and nRemark 19.3. Let R be a (commutative) ring and let A = n∈Z A and B = n∈Z B be graded (not necessarily commutative) R-algebras. L Define the graded tensor product A⊗gR B as the graded R-module A⊗R B with (A⊗R B)n = i+j=n Ai ⊗R B j endowed with the unique R-algebra structure such that ′

(a ⊗ b)(a′ ⊗ b′ ) = (−1)deg(b) deg(a ) aa′ ⊗ bb′ for homogeneous elements a, a′ of A and b, b′ of B. It is easy to check that if A and B are both (strictly) graded commutative, then A ⊗gR B is (strictly) graded commutative.

70

19 Local complete intersections

Now suppose that (A, d) and (B, d) are differential graded algebras with differential of degree −1. Then A ⊗gR B becomes a differential graded algebra with differential of degree −1 by endowing it with the structure of the tensor product of chain complexes (see Section (F.19) for the definition of the tensor product of cochain complexes). In other words, we have for a ∈ A homogeneous and for b ∈ B d(a ⊗ b) = d(a) ⊗ b + (−1)deg(a) a ⊗ d(b). If we are just interested in the underlying tensor product of chain complexes, we also write A ⊗ B instead of A ⊗g B. Example 19.4.L Let A beVa ring and Vq let L1 and L2 be A-modules. Then the isomorphism Vn p L2 ) induces an isomorphism of graded algebras (L1 ⊕ L2 ) ∼ = p+q=n ( L1 ⊗ (19.1.2)

(L1 ⊕ L2 ) ∼ =

V

V

L1 ⊗gA

V

L2 .

Remark 19.5. Let A be a ring and let M be an A-module. (1) The formation of the Koszul complex is functorial: Let L and L′ be A-modules, let u : L → A be a linear form and let w : L′ → L be an A-linear map. Then Vi w induces a homomorphism of complexes K• (u ◦ w, M ) → K• (u, M ), given by (w) ⊗ idM in degree i. (2) Let L1 and L2 be A-modules, ui : Li → A linear forms, and u := u1 +u2 : L1 ⊕L2 → A. Then it is easy to check that the isomorphism (19.1.2) is an isomorphism of differential graded algebras (19.1.3)

g K• (u) ∼ = K• (u1 ) ⊗A K• (u2 ).

(3) Let f = (f1 , . . . , fr ) with fi ∈ A. Then (2) implies that g g K• (f ) ∼ = K• (f1 ) ⊗A · · · ⊗A K• (fr ).

(4) The formation of the Koszul complex is compatible with arbitrary base change in → A aV linear form the following sense. Let A → A′ be a ring homomorphism, u : L V ∼ and u′ := idA′ ⊗u : A′ ⊗A L → A′ . Then the isomorphism A′ ⊗A L → (A′ ⊗A L) induces an isomorphism (19.1.4)

∼

A′ ⊗A K• (u, M ) → K• (u′ , A′ ⊗A M )

and hence homomorphisms of A′ -modules (19.1.5)

A′ ⊗A Hi (u, M ) → Hi (u′ , A′ ⊗A M ),

which are isomorphisms if A′ is a flat A-algebra. Remark 19.6. Let A be a ring and let V u: L → A V be a map of A-modules. Let ℓ ∈ L with image f = u(ℓ) ∈ A. Denote by hℓ : (L) → (L) the left multiplication by ℓ, i.e., hℓ (c) = ℓ ∧ c. Then by (19.1.1) the left multiplication by f satisfies f = hℓ ◦ d u + d u ◦ hℓ . In particular, it is homotopic to zero. Then left multiplication by f is also homotopic to zero on K• (u, M ) for every A-module M . As a consequence one sees that the ideal u(L) annihilates the homology groups Hi (K• (u, M )) for all i.

71 If u(L) = A, then we can take f = 1 and see that K• (u, M ) homotopic to zero. Remark 19.7. Sometimes it is useful that the Koszul complex is independent of the base ring in the following sense. Suppose that R is a ring and that A is an R-algebra. Let M be an A-module which we also can view as an R-module by restriction of scalars. Let g1 , . . . , gr ∈ R and denote by f1 , . . . , fr their images in A. Then the actions of gi and of fi on M are the same. Hence K• (g, M ) is simply the complex obtained K• (f , M ) by restriction of scalars. Conversely, one can obtain from the complex of R-modules K• (g, M ) the complex of A-modules K• (f , M ) as follows. Every a ∈ A yields by multiplication an R-linear map on M , and since K• (f , M ) is functorial in M , we recover the scalar multiplication of A on K• (f , M ). (19.2) Regular and completely intersecting sequences. We continue to denote by A a ring. Let f = (f1 , . . . , fn ) ∈ An and let M be an A-module. Recall the following definition (Definition B.60). Definition 19.8. The sequence f is called weakly M -regular if for all 1 ≤ i ≤ n the multiplication X X fj M ) fj M ) → M/( fi : M/( j 0. As for any finite sequence f there are only finitely many i such that Hi (f , A) ̸= 0, there exists g ∈ A with g ∈ / p such that Hi (f , A)g = 0 for all i. Then the image of f in Ag is completely intersecting and hence I |D(g) is generated by a completely intersecting sequence. We will also use the following weaker notion of regularity for an immersion. Definition 19.21. Let i : Z → X be an immersion defined by a quasi-coherent ideal I ⊆ OX 0 with X 0 ⊆ X open such that i(Z) is a closed subscheme of X 0 . Then i is called quasi-regular, if the following three conditions are satisfied. (a) The ideal I is of finite type.

75 (b) The conormal sheaf Ci = i∗ (I /I 2 ) is a finite locally free OZ -module. (c) The canonical homomorphism M SymOZ (I /I 2 ) → I d /I d+1 d≥0

is an isomorphism. Let z ∈ Z. Then i is called a quasi-regular at z if there exists an open neighborhood U of i(z) in X such that the restriction i−1 (U ) → U of i is quasi-regular. For a quasi-regular immersion i we call the finite locally free conormal sheaf Ci also the conormal bundle. Its dual N = Ni := C ∨ i is called the normal bundle. Remark 19.22. By Theorem 19.12 we have for an immersion i : Z → X the implications i regular ⇒ i completely intersecting ⇒ i quasi-regular. All properties are equivalent if X is locally noetherian (Corollary 19.13). More precisely, if i is given by a quasi-coherent ideal I ⊆ OX 0 with X 0 ⊆ X open such that i(Z) is a closed subscheme of X 0 and if f = (f1 , . . . , fn ) is a completely intersecting sequence of fi ∈ Γ(X, I ) which generates I , then the images f¯i of the fi in I /I 2 = Ci form a basis of the OZ -module Ci and there is an isomorphism of graded OZ -algebras M ∼ OZ [T1 , . . . , Tn ] → I d /I d+1 , Ti 7→ f¯i ∈ Γ(Z, I /I 2 ). d≥0

Definition 19.23. Let i : Z → X be a quasi-regular immersion of schemes. For z ∈ Z we call rkOZ,z (Ci )z the codimension of i in z. We also set dimz (i) := − rkOZ,z (Ci )z . As Ci is finite locally free if i is quasi-regular, the map Z → Z, z 7→ dimz (i) is locally constant. Let us look at regular immersions of small codimension. Remark 19.24. A completely intersecting immersion which is of codimension ≤ 1 in all points is a regular immersion, because completely intersecting sequences of length ≤ 1 are always weakly regular. An immersion is quasi-regular of codimension 0 in all points if and only if it is an open immersion. If the inclusion i : D → X of a closed subscheme is completely intersecting of codimension 1 in all points of D (Remark 11.27), then D is an effective Cartier divisor. Its conormal sheaf Ci is given by i∗ OX (−D). Example 19.25. Let X be a scheme, let E be a finite locally free OX -module of rank r, let V(E ) = Spec(Sym(E )) be the attached geometric vector bundle (Proposition 11.7), and let s0 : X → V(E ) be its zero section. Then s0 a regular immersion of codimension r with conormal sheaf Cs0 = E . L d Indeed, s0 is given by the ideal I := d≥1 Sym (E ) of Sym(E ) and hence Cs0 = 2 I /I = E , where the second inequality is induced by the projection I → Sym1 (E ) = E . To see that s0 is a regular immersion, we can work locally on X and hence assume that r E = OX . Then s0 can be identified with the zero section of ArX which is clearly regular, see Example 19.16. One has the following permanence properties for regular and completely intersecting immersions.

76

19 Local complete intersections

Proposition 19.26. (1) Let i : Z → Y and j : Y → X be regular immersions (resp. completely intersecting immersions). Then j ◦ i is a regular (resp. completely intersecting) immersion. For z ∈ Z one has dimz (j ◦ i) = dimz (i) + dimi(z) (j). (2) Let i : Z → X be an immersion and let g : X ′ → X be a flat morphism. If i is regular (resp. completely intersecting), then the base change i(X ′ ) : Z ×X X ′ → X ′ is a regular (resp. completely intersecting) immersion. (3) Let i : Z → X be an immersion and let g : X ′ → X be faithfully flat and quasi-compact. If i(X ′ ) is a regular (resp. completely intersecting) immersion, then i is a regular (resp. completely intersecting) immersion. We give the proof only for completely intersecting immersions. For the case of regular immersions we refer to [EGAIV] O (19.1.5). Proof. To show (1) we may assume that i and j are closed immersions and that X = Spec A is affine. Let f be a completely intersecting sequence in A generating an ideal I such that ¯ in A/I is completely intersecting Y ∼ = Spec A/I and let g be sequence in A whose image g ¯ is Z. Then there are quasi-isomorphisms of and such that the vanishing locus of g complexes of A-modules K• (f , g)

(19.1.3)

=

K• (f ) ⊗A K• (g) ∼ = A/I ⊗ K• (g)

(19.1.4)

=

K• (¯ g) ∼ = A/(f , g).

Therefore (f , g) is a completely intersecting sequence. This proves (1). To show (2) and (3), we again can assume that X = Spec A and i is a closed immersion. To show (2) we may also assume that X ′ is also affine. For (3) we remark that X ′ is quasi-compact because g is quasi-compact. Thus we may replace X ′ by the affine scheme ` ′ ′ ′ ′ ′ i Ui , where (Ui )i is a finite affine cover of X . Hence we may assume that X = Spec A ′ in both cases. Then A is a flat A-algebra in (2) and a faithfully flat A-algebra in (3). Hence (2) and (3) follow because the formation of H• (f ) is compatible with flat base change (Remark 19.5 (4)). The composition of two quasi-regular immersions is not quasi-regular in general (Exercise 19.12), see however Exercise 19.11. The analogous statements of (2) and (3) for quasi-regular immersions hold (Exercise 19.13). Proposition 19.27. Let T X be an S-scheme. For l = 1, . . . , r, let Xl ⊆ X be a closed r subscheme, and let Y := l=1 Xl be their schematic intersection. Assume that all the immersions Xl ,→ X are completely intersecting of codimension dlP , say, and that the r immersion i : Y → X is completely intersecting of codimension d := l=1 dl . Denote by CXl /X and CY /X the respective locally free conormal sheaves. Then r M CY /X ∼ i∗ CX /X . = l

l=1

Proof. We have a natural map i∗ CXl /X → CY /X for each l, because Lrthe ideal defining Xl inside X is contained in the ideal of Y , so we get a homomorphism l=1 i∗ CXl /X → CY /X . We can check that it is an isomorphism locally on X, and can hence assume that each Xl is defined by a completely intersecting sequence. Then our assumption on the codimension of Y implies that joining all these sequences gives a completely intersecting sequence (Proposition 19.18), which defines Y as a closed subscheme of X. Then all the conormal

77 sheaves we have to consider are free, with bases induced by the respective defining equations, and the claim follows; cf. Remark 19.22. (19.4) Regular immersions of flat and of smooth schemes. In the presence of flatness many of the notions introduced above fall together: Proposition 19.28. Let /X

i

Z f



S



g

be a commutative diagram of schemes, where i is an immersion and let z ∈ Z, s := f (z), and x := i(z). Assume that S and X are locally noetherian or that f and g are locally of finite presentation. Then the following assertions are equivalent. (i) f is flat in a neighborhood of z and i is a regular immersion at z. (ii) f is flat in a neighborhood of z and i is a completely intersecting at z. (iii) f is flat in a neighborhood of z and i is a quasi-regular immersion at z. (iv) There exists an open affine neighborhood U = Spec A of x in X and a regular sequence (f1 , . . . , fn ) in A such that i induces an isomorphism of i−1 (U ) with the closed subscheme Spec A/(f1 , . . . , fn ) and such that A/(f1 , . . . , fr ) is flat over S for all r = 0, . . . , n. (v) f is flat in a neighborhood of z, g is flat in a neighborhood of x and for all morphisms S ′ → S the base change i(S ′ ) : Z ×S S ′ → X ×S S ′ is regular in every point of Z ×S S ′ over z. (vi) g is flat in a neighborhood of x and the induced immersion on the fiber is : Zs → Xs is quasi-regular at z. The proof will show that if these equivalent conditions are satisfied, then dimz (i) = dimz (is ) = n with n as in (iv). We will give the proof only in the locally noetherian case. See [EGAIV] O (19.2.4) for the proof (using noetherian approximation) if f and g are locally of finite presentation. Proof. Assertions (i), (ii), and (iii) are equivalent by our assumption that X is locally noetherian (Remark 19.22), and trivially (iv) implies (i) and (v) implies (vi). To show the remaining assertions we may assume that S = Spec R, X = Spec A and Z = Spec A/I for noetherian rings R and A and an ideal I of A. If A/I is flat over R, which we may assume under all hypotheses except (vi), then for every R-algebra R′ the sequence 0 → I ⊗R R′ → A ⊗R R′ → A/I ⊗R R′ → 0 is exact (Proposition B.16). Hence for S ′ = Spec R′ we see that i(S ′ ) is given by the ideal I ⊗R R′ . We will write Ar := A/(f1 , . . . , fr ). (vi) ⇒ (iv). By assumption Ix /(ms Ix + Ix2 ) is a free Ax /(ms Ax + Ix )-module of finite rank. Let f = (f1 , . . . , fn ) be a sequence of elements fj ∈ I whose images in Ix /(ms Ix + Ix2 ) form a basis. Let gj be the image of fj in A/ms A. Then g = (g1 , . . . , gn ) generates Ix /ms Ix by Nakayama’s lemma. By Assumption and by Remark 19.22 we know that Ix /ms Ix is generated a regular sequence of length n. Hence g is a regular sequence in Ax /ms Ax (Proposition 19.18).

78

19 Local complete intersections

Replacing X = Spec A by a some open affine neighborhood of x, we may assume that g is a regular sequence in A/ms A. Again by Nakayama’s lemma, f generates I after passing to a smaller affine open neighborhood of x, if necessary. By hypothesis we may also assume that A is flat over S. By induction we may assume that for 0 ≤ r < n the f1 , . . . , fr form a regular sequence and that Ar := A/(f1 , . . . , fr ) is flat over R. To show that Ar+1 is flat over R and that the multiplication by fr+1 on Ar is injective we may assume that Ar is a local ring. But then we can conclude by Proposition G.2 because the multiplication by gr+1 on Ar /ms Ar is injective. (i) ⇒ (iv). We may assume that I is generated by a regular sequence (f1 , . . . , fn ) in A and that A/I is flat over R. By descending induction on r we may assume that Ar+1 is flat over R. Consider the sequence (*)

fr+1

0 → Ar −→ Ar −→ Ar+1 → 0

which is exact because (f1 , . . . , fn ) is regular. Then the local criterion for flatness Theorem B.51 shows that Ar is flat over R. (iv) ⇒ (v). We may assume that S ′ = Spec R′ and I is generated by a regular sequence (f1 , . . . , fn ) in A such that Ar := A/(f1 , . . . , fr ) is flat over S for all r = 0, . . . , n. Let R′ be an R-algebra. For all r = 0, . . . , n − 1 we have then Ar+1 ⊗R R′ = (A ⊗R R′ )/(f1 ⊗ 1, . . . , fr+1 ⊗ 1). As Ar+1 is flat over R, tensoring the sequence (*) with R′ yields an exact sequence fr+1 ⊗1 0 → Ar ⊗R R′ −→ Ar ⊗R R′ −→ Ar+1 ⊗R R′ → 0. This proves the claim. Note that for the proof of “(iv) ⇒ (v)” we did not use any finiteness conditions except that i is locally of finite presentation. Our next goal is the study of regular immersions between smooth schemes. We start by recalling (a geometric version of) a result from Commutative Algebra. Proposition 19.29. Let i : Z → X be an immersion of locally noetherian schemes. Let z ∈ Z such that OZ,z is regular. Then OX,i(z) is regular if and only if i is a regular immersion at z. Proof. By replacing X by an open neighborhood V of i(Z) such that i(Z) is closed in V we may assume that i is a closed immersion. By Proposition 19.20 we are reduced to Proposition G.19. Theorem 19.30. Let /X

i

Z f



S



g

be a commutative diagram of schemes, where i is an immersion and g is locally of finite presentation. Let z ∈ Z, x := i(z), and s := f (z). Assume that f is smooth at z. Then the following assertions are equivalent. (i) g is smooth at x. (ii) i is a regular immersion at z. (iii) i is a quasi-regular immersion at z.

79 (iv) The immersion is : Zs → Xs induced on fibers is a quasi-regular immersion at z. If these equivalent conditions are satisfied, one has dimz (f ) = dimz (i) + dimi(z) (g). In particular, any immersion of smooth S-schemes is regular and in particular completely intersecting. Proof. As f and g are locally of finite presentation in a neighborhood of z resp. x, they are both flat in a neighborhood of z resp. x if they are flat at z resp. x because the locus of flatness is open for morphisms locally of finite presentation (Theorem 14.44). Hence by Proposition 19.28 and as smoothness of flat morphisms can be checked on fibers (Corollary 18.57), we may assume that S = Spec k is a field. Then the equivalence of (iii) and (iv) is clear and the equivalence of (ii) and (iii) follows from Corollary 19.13. As the properties “regular immersion” and “smooth” can be checked after a faithfully flat quasi-compact base change, we may assume that k is algebraically closed. But then f is smooth at z (resp. g is smooth at x) if and only if OZ,z (resp. OX,x ) is regular (Theorem 6.28). Hence the equivalence of (i) and (ii) follows from Proposition 19.29. Corollary 19.31. Let f : X → S be a smooth morphism of schemes of relative dimension d. (1) Let i : S → X be a section of f . Then i is a regular immersion of codimension d. (2) Let g : Y → X be a morphism of S-schemes. Then the graph Γg : Y → Y ×S X is a regular immersion of codimension d. (3) The diagonal ∆ : X → X ×S X is a regular immersion of codimension d. Proof. Assertion (1) follows from Theorem 19.30 because every section is an immersion (Example 9.12). Assertion (2) follows from (1) because Γg is a section of the projection Y ×S X → Y which is smooth because smoothness is stable under base change. Finally, (3) is a special case of (2). If X → S is smooth of relative dimension n, then Ω1X/S is locally free of rank n Vn 1 and ΩnX/S = ΩX/S = det(Ω1X/S ) it its top exterior power. Here we set det(E ) := Vr E for a locally free module E of rank r, which is a line bundle. To calculate it, sometimes the following result is useful which relies on the remark (Exercise 7.29) that if 0 → E ′ → E → E ′′ → 0 is a short exact sequence of finite locally free modules, then det(E ) ∼ = det(E ′ ) ⊗ det(E ′′ ). Proposition 19.32. Let S be a scheme, let X → S and Y → S be smooth morphisms of constant relative dimension n and m, respectively. Let i : Y → X be a closed immersion of S-schemes. Then i is a regular immersion of codimension n − m and Vn−m  ∼ Ci ∨ ⊗OY i∗ ΩnX/S . Ωm Y /S = Proof. The immersion is regular of codimension m − n by Theorem 19.30. We have a short exact sequence of finite locally free OY -modules (Corollary 18.73) 0 → Ci → i∗ Ω1X/S → Ω1Y /S → 0 and hence we obtain i∗ ΩnX/S = det(i∗ Ω1X/S ) ∼ = det(Ci ) ⊗ det(Ω1Y /S ) which shows the claim.

80

19 Local complete intersections

Corollary 19.33. Let S be a scheme, let X be a smooth S-scheme, and for jT= 1, . . . , r let r Dj be an effective Cartier divisor on X. Assume that the embedding i : Y := j=1 Dj → X is regular of codimension r and that Y is smooth over S. Then for the top exterior power of Ω1Y /S we obtain det(Ω1Y /S )

∼ =

det(Ω1X/S )

⊗ OX

r X

!! Di

i=1

. |Y

Lr Proof. We have Ci ∼ = j=1 OX (−Dj )|Y by Proposition 19.27 and therefore we obtain N r det(Ci ) ∼ = j=1 OX (−Dj )|Y . Hence we conclude by Proposition 19.32. For instance, one often applies Proposition 19.32 and Corollary 19.33 to X = PnS in which case one has ΩnX/S = det(Ω1X/S ) = OX (−n − 1). (19.5) Blow-up of regularly immersed smooth subschemes. Remark 19.34. Let X be a scheme and let Z be a closed subscheme given by a inquasi-coherent ideal I of OX such that the inclusion i : Z → X is a completely L tersecting immersion (e.g., if i is a regular immersion). Recall that Proj( d I d ) is the blow up L BlZ (X) of X in the closed subscheme Z (Proposition 13.92) and that E := Proj( d I d /I d+1 ) ⊆ BlZ (X) is its exceptional divisor (Remark 13.94). Therefore Corollary 19.15 implies that we have isomorphisms of X-schemes BlZ (X) ∼ = P(I ),

E∼ = P(Ci ),

using that P(E ) ∼ = Proj(Sym(E )) for every quasi-coherent module E (Theorem 13.32). As Ci is a finite locally free OZ -module, the structure morphism E → Z is locally on Z isomorphic to Pn−1 Z , where n is the codimension of i. Proposition 19.35. Let f : X → S be a smooth morphism of schemes and let Z be a closed subscheme of X which is smooth over S. Then the blow-up BlZ (X) of X along Z is smooth over S. Proof. Denote by π : BlZ (X) → X the blow-up morphism and by E := π −1 (Z) the exceptional divisor. In particular, the immersion E → BlZ (X) is regular (of codimension 1). To show that BlZ (X) → S is smooth in every point x ˜ of BlZ (X) we distinguish two cases. If x ˜∈ / E then the isomorphism BlZ (X) \ E ∼ = X \ Z (Proposition 13.91 (3)) implies the smoothness at x ˜ because X is smooth over S. Now suppose that x ˜ ∈ E. By Theorem 19.30 we see that the inclusion i : Z → X is a regular immersion. Hence its conormal sheaf Ci is finite locally free and E ∼ = P(Ci ) as Z-schemes (Remark 19.34). In particular, E → Z is smooth and hence E is smooth over S. Now we can apply Theorem 19.30 to the regular immersion E → BlZ (X) to see that BlZ (X) → S is smooth at x ˜ ∈ E. Remark 19.36. Using Proposition 19.29 instead of Theorem 19.30 the same argument as in the proof of Proposition 19.35 shows that if one blows up a regular scheme X in a closed regular subscheme Z, then BlZ (X) is regular.

81

Local complete intersection and syntomic morphisms We now define the notion of a local complete intersection morphism as a morphism that can be locally factorized as a completely intersecting immersion followed by a smooth morphism. Flat local complete intersection morphisms will be called syntomic. (19.6) Local complete intersection morphisms. Definition 19.37. Let f : X → S be a morphism of schemes. (1) Let x ∈ X. Then f is called locally completely intersecting at x if there exists an i π open neighborhood U of x in X and a factorization of f |U of the form U −→ P −→ S, where i is a completely intersecting immersion and where π is a smooth morphism. (2) The morphism f is called a local complete intersection morphism (or short: an lci-morphism) if it is a locally completely intersecting at x for all x ∈ X. By definition, the set { x ∈ X ; f is locally completely intersecting at x} is open. An lci-morphism is locally of finite presentation. Every smooth morphism is an lci-morphism. The next result shows that the property of being an lci-morphism does not depend on the chosen factorization. Proposition 19.38. Let f : X → S be a morphism locally of finite presentation and let i

π

X −→ P −→ S

i′

π′

X −→ P ′ −→ S

and

be factorizations of f , where i and i′ are immersions and π and π ′ are smooth. Then i is a completely intersecting immersion if and only if i′ is completely intersecting. In this case for all x ∈ X one has dimx (i) + dimi(x) (π) = dimx (i′ ) + dimi′ (x) (π ′ ). The proposition shows in particular that the following notion is well defined. Definition 19.39. Let f : X → S be a morphism of schemes that is local complete intersection at x ∈ X. Let U be an open neighborhood of x such that there exists a factorization f |U = π ◦ i with i a completely intersecting immersion and π smooth. Then dimx (f ) := dimx (i) + dimi(x) (π) ∈ Z is called the relative dimension of f at x. To prove Proposition 19.38 we follow [Sta] 069E. The essential argument is the following lemma. Lemma 19.40. Let /X

i

Z j



S



f

be a commutative diagram of schemes, where i and j are immersions and where f : X → S is smooth. Then i is completely intersecting if and only if j is completely intersecting. In this case one has dimz (i) + dimi(z) (f ) = dimz (j) for all z ∈ Z.

82

19 Local complete intersections

Proof. (i). Assume that j is completely intersecting. The graph morphism Γi is a section of the smooth projection X ×S Z → Z and hence it is a regular immersion (Corollary 19.31) and in particular completely intersecting (Theorem 19.12). As X is smooth over S, the projection p : X ×S Z → X is a flat base change of the completely intersecting immersion j and hence it is again a completely intersecting immersion (Proposition 19.26 (2)). Therefore i = p ◦ Γi is completely intersecting (Proposition 19.26 (1)). (ii). We proceed in several steps. (I). Let us now assume that i is completely intersecting. The question whether j is completely intersecting is local on X and S. In particular we may assume that S = Spec R and X = Spec A is affine and that i and j are closed immersions. In particular j yields an isomorphism Z ∼ = Spec R/J for some ideal J ⊆ R. Writing A as a quotient of a polynomial ring, we obtain a closed immersion i1 : X → AnS which is automatically regular by Theorem 19.30 and in particular completely intersecting. Hence i1 ◦ i is completely intersecting (Proposition 19.26 (1)). Therefore we may assume that X = AnR . (II). Then i corresponds to a surjective R-algebra homomorphism φ : A = R[T1 , . . . , Tn ] → R/J. Choose sα ∈ R whose images in R/J are φ(Tα ). Set K := (T1 − s1 , . . . , Tn − sn ). Then the kernel of φ is the ideal I := K + JA. As A/K ∼ = R it suffices to show that j ′ : Z → V (K) is completely intersecting. The sequence (T1 − s1 , . . . , Tn − sn ) is regular (Example 19.16). Therefore K/K 2 is a free A/K-module with basis (Tα − sα )1≤α≤n . Hence K/K 2 ⊗A A/I ∼ = K/KI is free over A/I = R/J with the same basis. The composition of the canonical map u : K/KI → I/I 2 with the A/I-linear map I/I 2 → Ω1A/R ⊗A A/I maps the basis (Tα − sα )α to the basis of (dTα ⊗ 1)α . Hence this composition is an isomorphism. We deduce the existence of an A/I-linear map v : I/I 2 → K/KI with v ◦ u = id. Therefore u identifies K/KI with a direct summand of I/I 2 . j′ (III). Thus we now have closed immersions Z := V (I) −→ V (K) → X = Spec A, where A is some ring, such that V (I) → X and V (K) → X are completely intersecting and K/K 2 ⊗A A/I is a direct summand of I/I 2 . We claim that these hypotheses imply that j ′ is completely intersecting. Then the equality of relative dimensions follows because the relative dimension of V (K) → X is the negative of the relative dimension of X over S. We may assume that A is a local ring. Moreover we may assume that I is contained in the maximal ideal of A, otherwise our claim is trivial as the empty subscheme is always regularly embedded. As V (I) → X (resp. V (K) → X) is completely intersecting, I/I 2 (resp. K/K 2 ) is a finite free A/I-module (resp. A/K-module). As K/K 2 ⊗A A/I is a direct summand of I/I 2 we may choose f1 , . . . , fn ∈ K such that their images in K/K 2 are an A/K-basis and we may choose g1 , . . . , gs ∈ I such that the images of f1 , . . . , fn and of g1 , . . . , gs in I/I 2 are a basis of I/I 2 . By Nakayama’s lemma, f = (f1 , . . . , fn ) generates K and (f , g) = (f1 , . . . , fn , g1 , . . . , gs ) generates I. Then f is a completely intersecting sequence generating K and (f , g) is a completely intersecting sequence generating I (Proposition 19.18). ¯ in A/K = A/(f ) is completely intersecting. This It suffices to show that the image g follows from the quasi-isomorphism of chain complexes K• (¯ g)

(19.1.4)

=

A/I ⊗ K• (g) ∼ = K• (f ) ⊗A K• (g)

We now come to the proof of Proposition 19.38.

(19.1.3)

=

K• (f , g) ∼ = A/(f , g).

83 Proof. Let i′ be completely intersecting. By symmetry it suffices to show that i is completely intersecting. The morphism i′′ := (i, i′ ) : X → P ′′ := P ×S P ′ is a closed immersion into a smooth S-scheme (because the properties “closed immersion” and “smooth” are both stable under composition and under base change). Applying Lemma 19.40 to q i′′ i′ : X −→ P ′′ −→ P ′ , where q is the (smooth) second projection, shows that i′′ is comi′′ ′′ p pletely intersecting. Then applying Lemma 19.40 to i : X −→ P −→ P , where p is the (smooth) first projection shows that i is completely intersecting. Corollary 19.41. An immersion is an lci-morphism if and only if it is completely intersecting. Proposition 19.42. (1) The property “lci-morphism” is local on the source and local on the target. (2) If f : X → Y and g : Y → Z are lci-morphisms, then g ◦ f is an lci-morphism and dimx (g ◦ f ) = dimx (f ) + dimf (x) (g) for all x ∈ X. (3) If f : X → Y is an lci-morphism and Y ′ → Y is a flat morphism, then the base change f ′ : X ×Y Y ′ → Y ′ is an lci-morphism. The property “lci-morphism” is also stable under faithfully flat descent (Exercise 19.14). Proof. Assertion (1) holds by definition and (3) holds because the property “smooth” is stable under base change and the property “completely intersecting immersion” is stable under flat base change. Let us show (2): The question is local on X, Y , and Z. Hence we may assume that j i there are factorizations X −→ AnY → Y of f and Y −→ Am Z → Z of g, where i and j are completely intersecting immersions. We obtain a factorization of g ◦ f as i

X −→ AnY

j⊗idAm

−→ An+m → Z. Z

Then the composition (j ⊗ idAm ) ◦ i is a completely intersecting immersion because “completely intersecting immersion” is stable under composition and flat base change. This shows that g ◦ f is an lci-morphism. The assertion about the relative dimension follows because the relative dimension for completely intersecting immersions and for smooth morphisms is additive under composition. Remark 19.43. Let f : X → Y be an lci-morphism. Then X → Z, x 7→ dimx (f ) is a locally constant function because the analogous assertion holds for completely intersecting immersions (Proposition 19.26 (1)) and for smooth morphisms (Proposition 18.59). Corollary 19.44. Let f : X → Y be a morphism of S-schemes. Assume that Y → S is smooth. Then f is an lci-morphism if and only if X → S is an lci-morphism. Proof. If f is an lci-morphism, then the composition X → Y → S is an lci-morphism by Proposition 19.42 (2). Conversely, assume that X → S is an lci-morphism. Locally i on X we may factorize f in X −→ AnY → Y , where i is an immersion. The composition AnY → Y → S is smooth. Hence i is completely intersecting because X → S is an lci-morphism (Proposition 19.38).

84

19 Local complete intersections

(19.7) Complete intersection rings. There is also an absolute notion of complete intersection for locally noetherian schemes. We will see in Proposition 19.50 that every local ring of a scheme X locally of finite type over a field k is complete intersection in this absolute sense if and only if the structure morphism X → Spec k is a local complete intersection morphism. For the definition of a complete intersection ring we first introduce “the” Koszul complex of a local noetherian ring. Remark 19.45. Let A be a local noetherian ring, m its maximal ideal, k its residue field. Choose a sequence f = (f1 , . . . , fn ) in m whose image in m/m2 is a k-basis. The corresponding Koszul complex K• (f ) is independent of the choice f up to isomorphism. Indeed, Pnif g = (g1 , . . . , gn ) is another such sequence, then any n × n matrix L = (lij ) with gi = j=1 lji fj is invertible because modulo m it maps a k-basis to a k-basis. The A-linear ∼ isomorphism An → An corresponding to L then yields an isomorphism K• (g) → K• (f ). Moreover, mHi (f ) = 0 for all i because f generates m. Hence we may define for i ≥ 0 the numbers (19.7.1)

εi (A) := dimk Hi (f ),

which depend only on A. One can show that ε1 (A) ≥ dimk m/m2 − dim(A) for every local noetherian ring ([Mat2] 21.1). Definition 19.46. A local noetherian ring A with maximal ideal m and residue field k is called a complete intersection ring if ε1 (A) = dimk m/m2 − dim(A). If X is a scheme locally of finite type over a field k and if x ∈ X(k) is a k-rational point, then OX,x is a complete intersection ring if and only if dimk Tx (X) − dimx (X) = ε1 (OX,x ). Let us collect some properties of complete intersection rings. Remark 19.47. Let A be a local noetherian ring with maximal ideal m and residue class field k. (1) Definition B.73 and Corollary 19.13 show: A is regular ⇔ ε1 (A) = 0 ⇔ dimk m/m2 = dim(A). In particular, every regular local ring is a complete intersection ring. ˆ then m/m2 = m/ ˆ = mA, ˆ m ˆ2 (2) If Aˆ is the completion of A with maximal ideal m ˆ (Proposition B.39) and dim A = dim A (Proposition B.64). Moreover, a sequence in ˆ m ˆ 2 , hence if K• m that yields a basis of m/m2 is also a sequence in mAˆ generating m/ ˆ ˆ is the Koszul complex for A, the Koszul complex K• for A is isomorphic to Aˆ ⊗A K• . ˆ •) ∼ As Aˆ is flat over A, we also have Hi (K = Hi (K• ) ⊗A Aˆ for all i. Since Hi (K• ) is ˆ •) ∼ ˆ for all i. annihilated by m, this implies Hi (K H = i (K• ) and hence εi (A) = εi (A) In particular, A is a complete intersection ring if and only if Aˆ is a complete intersection ring. (3) If A is a complete intersection ring, then A is Gorenstein (Definition G.26) by [Mat2] Theorem 21.3. In particular A is Cohen-Macaulay (Proposition G.26). Hence we have the implications A regular ⇒ A complete intersection ⇒ A Gorenstein ⇒ A Cohen-Macaulay.

85 (4) Let A → B be a flat local homomorphism between local noetherian rings, and let m ⊂ A denote the maximal ideal. Then B is a complete intersection ring if and only if A and B/mB are complete intersection rings. ([Mat2], Remark on p. 182.) (19.8) Local complete intersection morphisms over a field. Remark 19.48. Let k be a field, A a finitely generated k-algebra, x ∈ X := Spec A and let f = (f1 , . . . , fr ) be a sequence contained in px . Let Z = Spec A/f A and z ∈ Z the point corresponding to px /f . Then Corollary B.61 implies that dimz (Z) ≥ dimx (X) − r with equality if f is a regular sequence. We will use the following result from Commutative Algebra ([Mat2] Theorem 21.2). Proposition 19.49. Let A be a local noetherian ring. (1) A is a complete intersection ring if and only if there exists a complete regular local ring R and an ideal I of R generated by an R-regular sequence such that Aˆ = R/I, where Aˆ denotes the completion of A. (2) Let R be a regular local ring such that A ∼ = R/a. Then A is a local intersection ring if and only if the ideal a is generated by a regular sequence. Proposition 19.50. Let k be a field, let g : X → Spec k be a k-scheme locally of finite type and let x ∈ X. Then the following assertions are equivalent. (i) g is locally completely intersecting at x. (ii) There exists an open affine neighborhood U = Spec A of x and an isomorphism of k-algebras A ∼ = k[T1 , . . . , Tn ]/(f1 , . . . , fc ) with dim A = n − c. (iii) OX,x is a complete intersection ring. In this case one has (19.8.1)

dimx (g) = dimx (X) = n − c

and U is equi-dimensional of dimension n − c. Proof. Every irreducible component of Spec k[T1 , . . . , Tn ]/(f1 , . . . , fc ) has dimension ≥ n− c (Corollary 5.33). Hence (ii) implies the last assertion. Then 19.8.1 holds by Remark 19.48 (and for all x ∈ U ). It remains to show that (i), (ii), and (iii) are equivalent. We may assume that X = Spec A is affine with A = R/I, where R is a smooth k-algebra of finite type, for instance R = k[T1 , . . . , Tm ]. Let i : X → Spec R be the corresponding closed immersion. Let p ⊂ A be the prime ideal corresponding to x and let q be the prime ideal of R corresponding to i(x). As R is regular (Theorem 6.28), Rq is a local regular ring (Proposition B.74) and Ap = Rq /Iq . (iii) ⇒ (i). If Ap = OX,x is a complete intersection ring, Iq is generated by a regular sequence f = (f1 , . . . , fc ) (Proposition 19.49). Replacing R by a localization Rg with g ∈ / q, we may assume that f1 , . . . , fc ∈ R and that they form a regular sequence of R (Proposition 19.20). Then i is a regular immersion into the spectrum of a smooth k-algebra and hence g is a local complete intersection morphism. (i) ⇒ (ii). We may assume that R = k[T1 , . . . , Tl ]. Passing to a principal open subset, we may assume A = Rh /I, where h ∈ R and I is generated by a regular sequence (f¯1 , . . . , f¯d ) (Proposition 19.49). We identify Rh with k[T1 , . . . , Tl+1 ]/(Tl+1 h − 1) and we choose fi ∈ k[T1 , . . . , Tl+1 ] whose images in Rh is f¯i . As f := Tl+1 h−1 is a regular element in k[T1 , . . . , Tl+1 ], we deduce from Proposition 19.26 (1) that (f1 , . . . fd , f ) is a regular sequence in k[T1 , . . . , Tl+1 ]. Hence A is a equi-dimensional of dimension (l + 1) − (d + 1).

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19 Local complete intersections

(ii) ⇒ (iii). After localization, (ii) implies that OX,x ∼ = B/J, where B is a regular local ring and where J is an ideal of B generated by c := dim B − dim OX,x elements. As regular rings are Cohen-Macaulay, Proposition G.20 implies that J is generated by a regular sequence. (19.9) Syntomic morphisms. Definition 19.51. A flat local complete intersection morphism is called syntomic. Every smooth morphism is syntomic and every syntomic morphism is flat and locally of finite presentation. Proposition 19.52. Let f : X → S be a morphism of schemes. Then the following assertions are equivalent. (i) f is syntomic. (ii) f is flat, locally of finite presentation and for all s ∈ S the fiber f −1 (s) is a local complete intersection over κ(s). (iii) For all x ∈ X there exist open affine neighborhoods Spec A of x in X and Spec R of f (x) in S such that A ∼ = R[T1 , . . . , Tn ]/(f1 , . . . , fc ) and such that fiber of Spec A → Spec R in f (x) has dimension n − c. Proof. All assertions are local on source and target, so we may assume that X = Spec A and S = Spec R are affine and that there exists a closed immersion X → AnR . Then the implications (iii) ⇒ (i) ⇔ (ii) follow from Proposition 19.28 applied to X → AnR → S (for (iii) ⇒ (ii) one also uses Proposition 19.50). Let us show (ii) ⇒ (iii). Let x ∈ X and set k := κ(f (x)) and write A = R[T1 , . . . , Tn ]/I for an ideal I, which is finitely generated because f is locally of finite presentation. By Proposition 19.50 we may assume (after possibly shrinking Spec A) that A ⊗R k = k[T1 , . . . , Tn ]/(f¯1 , . . . , f¯c ) is of dimension n − c, where f¯i are elements in the image I¯ of I in A ⊗R k. Choose fi ∈ I whose image in I¯ is f¯i and set A′ := R[T1 , . . . , Tn ]/(f1 , . . . , fc ). Let J := Ker(A′ → A). This is a finitely generated ideal because A and A′ are both of finite presentation over R. As A is flat over R, J ⊗R k → A′ ⊗R k is still injective (Proposition B.15), hence J ⊗R k = 0 by construction. Let q ⊂ B := R[T1 , . . . , Tn ] be the prime ideal corresponding to the image of x in AnR . Then Nakayama’s lemma (applied to the ring Bq , the finitely generated module Jq and the ideal mf (x) Bq ) implies that Jq = 0. As J is finitely generated, there exists g ∈ B, g ∈ / q such that Jg = 0. Replacing B by R[T1 , . . . , Tn+1 ] and (f1 , . . . , fc ) by (f1 , . . . , fc , gTn+1 − 1) we are done. Note that the proof shows, that if f : X → S is flat and locally of finite presentation and there exists s ∈ S such that f −1 (s) is a local complete intersection over κ(s) in a neighborhood of some z ∈ f −1 (s), then (iii) holds after replacing X by an open neighborhood of z. Proposition 19.53. The property “syntomic” is local on the source, local on the target, stable under composition, stable under base change, stable under fpqc descent and compatible with inductive limits.

87 Proof. Except for the stability under base change all these permanence properties hold because “lci-morphism” and “flat” have the same permanence properties. The stability under base change follows from Proposition 19.52 and the fact that the property “lcimorphism” is stable under flat base change and hence in particular under change of base fields for the fibers. Corollary 19.54. Let f : X → Y be a morphism of S-schemes. For s ∈ S let fs : Xs → Ys be the morphism induced by f on the fiber over s. Let X and Y be locally of finite presentation over S and let X be flat over S. Then f is syntomic if and only if fs is syntomic for all s ∈ S. Proof. If f is syntomic, then fs is syntomic for s ∈ S because “syntomic” is stable under base change (Proposition 19.53). Conversely, if all fs are syntomic, then for all s ∈ S and y ∈ Ys the fiber fs−1 (y) = f −1 (y) is syntomic. Moreover, as X is flat over S and fs is flat for all s ∈ S, the fiber criterion for flatness (Theorem 14.25) shows that f is flat. Therefore f is syntomic by Proposition 19.52. Remark 19.55. Let S = Spec R be an affine scheme, X = Spec R[T1 , . . . , Tn ]/(f1 , . . . , fc ). Then every irreducible component of every fiber of X → S has at least dimension n − c (Corollary 5.33). By semicontinuity of the fiber dimension (Theorem 14.112), U := { x ∈ X ; dimx f −1 (f (x)) = n − c } is therefore open in X (but possibly empty), and U → S is syntomic by Proposition 19.52. One important property of syntomic morphisms is that they can always be deformed locally: Theorem 19.56. Let X → S be a syntomic morphism of schemes and let S ,→ S˜ be a nil-immersion. Then for every point of X there exist an open neighborhood U ⊆ X of x ˜ ˜ such that U ˜ ט S ∼ and a syntomic S-scheme U = U. S

This theorem implies a similar result for smooth morphisms (see Exercise 19.22). Moreover, it can be generalized to arbitrary immersions S ,→ S˜ (Exercise 19.23). ˜ for an ideal I of R, ˜ and ˜ S = Spec R with R = R/I Proof. We may assume that S˜ = Spec R, X = Spec A are affine with A = R[T1 , . . . , Tn ]/(f1 , . . . , fc ) such that every non-empty fiber ˜ 1 , . . . , Tn ] of of X → S has dimension n − c (Proposition 19.52). Choose any lifts f˜i ∈ R[T ˜ 1 , . . . , Tn ]/(f˜1 , . . . , f˜c ). As the fibers of Spec A˜ → S˜ and of Spec A → S fi and set A˜ := R[T ˜ := Spec A˜ is syntomic over S. ˜ are the same, Proposition 19.52 implies that X

Exercises Exercise 19.1. Let k be a field, A = k[X, Y, Z]/(Z 2 , ZX, Z(Y − 1)) and let x and y be the image of X and Y in A. Show that (x, y) is a completely intersecting sequence in A but not a regular sequence. Show that (y, x) is a regular sequence in A. Exercise 19.2. Let A be a local ring such that A ∼ = R/I where R is a regular local ring and I ⊂ R an ideal. Show that there exists a regular local ring S with maximal ideal mS and an ideal J ⊂ S such that A ∼ = S/J and J ⊆ m2S . Moreover, show that I is generated by a regular sequence if and only if J is generated by a regular sequence.

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19 Local complete intersections

Exercise 19.3. Let R be a local regular ring with maximal ideal mR and residue field kR , let I ⊊ R be an ideal, A := R/I. Let x = (x1 , . . . , xd ) be a regular sequence of R generating mR (hence d = dim R). (1) Assume that I ⊆ m2R (cf. Exercise 19.2). Show that H1 (x, A) ∼ = I/mR I. (2) Show that the image of x in A is a minimal system of generators of the maximal ideal of A if and only if I ⊆ m2R . (3) Let mA be the maximal ideal and kA the residue field of A. Show that dimkA H1 (x, A) = dimkR (I/mR I) − d + dimkA (mA /m2A ). Exercise 19.4. Let A be a ring and let f = (f1 , . . . , fn ) ∈ An , n ≥ 1 an integer. For every A-module M we define a complex (with differential of degree +1) K • (f , M ) := Hom(K• (f ), M ). Denote its i-th cohomology by H i (f , M ). Assume that for all i = 1, . . . , n the multiplication by fi on the module fi−1

f1

ker(M −→ M ) ∩ · · · ∩ ker(M −→ M ) is surjective (then f is called M -coregular ). Show that Hi (f , M ) = 0 for i < n. Exercise 19.5. Let A be a ring, let L be a flat A-module, and let u : L → A be an A-linear map. Show that K• (u, −) is an exact functor from the category of A-modules to the category of chain complexes of A-modules. Deduce that (Hi (u, −))i form a homological δ-functor (here “homological” means that the boundary maps lower the degree). Remark : Using the derived category of A-modules and the derived tensor product introduced in Chapter 21, this exercise can be also interpreted as follows: Let L be any A-module also consider K• (u) as a cochain complex K(u) concentrated in degrees ≤ 0 by setting K i (u) := K−i (u) with the same differentials. Then M 7→ K(u) ⊗L A M defines a triangulated functor D(A) → D(A). Hence every distinguished triangle in D(A), e.g., given by an exact sequence of complexes of A-modules, yields a long exact cohomology sequence. If L is flat, then K(u) is a K-flat complex of A-modules and the derived tensor product K(u) ⊗L A M is given by the usual tensor products K(u) ⊗A M of complexes. Exercise 19.6. Let R be a ring, let M be an R-module, I := {1, . . . , n} and p ≥ 0 an integer. A map ω : I p → M is called alternating if it satisfies the following conditions. (a) For every permutation σ ∈ Sp and (a1 , . . . , ap ) ∈ I p one has ω(aσ(1) , . . . , aσ(p) ) = sgn(σ)ω(a1 , . . . , ap ). (b) For all (a1 , . . . , ap ) ∈ I p such that two of the ai are equal one has ω(a1 , . . . , ap ) = 0. We denote by Cnp (M ) the R-module of alternating maps I p → M . Now let A = R[X1 , . . . , Xn ] and assume M is an A-module (in other words, M is an R-module endowed with pairwise commuting R-linear endomorphisms fi of M defined by Xi ). Let X := (X1 , . . . , Xn ). V p (1) Consider the map HomA ( AI , M ) → Cnp (M ) that sends f to the alternating map (a1 , . . . , ap ) 7→ f (ea1 ∧ · · · ∧ eap ), where (ei )1≤i≤n is the standard basis of AI . Show that this map is an isomorphism of R-modules ∼

K i (X, M ) → Cnp (M ). By transport of structure one obtains a complex with differentials ∂ p : Cnp (M ) → Cnp+1 (M ). It is denoted by K • (f , M ). If fi is given by multiplication by an element of R for all i, then K • (f , M ) reduces to the complex defined in Exercise 19.4.

89 (2) Let M = R[T1 , . . . , Tn ] and fi the R-linear endomorphism P of M given by ∂/∂Ti . Show that attaching to ω ∈ Cnp (M ) the differential form a1 1 we write f = f (0) + f ′ (0)T + gT 2 for some g ∈ A[T ] and set b := −f ′ (0)−1 f (0). Then f (b) = g(b)b2 and hence f (b)n−1 = 0 since bn = 0. By induction hypothesis applied to f (b + T ) we find a nilpotent element a such that f (b + a) = 0. As b + a is nilpotent, this implies the claim. Corollary 20.18. Let A be a ring and let I ⊆ A be an ideal such that A = limn A/I n . Then (A, I) is a henselian pair. Proof. By Proposition 20.17, (A, J) is henselian if J contains only nilpotent elements. In particular, the pair (A/I n , I/I n ) is henselian for all n and this shows the claim as the henselian property is stable under limits (Proposition 20.16 (3)). Using henselian pairs one has the following variants of ´etale/smooth/unramified morphism. Definition 20.19. A morphism f : X → Y of schemes is said to be weakly smooth (resp. weakly unramified, resp. weakly ´etale) if for every henselian pair (A, I) and every commutative diagram /X Spec A/I  Spec A

 /Y

f

there exists at least (resp. at most, resp. exactly) one morphism Spec A → X that makes the resulting diagram commutative. We stress that the notions of being weakly smooth and weakly unramified are nonstandard. Remark 20.20. Let f : X → Y be a morphism of schemes. (1) If f is smooth (resp. unramified, resp. ´etale), then f is weakly smooth (resp. weakly unramified, resp. weakly ´etale). Indeed, for “smooth” this follows from Theorem 20.15 (iv). For “unramified” this follows from Lemma 20.11. Formally, this implies the assertion for “´etale”. (2) As any pair (A, I) with I nilpotent is henselian (Proposition 20.17), every weakly smooth (resp. weakly unramified, resp. weakly ´etale) morphism is formally smooth (resp. formally unramified, resp. formally ´etale). (3) Gabber and Ramero defined in [GaRa] O X a morphism f to be weakly unramified (resp. weakly ´etale) if ∆f : X → X ×Y X is flat (resp. if f and ∆f are flat). De Jong and Olander show in [dJOl] X Theorem 1, that a morphism is weakly ´etale in the sense of Gabber-Ramero if and only if it is weakly ´etale in the sense of Definition 20.19. They also show (loc. cit. Corollary 13) that every weakly unramified morphism in the sense of Gabber-Ramero is weakly unramified in the sense of Definition 20.19.

101

The ´ etale topology As explained above we will introduce the ´etale “topology” by replacing open subsets of a scheme X by ´etale morphisms U → X. Of course this will not lead to a topology in the usual sense. For instance it does not make sense to speak of the union of opens in the ´etale topology. But by defining the notion of a covering for the ´etale topology it makes sense to say that properties are “local for the ´etale topology”. Moreover we will consider for two ´etale morphisms U → X and V → X the fiber product U ×X V as the “intersection” of U → X and V → X. Then having the notions of finite intersections and coverings at our disposal we may also speak of “sheaves for the ´etale topology”. For instance the descent results on quasi-coherent modules in Section (14.16) will immediately imply that quasi-coherent modules define sheaves for the ´etale topology. We will also introduce the notion of a “point in the ´etale topology” and the stalk of a sheaf for the ´etale topology. It turns out that the stalk in the ´etale topology of the structure sheaf of a scheme X is always a henselian ring with separably closed residue field.

´ (20.5) Etale topology. Definition 20.21. Let X be a scheme. (1) An ´etale S covering of X is a family of ´etale morphisms (gi : Ui → X)i∈I such that X = i∈I g(Ui ). (2) If U = (Ui → X)i∈I and V = (Vj → X)j∈J are ´etale coverings of X, then V is called a refinement of U if there exists a map α : J → I and for each j ∈ J an X-morphism Vj → Uα(j) . Note that Vj → Uα(j) is automatically ´etale by Remark 18.36. If one replaces “´etale” by “inclusion of an open subscheme” in Definition 20.21, one obtains the usual topological notions. Remark 20.22. Let X be a quasi-compact scheme. Then for every ´etale covering U = (gi : Ui → X)i∈I of X there exists a finite refinement (Vj → X)j∈J of U such that Vj is affine for all j ∈ J. Indeed, choosing for each Ui an open affine covering, we can replace U by a refinement such that all Ui are affine. As gi (Ui ) is open in X (gi is flat and locally of finite presentation and hence open by Theorem 14.35) and as X is quasi-compact, there exists a finite subset S J of I such that X = j∈J g(Uj ). Remark 20.23. We say that a property P of a scheme is local for the ´etale topology if for any scheme X with an ´etale covering (Ui → X)i the scheme X has property P if and only if all Ui have property P. Examples for properties that are local for the ´etale topology on locally noetherian schemes are the properties “reduced”, “normal”, “regular”, “Cohen-Macaulay” and, more generally, the properties “(Rk )” and “(Sk )” (for fixed k) by Corollary 14.60 and Remark 14.61. Similarly one has the notion that a property P of a scheme morphism is local for the ´etale topology on the source or on the target.

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20 The ´etale topology

Definition 20.24. (1) P is local for the ´etale topology on the source if for all morphisms of schemes f : X → Y and for every ´etale covering (gi : Ui → X)i of X the morphism f satisfies P if and only if f ◦ gi satisfies P for all i. (2) P is local for the ´etale topology on the target if for all morphisms of schemes f : X → Y and for every ´etale covering (Vj )j of Y the morphism f satisfies P if and only if f × idVj : X ×Y Vj → Vj satisfies P for all j. Remark 20.25. Assume that the property “´etale” implies P and that for all morphisms f : X → Y and all ´etale surjective morphisms g : U → X the composition f ◦ g has property P if and only if f has property P. Then P is local on the source for the ´etale topology. Indeed, if (gi : Ui → X)i is F an ´etale covering of X, then one can pass to the induced ´etale surjective morphism g : i Ui → X whose restriction to Ui is gi . In view of Appendix C the following proposition yields plenty of properties that are local for the ´etale topology on the target. Proposition 20.26. Let P be a property of morphisms of schemes. Assume that P is local on the target for the Zariski topology, stable under ´etale base change and stable under faithfully flat descent (Definition 14.54). Then P is local for the ´etale topology on the target. Proof. Let f : X → Y be a morphism of schemes and let (Vj → Y )j∈J be an ´etale covering of Y . If f satisfies P, then its ´etale base change f(Yj ) : X ×Y Vj → Vj satisfies P for all j by hypothesis. Conversely, assume that f(Yj ) satisfies P for all j. As P is local on the target for the Zariski topology, we may assume that Y is affine. As P is stable under ´etale base change, we may replace (Vj → Y )j∈J by any refinement. Hence we may assume F by Remark 20.22 that J is finite and that Vj is affine. Then the morphism g : V := j Vj → Y , whose restriction to each Vj is the given Vj → X, is an ´etale surjective affine morphism. In particular, it is faithfully flat and quasi-compact. As P is local on the target for the Zariski topology, f(Vj ) satisfies P if and only if f(V ) satisfies P. As P is stable under faithfully flat descent, this implies that f has the property P. Example 20.27. The property “smooth” is local for the ´etale topology on the source (we may apply Remark 20.25 by Theorem 18.74) and on the target (Proposition 20.26). A morphism f : X → S is locally for the ´etale topology of the form AnS → S if and only if it is smooth (Corollary 18.57). Remark 20.28. One can also work with other Grothendieck topologies on the category of schemes by replacing “´etale” in Definition 20.21 by other properties P of scheme morphisms. To really get a Grothendieck topology one should assume that P is stable under composition and base change. Usually one also assumes that P is local for the Zariski topology on the source and on the target (to ensure that they define a Grothendieck topology which is finer than the Zariski topology). As one would like to glue morphisms of schemes given locally for such a topology one usually assumes that the topology is coarser than the fpqc-topology, cf. Section (14.18). Moreover, to have an analogue of Remark 20.22 it is useful to assume that P implies “open” (otherwise instead of a naive analogue of Definition 20.21 a more refined version should be used).

103 Standard choices for P are “syntomic” and “flat, locally of finite presentation” (usually abbreviated as fppf from the French fid`element plat de pr´esentation finie). For these choices one can use the analogues of Definition 20.21 and Definition 20.24, and Remark 20.22, Remark 20.25, and Proposition 20.26 hold with the same proof. See also Section (27.6) below where fppf-sheaves are studied in more detail. Another obvious choice would be P = “smooth” but Corollary 20.8 implies that every smooth covering can be refined by an ´etale covering. For a careful discussion of these matters we refer to the Stacks project [Sta], in particular Chapter 020K. (20.6) Lifting of ´ etale schemes. We consider the question what kind of morphisms should be considered as “homeomorphisms for the ´etale topology”. It will turn out that certainly universal homeomorphisms do the job (Theorem 20.30). Hence let us first recall the following characterization of universal homeomorphisms, which is essentially already contained in Volume I. Proposition 20.29. Let f : X → S be a morphism of schemes. The following assertions are equivalent. (i) f is a universal homeomorphism (i.e. for all morphisms of schemes S ′ → S the base change X ×S S ′ → S ′ of f is a homeomorphism). (ii) f is universally injective, integral, and surjective. If f is assumed to be of finite type, then this is Exercise 12.32 whose proof is sketched in Appendix D. Recall that according to Proposition 4.35, f is universally injective if and only if it is injective and all residue class extensions are purely inseparable. Proof. “(ii) ⇒ (i)”. As “surjective” is stable under base change, f is universally bijective. As f is integral and “integral” is stable under base change, f is universally closed by Proposition 12.12. Hence f is a universal homeomorphism. “(i) ⇒ (ii)”. Let f be a universal homeomorphism. Then f is universally injective and surjective. As f is injective and closed, f is affine (Exercise 12.3). As f is affine and universally closed, f is integral (Exercise 12.19). Examples for universal homeomorphisms are nil-immersions, the Frobenius morphism for a scheme in characteristic p (Exercise 4.17), and the projection X ⊗k K → X if X is a scheme over a field k and K/k is a purely inseparable extension (Corollary 5.46). ´ ´ We denote by (Et/X) the category of schemes ´etale over X. Morphisms in (Et/X) are morphisms of X-schemes, automatically ´etale by Remark 18.36. Theorem 20.30. Let i : X0 → X be a universal homeomorphism of schemes. Then the functor Y 7→ Y0 := Y ×X X0 yields an equivalence of categories ∼ ´ ´ Φ : (Et/X) −→ (Et/X 0 ).

The functor Φ is fully faithful under the weaker hypothesis that i is universally submersive (Corollary 14.43), separated and has geometrically connected fibers (Exercise 20.2). We will prove that Φ is essentially surjective only if i is a nil-immersion. For the general case see [SGA4] O Exp. VIII, Th´eor`eme 1.1.

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20 The ´etale topology

Proof. Φ is fully faithful. Let Y , Y ′ be ´etale X-schemes and let f0 : Y0 → Y0′ be a morphism of X0 -schemes. Let Γf0 ⊆ Y0 ×X0 Y0′ = (Y ×X Y ′ )0 be the graph of f0 . As Y0′ → X0 is unramified, Γf0 is an open subscheme of (Y ×X Y ′ )0 (Remark 18.30). As (Y ×X Y ′ )0 → Y ×X Y ′ is a homeomorphism, there exists a unique open subscheme Γ of Y ×X Y ′ such that Γ0 = Γf0 . Then the composition Γ → Y ×X Y ′ → Y is ´etale and a universal homeomorphism and hence an isomorphism by Proposition 18.76. In other words Γ is the graph of a necessarily unique morphism f : Y → Y ′ such that Φ(f ) = f0 . Φ is essentially surjective if i is a nil-immersion. As we already know that Φ is fully faithful and as i is a universal homeomorphism, gluing data for ´etale schemes over X and for ´etale schemes over X0 correspond to each other. Thus we may assume that X = Spec A and X0 = Spec A0 are affine and that we are given an affine ´etale X0 -scheme Y0 = Spec B0 , where B0 is a standard ´etale algebra (Definition 18.39). We have A0 = A/I, where I is an ideal consisting only of nilpotent elements. Write B0 = (A0 [T ]/f0 )g0 , where f0 , g0 ∈ A0 [T ] such that f0 is monic and such that f0′ is invertible in B0 . Choose polynomials f, g ∈ A[T ] lifting f0 and g0 , respectively, such that f is monic. Set B := (A[T ]/f )g . Then B ⊗A A0 ∼ = B0 and f ′ is invertible in B by Lemma 18.34. Hence B is a standard ´etale A-algebra which lifts B0 . (20.7) Sheaves in the ´ etale topology. We will now define what it means to be a sheaf for the ´etale topology on a scheme S. In Section (14.18) we sketched how to endow the category of all schemes over S with a “topology” and how to define sheaves for such topologies. Here we consider only schemes that are “open in the ´etale topology of S”, i.e., ´etale morphisms U → S. Accordingly, we will think of a sheaf (as defined below) as a sheaf on ´ S equipped with the ´etale topology. So let (Et/S) be the category of all ´etale S-schemes ´ U → S. A morphism (U → S) → (V → S) in (Et/S) is an S-morphism U → V which is automatically ´etale by Remark 18.36. ´ For morphisms j1 : U1 → V and j2 : U2 → V in (Et/S), their fiber product U1 ×V U2 is again an ´etale S-scheme because “´etale” is stable under base change and composition. ´ This clearly is a fiber product in the category (Et/S). If j1 and j2 are inclusions of open subschemes, U1 ×V U2 = U1 ∩ U2 . Hence we think in general of U1 ×V U2 as the “intersection” of two opens in the ´etale topology. With this in mind, the definition of a sheaf for the ´etale topology is verbatim as the definition of a sheaf on a usual topological space (Definition 2.18). Definition 20.31. Let S be a scheme. ´ (1) A presheaf (of sets) for the ´etale topology on S is a contravariant functor on (Et/S) with values in the category of sets. A morphism of presheaves is a morphism of ´ we often simply write s 7→ s|U for functors. For a morphism g : U → V in (Et/S) the map F (g) : F (V ) → F (U ). (2) A presheaf F on S for the ´etale topology is called a sheaf if for every ´etale S-scheme U → S and for every ´etale covering (gi : Ui → U )i∈I the diagram (20.7.1)

F (U )

ρ

/ Q F (Ui ) i∈I

σ σ

′

//

Q

F (Ui ×U Uj )

(i,j)∈I×I

is exact, where ρ is the map s 7→ (s|Ui )i and where σ and σ ′ are given by (si )i 7→ (si|Ui ×U Uj )i,j∈I and by (si )i 7→ (sj |Ui ×U Uj )i,j∈I , respectively. A morphism of sheaves is a morphism of presheaves.

105 In the same way one also defines (pre-)sheaves of groups or rings on S for the ´etale topology. Furthermore, given a sheaf of rings AS on S for the ´etale topology one has the obvious notion of an AS -module for the ´etale topology. Remark 20.32. Arguing similarly as in the proof of Proposition 20.26 one sees that a presheaf F for the ´etale topology on S is a sheaf if (20.7.1) is exact for all Zariski coverings (i.e., coverings (ji : Ui → U )i∈I , where ji is the inclusion of an open subscheme) and for all ´etale coverings of the form (g : V → U ), where g is an ´etale surjective morphism ´ of affine schemes in (Et/S). Proposition 20.33. Let S be a scheme and let F be a quasi-coherent OS -module. Consider F as a presheaf on S for the ´etale topology by defining F (U ) := Γ(U, g ∗ F ) for every ´etale morphism g : U → S. Then F is a sheaf for the ´etale topology. Proof. As F satisfies the sheaf condition of Zariski coverings, it suffices to check that the sequence (20.7.1) is exact for an ´etale covering of the form (g : U → V ) with U and V affine. As g is ´etale and surjective, g is faithfully flat. Hence Lemma 14.64 yields the exactness of (20.7.1) in this case. In fact the proof shows the (much stronger) assertion that F yields a sheaf for the fpqc-topology in the sense of Section (14.18). Example 20.34. Let f : X → S be a morphism of schemes and d ≥ 0 an integer. For every ´etale X-scheme U → X define F (U ) := Γ(U, ΩdU/S ). Then this is a sheaf for the ´etale topology on X. Indeed, for every ´etale X-scheme g : U → X we have g ∗ ΩdX/S = ΩdU/S (use Corollary 18.19 and that exterior powers are compatible with pullback g ∗ ). Hence F is simply the sheaf for the ´etale topology attached to the quasi-coherent OX -module ΩdX/S (Proposition 20.33). Remark 20.35. Once one has a notion of sheaf of abelian groups, one can define a cohomology theory by considering the right derived functors of the global section functor, and more generally the right derived functors of the direct image functor. For the ´etale topology, this gives rise to the theory of ´etale cohomology. While the starting point, that is the formalism of derived functors, is similar to the theory of cohomology of OX -modules and (quasi-)coherent sheaves (for the Zariski topology) that we will develop in the following chapters, it turns out that ´etale cohomology yields a theory that is “more topological” and closer to theories such as singular cohomology of topological spaces or de Rham cohomology of manifolds. We will not go into this any further, see for example [SGA4] O , [SGA4 21 ] O , [Mil1] O , [FrKi] O , [Tam] O or [Fu] O . (20.8) Points and stalks in the ´ etale topology. We now come to the stalk of the structure sheaf at a point in the ´etale topology. We first have to clarify what we mean by a “point”. There is the abstract notion of a point of a topos (i.e., of the category of sheaves on a category endowed with a Grothendieck topology), see [SGA4] O Exp. IV, §6. We will not need this abstract notion here. One can show ([Sta] 04HU) that for a scheme endowed with the ´etale topology the points in this abstract sense are given (up to isomorphism) by geometric points in the following sense.

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Definition 20.36. Let S be a scheme. A geometric point of S is a morphism Spec k → S, where k is an algebraically closed field. Such a geometric point Spec k → S is usually denoted by s¯ if s ∈ S is the image of Spec k → S. One also says that s¯ lies over s. The field k is then denoted by κ(¯ s). Of course there are different geometric points lying over the same (topological) point of S. If f : S → T is a morphism of schemes, we often denote the geometric point f ◦ s¯ simply by f (¯ s). More generally, every morphism Spec k → S, where k is separably closed, gives rise to a point for the ´etale topology on a scheme S in the abstract sense, and sometimes the notion “geometric point” is defined accordingly. Definition 20.37. Let S be a scheme, s ∈ S and let s¯ : Spec κ(¯ s) → S be a geometric point lying over s. An ´etale neighborhood of s¯ is a pair (U, u ¯), where U → S is an ´etale S-scheme and where u ¯ : Spec κ(¯ s) → U is a geometric point such that g(¯ u) = s¯. We will now define two notions of stalks of the structure sheaf, one where we consider only ´etale neighborhoods with trivial residue field extension and the second where we will consider ´etale neighborhoods whose residue field extension is a subextension of a given geometric point. We first define the corresponding categories of neighborhoods and then show that they are cofiltered to obtain a well behaved colimit. Let S be a scheme, s ∈ S and let s¯ : Spec κ(¯ s) → S be a geometric point lying over s. Let U (¯ s) be the category of ´etale neighborhoods (U, u ¯) of (S, s¯). A morphism (U, u ¯) → (V, v¯) in this category is an S-morphism h : U → V such h(¯ u) = v¯. Let U0 (s) be the full subcategory of U (¯ s) consisting of those ´etale neighborhoods (U, u ¯) such that if u ∈ U is the image of u ¯, the ´etale morphism U → S induces an isomorphism ∼ κ(s) → κ(u). Then the choice of u ∈ U already determines u ¯. Therefore U0 (s) can also be described as the category of pairs (U, u), where g : U → S is an ´etale S-scheme and where u ∈ U is a point with g(u) = s and κ(s) ∼ = κ(u). In particular it depends only on s and not on s¯. Lemma 20.38. The categories U (¯ s) and U0 (s) are cofiltered (Definition F.12). Proof. If (Ui , u ¯i ), i = 1, 2 are ´etale neighborhoods of s¯, then there exists an ´etale neighborhood (U, u ¯) and morphisms (U, u ¯) → (Ui , u ¯i ), namely (U, u ¯) = (U1 ×S U2 , (¯ u1 , u ¯2 )S ). Let h1 , h2 : (U, u ¯) → (U ′ , u ¯′ ) be two morphisms in U (¯ s). Then the inclusion j : U ′′ := Ker(h1 , h2 ) → U satisfies h1 ◦ j = h2 ◦ j. As h1 (¯ u) = h2 (¯ u), u ¯ factors through a geometric point u ¯′′ of U ′′ . As U ′ → S is unramified, U ′′ is an open subscheme of U (Remark 18.30). Hence it is ´etale over S and (U ′′ , u ¯′′ ) is an ´etale neighborhood of s¯. This shows that U (¯ s) is cofiltered. The same argument (using Lemma 4.28 to see that under the above constructions extensions of residue fields stay trivial) shows that U0 (s) is cofiltered. Definition 20.39. Let S be a scheme, let F be a presheaf on S for the ´etale topology, and let s¯ : Spec κ(¯ s) → S be a geometric point. The stalk of F in s¯ is defined as Fs¯ := colim F (U ). U (¯ s)

The colimit here “makes sense” because the category U (¯ s) admits a cofinal small category.

107 (20.9) Stalks of the structure sheaf: (strict) henselization. As every quasi-coherent module defines a sheaf for the ´etale topology (Proposition 20.33), this holds in particular for the structure sheaf of a scheme S. We will now study its stalk OS,¯s in a geometric point s¯ lying over a point s ∈ S. We also define (20.9.1)

h OS,s := colim OS (U ), U0 (s)

i.e., here we take the limit only over ´etale neighborhoods inducing a trivial residue field extension. As the category of Zariski open neighborhoods is a full subcategory of U0 (s), and as U0 (s) is a full subcategory of U (¯ s) we obtain ring homomorphism (20.9.2)

h OS,s −→ OS,s −→ OS,¯s .

Definition 20.40. A local henselian ring with separably closed residue field is called strictly henselian. Proposition 20.41. Let S be a scheme, let s ∈ S, let ms ⊂ OS,s be the maximal ideal, and let s¯ → S be a geometric point lying over s. h (1) Then OS,s is a henselian ring with residue field κ(s), and OS,¯s is a strictly henselian ring whose residue field is the separable closure of κ(s) in κ(¯ s). h is the (2) The homomorphisms (20.9.2) are local and faithfully flat. Moreover, ms OS,s h maximal ideal of OS,s , and ms OS,¯s is the maximal ideal of OS,¯s . Proof. As “´etale” is compatible with filtered inductive limits, we may replace S by Spec R, where R = OS,s . Let m be the maximal ideal of R and k its residue field. Let a ∈ Rh \ mRh . Then a is represented by a triple (U, u, aU ), where (U, u) ∈ U0 (s) and aU ∈ OU (U ). By shrinking U we may assume that U = Spec A is affine and that u is the only point of U lying over s. Then its fiber Us is isomorphic to Spec κ(u) because U → S is ´etale (Corollary 18.45). In other words, A/mA = κ(u) and hence pu = mA, where pu ⊂ A is the prime ideal corresponding to u. As a ∈ / mRh we find aU ∈ / mA = pu . Replacing U by the open neighborhood D(aU ) of u, we may assume that aU is invertible. Thus a is invertible in Rh . This shows that Rh is a local ring with maximal ideal mRh . s) Its residue field is the compositum of all residue field extensions κ(u) of k within κ(¯ hence equal to k because κ(u) = k for all (U, u) ∈ U0 (s). By restricting to the cofinal subcategory of U0 (s) of those (U, u) such that U is affine, one sees that Rh is a filtered colimit of flat R-algebras (because ´etale morphisms are flat). Therefore Rh is flat over R and even faithfully flat because R → Rh is local (Example B.18). Let us show that Rh is henselian. Let A be a standard ´etale Rh -algebra and set X = Spec A. By Remark 20.13 it suffices to show that X(Rh ) → X(k) is surjective. By Remark 18.41 there exists an affine ´etale neighborhood (Spec C, u) ∈ U0 (s) and a standard ´etale C-algebra A˜ such that A˜ ⊗C Rh = A. An element x ∈ X(k) yields a point ˜, u ˜ := Spec A˜ such that k = κ(˜ ˜) is an object in U0 (s). u ˜∈U u). As C is ´etale over R, (U Therefore by the definition of Rh there exists a C-algebra homomorphism A˜ → Rh or, equivalently, an element of X(Rh ) lifting x. The same arguments show the claims for OS,¯s using that for every (U, u ¯) ∈ U (¯ s), u ¯ lying over u ∈ U , the residue extension κ(s) → κ(u) is finite and separable.

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20 The ´etale topology

Definition and Remark 20.42. Let S be a scheme and let s ∈ S. h (1) The ring OS,s is called the henselization of S in s. In particular we have for any local ring R its henselization Rh (for S = Spec R, s its closed point). (2) Let s¯, s¯′ → S be two geometric points lying over s. Then, after possibly switching s¯ and s¯′ , we may assume that there exists an S-morphism s¯ → s¯′ . Composition with this morphism induces an equivalence of categories U (¯ s′ ) → U(¯ s). In particular, OS,¯s and OS,¯s′ are isomorphic. Therefore OS,¯s up to isomorphism (but not up to unique isomorphism) depends hs only on S and s and we denote it by OS,s and call it a strict henselization of OS,s . In particular we have defined for a local ring R a strict henselization Rhs . As usual we base the abbreviation “hs” on the French expression “henselis´e strict”. If R is a normal local ring, then Exercise 20.26 gives another description of Rh and of Rhs , see also Exercise 20.27. Remark 20.43. Let R be a local ring. Then R → Rh yields an isomorphism of the ch (with respect to the adic topologies given by maximal ideals). ˆ→R completions R Indeed, if R is noetherian, this follows from Corollary 18.66. For the general case we refer to [EGAIV] O (18.6.6). Many properties that are local for the ´etale topology can therefore be checked after a base change to a strictly henselian ring: Remark 20.44. Assume that P is a property of morphisms of schemes that is local for the ´etale topology on the target and that is compatible with filtered inductive limits. Let f : X → S be a scheme morphism such that for every s ∈ S the base change hs hs f(OS,s hs ) : X ×S Spec O S,s → Spec OS,s has property P. Then f has property P. As an example of this principle we prove the following variant of Zariski’s main theorem. Proposition 20.45. Let f : X → S be a morphism locally of finite type, x ∈ X, s = f (x). Assume that x is an isolated point of the fiber Xs . Then there exists (U, u) in U0 (s) such that every point x′ ∈ XU := X ×S U lying over x and over u has an open affine neighborhood V ′ in XU such that f(U )|V ′ : V ′ → U is a finite morphism. Moreover, if f is separated, then V ′ is open and closed in XU . We will give the proof only if f is locally of finite presentation. The general case then follows by a standard limit argument in the spirit of Chapter 10 (see [EGAIV] O (18.12.1)). Proof. If f is separated, f(U ) is separated. If j ′ : V ′ → XU denotes the inclusion, then if f(U ) ◦ j ′ is finite, j ′ is finite by the standard cancellation argument (Remark 9.11). Hence j ′ is closed. This proves the final assertion. As the question is local on X and on S, we may assume that S = Spec R and X = Spec A, where A is an R-algebra of finite presentation. By Zariski’s main theorem (Corollary 12.79) the points of X that are isolated in their fiber form an open subset of X. Thus by restriction to an affine open neighborhood of x we can assume that f is quasi-finite. h ˜ := X ×S T After base changing f with T := Spec OS,s → S we find for every point x ˜∈X ˜ ˜ such lying over the closed point of T an open and closed affine neighborhood V of x ˜ in X that f(T )| V˜ is finite (Proposition 20.5).

109 As V˜ is of finite presentation over T and as the property “finite” is compatible with h filtered inductive limits, we obtain the desired result because of the definition of OS,s . We conclude this section by giving examples of properties of rings that can be checked on the (strict) henselization. For later use we also consider the following definition. ˆ→ Definition 20.46. A local noetherian ring R is called a G-ring if all fibers of Spec R Spec R are geometrically regular. A quasi-excellent local noetherian ring is a G-ring by definition.

Proposition 20.47. Let R be a local ring. (1) The following assertions are equivalent. (i) R is noetherian (resp. normal, resp. reduced). (ii) Rh is noetherian (resp. normal, resp. reduced). (iii) Rhs is noetherian (resp. normal, resp. reduced). (2) Assume that R is noetherian. Then the following assertions are equivalent. (i) R is regular (resp. Cohen-Macaulay, resp. satisfies (Rk ), resp. satisfies (Sk ), resp. G-ring). (ii) Rh is regular (resp. Cohen-Macaulay, resp. satisfies (Rk ), resp. satisfies (Sk ), resp. G-ring). (iii) Rhs is regular (resp. Cohen-Macaulay, resp. satisfies (Rk ), resp. satisfies (Sk ), resp. G-ring). (3) If R is excellent, then Rh is excellent. For the proof we refer to [EGAIV] O 18.6 – 18.8. There is also a global analogue of the henselization of a local ring. For this consider the category of pairs (A, I) with A a ring and I ⊆ A an ideal. Morphisms (A, I) → (B, J) in this category are ring homomorphisms φ : A → B such that φ(I) ⊆ J. Then we have the following result (see [Sta] 0A02, [Sta] 0AGU, [Sta] 0F0L). Theorem 20.48. The inclusion functor from the full subcategory of henselian pairs (A, I) into the category of all pairs has a left adjoint (A, I) 7→ (Ah , I h ). The unit map A → Ah is flat, I h = IAh and A/I n ∼ = Ah /I n Ah for all n ≥ 1. Moreover, (Ah , I h ) depends only on A and the closed subset V (I) of Spec A. The construction of Ah is the same as in the local case: One defines Ah as the filtered colimit over all ´etale A-algebras B such that A/I → B/IB is an isomorphism and set I h = IAh . In particular, for local rings A and I their maximal ideal we recover the notion of henselization from Definition 20.42. (20.10) Unibranch schemes. Irreducibility is not local for the ´etale topology. To make this more precise, recall that for a scheme X and a point x ∈ X the irreducible components of X containing x are in bijection to the minimal prime ideals of OX,x . Hence x is contained only in a single irreducible component if and only if OX,x contains a unique prime ideal (if and only if Spec OX,x is irreducible).

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h hs As the ring homomorphisms OX,x → OX,x → OX,x are faithfully flat and in particular injective, one has implications hs h Spec OX,x is irreducible ⇒ Spec OX,x is irreducible ⇒ Spec OX,x is irreducible.

But the example of a node of a curve (Exercise 20.22) shows that the converse does not hold in general (even if OX,x is noetherian, reduced and of dimension 1). h We think of the irreducible components of Spec OX,x as “branches through x”. This leads us to the following definition. Definition 20.49. A local ring A is called unibranch (resp. geometrically unibranch) if Spec Ah (resp. Spec Ahs ) is irreducible. A scheme X is called unibranch (resp. geometrically unibranch) if OX,x is a unibranch (resp. geometrically unibranch) local ring for all x ∈ X. The following result relates minimal prime ideals of the (strict) henselization with maximal ideals of the normalization. For this we denote the set of minimal (resp. of maximal) prime ideals of a ring R by Min(R) (resp. by Max(R)). Lemma 20.50. Let A be a local integral domain with maximal ideal m and let B be its normalization. Let A → A˜ be a local ring homomorphism such that A˜ is henselian and such that A˜ is a filtered colimit of ´etale A-algebras (e.g., A˜ = Ah or A˜ = Ahs ). Let k˜ be ˜ Then there exist bijections ˜ := B ⊗A A. the residue field of A˜ and set B ˜ ˜ ←→ Min(B) ˜ ←→ Max(B) ˜ ←→ Spec(B ⊗A k). Min(A) ˜ is integral over A. ˜ Hence “going up” (Theorem B.56) Proof. As B is integral over A, B ˜ shows that the maximal ideals of B are the prime ideals lying over the maximal ideal of ˜ This proves the last bijection. A. Let us construct the first bijection. As A˜ is a filtered inductive limit of ´etale Aalgebras, it is flat over A. As flat homomorphisms are generizing (Lemma 14.9), every minimal prime ideal of A˜ lies over the unique minimal prime ideal of A. In other words ˜ = Min(A⊗ ˜ A K), where K is the field of fractions of A. As A˜ is flat over A, B ˜ is flat Min(A) ˜ ˜ over B and hence we also have Min(B) = Min(B ⊗B K) (as B is the normalization of A, ˜ we have B ˜ ⊗B K ∼ B has the same field of fractions as A). But by definition of B = A˜ ⊗A K. This yields the first bijection. ˜ and Max(B). ˜ If A˜ is the filtered It remains to construct a bijection between Min(B) colimit of ´etale A-algebras Aλ , then Bλ := B ⊗A Aλ is ´etale over B. As B is normal, Bλ ˜ = colim Bλ is normal. Hence for n ˜ the ˜ ∈ Max(B) is normal (Corollary 14.60), whence B ˜n˜ is local and normal and hence an integral domain. Hence there exists a local ring B ˜ contained in n ˜, which corresponds to the zero ideal in unique minimal prime ideal of B ˜ ˜ → Min(B). ˜ We claim that µ is bijective. Bn˜ . This defines a map µ : Max(B) Clearly, it is surjective as every minimal prime ideal is contained in some maximal ˜ Now B ˜ is ˜ and n ˜′ be distinct maximal ideal of B. ideal. To show the injectivity of µ let n ˜ ˜α . As A˜ is henselian, B integral over A˜ and hence a filtered colimit of finite A-algebras ˜α is a product of local rings. As the set idempotents in B ˜ is the filtered colimit each B ˜α , there exists an idempotent e ∈ B ˜ with value 0 in of the sets of idempotents of the B ′ ˜ n) and value 1 in κ(˜ n ). Therefore Spec B is the disjoint union of two open and closed κ(˜ ˜ ∈ Y and n ˜′ ∈ Y ′ . But then µ(˜ subschemes Y and Y ′ with n n) ∈ Y and µ(˜ n′ ) ∈ Y ′ and we ′ obtain that µ(˜ n) ̸= µ(˜ n ).

111 Proposition 20.51. Let A be a local integral domain with residue field k. (1) A is unibranch if and only if its normalization is a local ring. (2) A is geometrically unibranch if and only if its normalization is a local ring whose residue field is a purely inseparable extension of k. It is not difficult to generalize this Proposition to arbitrary local rings A (Exercise 20.20). Proof. Let B be the normalization of A. To see (1), we apply Lemma 20.50 to A˜ = Ah . As the residue field k˜ of A˜ is equal to k, we find a bijection between Min(Ah ) and Spec(B ⊗A k). As B is integral over A we deduce a bijection Min(Ah ) ↔ Max(B). This shows (1). Let us prove (2). Let Ahs be a strict henselization of A and let k sep be its residue field. Applying Lemma 20.50 to A˜ = Ahs we obtain a bijection Min(Ahs ) ↔ Spec(B ⊗A k sep ). Let (ni )i∈I be the family of maximal ideals of B and let ki = κ(ni ) their residue fields. As B is integral over A, ki is an algebraic extension of k and hence # Max(ki ⊗k k sep ) = [ki : k]sep ∈ Z≥1 ∪ {∞}. Therefore we obtain an equality in Z≥1 ∪ {∞} X X # Min(Ahs ) = # Spec(B ⊗A k sep ) = # Max(ki ⊗k k sep ) = [ki : k]sep . i∈I

i∈I

This shows in particular that A is geometrically unibranch if and only if #I = 1 (i.e., B is local) and [κB : k]sep = 1, where κB is the residue field of B. Corollary 20.52. Let X be an integral scheme and let π : X ′ → X be its normalization. (1) X is unibranch if and only if π is injective. In this case π is a homeomorphism. (2) X is geometrically unibranch if and only if π is universally injective. In this case π is a universal homeomorphism. Proof. We may assume that X = Spec A is affine. Then X ′ = Spec A′ is affine. For p ∈ Spec A the normalization of Ap is A′p (Proposition B.55) and the prime ideals of A′p over the maximal ideal pAp are the maximal ideals of A′p by “going up”. Hence π is injective (resp. universally injective) if and only if for all p ∈ Spec A the ring A′p is a local ring (resp. if and only if A′p is a local ring whose residue field is a purely inseparable extension of κ(p) (Proposition 4.35)), i.e., if and only if Ap is unibranch (resp. if and only if Ap is geometrically unibranch) for all p (Proposition 20.51). As π is surjective and integral, it is surjective and universally closed. Hence if π is (universally) injective, then π is a (universal) homeomorphism. Corollary 20.53. Every normal scheme is geometrically unibranch. The converse does not hold as the example of a cusp shows (Exercise 20.22). (20.11) Artin approximation. Let A be a local noetherian ring. Very often it is possible to construct solutions for polynomial equations in the completion Aˆ of A and one would like to deduce the existence ˆ More precisely, we of solutions in A which are arbitrarily close to the given solution in A. will consider the following property.

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Definition 20.54. A local noetherian ring A with maximal ideal m is said to have the approximation property if given polynomials f1 , . . . , fm ∈ A[T1 , . . . , Tn ], a solution a ˆ = (ˆ a1 , . . . , a ˆn ) in Aˆ of the system of equations f1 = · · · = fm = 0, and an integer c ≥ 1, there exists a solution a = (a1 , . . . , an ) ∈ An such that ai ≡ a ˆi (mod mc ) for all i = 1, . . . , n. Clearly, not every local noetherian ring has the approximation theory: Let 2 ̸= p ∈ Z be a prime. Then T 2 + 1 = 0 has no solution in Q and in particular not in Z(p) . But it has a solution in the completion Zp if (and only if) p ≡ 1 mod 4: As Zp is complete, it is henselian and it suffices to find a solution in Fp . But if p = 1 + 4n, then (2n)! is a solution of T 2 = −1 in Fp , as follows from Wilson’s theorem. In Remark 20.14 we have seen that A is henselian if and only if for certain polynomial equations any solution in the residue field of A lifts to a solution in A. As Aˆ is henselian (Example 20.3), and A and Aˆ have the same residue field, we see that every local noetherian ring with the approximation property is necessarily henselian. Moreover, in [Rot] O it is shown that any local noetherian ring with the approximation property is excellent (and henselian). Recall Popescu’s theorem (Theorem 10.76) that if Spec B → Spec A is a regular morphism of noetherian affine schemes, then B is a filtered colimit of smooth A-algebras. In particular Popescu’s theorem implies that a local noetherian ring A is a G-ring (Definition 20.46) if and only if Aˆ is a filtered colimit of smooth A-algebras. It is not difficult to see that if A is henselian and Aˆ is a filtered colimit of smooth A-algebras, then A has the approximation theory (cf. [Art5]). Finally, if A is excellent, then by definition A is a G-ring. Therefore Popescu’s theorem implies the following result. Theorem 20.55. Let A be a local noetherian ring. Then the following assertions are equivalent. (i) A has the approximation property. (ii) A is henselian and excellent. (iii) A is henselian and Aˆ is a filtered colimit of smooth A-algebras. Example 20.56. This result can in particular be applied to the case where A is the local ring of germs of analytic functions on a complex analytic space X in a point x. In fact, A is henselian by Exercise 20.15. Since quotients of excellent rings are excellent ([EGAIV] O (7.8.3) (ii)), to show that A is excellent it is enough to consider the case of a smooth point. But if x is a smooth point, then OX,x is the ring of convergent complex power series in dim(X) variables, and ObX,x is the ring of complex formal power series. It follows from [Mat1] Theorem 102 that OX,x is excellent. Corollary 20.57. Let X → S be a morphism locally of finite type of locally noetherian schemes, and let s ∈ S such that OS,s is a G-ring. Assume that the base change X ×S Spec ObS,s → Spec ObS,s of f has a section t0 . Then there exist an ´etale morphism g : U → S and u ∈ U with g(u) = s and κ(s) ∼ = κ(u) such that the base change X ×S U → U of f has a section t. Moreover, given an integer c ≥ 1 one can choose t such that the images of t and of t0 h h in X(OS,s /mcs OS,s ) = X(OS,s /mcs ) = X(ObS,s /mcs ObS,s ) coincide. It is easy to generalize this assertion from sections of an S-scheme X, i.e., values of the functor hX , to values of an arbitrary functor F : (Sch)S opp → (Sets) which is locally of finite presentation in the sense of Remark 18.61 (see [Art2] O Corollary (1.8)).

113 Proof. We may assume that S = Spec A′ and X = Spec A′ [T1 , . . . , Tn ]/(f1 , . . . , fm ) are affine. Let A := OS,s . Then a section of the base change f(Spec A) ˆ corresponds to a n ˆ ˆ ∈ A of f1 = · · · = fm = 0. As A is a G-ring, its henselization Ah is a solution a G-ring (Proposition 20.47) whose completion is Aˆ (Remark 20.43). Hence Ah has the approximation property and we find a section of the base change f(Spec Ah ) . The corollary follows from the definition of Ah (20.9.1). The hypothesis that OS,s is a G-ring is in practice very often satisfied: Recall from Section (12.12) that if R is a complete local noetherian ring (e.g., if R is a field) or if R a Dedekind ring whose field of fractions has characteristic zero (e.g., if R is the ring of integers in a number field), then every scheme S locally of finite type over R is excellent. In particular, every point s ∈ S satisfies the condition of the corollary. (20.12) Analytification of schemes over C. Let X be a scheme locally of finite type over C. It is possible to attach to X a complex analytic space X an whose underlying set is X(C). A similar construction is possible for other topological fields (such as Qp ) with the correct replacement for “complex analytic space” (such as rigid analytic space or adic space). Here we concentrate on the complex case. Let us first recall the definition of a complex analytic space. For n ≥ 0 we endow Cn with the analytic topology and for U ⊆ Cn open we denote by OU the sheaf of holomorphic functions on U , i.e., for V ⊆ U open, OU (V ) is the set of holomorphic maps V → C. Let U ⊆ Cn be an open subset, let f1 , . . . , fr : U → C be holomorphic functions, and let J ⊆ OU be the ideal generated by f1 , . . . , fr . Let X = { x ∈ U ; f1 (x) = · · · = fr (x) = 0 } and define OX := (OU /J )|X . Then (X, OX ) is a space locally ringed in C-algebras. Such a locally ringed space is called a standard analytic space and it is denoted by V(f1 , . . . , fr ). A complex analytic space is a space locally ringed in C-algebras (X, OX ) with the property for all x ∈ X there exists an open neighborhood U of x in X such that the locally ringed space (U, OX|U ) is isomorphic (over C) to a standard analytic space. A morphism of complex analytic spaces is a morphism of spaces locally ringed in C-algebras. Theorem 20.58. Let X be a scheme locally of finite type over C. Then there exists a complex analytic space X an and a morphism α : X an → X of spaces locally ringed in Calgebras such that for every complex analytic space Y and for every morphism φ : Y → X of spaces locally ringed in C-algebras there exists a unique morphism of analytic spaces f : Y → X an such that α ◦ f = φ. Moreover, α induces a bijection between the underlying topological set of X an and X(C) ∼ and for all x ∈ X an an isomorphism of the completed local rings ObX,α(x) → ObX an ,x . In particular α is flat. The complex analytic space X an is called the analytification of X. The idea of the proof is simple: Locally X is of the form X = Spec C[T1 , . . . , Tn ]/(f1 , . . . , fm ) and we set X an = V(f1 , . . . , fm ), where we consider fi as polynomial functions (and hence as holomorphic functions) on Cn . In particular one has (AnC )an = Cn with the usual structure of a complex manifold. If X is an arbitrary scheme locally of finite type over C, we glue the analytifications. One obtains the description of the underlying set of X an by applying

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20 The ´etale topology

the universal property of (X an , α) to the complex analytic space Spec C. For details we refer to [Ser2] O or [SGA1] O X Exp. XII, 1.1. Due to its universal property (X an , α) is unique up to unique isomorphism. The universal property of X an shows that for every morphism f : X → Y of schemes locally of finite type over C there exists a unique morphism of complex analytic spaces f an : X an → Y an making the following diagram commutative X  X an

f

f an

/Y  / Y an .

We obtain a functor X 7→ X an from the category of schemes locally of finite type over C to the category of complex analytic spaces. Remark 20.59. Many properties of X can be seen from properties of the attached complex analytic space X an . We refer to [SGA1] O X Exp. XII for details. For instance, a scheme locally of finite type over C is separated (resp. proper, resp. smooth) over C if and only if X an is Hausdorff (resp. X an is compact1 , resp. X an is a complex manifold2 ). Similarly, many properties are equivalent for a morphism f and its analytification f an (loc. cit.). For instance f is an isomorphism (resp. a closed immersion) if and only if f an is an isomorphism (resp. a closed immersion). Note that this does not imply that if X and Y are schemes locally of finite type over C such that X an and Y an are isomorphic, then X and Y are isomorphic (see [Har2] O Chap. VI, Example 3.2 for a counterexample). If one restricts to proper or projective schemes, one obtains better results. The main result is the following result, which we do not prove here (see [SGA1] O eor`eme X Exp. XII, Th´ 4.4 for a proof). Theorem 20.60. Let X be a scheme that is proper over C and let α : X an → X be its analytification. Then F 7→ F an := α∗ (F ) yields an equivalence of the category of coherent OX -modules with the category of coherent OX an -modules. Here we use that the notion of coherence is defined for OX -modules on an arbitrary ringed space (X, OX ) (Definition 7.45). This theorem is a special case of the GAGA theorems that will be discussed in Section (23.9) below. Properness is essential here (e.g., see Exercise 26.14). Similarly as for (locally noetherian) schemes, for every complex analytic space X there exists a bijective correspondence between closed analytic subspaces of X and coherent ideals of OX ([Ser2] O Proposition 1). Hence Theorem 20.60 also implies that if X is a proper C-scheme and Y is a closed analytic subspace of X an , then there exists a unique closed subscheme Y of X such that Y an = Y. As morphisms X → Y are given by their graphs Γf ⊂ X × Y , or equivalently, by the ideal sheaves of OX×Y defining Γf , Theorem 20.60 implies the following corollary (we refer to [SGA1] O X Exp. XII, Corollaire 4.5 for details). Corollary 20.61. The functor X 7→ X an from the category of schemes proper over C to the category of compact complex analytic spaces is fully faithful. 1 2

In this book, “compact” is defined to be quasi-compact and Hausdorff. Here, a manifold is not necessarily Hausdorff or second countable.

115 In particular two proper C-schemes X and Y are isomorphic if and only if the complex analytic spaces X an and Y an are isomorphic. The analytification of X = PnC is the usual complex projective space Pn (C). Hence every closed analytic subspace of Pn (C) is the analytification of a projective C-scheme: Corollary 20.62. (Chow’s theorem) The functor X 7→ X an yields an equivalence of the category of projective C-schemes and the category of complex analytic spaces X such that there exists a closed analytic embedding X ,→ Pn (C). On the other hand, the functor in Corollary 20.61 is far from being essentially surjective. We finish by explaining that ´etale morphisms of schemes correspond to local isomorphisms of complex analytic spaces. Remark 20.63. Let f : X → Y be a morphism of schemes locally of finite type over C. Then { x ∈ X ; f is ´etale in x} is open in X by definition. As the set of closed points in X is very dense (Proposition 3.35), f is ´etale if and only if it is ´etale in every closed point. Hence by Corollary 18.66, f is ´etale if and only if f induces isomorphisms ObY,f (x) → ObX,x for all x ∈ X(C). This in turn is equivalent to asking that f an induces ∼ isomorphisms ObY an ,f an (x) → ObX an ,x for all x ∈ X an . But this last condition holds if and only if f an : X an → Y an is a local isomorphism ([Gro5] O Proposition 1.9).

The ´ etale fundamental group of a scheme Our goal is to define a notion of fundamental group and fundamental groupoid for schemes which rests on the ´etale topology. Even though the ´etale topology is much more suitable at this point than the Zariski topology, there is no way to define a notion of path between two points of a scheme in a naive way. Let us begin by recalling two possible definitions of the fundamental groupoid of a topological space X from algebraic topology (see for instance [tDi] O or [May] O X ). The first definition of the fundamental groupoid Π(X) is as the category whose objects are the points of X and whose morphisms x → y are the homotopy classes [γ] of paths γ : [0, 1] → X such that γ(0) = x and γ(1) = y (homotopies of paths are always assumed to fix start and end point of the paths). For paths γ : x → y and δ : y → z the composition [δ] ◦ [γ] is defined as the homotopy class of the concatenation δ · γ obtained by first traversing γ and then δ, each with doubled velocity3 . Then the category Π(X) is indeed a groupoid, i.e., a category in which every morphism has an inverse: [γ]−1 is the homotopy class of the path γ − (t) = γ(1 − t). For a point x ∈ X, the fundamental group π1 (X, x) := AutΠ(X) (x) is the group of homotopy classes of closed paths with start point x in X. An equivalent description of Π(X) can be obtained as follows. Let (Cov(X)) be the category of covers of X (not to be confused with an open covering of X) in the topological sense. A trivial cover of X is a continuous map of the form of a projection X × E → X, where E is a non-empty discrete topological space. A cover is a continuous map p : Y → X 3

Classically, concatenation is often defined in the other order. Here we follow [May] O X which fits better into the categorical frame work explained here.

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20 The ´etale topology

which is locally on X isomorphic to a trivial cover. Then (Cov(X)) is the full subcategory of the category (Top/X) of topological spaces over X consisting of covers of X. Taking fibers gives rise to a natural functor F : Π(X) × (Cov(X)) → (Sets). Indeed, let x ∈ X be a point (i.e., an object in Π(X)) and let p : X ′ → X a cover. Then we define F (x, p) as the fiber p−1 (x). If [γ] : x → y is a morphism in Π(X) and if σ : X ′ → X ′′ is a morphism of covers p : X ′ → X and q : X ′′ → X, we define F ([γ], σ) : p−1 (x) → q −1 (y) as follows. For all c ∈ p−1 (x) there exists a unique path γ˜c : [0, 1] → X ′ such that γc (0) = c and p ◦ γc = γ. We set F ([γ], σ)(c) := σ(˜ γc (1)). This is well defined. The bi-functor F yields functors L : Π(X) → Func((Cov(X)), (Sets)),

T : (Cov(X)) → Func(Π(X), (Sets)),

where Func(·, ·) denotes the category of functors. From now on we assume that X satisfies certain additional connectedness assumptions (namely that X is path connected, locally path connected, and semilocally simply connected). Then T is an equivalence of categories and L is fully faithful ([tDi] O Theorem 3.3.2 and Proposition 3.4.1). Hence L yields an equivalence between Π(X) and the full subcategory of fiber functors, i.e., those functors that are of the form Fx : (p : Y → X) 7→ p−1 (x) for some point x ∈ X. In particular L induces for all x ∈ X an isomorphism (*)

∼

π1 (X, x) = AutΠ(X) (x) → Aut(Fx ).

For a group G let BG be the groupoid that has a single object ∗ and such that AutBG (∗) = G. Fix a point x ∈ X. Since X is path connected the inclusion Bπ1 (X, x) ,→ Π(X) sending ∗ to x is an equivalence. We obtain an equivalence of categories Func(Π(X), (Sets)) ≃ Func(Bπ1 (X, x), (Sets)) = (π1 (X, x)-Sets), where the right hand side denotes the category of sets endowed with a left action of π1 (X, x). Composing this equivalence with T we obtain an equivalence of categories εx : (Cov(X)) ≃ (π1 (X, x)-Sets) which attaches to a cover p : X ′ → X the π1 (X, x)-set p−1 (x). Under this equivalence a cover X ′ → X is path connected if and only if π1 (X, x) acts transitively on the associated π1 (X, x)-set. Let p : X ′ → X be a path connected cover, and choose an element c ∈ p−1 (x). We consider π1 (X ′ , c) as a subgroup of π1 (X, x) via the injective map p∗ : π1 (X ′ , c) → π1 (X, x) given by functoriality of the fundamental group. Then the corresponding π1 (X, x)-set is isomorphic to π1 (X, x)/π1 (X ′ , c) with the π(X, x)-action by left multiplication. A different choice of c replaces π1 (X ′ , c) by a conjugate subgroup. If π1 (X ′ , c) is a normal subgroup, then p : X ′ → X is called a Galois cover. In this case, the functor εx induces an isomorphism of the group of automorphisms of the cover X ′ over X and the quotient group π1 (X, x)/π1 (X ′ , c). ˜ → X which is simply connected and the above There exists a universal cover p : X construction yields in particular an isomorphism between the group of automorphism of a universal cover p and π1 (X, x).

117 Our goal is to develop a similar theory for connected schemes S. As there is no good notion of “path” in algebraic geometry, we will start by defining the algebraic version of a topological cover: ´etale covers. More precisely, what we will define should be seen as the analogue of a finite topological cover and this will be the reason why the theory is not entirely parallel to the topological theory. For every geometric point s¯ of S we obtain a fiber functor Fs¯ from the category of ´etale covers to the category of finite sets and the fundamental groupoid Π(S) of S will be defined as the category of all fiber functors. In analogy to (*) above we will define π1 (S, s¯) as the group of automorphisms of the fiber functor Fs¯. Then π1 (S, s¯) carries a natural topology which is in general non-discrete. It is the unique topology making π1 (S, s¯) into a topological group such that the stabilizers of c ∈ Fs¯(X), where X runs through the ´etale covers and c through the elements of Fs¯(X), form a basis of neighborhoods of the neutral element of π1 (S, s¯). It will turn out that with this topology π1 (S, s¯) is a profinite group. This is in contrast to the topological situation, where the existence of a universal cover implies that the topology on the topological fundamental group defined as above is the discrete topology. The main result of our discussion is an equivalence of the category of ´etale covers with the category of finite sets endowed with a continuous action by the profinite group π1 (S, s¯) (Theorem 20.101). At the end we will give several examples. In particular we show that this theory yields a vast generalization of classical Galois theory. We recall that according to our conventions, the empty topological space is not connected.

´ (20.13) Etale covers. Definition 20.64. Let S be a scheme. An ´etale cover of S is a finite ´etale morphism of schemes π : X → S. A morphism of ´etale covers of S is a morphism of S-schemes. ´ The category of ´etale covers of S is denoted by (FEt/S). If S = Spec R is affine, we also ´ ´ write (FEt/R) instead of (F Et/S). An ´ e tale cover X → S is called split if it is isomorphic `r as an S-scheme to i=1 S for some integer r ≥ 0. The notion of an “´etale cover” should not be confused with the concept of a “covering for the ´etale topology”. Moreover we do not assume that an ´etale cover is surjective (but see Remark 20.66 (1)). Example 20.65. (1) If K is a finite separable extension of a field k, then Spec K → Spec k is an ´etale cover. (2) Let R be a ring. Let f ∈ R[T ] be a monic polynomial such that f and f ′ generate R[T ] as ideal. Then Spec R[T ]/(f ) → Spec R is an ´etale cover by Example 18.47. This is in fact a generalization of (1), since K ∼ = k[T ]/(f ) for a separable polynomial f by the theorem of the primitive element (Proposition B.98). Let us collect some properties of ´etale covers that we have already seen. Remark 20.66. Let S be a scheme. (1) An ´etale cover π is finite, flat (Theorem 18.44), and locally of finite presentation and hence a finite locally free morphism (Proposition 12.19).

118

(2)

(3) (4)

(5) (6)

20 The ´etale topology In particular, π is open and closed. Therefore π is surjective if X ̸= ∅ and S is connected. It follows from (1) that the property to be an ´etale cover is stable under base change, under composition, is stable under faithfully flat descent, and is compatible with forming cofiltered limits of schemes with affine transition maps, since these permanence properties hold for the properties “finite”, “flat”, and “locally of finite presentation” (see Appendix C). Any morphism between ´etale covers of S is finite (Proposition 12.11) and ´etale (Remark 18.36). Let Z be any S-scheme, let π : X → S be an ´etale cover, and let f, g : Z → X be morphisms of S-schemes. If Z is connected and there exists a geometric point z¯ of Z such that f (¯ z ) = g(¯ z ), then f = g (Proposition 18.31). Every section of an ´etale cover is an open and closed immersion (Remark 18.30 (3) and Example 9.12). Every ´etale proper map is an ´etale cover. Indeed, quasi-compact ´etale maps are quasi-finite (Corollary 18.45) and quasi-finite and proper maps are finite by Zariski’s main theorem (Corollary 12.89).

As for every finite locally free morphism, we have the notion of the degree of a finite ´etale cover π : X → S, which is a locally constant function deg(X/S) := deg(π) : S → Z≥0 . Let us collect some properties of the degree which we have essentially already seen in Section (12.6). Remark 20.67. Let π : X → S be a finite locally free morphism (e.g., an ´etale cover). Then the OS -algebra π∗ OX is a locally free OS -module of finite type and X = Spec(π∗ OX ) since π is affine (Corollary 12.2). (1) The morphism π is an isomorphism if (and only if) deg(π) = 1. Indeed, if deg(π) = 1, then OS → π∗ OX is an isomorphism and hence π is an isomorphism. (2) If g : S ′ → S is a morphism of schemes, then the base change π ′ : X ×S S ′ → S ′ of π is finite locally free and deg(π ′ ) = deg(π) ◦ g. (3) If π ′ : X ′ → S is a second finite locally free morphism. Let X ⨿ X ′ be the disjoint union of X and X ′ . Then the morphism ϖ : X ⨿X ′ → S with ϖ|X = π and ϖ|X ′ = π ′ is finite locally free and deg(ϖ) = deg(π) + deg(π ′ ). (4) Let ϖ : X ′ → X be finite locally free. Then π ◦ ϖ is finite locally free. If deg(π) and deg(ϖ) are constant, then deg(π ◦ ϖ) = deg(π) deg(ϖ). The above properties of the degree function easily imply the following lemma. Lemma 20.68. Let π : X → S be an ´etale cover. Suppose that S and X are connected. Then every S-endomorphism of X is an automorphism. Proof. Let f be an S-endomorphism of X. Then f is an ´etale cover (Remark 20.66 (3)) with π ◦ f = π and hence deg(π) = deg(π ◦ f ) = deg(π) deg(f ). This implies that f is an isomorphism by Remark 20.67 (1). “Locally” every finite ´etale cover is split, more precisely: Proposition 20.69. Let S be a scheme, let π : X → S be a morphism, and let n ≥ 1 be an integer. Then the following assertions are equivalent.

119 (i) The morphism π is an ´etale cover of degree n. (ii) There exists `na faithfully flat quasi-compact morphism T → S and an isomorphism X ×S T ∼ = i=1 T of T -schemes. `n (iii) There exists a surjective ´etale cover T → S and an isomorphism X ×S T ∼ = i=1 T of T -schemes. Moreover, T → S can be chosen such that it is of constant degree n!. Proof. Clearly, (iii) implies (ii), and (ii) implies (i) because being an ´etale cover of a fixed degree can be checked after faithfully flat quasi-compact base change. Let us show that (i) implies (iii) by induction on n. For n = 1, π is an isomorphism and we can take T → S to be idS . Let n > 1. By base change, we know that the first projection X ×S X → X is an ´etale cover of degree n. Remark 20.66 (5), applied to X ×S X → X, shows that the diagonal X → X ×S X is an open and closed embedding. Hence we have X ×S X = X ⨿ X ′ for some open and closed subscheme X ′ of X ×S X. Its restriction X ′ → X is an ´etale cover of degree n − 1. Applying the induction `n−1 hypothesis, we find an ´etale cover T → X of degree (n − 1)! such that X ′ ×X T ∼ = i=1 T . Then the composition T → X → S is an ´etale cover of degree n! and X ×S T = X ×S X ×X T = (X ⨿ X ′ ) ×X T = T ⨿

n−1 a

T.

i=1

To characterize the ´etale covers among the finite locally free morphisms we introduce the trace form. Remark and Definition 20.70. Let π : X → S be a finite locally free morphism. Then A := π∗ OX is a finite locally free OS -algebra such that X = Spec A (Proposition 12.19). We define a trace homomorphism and a trace bilinear form as usual: For U ⊆ S open affine, Γ(U, A ) is a finitely generated projective Γ(U, OS )-module. For a ∈ Γ(U, A ) let ma : Γ(U, A ) → Γ(U, A ) be the multiplication by a. As A is a finite locally free OS module, we can define trπ (a) := tr(ma ) ∈ Γ(U, OS ) and τπ (a ⊗ a′ ) := tr(maa′ ) ∈ Γ(U, OS ) for a, a′ ∈ Γ(U, A ), see (7.20.8). We obtain the trace homomorphism of π, trπ := trX/S : A → OS , and the symmetric trace form of π τπ := τX/S : A ⊗OS A → OS . The bilinear form τπ induces an OS -linear map A → A ∨ := Hom OS (A , OS ) and τπ is called perfect if this OS -linear map is an isomorphism of OS -modules. For the formulation of the following proposition recall the ramification indices ex/s ′ ′′ and inertia indices fx/s and fx/s for quasi-finite morphisms that were introduced in Section (12.5). Proposition 20.71. Let π : X → S be a quasi-finite flat morphism locally of finite presentation. (1) Let x ∈ X and s := π(x). Then the following assertions are equivalent. (i) π is ´etale at x. ′ = 1. (ii) ex/s = fx/s 1 (iii) ΩX/S,x = 0.

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20 The ´etale topology

(2) Let π be a finite locally free morphism. Then the following assertions are equivalent. (i) π is an ´etale cover. ′ = 1. (ii) For all x ∈ X one has ex/π(x) = fx/π(x) (iii) Ω1X/S = 0. (iv) The trace form τπ is perfect. The last criterion generalizes the well known fact ([BouAII] O V, §8.2, Proposition 1) that a finite field extension is separable if and only if the attached trace form is non-degenerate and we will reduce to this case. Proof. Assertion (1) follows from Corollary 18.28 and Proposition 18.29 because a morphism locally of finite presentation is ´etale if and only if it is flat and unramified (Theorem 18.44). Moreover, the implications “(i) ⇔ (ii) ⇔ (iii)” in (2) follow from (1). It remains to show “(iii) ⇔ (iv)” in (2). Proposition 18.6 shows that Ω1X/S = 0 if and only if π is unramified. This can be checked on fibers (Proposition 18.29). On the other hand, the formation of the trace of an endomorphism of a finite locally free OS -module is compatible with arbitrary base change S ′ → S. Moreover, a homomorphism u : E → F of finite locally free OS -modules is an isomorphism if and only if the homomorphisms u ⊗ idκ(s) induced on fibers are all bijective (Corollary 8.12). Hence (iv) can also be checked on fibers. We thus may assume that S = Spec k for a field k. But then π is unramified if and only if X is isomorphic to the spectrum of a finite product of finite separable field extensions of k (by Proposition 18.24 and because π is quasi-compact). Therefore the equivalence of (iii) and (iv) follows from the characterization of separable extensions recalled before the proof. To apply the proposition, recall that if X and S are schemes of finite type over a noetherian ring, then a morphism π : X → S is quasi-finite and of finite presentation if and only if it has finite fibers. If in addition S is regular and X is Cohen-Macaulay, then π is automatically flat (Theorem 14.128). Finally, Exercise 20.17 gives a fiber criterion for a quasi-finite flat morphism of finite presentation to be finite locally free. Proposition 20.71 can be reformulated by introducing the discriminant and the different: Remark and Definition 20.72. We first define the discriminant. Let π : X → S be a finite locally free morphism. Let A := π∗ OX , and let u : A → A ∨ be the OS linear homomorphism corresponding to the trace form τπ . Then π is an ´etale cover if and only if det(u) : det(A ) → det(A ∨ ) is an isomorphism of line bundles (combine Proposition 20.71 and Corollary 8.12). As det(A ∨ ) = det(A )∨ , det(u) corresponds to an OS -linear homomorphism DX/S : det(A ) ⊗ det(A ) → OS which is surjective if and only if det(u) is an isomorphism (again by Corollary 8.12). We call the image of DX/S the discriminant of π and denote it by DX/S or by Dπ . Then DX/S is a quasi-coherent ideal of OS of finite type. As π is affine, the formation of π∗ OX commutes with arbitrary base change g : S ′ → S (Proposition 12.6). As the trace form and the determinant also commute with base change, we find g ∗ (DX/S ) = DX×S S ′ /S ′ . Hence Proposition 20.71 implies:

121 Corollary 20.73. Let π : X → S be finite locally free. Then a morphism g : S ′ → S factors through the open subscheme S \ V (DX/S ) if and only if the base change X ×S S ′ → S ′ is an ´etale cover. For the relation between the discriminant defined here and the discriminant of a monic polynomial we refer to Exercise 20.30. Remark and Definition 20.74. Let us now define the different and the branch locus. Let π : X → S be a quasi-finite morphism locally of finite presentation. The different of π is defined as the zero-th Fitting ideal of Ω1X/S , see Section (16.9). It is therefore a quasi-coherent ideal of OX of finite type. We denote it by dπ or dX/S . The closed subscheme V (dX/S ) ⊆ X is called the branch locus of π. By definition of the Fitting ideal one has Ω1X/S,x = 0 if and only if x ∈ X \ V (dX/S ). If π is in addition flat, then Proposition 20.71 shows that { x ∈ X ; π is ´etale in x} = X \ V (dX/S ). Sometimes the different is also defined as the annihilator of Ω1X/S . The resulting closed subscheme of X has the same underlying topological space as V (dX/S ), see Section (16.9). Taking the zero-th Fitting ideal has the advantage that the formation of the different is compatible with base change S ′ → S because this holds for the formation of Ω1X/S (Proposition 17.30) and for the formation of the Fitting ideals (Proposition 16.30). There exist other versions of the different ([Kun] O Chap. 8). All of them agree if π is also syntomic and ´etale over an open dense subscheme of S (loc. cit. Theorem 8.15). (20.14) Lifting of ´ etale covers. Theorem 20.75. Let R be a local henselian ring with residue field k. Then the functor ´ ´ F : (FEt/R) → (FEt/k),

X 7→ X ⊗R k,

is an equivalence of categories. Moreover, an ´etale cover Spec A → Spec R is the spectrum of a local R-algebra A if and only if A¯ := A ⊗R k is a (necessarily finite separable) field extension of k. In this case A is a standard ´etale R-algebra. We first show a lemma which in particular implies that F is fully faithful. Lemma 20.76. Let R be a local ring with maximal ideal m and residue field k. Then R is henselian if and only if for every finite morphism f : X → Spec R and for every ´etale morphism g : Y → Spec R the map HomR (X, Y ) → Homk (X ⊗R k, Y ⊗R k),

h 7→ h ⊗ idk

is bijective. Proof. Let S = Spec R. If we apply the condition to f = idS , Theorem 20.12 shows that R is henselian. Conversely, assume that R is henselian. We remark that Theorem 20.12 shows that the canonical map Y (R) → Y (R/m) is bijective for every ´etale R-scheme Y .

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20 The ´etale topology

Let f : X → S be a finite morphism. Then X = Spec A, where A is a finite product of local henselian finite R-algebras. Hence we may assume that X = Spec A for a local henselian finite R-algebra. Denote by K its residue field. Now let g : Y → Spec R be an ´etale morphism and set Y ′ := X ×S Y which is an ´etale A-scheme. Hence the remark above, applied to A in place of R, shows that HomS (X, Y ) = HomX (X, Y ′ ) = Y ′ (A) → Y ′ (A ⊗R k) = Homk (X ⊗R k, Y ⊗R k) is bijective. Proof. (of Theorem 20.75) As R is henselian, a finite R-algebra A is local if and only if A¯ = A ⊗R k is local. But an ´etale k-algebra is a finite product of finite separable field extensions. This shows the last assertion. It remains to show that for every (finite) ´etale k-algebra A0 there exists a finite ´etale R-algebra A such that A ⊗R k ∼ = A0 . We may assume that A0 = K is a finite separable extension field of k. By the theorem of the primitive element (Proposition B.98), K = k[T ]/(f0 ) for some monic polynomial f0 ∈ k[T ] which is separable, i.e. f0 is coprime to its derivative. Let f ∈ R[T ] be any monic polynomial with image f0 in k[T ] and set A := R[T ]/(f ). Then A is a finite free R-algebra with A ⊗R k ∼ = K. As (f ) + (f ′ ) = R[T ] ′ (Remark 20.6), the image of f in A is invertible. Therefore A = Af ′ is a standard ´etale R-algebra. Since every ´etale cover of Spec k is a finite disjoint union of spectra of finite separable field extensions of k, Theorem 20.75 implies the following corollary. Corollary 20.77. Let R be a local henselian ring and let X → Spec R be an ´etale cover. `n Then X ∼ = i=1 Spec Ai , where Ai is a finite standard ´etale algebra. If R is strictly henselian, then X is a finite disjoint union of copies of Spec R. Theorem 20.75 can be generalized to proper schemes over henselian rings, see Theorem 20.118 below. For an arbitrary Dedekind domain or for a connected normal curve the connected ´etale covers are simply the integral closures in a finite separable extension of the function field which is unramified in all points. More precisely: Example 20.78. Let S be a Dedekind scheme (i.e., S is noetherian, regular, irreducible, of dimension ≤ 1). Examples are S = Spec R for a Dedekind domain R and regular curves over a field. Let K := K(S) be its function field. (1) Let π : X → S be a connected ´etale cover. As X ̸= ∅ and S is connected, π is surjective (Remark 20.66 (1)). Since S is regular and π is ´etale, X is regular (Corollary 14.60) and dim X = dim S (Corollary 14.97). Hence X is a Dedekind scheme because X is connected. Because π is integral and X is normal, X is the integral closure of S in L := K(X). Let η = Spec K be the generic point of S. Then π −1 (η) = Spec L (Proposition B.55). As π is finite, L is a finite extension of K. Further, π is unramified, so L is a separable extension of K (Proposition 18.24). Let s ∈ S be a closed point. As π is ´etale, the fiber π −1 (s) is an ´etale κ(s)-scheme or, equivalently by Proposition 20.71, for all x ∈ π −1 (s) the ramification index ex/s is 1 and κ(x) is a separable field extension of κ(s).

123 (2) Conversely, let L be a finite separable extension of K and let X be the integral closure of S in L. Then X is a finite S-scheme (Proposition 12.53). Since S is a Dedekind scheme, π : X → S is dominant, X is reduced and X → S is flat (Proposition 14.14), so π is finite locally free. Moreover, π is an ´etale cover if and only if ex/π(x) = 1 and the extension κ(x)/κ(π(x)) is separable for all x ∈ X (Proposition 20.71). Example 20.79. Let K be a local field, i.e., a locally compact topological field with a non-discrete topology. Then the topology of K can be defined by an absolute value | · | : K → R≥0 ([BouAC] O VI, §9.3, Theorem 1). Assume that K is non-archimedean, i.e., | · | satisfies the strong triangle inequality |a + b| ≤ max{|a|, |b|} for all a, b ∈ K. Then K is isomorphic to a finite extension of the field Qp of p-adic numbers or to the field of Laurent series Fq ((T )) over a finite field (loc. cit.), and OK := { a ∈ K ; |a| ≤ 1 } is a subring of K which is a complete discrete valuation ring whose residue field k is a finite field. Let π ∈ OK be a uniformizing element. As OK is complete, it is henselian (Example 20.3). Hence (−) ⊗OK k is an equivalence between the category of local finite ´etale OK -algebras and the category of finite separable extensions of k (Theorem 20.75). Because k is finite, every extension of k is separable and for every integer n ≥ 1 there exists a unique finite extension of degree n (up to isomorphism). On the other hand, every local finite ´etale OK -algebra is the integral closure OL of OK in a finite separable extension L of K such that ℓ := OL /πOL is a field (Example 20.78). In algebraic number theory, one expresses this situation by saying that L is an unramified extension of K (e.g., [Neu] O II, §7). So by definition this means that the morphism Spec OL → Spec OK being unramified, which is much stronger than the condition that Spec L → Spec K is unramified in the sense of Definition 18.22. Hence Theorem 20.75 implies that for every integer n ≥ 1 there exists an unramified (in the sense of number theory) extension L with [L : K] = n which is unique up to isomorphism. By the “fundamental equality” (12.6.2) one has n = [ℓ : k] and hence AutK (L) ∼ = AutOK (OL ) ∼ = Gal(ℓ/k) ∼ = Z/nZ, because every K-automorphism of L preserves the ring OL of elements which are integral over OK . In particular L is a Galois extension of K. Remark 20.80. Let S be a scheme locally of finite type over C and let S an be its analytification (Section (20.12)). Then [SGA1] O eor`eme 5.1 says that ´etale X Exp. XII, Th´ covers of S correspond to finite covers of S an : Attaching to an ´etale cover π : X → S ´ of schemes its analytification π an yields an equivalence of the category (FEt/S) with the category of finite morphisms of analytic spaces X → S an that are isomorphisms locally for the analytic topology (i.e., a finite (not necessarily surjective) cover in the sense of classical topology). This result is called the Riemann existence theorem because it partially generalizes the classical Riemann existence theorem which asserts that on a compact Riemann surface there exist “many” meromorphic functions, so that it admits a closed embedding into a projective space. Compare Section (26.7). To relate the two versions, one can consider ´etale covers of open subschemes of the projective line. (20.15) Fibers of ´ etale covers and the fundamental groupoid. As recalled above, every finite locally free morphism π : X → S has a degree (12.6.1) which is a locally constant function S → Z≥0 . Moreover π is an isomorphism if and only

124

20 The ´etale topology

if deg(π) = 1 (Remark 20.67 (1)). This will be often used in the sequel, for instance to prove the following lemma which will often allow us to reduce to a connected cover. Lemma 20.81. Let S be a connected scheme, and let π : X → S be a finite locally free morphism. Then X is the finite disjoint union of open and closed connected subschemes. In particular, we see that for an ´etale cover X of a connected scheme S, the connected components of X agree with the open and closed subspaces of X. Proof. As S is connected, the degree of π (12.6.1) is constant. We show the claim by induction on deg(π). If deg(π) = 0 (resp. deg(π) = 1), then X = ∅ (resp. then π is an isomorphism) and the claim is clear. Now assume that X is not connected. Then X = X1 ⊔ X2 for non-empty open and closed subschemes Xi of X. As deg(π) = deg(π |X1 ) + deg(π |X2 ), the induction hypothesis implies that each Xi is the finite disjoint union of open and closed connected subschemes. This shows the lemma. Let S be a scheme and let s¯ : Spec κ(¯ s) → S be a geometric point. By abuse of notation we also denote by s¯ the scheme Spec κ(¯ s). If π : X → S is an ´etale cover, then its fiber Xs¯ := X ×S κ(¯ s) over s¯ is isomorphic to a finite disjoint union of copies of Spec κ(¯ s) and hence as a κ(¯ s)-scheme is determined by its underlying set, which can also be described as HomS (¯ s, X). We obtain the fiber functor (20.15.1)

´ Fs¯ : (FEt/S) → (sets),

X 7→ HomS (¯ s, X),

where we denote by (sets) the category of finite sets (morphisms are arbitrary maps of finite sets). Definition 20.82. Let S be a connected scheme. The full subcategory of the category of ´ all functors (FEt/S) → (Sets) consisting of functors that are isomorphic to a fiber functor Fs¯ for some geometric point s¯ of S is called the fundamental groupoid of S and denoted by Π(S). Below in Proposition 20.93 we will see that Π(S) is a connected groupoid, i.e., any morphism in Π(S) is an isomorphism and any two objects in Π(S) are isomorphic. We view Π(S) as the algebraic analogue of the topological fundamental groupoid of a path-connected space. More generally, one can define the fundamental groupoid Π(S) in the same way for all schemes in which all connected components are open (and hence open and closed). This is for instance the case if S is locally noetherian. Formally, the above definition also makes sense for arbitrary schemes S but it does not seem clear, whether this is conceptually the “right” definition. Let S be a connected scheme, so that the degree of any finite ´etale cover π : X → S is constant and stable under base change S ′ → S. In particular, for every geometric point s¯ of S we have (20.15.2)

deg(π) = #Fs¯(X).

If S is the spectrum of a separably closed field, the functor (20.15.1) is an equivalence of categories. More generally, Corollary 20.77 then implies:

125 Proposition 20.83. Let S be the spectrum of a strictly henselian ring and let s¯ be a ´ geometric point of S lying over the closed point of S. Then Fs¯ : (FEt/S) → (sets) yields an equivalence of categories. Definition 20.84. Let S be a connected scheme and let s¯ be a geometric point of S. A pointed ´etale cover is a pair (X, x ¯), where X is an ´etale cover of S and x ¯ ∈ Fs¯(X). A morphism of pointed ´etale covers (X, x ¯) → (Y, y¯) is a morphism of ´etale covers f : X → Y such that f ◦ x ¯ = y¯. Let Is¯ be the category of pointed ´etale covers (X, x ¯) → (S, s¯), where X is connected. Remark 20.85. (1) Let (X, x ¯) be an object in Is¯ and let Z → S be an ´etale cover. Then the map (20.15.3)

ιx¯ : Hom(FEt/S) (X, Z) → Fs¯(Z), ´

f 7→ f ◦ x ¯

is injective (Remark 20.66 (4)). (2) In particular, we see that for objects (X, x ¯) and (Y, y¯) there exists at most one morphism (X, x ¯) → (Y, y¯) in Is¯. ¯) and (Y, y¯) are objects in Is¯, set z¯ := (¯ x, y¯) : s¯ → (3) The category Is¯ is cofiltered: If (X, x X ×S Y and let Z be the connected component of X ×S Y such that z¯ factors through Z. Then (Z, z¯) is an object of Is¯ such that there exist morphisms (Z, z¯) → (X, x ¯) and (Z, z¯) → (Y, y¯). By (2) this implies also that we can always equalize morphisms. For every ´etale cover X → S a morphism (Z, z¯) → (Y, y¯) in Is¯ induces a map Hom(FEt/S) (Y, X) → Hom(FEt/S) (Z, X) compatible with the maps (20.15.3). We obtain ´ ´ a map (20.15.4)

(Z, X) → Fs¯(X) colim Hom(FEt/S) ´

(Z,¯ z )∈Is¯

which is functorial in Z. Theorem 20.86. The morphism (20.15.4) is an isomorphism of functors in X. Proof. The map (20.15.4) is injective because it is the filtered colimit of injective maps. If x ¯ ∈ Fs¯(X) then let X 0 be the connected component such that x ¯ factors through X 0 . Then (X 0 , x ¯) is an object of Is¯ and hence x ¯ lies in the image of (20.15.4). The theorem can be expressed by saying that Fs¯ is pro-representable by the cofiltered ´ diagram Is¯ → (FEt/S), (X, x ¯) 7→ X (see for instance [SGA4] O Exp. I, 8.10 for the notion of pro-representable functors). (20.16) Galois covers. Roughly speaking, a Galois cover is an ´etale cover X → S such that the induced map X/ AutS (X) → S is an isomorphism (cf. the analogy with topology). To make this more precise, we start with the following easy globalization of Proposition 12.27. Proposition 20.87. Let S be a scheme, let f : X → S be an affine morphism, and let G be a finite group of S-automorphisms of X. Then there exists a unique pair (Y, p), where Y is an S-scheme and p : X → Y is a morphism of S-schemes with p ◦ g = p for all g ∈ G, such that for every morphism of S-schemes q : X → Z with q ◦ g = q for all g ∈ G there exists a unique morphism q¯: Y → Z of S-schemes such that q¯ ◦ p = q.

126

20 The ´etale topology

Moreover, p is integral, surjective and has finite fibers. For x, x′ ∈ X one has p(x) = p(x′ ) if and only if there exists g ∈ G with g(x) = x′ . We denote Y by X/G and call it the quotient of X by G. Proof. As X is affine over S we have X = Spec f∗ OX (Proposition 12.1). Set Y := Spec(f∗ OX )G and let p : X → Y be the morphism of S-schemes corresponding to the inclusion (f∗ OX )G → f∗ OX . As taking G-invariants is compatible with flat base change (Remark 12.28), in particular with passing to open subschemes, we may assume that S is affine. Then all claims follow from Proposition 12.27. Remark 20.88. Globalizing Remark 12.28 we see that in the situation of Proposition 20.87 for every morphism S ′ → S of schemes there is a morphism (20.16.1)

(X ×S S ′ )/G −→ (X/G) ×S S ′ ,

which is an isomorphism if S ′ → S is flat. Proposition 20.89. Let f : X → S be an ´etale cover and let G be a finite group of S-automorphism of X. Then X/G is an ´etale cover of S. Moreover, for every geometric point s¯ of S one has Fs¯(X/G) = Fs¯(X)/G, where G acts by composition on Fs¯(X) = HomS (¯ s, X). In particular deg(X/G → S) = #(Fs¯(X)/G). Proof. As the formation of the quotient commutes with flat base change (Remark 20.88) hs → S for s ∈ S, we may assume and in particular with base change of the form Spec OS,s that S = Spec R, where R is a strictly henselian ring (Remark 20.44). But then X is isomorphic to a sum of copies of Spec R indexed by Fs¯(X), where s¯ is some geometric point of S lying over the closed point of s (Proposition 20.83), and G necessarily acts by permuting these copies. Hence X/G is also isomorphic to a finite sum of copies of Spec R, one for each G-orbit in Fs¯(X). Let S be a connected scheme, let s¯ be a geometric point, and let X → S be a connected ´etale cover. Then (20.15.3) shows that the action of Aut(FEt/S) (X) on Fs¯(X) is free. ´ Therefore one has (20.16.2)

#Aut(FEt/S) (X) ≤ #Fs¯(X) ´

and Aut(FEt/S) (X) is a finite group. ´ Definition and Remark 20.90. Let S be a connected scheme. A connected ´etale cover Z → S is called a Galois cover if the following equivalent conditions are satisfied. (i) There exists a geometric point s¯ of S such that # Aut(FEt/S) (Z) ≥ #Fs¯(Z). ´ (ii) For all geometric points s¯ of S the group Aut(FEt/S) (Z) acts simply transitively on ´ Fs¯(Z). (iii) # Aut(FEt/S) (Z) = deg(Z/S). ´ (iv) The morphism Z/ Aut(FEt/S) (Z) → S is an isomorphism. ´ Let us show the equivalence of these conditions. Set G := Aut(FEt/S) (Z). By Proposi´ tion 20.89, (iv) is equivalent to the transitivity of the G-action on Fs¯(Z) for one or for all s¯. As #G ≤ #Fs¯(Z) = deg(Z/S) such a transitive action is automatically simply transitive. Hence (ii), (iii), and (iv) are equivalent. Clearly, (ii) implies (i). Conversely, by ¯ ∈ Fs¯(Z) has a nontrivial stabilizer in G, hence the action Remark 20.85 (2) no element x of G on Fs¯(Z) must be simply transitive if # Aut(FEt/S) (Z) ≥ #Fs¯(Z). ´

127 Lemma 20.91. Let S be a connected scheme and let π : X → S and ϖ : Z → X be connected ´etale covers. If the composition Z → S is a Galois cover, then Z → X is a Galois cover. Proof. Let x ¯ be a geometric point of X. We show that H := Aut(FEt/X) (Z) acts tran´ sitively on Fx¯ (Z). Let s¯ be the image of x ¯ in S. Let z¯, z¯′ ∈ Fx¯ (Z) ⊆ Fs¯(Z). As Z → S is a Galois cover, there exists α ∈ Aut(FEt/S) (Z) sending z¯ to z¯′ . We have to show that ´ ϖ ◦ α = ϖ, i.e., Ker(ϖ ◦ α, ϖ) = Z. But Ker(ϖ ◦ α, ϖ) is an open and closed subscheme of the connected scheme Z (Remark 20.66 (4)), and it is non-empty because ϖ ◦ α and ϖ both send z¯ to x ¯. Let Is¯ be the category of pointed connected ´etale covers of (S, s¯) defined in Definition 20.84. Lemma 20.92. For every ´etale cover X → S there exists an object (Z, z¯) in Is¯ such that the injective map ιz¯ : Hom(FEt/S) (Z, X) → Fs¯(X) (20.15.3) is bijective and such that Z ´ is a Galois cover of S. Proof. Let Y := X Fs¯(X) be the fiber product over S of copies of X indexed by Fs¯(X). Let z¯ be the element of Fs¯(Y ) = Fs¯(X)Fs¯(X) whose x ¯-th component is x ¯ for x ¯ ∈ Fs¯(X). Let Z be the connected component of Y such that z¯ factors through Z. For x ¯ ∈ Fs¯(X) let px¯ : Z → X be the restriction to Z of the projection Y → X onto the x ¯-th coordinate. Then the map (20.15.3) sends px¯ to x ¯. This shows that the map ιz¯ in (20.15.3) is bijective. It remains to show that Z is a Galois cover of S. We show that Aut(FEt/S) (Z) acts ´ ′ transitively on Fs¯(Z). Let z¯ be any element of Fs¯(Z). As # Hom(FEt/S) (Z, X) = #Fs¯(X), ´ the injective map ιz¯′ is also bijective. This means that the coordinates of z¯′ , viewed as an element of Fs¯(Y ) = Fs¯(X)Fs¯(X) are precisely the elements of Fs¯(X). Therefore there exists an automorphism σ of the ´etale cover Y , permuting the factors, that sends z¯ to z¯′ . This automorphism maps the connected component Z of Y to some connected component Z ′ . From z¯′ ∈ Fs¯(Z) ∩ Fs¯(Z ′ ) we conclude Z = Z ′ . Lemma 20.92 implies that the full subcategory Gs¯ of objects (Z, z¯) of Is¯ such that Z is a Galois cover of S is a cofinal subcategory in Is¯. In particular, for every ´etale cover X of S the bijection (20.15.4) induces a bijection, functorial in X, (20.16.3)

∼

(Z, X) −→ Fs¯(X). colim Hom(FEt/S) ´

(Z,¯ z )∈Gs¯

Proposition 20.93. Let S be a connected. Then Π(S) is a connected groupoid, i.e., any morphism in Π(S) is an isomorphism and any two objects in Π(S) are isomorphic. Proof. Le s¯ and s¯′ be geometric points of S. Let φ : Fs¯ → Fs¯′ be a morphism of functors. Let (Z, z¯) in Gs¯ and set z¯′ := φZ (¯ z ). Then (Z, z¯′ ) ∈ Gs¯′ and for every ´etale cover X → S one has a diagram Hom(FEt/S) (Z, X) ´ ιz¯′

ιz¯

Fs¯(X)

w

φX

' / Fs¯′ (X),

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20 The ´etale topology

which is commutative since φ is a morphism of functors. Now given (X, x ¯), we choose (Z, z¯) as in Lemma 20.92, so that we obtain a diagram as above with ιz¯ bijective. As ιz¯′ is injective (Remark 20.85 (1)), it follows that φX is injective. Since #Fs¯(X) = deg(X/S) = #Fs¯′ (X) we see that φX is bijective. Hence φ is an isomorphism of functors. It remains to show that Hom(Fs¯, Fs¯′ ) ̸= ∅, where Hom(−, −) denotes morphisms of ´ ´ functors (FEt/S) → (Sets). For an ´etale cover Z → S define the functor hZ : (FEt/S) → Z (Sets), h (X) = Hom(FEt/S) (Z, X). Then (20.16.3) yields an isomorphism of functors ´ Fs¯ ∼ = colim hZ (Z,¯ z )∈Gs¯

and hence we have Hom(Fs¯, Fs¯′ ) ∼ = lim Hom(hZ , Fs¯′ ) = lim Fs¯′ (Z), (Z,¯ z)

(Z,¯ z)

where the second identity holds by the Yoneda lemma. The right hand side is a cofiltered limit of finite non-empty sets and therefore is non-empty. The proof shows that for geometric points s¯ and s¯′ of a connected scheme S one has a bijection (20.16.4)

∼

Isom(Fs¯, Fs¯′ ) →

lim (Z,¯ z )∈Gs¯

Fs¯′ (Z),

σ 7→ (σZ (¯ z ))(Z,¯z)∈Gs¯ .

(20.17) Profinite groups and the topology on the automorphism group of a functor. Let S be a connected scheme and let s¯ be a geometric point of S. Below we will define the algebraic fundamental group π1 (S, s¯) as the group of automorphisms of the fiber functor Fs¯. This group has a natural topology and with this topology it will be a profinite group. Let us now explain these notions briefly. We first recall the definition of a profinite group (e.g., see [NSW] O X (1.1.3)). Definition 20.94. A topological group G is called profinite if it satisfies the following equivalent conditions: (i) G is isomorphic (as a topological group) to cofiltered limit of finite discrete groups. (ii) G is compact4 and the unit element has a basis of neighborhoods consisting of open and closed normal subgroups. (iii) G is compact and totally disconnected. Remark 20.95. (1) Every closed subspace of a compact (resp. totally disconnected) space is again compact (resp. totally disconnected). Hence every closed subgroup of a profinite group is profinite. (2) As limits of compact (resp. totally disconnected) spaces are again compact (resp. totally disconnected), every limit of profinite groups is again profinite. In particular, products of profinite groups are profinite. Let us collect some properties of profinite groups that we will use later. 4

We define a topological space to be compact if it is quasi-compact and Hausdorff.

129 Lemma 20.96. Let G be a profinite group. (1) A subgroup of G is open if and only if it is closed and of finite index. (2) One has \ U = 1. U ⊆ G open normal subgroup

(3) A subgroup H ⊆ G is closed if and only if H is the intersection of all open subgroups of G containing H. Proof. (1). This is true for any compact group G. If H is open, then G/H is discrete and compact. Hence it is finite. The complement of H in G is a union of H-cosets and therefore open. Hence H is closed in G. Conversely, if H is closed and of finite index, then the complement of H is the union of finitely many closed H-cosets. Hence G \ H is closed and therefore H is open. (2). A topological group is Hausdorff if and only if the intersection of all open neighborhoods of the unit element e is {e}. A profinite group is Hausdorff and the open normal subgroups form a neighborhood basis of e. (3). By (1) the condition is sufficient. Conversely, let H be a closed subgroup of G. As every subgroup that contains an open subgroup is itself open, it suffices to show that \ \ HU = H( U ) = H, U ⊆ G open normal subgroup

U

T T where the second equality holds by (2). We have T to show that U (HU ) ⊆TH( U U ) since T the converse inclusion is clear. Let g ∈ U (HU ). Assume g ∈ / H( U U ), i.e. gH ∩ U UT= ∅. By compactness weTfind finitely many open normal subgroups U1 , . . . , Un n with Hg ∩ i=1 Ui = ∅. But U ′ := i Ui is itself an open normal compact subgroup of G and hence g ∈ HU ′ by assumption. This is a contradiction. Definition and Remark 20.97. Let G be a topological group. An action of G on a set X is called continuous if the map defining the action G × X → X is continuous, where we endow X with the discrete topology. This is the case if and only if for all x ∈ X the stabilizer Gx := { g ∈ G ; gx = x } is an open subgroup of X. If X is in addition finite, then the action is continuous if and only if there exists an open subgroup U of G such that U acts trivially on X. We denote by (G-sets) the category of finite sets with a continuous G-action, in which morphisms are G-equivariant maps. If G is profinite and acts on a finite set X, then a G-action on X is continuous if and only if there exists an open normal subgroup of G that acts trivially on X. As G is compact, every open subgroup has finite index. Let us also explain how to endow the group of automorphisms of a set-valued functor with a topology. Remark 20.98. Let C be a small category and let F : C → (sets) be a functor with values in the categories of finite sets. Then there is a canonical injective map Y Aut(F (X)). (20.17.1) Aut(F ) ,→ X∈Ob(C)

130

20 The ´etale topology

We endow the finite set Aut(F (X)) with the discrete topology and the right hand side with the product topology. Then it is easy to see that (20.17.1) identifies Aut(F ) with a closed subgroup of the right hand side. We endow Aut(F ) with the induced topology. Then Aut(F ) is a profinite group by Remark 20.95. (20.18) The ´ etale fundamental group. ´ Let S be a connected scheme, let s¯ be a geometric point, and let Fs¯ : (FEt/S) → (Sets) be the corresponding fiber functor. Then Aut(Fs¯) is a profinite group by Remark 20.98. Definition 20.99. The profinite group π1 (S, s¯) := Aut(Fs¯) is called the algebraic fundamental group of S with respect to s¯ (or the ´etale fundamental group). The topology on π1 (S, s¯) is by definition the unique topology making π1 (S, s¯) into a topological group such that the stabilizers of c ∈ Fs¯(X), where X runs through the ´etale covers and c through the elements of Fs¯(X), form a basis of neighborhoods of the neutral element of π1 (S, s¯). By (20.16.4) we have a homeomorphism (20.18.1)

π1 (S, s¯) ∼ =

lim (Z,¯ z )∈Gs¯

Fs¯(Z),

σ 7→ (σZ (¯ z ))(Z,¯z) . ∼

For a pointed Galois cover (Z, z¯) one has a bijection ιz¯ : Aut(FEt/S) (Z) → Fs¯(Z). A ´ morphism f : (Z, z¯) → (Z ′ , z¯′ ) in Gs¯ induces a map Fs¯(T ) → Fs¯(Z ′ ). Hence there is a unique map αf : Aut(FEt/S) (Z) → Aut(FEt/S) (Z ′ ) such that the following diagram with ´ ´ bijective vertical maps is commutative Aut(FEt/S) (Z) ´ (20.18.2)

αf

′ / Aut ´ (FEt/S) (Z ) ιz¯′

ιz¯

 Fs¯(Z)

z7→f ◦z

 / Fs¯(Z ′ ).

In other words, for u ∈ Aut(FEt/S) (Z) its image αf (u) is the unique automorphism of Z ′ ´ such that αf (u) ◦ f = f ◦ u. As the lower horizontal map is surjective, αf is surjective. Moreover, it is a group homomorphism: It suffices to check that for u1 , u2 ∈ Aut(FEt/S) (Z) ´ one has αf (u1 ◦ u2 ) ◦ f = αf (u1 ) ◦ αf (u2 ) ◦ f and both sides are equal to f ◦ u1 ◦ u2 . ∼ For (Z, z¯) ∈ Gs¯, the composition π1 (S, s¯) → Fs¯(Z) → Aut(FEt/S) (Z) sends σ ∈ π1 (S, s¯) ´ to the unique automorphism uσ of Z such that uσ ◦ z¯ = σZ (¯ z ). Therefore one has uσ◦σ′ = uσ′ ◦ uσ and by (20.18.1) one obtains an isomorphism of profinite groups (20.18.3)

∼

π1 (S, s¯) −→

lim (Z,¯ z )∈Gs¯

Aut(FEt/S) (Z)opp , ´

where the transition maps on the right hand side are all surjective. Here −opp denotes the opposite group, i.e., where the order of multiplication is reversed.

131 Remark 20.100. For two geometric points s¯1 and s¯2 of a connected scheme S the corresponding fiber functors Fs¯1 and Fs¯2 are isomorphic (Proposition 20.93). An iso∼ morphism Fs¯1 → Fs¯2 is called a path from s¯1 to s¯2 . Such a path yields an isomorphism ∼ ∼ π1 (S, s¯1 ) → π1 (S, s¯2 ) and two isomorphisms π1 (S, s¯1 ) → π1 (S, s¯2 ) obtained from different paths differ by an inner automorphism of π1 (S, s¯2 ) (or of π1 (S, s¯1 )). Hence π1 (S, s¯) does not depend on s¯ up to composition with inner automorphisms. For every ´etale cover X of S and every element σ ∈ π1 (S, s¯) there is an automorphism σX of Fs¯(X). This defines a continuous action of π1 (S, s¯) of Fs¯(X), i.e., we can consider Fs¯(X) as an object of (π1 (S, s¯)-sets), the category of finite sets with a continuous action by π1 (S, s¯). In this way we obtain a functor (20.18.4)

´ Ts¯ : (FEt/S) → (π1 (S, s¯)-sets).

By definition, the composition of Ts¯ with the forgetful functor (π1 (S, s¯)-sets) → (Sets) is the fiber functor Fs¯. Theorem 20.101. Let S be a connected scheme and let s¯ be a geometric point. Then the functor Ts¯ (20.18.4) is an equivalence of categories. Moreover: (1) The functor Ts¯ induces an equivalence between the category of connected ´etale covers of S and the category of finite sets with transitive continuous π1 (S, s¯)-action. (2) A connected ´etale cover Z → S is a Galois cover if and only if for one (or, equivalently, for all) t¯ ∈ Ts¯(Z) the stabilizer πt¯ := { σ ∈ π1 (S, s¯) ; σ(t¯) = t¯} is a normal subgroup of π1 (S, s¯). In this case πt¯ = π1 (Z, t¯) and Fs¯(Z) = π1 (S, s¯)/πt¯ ∼ (Z). = Aut(FEt/S) ´ Proof. Set π := π1 (S, s¯) and T := Ts¯. For an ´etale cover X of S we simply write Aut(X) instead of Aut(FEt/S) (X) und F (X) instead of Fs¯(X). ´ If X and Y are ´etale covers of S, then T (X ⊔ Y ) = T (X) ⊔ T (Y ). Therefore it suffices to show (1) and (2), and we restrict to connected covers and π-sets with transitive action from now on. (I) T is essentially surjective. Let Σ be a finite set with a continuous transitive π-action. As the action is continuous, the stabilizer of each point of Σ in π is an open subgroup. The intersection of these finitely many subgroups is again open and therefore contains an open normal subgroup H such that π/H ∼ = Aut(Z), where Z is a Galois cover. Hence Σ is of the form Aut(Z)/G for some finite subgroup G of Aut(Z). Then Z/G is an ´etale connected cover of S and T (Z/G) = Σ by Proposition 20.89. (II) T is fully faithful. We start with a remark. Let A be any group, G and H subgroups of A. Then every map φ : A/G → A/H that is equivariant with respect to the A-action by left multiplication is uniquely determined by φ(1 · G) ∈ A/H. Conversely σH ∈ A/H yields a well defined A-equivariant map A/G → A/H with 1 · G 7→ σH if and only if Gσ ⊆ σH. Now let X and Y be connected ´etale covers of S. Combining Remark 20.85 and Lemma 20.92 we see that there exists an object (Z, z¯) ∈ Gs¯ such that X = Z/G and Y = Z/H for finite subgroups G, H of Aut(Z) with T (X) = Aut(Z)/G and T (Y ) = Aut(Z)/H. By the above remark we may identify (*)

Hom(π-sets) (T (X), T (Y )) = { σH ∈ Aut(Z)/H ; Gσ ⊆ σH }.

132

20 The ´etale topology

Let f : X = Z/G → Y = Z/H be a morphism of ´etale covers. Choose an element z¯′ ∈ F (Z) whose image in F (Y ) is the image of z¯ under the composition F (f )

F (Z) −→ F (X) −→ F (Y ). By Remark 20.90 (ii) there exists σ ∈ Aut(Z) such that σ(¯ z ) = z¯′ . Clearly f is uniquely determined by the coset σH. Conversely, a given element σ ∈ Aut(Z) induces a well defined morphism X → Y if and only if Gσ ⊆ σH. This proves the fully faithfulness by (*). If (X, x ¯) is a pointed connected ´etale cover of (S, s¯), then we set π(X,¯x) := { σ ∈ π1 (S, s¯) ; σ(¯ x) = x ¯ }. Then Theorem 20.101 and its proof show the following Galois correspondence. Corollary 20.102. Attaching π(X,¯x) to (X, x ¯) yields a bijective map between the set of isomorphism classes of pointed connected ´etale covers and the set of open subgroups of π1 (S, s¯). Via this bijection, pointed Galois covers correspond to normal subgroups. Proof. The inverse map is given by sending an open subgroup U of π1 (S, s¯) to the finite π1 (S, s¯)-set π1 (S, s¯)/U and then applying the inverse of the functor Ts¯. (20.19) Functoriality of the fundamental group. To formulate functoriality statements for the fundamental group it is convenient to introduce the following notion. A pointed scheme is a pair (S, s¯) consisting of a scheme S and a geometric point s¯ of S. A morphism of pointed schemes f : (T, t¯) → (S, s¯) is a morphism of schemes f : T → S such that f (t¯) := f ◦ t¯ = s¯. By definition, this implies κ(t¯) = κ(¯ s). The fundamental group is functorial in the following sense. Definition and Remark 20.103. Let f : (T, t¯) → (S, s¯) be a morphism of connected pointed schemes. To emphasize the base scheme, we write Fs¯(X → S) instead of Fs¯(X) for an ´etale cover X of S, and likewise for the base change to T . One has Fs¯(X → S) = HomS (Spec κ(¯ s), X) = HomT (Spec κ(t¯), X ×S T ) = Ft¯(X ×S T → T ). Hence every isomorphism of fiber functors Ft¯ → Ft¯′ for geometric points t¯ and t¯′ of T yields an isomorphism Ff (t¯) → Ff (t¯′ ) and one obtains a functor f∗ : Π(T ) −→ Π(S) For every geometric point t¯ of T it induces a continuous group homomorphism f∗ : π1 (T, t¯) −→ π1 (S, f (t¯)). By composition the functor f∗ yields a functor f ∗ : (π1 (S, f (t¯))-sets) → (π1 (T, t¯)-sets) ´ and by construction one has an isomorphism of functors (FEt/S) → (π1 (T, t¯)-sets)

133 (20.19.1)

f ∗ ◦ Tf (t¯) ∼ = Tt¯ ◦ f ∗ ,

´ where the f ∗ on the right hand side denotes the base change functor f ∗ : (FEt/S) → ´ (FEt/T ). If g : U → T is a second morphism of connected schemes, then (f ◦ g)∗ ∼ = g∗ ◦ f ∗ .

(f ◦ g)∗ = g∗ ◦ f∗ , One has the following dictionary.

Proposition 20.104. Let f : (T, t¯) → (S, s¯) be a morphism of connected pointed schemes. Let ´ ´ f ∗ : (FEt/S) → (FEt/T ) be the base change functor that sends an ´etale cover X → S to X ×S T → T and let f∗ : π1 (T, t¯) → π1 (S, f (t¯)) be the morphism of topological groups induced by f . Let (S ′ , s¯′ ) (resp. (T ′ , t¯′ )) be a pointed connected ´etale cover of (S, s¯) (resp. of (T, t¯)) and let π(S ′ ,¯s′ ) (resp. π(T ′ ,t¯′ ) ) be the corresponding open subgroup of π1 (S, s¯) (resp. of π1 (T, t¯)) (Corollary 20.102). (1) The homomorphism f∗ is trivial if and only if f ∗ sends every connected ´etale cover to a split ´etale cover. (2) The homomorphism f∗ is an isomorphism if and only if f ∗ is an equivalence of categories. (3) There exists a morphism of pointed schemes f ′ : (T, t¯) → (S ′ , s¯′ ) (necessarily unique by Remark 20.66 (4)) such that the diagram

f

(T, t¯)

′

f

(S ′ , s¯′ ) :  / (S, s¯)

of pointed schemes commutes if and only if f∗ π1 (T, t¯) ⊆ π(S ′ ,¯s′ ) . In this case, π(S ′ ,¯s′ ) contains the normal subgroup generated by f∗ π1 (T, t¯) if and only if the ´etale cover T ×S S ′ of T is split. (4) The homomorphism f∗ is surjective if and only if f ∗ is fully faithful. (5) Suppose that f∗ is surjective. Then the subgroup π(T ′ ,t¯) contains Ker(f∗ ) if and only if there exists an ´etale cover X → S and an isomorphism of ´etale T -covers X ×S T ∼ = T ′. Proof. We set G := π1 (S, s¯), H := π1 (T, t¯), G′ := π(S ′ ,¯s′ ) and H ′ := π(T ′ ,t¯′ ) . We call a G-set trivial (resp. homogeneous) if G acts trivially (resp. transitively) on it. Every homogeneous G-set is isomorphic to a G-set of the form G/U for an open subgroup U ⊆ G. By (20.19.1) and Theorem 20.101 the functor f ∗ has some given property that is stable under equivalences of categories if and only if the functor Cf : (G-sets) → (H-sets) induced by f∗ : H → G has that same property.

134

20 The ´etale topology

Let us show (1). An ´etale cover of T is split if and only if the corresponding π1 (T, t¯)-set is trivial. Therefore f ∗ sends every connected ´etale cover to a split ´etale cover if and only if Cf sends every homogeneous G-set G/U , U ⊆ G some open subgroup, to a trivial H-set. This means that Im(f∗ ) ⊆ U for every open subgroup U of G and hence that f∗ is trivial by Lemma 20.96 (2). For (2) one has to show that f∗ is an isomorphism if and only if Cf is an equivalence of categories. The condition is clearly necessary. Conversely, if Cf is an equivalence of categories, then f∗ induces a bijection between open normal subgroups V of H and open normal subgroups U of G given by V 7→ f∗ (V ) and f∗ yields an isomorphism of groups ∼ H/V → G/f∗ (V ) all V ⊆ H open normal subgroup. Passing to the limit, we see that f∗ is an isomorphism of profinite groups. Let us show (3). The finite G-set corresponding to (S ′ , s¯′ ) is G/G′ . Then Im(f∗ ) (resp. the normal subgroup generated by Im(f∗ )) is contained in G′ if and only if the coset G′ in Cf (G/G′ ) has all of H as stabilizer (resp. if and only if Cf (G/G′ ) is the trivial H-set). We show (4). If f∗ is surjective, then Cf is fully faithful. Conversely, assume that f∗ is not surjective. Then there exists an open subgroup U ⊊ G such that the image of f∗ is contained in U (Lemma 20.96 (3)). Then G/U is a nontrivial finite G-set whose image under Cf is trivial. If Cf was fully faithful, we would find, denoting by ∗ the singleton with its trivial action ∼

∅ = Hom(G-sets) (∗, G/U ) −→ Hom(H-sets) (∗, Cf (G/U )) ̸= ∅, a contradiction. For (5) we have to show that the finite H-set H/H ′ is in the essential image of Cf if and only if H ′ contains Ker(f∗ ). The condition is clearly necessary. Conversely assume that H ′ contains Ker(f∗ ). Define an action of g ∈ G on H/H ′ by (g, hH ′ ) 7→ g˜hH ′ , where g˜ ∈ H with f∗ (˜ g ) = g which is possible because f∗ is surjective. This is well defined because Ker(f∗ ) ⊆ H ′ and because Ker(f∗ ) is a normal subgroup of H. There is the following invariance property for fundamental groups. Proposition 20.105. Let g : X → X ′ be a morphism of schemes that is a universal homeomorphism. Let x ¯ be a geometric point of X and let x ¯′ be its image in X ′ . Then the ′ ′ ′ ´ ´ ) → (FEt/X). If X or equivalently functor Y 7→ Y ×X ′ X yields an equivalence (FEt/X X ′ is connected it induces an isomorphism of profinite groups ∼

π1 (X, x ¯) −→ π1 (X ′ , x ¯′ ). In particular, the fundamental group of a connected scheme depends only on the underlying reduced scheme. Proof. It suffices to show the first assertion. By Theorem 20.30 it remains to show that an ´etale X ′ -scheme Y ′ is finite over X ′ if and only if Y := Y ′ ×X ′ X is finite over X. As being finite is stable under base change, the condition is clearly necessary. Conversely, suppose that Y is finite over X. As Y → Y ′ is a universal homeomorphism, it is surjective and integral by Proposition 20.29. Therefore Y ′ is affine over X ′ by Lemma 20.106 (1) below. Moreover, because the composition Y → Y ′ → X ′ is universally closed, Y ′ → X ′ is universally closed. Hence Y ′ is integral over X ′ by Lemma 20.106 (2) below. As Y ′ → X ′ is ´etale and in particular locally of finite type, Y ′ → X ′ is finite.

135 In the proof above we used the following facts about integral morphisms. Lemma 20.106. Let f : X → Y be a morphism of schemes. (1) Suppose that f is integral and surjective. Then X is affine if and only if Y is affine. (2) The morphism f is integral if and only if it is affine and universally closed. Proof. [Sketch] The first assertion is a generalization of Chevalley’s theorem, that we proved when Y is noetherian and f is finite (Theorem 12.39). The general case is deduced by noetherian approximation and by approximating integral morphisms by finite morphisms. For details we refer to [Sta] 05YU. The second assertion is Exercise 12.19. ´ If for a morphism of connected schemes X → S the base change functor (FEt/S) → ´ is fully faithful, then π1 (X, x ¯) → π1 (S, f (¯ x)) is surjective by Proposition 20.104. (FEt/X) This holds for instance in the following case, where for a scheme T we denote by (VB(T )) the category of locally free OT -modules of finite type. Proposition 20.107. Let f : X → S be morphism of schemes such that OS → f∗ OX is an isomorphism. (1) The functor (VB(S)) → (VB(X)), E 7→ f ∗ E , is fully faithful. (2) The functor T 7→ T ×S X from the category of finite locally free S-schemes to the category of finite locally free X-schemes is fully faithful. ´ ´ (3) The functor (FEt/S) → (FEt/X) is fully faithful. (4) Suppose that S is connected. Then X is connected and π1 (X, x ¯) → π1 (S, f (¯ x)) is surjective for every geometric point x ¯ of X. See also Lemma 24.67 below for a more precise version of Part (1) of the proposition. Proof. To show (1), we prove that for finite locally free OS -modules E and F the canonical map of OS -modules (*)

Hom OS (E , F ) → f∗ Hom OX (f ∗ E , f ∗ F )

is an isomorphism. Taking global sections of (*) we obtain (1). To show that (*) is an isomorphism, we may work locally on S. Hence we may assume E = OSn and F = OSm . As (*) is compatible with taking finite direct sums, we may assume E = F = OS , but then (*) holds by assumption. The functor E 7→ f ∗ E is also compatible with forming tensor products and hence induces a fully faithful functor from the category of finite locally free OS -algebras to the category of finite locally free OX -algebras. This shows (2) because the categories of finite locally free OY -algebras and of finite locally free Y -schemes are equivalent for every scheme Y (Proposition 12.19). Now (3) is an immediate corollary. If S is connected, i.e. 0 and 1 are the only idempotent elements in Γ(S, OS ), then X is connected as well because Γ(X, OX ) = Γ(S, OS ) by hypothesis. Hence (4) follows from (3) using Proposition 20.104 (4). Using properties of the Stein factorization (proved in Section (24.11) below) one obtains the following corollary. Corollary 20.108. Let f : X → S be a proper surjective morphism with geometri´ ´ → (FEt/X) is fully faithful and, for S connected, cally connected fibers. Then (FEt/S) π1 (X, x ¯) → π1 (S, f (¯ x)) is surjective.

136

20 The ´etale topology

´ ´ More generally, (FEt/S) → (FEt/X) is fully faithful if X → S is universally submersive with geometrically connected fibers (Exercise 20.2). f′

Proof. The Stein factorization of f is of the form X −→ S ′ −→ S where OS ′ → f∗′ OX is an isomorphism and S ′ → S is a universal homeomorphism (Proposition 24.56 below). We conclude by combining Proposition 20.107 and Proposition 20.105. (20.20) Fundamental groups of fields. Example 20.109. Let k be a field. Recall that for a k-algebra A the following assertions are equivalent (Remark 18.25 and Theorem 18.44). (i) Spec A is an ´etale cover of k. (ii) A is a finite separable k-algebra. (iii) A is isomorphic to a finite product of finite separable field extensions of k. In this case, Spec A is connected if and only if A is a separable field extension K of k. Such a connected ´etale cover Spec K → Spec k is a Galois cover (Definition 20.90) if and only if # Autk (K) = [K : k], i.e., if and only if K is a Galois extension of k. In this case we have a group isomorphism (20.20.1)

∼

opp Gal(K/k) → Aut(FEt/Spec , ´ k) (Spec K)

σ 7→ Spec σ.

A geometric point s¯ of Spec k corresponds to an algebraically closed extension field Ω of k. Let k sep be the separable closure of k in Ω. A Galois cover in the category of connected pointed Galois covers Gs¯ as in Section (20.18) is a finite Galois extension of k together with a k-embedding K → Ω or, equivalently, a k-embedding K → k sep . Taking the limit over all finite Galois extension K of k contained in k sep the isomorphism (20.20.1) induces an isomorphism of profinite groups Gal(k sep /k) = lim Gal(K/k) (20.20.2)

K

∼

opp → π1 (Spec k, s¯), = lim Aut(FEt/Spec ´ k) (Spec K) K

where the last isomorphism is (20.18.3). In this special case, Theorem 20.101 yields the following result. Corollary 20.110. There exists a contravariant equivalence T between the category of finite separable k-algebras A and the category of finite sets with continuous action by Gal(k sep /k), given by A 7→ T (A) := Homk (A, k sep ), where Gal(k sep /k) acts on T (A) by (σ, t) 7→ σ ◦ t. Moreover, A is a finite separable field extension of k if and only if the Gal(k sep /k)-action on T (A) is transitive. Let A be a finite separable k-algebra. For t ∈ T (A) denote by Γt the stabilizer { σ ∈ Gal(k sep /k) ; σ(t) = t }. This is an open subgroup of Gal(k sep /k). Assume that A = K is a finite separable field extension. Then T (K) = Homk (K, k sep ) can be identified with Gal(k sep /k)/Γt (where Gal(k sep /k) acts on the quotient by left multiplication). This identification depends on the choice of t. For t ∈ T (K) the field of elements in k sep fixed by Γt is given by (k sep )Γt = t(K).

137 Moreover, Γt is normal in Gal(k sep /k) for one t ∈ T (K) if and only if one has Γt = Γt′ for all t′ ∈ T (K) if and only if K is a Galois extension of k. In this case Gal(K/k) = opp ∼ Aut(FEt/Spec = Gal(k sep /k)/Γt . ´ k) (Spec K) (20.21) Examples from number theory. Proposition 20.111. Let R be a local henselian ring with residue field k and set S = Spec R. Let s¯ be a geometric point lying over the closed point s of S and let k sep be the separable closure of k in κ(¯ s). Then by functoriality the closed immersion s → S yields an isomorphism ∼

Gal(k sep /k) = π1 (Spec k, s¯) −→ π1 (S, s¯). Proof. This follows from Example 20.109 and the fact that the functor X 7→ X ⊗R k ´ ´ yields an equivalence (FEt/R) → (FEt/k), see Theorem 20.75. Example 20.112. This proposition can in particular be applied if R is the ring of integers of a non-archimedean local field (Example 20.79). In this case, its residue field k is a finite Q b := limn Z/nZ ∼ field and hence Gal(k sep /k) ∼ = p Zp , where p runs through all prime =Z b numbers in Z (e.g., [BouAII] O V, §12, No.3), and hence π1 (Spec R, s¯) ∼ = Z. Example 20.113. Let S be a Dedekind scheme with function field K. Fix an algebraic closure of K and let s¯ be the corresponding geometric point of S. By Example 20.78 there is a bijective correspondence between ´etale covers of S and finite separable extensions of K that are unramified at all points of S. This correspondence is given by attaching to an ´etale cover X → S the function field LX of X and by attaching to a finite separable extension L of K the normalization of S in L. Every S-automorphism of X induces a K-automorphism of LX . Conversely, every K-automorphism of LX extends uniquely to an S-automorphism of X because of the functoriality of the normalization (Proposition 12.44). Therefore we see that π1 (S, s¯) can ¯ be identified with the profinite Galois group of the maximal subextension L of K in K which is unramified at all points of S. A typical example from number theory is the following. Let K be a number field, i.e., a finite extension of Q, let OK be its ring of integers and let T ⊂ Spec OK be a finite closed subset. Set U := (Spec OK ) \ T . Then π1 (U, s¯) is the profinite Galois group of the ¯ which is unramified outside T . maximal subextension KT of K in K In the special case K = Q, OK = Z and T = ∅, it is known that KT = Q by Minkowski’s theorem ([Neu] O III, (2.18)). Hence π1 (Spec Z, s¯) = 1. In this sense Spec Z is simply connected. (20.22) The fundamental group of P1 . We will now compute the fundamental group of P1k for a field k. Theorem 20.114. Let k sep be a separable closure of k and let x ¯ ∈ P1 (k sep ). Then the structure morphism P1k → Spec k induces an isomorphism of profinite groups ∼

π1 (P1k , x ¯) −→ π1 (Spec k, k sep ) = Gal(k sep /k).

138

20 The ´etale topology

In particular the fundamental group of P1k is trivial if and only if k is separably closed. ´ Proof. We show that T 7→ T ×k P1k is an equivalence of categories (FEt/Spec k) → 1 1 ´ (FEt/Pk ). As Γ(Pk , OP1k ) = k (Example 11.45), by Proposition 20.107 we know already that the functor is fully faithful. Its essential image consists of those finite ´etale morphisms g : T˜ → P1k such that the finite locally free module g∗ OT˜ is isomorphic to OPn1 for some n. k Let g : T˜ → P1k be finite ´etale and set A := g∗ OT˜ . Any finite locally free module A is a finite direct sum of line bundles O(di ) for unique integers d1 ≥ · · · ≥ dn (Theorem 11.53, see also Section (26.22)). As the trace form is perfect (Proposition 20.71), we find A ∼ =A∨ ∨ as locally free modules. Because O(d) = O(−d), this implies (*)

di + dn+1−i = 0,

i = 1, . . . , n.

In particular d1 ≥ 0 and dn = −d1 . Let 0 ̸= s ∈ Γ(P1k , O(d1 )) ⊆ Γ(P1k , A ). Now Γ(P1k , A ) = Γ(T˜, OT˜ ) is reduced because ˜ T is reduced. Hence s2 = ̸ 0. Assume that d1 > 0. If we restrict the multiplication m : A ⊗ A → A to O(d1 ) ⊗ O(d1 ) = O(2d1 ), this corresponds to a global section of A (−2d1 ) which has to be zero, because A (−2d1 ) is the direct sum of line bundles O(e) with e < 0. But this contradicts s2 = ̸ 0. Hence d1 = 0 and therefore di = 0 for all i by (*). (20.23) Algebraic and analytic fundamental group. Let S be a connected scheme locally of finite type over C and let α : S an → S be its analytification (Section (20.12)). Let s ∈ S an and let s¯ be a geometric point of S lying over α(s). Let us compare π1 (S an , s) and π1 (S, s¯). To do this we first recall that if G is any group, then the normal subgroups of G of finite ˆ := limi∈I G/Hi index form a filtered inductive system (Hi )i∈I and the profinite group G is called the profinite completion of G, see also Exercise 20.40. Now ´etale covers of S correspond bijectively to finite covers of S an in a functorial way by the Riemann existence theorem (Remark 20.80). As every finite cover of S an is a quotient of the universal cover of S an by a subgroup of finite index of π1 (S an , s), we hence obtain an isomorphism of profinite groups ∼ π1 (S, s¯) → π1\ (S an , s).

(20.24) The fundamental exact sequence of fundamental groups. Theorem 20.115. Let k be a field, let k¯ be an algebraic closure of k, and let k sep be the ¯ Let X be a geometrically connected qcqs k-scheme and set separable closure of k in k. ¯ Let x ¯ := X ⊗k k. ¯ and denote its image in X also by x X ¯ be a geometric point of X ¯. Then the sequence (20.24.1)

¯ x 1 −→ π1 (X, ¯) −→ π1 (X, x ¯) −→ Gal(k sep /k) −→ 1

is exact. Here we identify Gal(k sep /k) with π1 (Spec k, s¯) (Example 20.109), where s¯ is the image of x ¯ in Spec k, and the maps in the exact sequence arise by functoriality from the ¯ → X → Spec k. morphisms X

139 ¯ Then Spec k p → Spec k and X p := Proof. Let k p be the perfect closure of k in k. X ⊗k k p → X are universal homeomorphisms. Replacing k by k p and X by X p , by Proposition 20.105 we may assume that k is perfect. Then k¯ is the filtered colimit of finite Galois subextensions ki of k. Set Xi := X ⊗k ki and ¯ is an ´etale covering, then there exists denote the image of x ¯ in Xi again by x ¯. If Y → X ¯ = Y and any two an i and a morphism of finite presentation Yi → Xi such that Yi ×Xi X such schemes are isomorphic after possibly enlarging i (Theorem 10.66 and Remark 10.68). As the properties “finite” and “´etale” are compatible with inductive limits, Yi → Xi is an ´etale cover for i large enough. Using the notion of colimit of categories (which we do not define here), this can be expressed by saying that there is an equivalence of categories ∼ ´ ´ colimi (FEt/X i ) = (FEt/X). It follows formally that the canonical homomorphism of profinite groups (*)

¯ x π1 (X, ¯) −→ lim π1 (Xi , x ¯) i

is an isomorphism. On the other hand Spec ki → Spec k is a torsor (for the ´etale or for the fppf-topology) under the Galois group Gal(ki /k) (cf. Definition 14.84). Hence Xi → X is a Galois cover with automorphism group Gal(ki /k). By Theorem 20.101 (2) we have an exact sequence 1 −→ π1 (Xi , x ¯) −→ π1 (X, x ¯) −→ Gal(ki /k) −→ 1. Passing to the limit over i and using (*) we obtain the desired sequence. It is exact as a limit of exact sequences of compact topological groups (recall that a limit of non-empty compact spaces is non-empty and apply this to the fibers of π1 (X, x ¯) −→ Gal(k sep /k)). Remark 20.116. Let X be as in Theorem 20.115. Every k-valued point of X defines a section of the exact sequence (20.24.1) by functoriality. The famous section conjecture by Grothendieck asserts that for k/Q finitely generated and X a geometrically connected, smooth projective curve of genus ≥ 2 (see Section (26.8)) every such section arises from a k-rational point. There is the following variant of Theorem 20.115 for arbitrary base schemes and proper morphisms. It uses that the Stein factorization yields an ´etale cover for flat proper finitely presented morphisms with geometrically reduced fibers (see Theorem 24.61 below). Proposition 20.117. Let f : X → S be a flat proper morphism of finite presentation with geometrically connected and geometrically reduced fibers. Assume that S is connected. Let x ¯ be a geometric point of X and let s¯ be its image in S. Then there is an exact sequence π1 (Xs¯, x ¯) −→ π1 (X, x ¯) −→ π1 (S, s¯) −→ 1 of profinite groups. Note that the hypotheses imply that X and Xs¯ are also connected. Proof. We have already seen in Corollary 20.108 that π1 (X, x ¯) −→ π1 (S, s¯) is surjective. ´ ´ ´ As the composition (FEt/S) → (FEt/X) → (FEt/X etale covering of S to s¯) sends any ´ a split covering, the composition π1 (Xs¯) → π1 (X) → π1 (S) is trivial.

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20 The ´etale topology

It remains to show that if Y → X is an ´etale covering with Y connected and such that Y ×X Xs¯ = Ys¯ has a section over Xs¯, then there exists an ´etale covering T → S and an ∼ X-isomorphism Y −→ X ×S T (Proposition 20.104). To see this let Y → T → S be the Stein factorization of Y → S. Then T → S is finite ´etale by Theorem 24.61 below. We claim that the X-morphism u : Y → X ×S T is an isomorphism. As u is a morphism of ´etale covers, it is finite ´etale (Remark 20.66 (3)). Because Y is connected and Y → T is surjective, T is connected. As X ×S T → T is closed with connected fibers, X ×S T is connected as well by an elementary topological argument (see Lemma 24.52 below for a more general statement and its proof). Hence the degree of u is a constant function and it suffices to show that it equals 1 in a single point. This can be done after base change s¯ → S, where it follows from the assumption that Ys¯ has a section over Xs¯. (20.25) Specialization of fundamental groups. Let R be a local henselian ring with residue field k. In Theorem 20.75 we showed that ´ ´ the base change U 7→ U ⊗R k is an equivalence of categories (FEt/R) → (FEt/k). This result can be generalized considerably. Theorem 20.118. Let (R, I) be a henselian pair (Definition 20.15). Let X → Spec R be a proper morphism and set X0 := X ⊗R R/I. Then the functor ´ ´ (FEt/X) −→ (FEt/X 0 ),

U 7→ U ×X X0

is an equivalence of categories. In particular, if X and X0 are connected, then for every geometric point x ¯ of X0 the inclusion X0 → X yields an isomorphism ∼

π1 (X0 , x ¯) −→ π1 (X, x ¯). We will give a proof of this theorem below (Corollary 24.111) in the case that R is an I-adically complete noetherian ring, as an application of Grothendieck’s existence theorem. The general case is reduced to this special case by Artin approximation (Section (20.11)) and by noetherian approximation. We refer to [Sta] 0GS2 for details. We can use the theorem to show the following invariance of fundamental groups. Proposition 20.119. Let k → K be an extension of separably closed fields and let X be a proper scheme over k. Then the functor ´ ´ F : (FEt/X) −→ (FEt/X K ),

U 7→ UK := U ⊗k K

is an equivalence of categories. In particular, if X is connected, for every geometric point ∼ x ¯ of X one has an isomorphism π1 (XK , x ¯) → π1 (X, x ¯). Note that if X is connected, then it is geometrically connected because k is separably closed (Proposition 5.53). Hence XK is connected as well. Proof. By Proposition 20.105 we may assume that K and k are algebraically closed. The k-algebra K is the filtered union of its finitely generated k-subalgebras R. We first show that F is essentially surjective. Let U be a finite ´etale scheme over XK . Then we can find a finitely generated k-subalgebra R of K and an R-scheme UR whose base change to XK is U (Section (10.17)). As the properties “finite” and “´etale” are compatible with inductive limits (Proposition 12.11, and Corollary 18.43) we may assume after enlarging R that UR is finite ´etale over XR .

141 Let m ⊂ R be a maximal ideal, then R/m = k since R is a finitely generated k-algebra and k is algebraically closed. Let U0 := UR ⊗R k be the special fiber of UR . Let Rh be the henselization of Rm . We may choose a homomorphism of R-algebras Rh → K. Indeed, let L be the field of fractions of Rm which is a subfield of K. As Rh is the colimit of ¯ localizations of ´etale Rm -algebras, we may view Rh as a subring of an algebraic closure L ¯ of L. As K is algebraically closed, we find an L-embedding L → K and in particular a homomorphism of R-algebras Rh → K. By Theorem 20.118 we have U0 ⊗k Rh ∼ = UR ⊗R Rh . Hence we see that U is isomorphic to the base change of U0 . This proves that F is essentially surjective. To show that F is fully faithful we first remark that F is faithful because k → K is faithfully flat (Proposition 14.70). Now let U and U ′ be finite ´etale schemes over X and ′ let πK : UK → UK be a morphism of finite ´etale schemes over XK . As above we find a h k-algebra R that is a local henselian ring with residue field k, a k-algebra homomorphism Rh → K, and a morphism πRh : URh → UR′ h whose base change to XK is π. We set π := πRh ⊗Rh idk . Again using Theorem 20.118 we see that URh ⊗Rh k ∼ = U ′ and that = U and UR′ h ⊗Rh k ∼ via these isomorphisms we have π ⊗k idRh = πRh and therefore π ⊗k idK = πK . Remark 20.120. The assertion in Proposition 20.119 does not hold for non-proper schemes over a separably closed field k in general (Exercise 20.44). Following essentially Kedlaya [Ked] O X §4.1 let us say that a k-scheme X is π1 -proper if for every separably closed extension K of k the pullback via the projection X ⊗k K → X induces an equivalence ´ ´ of categories (FEt/X) −→ (FEt/X ⊗k K). Then Proposition 20.119 means that every proper k-scheme is π1 -proper. Moreover, in loc. cit. the following assertions are shown. (1) If char(k) = 0, then any k-scheme X is π1 -proper. (2) A connected qcqs k-scheme X is π1 -proper if and only if for every connected k-scheme Y and every geometric point z¯ of (the connected scheme) X ×k Y the map induced by functoriality π1 (X ×k Y, z¯) −→ π1 (X, z¯) × π1 (Y, z¯) is an isomorphism of topological groups. The above results allow us to define specialization maps for fundamental groups as follows. Let S be a scheme, let f : X → S be a proper morphism with geometrically connected fibers. Let s, η ∈ S be points such that s is a specialization of η, i.e., s ∈ {η}. Hence η corresponds to a point of Spec OS,s . Let s¯ and η¯ be geometric points lying over s and η, respectively. Let OS,¯s be the strict henselization of OS,s with respect to s¯ (Def. 20.39). Its residue field is a separable closure κ(s)sep of κ(s) in κ(¯ s). Choose an S-morphism η¯ → Spec(OS,¯s ). Let x ¯ and x ¯′ be geometric sep points of Xκ(s) and Xη¯. Then one defines a homomorphism of pro-finite groups, the specialization map, as the composition ∼

∼

∼

π1 (Xη¯, x ¯′ ) −→ π1 (XOS,¯s , x ¯) −→ π1 (XOS,¯s , x ¯′ ) −→ π1 (Xκ(s)sep , x ¯′ ) −→ π1 (Xs¯, x ¯′ ), where the first map is given by functoriality, the second by the choice of a path from x ¯ to x ¯′ (Remark 20.100), the third is the isomorphism of Theorem 20.118, and the last isomorphism is given by Proposition 20.119. For smooth proper morphisms the specialization map is surjective. More precisely, let p be a prime number or p = 1. We define for a profinite group π it maximal prime-to-pquotient

142

20 The ´etale topology ′

π (p ) := lim π/U, U

where U runs through the open normal subgroups U of π such that π/U has order prime ′ to p. Hence π (1 ) = π. Then there is the following result, for whose proof we refer to [Sta] 0C0Q and 0C0R. Theorem 20.121. Let f : X → S be a smooth proper morphism of schemes with geometrically connected fibers. Let s, η ∈ S with s a specialization of η. Let p be the characteristic exponent of κ(s) (i.e., p = 1 if the characteristic of κ(s) is zero and otherwise p = char(κ(s))). Let x ¯ be a geometric point of Xs and let x ¯′ be a geometric point of Xη . Then the specialization map π1 (Xη¯, x ¯′ ) −→ π1 (Xs¯, x ¯) is surjective and induces an isomorphism ′

′

π1 (Xη¯, x ¯′ )(p ) −→ π1 (Xs¯, x ¯)(p ) .

Exercises Exercise 20.1. Let A be a ring. For a polynomial f ∈ A[T ] let c(f ) be the ideal in A generated by the coefficients of f . It is called the content of f 5 . A polynomial f is called primitive if c(f ) = A. Let f, g ∈ A[T ] be polynomials, n := deg(g). Show the following assertions. p (1) c(f g) ⊆ c(f )c(g) ⊆ c(f g). Deduce that f and g are primitive if and only if f g is primitive. (2) c(f )n+1 c(f g) = c(f )n c(f g). Remark : This result is called the Dedekind-Mertens lemma. (3) Let A = k[x, y] for a field k and set f = xT + y and g = xT − y. Show that c(f g) ̸= c(f )c(g). Exercise 20.2. Let h : X0 → X be a morphism of schemes. For every X-scheme Y let Y0 := X0 ×X Y be its base change with h. We also set X00 := X0 ×X X0 and denote by Y00 the base change of Y to X00 (with respect to X00 → X0 → X, which is independent of the choice which projection X00 → X0 is used). Let Y , Y ′ be X-schemes. (1) Let h be surjective and assume that Y ′ is unramified over X. Show that the base change with h yields an injective map HomX (Y, Y ′ ) → HomX0 (Y0 , Y0′ ). (2) Let h be universally submersive and assume that Y ′ is ´etale over X. Show that the diagram HomX (Y, Y ′ )

h∗

/ HomX (Y0 , Y0′ ) 0

pr∗ 1 pr∗ 2

// Hom

′ X00 (Y00 , Y00 )

is exact. 5

This notion of content is not the same as the content of a polynomial over a factorial ring A as for instance defined in [BouAC] O , except if A is a principal domain.

143 (3) Assume that h is universally submersive, separated, and has geometrically connected fibers. Assume that Y and Y ′ are ´etale over X. Show that h∗

HomX (Y, Y ′ ) −→ HomX0 (Y0 , Y0′ ) is bijective. Exercise 20.3. Let p ∈ Z be a prime number and let Z(p) the localization in (p). Show that Z(p) is not henselian. Exercise 20.4. Let A be a local ring, let f ∈ A[T ] be a monic polynomial, and let B = A[T ]/(f ). Show that there exists a bijective correspondence between decompositions f = gh as a product of two monic polynomials g, h ∈ A[T ] such that (g) + (h) = A[T ] and decompositions B = C × D as a product of A-algebras, which is given by sending ∼ (g, h) to the decomposition B → A[T ]/(g) × A[T ]/(h). Exercise 20.5. Let A be a local ring with maximal ideal m and residue field k. For f ∈ A[T ] let f¯ be the image of f in k[T ]. Show that the following assertions are equivalent. (i) A is henselian. ¯ of the image f¯ of f in k[T ] such that f¯′ (¯ a) ̸= 0 (ii) For every f ∈ A[T ] and every root a there exists a unique a ∈ A whose image in k is a ¯ and such that f (a) = 0. (iii) For every monic f ∈ A[T ] such that f¯(0) = 0 and f¯′ (0) ̸= 0 there exists a ∈ m such that f (a) = 0. (iv) For every primitive f ∈ A[T ] and every factorization f¯ = g0 h0 in k[T ] such that g0 and h0 are prime to each other there exists a factorization f = gh in A[T ] such that ¯ = h0 . deg g = deg g0 , g¯ = g0 , and h Exercise 20.6. Let K be a field, v a valuation on K (not necessarily discrete), and let A be the valuation ring of v. Show that A is henselian if and only if for every algebraic extension K ′ of K any two extensions of v to K ′ are equivalent. Exercise 20.7. Let A be a local ring with maximal ideal m and let I ⊂ A be an ideal. (1) Assume first that I consists of nilpotent elements. Show that A is henselian if and only if A/I is henselian. Hint: Use that the inclusion Spec B/IB → Spec B is a homeomorphism for every A-algebra B. (2) More generally, assume that A is I-adically complete (Definition B.39). Show that A is henselian if and only if A/I is henselian. Hint: Use that every finite free A-algebra B is complete for the I-adic topology and apply (1) to B/I n B. Alternatively, use Exercise 20.5 and construct (with the notations there) a mod I n successively. (3) Deduce that if A is m-adically complete, then A is henselian. Exercise 20.8. Let R be a local ring, m its maximal ideal, k its residue field, S = Spec R, s ∈ S its closed point. Show that the following assertions are equivalent. (i) R is henselian. (ii) Every R-algebra A of finite type is isomorphic to a product A = B × A1 × · · · × An , where the Ai are finite local R-algebras and all irreducible components of Spec B ⊗R k have dimension ≥ 1. (iii) For every finitely generated R-algebra C and every prime ideal p lying over m such that Cp /mCp is a finite k-algebra, Cp is a finite R-algebra.

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20 The ´etale topology

(iv) For every morphism f : X → S locally of finite type and for every x in the fiber Xs such that κ(s) = OX,x /ms OX,x there exists an open neighborhood U of x in X such that f |U : U → S is an open immersion. Exercise 20.9. Let (Rα , φβα ) be a filtered inductive system of local rings, such that all homomorphisms φβα are local. Let R be its inductive limit. (1) Show that R is local and that the ring homomorphisms Rα → R are local. (2) Show that if Rα is henselian for all α, then R is henselian. Exercise 20.10. Let A be a local henselian ring with residue field k and let B be an A-algebra that is integral over A. (1) Show that the canonical map B → B ⊗A k induces a bijection {idempotents of B} ↔ {idempotents of B ⊗A k}. (2) Show that for every maximal ideal n of B the localization Bn is integral over A and henselian. Hint: Write B as the inductive limit of its finite A-subalgebras and use Exercise 20.9. Exercise 20.11. Let A be a local ring. Show that A is henselian if and only if for every finite A-algebra B and for every maximal ideal n of B the localization Bn is integral over A. Exercise 20.12. Let R be a local ring and let ι : R → Rh be its henselization. Show that for every local henselian ring A and every local homomorphism φ : R → A there exists a unique local homomorphism φh : Rh → A such that φh ◦ ι = φ. Exercise 20.13. Let φ : A → B be a local homomorphism of local rings. (1) Show that there exists a unique (necessarily local) homomorphism φh : Ah → B h making the following diagram commutative / Ah

A φ

φh

 / Bh

 B

Hint: Exercise 20.12. (2) Let k sep be a separable closure of the residue field k of A, let ℓsep be a separable closure of the residue field ℓ of B, let Ahs and B hs be the corresponding strict henselizations, and let ι : k sep → ℓsep be a homomorphism which extends the homomorphism k → ℓ induced by φ. Show that there exists a unique (necessarily local) homomorphism φhs : Ahs → B hs making the following diagram commutative A φ

 B

/ Ahs 

φhs

/ B hs

/ k sep 

ι

/ ℓsep

Assertion (2) means that one should see strict henselization as a functor on the category of pairs consisting of a local ring R and a separable closure of the residue field of R.

145 Exercise 20.14. Let R be a local ring and fix a separable closure k sep of the residue field of R. (1) Assume that R is henselian (resp. strictly henselian). Show that R → Rh (resp. R → Rhs ) is an isomorphism. (2) Show that Rhs ∼ = (Rh )hs . Here all strict henselizations are formed with respect to k sep . Exercise 20.15. Consider the following examples of a locally ringed space (X, OX ). (1) X is a topological space and OX is the sheaf of continuous R-valued functions on X. (2) X is a real C α -manifold (α ∈ N ∪ {∞}) and OX is the sheaf of R-valued C α -functions on X. (3) X is a complex analytic space (e.g., a complex manifold) and OX is the sheaf of complex analytic C-valued functions on X. Show that in all cases for all x ∈ X the local ring OX,x is henselian. Exercise 20.16. Let K be a field endowed with an absolute value | · | : K → R≥0 (i.e., for all a, b ∈ K one has: P |a| = 0 ⇔ a = 0, |ab| = |a||b|, and |a + b| ≤ |a| + |b|). A ai1 ...in T1i1 · · · Tnin ∈∈ K[[T1 , . . . , Tn ]] is called convergent if formal power series f = >0 there exist c1 , . . . , cn , M ∈ R such that |ai1 ...in |ci11 · · · cinn ≤ M for all (i1 , . . . , in ) ∈ Nn . Let K⟨⟨T1 , . . . , Tn ⟩⟩ be the subset of convergent power series in K[[T1 , . . . , Tn ]]. Show that K⟨⟨T1 , . . . , Tn ⟩⟩ is a regular noetherian henselian local ring whose completion is K[[T1 , . . . , Tn ]]. Hint: This is a difficult exercise; see [Nag] §45. Exercise 20.17. Let f : X → Y be a separated quasi-finite flat morphism of finite presentation. (1) Show that the degree function deg(f ) : Y → Z,

y 7→ dimκ(y) Γ(Xy , OXy )

is lower semicontinuous and constructible, i.e., { s ∈ S ; deg(f )(s) ≥ n } is open and constructible in S for all n ∈ Z. (2) Show that f is finite locally free if and only if the function y 7→ dimκ(y) Γ(Xy , OXy ) is locally constant. Hint: For the constructibility reduce to Y noetherian. Then it remains to show that deg(f ) jumps up under generization. For this reduce to the case that Y is the spectrum of a henselian local ring. Exercise 20.18. Let f : X → S be a quasi-finite morphism. For every geometric point s¯ lying over a point s of S we call n(s) := # HomS (¯ s, X) the number of geometric points of f −1 (s). P ′′ ′′ , where fx/s is the inertia index defined in Sec(1) Show that n(s) = x∈f −1 (s) fx/s tion (12.5). (2) Let f in addition be flat and of finite presentation. Show that n : S → Z is constructible and lower semicontinuous. (3) Let f be quasi-compact, separated, and ´etale. Show that f is an ´etale cover if and only if the map S → Z, s 7→ n(s) is locally constant. (4) Let f be finite locally free. Show that s 7→ n(s) is locally constant if and only if there exist a factorization of f fr fe f : X −→ X ′ −→ S,

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20 The ´etale topology

with fr a finite locally free universal homeomorphism and fe finite ´etale. In this case the factorization is unique up to unique isomorphism, is functorial in X → S, and is compatible with base change S ′ → S. Hint: Exercise 20.17. Exercise 20.19. Let A be a reduced local noetherian ring and let B be its normalization (i.e., B = { a ∈ Frac A ; a is integral over A}, where Frac A denotes the total ring of fractions of A). Show that B is unramified over A if and only if the irreducible components of Spec Ahs are normal. Exercise 20.20. Let A be a local integral domain with residue field k and let Ared be the quotient of A by its nilradical. (1) Show that A is unibranch if and only if Ared is an integral domain and its normalization (see Exercise 20.19) is local. (2) Show that A is geometrically unibranch if and only if Ared is an integral domain and its normalization is a local ring whose residue field is a purely inseparable extension of k. Hint: Show first that (Ah )red = (Ared )h and (Ahs )red = (Ared )hs (with respect to some chosen separable closure of k). Exercise 20.21. Let k be a field and let 0 ̸= f ∈ k[T, U ] with f (0, 0) = 0. Set X = Spec k[T, U ]/(f ) and x := (0, 0) ∈ X(k). Let f ∗ be the leading term of f and let n be its degree (Exercise 6.10). Assume n ≥ 1. hs has at most n irreducible components (i.e., there are at most n (1) Show that Spec OX,x “geometric branches” passing through x). hs has exactly n irreducible components if and only if every (2) Show that Spec OX,x irreducible component of Spec Ahs is normal (cf. Exercise 20.19). Assume n ≥ 2. If the above condition is satisfied, then x is called a non-cuspidal singularity of X; otherwise x is called a cuspidal singularity. (3) Write f ∗ over an algebraic closure k¯ of k as a product f ∗ = ℓ1 · · · ℓn of homogeneous polynomials of degree 1. Show that if the lines V (ℓi ) ⊂ A2k¯ are pairwise distinct, then hs OX,x is isomorphic to the strict henselization of the tangent cone of X in x (i.e., X and its tangent cone are locally for the ´etale topology isomorphic). This generalizes Exercise 6.11. Exercise 20.22. Let k be a field of characteristic ̸= 2. (1) Let R be the local ring in (0, 0) of V (Y 2 − X 2 (X + 1)) ⊂ A2k . Show that R is an integral domain but that Spec Rh has two irreducible components. In particular R is not unibranch. (2) Let R be the local ring in (0, 0) of V (Y 2 − X 3 ) ⊂ A2k . Show that R is geometrically unibranch but not normal. Exercise 20.23. Let A be a henselian local noetherian ring and let Aˆ be its completion. Show that B 7→ B ⊗A Aˆ yields an equivalence between the category of finite ´etale ˆ A-algebras and the category of finite ´etale A-algebras. Exercise 20.24. Let X be a scheme, let G be a finite group of automorphisms of X, let x ∈ X. If g(x) = x for some g ∈ G, then g induces an automorphism gκ(x) of κ(x). The subgroup Ix := { g ∈ G ; g(x) = x, gκ(x) = idκ(x) } is called the inertia subgroup of G at x.

147 Now assume that there exists an affine morphism X → S such that G acts by Sautomorphisms. Let H be a subgroup of G and set Y := X/H and Z := X/G (cf. Proposition 20.87). Let p : X → Y and q : Y → Z be the canonical morphisms. Consider the following assertions. (i) Ix ⊆ H. (ii) There exists an open neighborhood V of p(x) in Y such that q |V : V → Z is ´etale. (iii) There exists an open neighborhood V of p(x) in Y such that q |V : V → Z is unramified. Show the implications “(i) ⇒ (ii) ⇒ (iii)”. Show that all assertions are equivalent if X is an integral scheme. Hint: One may assume S = Z. To show “(i) ⇒ (ii)” reduce to the case that Z = Spec R is the spectrum of a strictly henselian ring. Then prove that R → OY,p(x) can be identified Ix Ix ∩H with OX,x → OX,x . To show “(iii) ⇒ (i)” consider Ug := Ker(p, p ◦ g) for g ∈ Ix . Show that Ug is closed in X and contains an open neighborhood of x. Remark : The existence of an affine morphism X → S such that G acts by Sautomorphisms is only needed to ensure the existence of X/G and X/H. One can show that such a quotient, satisfying all the properties stated in Proposition 20.87, exists if every G-orbit is contained in an open affine subscheme of X ([SGA3] O X Exp. V, Th´eor`eme 4.1). This conditions is for instance satisfied if X is qcqs and there exists an ample line bundle on X (Proposition 13.49). The same remark applies to Exercise 20.25. Exercise 20.25. Let X be a scheme and let G be a finite group of automorphisms of X. Assume that there exists an affine morphism X → S such that G acts by Sautomorphisms. (1) Assume that one has Ix = 1 for all x ∈ X, where Ix is the inertia group (Exercise 20.24). Show that X → X/G is finite ´etale. (2) Assume that X is connected, that G acts faithfully on X, and that X → X/G is unramified. Show that Ix = 1 for all x ∈ X. Exercise 20.26. Let R be a local normal ring with field of fractions K. Let K sep be a separable closure of K. Then Γ := Gal(K sep /K) acts on the integral closure A of R in K sep . Fix a maximal ideal m of A, let D := { σ ∈ Γ ; σ(m) = m } be the decomposition group and let I := { σ ∈ D ; σ induces on κ(m) the identity} be the inertia group of m. Set B := AD , n := m ∩ B, and B ′ = AI , n′ = m ∩ B ′ . Show Rh ∼ = Bn ,

Rhs ∼ = Bn′ ′ .

Hint: Exercise 20.24 Exercise 20.27. Let R be a local noetherian integral domain with field of fractions K. ˆ be the completion of R. Show that Rh is the subring of Let Rh be the henselization and R ˆ ˆ R consisting of elements f ∈ R that are algebraic over K, i.e., there exist a0 , . . . , an ∈ K with n > 0 and an ̸= 0 such that an f n + an−1 f n−1 + · · · + a1 f + a0 = 0. Exercise 20.28. Let S be a Q-scheme and let f : X → S be a smooth morphism. Show that the De Rham complex Ω•X/S is acyclic in degrees > 0. Hint: Exercise 19.7 Exercise 20.29. Let A be a ring and let B be a finite A-algebra. Show that Spec B → Spec A is an ´etale cover if and only if B is a projective A-module and a projective B ⊗A B-module.

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20 The ´etale topology

Exercise 20.30. Let R be a ring. (1) Let A be a finite R-algebra which is free as an R-module with basis (x1 , . . . , xn ). Show that the discriminant DA/R is a principal ideal of R generated by det(trA/R (xi xi ))i,j . (2) Let f ∈ R[T ] be a monic polynomial and set A := R[T ]/(f ). Show that DA/R is a principal ideal of R generated by the discriminant Df of f (in the sense of Section (B.20), or [BouAII] O IV §6, No. 7). (3) Let f ∈ R[T ] be a monic polynomial. Show that R[T ]/(f ) is a finite ´etale R-algebra if and only if Df is a unit in R. Exercise 20.31. Let (A, I) be a henselian pair. Show that P 7→ P/IP induces a bijection between the set of isomorphism classes of finite projective A-modules and the set of isomorphism classes of finite projective A/I-modules. Exercise 20.32. A pair (A, I) consisting of a ring A and an ideal I ⊆ A is called a Zariski pair if I is contained in the Jacobson radical of A. Show that the inclusion functor from the full subcategory of Zariski pairs into the category of all pairs (A, I) consisting of a ring A and an ideal I ⊆ A has a left adjoint functor which is given by (A, I) 7→ (S −1 A, S −1 I), where S := 1 + I. Exercise 20.33. Show that the inclusion functor from the full subcategory of henselian pairs to the category of all pairs (A, I) consisting of a ring A and an ideal I ⊆ A has a left adjoint functor (A, I) 7→ (A, I)h , called henselization of the pair (A, I). Hint: Consider the category C of ´etale ring homomorphism A → B that induce an ∼ isomorphism A/IA → B/IB. Show that C is filtered. Set Ah := colimB∈C B and I h := IAh and show that (A, I) 7→ (Ah , I h ) gives the desired left adjoint functor. Show the following properties of the henselization (Ah , I h ) of a pair (A, I). (1) If A is a local ring with maximal ideal m, then (Ah , mh ) is the henselization defined in Definition 20.42. (2) The ring homomorphism A → Ah is flat. It is faithfully flat if and only if (A, I) is a Zariski pair (Exercise 20.32). (3) If I, J ⊆ A are ideals with V (I) = V (J), then (A, I)h ∼ = (A, J)h . ∼ h (4) For all n ≥ 1 the map A → A induces isomorphisms A/I n → Ah /(I h )n . Exercise 20.34. Let X and S be integral schemes and let f : X → S be a finite dominant morphism of finite presentation. Let S be unibranch. Show that f is open. Hint: Use the going down property (Theorem B.54 (3)). Exercise 20.35. Let S be an integral geometrically unibranch scheme and let f : X → S be a dominant morphism. Assume that X is connected. Show that f is unramified if and only if f is ´etale. Hint: Reduce to the case that S = Spec R and X = Spec A such that R ,→ A is injective and such that A is the quotient of a standard ´etale R-algebra. Then reduce to the case that R is strictly henselian. Exercise 20.36. Let X be a scheme, U ⊆ X a non-empty open subscheme. Show that the ´ ´ functor (FEt/X) → (FEt/U ) that sends X ′ to X ′ ×X U is fully faithful in the following two cases. (1) The scheme X is the spectrum of a local geometrically unibranch ring and U is the complement of its closed point. (2) The scheme X is geometrically unibranch and has a noetherian underlying topological space and U is dense in X.

149 Exercise 20.37. Let π : X → S be a finite locally free morphism of locally noetherian schemes and let Z be the branch locus. Show that one has “purity of the branch locus”, i.e, each irreducible component of the branch locus of π has codimension 1 in X. Remark : One also has purity of the branch locus if X and S are integral and locally noetherian, S is regular, X is normal, and f is quasi-finite and dominant ([SGA2] O X Exp. X, Th´er`eme 3.4). Exercise 20.38. Let k be an algebraically closed field of characteristic p > 0 and let q = pn for some integer n ≥ 1. Show that the k-algebra homomorphism k[T ] → k[T ], T 7→ T q − T , defines a finite ´etale Galois cover A1k → A1k , called Artin-Schreier cover with Galois group (Z/pZ)n . Exercise 20.39. Let X be a scheme locally of finite type over a field k. (1) Show that there exists an ´etale k-scheme π0 (X) and a morphism qX : X → π0 (X) with the following universal property. For every morphism f : X → Y of k schemes, where Y is an ´etale k-scheme, there exists a unique morphism g : π0 (X) → Y such that f = g ◦ qX . (2) Show that qX is faithfully flat and the fibers of qX are the connected components of X. (3) Show that for every morphism f : X → Y of k-schemes locally of finite type there exists a unique morphism π0 (f ) : π0 (X) → π0 (Y ) such that qy ◦ f = π0 (f ) ◦ g and that we obtain a functor π0 from the category of k-schemes locally of finite type to the category of ´etale k-schemes. (4) Let K be a field extension of k. Show that there exists an isomorphism of K-schemes, functorial in X, ∼ π0 (X ⊗k K) → π0 (X) ⊗k K. (5) Show that for two k-schemes X and Y locally of finite type over k, the canonical morphism π0 (X ×k Y ) → π0 (X) ×k π0 (Y ) is an isomorphism, functorial in X and Y . (6) Show that X is geometrically connected over k if and only if π0 (X) = Spec k. The scheme π0 (X) is called the scheme of connected components of X. Exercise 20.40. (1) Show that the inclusion functor from the category of profinite groups into the category ˆ where of topological groups admits a left adjoint functor G 7→ G, ˆ = lim G/U G U

where U ⊆ G runs through the open normal subgroups of finite index. The profinite ˆ is called the profinite completion of G. group G (2) Let G be a topological group and let F : (G-sets) → (Sets) be the forgetful functor. ∼ ˆ→ Show that one has a functorial isomorphism G Aut(F ) of profinite groups. Exercise 20.41. Let S be a connected scheme, let s¯ be a geometric point of S, let X → S be an ´etale cover, and let F be the corresponding finite set with continuous π1 (S, s¯)-action. (1) Show that the action of π1 (S, s¯) on F is trivial if and only if X → S is a split ´etale cover. (2) Show that there exists a Galois cover T → S such that X ×S T → T is a split ´etale cover.

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20 The ´etale topology

Exercise 20.42. Let Γ be a profinite group, let (t-Γ-sets) be the full subcategory of (Γ-sets) consisting of finite sets with transitive Γ-action. Let V t : (t-Γ-sets) → (sets) and V : (Γ-sets) → (sets) be the forgetful functors. ∼ (1) Show that the inclusion (t-Γ-sets) → (Γ-sets) yields an isomorphism Aut(V) → t Aut(V ). (2) Let ∆ be an open normal subgroup of Γ. Show that every map Γ/∆ → Γ/∆ in (sets) commuting with all automorphisms in (Γ-sets) of Γ/∆ is given by left multiplication by some element of Γ/∆. (3) Let ∆ as in (2) and for σ ∈ Aut(V t ) let γ∆ ∈ Γ/∆ be the coset such that σΓ/∆ (γ ′ ∆) = γγ ′ ∆. Show that one obtains a surjective group homomorphism Aut(V t ) → Γ/∆ ∼ which yields an isomorphism Aut(V t ) → lim∆ Γ/∆. Exercise 20.43. Let k be a separably closed field and let X be a connected k-scheme. Let k → K be a separably closed extension of k and let x ¯ be a geometric point of X ⊗k K. Show that the canonical map π1 (X ⊗k K, x ¯) −→ π1 (X, x ¯) is surjective. Exercise 20.44. Let k be an algebraically closed field of characteristic p > 0 and set X = A1k . Show that the Artin-Schreier construction (Exercise 20.38) provides for any geometric point x ¯ of A1k an isomorphism between P the group of continuous group homomorphisms 1 ¯) → Z/pZ and the group { i≥0 ai T i ∈ k[T ] ; ai = 0 if p divides i}. Deduce that π1 (Ak , x for a non-trivial algebraically closed extension k → K and for any geometric point x ¯ of A1K the surjective map π1 (X ⊗k K, x ¯) −→ π1 (X, x ¯) (Exercise 20.43) is not injective.

21

Cohomology of OX -modules

Content – Categories of abelian sheaves and of OX -modules – Cohomology and derived direct image ˇ – Cech cohomology – Derived inverse image, Hom sheaves, and tensor products – Relations between derived functors – Perfect and pseudo-coherent complexes In this chapter we develop a “four functor formalism” for the category of modules over a ringed space. For every ringed space (X, OX ) we will denote by D(X) the (unbounded) derived category of the abelian category of OX -modules. More precisely, we will define for every morphism f : (X, OX ) → (Y, OY ) of ringed spaces adjoint functors Lf ∗ : D(Y ) o

/ D(X) : Rf ∗

where Rf∗ is the right derived functor of the direct image functor and Lf ∗ is the left derived functor of the inverse image functor. The notation implies that the functor on the left Lf ∗ is left adjoint to Rf∗ (and that Rf∗ is right adjoint to Lf ∗ ). Moreover, for every ringed space (X, OX ) we will define bi-functors “derived tensor product” and “derived inner Hom” − ⊗L OX − : D(X) × D(X) −→ D(X), R Hom OX (−, −) : D(X)opp × D(X) −→ D(X), such that (−) ⊗L OX G is left adjoint to R Hom OX (G , −) for every OX -module G . We will show that these functors satisfy several properties, e.g., functoriality of Rf∗ and Lf ∗ (i.e., R(g ◦ f )∗ = Rf∗ ◦ Rg∗ and L(g ◦ f )∗ = Lf ∗ ◦ Lg ∗ ), symmetry and associativity of ∗ − ⊗L OX −, and compatibility of Lf with derived tensor product. Ideally one would like to extend this “four functor formalism” to a six functor formalism in the sense of Grothendieck, see Section (21.30) below for further remarks what this means and why this is not possible without enlarging the category of schemes. A special case of f∗ (namely if Y consists of a single point with OY (Y ) = Γ(X, OX )) is the global section functor Γ(X, −). In this case Rf∗ identifies with the derived functor RΓ(X, −) : D(X) −→ D(Γ(X, OX )), where D(A) denotes the derived category of the abelian category of A-modules for a ring A. Its cohomology modules define the sheaf cohomology of a (complex of) OX -module F by H i (X, F ) := H i RΓ(X, F ), i ∈ Z. © Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_6

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21 Cohomology of OX -modules

The chapter develops the theory in reverse order as stated above. We start by proving that the abelian category of OX -modules is a Grothendieck abelian category which in particular ensures that all additive functors defined on it can be right derived (Corollary F.187). Then we study cohomology and, more generally, right derived image. Next we introduce ˇ the formalism of Cech cohomology, which is sometimes easier to compute than sheaf ˇ cohomology, and we explain the relation between Cech cohomology and sheaf cohomology. The next two parts of the chapter define the other three functors and study formal relations between the four functors. In other words, they develop the “four functor formalism”. We conclude the chapter with defining and studying finiteness conditions for objects in D(X). Everything in this chapter is quite formal and will be done for arbitrary ringed spaces 1 . Only in the following chapters, we will prove deeper results that require the ringed spaces to be schemes such as the projection formula or compatibility of cohomology with base change. Hence we suggest to start with the definition of cohomology and derived direct image and then to come back to the other constructions, definitions, and results only later when they are needed. Ringed spaces and rings Let A be a ring. Every construction or property P defined for complexes of OX -modules for an arbitrary ringed space (X, OX ) yields the notion of property P for complexes of A-modules as follows. The category of complexes of A-modules is isomorphic to the category of complexes of OX -modules, where (X, OX ) is the ringed space consisting of a point with Γ(X, OX ) = A. This ringed space is sometimes denoted by (∗, A). Then we can identify the category of A-modules and the category of OX -modules via taking global sections over X. We obtain in particular an identification of their derived categories, D(A) and D(X). Hence one defines a complex of A-modules E to have property P if E considered as a complex of OX -modules has the property P. Similarly, a ring homomorphism φ : A → B corresponds to a morphism of ringed spaces (∗, B) → (∗, A), again denoted by φ. More precisely, attaching to a ring A the ringed space (∗, A) defines a contravariant fully faithful functor from the category of rings into the category of ringed spaces. Then pullback becomes the base change functor M 7→ φ∗ (M ) = B ⊗A M , and pushforward becomes the functor N 7→ φ∗ N of restriction of scalars, i.e., of viewing a B-module as an A-module via φ. In this way, all definitions and constructions for ringed spaces specialize to those for rings. When we introduce a notion for ringed spaces (or morphisms of them) we will always define the same notion for rings (or ring homomorphisms) in this manner. Of course, there is a different definition for an A-module or a complex of A-module to have a property P defined for modules or complexes of modules over arbitrary ringed ˜ of quasi-coherent OS -modules spaces: One could define that the associated complex E should have property P, where S = Spec A. This will never be our definition if not explicitly stated otherwise. Nevertheless, we will see that this often (but not always) gives the same notion. Let us give some examples. (1) For the properties “of finite type”, “of finite presentation”, and “flat” of an A-module E we have seen that E has this property if and only if the quasi-coherent OS -module ˜ has this property (Proposition 7.26 and Remark 7.39). E 1

In fact, almost all of the results generalize to modules over arbitrary ringed topoi, often with the same proofs.

153 (2) Let E be a complex of A-modules. Then E is K-flat (see Definition 21.91 below) if ˜ is K-flat (Lemma 22.39). and only if E (3) For the properties “perfect”, “pseudo-coherent”, and “of finite tor dimension” defined ˜ below, we will see in Lemma 22.44 that E has one of these property if and only if E has the same property. (4) But note that if E is K-injective (Definition F.179), even if E is concentrated in ˜ is not a K-injective object in degree 0 and hence an injective A-module, then E D(Spec A) in general (see Caveat 22.77 (3) below). Notation If X is a topological space, we denote by (PSh(X)) the category of presheaves (of sets) on X and by (Sh(X)) the category of sheaves of sets on X. For a ringed space (X, OX ) let (OX -Mod) be the abelian category of OX -modules. We work in this general setting because this allows us to treat the case OX = Z (the constant sheaf), the case that X consists of a single point and OX (X) = A for a given ring A, and the case of a scheme simultaneously. In the first case, (OX -Mod) is the category of all abelian sheaves on X and in this case we also write (Ab(X)) instead of (OX -Mod). In the second case, (OX -Mod) can be identified with the category of A-modules. For more technical reasons, we will also consider the abelian category of presheaves of OX -modules which we denote by (P-OX -Mod). Objects are presheaves F of abelian groups together with a scalar multiplication OX × F → F such that F (U ) is an OX (U )module for all U ⊆ X open. As a special case we obtain the category (PAb(X)) of abelian presheaves on X. We denote by C(X) := C(OX -Mod) the category of complexes of OX -modules (Section (F.14)), by K(X) := K(OX -Mod) the category of complexes up to homotopy (Section (F.15)), and by D(X) := D(OX -Mod) the derived category of the category of OX -modules (Section (F.37)). For ∗ ∈ {b, +, −} we denote by C ∗ (X), K ∗ (X), and D∗ (X) the full subcategories of (left or right) bounded complexes. We also identify D∗ (X) with the full subcategory of D(X) of (left or right) cohomologically bounded complexes (Proposition F.154). Similarly, if R is a ring we write C ∗ (R), K ∗ (R), and D∗ (R) instead of C ∗ (R-Mod), K ∗ (R-Mod), and D∗ (R-Mod) for ∗ ∈ {∅, +, −, b}.

Categories of abelian sheaves and of OX -modules Our goal in the next sections is to prove that the category (OX -Mod) of modules on a ringed space (X, OX ) is a Grothendieck abelian category (Definition F.54). For this one has to show that direct sums exist (which we have already seen in Section (7.4)), that filtered colimits are exact (which is easily seen on stalks using that filtered colimits in the category of modules over a ring are exact), and that (OX -Mod) has a generator. We will see that a system of generators is given by the proper direct images of the structure sheaves of open subsets. Therefore we first study the functor of proper direct image for inclusions of locally closed subspaces.

21 Cohomology of OX -modules

154 (21.1) Sheaves of sections with proper support.

Let X be a topological space and let F be a presheaf of abelian groups on X. Recall that we define for s ∈ F (U ), U ⊆ X open, the support Supp(s) = { x ∈ U ; sx ̸= 0 }. Then Supp(s) is always closed in U . Definition 21.1. Let X be a topological space and let W ⊆ X be a locally closed subspace. Let j : W → X be the inclusion. For a sheaf G of abelian groups on W we define a sheaf j! G on X by j! G (U ) := { s ∈ G (W ∩ U ) ; Supp(s) is closed in U } and call it the proper direct image of G . Clearly, this construction is functorial and we obtain a functor j! : (Ab(W )) −→ (Ab(X)). Remark 21.2. Let j : W → X be the inclusion of a locally closed subspace W of a topological space X. Let G be a sheaf of abelian groups on W . (1) By definition j! G is a subsheaf of j∗ G . If W is closed in X, then j! G = j∗ G . (2) For x ∈ X one has ( Gx , if x ∈ W ; (21.1.1) (j! G )x = 0, if x ∈ X \ W . In particular, j! is an exact functor and fully faithful (Proposition 2.23). (3) We have (21.1.2)

j −1 j! G = G .

If F is an abelian sheaf on X with Fx = 0 for x ∈ X \ W , then the adjunction morphism F → j∗ j −1 F factors through an isomorphism (21.1.3)

∼

F → j! j ∗ F .

This shows that the functor j! induces an equivalence between (Ab(W )) and the full subcategory of (Ab(X)) consisting of sheaves F with Fx = 0 for all x ∈ X \ W . The inverse functor is induced by j −1 . Remark 21.3. Let X be a topological space, let Z ⊆ X be a closed subspace and let U = X \ Z be its complement. Denote by i : Z → X and j : U → X the inclusions. Then one has for every abelian sheaf F on X a functorial exact sequence 0 −→ j! j −1 F −→ F −→ i∗ i∗ F → 0, where the morphisms are given by adjunction. The exactness is seen on stalks. If (X, OX ) is a ringed space, W is open in X, and G is an OW -module, then j! G is an OX -submodule of j∗ G and we obtain a functor j! : (OW -Mod) −→ (OX -Mod). There is a general construction of a proper direct image with respect to an arbitrary continuous map (Exercise 21.1) which we will not use.

155 We will mainly use the case that j is an open immersion. Then the functor j! can alternatively be described as the extension by zero as follows. For this we define the following functor of presheaves. Let X be a topological space and let j : U → X be the inclusion of an open subspace. Let G be a presheaf of abelian groups on U and define a presheaf j!p G of abelian groups on X by ( G (V ), if V ⊆ U ; p , V ⊆ X open. (21.1.4) (j! G )(V ) := 0, otherwise We obtain a functor j!p : (PAb(U )) → (PAb(X)) from the category of abelian presheaves on U to the category of abelian presheaves on X. If (X, OX ) is a ringed space, then j!p induces a functor j!p : (P-OU -Mod) → (P-OX -Mod). Proposition 21.4. Let X be a topological space and let j : U → X be the inclusion of an open subspace. (1) Let G be a sheaf of abelian group on U . Then there is a functorial isomorphism ∼

(j!p G )# −→ j! (G ), where ( )# denotes the sheafification functor. (2) The functor j! is left adjoint to j −1 : (Ab(X)) → (Ab(U )). Proof. By definition there is a functorial monomorphism of presheaves j!p G → j! G that induces an isomorphism on all stalks. This shows (1). For an abelian sheaf G on U and an abelian sheaf F on X one has functorial bijective maps p Hom(Ab(X)) (j! G , F ) ∼ = Hom(PAb(X)) (j! G , F ) ∼ = Hom(PAb(U )) (G , F |U ) = Hom(Ab(U )) (G , j −1 F ),

where the first bijection is due to (1). This shows (2). Remark 21.5. Let (X, OX ) be a ringed space, and let j : U → X be the inclusion of on open subspace. Then define the presheaf of OX -modules (21.1.5)

OUp := j!p (OX |U )

and its sheafification (21.1.6)

OU ⊆X := j! (OX |U ).

Then for every presheaf of OX -modules F , evaluation in 1 ∈ OUp (U ) = OX (U ) yields a functorial isomorphism of OX (U )-modules (21.1.7)

∼

Hom(P-OX -Mod) (OUp , F ) −→ F (U ).

If F is a sheaf, then by the universal property of the sheafification we also obtain an isomorphism (21.1.8)

∼

Hom(OX -Mod) (OU ⊆X , F ) −→ F (U )

which is functorial in F . Both these isomorphisms are compatible with restriction to smaller open subsets of X.

156

21 Cohomology of OX -modules

(21.2) Categories of sheaves on a ringed space. We aim to show that (OX -Mod) and (P-OX -Mod) are Grothendieck abelian categories (Definition F.54). By the following remark, it suffices to show that (P-OX -Mod) is a Grothendieck abelian category. Remark 21.6. The abelian category (OX -Mod) is a Giraud subcategory (Definition F.60) of (P-OX -Mod). Indeed, the inclusion functor ι : (OX -Mod) → (P-OX -Mod) has a left adjoint functor given by sheafification. It remains to show that sheafification is an exact functor. A sequence 0 → F ′ → F → F ′′ → 0 of presheaves of OX -modules is exact in (P-OX -Mod) if and only if for all U ⊆ X open the sequence 0 → F ′ (U ) → F (U ) → F ′′ (U ) → 0 is exact. As filtered colimits are exact in the category of abelian groups, this implies that the sequence of stalks 0 → Fx′ → Fx → Fx′′ → 0 is exact for all x ∈ X. Hence sheafification preserves stalks and one can check exactness of a sequence of OX -modules on stalks, sheafification is an exact functor. Proposition 21.7. The categories (OX -Mod) and (P-OX -Mod) are Grothendieck abelian categories. In particular (Theorem F.185), for every complex F of OX -modules there exists a quasi-isomorphism F → I , where I is K-injective with injective components. If there exists N ∈ Z such that H n (F ) = 0 for all n < N , then I can be chosen such that I n = 0 for all n < N . By our definition, the structure sheaf OX of a ringed space is always a sheaf of commutative rings. But the proposition also holds for categories of (pre-)sheaves of left OX -modules if OX is a sheaf of not necessarily commutative rings (with the same proof). Proof. As (OX -Mod) is a Giraud subcategory of (P-OX -Mod), it suffices to show that (P-OX -Mod) is a Grothendieck abelian category (Proposition F.61). Direct sums and filtered colimits in (P-OX -Mod) are formed section-wise, hence they exist and are exact. It remains to show that (P-OX -Mod) has a generator. We use Remark F.55. For U ⊆ X consider the presheaf of OX -modules OUp (21.1.5). If α : F → G is a non-zero morphism and s ∈ F (U ) is a section over some open set U such that α(s) ̸= 0, this section s corresponds to a morphism OUp → F (21.1.7) whose composition with α is non-zero. We conclude by Remark F.55. Remark 21.8. The proof shows that the family of presheaves OUp , for U ⊆ X open, forms a system of generators of (P-OX -Mod). Moreover, Proposition F.61 shows that their sheafifications OU ⊆X form a system of generators of (OX -Mod). This is also easily seen directly as in the proof of Proposition 21.7. Hence the same proof allows to show directly that (OX -Mod) is a Grothendieck abelian category without invoking the formalism of Giraud subcategories. In fact, the argument in the proof shows that the sheaves OU ⊆X where U runs through a fixed basis of the topology already generate (OX -Mod). Proposition 21.7 implies that all functors on (OX -Mod) are right derivable by Corollary F.187: Corollary 21.9. Let B be an abelian category, and let F : (OX -Mod) → B be an additive functor. We also denote by F the induced functor K(X) = K(OX -Mod) → D(B).

157 (1) Then F admits a right derived functor RF : D(X) → D(B), and for every complex F of OX -modules one has RF (F ) = F (IF ), where F → IF is a quasi-isomorphism to a K-injective complex of OX -modules. (2) The restriction of F to K + (X) also admits a right derived functor R+ F : D+ (X) → D+ (B), and R+ F is the restriction of RF to D+ (X). The second assertion follows from Corollary F.178. Again by Theorem F.185, Condition (Ac) (see F.198) is satisfied for every additive functor F : (OX -Mod) → B of abelian categories (Example F.199). Hence we obtain the following version of Corollary 21.9 in the language of the higher derived functor Ri F = H i ◦ RF : (OX -Mod) → B. Corollary 21.10. Let B be an abelian category, and let F : (OX -Mod) → B be a left exact functor. (1) The higher right derived functors (Ri F )i≥0 exist and form a universal δ-functor with R0 F = F . In particular, if 0 → F ′ → F → F ′′ → 0 is a short exact sequence of OX -modules, then one has a long exact sequence (21.2.1)

0 → F (F ′ ) → F (F ) → F (F ′′ ) → R1 F (F ′ ) → R1 (F ) → R1 F (F ′′ ) → R2 F (F ′ ) → . . .

(2) An OX -module A is right F -acyclic if and only if Ri F (A ) = 0 for all i > 0. Every injective OX -module is right F -acyclic. Proof. By Remark F.200 one has Ri F = 0 for i < 0 and R0 F = F because F is left exact. Injective OX -modules are right F -acyclic for every additive functor F (Example F.203). Hence the remaining assertions follow from Proposition F.204. (21.3) Restriction to open subsets. Let (X, OX ) be a ringed space and let j : U → X be the inclusion of an open subset. As the functor j −1 from the category of OX -modules to the category of OU -modules is exact, it induces a functor j −1 : D(X) → D(U ) which we usually denote by F 7→ F |U . Remark 21.11. Let (Ui )i∈I be an open covering of X. (1) Let F be a complex of OX -modules. If F |Ui = 0 in D(Ui ) for all i, then F = 0 in D(X). Indeed, F = 0 if and only if H p (F ) = 0 for all p ∈ Z. Now we use that the formation of H p (F ) commutes with restriction to open subsets. (2) Let v : F → G be a morphism in D(X). As being an isomorphism can be checked after applying H p ( ) (Remark F.151 (2)), v is an isomorphism if and only if v |Ui is an isomorphism in D(Ui ) for all i. (3) In general, there exist non-zero morphisms in D(X) whose restrictions to each Ui are zero (Exercise 22.1). Lemma 21.12. Let (X, OX ) be a ringed space. (1) Let U ⊆ X be an open subset. If I is a K-injective complex of OX -modules (resp. an injective OX -module), then I |U is a K-injective complex of OX |U -modules (resp. an injective OX |U -module). (2) Conversely, let I be an OX -module and let (Ui )i be an open covering such that I |Ui is an injective OUi -module for all i. Then I is an injective OX -module.

158

21 Cohomology of OX -modules

Proof. Let j : U → X be the inclusion such that j −1 F = F |U . The functor j −1 has an exact left adjoint functor, namely j! (Proposition 21.4 (2)). This shows the first assertion by Corollary F.183. Let us show the second assertion. The OX -modules OU ⊆X for U open and contained in some Ui form a system of generators of (OX -Mod) (Remark 21.8). Hence it is enough to show that for every monomorphism G → OU ⊆X for U contained in some Ui every map u : G → I can be extended to OU ⊆X (Proposition F.63). Denote by j : U → X the inclusion. As Gx = 0 for x ∈ X \ U , we have G = j! G ′ for some OU -module G ′ and u corresponds to a map G ′ → I |U that can be extended by hypothesis and by (1) to a map OU → I |U or, equivalently, to a map OU ⊆X → I .

Cohomology and derived direct image By now we have seen (Corollary 21.9) that we can right derive all additive functors defined on the category of OX -modules for an arbitrary ringed space (X, OX ). In particular we obtain the right derived functors of the direct image functor and the global section functor, i.e., higher direct images and cohomology, respectively. The right derivations of both functors determine each other. In fact, the global section functor can be viewed as a very special case of the direct image functor (Remark 21.25 below). Conversely, higher direct images can be calculated via cohomology (Proposition 21.27 below). By definition one can calculate the cohomology of an OX -module, or even of a whole complex of OX -modules, F by choosing a quasi-isomorphism to a Γ(X, −)-acyclic complex A and applying the global section functor to A . Here one can take for A for instance a K-injective complex as K-injective complexes are acyclic for all additive functors. But usually it is difficult to have a good description of A in terms of F . Hence we will define a different class of Γ(X, −)-acyclic complexes, namely bounded below complexes consisting of flasque sheaves. These complexes are also acyclic for the direct image functor and every OX -module has a rather concrete resolution by flasque sheaves (the Godement resolution, Remark 21.31). Moreover, we can use flasque sheaves to show a number of results (such as that H 1 defined via derived functors coincides with the definition of H 1 via torsors given in Volume I, the Leray spectral sequence, or the Mayer-Vietoris sequence). In the next part of this chapter, we will develop even better techniques to give concrete calculations ˇ of some cohomology groups using Cech cohomology. In Section (21.10) we will apply the techniques developed so far to compute some cohomology groups of line bundles on the projective line. The awkwardness of the arguments there is also thought as a motivation to develop the theory further in the following chapters. The remainder of this part of the chapter is devoted to some further general results about cohomology. We study the question under what hypotheses cohomology and higher direct images commute with filtered colimits. Although this might seem to be a rather technical question, it will be very useful in the sequel, for instance for the proof of the projection formula in Section (22.19) or in Chapter 25 on Grothendieck duality. We also state a general vanishing result of sheaf cohomology in degrees > dim(X) if X is a spectral space, i.e., the underlying topological space of a qcqs scheme, and prove this if the topological space X is noetherian (a result due to Grothendieck). We conclude by

159 defining cohomology with support in a closed subset. For us this will be only a technical tool and we show only some very basic properties about it. (21.4) Cohomology as derived functor. Let (X, OX ) be a ringed space and set A := Γ(X, OX ). Consider the global section functor Γ := Γ(X, −) : (OX -Mod) → (A-Mod). This is a left exact functor of abelian categories. In the special case that OX is the constant sheaf ZX the category (OX -Mod) is the category of abelian sheaves on X. In this case we also set A := Z and consider Γ(X, −) as a functor from the category of abelian sheaves to the category of abelian groups. The right derived functor is denoted by RΓ(X, −) : D(X) → D(A). It exists by Corollary 21.9. Definition 21.13. Let F be a complex of OX -modules. For i ∈ Z, the i-th cohomology of X with coefficients in F or simply the i-th cohomology of F is the A-module H i (X, F ) := Ri Γ(X, F ). This definition can in particular be applied to an OX -module F which we always consider as a complex of OX -modules concentrated in degree 0. If one considers H i (X, F ) for a complex F not concentrated in a single degree, then i H (X, F ) is often also called the i-th hypercohomology of F . We will not use this terminology. Caveat 21.14. For a complex F of OX -modules one has to distinguish between H i (F ) = Ker(F i → F i+1 )/ Im(F i−1 → F i ), which is an OX -module, and H i (X, F ), which is an A-module. Moreover, usually one has Γ(X, H i (F )) ̸= H i (X, F ). For instance, if F is concentrated in degree 0, then Γ(X, H i (F )) = 0 for all i ̸= 0 but very often H i (X, F ) ̸= 0 for some i > 0. The general results on derived functors collected in Appendix F specialize to the current situations as follows. Remark 21.15. For any complex F of OX -modules, we can compute H i (X, F ) as follows by Theorem F.173: Choose a quasi-isomorphism F → AF to a right Γ-acyclic complex AF . For instance, AF can be chosen to be a K-injective complex. Then

i−1 i H i (X, F ) = Ker Γ(X, AiF ) → Γ(X, Ai+1 F ) / Im Γ(X, AF ) → Γ(X, AF ) . If F is a single OX -module, such a quasi-isomorphism can be chosen to be an exact sequence of OX -modules 0 → F → A0F → A1F → . . . , where for all n ≥ 0 the OX -module AnF is right Γ-acyclic, i.e., H i (X, AFn ) = 0 for all i > 0. In Section (21.7) below we will give an example of a useful class of right Γ-acyclic OX -modules.

21 Cohomology of OX -modules

160

Remark 21.16. Let F be a complex of OX -modules. (1) If F is an OX -module, considered as a complex concentrated in degree 0, then one has H 0 (X, F ) = Γ(X, F ) (Remark F.200 (3)). (2) If there exists n ∈ Z with H i (F ) = 0 for all i < n, then H i (X, F ) = 0 for all i < n (Remark F.200 (1)). (3) Considered as functors H i (X, −) : (OX -Mod) → (A-Mod) the family (H i (X, −))i≥0 forms a universal δ-functor by Remark F.205. (4) Every short exact sequence of complexes of OX -modules 0 → F → G → H → 0 yields in D(X) a distinguished triangle (F.37.3) and hence by Remark F.167 there is a long exact cohomology sequence, functorial in the short exact sequence (21.4.1) . . . −→ H i (X, F ) −→ H i (X, G ) −→ H i (X, H ) −→ H i+1 (X, F ) −→ . . . (5) As the derived functor RΓ is triangulated and in particular preserves shifts, one has for all i, n ∈ Z (21.4.2)

H i (X, F [n]) = H i+n (X, F ).

Every complex F of OX -modules can also be considered as a complex F ab of abelian sheaves. Below (Corollary 21.43 for bounded below complexes, and Corollary 21.116 in general) we will see that the underlying abelian group of the A-module H i (X, F ) is the group H i (X, F ab ) for all i. There is a different way to express cohomology, namely via Ext groups (Section (F.52)) as we explain now. Remark 21.17. Let U ⊆ X be an open subset. Recall from Remark 21.5 that for every OX -module F there is a isomorphism of Γ(U, OX )-modules ∼

Hom(OX -Mod) (OU ⊆X , F ) −→ Γ(U, F ) which is functorial in F and compatible with restriction to smaller open subsets of X. Hence we obtain a triangulated isomorphism of derived functors (21.4.3)

∼

R HomOX (OU ⊆X , −) −→ RΓ(U, −).

Using H i ◦ R Hom = Exti (F.52.5), we get isomorphisms (21.4.4)

∼

ExtiOX (OU ⊆X , F ) −→ H i (U, F ).

which are functorial in the complex F of OX -modules. (21.5) Cohomology and restriction to open subspaces. Let (X, OX ) be a ringed space and let U ⊆ X be open. If F is a complex of OX -modules, we can restrict each component of F to U and obtain a complex F |U of (OX |U )-modules. Lemma 21.18. Denote by RΓ(U, −) the right derived functor of the functor (OX -Mod) → (Γ(U, OX )-Mod), F 7→ F (U ). Then we have for every complex of OX -modules F a functorial isomorphism (21.5.1)

∼

RΓ(U, F |U ) −→ RΓ(U, F ).

In particular H i (U, F |U ) = H i (U, F ) for all i ∈ Z. In the sequel will usually identify RΓ(U, F |U ) and RΓ(U, F ).

161 Proof. Let F be a complex of OX -modules and let F → I be a quasi-isomorphism to a K-injective complex. Then RΓ(U, F |U ) = Γ(U, I |U ) = Γ(U, I ) = RΓ(U, F ). where the first equality holds by Lemma 21.12. Remark 21.19. Let (X, OX ) be a ringed space and let v : F → G be a morphism in D(X). Let B be a basis of open subsets of X. Then v is an isomorphism (resp. v = 0) if and only the induced map RΓ(U, F ) → RΓ(U, G ) is an isomorphism (resp. is zero) for all U ∈ B. Indeed, the condition is clearly necessary. To show that it is sufficient we may replace F and G by in D(X) isomorphic objects and can assume that both are K-injective. But then RΓ(U, F ) = F (U ) and RΓ(U, G ) = G (U ) and v can be represented by a morphism in K(X) (Prop. F.179 (7)). This is an isomorphism (resp. is zero) if F (U ) → G (U ) is an isomorphism (resp. is zero) for U in B. Remark 21.20. Let (X, OX ) be a ringed space. For U ⊆ V ⊆ X open, restriction of sections defines a morphism of additive functors Γ(V, −) → Γ(U, −). Hence we obtain a morphism of triangulated functors RΓ(V, −) → RΓ(U, −) (Remark F.166) and in particular for all i ∈ Z homomorphisms of Γ(V, OX )-modules H i (V, F ) −→ H i (U, F )

(21.5.2)

functorial in complexes F of OX -modules. In particular we obtain a presheaf H i (F ) : U 7→ H i (U, F ) of OX -modules. These presheaves are in fact the higher derived functors of some functor: Remark 21.21. Consider the inclusion ι from the category of OX -modules to the category of presheaves of OX -modules. This is a left exact functor. For all i ∈ Z and every complex F of OX -modules there is an isomorphism (21.5.3)

∼

Ri ι(F ) −→ H i (F ).

of cohomological functors in F . Indeed, choose a K-injective resolution F → I . Then one has for U ⊆ X open (∗)

Ri ι(F )(U ) = H i (I )(U ) = H i (I (U )) = H i (U, F ), where for (∗) it is important to note that one takes cohomology as presheaves. Lemma 21.22. Let X be a ringed space, let F be a complex of OX -modules, and let n ∈ Z. Then the sheafification of the presheaf H n (F ) is the n-th cohomology sheaf H n (F ) of F . Proof. Let F → I be a K-injective resolution. For all U ⊆ X open one has by Lemma 21.18 H n (U, F ) = H n (. . . → I i−1 (U ) → I i (U ) → I i+1 (U ) → . . .). Hence the sheafification of U 7→ H n (U, F ) is given by H n (I ) = H n (F ).

21 Cohomology of OX -modules

162

Corollary 21.23. Let X be a ringed space and let F be a complex of OX -modules. Let n ∈ Z such that H n (F ) = 0 and let ξ ∈ H n (X, F ). Then there exists an open covering (Ui )i of X such that ξ |Ui = 0 for all i. (21.6) Higher direct images as derived functors. The higher direct image functors provide relative versions of cohomology groups. Let f : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. Then the direct image is a left exact functor f∗ : (OX -Mod) −→ (OY -Mod). As all additive functors on (OX -Mod) it has a right derived functor (Corollary 21.9). Definition 21.24. The right derived functor Rf∗ : D(X) → D(Y ) of the direct image functor f∗ is called derived direct image functor. If F is a complex of OX -modules, the OY -module Ri f∗ F (i ≥ 0) is called the i-th higher direct image of F (under f ). Remark 21.25. Taking cohomology is in fact a special case of forming higher derived images: Let (X, OX ) be a ringed space and consider the ringed space (∗, Γ(X, OX )) consisting of one point ∗ with structure sheaf given by O∗ (∗) := Γ(X, OX ). Modules over this ringed space are simply Γ(X, OX )-modules. If we denote by f : X → ∗ the obvious morphism of ringed spaces, then f∗ F = Γ(X, F ) for every OX -module F . Therefore Rf∗ F = RΓ(X, F ) and Ri f∗ F = H i (X, F ) for every complex F of OX -modules and for all i ∈ Z. Remark 21.26. Let f : X → Y be a morphism of ringed spaces and let F be a complex of OX -modules. (1) Every exact sequence 0 → F ′ → F → F ′′ → 0 of complexes of OX -modules yields a long exact sequence of higher direct images (21.6.1)

. . . −→ Ri−1 f∗ F ′′ −→ Ri f∗ F ′ −→ Ri f∗ F −→ Ri f∗ F ′′ −→ . . .

(2) Suppose there exists n ∈ Z such that H i (F ) = 0 for all i < n. Then Ri f∗ F = 0 for all i < n (Remark F.200 (1)). (3) Let F be an OX -module (as always considered as a complex concentrated in degree 0). Then Ri f∗ F = 0 for all i < 0 by (2) and R0 f∗ F = f∗ F (Remark F.200 (3)). Proposition 21.27. Let f : X → Y be a morphism of ringed spaces, and let F be a complex of OX -modules. For all i ∈ Z, the OY -module Ri f∗ F is the sheaf associated to the presheaf V 7→ H i (f −1 (V ), F ), V ⊆ Y open (with restriction maps given by (21.5.2)). Proof. Consider the commutative diagram of functors (OX -Mod)

/ (P-OX -Mod)

ι

f∗p

f∗

 (OY -Mod) o

(−)

#

 (P-OY -Mod),

163 where ι denotes the inclusion, f∗p the direct image of presheaves of OX -modules, and (−)# the sheafification functor. Note that f∗p and (−)# are both exact functors. Hence we obtain by Remark F.175 isomorphisms of triangulated functors Rf∗ = R((−)# ◦ f∗p ◦ ι) ∼ = (−)# ◦ f∗p ◦ Rι. Applying H i one obtains the claim by Remark 21.21. Derived direct image also commutes with restriction to open subsets: Lemma 21.28. let f : X → Y be a morphism of ringed spaces, let V ⊆ Y be open and denote by fV : f −1 (V ) → V the restriction of f . Then there is for all complexes F of OX -modules a functorial isomorphism in D(V ) ∼

Rf∗ F |V −→ R(fV )∗ (F |f −1 (V ) ). Proof. Let F → IF be a K-injective resolution. As (IF )|f −1 (V ) is again K-injective (Lemma 21.12) we find Rf∗ (F )|V = f∗ (IF )|V = (fV )∗ (IF )|f −1 (V ) = R(fV )∗ (F |f −1 (V ) ). (21.7) Flasque Sheaves. In this section we will study “flasque sheaves” and we will show that they are right Γ-acyclic. This yields a – somewhat – more concrete way to calculate cohomology via flasque sheaves. Definition 21.29. Let X be a topological space. A sheaf F on X is called flasque or flabby, if all restriction maps Γ(U, F ) → Γ(V, F ) for V ⊆ U ⊆ X open, are surjective. Remark 21.30. (1) Let f : X → Y be a continuous map of topological spaces and let F be a flasque sheaf on X. Then f∗ F is a flasque sheaf on Y . (2) Let X be an irreducible topological space, and let A be a set. Then the constant sheaf A attached to A is flasque. Indeed, since X is irreducible, all non-empty open subsets are connected. Hence the presheaf U 7→ A (for ∅ = ̸ U ⊆ X open) is a sheaf, so this is the sheaf A. The restriction maps between the sections on non-empty open subsets of X are simply the identity maps idA . Remark 21.31. (Godement resolution) Q Let F be a sheaf on a topological space X. Define a sheaf F [0] on X by F [0] (U ) := x∈U Fx for U ⊆ X open, where the restriction maps are given by the projections. The sheaf F [0] is flasque and there is an injective morphism of sheaves ιF : F ,→ F [0] ,

F (U ) ∋ s 7→ (sx )x∈U ∈ F [0] (U ),

U ⊆ X open.

Every morphism φ : F → G of sheaves induces a morphism of flasque sheaves φ[0] : F [0] → Q [0] G [0] with φU = x∈U φx . We obtain a functor ( )[0] from the category of sheaves to the full subcategory of flasque sheaves. The morphism ιF is functorial in F .

21 Cohomology of OX -modules

164

Now suppose that F is an OX -module. Then F [0] is an OX -module and we obtain a functor ( )[0] from the category of OX -modules to the full subcategory of flasque OX -modules. The morphism ιF is a functorial homomorphism of OX -modules. Applying Lemma F.194 we find for every bounded below complex F of OX -modules a quasi-isomorphism F → GF , functorial in F , where GF is a bounded below complex whose components are flasque sheaves. This resolution of F is called the Godement resolution. Lemma 21.32. Let (X, OX ) be a ringed space. Every injective OX -module is flasque. Proof. Let I be an injective OX -module. By Remark 21.31 there exists an injective homomorphism of OX -modules i : I → G where G is flasque. As I is injective, this makes I into a direct summand of G . Hence I is flasque. Recall that for any sheaf G of (not necessarily abelian) groups we defined the notion of a G -torsor (Section (11.5)). We denoted by H 1 (X, G ) the pointed set of isomorphism classes of G -torsors. If G is an abelian sheaf, then H 1 (X, G ) has the structure of an abelian group. Below (Proposition 21.40) we will show that this group is functorially isomorphic to the cohomology H 1 (X, G ) defined via derived functors. Until then, we will 1 denote the first cohomology defined via torsors by HTors (X, G ). Lemma 21.33. Let X be a topological space and let G be a flasque sheaf of groups on X. Then every G -torsor is trivial. Proof. Let T be a G -torsor, let (U S i )i∈I be an open covering such that there exist ti ∈ T (Ui ) for all i. For J ⊆ I set UJ := i∈J Ui . Then UI = X. Define E := { (t, J) ; J ⊆ I, t ∈ T (UJ ) }. Then E ̸= ∅ because (∗, ∅) ∈ E. It is partially ordered by (t, J) ≤ (t′ , J ′ ) if J ⊆ J ′ and t′ |UJ = t. As T is a sheaf, every totally ordered subset of E has an upper bound in E. Hence there exists a maximal element (t, J) in E by Zorn’s lemma. It suffices to show that J = I. Assume there exists i ∈ I \ J and let g ∈ G (UJ ∩ Ui ) with t|UJ ∩Ui = gti|UJ ∩Ui . As G is flasque, we can extend g to g˜ ∈ G (X). Replacing ti by g˜|Ui ti we may assume t|UJ ∩Ui = ti|UJ ∩Ui and hence we can glue t and ti to a section over UJ∪{i} . This contradicts the maximality of (t, J). Corollary 21.34. Let X be a topological space, and let 1

/F

α

/G

β

/H

/1

be an exact sequence of sheaves of groups on X. (1) Suppose that F is flasque. Then for every open subset U ⊆ X the sequence 1 → F (U ) → G (U ) → H (U ) → 1 is exact. (2) Suppose that F and G are flasque. Then H is flasque. Proof. To prove part (1), we may assume that U = X. By Proposition 11.14 one has the 1 exact sequence of non-abelian cohomology 1 → F (X) → G (X) → H (X) → HTors (X, F ) 1 and HTors (X, F ) = 1 by Lemma 21.33. By an easy diagram chase, we obtain part (2) as a consequence of part (1).

165 Proposition 21.35. Let (X, OX ) be a ringed space and let U ⊆ X be open. Then flasque OX -modules are right Γ(U, −)-acyclic, i.e., for all flasque OX -modules I and i > 0 we have H i (U, I ) = 0. Moreover, for every bounded below complex F of OX -modules there exists a quasi-isomorphism F → AF , where AF is a bounded below complex whose components are flasque OX -modules. One can choose AF to depend functorially on F , for instance the Godement resolution (Remark 21.31). Proof. As restrictions of flasque sheaves to open subsets are again flasque, we can assume that U = X. We can apply Proposition F.207: Condition (a) there is satisfied by Remark 21.31, Condition (b) is clear, and Condition (c) is satisfied by Corollary 21.34. Below, we will use the following technical remark. Remark 21.36. In some cases, we can enlarge the class of flasque sheaves slightly: Let X be a topological space such that the set B of open quasi-compact subsets of X is a basis of the topology of X that is stable under finite intersections (in particular X is quasi-compact). Let us call a sheaf F in X quasi-flasque if F (X) → F (U ) is surjective for all U ∈ B. Then the same proof as in Lemma 21.33, using only finite coverings (Ui )i with Ui ∈ B for all i, shows that for a quasi-flasque sheaf of groups G every G -torsor is trivial. Then the same arguments as above show that bounded below complexes of quasi-flasque OX -modules are right Γ-acyclic. Corollary 21.37. Let f : X → Y be a morphism of ringed spaces. Then flasque OX modules are right f∗ -acyclic. Proof. By Proposition F.204 we have to show that Ri f∗ F = 0 for all flasque OX -modules F and all i > 0. This follows by Proposition 21.27 from Proposition 21.35. Hence we can calculate cohomology and, more generally (Remark 21.25), higher direct images of bounded below complexes by flasque resolutions: Upshot 21.38. Let X be a ringed space and let F be a bounded below complex of OX -modules. Let F → AF be any quasi-isomorphism where AF is a bounded below complex with flasque components (these exist, for instance the Godement resolution). Then for all i ∈ Z one has
Ker Γ(X, AiF ) → Γ(X, Ai+1 F ) i
. H (X, F ) = i Im Γ(X, Ai−1 F ) → Γ(X, AF ) More generally, if f : X → Y is a morphism of ringed spaces, then for all i ∈ Z one has
Ker f∗ AiF → f∗ Ai+1 F
R i f∗ F = . i Im f∗ Ai−1 F → f∗ AF Example 21.39. Let k a field, let X := A1k be the affine line over k, and let U := A1k \{0, 1}. Denote by j : U ,→ X the inclusion. Let us compute the cohomology group H 1 (A1k , j! ZU ). Consider the exact sequence 0 → j! ZU → ZX → i0,∗ Z ⊕ i1,∗ ZX → 0

21 Cohomology of OX -modules

166

of abelian sheaves on X, where ix the inclusion Spec κ(x) ,→ X, and the homomorphism ZX → i0,∗ Z ⊕ i1,∗ Z is the direct sum of the natural homomorphisms. The exactness is easily checked on stalks. The sheaves ZX and i0,∗ Z ⊕ i1,∗ Z are flasque, hence this is an acyclic resolution of j! ZU which we can use to compute its cohomology. Passing to global sections of this resolution, we obtain a complex m7→(m,m) 0 −→ Z −−−−−−−−→ Z2 −→ 0 with Z sitting in degree 0. This shows that H 1 (X, j! ZU ) ∼ = Z and H n (X, j! ZU ) = 0 for all n ̸= 1. (21.8) Applications of flasque resolutions. Our main applications of the results of the previous section will be (1) that the H 1 defined via derived functors and the H 1 defined via torsors in Section (11.5) coincide (Proposition 21.40), ∼ (2) an isomorphism R(g∗ ◦ f∗ )F → (Rg∗ ◦ Rf∗ )F for bounded below complexes F (Proposition 21.41) and the corollary that the cohomology of a bounded below complex of OX -modules is the same as the cohomology of the underlying complex of abelian sheaves (Corollary 21.43), (3) the Leray spectral sequence (Corollary 21.45) as an application of (2), (4) and the Mayer-Vietoris sequence (Proposition 21.47) in the next section. Later (Proposition 21.115) we will prove (2) also for arbitrary complexes and the proof there will be independent of the result obtained here. Proposition 21.40. Let X be a topological space and let F be a sheaf of abelian groups which we may consider a ZX -module. Then there is a functorial isomorphism 1 H 1 (X, F ) ∼ (X, F ). = HTors 1 Hence from now on we will not distinguish between H 1 (X, F ) and HTors (X, F ).

Proof. By Remark 21.31 there exists a functorial exact sequence of sheaves of abelian groups 0 → F → A 0 → C → 0 with A 0 flasque. Then H 1 (X, A 0 ) = 0 by Proposi1 tion 21.35 and HTors (X, F 0 ) = 0 by Lemma 21.33. Hence the long exact cohomology i 1 sequence for H (X, −) and the long exact sequence for HTors (X, −) imply that 1 H 1 (X, F ) = HTors (X, A ) = Coker(A 0 (X) → C (X)).

Proposition 21.41. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Then the canonical morphism R(g∗ ◦ f∗ ) → R(g∗ ) ◦ R(f∗ ) of derived functors D+ (X) → D+ (Z) is an isomorphism. Proof. We apply Proposition F.211. As every module over a sheaf of rings can be embedded into a flasque module, it suffices to show that if I is a flasque OX -module, then f∗ I is right g∗ -acyclic. But f∗ I is clearly a flasque OY -module and hence right g∗ -acyclic (Corollary 21.37). Let f : X → Y be a morphism of ringed spaces, and set A := Γ(X, OX ) and B := Γ(Y, OY ). The morphism f yields a ring homomorphism φ : A → B which allows to consider every B-module M as an A-module φ∗ (M ). This yields an exact functor φ∗ : (B-Mod) −→ (A-Mod).

167 By the definition of the direct image we have Γ(Y, −) ◦ f∗ = φ∗ ◦ Γ(X, −). Let Z be the ringed space consisting of one point ∗ with OZ (∗) = A and g : Y → Z the canonical morphism. Then we can identify g∗ = Γ(Y, −). Hence we deduce from Proposition 21.41: Corollary 21.42. Let f : X → Y be a morphism of ringed spaces. There is an isomorphism of functors D+ (X) → D+ (Γ(Y, OY )) φ∗ ◦ RΓ(X, −) ∼ = RΓ(Y, −) ◦ Rf∗ . We use that R(φ∗ ◦ Γ(X, −)) = φ∗ ◦ RΓ(X, −) because φ∗ is exact (Remark F.175). Applying Corollary 21.42 to the unique morphism of ringed spaces (X, OX ) → (X, ZX ) whose underlying topological map is the identity we deduce: Corollary 21.43. Let X be a ringed space and let F be a bounded below complex of OX modules. Denote by F ab the underlying complex of abelian sheaves. Then the underlying abelian group of the Γ(X, OX )-module H i (X, F ) is H i (X, F ab ) for all i ∈ Z. Below (Section (21.23)) we will prove by totally different techniques the assertions in Proposition 21.41 (and hence Corollaries 21.42 and 21.43) also for unbounded complexes. There is one important special case where we can do this right now. Remark 21.44. Let f : X → Y and g : Y → Z be continuous maps of topological spaces. Then the direct image functor f∗ : (Ab(X)) → (Ab(Y )) of abelian sheaves has an exact left adjoint, namely f −1 . Hence f∗ preserves K-injective complexes (Corollary F.183) and therefore the canonical morphism of functors D(Ab(X)) → D(Ab(Z)) R(g∗ ◦ f∗ ) −→ Rg∗ ◦ Rf∗ is an isomorphism (Proposition F.176). Applying this to the case that Z consists of a single point we obtain an isomorphism of functors D(Ab(X)) → D(Z) (21.8.1)

∼

RΓ(Y, −) ◦ Rf∗ −→ RΓ(X, −).

If f = i is the inclusion of a closed subspace, then i∗ is exact and we obtain by Remark F.177 for every complex F of abelian sheaves on X a functorial isomorphism (21.8.2)

∼

R(g |X )∗ F −→ Rg∗ (i∗ F ).

We can also apply this in the case where g is the map to a point such that we can identify g∗ and Γ(Y, −). Then (21.8.2) becomes an isomorphism in D(Z) (21.8.3)

∼

RΓ(X, F ) −→ RΓ(Y, i∗ F )

which is functorial for every complex F of abelian sheaves on X. Passing to cohomology we obtain for all p ∈ Z isomorphisms of abelian groups (21.8.4)

∼

H p (X, F ) −→ H p (Y, i∗ F ).

Corollary 21.42 and Proposition 21.41 also imply the existence of a Grothendieck spectral sequence (Proposition F.212):

21 Cohomology of OX -modules

168

Corollary and Definition 21.45. Let f : X → Y be a morphism of ringed spaces and let F be a bounded below complex of OX -modules. (1) There is a convergent spectral sequence of Γ(Y, OY )-modules, called the Leray spectral sequence for the morphism f , E2p,q = H p (Y, Rq f∗ F ) =⇒ H p+q (X, F )

(21.8.5)

which is functorial in F . (2) Let g : Y → Z be a morphism of ringed spaces. Then there is a convergent spectral sequence of OZ -modules E2p,q = Rp g∗ (Rq f∗ F ) =⇒ Rp+q (g ◦ f )∗ F

(21.8.6)

which is functorial in F . If N ∈ Z is an integer such that H n (F ) = 0 for all n < N , then both spectral sequences are concentrated in degrees q ≥ N and p ≥ 0. If f = idX , then Rf = idD(X) and hence Rq (idX )∗ F = H q (F ). Hence we obtain the following Corollary. Corollary 21.46. Let (X, OX ) be a ringed space and let F be a bounded below complex of OX -modules. Then there is a convergent spectral sequence of Γ(X, OX )-modules, sometimes called hypercohomology spectral sequence, (21.8.7)

E2p,q = H p (X, H q (F )) =⇒ H p+q (X, F )

which is functorial in F . If f : X → Y is a morphism of ringed spaces, then there is a convergent spectral sequence of OY -modules E2p,q = Rp f∗ H q (F ) =⇒ Rp+q f∗ F .

(21.8.8)

(21.9) Mayer-Vietoris sequences and gluing of complexes. As an application of Lemma 21.32, we obtain the following useful exact sequence. Proposition 21.47. (Mayer-Vietoris sequence) Let X be a ringed space and suppose that X = U ∪ V for two open subsets U, V ⊆ X. Let F be a complex of OX -modules. Then there exists a long exact cohomology sequence (21.9.1)

. . . −→H n (X, F ) −→ H n (U, F ) ⊕ H n (V, F ) −→ H n (U ∩ V, F ) −→H n+1 (X, F ) −→ . . . ,

which is functorial in F . Proof. By Proposition 21.7 there exists a quasi-isomorphism F → I , where I is a K-injective complex whose components are all injective and hence flasque OX -modules. Therefore the sequence of complexes (*)

s7→(s| U ,s| V )

(s,t)7→s| U ∩V −t| U ∩V

0 → I (X) −−−−−−−−−→ I (U ) ⊕ I (V ) −−−−−−−−−−−−−−−→ I (U ∩ V ) → 0

is exact. By Lemma 21.18 we have H n (W, F ) = H n (I (W )) for all W ⊆ X open. Hence taking cohomology of (*) gives the result by the Snake Lemma F.48.

169 If F is a bounded below complex, one could have worked in the proof with I the Godement resolution of F (and hence avoid the usage of Theorem 21.7). In this case the Mayer-Vietoris sequence is also a special case of a hypercohomology spectral sequence ˇ attached to the Cech double complex (see Exercise 21.10). Proposition 21.48. (Mayer-Vietoris triangle) Let X be a ringed space, and let U, V ⊆ X be open subspaces with U ∪ V = X. Let jU , jV , jU ∩V denote the inclusions of these open subspaces into X. (1) For every OX -module F , there is a short exact sequence 0 → F → jU,∗ (F|U ) ⊕ jV,∗ (F|V ) → jU ∩V,∗ (F|U ∩V ) → 0, of OX -modules, and these sequences are functorial in F . (2) There is an exact triangle (21.9.2)

F → RjU,∗ jU∗ F ⊕ RjV,∗ jV∗ F → RjU ∩V,∗ jU∗ ∩V F →

for every F ∈ D(X). Proof. For (1), note that the sheaf property of F immediately implies the exactness of the sequence 0 → F → jU,∗ (F|U ) ⊕ jV,∗ (F|V ) → jU ∩V,∗ (F|U ∩V ), in fact that sequence is even exact after passing to sections on any open subscheme of X. It therefore only remains to prove the surjectivity of the final arrow. This follows from PropositionS21.47 and the fact that for every element η ∈ H 1 (X, F ), there exists an open cover X = i Ui such that all restrictions η|Ui vanish (Corollary 21.23). Using part (1), part (2) follows by choosing a K-injective resolution of F and computing the derived pushforwards using the restrictions of this resolution to U , V and U ∩ V respectively. In fact, by Lemma 21.12 these restrictions are K-injective resolutions of the corresponding restrictions of F . Reverse engineering from (21.9.2) we obtain the following result about gluing of complexes. Proposition 21.49. We keep the notation of Proposition 21.48. Let FU ∈ D(U ) and ∼ FV ∈ D(V ) and let α : FU |U ∩V → FV |U ∩V be an isomorphism. Then there exists an object F ∈ D(X) and isomorphisms F |U ∼ = FU and F |V ∼ = FV . Proof. Set FU ∩V := (FU )U ∩V ∼ = (FV )|U ∩V . We have natural maps RjU ∗ FU −→ RjU ∩V ∗ FU ∩V , RjV ∗ FV −→ RjU ∩V ∗ FU ∩V , and taking the difference of these maps, a map RjU ∗ FU ⊕ RjV ∗ FV −→ RjU ∩V,∗ FU ∩V which we complete to an exact triangle +

F −→ RjU ∗ FU ⊕ RjV ∗ FV −→ RjU ∩V,∗ FU ∩V −→ in D(X). We have (RjU,∗ FU )|U ∼ = FU and the restriction of the map RjV,∗ FV → RjU ∩V ∗ (FU ∩V ) to U is an isomorphism. Hence the first map of the triangle induces F |U ∼ = FU . Likewise, we obtain F |V ∼ = FV .

21 Cohomology of OX -modules

170 (21.10) Cohomology groups — a first example.

Let k be an algebraically closed field, let k be an integral proper smooth curve over k, and let x ∈ C be a closed point. For any d ∈ Z we can consider the divisor d[x] on C and the associated line bundle O(d[x]). We want to investigate the cohomology groups H i (C, O(d[x])). Let K denote the constant sheaf associated with the field K(C) of rational functions on C. Since this is a constant sheaf, it is flasque. We can identify O(d[x]) as the subsheaf of K consisting of those sections which have a pole of order ≤ d at x and have no pole elsewhere. We obtain an embedding O(d[x]) ,→ K . Let us analyze the quotient K /O(d[x]). Its stalk at a point z ∈ C can be identified with the quotient Kz /O(d[x])z . If z is the generic point of C, then this quotient vanishes. ̸ x the stalk O(d[x])z can be identified, as a subring of Kz = K(C), For a closed point z = with OC,z . For z = x, the stalk O(d[x])x consists of all functions in K(C) which have a pole of order ≤ d at x. Since every rational function on C is regular at almost all closed points, we have a homomorphism from K /O(d[x]) into the direct sum M ιz,∗ Kz /O(d[x])z , z∈C(k)

the direct sum over all closed points of C, where we consider each stalk as a sheaf on the corresponding one-point space and denote by ιz the inclusion {z} ,→ C. In this way we obtain an exact sequence M ιz,∗ Kz /O(d[x])z → 0. 0 → O(d[x]) → K → z∈C(k)

The exactness can be checked L on stalks, where it is obvious. Since the sheaves K and z∈C(k) ιz,∗ Kz /O(d[x])z are flasque (use Example 21.30 and Lemma 21.52), we can use it to compute the cohomology groups H n (C, O(d[x])). We see immediately that H n (C, O(d[x])) = 0 for n > 2 (compare Grothendieck’s vanishing theorem, Theorem 21.57). Furthermore, we have an exact sequence M Kz /O(d[x])z → H 1 (C, O(d[x])) → 0. 0 → H 0 (C, O(d[x])) → K(C) → z∈C(k)

The global sections H 0 (C, O(d[x])) are those rational functions on C which are regular outside x and have a pole of order ≤ d at x; of course, this is part of the definition of the line bundle O(d[x]). So the interesting cohomology group in this case is the group H 1 (C, O(d[x])). The question whether H 1 (C, O(d[x])) = 0, or equivalently whether the map in the middle is surjective, is called the Cousin problem for C and O(d[x]). In the case C = P1 (C), d = 0, in the context of complex geometry, one can interpret the middle map as attaching to a meromorphic function on P1 (C) its principal parts in the (finitely many) poles of f . Then the positive solution of the Cousin problem is the Theorem of Mittag-Leffler. For an arbitrary algebraically closed field k and C = P1k (now again considered as a scheme), we can use the above sequence to compute all the cohomology groups H 1 (P1 , O(d)). See also Theorem 22.22 for a generalization to Pnk .

171 Proposition 21.50. With notation as above, for C = P1k we obtain  0 if d ≥ 0, dimk H 1 (P1 , O(d)) = −d − 1 if d < 0. Proof. In this case, K(C) = k(T ), the field of rational functions in one variable over k, and we need to compute the cokernel of the natural map M k(T ) −→ k(T )/k[T ](T −a) ⊕ k(T )/T d k[T −1 ](T −1 ) . a∈k

The right hand side is generated by the powers T i , i > d, over k. Considering the images of the elements (T −a)−i and T i for a ∈ k, i ≥ 0, we see that this map is surjective for d = −1, and a fortiori for d > −1. Similarly, for d < −1, the elements T i ∈ k(T )/T d k[T −1 ](T −1 ) for i = −1, . . . , d + 1 induce a k-basis of the cokernel. In particular, we see that O(−1) is the only line bundle on P1 which has vanishing H 0 and H 1 . Below in Corollary 25.129 we will prove Serre duality for a general proper smooth curve as an application of general Grothendieck duality. It implies that dimk H 1 (C, O(d[x])) = dimk H 0 (C, O(−d[x] + KC ), where KC is a canonical divisor (any divisor on C such that O(KC ) ∼ = Ω1C/k ). If C = P1k , then O(KC ) = O(−2) and hence dimk H 1 (P1k , O(d)) = 0 1 dimk H (Pk , O(−d − 2) which again shows Proposition 21.50 since we already computed dim H 0 (P1k , O(d)) = d + 1 in (11.14.5). Coming back to the general case, we find the following criterion when the curve C is isomorphic to the projective line. Compare Section (26.16). Proposition 21.51. In the above situation, we have H 1 (C, OC ) = 0 if and only if C∼ = P1k . Proof. We have already seen that H 1 (P1k , OP1k ) = 0. Conversely, if H 1 (C, OC ) = 0 then for any two points x0 ̸= x1 ∈ C(k) we find f ∈ K(C) such that div(f ) = [x1 ] − [x0 ]. We may consider f as a finite surjective morphism C → P1k , see Corollary 15.22. Let us show that this morphism is an isomorphism. By Theorem 15.21 it is enough to show that the inclusion K(P1k ) ⊆ K(C) of function fields given by f is an equality. It follows from the definition of f that f ∗ [0] = [x1 ], hence [K(C) : K(P1k )] = deg(f ) = 1 by Proposition 15.30. We will come back to the cohomology of curves in much greater detail in Chapter 26. (21.11) Compatibility with colimits. In this section we will give criteria when cohomology commutes with filtered colimits. This is not true in general, not even for global sections (see Exercise 7.1 considering a direct sum as a filtered colimit of finite direct sums). But it holds under the following hypothesis for a topological space X: (COH) The space X is quasi-compact and has a basis B consisting of quasi-compact open subsets that is stable under finite intersections2 . 2

Formally, the condition “X quasi–compact” is superfluous because we demanded that the empty intersection is part of the basis.

21 Cohomology of OX -modules

172

Such spaces are called coherent in [SGA4] O Exp. VI, 2.2. We will not use this terminology. If X satisfies (COH), then every open quasi-compact subspace of X satisfies (COH). Examples for spaces satisfying (COH) are underlying topological spaces of qcqs schemes (in particular, of affine schemes, and of noetherian schemes). Lemma 21.52. Let X be a topological space satisfying (COH). Let I be a filtered category and let F : I → (Sh(X)), i 7→ Fi , be a diagram of sheaves on X. Then the canonical map φ : colim Fi (X) −→ (colim Fi )(X) i

i

is bijective. Proof. (i). We first show that for every quasi-compact topological space the map φ is injective. Let s ∈ Fi (X) and s′ ∈ Fi′ (X ′ ) be sections that have the same image in the right hand side. As I is filtered, we find an open covering (Uα )α∈A of X, for all α an iα in I and morphisms κα : i → iα and κ′α : i′ → iα such that Fκα (s) = Fκ′α (s′ ). As X is quasi-compact, we can choose A to be finite. As I is filtered, we can find i∞ in I and morphisms iα → i∞ for all α such that the compositions κ

α ν : i −→ iα −→ i∞

and

κ′

α ν ′ : i′ −→ iα −→ i∞

are independent of α. Then Fν (s)|Uα = Fν ′ (s′ )|Uα for all α and hence Fν (s) = Fν ′ (s′ ) which shows that the images of s and s′ in colimi Fi (X) are equal. (ii). It remains to show that φ is surjective if the Condition (COH) is satisfied. Let s be an element of the right hand side. By assumption there exists a finite covering (Uα )α of X such that Uα ∩ Uα′ is quasi-compact for all α, α′ and for each α an iα in I and sα ∈ Fiα (Uα ) such that s|Uα is the image of sα . As Uα ∩ Uα′ is quasi-compact, we can apply (i) and find iαα′ in I and morphisms καα′ : iα → iαα′ and κ′αα′ : iα′ → iαα′ such that the restrictions of Fκαα′ (sα ) and Fκ′αα′ (sα′ ) to Uα ∩ Uα′ are equal. As I is filtered, we can choose i in I and morphisms iαα′ → i such that the images of sα in Fi (Uα ) glue to a section in Fi (X). This yields a preimage of s. Corollary 21.53. Let (X, OX ) be a ringed space. Suppose that X has a basis of open subspaces satisfying (COH) (e.g., if X is the underlying topological space of a scheme). Then the Grothendieck abelian category (OX -Mod) is locally finitely generated (Definition F.66). Proof. By Remark 21.8 it suffices to show that the OX -modules of the form OU ⊆X for U ⊆ X open and satisfying (COH) are finitely generated. As HomOX (OU ⊆X , F ) = F (U ) for every OX -module F , this follows from Lemma 21.52. Let X be a ringed space such that the underlying topological space of X satisfies (COH) and let I be a filtered category. Suppose that F is an I-diagram of OX -modules. Then the underlying sheaf of the colimit of F in the category of OX -modules is the colimit of the underlying diagram of sheaves. Hence in this situation we have the bijectivity of φ also for filtered colimits of OX -modules. Hence we obtain an isomorphism of functors (21.11.1)

I

colim ◦Γ(X, −) ∼ = Γ(X, −) ◦ colim : (OX -Mod) −→ (Γ(X, OX )-Mod), I

I

I

I

where we extend Γ(X, −) to a functor (OX -Mod) → (Γ(X, OX )-Mod) by composiI I tion and where colimI : (OX -Mod) → (OX -Mod) and colimI : (Γ(X, OX )-Mod) → (Γ(X, OX )-Mod) are the exact functors of functors of Grothendieck categories given by forming filtered colimits. Right derivation of (21.11.1) now yields:

173 Proposition 21.54. Let X be a ringed space such that the underlying topological space of X satisfies (COH). Let I be a filtered category and let F : I → C(OX -Mod), i 7→ Fi be an I-diagram. Suppose that there exists an n ∈ Z (independent of i) such that H p (Fi ) = 0 for all p < n and for all i ∈ I. Then there is an isomorphism in D(Γ(X, OX )) ∼

colim RΓ(X, Fi ) −→ RΓ(X, colim Fi ). i

i

which is functorial in F . In particular, we obtain for p ∈ Z an isomorphism of Γ(X, OX )-modules ∼

colim H p (X, Fi ) −→ H p (X, colim Fi ), i

i

which is functorial in F . The proposition applies if X is a qcqs scheme (for instance a noetherian scheme). Proof. One has colim ◦RΓ(X, −) ∼ = R(colim ◦Γ(X, −)) ∼ = R(Γ(X, −) ◦ colim), I

I

I

where the first isomorphism holds by Remark F.175 because filtered colimits are exact and the second isomorphism is obtained by right derivation of (21.11.1). It remains to show that R(Γ(X, −) ◦ colim) = RΓ(X, −) ◦ colim . I

I

I

Hence we have to show that for every complex F in (OX -Mod) satisfying the hypothesis above there exists a quasi-isomorphism F → I such that colimi I is right Γ(X, −)acyclic (Remark F.177). (I). We first remark that filtered colimits of flasque sheaves Gi are quasi-flasque3 (Remark 21.36): If U ⊆ X is open quasi-compact, then (colim Gi )(X) = colim Gi (X) −→ colim Gi (U ) = (colim Gi )(U ) i

i

i

i

is surjective, where the equalities hold by Lemma 21.52. (II). As the Godement resolution is functorial, we obtain a quasi-isomorphism F → I , I with I in (OX -Mod) , where Ii is the Godement resolution of Fi . In particular, for all i ∈ I the OX -modules Iim are flasque for all m and Iim = 0 for m < n. By Step (I), colimi I is a bounded below complex with quasi-flasque components hence it is right Γ(X, −)-acyclic by Remark 21.36. Corollary 21.55. Let f : X → Y be a morphism of ringed spaces such that f −1 (V ) satisfies Condition (COH) for all V in some basis B of the topology of Y . Let I and F be as in Proposition 21.54. Then one has for all p ∈ Z an isomorphism of OY -modules ∼

colim Rp f∗ Fi −→ Rp f∗ colim Fi , i

i

which is functorial in F . The corollary applies if f is a qcqs morphism of schemes. 3

If X is noetherian, one can simply work with flasque sheaves because on a noetherian space all open subspaces are quasi-compact.

21 Cohomology of OX -modules

174

Proof. By Proposition 21.27 both sides are the sheaves associated to the presheaf on B V 7→ colim H p (f −1 (V ), Fi ) i

21.54

=

H p (f −1 (V ), colim Fi ). i

In fact, the same argument as in the proof of Proposition 21.54, using that quasiflasque OX -modules are also right f∗ -acyclic by Proposition 21.27, shows that there is an isomorphism in D(Y ) (21.11.2)

∼

colim(Rf∗ Fi ) −→ Rf∗ (colim Fi ). i

i

Finally, as direct sums are filtered colimits of finite direct sums, we deduce: Corollary 21.56. Let X be a ringed space and let (Fi )i be a family of complexes such that there exists an n ∈ Z with H m (Fi ) = 0 for all m < n and all i. (1) Suppose that X satisfies Condition (COH). Then for all p ∈ Z the functorial homomorphism of Γ(X, OX )-modules M M H p (X, Fi ) −→ H p (X, Fi ) i

i

is an isomorphism. (2) Let f : Y → Y be a morphism of ringed spaces such that f −1 (V ) satisfies Condition (COH) for all V in some basis B of the topology of Y . Then for all p ∈ Z the functorial homomorphism of OY -modules M M (Rp f∗ Fi ) −→ Rp f∗ ( Fi ) i

is an isomorphism. Below (Lemma 22.82), we will prove a similar result for qcqs schemes and families of unbounded complexes of quasi-coherent modules. (21.12) The Grothendieck-Scheiderer vanishing theorem. Recall that the underlying topological space of a qcqs scheme has the following properties. It is quasi-compact, it has a basis of quasi-compact open subsets stable under finite intersections, and every closed irreducible subspace has a unique generic point. Such spaces are called spectral . It can be shown ([Hoc] O ) that every spectral space is homeomorphic to the underlying topological space of Spec A for some ring A. Note that the underlying topological space of every qcqs (and in particular every noetherian) scheme is spectral. Theorem 21.57. (Grothendieck-Scheiderer) Let X be a spectral topological space (for instance a noetherian space) of Krull dimension d. Then for every abelian sheaf F on X and for all n > d, we have H n (X, F ) = 0. We explain the proof only for noetherian spaces. For the general case see [Sche] O . Proof. The proof consists of a series of reduction steps, which will eventually show that the following lemma implies the desired result.

175 Lemma 21.58. Suppose that X is an irreducible spectral space of dimension d. For every inclusion j : U ,→ X of an open subset and every n > d we have H n (X, j! (ZU )) = 0. ̸ ∅. We argue by induction on d. If d = 0, then X consists Proof. We may assume that U = of a point and the assertion is trivial. Let Z denote the complement of U in X, and let i : Z → X denote the inclusion. We have a short sequence of abelian sheaves on X 0 → j! (ZU ) → ZX → i∗ (ZZ ) → 0 which is seen to be exact by looking at stalks. Since Z is a proper closed subset of the irreducible space X, we have dim Z < dim X. Using our induction hypothesis, we obtain that H n (X, i∗ ZZ ) ∼ = H n (Z, ZZ ) = 0,

for all n ≥ d.

Here the first isomorphism is (21.8.4). Furthermore, every constant sheaf on an irreducible space is flasque (Example 21.30), hence H n (X, ZX ) = 0 for all n > 0. Now the statement of the lemma follows from the long exact cohomology sequence corresponding to the short exact sequence above. It remains to reduce the general situation to the statement of the lemma. Again we use induction on the dimension of X. Hence we may assume that the result holds for all noetherian topological spaces of dimension < d. We start by showing that it is enough to consider the case where X is irreducible. In general, X has finitely many irreducible components, and we proceed by induction on the number of irreducible components. Let Z ⊂ X denote one of the irreducible components of X. This is a closed subset and we denote by U ⊂ X its open complement. We have a short exact sequence 0 → j! (F|U ) → F → i∗ (F|Z ) → 0, where i and j denote the inclusions of Z and U into X, respectively. Furthermore, the closure U of U has one irreducible component less than X, so by the induction hypothesis and by (21.8.4) we have for all n > d H n (X, i∗ (F|Z )) = H n (Z, F|Z ) = 0 and H n (X, j! (F|U )) = H n (U , j! (F|U )) = 0, where we denote the inclusion U ,→ U by j, as well. The long exact cohomology sequence for the short exact sequence above now implies that H n (X, F ) = 0, as desired. Now assume that X is irreducible. We write F = colim FI , I

where I runs over the set, ordered by inclusion, of all finite sets I = {(U1 , s1 ), . . . , (Ur , sr )} with Uµ ⊆ X open, sµ ∈ Γ(Uµ , F ), and for such a set I, FI is the abelian subsheaf of F generated by the sections sµ . In other words, if sµ corresponds to the map αµ : jµ,! ZUµ → F (21.1.8), then FI is the image of the sum of the αµ . By Proposition 21.54 it is enough to show that the cohomology of the sheaves FI vanishes in degrees > d.

21 Cohomology of OX -modules

176 If I ′ ⊂ I, then we have an exact sequence

0 → FI ′ → FI → FI /FI ′ → 0, and the quotient FI /FI ′ is generated by the sections in I \ I ′ . If the cohomology of the terms on the left and on the right vanishes in degrees > d, then the same holds for the cohomology of FI because of the long exact cohomology sequence. By induction, it is hence enough to consider the case where F is generated by a single section s ∈ Γ(U, F ) for some open U ⊆ X, i.e., we may assume that we have an exact sequence 0 → G → j! (ZU ) → F → 0 for some open j : U ,→ X and some sheaf G on X. It is then enough to show the desired cohomology vanishing for G and for j! (ZU ). For the latter one, we can simply invoke Lemma 21.58 above. If G = 0, then there is nothing to do. Otherwise, there is a non-zero stalk Gx for some x ∈ U , and for all x ∈ U we have Gx ⊆ j! (ZU )x = Z, so we can identify Gx = mx Z for an integer mx ≥ 0. Let m be the minimum of all non-zero mx . Picking x ∈ U with mx = m and lifting the element mx ∈ Gx to a suitable open neighborhood of x, we see that there exists a non-empty open U ′ ⊆ U such that G|U ′ = mj! (ZU )|U ′ ∼ = ZU ′ . We obtain a short exact sequence 0 → j!′ (ZU ′ ) → G → G /j!′ (ZU ′ ) → 0, where j ′ : U ′ ,→ X denotes the inclusion. Now for the left term we have cohomology vanishing by the lemma, and for the right term we have it by induction, since it is supported on X \ U ′ . Hence the claim follows for G by invoking once again the long exact cohomology sequence. (21.13) The local cohomology triangle. In this section let X be a topological space and let i : Z ,→ X be the inclusion of a locally closed subset. Definition and Remark 21.59. Let F be an abelian sheaf. We define the group of sections of F with support in Z as follows. Suppose first that Z is closed in X. Then set ΓZ (X, F ) := { s ∈ Γ(X, F ) ; Supp(s) ⊆ Z } = Ker(Γ(X, F ) → Γ(X \ Z, F )). If Z is only locally closed, we choose an open subset V ⊆ X such that Z is closed in V and define ΓZ (X, F ) := ΓZ (V, F |V ). We claim that ΓZ (V, F |V ) does not depend on the choice of V . Indeed, let V ′ ⊆ X be another open subset such that Z ⊆ V ′ is closed. Replacing V ′ by V ∩ V ′ we may assume that V ′ ⊆ V . We show that the restriction F (V ) → F (V ′ ) induces an isomorphism ∼ ΓZ (V, F |V ) → ΓZ (V ′ , F |V ′ ). Indeed let s ∈ ΓZ (V, F |V ) with s|V ′ = 0, then s = 0 because s|V \Z = 0 by definition and (V ′ , V \ Z) is an open covering of V . Conversely, an element s′ ∈ ΓZ (V ′ , F |V ′ ) and 0 ∈ F (V \ Z) glue to a section s ∈ F (V ) such that s|V ′ = s′ and s ∈ ΓZ (V, F |V ). We obtain for every locally closed subset Z of X a left exact functor (21.13.1)

ΓZ : F 7→ ΓZ (X, F )

177 from the category of abelian sheaves to the category of abelian groups. Remark 21.60. Let F be an abelian sheaf on X. We define an abelian sheaf on Z by (i! F )(W ) := ΓW (X, F ),

W ⊆ Z open

and obtain a functor (21.13.2)

i! : (Ab(X)) → (Ab(Z)).

For U ⊆ X open, the restriction of the canonical homomorphism Γ(U, F ) → Γ(U ∩ Z, i−1 F ) to ΓU ∩Z (U, F |U ) ⊆ Γ(U, F ) is injective, we can view i! F as a subsheaf of i−1 (F ). We have (21.13.3)

Γ(X, i∗ i! F ) = ΓZ (F )

Proposition 21.61. For every inclusion i : Z → X of a locally closed subset, the functor i! is right adjoint to the functor i! (Definition 21.1). Proof. For an abelian sheaf F on X let F Z be the sheaf on X with F Z (U ) = { s ∈ Γ(U, F ) ; Supp(s) ⊆ Z } for U ⊆ X open. Then i! F = i−1 F Z . One has (F Z )x = 0 for x ∈ / Z. Hence Remark 21.2 (3) shows that i! i! F = F Z . In particular we obtain a monomorphism i! i! F ,→ F and any morphism of abelian sheaves G → F , where G is an abelian sheaf on X with Gx = 0 for x ∈ / Z factors through this monomorphism. Hence we obtain for every sheaf E on Z functorial bijections Hom(i! E , F ) = Hom(i! E , i! i! F ) = Hom(E , i! F ), where the second equality holds because i! is fully faithful (Remark 21.2 (2)). In particular, i! is left exact. Moreover, it has an exact left adjoint functor and hence preserves K-injective complexes (Proposition F.183). For its derived functor, we have the following local cohomology triangle. Proposition 21.62. Let X be a topological space, let i : Z → X be the inclusion of a closed subset, and let j : U := X \ Z ,→ X be the inclusion of the open complement of Z. Then for every complex F of abelian sheaves on X there is a distinguished triangle (21.13.4)

i∗ Ri! F −→ F −→ Rj∗ j −1 F −→ i∗ Ri! F [1]

in D(Ab(X)), which is functorial in F . Proof. Let F → I be a quasi-isomorphism to a K-injective complex with injective components (Theorem 21.7). Then i∗ Ri! F = i∗ i! I . Moreover, j −1 I is again a K-injective complex with injective components (Lemma 21.12) and hence Rj∗ j −1 F = j∗ (I |U ). As injective abelian sheaves are flasque (Lemma 21.32) there is an exact sequence of complexes of abelian sheaves 0 −→ i∗ i! I −→ I −→ j∗ (I |U ) −→ 0 which yields the distinguished triangle (21.13.4). The same argument also shows:

21 Cohomology of OX -modules

178

Corollary 21.63. For every complex F of abelian sheaves on X one has a distinguished triangle in D(Z) (21.13.5)

RΓZ (X, F ) −→ RΓ(X, F ) −→ RΓ(U, F |U ) −→ RΓZ (X, F )[1]

which is functorial in F . Writing HZn (X, F ) = H n (RΓZ (X, F )), we obtain a long exact sequence (21.13.6)

· · · → HZn (X, F ) → H n (X, F ) → H n (U, F |U ) → HZn+1 (X, F ) → . . .

which is functorial in F . We also could have obtained (21.13.5) by applying the triangulated functor RΓ(X, −) to the distinguished triangle (21.13.4) using the isomorphism RΓ(X, −)◦Rj∗ = RΓ(U, −), see (21.8.1). Indeed, as i∗ and i! both have an exact left adjoint, namely i−1 and i! respectively, they preserve K-injective complexes (Proposition F.183) and hence RΓ ◦ i∗ ◦ Ri! = R(Γ ◦ i∗ ◦ i! ) = RΓZ by Proposition F.176.

ˇ Cech cohomology Although sheaf cohomology has very nice formal properties that often can be used to relate various cohomology groups to each other, and thus to compute unknown cohomology groups from other, known ones, it is usually not easy to compute examples of cohomology groups using injective or flasque resolutions. For computing specific cohomology groups, ˇ often the point of view of Cech cohomology is useful. For instance, this is what we will use to compute the cohomology groups of line bundles on projective space, see Section (22.6) which is the starting point of one of the central finiteness theorems in Algebraic Geometry, namely that higher direct images of coherent modules under proper morphisms between locally noetherian schemes are again coherent. Let X be a ringed space and let U = (Ui )i∈I be an open covering of X. We will define ˇ ˇ n (U , F ) of an OX -module F with respect to U . These the Cech cohomology groups H groups admit a rather concrete description as the cohomology of a complex in which only the module F and its sections over intersections of the open subsets Ui enter (and not some usually difficult to handle flasque or injective resolution). One can even ease the ˇ complex as bookkeeping further by working with the alternating or the ordered Cech explained in Section (21.15). ˇ Then we relate Cech cohomology to usual cohomology. This happens in two steps. We ˇ n (X, F ) of Cech ˇ first define a version H cohomology dependent only on X and F but not on U be forming the colimit over all open coverings, where morphisms of coverings ˇ n (X, F ) and sheaf cohomology are given by refinements of coverings. Then we relate H H n (X, F ) by a spectral sequence. This allows us to prove theorems of Leray and Cartan (Corollary 21.82 and Proposition 21.83) that give criteria on X, F , and U that ensure ˇ n (U , F ) ∼ that H = H n (X, F ). In the next chapter we will use these criteria to prove that for any quasi-compact separated scheme X, any quasi-coherent OX -module F , and any open affine covering U ˇ n (U , F ) ∼ of X one has H = H n (X, F ) for all n. This allows us to calculate the cohomology of quasi-coherent modules in many “reasonable situations”.

179 ˇ n (X, F ) does not coincide with sheaf cohomology. The problem is that In general, even H ˇ in higher degrees of the Cech complex we allow only sections over intersections of the open subsets of our given open covering. It is possible to generalize this construction arriving at the notion of a hypercovering and the cohomology of F with respect to a hypercovering. The colimit over all hypercoverings is indeed isomorphic to sheaf cohomology. We will however not explain the formalism of hypercoverings since for quasi-coherent modules on ˇ most schemes it suffices to consider Cech cohomology. ˇ (21.14) The Cech complex. Let X be a ringed space, let U = (Ui )i∈I be a covering of X by open subsets, and let F be a presheaf of OX -modules. We write for i = (i0 , . . . , in ) ∈ I n+1

Ui := Ui0 ···in := Ui0 ∩ · · · ∩ Uin and define the Γ(X, OX )-module Cˇ n (U , F ) =

Y

Γ(Ui , F ).

i∈I n+1

We have Γ(X, OX )-linear maps d : Cˇ n (U , F ) → Cˇ n+1 (U , F ),

(si )i∈I n+1 7→

n+1 X ν=0

! ν

(−1) sj(bν )|Uj

, j∈I n+2

where we denote by j(b ν ) the (n + 1)-tuple obtained by omitting the ν-th entry from j. It is easy to check that d ◦ d = 0. ˇ , F ) is called the Cech ˇ Definition 21.64. The complex C(U complex of F with respect ˇ n (U , F ) and are called the to the covering U . Its cohomology groups are denoted by H ˇ Cech cohomology groups of F with respect to the covering U . ˇ The Cech complex and its cohomology groups clearly behave functorially in F . In ˇ particular we obtain a functor C(U , −) from the category of presheaves of OX -modules to the category of complexes of Γ(X, OX )-modules. As we will see later (Section (21.17)), in certain favorable situations, the cohomology ˇ of the Cech complex coincides with the cohomology in the sense of derived functors. This will allow us to explicitly compute cohomology groups much more easily than using acyclic resolutions. Lemma 21.65. Let X be a ringed space, and let F be a presheaf of OX -modules. The following are equivalent: (i) The presheaf F is a sheaf. (ii) For all open subsets U ⊆ X and all open coverings U of U , the natural map ˇ 0 (U , F ) is an isomorphism. Γ(U, F ) → H Proof. This follows immediately from the definitions, because we can express the sheaf property by requiring that for all coverings U = (Ui )i of open subsets U ⊆ X, the sequence Y Y 0 → Γ(U, F ) → Γ(Ui , F ) → Γ(Uij , F ) i

i,j

21 Cohomology of OX -modules

180

ˇ is exact, where the second map is given by (si )i 7→ (sj |Uij − si|Uij )i,j , i.e., by the Cech differential. Remark 21.66. If F is a sheaf of abelian groups on a topological space X, then ˇ 1 (U , F ) is the same abelian group as defined in (11.5). Hence it can be identified with H the group of isomorphism classes of F -torsors T such that T |Ui is trivial for all i ∈ I (Proposition 11.13). ˇ (21.15) The alternating and the ordered Cech complex. ˇ There are two useful variants of the Cech complex which have the same cohomology ˇ ˇ groups: The alternating Cech complex and the ordered Cech complex. As before, let X be a ringed space, and let F be a presheaf of OX -modules. We also fix an open covering U = (Ui )i∈I of X. ˇ Definition 21.67. The alternating Cech complex is given by n Cˇalt (U , F ) := {(si )i ∈ Cˇ n (U , F ); si = 0 whenever two entries of i are equal, sσ(i) = sgn(σ)si for every permutation σ ∈ Sn+1 },

ˇ where the differential d is given by restricting the differential of the Cech complex to n (U , F ). Cˇalt Now let U = (Ui )i∈I be an open covering, and suppose that a total order < of the index set I is given. ˇ Definition 21.68. The ordered Cech complex is given by Y n Cˇord (U , F ) = Γ(Ui0 ···in , F ), i0 0. Fix i0 ∈ I with W ⊆ Ui0 (if such an index does not exist, then Z(U )(W ) = 0 and nothing needs to be done). Define a “homotopy between id and 0 in degrees > 0” by h : Z(U )n (W ) −→ Z(U )n+1 (W ), ( 0, if i0 ̸= i0 ; h(s)i0 ...in := si1 ...in+1 , if i0 = i0 for n ≥ 0. Then d ◦ h + h ◦ d = id in degrees ≥ 0 which shows the exactness of Z(U )(W ) in degrees > 0.

21 Cohomology of OX -modules

184

Lemma 21.77. For all bounded below complexes F of presheaves of OX -modules there is a functorial isomorphism in D+ (Γ(X, OX )) ∼ ˇ ˇ 0 (U , F ), C(U , F ) −→ RH

(21.17.3)

where the right hand side is the right derived functor D+ (P-OX -Mod) → D+ (Γ(X, OX )) ˇ 0 (U , −) : (P-OX -Mod) → (Γ(X, OX )-Mod). of the left exact functor H Proof. By Corollary F.186 we have D+ (P-OX -Mod) ∼ = K + (I), where I is the full subcategory of injective objects in (P-OX -Mod). Therefore it suffices to construct the isomorphism (21.17.3) for F = I a bounded below complex of injective objects. By Lemma 21.76, ˇ 0 (U , I ) −→ (· · · → Cˇ i (U , I ) → Cˇ i+1 (U , I ) → . . . ) H

(*)

is a quasi-isomorphism of complexes with components in C + (Γ(X, OX )), where we consider the left hand side as a complex with components in C + (Γ(X, OX )) concentrated in degree 0. Hence we have a functorial quasi-isomorphism in K + (Γ(X, OX )) ˇ 0 (U , I ) = H ˇ 0 (U , I ) −→ C(U ˇ RH , I ), which is induced by applying the total complex functor to the quasi-isomorphism (∗) (Lemma F.113). ˇ 0 (U , F ) = Γ(X, F ) (Lemma 21.65) and hence an For an OX -module F we have H isomorphism of functors (21.17.4)

∼ ˇ 0 (U , −) ◦ ι −→ H Γ(X, −) : (OX -Mod) → (Γ(X, OX )-Mod).

As the inclusion ι : (OX -Mod) → (P-OX -Mod) has an exact left adjoint functor, namely sheafification, its extension K + (OX -Mod) → K + (P-OX -Mod) preserves bounded below K-injective complexes (Corollary F.183) and hence by Proposition F.176 we obtain an isomorphism of functors (21.17.5)

∼

ˇ 0 (U , −) ◦ Rι : D+ (OX -Mod) → D+ (Γ(X, OX )). RΓ(X, −) −→ RH

Using Lemma 21.77 we easily derive the main result of this section. Theorem 21.78. Let X be a ringed space and let U be an open covering of X. Let F be a bounded below complex of OX -modules. (1) There is a natural functorial morphism (in D+ (Γ(X, O))) (21.17.6)

ˇ C(U , F ) → RΓ(X, F ).

Passing to cohomology, we obtain homomorphisms (21.17.7)

ˇ n (U , F ) → H n (X, F ) H

for all n ∈ Z

that are functorial in F . (2) There is a converging spectral sequence (21.17.8)

ˇ p (U , H q (F )) ⇒ H p+q (X, F ), E2p,q = H

which is functorial in F . Here H q (F ) is the presheaf defined in Remark 21.20.

185 Proof. The spectral sequence in (2) is the Grothendieck spectral sequence (Proposition F.212) attached to (21.17.4): By (21.5.3) we have Rq ι(F ) = H q (F ) and by ˇ 0 (U , −) = Cˇ p (U , −). Lemma 21.77 we have Rp H The morphism (21.17.6) is the composition of functors ∼ ˇ ˇ 0 (U , −) ◦ ι −→ RH ˇ 0 (U , −) ◦ Rι ∼ C(U , −) → RH = RΓ(X, −),

where the first isomorphism holds by Lemma 21.77 and the last isomorphism is (21.17.4). Note that the isomorphism (21.17.5) exists also for unbounded derived functors (with the same argument) but that we proved Lemma 21.77 and hence Theorem 21.78 only for bounded below complexes. This suffices for our purposes because we are interested in ˇ Cech cohomology only to compute cohomology in concrete situations, where all complexes are bounded below (and usually are even concentrated in degree 0). Remark 21.79. Let F be an OX -module. Then H 0 (F ) = F . The spectral sequence (21.17.8) is a first quadrant spectral sequence (i.e., E2p,q = 0 if p < 0 or q < 0). The proof of Theorem 21.78 shows that the morphism (21.17.6) is the edge morphism ˇ p (U , F ) −→ H p (X, F ). E2p,0 = H The formation of the spectral sequence (21.17.8) is compatible with passing to refinements and hence induces for all OX -modules F a functorial converging spectral sequence (21.17.9)

ˇ p (X, H q (F )) ⇒ H p+q (X, F ). E2p,q = H

In particular the edge morphisms are functorial homomorphism of Γ(X, OX )-modules (21.17.10)

ˇ n (U , F ) −→ H n (X, F ), ˇ n (X, F ) = colim H H U

n ≥ 0.

Lemma 21.80. Let G be a presheaf of OX -modules whose sheafification is 0. Then ˇ 0 (X, G ) = 0. H Proof. If (si )i ∈ Cˇ 0 (U , G ), then locally all sections si are zero, hence there exists a refinement V of U such that the image of (si )i in Cˇ 0 (V , G ) is zero. Hence colimU Cˇ 0 (U , G ) = 0 ˇ 0 (X, G ) = 0. and in particular H Corollary 21.81. Let X be a ringed space, and let F be an OX -module. Then the functorial homomorphism (21.17.10) is an isomorphism for n = 0, 1, and is injective for n = 2. Proof. The case n = 0 easily follows from the definitions (see Lemma 21.65). For the other cases we use the exact sequence in low degrees (Proposition F.105) for the spectral sequence (21.17.9) and get for every open covering U of X an exact sequence ˇ 1 (U , F ) −→ H 1 (X, F ) −→ H ˇ 0 (U , H 1 (F )) −→ H ˇ 2 (U , F ) −→ H 2 (X, F ). 0 −→ H ˇ 0 (X, H 1 (F )) = 0. Hence we have to show that H 1 But H (F ) is a presheaf whose sheafification is zero (Lemma 21.22). We conclude by Lemma 21.80.

21 Cohomology of OX -modules

186

The same proof shows that the corollary also holds for complexes F of OX -modules concentrated in degrees ≥ 0 (to obtain the exact sequence of lower terms) such that H 1 (F ) = 0 (to ensure that H 1 (F ) is a presheaf whose sheafification is zero). For an abelian sheaf F and for n = 1 one could also have shown that (21.17.10) is equal ∼ ˇ 1 (X, F ) → H 1 (X, F ) of (11.5.2), where we identify H 1 (X, F ) to the isomorphism H with the group of isomorphism classes of F -torsors (Proposition 21.40). Corollary 21.82. (Leray) Let X be a ringed space and let U = (Ui )i∈I be an open covering of X. Let F be an OX -module such that for all q > 0, p ≥ 0 and i0 , . . . , ip ∈ I, H q (Ui0 ···ip , F ) = 0. ˇ n (U , F ) → H n (X, F ) is an isomorphism for all Then the natural homomorphism H n ≥ 0. Proof. The hypothesis means that Cˇ p (U , H q (F )) = 0 if q > 0. In particular for the spectral sequence (21.17.8) we see that (*)

E2p,q = 0

for all (p, q) with q ̸= 0.

Hence the spectral sequence degenerates at E2 which implies that H n (X, F ) has a filtration whose graded pieces are the E2p,q with p + q = n. Hence (*) implies that the ˇ n (U , F ) → H n (X, F ) is an isomorphism. edge morphism H Proposition 21.83. (Cartan) Let X be a ringed space and let B be a basis of the topology of X such that U, U ′ ∈ B implies U ∩ U ′ ∈ B. Let F be an OX -module. Suppose that ˇ p (U, F ) = 0 H

for all p > 0 and all U ∈ B.

Then (1) for U ∈ B one has H n (U, F ) = 0 for all n > 0, (2) for every open covering U of X consisting of open sets in B the canonical morphism ˇ n (U , F ) → H n (X, F ) is an isomorphism for all n ≥ 0, and H ˇ n (X, F ) → H n (X, F ) is an isomorphism for all (3) the canonical homomorphism H n ≥ 0. Proof. We prove (1) by induction on n ≥ 1. Hence we may assume that H q (U, F ) = 0 for all 0 < q < n and all U ∈ B. As every open covering has a refinement by an open covering (Ui )i with Ui ∈ B (because B is a basis of the topology) and as Ui0 ...ip ∈ B (because B is stable under finite nonempty intersections), the induction hypothesis implies ˇ C(X, H q (F )) = 0 for all 0 < q < n. Hence E2p,q = 0 for 0 < q < n and all p in the spectral sequence (21.17.9). Hence we obtain from (F.22.5) an exact sequence ˇ n (X, F ) −→ H n (X, F ) −→ H ˇ 0 (X, H n (F )), 0 −→ H and the last term is 0 by Lemma 21.80 because the sheafification of H n (F ) is zero by Lemma 21.22. Replacing X by U ∈ B, F by F |U , and B by { U ′ ∈ B ; U ′ ⊆ U } we ˇ n (U, F ) −→ H n (U, F ) is bijective for all U ∈ B which shows the deduce that 0 = H claim. Now (2) follows by applying Corollary 21.82 using again that B is stable under finite intersections (with non-empty index set). As open coverings consisting of elements in B are cofinal in the set of all open coverings, also (3) follows.

187

Derived inverse image, Hom sheaves, and tensor products We will now define the other functors in our “formalism of four functors” explained in the introduction of this chapter. More precisely we are going to derive the Hom- and the Hom-functor, the tensor product, and the functor f ∗ for a morphism f of ringed spaces. There are two main difficulties to overcome. The first one is that the Hom-functor and the tensor product are both bi-functors, i.e., functors in two variables and we have to explain in which variable we derive the functors, whether we could also derive these functors in the other variable, and whether we obtain in this case the same functor. To solve these problems we will check that we can apply the general results of Section (F.51). The second difficulty is that the tensor product and the functor f ∗ are right exact functors. Hence we want to take their left derivation. If we wanted simply to dualize our general construction of right derived functors, we would need for every complex F of OX -modules a K-projective resolution P → F . But in the category of OX -modules such K-projective resolutions usually do not exist, even for standard schemes such as the projective line X = P1k over a field k (see Exercise 21.5). Hence we need a larger class of complexes that are acyclic for the tensor product and the functor f ∗ . These will be the class of K-flat complexes that we will introduce in Section (21.19). This problem also would occur if we wanted to derive the Hom-functor in the first variable. Here we cannot use K-flat resolutions as these are in general not acyclic for the Hom-functor in the first variable. In fact, the only complexes that are acyclic for the Hom-functor in the first variable are by definition the K-projective complexes (see the dual of Definition F.179). If we consider the category of modules over a ring A (corresponding modules over a ringed spaces whose underlying topological space consists of a single point), then every complex of A-modules has a K-projective resolution (Theorem F.189) and we can also derive the Hom-functor in the first variable. Then the results of Section (F.51) will show that this derived functor coincides with the one obtained by deriving in the second variable. (21.18) Derived functor of Hom and Hom. In this section, we always denote by (X, OX ) a ringed space. Recall from Section (F.52), that for any abelian category A with enough K-injective objects, we have a derived Hom-functor R HomA (−, −) : D(A)opp × D(A) → D(AbGrp). For p ∈ Z one has for all complexes F and G in D(A) ExtpA (F, G) = H p (R HomA (F, G)) = HomD(A) (F, G[p]), where the second identity holds by (F.52.5). Remark 21.84. If A is the category of OX -modules, then for complexes F and G of OX -modules the Hom complex (Section (F.18)) is a complex of Γ(X, OX )-modules and we obtain a derived Hom-functor (21.18.1)

R HomOX (−, −) : D(X)opp × D(X) → D(Γ(X, OX )).

21 Cohomology of OX -modules

188

The construction in Section (F.52) shows that R HomOX (F , G ) is calculated as follows. One chooses a quasi-isomorphism G → I , where I is a K-injective complex of OX -modules. Then R HomOX (F , G ) is represented by the complex HomOX (F , I ) of Γ(X, OX )-modules. If there exists a quasi-isomorphism P → F for a K-projective complex of OX -modules (which is always the case if the category of OX -modules can be identified with the category of modules over a ring), then R HomOX (F , G ) can also be represented by HomOX (P, G ). Often, K-projective resolutions are more manageable if they exist. Consider the following example. Example 21.85. Let A be a ring and let I be an ideal generated by a completely intersected sequence f = (f1 , . . . , fr ) in A. Then the Koszul complex K(f ), considered as cochain complex sitting in degrees [−r, 0], defines a projective resolution K(f ) → A/I. Hence for every complex M of A-modules we can calculate R HomA (A/I, M ) by the complex HomA (K(f ), M ). We want to define also a sheaf version of R Hom, i.e., a derived functor attached to the Hom functor (see Section (7.4)). For this we first define a sheaf version of the Hom-complex in the same way as in Section (F.18). Definition and Remark 21.86. Let F and G be complexes of OX -modules. Setting Y Hom iOX (F , G ) := Hom OX (F k , G k+i ), k∈Z

di ((uk )k ) = (dk+i ◦ uk − (−1)i uk+1 ◦ dkF )k , G defines a complex Hom OX (F , G ) of OX -modules. The construction is obviously functorial in F and G , and we obtain extensions of the bi-functor (21.18.2)

Hom OX (−, −) : (OX -Mod)opp × (OX -Mod) −→ (OX -Mod)

to a bi-functor Hom OX (−, −) : C(X)opp × C(X) −→ C(X) that induces a triangulated bi-functor (21.18.3)

Hom OX (−, −) : K(X)opp × K(X) −→ K(X).

We have (21.18.4)

Γ(X, Hom OX (F , G )) = HomOX (F , G ) ∈ C(Γ(X, OX ))

for all complexes F and G of OX -modules. To derive this functor, we show that Condition (ACII) of Section (F.51) is satisfied. In other words we have to show: Lemma 21.87. Let I be a K-injective complex in K(X). Then the functor Hom OX (−, I ) sends quasi-isomorphisms to quasi-isomorphisms in K(X) (equivalently, to isomorphisms in D(X)). Proof. We have to show that Hom OX (−, I ) sends exact complexes to exact complexes (Example F.169). Let F be a complex of OX -modules. For i ∈ Z the cohomology sheaf H i (Hom OX (F , I )) is the sheaf associated to the presheaf

189 U 7→ H i (Γ(U, Hom OX (F , I )) (*)

= H i (HomOU (F |U , I |U )) = HomK(U ) (F |U , I |U [i]),

where the last equality is by (F.18.2). Now I |U (and hence I |U [i]) is K-injective by Lemma 21.12. Therefore, if F is an exact complex, then the presheaf (*) is zero by Proposition F.179 (2) and Hom OX (F , I ) is exact. Definition and Remark 21.88. Now Section (F.51) shows that the functor (21.18.3) induces a triangulated bi-functor (21.18.5)

R Hom OX : D(X)opp × D(X) → D(X),

called the derived inner Hom functor . Hence the partial functors R Hom OX (−, G ) and R Hom OX (F , −) are triangulated functors (D(X)opp viewed as triangulated category as in Section (F.30)), and in particular we are given for all p ∈ Z isomorphisms (21.18.6)

R Hom OX (F , G [p]) ∼ = R Hom OX (F , G )[p] ∼ = R Hom OX (F [−p], G ),

functorial in both variables. If we choose K-injective resolutions G → IG of every complex G , then by definition (21.18.7)

R Hom OX (F , G ) = Hom OX (F , IG )

and we obtain a morphism of functors in F and G (21.18.8)

Hom OX (F , G ) −→ Hom OX (F , IG ) = R Hom OX (F , G ).

As the restriction of a K-injective complex to an open subset is again K-injective (see Lemma 21.12), for all U ⊆ X open we have

(21.18.9)

R Hom OX (F , G )|U = Hom OX (F , IG )|U = Hom OU (F |U , IG |U ) = R Hom OU (F |U , G |U ).

We have a functorial isomorphism in D(X) (21.18.10)

R Hom OX (OX , G ) = G

since both sides are represented by Hom OX (OX , IG ) = IG . Finally, if (X, OX ) = (∗, A) for a commutative ring A, we identify D(X) and D(A) and have R Hom OX (F, G) = R HomA (F, G) for F, G ∈ D(A). Again, it is often useful to calculate R Hom via the first variable. For this one has the following result. Lemma 21.89. Let F and G be complexes of OX -modules. Suppose that G is bounded below, that F is bounded above and that F n is a direct summand of a finite free OX -module for all n. Then R Hom OX (F , G ) = Hom OX (F , G ). For a similar assertion with G arbitrary and F bounded see Lemma 21.144 below.

21 Cohomology of OX -modules

190

Proof. Let G → I be a K-injective resolution. As G is bounded below, we may assume that I is bounded below (Theorem F.185). Then R Hom(F , G ) = Hom(F , I ) and it suffices to show that the canonical map u : Hom(F , G ) → Hom(F , I ) is a quasiisomorphism. Since F is bounded above and I is bounded below, we have Y M Hom(F , I )n = Hom(F p , I p+n ) = Hom(F p , I p+n ). p∈Z

p∈Z

Similarly, Hom(F , G )n is in each degree the direct sum of Hom(F p , G p+n ). Therefore both complexes are the total complex associated to a double complex and u comes from the canonical map (Hom(F p , G q ))p,q → (Hom(F p , I q ))p,q of double complexes. As the formation of the total complex preserves quasi-isomorphisms in each direction (Lemma F.113), it suffices to show that for a fixed p ∈ Z the map Hom(F p , G ) → Hom(F p , I ) is a quasi-isomorphism. This holds because F p is the direct summand of a finite free OX -module. (21.19) K-flat complexes. Let (X, OX ) be a ringed space. Recall from Section (F.19) that for complexes F and G of OX -modules the tensor complex F ⊗OX G is defined, that we obtain functors (21.19.1) (21.19.2)

− ⊗ − = − ⊗OX − : C(X) × C(X) −→ C(X), − ⊗ − = − ⊗OX − : K(X) × K(X) −→ K(X), ∼

∼

and there exist functorial isomorphism F ⊗G → G ⊗F and (F ⊗G )⊗H → F ⊗(G ⊗H ). Moreover, (21.19.2) is a triangulated bi-functor (Example F.214). As passing to stalks commutes with tensor products and with direct sums, there is for all x ∈ X a functorial isomorphism of complexes of OX,x -modules (21.19.3)

(F ⊗OX G )x ∼ = Fx ⊗OX,x Gx

for all complexes F and G of OX -modules. Remark 21.90. Let F , G , and H be complexes of OX -modules. (1) A sheafified version of Proposition F.93 shows that there is a functorial isomorphism of complexes of OX -modules (21.19.4)

∼

Hom OX (F ⊗OX G , H ) −→ Hom OX (F , Hom OX (G , H )).

(2) By applying Z 0 (−) ◦ Γ(X, −) to (21.19.4) one sees by (F.18.2) that for a fixed complex G the tensor functor − ⊗ G : C(X) → C(X) is left adjoint to the functor Hom OX (G , −) : C(X) → C(X). In particular we see that − ⊗ F commutes with arbitrary colimits. By symmetry we see that also the functor F ⊗ − commutes with arbitrary colimits. We obtain the same assertions for the induced functors − ⊗ F : K(X) → K(X) and Hom OX (G , −) : K(X) → K(X), now by applying H 0 (−) ◦ Γ(X, −) to (21.19.4). We now want to construct the left derivation of the tensor functor. The following class of complexes will be acyclic for the tensor functor, see Lemma 21.96 below.

191 Definition 21.91. A complex P of OX -modules is called K-flat, if for every exact complex F of OX -modules, the tensor complex F ⊗OX P is exact (or equivalently: P ⊗OX F is exact). Applying this definition to the case where he underlying topological space of X consists of a single point, we obtain as a special case that for a ring A a complex P of A-modules is called K-flat, if for every exact complex F of A-modules, the tensor complex P ⊗A F is exact. Remark 21.92. (1) An OX -module F is flat if and only if F , considered as a complex concentrated in degree 0, is K-flat. (2) A complex F of OX -modules is K-flat if and only if for every quasi-isomorphism of complexes G1 → G2 the induced map F ⊗ G1 → F ⊗ G2 is again a quasi-isomorphism (Example F.169). (3) The associativity of the tensor complex (Remark F.92) implies that if P and Q are K-flat complexes of OX -modules then the tensor complex P ⊗OX Q is a K-flat complex. (4) A complex P of OX -modules is K-flat if and only if for every x ∈ X the complex Px of OX,x -modules is K-flat. Indeed, the condition is sufficient because exactness of complexes can be checked on stalks and because tensor products commute with forming stalks (21.19.3). Conversely, suppose that P is K-flat, let x ∈ X, and let F be an exact complex of OX,x -modules. Let ix : ({x}, OX,x ) → (X, OX ) be the canonical morphism of ringed spaces. Then ix,∗ F is an exact complex of OX -modules and hence P ⊗OX ix,∗ F is exact. Therefore (P ⊗OX ix,∗ F )x = Px ⊗OX,x F is exact. This shows that Px is K-flat. (5) As tensoring with a fixed complex F is a triangulated functor, the long exact cohomology sequence (21.4.1) shows that the K-flat complexes form a triangulated subcategory of K(X). (6) Every filtered colimit of K-flat complexes in C(X) is again K-flat. Indeed this follows because arbitrary colimits commute with tensor products (Remark 21.90) and filtered colimits commute with taking cohomology because filtered colimits are exact (Proposition 21.7). Lemma 21.93. Let P be a bounded above complex of flat OX -modules. Then P is a K-flat complex. Proof. We can write P = colimm σ ≥−m P and hence can assume by Remark 21.92 (6) that P is a bounded complex of flat modules, say of length m − k + 1: P = (. . . → 0 → P k → . . . → P m → 0 → . . .). Then the termwise split sequence 0 → σ ≥m P → P → σ ≤m−1 P → 0 yields a distinguished triangle σ ≥m P → P → σ ≤m−1 P → σ ≥m P[1]. By induction on the length of the complex using Remark 21.92 (1) we can assume that σ ≥m P and σ ≤m−1 P are K-flat. Hence P is K-flat by Remark 21.92 (5). Proposition 21.94. For each complex F of OX -modules there exists a K-flat complex PF together with a quasi-isomorphism αF : PF → F , such that PF and αF depend functorially on F and are compatible with shifts: PF [1] = PF [1] and αF [1] = αF [1].

21 Cohomology of OX -modules

192

Proof. Let F be an OX -module. We will construct a functorial epimorphism P 0 (F ) → F with P 0 (F ) a flat OX -module. For this recall from (21.1.6) the OX -module OU ⊆X = j! (OX |U ) for an open subspace j : U ,→ X. Its stalk in x ∈ X is OX,x for x ∈ U and zero otherwise. In particular, it is a flat OX -module. Define a flat OX -module M M OU ⊆X P 0 (F ) := U ⊆X open s∈F (U )

and a homomorphism P 0 (F ) → F whose restriction to the direct summand indexed by (U, s) is the homomorphism OU ⊆X → F corresponding to s via (21.1.8). Therefore every local section of F is in the image and in particular P 0 (F ) → F is an epimorphism. Now we apply Corollary F.197 and see that every complex F of OX -modules has a functorial left resolution PF → F , where PF is a filtered colimit of bounded above complexes with flat components. Hence PF is K-flat by Lemma 21.93 and Remark 21.92 (6). Remark 21.95. The proof of Proposition 21.94 shows in fact the following more precise assertions. (1) One can choose PF to be the colimit of an inductive system (Pn )n∈N of bounded above complexes such that every component of each Pn is a direct sum of OX -modules of the form OU ⊆X . (2) If there exists n ∈ Z such that H i (F ) = 0 for all i > n, one has a quasi-isomorphism τ ≤n F → F and hence there exists a quasi-isomorphism Pτ ≤n F → F such that every component of Pτ ≤n F is a direct sum of OX -modules of the form OU ⊆X and such that Pτi ≤n F = 0 for all i > n. (21.20) Derived tensor product and Tor sheaves. We now want to derive the bi-functor given by the tensor product. We check that (the dual versions of) Conditions (ACI) and (ACII) of Section (F.51) are satisfied using the class of K-flat complexes. We already know that there exist functorial quasi-isomorphisms PF → F for all complexes F with PF a K-flat complex (Proposition 21.94). 1 : K(X) → D(X) Lemma 21.96. Let F be a complex of OX -modules. Denote by TF 2 : K(X) → D(X)) the composition of the functor (−) ⊗ F : K(X) → K(X) (resp. by TF (resp. of the functor F ⊗(−) : K(X) → K(X)) with the localization K(X) → D(X). 1 2 (1) For every K-flat complex P the functors TP and TP send quasi-isomorphisms to isomorphisms. (2) For every exact K-flat complex P the complex P ⊗ F is exact for all complexes F of OX -modules. 1 2 (3) Every K-flat complex P of OX -modules is left acyclic for TF and TF . 1 Proof. (1). By symmetry, it suffices to consider T 1 . The functor TP sends quasiisomorphisms to isomorphisms if and only if it maps exact complexes to the zero complex in D(X) (Example F.169). This shows Assertion (1). (2). Indeed, choose a quasi-isomorphism Q → F (Proposition 21.94). As P is K-flat, we may replace F by Q using (1). But then P ⊗ Q is exact because P is exact. (3). Let Q ′ → P be a quasi-isomorphism. By Proposition 21.94 there exists a quasi1 isomorphism Q → Q ′ with Q a K-flat complex. It suffices to show that by applying TF to the quasi-isomorphism Q → P we obtain an isomorphism in D(X).

193 Let C be the mapping cone of the quasi-isomorphism Q → P. Then C is exact (Remark F.85) and K-flat (Remark 21.92 (5)). Therefore C ⊗ F is again exact by (2). As (−) ⊗ F preserves distinguished triangles, we have a distinguished triangle Q ⊗ F −→ P ⊗ F −→ C ⊗ F −→ (Q ⊗ P)[1]. From its associated long cohomology sequence (Example F.138) we see that the homomorphism Q ⊗ F → P ⊗ F is a quasi-isomorphism, i.e., an isomorphism in D(X). We can now apply the general formalism of Section (F.51) and obtain left derivations of the tensor product. Proposition/Definition 21.97. Let (X, OX ) be a ringed space. The derived tensor product is the triangulated bi-functor ⊗L := ⊗L OX : D(X) × D(X) → D(X) which maps a pair (G , H ) of a complex G and a complex H with K-flat resolution PH → H to the tensor product G ⊗ PH (considered as an object of D(X)). For fixed G , the functors G ⊗L − and − ⊗L G are the left derived functors of the tensor product functors G ⊗ − and − ⊗ G . There is a natural equivalence between these two functors. The derived tensor product is associative in the obvious sense. Definition 21.98. Let F and G be complexes of OX -modules. For n ∈ Z we define the n-th Tor sheaf as the OX -module −n X Tor n (F , G ) := Tor O (F ⊗L G ). n (F , G ) := H

Remark 21.99. Let F and G be complexes of OX -modules. (1) Given a short exact sequence 0 → G ′ → G → G ′′ → 0 of complexes of OX -modules or, more generally, given a distinguished triangle G ′ → G → G ′′ → in D(X), there is a long exact Tor sheaf sequence

(21.20.1)

. . . −→ Tor n+1 (F , G ′ ) −→ Tor n+1 (F , G ) −→ Tor n+1 (F , G ′′ ) −→ Tor n (F , G ′ ) −→ Tor n (F , G ) −→ Tor n (F , G ′′ ) −→ Tor n−1 (F , G ′ ) −→ . . .

obtained as the long exact cohomology sequence of the distinguished triangle F ⊗L G ′ → F ⊗L G → F ⊗L G ′′ →. (2) Suppose there are m, n ∈ Z such that H i (F ) = 0 and H j (G ) = 0 for all i > n and all j > m. Then Tor k (F , G ) = 0 for all k < −n − m. i Indeed, we can choose a K-flat resolution PF → F with PF = 0 for all all i > n L (Remark 21.95 (2)). Then F ⊗ G = PF ⊗ G . By Lemma 21.96 (1) we may also replace G by the complex τ ≤m G since τ ≤m G → G is a quasi-isomorphism. But then (PF ⊗ G )k = 0 for all k > n + m by the definition of the tensor product of complexes. (3) Let F and G be OX -modules. Then Tor n (F , G ) = 0 for all n < 0 by (2) and Tor 0 (F , G ) = F ⊗OX G by the dual version of Remark F.200 (3) because ⊗ is right exact in each variable as functor (OX -Mod) → (OX -Mod).

21 Cohomology of OX -modules

194

Proposition 21.100. Let F be an OX -module. Then the following are equivalent. (i) F is a flat OX -module. X (ii) Tor O 1 (F , G ) = 0 for every OX -module G . OX (iii) Tor i (F , G ) = 0 for all i ≥ 1 and every complex of OX -modules G concentrated in degrees ≥ 0. Proof. If F is flat, then it is K-flat as a complex concentrated in degree 0. Hence for every complex of OX -modules we have F ⊗L G = F ⊗ G which has only zero coefficients in degrees ≤ 0 if G is concentrated in degrees ≥ 0. This shows “(i) ⇒ (iii)”. Clearly, (iii) implies (ii). Suppose that (ii) holds. If 0 → G ′ → G → G ′′ → 0 is an exact sequence of OX -modules, then by Remark 21.99 one obtains a long exact sequence · · · → Tor 1 (F , G ′′ ) → F ⊗ G ′ → F ⊗ G → F ⊗ G ′′ → 0 with Tor 1 (F , G ′′ ) = 0. In other words, F ⊗OX (−) is an exact functor, i.e., F is flat. Remark 21.101. If X consists of a single point and A := Γ(X, OX ), we identify the category of OX -modules with the category of A-modules and we obtain a triangulated bi-functor ⊗L A : D(A) × D(A) −→ D(A). For complexes F and G of A-modules we define Tor A-modules −i L TorA i (F, G) := H (F ⊗A G).

Suppose that F and G are A-modules. Then the above general definitions show that TorA K • of−1 one of the modules, i (F, G) can be calculated as follows. Choose a flat resolution −2 d −2 d say of F , in other words an exact sequence · · · −→ K −→ K −1 −→ K 0 −→ F −→ 0 with K i flat. Then TorA i (F, G) is the cohomology in degree −i of the complex d−2 ⊗id

d−1 ⊗id

· · · −→ K −2 ⊗A G −−−−−→ K −1 ⊗A G −−−−−→ K 0 ⊗A G −→ 0 −→ · · · By Proposition 21.100 an A-module M is flat if and only if TorA 1 (M, N ) = 0 for every A-module N . It is even enough to check that TorA 1 (M, A/I) = 0 for every finitely generated ideal I of A. Indeed, the exact sequence 0 → I → A → A/I → 0 yields the exact sequence 0 = Tor1 (M, A/I) −→ M ⊗ I −→ M. Hence M ⊗ I → M is injective for every finitely generated ideal and we conclude by Proposition B.15. Remark 21.102. Let x ∈ X. Then forming the stalk (−)x is an exact functor which commutes with forming tensor products and which sends K-flat complexes of OX -modules to K-flat complexes of OX,x -modules (Remark 21.92 (4)). Hence one has (21.20.2)

L (F ⊗L OX G )x = Fx ⊗OX,x Gx

for all complexes F and G of OX -modules. Again, since (−)x is exact, it commutes with passage to cohomology and we obtain a functorial isomorphism (21.20.3)

OX,x X Tor O (Fx , Gx ), p (F , G )x = Torp

p ∈ Z.

195 Remark 21.103. Let F and G be complexes in D− (X). Replacing F and G by Kflat bounded above complexes and applying Lemma F.112 yields two bounded spectral sequences, functorial in F and G , (21.20.4)

pq I E2 pq II E2

q n L = H p (F ⊗L OX H (G )) ⇒ H (F ⊗OX G ), n L = H q (H p (F ) ⊗L OX G ) ⇒ H (F ⊗OX G ).

Both induced filtrations on the limit term are finite. Remark 21.104. Let F and G be complexes in D(X). For all p, q ∈ Z there is an OX -linear map (21.20.5)

∪ : H p (F ) ⊗OX H q (G ) −→ H p+q (F ⊗L OX G )

defined as follows. We replace F or G by a K-flat complex. Then F ⊗L OX G = F ⊗OX G . Let α (resp. β) be a local section of H p (F ) (resp. of H q (G )) over an open subset U ⊆ X, represented locally by ai ∈ Ker(F p (Ui ) → F p+1 (Ui )) and bi ∈ Ker(G q (Ui ) → G q+1 (Ui )) for some open covering (Ui )i of U . Then ai ⊗ bi ∈ Ker((F ⊗ G )p+q (Ui ) −→ (F ⊗ G )p+q+1 (Ui )) yields a class ai ∪ bi ∈ H p+q (F ⊗ G )(Ui ). Then it is straightforward to check that the ai ∪ bi agree on all intersections Ui ∩ Uj and are the restriction of a unique element α ∪ β ∈ H p+q (F ⊗ G )(U ) which does not depend on the choice of (Ui )i or of the ai and bi . (21.21) Ext sheaves. Let X be a ringed space. Recall from Section (F.52) that we defined for complexes F and G of OX -modules and for all i ∈ Z the Γ(X, OX )-modules (21.21.1)

ExtiOX (F , G ) = HomD(X) (F , G [i]) ∼ = H i (R HomOX (F , G )).

Similarly to the second description, we define the Ext sheaves as the derived functors of the sheaf Hom functor: (21.21.2)

Ext i (F , G ) := H i (R Hom OX (F , G )).

There the following easy properties of Ext sheaves. Remark 21.105. (1) Let b, c ∈ Z. If F is in D(−∞,b] (X) and G is in D[c,∞) (X), then for p < c − b one has ExtpOX (F , G ) = 0 and Ext pOX (F , G ) = 0 by Remark F.218. (2) If U ⊆ X is an open subspace, then (21.21.3)

Ext iOX (F , G )|U = Ext iOU (F |U , G |U )

for all i ∈ Z by (21.18.9). For every OX -module F , one has an equality Γ(X, −) ◦ Hom(F , −) = Hom(F , −) as functors (OX -Mod) → (Γ(X, OX )-Mod). To obtain a derived version we will use the following lemma.

21 Cohomology of OX -modules

196

Lemma 21.106. Let (X, OX ) be a ringed space, let I be a K-injective complexes of OX -modules, and let P be a K-flat complex of OX -modules. Then Hom OX (P, I ) is a K-injective complex of OX -modules. Proof. Let F be an exact complex of OX -modules. By Remark 21.90 we have a functorial isomorphism HomK(X) (F , Hom OX (P, I )) ∼ = HomK(X) (F ⊗ P, I ) and F ⊗ P is exact because P is K-flat. We conclude by Proposition F.179 (2). Proposition 21.107. There is an isomorphism of triangulated bi-functors (21.21.4)

∼

R HomOX (−, −) −→ RΓ R Hom OX (−, −).

Proof. By Proposition F.176 it suffices to show that for a fixed complex F in C(X) the functor Hom OX (F , −) sends a K-injective complex I to a right Γ-acyclic complex. By Lemma 21.87 we may replace F by a K-flat resolution which exists by Proposition 21.94. Then Lemma 21.106 shows that HomOX (F , I ) is K-injective and in particular acyclic for every functor. In particular we find that for all complexes F and G of OX -modules we have (21.21.5)

H p (X, R Hom OX (F , G )) = ExtpOX (F , G )

for all p ∈ Z. Restricting the isomorphism (21.21.4) to complexes F that are bounded below in the opposite category and to bounded below complexes G we obtain the following Grothendieck spectral sequence by Proposition F.212. Corollary 21.108. (Local-to-global Ext spectral sequence) Let (X, OX ) be a ringed space, let F and G be complexes of OX -modules. Suppose that F is bounded above and that G is bounded below. Then we have a convergent spectral sequence E2pq = H p (X, Ext qOX (F , G )) ⇒ Extp+q OX (F , G ), which is functorial in F (contravariant) and G (covariant). (21.22) Derived functor of the inverse image functor. In this section we denote by f : X → Y a morphism of ringed spaces. We want to define the left derived functor of the pullback functor f ∗ . We first convince ourselves that K-flat complexes are acyclic for f ∗ . Remark 21.109. Let Q be a K-flat complex of OY -modules. (1) As we can test K-flatness on stalks (Remark 21.92 (4)) we see that its pullback f ∗ Q is a K-flat complex of OX -modules: Let x ∈ X. If G is an exact complex of OX,x -modules, then f ∗ (Q)x ⊗OX,x F = Qf (x) ⊗OY,f (x) F is again an exact complex of OX,x -modules.

197 (2) If Q is in addition exact, then f ∗ (Q)x = Qf (x) ⊗OY,f (x) OX,x is exact for all x ∈ X because of Lemma 21.96 (2). Hence f ∗ (Q) is also exact. Now the same argument as in the proof of Lemma 21.96 (3) shows that K-flat complexes are left f ∗ -acyclic. Therefore, using K-flat resolutions, we obtain the derived functor Lf ∗ of the inverse image functor f ∗ : Proposition/Definition 21.110. The pullback functor f ∗ : (OY -Mod) → (OX -Mod) has a left derived functor Lf ∗ : D(Y ) → D(X) which maps a complex G of OY -modules with K-flat resolution PG → G to f ∗ PG . We define for i ∈ Z Li f ∗ (G ) := H −i Lf ∗ (G ). Of course, if the morphism f is flat, then f ∗ is an exact functor and hence Lf ∗ = f ∗ (Example F.169). Remark 21.111. Let G be a complex of OY -modules. (1) For every exact sequence 0 → G ′ → G → G ′′ → 0 in C(Y ) or, more generally, for every distinguished triangle G ′ → G → G ′′ → in D(Y ) we obtain a long exact sequence of OX -modules (21.22.1)

. . . −→ Ln+1 f ∗ G ′′ −→ Ln f ∗ G ′ −→ Ln f ∗ G −→ Ln f ∗ G ′′ −→ . . .

(2) Suppose there exists n ∈ Z such that H i (G ) = 0 for all i > n. Then Li f ∗ G = 0 for all i < −n. (3) In particular, if G is an OY -module, then Li f ∗ G = 0 for i < 0 and L0 f ∗ G = f ∗ G . Proposition 21.112. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Then there is a natural isomorphism L(g ◦ f )∗ ∼ = Lf ∗ ◦ Lg ∗ of functors D(Z) → D(X). Proof. By Remark 21.109 (2), K-flat complexes are left acyclic for the inverse image functor. Hence it suffices to show that g ∗ sends K-flat complexes to K-flat complexes by Proposition F.176. This we have seen in Remark 21.109 (1). (21.23) Derived direct image and composition. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Recall that for bounded below complexes F of OX -modules we already have seen that the functorial morphism ∼

R(g∗ ◦ f∗ )(F ) −→ (Rg∗ ◦ Rf∗ )(F ) is an isomorphism (Proposition 21.41). We now want to extend this result to unbounded complexes. As f ∗ is left adjoint to f∗ we obtain by Proposition F.191 triangulated adjoint pairs of functors (Lf ∗ , Rf∗ ), (Lg ∗ , Rg∗ ), and (L(f ∗ ) ◦ L(g∗ ), R(g∗ ◦ f∗ )), where we identify L(f ∗ ) ◦ L(g∗ ) = L(f ∗ ◦ g ∗ ) by Proposition 21.112. One can show that this yields purely formally using Proposition F.191 an isomorphism of triangulated functors

21 Cohomology of OX -modules

198 ∼

R(g∗ ◦ f∗ ) −→ Rg∗ ◦ Rf∗ . But for concrete calculations it is sometimes helpful to know that this isomorphism is obtained by right G-acyclicity of f∗ F for K-injective complexes F (see Proposition F.176). To show this we start with a lemma that will also be used later. Lemma 21.113. Let f : X → Y be a morphism of ringed spaces. Let G be a K-flat complex of OY -modules and let F be a K-injective complex of OX -modules. Then the natural morphism in D(Γ(Y, OY )) γ : HomOY (G , f∗ F ) −→ R HomOY (G , f∗ F ) is an isomorphism. Proof. By the last assertion of Proposition F.191, the morphisms H i (γ) are isomorphisms
(F.18.2) H i HomOY (G , f∗ F ) = HomK(Y ) (G , f∗ F [i]) −→ HomD(Y ) (G , f∗ F [i])
(F.52.5) = ExtiOY (G , f∗ F [i]) = H i R HomOY (G , f∗ F [i]) for all i ∈ Z. Hence γ is an isomorphism in D(Γ(Y, OY )). Proposition 21.114. Let f : X → Y be a morphism of ringed spaces. Let V ⊆ Y be open. (1) For every K-injective complex F of OX -modules f∗ F is right Γ(V, −)-acyclic. (2) The canonical morphism of functors D(X) → D(Γ(V, OY )) RΓ(f −1 (V ), −) −→ RΓ(V, −) ◦ Rf∗ is an isomorphism. Proof. Assertion (2) follows from (1) by Proposition F.176. To show (1) recall that we have Γ(V, −) = HomOY (OV ⊆Y , −),

RΓ(V, −) = R HomOY (OV ⊆Y , −)

by Remark 21.17. As OV ⊆Y is a flat OY -module, i.e., a K-flat complex concentrated in degree 0, the map Γ(V, f∗ F ) → RΓ(V, f∗ F ) is an isomorphism by Lemma 21.113. This shows that f∗ F is right Γ(V, −)-acyclic by Corollary F.174 (1). Proposition 21.115. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. (1) For every K-injective complex F of OX -modules, f∗ F is right g∗ -acyclic. (2) The canonical morphism of functors D(X) → D(Z) R(g∗ ◦ f∗ ) −→ Rg∗ ◦ Rf∗ is an isomorphism. Proof. Let s : f∗ F → G be a quasi-isomorphism in K(Y ) and let t : G → I be a quasi-isomorphism to a K-injective complex in K(Y ). For (1) it suffices to show that (*)

g∗ (t ◦ s) : g∗ f∗ F → g∗ I

199 is a quasi-isomorphism in K(Z). Let W ⊆ Z be open. Then f∗ F is right Γ(g −1 (W ), −)acyclic by Proposition 21.114 and I is right Γ(g −1 (W ), −)-acyclic because it is K-injective. Hence Γ(g −1 (W ), −) sends t ◦ s to a quasi-isomorphism (g∗ f∗ F )(W ) = (f∗ F )(g −1 (W )) −→ I (g −1 (W ) = (g∗ I )(W ) by Lemma F.172. This shows that (*) is a quasi-isomorphism. Again, Assertion (2) follows from (1) by Proposition F.176. By applying Proposition 21.114 and Proposition 21.115 to f : (X, OX ) → (X, ZX ) we obtain an unbounded version of Corollary 21.43: Corollary 21.116. Let X be a ringed space and let F be a complex of OX -modules. Denote by F ab the underlying complex of abelian sheaves. Then the underlying abelian group of the Γ(X, OX )-module H i (X, F ) is H i (X, F ab ) for all i ∈ Z. More generally, let g : X → Z be a morphism of ringed spaces. Then the higher direct images Ri f∗ F of the complex F of OX -modules and of abelian sheaves are identical as abelian sheaves.

Relations between derived functors For any ringed space X and any morphism f : X → Y of ringed spaces we have now defined functors Rf∗ , Lf ∗ , R Hom OX (−, −), and − ⊗L OX −. Next we want to show relations between this functors. For instance, it follows formally from the fact that f ∗ is left adjoint to f∗ that Lf ∗ is left adjoint to Rf∗ (Proposition F.191), in other words, we have a functorial isomorphism of Γ(Y, OY )-modules (*)

HomD(X) (Lf ∗ G , F ) ∼ = HomD(Y ) (G , Rf∗ F )

for F ∈ D(X) and G ∈ D(Y ). We want to “upgrade” this isomorphism to a functorial isomorphism in D(Y ) (**)

Rf∗ Hom OX (Lf ∗ G , F ) ∼ = Hom OY (G , Rf∗ F ).

Here “upgrade” should mean the following. If we apply RΓ(Y, −) to (**), we obtain a functorial isomorphism in D(Γ(Y, OY )) R HomOX (Lf ∗ G , F ) ∼ = R HomOY (G , Rf∗ F ) since we have RΓ(Y, −) ◦ Rf∗ = RΓ(X, −) by Proposition 21.114 (2) and RΓ ◦ R Hom = R Hom by Proposition 21.107. If we further apply H 0 (−) we then should obtain (*) by (F.52.6). We also want a similar upgrade to the ⊗-Hom-adjunction. This part starts by showing that Lf ∗ respects derived tensor products which is easy to see since both are defined via K-flat resolutions. Then we give a general recipe how to upgrade an adjunction to a sheaf theoretic version. This is then applied to the both cases mentioned above.

21 Cohomology of OX -modules

200

After this, we add some further formal constructions. We define for every commutative square of ringed spaces a derived base change morphism. The deep question when this base change morphism is an isomorphism will be only studied for schemes in the last part of the next chapter. The same holds for the projection formula which should be part of our “formalism of four functors”. We also define the cup product, again postponing any non-formal questions to the next two chapters. This part is concluded with a remark what it would mean to extend our “formalism of four functors” to a quasi-coherent six-functor formalism for schemes in the sense of Grothendieck – and why for schemes the best one can hope for is a “five-functor formalism”. This fifth functor, the exceptional inverse image, will be defined and studied in Chapter 25 on Grothendieck duality. (21.24) Derived inverse images and ⊗L . Let f : X → Y be a morphism of ringed spaces. Recall that for OY -modules F and G ∼ one has a functorial isomorphism f ∗ F ⊗OX f ∗ G → f ∗ (F ⊗OY G ). As f ∗ commutes with direct sums we also obtain isomorphisms (21.24.1)

∼

f ∗ F ⊗OX f ∗ G −→ f ∗ (F ⊗OY G ).

of complexes of OX -modules, for all complexes F and G of OY -modules. Proposition 21.117. Let f : X → Y be a morphism of ringed spaces. Then there is a unique isomorphism of triangulated bi-functors in F , G ∈ D(Y ) ∼

∗ L ∗ Lf ∗ (F ⊗L OY G ) −→ Lf F ⊗OX Lf G

which is compatible with the corresponding non-derived isomorphism, i.e., such that the following diagram commutes f ∗ (F ⊗OY G ) o (21.24.2)

Lf ∗ (F ⊗L OY G )

(21.24.1) ∼ =

 f ∗ F ⊗OX f ∗ G o



∼ =

∗ Lf ∗ F ⊗L OX Lf G

Proof. We may assume that F and G are K-flat. Then f ∗ F and f ∗ G are K-flat (Remark 21.109 (1)) and f ∗ F ⊗ f ∗ G is K-flat (Remark 21.92 (3)). Therefore the horizontal arrows in (21.24.2) are isomorphisms. (21.25) Adjointness and derived functors. Let us consider the following situation. Let f : X → Y be a morphism of ringed spaces and let Φ : (OX -Mod) → (OY -Mod) and Ψ : (OY -Mod) → (OX -Mod) be additive functors. Suppose that for all OX -modules F and OY -modules G there is a functorial isomorphism of OY -modules f∗ Hom OX (Ψ(G ), F ) ∼ = Hom OY (G , Φ(F )). Applying Γ(Y, −) one sees that in particular Ψ is a left adjoint functor of Φ. By Remark F.190 we obtain for F in C(X) and G in C(Y ) a functorial isomorphism of complexes of OY -modules

201 f∗ Hom OX (Ψ(G ), F ) ∼ = Hom OY (G , Φ(F )).

(21.25.1)

Now let us also suppose that K-flat complexes OY -modules are left Ψ-acyclic. Then we can apply Proposition F.191 and see that LΨ : D(Y ) → D(X) is left adjoint to RΦ : D(X) → D(Y ), i.e., we obtain a functorial isomorphism HomD(X) (LΨ(G ), F ) ∼ = HomD(Y ) (G , RΦ(F ))

(21.25.2)

of Γ(Y, OY )-modules. One would like to “sheafify and derive” this isomorphism as follows. Remark 21.118. In the situation above suppose that the canonical morphisms f∗ Hom OX (Ψ(G ), F ) −→ Rf∗ Hom OX (Ψ(G ), F ), Hom OY (G , Φ(F )) −→ R Hom OY (G , Φ(F )) are both isomorphisms if F is a K-injective complex and G is a K-flat complex. (1) Then there is for all F in D(X) and G in D(Y ) an isomorphism in D(Y ) (21.25.3)

ρ : Rf∗ R Hom OX (LΨ(G ), F ) ∼ = R Hom OY (G , RΦ(F ))

which is triangulated functorial in F and in G and which is the unique such isomorphism such that for all F and G the following diagram commutes f∗ Hom OX (Ψ(G ), F ) (21.25.4)

(21.25.1) ∼ =

 Hom OY (G , Φ(F ))

/ Rf∗ R Hom O (LΨ(G ), F ) X ρ

 / R Hom O (G , RΦ(F )). Y

Indeed, replacing F by a K-injective resolution and G by a K-flat resolution we may assume that F is K-injective and G is K-flat. Then the upper horizontal line in (21.25.4) is the composition of isomorphisms (a)

(b)

f∗ Hom OX (Ψ(G ), F ) −→ Rf∗ Hom OX (Ψ(G ), F ) −→ Rf∗ R Hom OX (Ψ(G ), F ) (c) −→ Rf∗ R Hom OX (LΨ(G ), F ), where (a) is an isomorphism by hypothesis, (b) is an isomorphism because F is Kinjective, and (c) is an isomorphism because G is K-flat. Similarly, the lower horizontal line in (21.25.4) is the composition ∼ Hom OY (G , Φ(F )) −→ R Hom OY (G , Φ(F )) ∼ = Hom OY (G , RΦ(F )),

the first morphism is an isomorphism by hypothesis and the second because F is K-injective. (2) Next, we can apply RΓ(Y, −) to (21.25.3). As RΓ(Y, −) ◦ Rf∗ = RΓ(X, −) (Proposition 21.114) and RΓ ◦ R Hom = R Hom (Proposition 21.107) we obtain an isomorphism in D(Γ(Y, OY )) (21.25.5)

σ : R HomOX (LΨ(G ), F ) ∼ = R HomOY (G , RΦ(F ))

which is triangulated functorial in F in D(X) and G in D(Y ) and which is the unique such isomorphism such that for all F and G the following diagram commutes

21 Cohomology of OX -modules

202

/ R HomO (LΨ(G ), F ) X

HomOX (Ψ(G ), F ) ∼ =

(21.25.6)

ρ

 / R HomO (G , RΦ(F )). Y

 HomOY (G , Φ(F ))

We will now specialize the previous remark to adjointness of R Hom and ⊗L and to Rf∗ and Lf ∗ . (21.26) Relations between R Hom and ⊗L . Let X be a ringed space. By Section (7.5) one has for arbitrary OX -modules F , G , and H functorial isomorphisms of OX -modules ∼

Hom OX (F ⊗OX G , H ) −→ Hom OX (F , Hom OX (G , H )) and functorial homomorphisms Hom OX (F , G ) ⊗OX H −→ Hom OX (F , G ⊗OX H ). We will now construct derived versions of these isomorphisms and homomorphisms. Recall that the second map is an isomorphisms if F is finite locally free (Proposition 7.7). Theorem 21.119. (Adjointness of R Hom and ⊗L ) Let X be a ringed space, and let F , G , H ∈ D(OX -Mod). (1) There is an isomorphism R Hom(F ⊗L G , H ) ∼ = R Hom(F , R Hom(G , H )),

(21.26.1)

which is triangulated-functorial in F , G and H . (2) There is a natural isomorphism R Hom(F ⊗L G , H ) ∼ = R Hom(F , R Hom(G , H )),

(21.26.2)

which is triangulated-functorial in F , G and H . Proof. Replacing G by a K-flat complex isomorphic to G in D(X), we may assume that G is K-flat. We apply Remark 21.118 to f = idX , Φ = Hom(G , −) and Ψ = − ⊗ G . Hence it suffices to show that Hom(F , Hom(G , I )) −→ R Hom(F , Hom(G , I )) is an isomorphism if I is K-injective. This holds because Hom(G , I ) is K-injective (Lemma 21.106). By Remark 21.118, the isomorphism (21.26.1) is compatible with the non-derived adjointness, i.e., it is the unique such isomorphism making the following diagram commutative Hom(F ⊗ G , H ) ∼ =

 Hom(F , Hom(G , H ))

/ R Hom(F ⊗ G , H )

/ R Hom(F ⊗L G , H )

 / R Hom(F , Hom(G , H ))

 / R Hom(F , R Hom(G , H )),

α

203 where the left vertical arrow is the isomorphism (21.19.4) and the horizontal arrows are induced by the canonical morphism F → RF and LF → F for functors F that are right derivable and left derivable, respectively. Similarly, the isomorphism (21.26.2) is compatible with non-derived adjointness. Remark 21.120. Let X be a ringed space, and let F , G , H ∈ D(X). (1) Applying the functor H 0 (−) to (21.26.2) we obtain by (F.52.6) a functorial isomorphism of Γ(X, OX )-modules HomD(X) (F ⊗L G , H ) ∼ = HomD(X) (F , R Hom(G , H )).

(21.26.3)

This shows in particular that for every complex G in D(X) the functor (−) ⊗L G is left adjoint to the functor R Hom(G , −). (2) In particular, (−)⊗L G commutes with arbitrary colimits and R Hom(G , −) commutes with arbitrary limits if they exist. For instance arbitrary products and direct sums exist by Lemma F.188. The functor R Hom OX also sends direct sums in D(X) in the first component to products. Corollary 21.121. Let X be a ringed space, H be an object of D(X), and let (Gi )i be a family of objects in D(X). Then there is a functorial isomorphism Y M R Hom OX (Gi , H ). (21.26.4) R Hom OX ( Gi , H ) ∼ = i

i

of objects in D(X). Proof. Let F be an object in D(X). Then we have functorial isomorphisms Y Y HomD(X) (F , R Hom(Gi , H )) = HomD(X) (F , R Hom(Gi , H )) i

i

=

Y

HomD(X) (F ⊗L Gi , H )

i

= HomD(X) (F ⊗L

M

Gi , H )

i

= HomD(X) (F , R Hom(

M

Gi , H ))

i

and the claim follows from the Yoneda Lemma. Remark 21.122. Let X be a ringed space, and let E , F , and G be complexes in D(X). We will construct a functorial map (21.26.5)

L R Hom OX (E , F ) ⊗L OX G −→ R Hom OX (E , F ⊗ G )

in D(X) functorial in E , F , and G . We may replace F by a K-injective complex I and G by a K-flat complex K . Also choose a quasi-isomorphism of the tensor complex I ⊗ K to a K-injective complex J . It is easy to see that there is a functorial map of complexes Hom OX (E , I ) ⊗OX K −→ Hom OX (E , I ⊗ K ).

21 Cohomology of OX -modules

204

The left hand side computes R Hom OX (E , F ) ⊗L OX G . Now by composing with the map Hom OX (E , I ⊗ K ) → Hom OX (E , J ) we get the desired map because Hom OX (E , J ) computes R Hom OX (E , F ⊗L G ). Remark 21.123. Let X be a ringed space and let E , F , and G be complexes in D(X). We will construct a functorial “derived composition” map in D(X) R Hom(F , G ) ⊗L R Hom(E , F ) −→ R Hom(E , G ).

(21.26.6)

For this replace F and G be a K-injective complexes I and J , respectively. Choose also a quasi-isomorphism K → Hom(E , I ) with K a K-flat complex. Consider the composition Hom(I , J ) ⊗ K −→ Hom(I , J ) ⊗ Hom(E , I ) −→ Hom(E , J ), where the second map is given by composition. The source calculates R Hom(F , G ) ⊗L R Hom(E , F ) and the target calculates R Hom(E , G ). Setting E = OX and using that R Hom OX (OX , F ) = F (21.18.10) we deduce from (21.26.6) in particular a functorial derived evaluation map (21.26.7)

R Hom(F , G ) ⊗L F −→ G .

Remark 21.124. Let X be a ringed space and E , G ∈ D(X) be complexes. Then there is a functorial map (21.26.8)

G −→ R Hom OX (R Hom OX (G , E ), E ).

Indeed, the evaluation map R Hom(G , E ) ⊗L G → E (21.26.7) corresponds to (21.26.8) by applying H 0 to (21.26.2). If F is a further complex in D(X), then we obtain by composing (21.26.8) with the derived composition map (21.26.6) a functorial map R Hom OX (E , F ) ⊗L OX G (21.26.9)

−→ R Hom OX (E , F ) ⊗L OX R Hom OX (R Hom OX (G , E ), E ) −→ R Hom OX (R Hom OX (G , E ), F ).

Remark 21.125. Let f : X → Y be a morphism of ringed spaces, let F and G be complexes in D(Y ). We will construct a functorial map (21.26.10)

Lf ∗ R Hom OY (F , G ) −→ R Hom OX (Lf ∗ F , Lf ∗ G )

in D(X). By adjointness of R Hom and derived tensor product (21.26.3) it suffices to construct a map ∗ L ∗ ∗ Lf ∗ (R Hom OY (F , G ) ⊗L OY F ) = Lf R Hom OY (F , G ) ⊗OX Lf F −→ Lf G ,

where the first equality holds by Proposition 21.117. This is done by applying Lf ∗ to (21.26.7).

205 (21.27) Relations between Rf∗ and Lf ∗ . Next we look at the adjoint pair (f ∗ , f∗ ). Proposition F.191 immediately implies that Lf ∗ is left adjoint to Rf∗ : Proposition 21.126. Let f : X → Y be a morphism of ringed spaces. Let F ∈ D(X) and G ∈ D(Y ). There is a unique isomorphism HomD(X) (Lf ∗ G , F ) ∼ = HomD(Y ) (G , Rf∗ F ), which is triangulated functorial in F and G and compatible with non-derived adjointness. Remark 21.127. The unit and counit of the adjunction are functorial morphisms G → Rf∗ Lf ∗ G ,

Lf ∗ Rf∗ F → F

which are usually not isomorphisms. However, if f = j : U → Y is the embedding of an open subspace, then the counit Lj ∗ Rj∗ F → F is an isomorphism for all F in D(U ). Indeed, if F → I is a K-injective resolution, then Lj ∗ Rj∗ F ∼ = F, =I ∼ = Lj ∗ j∗ I = j ∗ j∗ I ∼ where we have Lj ∗ = j ∗ because j is flat. We also apply Remark 21.118 to (f ∗ , f∗ ). Theorem 21.128. Let f : X → Y be a morphism of ringed spaces. Let F ∈ D(X), G ∈ D(Y ). (1) There is a unique isomorphism Rf∗ R Hom X (Lf ∗ G , F ) ∼ = R Hom Y (G , Rf∗ F ) which is compatible with the non-derived adjointness of f∗ and f ∗ and which is triangulated functorial in F and G . (2) There is a unique isomorphism R HomOX (Lf ∗ G , F ) ∼ = R HomOY (G , Rf∗ F ) which is compatible with the non-derived adjointness of f∗ and f ∗ and which is triangulated functorial in F and G . Proof. We apply Remark 21.118 to the functors Φ = f∗ and Ψ = f ∗ . Hence we have to show that for F a K-injective complex and G a K-flat complex the morphisms (*) (**)

f∗ Hom OX (f ∗ (G ), F ) −→ Rf∗ Hom OX (f ∗ (G ), F ), Hom OY (G , f∗ (F )) −→ R Hom OY (G , f∗ (F ))

are isomorphisms. As f ∗ G is again K-flat (Remark 21.109), Lemma 21.106 shows that Hom OX (f ∗ (G ), F ) is K-injective. Hence (*) is an isomorphism. Let f∗ F → I be a K-injective resolution, so R Hom OY (G , f∗ (F )) = Hom OY (G , I ) and to see that (**) is an isomorphism it suffices to see that for all open subsets V ⊆ Y the morphism (**) yields an isomorphism

21 Cohomology of OX -modules

206

α : HomOV (G |V , f∗ F |V ) = Γ(V, Hom OY (G , f∗ F )) −→ Γ(V, Hom OY (G , I )) = HomOV (G |V , I |V ). Setting g := f |f −1 (V ) : f −1 (V ) → V this follows from the commutative diagram / HomO (G |V , I |V ) V

α

HomOV (G |V , f∗ F |V )

∼ = β

HomOV (G |V , g∗ (F |f −1 (V ) ))

 / RHomO (G |V , g∗ (F |f −1 (V ) )) V

γ ∼ =

where β is an isomorphism because g∗ (F |f −1 (V ) ) = (f∗ F )|V → I |V is a K-injective resolution (Lemma 21.12) and γ is an isomorphism by Lemma 21.113, again using that restrictions of K-injective complexes to open subspaces are K-injective. (21.28) The base change morphism. We consider a commutative diagram X′ (21.28.1)

u′

/X

u

 /Y

f′

 Y′

f

of morphisms of ringed spaces. Definition and Remark 21.129. Given a diagram as above, there exists a canonical base change morphism of functors D(X) → D(Y ′ ) Lu∗ Rf∗ −→ Rf∗′ L(u′ )∗ . defined in the following equivalent ways. (i) By adjunction between L(f ′ )∗ and Rf∗′ (Proposition 21.126) and because forming the derived pullback is compatible with composition (Proposition 21.112), specifying the base change morphism is the same as specifying a morphism L(u′ )∗ Lf ∗ Rf∗ = L(f ′ )∗ Lu∗ Rf∗ → L(u′ )∗ , and to get this morphism, we can simply apply the functor L(u′ )∗ to the adjunction counit map Lf ∗ Rf∗ → id. (ii) By adjunction between Lu∗ and Ru∗ , defining the base change morphism is equivalent to specifying a morphism Rf∗ → Ru∗ Rf∗′ Lu′∗ = Rf∗ Ru′∗ Lu′∗ which we can get by applying Rf∗ to the adjunction unit map id → Ru′∗ Lu′∗ . It is a non-trivial fact that these definitions are equivalent. For the proof we refer O Exp. XVII, 2.1.3. to [Lip2] O X (3.7.2) or, for a much more general statement, to [SGA4] It is an interesting question, under which conditions the canonical base change morphism is an isomorphism. For morphisms of schemes we will study this question in Theorem 22.99. (21.29) Cup Product. Let f : X → Y be a morphism of ringed spaces. For complexes F and G of OX -modules we will construct a morphism in D(Y ), called the relative cup product

207 (21.29.1)

L Rf∗ F ⊗L OY Rf∗ G −→ Rf∗ (F ⊗OX G )

as follows. By definition it is the map adjoint to the composition ∼

∗ L ∗ L Lf ∗ (Rf∗ F ⊗L OY Rf∗ G ) −→ Lf Rf∗ F ⊗OX Lf Rf∗ G −→ F ⊗OX G ,

where the first isomorphism is given by compatibility of derived pullback with tensor product (Proposition 21.117) and the second morphism is induced by the counit Lf ∗ ◦ Rf∗ −→ id. If Y = (∗, A), i.e., the underlying topological space of Y consists of a single point and A is the ring of global sections of Y , then (21.29.1) is a morphism in D(A), simply called cup product (21.29.2)

L RΓ(X, F ) ⊗L A RΓ(X, G ) −→ RΓ(X, F ⊗OX G ).

It induces for all p, q ∈ Z an A-bilinear map, also called cup product (21.29.3)

∪ : H p (X, F ) × H q (X, G ) −→ H p+q (X, F ⊗L OX G )

as follows. By Remark 21.104 we have an A-linear map H p (X, F ) ⊗A H q (X, G ) −→ H p+q (RΓ(X, F ) ⊗L A RΓ(X, G )) which we compose with (21.29.2) to obtain (21.29.3). It can also be described as follows (we leave it as Exercise 21.27 to check this). Fix cohomology classes α ∈ H p (X, F ) = ExtpOX (OX , F ) = HomD(X) (OX [−p], F ), β ∈ H q (X, G ) = HomD(X) (OX [−q], G ), where the equalities are given by (21.4.4) and by our definition of the Ext modules. Using OX [−p] ⊗L OX [−q] = OX [−p − q], the functoriality of the derived tensor product yields an element p+q α ∪ β ∈ HomD(X) (OX [−p − q], F ⊗L (X, F ⊗L OX G ) = H OX G ).

Remark 21.130. Let f : X → Y be a morphism of ringed spaces and let B be a differential graded OY -algebra over X (Section (17.9)). Considering them as complexes of f −1 (OY )-modules we obtain a homomorphism in D(X, f −1 (OY )) B ⊗L f −1 (OY ) B −→ B ⊗f −1 (OY ) B → B, where the first map is the canonical map from the derived tensor product to the usual tensor product of complexes of OX -modules and where the second map is the multiplication on B. Viewing f as a morphism of ringed spaces (X, f −1 (OY )) → (Y, OY ), we can apply Rf∗ to this map and precompose with the relative cup product (21.29.1). We obtain a multiplication m : Rf∗ B ⊗L OY Rf∗ B −→ Rf∗ B. By Remark 21.104 we also deduce for all p, q ∈ Z maps of OY -modules Rp f∗ B ⊗ Rq f∗ B −→ Rp+q f∗ B. The of the tensor product of complexes (F.19.3) implies that this defines on L associativity p the structure of a Z-graded OY -algebra. If the differential graded algebra R f B ∗ p∈Z L B is (strictly) graded commutative, then (F.19.2) shows that p∈Z Rp f∗ B is (strictly) graded commutative.

21 Cohomology of OX -modules

208

If B is strictly graded commutative, then the identity of R1 f∗ B induces a homomorphism of strictly graded commutative graded OY -algebras M V• 1 (21.29.4) R f B −→ R p f∗ B ∗ OY p∈Z

by Proposition 17.50. If Y = (∗, A) for a ring A, then the above maps correspond to a multiplication map RΓ(X, B) ⊗L R RΓ(X, B) −→ RΓ(X, B) L in D(A) and induce the structure of a graded A-algebra on p∈Z H p (X, B). It is (strictly) graded commutative if B is it. Example 21.131. Let X be a ringed space, let A be a ring, and let A → Γ(X, OX ) be a ring homomorphism, i.e., we are given a morphism of ringed spaces X → (∗, A). We can consider OX as a strictly graded commutative differential graded A-algebra over X concentrated in degree 0. By Remark 21.130 we obtain the structure of a strictly graded commutative graded A-algebra on M H • (X, OX ) := H p (X, OX ) p≥0

and a canonical map of graded A-algebras (21.29.5)

V•

H 1 (X, OX ) −→ H • (X, OX ).

(21.30) Formalism of six functors. Until now we have studied four functors on derived categories: For a morphism f : X → Y we have defined an adjoint pair Lf ∗ : D(Y ) o

/ D(X) : Rf ∗

and for every ringed space X bi-functors ⊗L : D(X) × D(X) −→ D(X),

R Hom : D(X)opp × D(X) → D(X)

such that ⊗L is associative and symmetric and with unit (OX in this case) up to natural isomorphism which are subject to certain coherence conditions and such that (−) ⊗L G is left adjoint to R Hom(G , −). Moreover Lf ∗ is monoidal, i.e., compatible with derived tensor product and preserving the unit object (again up to natural isomorphisms satisfying certain coherence conditions). Functors analogous to these exist for many cohomology theories, and their natural properties and compatibilities are codified by Grothendieck’s Six Functor Formalism. For instance ´etale cohomology and the theory of D-modules admit all six functors, for other theories such as cohomology for quasi-coherent sheaves as discussed in this book, only part of the formalism can be realized. The six functor formalism predicts, in addition to the four functors listed above, the existence of another adjoint pair f! : D(X) o

/ D(Y ) : f !

209 for “some” morphisms f : X → Y of ringed spaces. In our context, that of Algebraic Geometry, “some” should mean morphisms of schemes that are “sufficiently finite”, e.g., separated morphisms of finite type between noetherian schemes. Note that it is not required that f ! or f! are derived functors of functors between categories of modules over the structure sheaves. The functor f! is called direct image with proper support. The functor f ! is called exceptional inverse image and is usually not the (left) derived functor of any functor (OY -Mod) → (OX -Mod). These six functors should satisfy the following properties, which we deliberately formulate somewhat vaguely. (1) The formation of Lf ∗ , Rf∗ , f ! and of f! is compatible with compositions of morphisms. (2) There is a morphism of functors αf : f! → Rf∗ , compatible with compositions, which is an isomorphism if f is proper. (3) For any ´etale “sufficiently finite” morphism f : X → Y there exists an isomorphism, ∼ Lf ∗ −→ f ! , compatible with compositions. (4) For any “sufficiently finite” cartesian diagram of morphisms of schemes X′

f′

g′

 X

f

/ Y′  /Y

g

there exist natural base change isomorphisms ∼

(BC1)

Lg ∗ ◦ f! −→ f!′ ◦ Lg ′∗ ,

(BC2)

Rg∗′ ◦ f ′! −→ f ! ◦ Rg∗ .

∼

(5) For any “sufficiently finite” morphism f : X → Y there exist natural isomorphisms (Projection formula)

∼

(f! F ) ⊗L G −→ f! (F ⊗L Lf ∗ G ), ∼

Hom OY (f! F , G ) −→ Rf∗ Hom OX (F , f ! G ), ∼

f ! Hom OY (G , G ′ ) −→ Hom OX (Lf ∗ G , f ! G ′ ). (6) For any closed immersion i : Z → X with complementary open immersion j : U := X \ Z → X the units and counits, respectively, of the relevant adjunctions yield a distinguished triangle j! j ! −→ id −→ i∗ Li∗ −→ j! j ! [1] For noetherian schemes X and for (bounded below) complexes of OX -modules with quasi-coherent cohomology we will establish a “formalism of five functors”, i.e., we construct all functors as above except f! and show that they satisfy these properties. The construction of f ! will be given in Chapter 25. By construction (Definition 25.61), f ! will also be right adjoint to Rf∗ for proper morphisms f of noetherian schemes. Let us explain why we cannot expect a functor f! in this setting. In fact the above properties would determine f! for every separated finite type morphism f : X → Y of noetherian scheme because by Nagata compactification (Theorem 12.70) we can factorize f as an open immersion j followed by a proper morphism f¯. Then we necessarily have f! = f¯! ◦ j! = Rf¯∗ ◦ j! by (2), where j! is left adjoint to j ! = Lj ∗ = j ∗ by (3). So f! would be determined by the functors that we already constructed if it existed.

21 Cohomology of OX -modules

210

But this cannot be the case since j ∗ cannot be right adjoint to another functor. Otherwise j ∗ would commute with arbitrary limits. In particular, for any index set I, for any ring A and for every f ∈ A we would have ! Y Y A ⊗A Af = Af . I

I

But on the left side powers of f in the denominator are bounded independent of i ∈ I and on the right side these powers can get arbitrarily large. Hence we cannot expect such an equality and a full 6-functor formalism for quasi-coherent sheaves4 . Note that we have already shown (1) for Rf∗ and Lf ∗ (Proposition 21.115 and Proposition 21.112) and that we already constructed a morphism as in (BC1) (Proposition 21.129) with f! replaced by Rf∗ (as we would expect at least for proper morphisms for which f! = Rf∗ should hold). Finally, we already established a variant of a distinguished triangle as in (6) for abelian sheaves on topological spaces (Proposition 21.62).

Perfect and pseudo-coherent complexes We now define finiteness properties in the derived category of OX -modules: Perfect and pseudo-coherent complexes. Here one can think of a perfect complex as being the derived analogue of an OX -module that is finite locally free (see Remark 21.149 below for a justification of this slogan) and of a pseudo-coherent complex as the derived analogue of being “sufficiently finitely generated”. We will also introduce the notion to be of finite tor-amplitude. One of the main results in this part will be that a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-amplitude (Theorem 21.174). For instance, if A is a ring, then an A-module M considered as an object in D(A) is perfect (resp. pseudo-coherent) if and only if M admits a finite left resolution (resp. a left resolution) by finite projective modules. It is of finite tor-amplitude if and only if it admits a finite left resolution by flat modules. (21.31) Perfect complexes. We fix a ringed space (X, OX ). Our basic finiteness condition for a complex will be to consist only of direct summands of finite free modules. Let us recall some properties of such modules. Remark 21.132. Let F be a direct summand of a finite free OX -module. (1) The OX -module F is flat and of finite presentation. Conversely, one can show that every flat OX -module of finite presentation is locally on X a direct summand of a finite free OX -module (Exercise 21.28). (2) If (X, OX ) is a locally ringed space, F is finite locally free (Proposition 7.41). (3) Let A be a ring. Then a direct summand of a finite free A-module is the same as a finite projective A-module.

4

By enlarging the category of schemes it is possible to define a functor f! using the formalism of condensed mathematics, see [Scho] X and [Man] X 2.9.

211 Definition 21.133. Let F be a complex of OX -modules. (1) The complex F is called strictly perfect if F p is zero for all but finitely many p and F p is a direct summand of a finite free OX -module for all p. (2) The complex F is called perfect if there exists an open covering (Ui )i of X such that for all i there exists a strictly perfect complex Ei of OUi -modules and a quasi-isomorphism Ei → F |Ui . The restriction of a (strictly) perfect complex to an open subspace is again (strictly) perfect. We want to define an object in the derived category to be perfect if it is perfect as a complex of OX -modules. But a priori this definition is problematic because it is not clear that being perfect is preserved under isomorphisms in D(X). To see that this is indeed the case we need the following preparative lemma. Lemma 21.134. Let E , F , and G be complexes of OX -modules, where E is strictly perfect. Let m ∈ Z. (1) Let α : E → F be a morphism of complexes. Suppose that H p (F ) = 0 for p ≥ m and that E p = 0 for p < m. Then there exists an open covering (Ui )i of X such that α|Ui is homotopic to 0. (2) Let α : E → G and u : F → G be maps in C(X). Suppose that E p = 0 for all p < m and that u is an m-isomorphism (Definition F.159). Then locally on X there exists a map β : E → F such that α and u ◦ β are homotopic. (3) For any morphism α : E → F in D(X) there exists an open covering (Ui )i of X such that α|Ui is given by a morphism of complexes E |Ui → F |Ui in C(Ui ). / [a, b]. We Proof. Let us show (1). Let a ≤ b be integers such that E p = 0 for all p ∈ show (1) by induction on b − a. Suppose that a = b =: n. Then E = E n [−n] for a direct summand E n of a finite free OX module and an integer n ≥ m. We can consider α as a morphism E n → Ker(F n → F n+1 ). We have to find locally on X a map h : E n → F n−1 that lifts α. For this we may r assume that E n = OX is a free module. Then α corresponds to r global sections of Ker(F n → F n+1 ). As H n (F ) = 0, the map F n−1 → Ker(F n → F n+1 ) is surjective. Hence the r global sections can be locally lifted to F n−1 defining h locally. In general there is a split exact sequence of complexes 0 −→ E b [−b] −→ E −→ σ ≤b−1 E −→ 0 yielding a distinguished triangle in K(X). As HomK(X) (−, F ) is a cohomological functor (Example F.137), we have an exact sequence HomK(X) (σ ≤b−1 E , F ) −→ HomK(X) (E , F ) −→ HomK(X) (E b [−b], F ). As we have seen above, the composition E b [−b] → F is locally zero in K(X). Hence we may assume that α factors through a map σ ≤b−1 E → F which is locally zero by induction hypothesis. This shows (1). To prove (2) let Cu be the mapping cone of u. Then H p (Cu ) = 0 for all p ≥ m (Remark F.159). As we have for every open subset V ⊆ X a distinguished triangle u

+1

G |V −→ F |V −→ Cu|V −→, it suffices to show that locally on X the composition E → F → Cu is homotopic to 0. This follows from (1).

212

21 Cohomology of OX -modules

To show (3) it suffices to show that we can choose (Ui )i such that α|Ui is given by a morphism in K(Ui ). By the construction of the derived category, α is given by s−1 β, where s : F → G is a quasi-isomorphism and β : E → G is a map of complexes. Hence we are done by (2). Remark 21.135. If u in Lemma 21.134 (2) is in addition surjective, then one can show that one can find β such that u ◦ β = α, see [Sta] 0649. Now we can see that all plausible definitions for an object in the derived category D(X) to be perfect are in fact equivalent. Definition and Lemma 21.136. Let X be a ringed space. An object E in D(X) is called perfect, if it satisfies the following equivalent conditions. (i) It is perfect as a complex of OX -modules. (ii) Every complex of OX -modules F such that F ∼ = E in D(X) is perfect. (iii) There exists an open covering (Ui )i of X and for all i strictly perfect complexes Ei of OUi -modules such that E |Ui ∼ = Ei in D(Ui ). Proof. Clearly one has “(ii) ⇒ (i) ⇒ (iii)”. So suppose that (iii) holds and let F be a complex of OX -modules which is isomorphic to E in D(X). By Lemma 21.134 the isomorphisms Ei ∼ = E |Ui ∼ = F |Ui can be represented, after refining the open covering if necessary, by maps of complexes αi : Ei → F |Ui that are necessarily quasi-isomorphisms. Although one may think of a perfect complex as the derived version of a finite locally free module, note that even if a perfect complex F is concentrated in degree 0, then F 0 is not necessarily finite locally free (see Example 21.138 (1) below). Remark 21.137. Let X be a ringed space. (1) Let (Ui )i be an open covering of X. A complex E in D(X) is perfect if and only if E |Ui is perfect for all i ∈ I. Indeed, this follows from Definition 21.136 (iii) since restrictions of strictly perfect complexes to open subspaces are again strictly perfect. (2) If E is a perfect complex in D(X), then E [n] is perfect for all n ∈ Z. Example 21.138. (1) Let A be a ring made into a ringed space with underlying topological space consisting of a single point and A being the global section of the sheaf of rings. Then the category of A-modules is isomorphic to the category of modules over this ringed space. A complex of A-modules F is then perfect if and only if it is isomorphic in D(A) to a finite complex E consisting of finitely generated projective A-modules. In fact ∼ Lemma 21.134 (3) shows that every isomorphism E → F in D(A) is induced by a quasi-isomorphism of complexes E → F . For instance, every A-module considered as a complex concentrated in degree 0 that has a finite resolution by finite projective modules is perfect. (2) Let (X, OX ) be a locally ringed space (e.g. a scheme). Then an object E in D(X) is perfect if and only if there exists an open covering (Ui )i of X such that E |Ui can be represented by a finite complex of finite free OUi -modules (Remark 21.132 (2)). (3) If X = Spec(A), then we will see in Lemma 22.44 below that a complex of A-modules is perfect if and only if the associated complex of quasi-coherent OX -modules is perfect.

213 If one wants to show some assertion for perfect complexes, one can often work locally and then can reduce to show such an assertion for strictly perfect complexes. In this case the following result is often useful. Proposition 21.139. Let X be a ringed space and let D be a triangulated subcategory of D(X) that contains OX and is stable under direct summands in D(X). Then D contains every strictly perfect complex. Proof. The hypotheses imply immediately that M [n] is in D for every direct summand M of a finite free OX -module and for all n ∈ Z. Now let F be a strictly perfect complex, say concentrated in degrees [a, b]. We show by induction on b − a that F is contained in D. For b = a we have F = M [−a] for some direct summand M of a finite free module. In general, the exact sequence of complexes 0 → σ >a F → F → F a [−a] → 0 yields a distinguished triangle σ >a F → F → F a [−a] →. Then σ >a F is contained in D by induction hypothesis and F a [−a] is contained in D by the initial remark. Hence F is in D since D is triangulated. If A is a ring, then every perfect complex in D(A) is isomorphic to a strictly perfect complex. Hence we obtain the following useful corollary. Corollary 21.140. Let A be a ring and let D be a triangulated subcategory of D(A) that is closed under isomorphisms, stable under direct summands and that contains A. Then D contains every perfect complex in D(A). Often one can use Proposition 21.139 in the following form. Corollary 21.141. Let X be a ringed space, let T be any triangulated category, let Φ, Ψ : D(X) → T be triangulated functors and let Φ → Ψ be a morphism of triangulated functors. Suppose F (OX ) → G (OX ) is an isomorphism. Then Φ(F ) → Ψ(F ) is an isomorphism for all strictly perfect complexes F . Proof. Let D be the full subcategory of D(X) of complexes F such that Φ(F ) → Ψ(F ) is an isomorphism. As Φ and Ψ are triangulated, D is a triangulated subcategory. Indeed, it is additive and stable under shifts and the five lemma in triangulated categories (Lemma F.119) shows that if two vertices of a distinguished triangle are in D then so is the third. Moreover, it is stable under direct summands and it contains OX by hypothesis. One concludes by Lemma 21.139. Remark 21.142. Let f : X → Y be a morphism of ringed spaces. If E is a perfect complex on Y , then Lf ∗ E is a perfect complex on X. Indeed, the question is local on Y and hence we can assume that E is a strictly perfect complex. Such a complex is K-flat by Lemma 21.93. Hence one has Lf ∗ E = f ∗ E . As the pullback of a strictly perfect complex is strictly perfect, Lf ∗ E is again perfect. Remark 21.143. Let E and F be perfect complexes of OX -modules. Then E ⊗L OX F is a perfect complex in D(X). Indeed, we may assume that E and F are strictly perfect and in particular K-flat. Then E ⊗L F = E ⊗ F is also strictly perfect.

21 Cohomology of OX -modules

214 (21.32) The dual of a perfect complex.

As for finite locally free modules we will define the notion of the dual of a perfect complex, and a canonical isomorphism with its double dual. As a preparation we need the following lemma. Lemma 21.144. Let X be a ringed space and let E and F be complexes of OX -modules. Suppose that E is strictly perfect. Then R Hom OX (E , F ) is represented by the complex Hom OX (E , F ) (Definition 21.86). For complexes of modules over a ring A, which is a special case (Example 21.138), there is nothing to prove because in this case strictly perfect complexes are K-projective (by the dual of Remark F.181 (2)) and we can calculate R Hom with K-projective resolutions in the first argument. But in general even a finite free OX -module is not a projective object in the category of OX -modules (Exercise 21.5) and hence a strictly perfect complex is typically not a K-projective complex. Proof. Let F → I be a quasi-isomorphism of F with a K-injective complex I . Then the complex Hom OX (E , I ) represents R Hom OX (E , F ). Hence it suffices to show that ι : H := Hom OX (E , F ) −→ H ′ := Hom OX (E , I ) is a quasi-isomorphism. Replacing E by shifts it suffices to show that ι induces an isomorphism after applying H 0 . For U ⊆ X open we have H 0 (H (U )) = Hom K(X) (E |U , F |U ) by (F.18.2) and H 0 (H ′ (U )) = Hom D(X) (E |U , F |U ) by (F.52.5). Now Lemma 21.134 (3) shows that the sheafifications of U 7→ H 0 (H (U )) and of U 7→ H 0 (H ′ (U )) are equal. Lemma 21.145. Let X be a ringed space, and let E , F , and G be complexes in D(X) and suppose that E or G is perfect. Then the functorial map (21.26.5) is an isomorphism ∼

L R Hom OX (E , F ) ⊗L OX G −→ R Hom OX (E , F ⊗ G )

Proof. We can work locally on X and hence can assume that E or G is strictly perfect. As both sides are triangulated functors in E and G , we can even assume that E = OX or G = OX by Corollary 21.141. In this case, the assertion is clear. Definition 21.146. Let X be a ringed space. For a complex E in D(X) we call E ∨ := R Hom OX (E , OX ) its dual. Remark 21.147. Let X be a ringed space. For every complex E in D(X) there is functorial map (21.32.1)

E −→ (E ∨ )∨ .

Indeed, by applying H 0 to (21.26.2) one sees that it suffices to construct a functorial map E ∨ ⊗L E → OX . But this is a special case of (21.26.7). Proposition 21.148. Let X be a ringed space and let E be a perfect complex in D(X). (1) The dual complex E ∨ is perfect and the functorial map E → (E ∨ )∨ is an isomorphism.

215 (2) For all complexes G there is an isomorphism ∼

E ∨ ⊗L OX G −→ R Hom OX (E , G )

(21.32.2)

in D(X), functorial in G and E . (3) For all complexes F and G there is an isomorphism ∼

L R Hom OX (E ∨ ⊗L OX F , G ) −→ R Hom OX (F , E ⊗OX G )

(21.32.3)

in D(X), functorial in E , F , and G . Proof. The morphism (21.32.2) is given by the functorial map (21.26.5). Hence all assertions can be checked locally on X. Hence we can assume that E is given by a strictly perfect complex. By Lemma 21.144 the dual E ∨ is then given by the complex Hom(E , OX ) which is again a strictly perfect complex. Hence E ∨ is perfect. Moreover E → (E ∨ )∨ is an isomorphism as this is the case for strictly perfect complexes. This shows (1). Assertion (2) is a special case of Lemma 21.145. Now (3) is obtained from the following functorial isomorphisms ∨ ∼ R Hom OX (E ∨ ⊗L OX F , G ) = R Hom OX (F , R Hom OX (E , G )) ∼ R Hom O (F , E ⊗L G ), = X

OX

where the first isomorphism is (21.26.1) and the second holds by (2) and (1). Definition and Remark 21.149. Let X be a ringed space and let E be in D(X). One says that E is dualizable if there exists a dual, i.e., an object F in D(X) and morphisms η : OX → E ⊗L F and ϵ : F ⊗L E → OX such that the following diagrams commute E

η⊗L id

id

/ E ⊗L F ⊗L E )  E

id ⊗L ϵ

and

F

id ⊗L η

id

/ F ⊗L E ⊗L F ϵ⊗L id

)  F.

Then Proposition 21.148 shows that every perfect complex E is dualizable and that E ∨ = R Hom OX (E , OX ) is a dual. Conversely, one can show that every dualizable object in D(X) is perfect ([Sta] 0FPV, see also Exercise 25.1 if X is a scheme). Hence in D(X) a complex is perfect if and only if it is dualizable. This gives a more concrete meaning to the slogan that being perfect is the “derived analogue” of being finite locally free since dualizable objects in the category of OX -modules are precisely those that are locally direct summands of a finite free OX -module (Exercise 21.28). In Proposition 25.16 below, we will also see a further categorical characterization of perfect complexes: If X is qcqs scheme, then a complex E is perfect if and only if E is a compact object (Definition 25.1 below) of Dqcoh (X), the full subcategory of D(X) of complexes all of whose cohomology objects are quasi-coherent. This characterization is rather particular to qcqs schemes and does not generalize to more general situations such as ringed spaces or qcqs algebraic stacks.

21 Cohomology of OX -modules

216 (21.33) Pseudo-coherent complexes.

Let (X, OX ) be a ringed space. Let m ∈ Z. Recall from Definition F.159 that an misomorphism E → F is a morphism in D(X) sitting in a distinguished triangle E −→ + F −→ C −→ with H p (C ) = 0 for all p ≥ m, i.e., H m (E ) → H m (F ) is surjective and ≥m+1 τ E → τ ≥m+1 F is an isomorphism in D(X). Definition and Lemma 21.150. Let (X, OX ) be a ringed space. Let m ∈ Z. (1) A complex F of OX -modules is called m-pseudo-coherent if there exists an open covering (Ui )i of X and for each i an m-isomorphism Ei → F |Ui , where Ei is a strictly perfect complex on Ui . The complex F is called pseudo-coherent if it is m-pseudo-coherent for all m ∈ Z. (2) An object E in D(X) is called m-pseudo-coherent if it satisfies the following equivalent conditions. (i) The complex E of OX -modules is m-pseudo-coherent. (ii) Every complex F of OX -modules such that F ∼ = E in D(X) is m-pseudo-coherent. (iii) There exist an open covering (Ui )i of X, strictly perfect complexes Ei on Ui , and m-isomorphisms Ei → E |Ui in D(Ui ). The object E in D(X) is called pseudo-coherent if it is m-pseudo-coherent for all m ∈ Z. (3) An OX -module is called m-pseudo-coherent (resp. pseudo-coherent) if it is m-pseudocoherent (resp. pseudo-coherent) considered as a complex concentrated in degree 0. Proof. The equivalence of the conditions in (2) is proved as in Lemma 21.136. Any perfect complex is pseudo-coherent. In general, the converse does not hold (Exercise 22.36). In Theorem 21.174 below we will give a criterion for a pseudo-coherent complex to be perfect. Remark 21.151. Let m ∈ Z. Let F be an m-pseudo-coherent object in D(X). (1) The object F is n-pseudo-coherent for all n ≥ m. (2) For all i ∈ Z the shifted complex F [i] is (m − i)-pseudo-coherent. (3) Let X be quasi-compact. Then every m-pseudo-coherent complex on X lies in D− (X). Lemma 21.152. Let E be a bounded above complex of direct summands of finite free OX -modules. Then there exists a homotopy equivalence to a bounded above complex of finite free OX -modules. Proof. Let n ∈ Z such that E p = 0 for all p > n. Choose F n = E n ⊕ G n with F n a finite free OX -module and for p < n inductively F p = E p ⊕ G p+1 ⊕ G p with F p finite free for all p. Set also F p = 0 for p > n. For p < n define maps F p → F p+1 as the sum of the given map E p → E p+1 , of the identity G p+1 → G p+1 , and of the zero map on G p . This makes F into a complex and the projection F → E is a homotopy equivalence. Proposition 21.153. Let F be an OX -module viewed as an object in D(X). Let r ≥ 0 be an integer. Then F is (−r)-pseudo-coherent if and only if locally on X there exists an exact sequence of OX -modules (*)

E −r −→ E −r+1 −→ · · · −→ E 0 −→ F −→ 0,

where E i is a finite free OX -module. For r = 0 (resp. r = 1) this means that F is of finite type (resp. of finite presentation).

217 Proof. Suppose that F is (−r)-pseudo-coherent. We can work locally on X and hence can assume that there exists a strictly perfect complex E and a (−r)-isomorphism E → F . We claim that we may assume that E p = 0 for all p > 0. Indeed, let n be the largest integer such that E n ̸= 0 and assume that n > 0. As H n (E ) = 0, we see that E n−1 → E n is surjective. Hence locally on X we can write E n−1 = E ′ ⊕ E n and replace E by · · · → E n−3 → E n−2 → E ′ → 0. Hence induction on n shows our claim. Now the −r-isomorphism E → F yields a resolution as in (*) with E p a direct summand of a finite free OX -module for p = −r, . . . , 0. By Lemma 21.152 we may assume that E p is a finite free OX -module for p = −r, . . . , 0. Conversely, suppose that a resolution as in (*) exists, then E → F is a (−r)-isomorphism. Proposition 21.154. Let m ∈ Z and let u

v

w

E −→ F −→ G −→ E [1] be a distinguished triangle in D(X). If E and G are m-pseudo-coherent, then F is m-pseudo-coherent. By rotating the triangle one also sees: (1) If E is (m + 1)-pseudo-coherent and F is m-pseudo-coherent, then G is m-pseudocoherent. (2) If F is m-pseudo-coherent and G is (m − 1)-pseudo-coherent, then E is m-pseudocoherent. Proof. By rotating the triangle it suffices to show that if E is (m + 1)-pseudo-coherent and F is m-pseudo-coherent, then G is m-pseudo-coherent. This can be shown locally on X. Hence we may assume that there exist strictly perfect complexes K and L , an (m + 1)-isomorphism α : K → E and an m-isomorphism β : L → F . Replacing K by σ ≥m+1 K we may assume that K p = 0 for p < m + 1. Working locally and using Lemma 21.134, we can first assume that the composition K → E → F and the map β are given by maps of complexes and then that there exists a map of complexes u ˜ : K → L such that β ◦ u ˜ = u ◦ α in D(X). The cone Cu˜ of u ˜ is strictly perfect by definition of a cone. By the axioms of a triangulated category there exists a morphism (K → L → Cu˜ →) −→ (E → F → G →) of distinguished triangles. Looking at the induced map on long exact cohomology sequences the Four Lemma F.46 shows that Cu˜ → G is an m-isomorphism. Corollary 21.155. Let E → F → G → be a distinguished triangle in D(X). If two of the three complexes are pseudo-coherent, then the third is pseudo-coherent. Hence the full subcategory of pseudo-coherent complexes is a triangulated subcategory of D(X). Proposition 21.156. Let K and L be complexes in D(X) and let m ∈ Z. Then K and L are m-pseudo-coherent (resp. pseudo-coherent) if and only if K ⊕ L is m-pseudo-coherent (resp. pseudo-coherent).

218

21 Cohomology of OX -modules

Proof. The distinguished triangle K → K ⊕ L → L → shows the necessity of the condition by Proposition 21.154. Suppose that K ⊕ L is m-pseudo-coherent. It suffices to show that L is m-pseudo-coherent. Working locally on X, we may assume that K ⊕ L is bounded above. Then K and L are bounded above. Hence L [N ] is m-pseudo-coherent for large N . Using the direct sum of the distinguished triangles K → K → 0 → and L → L → L ⊕ L [1] → we see by Proposition 21.154 that L ⊕ L [1] is m-pseudo-coherent. Then for every n ≥ 0 the shift L [n] ⊕ L [n + 1] is (m − n)-pseudo-coherent and in particular m-pseudo-coherent by Remark 21.151. As L [N ] is m-pseudo-coherent for large N the distinguished triangle L [n + 1] → L [n] ⊕ L [n + 1] → L [n] shows that L [n] is m-pseudo-coherent for all n ≥ 0. In particular, L is m-pseudo-coherent. Proposition 21.157. Let f : X → Y be a morphism of ringed spaces, let m ∈ Z, and let E be a complex in D(Y ). If E is m-pseudo-coherent, then Lf ∗ E is m-pseudo-coherent. In particular, if E is pseudo-coherent, then Lf ∗ E is pseudo-coherent. Proof. Working locally on Y we may assume there exists a strictly perfect complex K of OY -modules and an m-isomorphism K → E in D(Y ). As Lf ∗ K is strictly perfect (Remark 21.142), it suffices to show that for every m-isomorphism u : L → M in D(Y ) its derived pullback Lf ∗ u is again an m-isomorphism. Indeed, we complement to a distinguished triangle u + L −→ M −→ C −→ . Then H p (C ) = 0 for all p ≥ m and hence H p (Lf ∗ C ) = 0 for all p ≥ m (Remark 21.111). As Lf ∗ preserves distinguished triangles, this shows that Lf ∗ u is an m-isomorphism. The question whether the derived image of pseudo-coherent complexes is again pseudocoherent is one of the fundamental finiteness questions in Algebraic Geometry. We will for instance see that this is the case for proper morphisms between locally noetherian schemes (Section (23.6)). Lemma 21.158. Let X be a ringed space and let E be a locally bounded above complex of OX -modules such that E i is a pseudo-coherent OX -module for all i ∈ Z. Then E is pseudo-coherent. Proof. Fix m ∈ Z. We will show that if E i is (m − i)-pseudo-coherent for all i, then E is m-pseudo-coherent. We may work locally on X and hence can assume that E is bounded above. To check that E is m-pseudo-coherent, we may replace E by σ ≥m−1 E and hence can assume that E is bounded. By hypothesis, E i [−i] is m-pseudo-coherent for all i. We conclude by induction on the length of the complex using the distinguished triangle σ ≥i+1 E → σ ≥i E → E i [−i] → and Proposition 21.154. By definition, for any pseudo-coherent complex F for all m ∈ Z there locally exists an m-isomorphism E → F for some strictly perfect complex E , depending on m. Hence the following result is sometimes useful to prove that certain functors are isomorphic on pseudo-coherent complexes. Proposition 21.159. Let X be a ringed space and let A be an abelian category. Let Φ, Ψ : D(X) → D(A) be triangulated functors and let η : Φ → Ψ be a morphism of triangulated functors. Suppose that the following conditions hold. (a) η(OX ) : Φ(OX ) → Ψ(OX ) is an isomorphism. (b) There exists an integer N such that Φ and Ψ map D b, then K has tor-amplitude in (−∞, b] relative to f . In particular, every morphism has tor-amplitude in (−∞, 0]. Indeed, there exists a quasi-isomorphism from a K-flat complex K ′ concentrated in degrees ≤ b to K . Hence for p > m one has H p (K ⊗L Lf ∗ F ) = H p (K ′ ⊗Lf ∗ F ) = 0 for all OX -modules F because Lf ∗ F has only cohomology in degrees ≤ 0. (3) If K has tor-amplitude in [a, b] relative to f , then K is in D[a,b] (X) because ∗ / [a, b]. H p (K ) = H p (K ⊗L OX Lf OY ) = 0 for p ∈ (4) If K is a complex of flat OX -modules concentrated in degrees [a, b], then K is K-flat by Lemma 21.93 and hence K ⊗L OX F = K ⊗OX F is concentrated in degrees [a, b] for every OX -module F . Therefore K has tor-amplitude in [a, b]. There is also a converse, see Proposition 21.169 below. (5) An OX -module has tor-amplitude in [0, 0] if and only if it is flat by Proposition 21.100. A morphism f of ringed spaces is flat if and only if it has tor-dimension ≤ 0. (6) Every strictly perfect complex concentrated in degrees [a, b] has tor-amplitude in [a, b] by (4). Hence every perfect complex is locally of finite tor-dimension. Proposition 21.166. Let K → L → M → be a distinguished triangle in D(X) and suppose that K and M have tor-amplitude in [a, b]. Then L has tor-amplitude in [a, b]. By rotation of distinguished triangles, one obtains also the implications (21.35.1)

tor-amp K ⊆ [a + 1, b + 1], tor-amp L ⊆ [a, b] tor-amp L ⊆ [a, b], tor-amp M ⊆ [a − 1, b − 1]

⇒ ⇒

tor-amp M ⊆ [a, b], tor-amp K ⊆ [a, b]

Proof. This follows from the long exact cohomology sequence associated to the distinguished triangle +1 K ⊗L F −→ L ⊗L F −→ M ⊗L F −→ for every OX -module F .

21 Cohomology of OX -modules

222

Lemma 21.167. Let [a, b] and [c, d] be bounded intervals in Z and let t ≥ 0 be an integer. Let f : X → Y be a morphism of ringed spaces of tor-dimension ≤ t and let K be a complex of OX -modules with tor-amplitude in [a, b]. Let G be a complex of OY -modules ∗ with H i (G ) = 0 for all i ∈ / [c, d]. Then H j (K ⊗L / [a + c − t, b + d]. OX Lf G ) = 0 for all j ∈ Proof. We claim that H j (Lf ∗ G ) = 0 for all j ∈ / [c − t, d]. Indeed, by induction on d − c using the distinguished triangle H c (G )[−c] −→ G −→ τ ≥c+1 G −→ and shifting, one can assume that G is an OY -module, i.e., c = d = 0. Then the claim holds by definition. The claim implies that it suffices to show the lemma for f = id. This can again be done by induction on d − c as above. Proposition 21.168. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Suppose that f has tor-dimension ≤ t. Let K be a complex of OX -modules and let L be a complex of OY -modules with tor-ampf K ⊆ [a, b] and tor-ampg L ⊆ [c, d] for bounded intervals [a, b] and [c, d] of Z. Then tor-ampg◦f (K ⊗L Lf ∗ L ) ⊆ [a + c − t, b + d]. We have the following special cases. (1) Taking f = idX , we see that for two complexes K and L in D(X) one has (21.35.2)

tor-ampg K ⊆ [a, b], ⇒ tor-ampg (K

tor-ampg L ⊆ [c, d] ⊗L OX

L ) ⊆ [a + c, b + d].

(2) Taking K = OX and [a, b] = [0, 0], we see that for every complex L in D(Y ) one has (21.35.3)

tor-ampg L ⊆ [c, d] ⇒ tor-ampg◦f (Lf ∗ L ) ⊆ [c − t, d]

If in addition L = OY one deduces (21.35.4)

tor-dim f ≤ t, tor-dim g ≤ s ⇒ tor-dim g ◦ f ≤ s + t.

Proof. Let F be an OZ -module. As the cohomology of L ⊗L Lg ∗ F is concentrated in degrees [c, d], the cohomology of K ⊗L Lf ∗ L ⊗L L(g ◦ f )∗ F = K ⊗L Lf ∗ (L ⊗L Lg ∗ F ) is concentrated in degrees [a + c − t, b + d] by Lemma 21.167. Proposition 21.169. Let K be a complex in D(X) and let a ≤ b be integers. The following assertions are equivalent. (i) The complex K has tor-amplitude in [a, b]. (ii) There exists a complex E of flat OX -modules with E p = 0 for p ∈ / [a, b] and an isomorphism E ∼ = K in D(X). Proof. We have already seen in Remark 21.165 (4) that (ii) implies (i). Conversely, suppose that (i) holds. As H p (K ) = 0 for all p > b (Remark 21.165 (3)) there exists a complex G consisting of flat OX -modules with G p = 0 for p > b and an ∼ isomorphism G → K in D(X) by Remark 21.95 (2). As H p (G ) = H p (K ) = 0 for all p < a we have G ∼ = τ ≥a G =: E in D(X). It remains to show that E a is flat. There is a distinguished triangle E ′ → E → E a [−a] →, where E ′ has flat components that are zero outside degrees [a + 1, b]. Now apply the triangulated functor F ⊗L (−) for an OX -module F . We obtain an exact sequence

223 H a−1 (F ⊗L E ) → H a−1 (F ⊗L E a [−a]) → H a (F ⊗L E ′ ) As E has tor-amplitude in [a, b] the left module is zero. As E ′ is K-flat, we have F ⊗L E ′ = F ⊗ E ′ and hence the right module is zero because E ′ is concentrated in degrees > a. Therefore H a−1 (F ⊗L E a [−a]) = H −1 (F ⊗L E a ) = Tor 1 (F , E a ) = 0 for all F . Hence E a is flat (Proposition 21.100). Corollary 21.170. Let f : X → Y be a morphism of ringed spaces. Let K be a complex in D(Y ). If K has tor-amplitude in [a, b], then Lf ∗ K has tor-amplitude in [a, b]. Proof. By Proposition 21.169 we may assume that K is a complex of flat modules concentrated in degrees [a, b]. Then Lf ∗ K = f ∗ K because K is K-flat, and f ∗ K is a complex of flat OX -modules concentrated in degrees [a, b]. Hence Lf ∗ K has tor-amplitude in [a, b]. Tor-amplitude can be checked locally in the following sense. Lemma 21.171. Let f : X → Y be a morphism of ringed spaces, let K be a complex of OX -modules, and let a ≤ b be integers. Then K has tor-amplitude in [a, b] relative to f if and only if for every x ∈ X the complex Kx of OX,x -modules has tor-amplitude in [a, b] relative to fx# : OY,f (x) → OX,x . Proof. Consider the commutative diagram of ringed spaces (*)

({x}, OX,x ) fx#



({f (x)}, OY,f (x) )

ix

/X

if (x)

 / Y.

f

Then ix and if (x) are flat and we have Kx = i∗x K = Li∗x K . Suppose that K has tor-amplitude in [a, b] relative to f . By (21.35.3) we have tor-ampif (x) ◦fx# (Kx ) = tor-ampf ◦ix (Kx ) ⊆ [a, b]. Every OY,f (x) -module G is of the form i∗f (x) G for some OY -module G , for instance G = if (x),∗ G. Therefore tor-ampfx# Kx ⊆ [a, b]. Conversely, suppose that tor-ampfx# Kx ⊆ [a, b] for all x ∈ X. Let F be an OY -module. The commutative diagram (*) shows (Lf ∗ F )x = Lfx#∗ Ff (x) . As forming stalks is exact and commutes with derived tensor products (21.20.2), we have for i ∈ / [a, b] that ∗ i L ∗ i L #∗ H i (K ⊗L OX Lf F )x = H (Kx ⊗OX,x (Lf F )x ) = H (Kx ⊗OX,x Lfx Ff (x) ) = 0,

hence K has tor-amplitude in [a, b]. Lemma 21.172. Let A → B be a faithfully flat ring homomorphism and let E be a complex of A-modules. Then E has tor-amplitude in [a, b] if and only if the complex of B-modules E ⊗A B has tor-amplitude in [a, b]. Proof. The ring homomorphism A → B corresponds to a morphism of ringed spaces f : (∗, B) → (∗, A), where ∗ denotes a singleton. As A → B is flat, the functor f ∗ = B ⊗A (−) is exact and hence Lf ∗ = f ∗ . Hence the condition is necessary by Corollary 21.170.

224

21 Cohomology of OX -modules

Conversely, suppose that E ⊗A B has tor-amplitude in [a, b] and let M be an A-module. As A → B is flat, taking cohomology commutes with (−)B := (−) ⊗A B and we have A B L (M ⊗L A E)B = MB ⊗B EB . Therefore Torp (M, E)B = Torp (MB , EB ). By hypothesis TorB / [a, b] and hence TorA / [a, b] because A → B p (MB , EB ) = 0 for p ∈ p (M, E) = 0 for p ∈ is faithfully flat. (21.36) Pseudo-coherent complexes of finite tor-dimension are perfect. Let (X, OX ) be a ringed space. The goal of this section is to show that pseudo-coherent complexes of finite tor-dimension are perfect. This follows from the following more precise lemma. Lemma 21.173. Let E be an object in D(X) and let a ≤ b be integers. If E has tor-amplitude in [a, b] and is (a − 1)-pseudo-coherent, then E is perfect. Proof. Working locally we can assume that there exists a strictly perfect complex K and an (a − 1)-isomorphism α : K → E . Replacing K by σ ≥a−1 K we may assume that K p = 0 for p < a − 1. By assumption we have H p (E ) = 0 for p < a. Hence we may replace E by τ ≥a E and can assume that E p = 0 for all p < a. Complete α to a distinguished triangle C → K → E →. Looking at the long exact cohomology sequence and using that H p (α) is an isomorphism for p ≥ a one sees that H a−1 (C ) = Ker(K a−1 → K a ) =: C and that H p (C ) = 0 for all p = ̸ a − 1. Hence C = C[1 − a] = τ ≤a−1 K . Thus the distinguished triangle τ ≤a−1 K → K → τ ≥a K → (Proposition F.157) shows that we have an isomorphism τ ≥a K ∼ = E in D(X). It therefore suffices to show that L := τ ≥a K is strictly perfect. Now L is a finite complex concentrated in degrees ≥ a and L p = K p is a direct summand of a finite free OX -module for p > a. One has L a = Coker(K a−1 → K a ). In particular L a is an OX -module of finite presentation. There is a distinguished triangle L ′ → L → L a [−a] →, where L ′ is a finite complex concentrated in degrees > a with flat components. In particular it is K-flat. Applying the functor F ⊗L (−) for an arbitrary OX -module F , the long exact cohomology sequence yields an exact sequence H a−1 (L ⊗L F ) −→ H −1 (L a ⊗L F ) −→ H a (L ′ ⊗ F ). Now the left cohomology vanishes because L ∼ = E has tor-amplitude in [a, b] and the right cohomology vanishes because the a-th component of L ′ ⊗ F is zero. This shows that Tor1 (L a , F ) = H −1 (L a ⊗L F ) = 0 for all F . Hence L a is a flat OX -module. Being flat and of finite presentation, L a is locally a direct summand of a finite free OX -module (see Proposition 7.41 if (X, OX ) is a locally ringed space, Proposition B.29 if X consists of a single point, and Exercise 21.28 in general). As perfect complexes are pseudo-coherent and locally of finite tor-dimension (Remark 21.165 (6)) we obtain the following result. Theorem 21.174. Let E be an object in D(X). Then E is perfect if and only if E is pseudo-coherent and locally on X has finite tor-dimension. Proposition 21.175. Let K → L → M → be a distinguished triangle in D(X) and suppose that two of the three complexes are perfect. Then the third is perfect.

225 Proof. By Theorem 21.174 this follows from the analogue assertions for pseudo-coherent complexes (Proposition 21.154) and complexes of finite tor-amplitude (Proposition 21.166). Proposition 21.176. Let K and L be complexes in D(X). Then K and L are perfect if and only if K ⊕ L is perfect. Proof. It is clear that K and L are locally of finite tor-dimension if and only if K ⊕ L is locally of finite tor-dimension. The analogous assertion holds for pseudo-coherence (Proposition 21.156). Hence we conclude by Theorem 21.174. We finish the section by recording two lemmas for later use. Lemma 21.177. Let X be a ringed space. (1) If X is quasi-compact and E in D(X) is perfect, then E ∈ Db (X). (2) Let E be in Db (X). If H i (E ) is perfect (as a complex concentrated in degree 0) for all i ∈ Z, then E is perfect. The converse to the second assertion does not hold in general, see Exercise 22.36. Proof. The first assertion follows from the Definition 21.136 (iii). Let us show the second assertion. We may assume that E is cohomologically bounded, say H i (E ) = 0 for i ∈ / [a, b] for some integers a ≤ b. We proceed by induction an b − a. Consider the distinguished triangle H a (X)[−a] −→ E −→ τ ≥a+1 E −→ . Then τ ≥a+1 E has cohomology concentrated in degrees [a + 1, b] and H i (τ ≥a+1 E ) = H i (E ) for all i ≥ a + 1. Hence τ ≥a+1 E is perfect by induction hypothesis. Moreover, H a (X)[−a] is perfect by assumption. Therefore we conclude by Proposition 21.175. Lemma 21.178. Let A → B be a faithfully flat ring homomorphism, and let E be a complex of A-modules. Then E is perfect if and only if the complex of B-modules E ⊗A B is perfect. Proof. Again by Theorem 21.174 this follows from the analogues statements for pseudocoherent complexes (Proposition 21.163) and complexes of finite tor-dimension (Proposition 21.163).

Exercises Exercise 21.1. Recall (for instance from [Wed] O Chap. 1) the following properties of continuous maps. A continuous map of topological spaces f : X → Y is called proper if it satisfies the following equivalent conditions. (i) For every topological space Z the map f × idZ : X × Z → Y × Z is closed. (ii) The map f is closed and for every quasi-compact subspace V of Y the preimage f −1 (V ) is a quasi-compact subspace of X. The map f is called separated if it satisfies the following equivalent conditions. (i) The diagonal ∆f : X → X ×Y X is a closed topological embedding.

21 Cohomology of OX -modules

226

(ii) For any relatively Hausdorff subspace B of Y (i.e., any two distinct points in B have disjoint neighborhoods in Y ) the preimage f −1 (B) is a relatively Hausdorff space in X. Let f : X → Y be a continuous map of topological spaces. For an abelian sheaf F on X define a presheaf f! F on Y by f! F (V ) := { s ∈ F (f −1 (V )) ; f : Supp(s) → U is proper}. (1) Show that f! F is an abelian sheaf on Y and that one obtains a functor f! : (Ab(X)) → (Ab(Y )). Show that if f is proper, then f! = f∗ . Show that if f = j is the inclusion of a locally closed subspace, then j! is the functor defined in Definition 21.1. (2) Let f : X → Y and g : Y → Z continuous maps of topological spaces and suppose that g is separated. Show that the identity of functors (g ◦ f )∗ = g∗ ◦ f∗ restricts to an identity (g ◦ f )! = g! ◦ f! . Exercise 21.2. We continue to use the notions from Exercise 21.1. Consider a cartesian diagram in the category of topological spaces W g

 Z

/Y

q

f

p

 / X.

(1) Show that the identity f∗ ◦ q∗ = p∗ ◦ g∗ induces a morphism of functors f! ◦ q∗ → p∗ ◦ g! . (2) Consider the morphisms of functors f! → f! q∗ q −1 → p∗ g! q −1 , where the first morphism is given by the adjunction (q −1 , q∗ ) and the second is given by (1). By the adjunction (p−1 , p∗ ) we obtain a morphism β : p−1 f! −→ g! q −1 of functors (Ab(Y )) → (Ab(Z)). Show that if f is proper and separated, then g is ∼ also proper and separated, and β is an isomorphism p−1 f∗ −→ g∗ q −1 . (3) Let f again be proper and separated. Show that β induces an isomorphism ∼

p−1 ◦ Rf∗ −→ Rg∗ ◦ q −1 of functors D+ (Ab(Y )) → D+ (Ab(Z)). The conclusions in (2) and (3) are called the proper base change theorems. Exercise 21.3. Let X be an irreducible topological space and let G be a constant abelian sheaf. Show that H p (X, G) = 0 for all p > 0. Exercise 21.4. Let S 1 be the unit circle with its usual topology and let Z be the constant sheaf of the abelian group of integers. Show that H 1 (S 1 , Z) ∼ = Z.

227 Exercise 21.5. Let k be a field and let X be an integral scheme of finite type over k of positive dimension. Show that there exists no surjective homomorphism P → OX of OX -modules where P is a projective objective in the category of OX -modules. Hint: Let P be a projective object in the category of OX -modules and let u : P → OX be a map of OX -modules. Then necessarily u = 0. To see this, consider for ∅ ̸= V ⊆ X open the OX -module OV ⊆X (21.1.6). Then OV ⊆X (U ) = 0 for all U ⊆ X open with U ⊈ V . Now choose a closed point x ∈ X and set W := X \ {x}. Let U be any open neighborhood of x. Choose a strictly smaller open neighborhood V of x. Then u factors through the surjective map OV ⊆X ⊕ OW ⊆X → OX and hence is zero on sections over U . This shows ux = 0 which implies u = 0. Remark : If X = Spec A is affine, every quasi-coherent OX -module admits a surjection from a projective object in the category of quasi-coherent OX -modules since in this case QCoh(X) is equivalent to the category of A-modules. But one can show that for X = P1k there are no non-zero projective objectives in the abelian category of quasi-coherent OX -modules (e.g., [EEGRO] O 2.3). Exercise 21.6. Let X be a topological space, x ∈ X a point, and denote by ix : {x} → X the inclusion. Let A be an abelian group considered as abelian sheaf on {x}. Show that H p (X, ix,∗ A) = 0 for all p > 0. Exercise 21.7. Let (X, OX ) be a ringed space and let Z ⊆ X be a closed subset. Consider the functor ΓZ (X, −) : (OX -Mod) → (Γ(X, OX )-Mod), F 7→ ΓZ (X, F ) := { s ∈ F (X) ; Supp(s) ⊂ Z }. Show that this functor is left exact and has a right derived functor RΓZ (X, −) : D(X) → D(Γ(X, OX )). For F in D(X) and i ∈ Z one defines HZi (X, F ) := H i RΓZ (X, −) the cohomology of F with support in Z. Moreover, show the following assertions. (1) Set U := X \ Z. Show that there is a functorial distinguished triangle in D(Γ(X, OX )) +1

RΓZ (X, F ) −→ RΓ(X, F ) −→ RΓ(U, F ) −→ . (2) For F in D(X) let F ab be the underlying complex of abelian sheaves. Show that ∼ there is a functorial isomorphism RΓZ (X, F ) −→ RΓZ (X, F ab ) in D(Z). Hint: Use (1) Exercise 21.8. Let X be ringed space, let F be a complex of OX -modules. Suppose that there exists a basis B of the topology of X and an integer d ≥ 0 such that H p (U, H q (F )) = 0 for all U ∈ B, p > d and q < 0. (1) Show that the canonical map F −→ R limn τ ≥n F is an isomorphism. (2) Let (In )n be a system of bounded below complexes of injective OX -modules that form a resolution of (τ ≥−n F )n as in Lemma F.195. Show that the limit map F = limn τ ≥−n F → lim In (F.47.2) is a quasi-isomorphism. Hint: Exercise F.39 Exercise 21.9. Let X be a ringed space, let U = (Ui )i∈I be an open covering and let F be an OX -module. For every i ∈ I n+1 we denote by ji : Ui → X the inclusion. For n ≥ 0 define an OX -module

21 Cohomology of OX -modules

228 Cˇn (U , F ) :=

Y

(ji )∗ ji−1 F

i∈I n+1

and define differentials as in Section (21.14) to obtain a complex Cˇ• (U , F ) of OX modules. More generally, let F be a bounded below complex of OX -module. Then Cˇ• (U , F ) is defined as the total complex of the double complex (Cˇp (U , F q ))p,q . For F a bounded below complex let F → Cˇ0 (U , F ) be the map of complexes of OX -modules which is in each component the canonical map F q → (ji )∗ ji−1 F q . Show that this maps yields a quasi-isomorphism F → Cˇ• (U , F ) in C(X). Exercise 21.10. Let X be a ringed space, let U be an open covering of X, and let F be a bounded below complex of OX -modules. (1) Show that if one applies the first hypercohomology spectral sequence (F.49.5) to the double complex (Cˇp (U , F q ))p,q (Exercise 21.9) one obtains a convergent spectral sequence E1pq = H q (X, Cˇp (U , F )) ⇒ H n (X, F ) Hint: Use Exercise 21.9. (2) Show that if U = (U, V ) consists of only two open subsets, then the spectral sequence degenerates at E2 and yields the Mayer-Vietoris sequence for F and (U, V ). Exercise 21.11. Let X be a ringed space and let U be a finite open covering of X. Show that for every complex F of OX -modules there is a functorial map in D(Γ(X, OX )) Tot(Cˇalt (U , F )) −→ RΓ(X, F ). Show that this map is an isomorphism if there exists a basis B of the topology of X that contains all finite intersections of the members of U and such that for all U ∈ B, q ∈ Z, and p > 0 one has H p (U, F q ) = H p (U, Coker(F q−1 → F q ) = H p (U, H q (F )) = 0. Hint: Prove the assertions first for F bounded below, then use Exercise 21.8. Exercise 21.12. Let X be a ringed space and let F and G be OX -modules. Show that X one has Supp Tor O p (F , G ) ⊆ Supp(F ) ∩ Supp(G ) for all p ≥ 0. Exercise 21.13. Let f : X → Y and g : Y → Z be morphisms of ringed spaces, and let G be a complex of OY -modules. Construct by adjunction a functorial morphism Rg∗ G → R(g ◦ f )∗ Lf ∗ G in D(Z). Exercise 21.14. Let X be a ringed spaces. Show that every direct sum and every filtered colimit of K-flat complexes of OX -modules is again K-flat. Are arbitrary colimits of K-flat complexes again K-flat? Exercise 21.15. Let X be a ringed space. For every complex of OX -modules G consider the functor TG : K(X) → D(X), E 7→ G ⊗OX E . Let F be a complex of OX -modules. Show that the following assertions are equivalent. (i) F is K-flat. (ii) F is left-TG -acyclic for all complexes G of OX -modules. (iii) F is left-TG -acyclic for all exact complexes G of OX -modules.

229 Exercise 21.16. Let X be a ringed space and let F and G be complexes of OX -modules. Suppose that for all a, b, c, d ∈ Z the complex τ ≤a τ ≥b F ⊗ τ ≤c τ ≥d G is exact. Show that F ⊗ G is exact. Hint: Use that F = colima τ ≤a F . Exercise 21.17. Let X be a ringed space and let F be a complex of OX -modules such that H n (F ) is flat for all n ∈ Z and such that the image of F n → F n+1 is a direct summand of F n+1 for all n. Show that F is K-flat. Hint: Use Exercise 21.16 to reduce to the case that F is a bounded complex. Exercise 21.18. Let A be a ring and let F be a complex of R-modules. Show that the following assertions are equivalent. (i) The complex F is K-flat and exact. (ii) The tensor complex F ⊗A E is exact for every complex E of A-modules. (iii) The tensor complex F ⊗A M is exact for every A-module M of finite presentation. Exercise 21.19. Let A be a ring, let a and b be two ideals of A. Show that ∼ TorA 1 (A/a, A/b) = (a ∩ b)/ab. Exercise 21.20. Let A be a ring, let f = (f1 , . . . , fr ) be a sequence of elements in A, and let K(f ) be the corresponding Koszul complex considered as a cochain complex sitting in degrees [−r, 0]. Let Z[T1 , . . . , Tr ] be the unique ring homomorphism sending Ti to fi . Show that K(f ) ∼ = A ⊗L Z Z[T1 , . . . , Tr ]/(T1 , . . . , Tr ). Exercise 21.21. Let A be a ring, let f = (f1 , . . . , fr ) be a completely intersecting sequence, and let I ⊆ A be the ideal generated by f . Show that one has for every complex E in D(A) and for all p ∈ Z p Tor−p A (A/I, E) = H (K(f ) ⊗R E),

where K(f ) is the Koszul complex associated to f considered as a cochain complex sitting in degrees [−r, 0]. Exercise 21.22. Let X be a ringed space and let (Fi )i be a filtered diagram of complexes of OX -modules. (1) Show that for every complex G of OX -modules there is a functorial isomorphism ∼ colimi Tor n (Fi , G ) → Tor n (colimi Fi , G ) for all n ∈ Z. (2) Let f : Y → X be a morphism of ringed spaces. Show that for all n ∈ Z there exists ∼ a functorial isomorphism colimi Ln f ∗ Fi −→ Ln f ∗ colimi Fi . Exercise 21.23. Let A be a ring and let E and F be complexes of R-modules. (1) Let α ∈ H p (E) and β ∈ H q (F ) be represented by elements a ∈ Ker(E p → E p+1 ) and b ∈ Ker(F q → F q+1 ). Let α · β be the class of a ⊗ b in H p+q (E ⊗A F ). Show that this yields a well defined A-linear map (*)

H p (E) ⊗R H q (F ) −→ H p+q (E ⊗A F ).

(2) Show that there is a unique A-linear functorial map (**)

H p (E) ⊗R H q (F ) −→ H p+q (E ⊗L A F ).

whose composition with the map induced by E ⊗L A F → E ⊗A F is (*).

21 Cohomology of OX -modules

230

Exercise 21.24. Let φ : R → A be a ring homomorphism and let E and F be in D(R). (1) Show that there is a functorial isomorphism in D(A) L L L L ∼ (E ⊗L R A) ⊗A (F ⊗R A) = (E ⊗R F ) ⊗R A

and use Exercise 21.23 to obtain A-linear maps (*)

R R L TorR p (E, A) ⊗A Torq (F, A) −→ Torp+q (E ⊗ F, A).

(2) Let R → B be a second ring homomorphism. Show that (*) for E = F = B yields a map R R R TorR p (B, A) ⊗A Torq (B, A) −→ Torp+q (B ⊗R B, A) −→ Torp+q (B, A),

where the second map is induced by the multiplication B ⊗R B → B. Show that this makes TorR • (B, A) into a strictly graded commutative graded A-algebra. Exercise 21.25. Let A be a ring and let I be an ideal. ∼ (1) Show that the isomorphism I/I 2 → TorA 1 (A/I, A/I) (Exercise 21.19) induces a map of graded A/I-algebras (Exercise 21.24) (*)

V

A I/I

2

−→ TorA • (A/I, A/I).

(2) Show that (*) is an isomorphism if I is generated by a completely intersecting sequence. Hint: Reduce to the case that I is generated by one element. Exercise 21.26. Let X be a locally ringed space and let L be an object of D(X). Show that the following assertions are equivalent. (i) The object L is invertible in D(X), i.e., the functor D(X) → D(X), E 7→ L ⊗L E is an equivalence of categories. (ii) There exists an open covering (Ui )i of X and for each i an integer ni such that L |Ui ∼ = Li [ni ] for some invertible OUi -module Li . Show that every invertible object in D(X) is a perfect complex. Exercise 21.27. Show that the definitions of the cup product in Section (21.29) agree. Exercise 21.28. Let (X, OX ) be a ringed space. We define an OX -module E to be dualizable if it has a dual F , where we use the same definition as in Definition 21.149 replacing derived tensor products by usual tensor products. Let E be an OX -module. Show that the following assertions are equivalent. (i) E is dualizable. (ii) E is locally the direct summand of a finite free OX -module. (iii) E is of finite presentation and flat. Exercise 21.29. Let X be a ringed space. Let m ∈ Z. A complex F of OX -modules is called strictly m-pseudo-coherent if F i is a direct summand of a finite free OX -module for i ≥ m and F i = 0 for i sufficiently large. A complex F is called strictly pseudo-coherent if it is a bounded above complex of direct summands of finite free OX -modules. (1) Let F be a strictly m-pseudo-coherent complex of OX -modules with H i (F ) = 0 for i > m. Show that H m (F ) is an OX -module of finite type. (2) Show that the following conditions for a complex F of OX -modules are equivalent.

231 (i) The complex F is m-pseudo-coherent. (ii) Locally on X there exists a quasi-isomorphism E → F of complexes with E strictly m-pseudo-coherent. (iii) Locally on X there exists an isomorphism E → F in D(X) with E strictly m-pseudo-coherent. Exercise 21.30. Let X be a ringed space and let F and G be in D(X). (1) Suppose that there exist a, b ∈ Z such that H i (F ) = 0 for i > a and H j (G ) = 0 for j > b. Let K and L be a strictly perfect complexes and let K → F be an m-isomorphism and L → G be an n-isomorphism, m, n ∈ Z. Show that the canonical map K ⊗ L → F ⊗L OX G is a t-isomorphism for t = max(m + a, n + b). (2) Show that if F and G are pseudo-coherent, then F ⊗L OX G is pseudo-coherent. Exercise 21.31. Let R be a ring. Show that a complex E ∈ D− (R) is pseudo-coherent if and only if the functors ExtiR (E, −) : (R-Mod) → (R-Mod) commute with filtered colimits for all i ∈ Z. Hint: Use that an R-module M is of finite presentation if and only if HomR (M, −) commutes with filtered colimits. Exercise 21.32. Let f : X → Y be a morphism of ringed spaces and let t ≥ 0 be an integer. Show that the following assertions are equivalent. (i) The morphism f has tor-dimension ≤ t. (ii) For every OY -module F one has Lt+1 f ∗ F = 0. (iii) For all x ∈ X there exists an exact sequence of OY,f (x) -modules 0 −→ Pt −→ Pt−1 −→ · · · −→ P1 −→ P0 −→ OX,x −→ 0 with Pi a flat OY,f (x) -module for all 0 ≤ i ≤ t. Exercise 21.33. Let X be a ringed space and let F be an OX -module. The tor-dimension or flat dimension of F is defined as tor-dim(F ) := inf{ n ∈ Z ; tor-amp F ⊆ [−n, 0] }. In particular, we have tor-dim(F ) = −∞ if and only if F = 0. Otherwise tor-dim(F ) ≥ 0. As a special case (where X consists of a single point) we obtain the notion of the tordimension of an A-module M for a ring A. (1) Let n ≥ 0. Show that the following assertions are equivalent. (i) tor-dim F ≤ n. X (ii) Tor O n+1 (F , −) is the zero functor. OX (iii) Tor n (F , −) is left exact. (iv) For any exact sequence of OX -modules 0 → F ′ → En−1 → · · · → E0 → F → 0 with each Ei flat, F ′ is flat. (v) There exists an exact sequence of OX -modules 0 → En → · · · → E0 → F → 0 with all Ei flat OX -modules. (2) Let (Ui )i∈I be an open covering of X. Show that tor-dim F = sup{ tor-dimOX,x Fx ; x ∈ X } = sup{ tor-dim F |Ui ; i ∈ I }.

22

Cohomology of quasi-coherent modules

Content ˇ – Cohomology of quasi-coherent modules and Cech cohomology – Derived categories of quasi-coherent modules – Finiteness properties of complexes on schemes – Projection formula, base change and the K¨ unneth formula We now study the cohomology of complexes of quasi-coherent modules on schemes. The first main result is the vanishing of higher cohomology of quasi-coherent modules on affine schemes (Theorem 22.2) and a relative version for affine morphisms (Corollary 22.5). This ˇ follows quite formally from the vanishing of Cech cohomology for open coverings of affine schemes by principal open subschemes (Lemma 22.1). ˇ This lemma also implies that cohomology and Cech cohomology for open affine coverings are the same for quasi-coherent modules on a separated scheme (Theorem 22.9). This allows us to calculate several important examples of cohomology for quasi-coherent modules. We explain a general procedure how to do this on quasi-affine schemes and on projective spectra using the Koszul complex (Sections (22.4)–(22.6)) and apply this to calculate the cohomology of twisted line bundles on projective space (Theorem 22.22). Next we study complexes of quasi-coherent modules on a scheme X. We show that the quasi-coherent modules form a Grothendieck abelian category (Theorem 22.32). If X is quasi-compact and separated, then its derived category can be identified with the full subcategory Dqcoh (X) of all F in D(X) such that all H p (F ) are quasi-coherent (Theorem 22.35). In particular, for a ring R we can identify D(R) and Dqcoh (Spec R). We show similar results for locally noetherian schemes and coherent modules (Theorem 22.42). For any scheme X, every object in Dqcoh (X) is the homotopy limit of bounded below complexes (Lemma 22.24). This allows to prove many results on unbounded complexes by reducing to bounded below complexes. For instance, we use this result to show that for a qcqs morphism f : X → Y of schemes the functor Rf∗ maps Dqcoh (X) to Dqcoh (Y ) (Theorem 22.31). In particular Rp f∗ F is quasi-coherent for every quasi-coherent OX module F and for all p (Theorem 22.27). In the next part of the chapter we more closely study how certain properties of complexes which we have introduced in the setting of ringed spaces behave on schemes. For instance the properties to be pseudo-coherent, to be perfect, to be of bounded tor-amplitude, and to be of bounded injective amplitude. Finally, we prove two central results, the projection formula (Section (22.19)) and the derived base change theorem (Theorem 22.99).

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_7

233

ˇ Cohomology of quasi-coherent modules and Cech cohomology

(22.1) Quasi-coherent cohomology of affine schemes. Let us start by rephrasing a result obtained in Volume I: Lemma 22.1. Let X be an affine scheme, and let F be a quasi-coherent OX -module. ˇ 0 (X, F ) = Γ(X, F ) and all higher Cech ˇ Then H cohomology groups vanish: ˇ i (X, F ) = 0 H

for all i > 0.

ˇ 0 (U , F ) = Γ(X, F ), and H ˇ i (U , F ) = 0 for Proof. It is clearly enough to check that H i > 0 for all finite coverings U = (D(fi ))i of X by principal open affine subschemes. This statement can be checked directly, and we have actually done so in Lemma 12.33. Another ˇ way to obtain this result is the following: Evaluating the definition of the Cech complex in this situation shows that we have to prove that the complex M M 0→M → M fi → Mfi fj → · · · i

i,j

is exact, where M := Γ(X, F ). But the exactness of this complex is a standard result in the theoryQ of faithfully flat descent, applied to the faithfully flat ring homomorphism Γ(X, OX ) → i Γ(D(fi ), OX ). See Lemma 14.64. Note that this result, as explained in Corollary 12.34, implies that for every exact sequence 0 → F ′ → F → F ′′ → 0 of OX -modules with F ′ quasi-coherent, the sequence 0 → Γ(X, F ′ ) → Γ(X, F ) → Γ(X, F ′′ ) → 0 is exact. Since every (quasi-coherent) OX -module F ′ can be embedded into an injective OX -module F , applying this reasoning to the short exact sequence thus obtained (with F ′′ = F /F ′ ), we conclude that H 1 (X, F ′ ) = 0. (We can also get this result from the ˇ 1 and H 1 , see Corollary 21.81). But we can get more: equality of H Theorem 22.2. Let X be an affine scheme, and let F be a quasi-coherent OX -module. Then the higher cohomology groups vanish: H i (X, F ) = 0

for all i > 0.

Proof. We apply Cartan’s theorem, Proposition 21.83, to the basis of the topology of X consisting of affine open subsets of X. By Lemma 22.1, the hypotheses of that theorem are satisfied, and we obtain the desired result. Recall that the vanishing of higher cohomology groups of quasi-coherent sheaves (or just the vanishing of H 1 for all ideal sheaves) characterizes affine schemes, Theorem 12.35.

234

22 Cohomology of quasi-coherent modules

Remark 22.3. The above theorem is not true without the condition that F be quasicoherent. For an example, see Exercise 22.3 Of course, it is also easy to find non-vanishing cohomology groups H 1 (X, F ) with X × affine where F is not even an OX -module, but just an abelian sheaf, e. g. for F = OX × 1 we have an identification H (X, OX ) = Pic(X), and this group is usually non-trivial. Corollary 22.4. Let X be a scheme, and let 0→F →G →H →0 be a short exact sequence of OX -modules. If two of the three modules are quasi-coherent, then so is the third. Proof. We know already that kernels and cokernels of morphisms of quasi-coherent modules are again quasi-coherent (Corollary 7.19). Hence it suffices to show that if F and H are quasi-coherent, then so is G . Since quasi-coherence is a local property, we may assume that X is affine. We have a commutative diagram 0

/ Γ(X, F )∼

/ Γ(X, G )∼

/ Γ(X, H )∼

/0

0

 /F

 /G

 /H

/0

with exact rows; for the top row, we use Theorem 22.2 (or Corollary 12.34) and that e· is exact. The outer two vertical maps are isomorphisms, hence so is the middle one. Corollary 22.5. Let f : X → Y be an affine morphism of schemes, and let F be a quasi-coherent OX -module. Then Ri f∗ F = 0 for all i > 0. Proof. In view of Theorem 22.2, this is an immediate consequence of the description of Ri f∗ F as the sheaf associated to the presheaf V 7→ H i (f −1 (V ), F ), see Proposition 21.27. Corollary 22.6. Let f : X → Y be an affine morphism of schemes, and let F be a quasi-coherent OX -module. (1) We have functorial isomorphisms (22.1.1)

H p (X, F ) ∼ = H p (Y, f∗ F )

for all p ≥ 0.

(2) Let g : Y → Z be a morphism of schemes, then we have functorial isomorphisms (22.1.2)

Rp (g ◦ f )∗ F ∼ = Rp g∗ f∗ F

for all p ≥ 0.

Proof. For (1) consider the Leray spectral sequence (21.8.5): E2p,q = H p (Y, Rq f∗ F ) ⇒ H p+q (X, F ). By Corollary 22.5 we have E2pq = 0 for q ̸= 0. Hence the spectral sequence degenerates at E2 and the edge morphism E2p0 → H p (X, F ) is an isomorphism for all p ≥ 0. For (2) use the same argument with the relative Leray spectral sequence (21.8.6). For a corresponding statement in terms of derived categories, see Proposition 22.33.

235 Remark 22.7. The argument in the proof of Corollary 22.6 shows that for every morphism of ringed spaces f : X → Y and every OX -module F one has isomorphisms as in (22.1.1) and in (22.1.2) (for g : Y → Z any morphism of ringed spaces) if Rq f∗ F = 0 for q > 0. Remark 22.8. Theorem 22.2 and Corollary 22.5 are central results in the theory of the cohomology of quasi-coherent modules on schemes. To prove these results we first ˇ showed the vanishing of the higher Cech cohomology (Lemma 22.1, in fact we gave two proofs for this lemma, see also Remark 22.19 below for a third proof) and then used the ˇ spectral sequence linking Cech cohomology and cohomology in form of Cartan’s theorem ˇ (Proposition 21.83). Let us now sketch an argument that avoids Cech cohomology beyond degree 1 and spectral sequences which we learned from Johannes Ansch¨ utz. In fact we show the following two assertions by induction on p ≥ 1 simultaneously. (Ap ) One has H i (X, F ) = 0 for 1 ≤ i ≤ p for all affine schemes X and all quasi-coherent OX -modules F . (Bp ) One has Ri f∗ F = 0 for 1 ≤ i ≤ p for all affine morphisms f : X → Y of schemes and for all quasi-coherent OX -modules F . Assertion (Ap ) implies (Bp ) for all p by the same argument as in in the proof of Corollary 22.5. Assertion (A1 ) follows from Lemma 12.33 for p = 1 (here we use that cohomology ˇ agrees with Cech cohomology in degree 1 by Proposition 11.13 and Proposition 21.40). It remains to show (Ap ) for p ≥ 2 assuming that we already have proved (Ap−1 ) and (Bp−1 ). Let ξ ∈ H p (X, F ). By Corollary 21.23 there exists an affine open covering (Uk )1≤k≤n of X such that ξ |Uk = 0 for all Lk. Let jk : Uk → X be the inclusion which is an open affine immersion. Complete F → k Rjk,∗ (F |Uk ) to a distinguished triangle F −→

M

+1

Rjk,∗ (F |Uk ) −→ G −→

k

and consider the associated long exact cohomology sequence ··· →

n M

a

H p−1 (Uk , F |Uk ) → H p−1 (X, G ) → H p (X, F ) −→

k=1

n M

H p (Uk , F |Uk ) → · · · .

k=1

By induction hypothesis, we have H p−1 (Uk , F |Uk ) = 0. By assumption, we have a(ξ) = 0. Hence it suffices to show that H p−1 (X, G ) = 0. For this consider the distinguished triangle +1

τ ≤0 G −→ G −→ τ ≥1 G −→ . L Then τ ≤0 G is the quasi-coherent module Coker(F → k jk,∗ (F |Uk )) considered as a complex concentrated in degree 0. Moreover, since by induction hypothesis we have Ri jk,∗ (F |Uk ) = 0 for 1 ≤ i < p, we have τ ≥1 G = τ ≥p G . Therefore the long exact cohomology sequence attached to (*) shows that

(*)

H p−1 (X, G ) ∼ = H p−1 (X, τ ≤0 G ) = 0, where the second equality holds by induction hypothesis.

236

22 Cohomology of quasi-coherent modules

ˇ (22.2) Cohomology versus Cech Cohomology. Theorem 22.9. Let X be a separated scheme, and let F be a quasi-coherent OX module. Then for every open covering U of X by open affine subschemes, the canonical homomorphisms ˇ i (X, F ) → H i (X, F ) H i (U , F ) → H are isomorphisms for all i ≥ 0. Proof. In view of Lemma 22.1, this follows immediately from Cartan’s Theorem, Proposition 21.83, using that in a separated scheme the intersection of two open affine subschemes is again affine (Proposition 9.15). Corollary 22.10. Let X be a separated scheme which can be covered by n affine open subschemes. Then for all i ≥ n and all quasi-coherent OX -modules F , H i (X, F ) = 0. ˇ Proof. Use the above identification of cohomology with Cech cohomology and the correˇ sponding fact for Cech cohomology, which is a direct consequence of the fact that we can ˇ compute Cech cohomology using alternating cocycles, see Corollary 21.70. Remark 22.11. Let X be a separated scheme, let U be an affine open cover of X and let 0 → F ′ → F → F ′′ → 0 be a short exact sequence of quasi-coherent OX -modules. The vanishing of higher degree cohomology of quasi-coherent modules on affine schemes implies that the sequence 0 → C • (U , F ′ ) → C • (U , F ) → C • (U , F ′′ ) → 0 ˇ of Cech complexes is exact. It hence gives rise to a long exact cohomology sequence. ˇ It follows from the comparison between Cech cohomology and usual cohomology (Theorem 21.78) that under the identification of Theorem 22.9 this long exact sequence coincides with the long exact sequence for usual cohomology. This gives a handle on actually computing the maps in the long exact cohomology sequence. Remark and Definition 22.12. The proof of Theorem 22.9 shows that the conclusions of Theorem 22.9 and Corollary 22.10 also hold if X has an affine diagonal (i.e. the intersection of any two open affine subschemes is again affine) instead of assuming that X is separated. Such schemes are called semiseparated . See Exercise 22.4 for more details. For instance, let k be a field and let X be the k-scheme obtained from gluing two copies of Ank along Ank \ {0} via the identity. Then X is not separated for n ≥ 1. It is ˇ semiseparated if and only if n = 1. Indeed, one can show that Cech cohomology does not agree with cohomology for n ≥ 2 (cf. Exercise 22.5). (22.3) Elementary examples: P1 and A2 \ {0}. Let us give some examples of (and references to) explicit computations of cohomology ˇ groups using Cech cohomology.

237 Example 22.13. Let R ̸= 0 be a ring, and consider the standard covering P1R = Proj(R[T0 , T1 ]) = U0 ∪ U1 ,

Ui = D+ (Ti ).

By Theorem 22.9 we can compute the cohomology groups of quasi-coherent sheaves using ˇ the alternating Cech complex for this covering, in particular for d ∈ Z the R-modules i 1 H (PR , O(d)) are the cohomology modules of the sequence 0 → H 0 (U0 , O(d)) × H 0 (U1 , O(d)) → H 0 (U0 ∩ U1 , O(d)) → 0. As in Example 11.45 we can consider the components of the sequence as R-submodules of R(T0 , T1 ) and can by (11.14.4) identify the above sequence with 0 → T0d R[T1 /T0 ] × T1d R[T0 /T1 ] → T0d R[T0 /T1 , T1 /T0 ] = T1d R[T0 /T1 , T1 /T0 ] → 0, (f, g) 7→ f − g. We obtain that H 0 (P1R , O(d)) and H 1 (P1R , O(d)) are free R-modules of rank  d + 1 if d ≥ 0, 0 1 rk H (PR , O(d)) = 0 if d < 0, and rk H 1 (P1R , O(d)) =



0 if d ≥ −1, −d − 1 if d < −1.

For H 0 (P1R , O(d)) this reproduces the results of Example 11.45. Together with some more careful bookkeeping, the same method can be used to compute all cohomology groups H i (PrR , O(d)). See Theorem 22.22 below. Example 22.14. Let R = ̸ 0 be a ring, A = R[T, S], and let X = A2R \ {0} ⊂ A2R = Spec A. We have the affine open cover U = (D(T ), D(S)) of X, and the corresponding alternating ˇ Cech complex for OX is d

0 → AT × AS −→ AT S → 0, Hence

d(f, g) = g − f.

ˇ 0 (U , OX ) = A, H 0 (X, OX ) = H M ˇ 1 (U , OX ) = H 1 (X, OX ) = H RT −i S −j . i,j>0

In particular, both are of infinite rank over R and we see (again, at least for R = k an algebraically closed field, cf. Exercise 1.13) that X is not affine because otherwise we would have H 1 (X, OX ) = 0 by Theorem 22.2. We also see that the cohomology, even for “nice” sheaves such as the structure sheaf, may be not finitely generated. In fact, H 1 (X, OX ) is not even finitely generated as a Γ(X, OX )-module. We will see later (Corollary 23.18) that for R a noetherian ring, a proper R-scheme Z, and a coherent OZ -module F , all cohomology groups H i (Z, F ) are finitely generated R-modules.

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22 Cohomology of quasi-coherent modules

ˇ (22.4) The extended ordered Cech complex and Koszul complexes. The examples in Section (22.3) are a special case of a general procedure to calculate the ˇ Cech cohomology of quasi-affine schemes U and of projective spectra. This is based on ˇ identifying the extended ordered Cech complex for an open covering by principal open affine subsets with a colimit of Koszul complexes (Section (19.1)), as we explain now. We put ourselves in the following situation. Let A be a ring, X = Spec A, and let ˇ f1 , . . . , fr be elements of A. The ordered Cech complex of the structure sheaf associated to the open covering U := (D(fi ))i of U := X \ V (f1 , . . . , fr ), augmented by A, then ˇ yields the extended ordered Cech complex Y Y Afi1 fi2 −→ . . . −→ Af1 f2 ···fr , Afi1 −→ (22.4.1) CA (f1 , . . . , fr ) : A −→ i1 0 one has (22.4.3)

ˇ p (U , F ) = H p+1 (CA (f ) ⊗A M ), H p (U, F ) = H

where the first equality holds by Theorem 22.9. Note that CA (fi ) is the complex A → Afi , concentrated in degrees 0 and 1, and that (22.4.4)

CA (f1 , . . . , fr ) = CA (f1 ) ⊗A · · · ⊗A CA (fr ).

We now relate the complex CA (f1 , . . . , fr ) of (22.4.1) to the Koszul complex. For a sequence of elements f = (f1 , . . . , fr ) in A we view the attached Koszul complex as a cochain complex V K(f ) = K • (f ) in degrees [−r, 0]. Hence the entry of K(f ) in degree −p p (Ar ) and the differential K −p (f ) → K −p+1 (f ) is given by is the A-module x1 ∧ · · · ∧ xp 7→

p X i=1

(−1)i+1

r X


fj xi,j x1 ∧ · · · ∧ xbi ∧ · · · ∧ xp .

j=1 fi

For all i we have that K(fi ) is the complex A −→ A concentrated in degrees −1 and 0 and L L (22.4.5) K(f ) = K(f1 ) ⊗A K(f2 ) ⊗A · · · ⊗A K(fr ) = K(f1 ) ⊗L A K(f2 ) ⊗A · · · ⊗A K(fr ),

where the second equality holds because K(fi ) is K-flat. We also consider the dual Koszul complex K(f )∨ := R HomA (K(f ), A) = HomA (K(f ), A), where the equality holds because K(f ) is a K-projective complex of A-modules. For all i −fi one has K(fi )∨ = A −→ A, now concentrated in degrees 0 and 1, i.e., K(fi )∨ ∼ = K(fi )[−1] (recall that by definition, also the shift by i introduces a sign (−1)i ). By (22.4.5) one deduces (22.4.6)

K(f )∨ = K(f1 )∨ ⊗A · · · ⊗A K(fr )∨ ∼ = K(f )[−r].

Remark 22.15. Set f n := (f1n , . . . , frn ) for n ≥ 1.

239 (1) The functoriality of the Koszul complex (Remark 19.5) then yields an Zopp ≥1 -diagram of complexes · · · → K(f 3 ) → K(f 2 ) → K(f ). If (e1 , . . . , er ) denotes the standard basis of Ar , then in degree −p the transition maps are given by

Vp

Vp

Ar −→ K −p (f n ) = Ar , K −p (f n+1 ) = ei1 ∧ · · · ∧ eip 7−→ ((−1)p fi1 · · · fip )ei1 ∧ · · · ∧ eip . (2) Passing to dual Koszul complexes and using (22.4.6), we obtain an Z≥1 -diagram of complexes K(f n )[−r] → K(f n+1 )[−r] of complexes sitting in degrees [0, r] by functoriality. The transition maps are given by   Y f i  ε i1 ∧ · · · ∧ ε ip , εi1 ∧ · · · ∧ εip 7−→  i∈{i / 1 ,...,ip }

where (ε1 , . . . , εr ) now denotes the dual basis of the standard basis. In particular, the transition maps are the identity in degree 0 and equal to f1 · · · fr in degree r. Using f f f that Af = colim(A −→ A −→ A −→ · · · ) one sees that (22.4.7)

colim K(f n )∨ = colim K(f1n )[−1] ⊗ · · · ⊗ colim K(frn )[−1] = (A → Af1 ) ⊗ · · · ⊗ (A → Afr ) = CA (f1 , . . . , fr )

ˇ is the extended ordered Cech complex defined in (22.4.1). By (22.4.6) we obtain from (22.4.7) the following result. Proposition 22.16. There exists an isomorphism of complexes of A-modules (22.4.8)

colim K(f n )[−r] ∼ = colim K(f n )∨ = CA (f1 , . . . , fr ).

(22.5) Example: Cohomology of quasi-affine schemes. We can now use (22.4.8) to compute some cohomology groups for quasi-affine schemes. Example 22.17. Let U be a quasi-affine scheme, let A := Γ(U, OU ) and let j : U → Spec A be the canonical open quasi-compact schematically dominant immersion (Proposition 13.80). As U is quasi-compact, there exists a finite family f = (f1 , . . . , fr ) of elements of A such that U := (D(fi ))i is an affine open covering of U . Let F be a quasi-coherent ˜ for an OU -module. Then j∗ F is quasi-coherent (Corollary 10.27) and hence j∗ F ∼ =M A-module M . We have (22.5.1)

H 0 (U, F ) = H 0 (Spec A, j∗ F ) = M.

Now assume that p > 0. If r = 1, then U is affine and hence A = Af1 , i.e., f1 is a unit and H p (U, F ) = 0. Hence let us assume from now on that r ≥ 2. ˇ We can calculate Cech cohomology of F by (22.4.3) and hence get by (22.4.8) ˇ p (U , F ) H p (U, F ) = H = H p+1 (CA (f ) ⊗A M ) (22.5.2)

= H p+1 (colim K(f n )[−r] ⊗A M ) = colim H n

n p−r+1

(K(f n , M ))

for p > 0.

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22 Cohomology of quasi-coherent modules

Now suppose that f is M -completely intersecting. Then f n is M -completely intersecting (Proposition 19.17) and hence H q (K(f n , M )) = 0 for q < 0 and H 0 (K(f n , M )) = M/f n M := M/(f1n , . . . , frn )M . Hence in this case by (22.5.2) and (22.5.1) we obtain   if p = 0; M, (22.5.3) H p (U, F ) = 0, if p ̸= 0, r − 1;   colimn M/f n M, if p = r − 1, where the transition maps in the colimit are given by multiplication by f1 f2 · · · fr . Example 22.18. Let R be a ring and let A = R[T1 , . . . , Tr ] with r ≥ 2. We apply (22.5.3) to the family (T1 , . . . , Tr ) which S is regular and in particular completely intersecting (Theorem 19.12). One has U = i D(Ti ) = ArR \ {0} and Γ(U, OU ) = A. Hence we obtain   if p = 0; A, p H (U, OU ) = 0, if p ̸= 0, r − 1;   colimn A/(T1n , . . . , Trn )A, if p = r − 1, Let us make the term colimn A/(T1n , . . . , Trn )A more explicit. For any subset I ⊆ Zr , we denote the free R-submodule of R[T1±1 , . . . , Tr±1 ] generated by T1i1 T2i2 · · · Trir for (i1 , . . . , ir ) ∈ I by M I . Then M I is an A-submodule if and only if I is stable under addition with elements in Nr . Set Zr+ = { (i1 , . . . , ir ) ∈ Zr ; ∃ j : ij ≥ 0 }. r

r

r

As an A-module, we can identify A/(T1n , . . . , Trn )A with M [−n,∞) /(M [−n,∞) ∩ M Z+ ) such that the transition map A/(T1n , . . . , Trn )A → A/(T1n+1 , . . . , Trn+1 )A given by mulr r tiplication by T1 · · · Tr is induced by the inclusion M [−n,∞) ,→ M [−n−1,∞) . Then colimn A/(T1n , . . . , Trn )A becomes the A-module r

r

H r−1 (U, OU ) = M Z /M Z+ , =

R[T1±1 , . . . , Tr±1 ]

X

±1 ±1 R[T1±1 , . . . , Ti−1 , Ti , Ti+1 , . . . , Tr±1 ]

i r

which we can identify as an R-module with M (−∞,−1] , i.e., the free R-module generated by the monomials T1i1 · · · Trir with ij < 0 for all j = 1, . . . , r. ˇ p (Spec A, F ) = 0 for all Remark 22.19. Example 22.17 gives also a new proof that H ˜ quasi-coherent sheaves F = M on Spec A and all p > 0: If (D(fi ))i is an open covering of Spec A, i.e., f1 , . . . , fr generate the unit ideal, then K(f n ) is homotopy equivalent to 0 by Remark 19.6. Hence K(f n , M ) is homotopy equivalent to 0 and we conclude by (22.5.2). (22.6) Example: Cohomology of twisted line bundles on projective spectra and on projective space. Example 22.13 can also be considered as a special case of the technique to use the Koszul ˇ complex to calculate (Cech ) cohomology.

241 L Example 22.20. Let R be a ring and let A = d≥0 Ad be a graded R-algebra with A0 = R which is generated by a finite family f = (f1 , . . . , fr ) of elements fi ∈ A1 of degree 1. Set X := Proj A. This is a projective scheme over R with an open affine covering U := (D+ (fi ))i (Proposition 13.12). Hence we can calculate the cohomology of ˇ quasi-coherent OX -modules as Cech cohomology for the covering U . We would like to compute H p (X, OX (d)) for all d ∈ Z. As X is separated, we have ˇ p (U , F ) for every quasi-coherent OX -module F by Theorem 22.9. MoreH p (X, F ) = H over, cohomology commutes with direct sums L (Corollary 21.56) because X is qcqs. Hence we can compute the cohomology of F := d∈Z OX (d), and split up the result according to d afterwards. Recall that we denote by Afi the graded localization of A by fi and by A(fi ) the subring of degree 0 elements (Section (13.1)). By definition of OX (d) (Section (13.4)) we have an ˇ isomorphism Γ(D+ (fi ), F ) = Afi of graded A(fi ) -modules. Therefore the ordered Cech complex of F with respect to U is given by the complex Y Y Afi1 fi2 −→ . . . −→ Af1 f2 ···fr −→ 0 Afi1 −→ (*) 0 −→ i1

i1 0, (22.6.1)

H p (X, OX (d)) = colim H p−r+1 (K(f n ))d , n

where (−)d denotes the R-submodule of homogeneous elements of degree d. Now suppose that f is completely intersecting and that r ≥ 2, then we obtain, again as in Remark 22.17, that ( 0, if p ̸= 0, r − 1; p (22.6.2) H (X, OX (d)) = n colimn (A/f A)d , if p = r − 1, Moreover, we have H 0 (CA (f1 , . . . , fr )) = H 1 (CA (f1 , . . . , fr )) = 0 in this case because ˇ K(f n ) is acyclic in degrees −r and −r + 1. Hence the cohomology of the ordered Cech 0 0 ≥0 complex (*) in degree 0 is A. Therefore H (X, F ) = H (σ (CA (f1 , . . . , fr )[1]) = A and (22.6.3)

H 0 (X, OX (d)) = Ad .

Similarly as in Example 22.20 one can also compute H p (X, F (d)) for every quasicoherent module F on Proj A, see Exercise 22.6. All of these ideas can be further generalized, see Exercise 22.7. Remark 22.21. The similarity of the results in Example 22.20 and in Example 22.17 is no happenstance. Let A be a graded R-algebra as in Example 22.20, let f1 , . . . , fr ∈ A1 be generators of the R-algebra A, and set X = Proj(A). Suppose that (f1 , . . . , fr ) is completely intersecting with r ≥ 2. Then M (*) A = Γ∗ (OX ) = Γ(X, OX (d)) d

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22 Cohomology of quasi-coherent modules

by (22.6.3). Let C 0 = Spec A \ V (f1 , . . . , fr ) be the pointed cone of A (Section (13.9)). The canonical morphism π : C 0 → Proj(A) is a surjective morphism such that π −1 (D+ (fi )) = D(fi ) ∼ D+ (fi ) ×R (A1R \ {0}) =L (Proposition 13.37). In particular, π is affine. Moreover π∗ OC 0 = d∈Z OX (d) by (*) and hence we find by Corollary 22.6 for all p ≥ 0 an isomorphism M H p (C 0 , OC 0 ) = H p (X, OX (d)). d∈Z

As an application we can calculate the cohomology of the twisted line bundles on projective space. Theorem 22.22. Let R be a ring, let r > 0 and consider PrR = Proj R[T0 , . . . , Tr ]. (1) For all d ∈ Z and all i ̸= 0, r, we have H i (PrR , OPrR (d)) = 0. (2) The canonical homomorphism of graded R-algebras M R[T0 , . . . , Tr ] → H 0 (PrR , OPrR (d)) d∈Z

is an isomorphism. In particular, H 0 (PrR , OPrR (d)) = 0 for d < 0. (3) The A-module H r (PrR , OPrR (d)) is free, and is isomorphic to the A-submodule of p0 −1 −1 pr R[T P 0 , . . . , Tr ] generated rby rall monomials T0 · · · Tr with pi < 0 for all i and r i pi = d. In particular, H (PR , OPR (d)) = 0 for d > −r − 1. The theorem can be generalized to arbitrary base schemes and projective bundles, see Theorem 22.86 below. Proof. As (T0 , . . . , Tr ) is a regular sequence (of length r + 1) in R[T0 , . . . , Tr ], it is completely intersecting (Theorem 19.12). Hence (2) (which we also had already seen in Example 13.16) follows from (22.6.3). Assertions (1) and (3) are implied by (22.6.2), where for the description of H r (PrR , OPrR (d)) we argue as in Example 22.18. Instead of referring to (22.6.2) we could also have used the pointed affine cone π : Ar+1 R \ {0} → PrR and Example 22.18. Corollary 22.23. Let R be a ring, and let r > 0. The A-module H r (PrR , OPrR (−r − 1)) is free of rank 1, and for every d ∈ Z there is a perfect pairing (*)

H 0 (PrR , OPrR (−r − 1 − d)) × H r (PrR , OPrR (d)) → H r (PrR , OPrR (−r − 1)) ∼ = R.

Note that ω := OPrR (−r − 1) = ΩrPr /R as one sees by applying the determinant to the R short exact sequence (17.7.3) (Remark 17.58). This line bundle is in fact the relative ∼ dualizing sheaf for PrR , see Section (25.9) below. The isomorphism R → H r (PrR , OPrR (−r − 1)) is canonical (Theorem 22.86 below) and the pairing (*) is a special case of Grothendieck duality (Proposition 25.55).

243

Derived categories of quasi-coherent modules Similarly as in (F.41), we denote for every scheme X by Dqcoh (X) the full additive subcategory of D(X) consisting of objects E such that H i (E) is a quasi-coherent OX module for all i. We now will show two central results. The first (Theorem 22.31) shows that if f : X → Y is a qcqs morphism of schemes, then Rf∗ maps Dqcoh (X) to Dqcoh (Y ). A special case is the assertion that for every quasi-coherent module F on X all higher direct images Ri f∗ F are again quasi-coherent. In fact, we will prove the special case first (Theorem 22.27). Deducing Theorem 22.31 from this special case is not difficult for bounded below complexes. In the general case one has to approximate complexes by bounded below complexes. This technique is explained in Section (22.7). The second central result (Theorem 22.35) is that for quasi-compact separated1 schemes X the natural functor from the derived category of the abelian category of quasi-coherent OX -modules to Dqcoh (X) is an equivalence of categories. In particular one obtains for every ring A an equivalence D(A) ∼ = Dqcoh (Spec A). In the end we will show that this equivalence is compatible with derived tensor products and derived pullback. (22.7) Homotopy Limits. For the notion of homotopy limit, indexed by N, see Definition F.228. Since products exist in D(X) (Lemma F.188), all homotopy limits exist. Definition and Lemma 22.24. Let X be a scheme. Then Dqcoh (X) is left complete, i.e., for every E ∈ Dqcoh (X) the natural morphism ∼

E → holim τ ≥−n E is an isomorphism in D(X). Proof. We write Kn := τ ≥−n E and K := holim Kn . The natural maps E → Kn induce a morphism E → K via the exact triangle defining holim Kn , and we want to show that ∼ this is an isomorphism, i.e., that H m (E) → H m (K) for all m. ∼ (I). We claim that for all m, there exists an integer n(m) ≥ −m such that H m (U, K) → m H (U, Kn(m) ) for all affine open subschemes U ⊆ X. In fact, we have distinguished triangles H −n−1 (E)[n + 1] → Kn+1 → Kn → . Since E has quasi-coherent cohomology sheaves and U is affine, by Theorem 22.2 we have H i (U, H n (E)) = 0 for all n and all i > 0, so using the long exact cohomology sequence for this triangle, we find n(m) such that for all n ≥ n(m), H m (U, Kn+1 ) → H m (U, Kn ) and H m−1 (U, Kn+1 ) → H m−1 (U, Kn ) are isomorphisms. Then R1 limn H m−1 (RΓ(U, Kn )) = 0 by Lemma F.227, and Lemma F.233 together with the isomorphisms H m (U, Kn+1 ) → H m (U, Kn ) gives ∼

H m (holim RΓ(U, Kn )) → lim H m (RΓ(U, Kn )) = lim H m (U, Kn ) ∼ = H m (U, Kn(m) ). Since H m (holim RΓ(U, Kn )) = H m (RΓ(U, holim Kn )) = H m (U, K) by Lemma F.230, we obtain the desired isomorphisms. This shows our claim. 1

We will see that it is possible to relax the condition to be separated somewhat.

244

22 Cohomology of quasi-coherent modules

(II). Now let x ∈ X. We claim that the map H m (K)x → H m (Kn(m) )x on stalks is injective. To see this, let γ ∈ Ker(H m (K)x → H m (Kn(m) )x ). Since H m (K) is the sheafification of the presheaf U 7→ H m (U, K) (Lemma 21.22), we can lift γ to an element γ˜ ∈ H m (U, K) for some sufficiently small affine open U ⊆ X. Since γ maps to 0 in H m (Kn(m) )x , it follows that after shrinking U , if necessary, γ˜ ∈ Ker(H m (U, K) → H m (U, Kn(m) )) which is 0 by Step (I), whence γ = 0. (III). Now consider the maps H m (E)x → H m (K)x → H m (Kn(m) )x . Since n(m) ≥ −m, H m (E) → H m (Kn(m) ) is an isomorphism, so the composition of the two maps above is an isomorphism. By Step (ii), we also know that the second map is injective. It follows that H m (E)x → H m (K)x is an isomorphism for all x, whence H m (E) → H m (K) is an isomorphism, as desired. Lemma 22.25. Let X be a scheme, let F : (OX -Mod) → (AbGrp) be an additive functor which commutes with countable products, and let N ≥ 0 such that Ri F (F ) = 0 for all i ≥ N and all quasi-coherent OX -modules F . Then for all E ∈ Dqcoh (X) and all i ∈ Z, the natural morphism Ri F (E) → Ri F (τ ≥i−N +1 E) is an isomorphism. Proof. Since the claimed statement is compatible with shifts, we may assume that i = 0. By Lemma 22.24, we have E ∼ = holim(τ ≥−n E). By Lemma F.230 and the following remark, the derived functor RF commutes with derived limits, so RF (E) = holim RF (τ ≥−n E). As a consequence, Lemma F.233 gives us a short exact sequence 0 → R1 lim R−1 F (τ ≥−n E) → R0 F (E) → lim R0 F (τ ≥−n E) → 0. Claim. The morphism Rj F (τ ≥−n E) → Rj F (τ ≥−n+1 E) is an isomorphism whenever j + n ≥ N . Note that the claim implies the lemma. Namely, using the claim for j = −1 and n sufficiently large, we see that the inverse system (R−1 F (τ ≥−n E))n satisfies the MittagLeffler property, and as a consequence the R1 lim term on the left of the above short exact sequence vanishes (Lemma F.227). Furthermore, for j = 0 we see that the inverse system (R0 F (τ ≥−n E))n becomes constant at n = N − 1, so the term on the right of the short exact sequence can be identified with R0 F (τ ≥−N +1 E), as desired. It remains to prove the claim. To this end, we consider the exact triangle (Proposition F.157) H −n (E)[n] → τ ≥−n E → τ ≥−n+1 E → relating E with its truncations. By the assumptions on F , we have Rj F (H −n (E)[n]) = Rj+n F H −n (E) = 0 for j + n ≥ N, so the long exact cohomology sequence for RF proves the claim.

245 (22.8) Quasi-coherence of higher direct images. We have already seen (Corollary 10.27) that the direct image f∗ F of a quasi-coherent OX -module under a qcqs morphism f : X → Y is quasi-coherent. The same is true for the higher direct images. To prove this, we start with a lemma. Lemma 22.26. Let X be a qcqs scheme, and let F be a quasi-coherent OX -module. Let t ∈ Γ(X, OX ), Xt = {x ∈ X; t(x) ̸= 0}, and i ≥ 0. Then we have a functorial isomorphism H i (X, F )t → H i (Xt , F ). Proof. Here, as usual, we write H i (Xt , F ) as short-hand notation for H i (Xt , F|Xt ). For i = 0, this lemma is a special case (L = OX ) of Theorem 7.22 (and the above lemma also admits a corresponding generalization, see [EGAIII] O (1.4.5) or Exercise 22.7). We start with the following remark: Let R be a ring, t ∈ R, and M an R-module. Then the localization Mt is naturally isomorphic to the colimit colim M of the system M → M → M → · · · whose transition maps are all given by multiplication by t. For example, we can view Γ(Xt , OX ) = Γ(X, OX )t as colimt· Γ(X, OX ). Passing to the context of sheaves, let j : Xt → X be the inclusion, and let F be a quasicoherent sheaf on X. Then j is an open affine immersion. We claim that j∗ j ∗ F = colimt· F . In fact, both sides are quasi-coherent OX -modules, and the sections over some open affine subset U ⊆ X are H 0 (Ut| U , F ) and colimt· H 0 (U, F ) (in the second case, use Lemma 21.52), and the claim follows from the above remark and the case i = 0 of the lemma, which we know is true. Combining this with Corollary 22.6, applied to the affine morphism j, we see that H i (Xt , F ) = H i (X, j∗ j ∗ F ) = H i (X, colim F ). t·

Since cohomology commutes with inductive limits (Proposition 21.54), we also have H i (X, colim F ) = colim H i (X, F ) = H i (X, F )t . t·

t·

Note that all maps we considered are Γ(X, OX )-module homomorphisms. Because both H i (Xt , F ) and H i (X, F )t are modules over Γ(X, OX )t = Γ(Xt , OX ), the resulting identification is actually an isomorphism of Γ(X, OX )t -modules. Theorem 22.27. Let f : X → Y be a qcqs morphism of schemes, and let F be a quasicoherent OX -module. Then for all i ≥ 0, the OY -module Ri f∗ F is quasi-coherent. If Y is affine, then Ri f∗ F = H i (X, F )∼ . Proof. Since taking higher direct images is local on Y (Lemma 21.28), it is enough to prove that Ri f∗ F = H i (X, F )∼ provided that Y = Spec A is affine. As in Remark 21.20, we denote by H i (F ) the presheaf U 7→ H i (U, F|U ) on X. Recall (Proposition 21.27) that Ri f∗ F equals the sheafification of f∗ H i (F ). Now let s ∈ A. We obtain Γ(D(s), f∗ H i (F )) = H i (Xφ(s) , F ) = H i (X, F )s = Γ(D(s), H i (X, F )∼ ),

246

22 Cohomology of quasi-coherent modules

where φ : A → Γ(X, OX ) denotes the ring homomorphism given by f , where one defines Xφ(s) = {x ∈ X; φ(s)(x) ̸= 0} = f −1 (D(s)), and the second equality follows from Lemma 22.26. These equalities are compatible with restriction maps with respect to inclusions D(t) ⊆ D(s). We see that the presheaf f∗ H i (F ) equals the sheaf H i (X, F )∼ on the basis (D(s))s∈A of open subsets of Y , and hence its sheafification Ri f∗ F is H i (X, F )∼ . For a different proof for Y affine and X noetherian, see Exercise 22.18. Lemma 22.28. Let Y be a quasi-compact scheme, and let f : X → Y be a qcqs morphism. There exists N such that Ri f∗ F = 0 for all i ≥ N and all quasi-coherent OX -modules F . If Y is affine and f is separated, one can take N to be the minimal number r such that X can be covered by r open affine subschemes. Proof. We may assume that Y is affine. By Theorem 22.27 it is enough to show that there exists N with H i (X, F ) = 0 for all i ≥ N and all quasi-coherent F . If X is separated, then this follows from Corollary 22.10. ˇ To deal with the general case, we can use the spectral sequence from Cech to usual cohomology, Theorem 21.78, for a covering U of X by finitely many affine open subschemes, ˇ p (U , H q (F )) ⇒ H p+q (X, F ). E2p,q = H ˇ By Proposition 21.69 we can compute the E2p,q -terms using the alternating Cech complex, hence E2p,q = 0 whenever p is sufficiently large, independent of F . On the other hand, all intersections of subsets in U are separated, so that we know, by the remark in the ˇ beginning of the proof, that the lemma applies to them. Therefore the (alternating) Cech q complex for U and H (F ) vanishes whenever q is sufficiently large. Altogether we see that E2p,q = 0 for p + q sufficiently large and the desired result follows. Proposition 22.29. Let f : X → S be a qcqs morphism between quasi-compact schemes. There exists an integer N ≥ 0 (depending on f ) such that for all integers b ∈ Z and for all E ∈ Dqcoh (X) with H j (E) = 0 for j > b we have Ri f∗ E = 0 for all i ≥ N + b. The proof will show that one can take the same N as in Lemma 22.28. Proof. By shifting the degree of E it suffices to show the existence of N for b = 0. Let N be as in Lemma 22.28. First assume that E is bounded below. We then have a spectral sequence (Corollary 21.46) E2ij = Ri f∗ H j (E) =⇒ Ri+j f∗ E. Since the H j (E) are quasi-coherent, Ri f∗ H j (E) = 0 whenever i ≥ N by Lemma 22.28. By assumption we have H j (E) = 0 whenever j > 0. Therefore the spectral sequence can yield a non-zero term only for i + j < N . This proves the proposition for bounded below complexes. To prove the general case, it is enough to show that H i (U, Rf∗ E) = 0 for all i ≥ N and U ⊆ S affine: Indeed, by Lemma 21.22, this implies that the stalks H i (Rf∗ E)x all vanish, whence Ri f∗ E = 0, as desired. We want to put ourselves into a situation where we can apply Lemma 22.25 to the functor F = Γ(f −1 (U ), −). First note that F commutes with arbitrary products.

247 For i ≥ N and F a quasi-coherent OX -module, we have (using Theorem 22.27 and Lemma 21.28) H i (f −1 (U ), F ) = H 0 (U, (Ri f∗ F )|U ) = 0

for i ≥ N.

We can identify RΓ(U, Rf∗ E) = RΓ(f −1 (U ), E) by Proposition 21.114 and in particular H j (U, Rf∗ E) = H j (f −1 (U ), E) for all j. We thus get from Lemma 22.25, applied to the functor F = Γ(f −1 (U ), −), for all i ≥ N that H i (U, Rf∗ E) = H i (f −1 (U ), E) = H i (f −1 (U ), τ ≥i−N +1 E) = H i (U, Rf∗ (τ ≥i−N +1 E)) = 0 as we have already proved the proposition for bounded below complexes. As this holds for all open affine subschemes U ⊆ S, this show that H i (Rf∗ E) = Ri f∗ E = 0 for i ≥ N by Lemma 21.22. Corollary 22.30. Let f : X → S be a qcqs morphism between quasi-compact schemes. Then there exists N ≥ 0 such that for all E ∈ Dqcoh (X) and all i ∈ Z, the natural morphism Ri f∗ E → Ri f∗ (τ ≥i−N +1 E) is an isomorphism. Proof. Let N be as in Proposition 22.29, and consider the exact triangle (see Proposition F.157) τ ≤−N E → E → τ ≥−N +1 E → By Proposition 22.29, the cohomology of Rf∗ τ ≤−N E vanishes in degrees ≥ −N + N = 0, hence the long exact cohomology sequence gives that Ri f∗ E coincides with Ri f∗ τ ≥−N +1 E for all i ≥ 0. Applying this to all shifts of E gives the result. Theorem 22.31. Let f : X → Y be a qcqs morphism of schemes. Then Rf∗ induces functors Rf∗ : Dqcoh (X) → Dqcoh (Y ), ≥n ≥n Rf∗ : Dqcoh (X) → Dqcoh (Y )

for each n ∈ Z.

If Y is quasi-compact, then there exists an N ≥ 0 such that Rf∗ induces a functor [a,b]

[a,b+N ]

Rf∗ : Dqcoh (X) → Dqcoh

(Y )

for all integers a ≤ b.

Proof. For E bounded below, we use the spectral sequence Ri f∗ H j (E) =⇒ Ri+j f∗ E as in the proof of Proposition 22.29, and apply Theorem 22.27 together with the fact that extensions of quasi-coherent OY -modules are quasi-coherent (Corollary 22.4). For the general case, we use Corollary 22.30 for the functor f∗ and see that Ri f∗ E is isomorphic to a higher direct image of some bounded below complex, hence quasi-coherent by the above.

248

22 Cohomology of quasi-coherent modules

(22.9) The category of quasi-coherent OX -modules and its derived category. Let X be a scheme. Denote by (QCoh(X)) the abelian category of quasi-coherent OX modules. It is a plump (Definition F.43) abelian subcategory of (OX -Mod) (Corollary 22.4) which is stable under tensor products. If X = Spec A is affine, then (QCoh(X)) is equivalent to the category of A-modules. By reduction to the affine case, it follows that the category (QCoh(X)) has finite limits, and all colimits, and the inclusion ι : QCoh(X) → (OX -Mod) preserves finite limits and all colimits, for an arbitrary scheme X. Returning to the affine case X = Spec A for a moment, the equivalence between (QCoh(X)) and the category of A-modules also gives us that this category has a generator (namely, OX ), enough injectives, and all limits. In order to generalize this, we look at this from the following point of view: Every quasi-coherent OX -module on X has the form M ∼ for some A-module M . Let G be an arbitrary OX -module. It is easy to check that Hom(OX -Mod) (M ∼ , G ) ∼ = HomA (M, Γ(X, G )) = HomQCoh(X) (M ∼ , Γ(X, G )∼ ), functorially in M and G , i.e., the functor Q : G 7→ Γ(X, G )∼ is the right adjoint functor of the inclusion ι. Since Q is a right adjoint functor, it preserves all limits. This implies that the category QCoh(X) has all limits — just take the limit in (OX -Mod) and apply Q. As ι is exact, Q sends injectives in (OX -Mod) to injectives in QCoh(X). We obtain that QCoh(X) has enough injectives. As the following theorem states, the existence of a right adjoint to ι generalizes to arbitrary schemes. Theorem and Definition 22.32. Let X be a scheme. (1) The category QCoh(X) has all limits. It is a Grothendieck abelian category (Def. F.54) and in particular has K-injective resolutions (Theorem F.185). (2) The inclusion ι : QCoh(X) → (OX -Mod) admits a right adjoint functor Q, called the coherator. The adjunction morphism id → Q ◦ ι is an isomorphism. Proof. We give the proof of (1) only in the case that X is qcqs. (See below for remarks and references concerning the general case.) The key point is to show the existence of a generator. This can be seen as a set-theoretic problem: In fact, if we could form the direct sum of all objects in the category, it would obviously have the property of a generator. Now if X is qcqs, every quasi-coherent sheaf on X is a filtered colimit of finitely presented OX -modules by Corollary 10.50. The family of isomorphism classes of all finitely presented OX -modules is a set, and hence the direct sum of a set of representatives of these isomorphism classes exists and is a generator. We have already noted that all colimits exist in QCoh(X). Furthermore, filtered colimits are exact, since the stalk of a colimit is equal to the colimit of the stalks at the given point. Every Grothendieck abelian category has enough K-injective complexes, see Theorem F.185. For the general case, one needs to prove a replacement for the above result that quasicoherent OX -modules for X qcqs are filtered colimits of finitely presented modules, where roughly speaking “finite” is replaced by “less than some suitable cardinal κ”. See [EnEs] O Corollary 3.5 (Gabber’s Lemma). For (2), the existence of an adjoint functor follows from an appropriate version of the adjoint functor theorem (Corollary F.57).

249 In terms of derived categories, we can now restate the above result on vanishing of higher direct images of quasi-coherent modules under affine morphisms as follows. Proposition 22.33. Let f : X → Y be an affine morphism. The push-forward functor f∗ : (QCoh(X)) → (QCoh(Y )) is exact and hence it “is” its own derived functor f∗ : D(QCoh(X)) → D(QCoh(Y )). We obtain a commutative diagram D(QCoh(X))

ιX

/ D(X)

ιY

 / D(Y ),

f∗

 D(QCoh(Y ))

Rf∗

where the horizontal arrows are the natural inclusion functors. This property in fact characterizes affine morphisms among qcqs morphisms, see Exercise 22.11. Proof. We know by Corollary 22.5 that f∗ is exact on the category of quasi-coherent OX -modules. This means that we can compute its derived functor by applying f∗ to all terms of any complex representing the object of the derived category at hand. (This is what we mean when we say that f∗ is its own derived functor.) To prove the commutativity of the diagram, we will show that for all complexes F of quasi-coherent OX -modules the natural morphism f∗ F → Rf∗ F is a quasi-isomorphism, hence an isomorphism in D(Y ). (Note that this is a direct consequence of Corollary 22.5 in case F is concentrated in degree 0.) Since this morphism is compatible with shifts, it is enough to show that for all complexes F of quasi-coherent OX -modules, the induced morphism H 0 (f∗ F ) → R0 f∗ F is an isomorphism. By Corollary 22.30 we can reduce to the case that F is bounded below. Then we can apply Proposition F.204 which says that a bounded below complex consisting of objects which are all acyclic for some functor is itself acyclic for that functor. Since higher direct images of quasi-coherent sheaves under f vanish, we conclude that F is f∗ -acyclic, or equivalently, that the natural morphism f∗ F → Rf∗ F is a quasi-isomorphism (see Corollary F.174). In particular, H 0 (f∗ F ) → R0 f∗ F is an isomorphism, as desired. Proposition 22.34. Let f : X → Y be a flat qcqs morphism of schemes. Let RQX be the right derived functor of the coherator QCoh(X) → (OX -Mod) (see Theorem 22.32), and correspondingly for Y . Then we have a commutative diagram (up to natural isomorphism) D(QCoh(X)) o

RQX

RQCoh(X) f∗

 D(QCoh(Y )) o

D(X) Rf∗

RQY

 D(Y ),

where RQCoh(X) f∗ denotes the right derived functor of the direct image functor QCoh(X) → QCoh(Y ).

250

22 Cohomology of quasi-coherent modules

Proof. Since f is qcqs, it gives rise to a functor f∗ : QCoh(X) → QCoh(Y ). Since QCoh(X) is a Grothendieck abelian category (Theorem 22.32), the right derived functor exists (Corollary F.187). Now consider the inverse image functor f ∗ . Since f is flat by assumption, f ∗ is exact and is hence is equal to its own derived functor f ∗ = Lf ∗ , regardless of whether we consider f on all of (OX -Mod), or as a functor QCoh(Y ) → QCoh(X). Denoting by ιX : DQCoh(X) → D(X) the inclusion functor, and likewise for Y , we have f ∗ ιY (F ) = ιX (f ∗ F ). Using this, and adjunction between Rf∗ and Lf ∗ (Theorem 21.126), one easily checks that both compositions D(X) → DQCoh(Y ) in the above diagram are right adjoint to the functor DQCoh(Y ) → D(X), F 7→ f ∗ ιY (F ) = ιX (f ∗ F ). (22.10) Comparison of Dqcoh (X) and D(QCoh(X)). Recall that we denote by Dqcoh (X) the full additive subcategory of D(X) consisting of objects E such that H i (E) is a quasi-coherent OX -module for all i. The goal of this section is the proof of the following theorem: Theorem 22.35. Let X be a quasi-compact and semiseparated (Definition 22.12) scheme. The natural functor ιX : D(QCoh(X)) → Dqcoh (X) is an equivalence of categories. The proof will proceed in two steps. First we will consider the case where X = Spec R is affine. For most applications of Theorem 22.35 in the sequel, this special case will be sufficient. In the second step we will then globalize from affine schemes to quasi-compact semiseparated schemes using the right derived functor of the coherator (Definition 22.32). The first step is the following lemma. For X = Spec R, we identify the categories f, and as a consequence, we (R-Mod) and (QCoh(X)) via the usual construction M 7→ M get an identification of derived categories D(R) = D(QCoh(X)). Lemma 22.36. Let X = Spec R be an affine scheme. The functor ιX : D(R) = D(QCoh(X)) → Dqcoh (X) is an equivalence of categories with quasi-inverse functor RΓ(X, −). Proof. For every OX -module E, Γ(X, E) is an R-module, and we denote by Ψ the restriction of the derived functor of Γ(X, −) to Dqcoh (X): Ψ = RΓ(X, −)|Dqcoh (X) : Dqcoh (X) −→ D(R) = D(QCoh(X)). We have to show that ιX and Ψ are quasi-inverse to each other. Let us first show that the functor ιX is left adjoint to Ψ. Write Y for the one-point ringed space ({pt}, R). Then (OY -Mod) = (R-Mod) in an obvious way, and D(Y ) = D(R-Mod). We have an obvious morphism f : X → Y of ringed spaces. Since f∗ = Γ(X, −), we get Ψ(E) = Rf∗ (E) for all E ∈ Dqcoh (X). Furthermore, for an R-module M , we have f = f ∗ M , so we have ιX = f ∗ = Lf ∗ , the latter equality following from the fact that M the functor e· is exact. Now Proposition 21.126 implies that ιX is left adjoint to Ψ. This adjunction gives rise to natural maps ^E) = ιX (Ψ(E)) → E, RΓ(X,

E ∈ Dqcoh (X),

251 and f), M → Ψ(ιX (M )) = RΓ(X, M

M ∈ D(R-Mod).

Here we denote the derived functor of e· by the same symbol. Since the original functor is exact, it just means applying e· to each term of any representative. To conclude the proof, we will show that both these adjunction morphisms are isomorphisms (in the respective derived categories), for all E and M . In other words, choosing complexes representing E and M , respectively, and considering the above morphisms as morphisms of complexes, we have to show that they are quasi-isomorphisms. Since they are compatible with shifts, it is enough to prove that applying H 0 (−) yields isomorphisms. By Lemma 22.25 applied to the functor Γ(X, −) and N = 1 (Theorem 22.2), we have R0 Γ(X, E) = R0 Γ(X, τ ≥0 E) (note that this is immediate if E is concentrated in a single degree; the lemma requires quite some work in order to obtain the result for unbounded complexes). Since τ ≥0 E is bounded below, the spectral sequence in Corollary 21.46 then shows that R0 Γ(X, E) = Γ(X, H 0 (E)). Using this together with the exactness of e· and the assumption that H 0 (E) is quasicoherent, we can compute ^E)) = R0^ H 0 (RΓ(X, Γ(X, E) = Γ(X,^ H 0 (E)) = H 0 (E). Similarly, we get 0 (M )) = Γ(X, H 0 (M f)) = R0 Γ(X, M f) = H 0 (RΓ(X, M f)). H 0 (M ) = Γ(X, H^

One checks that these identifications coincide with the adjunction maps. Recall the coherator Q : (OX -Mod) → QCoh(X), the right adjoint of the inclusion functor QCoh(X) → (OX -Mod). As a right adjoint, Q is left exact, and passing to its derived functor, we obtain a functor RQ : D(X) → DQCoh(X), which is right adjoint to the inclusion DQCoh(X) → D(X) (Proposition F.191), and a fortiori its restriction to Dqcoh (X) is right adjoint to the functor ιX : DQCoh(X) → Dqcoh (X). The previous lemma gives us an explicit description of Q (on Dqcoh (X)) in case X is affine. Proof. [Proof of Theorem 22.35] It remains to extend the result to the general case. We again consider the derived version RQ = RQX : Dqcoh (X) → DQCoh(X) of the coherator, and will show that the natural morphisms obtained from the adjunction, ιX (RQX (E)) → E

and

F → RQX (ιX (F )),

are isomorphisms for all E ∈ Dqcoh (X), F ∈ DQCoh(X). This implies that ιX is an equivalence with inverse RQX . The proofs for the two maps proceed in a similar way: In both cases, we do induction on the minimum number n such that E (or F , resp.) is supported on a closed subscheme of X which can be covered by n affine open subschemes. Here we say that an object of a derived category is supported on a closed subscheme if this is true for all its cohomology objects. Let us first consider the map ιX (RQX (E)) → E. In case n = 0, the object E is supported on the empty subscheme, and is hence = 0, so that the claim becomes trivial. To proceed, assume that we can write X = U ∪ U ′ where U ⊆ X is open and affine, and U ′ ⊆ X is open and can be written as the union of n − 1 open affines, so that the induction hypothesis applies to U ′ . Denote by j the inclusion of U into X. Let i : Y := X \ U ,→ X be the closed embedding of the complement of U , endowed with the reduced scheme structure, into X. Consider the local cohomology exact triangle (Proposition 21.62)

252

22 Cohomology of quasi-coherent modules i∗ Ri! E → E → Rj∗ j ∗ E → i∗ Ri! E[1].

The morphism j is affine, since X is semiseparated, therefore Rj∗ j ∗ E ∈ Dqcoh (X). Thus two of the three terms are in Dqcoh (X), and it follows that the term E ′ := i∗ Ri! E on the left is too. Moreover, E ′ is supported on Y ⊆ U ′ , so the induction hypothesis ensures that the map ιX (RQX (E ′ )) → E ′ is an isomorphism. (In fact, we do not need the exact description of E ′ in terms of Ri! — it is enough to see that the morphism E → Rj∗ j ∗ E becomes an isomorphism when restricted to U , whence E ′ is supported on Y .) Let us show that for E ′′ := Rj∗ j ∗ E, the morphism ιX (RQX (E ′′ )) → E ′′ is an isomorphism, as well. This implies that for E we also get an isomorphism, as desired. In fact, we can write this map as the composition of a chain of isomorphisms, ιX (RQX (E ′′ )) = ιX RQX (Rj∗ j ∗ E) ∼ = ιX j∗ RQU j ∗ E ∼ = Rj∗ ιU RQU j ∗ E ∼ = Rj∗ j ∗ E = E ′′ , where we have used Proposition 22.34 for the first, Proposition 22.33 for the second, and the affine case of the theorem for the third isomorphisms (not counting the equalities in the beginning and the end). Now let us consider the morphisms F → RQX (ιX (F )) for F ∈ DQCoh(X). As before, we define n as the minimum number such that there exists a closed subscheme in X which can be covered by n affine open subschemes and such that all cohomology objects of F are supported on this closed subscheme. If n = 0, then F = 0, and the above morphism is an isomorphism for trivial reasons. For n > 0, writing U , Y , U ′ , etc., as before, we consider a similar exact triangle as in the first case, now in DQCoh(()X): F ′ → F → j∗ j ∗ F → F ′ [1]. Since j is affine, we are in the situation of Proposition 22.33. When we restrict the morphism F → j∗ j ∗ F to U , we obtain an isomorphism. This shows that F ′ is supported on Y ⊆ U ′ , so the induction hypothesis applies to F ′ . It remains to prove that for F ′′ := j∗ j ∗ F we also obtain an isomorphism F ′′ → RQX (ιX (F ′′ )). We have F ′′ = j∗ j ∗ F ∼ = j∗ RQU ιU j ∗ F ∼ = RQX Rj∗ ιU j ∗ F ∼ = RQX ιX j∗ j ∗ F = RQX (ιX (F ′′ )) using the affine case of the theorem for the first, Proposition 22.34 for the second, and Proposition 22.33 for the third isomorphism. Corollary 22.37. Let X, Y be quasi-compact semiseparated schemes, and let f : X → Y be a qcqs morphism. Let RQCoh(X) f∗ : D(QCoh(X)) → D(QCoh(Y )) denote the derived functor of f∗ : QCoh(X) → QCoh(Y ). Then the diagram D(QCoh(X))

ιX

/ D(X)

ιY

 / D(Y )

RQCoh(X) f∗

 D(QCoh(Y ))

Rf∗

is commutative (up to natural isomorphism). Proof. We have a canonical map RQCoh(X) f∗ E → Rf∗ E for E ∈ D(QCoh(X)), and we need to check that this is an isomorphism.

253 We first consider the case that X is affine. Since Y has affine diagonal, it follows that the morphism f is affine, and Proposition 22.33 yields the desired diagram. Now let X be general and suppose j : U → X is the inclusion of an affine open subscheme. We obtain a diagram ιU / D(U ) D(QCoh(U )) j∗

 D(QCoh(X))

Rj∗

ιX

 / D(X)

ιY

 / D(Y )

RQCoh(X) f∗

 D(QCoh(Y ))

Rf∗

where the upper square and the full rectangle are commutative by the affine case. By Proposition 21.115, the composition Rf∗ ◦Rj∗ can be identified with R(f ◦j)∗ . Furthermore, the composition RQCoh(X) f∗ ◦ j∗ can be identified with (f ◦ j)∗ (again, in the sense of Proposition 22.33). In fact, to see this, it is enough to see that j∗ preserves the property of being K-injective (Proposition F.176). This follows from the fact that it admits an exact left adjoint, the inverse image j ∗ , see Corollary F.183. From this diagram we obtain the claim for all objects in DQCoh(X) of the form j∗ F for some j and F ∈ DQCoh(U ). Next, let E ∈ DQCoh(X) ∼ = Dqcoh (X) be supported on a closed subscheme of X which can be covered by n affine open subschemes U1 , . . . Un . We may assume n ≥ 1 and write U := U1 , and j : U → X, i : X \ U → X for the inclusions. By induction, we may assume that the result holds for all E which are supported on a closed subscheme of X which can be covered by n − 1 open subschemes, such as X \ U . Consider the exact triangle (Proposition 21.62) i∗ Ri! E → E → j∗ j ∗ E → By the above, and by induction, the claim holds for the left and right hand terms of the triangle, and it follows that it holds for the middle term E as well. Remark 22.38. (1) If X is a noetherian scheme, then X is qcqs but not necessarily semiseparated (Remark 22.12). Hence Theorem 22.35 does not apply to arbitrary noetherian schemes. Nevertheless, one can show that for X noetherian the natural functor D(QCoh(X)) → Dqcoh (X) is an equivalence of categories whose quasi-inverse is given by the derived coherator functor DQX (Exercise 22.20). (2) On the other hand, an example by Verdier (see [SGA6] O II, App. I) shows that for an arbitrary qcqs scheme X, the functor D(QCoh(X)) → Dqcoh (X) need not be fully faithful. (22.11) Derived tensor product and pullback of quasi-coherent complexes. ˜ from A-modules to OX Let X = Spec A be an affine scheme. The functor M 7→ M modules is exact, commutes with tensor products, and with direct sums (Corollary 7.19). Therefore for all complexes E and F of A-modules for the tensor complexes we find that (22.11.1)

˜ ⊗O F˜ . (E ⊗A F )∼ = E X

254

22 Cohomology of quasi-coherent modules

Lemma 22.39. Let X = Spec A be an affine scheme and let E be a complex of A-modules. ˜ is a K-flat complex of OX -modules. Then E is K-flat if and only if E ˜ is a K-flat complex of OX -modules, then E ˜ ⊗O F is exact for every exact Proof. If E X complex F of OX -modules, in particular for those that are of the form F˜ for an exact complex F of A-modules. Therefore E is K-flat by (22.11.1). ˜x = Ep is a K-flat complex of OX,x -modules for all Conversely, if E is K-flat, then E x ˜ is K-flat by Remark 21.92 (4). x ∈ X (Remark 21.109 (1)). Hence E Proposition 22.40. Let X be a scheme. (1) Let E and F be complexes in Dqcoh (X). Then E ⊗L OX F is in Dqcoh (X). (2) Let f : X ′ → X be a morphism of schemes and let E be a complex in Dqcoh (X). Then Lf ∗ E is in Dqcoh (X ′ ). Proof. The assertions are local on X and the formation of ⊗L and of Lf ∗ is compatible with restriction to open subsets. Hence we may assume that X = Spec A and X ′ are ˜ and F = F˜ for complexes E and F of A-modules (Lemma 22.36). affine. Then E = E Hence it suffices to show the following lemma. Lemma 22.41. Let A be a ring, X = Spec A. (1) Let E und F be complexes of A-modules. Then ∼ ˜ L ˜ (E ⊗L A F ) = E ⊗ OX F .

(2) Let E be a complex of A-modules and let A → B be a ring homomorphism corresponding to a morphism of schemes f : Spec B → Spec A. Then ∼ ∗ ˜ (E ⊗L A B) = Lf E, ∼ where (E ⊗L denotes the complex of quasi-coherent modules on Spec B correA B) sponding to the complex E ⊗L A B in D(B).

˜ is a quasi-isomorphism with Proof. Let P → E be a K-flat resolution of E. Then P˜ → E ˜ P K-flat by Lemma 22.39. Therefore ∼ ∼ ˜ ˜ L ˜ ˜ (E ⊗L A F ) = (P ⊗A F ) = P ⊗OX F = E ⊗OX F .

The second assertion is proved in the same way. (22.12) Derived categories of coherent modules on noetherian schemes. Let X be a locally noetherian scheme and let (Coh(X)) be the category of coherent OX modules, which is a plumb subcategory (Definition F.43) of the category of OX -modules (Proposition 7.47), in particular, it is an abelian category. We denote by Dcoh (X) the full subcategory of complexes F in D(X) such that H p (F ) is coherent for all p ∈ Z. It is a ? triangulated subcategory (Section (F.41)). As usual, we set Dcoh (X) := D? (X) ∩ Dcoh (X) for ? ∈ {+, −, b, I}, where I ⊆ Z is some interval. We introduce similar notation in the affine case. If A is a noetherian ring, then we denote by (Coh(A)) the abelian category of finitely generated A-modules, which is also a plump subcategory of the category of A-modules, and Dcoh (A) denotes the triangulated subcategory of D(A) of complexes M such that H p (M ) is a finitely generated A-module ? (A) for ? ∈ {+, −, b, I} as above. for all p ∈ Z. We define Dcoh

255 Theorem 22.42. Let X be a noetherian scheme. Then the natural functors − D− (Coh(X)) −→ Dcoh (X)

and

b Db (Coh(X)) −→ Dcoh (X)

are equivalences of categories. Proof. All of the above categories are by definition full subcategories of D(X). Hence the two functors are fully faithful. Moreover, if the first functor is essentially surjective, − so is the second. Hence it suffices to show that D− (Coh(X)) −→ Dcoh (X) is essentially surjective. − Let F be in Dcoh (X). By Theorem 22.35 (if X is semiseparated) or Remark 22.38 (in general) we may assume that F has quasi-coherent components. We use Proposition F.160, applied to A the abelian category of quasi-coherent OX -modules, C the category of complexes of OX -modules with coherent cohomology, and to D the category of coherent OX -modules. Hence it suffices to show that if F is a coherent quotient of a quasicoherent OX -module G , then there exists a coherent OX -submodule H of G such that the composition H ,→ G → F is surjective. By Corollary 10.50, G is the filtered colimit of its coherent submodules. We can choose for H one of these by Lemma 10.47. Corollary 22.43. Let A be a noetherian ring and write X = Spec A. Then one has for ∗ ∈ {b, −} a commutative diagram of triangulated equivalences D∗ (Coh(A))

∼ =

∼ =

 D∗ (Coh(X))

∼ =

/ D∗ (A) coh 

∼ =

/ D∗ (X), coh

where the horizontal maps are given by the inclusion functors and the vertical maps by ˜. M 7→ M

Finiteness properties of complexes on schemes In the next sections, we study the finiteness properties for complexes on a ringed space which we have introduced in Chapter 21, specifically pseudo-coherence and perfectness, in the case of schemes. One of the important results is Corollary 22.47 which says that a complex of OX -modules on a scheme X is pseudo-coherent if and only if locally on X it is isomorphic in D(X) to a bounded above complex of finite free modules. For noetherian schemes, the situation is particularly simple, see Sections (22.15) and (22.18). (22.13) Perfect and pseudo-coherent complexes on schemes. We revisit different finiteness conditions for complexes on ringed spaces given in Chapter 21 and study them for complexes on schemes. Recall, for a complex of modules on a ringed space or over a ring, the notion of being perfect (Definition 21.136), to be pseudo-coherent (Definition 21.150) or of having tor-amplitude in an interval [a, b] (Definition 21.164).

256

22 Cohomology of quasi-coherent modules

By definition, given a ringed space X and an open covering (Ui )i of X, a complex F of modules on X is perfect (resp. pseudo-coherent, resp. of tor-amplitude in [a, b]) if and only if F |Ui has this property for all i. Moreover, recall (Theorem 21.174) that a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-dimension (i.e., locally of tor-amplitude in some bounded interval [a, b]). Using the stability of these notions under faithfully flat descent (Lemma 21.163 and Lemma 21.172) one shows that for affine schemes the notions defined for a ring A and the ringed space Spec A are equivalent: Lemma 22.44. Let A be a ring, X = Spec A, and let E be a complex of A-modules. (1) Let m ∈ Z. The complex E is m-pseudo-coherent if and only if the associated complex ˜ of quasi-coherent OX -modules is m-pseudo-coherent. In particular, E is pseudoE ˜ is pseudo-coherent. coherent if and only if E (2) Let a, b ∈ Z with a ≤ b. The complex E has tor-amplitude in [a, b] if and only if the ˜ of quasi-coherent OX -modules has tor-amplitude in [a, b]. associated complex E ˜ of quasi-coherent (3) The complex E is perfect if and only if the associated complex E OX -modules is perfect. Proof. If E is m-pseudo-coherent, there exists an m-isomorphism of a finite complex of finite projective modules to E. Hence there exists an m-isomorphism of a finite complex ˜ Then Em is perfect which shows that E ˜ is of finite locally free OX -modules Em to E. m-pseudo-coherent. ˜ is m-pseudo-coherent. Then there exists a finite principal Conversely, assume that E open covering (D(fi ))i of X and an m-isomorphism of a finite complex of finite free ˜ |D(f ) = (E ⊗A Af )∼ . Hence E ⊗A Af is m-pseudo-coherent for OD(fi ) -modules Q to E i i i all i. As A → i Afi is faithfully flat, this implies that E is m-pseudo-coherent by Lemma 21.163. The proof for the other properties is similar using the stability of both notions under faithfully flat descent (Lemma 21.172 and Lemma 21.178). Remark 22.45. Every strictly perfect complex on a scheme X has quasi-coherent cohomology sheaves. As quasi-coherence is a local property, it follows that every perfect complex is in Dqcoh (X). Hence also every pseudo-coherent complex on X lies in Dqcoh (X). Therefore Lemma 22.44 shows that if X = Spec A is an affine scheme, then the equivalence of D(A) and Dqcoh (X) (Lemma 22.36) induces an equivalence between the full subcategory of D(A) of perfect (resp. pseudo-coherent) complexes of A-modules and the full subcategory of D(X) of perfect (resp. pseudo-coherent) complexes of OX -modules. This implies that we have the following description of perfect and pseudo-coherent complexes on affine schemes. Remark 22.46. Let X = Spec A be an affine scheme. (1) Every perfect complex is isomorphic in D(X) to a strictly perfect complex. More precisely, for every perfect complex F on X there exists a finite complex E of finite locally free OX -modules and a quasi-isomorphism of complexes E → F (Example 21.138 (1)). See also Proposition 22.53 below how to bound E in terms of the tor-amplitude of F . (2) Every pseudo-coherent complex F is isomorphic in D(X) to a bounded above complex E of finite free modules. If m is an integer such that H i (F ) = 0 for all i > m, then we may choose E such that E i = 0 for all i > m (Proposition 21.162).

257 As perfectness and pseudo-coherence can be checked locally, we obtain the following corollary. Corollary 22.47. Let X be a scheme. (1) A complex of OX -modules is perfect if and only if it is locally on X isomorphic in D(X) to a bounded complex of finite locally free OX -modules. (2) A complex of OX -modules is pseudo-coherent if and only if it is locally on X isomorphic in D(X) to a bounded above complex of finite free OX -modules. If X is quasi-compact and E is a pseudo-coherent complex of OX -modules, then there exists b ∈ Z such that H i (E) = 0 for all i > b. Furthermore, E is then of tor-amplitude in (−∞, b]. Example 22.48. Let k be a field. A perfect complex E over k is isomorphic in D(k) to a finite complex of finite-dimensional vector spaces whose differentials are zero. Indeed, we may assume that E is a finite complex of finite-dimensional vector spaces, say concentrated in degrees [a, b]. As all exact sequences of k-vector spaces split, we can find an isomorphism of complexes E ∼ = τ ≥b E ⊕ F , where F is the cokernel of the inclusion ≥b b E→τ E=HL (E)[−b]. We get E ∼ = F ⊕ H b (E)[−b] in D(k). Proceeding inductively, p we see that E ∼ H (E)[−p]. = p This argument can be vastly generalized, see Exercise F.31. Example 22.49. Let X be a scheme and let E be a perfect complex. Then E is isomorphic to a finite locally free OX -module (considered as a complex concentrated in degree 0) if and only if E has tor-amplitude in [0, 0]. Indeed, the condition is clearly necessary. Conversely, if E has tor-amplitude in [0, 0], then H p (E ) = 0 for p = ̸ 0 and hence E ∼ = H 0 (E )[0]. As E is pseudo-coherent, H 0 (E ) is 0 of finite presentation. As H (E ) has tor-dimension ≤ 0, it is also flat. Hence H 0 (E ) is finite locally free. Example 22.50. Let X be a scheme, and let i : Z → X be a completely intersecting closed immersion (e.g., a regular closed immersion). Then i∗ OZ is a perfect object in D(X). Indeed, by definition, locally on X there exists a completely intersecting family of sections f1 , . . . , fr of OX such that Z = V (f1 , . . . , fr ). Then there exists a quasi-isomorphism from the Koszul complex attached to (f1 , . . . , fr ), which is a finite complex of finite free OX -modules, to i∗ OZ . By definition a complex in D(X) is of tor-amplitude in [a, b] relative to some morphism ∗ f : X → Y of schemes if H p (E ⊗L / [a, b] and for every OY -module OX Lf G ) = 0 for all p ∈ G . The next lemma shows that if E ∈ Dqcoh (X) and if Y is quasi-separated, then it suffices to check the condition only for quasi-coherent OY -modules G . Lemma 22.51. Let f : X → Y be a morphism of schemes and let a ≤ b be integers. Suppose that Y is quasi-separated (e.g., if Y is locally noetherian). Then a complex E in Dqcoh (X) is of tor-amplitude in [a, b] relative to f if and only if for all quasi-coherent ∗ OY -modules G and for all p ∈ / [a, b] one has H p (E ⊗L OX Lf G ) = 0. Proof. The condition is clearly necessary. For the converse implication, it suffices to show that for all open affines U ⊆ X and V ⊆ Y with f (U ) ⊆ V , E |U is of tor-amplitude in [a, b] relative to the restriction g : U → V of f (Lemma 21.171). The functor (−)∼ commutes with derived tensor products (Lemma 22.41). By Lemma 22.44 it suffices to ∗ show that H p (E |U ⊗L / [a, b] and all quasi-coherent OV -modules G . OU Lg G ) = 0 for all p ∈

258

22 Cohomology of quasi-coherent modules

∗ ′ p L For every quasi-coherent OY -modules G ′ one has H p (E ⊗L OX Lf G )|U = H (E |U ⊗OU Lg ∗ G ′ |V ). Therefore it suffices to show that every quasi-coherent OV -module G is isomorphic to the restriction of a quasi-coherent OY -module G ′ . As Y is quasi-separated, the inclusion j : V → Y is quasi-compact (and quasi-separated). Hence one can take G ′ := j∗ G which is quasi-coherent by Corollary 10.27.

Proposition 22.52. Let f : X → Y be a morphism of schemes. Let E be a complex of OY -modules. If E is perfect (resp. pseudo-coherent), then Lf ∗ E is perfect (resp. pseudocoherent). The converse holds if f is faithfully flat and quasi-compact. Proof. The first assertion is a special case of Remark 21.142 and Proposition 21.157. Now suppose that f is faithfully flat and quasi-compact and that Lf ∗ E is perfect (resp. pseudo-coherent). To see whether E has the same property, we may assume that Y = Spec A is affine. `As f is quasi-compact, we find a finite open affine covering (Ui )i of X. Then X ′ := i Ui is affine, f ′ : X ′ → Y is faithfully flat, and Lf ′∗ E is perfect (resp. pseudo-coherent). Hence we can assume that X = Spec B is also affine. Then we can apply Lemma 21.178 (resp. Lemma 21.163) by Remark 22.45. Perfect complexes (resp. pseudo-coherent complexes) on affine schemes can always be represented by finite (resp. bounded above) complexes of finite locally free modules. This is a key point for the theory of cohomology and base change. See Section (23.28) below. For perfect complexes the degrees in which these complexes of finite locally free modules are non-zero is given by the tor-amplitude as follows. Proposition 22.53. Let X = Spec A be an affine scheme, let F be a complex of OX modules, and let a ≤ b be integers. Then F is pseudo-coherent and of tor-amplitude in [a, b] if and only if F is isomorphic in D(X) to a complex E of finite locally free OX -modules concentrated in degrees [a, b]. A complex F satisfying these equivalent conditions is perfect. It follows from Exam∼ ple 21.138 (1) that the isomorphism E → F in D(X) is induced by a quasi-isomorphism E → F of complexes. Proof. The condition is clearly sufficient. Suppose that F is pseudo-coherent and of tor-amplitude in [a, b]. We may pass to the corresponding pseudo-coherent complex E of A-modules which is of tor-amplitude in [a, b]. In particular, H p (E) = 0 for all p ∈ / [a, b]. By Proposition 21.162 we may assume that E is a complex of finite free A-modules with E i = 0 for i > b. As E → τ ≥a E is an isomorphism in D(X), it suffices to show that C := Coker(da−1 : E a−1 → E a ) is a finite projective A-module. Since E a−1 and E a are of finite type, it is of finite presentation. Therefore it suffices to show that C is flat. As H p (E) = 0 for p < a, the map σ ≤a E → C[−a] is a quasi-isomorphism. As σ ≤a E and E are K-flat, we find for all A-modules M that a−1 a−1 ≤a TorA (C[a] ⊗L (σ E ⊗A M ) = H a−1 (E ⊗L 1 (C, M ) = H A M) = H A M) = 0

because E is of tor-amplitude ≥ a. Therefore C is a flat A-module. Locally, one can say even more.

259 Proposition 22.54. Let X be a scheme and let E be a perfect complex of OX -modules. Let x ∈ X be a point, let ix : Spec κ(x) → X be the canonical map, and set dp := dimκ(x) H p (E ⊗L κ(x)), where E ⊗L κ(x) := Li∗x E is the fiber of E in x. Then there exists an open neighborhood U of x and a complex F of finite free OU modules of the form d

d

d

. . . −→ OUp−1 −→ OUp −→ OUp+1 −→ . . . such that E |U is isomorphic to F in D(U ). Moreover, if (U1 , F1 ) and (U2 , F2 ) are two such pairs, then there exist an open neighborhood V ⊆ U1 ∩ U2 of x and an isomorphism of complexes of OV -modules F1|V ∼ = F2|V . Only finitely many of dp are non-zero, thus F is a finite complex. We have an isomorphism E ⊗L κ(x) ∼ = F ⊗ κ(x) in D(κ(x)) and F ⊗ κ(x) is a finite acyclic complex since the dimension of its components equals the dimension of its cohomology vector spaces. Proof. We can pass to stalks by Remark 22.55 below. Then we are in the situation that we are given a perfect complex E over a local ring A. Let k be its residue field. Then dp = dimk H p (E ⊗L A k). We may assume that E is a finite complex of finite free A-modules. k = E ⊗A k is a perfect complex over a field and therefore it is isomorphic to a Now E ⊗L A ¯ ′ with zero differentials of the form · · · → k dp → k dp+1 → . . . (Example 22.48). complex E ¯ ′ → E ⊗A k of complexes This isomorphism in D(k) is given by a quasi-isomorphism u ¯: E by Lemma 21.134. Let C¯ be the cone of u ¯ which is an acyclic complex because u ¯ is a quasi-isomorphism. Let C be an acyclic complex of finite free A-modules lifting C¯ with ¯ Then the natural map C → C¯ is a surjective quasi-isomorphism of complexes C ⊗A k ∼ = C. of A-modules and using Remark 21.135 we obtain a map of complexes v : E → C making E  E ⊗A k

v

/C  / C¯

commutative. Then v i : E i → C i is injective with free cokernel as this holds after base change to k (Proposition 8.10). Now we set F := Coker(v)[−1]. We showed that F is a complex of finite free A-modules. v As v is injective, we have a distinguished triangle E −→ C −→ F [1] −→. Rotating the triangle, we obtain a morphism F → E in D(A) which is an isomorphism because C is ¯ ¯ ′ by definition of acyclic. Finally, F ⊗A k is Coker(E ⊗A k → C)[−1] which is equal to E the cone. In particular F p has rank dp for all p. It remains to show that F is unique up to isomorphism. If there is another complex G with Gp free of rank dp that is isomorphic to E in D(A), we find again by Lemma 21.134 a quasi-isomorphism of complexes w : F → G representing the isomorphism F ∼ =E∼ = G. The induced map w ⊗ 1 : F ⊗A k → G ⊗A k is a quasi-isomorphism of acyclic complexes and hence an isomorphism. Therefore w is an isomorphism by Corollary 8.12. Remark 22.55. Let A = colimλ Aλ be a filtered colimit of rings. Then the triangulated category of perfect complexes over A is the “colimit of the triangulated categories of perfect complexes over Aλ ”. If E is a perfect complex in D(A) of tor-amplitude in some finite interval [a, b], then there exists a λ and a perfect complex Eλ in D(Aλ ) of tor-amplitude ∼ in [a, b] such that Eλ ⊗L Aλ A = E.

260

22 Cohomology of quasi-coherent modules

Indeed, we have a similar assertion for the category of finitely generated projective modules by Section (10.16). As perfect complexes can be represented by finite complexes of finitely generated projective modules, which are concentrated in degrees [a, b] if E is of tor-amplitude in [a, b], and morphisms in the derived category between such complexes can be represented by morphisms of complexes (Lemma 21.134), this implies the claim. (22.14) The resolution property and representing perfect complexes. Definition 22.56. We say that a scheme X satisfies the resolution property if every quasi-coherent OX -module of finite type is the quotient of a finite locally free OX -module. Any affine scheme has the resolution property since every finitely generated module over a ring is the quotient of a finitely generated free module. More generally, one has: Proposition 22.57. Let X be a qcqs scheme such that there exists an ample line bundle L on X. Then X has the resolution property. Proof. If F is a quasi-coherent OX -module of finite type, then there exists an n ≥ 0 such that F ⊗ L ⊗n is generated by its global sections. Hence we find a surjection N OX → F ⊗ L ⊗n for some N (Lemma 10.47). In other words, F is a quotient of the finite locally free OX -module (L ⊗−n )N . Every perfect complex on an affine scheme X is (globally) isomorphic in D(X) to a finite complex of finite locally free OX -modules, i.e., to a strictly perfect complex. This can be generalized as follows. Proposition 22.58. Let X be a quasi-compact and semiseparated2 scheme that has the resolution property. Then every perfect complex on X is isomorphic in D(X) to a strictly perfect complex. The proof will show that if E is a perfect complex of tor-amplitude in [a, b], then there exists a strictly perfect complex F concentrated in degrees [a, b] and an isomorphism ∼ F → E in D(X). Proof. Let E be a perfect complex in D(X). Then E lies in Dqcoh (X), so we can represent E by a complex E of quasi-coherent OX -modules (Theorem 22.35). As X is quasi-compact, it has bounded tor-amplitude, say in [a, b] for integers a ≤ b. In particular H i (E) = 0 for i ∈ / [a, b]. Replacing E by τ ≤b τ ≥a E we may assume that E is concentrated in degrees [a, b]. We claim that there exists a quasi-isomorphism F → E with F a complex of finite locally free modules concentrated in degrees [a, b]. We now prove the claim by induction on b − a. If a = b, then E ∼ = H a (E )[−a] and a H (E ) is finite locally free (Example 22.49). Now let b > a. Next we show that H b (E ) is a quasi-coherent OX -module of finite type. Indeed, this can be checked locally on X, so that we may assume that X is affine and that E is a complex of finite locally free OX -modules concentrated in degrees [a, b] (Proposition 22.53). In this case H b (E ) is a quotient of E b and hence of finite type. By the resolution property, we may choose a finite locally free OX -module F b and a surjection F b → E b . This surjection defines a map of complexes u : F b [−b] → E . Let Cu be its cone, so that we have a distinguished triangle 2

A result by Totaro ([To] O X 1.3) shows that the condition to be semiseparated automatically follows from the resolution property.

261 +1

F b [−b] −→ E −→ Cu −→ . Then Cu is concentrated in degrees [a, b] by definition of the cone and since b > a, it has tor-amplitude in [a, b] by Proposition 21.166 and has H b (Cu ) = 0 as u induces a surjection H b (F b [−b]) = F b → H b (E ) by construction. Hence Cu has tor-amplitude in [a, b − 1] (Remark 21.165 (2)). Therefore, we can apply the induction hypothesis to Cu and find a complex H of finite locally free OX -modules concentrated in degrees [a, b − 1] and, by the definition of Cu , a quasi-isomorphism ···

···

/ H b−2

/ H b−1



 L b−1

/ E b−2

(v,w)

/E

Fb

/0

/ ···

 / Eb

/ ··· . w

It induces a homomorphism of complexes (H a −→ · · · −→ H b−1 −→ F b ) −→ E and an easy diagram chase shows that this is a quasi-isomorphism. Corollary 22.59. Let X be a qcqs scheme such that there exists an ample line bundle L on X. Then every perfect complex on X is isomorphic in D(X) to a strictly perfect complex. Proof. The existence of an ample line bundle implies that X is separated (Proposition 13.48). Hence we can apply Proposition 22.58 by Proposition 22.57. Remark 22.60. Proposition 22.57 and Corollary 22.59 hold more generally if X has an ample family of line bundles (Exercise 23.19 and Exercise 23.20), e.g., if X is noetherian, semiseparated and locally factorial (Exercise 23.17). (22.15) Pseudo-coherent complexes on noetherian schemes. Let X be a scheme. Recall (Proposition 21.153) that an OX -module F , considered as a complex concentrated in degree 0, is pseudo-coherent if and only if locally on X there exists an exact sequence of OX -modules . . . −→ E −2 −→ E −1 −→ E 0 −→ F −→ 0, where E i is a finite free OX -module for all i ≤ 0. From this one easily deduces that on a locally noetherian scheme X an OX -module is pseudo-coherent if and only if it is coherent. More generally, for noetherian schemes we have the following criterion when a complex is pseudo-coherent. Proposition 22.61. Let X be a noetherian scheme. Then a complex E in D(X) is − pseudo-coherent if and only if E is in Dcoh (X). Proof. Let E be pseudo-coherent and m ∈ Z. As X is quasi-compact, we find a finite open covering (Ui )i and for all i a bounded complex Ki of finite locally free OUi -modules such that there exists an (m − 1)-isomorphism Ki → E |Ui . In particular H p (E )|Ui ∼ = H p (Ki ) m for all p ≥ m and all i. This shows that H (E ) is coherent and that E is in D− (X).

262

22 Cohomology of quasi-coherent modules

Conversely, suppose that E is in D− (X) with coherent cohomology. Let m ∈ Z and x ∈ X. We now apply Proposition F.160 to A the category of OX,x -modules, D the full subcategory of finite free OX,x -modules, and C the category of complexes of OX,x -modules with bounded above cohomology and finitely generated cohomology modules. Hence we obtain a bounded above complex Fx of finite free OX,x -modules and a quasi-isomorphism Fx → Ex . Then σ ≥m Fx is strictly perfect and σ ≥m Fx → Ex is an m-isomorphism. To extend this map to an open neighborhood of x we now repeatedly use Proposition 7.27 and that coherent modules are the same as modules of finite presentation over noetherian schemes (Proposition 7.46) as follows. Replacing X by an open neighborhood of x we may assume that there exists a strictly perfect complex F whose stalk in x is σ ≥m Fx . After shrinking X further we may assume that σ ≥m Fx → Ex is induced by a map of complexes F → E which we may assume after shrinking X once more to be an m-isomorphism because H p (F ) and H p (E ) are coherent. Corollary 22.62. Let A be a noetherian ring. Then a complex E in D(A) is pseudo− coherent if and only if E ∈ Dcoh (A). (22.16) Derived tensor products of pseudo-coherent and perfect complexes. Let X be a scheme. Proposition 22.63. Let E and F be complexes in D(X). If E and F are pseudo-coherent (resp. perfect), then E ⊗L OX F is pseudo-coherent (resp. perfect). Proof. For perfect complexes, we have already seen the result for general ringed spaces in Remark 21.143. Let E and F be pseudo-coherent. To show that E ⊗L F is pseudo-coherent, we may assume that X is affine. Then we may assume that E and F are bounded above complexes of finite free OX -modules (Remark 22.46). Then E ⊗L F = E ⊗ F is represented by a bounded above complex of finite free modules and hence it is pseudo-coherent. Proposition 22.63 also holds if X is a ringed space but the proof for pseudo-coherent complexes gets more complicated since we cannot use Remark 22.46. Instead one can argue as sketched in Exercise 21.30. Lemma 22.64. Let X be quasi-compact and let E be a perfect complex in D(X). b (1) Let F ∈ Db (X). Then E ⊗L OX F ∈ D (X). b b (X). Then E ⊗L (2) Let X be noetherian and let F be in Dcoh OX F ∈ Dcoh (X). Proof. As X is quasi-compact, both assertions are local on X and hence we may assume that E is finite complex of free OX -modules. Then E is K-flat and hence E ⊗L OX F = E ⊗OX F is bounded. This implies (1). To see (2) we use that F can be represented by a bounded complex of coherent OX -modules (Theorem 22.42). Then E ⊗OX F is also a bounded complex of coherent OX -modules. (22.17) Derived Hom and Ext on schemes. Recall the notion of Ext groups and of Ext sheaves on ringed spaces (Sections (F.52), (21.21)).

263 Proposition 22.65. Let X be a scheme, and let F , G be complexes of OX -modules. Assume that F is pseudo-coherent and that G is bounded below. Then we have natural isomorphisms Ext i (F , G )x ∼ = ExtiOX,x (Fx , Gx ) for all i ∈ Z. The same assertion also holds if F is perfect and G is arbitrary (Exercise 22.38). Proof. Since the Ext functor is compatible with restriction to open subsets of X (21.21.3), we may assume that X = Spec A is affine. By Remark 22.46 we may assume that F is a bounded above complex with F i free of finite rank for all i. As G is bounded below we have R Hom(F , G ) = Hom(F , G ) by Lemma 21.89. We also have R HomOX,x (Fx , Gx ) = HomOX,x (Fx , Gx ) since Fx , being a bounded above complex of free OX,x -modules, is K-projective. As the exact localizing functor (−)x commutes with taking cohomology, it suffices to show that the canonical map Hom(F , G )x ∼ = HomOX,x (Fx , Gx ) of complexes of OX,x -modules L is an isomorphism. As F is bounded above and G is bounded below, Hom(F , G )n ∼ = p Hom(F p , G p+n ). As localization commutes with direct sums, it suffices to show that Hom(F p , G q )x → HomOX,x (Fxp , Gxq ) is an isomorphism for all p and q. But this holds because F p is finite locally free and thus of finite presentation (Proposition 7.27). Proposition 22.66. Let P be a property of OX -modules for all schemes X satisfying the following conditions. (a) For every scheme X the subcategory CX of all OX -modules that have P is plump (Definition F.43). (b) For every scheme X and every open covering (Ui )i of X an OX -module F has P if and only if F |Ui has P for all i. (c) For every scheme X all OX -modules that have P are quasi-coherent. Let X be a scheme and set C := CX . Let DC (X) ⊆ Dqcoh (X) be the triangulated subcategory of all complexes E in D(X) such that one has H p (E) ∈ C for all p ∈ Z. Let F ∈ D(X) and G ∈ DC (X). Suppose that one of the following hypotheses is satisfied. (1) F is perfect. (2) F is pseudo-coherent and G is locally bounded below. Then R Hom OX (F , G ) ∈ DC (X), i.e. Ext nOX (F , G ) ∈ C for all n ∈ Z. Proof. As the formation of Ext commutes with restriction to open subsets (21.21.3), by (b) we may work locally on X. In particular, we may assume that X = Spec A for a ring A. By (c) we have DC (X) ⊆ Dqcoh (X) ∼ = D(A). Let D = D(X, G ) be the full subcategory of D(X) of complexes F such that Ext nOX (F , G ) is in C for all n. As C is a plump subcategory of the category of OX -modules, D is a triangulated subcategory of D(X). Clearly, OX is contained in D since R Hom OX (OX , G ) = G . Let us first show that every perfect complex F is in D. As X is affine, F is represented by a strictly perfect complex. Hence we conclude by Proposition 21.139.

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22 Cohomology of quasi-coherent modules

Now suppose that G is locally bounded below. As X now is quasi-compact, one has G ∈ D+ (X). We have to show that every pseudo-coherent complex F is in D. As X is affine, there exists for all m ∈ Z an m-isomorphism E → F with E strictly perfect (Remark 22.45). Let a ∈ Z be such that G ∈ D≥a (X) (i.e. H p (G ) = 0 for all p < a). If F is a complex with H p (F ) = 0 for all p ≥ n, then R Hom(F , G ) is acyclic in degrees ≤ a − n (use the final statement of Theorem F.185). Hence we can apply Remark 21.160 to conclude. The properties P of being quasi-coherent on a scheme or of being coherent on a locally noetherian scheme both satisfy the hypotheses in Proposition 22.66. Hence we obtain the following corollary. Corollary 22.67. Let X be a scheme and let F and G be in D(X). Suppose that one of the following hypotheses is satisfied. (a) F is perfect. (b) F is a pseudo-coherent complex and G is locally bounded below. Then one has the following assertions. (1) Let G be in Dqcoh (X). Then R Hom OX (F , G ) is in Dqcoh (X). (2) Let X be locally noetherian and let G be in Dcoh (X). Then R Hom OX (F , G ) is in Dcoh (X), i.e., Ext pOX (F , G ) is coherent for all p ∈ Z. Assertion (2) can be generalized to certain non-noetherian schemes, see Exercise 22.33 for details. Using the local-to-global spectral sequence for Ext (Corollary 21.108) and the vanishing of cohomology of quasi-coherent sheaves on affine schemes, as a further corollary to Corollary 22.67 (1) we obtain that if hypothesis (a) or (b) is satisfied and G is in Dqcoh (X), then for every affine open subscheme U ⊆ X and every i ∈ Z one has (22.17.1)

Γ(U, Ext i (F , G )) = ExtiA (Γ(U, F ), Γ(U, G )).

Proposition 22.68. Let A be a ring, X = Spec A, and let M, N ∈ D(A). Suppose that M is pseudo-coherent and that N is bounded below. Then there is a functorial isomorphism ∼

˜,N ˜ ). R HomA (M, N )∼ −→ R Hom OX (M ∼ Ext i (M ˜,N ˜ ) for all i ∈ Z. In particular one sees that ExtiA (M, N )∼ = OX One has a similar result if M is perfect and N is an arbitrary complex (Exercise 22.37). Proof. We may replace M and N , respectively, by complexes isomorphic to them in D(A), and hence may assume that they are bounded above, and bounded below, respectively, and that every term of M is a finite free A-module (Proposition 21.162). It follows from Lemma 21.89, applied to the ringed spaces X and the one-point space with structure ˜ −q , N ˜ p) ∼ sheaf A, that it is enough to show that Hom OX (M = HomA (M −q , N p )∼ for all p, q. But this is clear since M −q is a finitely generated free A-module. − + Corollary 22.69. Let A be a noetherian ring, let M ∈ Dcoh (A) and N ∈ Dcoh (A). Then p ExtA (M, N ) is a finitely generated A-module for all p ∈ Z.

Proof. The complex M is pseudo-coherent by Corollary 22.62 and we conclude by Corollary 22.67 (2) and Proposition 22.68.

265 Proposition 22.70. Let f : X ′ → X be a morphism of schemes. Let F , G ∈ D(X). Assume that we are in one of the following situations: (1) The complex F is perfect. (2) The morphism f : X ′ → X has locally finite tor-dimension (Definition 21.164), F is pseudo-coherent, and G is locally bounded below. Then the natural morphism (21.26.10) Lf ∗ R Hom X (F , G ) → R Hom X ′ (Lf ∗ F , Lf ∗ G ) is an isomorphism. Proof. As the formation of R Hom commutes with restriction to open subsets (21.18.9) we can work locally on X. In particular, we can assume that X = Spec A is affine. Fix G ∈ D(X). We consider the contravariant triangulated functors Φ, Ψ : D(X) → D(X ′ ) given by Φ(F ) = Lf ∗ R Hom X (F , G ) and Ψ(F ) = R Hom X ′ (Lf ∗ F , Lf ∗ G ). Then (21.26.10) yields a morphism η : Φ → Ψ of triangulated functors. Clearly, η(OX ) is an isomorphism. To show (1) it suffices to show that η(F ) is an isomorphism for every strictly perfect complex. This follows from Corollary 21.141. Let us show (2). By working locally on X it suffices to show that η(F ) is an isomorphism for all F ∈ D(X) such that for all m ∈ Z there exists an m-isomorphism E → F for some strictly perfect complex E . As X is now quasi-compact, f is of finite tor-dimension, say ≤ t, and G is bounded below, say G ∈ D≥a (X). Then Lf ∗ G ∈ D≥a−t (X ′ ). The functors Φ and Ψ are contravariant and both map D≤m (X) to D≥a−m−t (X ′ ). Hence we can conclude by Proposition 21.159 and the remark following it. We can apply Proposition 22.70 in particular to morphisms f that are flat, i.e., of tor-dimension ≤ 0. For instance we obtain the following affine variant of Proposition 22.70 using that under the given hypothesis the equivalence D(A) ∼ = Dqcoh (Spec A) is compatible with derived inverse images by Lemma 22.41 and derived Hom by Proposition 22.68. Corollary 22.71. Let A be a ring, let A′ be a flat A-algebra, let F ∈ D(A) be pseudocoherent, and let G ∈ D+ (A). Then the natural morphism ′ L ′ L ′ R HomA (F, G) ⊗L A A → R HomA′ (F ⊗A A , G ⊗A A )

is an isomorphism. Lemma 22.72. Let X be a scheme and let E , F and G be complexes in D(X). Then the functorial map (21.26.9) is an isomorphism (22.17.2)

∼

R Hom OX (E , F ) ⊗L OX G −→ R Hom OX (R Hom OX (G , E ), F )

in the following two cases. (1) G is perfect. (2) G is pseudo-coherent, E is locally bounded below and F is locally of finite injective dimension, i.e., there exists an open covering (Ui )i of X such that F |Ui is of finite injective dimension (Definition F.221). Proof. The map (22.17.2) is an isomorphism if G = OX . The formation of R Hom and of the derived tensor product are compatible with restriction to open subsets, thus this is a local question. Hence we can assume for (1) that G is strictly perfect. Then (1) follows from Corollary 21.141.

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22 Cohomology of quasi-coherent modules

To show (2) it suffices by Proposition 21.159 that both sides of (22.17.2) as functors in G are cohomologically bounded above if E is bounded below and F is of finite injective dimension. But then F can be represented by a finite complex I of injective OX -modules and H := R Hom OX (E , F ) is represented by Hom OX (E , I ) which is zero in degrees ≫ 0. Hence H i (H ) = 0 for i ≫ 0 and hence the left hand side H ⊗L (−) of (22.17.2) is cohomologically bounded above. A similar argument shows that the right hand side H ⊗L (−) of (22.17.2) is cohomologically bounded above. (22.18) Injective modules on locally noetherian schemes. We study injective modules on schemes. Recall that we can show injectivity locally (Lemma 21.12). Hence if X is a scheme, then an OX -module I is injective if and only if I |Ui is injective for all i for some affine open covering (Ui )i of X. Even if X = Spec A is affine, then an injective OX -module is in general not quasicoherent, so that it does not correspond to an A-module. Even worse, if M is an injective ˜ will be an OX -module which is injective in QCoh(X) but it will in A-module, then M general not be injective in the category of all OX -modules (see Caveat 22.77 (3) below). The situation becomes better in the noetherian case as we explain in the following. Some proofs will be only sketched, for details we refer to [Har1] O II §7. Proposition 22.73. Let X be a locally noetherian scheme. Then (OX -Mod) is a locally noetherian Grothendieck category (see Section (F.13)). By Remark 21.8 it suffices to show that the extension by zero OU ⊆X (21.1.6) is noetherian for every affine open U ⊆ X. Replacing X by U we may assume that X = Spec A is affine and it suffices to show that OX is a noetherian object in the category of all OX -modules, i.e., that every ascending sequence J1 ⊆ J2 ⊆ . . . of ideals of OX becomes stationary. If all these ideal were quasi-coherent, they would correspond to ideals of A and the claim would be clear since A is noetherian. In general one has to argue differently, see [Har1] O II 7.8. Remark 22.74. Proposition 22.73 implies by Proposition F.67 and Corollary 21.53 that on a locally noetherian scheme every direct sum of injective OX -modules is again injective and that every injective OX -module is a direct sum of indecomposable injective OX -modules in a unique way (up to order). In [Har1] O II 7.11 it is shown that the indecomposable injective OX -modules are precisely the OX -modules J (x, x′ ) that are defined as follows. Let x ∈ X and let x′ ∈ {x} be a specialization of x. Let jx : Spec(OX,x ) → X and ix′ : {x′ } → X be the inclusions. Let Ix be an injective hull of κ(x) over OX,x (Definition F.64) and set ˜ J (x, x′ ) := ix′ ,∗ i−1 x′ jx,∗ (Ix ),

J (x) := J (x, x).

Moreover, J (x, x′ ) is quasi-coherent if and only if x = x′ . To see that on a locally noetherian scheme X, the embedding QCoh(X) → (OX -Mod) preserves injective objects, we will use the following lemma. Lemma 22.75. Let X be a locally noetherian scheme, and let F be a quasi-coherent OX module. Then there exists a quasi-coherent OX -module G which is injective in (OX -Mod) together with a monomorphism F ,→ G .

267 In fact, one can take for G the injective hull of F in (OX -Mod) (Definition F.64). By Remark 22.74 it is a direct sum of OX -modules of the form J (x, x′ ) and one shows that one necessarily has x = x′ since F is quasi-coherent. But this implies that the injective hull is a direct sum of quasi-coherent OX -modules and hence quasi-coherent (see [Har1] O II 7.18 for details). Proposition 22.76. Let X be a locally noetherian scheme. Then the inclusion functor QCoh(X) → (OX -Mod) sends injectives to injectives. Hence we see that a quasi-coherent OX -module is injective in the category of all OX -modules if and only if it is injective in QCoh(X). Proof. Let F ∈ QCoh(X) be an injective object. As in the lemma, let ι : F ,→ G be a monomorphism from F into a quasi-coherent OX -module G which is injective as an OX -module. Since F is injective, there is a retraction ρ : G → F with ρ ◦ ι = id. Hence F is a direct summand of G , in (QCoh(X)) as well as in (OX -Mod), and as a direct summand of an injective (in (OX -Mod)) is injective, as well. Caveat 22.77. (1) An infinite product in (OX -Mod) of quasi-coherent OX -modules is not necessarily quasi-coherent. (2) The coherator Q is not exact in general. (3) There are examples of affine schemes with underlying noetherian topological space of finite dimension, for which the inclusion (QCoh(X)) → (OX -Mod) does not send injectives to injectives (not even to flasque sheaves). Cf. Verdier’s example in [SGA6] O II App. I. So for a locally noetherian scheme X, we can compute the value of right derived functors on a quasi-coherent OX -module using an injective resolution in (QCoh(X)), while for general X this will not necessarily give the right result. For quasi-coherent OX -modules on a locally noetherian scheme X, injectivity behaves very well. Proposition 22.78. Let X be a locally noetherian scheme and let F be a quasi-coherent OX -module. Then the following assertions are equivalent. (i) F is an injective OX -module. (ii) For every affine open covering (Ui )i of X and for all i one has that F |Ui ∼ = I˜i for some injective Γ(Ui , OX )-module Ii . (iii) There exists an affine open covering (Ui )i of X such that for all i, F |Ui ∼ = I˜i for some injective Γ(Ui , OX )-module Ii . (iv) For every x ∈ X the stalk Fx is an injective OX,x -module. (v) For every OX -module G and for all i > 0 one has Ext iOX (G , F ) = 0. (vi) For every coherent ideal J ⊆ OX one has Ext 1OX (OX /J , F ) = 0. Proof. We will proceed as follows: (ii) ks  (iii) ^f

9A (i) +3 (iv)

+3 (v)  +3 (vi).

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22 Cohomology of quasi-coherent modules

(i) ⇒ (ii). Let (Ui )i be any open affine covering, Ui = Spec Ai . Then F |Ui is a quasicoherent OUi -module that is injective in the category of OUi -modules (Lemma 21.12) and in particular it is injective in the category of quasi-coherent OUi -modules. Hence it corresponds to an injective Ai -module. (ii) ⇒ (iii). This is clear. (iii) ⇒ (i). The hypothesis implies that F is locally on X an injective module in the category of quasi-coherent OX -modules. Hence it is locally injective in the category of all OX -modules by Proposition 22.76. Then Lemma 21.12 implies that F is an injective OX -module. (iii) ⇒ (iv). This follows from Proposition G.23. (iv) ⇒ (vi). As O/J is coherent, the formation of Ext commutes with passage to stalks (Proposition 22.65) and Ext1OX,x (Ox /JX , Fx ) = 0 since Fx is injective. (i) ⇒ (v). As F is injective, we find R Hom OX (G , F ) = Hom OX (G , F ) which is a complex concentrated in degree 0. (v) ⇒ (vi). This is clear. (vi) ⇒ (iii). Assume first that X = Spec A is affine. Let I˜ = F . By Proposition G.21 it suffices to show that Ext1A (A/a, I) = 0 for every ideal a of A. As Ext1A (A/a, I)∼ = Ext 1A (OX /˜ a, F ) one concludes by Proposition 22.68. The general case follows since the formation of Ext commutes with restriction to open subsets (21.21.3) and since coherent ideals on an open subset of a locally noetherian scheme can be extended to a coherent ideal on all of X (see Proposition 10.48 if X is noetherian and [EGAInew] (6.9.7) in general). Now we can use similar arguments to show that also the notion of injective dimension (Definition F.221) behaves well on locally noetherian schemes. Proposition 22.79. Let X be a locally noetherian scheme and let F be a complex in Dqcoh (X). Let a ≤ b be integers. Then the following assertions are equivalent. (i) F has injective amplitude contained in [a, b] in D(X). (ii) F is isomorphic in Dqcoh (X) to a finite complex of injective quasi-coherent OX modules concentrated in degrees [a, b]. (iii) For every open affine covering (Ui )i of X one has for all i that F |Ui ∼ = I˜i for a complex Ii of injective Γ(Ui , OX )-modules concentrated in degrees [a, b]. (iv) There exists an open affine covering (Ui )i of X such that for all i, F |Ui ∼ = I˜i for some complex Ii of injective Γ(Ui , OX )-modules concentrated in degrees [a, b]. (v) For every x ∈ X the stalk Fx is a complex of OX,x -modules of injective amplitude contained in [a, b]. (vi) For every OX -module G one has Ext iOX (G , F ) = 0 for all i ∈ / [a, b]. (vii) For every coherent ideal J ⊆ OX -module one has Ext iOX (OX /J , F ) = 0 for all i∈ / [a, b]. Proof. Here we proceed as follows: (i) ks

+3 (ii) fn  (vi)

+3 (iii)

+3 (iv)

+3 (v)  +3 (vii).

(ii) ⇒ (i). By Proposition 22.76, F is isomorphic to a complex of injective OX -modules concentrated in degrees [a, b]. Hence one concludes by Proposition F.222.

269 (i) ⇒ (ii). The proof of Proposition F.222 shows that F ∈ D[a,b] (X). By Lemma 22.75, ∼ we find then an isomorphism F → J • such that J i = 0 for i < a and such that J i is a quasi-coherent injective OX -module. Then F ∼ = τ ≤b F ∼ = τ ≤b J • and all components ≤b • of τ J are quasi-coherent. The proof of Proposition F.222 shows that all components of τ ≤b J • are injective. (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) and (ii) ⇒ (vi). This follows as in Proposition 22.78. (v) ⇒ (vii). The hypothesis implies that Fx is in D+ (OX,x ) for all x and hence F ∈ D+ (X). Hence one can again apply Proposition 22.65 and argue as in the proof of Proposition 22.78. (vi) ⇒ (vii). This is clear. (vii) ⇒ (ii). As R Hom OX (OX , F ) = F (21.18.10), we have Ext iOX (OX , F ) = H i (F ). [a,b] Hence (vii) shows that F ∈ Dqcoh (X). As in the proof of the implication “(i) ⇒ (ii)”, we see that F is isomorphic in Dqcoh (X) to a complex I of quasi-coherent OX -modules concentrated in degrees [a, b] and with I i injective for all i = a, . . . , b − 1. To show that I b is injective one can argue as in the proof of Proposition F.222, using Ext instead of Ext and that we have already seen that (ii) implies (vii). Then the argument shows that Ext 1OX (OX /J , I b ) = 0 for every coherent ideal J of OX . This implies that I b is injective by Proposition 22.78. Corollary 22.80. Let A be a noetherian ring, let X = Spec A, and let a ≤ b be integers. ˜ has injective Then a complex M in D(A) has injective amplitude in [a, b] if and only if M amplitude contained in [a, b] in D(X).

Projection formula, base change and the K¨ unneth formula The projection formula is a compatibility between pullbacks and pushforwards which holds in many contexts. The simplest prototype is the statement f (F ∩ f −1 (G)) = f (F ) ∩ G for a map f : X → Y of sets and subsets F ⊆ X, G ⊆ Y . Another particularly simple instance is Proposition 22.81 (3). Below we will prove a derived version of the projection formula. Afterwards we will come back to the base change morphism (Definition 21.129) for a cartesian diagram of schemes and investigate when it is an isomorphism for all objects in Dqcoh (X). As we will see, this is closely related to the K¨ unneth morphism, which relates, roughly speaking, cohomology on a fiber product and cohomology of the factors of the product. (22.19) The projection formula. We will show two versions of projection formulas. The first is a non-derived version which for general quasi-coherent modules requires rather strong hypotheses, see (3) and (4) in the following proposition. The second version will be a derived version that will hold in (almost) complete generality, see Proposition 22.84 (3) below. Proposition 22.81. (Projection formula) Let f : X → Y be a morphism of ringed spaces, let F be an OX -module, and let G be an OY -module. There is a natural homomorphism (f∗ F ) ⊗OY G → f∗ (F ⊗OX f ∗ G )

270

22 Cohomology of quasi-coherent modules

which is functorial in F and G . This homomorphism is an isomorphism in the following cases: (1) If G is locally free of finite rank. (2) If f is a homeomorphism onto a closed subspace of Y . (3) If f is an affine morphism of schemes and F , G are quasi-coherent. (4) If f is a qcqs morphism of schemes, F and G are quasi-coherent, and G is flat over OY . Often, Case (4) is the most interesting case. Proof. The homomorphism is obtained by adjunction from the homomorphism f ∗ (f∗ F ⊗ G ) ∼ = (f ∗ f∗ F ) ⊗ f ∗ G → F ⊗ f ∗ G . To check that this is an isomorphism, we may work locally on Y . Case (1). We may assume that G is free, or even that it is isomorphic to OY , by passing to a direct sum afterwards. In this case it is easy to check that the homomorphism in question is an isomorphism. Case (2). We check this on stalks. Outside the image of f , the stalks of direct images under f vanish, hence the stalks of both the left and the right hand side vanish. For y = f (x), the stalks are Fx ⊗OY,y Gy

and

Fx ⊗OX,x (OX,x ⊗OY,y Gy ).

The natural map clearly induces an isomorphism between those, so we are done. Case (3). We may assume that Y is affine. In this case, X is affine, as well, and the assertion is easily checked using the description of the functors f∗ , f ∗ for a morphism between affine schemes (Proposition 7.24). Case (4). We may assume that Y is affine. We want to use Lemma 10.26 with the property P(U ), for U ⊆ X open quasi-compact, to hold if for every quasi-coherent OU -module F ′ the homomorphism (f |U )∗ F ′ ⊗OY G → (f |U )∗ (F ′ ⊗OU (f |U )∗ G ) is an isomorphism. We want to show that P(X) holds. If U ⊆ X is open affine, then f |U is affine and P(U ) holds by (3). Now let U ⊆ X be quasi-compact open and let (Ui )i be a finite open affine covering of U . Consider for a quasi-coherent OU -module F ′ the following commutative diagram / Q fij,∗ (F ′ ) ⊗ G / Q fi,∗ (F ′ ) ⊗ G / (f∗′ F ′ ) ⊗ G 0 i i,j |Ui |Uij (*) 0

 / f∗′ (F ′ ⊗ f ∗ G ) /

Q

i

 ′ ⊗ fi∗ G ) fi,∗ (F|U i /

Q

i,j

 ∗ ′ ⊗ fij G ), fij,∗ (F|U ij

where we write f ′ := f |U , Uij := Ui ∩ Uj , fi := f|Ui , fij := f|Uij , as usual. Both rows are exact (see (10.3.2) and use the flatness assumption on G ). We have already seen that P(Ui ) holds for all i and hence if P(Uij ) holds for all i, j, then P(U ) holds. Now Lemma 10.26 implies that P(X) holds. We now come to the derived version of the projection formula. For its proof we will use the following two lemmas.

271 Lemma 22.82. Let f : X → Y be a qcqs morphism. The functor Rf∗ : Dqcoh (X) → Dqcoh (Y ) commutes with arbitrary direct sums. Note that this property is not true when the assumptions on f are dropped; in fact, even the global sections functor Γ(X, −) does not commute with coproducts in general. Also, a coproduct of K-injective objects is not K-injective in general. L Proof. Given a direct sum E = i Ei in Dqcoh (X), we have a natural morphism M Rf∗ Ei → Rf∗ E. i

We need to show that it induces isomorphisms after passing to cohomology objects. This can be done locally on Y . Hence we may assume Y is affine. Since the morphism is compatible with shifts, it is enough to L prove that we obtain an isomorphism after applying H 0 , i.e., we have to show that i R0 f∗ Ei → R0 f∗ E is an isomorphism (using that applying H 0 commutes with direct sums). Using Corollary 22.30, we see that we can compute these R0 f∗ terms as R0 f∗ of bounded below complexes in Dqcoh (X), allowing us to reduce to the case of bounded below complexes. For those, we can use the spectral sequence Rp f∗ H q (−) ⇒ Rp+q f∗ − (see Corollary 21.46). Again using that taking cohomology objects of a complex commutes with direct sums, this leaves us with showing that higher direct images of quasi-coherent sheaves commute with direct sums, which we have already seen in Corollary 21.56 (2). Lemma 22.83. Let A be a ring, and let P be a property of objects in D(A) such that the following assertions hold. (a) The ring A, considered as a complex concentrated in degree 0, has property L P. (b) If P holds for objects Ei ∈ D(A), i ∈ I, then it holds for the direct sum i Ei . (c) If E → F → G → is an exact triangle in D(A) and P holds for two of its terms, then it holds for all of them. Then all objects of D(A) have the property P. In particular, if T is a triangulated subcategory of D(A) containing A that is stable under direct sums, then T = D(A). ∼

Proof. Considering the empty direct sum, we see that 0 has property P. If E −→ F is an isomorphism in D(R) and E has P, then F has P using the distinguished triangle E → F → 0 →. For E ∈ D(A) we have an exact triangle E → 0 → E[1] → (obtained by shifting the exact triangle E → E → 0 →), so (c) implies that P is invariant under shifts. Let M be a complex of A-modules. To finish the proof, we will construct a quasiisomorphism P → M of complexes of A-modules such that there exists a filtration 0 = F0 ⊆ F1 ⊆ F2 ⊆ · · · ⊆ P S by subcomplexes with P = i Fi and such that each inclusion Fi ⊆ Fi+1 is termwise split, and each subquotient Fi+1 /Fi is isomorphic, as a complex, to a direct sum of shifts of A. The lemma follows from this: In fact, since P → M is a quasi-isomorphism, it is enough to check property P for P . We have a short exact sequence M M 0→ Fi → Fi → P → 0 i≥0

i≥0

272

22 Cohomology of quasi-coherent modules

of Lcomplexes, where the first map is given by the direct sum of the maps Fi → Fi ⊕ Fi+1 ⊂ i Fi , f 7→ (f, −f ). It is termwise split, as is easily checked using that the inclusions Fi → Fi+1 are termwise split, and hence we obtain an exact triangle (even in K(R), see F.123) M M Fi → P → Fi → i

i

Now it follows easily from the properties of the Fi and the assumptions in the lemma that P has property P. To construct P , first note that we can find a (degree-wise) surjective morphism of complexes Q → M such that there is a short exact sequence 0 → Q′ → Q → Q′′ → 0 of complexes where Q′ and Q′′ are direct sums of shifts of A (as complexes). In fact, for m ∈ M i , the complex (*)

··· → 0 → A → A → 0 → ···

with A in degrees i and i + 1 and differential the identity maps to M by mapping 1 to m in degree i, and mapping 1 to d(m) in degree i + 1. Each complex as in (*) contains A[−i − 1] as subcomplex and A[−i] as quotient complex. Taking a direct sum of complexes as in (*), we can construct the desired surjection Q → M . Moreover, by adding further copies of shifts of A, we can ensure that in addition to the above properties, the map Ker(dQ ) → Ker(dM ) is (degree-wise) surjective, and we will always assume that this is true when we apply this construction below. Now define families of complexes Mi and Pi by induction as follows: Let M0 = M . For any i where Mi is defined, choose a surjection Pi → Mi as above, and let Mi+1 be its kernel. We obtain an exact complex of complexes · · · → P2 → P1 → P0 → M → 0. • We define P as the total complex P := Tot⊕ (P−• ) (where the lower index is the row index, i.e., each Pi is a horizontal complex). We now define the filtration. By construction, for each Pi we have a short exact sequence

0 → Pi′ → Pi → Pi′′ → 0, where Pi′ and Pi′′ are direct sums of shifts of A. We let M ′ F2i P := Pi , F2i+1 P := F2i P ⊕ Pi+1 . 0≤j≤i

One sees that the inclusions are termwise split and that the subquotients of this filtration are direct sums of shifts of A. It remains to show that P → M is a quasi-isomorphism. This follows from Lemma F.114. Proposition 22.84. (Derived projection formula) Let f : X → Y be a morphism of ringed spaces. Let F ∈ D(X), G ∈ D(Y ). There is a natural morphism in D(Y ), L ∗ (Rf∗ F ) ⊗L OY G → Rf∗ (F ⊗OX Lf G ),

which is functorial in F and G . This morphism is an isomorphism in any of the following situations:

273 (1) If G is perfect (Definition 21.136). (2) If f is a homeomorphism from X onto a closed subspace of Y . (3) If f : X → Y is a qcqs morphism of schemes and F ∈ Dqcoh (X), G ∈ Dqcoh (Y ). Proof. As in the non-derived case, the existence of the map in question follows from the adjunction of Lf ∗ and Rf∗ , Proposition 21.126. Namely, we have the counit of the adjunction Lf ∗ Rf∗ F → F . It gives rise to Lf ∗ (Rf∗ F ⊗L G ) ∼ = (Lf ∗ Rf∗ F ) ⊗L Lf ∗ G → F ⊗L Rf∗ G , where the first isomorphism is given by Proposition 21.117. Using adjunction again, we obtain the desired morphism. To check whether this is an isomorphism,Lwe may work locally on Y , see Lemma 21.28. Gi if and only if it holds for all Gi To show Note that the statement holds for G = this, observe that Lf ∗ , Rf∗ and ⊗L are compatible with direct sums. For Lf ∗ , this is a formal consequence of the fact that it is left adjoint to some functor, namely Rf∗ . For the other two see Lemma 22.82 and Remark 21.120 (2). Similarly, if G1 → G2 → G3 → is an exact triangle in D(Y ), and the statement holds for two of the terms of this triangle (in the role of G ), then it holds for the third one. This follows from the general fact that exactness of triangles is preserved by derived functors. Case (1). After shrinking Y , if necessary, we may assume that G is represented by a strictly perfect complex. Using the “stupid truncation” functor (see (F.14.3)) to cut off the first non-zero term, and the compatibility with exact triangles discussed above, we inductively reduce to the case that G is a single direct summand of a finite free OY -module. Then we may even assume that G is a finite free module and finally that G = OY because of the compatibility of the statement with direct sums. In this case, the assertion is obviously true. Case (2). In this case, the functor f∗ is exact, as is easy to check by looking at stalks, and thus we can compute the functor Rf∗ by applying f∗ to any representative. To make explicit the computation of Lf ∗ and ⊗L , choose a K-flat representative G of G . Then Lf ∗ G is represented by f ∗ G . We also fix an arbitrary representative F of F , so the left hand side of the morphism above is represented by Tot((f∗ F ) ⊗ G ). Similarly the right hand side is represented by f∗ (Tot(F ⊗ f ∗ G )). Since f∗ commutes with direct sums, we are reduced to showing that the natural map (f∗ F ) ⊗ G → f∗ (F ⊗ f ∗ G ) is an isomorphism for an OX -module F and an OY -module G . This is just the usual projection formula Proposition 22.81 (2). Case (3). We may assume that Y is affine, say S = Spec A. Because of Theorem 22.35, we may assume that G is a complex of quasi-coherent OY -modules. Because of the e compatibilities discussed above, we can apply Lemma 22.83 and reduce to the case G = A. But in this case, the above morphism is just the identity morphism of Rf∗ F . Example 22.85. See Exercise 22.24 for an example illustrating that in Case (3) the condition that G ∈ Dqcoh (Y ) cannot be dropped.

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22 Cohomology of quasi-coherent modules

(22.20) Example: Cohomology of projective bundles. Let us globalize our results on higher direct images of Serre’s twisting line bundles on projective spaces (Theorem 22.22). Theorem 22.86. Let S be a scheme, let E be a finite locally free OS -module of constant rank r + 1, let X := P(E ) be the corresponding projective bundle, and let π : X → S be the structure map. (1) Then one has Ri π∗ OX (d) = 0 for i ̸= 0, r and for all d ∈ Z. (2) The canonical map (13.7.2) M Sym(E ) −→ π∗ OX (d) d∈Z

is an isomorphism of graded OS -algebras. (3) One has (22.20.1)

Rr π∗ OX (d) ∼ = (Sym−r−1−d E ⊗OS det(E ))∨ . ∼

(4) Set ω := π ∗ (det E )(−r − 1). There exists a canonical isomorphism OS −→ Rr π∗ (ω) and for all d ∈ Z a perfect pairing (22.20.2)

Rr π∗ OX (d) ⊗OS π∗ ω(−d) −→ Rr π∗ (ω) = OS .

Vr+1 Here det(E ) = E is the top non-zero exterior power of E , a line bundle. Note that ω ∼ = ΩrP(E )/S as one sees by applying the determinant to the short exact sequence (17.7.3) (Exercise 7.29). We refer to Section (25.9) for the relation of Assertion (4) with Grothendieck duality. If S = Spec R is affine and E = OSr+1 , then P(E ) = PrR and we have already seen all assertions in Theorem 22.22 and Corollary 22.23. The proof will reduce to this case. To achieve this, we will use the following global version of the Koszul complex, see also Exercise 19.8. Remark 22.87. Let X be a scheme, let L be an OX -module, and let u : L → OX be an OX -linear map. As in Section (19.1) one constructs an attached Koszul complex K(u), viewed Vp as a cochain complex of OX -modules concentrated in degrees ≤ 0 with L for p ≥ 0. K(u)−p = From now on we suppose that L is locally
free of finite rank r. Then K(u)p = 0 for r p∈ / [−r, 0] and K(u)p is locally free of rank −p for −r ≤ p ≤ 0. For every quasi-coherent OX -module F we also set K(u, F ) = K(u) ⊗OX F in this case. Let I ⊆ OX be the image of u. If V ⊆ X is an open affine subscheme and f ∈ I (V ) a section, we can choose ℓ ∈ L (V ) with V u(ℓ) = f since V is affine. As in Remark 19.6, one sees that the multiplication by ℓ on L induces a homotopy between the multiplication by f and 0. In particular, for all p one has (22.20.3)

I H p (K(u, F )) = 0.

We have τ ≥0 K(u, F ) = H 0 (K(u, F )) = F /I F . We say that u is completely intersecting for F if the functorial map K(u, F ) → F /I F is a quasi-isomorphism. A trivial example is the case that u is surjective. Then (22.20.3) shows that K(u, F ) is homotopy equivalent to 0.

275 Proof. [of Theorem 22.86] To see (1) we may work locally on S and hence can assume that S = Spec R and that E is quasi-coherent module corresponding to the free R-module Rr+1 . Then P(E ) = PrR and (1) follows from Theorem 22.22. For (2) note that P(E ) = Proj(Sym(E )) by Theorem 13.32. Hence L by (13.7.2) there is a functorial homomorphism of graded OS -algebras Sym(E ) −→ d∈Z π∗ (OX (d)) which is easily seen to be a homomorphism of graded algebras. To see that it is an isomorphism, we can again work locally on S and hence use Theorem 22.22. Similarly, we can check that Rr π∗ OX (d) = 0 for d > −(r + 1) locally on S and again get the result from Theorem 22.22. To show (3) for d ≤ −r−1 and (4) we twist the canonical surjective map π ∗ (E ) → OX (1) by −1 and obtain a surjective map u : π ∗ (E )(−1) → OX . The corresponding Koszul complex K(u) is the complex of locally free OX -modules

Vp

0 → π ∗ (det E )(−r − 1) −→ . . . −→ π ∗ ( E )(−p) −→ . . . −→ π ∗ E (−1) −→ OX −→ 0, Vp where π ∗ ( E )(−p) sits in degree −p. As u is surjective, it is = 0 in the derived category D(X) (Remark 22.87). Hence Rπ∗ K(u) = 0 and the first hypercohomology spectral sequence for π∗ (F.49.3) takes the form E1pq = Rq π∗ (π ∗ (

V−p

E )(p)) =

V−p

E ⊗OS Rq π∗ (OX (p)) ⇒ 0,

V−p E = 0 for p < −(r + 1) where the second equality holds by the projection formula. As pq or p > 0, it follows from what we have already seen that E1 = 0 unless (p, q) is (0, 0) or (−r − 1, r). Therefore there exists a unique nonzero differential in the spectral sequence, 0,0 −r−1,r → Er+1 : Er+1 namely d−r−1,r which is necessarily an isomorphism since the spectral r+1 pq sequence converges to 0. As all differentials on level < r + 1 are zero, we have E1pq = Er+1 . −r−1,r yields an isomorphism Hence dr+1 ∼

E1−r−1,r = Rr π∗ (ω) −→ E10,0 = OS . This yields (22.20.1) for d = −r − 1. To show (22.20.1) for d < −r − 1, note that π∗ ω(−d) = det(E ) ⊗ Sym−r−1−d (E ) by the projection formula. Therefore it suffices to show the existence of the perfect pairing (22.20.2). For this consider the map π ∗ Sym−r−1−d E ⊗OX OX (d) −→ OX (−r − 1). If we apply Rr π∗ and use the projection formula, we obtain a map Sym−r−1−d E ⊗OS Rr π∗ OX (d) −→ Rr π∗ OX (−r − 1) which yields by tensoring with det(E ) the pairing (22.20.2). To show that this pairing is perfect, we may again work locally and can conclude by Corollary 22.23. Theorem 22.86 will enable us (Remark 24.71 below) to calculate the higher direct images of an arbitrary line bundle M on P(E ) using that M ∼ = π ∗ (L )(d) for some line bundle L on S and some integer d ∈ Z, see Proposition 24.69 below.

276

22 Cohomology of quasi-coherent modules

(22.21) Special case of base change: Non-derived flat base change. Consider a commutative diagram of schemes X′ (22.21.1)

u′

/X

u

 / S.

f′

 S′

f

If the diagram is cartesian and X = Spec B, S = Spec A and S ′ = Spec A′ are all affine, we have isomorphisms ∼ A′ ⊗A M → (A′ ⊗A B) ⊗B M for every B-module M . In terms of quasi-coherent sheaves, we can rephrase this by saying that we have isomorphisms ∼ u∗ f∗ F → (f ′ )∗ (u′ )∗ F for every quasi-coherent OX -module F . Recall (Proposition 12.6), that more generally we have Lemma 22.88. Consider a diagram (22.21.1) of schemes where f is affine or f is qcqs and u is flat. Assume that the diagram is cartesian. For every quasi-coherent OX -module F , there is a natural isomorphism ∼

u∗ f∗ F → (f ′ )∗ (u′ )∗ F . Let us now pass to the derived version of the base change morphism for the commutative diagram (22.21.1) which we constructed in Definition 21.129. The base change morphism is a natural morphism (22.21.2)

θ : Lu∗ Rf∗ → R(f ′ )∗ L(u′ )∗

of functors D(X) → D(S ′ ). We will give in Theorem 22.99 below a criterion when θ is an isomorphism if the functors are restricted to Dqcoh (X). Remark 22.89. It follows from the compatibility of derived direct and inverse images with restriction to open subsets, that the base change θ commutes with restriction to open subschemes first on S and then on S ′ and X. In other words, let U ⊆ S, U ′ ⊆ u−1 (U ) and V ⊆ f −1 (U ) be open affine subsets, and set V ′ := u′−1 (V ) ∩ f ′−1 (U ′ ) (we do not assume the diagram to be cartesian). Then the base change morphism ϑ for the commutative diagram /V V′  U′

 /U

satisfies ϑ ◦ (−|V ) = (−|U ′ ) ◦ θ as morphisms of functors D(X) → D(U ′ ). Let us describe θF : Lu∗ Rf∗ F → R(f ′ )∗ L(u′ )∗ F for F ∈ Dqcoh (X) if S = Spec R, ′ S = Spec R′ and X = Spec A are affine. Then F ∼ = F˜ for a complex F of A-modules (Lemma 22.36). Choose a quasi-isomorphism P → F , where P is a colimit of bounded above complexes of free A-modules (which is possible by Remark 21.95 (1) applied to the ringed space (∗, A)). Then Lu′∗ F is represented by u′∗ P˜ (Lemma 22.39). Moreover, as f is affine and hence f∗ is exact on quasi-coherent modules, we have Rf∗ F ∼ = f∗ P˜ . Now θF is the composition

277 Lu∗ Rf∗ F ∼ = Lu∗ f∗ P˜ → u∗ f∗ P˜ −→ f∗′ u′∗ P˜ → Rf∗′ u′∗ P˜ ∼ = Rf∗′ Lu′∗ F , where the longer arrow in the middle is given by the non-derived base change map (12.2.2) extended to complexes of OX -modules. Suppose that u is flat, so that Lu∗ = u∗ (Example F.169). Passing to cohomology objects, in this case we obtain homomorphisms (22.21.3)

u∗ Ri f∗ F → Ri (f ′ )∗ (u′ )∗ F

for every F ∈ D(X) and i ≥ 0. For their construction, as before, it is not necessary to assume that the diagram above is cartesian. One can construct a homomorphism like this also without the flatness hypothesis, see Section (23.28) below. Although the base change homomorphism is not in general an isomorphism (Exercise 22.42), it is so under certain circumstances. Interestingly, there are several quite different kinds of conditions one can impose to ensure this. At this point we prove a relatively simple case directly. See Theorem 23.140 for further important cases which are particularly significant because there u is not required to be flat. Proposition 22.90. Consider a cartesian diagram (22.21.1) of morphisms of schemes. If the morphism u is flat and the morphism f is qcqs, then for every complex of quasi-coherent OX -modules and every i ∈ Z the base change morphism induces an isomorphism u∗ R i f ∗ F ∼ = Ri (f ′ )∗ (u′ )∗ F . This generalizes Lemma 22.88 to higher direct images. Note that the base change morphism is also an isomorphism if f is affine because then all higher direct images vanish (Corollary 22.5). We give here a direct proof in the case that f is separated and that F is a quasi-coherent OX -module and refer to Remark 22.108 below, where we will deduce the general result from the much more general base change Theorem 22.99. ˇ Proof. Under our assumption that f is separated, we can compute cohomology as Cech cohomology, allowing for a simpler proof than in the general case. We can work locally on S ′ and on S, and hence assume that S = Spec A and S ′ = Spec A′ are affine. Then u and hence u′ are affine. Moreover, X and X ′ are quasi-compact and separated since we assumed f and hence f ′ to be so. In this situation, the base change morphism (22.21.3) is given by a homomorphism of A′ -modules (*)

∼

H i (X, F ) ⊗A A′ → H i (X ′ , F ′ ),

S where we set F ′ := (u′S )∗ F . Choose a finite affine open cover X = i Ui . We obtain an affine open cover X ′ = i (u′ )−1 (Ui ) and can compute the two cohomology groups above ˇ using Cech cohomology for this covering (Theorem 22.9). As we have F (V ) ⊗A A′ = F (u′−1 (V ), F ′ ) for every open affine subscheme V ⊆ X ˇ and as tensor products commute with finite products, we find for the Cech complexes Cˇ • ((Ui )i , F ) ⊗A A′ ∼ = Cˇ • (((u′ )−1 (Ui )i , F ′ ). As A′ is a flat A-algebra, applying cohomology H i (−) commutes with the exact functor − ⊗A A′ . This yields an isomorphism (*). We omit the check that this isomorphism is given by (22.21.3).

278

22 Cohomology of quasi-coherent modules

Corollary 22.91. Let f : X → Spec R be a qcqs morphism and let R → R′ be a flat map of rings. Then for every quasi-coherent OX -module F the base change morphism induces for all i ≥ 0 isomorphisms H i (X, F ) ⊗R R′ ∼ = H i (XR′ , FR′ ), where XR′ := X ⊗R R′ and FR′ is the pullback of F to XR′ . Corollary 22.92. Let A be a ring, let X an A-scheme, and B a faithfully flat A-algebra. Then X is affine if and only X ×Spec A Spec B is affine. We give a cohomological proof here. Note that X is affine if and only if the morphism X → Spec A is affine. Hence the corollary is also a special case of the fact that being “affine” is a stable under fpqc-descent (Proposition 14.53). Proof. Assume that X ×Spec A Spec B is affine. We want to show that X is affine. Note that X is qcqs by Proposition 14.51. Using Serre’s criterion for affineness, Theorem 12.35, we see that it is enough to show that H 1 (X, F ) = 0 for all quasi-coherent OX -modules F . By Corollary 22.91 and since X ⊗A B is affine by assumption, we have H 1 (X, F ) ⊗A B ∼ = H 1 (X ⊗A B, FB ) = 0, where FB denotes the pullback of F . Since B is faithfully flat over A, the result follows. (22.22) Tor-independence. Definition and Lemma 22.93. Two morphisms of schemes X (22.22.1) S

′

u

 /S

f

are said to be tor-independent, if the following equivalent assertions are satisfied. (i) For all s′ ∈ S ′ and x ∈ X with same image s ∈ S one has OS,s

Tori ′

(OS ′ ,s′ , OX,x ) = 0

for all i > 0.

′

(ii) For all s ∈ S and x ∈ X with same image s ∈ S there exist open affine neighborhoods Spec R ⊆ S of s, Spec A ⊆ f −1 (Spec R) of x, and Spec R′ ⊆ u−1 (Spec R) of s′ such that ′ TorR for all i > 0. i (R , A) = 0 (iii) For all open affine subschemes Spec R ⊆ S, Spec A ⊆ f −1 (Spec R), and Spec R′ ⊆ u−1 (Spec R) one has ′ TorR i (R , A) = 0

for all i > 0.

Below (Definition 22.96) we will define tor-independence for complexes of quasicoherent modules such that the complexes OX and OS ′ , concentrated in degree 0, are tor-independent if and only if f and u are tor-independent (Remark 22.97 (1)). Then the equivalences of the conditions above will be a special case of Lemma 22.96. Hence, we do not give a proof here.

279 It is clear from the definition that tor-independence can be checked locally on S, S ′ and X. Furthermore, interchanging the roles of f and u does not affect the property of tor-independence. There are many examples of morphisms that are not tor-independent. For instance every pair of closed immersions Y → X and Z → X given by quasi-coherent ideals I , J ⊆ OX with I J = ̸ I ∩ J is not tor-independent (Exercise 21.19). A trivial but important example of tor-independent morphisms is the following. Remark 22.94. If in the situation of Definition 22.93 one of the morphisms u or f is flat, then u and f are tor-independent. An example of tor-independent closed immersions is the following. Example 22.95. Let R be a ring and let f = (f1 , . . . , fr ) be a completely intersecting sequence in R (Definition 19.10), choose some 1 ≤ s ≤ r and set R′ = R/(f1 , . . . , fs−1 ) and A = R/(fs , . . . , fr ). Then Spec A and Spec R′ are tor-independent over Spec R. Indeed, in D(R) one has isomorphisms ∼ ∼ ∼ R ′ ⊗L R A = K• (f1 , . . . , fs−1 ) ⊗R K• (fs , . . . , fr ) = K• (f1 , . . . , fr ) = R/(f1 , . . . , fr ) for the first isomorphism using that Koszul complexes are K-flat, and we obtain ( R/(f1 , . . . , fr ), if i = 0, R ′ Tori (R , A) = 0, if i > 0. One can also define tor-independence more generally for complexes. The condition that ′ TorR i (R , A) = 0 for i > 0 can be rephrased as the condition that the canonical map ′ L R ⊗R A → R′ ⊗R A is an isomorphism. This makes the following definition plausible. Definition and Lemma 22.96. Let f : X → S and u : S ′ → S be morphisms of schemes and let F (resp. G ) be a complex of quasi-coherent OX -modules (resp. of quasi-coherent OS ′ -modules). Then F and G are said to be tor-independent over S if the following equivalent conditions are satisfied. (i) For all s′ ∈ S ′ and x ∈ X with same image s ∈ S the canonical homomorphism Fx ⊗ L OS,s Gs′ −→ Fx ⊗OS,s Gs′ is an isomorphism. (ii) For all s′ ∈ S ′ and x ∈ X with same image s ∈ S there exist open affine neighborhoods Spec R ⊆ S of s, Spec A ⊆ f −1 (Spec R) of x, and Spec R′ ⊆ u−1 (Spec R) of s′ such that the canonical homomorphism F ⊗L R G −→ F ⊗R G is an isomorphism, where F (resp. G) is the complex of A-modules (resp. of R′ modules) corresponding to F |Spec A (resp. to G |Spec R′ ). (iii) For all open affine subschemes Spec R ⊆ S, Spec A ⊆ f −1 (Spec R), and Spec R′ ⊆ u−1 (Spec R) the canonical homomorphism F ⊗L R G −→ F ⊗R G is an isomorphism, where F and G are as in (ii).

280

22 Cohomology of quasi-coherent modules

Proof. Clearly, (iii) implies (ii). To see that (ii) implies (i) let p ⊂ R, p′ ⊂ R′ and q ⊂ A be the prime ideals corresponding to s, s′ , and x, respectively. One has isomorphisms (*)

L L ′ Fx ⊗ L Rp Gs′ = Fx ⊗R Gs = Aq ⊗ F ⊗R G ⊗ Rp′ ,

Fx ⊗Rp Gs′ = Fx ⊗R Gs = Aq ⊗ F ⊗R G ⊗ Rp′ ′

where the second equality in the first line holds since localizations are flat modules. Hence if (ii) holds, then Fx ⊗L R p G s′ = Fx ⊗ R p G s′ . It remains to show that (i) implies (iii). The map F ⊗L R G → F ⊗R G is an isomorphism in D(A ⊗R R′ ) if and only if it is an isomorphism on stalks, i.e., after localization in all prime ideals q′ ⊂ R′ ⊗R A. Let p, p′ , and q be the inverse images of q′ in R, R′ , and A. Then L (*) shows that (F ⊗L R G)q′ → (F ⊗R G)q′ is a localization of Fx ⊗Rp Gs′ → Fx ⊗Rp Gs′ , which is an isomorphism by hypothesis. Remark 22.97. Let f : X → S, u : S ′ → S, and F and G be as in Definition 22.96. (1) If F and G are quasi-coherent modules, considered as complexes concentrated in O degree 0, then F and G are tor-independent if and only if Tori S,s (Gs′ , Fx ) = 0 for all i > 0 and for all s′ ∈ S ′ and x ∈ X with the same image s in S. In particular, the morphisms f and u are tor-independent if and only if OX and OS ′ are tor-independent over S. (2) Let F be K-flat as complex of f −1 OS -modules, e.g., if F is a bounded above complex of quasi-coherent modules that are flat over S (Lemma 21.93). Then F and G are tor-independent over S for all G . Indeed, we may assume that S = Spec R, S ′ = Spec R′ , and X = Spec A are affine. Then F corresponds to a complex F of A-modules that is K-flat as complex of ′ R-modules (Lemma 22.39). Hence F ⊗L R G = F ⊗R G for every complex G in D(R ). (22.23) Base change and K¨ unneth formula: Main Theorem. In this section, we will more or less follow Lipman [Lip2] O X 3.10 with some generalizations (e.g. Proposition 22.102) needed for the general theory of base change of higher direct images in Section (23.28). Let X′ (22.23.1)

u′

/X

u

 /S

f′

 S′

f

be a commutative diagram of schemes. We write g = f ◦ u′ = u ◦ f ′ for the composition and define the K¨ unneth map for E ∈ Dqcoh (S ′ ), F ∈ Dqcoh (X) as the composition η : Ru∗ E ⊗L Rf∗ F → Rg∗ Lg ∗ (Ru∗ E ⊗L Rf∗ F ) (22.23.2)

→ Rg∗ (L(f ′ )∗ Lu∗ Ru∗ E ⊗L L(u′ )∗ Lf ∗ Rf∗ F ) → Rg∗ (L(f ′ )∗ E ⊗L L(u′ )∗ F ).

Here we use the adjunction morphisms in the first and in the last step, and the compatibility between derived tensor product and derived pullback in the middle step. This construction is functorial in E and in F .

281 Example 22.98. Assume that in the above situation S = Spec k is the spectrum of a field. Then in particular u and f , and hence u′ and f ′ are flat. In this case, we can compute derived pullback as usual pullback and derived tensor product over OS as usual tensor product of complexes, using representatives of E and F . Then the K¨ unneth morphism simplifies to give us a morphism η : Ru∗ E ⊗ Rf∗ F → Rg∗ ((f ′ )∗ E ⊗ (u′ )∗ F ). Let E → I and F → J be K-injective resolutions. Then Ru∗ E⊗Rf∗ F = u∗ I ⊗v∗ J. Passing to cohomology objects we obtain by Proposition F.95 for all n ∈ Z a homomorphism i M

H i (S ′ , E) ⊗k H j (X, F ) ∼ = H n (u∗ I ⊗ f∗ J) −→ H n (X ′ , (f ′ )∗ E ⊗ (u′ )∗ F ),

i+j=n

where we identify the categories of OSpec k -modules and of k-vector spaces. We will see later (Corollary 22.110) that these homomorphisms always are isomorphisms. The main result in this section is the following theorem. Theorem 22.99. Assume that the diagram (22.23.1) is cartesian, that all the schemes in the diagram are quasi-separated and that f and u are qcqs. Write g := f ◦ u′ = u ◦ f ′ . The following are equivalent: (i) The base change morphism θ : Lu∗ Rf∗ → R(f ′ )∗ L(u′ )∗ (22.21.2) is an isomorphism of functors Dqcoh (X) → Dqcoh (S ′ ). (ii) The K¨ unneth morphism (22.23.2) for this diagram ′∗ L ′∗ η : Ru∗ E ⊗L OS Rf∗ F −→ Rg∗ (Lf E ⊗OX ′ Lu F )

is an isomorphism for all E ∈ Dqcoh (S ′ ), F ∈ Dqcoh (X). (iii) The schemes X and S ′ are tor-independent over S (Definition 22.93). We will prove the theorem in the next sections. The proof will show that for some of the implications, the hypotheses can be further weakened, see Remark 22.106 below. The following remark will show that (i) implies (ii). Remark 22.100. With the notation of Theorem 22.99 (ii), we have a commutative diagram (22.23.3)

Ru∗ E ⊗L Rf∗ F 

∼ =

Ru∗ (E ⊗L Lu∗ Rf∗ F )

η

/ Rg∗ (L(f ′ )∗ E ⊗L L(u′ )∗ F ) O ∼ =

/ Ru∗ (E ⊗L R(f ′ )∗ L(u′ )∗ F ),

where the lower horizontal arrow is obtained from the base change morphism by applying E ⊗L − and Ru∗ , and the vertical isomorphisms come from the derived projection formula Proposition 22.84. The commutativity of the diagram follows formally from the properties of derived pullback and derived direct image (see [Lip2] O X (3.10.2.3) for details). From the diagram, we obtain immediately that (i) implies (ii) in Theorem 22.99. Furthermore, whenever Ru∗ is conservative (e.g., if u is affine, see Lemma 22.101 below), we even get that (ii) and (i) are equivalent. If u is an open immersion, the base change morphism is an isomorphism (Lemma 21.28). Hence we conclude that the K¨ unneth morphism is an isomorphism in this case.

282

22 Cohomology of quasi-coherent modules

(22.24) Tor-independence implies base change. To show the other implications of Theorem 22.99 we will use the following lemmas. Lemma 22.101. Let f : X → S be a qcqs morphism of schemes. If f is an open immersion, or if f is affine, then the functor Rf∗ : Dqcoh (X) → Dqcoh (S) is conservative (Definition F.3). The lemma in particular shows that Rf∗ : Dqcoh (X) → Dqcoh (S) is conservative for every quasi-affine morphism f : X → S (Exercise 22.23). Proof. If f is an open immersion, then the adjunction morphism Lf ∗ Rf∗ E → E is an isomorphism for all E (Remark 21.127) and we can identify α = Lf ∗ Rf∗ α, hence α is an isomorphism if Rf∗ α is an isomorphism. Now assume that f is affine, and let α : F → G be a morphism in Dqcoh (X). Since the statement if local on S, we may assume that S and hence X are affine. In fact, by Proposition 22.33, we can equivalently consider complexes F , G of quasi-coherent OX -modules, and compute Rf∗ by applying f∗ to all terms (of a fixed representative). The fact that α, or Rf∗ α, is an isomorphism can equivalently be expressed by saying that the respective morphism fits into an exact triangle with third term an exact complex. Thus it is enough to show that a complex E of quasi-coherent OX -modules such that f∗ E is exact, is itself exact. But the functor f∗ is exact on the category of quasi-coherent OX -modules, so H i (f∗ E) = f∗ (H i (E)), and if F is any quasi-coherent OX -module with f∗ F = 0, then F = 0. It follows that Rf∗ is indeed conservative. We now come to the proof that (iii), i.e., tor-independence, implies (i), i.e., the base change formula. This will be the special case F = OX and G = OS ′ of Proposition 22.102 below. To formulate this more general result we introduce the following notation. Consider a commutative diagram of morphism of schemes as in (22.23.1). Let F (resp. G ) be a complex of quasi-coherent OX -modules (resp. of quasi-coherent OS ′ -modules), and let E be in Dqcoh (X). We construct a general base change map in D(S ′ ) (22.24.1)

∗ L ′ ′∗ ′∗ L ∗ G ⊗L OS ′ Lu Rf∗ (F ⊗OX E) −→ Rf∗ (f G ⊗OX ′ u F ⊗OX ′ Lu E),

which is functorial in G , F and E as follows. It is the composition in D(S ′ ) η : G ⊗L Lu∗ Rf∗ (F ⊗L E) −→ G ⊗L Rf∗′ Lu′∗ (F ⊗L E) ∼

−→ Rf∗′ Lf ′∗ G ⊗L Lu′∗ (F ⊗L E)





∼ −→ Rf∗′ Lf ′∗ G ⊗L Lu′∗ F ⊗L Lu′∗ E)
−→ Rf∗′ f ′∗ G ⊗OX ′ u′∗ F ⊗L Lu′∗ E) , where the first map is given by the usual base change map, the second by the projection formula, the third by compatibility of derived pullback and derived tensor product (Proposition 21.117), and the last by the canonical map Lf ′∗ G ⊗L Lu′∗ F → f ′∗ G ⊗ u′∗ F . Proposition 22.102. In the situation above suppose that Diagram (22.23.1) is cartesian, that f is qcqs, and that F and G are tor-independent over S (Definition 22.96). Then the base change map (22.24.1) is an isomorphism in Dqcoh (S ′ ).

283 Proof. To show that the base change homomorphism is an isomorphism, we can work locally S and on S ′ , and therefore we can assume that S and S ′ are affine. Then u, and hence u′ , are affine morphisms. Let ∗ θ0 : Lf ∗ (u∗ G ) ⊗L OX F −→ f (u∗ G ) ⊗OX F

be the canonical morphism in Dqcoh (X). We will show the following assertions (using the assumption that S and S ′ are affine). These assertions prove the proposition. (1) The base change map (22.24.1) is an isomorphism if and only if the map

(*) Rf∗ Lf ∗ (u∗ G ) ⊗L F ⊗L E −→ Rf∗ f ∗ (u∗ G ) ⊗OX F ⊗L E induced from θ0 by functoriality is an isomorphism. (2) The morphism θ0 is an isomorphism if and only if F and G are tor-independent. The proof of the first assertion will not need F and G to be tor-independent. To show Assertion (1), note that by Lemma 22.101 (and noting that the terms involved have quasi-coherent cohomology) the base change map is an isomorphism if and only if it is an isomorphism after applying Ru∗ . If we apply Ru∗ to the left side of (22.24.1) we obtain Ru∗ (G ⊗L Lu∗ Rf∗ (F ⊗L E) = Ru∗ G ⊗L Rf∗ (F ⊗L E) = u∗ G ⊗L Rf∗ (F ⊗L E) = Rf∗ (Lf ∗ (u∗ G ) ⊗L F ⊗L E), where the first and the last equality hold by the projection formula and where the second equality holds since u is affine and G is a complex of quasi-coherent modules (Proposition 22.33). If we apply Ru∗ to the right side of (22.24.1) we obtain

Ru∗ Rf∗′ (f ′∗ G ⊗ u′∗ F ⊗L Lu′∗ E) = Rf∗ Ru′∗ (f ′∗ G ⊗ u′∗ F ⊗L Lu′∗ E)
= Rf∗ Ru′∗ (f ′∗ G ⊗ u′∗ F ) ⊗L E)
= Rf∗ u′∗ (f ′∗ G ⊗ u′∗ F ) ⊗L E
= Rf∗ u′∗ (f ′∗ G ) ⊗ F ⊗L E
= Rf∗ f ∗ (u∗ G ) ⊗ F ⊗L E . Here the first equality holds since u ◦ f ′ = f ◦ u using Proposition 21.115, the second by the projection formula. The remaining equalities holds since u′ and u are affine, where for the last equality one uses also Lemma 22.88. One checks, using Remark 21.129, that via these identifications the base change map is identified with the map in Assertion (1). This shows (1). To see that Assertion (2) holds, we may work locally on X and hence assume that X = Spec A is affine. Let S = Spec R and S ′ = Spec R′ . We consider F as a complex F of A-modules and G as complex G of R′ -modules. By Lemma 22.41, the source of θ0 becomes (the complex of quasi-coherent OX -modules associated with) the complex of A-modules L L G ⊗L R A ⊗A F = G ⊗R F, and the target of θ0 becomes G ⊗R A ⊗A F = G ⊗R F. This shows Assertion (2).

284

22 Cohomology of quasi-coherent modules

In the next chapter we will use the proposition in the following form. Corollary 22.103. In the situation above suppose that Diagram (22.23.1) is cartesian and that f is qcqs. Let F be a bounded above complex of quasi-coherent modules that are flat over S and let E be in Dqcoh (X). Then there is a functorial isomorphism in Dqcoh (S ′ ) (22.24.2)

∼

′ ′∗ L ∗ Lu∗ Rf∗ (F ⊗L OX E) −→ Rf∗ (u F ⊗OX ′ Lu E).

Proof. This is the special case G = OS ′ of Proposition 22.102 using that OS and F are tor-independent by Remark 22.97 (2). The proof shows that instead of assuming that F is bounded above with S-flat components, it would be sufficient that F and OS ′ are tor-independent over S. (22.25) K¨ unneth isomorphism implies tor-independence. To finish the proof of Theorem 22.99, it remains to show that if the K¨ unneth map is an isomorphism, then f und u are tor-independent. For this we show first that if the K¨ unneth map is an isomorphism, then it is also locally an S isomorphism. S ′ Consider the commutative diagram (22.23.1). Let S = i∈I Si , u−1 (Si ) = j∈Ji Sij , S −1 f (Si ) = ℓ∈Li Xiℓ be open covers, such that all the corresponding open immersions ′ , and Xiℓ are affine). We also define are quasi-compact morphisms (e.g., if all Si , Sij ′ ′−1 ′ ′−1 ′ Xijℓ := f (Sij ) ∩ u (Xiℓ ). Then the Xijℓ form an open covering of X ′ and the open ′ → X ′ are quasi-compact and in particular qcqs because all immersions immersions Xijℓ are separated. We obtain for every i ∈ I, j ∈ Ji , ℓ ∈ Li , a commutative diagram Xijℓ (22.25.1)

f′



′ Sij

u′

/ Xiℓ f

u

 / Si .

If (22.23.1) is cartesian, then (22.25.1) is cartesian (Corollary 4.19). Lemma 22.104. Assume that the K¨ unneth morphism for diagram (22.23.1) is an isomorphism for all E ∈ Dqcoh (S ′ ), F ∈ Dqcoh (X). Then the K¨ unneth morphism for (22.25.1) ′ is an isomorphism for all objects in Dqcoh (Sij ) and Dqcoh (Xiℓ ). As we can check (iii) of Theorem 22.99 locally, the theorem shows that a converse in the following sense is true if diagram (22.23.1) is cartesian: If the K¨ unneth morphisms are ′ isomorphisms for all objects in Dqcoh (Sij ) and Dqcoh (Xiℓ ) and the (then automatically cartesian) diagram (22.25.1), then the K¨ unneth morphisms for all objects in Dqcoh (S ′ ) and Dqcoh (X) and the original cartesian diagram are isomorphisms. Proof. [Sketch of proof] (I). First, we check that we may pass to open subsets U ⊆ S (and correspondingly replace X by XU := X ×S U , S ′ by U ′ := S ′ ×S U and X ′ by XU′ := X ′ ×S U ) whenever the corresponding open immersion i : U → S is quasi-compact. In fact, denote by i′ : U ′ → S ′ and j : XU → X the morphisms obtained by base change from i. Let η be the K¨ unneth morphism for the diagram

285 XU′

/ XU

 U′

 /U

f

and E ∈ Dqcoh (U ′ ), F ∈ Dqcoh (XU ). Then it is not difficult albeit quite tedious to check (see [Lip2] O X Lemma (3.10.3.4) for the details, note that there the diagram (22.23.1) is assumed to be cartesian but this is not used in the proof) that Ri∗ η is isomorphic to the K¨ unneth isomorphism for the original diagram and for Ri′∗ E and Rj∗ F (which lie in Dqcoh (S ′ ) and Dqcoh (X), resp., by Theorem 22.32). Hence by our assumption, Ri∗ η is an isomorphism. Since i is an open immersion, this implies that η is an isomorphism (Lemma 22.101), as desired. (II). Second, given a diagram U ′ ×S ′ X ′

/ X′

 U′

 / S′

u′

/X

u

 /S

f′

f

where both squares have the property that the K¨ unneth morphism always is an isomorphism, then the rectangle also has this property (cf. [Lip2] O X , Lemma 3.10.3.2). (III). Finally, if u in (22.23.1) is an open immersion, then the K¨ unneth morphism is an isomorphism (Remark 22.100). Using (II) this allows us to pass to open subsets of S ′ . Since the K¨ unneth morphism is symmetric with respect to switching the roles of f and of u, by the same argument we obtain that we may pass to open subsets of X. Proof. [Proof of Theorem 22.99] We have already seen that (ii) implies (i) (Remark 22.100) and that (iii) implies (i) (Proposition 22.102). It remains to show that (ii) implies (iii). By Lemma 22.104 we know that (ii) for the original diagram implies that (ii) holds for all diagrams attached to suitable coverings of S, S ′ , X. Since we can check (iii) locally on S, S ′ and X, this means that we may assume, without loss of generality, that S, S ′ and X are affine schemes, say S = Spec A, S ′ = Spec A′ , X = Spec B. Moreover, since the diagram is cartesian, X ′ = X ×S S ′ = Spec(A ⊗R R′ ) is also affine. Then u is affine and hence we have already seen that (ii) and (i) are equivalent (Remark 22.100). Hence it suffices to show that the diagram (22.23.1) is tor-independent if the base change morphism θOX : Lu∗ Rf∗ OX → Rf∗′ Lu′∗ OX is an isomorphism. As f and f ′ are affine, we may identify by Proposition 22.33 ∼ Lu∗ Rf∗ OX = Lu∗ f∗ OX = (R′ ⊗L R A)

and R(f ′ )∗ L(u′ )∗ OX = (f ′ )∗ L(u′ )∗ OX = (f ′ )∗ (u′ )∗ OX = u∗ f∗ OX = (R′ ⊗R A)∼ , where we have also used that L(u′ )∗ OX = (u′ )∗ OX (since OX is flat over itself), and in the final step the non-derived base change homomorphism u∗ f∗ OX → (f∗′ )(u′ )∗ OX which is an isomorphism by Lemma 22.88.

286

22 Cohomology of quasi-coherent modules

Since the non-derived and derived base change homomorphisms are compatible (Remark 22.89), one concludes that θOX is an isomorphism if and only if the natural map ′ A′ ⊗L A B → A ⊗A B

is an isomorphism, which means that f and u are tor-independent. Remark 22.105. The condition that the K¨ unneth morphism is an isomorphism (or the condition about the tor-independence) does not change when the roles of f and u in the base change diagram are exchanged; this actually was used in the proof. In particular it follows that the base change homomorphism for the cartesian diagram (22.23.1) is a functorial isomorphism if and only if the same holds for the diagram obtained by exchanging f and u. Remark 22.106. The proof of Theorem 22.99 shows that for some of the implications some of the hypotheses are superfluous: (1) The implications “(iii) ⇒ (i) ⇒ (ii)” hold without assuming that all schemes are quasi-separated. (2) The implication “(iii) ⇒ (i)” holds without assuming that u is qcqs. (3) If Diagram (22.23.1) is merely commutative, but not necessarily cartesian, the implication “(i) ⇒ (ii)” still holds, and (i) and (ii) are even equivalent if u is affine. (4) Suppose again that the diagram (22.23.1) is merely commutative. Let f be affine and f ′ qcqs. Suppose that the base change morphism θOX : Lu∗ Rf∗ OX → Rf∗′ Lu′∗ OX is an isomorphism. As f∗ is exact for quasi-coherent modules and OX is K-flat as OX -module, we obtain an isomorphism Lu∗ f∗ OX ∼ = Rf∗′ u′∗ OX = Rf∗′ OX ′ . The left hand side has cohomology in degrees ≤ 0, the right hand side has cohomology in degrees ≥ 0. This shows that one has isomorphisms of complexes in Dqcoh (S ′ ) Lu∗ f∗ OX ∼ = u∗ f∗ O X ∼ = f∗′ OX ′ ∼ = Rf∗′ OX ′ , in other words, one has S (a) TorO i (OX , OS ′ ) = 0 for all i > 0, i ′ (b) R f∗ OX ′ = 0 for all i > 0, (c) u∗ f∗ OX ∼ = f∗′ OX ′ . ′ If f is also affine, then (b) holds a priori (Corollary 22.5) and (c) shows that Diagram (22.23.1) is automatically cartesian by the equivalence of quasi-coherent OS ′ -algebras and affine morphisms with target S ′ (Corollary 12.2). More generally, if f ′ is only quasi-affine, then (b) implies that f ′ is affine (Exercise 22.12) and the argument above again shows that (22.23.1) is cartesian. Corollary 22.107. Let X′ (22.25.2)

u′

/X

u

 /S

f′

 S′

f

be a cartesian diagram of schemes. Assume that f is qcqs, and that at least one of f and u is flat.

287 (1) The base change morphism Lu∗ Rf∗ → R(f ′ )∗ L(u′ )∗ is an isomorphism of functors Dqcoh (X) → Dqcoh (S ′ ). unneth morphism (22.23.2) for this diagram is an (2) If u is addition qcqs, then the K¨ isomorphism for all E ∈ Dqcoh (S ′ ), F ∈ Dqcoh (X). Proof. If f or u are flat, then they are tor-independent. Hence the corollary follows from Theorem 22.99 and Remark 22.106 (1). Remark 22.108. If u is flat in the cartesian diagram (22.25.2), the functors u∗ and u′∗ are exact, and the base change morphism becomes a functorial isomorphism ∼

u∗ Rf∗ E −→ R(f ′ )∗ (u′ )∗ E

(22.25.3)

for every E ∈ Dqcoh (X). Applying H i we obtain an isomorphism ′ ∼ u∗ R i f∗ E ∼ = H i (u∗ Rf∗ E) −→ H i (Rf∗′ u′ ) = Ri f∗′ u′∗ E,

where the first isomorphism holds because u∗ is exact and hence commutes with H i (−). This proves Proposition 22.90 in general. Corollary 22.109. Let f : X → S be a flat qcqs morphism, let s ∈ S be a point, let Xs = f −1 (s) be the fiber in s, and let i : Xs → X be the inclusion. Then for all E in Dqcoh (X) one has a functorial isomorphism in the derived category of κ(s)-vector spaces ∼

∗ κ(s) ⊗L OS Rf∗ E −→ RΓ(Xs , Li E)

An important special case is the situation over a field k. For k-schemes X and Y and complexes of OX - and OY -modules F , G , we denote by F ⊠ G the tensor product p∗1 F ⊗OX×k Y p∗2 G on X ×k Y . Corollary 22.110. Let k be a field, and let X, Y be qcqs k-schemes. Then for all complexes F , G of quasi-coherent modules over OX , and OY resp., the K¨ unneth morphism induces an isomorphism (22.25.4)

i M

∼

H j (X, F ) ⊗k H i−j (Y, G ) → H i (X ×k Y, F ⊠ G ).

j=0

Proof. As we have discussed in Example 22.98, in this case the K¨ unneth morphism gives rise to homomorphisms as above. Since k is a field, the base change diagram is tor-independent, so by Theorem 22.99 the K¨ unneth morphism is an isomorphism, which gives us the desired conclusion. Remark 22.111. Let X and Y by qcqs schemes over a field k. Then the cup product makes M H • (X, OX ) := H i (X, OX ) i≥0

into a strictly graded commutative graded k-algebra (Example 21.131). The K¨ unneth isomorphism (22.25.4) yields an isomorphism of graded k-vector spaces ∼

κ : H • (X, OX ) ⊗k H • (Y, OY ) −→ H • (X ×k Y, OX×k Y ). It is not difficult to check (Exercise 22.45) that the map κ is an isomorphism of graded algebras if one endows the left hand side with the structure of the graded tensor product in the sense of Definition 19.3.

288

22 Cohomology of quasi-coherent modules

Exercises Exercise 22.1. Let X be a ringed space and let F be a complex of OX -modules. Show that H p (X, F ) = HomD(X) (OX [−p], F ). Deduce that there are schemes X and non-zero morphisms u between objects in D(X) such that the restriction of u to every open affine subscheme is zero. Exercise 22.2. Give a different proof of the fact that for a quasi-coherent OX -module F on an affine scheme X we have H 1 (X, F ) = 0, by writing the H 1 as Ext1 (OX , F ) and using the Yoneda description of Ext1 (see Remark F.219). Exercise 22.3. Let X be an integral affine scheme with more than one closed point, let z ∈ X be a closed point, Z = {z}, U = X \ Z. Denote by i : Z ,→ X, j : U ,→ X the inclusions. Recall that i−1 OX is a sheaf on the one-point space Z which we can identify with the stalk OX,z . Show that we have a short exact sequence 0 → j! OU → OX → i∗ i−1 OX → 0. Show that neither of the outer two terms is quasi-coherent, and that H 1 (X, j! OU ) ̸= 0. Exercise 22.4. A morphism of schemes f : X → Y is called semiseparated if the diagonal ∆f : X → X ×Y X is an affine morphism. A scheme X is called semiseparated if X → Spec Z is semiseparated. (1) Show that for a scheme X the following assertions are equivalent. (i) X is semiseparated. (ii) For all open affine subschemes U, V ⊆ X the intersection U ∩ V is affine. (iii) There exists an open affine covering (Ui )i such that Ui ∩ Uj is affine for all i, j. (2) Show that every separated morphism is semiseparated. In particular, every monomorphism is semiseparated. Show that every semiseparated morphism is quasi-separated. (3) Show that the composition of two semiseparated morphisms is again semiseparated. (4) Show that if f : X → Y is a semiseparated morphism, then for every morphism of schemes Y ′ → Y the base change X ×Y Y ′ → Y ′ is semiseparated. (5) Show that if the composition g ◦ f of scheme morphisms is semiseparated, then f is semiseparated. Remark : Totaro has shown ([To] O X 8.1) that for a smooth scheme X of finite type over a field the following assertions are equivalent. (i) X is semiseparated. (ii) Every coherent OX -module is the quotient of a finite locally free OX -module. (iii) X has an open affine covering by sets of the form { x ∈ X ; s(x) ̸= 0 } with s a section of a line bundle on X. Exercise 22.5. Let R be a ring, and let X be the R-scheme obtained from gluing two copies of A2R along A2R \ {0} via the identity. Thus X is the “affine plane with doubled origin”. Let U be the open affine covering of X given by the two copies of A2R . Show that ˇ 2 (U , OX ) → H 2 (X, OX ) is not an isomorphism. H Exercise 22.6. Let R be a ring, let A be a graded R-algebra, let f = (f1 , . . . , fr ) be a finite family of homogeneous elements of A+ , and let M be a graded A-module. Set S ˜ be the quasi-coherent OX -module attached to X := Proj A, U := i D+ (fi ), and let M M.

289 (1) Show that for all p > 0 there exists an isomorphism of graded A-modules, functorial in M , M ∼ ˜ (d)) −→ H p (U, M colim H p−r+1 (K(f n ), M ). n

d∈Z

(2) Show that there exists an exact sequence, functorial in M , M ˜ (d)) 0 −→ colim H −r (K(f n ), M ) −→ M −→ H 0 (U, M n

d∈Z

−→ colim H −r+1 (K(f n ), M ) −→ 0. n

Remark : If A is generated by finitely many elements in degree 1, then every quasi-coherent ˜ for some graded A-module M (Theorem 13.20). OX -module is of the form M Exercise 22.7. Let X be a qcqs scheme, let L be a line bundle on LX, and let F be ⊗m a quasi-coherent O -module. We set F (m) := F ⊗ L , S := X m∈Z OX (m), and L ⊗m M := ) as a homogeneous element in the m∈Z F (m). We consider L f ∈ Γ(X, L graded ring S := Γ(X, S ) = m∈Z Γ(X, OX (m)). Let Xf be the open quasi-compact subscheme of X of points x ∈ X such that f (x) ̸= 0. (1) Show that for all k ≥ 0 the structure of a graded S -module on M endows M H k (X, M ) = H k (X, F (m)) m∈Z

with the structure of a graded S-module. (2) Show that for all k ≥ 0 there exists an isomorphism of graded S-modules H k (Xf , M ) ∼ = H k (X, M )f . S (3) For i = 1, . . . , r let fi ∈ Γ(X, L ⊗mi ) and set U := i Xfi . Show that there exists an isomorphism of graded complexes M M
Cˇord ((Xfi )i , F (m)) ∼ F (m)))[1 − r] = σ ≥0 colim K(f n , Γ(U, n

m∈Z

m∈Z

(4) Suppose that X is semiseparated (Definition 22.12) and that Xfi is affine for all i. Show that for p > 0 one has an isomorphism of graded S-modules M M
H p (U, F (m)) = colim H p−r+1 K(f n , Γ(U, F (m))) . m∈Z

n

m∈Z

Exercise 22.8. Let X be a separated scheme that can be covered by n + 1 open affine schemes. Let b ∈ Z and let E be a complex in Dqcoh (X), that is acyclic in degrees > b. Show that H p (X, E) = 0 for all p > n + b. Exercise 22.9. Let R = ̸ 0 be a ring and n ≥ 1. Show that PnR cannot be covered by n open affine subschemes. Exercise 22.10. Let A be a ring and let I = (f1 , . . . , fr ) and J = (g1 , . . . , gs ) be finitely generated ideals of A with the same radical. Show that the natural morphisms of extended ˇ ordered Cech complexes CA (f1 , . . . , fr ) ←− CA (f1 , . . . , fr , g1 , . . . , gs ) −→ CA (g1 , . . . , gs ) are quasi-isomorphisms.

290

22 Cohomology of quasi-coherent modules

Exercise 22.11. Let f : X → S be a qcqs morphism of schemes. Show that the following assertions are equivalent. (i) f is affine. (ii) The functor f∗ : QCoh(X) → QCoh(S) is exact. (iii) f∗ : QCoh(X) −→ QCoh(S) commutes with arbitrary colimits. (iv) f∗ : QCoh(X) −→ QCoh(S) has a right adjoint functor. (v) One has Ri f∗ F = 0 for every quasi-coherent OX -module F and for all i > 0. (vi) One has R1 f∗ I = 0 for every quasi-coherent ideal I ⊆ OX of finite type. Exercise 22.12. Let f : X → S be a quasi-affine morphism. Show that f is affine if and only if Ri f∗ OX = 0 for all i > 0. Hint: It suffices to show that a quasi-affine scheme X is affine if H i (X, OX ) = 0 for all i > 0. Set A := Γ(X, OX ) and let j : X → Spec A be the canonical quasi-compact open immersion. Cover X by finitely many principal open subsets D(fi ) of Spec A. Use the ˇ Cech complex of the covering and the hypothesis to conclude that Γ(X, OX ) is a flat A-module and that for every morphism f : Spec B → Spec A of affine schemes one has Γ(f −1 (X), Of −1 (X) ) = Γ(X, OX ) ⊗A B. Deduce that the fi generate the unit ideal of A and conclude by Serre’s affineness criterion (Theorem 12.35). Exercise 22.13. Let f : X → S be a quasi-separated and quasi-compact morphism of schemes. Let F be a complex of quasi-coherent OX -modules such that all components F n are right acyclic for f∗ . Show that F is right acyclic for f∗ . Exercise 22.14. Let A be a noetherian ring, and let M be an injective A-module. ˜ is flasque OSpec A -module. (1) Prove that M ˜ ) and Exercise F.27. Hint: Use noetherian induction on Supp(M ˜ (2) Prove the more difficult assertion that M is an injective OSpec A -module. Remark : See [Har1] O II 7.14. Exercise 22.15. Let X be a noetherian scheme, and let F be a quasi-coherent OX module. Show that there exists a quasi-coherent OX -module G , which is injective in (OX -Mod), together with a monomorphism F ,→ G . Hint: Exercise 22.14. Exercise 22.16. Let X be a noetherian scheme. Show that the inclusion of abelian categories QCoh(X) → (OX -Mod) sends injective objects to injective objects. Hint: Exercise 22.15. Remark : See Caveat 22.77 (3). Exercise 22.17. Let A be a discrete valuation ring, π a uniformizing element, X = Spec A, n and Q let In ⊆ OX be the quasi-coherent ideal corresponding to the ideal (π ). Show that I (product in the category of O -modules, as usual) is not a quasi-coherent OX X n n module. What is the product of the In in the category of quasi-coherent OX -modules? What is its generic fiber? Remark : Cf. Exercise 7.12. Exercise 22.18. Give a different proof of Theorem 22.27 under the assumption that Y is affine and X is noetherian, by showing that (H i (X, F )∼ )i is an effaceable (hence universal) δ-functor QCoh(X) → (OY -Mod).

291 Hint: Use Exercise 22.15. (Note that for the whole proof we need to work inside QCoh(X) since the statement fails for non-quasi-coherent F even for i = 0 and f = id, so effaceability amounts to showing that every quasi-coherent OX -module F can be embedded into a quasi-coherent OX -module G with H i (X, G )∼ = 0 for all i > 0.) Exercise 22.19. Give a direct proof of Lemma 22.82 in case f is semiseparated (Exercise 22.4), i.e., prove the following: Let f : X → Y be a semiseparated morphism. The functor Rf∗ : Dqcoh (X) → Dqcoh (Y ) commutes with arbitrary direct sums. Hint: We may assume that S is affine. Do an induction over the number of affine open subschemes needed to cover X. Use the Mayer-Vietoris triangle, Proposition 21.48. Exercise 22.20. Let X be a noetherian scheme. Show that the functor DQCoh(X) → Dqcoh (X) is a triangulated equivalence with quasi-inverse given by RQX , where QX is the coherator. Hint: Exercise 22.13 Exercise 22.21. Let p be a prime number and let X be a scheme of characteristic p (i.e., pOX = 0). Let I ⊆ OX be a quasi-coherent ideal such that I n = 0 for some n ≥ 1 and let i : Y = V (I ) → X be the corresponding closed immersion (e.g., if X is noetherian and I is the nilradical, then such an n exists; in this case Y = Xred ). Let e ≥ 1 be an integer such that pe ≥ n. e (1) Show that the map OY → OX given on local sections by s 7→ sp is well defined and that it induces a group homomorphism Ni : Pic(Y ) → Pic(X) such that i∗ Ni (L ) = e L ⊗p for all L ∈ Pic(Y ). (2) Suppose that X is qcqs. Show that L ∈ Pic(Y ) is ample if and only if Ni (L ) is ample. Hint: Use arguments as in Remark 12.25 and Proposition 13.66. Exercise 22.22. Let f : X → S be a quasi-affine morphism. Show that f can be written as a composition f = g ◦ j, where g is affine and j is an open quasi-compact schematically dominant morphism. Hint: Consider for j the canonical map X → Spec f∗ OX . Exercise 22.23. Let f : X → S be a quasi-affine morphism of schemes. Show that the functor Rf∗ : Dqcoh (X) → Dqcoh (S) is conservative. Hint: Exercise 22.22 Exercise 22.24. Let R be a local noetherian ring of dimension 2, let S = Spec R, let s ∈ S be the closed point, and let j : U := S \ {s} → S be the inclusion. Set G := j! OU and F := OU . (1) Show that the stalk of R1 j∗ F ⊗ G at s in 0 and that the stalk of R1 j∗ (F ⊗ j ∗ G ) = R1 j∗ OU is H 1 (U, OU ). (2) Show that G is not quasi-coherent and that H 1 (U, OU ) ̸= 0. 2 Hint: One has H 1 (U, OU ) = H{s} (R), where H{s} denotes the local cohomology supported in {s}. Exercise 22.25. Let A be a ring and let M be an A-module. Show the following assertions. Q Q (1) M is finitely generated if and only if the map M ⊗ i Ni → i (M ⊗ Ni ) is surjective for every family (Ni )i of A-modules.

292

22 Cohomology of quasi-coherent modules

Q Q (2) M is of finite presentation if and only if the map M ⊗ i Ni → i (M ⊗ Ni ) is bijective for every family (Ni )i of A-modules. Q Remark : One can also characterize the A-modules M for which the map M ⊗ i Ni → Q i (M ⊗ Ni ) is injective for every family (Ni )i of A-modules. These are the so-called Mittag-Leffler modules, see [Sta] 059M. Exercise 22.26. Let A be ring and let E be in D− (A). Show that the following assertions are equivalent. (i) E is pseudo-coherent. Q Q L (ii) The canonical map E ⊗L A ( i Mi ) → i (E ⊗A Mi ) is an isomorphism in D(A) for every family (Mi )i of A-modules. I I (iii) For every set I, the canonical map E ⊗L A A → E is an isomorphism in D(A). Exercise 22.27. Let A be a ring. An A-module M is called coherent if M is finitely generated and every finitely generated A-submodule is of finite presentation (see also Section (7.19)). Let M1 → M2 → M 3 → M4 → M5 be an exact sequence of A-modules. Show that if M1 , M2 , M4 , and M5 are coherent, then M3 is coherent. Deduce that the full subcategory of (A-Mod) of coherent A-modules is plump, in particular it is an abelian category. Show that it is a Serre subcategory if and only if A is noetherian. Exercise 22.28. Let A be a ring. A ring A is called coherent if it is coherent as an A-module (Exercise 22.27), i.e., if every finitely generated ideal of A is an A-module of finite presentation. (1) Show that the following properties are equivalent. (i) A is coherent. (ii) Every A-module of finite presentation is coherent. (iii) For every short exact sequence 0 → M ′ → M → M ′′ → 0 of A-modules such that two of the modules M , M ′ , M ′′ are of finite presentation, the third is of finite presentation. (iv) The full subcategory of (A-Mod) of finitely presented A-modules is a plump subcategory (in particular, it is an abelian category). (v) Every finitely generated submodule of an A-module of finite presentation is itself of finite presentation. (vi) For every complex M of A-modules such that M i is of finite presentation for all i ∈ Z, H i (M ) is of finite presentation for all i ∈ Z. (vii) Arbitrary products of flat A-modules are flat. Hint: To show that (i) and (vii) are equivalent one can use Exercise 22.25. (2) Show that every noetherian ring is coherent. (3) Let A be a coherent ring and let S ⊆ A be a multiplicative subset. Show that S −1 A is a coherent. (4) Let A be a coherent ring and let B be an A-algebra that is of finite presentation as an A-module (e.g., B = A/I for I a finitely generated ideal in A). Show that B is coherent. (5) Let A → B be a faithfully flat ring homomorphism and let B be coherent. Show that A is coherent. (6) For every filtered diagram of coherent rings with flat transition maps its colimit is coherent.

293 (7) Let R be a noetherian ring. Show that any polynomial algebra over R (in possibly infinitely many variables) is coherent. Remark : An analogous assertion for coherent rings, even for one variable, does not hold (Exercise 22.32). (8) Let R be a Dedekind domain, let K be an algebraic extension of the field of fractions of R and let A the integral closure of R in K. Show that A is coherent. (9) Let k be a field, let A = k[T1 , T2 , . . . ] be the polynomial ring in countably many variables and let I be the ideal in A generated by x1 xi for i = 2, 3, . . . . Show that A is coherent but A/I is not coherent. Exercise 22.29. Let A be a ring and let M be an A-module. Consider the following finiteness properties for M . (i) M is coherent (Exercise 22.27). (ii) M is pseudo-coherent. (iii) M is of finite presentation. (iv) M is finitely generated. Show the following assertions. (1) In general, one has the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv). (2) If A is noetherian, then all properties are equivalent. (3) If A is coherent (Exercise 22.28), then (i) ⇔ (ii) ⇔ (iii). (4) For each of the implications (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) give an example of a ring A and an ideal I ⊆ A such that this implication does not hold for the A-module A/I. Exercise 22.30. A scheme X is called coherent 3 if its structure sheaf OX is a coherent OX -module (Definition 7.45). (1) Show that an affine scheme X = Spec A is coherent if and only if A is a coherent ring ˜ (Exercise 22.28). More generally, show that for every ring A the functor M 7→ M yields an equivalence from the category of coherent A-modules to the category of coherent OX -modules. (2) Show that any locally noetherian scheme is coherent. (3) Let X be a coherent scheme. Show that an OX -module is coherent if and only if it is of finite presentation. (4) Show that a scheme X is coherent if and only if the category of OX -modules of finite presentation, considered as a full subcategory of the category of all OX -modules, is abelian. Show that in this case it is a plump subcategory. (5) Let f : X → Y be a faithfully flat quasi-compact morphism of schemes and suppose that X is coherent. Show that Y is coherent. Hint: Exercise 22.28 − Exercise 22.31. Let X be a coherent scheme (Exercise 22.30). Let Dcoh (X) be the full subcategory of D(X) consisting of bounded above complexes E such that H i (E) is a coherent OX -module (equivalently, an OX -module of finite presentation) for all i. (1) Show that an OX -module F is of finite presentation if and only if it is pseudo-coherent (considered as complex concentrated in degree 0). (2) Suppose that X is quasi-compact. Show that a complex E in D(X) is pseudo-coherent − if and only if E ∈ Dcoh (X). 3

Some authors call a scheme coherent if it is qcqs. This yields a totally different notion. For instance, not every affine scheme, which is always qcqs, is coherent in the sense defined here.

294

22 Cohomology of quasi-coherent modules

Exercise 22.32. Show that the following assertions for a ring A are equivalent. (i) The set of integers n ≥ 0 such that A[T1 , . . . , Tn ] is coherent (Exercise 22.28) is unbounded. (ii) Every A-algebra of finite presentation is coherent. (iii) Every polynomial algebra over A (in possibly infinitely many variables) is coherent. If A satisfies these equivalent conditions, A is said to be universally coherent . Moreover, a scheme X is called universally coherent if every X-scheme locally of finite presentation is coherent (Exercise 22.30). Show the following assertions. (1) Let A be a ring. Show that A is universally coherent if and only if Spec A is a universally coherent scheme. (2) Every absolutely flat ring A (i.e., every A-module is flat) is universally coherent. Deduce that reduced rings of dimension 0 (e.g., arbitrary products of fields) are coherent. (3) Let A = QN [[T1 , T2 ]]. Show that A is coherent but not universally coherent. Exercise 22.33. Let X be a coherent scheme (Exercise 22.30), let F ∈ D(X) be pseudocoherent and let G ∈ D(X) be a locally bounded below complex such that H p (G ) is coherent for all p ∈ Z. Show that Ext nOX (F , G ) is coherent for all n ∈ Z. Exercise 22.34. Let X be a quasi-compact semiseparated (Definition 22.12) scheme. Show that for every complex F in Dqcoh (X) there exists a complex I that is K-injective in Dqcoh (X) and a quasi-isomorphism F → I . Exercise 22.35. Let X be a quasi-compact semiseparated (Exercise 22.4) scheme. Show that for every complex F in Dqcoh (X) there exists a K-flat complex P of quasi-coherent modules and a quasi-isomorphism P → F . Exercise 22.36. (1) Let R be a principal ideal domain. Show that every finitely generated R-module, considered as a complex in degree 0, is perfect. (2) Give an example of a noetherian ring R and an R-module M such that M , considered as a complex concentrated in degree 0, is pseudo-coherent (in particular M is a finitely generated R-module) but not perfect. (3) Give an example of a ring R and of a perfect complex E in Db (R) such that for some i the cohomology H i (E) (considered as a complex concentrated in degree 0) is not perfect. Z/p2 Z

Hint: For (b), one could compute Tori

(Z/pZ, Z/pZ) for all i.

Remark : Part (a) can be vastly generalized, see Proposition 23.55. Exercise 22.37. Let X = Spec A be an affine scheme and let M and N be complexes of A-modules. Suppose that M is perfect. Show that there is a functorial isomorphism in D(X) ^ ˜ ˜ R Hom A (M, N ) = R Hom O (M , N ). X

Exercise 22.38. Let X be a scheme, let F and G be complexes of OX -modules. Suppose that F is perfect. (1) Let f : X ′ → X be a morphism of schemes. Show that one has a natural isomorphism ∼

Lf ∗ R Hom OX (F , G ) → R Hom OX ′ (Lf ∗ F , Lf ∗ G )

295 (2) Show that for all x ∈ X one has a natural isomorphism ∼

R Hom OX (F , G )x → R HomOX,x (Fx , Gx ). Deduce that one has for all p ∈ Z a functorial isomorphism ∼

Ext pOX (F , G )x → ExtpOX,x (Fx , Gx ). Exercise 22.39. Let f : X → Y be a qcqs morphism of schemes, let F be in D+ (X) and let G be in Dqcoh (Y ) be locally of finite tor-dimension. Show that the projection formula map L ∗ Rf∗ F ⊗L OY G −→ Rf∗ (F ⊗OX Lf G ) is an isomorphism. Hint: Assume that Y is affine, replace G by a bounded complex of flat quasi-coherent modules using Exercise 22.35 and reduce to the case that G is a single flat quasi-coherent module. Write G as colimit of finite free modules (Theorem G.3). Exercise 22.40. Let X → S, Y → S be morphisms of schemes, let F (resp. G ) be a complex of quasi-coherent OX -modules (resp. OY -modules), and let S ′ → S be a flat morphism. Let v : X ×S S ′ → X and w : Y ×S S ′ → Y be the projections. Show that if F and G are tor-independent over S and S ′ → S is flat, then v ∗ F and w∗ G are tor-independent over S ′ . Exercise 22.41. Let A be a noetherian local ring and let k be its residue field. Show that the diagram id / Spec k Spec k i

id

 Spec k

i

 / Spec A

is cartesian, where i denotes the canonical map. Show that this diagram is tor-independent if and only if A = k. Hint: Exercise 21.19. Exercise 22.42. Show by an example that the base change homomorphism in Definition 21.129 is not an isomorphism, in general. Hint: Exercise 22.41 Exercise 22.43. Let u : S ′ → S be a morphism of affine schemes and let f : X → S be a quasi-affine morphism and form the cartesian diagram X′

u′

/X

u

 / S.

f′

 S′

f

Let F be a complex of quasi-coherent OX -modules and let G be a complex of quasicoherent OS ′ -modules. Show that F and G are tor-independent over S if and only if the base change morphism ∗ ′ ′∗ ′∗ G ⊗L OS ′ Lu Rf∗ F −→ Rf∗ (f G ⊗OX ′ u F )

is an isomorphism. Hint: Exercise 22.23.

296

22 Cohomology of quasi-coherent modules

Exercise 22.44. Show the following variant of the projection formula. Let f : X → S be a qcqs morphism of schemes, let F (resp. G ) be a complex of quasi-coherent OX -modules (resp. OS -modules) such that F and G are tor-independent (e.g., if F or G is bounded above with components that are flat over S). Show that there is a functorial isomorphism ∼

∗ G ⊗L OS Rf∗ F −→ Rf∗ (f G ⊗OX F ).

Exercise 22.45. Let f : X → S be a morphism of ringed spaces, let g : X ×S X → S be the structure morphism, and let ∆ : X → X ×S X be the diagonal. Consider for F and G in D(X) the K¨ unneth map (22.23.2) η : Rf∗ F ⊗L Rf∗ G → Rg∗ (F ⊠L OX×

SX

G)

for S ′ = X and u = f . Compose it with Rg∗ applied to the adjunction unit id → R∆∗ ◦L∆∗ and show that the resulting morphism coincides with the relative cup product (21.29.1). Exercise 22.46. Let f : X → Y be a morphism of schemes and let t ≥ 0 be an integer. Show that the following assertions are equivalent. (i) The morphism f has tor-dimension ≤ t. (ii) For every quasi-coherent OY -module F one has Lt+1 f ∗ F = 0. If Y is qcqs, show that these assertions are equivalent to (iii) For every quasi-coherent ideal J ⊆ OY of finite type one has Lt+1 f ∗ (OY /J ) = 0. Hint: Exercise 21.32 Exercise 22.47. Let A be a ring and let I ⊆ A be a finitely generated ideal. An A-module M is called an I-power torsion module if for all m ∈ M there exists an n ∈ N such that I n m = 0. Denote by I ∞ -Tors the full subcategory of I-power torsion modules of the category of A-modules. (1) Show that an A-module is I-power torsion if and only if its support is contained in V (I). Deduce that the subcategory I ∞ -Tors depends only on V (I). (2) Show that I ∞ -Tors is a Serre subcategory of the category of A-modules. (3) Show that for an object K of D(A) the following assertions are equivalent. (i) K ⊗L A A/I = 0. (ii) K ⊗L A M = 0 for every I-power torsion A-module M . b i (iii) K ⊗L A N = 0 for every object N in D (A) such that H (N ) is I-power torsion for all i ∈ Z.

23

Cohomology of projective and proper schemes

Content – Cohomology of projective schemes – Coherence of higher direct images – Numerical intersection theory, Euler characteristic, and Hilbert polynomial – The Grothendieck-Riemann-Roch theorem – Cohomology and base change – Hilbert polynomials and flattening stratification This chapter studies the question when cohomology or, more generally, derived direct images satisfy certain finiteness conditions (in the first two parts) and several applications of it. The first central result is the coherence of higher direct images of coherent sheaves under proper morphisms f : X → Y of locally noetherian schemes (Theorem 23.17) and its derived version that Rf∗ preserves pseudo-coherent complexes. From this we deduce b b that it maps Dcoh (X) to Dcoh (Y ) if Y is noetherian (Remark 23.24) and Dcoh (X) to Dcoh (Y ) (Corollary 23.25). The strategy for the proof of the coherence of higher direct images is to reduce the question to a similar and more precise version for projective morphisms (Theorem 23.1) using Chow’s lemma. The result for projective morphisms will follow easily from the calculation of the higher direct images of Serre’s twisting line bundles along the projection to the base (Theorem 22.22). We will also prove Serre’s criterion for a line bundle to be ample (Theorem 23.6). In the remaining parts of this chapter, we use these finiteness results to discuss further topics. We introduce Grothendieck groups and K-groups of schemes and develop numerical intersection theory for proper schemes over a field. In the following part, we will discuss K-theory as a special case of a cohomology theory. Relating K-theory to other cohomology theories will result in the Grothendieck-Riemann-Roch theorem, a vast generalization of the Theorem of Riemann-Roch for curves figuring in Chapters 15 and 26. These two parts have more exemplary character and we do not strive for greatest possible generality. In the part on numerical intersection theory we give full proofs hoping to convince the reader that the language of derived categories simplifies the exposition considerably. In the part about the Grothendieck-Riemann-Roch theorem we will only explain (Section (23.23)) how the theorem follows formally from the fact that K-theory is the initial multiplicative cohomology theory (Theorem 23.108) but we will give only a reference for the proof of this universal property of K-theory. The last two parts of the chapter then develop the theory of cohomology and base change and apply this to the behavior of intersection numbers and Hilbert polynomials in flat families. In the last section we prove the existence of a flattening stratification, indexed by the Hilbert polynomial, for arbitrary projective morphisms of finite presentation.

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_8

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23 Cohomology of projective and proper schemes

Cohomology of projective schemes In this part we start with showing that for projective morphisms of noetherian schemes higher direct images of coherent modules are again coherent. Moreover, we show that after tensoring with a sufficiently high power of an ample line bundle, projective morphisms “behave as affine morphisms”, at least for noetherian schemes and for coherent modules. Examples are the vanishing of higher direct images (Theorem 23.1) and the exactness if the direct image functor (Corollary 23.3). In fact, these kind of behavior characterizes ample line bundles (Theorem 23.6). In the final section of this part we apply these results to classify coherent modules on projective spectra of graded algebras. (23.1) Coherence of direct images under projective morphisms and Serre’s vanishing criterion. The fundamental theorem on cohomology of coherent modules for projective schemes is the following result. Theorem 23.1. Let f : X → S be a proper morphism between locally noetherian schemes and let L be a line bundle on X that is ample for f . Let F be a coherent OX -module and set F (n) = F ⊗ L ⊗n for n ∈ Z. (1) For all i ≥ 0 the higher direct image Ri f∗ F is a coherent OS -module. (2) If S is noetherian, there exists n0 ≥ 0 such that Ri f∗ F (n) = 0

for all n ≥ n0 and i > 0.

(3) If S is noetherian, there exists n1 ≥ 0 such that the canonical homomorphism f ∗ (f∗ F (n)) → F (n) is surjective for all n ≥ n1 . Proof. All assertions can be checked locally on S (for (2) and (3) we use that S is quasi-compact). Hence we may assume S = Spec A for a noetherian ring. Then X → S is projective (Corollary 13.72) and using Theorem 22.27 it suffices to show the following corollary. Corollary 23.2. Let X be a projective scheme over a noetherian ring A, and let L be an ample invertible OX -module. Let F be a coherent OX -module and set F (n) := F ⊗ L ⊗n for n ∈ Z. (1) For all i ≥ 0 the cohomology modules H i (X, F ) are finitely generated A-modules. (2) There exists n0 ∈ Z such that H i (X, F ⊗ L ⊗n ) = 0

for all n ≥ n0 and i > 0.

(3) There exists n1 ∈ Z such that F (n) is generated by its global sections for all n ≥ n1 . Proof. To see that (3) holds, recall (Summary 13.71) that we have X ∼ = Proj S where S is a graded A-algebra generated by finitely many elements in S1 and we conclude by Proposition 13.22. We now prove (1) and (2) simultaneously. It is enough to show the statement after replacing L with some positive tensor power L ⊗d (because as the bound n0 for L and F we can then take d times the maximum of the bounds for L ⊗d and F ⊗ L ⊗j , j = 0, . . . , d − 1). Therefore we may assume that L is very ample (Theorem 13.59). By Proposition 13.56, there exists a closed immersion i : X → PN A for some N , with ∼ (1) L . Then i ∗ O PN = A

299 (1)⊗n = i∗ (F )(n) (1))⊗n ) = i∗ (F ) ⊗ OPN i∗ (F (n)) = i∗ (F ⊗ (i∗ OPN A A by the projection formula. The isomorphism (21.8.4) (or Corollary 22.6) shows that N H i (X, F (n)) = H i (PN A , i∗ (F )(n)), whence it is enough to show (1) and (2) for X = PA and L = OX (1). We now proceed by descending induction on i. In degrees i > N , the cohomology groups of all quasi-coherent sheaves vanish (Corollary 22.10). Assume we have proved the claims for degrees > i and for all F . By Theorem 22.22 we know that (1) and (2) hold if F is a direct sum of OX -modules of the form OX (di ). As we already proved (3), we know that there exists a surjection r OX → F (n1 ) for some r ≥ 0 and hence a surjection φ : G := OX (−n1 )r → F . Let E = Ker(φ) which is coherent since X is noetherian. As tensoring with a line bundle is exact, we obtain for all n ∈ Z an exact sequence 0 → E (n) → G (n) → F (n) → 0. We know that (1) and (2) hold for G (n) in all cohomology degrees and hold for E (n) in cohomology degrees > i. From the long exact cohomology sequence we obtain an exact sequence of A-modules i N i+1 H i (PN (PN A , G (n)) −→ H (PA , F (n)) −→ H A , E (n))

The outer terms are finitely generated A-modules, hence H i (PN A , F (n)) is finitely generated for all n ∈ Z since A is noetherian. In particular, H i (PN , F ) is finitely generated. A i+1 N Moreover, H i (PN , G (n)) = 0 and H (P , E (n)) = 0 for large n. Hence we obtain A H i (PN A , F (n)) = 0 for large n. Corollary 23.3. Let S be a noetherian scheme. With the notation of Theorem 23.1 let F → G → H be an exact sequence of coherent OX -modules. Then there exists an integer N such that for all n ≥ N the sequence f∗ F (n) → f∗ G (n) → f∗ H (n) is exact. Proof. This is a routine argument by splitting the exact sequence into short exact sequences as follows. Let K , I , C be the kernel, the image, the cokernel of F → G , respectively. Then I is the kernel and C is the image of G → H . Let D be the cokernel of G → H . These are all coherent OX -modules since X is noetherian. As F (n) is an exact functor in F for all n it suffices to show that for large n the sequences 0 → f∗ (K (n)) → f∗ (F (n)) → f∗ (I (n)) → 0 0 → f∗ (I (n)) → f∗ (G (n)) → f∗ (C (n)) → 0 0 → f∗ (C (n)) → f∗ (H (n)) → f∗ (D(n)) → 0 are exact. In other words, we may assume that 0 → F → G → H → 0 is exact. The cohomology sequence is then of the form 0 → f∗ (F (n)) → f∗ (G (n)) → f∗ (H (n)) → R1 f∗ F (n) → · · · , and R1 f∗ F (n) = 0 for large n by Theorem 23.1. (23.2) Serre’s ampleness criterion. We can now extend the results of Theorem 12.35 and Proposition 13.47, which relate ampleness of a line bundle L to the vanishing of cohomology groups. We start with a purely topological lemma (Exercise 3.13) that will also be useful at other occasions.

300

23 Cohomology of projective and proper schemes

Lemma 23.4. Let S be a quasi-compact Kolmogorov space (e.g., the underlying topological space of a quasi-compact scheme). Then for every point s ∈ S there exists a specialization s0 of s that is a closed point in S. In particular, S is the only open subset of S containing all closed points. Proof. Consider S := { {x} ; x ∈ S is a specialization of s}. It is non-empty. TLet T ⊆ S ̸ ∅. be a subset that is totally ordered by inclusion. As S is quasi-compact, Z0 := Z∈T Z = Choose x0 ∈ Z0 . Then Z0 = {x0 } and we see that Z0 ∈ S is a lower bound for T . By Zorn’s lemma, S has a minimal element Z. As S is Kolmogorov, Z = {x} for some specialization x of s and x ∈ S is a closed point. Now we first state and prove the local case of Serre’s ampleness criterion. Lemma 23.5. Let A be a noetherian ring, and let X be a scheme which is proper over Spec A. For an invertible OX -module L , the following are equivalent: (i) The sheaf L is ample. (ii) For every coherent OX -module F , there exists n0 ≥ 0 such that H i (X, F ⊗ L ⊗n ) = 0 for all n ≥ n0 , i > 0. (iii) For every coherent ideal sheaf I ⊆ OX , there exists n0 ≥ 0 such that H 1 (X, I ⊗ L ⊗n ) = 0 for all n ≥ n0 . Proof. We have shown that (i) implies (ii) in Corollary 23.2 above, and (ii) trivially implies (iii). Now let us show that (iii) implies (i). By Proposition 13.47 it is enough to show that for every point x ∈ X and every affine open neighborhood U ⊆ X, there exist N ≥ 0 and h ∈ Γ(X, L ⊗N ) with x ∈ Xh ⊆ U (note that then Xh is necessarily affine by Example 12.4 (4)). By Lemma 23.4 we may assume that x is a closed point. So consider x and U as above. Let Z = X \ U , viewed as a reduced closed subscheme of X. Similarly, we view Z ∪ {x} as a reduced closed subscheme of X. Let J be the quasi-coherent ideal sheaf corresponding to Z, and let I be the quasi-coherent ideal sheaf given by Z ∪ {x}. As X is noetherian, I and J are coherent. We obtain a short exact sequence 0 → I → J → κ(x) → 0. By assumption, there exists n > 0 with H 1 (X, I ⊗ L ⊗n ) = 0, so from the long exact cohomology sequence for the above short exact sequence twisted by L ⊗n , we get that the map H 0 (X, J ⊗ L ⊗n ) → κ(x) is surjective. Taking any global section of J ⊗ L ⊗n which does not map to 0, we obtain a section h of L ⊗n which vanishes on X \ U , but does not vanish at x. Theorem 23.6. Let f : X → Y be a proper morphism between noetherian schemes, and let L be an invertible OX -module. The following are equivalent: (i) The sheaf L is relatively ample for f (see Definition 13.60). (ii) For every coherent OX -module F , there exists n0 ≥ 0 such that Ri f∗ (F ⊗ L ⊗n ) = 0 for all n ≥ n0 , i > 0. (iii) For every coherent ideal sheaf I ⊆ OX , there exists n0 ≥ 0 such that R1 f∗ (I ⊗ L ⊗n ) = 0 for all n ≥ n0 . Proof. Working locally on Y and using Theorem 22.27, we reduce to Lemma 23.5.

301 We want to use the cohomological characterization of ampleness to prove a criterion for ampleness of a line bundle, provided that the pullback under a finite surjective morphism is ample (see Proposition 23.8 for the precise statement). The key ingredient that is still missing is the following lemma. Lemma 23.7. Let X be a quasi-compact scheme, and let i : X ′ → X be a closed immersion that is given by a nilpotent quasi-coherent ideal I . Let L be an invertible OX -module such that i∗ L is ample. Then L is ample. If X is noetherian, then Xred is a closed subscheme in X defined by a nilpotent ideal. Proof. By induction, we may assume that I 2 = 0. We will show that for f running through all sections of tensor powers L ⊗k of L , those open subsets Xf of X which are affine form a basis of the topology of X. By Proposition 13.47 this is equivalent to L being ample. By assumption the analogous statement is true when f runs through sections of (L|X ′ )⊗k , n ≥ 1; note that X and X ′ have the same topological space. Recall that the assumption implies that X ′ is separated (Proposition 13.48); it follows from Proposition 9.13 (4) that X is separated, too. Also recall that an open subset U of X ′ = X defines an affine open subscheme of X ′ if and only if it defines an affine open subscheme of X (Lemma 12.38). Therefore we have reduced to proving the following Claim. Let n ≥ 1, and let g ∈ Γ(X ′ , (L|X ′ )⊗n ) such that (X ′ )g is affine. Then there exists m ≥ 1 such that g ⊗(m+1) is the image under pullback of a section of L ⊗(m+1)n . Proof of claim. Consider the exact sequence 0 → I ⊗ L ⊗n → L ⊗n → OX ′ ⊗OX L ⊗n → 0 of sheaves on X = X ′ , which gives us an exact sequence / Γ(X, (L|X ′ )⊗n )

Γ(X, L ⊗n )

∂

/ H 1 (X, I ⊗ L ⊗n ).

Since I 2 = 0, we can consider I as an OX ′ -module. Tensoring by sections of (L|X ′ )⊗k induces maps on cohomology which we will again denote as tensor products. Let us first show that g ⊗m ⊗ ∂(g) = 0 in H 1 (X, I ⊗ L ⊗(m+1)n ) for m sufficiently large. Indeed, writing g ′ = g|(X ′ )g for the restriction of g to the affine scheme (X ′ )g , we have ∂(g)|(X ′ )g = ∂(g ′ ) = 0 since H 1 ((X ′ )g , I ⊗ L ⊗n ) = 0 by Theorem 22.2 and since I is quasi-coherent. Let us spell out explicitly what that means, viewing the H 1 cohomology ˇ groups as Cech cohomology groups for a finite affine open cover (Ui )i of X. Lifting each g|Ui to an element gi ∈ Γ(Ui , L ⊗n ), ∂(g) is described by the cocycle (gj |Uij − gi |Uij )i,j (these differences lying in Γ(Uij , I ⊗ L ⊗n )). Here we use the notation Uij = Ui ∩ Uj . By separatedness, all Ui ∩ (X ′ )g are affine, too. Thus ∂(g ′ ) = 0 means that there exist hi ∈ Γ(Ui′ , I ⊗ L ⊗n ) with gj |U ′ − gi |Uij′ = hj |U ′ − hi|Uij′ ij

ij

Ui′

′

′ for all i, j, where we write = Ui ∩ (X )g , Uij = Uij ∩ (X ′ )g . Now we can apply m Theorem 7.22 which says that g ⊗ hi lifts to Γ(X, I ⊗ L ⊗(m+1)n ). Choosing an m which works for all i simultaneously, we find that g ⊗m ⊗ ∂(g) = 0, as desired. Next, we use that the boundary maps ∂ = ∂k : Γ(X, (L|X ′ )⊗k ) → H 1 (X, I ⊗ L ⊗k ) satisfy a Leibniz rule

302

23 Cohomology of projective and proper schemes ∂(s ⊗ t) = ∂(s) ⊗ t + s ⊗ ∂(t) ∈ H 1 (X, I ⊗ L ⊗(k+ℓ) )

for s ∈ Γ(X, (L|X ′ )⊗k ), t ∈ Γ(X, (L|X ′ )⊗ℓ ). This can again be checked by an explicit ˇ cohomology groups. With notation similar to the above, the computation with Cech crucial point is the identity (si ⊗ ti ) − (sj ⊗ tj ) = (si − sj ) ⊗ ti + sj ⊗ (ti − tj ) on Uij where we have omitted the restriction to Uij from the notation everywhere. Using this, we compute ∂(g ⊗(m+1) ) = (m + 1)g ⊗m ⊗ ∂(g) = 0 which shows that g ⊗(m+1) is in the image of the pullback map Γ(X, L ⊗(m+1)n ) → Γ(X, (L|X ′ )⊗(m+1)n ). Proposition 23.8. Let f : X → Y be a proper morphism between noetherian schemes, and let g : X ′ → X be a surjective finite morphism. Let L be an invertible OX -module. The following are equivalent: (i) The sheaf L is relatively ample for f . (ii) The sheaf g ∗ L is relatively ample for f ◦ g. We also proved that if g is finite locally free and surjective, then L is ample if and only if g ∗ L is ample (Proposition 13.66). The proof here is entirely different and uses that X and X ′ are proper over some noetherian base scheme Y . Proof. It is easy to see that (i) implies (ii), and for this implication it suffices that g is quasi-affine; see Proposition 13.83. Now let us assume that g ∗ L is relatively ample for f ◦ g. We know then that R1 f∗ (g∗ (H ) ⊗ L ⊗n ) = R1 f∗ (g∗ (H ⊗ g ∗ L ⊗n )) = R1 (f ◦ g)∗ (H ⊗ g ∗ L ⊗n ) = 0 for every coherent OX ′ -module H and for sufficiently large n (depending on H ). Here we use the projection formula Proposition 22.81 and Theorem 23.6. To show that L is f -ample, we may work locally on Y and hence assume that Y is affine. Then the notion of relative ampleness coincides with the notion of ampleness. ′ → Xred is We can furthermore assume that X is reduced: In fact, the morphism Xred ′ still finite and surjective, the pullback of g ∗ L to Xred is ample by the easy implication of the proposition, and if we know that the pullback of L to Xred is ample, then L is ample ′ by the previous Lemma 23.7. Thus we may replace X ′ by Xred , X by Xred and g by gred . We now show that L|Z is ample by noetherian induction on the closed subscheme Z ⊆ X. This means that in order to show that L is ample on X, we may assume that for every proper closed subscheme Z ⊊ X, the restriction L|Z is ample. If X is not irreducible, let i : Z ,→ X be the inclusion of an irreducible component of X. Let F be a coherent OX -module and consider the natural homomorphism ρ : F → i∗ i∗ F . Then we have a short exact sequence 0 → Ker(ρ) → F → Im(ρ) → 0.

303 But i∗ i∗ F , and a fortiori Im(ρ) has support in Z. On the other hand, the generic point of Z is not contained in the support of Ker(ρ), since i is an isomorphism on a non-empty open subscheme of Z (here we use that X and Z are reduced). Hence both Ker(ρ) and Im(ρ) have support properly contained in X, which means that R1 f∗ (Ker(ρ) ⊗ L ⊗n ) = 0 (and similarly for Im(ρ)), for sufficiently large n by our noetherian induction hypothesis. The long exact cohomology sequence then implies that the same holds for F . It remains to consider the case where X is irreducible (and hence integral). By replacing, if necessary, X ′ by one of its irreducible components which surject onto X, we may assume that X ′ is also integral. We now apply the reasoning used in step (iii) of the proof of Theorem 12.39: Using Proposition 7.27, we see that OX ′ |V is free over OV for a suitable non-empty open V ⊆ X. This allows us to define an OX -module homomorphism n u : OX → g∗ OX ′ which induces an isomorphism after restriction to a suitable open W ⊆ X. Now let I ⊆ OX be a coherent sheaf of ideals. Composition with u yields a homomorphism n v : G := Hom OX (g∗ OX ′ , I ) → Hom OX (OX ,I) = In of OX -modules whose restriction to W is an isomorphism. Since X is integral, all restriction maps for I are injective, and hence v is injective. Furthermore, the cokernel of v is supported on a proper closed subscheme of X, so by induction hypothesis we have R1 f∗ (Coker(v)⊗L ⊗n ) = 0 for all sufficiently large n. In view of the long exact cohomology sequence, it is then enough to show that R1 f∗ (G ⊗ L ⊗n ) = 0 for n sufficiently large. But G is an g∗ OX ′ -module, and hence of the form g∗ H for a coherent OX ′ -module H (use Proposition 12.5 and note that H is coherent since this is true for G ). As pointed out in the beginning of the proof, this implies R1 f∗ (G ⊗ L ⊗n ) = 0 for n sufficiently large, as desired. Corollary 23.9. Let f : X → Y be a proper morphism between noetherian schemes, and let L be an invertible OX -module. The following are equivalent: (i) The sheaf L is relatively ample for f . (ii) The restriction of L to each irreducible component Z of X (considered as a reduced closed subscheme) is relatively ample for f|Z . Proof. If Z1 , . . . , Zr are the irreducible components of X with the reduced ` scheme structure, then we can apply Proposition 23.8 to the surjective finite morphism i Zi → X. (23.3) Coherent modules on projective spectra revisited. L In this section we fix the following notation. Let Y be a scheme, let S = d≥0 Sd be a graded quasi-coherent OY -algebra. Set X := Proj(S ) and let π : X → Y be the structure morphism, which is separated. For every graded quasi-coherent S -module M let M˜ be the associated quasi-coherent OX -module (Sections (13.4) and (13.7)). Lemma 23.10. The covariant functor M 7→ M˜ from the category of graded quasicoherent S -modules to the category of quasi-coherent OX -modules is exact and commutes with colimits.

304

23 Cohomology of projective and proper schemes

Proof. We may assume that Y = Spec A is affine. Let S be the graded A-algebra corresponding to S . The assertions can also be checked locally on X. As X can be covered by open subschemes of the form D+ (f ) for f ∈ S homogeneous of degree > 0, it ˜ |D (f ) = (M(f ) )∼ from the category of graded suffices to show that the functor M 7→ M + A-modules to the category of quasi-coherent OD+ (f ) -modules has the asserted properties. But this functor is the composition of the functors M 7→ Mf , of passing to degree 0, and of (−)∼ and each of these functors is exact and commutes with colimits. From now on we also assume that S is generated by S1 and that S1 is an OY -module of finite type. Then π : X → Y is projective (Summary 13.71). One has Serre’s twisting line bundles OX (n) for all n ∈ Z. If F is a quasi-coherent OX -module we set M Γ∗ (F ) := π∗ F (n), n∈Z

where F (n) = F ⊗OX OX (n). Then Γ∗ (F ) is a graded quasi-coherent S -module and by Theorem 13.29 one has a functorial isomorphism of quasi-coherent OX -modules ∼

β : Γ∗ (F )∼ → F . Recall (cf. Exercise 13.2) also the following definitions. Definition 23.11. Let M and N be graded quasi-coherent S -modules. (1) The graded module M is said L to satisfy (TF) (resp. (TN)) if there exists an integer N such that the S -module n≥N Mn is of finite type (resp. is zero). (2) A homomorphism u : M → N of graded S -modules of degree 0 is called (TN)injective (resp. (TN)-surjective) if Ker(u) (resp. Coker(u)) satisfies (TN). One calls u a (TN)-isomorphism if it is (TN)-injective and (TN)-surjective. Proposition 23.12. Let M and N be graded quasi-coherent S -modules. (1) If M satisfies (TN), then M˜ = 0. Conversely, if M satisfies (TF) and M˜ = 0, then M satisfies (TN). (2) If M satisfies (TF), then M˜ is an OX -module of finite type. (3) If a homomorphism u : M → N is (TN)-injective (resp. (TN)-surjective, resp. a (TN)-isomorphism), then u ˜ : M˜ → N˜ is injective (resp. surjective, resp. an isomorphism). L In the proof we use only the assumption that d>0 Sd is an ideal of S of finite type (cf. Lemma 13.9). Proof. All assertions are local on Y and hence we can assume that Y = Spec A. Then S corresponds to a graded A-algebra S such that S1 generates S and is an A-module of finite type. Let M and N be the graded S-modules corresponding to M and N . If M satisfies (TN), then M(f ) = 0 for every homogeneous element of S of degree > 0. ˜ = 0. As M 7→ M ˜ is exact (Lemma 23.10), (3) follows. Hence M ˜ is of finite type, we Let us show (2). Suppose that M satisfies (TF). To see that M L choose an integer N such that M ′ := n≥N Mn is a finitely generated S-module. Then the inclusion M ′ → M is a (TN)-isomorphism and hence we may by (3) assume that ˜ is of finite type it suffices to see that for f ∈ S M is finitely generated. To see that M homogeneous of degree d > 0 the module M(f ) = M (d) /(f − 1)M (d) is a finitely generated module over S(f ) = S (d) /(f − 1)S (d) . This follows from Lemma 13.10 (2), so we get (2).

305 ˜ = 0. As in the proof of (2) we may Now suppose that M satisfies (TF) and that M assume that M as an S-module is generated by finitely many homogeneous L elements x1 , . . . , xt . Let f1 , . . . , fr ∈ S be homogeneous elements that generate the ideal d>0 Sd . By assumption one has M(fi ) = 0 for all i. Hence we can find an integer n such that fin xj = 0 for all i and j. This implies the existence of an integer m such that Sk xj = 0 for all k > m and all j. If d is the maximal degree of the xj , then we conclude that Mk = 0 for all k > d + m. Proposition 23.13. Let Y be a noetherian scheme and let M be a graded quasi-coherent S -module satisfying (TF). Then the canonical functorial homomorphism (13.5.4) α : M −→ Γ∗ (M˜) is a (TN)-isomorphism. Proof. As π : X → Y is projective, X is noetherian as well. We have to show that Ker(α) and Coker(α) is zero in large degrees. As Y is quasi-compact, we may assume that Y = Spec A is affine for a noetherian ring A. Then S corresponds to a graded A-algebra S and M to a graded S-module M . By our general assumption, S1 is a finitely generated A-module that generates S. As in the proof of Proposition L 23.12 we may assume that M is a finitely generated S-module by replacing M with n≥N Mn for some large N . Note ˜ is a coherent OX -module by Proposition 23.12 (2). that M We claim that it suffices to show the proposition for M = S. Indeed, as M is finitely generated and A is noetherian, we find an exact sequence L′ → L → M → 0, where L and L′ are direct sums of graded S-modules of the form S(m) for some m ∈ Z. The ˜′ → L ˜→M ˜ → 0 is exact by Lemma 23.10. If the result holds for S, it also sequence L holds for S(m) and hence for L and L′ . For each n ∈ Z we have a commutative diagram of A-modules / Ln

L′n αn

 ˜ ′ (n)) Γ(X, L

αn

 ˜ / Γ(X, L(n))

/ Mn

/0

αn

 ˜ (n)) / Γ(X, M

/ 0.

The upper row is exact by definition and the lower row is exact for large n by Corollary 23.3. Hence our claim follows by the five lemma. Note that α is an isomorphism for S = A[T0 , . . . , Tr ] by Example 13.16. This shows the proposition for X = PrA . In general, S is a quotient of S ′ := A[T1 , . . . , Tr ]. Let i : X → PrA be the corresponding ˜ ˜ (n), where M denotes S considered as a graded closed immersion. Then i∗ (S(n)) =M S ′ -module. As we have already seen the proposition for S ′ , we obtain that ˜ (n)) = Γ(X, S(n)) ˜ Sn = Mn −→ Γ(PrA , M is bijective for large n. Lemma 23.14. Let Y be qcqs and let F be a quasi-coherent OX -module of finite type. Then there exists a graded quasi-coherent S -module M of finite type such that M˜ ∼ = F.

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23 Cohomology of projective and proper schemes

Proof. Set N := Γ∗ (F ). Then Nn is a quasi-coherent OY -module for every n ∈ Z. As Y is qcqs, we can write Nn as the filtered union of its finitely generated quasi-coherent submodules Pn,λn (Corollary 10.50). Then Qλn := S · Pn,λn ⊆ N is a graded quasicoherent S -module of finite type for all n and all λn . If Φ denotes the set of all finite sums of submodules of N of the form Qλn , then N is the filtered union of the S -submodule in Φ which are all of finite type. By Lemma 23.10 we see that F ∼ = N˜ is then the filtered union of the submodules N˜µ ˜ for Nµ ∈ Φ. Hence F = Nµ for some Nµ ∈ Φ by Lemma 10.47. Corollary 23.15. Let Y be a noetherian scheme and let F be a quasi-coherent OX module. Then F is coherent if and only if the graded quasi-coherent S -module Γ∗ (F ) satisfies (TF). Proof. If Γ∗ (F ) satisfies (TF), then F ∼ = Γ∗ (F )∼ is a quasi-coherent OX -module of finite type (Proposition 23.12 (2)) and hence it is coherent since X is noetherian. Conversely, if F is coherent, then there exists a graded quasi-coherent S -module M of finite type such that M˜ ∼ = F by Lemma 23.14. By Proposition 23.13, Γ∗ (F ) is (TN)-isomorphic to M . Therefore Γ∗ (F ) satisfies (TF). Remark 23.16. Proposition 23.13 and Corollary 23.15 can be reformulated as follows. Suppose that Y is noetherian. Then the functors F 7→ Γ∗ (F ) and M 7→ M˜ define an equivalence between the category of coherent OX -modules and the quotient category (Exercise F.14) ModTF (S )/ModTN (S ), where ModTF (S ) (resp. ModTN (S )) denotes the full abelian subcategory of the category of graded quasi-coherent S -modules consisting of modules satisfying (TF) (resp. (TN)).

Coherence of higher direct images for proper morphisms In this part we prove that for proper morphisms f : X → Y of locally noetherian schemes the higher direct images of a coherent module is again coherent. This is deduced from the analogue statement for projective morphism using a d´evissage argument and the Lemma of Chow. Thereafter, it is a rather formal argument to generalize this result by showing that Rf∗ preserves pseudo-coherent complexes and to deduce that Rf∗ maps Dcoh (X) to b b (Y ). (X) to Dcoh Dcoh (Y ) and, if Y is noetherian, maps Dcoh Next we study the question under which hypotheses Rf∗ preserves pseudo-coherent (resp. perfect) complexes and give some general criteria. We conclude the part by showing some finiteness result of Ext groups and by explaining the GAGA principle which links cohomology of coherent modules on proper schemes over C and their analytic counterparts on the attached compact complex analytic spaces. (23.4) Finiteness of higher direct images under proper morphisms. Using the technique of d´evissage (Lemma 12.63), we can generalize the result that higher direct images under projective morphisms of coherent modules are again coherent (Theorem 23.1) to the case of proper morphisms.

307 Theorem 23.17. Let S be a locally noetherian scheme and let f : X → S be a proper morphism of schemes. If F is a coherent OX -module, then for all i ≥ 0, the higher direct image Ri f∗ F is a coherent OS -module. Proof. First note that we know that Ri f∗ F is quasi-coherent by Theorem 22.27. Recall that a quasi-coherent module on a locally noetherian scheme is coherent if and only if it is of finite type (Proposition 7.46). To show that Ri f∗ F is coherent, we can work locally on S, so we may assume that S is noetherian. Then X is also noetherian because f is proper and in particular of finite type. Let C be the full subcategory of the category of coherent OX -modules of all F which satisfy the conclusion of the theorem. We want to show that C is all of (Coh(X)). By Lemma 12.63 it suffices to show the following assertions. (1) Let 0 → F ′ → F → F ′′ → 0 be an exact sequence of coherent OX -modules. If two of the modules are in C, then the third is contained in C. (2) C is closed under taking direct summands. (3) For every integral closed subscheme Y ⊆ X with generic point η, there exists a coherent OY -module G with non-vanishing stalk Gη and such that all higher direct images Ri f∗ G are coherent. Assertion (1) follows from the long exact cohomology sequence and the fact that if E1 → E2 → E3 is an exact sequence of quasi-coherent OS -modules and E1 and E3 are of finite type, then E2 is of finite type. Indeed, we can assume that S = Spec R for a noetherian ring and hence that the exact sequence corresponds to an exact sequence M1 → M2 → M3 for R-modules where M1 and M3 are finitely generated. Then M2 is finitely generated R-module since R is noetherian. Hence its attached quasi-coherent module E2 is coherent. Assertion (2) is clear since direct summands of coherent modules are coherent and Ri f∗ is additive and hence compatible with finite direct sums. It remains to show (3). We may as well replace X by Y . In other words, we may assume that X is integral, with generic point η, and need to show that there exists a coherent OX -module G such that Gη ̸= 0 and all Ri f∗ G are coherent OS -modules. By Chow’s Lemma (Theorem 13.100), there exists a surjective projective morphism π : X ′ → X such that the composition g := f ◦ π is also projective and such that π is an isomorphism over a non-empty open subscheme of X. We will find an OX ′ -module G ′ such that G := π∗ (G ′ ) satisfies Gη ̸= 0 and has coherent higher direct images. Note that G is coherent by Theorem 23.1. Let L be a π-ample invertible OX ′ -module. For n sufficiently large, we have that G ′ := L ⊗n satisfies Ri π∗ G ′ = 0 for all i > 0 by Theorem 23.1. Since π is an isomorphism over a non-empty open subscheme of X, (π∗ G ′ )η ̸= 0. By Remark 22.7 and Remark 21.44 we find that Ri f∗ G = Ri f∗ (π∗ G ′ ) ∼ = Ri (f ◦ π)∗ (G ′ ) = Ri g∗ G ′ , which is coherent for all i ≥ 0 by Theorem 23.1. Corollary 23.18. Let X be proper over an affine scheme S = Spec R with R noetherian, and let F be a coherent OX -module. Then for all i, H i (X, F ) is a finite generated R-module. Proof. This follows from Theorem 23.17 by Theorem 22.27.

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23 Cohomology of projective and proper schemes

Remark 23.19. The analogue of Theorem 23.17 in complex analysis is also true: It is a theorem by Grauert that the higher direct images of a coherent sheaf under a proper morphism of complex spaces are coherent. The proof is considerably more difficult in this setting because of the intricate analytic problems it poses. (23.5) Finiteness of cohomology of modules with proper support. In this section we explain how to weaken the hypothesis in the above finiteness results that the morphism under which direct images are taken is proper. Remark and Definition 23.20. Let X be a scheme and let F be a quasi-coherent OX -module of finite type. Then its annihilator I := Ann(F ) is a quasi-coherent ideal and Z := V (I ) is a closed subscheme whose underlying topological space is Supp(F ) (Proposition 7.35). In the sequel, we will endow Supp(F ) with this structure of a closed subscheme. One has I F = 0 and hence F → i∗ (i∗ F ) is an isomorphism, where i : Z → X is the inclusion. Definition and Lemma 23.21. Let X → S be a morphism locally of finite type and let F be a quasi-coherent OX -module of finite type. Then F is said to have proper support over S, if the following equivalent conditions hold. (i) There exists a structure of closed subscheme of X on Supp(F ) such that the composition Supp(F ) → X → S is proper. (ii) For every structure of closed subscheme of X on Supp(F ) the composition Supp(F ) → X → S is proper. (iii) There exists a closed immersion i : Z → X such that the composition Z → X → S is proper and such that F ∼ = i∗ G for some quasi-coherent OZ -module of finite type. Proof. Assertions (i) and (ii) are equivalent by Proposition 12.58 and clearly (iii) implies (i). Finally (ii) implies (iii) by Remark 23.20. Corollary 23.22. Let S be a locally noetherian scheme, let f : X → S be a morphism locally of finite type, and let F be coherent OX -module with proper support over S. Then Rp f∗ F is a coherent OS -module for all p ≥ 0. If S = Spec R is affine, then H p (X, F ) is a finitely generated R-module for all p ≥ 0. Proof. Coherence can be checked locally hence it suffices to show the second assertion. By Lemma 23.21 there exists a closed immersion i : Z → X such that g : Z → S is proper and such that F is of the form i∗ G for a coherent OZ -module G . We have H p (X, F ) = H p (Z, G ) by Corollary 22.6 and hence conclude by Corollary 23.18. (23.6) Derived image of pseudo-coherent complexes. Let X be a locally noetherian scheme. Recall that for ? ∈ {+, −, b, ∅} we denote by ? Dcoh (X) the triangulated subcategory of complexes F in D? (X) such that H p (F ) is − coherent for all p ∈ Z. For instance, if X is noetherian, then Dcoh (X) is the category of pseudo-coherent complexes on X by Proposition 22.61. Proposition 23.23. Let f : X → Y be a proper morphism between locally noetherian schemes. Let E ∈ D(X) be pseudo-coherent. Then Rf∗ E is pseudo-coherent.

309 Proof. Since the question is local on Y , we may assume that Y is noetherian. Then X is noetherian because f is of finite type. Let us first consider the case that the cohomology of E is bounded, i.e., that H i (E) = 0 for all i with |i| larger than some bound (depending on E). By assumption, all H i (E) are coherent, so Theorem 23.17 shows that all Rj f∗ H i (E) are coherent. Furthermore, we know that Rj f∗ H i (E) = 0 whenever j is larger than some constant N (depending only on f ) by Proposition 22.29. Now the hypercohomology spectral sequence (Corollary 21.46), Rj f∗ H i (E) =⇒ Ri+j f∗ E, implies that all cohomology sheaves of Rf∗ E are coherent, and that the cohomology of Rf∗ E is bounded above, whence Rf∗ E is pseudo-coherent (Proposition 22.61). Now we prove the general case. For an integer n, consider an exact triangle E → τ ≥n E → C → where E → τ ≥n E is the natural morphism to the truncation. Now τ ≥n E is pseudocoherent and bounded, and hence Rf∗ τ ≥n E is pseudo-coherent by what we have already proved. Furthermore, we have that τ ≥n−1 C is exact, so using Proposition 22.29 again, we have Ri f∗ C = 0 for all i ≥ n + N − 1 for some N depending only on the morphism f . This shows that τ ≥n+N Rf∗ E ∼ = τ ≥n+N Rf∗ τ ≥n E, so τ ≥n+N Rf∗ E is pseudo-coherent for all n. It follows that Rf∗ E is pseudo-coherent. Remark 23.24. The proof of Proposition 23.23 shows that if f : X → Y is a proper b morphism of noetherian schemes, then Rf∗ sends objects in Dcoh (X) to objects in b Dcoh (Y ). Corollary 23.25. Let f : X → Y be a proper morphism between locally noetherian schemes. Let E ∈ Dcoh (X) be a complex with coherent cohomology. Then Rf∗ E ∈ Dcoh (Y ), i.e., Rp f∗ E is coherent for all p ∈ Z. Proof. We may work locally on Y and hence can assume that Y noetherian. Then X is also noetherian. Fix p ∈ Z. Then τ ≤p E is pseudo-coherent by Proposition 22.61. Consider a distinguished triangle + τ ≤p E −→ E −→ C −→ . The long exact cohomology sequence shows that C ∈ D≥p+1 (X) and hence Rf∗ C ∈ D≥p+1 (Y ). Applying Rf∗ to the distinguished triangle, the long exact cohomology se∼ quence yields an isomorphism Rp f∗ (τ ≤p E) → Rp f∗ (E). As Rf∗ (τ ≤p E) is pseudo-coherent by Proposition 23.23, Rp f∗ (τ ≤p E) is coherent. Therefore Rp f∗ (E) is coherent. Proposition 23.23 can be generalized to non-noetherian situations, see Corollary 23.136 O below, Kiehl’s paper [Kie] O , [Lip2] O X Corollary 4.3.3.2, or [FuKa] X I, 8.1.3. (23.7) Morphisms of finite tor-dimension. After we gave in Proposition 23.23 a criterion when Rf∗ preserves pseudo-coherent complexes (see also Definition 23.31 below), we now study the question when Rf∗ preserves complexes of bounded tor-amplitude. In the next section we will combine these results and give criteria when Rf∗ preserves perfect complexes using that a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-dimension (Theorem 21.174).

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23 Cohomology of projective and proper schemes

Let t ≥ 0 be an integer. Recall (Definition 21.164) that a morphism f : X → Y of schemes is said to be of tor-dimension ≤ t if H i (Lf ∗ G ) = 0 for all i < −t and for all OY -modules G . If Y is quasi-separated (e.g., if Y is locally noetherian), then it suffices to check this condition for quasi-coherent OY -modules G (Lemma 22.51). The morphism f is called locally of finite tor-dimension if there exists an open covering (Vi )i of Y such that the restriction f −1 (Vi ) → Vi has tor-dimension ≤ ti for some ti ≥ 0. Moreover, one has the following properties. (1) The condition that f has tor-dimension ≤ t can be checked Zariski locally on X and on Y (Lemma 21.171). (2) If f : Spec B → Spec A is a morphism of affine schemes, then f is of tor-dimension ≤ t if and only if B is of finite tor-dimension ≤ t as an A-module (Lemma 22.44). (3) If f is of tor-dimension ≤ t and g : Y → Z is of tor-dimension ≤ s, then g ◦ f is of tor-dimension ≤ t + s (21.35.4). Example 23.26. (1) Remark 21.165 (5) shows that a morphism of schemes is flat if and only if it is of tor-dimension ≤ 0. (2) If i : Y → X is a completely intersecting immersion of codimension c, then i is of tor-dimension ≤ c. Indeed, as we can check the condition locally on X and Y , we may assume that X = Spec A and Y = Spec A/I for a ring A and an ideal I ⊆ A that is generated by a completely intersecting sequence f = (f1 , . . . , fc ). Then we have a quasi-isomorphism between A/I and the Koszul complex K(f ) defined by f which we view as a complex of free A-modules concentrated in degrees [−c, 0]. Hence for every A-module M , ˜ ∈ Dqcoh (Y ) corresponds to the complex M ⊗L A/I = M ⊗A K(f ) that is Li∗ M A concentrated in degrees [−c, 0]. (3) Every locally completely intersecting morphism of locally of finite tor-dimension. Indeed, as this can be checked locally, we can assume that the morphism can be factorized as a completely intersecting immersion followed by a smooth (and in particular flat) morphism. Hence we can use (2) and (1). (4) Let A be a ring and let a ∈ A be a regular element that is not a unit. Then the closed immersion Spec A/(a) → Spec A/(a2 ) is not of finite tor-dimension. Indeed A/(a) has a quasi-isomorphism from the K-flat complex of A/(a2 )-modules ·a

·a

·a

· · · −→ A/(a2 ) −→ A/(a2 ) −→ A/(a2 ) −→ 0 −→ . . . , concentrated in degrees ≤ 0. Hence A/(a) ⊗L A/(a2 ) A/(a) is represented by the complex 0

0

0

· · · −→ A/(a) −→ A/(a) −→ A/(a) −→ 0 −→ . . . , which has cohomology in every degree ≤ 0. Lemma 23.27. Let f : X → Y be a morphisms of finite tor-dimension and let /X

X′ f′

 Y′

f

u

 /X

be a cartesian diagram of schemes such that u is flat. Then f ′ is of finite tor-dimension.

311 Proof. As we can check finite tor-dimension Zariski locally on source and target we may assume that all schemes are affine, say Y = Spec A, X = Spec B, Y ′ = Spec A′ . By hypothesis, B has finite tor-dimension as an A-module and hence B ⊗A A′ has finite tor-dimension as an A′ -module by Corollary 21.170, applied to u and using that Lu∗ = u∗ since u is flat. Proposition 23.28. Let Y be a qcqs scheme and let f : X → Y be a qcqs morphism of tor-dimension ≤ t. Let N ≥ 0 an integer such that Ri f∗ F = 0 for all i ≥ N and for all quasi-coherent OX -modules F . Let E in Dqcoh (X) be of tor-amplitude in [a, b]. Then Rf∗ E is of tor-amplitude in [a − t, b + N ]. Such an integer N always exists by Lemma 22.28. [a−t,b+N ] Proof. By Lemma 22.51 it suffices to show that Rf∗ E ⊗L (Y ) for every OY G ∈ D L L ∼ quasi-coherent OY -module G . But Rf∗ E ⊗OY G = Rf∗ (E ⊗OX Lf ∗ G ) by the projection [a−t,b] ∗ formula. One has E ⊗L OX Lf G ∈ Dqcoh (X) by Lemma 21.167 and Proposition 22.40. [a−t,b+N ]

∗ Therefore Rf∗ (E ⊗L OX Lf G ) ∈ Dqcoh

(Y ) by Proposition 22.29.

Corollary 23.29. Let f : X → Y be a qcqs morphism of schemes which is locally of finite tor-dimension, and let E in Dqcoh (X) be locally of finite tor-dimension. Then Rf∗ E ∈ Dqcoh (Y ) is locally of finite tor-dimension. (23.8) Derived image of perfect complexes.

Proposition 23.30. Let Y be a locally noetherian scheme and let f : X → Y be a proper morphism locally of finite tor-dimension. Let E be a perfect complex on X. Then Rf∗ E is perfect. There are in fact several nice characterizations of morphisms f such that Rf∗ preserves perfect objects (see Remark 25.39 below). Proof. Recall that a complex is perfect if and only if it is pseudo-coherent and has locally finite tor-dimension (Theorem 21.174). The complex Rf∗ E is pseudo-coherent by Proposition 23.23 and has finite tor-dimension locally on Y by Corollary 23.29. The proof uses Theorem 21.174, which holds for arbitrary ringed spaces, Corollary 23.29, which holds for every qcqs morphism, and it uses the criterion Proposition 23.23 for the derived direct image of pseudo-coherent complexes to be pseudo-coherent. Let us coin the following (non-standard) notion. Definition 23.31. A morphism f : X → Y of schemes is called cohomologically proper1 if f is qcqs and if Rf∗ E is pseudo-coherent for every pseudo-coherent complex E in D(X). Then the same proof as in Proposition 23.30 yields the following result. 1

Lipman calls such morphisms “quasi-proper” which however could be confusing since in general a proper morphism need not be quasi-proper (Exercise 23.8). A notion of a cohomologically proper morphism has also been defined by Kubrak and Prikhodko for morphisms of locally noetherian algebraic stacks in [KuPr] O X which, if specialized to morphisms of locally noetherian schemes, is equivalent to the property of being proper and hence to the notion given here (Remark 23.33).

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23 Cohomology of projective and proper schemes

Proposition 23.32. Let f : X → Y be a cohomologically proper morphism locally of finite tor-dimension. If E is a perfect complex on X, then Rf∗ E is a perfect complex on Y. Remark 23.33. Proposition 23.23 just says that any proper morphism between locally noetherian schemes is cohomologically proper. The converse also holds for quasi-compact separated morphisms between locally noetherian schemes ([Hal] X 3.10, see also Exercise 26.6 if the morphism is in addition of finite type). Let us give a criterion for affine morphisms to be cohomologically proper. Proposition 23.34. Let f : X → Y be an affine morphism of schemes. Then f is cohomologically proper if and only if f∗ OX is pseudo-coherent (i.e., f∗ OX considered as a complex of OY -modules concentrated in degree 0 is pseudo-coherent). These equivalent hypotheses imply that f is finite and of finite presentation by Proposition 21.153, in particular that f is proper. Proof. As f is affine, we have Rf∗ OX = f∗ OX and as OX is pseudo-coherent, the condition is clearly necessary. Now assume that f∗ OX is pseudo-coherent and let E be a pseudo-coherent complex in D(X). To show that f∗ E is pseudo-coherent, we may work locally on Y and hence can assume that Y = Spec A is affine. Then X is also affine, say X = Spec B and B is a pseudo-coherent A-module. The complex E corresponds to a pseudo-coherent complex E of B-modules (Remark 22.45) and we may represent E by a bounded above complex of finitely generated free B-modules (Proposition 21.162). As finite direct sums of pseudocoherent complexes are again pseudo-coherent (Proposition 21.156) we deduce that all components of E are pseudo-coherent A-modules. Hence E is pseudo-coherent in D(A) by Lemma 21.158. There are further important examples of cohomologically proper morphisms. Below in Corollary 23.136 we will see that flat proper morphisms of finite presentation between arbitrary schemes are cohomologically proper. Exercise 23.10 also gives a criterion for projective morphisms to be cohomologically proper. More generally, one calls a morphism f : X → Y of schemes pseudo-coherent if there exists an open covering (Uλ )λ of X such that for all λ the restriction f |Uλ can be factorized iλ into Uλ −→ Pλ → S, where Pλ → S is smooth and iλ is a closed immersion such that iλ∗ OUλ is a pseudo-coherent OPλ -module. Then one has the following result for whose proof we refer to [Lip2] O X (4.3.3.2). Proposition 23.35. Let f : X → Y be a proper pseudo-coherent morphism of schemes. Then f is cohomologically proper. Corollary 23.36. Let f : X → Y be a proper lci-morphism and let E be a perfect complex on X, then Rf∗ E is perfect. Proof. Example 23.26 (2) and (3) show that f is of finite tor-dimension and pseudocoherent. Hence we conclude by Proposition 23.35 and Proposition 23.32. Finally, [FuKa] O X I.8.1.4 implies the following result, which generalizes the result that proper morphisms between locally noetherian schemes are cohomologically proper.

313 Proposition 23.37. Let Y be a universally coherent scheme (Exercise 22.32) and let f : X → Y be a proper morphism of finite presentation. Then f is cohomologically proper. Proof. This can be checked locally on Y and hence we may assume that Y is quasi-compact. Then X is quasi-compact and coherent and hence a complex in D(X) is pseudo-coherent if and only if it is in D− (X) (Exercise 22.31). Now we can apply loc. cit. which ensures that Rf∗ sends D− (X) to D− (Y ). Corollary 23.38. Let A be a noetherian ring and let X → Spec A be a proper morphism. Let F and G be complexes in D(X). Suppose that one of the following hypotheses is satisfied. (1) F is perfect and G is in Dcoh (X). − + (2) One has F ∈ Dcoh (X) and G ∈ Dcoh (X). Then ExtnOX (F , G ) is a finitely generated A-module for all n ∈ Z. Proof. By Corollary 22.67, H n (R Hom OX (F , G )) is coherent for all n ∈ Z. Hence Extn (F , G ) = H n (RΓ(X, R Hom OX (F , G ))) is a finitely generated A-module for all n because f is proper (Corollary 23.25). (23.9) GAGA. Recall the notion of a complex analytic space and of the analytification from Section (20.12). Let X be a scheme that is locally of finite type over C, let X an be its analytification, which is a complex analytic space, and let α : X an → X be the canonical flat morphism of locally ringed spaces (Theorem 20.58). For F ∈ Dcoh (X) we call F an := Lα∗ F its analytification. Since α is flat, for every complex F of OX -modules, its derived inverse image Lα∗ F is represented by the complex α∗ F . In Theorem 20.60 we have seen that pullback by α yields an equivalence of the category of coherent OX -modules and the coherent OX an -modules if X is proper over C. This extends to derived categories (e.g., [Hal] X Theorem A). Theorem 23.39. Let X be a scheme that is proper over C with analytification α : X an → X. Then Lα∗ induces triangulated equivalences of triangulated categories ∼

∼

− − Dcoh (X) −→ Dcoh (X an ),

b b Dcoh (X) −→ Dcoh (X an ).

− yielding for all F in Dcoh (X) and for all p ∈ Z an isomorphism

(23.9.1)

∼

H p (X, F ) −→ H p (X an , F an ).

There is also a relative version of (23.9.1). Let f : X → Y be a morphism of schemes locally of finite type over C. Consider the commutative diagram of morphisms of locally ringed spaces αX /X X an f an

f

 / Y. Y an By Proposition 21.129 one obtains a base change morphism 

αY

b : LαY ◦ Rf∗ −→ Rf∗an ◦ LαX of functors D(X) → D(Y an ).

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23 Cohomology of projective and proper schemes

b Theorem 23.40. Let f be proper. Then b induces an isomorphism of functors Dcoh (X) → b b Dcoh (Y an ) and for every F in Dcoh (X) and for all p ∈ Z one has a functorial isomorphism ∼

(Rp f∗ F )an −→ Rp f∗an F an .

Numerical intersection theory, Euler characteristic, and Hilbert polynomial In the next few sections, we will introduce the zeroth K-group of schemes, discuss the Euler characteristic and Hilbert polynomial of coherent sheaves, and study the basics of numerical intersection theory, i.e., we will attach to a family of line bundles and a coherent sheaf an integer. If the line bundles are the line bundles attached to effective divisors, and the coherent sheaf is the structure sheaf of a closed subscheme of the ambient scheme, we think of this number as the number of points in the intersection of the divisors and the closed subscheme, counted “correctly”, i.e., with multiplicities. The formalism of the Euler characteristic, which is based on the cohomology of coherent sheaves, provides a relatively easy way to give a general definition which can then also be applied to cases where the geometric intuition fails or is problematic, e.g., taking the d-fold self-intersection of a divisor on a d-dimensional scheme. For a more sophisticated approach of intersection theory, one can set up a theory of algebraic cycles and Chow groups, generalizing the notion of Weil divisor to higher codimensions, and define intersection products in this setting. See for instance [Ful2] O . (23.10) Grothendieck group of abelian categories and triangulated categories. Recall that a category is called essentially small if it is equivalent to a small category. Definition 23.41. (1) Let A be an essentially small abelian category. The Grothendieck group K0 (A) is defined as the abelian group with generators [X] for each isomorphism class X of objects in A and relations of the form [X] = [X ′ ] + [X ′′ ] for each exact sequence 0 → X ′ → X → X ′′ → 0 in A. (2) Let T be an essentially small triangulated category. The Grothendieck group K0 (T ) of T is the abelian group with generators [X] for each isomorphism class X of objects in T and relations of the form [X] = [X ′ ] + [X ′′ ] for each distinguished triangle X ′ → X → X ′′ → in T . The definition of K0 (A) can be generalized to exact categories (Exercise F.15), see Exercise 23.16.

315 Remark 23.42. Let T be an essentially small triangulated category. Then in K0 (T ) one has [X[1]] = −[X] for all objects X in T . + Indeed, the distinguished triangle 0 −→ 0 −→ 0 −→ shows that [0] = 0 in K0 (T ), and idX + rotating X −→ X −→ 0 −→ yields the relation 0 = [0] = [X] + [X[1]]. Proposition 23.43. Let A be an essentially small abelian category. Then the map Φ : K0 (A) −→ K0 (Db (A)),

[M ] 7→ [M [0]]

is an isomorphism of abelian groups. Its inverse is given by X (−1)i [H i (X)]. Ψ : [X] 7→ i∈Z

Proof. It is clear that Φ is well defined, and Ψ is well defined by the long exact cohomology sequence attached to a distinguished triangle. Moreover, Ψ ◦ Φ = id by definition. It remains to show that Φ is surjective. By Remark 23.42, the class of M [i] is in the image of Φ for every object M in A and [a,c] b all i ∈ Z. Let X be in DA (A′ ) and a ≤ c such that X ∈ DA (A′ ). To show that [X] is in the image of Φ, we use the distinguished triangle τ ≤c−1 X → X → H c (X)[−c] → [a,c−1]

to argue by induction on c−a applying the induction hypothesis to τ ≤c−1 X ∈ DA

(A′ ).

(23.11) Grothendieck group of noetherian schemes. Definition and Lemma 23.44. Let X be a noetherian scheme. Then (23.11.1)

b K0′ (X) := K0 (Coh(X)) = K0 (Dcoh (X)),

is called the Grothendieck group of X. For each r ≥ 0 let K0′ (X)r ⊆ K0′ (X) be the subgroup generated by those coherent OX -modules F with dim Supp(F ) ≤ r. If X = Spec A is affine with a noetherian ring A, we write K0′ (A) and K0′ (A)r instead of K0′ (Spec A) and K0′ (Spec A)r . The group K0′ (X) is also sometimes denoted as G0 (X). Proof. We have to show the equality in (23.11.1), but K0 (Coh(X)) = K0 (Db (Coh(X))) b by Proposition 23.43, and Db (Coh(X)) is equivalent to Dcoh (X) by Theorem 22.42. We obtain an ascending filtration K0′ (X)0 ⊆ K0′ (X)1 ⊆ . . . We also set K0′ (X)r := 0 for r < 0 and define grr K0′ (X) := K0′ (X)r /K0′ (X)r−1 ,

r ∈ Z.

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23 Cohomology of projective and proper schemes

b Remark 23.45. The class of a complex F ∈ Dcoh (X) is contained in K0′ (X)r if i dim Supp H (F ) ≤ r for all i ∈ Z. Indeed, by Proposition 23.43 we know that [F ] = P i i ′ (X). (−1) [H (F )] in K 0 i 0 The converse does not hold as the example of the complex 0 −→ OX −→ OX −→ 0 shows, whose class in K0 (X) is [OX ] − [OX ] = 0 but which has cohomology that has support on all of X.

Remark 23.46. Let f : X → Y be a proper morphism of noetherian schemes. Then b b the triangulated functor Rf∗ : Dcoh (X) → Dcoh (Y ) (Remark 23.24) induces a group homomorphism (23.11.2)

f∗ : K0′ (X) → K0′ (Y ).

Let F be a coherent OX -module. (1) By Proposition 23.43 one has (23.11.3)

f∗ [F ] =

X

(−1)i [Ri f∗ F ].

i≥0

(2) One has Supp Ri f∗ F ⊆ f (Supp F ) and dim f (Supp F ) ≤ dim Supp F because f is closed. Therefore we see that for all r one has f∗ (K0′ (X)r ) ⊆ K0′ (Y )r . In particular, f∗ induces for all r ≥ 0 a homomorphism (23.11.4)

f∗ : grr K0′ (X) −→ grr K0′ (Y ).

(3) If f is finite (e.g., if f is a closed immersion), then Ri f∗ F = 0 for i > 0 (Corollary 22.5) and we see that f∗ [F ] = [f∗ F ] in this case by (23.11.3). By d´evissage we obtain the following result. Proposition 23.47. Let X be a noetherian scheme and let r ≥ 0. Let φ, ψ : K0′ (X)r → A be group homomorphisms with values in an abelian group A such that φ(i∗ OZ ) = ψ(i∗ OZ ) for each integral closed subscheme i : Z → X of dimension ≤ r. Then φ = ψ. Proof. Considering φ − ψ we may assume that ψ = 0. We want to show that the class of any coherent OX -module F with support of dimension ≤ r is in the kernel of φ. We equip Supp(F ) with the structure of a closed subscheme by setting Supp(F ) = V (Ann(F )) (Remark 23.20) and let ι : Supp(F ) → X be the inclusion. The image of ι∗ : K0′ (Supp(F )) → K0′ (X) is contained in K0′ (X)r and contains the class of F = ι∗ (ι∗ F ). Hence it suffices to show that φ ◦ ι∗ = 0. Let Σ be the kernel of φ ◦ ι∗ . By hypothesis, Σ contains i∗ OZ for every closed integral subscheme i : Z → Supp(F ). Moreover, if 0 → G ′ → G → G ′′ → 0 is an exact sequence of coherent modules on Supp(F ) and two of the three modules are in Σ, then the third is in Σ. Hence we conclude by Lemma 12.63. Corollary 23.48. Let X be a noetherian scheme, let F be a coherent sheaf on X, and r := dim Supp F . Let Z1 , . . . , Zm be the irreducible components of Supp F of dimension r, considered as integral subschemes of X, and let ηi ∈ Zi be the generic point. Then one has

317 [F ] ≡

m X

lgOX,η (Fηi ) [OZi ] mod K0′ (X)r−1 i

i=1

in grr K0′ (X) = K0′ (X)r /K0′ (X)r−1 . Proof. Both sides of the equation can be viewed as group homomorphisms K0′ (X)r → grr K0′ (X). They agree if F = i∗ OZ for a closed integral subscheme i : Z → X. Hence we can conclude by Proposition 23.47. Corollary 23.49. Let X be a noetherian scheme and let E be a locally free OX -module of constant rank e ≥ 1. Then F 7→ E ⊗ F induces an endomorphism of abelian groups K0′ (X) → K0′ (X) which preserves K0′ (X)r and induces on grr K0′ (X) the multiplication by e for all r ≥ 0. Proof. As tensoring with E is an exact functor, it induces an endomorphism of abelian groups K0′ (X) → K0′ (X). Because Supp(E ⊗ F ) = Supp F for every coherent OX module, it preserves K0′ (X)r . Let F be a coherent module with dim Supp F ≤ r and let Z1 , . . . , Zm be the irreducible components of Supp F of dimension r with generic points ηi . Then lgOZ ,η (E ⊗ F )ηi = e lgOZ ,η Fηi . By Corollary 23.48, then one has modulo i i i i K0′ (X)r−1 that [E ⊗ F ] ≡ e[F ]. (23.12) K-groups of quasi-compact schemes. Let X be a scheme, then the full subcategory (Perf(X)) of D(X) consisting of perfect complexes is a triangulated subcategory by Proposition 21.175. Definition 23.50. Let X be a quasi-compact scheme. Then we set K0 (X) := K0 (Perf(X)). and call it the (zero-th) K-group of X. If X = Spec A, then we define the K-group K0 (A) := K0 (Spec A) of the ring A. If X is not necessarily quasi-compact, one defines K0 (X) to be the K-group of all perfect complexes of (globally) finite tor-amplitude [ThTr] O . We will not use this generality in the sequel. Moreover, one can also define higher K-groups but here we use only the 0-th K-group. Nevertheless we keep the index 0 to distinguish the K-group from K(X), the triangulated category of complexes of OX -modules up to homotopy, and to remind the reader that this is only the top layer of a much deeper story. Under certain circumstances, one can define K0 (X) also just in terms of finite locally free modules as the following remark shows. Remark 23.51. Let X be a quasi-compact scheme and let E be a strictly perfect complex, i.e., a finite complex of finite locally free OX -modules, say concentrated in degrees [a, b]. Then 0 = σ ≥b+1 E ⊆ σ ≥b E ⊆ · · · σ ≥a+1 E ⊆ σ ≥a E = E is a filtration by strictly perfect complexes such that σ ≥i+1 E /σ ≥i E ∼ = E i [−i]. As short exact sequences of complexes define distinguished triangles, we see that in K0 (X) we have

318

23 Cohomology of projective and proper schemes

[E ] =

b X

(−1)i [E i ].

i=a

Now suppose that X is semiseparated and has the resolution property (Definition 22.56), e.g., if X admits an ample line bundle (Proposition 22.57). Then every perfect complex on X can be represented by a strictly perfect complex (Proposition 22.58). Hence K0 (X) is generated as abelian group by the classes [F ], where F is a finite locally free OX module. One can show that the only relations between these generators are of the form [F ] = [F ′ ] + [F ′′ ] for short exact sequences 0 → F ′ → F → F ′′ → 0, see Exercise 23.23. See also Exercises 23.19 and 23.17 for further examples when the resolution property is satisfied. The K-group K0 (X) has a commutative ring structure and the Grothendieck group K0′ (X) has the structure of a module over it: Remark and Definition 23.52. Let X be a quasi-compact scheme. (1) We define on the abelian group K0 (X) the structure of a commutative ring by [E ] ⊗ [F ] := [E ⊗L OX F ]

K0 (X) ⊗Z K0 (X) −→ K0 (X),

for perfect complexes E and F in X. Indeed, E ⊗L OX F is perfect by Remark 21.143 L and since − ⊗ − is triangulated in both components we obtain an induced Z-bilinear map K0 (X) × K0 (X) → K0 (X) which is clearly associative and commutative. The unit object is the class of the structure sheaf. (2) Let X be a noetherian scheme. By Lemma 22.64, [E ] ⊗ [F ] 7→ [E ⊗L OX F ]

K0 (X) ⊗Z K0′ (X) −→ K0′ (X),

defines on the abelian group K0′ (X) the structure of a K0 (X)-module. Remark and Definition 23.53. Let f : X → Y be a morphism of quasi-compact schemes. Then sending a perfect complex E on Y to the perfect complex Lf ∗ E defines a homomorphism of rings (23.12.1)

f ∗ : K0 (Y ) → K0 (X).

Indeed, since Lf ∗ is a triangulated functor, it induces a homomorphism of abelian groups on K-groups which is a ring homomorphism by Proposition 21.117. Note that the class of finite locally free modules E , or more generally of strictly perfect complexes E , on Y is sent to the class of f ∗ E since Lf ∗ E = f ∗ E as E is K-flat in this case. Remark 23.54. Let f : X → Y be a proper morphism of noetherian schemes. By f ∗ : K0 (Y ) → K0 (X) we can view every K0 (X)-module as K0 (Y )-module. Then the projection formula (Proposition 22.84) shows that f∗ : K0′ (X) → K0′ (Y ) is a homomorphism of K0 (Y )-modules. More generally, let /Y

f

X p



S



q

319 be a commutative diagram of proper morphisms of noetherian schemes. Then for α ∈ K0′ (X) and β ∈ K0 (Y ) one has (23.12.2)

q∗ (β · f∗ α) = q∗ (f∗ (f ∗ β · α)) = p∗ (f ∗ β · α).

b Let X be a noetherian scheme. Then every perfect complex is in Dcoh (X) and the b inclusion (Perf(X)) → Dcoh (X) induces a group homomorphism

K0 (X) → K0′ (X). b Proposition 23.55. Let X be a regular noetherian scheme. Then every object in Dcoh (X) ′ is perfect. In particular, the map K0 (X) → K0 (X) is an isomorphism.

Proof. As the condition of being perfect can be checked locally, we may assume that b X = Spec A is affine. Let E be in Dcoh (X). By Lemma 21.177 it suffices to show that H i (E ) is perfect. Hence it suffices to show that every finitely generated A-module M has a finite left resolution 0 → F−n → · · · F0 → M → 0 by finitely generated projective modules Fi . But this holds because A is regular (Proposition G.5). Remark 23.56. If X is noetherian regular and semiseparated, then for every perfect complex F of tor-amplitude in [a, b] there exists a complex E of finite locally free OX -modules concentrated in degrees [a, b] and a quasi-isomorphism E → F (see [Sta] 0F8A, 0F8E (and its proof)). Hence if X is in addition of finite dimension n, then every coherent OX -module is isomorphic in D(X) to a complex of finite locally free OX -modules concentrated in degrees [−n, 0]. (23.13) Chern classes of line bundles on noetherian schemes. In this section X denotes a noetherian scheme. We will now define Chern classes. These are invariants attached to line bundles, and more generally to vector bundles, which lie in the K-group K0 (X) of the underlying scheme X, and are an important tool in the study of line bundles and vector bundles on X. Definition 23.57. Let L be a line bundle on X. The first Chern class of L is defined as c1 (L ) = [OX ] − [L ⊗−1 ] ∈ K0 (X). The Chern class induces an endomorphism of the abelian group K0′ (X) given by (23.13.1)

[F ] 7→ c1 (L ) · [F ] = [F ] − [L ⊗−1 ⊗ F ],

and sometimes this endomorphism is called the first Chern class of L , as well. Proposition 23.58. Let L be a line bundle on X. (1) Then c1 (L ) · K0′ (X)r ⊆ K0′ (X)r−1 for all r ≥ 0. (2) For any two line bundles L1 and L2 the first Chern class endomorphisms c1 (L1 ) and c1 (L2 ) commute with each other. (3) If L = OX (D) for an effective Cartier divisor i : D → X, then i∗ OD is perfect and for all α ∈ K0′ (X) one has c1 (L ) · α = i∗ OD ⊗L OX α, where the right hand side is the multiplication defined in Definition 23.52 (2).

320

23 Cohomology of projective and proper schemes

(4) For each r ≥ 0 and α ∈ K0′ (X)r the map L 7→ c1 (L ) · α mod K0′ (X)r−2

Pic(X) −→ grr−1 K0′ (X), is a group homomorphism.

Proof. By Corollary 23.49 one has c1 (L ) · F ≡ 0 modulo K0′ (X)r−1 for all [F ] ∈ K0′ (X)r . This shows (1). Assertion (2) holds by commutativity of the tensor product, and Assertion (3) follows from the exact sequence 0 → L ⊗−1 → OX → i∗ OD → 0. Finally, we have for L1 , L2 ∈ Pic(X) the formal equality (23.13.2)

c1 (L1 ⊗ L2 ) · α = c1 (L1 ) · α + c1 (L2 ) · α − c1 (L1 ) · c1 (L2 ) · α.

By (1) one has c1 (L1 ) · c1 (L2 ) · α ∈ K0′ (X)r−2 which shows (4). (23.14) Euler characteristic of schemes over a field. Let k be a field and let X → Spec k be a proper morphism. b Definition and Remark 23.59. For every complex F in Dcoh (X) we define its Euler characteristic X χ(F ) := χ(X, F ) := (−1)i dimk H i (X, F ). i∈Z

The sum on the right side is finite by Remark 23.24. The long exact cohomology sequence shows that χ induces a group homomorphism χ : K0′ (X) −→ Z. Note that χ(F ) also depends on the ground field. Hence we sometimes write χk (F ). With the identification K0′ (k) = Z of Example 23.60, the map χ can be identified with the map induced by the derived direct image of the structure morphism X → Spec k on K-groups (Exercise 23.14). Example 23.60. Let k be a field. Then K0 (k) = K0′ (k) and ∼

χ : K0 (k) −→ Z is an isomorphism of rings. Indeed, clearly it is an isomorphism of abelian groups, as every vector space is free. To check that χ(E ⊗L k F ) = χ(E)χ(F ) one can assume that E and F are acyclic complexes of finite-dimensional k-vector spaces (Example 22.48), then E ⊗L F = E ⊗k F is acyclic as well and X X χ(E)χ(F ) = ( (−1)i dim E i )( (−1)j dim F j ) i

j

=

X X

=

X

(−1)

i+j

dim E i dim F j

n i+j=n

n

(−1)n

X i+j=n

= χ(E ⊗L k F ).

dim(E i ⊗ F j )

321 Example 23.61. Let X = Prk and n ∈ Z. Then one has by Theorem 22.22   n+r χ(OX (n)) = . r Here we define for every α ∈ R (or, more generally, in any Q-algebra)   α α(α − 1) · · · (α − r + 1) (23.14.1) := . r r! Relations within K0′ (X) yield identities for the Euler characteristic: Remark 23.62. Let X → Spec k be a proper morphism. b (X). Then χ(F [1]) = −χ(F ). (1) Let F ∈ Dcoh b (2) Let F ′ → F → F ′′ → be a distinguished triangle in Dcoh (X) (e.g., given by a short exact sequence 0 → F ′ → F → F ′′ → 0 of coherent OX -modules), then χ(F ) = χ(F ′ ) + χ(F ′′ ). b (X). Then (3) Let F ∈ Dcoh

χ(F ) =

X

(−1)i χ(H i (F )).

i∈Z

Corollary 23.63. Let f : X → Y be a morphism of proper k-schemes. Then for every complex F ∈ Db (X) one has X χ(X, F ) = χ(Y, Rf∗ F ) = (−1)i χ(Y, Ri f∗ F ). i∈Z

Proof. One has RΓ(X, F ) = RΓ(Y, Rf∗ F ) (Proposition 21.114) and hence we conclude by Remark 23.62 (3). b Lemma 23.64. Let X → Spec k be a proper morphism and let F be in Dcoh (X). Let K be a field extension, set XK := X ⊗k K and let FK be the (derived) pullback of F to XK . Then χK (FK ) = χk (F ).

If F a coherent OX -module, then FK is the usual (non-derived) pullback of F to XK since XK → X is flat. Proof. This follows from H i (X, F ) ⊗k K = H i (XK , FK ) by Corollary 22.91. Proposition 23.65. (Projection formula) Let f : X → Y be a morphism of proper k-schemes. For β ∈ K0 (Y ) and α ∈ K0′ (X) one has χ(Y, β · f∗ α) = χ(X, f ∗ β · α). Proof. This is the special case S = Spec k of (23.12.2). Remark 23.66. All definitions and results in this section also hold for proper schemes over a local Artinian ring R by replacing “dimension of k-vector spaces” by “length of R-modules”.

322

23 Cohomology of projective and proper schemes

(23.15) Numerical intersection number for proper schemes over a field. We now aim to define intersection numbers and study their properties. We will use the following formal lemma. Lemma 23.67. In Z[Y ±1 ][[1 − Y −1 ]] one has for all m ∈ Z the identity X m + i − 1 m Y = (1 − Y −1 )i . i i≥0

Recall that by definition (23.14.1) we have   m+i−1 (m + i − 1)(m + i − 2) · · · (m + 1)m . = i! i Proof. In Z[(1 + X)±1 ][[X]] we have for all n ∈ Z the formal identity X n  (1 + X)n = X i. i i≥0

Substituting m := −n and Y := (1 + X)−1 and using     −m m+i−1 = (−1)i i i we obtain the desired identity. Let X be a proper scheme over a field k and let L be a line bundle. Then χ(L ⊗ F ) is additive in exact sequences in coherent OX -modules F and therefore the map F 7→ χ(L ⊗ F ) factors through K0′ (X). We can describe the resulting group homomorphism as K0′ (X) −→ Z, α 7→ χ(L · α), where L · α is the scalar multiplication of α by the class of L in K0 (X), see Remark 23.52 (2). Theorem 23.68. Let X be a proper k-scheme, let t ≥ 0 be integers, and let L1 , . . . , Lt be line bundles on X. Then for all m1 , . . . , mt ∈ Z and for all α ∈ K0′ (X) one has

=

X i1 ,...,it ≥0

χ(L1⊗m1 ⊗ · · · ⊗ Lt⊗mt · α)     m1 + i1 − 1 mt + i t − 1 χ(c1 (L1 )i1 · · · c1 (Lt )it · α) ··· i1 it

Proof. By Lemma 23.67 we have the identity (*)

Y1m1

· · · Ytmt

=

X i1 ,...,it ≥0

(1 −

Y1−1 )i1

· · · (1 −

Yt−1 )it



   m1 + i1 − 1 mt + it − 1 ··· i1 it

of formal power series. We let Yj act on K0′ (X) by Yj · α := Lj · α. Then c1 (Lj ) · α = (1 − Yj−1 ) · α and hence 1 − Yj−1 acts nilpotently by Proposition 23.58 (1). Then applying (*) and taking Euler characteristics yield the theorem.

323 If α ∈ K0′ (X)r for some r ≥ 0 and i1 + · · · + it > r, then χ(c1 (L1 )i1 · · · c1 (Lt )it · α) = 0 by Proposition 23.58 (1). In particular the sum in Theorem 23.68 is finite. We obtain the following corollary. Corollary and Definition 23.69. Let X be a proper scheme over a field k, let t ≥ 0, b (X). Then the function let L1 , . . . , Lt be line bundles on X, and let F ∈ Dcoh σF : (n1 , . . . , nt ) 7→ χ(L1⊗n1 ⊗ · · · ⊗ Lt⊗nt ⊗L OX F ) is a numerical polynomial, i.e., a polynomial function given by a (necessarily unique) polynomial with rational coefficients, again denoted by σF , that is integer-valued on integers. It is of total degree ≤ r if F ∈ K0′ (X)r . The polynomial σF is called the Snapper polynomial of F and L1 , . . . , Lt . For more properties of numerical polynomials see Exercise 23.25. Definition 23.70. Let k be a field and let X be a proper k-scheme. Let t ≥ 0 be an b integer, let L1 , . . . , Lt be line bundles on X, and let F be an object in Dcoh (X) such ′ i that [F ] ∈ K0 (X)t (e.g., if dim Supp(H (F )) ≤ t for all i by Remark 23.45). Then (L1 · · · Lt · F ) := χ(c1 (L1 ) · · · c1 (Lt ) · F ) is called the intersection number of L1 , · · · , Lt on F . If i : Z → X is a closed subscheme of dimension ≤ t, then we set (L1 · · · Lt · Z) := (L1 · · · Lt · i∗ OZ ). In the special case that L1 = · · · = Lt = L we write (L t · F ) or (L t · Z). If Z = X with dim(X) ≤ t, we also write (L1 · · · Lt ) (resp. (L t )) instead of (L1 · · · Lt · X) (resp. (L t · X)). If Li = OX (Di ) for Cartier divisors Di , one also writes (D1 · · · Dt · F ) instead of (L1 · · · Lt · F ). Theorem 23.68 shows that (23.15.1)

(L1 · · · Lt · F ) = coefficient of n1 n2 · · · nt in σF ,

where σF is the degree ≤ t Snapper polynomial of Definition 23.69. The formation of the Snapper polynomial is invariant under changing the base field by Lemma 23.64. In particular: Remark 23.71. In the situation of Definition 23.70, let K be a field extension of k, and denote Li,K and FK the pullback of Li and F to X ⊗k K. Then (L1 · · · Lt · F ) = (L1,K · · · Lt,K · FK ). Corollary 23.72. In the situation of Definition 23.70 the following assertions hold. (1) If F ∈ K0′ (X)t−1 , then (L1 · · · Lt · F ) = 0 and hence there is a well defined group homomorphism grt K0′ (X) → Z, [F ] 7→ (L1 · · · Lt · F ). In particular (L1 · · · Lt ) = 0 if t > dim(X). (2) The intersection number is symmetric and Z-linear in each Lj .

324

23 Cohomology of projective and proper schemes

(3) Let F be a coherent OX -module and let Z1 , . . . , Zm be the irreducible components of Supp F of dimension t, considered as integral subschemes of X, and let ηi ∈ Zi be the generic point. Then one has (23.15.2)

(L1 · · · Lt · F ) =

m X

lgOX,η (Fηi ) (L1 · · · Lt · Zi ). i

i=1

Proof. Assertion (1) follows from Proposition 23.68 since multiplication by the first Chern class of line bundles and the Euler characteristic are additive. Assertion (2) follows from Proposition 23.58. Finally, (3) follows by (1) from Corollary 23.48. Proposition 23.73. Let f : X → Y be a morphism of proper k-schemes, let L1 , . . . , Lt b be line bundles on Y and let F be an object in Dcoh (X) with [F ] ∈ K0′ (X)t . Then one has (f ∗ L1 · · · f ∗ Lt · F ) = (L1 · · · Lt · Rf∗ F ). Proof. For every line bundle L on Y we have c1 (f ∗ L ) · [F ] = ([OX ] − [f ∗ L ⊗−1 ]) · [F ] = f ∗ ([OY ] − [L ⊗−1 ]) · [F ] and hence the equality follows from the projection formula (Proposition 23.65). If f is a closed immersion, then Rf∗ F = f∗ F . In particular for every closed immersion i : Z → Y and for all line bundles L1 , . . . , Lt on Y with t ≥ dim Z we obtain (23.15.3)

(i∗ L1 · · · i∗ Lt · Z) = (L1 · · · Lt · Z).

Example 23.74. Let X = Prk . Then every line bundle on X is of the form OX (d) for an integer d ∈ Z (Example 11.45). Let Li = OX (di ) with di ∈ Z for i = 1, . . . , r. Then the Snapper polynomial is of the form σOX (n1 , . . . , nr ) = χ(OX (d1 )⊗n1 ⊗ · · · ⊗ OX (dr )⊗nr ) = χ(OX (d1 n1 + · · · + dr nr ))   d 1 n1 + · · · + d r nr + r = , r where the last equality holds by Example 23.61. In particular (OX (d1 ) · · · OX (dr ) · Prk ) = d1 d2 · · · dr . Example 23.75. Let k be a field. (0) Let X → Spec k be a proper scheme, and let F be a coherent OX -module with dim Supp F = 0. Then one has (F ) = χ(X, F ) = dimk Γ(X, F ) for the intersection of F with 0 line bundles. (1) Let C → Spec k be a proper curve (i.e. C → Spec k is proper and all irreducible components of C have dimension 1). Let L be a line bundle on C. Then (L · C) = χ(OC ) − χ(L ⊗−1 ) = deg(L ), where the last equality is the definition of deg(L ) given in Section (15.9). Using deg(L −1 ) = − deg(L ) it follows that for all n ∈ Z one has (23.15.4)

χ(L ⊗n ) = n deg(L ) + χ(OC ).

325 (2) Now let X → Spec k be a proper surface (i.e. X → Spec k is proper and all irreducible components of X have dimension 2). Let C, D ⊆ X be effective Cartier divisors with no common irreducible component. Then Z := C ∩ D is either empty or finite of dimension 0 and X (C · D · X) = dimk (OZ,z ) = dimk (Γ(Z, OZ )). z∈Z

Indeed, as C and D are equi-dimensional of dimension 1, it is clear that Z has dimension ≤ 0. Hence it is finite and the second equality holds (Proposition 5.20). Consider the exact sequence 0 −→ OX (−D) ⊗ OC → OC → OZ → 0. Then dimk Γ(Z, OZ ) = χ(OZ ) = χ(OC ) − χ(OX (D)⊗−1 ) (∗) = χ(c1 (OX (D)) · OC ) = χ(c1 (OX (D))c1 (OX (C))OX ) = (C · D · X), where (*) holds by Proposition 23.58 (3). This shows that for X = P2k the intersection number (C · D · X) equals the intersection number i(C, D) defined in Section (5.14) in a rather ad hoc way. We obtain the following quick proof of B´ezout’s theorem 5.61 using that (C · D · X) depends only on OX (C) and OX (D) and is bilinear in the line bundles: If C = V+ (f ) and D = V+ (g) for non-constant homogeneous polynomials f, g ∈ k[T0 , T1 , T2 ] with deg(f ) =: n and deg(g) =: m. Then OX (C) ∼ = OX (n) and OX (D) ∼ = OX (m). If m = n = 1, then we can choose C and D to be different lines, and the intersection number is 1. In general one has (C · D · P2k ) = (O(n) · O(m) · P2k ) = (O(1)⊗n · O(1)⊗m · P2k ) = nm. We could also have used Example 23.74. Remark and Definition 23.76. Let f : X → Y be a morphism between integral k-schemes that are proper over k. Then f is proper and hence f surjective

⇐⇒

dim Y = dim f (X).

If f is surjective and dim X = dim Y , then the function field K(X) is a finite extension of K(Y ). The degree of f is defined to be the following non-negative integer: ( [K(X) : K(Y )], if dim(X) = dim(Y ) = dim(f (X)); deg(f ) := 0, otherwise. If f is finite locally free, then the degree defined in Definition 23.76 is the same as the degree defined for finite locally free morphisms in (12.6.1).

326

23 Cohomology of projective and proper schemes

Proposition 23.77. Let k be a field and let f : X → Y be a morphism of proper schemes over k. Let Z ⊆ X be an integral closed subscheme of dimension d and let L1 , . . . , Ld be line bundles on Y . Then (f ∗ L1 · · · f ∗ Ld · Z) = deg(f |Z : Z → f (Z))(L1 · · · Ld · f (Z)), where f (Z) is considered as closed integral subscheme of Y . Proof. By (23.15.3) we may replace X by Z and Y by f (Z) and hence may assume that X is integral of dimension d, Y is integral, and that f is surjective. In particular, dim Y ≤ d. We have (f ∗ L1 · · · f ∗ Ld · X) = (L1 · · · Ld · Rf∗ OX )

(*)

by Proposition 23.73. If dim Y < d, then deg(f ) = 0 and the right hand side of (*) is zero by Corollary 23.72 (1). Hence we can assume that dim Y = d. Then f is finite over a non-empty open subscheme V of Y . Therefore one has for all i > 0 that Ri f∗ F |V = 0 (Corollary 22.5) and hence dim Supp Ri f∗ F < d. This shows that, denoting by η ∈ Y the generic point, (1)

(L1 · · · Ld · Rf∗ OX ) = (L1 · · · Ld · f∗ OX ) (2) = dimK(Y ) (f∗ OX )η (L1 · · · Ld · Y ). where (1) and (2) hold by Corollary 23.72 (1) and (3). As deg(f ) = dimK(Y ) (f∗ OX )η , this shows the claim. (23.16) Asymptotic Riemann Roch theorem. Let k be a field. Proposition 23.78. (Asymptotic Riemann-Roch theorem) Let X be a proper k-scheme b (X), and let L be a line bundle on X. Then of dimension d, let F be in Dcoh χ(L ⊗n ⊗L F ) =

(L d · F ) d n + g(n), d!

for all n ∈ Z, where g(n) is a polynomial of degree < d which depends only on X, F , and the line bundle L . Proof. We use Theorem 23.68 with t = 1. By definition, χ(c1 (L )i · [F ]) = (L i · F ), so   d X n+i−1 (L i · F ) . χ(L ⊗n ⊗L F ) = i i=0 We see that the left hand side is a sum of polynomials in n of degree ≤ i for i = 0, . . . , d and that the d-th summand is   n+d−1 (L d · F ) d (L d · F ) = n + terms of degree < d. d d! Proposition 23.79. Let k be a field and let X be a proper k-scheme. Let F = ̸ 0 be a coherent OX -module and set t := dim Supp F . Let L1 , . . . , Lt be ample line bundles on X. Then (L1 · · · Lt · F ) > 0.

327 Proof. By Corollary 23.72 (3) it suffices to show that (L1 · · · Lt · Z) > 0 for every closed integral subscheme Z of dimension t of X. We use induction on t, the case t = 0 holding by Example 23.75 (0). Suppose t > 0. By (23.15.3), we may assume that Z = X. In particular, X is integral. By the multilinearity of the intersection number we may replace Lt by some positive power, hence we can assume that there exists 0 ̸= s ∈ Γ(X, Lt ) such that its non-vanishing locus Xs is affine since Lt is ample. This section is automatically regular because X is integral and its vanishing locus V (s) is non-empty because otherwise Xs = X is affine and proper over k which would imply that X is finite (Corollary 12.89), contradicting that t > 0. Hence i : V (s) → X is an effective Cartier divisor with OX (V (s)) ∼ = Lt . By Proposition 23.58 (3) and using the induction hypothesis we then find (L1 · · · Lt · X) = (L1 · · · Lt−1 · i∗ OV (s) ) = (i∗ L1 · · · i∗ Lt−1 · OV (s) ) > 0, where we use (23.15.3) for the second equality. We will prove the converse to this statement, the Nakai-Moishezon criterion, in Section (23.90). (23.17) The degree of a closed subscheme. Let k be a field. Definition 23.80. Let X be a proper scheme over k. Let L be a line bundle on X. For a closed subscheme Z of X the integer (23.17.1)

degL (Z) := (L dim Z · Z)

is called the degree of Z with respect to L . We also call (23.17.2)

deg(L ) := degL (X)

the degree of L . If X = Prk for some r ≥ 1, then (23.17.3)

deg(Z) := degOX (1) (Z)

is simply called the degree of Z. Hence for every projective scheme Z over k and for every closed immersion i : Z → Prk we have the notion of the degree of (Z, i) by identifying Z with a closed subscheme of Prk via i. Note that the degree of (Z, i) depends on i, see Exercise 23.36. The degree is preserved by base change in the following sense. Remark 23.81. Let k be a field, X be a proper k-scheme, Z ⊆ X a closed subscheme, and let L be a line bundle on X. Let k ′ be a field extension of k, and set X ′ := X ⊗k k ′ , Z ′ := Z ⊗k k ′ , and let L ′ be the pullback of L to X ′ . Then Remark 23.71 shows that degL ′ (Z ′ ) = degL (Z). Remark 23.82. If L is an ample line bundle on a proper k-scheme and Z ⊆ X is a non-empty closed subscheme, then degL (Z) > 0 by Proposition 23.79. By Serre’s vanishing theorem we obtain the following special case of the asymptotic Riemann-Roch theorem.

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23 Cohomology of projective and proper schemes

Proposition 23.83. Let X be a proper k-scheme of dimension d and let L be an ample line bundle on X. Then Γ(X, L ⊗n ) =

degL (X) d n + O(nd−1 ). d!

Proof. This follows from Proposition 23.78 and the fact that H i (X, L ⊗n ) = 0 for large n and i > 0 because L is ample (Theorem 23.2). Proposition 23.84. Let X and Y be integral proper k-schemes and let f : X → Y be a finite dominant morphism. Then for every line bundle L on Y one has the equality deg(f ∗ L ) = deg(f ) deg(L ) Proof. This follows from Proposition 23.77. Remark 23.85. Let k be a field and let X be a proper scheme over k of dimension d. Let L be a line bundle on X. Then deg(L ⊗k ) = k d deg(L ),

for all k ∈ Z

by Corollary 23.72 (2). We have by now introduced at several occasions the notion of degree of a line bundle or the degree of a subvariety. Let us show in the remainder of this section that all these notions coincide. Remark 23.86. If X is a proper curve over k and L is a line bundle on X, then deg(L ) is the degree defined in Section (15.9) by Example 23.75 (1). Example 23.87. Let X = Prk and let f ∈ k[T0 , . . . , Tr ] be homogeneous of degree d ≥ 1. Then deg(V+ (f )) = (OX (1)r−1 · OV+ (f ) ) = (OX (1)r−1 · OX (d)) = d by Example 23.74. Remark 23.88. Let X be a proper k-scheme and let Z ⊆ X be a closed subscheme of dimension 0. Then f : Z → Spec k is finite and hence finite locally free and by Example 23.75 (0) for every line bundle L on X one has degL (Z) = dimk Γ(Z, OZ ) = deg(f ), where the second equality holds by the definition of the degree in Definition (12.6.1). Recall that in Section (14.31) we defined the degree of a closed subscheme X of Pnk X as follows. Let d := dim(X). We defined the incidence k-scheme H X = Hn−d such that for any field extension k ′ of k the k ′ -valued points H X (k ′ ) are the set of pairs (x, Λ) where x ∈ X(k ′ ) and Λ ⊆ Pnk′ is a linear subspace of dimension n − d with x ∈ Λ (see Remark 13.86 for a description of H X as a k-scheme). The scheme H X is projective, and it is irreducible if and only if X is irreducible (Proposition 14.134). Let L = Ln−d be the scheme of linear subspaces of Pnk of dimension n − d and let q : H X → L,

(x, Λ) 7→ Λ

329 be the projection. Hence for every field extension k ′ and for every linear subspace Λ ∈ L(k ′ ) of Pnk′ one has q −1 (Λ) = X(k ′ ) ∩ Λ(k ′ ). Then q is generically finite, i.e., the fiber of q over the generic point η of L is finite (Proposition 14.135), and we defined deg(X) as the degree of the finite κ(η)-scheme Xd := q −1 (η), i.e., as dimκ(η) H 0 (Xd , OXd ). This can be expressed more geometrically as follows: As the property “finite” is a constructible property (Proposition 10.96), we find an open neighborhood U ⊆ L of η such that the restriction q −1 (U ) → U of q is finite. By generic flatness (Corollary 10.85), we can shrink U such that q −1 (U ) → U is finite and flat and hence finite locally free (Proposition 12.19). As L is irreducible, U is irreducible, and q −1 (U ) → U has constant rank equal to deg(X). In other words, there exists an open dense subscheme U of L such that for every field extension k ′ and every linear subspace Λ ⊆ Pnk′ corresponding to a k ′ -valued point of U , we have that deg(X) is the k ′ -dimension of H 0 (Xk′ ∩ Λ, OXk′ ∩Λ ), i.e., the “number of points of Xk′ ∩ Λ counted with multiplicities”. Note that it is crucial to consider extension fields of k here since in general we could have U (k) = ∅. As L is isomorphic to a Grassmannian and hence has an open covering by affine spaces, this can only happen if k is finite. More classically this is expressed by writing Λ as the intersection of d hyperplanes H1 , . . . , Hd and by saying that deg(X) is the “number of points of Xk′ ∩ H1 ∩ · · · ∩ Hd counted with multiplicities for hyperplanes Hi in general position”. Proposition 23.89. For a closed subscheme X ⊆ Pnk of projective space over a field k, the two notions of degree defined in (23.17.3) and in Section (14.31) coincide. Proof. For the moment, we denote the degree defined in (23.17.3) by deg′ (X). We keep the above notation, in particular we choose U as above. As deg′ (X) is compatible with base change k → k ′ (Remark 23.81), we may assume that k ′ = k is algebraically closed. Then we are given hyperplanes H1 , · · · , Hd in Pnk such that Xd := X ∩ H1 ∩ · · · ∩ Hd has dimension 0. This implies that Xi := X ∩ H1 ∩ · · · ∩ Hi has dimension d − i for all 0 ≤ i ≤ d. By definition we have deg(X) = dimk H 0 (Xd , OXd ). Since every noetherian scheme has only finitely many associated points, we even can find hyperplanes such that in addition for all i ≥ 1 the hyperplane Hi does not contain any associated point of Xi−1 = X ∩ H1 ∩ · · · ∩ Hi−1 . Then Xi is a Cartier divisor in Xi−1 whose line bundle is the restriction of OPnk (1) to Xi−1 . Thus we see by (23.15.3) and Proposition 23.58 (3) that c1 (OPnk (1)) · OXi−1 = OXi ⊗L OX

i−1

OXi−1 = OXi

and hence deg′ (X) = χ(c1 (OPnk (1))d · OX ) = χ(OXd ) = dimk H 0 (Xd , OXd ) = deg(X). (23.18) The Nakai-Moishezon criterion for ampleness. The following theorem is a useful criterion to check ampleness of a line bundle in terms of intersection numbers. We will use it in Section (25.31) to prove that every smooth proper surface over a field, i.e., every smooth proper scheme over a field that is equi-dimensional of dimension 2, is projective. Theorem 23.90. (Nakai-Moishezon criterion for ampleness) Let k be a field and let X be a proper k-scheme. A line bundle L on X is ample if and only if for every integral closed subscheme Z ⊆ X with d := dim Z > 0, one has (L d · Z) > 0.

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23 Cohomology of projective and proper schemes

If X is a proper curve over k, the test objects Z are the irreducible components of X with the reduced scheme structure, and (L · Z) = deg(L|Z ) by Example 23.75 (1). Compare Proposition 26.57. See also the version for surfaces in Theorem 25.146 below. For globally generated line bundles, it suffices to check the criterion for integral closed curves (Exercise 26.7). Proof. If L is ample, then Proposition 23.79 shows that the intersection numbers in question are all positive. To show the converse, we proceed in several steps. (I). We may reduce to the case that X is integral. In fact, L satisfies the numerical criterion of the theorem if and only if its restriction to each irreducible component (with the reduced scheme structure) of X satisfies the condition, because the test objects Z are the same, and (L d · Z) depends only on Z and the restriction L|Z (see (23.15.3)). On the other hand, by Corollary 23.9 we may check ampleness of a line bundle by restricting it to each of the reduced irreducible components. (II). Now assume that X is integral and that (L d · Z) > 0 for all integral closed subschemes Z ⊆ X, d = dim Z. By induction we may assume that the theorem holds for all proper k-schemes of smaller dimension. (Note that for dim X = 0, the statement is clearly true.) In particular, the induction hypothesis shows that L|W is ample on W for every closed subscheme W ⊊ X. Note that it is enough to show that some positive power of L is ample, so we may replace L by a positive power whenever that is convenient. Let us show that some positive power of L has a non-trivial global section (in other words, some power of L is the line bundle attached to an effective divisor on X). The key point is to prove the following Claim. For all q ≥ 2 and all n ≫ 0, we have dimk H q (X, L n ) = dimk H q (X, L n−1 ). Of course, we expect this to hold, and more precisely expect these cohomology groups to vanish if n is large by Theorem 23.6. Once the claim is proved, we find n0 such that X χ(L n ) = dimk H 0 (X, L n ) − dimk H 1 (X, L n ) + (−1)i dimk H i (X, L n0 ) i≥2

for all n ≥ n0 , with the sum on the very right independent of n. But the asymptotic Riemann-Roch theorem, Proposition 23.78, together with our assumption on L for Z = X, shows that χ(L n ) tends to infinity for n → ∞. This is only possible if H 0 (X, L n ) ̸= 0 for n large. Proof of claim. Let I , J ⊂ OX be non-zero quasi-coherent ideal sheaves such that I ⊗L ∼ = J . To see that such ideal sheaves exist, we can identify L with an OX submodule of the constant sheaf KX of rational functions on X (see Proposition 11.29). This also gives us an embedding L −1 ⊂ KX , and we may then set I = L −1 ∩ OX , and J := I L ⊆ OX . Now consider the exact sequences 0 → I ⊗ L n → L n → OX /I ⊗ L n → 0, 0 → J ⊗ L n−1 → L n−1 → OX /J ⊗ L n−1 → 0. By definition of I and J , the terms on the left are isomorphic. We may view the terms on the right as powers of the restriction of L to the closed subscheme defined by I and J , respectively. These restrictions are ample by induction hypothesis, so for n sufficiently large the cohomology of the terms on the right vanishes in all positive degrees. Comparing the two long exact cohomology sequences then proves the claim.

331 (III). Let us show that some positive power of L is generated by its global sections. By the previous step we may assume that L ∼ = OX (D) for an effective Cartier divisor D on X. Consider the short exact sequence 0 → L n−1 → L n → L n ⊗ OD → 0. By induction hypothesis, L|D is ample, and hence H 1 (X, L n ⊗OD ) vanishes for sufficiently large n. In that case we obtain an exact sequence H 0 (X, L n ) → H 0 (X, L n ⊗ OD ) → H 1 (X, L n−1 ) → H 1 (X, L n ) → 0. In particular, dimk H 1 (X, L n ) ≤ dimk H 1 (X, L n−1 ), and since these k-vector spaces are finite-dimensional, we get equality for n sufficiently large. If equality holds, the map H 0 (X, L n ) → H 0 (X, L n ⊗ OD ) in the above exact sequence is surjective. We know already that L|D is ample on D, and increasing n further, if required, L n ⊗ OD is generated by global sections. Lifting a generating family to H 0 (X, L n ) and adding a section in H 0 (X, L n ) which has zero set D (and hence does not vanish anywhere on X \ D), we obtain a family of global sections of L n generating this line bundle. (IV). We can now conclude. As we have shown, we may assume that L is globally generated, and hence defines a morphism f : X → PN k into some projective space with ∼ L f ∗ O PN (1) . Then f is proper, since X is proper. Furthermore, f is quasi-finite because = k otherwise there would exist an integral curve C ⊆ X which is mapped under f to a point. However, then (L · C) = (f ∗ OPN (1) · C) = 0 by Proposition 23.77, in contradiction to k our assumption. Thus f is quasi-finite and proper, hence finite (Corollary 12.89). Since L is isomorphic to the pullback of an ample line bundle under a finite morphism, it is itself ample by Proposition 13.83. (23.19) Hilbert polynomials of proper schemes over a field. In this section we fix a field k, a proper k-scheme X, and an ample line bundle L on X, in particular, X is projective over k. For every complex F of OX -modules we set F (n) := F ⊗OX L ⊗n for all n ∈ Z. b Proposition and Definition 23.91. In the above situation, let F be in Dcoh (X). Then

ΦX,L ,F (n) := ΦF (n) := χ(X, F (n)) is a numerical polynomial in n, called the Hilbert polynomial of F and of (X, L ). (1) If F is a coherent OX -module, one has d := deg ΦX,L ,F = dim Supp(F ) and ΦF has leading term

(L d · F ) . d!

Moreover, (L d · F ) is a positive integer if F ̸= 0. (2) The map F 7→ ΦF induces a homomorphism of abelian groups (23.19.1)

Φ : K0′ (X) −→ Q[T ].

If F = OX , we simple write ΦX or ΦX,L .

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23 Cohomology of projective and proper schemes

b Proof. The Hilbert polynomial is additive in distinguished triangles of objects in Dcoh (X) because the Euler characteristic has this property and tensoring with line bundles is a triangulated functor. Therefore we obtain (2). Let us show (1). Let F be a coherent OX -module, endow Z := Supp F with the scheme structure given by the annihilator ideal of F (Definition 23.20), and let i : Z → X be the inclusion. Then F = i∗ i∗ F and χ(X, F (n)) = χ(Z, i∗ F (n)) by Proposition 23.65. Hence we may assume that X = Supp F . By Proposition 23.78 we already know that ΦF is a numerical polynomial of degree ≤ d := dim X and that its degree d coefficient is (L d · F )/d!. If F = ̸ 0 is a coherent OX -module, it is positive by Proposition 23.79.

Remark 23.92. If F is a coherent OX -module, then the vanishing of higher cohomology of F (n) for large n (Theorem 23.2) shows that one has ΦF (n) = dimk H 0 (X, F (n)),

n ≫ 0.

L Example 23.93. Let S = d≥0 Sd be a graded k-algebra of finite type that is generated by S1 , let X := Proj(S) and fix the very ample line bundle L = OX (1). Let M be a ˜ be the associated coherent OX -module. graded S-module of finite type and let F := M Then there exists an N ≥ 1 such that one has (23.19.2)

ΦF (n) = dimk Mn ,

for all n ≥ N .

Indeed, for large n one has dimk Mn = dim H 0 (X, F (n)) (Section (13.5)) and one can use Remark 23.92. The map n 7→ dimk Mn is sometimes called the Hilbert function of M . Now let f ∈ S be a homogeneous regular element of degree d and let H := V+ (f ) be the corresponding hypersurface in X. Then from the closed subscheme exact sequence 0 −→ OX (−d) −→ OX −→ OH −→ 0 one obtains by (23.19.1) (23.19.3)

ΦH (n) = ΦX (n) − ΦX (n − d).

If we apply the example to S = k[T0 , . . . , Tr ] so that X = Prk , one gets: Example 23.94. Let r ≥ 1 be an integer. Then   r+n ΦPrk (n) = . r This also follows from Example 23.61. If H ⊆ Prk is a hypersurface of degree d ≥ 1, then (23.19.3) shows     r+n r+n−d ΦH (n) = − . r r Example 23.95. Let C be a proper curve over k and let L be an ample line bundle on C (we will see in Section (26.5) below that such a line bundle always exists). By (23.15.4) the Hilbert polynomial ΦC,L is the numerical polynomial deg(L )T + χ(OC ) ∈ Q[T ].

333

The Grothendieck-Riemann-Roch theorem In this part of the chapter we will briefly explain Grothendieck’s variant of the RiemannRoch theorem, mostly without proof. To keep the exposition as simple as possible, we do not strive for a general version but work only with smooth quasi-projective varieties over a field. More precisely, in this section we denote by k a field and by Sm = Smk the full subcategory of the category of k-schemes consisting of smooth quasi-projective k-schemes. Every morphism in Smk is then quasi-projective ([EGAII] O (5.3.4)) and locally of complete intersection (Corollary 19.44). In particular, every closed immersion is regular. Often the Grothendieck-Riemann-Roch theorem is formulated as a compatibility statement between K-theory and Chow theory. We will not introduce Chow groups here but choose a more axiomatic approach following Grothendieck [Gro2] O , Panin [Pan] O and O Navarro [Nav] O X . Following in particular the article [Nav] X very closely, we make the following definition. (23.20) Cohomology theories. The following definition axiomatizes some nice properties that a “cohomology theory” should have. Note however that these properties are inspired from algebraic topology more than from algebraic geometry. Cohomology of coherent sheaves does not satisfy these conditions and thus is not a cohomology theory in the sense of this definition. We will give examples in Section (23.22). Definition 23.96. A cohomology theory over k is a functor A : Smopp −→ (Ring), k

X 7→ A(X),

f 7→ f ∗ ,

endowed for any proper morphism f : X → Y in Smk with a functorial morphism f∗ : A(X) → A(Y ) of A(Y )-modules, called direct image, i.e., one has (idX )∗ = idA(X) , (f ◦ g)∗ = f∗ ◦ g∗ and the projection formula (23.20.1)

f∗ (f ∗ (y)x) = yf∗ (x),

x ∈ A(X), y ∈ A(Y ).

If L is a line bundle on X, its zero section yields the zero section s0 : X → V(L ∨ ) ∗ (Section (11.3)). Then cA 1 (L ) := s0 (s0∗ (1)) ∈ A(X) is called the Chern class of L . These data are supposed to satisfy the following conditions. ` (a) For X1 , X2 in Smk with natural ` immersions ij : Xj → X1 X2 , j = 1, 2, the ring homomorphism (i∗1 , i∗2 ) : A(X1 X2 ) → A(X1 ) × A(X2 ) is an isomorphism. (b) Strong homotopy invariance: Let π : E → X be a vector bundle torsor (i.e., there exists a geometric vector bundle E0 → X such that E is an E0 -torsor over X for the Zariski topology). Then π ∗ : A(X) → A(E) is an isomorphism. (c) If i : Y → X is a smooth closed subscheme and j : X \ Y → X the open immersion of the complement, then the sequence i

j∗

∗ A(Y ) −→ A(X) −→ A(X \ Y )

is exact.

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23 Cohomology of projective and proper schemes

(d) Let Y′

i′

/ X′

i

 /X

g

f

 Y

be a cartesian diagram of k-schemes that are in Smk such that i : Y → X is a closed immersion and such that the natural homomorphism of conormal sheaves g ∗ Ci → Ci′ is an isomorphism, i.e., f is transversal for Y (see Exercises 18.27 and 18.28). Then one has f ∗ ◦ i∗ = i′∗ ◦ g ∗ . (e) Let X be in Smk and let E be a finite locally free OX -module of constant rank r and let π : P(E ) → X be the corresponding projective bundle. Then for every morphism f : Y → X in Smk one has a commutative diagram A(P(E ))

φ∗

π∗

 A(X)

/ A(P(f ∗ E )) ϖ∗

f

∗

 / A(Y ),

where φ : P(f ∗ (E )) = P(E ) ×X Y → P(E ) and ϖ : P(f ∗ (E )) → Y are the projections. Moreover, let xE := cA 1 (OP(E ) (1)) be the first Chern class of the tautological bundle OP(E ) (1) on P(E ). Then A(P(E)), made into an A(X)-algebra via π ∗ : A(X) → A(P(E)), is a free A(X)-module with basis (1, xE , x2E , . . . , xr−1 E ). If A and A˜ are cohomology theories over k, then a morphism of cohomology theories φ : A → A˜ is a morphism of contravariant functors preserving direct images for proper morphisms. A cohomology theory A is called multiplicative (resp. additive) if one has A A A A cA 1 (L1 ⊗ L2 ) = c1 (L1 ) + c1 (L2 ) − c1 (L1 )c1 (L2 )

(resp. A A cA 1 (L1 ⊗ L2 ) = c1 (L1 ) + c1 (L2 )

)

for all line bundles L1 and L2 on a scheme X in Smk . A cohomology L theory A is graded if A(X) is endowed with the structure of a graded ring A(X) = d≥0 Ad (X) such that f ∗ is homogeneous of degree 0 for every morphism f in Smk and such that f∗ is homogeneous of degree dim Y − dim X for every proper morphism f : X → Y between irreducible schemes X and Y in Smk . Remark and Definition 23.97. Let A be a cohomology theory over k. (1) Let i : Y → X be a closed immersion in Smk . Then i is a regular immersion (Theorem 19.30) and [Y ]A := i∗ (1) ∈ A(X) is called the fundamental class of Y in A(X). (2) Property (a) implies that A(∅) = 0. (3) The hypotheses in (c) are satisfied if Y ′ = ∅. In this case the condition simply means that f ∗ ◦ i∗ = 0.

335 (4) We can apply (c) to the situation of (a). As both i1 and i2 are transversal to i1 and i2 , one has i∗2 i1∗ = 0, i∗1 i2∗ = 0, i∗1 i1∗ = id, i∗2 i2∗ = id. Therefore the inverse of the isomorphism (i∗1 , i∗2 ) is given by a ∼ i1∗ + i2∗ : A(X1 ) × A(X2 ) → A(X1 X2 ). ` ` Hence we find for any proper morphism f = f1 f2 : X1 X2 → Z that f∗ = f1∗ + f2∗ : A(X1 ) × A(X2 ) → A(Z). (5) The formation of Chern classes is compatible with pullback: Let f : Y → X be a morphism in Smk and let L be a line bundle on X. Consider the cartesian diagram Y

s′0

/ Y ×X V(L ∨ ) = V(f ∗ L ∨ ) p2

f

 X

 / V(L ∨ ),

s0

where s′0 denotes the zero section of V(f ∗ L ∨ ) → Y . It shows that p2 is transversal for s0 . We therefore find ∗ ∗ ′∗ ∗ ′∗ ′ A ∗ f ∗ cA 1 (L ) = f s0 s0∗ (1) = s0 φ s0∗ (1) = s0 s0∗ (1) = c1 (f L ).

Remark 23.98. Let i : D → X be a smooth Cartier divisor in X, i.e., i is a regular closed immersion of codimension 1 given by an invertible ideal I ⊆ OX . The canonical section of OX (D) = I ⊗−1 (Remark 11.33) is a global regular section sD ∈ Γ(X, OX (D)) whose vanishing locus is D. We view it as a section sD : X → V(I ) = Spec(Sym(I )) of the geometric line bundle p : V(I ) → X. It corresponds to the unique homomorphism of OX -algebras s∗D : Sym I → OX whose restriction to I = Sym1 (I ) is the inclusion I ,→ OX . Then sD is transversal to the zero section s0 : X → V(I ) as we have a cartesian diagram i /X D sD

i

 X

s0

 / V(I ).

Therefore we find (23.20.2)

c1 (OX (D)) = s∗0 (s0∗ (1)) = s∗D (s0∗ (1)) = i∗ (i∗ (1)) = i∗ (1) = [D],

where the second equality holds since s∗0 and s∗D are both inverse maps to the isomorphism p∗ , hence they are equal. (23.21) Chern classes of vector bundles. The following proposition, called the splitting principle shows that given a vector bundle E on X in Smk , we can find a smooth proper cover X ′ of X such that the class [E ] ∈ K0 (X ′ ) splits as a sum of classes of line bundles. This is very useful to reduce statements about vector bundle to the case of line bundles which typically is much easier to handle. More precisely, we have the following result.

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23 Cohomology of projective and proper schemes

Proposition 23.99. (Splitting principle) Given X in Smk and a vector bundle E on X of rank r, there exists a proper smooth morphism π : X ′ → X satisfying the following properties. (i) There exists a filtration of OX ′ -modules π∗ E = E 0 ⊃ E 1 ⊃ · · · ⊃ E r = 0 such that Ll := E r−l /E r−l+1 is a line bundle for l = 1, . . . , r. (ii) The homomorphism π ∗ : A(X) → A(X ′ ) is injective for all cohomology theories A on Smk . Proof. We proceed by induction on r, the cases r = 0, 1 being trivial. Let ϖ : P(E ) → X be the projective bundle of E . The universal line bundle on P(E ) sits in an exact sequence (see Section (13.8)) 0 −→ E ′ −→ ϖ∗ E −→ OP(E ) (1) −→ 0, of vector bundles on P(E ), where E ′ is a vector bundle of rank r−1. The map ϖ∗ : A(X) → A(P(E )) is injective for all cohomology theories A by Definition 23.96 (c). Now we conclude by the induction hypothesis. Definition 23.100. Let A be a cohomology theory over k. Let X be in Smk and let E be a vector bundle over X of rank r. Let xE = cA 1 (OP(E ) (1)) ∈ A(P(E )) be the first Chern class of the universal line bundle on P(E ). Multiplication by xE defines an endomorphism mE of the free A(X)-module A(P(E )). The characteristic polynomial of mE is denoted by r A r−1 χA + · · · + (−1)r cA E = T − c1 (E )T r (E ) ∈ A(X)[T ] A and cA i (E ) ∈ A(X) is called the i-th Chern class of E . One also sets c0 (E ) := 1 and A ci (E ) := 0 for i > r. The polynomial −1 A 2 A r cA (E ) := (−T )r χA ) = 1 + cA E (−T 1 (E )T + c2 (E )T + · · · + cr (E )T

is called the Chern polynomial of E . If E = L is a line bundle, then P(L ) = X and OP(L ) (1) = L and hence cA 1 (L ) as defined in Definition 23.100 equals cA 1 (L ) as given by the cohomology theory. The Chern polynomial of L has the form cA (L ) = 1 + cA 1 (L )T. Remark 23.101. Let A be a cohomology theory. For any morphism f : X → Y in Smk ∗ ∗ A and for every vector bundle E on Y one has cA i (f E ) = f ci (E ). ∗ Indeed, this follows easily because X ×Y P(E ) = P(f E ) and since first Chern classes of line bundles are compatible with pullback (Remark 23.97 (5)). Proposition 23.102. Let A be a cohomology theory. For every exact sequence 0 → E ′ → E → E ′′ → 0 of vector bundles on a scheme in Smk one has χA (E ) = χA (E ′ )χA (E ′′ ), cA (E ) = cA (E ′ )cA (E ′′ ), X ′ A ′′ cA cA n (E ) = i (E )cj (E ). i+j=n

One can prove cA (E ) = cA (E ′ )cA (E ′′ ) by induction on the rank of E ′ . The other two identities are then immediate corollaries. We refer to [Nav] O X 1.2 for details.

337 r Example 23.103. Let X be in Smk and let E = OX be the trivial vector bundle of rank r r. As cA (O ) = 0, an induction on r using Proposition 23.102 shows that cA X 1 i (OX ) = 0 for all i > 0. r If 0 → E ′ → OX → L → 0 is an exact sequence, where L is a line bundle, then ′ A ′ Proposition 23.102 shows that for all n > 0 one has 0 = cA n (E ) + cn−1 (E )c1 (L ) and hence ′ i ′ i cA 0 ≤ i ≤ n. n (E ) = (−1) cn−i (E )c1 (L )

In particular, one obtains for i = n = r that c1 (L )r = 0. Such an exact sequence exists for every line bundle that is generated by finitely many global sections, so we see that the Chern classes of such line bundles are nilpotent. See Proposition 23.106 for a generalization. Remark and Definition 23.104. Choose π : X ′ → X and line bundles L1 , . . . , Lr on X ′ as in the splitting principle (Proposition 23.99). Then Proposition 23.102 shows π ∗ χA (E ) =

r Y

(T − cA 1 (Ll )),

π ∗ cA (E ) =

l=1

r Y

′ (1 + cA 1 (Ll )T ) ∈ A(X )[T ].

l=1

′ ∗ A The elements αl := cA 1 (Ll ) ∈ A(X ) are called Chern roots of E . Then π ci (E ) is the i-th elementary symmetric polynomial of the Chern roots X (23.21.1) π ∗ cA α l1 · · · α li . i (E ) = 1≤l1 1. As C is proper, smooth, and geometrically connected, we have dim H 0 (C, OC ) = 1 by Proposition 12.66. Setting g := dim H 1 (C, OC ) we therefore have χ(C, OC ) = 1 − g. Hence (23.23.6) yields Z 1 1 1 1 − g = χ(C, OC ) = (1 + c1 (TC/k )) = deg(TC/k ) = − deg(Ω1C/k ) 2 2 2 C

and hence deg(TC/k ) = 2 − 2g. Now we can again apply Hirzebruch-Riemann-Roch and get for every line bundle L on C Z 1 χ(C, L ) = (1 + c1 (TC/k ))(1 + c1 (L )) 2 C Z 1 = (1 + c1 (TC/k ) + c1 (L )) 2 C

1 deg(TC/k ) + deg(L ) 2 = 1 − g + deg(L ).

=

This is a variant of the Riemann-Roch theorem in the special case that C is smooth, see Section (26.11) below.

343

Cohomology and base change We now come to the question how cohomology is compatible with base change. Let X′

g′

f′

 S′

g

/X  /S

f

be a cartesian diagram of schemes. Then for every complex E in Dqcoh (X) there is a functorial base change morphism (Proposition 21.129) Lg ∗ Rf∗ E −→ Rf∗′ L(g ′ )∗ E. If f is qcqs, and f and g are tor-independent, e.g., if f is flat, then the base change morphism is an isomorphism by Theorem 22.99 and Remark 22.106. Unfortunately, this gets more complicated if one is interested in non-derived base change for higher direct images, i.e., in the question whether for a quasi-coherent module F and an integer i ≥ 0 one has an isomorphism ∼ g ∗ Ri f∗ F −→ Ri f∗′ g ′∗ F . This is the case if g is flat by Proposition 22.90. But a case which is not covered by this result but which is geometrically very interesting and where something useful can be said is the case where S ′ is the spectrum of a residue class field of a point of S. We approach this problem in two main steps. We gave already several criteria when Rf∗ E is pseudo-coherent or even perfect. Hence first we study the “Linear Algebra” problem how pseudo-coherent or perfect complexes on S and their cohomology are compatible with derived and non-derived pullback. It will turn out that passage to fibers, i.e., to residue fields of points of S, is the crucial case to understand. This will be done in Sections (23.24)–(23.26). Then we apply these “Linear Algebra” results to the special case that the perfect complex is the derived direct image Rf∗ E of a complex E on X in Sections (23.27) and (23.28). The main result will be Theorem 23.140 and its many applications. (23.24) Semicontinuity of Betti numbers for pseudo-coherent complexes. Recall (Remark 21.138) that a complex of OS -modules E on a locally ringed space S is called perfect if it is locally on S isomorphic in D(S) to a finite complex of finite free OS -modules. A complex E is called pseudo-coherent if locally on S for all m there exists +1 a distinguished triangle P −→ E −→ C −→ in D(S) with P perfect and H p (C) = 0 for all p ≥ m. If S is a scheme, then a complex E is pseudo-coherent if and only if it is locally on S isomorphic in D(S) to a bounded above complex of finite free OS -modules (Corollary 22.47). For more basic facts about perfect and pseudo-coherent complexes on schemes see Sections (22.13) and (22.15). Let S be a locally ringed space, let s ∈ S be a point, and let (23.24.1)

is : Spec κ(s) → S

344

23 Cohomology of projective and proper schemes

be the canonical morphism of locally ringed spaces with image {s}. Let E be a complex in D(S). Then we set L ∗ E ⊗L OS κ(s) := E ⊗ κ(s) := Lis E.

(23.24.2)

Definition and Remark 23.116. Let S be a locally ringed space. (1) Fix i ∈ Z and let E be a pseudo-coherent complex. Then the function (23.24.3)

bi := biE : S → Z,

s 7→ dimκ(s) H i (E ⊗L OS κ(s))

is called the Betti function for E. As E ⊗L κ(s) is a pseudo-coherent complex in D(κ(s)), bi (s) is indeed a finite number ≥ 0. (2) Let E be a perfect complex in D(S). Then the function X (23.24.4) χE : S → Z, s 7→ χ(E ⊗L κ(s)) = (−1)i dimκ(s) H i (E ⊗L κ(s)) i

is called the Euler characteristic (function) of E. As the perfect complex E ⊗L κ(s) is isomorphic in D(κ(s)) to a finite complex of finite-dimensional vector spaces, the sum is finite. Proposition 23.117. Let S be a scheme, and let E ∈ D(S) be pseudo-coherent. (1) Let i ∈ Z. The Betti function bi : S → Z is upper semicontinuous and constructible, i.e., for all n the subsets of the form {s ∈ S; bi (s) ≥ n}

(23.24.5)

are closed and constructible. (2) For all integers i ∈ Z and n ≥ 0 the set {s ∈ S; bi (s) = n}

(23.24.6)

is locally closed and constructible. (3) Suppose that E is perfect. Then the Euler characteristic χE : S → Z is locally constant on S. Recall from Section (10.10) that a closed subset of a qcqs scheme S is constructible if and only if its open complement is quasi-compact. A locally closed subset of S is constructible if and only if it is the intersection of a closed constructible and an open quasi-compact subset. In a general scheme S, a subset C is constructible if and only if there exists an open covering (Ui )i by qcqs schemes such that C ∩ Ui is constructible in Ui for all i. Proof. All assertions can be checked locally on S and after passing to an isomorphic complex in the derived category. We may therefore assume that E is a bounded above complex of free O-modules of finite rank, ···

/ E i−1

di−1

/ Ei

di

/ E i+1

Since the E i are free, E ⊗L OS κ(s) = E ⊗OS κ(s) and

di+1

/ ··· .

345 bi (s) = dim H i (E ⊗OS κ(s)) (23.24.7)

= dim Ker(di ⊗ κ(s)) − dim Im(di−1 ⊗ κ(s)) = dim Coker(di ⊗ κ(s)) + dim Coker(di−1 ⊗ κ(s)) − dim E i−1 ⊗ κ(s)

where dim E i−1 ⊗ κ(s) = rk E i−1 is independent of s. If E is perfect, then we can assume that E is bounded. Taking the alternating sum of these terms, the dimensions of the cokernels cancel, and (3) follows. In general, Corollary 7.31 and the remark following it shows that the set (*)

{s ∈ S; dimκ(s) Coker(di ) ⊗ κ(s) ≥ r}

is closed for every r ∈ Z. Since Coker(di ) ⊗ κ(s) ∼ = Coker(di ⊗ κ(s)), this implies (1) by (23.24.7). Assertion (2) follows from (1). It remains to show that the sets in (23.24.5) and (23.24.6) are constructible. We fix i ∈ Z. As being constructible can be checked locally, we may assume that S = Spec A is affine. It suffices to show that (23.24.5) is constructible. Again by (23.24.7) it suffices to show that (*) is constructible. But this is the locus where the s-th exterior power of di vanishes, where s = rk(E i+1 ) − r. This is a closed constructible subset of S as it is defined by a finitely generated ideal of A. Remark 23.118. Let S be a scheme. From Example 23.60 one deduces the following assertions. (1) If E ′ → E → E ′′ → is a distinguished triangle of perfect complexes on S, then χE = χE ′ + χE ′′ . (2) For perfect complexes E and F on S one has χE⊗L F = χE χF . The Betti function and the Euler characteristic are compatible with derived pull back in the following sense. Proposition 23.119. Let f : S ′ → S be a morphism of schemes and let E be a pseudocoherent complex in D(S). Then one has for all i ∈ Z that biLf ∗ E = biE ◦ f. If E is perfect, then one also has χLf ∗ E = χE ◦ f. Proof. It suffices to show the first equality. For s′ ∈ S ′ set s := f (s′ ). Let g : Spec κ(s′ ) → Spec κ(s) be the morphism corresponding to the field extension κ(s) → κ(s′ ) induced by f . As g is flat, we have Lg ∗ = g ∗ and H i (g ∗ V ) = H i (V ) ⊗κ(s) κ(s′ ) for every complex V in D(κ(s)). We have f ◦ is′ = is ◦ g with is and is′ defined in (23.24.1). Therefore biLf ∗ E (s′ ) = dimκ(s′ ) (H i (Lg ∗ Li∗s E)) = dimκ(s) (H i (i∗s E)) = biE (s). (23.25) Base change for pseudo-coherent complexes. For A a ring, and E ∈ Dqcoh (Spec A) = D(A), we introduce the following notation: T i := TEi : (A-Mod) → (A-Mod),

M 7→ H i (E ⊗L M )

346

23 Cohomology of projective and proper schemes

Considering this functor on the category of all A-modules (in contrast to only looking at A-algebras) is a crucial point in the theory of cohomology and base change developed below. More generally, whenever S is a not necessarily affine scheme and E ∈ Dqcoh (S), then we have by Proposition 22.40 a functor T i := TEi : QCoh(S) → QCoh(S),

M 7→ H i (E ⊗L M )

which we denote by the same symbol. These functors are compatible with restriction to open subschemes, and if S is affine, then we recover the previous version (up to taking global sections and applying the e· -construction, respectively). Whenever 0 → M1 → M2 → M3 → 0 is a short exact sequence of quasi-coherent OS -modules, we obtain a long exact sequence · · · → T i (M1 ) → T i (M2 ) → T i (M3 ) → T i+1 (M1 ) → · · · . More precisely, the family (Ti )i is a δ-functor (Definition F.201). Returning to the affine situation S = Spec A, E ∈ Dqcoh (S), for any A-module M we have a natural morphism β i (M ) : H i (E) ⊗ M → H i (E ⊗L M ). In fact, the existence of β i follows from the following general lemma, applied to T = T i . Lemma 23.120. Let A be a ring, and let T : (A-Mod) → (A-Mod) be an A-linear functor (Definition F.28). (1) For all A-modules M we have natural homomorphisms, functorial in M , β(M ) : T (A) ⊗A M → T (M ). (2) The morphism of functors β is an isomorphism if and only if T is right exact and commutes with arbitrary direct sums. The functor T is right exact and commutes with arbitrary direct sums if and only if it commutes with arbitrary colimits (Proposition F.19). Proof. The functor T induces an A-module homomorphism M = HomA (A, M ) → HomA (T (A), T (M )), which by the adjunction between Hom and ⊗ gives the desired homomorphism. To prove part (2), first note that the functor M 7→ T (A) ⊗A M clearly is right exact and commutes with direct sums. Conversely, assume that T has these properties. Clearly, β(A) is an isomorphism, and since T commutes with direct sums, the same holds for every free A-module. For every A-module M one can choose an exact sequence E ′ → E → M → 0 with E and E ′ free A-modules. Using the right exactness, we obtain that T (M ) is an isomorphism. Similarly, we have a version for schemes S which are not necessarily affine. Let E be in Dqcoh (S). With notation as before, we obtain a morphism of functors (OS -Mod) → (OS -Mod) (23.25.1)

β i (M ) : H i (E) ⊗OS M → H i (E ⊗L M ).

347 If g : S ′ → S is a morphism of schemes, we obtain a homomorphism of quasi-coherent OS ′ -modules (23.25.2)

g ∗ H i (E) −→ H i (Lg ∗ E).

If S = Spec R and S ′ = Spec R′ and we consider E as a complex of R-modules (Lemma 22.36), then this corresponds to the canonical homomorphism of R-modules ′ H i (E) ⊗R R′ −→ H i (E ⊗L R R ).

If s ∈ S is a point, we obtain in particular the base change map (23.25.3)

β i (κ(s)) : H i (E) ⊗OS κ(s) → H i (E ⊗L κ(s)).

We will use the following two linear algebra lemmas which we will prove simultaneously. Lemma 23.121. Let R be a ring and let d : M → N be an R-linear map of finitely generated projective R-modules. Then the following assertions are equivalent. (i) Coker(d) is a finitely generated projective R-module. (ii) Im(d) is a direct summand of N and for every R-module Q one has Im(d) ⊗R Q = Im(d ⊗ idQ ). (iii) Im(d) is a finitely generated projective R-module and for every s ∈ Spec R one has Im(d) ⊗R κ(s) = Im(d ⊗ idκ(s) ). (iv) Ker(d) is a direct summand of M and for every R-module Q one has Ker(d) ⊗R Q = Ker(d ⊗ idQ ). (v) For every s ∈ Spec R the map Ker(d) ⊗R κ(s) → Ker(d ⊗ idκ(s) ) is surjective. Lemma 23.122. Let R be a ring, let d : M → N be an R-linear map of finitely generated projective R-modules. Let s ∈ Spec R. Then Ker(d) ⊗ κ(s) → Ker(d ⊗ κ(s)) is surjective if and only if there exists f ∈ R with s ∈ D(f ) such that Coker(d)f is finitely generated projective (and hence df : Mf → Nf satisfies the equivalent conditions of Lemma 23.121). Proof. We start with the proof of Lemma 23.121. If (i) holds, the exact sequence 0 → Im(d) → N → Coker(d) → 0 is split and hence stays exact after tensoring with any module Q. As Coker(d ⊗ idQ ) = Coker(d) ⊗ Q, this implies (ii). Clearly, one has “(ii) ⇒ (iii)”. If Im(d) is projective, then the exact sequence 0 → Ker(d) → M → Im(d) → 0 splits and if the formation of Im(d) commutes with tensoring by Q (resp. by κ(s)), the same holds for the formation of Ker(d). Hence (ii) implies (iv), and (iii) implies (v). Clearly, (iv) implies (v). It remains to show that (v) implies (i). For this it suffices to show that the condition in Lemma 23.122 is necessary. Hence suppose that Ker(d) ⊗ κ(s) → Ker(d ⊗ κ(s)) is surjective. As Coker(d) is of finite presentation, it suffices to show that the localization of Coker(d) in ps is a free module (Proposition 7.27). Therefore we may assume that R is a local ring, and s is the closed point of Spec R. Then Ker(d) → Ker(d ⊗ κ(s)) is surjective and M and N are free of ranks m and n, say. Set κ := κ(s) and r := rk(d ⊗ κ). We will show that Im(d) is a direct summand of N of rank r. Then Coker(d) is isomorphic to a direct summand of N and hence projective. Let d¯: M/ Ker(d) → N be the induced injective map. By Proposition 8.10 it suffices to show that d¯ ⊗ idκ is injective.

348

23 Cohomology of projective and proper schemes

As Ker(d) ⊗ κ → Ker(d ⊗ κ) is surjective, we may choose x1 , . . . , xt ∈ Ker(d) that map to a basis of Ker(d ⊗ κ). We also choose xt+1 , . . . , xm in M whose images in the quotient (M ⊗ κ)/ Ker(d ⊗ κ) are a basis. Then x1 , . . . , xm generate M by Nakayama’s lemma and hence form a basis because M is free of rank m = dimκ (M ⊗ κ). As x1 , . . . , xt ∈ Ker(d), for i > t, the images x ¯i of xi in M/ Ker(d) generate this module. Now d¯ ⊗ κ maps x ¯i ⊗ 1 to (d ⊗ κ)(xi ⊗ 1) for i > t and these elements are linearly independent. This shows that d¯ ⊗ κ is injective. This concludes the proof of the necessity of the condition in Lemma 23.122 and of all equivalences in Lemma 23.121. The sufficiency in Lemma 23.122 now follows from applying Lemma 23.121 to the map df = d ⊗ idRf : Mf → Nf of Rf -modules. The central local result in this section is the following criterion for splitting up a pseudo-coherent complex as a direct sum of complexes concentrated in degrees below and above some degree, respectively. Proposition 23.123. Let R be a ring, let E be a pseudo-coherent complex of R-modules, and fix i ∈ Z. Let s ∈ Spec R be a point such that the base change map (23.25.4)

β i (κ(s)) : H i (E) ⊗ κ(s) → H i (E ⊗L κ(s))

is surjective. Then there exists f ∈ R such that s ∈ U := D(f ), and such that we have, setting E |U := E ⊗R Rf : (1) We have a decomposition in D(Rf ) E|U ∼ = τ ≤i (E|U ) ⊕ τ ≥i+1 (E|U ).

(23.25.5)

(2) The truncated complex τ ≥i+1 (E|U ) is a perfect complex of Rf -modules. If E is of tor-amplitude in (−∞, b] where b ∈ Z, then τ ≥i+1 (E|U ) is of tor-amplitude in [i + 1, b]. The complex τ ≤i (E|U ) is pseudo-coherent. It is perfect if E |U is perfect. (3) The Rf -linear map β i (M ) : H i (E |U ) ⊗ M → H i (E |U ⊗L M ) is an isomorphism for every Rf -module M . Therefore the functor T i : (Rf -Mod) → (Rf -Mod),

M 7→ H i (E |U ⊗L Rf M )

commutes with arbitrary colimits and in particular is right exact. Conversely, if the functor T i commutes with base change for M = κ(s), i.e., if the natural map yields an isomorphism H i (E |U ) ⊗ M ∼ = H i (E |U ⊗L M ) for M = κ(s), this says precisely that (23.25.4) is an isomorphism. Proof. Replacing E by an isomorphic complex in D(R), we may assume that E is a bounded above complex of finite free modules. Then E is K-flat and hence E ⊗L RM = E ⊗R M for every R-module M . Moreover, if b is an integer such that E is of tor-amplitude in (−∞, b] (e.g., an integer b such that H p (E) = 0 for all p > b), we may assume that E i = 0 for all i > b (Remark 22.46). By hypothesis, the map di

Ker(E i −→ E i+1 ) ⊗ κ(s) → Ker(di ⊗ κ(s))/ Im(di−1 ⊗ κ(s))

349 is surjective. Hence we may choose y1 , . . . , yt ∈ Ker(di ) that map to a basis of Ker(di ⊗ κ)/ Im(di−1 ⊗ κ). Choose also x′1 , . . . , x′r ∈ E i−1 such that the images of di−1 (x′k ) in E i ⊗ κ form a basis of Im(di−1 ⊗ κ). Then the di−1 (x′k ) and the yl are contained in Ker(di ) and generate Ker(di ⊗ κ(s)) and hence Ker(di ) ⊗ κ(s) → Ker(di ⊗ κ(s)) is surjective. Therefore we can use Lemma 23.122. It shows that, after replacing Spec R by an open affine neighborhood of s, if necessary, Im(di ) and Coker(di ) are free. In particular τ ≥i+1 E is perfect of tor-amplitude in [i + 1, b]. The projection E i+1 → Coker(di ) admits a section, yielding a homomorphism of complexes τ ≥i+1 (E) → E. Hence there is a quasiisomorphism of complexes τ ≤i (E) ⊕ τ ≥i+1 (E) → E and therefore E has a decomposition as in (1). Moreover, as direct summands of pseudo-coherent (resp. perfect) complexes are again pseudo-coherent (resp. perfect) by Proposition 21.156 (resp. by Proposition 21.176), τ ≤i E is again pseudo-coherent (resp. perfect if E is perfect). It remains to show (3). We have H i (E) = Coker(di−1 : E i−1 → Ker(di )). As we have already seen that Im(di ) = E i / Ker(di ) is free, Ker(di ) is a direct summand of E i and hence its formation commutes with ⊗M . As this is also true for formations of cokernels, we see that H i (E ⊗ M ) = H i (E) ⊗ M . The last assertion follows from Lemma 23.120. Corollary 23.124. Let R be a ring, and let E be a pseudo-coherent complex in D(R). Let i ∈ Z be such that the base change map β i (κ(s)) : H i (E) ⊗ κ(s) → H i (E ⊗L κ(s)) is surjective for every closed point s ∈ Spec R. Then the base change homomorphism β i (M ) : H i (E) ⊗R M → H i (E ⊗L R M ) is an isomorphism for every R-module M . Corollary 23.125. Let S be a scheme, let E ∈ D(S) be pseudo-coherent, and fix i ∈ Z. Let s ∈ S such that H i (E ⊗L κ(s)) = 0. Then there exists an affine open neighborhood U of s such that H i (E)|U = 0 and such that (23.25.6)

E |U ∼ = τ ≤i−1 E |U ⊕ τ ≥i+1 E |U

with τ ≥i+1 E perfect of tor-amplitude in [i + 1, ∞). See also Lemma 27.227 for a more general statement. Proof. By Proposition 23.123 we can write E = τ ≤i E ⊕ τ ≥i+1 E with τ ≥i+1 E perfect of tor-amplitude in [i + 1, ∞), τ ≤i E pseudo-coherent, and (*)

H i (E ⊗L κ(s)) = H i (E) ⊗ κ(s) = 0

after replacing S be an open affine neighborhood of s. Therefore τ ≤i E can be represented by a complex L of finite free modules with Lp = 0 for p > i (Remark 22.46). But then H i (E) = H i (L) = Coker(Li−1 → Li ) is of finite presentation and hence zero in an open affine neighborhood U of s by (*). Hence τ ≤i E |U → τ ≤i−1 E |U is an isomorphism in D(U ) and (23.25.6) follows from Proposition 23.123 (1). Proposition 23.126. Let S be a scheme, let E be a pseudo-coherent complex on S and let a ≤ b be integers. (1) Suppose that E is locally bounded below (e.g., if E is perfect) and let s ∈ S. Then E is perfect of tor-amplitude in [a, b] in an open neighborhood of s if and only if H i (E ⊗L κ(s)) = 0 for all i ∈ / [a, b].

350

23 Cohomology of projective and proper schemes

(2) The complex E is perfect of tor-amplitude contained in [a, b] if and only if H i (E ⊗L κ(s)) = 0 for all i ∈ / [a, b] and for all s ∈ S. Proof. For both assertions, the condition are clearly necessary. Let us show that in (1) the condition is sufficient. Working locally on S, we may assume that E ∈ Db (S) since E is pseudo-coherent and locally bounded below. Since H i (E ⊗L κ(s)) = 0 for all i ∈ / [a, b], by Corollary 23.125, we may replace S by an open neighborhood of s such that H i (E) = 0 for all i ∈ / [a, b] and such that E = τ ≥a E = τ ≤b τ ≥a E is perfect of tor-amplitude in [a, b]. Now we show the sufficiency in (2). Corollary 23.125 implies that H i (E)s = 0 for all i∈ / [a, b] and all s ∈ S. Hence E ∈ D[a,b] (S). Now we can apply (1) as we can check the condition on the tor-amplitude locally on S by Lemma 21.171. Proposition 23.127. Let S be a scheme and let E ∈ D(S) be pseudo-coherent. Let s0 ∈ S be a point. Assume that the base change homomorphism β i (κ(s0 )) is surjective. Then there exists an open affine neighborhood U of s0 such that β i (G ) : H i (E |U ) ⊗OU G → H i (E |U ⊗L OU G ) is an isomorphism for every quasi-coherent OU -module G . Moreover, the following assertions are equivalent: (i) The homomorphisms β i−1 (κ(s0 )) is surjective. (ii) There exists an open affine neighborhood V of s0 such that the OV -module H i (E |V ) is finite locally free. In this case, the Betti function s 7→ dimκ(s) H i (E ⊗L κ(s)) is constant on an open neighborhood of s0 . Proof. We may assume without loss of generality that S = Spec R is affine. We denote the pseudo-coherent complex of R-modules corresponding to E again by E. Then the first assertion follows from Proposition 23.123. Replacing S by U , we may assume (*)

H i (E) ⊗R M = H i (E ⊗L R M)

for every R-module M . Hence T i = H i (E ⊗L R −) is right exact and it is left exact if and only if T i−1 is right exact. If β i−1 (κ(s0 )) is surjective, then passing to an open affine neighborhood of s0 , we may assume that T i−1 is in fact right exact, so T i is exact. Therefore H i (E) is flat. By Proposition 23.123, τ ≤i E is a direct summand of E and hence is again pseudo-coherent. The map β i−1 (κ(s0 )) is still surjective for τ ≤i E. Hence we can apply Proposition 23.123 again for i − 1 and τ ≤i E. Then we see that τ ≥i τ ≤i E = H i (E)[−i] is a perfect complex (concentrated in degree i). In particular, H i (E) is pseudo-coherent and therefore of finite presentation by Proposition 21.153. Hence H i (E) is finitely generated and projective (Corollary 7.42). Conversely, if H i (E) is flat, then T i = H i (E) ⊗ − is exact, so T i−1 is right exact. Since it also commutes with direct sums (Remark 21.120), it follows from Lemma 23.120 that β i−1 is an isomorphism of functors. Finally the i-th Betti function is locally constant since bi (s) = dimκ(s) H i (E ⊗L κ(s)) = dimκ(s) H i (E) ⊗ κ(s) for all s ∈ S by (*). Proposition 23.127 has the following converse if S is reduced.

351 Proposition 23.128. Let S be a reduced scheme, let E be a pseudo-coherent complex in D(S), and fix i ∈ Z. Suppose that the Betti function bi : s 7→ dimκ(s) H i (E ⊗L κ(s)) is locally constant on S. Then the base change homomorphisms β j (G ) : H j (E) ⊗OS G −→ H j (E ⊗L OS G ) are isomorphisms for j = i, i − 1 and for all quasi-coherent OS -modules G , and H i (E) is finite locally free. Proof. We may assume that S = Spec R is affine and that E is a bounded above complex of finite free OS -modules. Looking back to the proof of Proposition 23.117, in particular at (23.24.7), we see that the sum of the upper semicontinuous maps s 7→ dim Coker(di ) ⊗ κ(s)

and

s 7→ dim Coker(di−1 ) ⊗ κ(s).

is locally constant since bi is locally constant. Therefore these two maps are also lower semicontinuous and hence locally constant on S. Since S is reduced by assumption, we obtain that Coker(di ) and Coker(di−1 ) are finite locally free (Corollary 11.19). As Coker(di ) is finite locally free, for every quasi-coherent OS -module G the short exact sequence 0 → Im(di ) ⊗ G → E i+1 ⊗ G → Coker(di ) ⊗ G → 0 splits. Hence Im(di ) is finite locally free. Since Coker(di ⊗ G ) ∼ = Coker(di ) ⊗ G , the i i i sequence shows that Im(d ) ⊗ G = Im(d ⊗ G ). As Im(d ) is finite locally free, the short exact sequence 0 → H i (E) ⊗ G → Coker(di−1 ) ⊗ G → Im(di ) ⊗ G → 0 is split. As the formation of Im(di ) and Coker(di−1 ) commute with − ⊗ G this also holds for the formation of H i (E) which shows that β i (G ) is an isomorphism for all G . Moreover, the second exact sequence of G = OS shows that H i (E) is finite locally free because Coker(di−1 ) is finite locally free. Hence Proposition 23.127 implies that β i−1 (G ) is an isomorphism for all quasi-coherent modules G . Remark 23.129. Let S be a scheme, let E be a complex in D(S), and let i ∈ Z. Suppose that for every quasi-coherent OS -module G the base change homomorphism β i (G ) : H i (E) ⊗OS G −→ H i (E ⊗L OS G ) is an isomorphism. Then for every morphism of schemes g : S ′ → S one has functorial isomorphisms of OS ′ -modules (23.25.7)

g ∗ H i (E) ∼ = H i (Lg ∗ E).

Indeed, to check that the functorial homomorphism g ∗ H i (E) → H i (Lg ∗ E) is an isomorphism we can assume that S = Spec R and S ′ = Spec R′ are affine. Then we can apply the hypothesis to the quasi-coherent OS -algebra G = g∗ OS ′ .

352

23 Cohomology of projective and proper schemes

(23.26) Subschemes classifying properties of perfect complexes. If E is a perfect complex of tor-amplitude in an interval [a, b], then the condition that H a (E) or H b (E) are locally free of a fixed rank are represented by locally closed subschemes, more generally: Proposition 23.130. Let S be a scheme, let E be a perfect complex in D(S) of toramplitude in an interval [a, b], and let I ⊆ [a, b] be an interval containing a or b. Fix a map r : I → Z≥0 , i 7→ ri . Then there exists a unique locally closed subscheme j : Z = Zr → S such that a morphism f : T → S factors through Z if and only if for all morphisms g : T ′ → T and for all i ∈ I the OT ′ -module H i (L(f ◦ g)∗ E) is locally free of rank ri . Moreover, (1) the immersion j : Z → X is of finite presentation, (2) as a set one has (23.26.1)

Z = { s ∈ S ; bi (s) = ri for all i ∈ I}, ¯

f j (3) if f : T → S factors as T −→ Z −→ X, then H i (Lf ∗ E ⊗OT G ) = f¯∗ H i (Lj ∗ E) ⊗OT G for all i ∈ I and for all quasi-coherent OT -modules G .

Proof. Since Z is characterized by a universal property, the uniqueness assertion is clear. Hence, we may work locally on S and can assume that S = Spec R is affine. By Proposition 22.53, we may assume that E is given by a complex of finitely generated projective R-modules concentrated in degrees [a, b], which we again denote by E. Again working locally on S, we may assume that E i is a free R-module of rank ni , say, for all i ∈ Z. Note that then Lf ∗ E = f ∗ E for every morphism f : T → S. By hypothesis, the interval I is of the form [a, b′ ] for some b′ ∈ [a, b] or of the form [a′ , b] for some a′ ∈ [a, b]. We first consider the case that I = [a, b′ ]. The condition that da

H a (E) = Ker(E a −→ E a+1 ) is locally free and that its formation commutes with base change is equivalent to the condition that Coker(da ) is locally free by Lemma 23.121. In this case, H a (E) has rank ra if and only if Coker(da ) has rank ta := na+1 − na + ra . Furthermore, in this case Im(da ) is a direct summand of E a+1 , it is locally free of rank na+1 − ta = na − ra and its formation commutes with base change (again by Lemma 23.121). Hence we now can apply Lemma 23.121 to E a+1 / Im(da ) → E a+2 and see that H a+1 (E) is locally free of rank ra+1 and its formation commutes with base change, and hence it is a direct summand of E a+1 / Im(da ), if and only if Coker(da+1 ) is locally free of rank ta+1 := (na+2 − na+1 + na ) + (ra+1 − ra ). Proceeding by induction one sees that for i ≥ a the R-module H i (E) is locally free of rank ri and that its formation commutes with base change if and only if Coker(di ) is locally free of rank ti :=

i−a X j=−1

(−1)j+1 ni−j +

i−a X

(−1)j ri−j .

j=0

In fact Lemma 23.121 shows that in this case, the formation of Ker(di ) and of Im(di−1 ) also commutes with tensoring by any R-module Q. Hence for every R-module Q we have (*)

H i (E ⊗R Q) = H i (E) ⊗R Q.

353 By Theorem 11.18 there is a subscheme Zi = F=ti (Coker(di )) such that a morphism f : T → S factors though Zi if and only of f ∗ Coker(di ) is locally free of T rank ti . As formation of Coker(di ) always commutes with base change, we can take Z = i∈I Zi , the scheme-theoretic intersection. To construct Z in the case that I = [a′ , b], one proceeds similarly by starting with db−1

H b (E) = Coker(E b−1 −→ E b ) and shows inductively for i = b, b − 1, . . . , a′ that H i (E) is locally free of rank ri and that its formation commutes with base change if and only if Coker(di−1 ) is locally free of rank si :=

b−i X

(−1)j−1 ni+j +

(−1)j ri+j .

j=0

j=1

T

b−i X

i−1

Hence one can set Z := a′ ≤i≤b F=si (Coker(d ). As Coker(di ) is of finite presentation, all immersions F=r (Coker(di )) → S are of finite presentation (see the remark after Theorem 11.18), hence Z → S is of finite presentation. A point s ∈ S is contained in Z if and only if H i (E ⊗OS κ(s)) is of rank ri and its formation commutes with base change T → Spec κ(s). But this base change is flat and hence the second conditions holds automatically. This shows (23.26.1). The last assertion holds by construction of Z using (*). Proposition 23.131. Let S be a scheme and let E be a perfect complex in D(S). Let a ≤ b be integers. Then there exists a unique open constructible subscheme U of S such that a morphism f : T → S factors through U if and only if Lf ∗ E is of tor-amplitude in [a, b]. Proof. The uniqueness assertion is clear. Hence we can assume that S = Spec R and that E is of some tor-amplitude contained in a finite interval I (Theorem 21.174). Then Lf ∗ E also has tor-amplitude in I for every morphism f : T → S. By using Proposition 23.126 and using that the formation of the Betti function is compatible with inverse image (Proposition 23.119) we see that the subscheme U := { s ∈ S ; rk H i (E ⊗L κ(s)) ≤ 0 for all i ∈ I \ [a, b]}, which is open and constructible by Proposition 23.117, has the desired properties. Corollary 23.132. Let S be a scheme, and let E ∈ D(S) be perfect. Let a, r ∈ Z with r ≥ 0. There exists a unique open subscheme U ⊆ S such that a morphism f : T → S factors through U if and only if Lf ∗ E is isomorphic to a locally free OT -module of rank r placed in degree a. Proof. We apply Proposition 23.131 to the interval [a, a] using that a perfect complex is of tor-amplitude in [a, a] if and only if it is isomorphic to a finite locally free module placed in degree a (Proposition 22.53). (23.27) Criteria when direct images are perfect and commute with base change. Let f : X → S be a morphism of schemes, let E ∈ D(X) and let F be a complex of OX -modules. Then we say that the formation of Rf∗ (E ⊗L OX F ) commutes with base change if the following holds. For every morphism S ′ → S of schemes let X ′ = X ×S S ′ , and consider the cartesian diagram

354

23 Cohomology of projective and proper schemes g′

X′ f′

 S′

/X f

g

 / S.

Then the following natural morphism is an isomorphism ∼

′ ′ ∗ L ′ ∗ θ : Lg ∗ Rf∗ (E ⊗L OX F ) → Rf∗ (L(g ) E ⊗OX ′ (g ) F )

in D(S ′ ). Note that we do not use L(g ′ )∗ F in the term on the right hand side of the isomorphism θ. See Exercise 23.42 for an example where the variant of Theorem 23.133 below with L(g ′ )∗ F in place of (g ′ )∗ F fails. We have seen in Corollary 22.103 that Rf∗ (E ⊗L OX F ) commutes with base change if E ∈ Dqcoh (X) and F is a bounded above complex of quasi-coherent OX -modules that are flat over S. We will now study criteria, when Rf∗ (E ⊗L OX F ) is in addition perfect or at least pseudo-coherent. Theorem 23.133. Let f : X → S be a proper morphism of finite presentation (e.g., S locally noetherian and f proper). Let E ∈ D(X) be perfect (resp. pseudo-coherent), and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Then Rf∗ (E⊗L OX F ) ∈ D(OS -Mod) is perfect (resp. pseudo-coherent) and the formation F ) commutes with base change. of Rf∗ (E ⊗L OX We will prove the theorem only if S is locally noetherian and sketch the proof for general S if E is perfect. Then one can show the result for pseudo-coherent complexes by approximating pseudo-coherent complexes by perfect complexes. We refer to [Sta] 0CSC for this. Proof. The formation of Rf∗ (E ⊗L OX F ) commutes with base change by Corollary 22.103. F To show that F := Rf∗ (E ⊗L ) is perfect (resp. pseudo-coherent), we may assume OX that S = Spec R is affine. (I). Let us first assume that R is noetherian and that E is pseudo-coherent. Then X is noetherian and F is pseudo-coherent (Proposition 22.61). Hence E ⊗L OX F is pseudocoherent (Proposition 22.63). Therefore F is pseudo-coherent by Proposition 23.23. (II). Now suppose that E is perfect and R arbitrary. We will show that F is of finite tor-amplitude. As X is quasi-compact, there exist integers a ≤ b such that locally on X, the complex E can be represented by a complex of finite free modules E which is concentrated in degrees [a, b] (Proposition 22.53). Suppose that F is concentrated in degrees [c, d] for integers c ≤ d. Then E ′ := E ⊗L OX F = E ⊗OX F is locally on X represented by a complex concentrated in degrees [a + c, b + d] such that each term is a direct sum of the F i . In particular it is represented by a complex whose terms are flat over S. As tor-amplitude can be checked locally (Lemma 21.171), this implies that E ′ has tor-amplitude in [a + c, b + d] as an object of D(X, f −1 OS ) (Proposition 21.169). Let us show that F = Rf∗ (E ′ ) is of tor-amplitude in [a + c, b + d + N ], where N ≥ 1 is some number such that X can be covered N + 1 open affine subschemes. By Lemma 22.51, it is enough to show that H i (F ⊗L / [a + c, b + d + N ] and all OS K ) = 0 for all i ∈ quasi-coherent OS -modules K . For the upper bound b + d + N it suffices to show that H i (F ) = 0 for all i > b + d + N (Remark 21.165 (2)). This follows from Corollary 22.10.

355 To obtain the lower bound a + c, we apply the projection formula (Proposition 22.84) and obtain ′ L ′ L ∗ ′ L −1 ∼ ∼ F ⊗L K ). OS K = Rf∗ (E ) ⊗OS K = Rf∗ (E ⊗OX Lf K ) = Rf∗ (E ⊗f −1 OS f

As E ′ is represented by a complex concentrated in degrees ≥ a + c that is K-flat as −1 K is concentrated in a complex of f −1 OS -modules, the cohomology of E ′ ⊗L f −1 OS f degrees ≥ a + c for all quasi-coherent OS -modules K , and Rf∗ preserves this property. (III). Combining Steps (I) and (II) proves that F is perfect if E is perfect and R noetherian by Theorem 21.174. (IV). Now we show that F is perfect if E is perfect and if R is arbitrary. For this write R as a filtered colimit of noetherian rings Rλ . By the techniques explained in Chapter 10 (for details we refer to [Sta] 0A1H, see also Remark 22.55), we find for some λ a proper morphism fλ : Xλ → Sλ := Spec Rλ such that X ∼ = S ×Sλ Xλ , a perfect complex Eλ on Xλ whose derived pullback to X is E, and a bounded complex Fλ of coherent OXλ -modules that are flat over Sλ whose pullback to X is F . If E is of tor-amplitude contained in [a, b] (resp. F is concentrated in degrees [c, d]), we may assume that Eλ (resp. Fλ ) has the same property. If N is an integer such that X can be covered by N + 1 open affine subschemes, then we may assume that Xλ can be covered by N + 1 open affine subschemes. We have already shown in Step (III) that Rfλ,∗ (Eλ ⊗L Fλ ) is perfect of tor-amplitude in [a + c, b + d + N ]. Moreover, its formation commutes with base change and hence F is its derived pullback under S → Sλ . This show that F is perfect of tor-amplitude in [a + c, b + d + N ]. With only a little more work, the assumption that f be proper can be replaced by requiring that the support of the terms of the complex F are proper over S (Definition 23.21), see [Sta] 0A1H and 0CSC. Remark 23.134. Suppose that E is perfect of tor-amplitude in an interval [a, b] and that F is concentrated in degrees [c, d] in the situation of Theorem 23.133. Then the proof shows that Rf∗ (E ⊗L OX F ) is perfect of tor-amplitude ≥ a + c. Moreover, if S is affine and X can be covered by N + 1 open affine subschemes, then the tor-amplitude of Rf∗ (E ⊗L OX F ) is contained in [a + c, b + d + N ]. For instance, if F is a module of finite presentation that is flat over S, then the tor-amplitude of Rf∗ F is contained in [0, N ]. Let us record the following special cases of the previous result: Corollary 23.135. Let f : X → S be a proper morphism of finite presentation and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Then Rf∗ F is perfect and for all cartesian diagrams of schemes X′

g′

f′

 S′

/X f

g

 / S.

one has Lg ∗ Rf∗ F = Rf∗′ ((g ′ )∗ F ).

356

23 Cohomology of projective and proper schemes

Corollary 23.136. Let f : X → S be a flat proper morphism of finite presentation and let E be a perfect (resp. pseudo-coherent) complex on X. Then Rf∗ E is a perfect (resp. pseudo-coherent) complex in D(S) and for all cartesian diagrams of schemes X′

g′

f′

/X f

 S′

g

 / S.

one has Lg ∗ Rf∗ (E) = Rf∗′ (L(g ′ )∗ E). Corollary 23.137. Let S = Spec A be an affine noetherian scheme, and let f : X → S be a proper morphism. Let F be a coherent OX -module which is flat over S. Then there exists a finite complex K • of finitely generated projective A-modules and isomorphisms, functorial in the A-algebra B, ∼

H i (X ×S Spec B, F ⊗A B) → H i (K • ⊗A B),

for all i ≥ 0.

For a direct proof of this version of the result see [Mum1] § 5. By Remark 23.134, we can even find K • such that H i (K • ) = 0 for all i ∈ / [0, N ], where N ≥ 0 is the minimal number such that X can be covered by N + 1 open affine subschemes. (23.28) Semicontinuity theorems and base change of higher direct images. We fix a cartesian diagram of schemes X′ (23.28.1)

g′

f′

 S′

/X f

g

 / S.

Let f be qcqs and let F be a complex of quasi-coherent OX -modules. Then we have a functorial homomorphism of OS ′ -modules (23.28.2)

g ∗ Ri f∗ F −→ Ri f∗′ g ′∗ F .

It is the composition of the following two morphisms. (1) The base change morphism (23.28.3)

β : g ∗ Ri f∗ F → H i (Lg ∗ Rf∗ F )

given by (23.25.2) applied to Rf∗ F which is in Dqcoh (S) since f is qcqs (Theorem 22.31). (2) The homomorphism (23.28.4)

γ : H i (Lg ∗ Rf∗ F ) −→ H i (Rf∗′ g ′∗ F ) = Ri f∗′ g ′∗ F ,

obtained by applying H i (−) to the composition Lg ∗ Rf∗ F −→ Rf∗′ Lg ′∗ F −→ Rf∗′ g ′∗ F , where the first map is the derived base change morphism (Definition 21.129) and the second is induced by the functorial map Lg ′∗ F → g ′∗ F .

357 If g is flat, the construction of (23.28.2) given here specializes to the construction of (22.21.3) and we already showed in Proposition 22.90 and Remark 22.108 that (23.28.2) is an isomorphism in this case. Definition 23.138. Let i ∈ Z. Then we say that the formation of Ri f∗ F commutes with base change if (23.28.2) is an isomorphism for every morphism of schemes g : S ′ → S. If U ⊆ S is an open subscheme, then we say that the formation of Ri f∗ F |U commutes with base change if the formation of Ri fU,∗ F |f −1 (U ) commutes with every base change U ′ → U . Here fU : f −1 (U ) → U is the restriction of f . In this section we will focus on the following special situation in which we know Rf∗ F to be perfect by Theorem 23.133 and γ (23.28.4) to be an isomorphism by Corollary 22.103. (S) Let f : X → S be a proper morphism of finite presentation. Let F be a bounded complex of OX -modules of finite presentation that are flat over S. These hypotheses are for instance satisfied if S is locally noetherian, f is proper, and F is a coherent OX -module that is flat over S. We will also show in Theorem 23.159 below that for every OX -module F of finite presentation there exists a stratification on S by locally closed subschemes, indexed by the Hilbert polynomial with respect to some relatively ample line bundle, such that the restriction of F to the inverse image of each stratum is flat over S. Instead of (S) one can also consider other hypotheses to apply our general results of the previous sections, see Remark 23.149 below. By the construction of the base change map (23.28.2), the formation of Ri f∗ F commutes with base change if and only if the base change map β (23.28.3), which was based on (23.25.2), is an isomorphism. Conditions when β is an isomorphism were studied in detail in Section (23.25) for the perfect complex Rf∗ F . The results there show that the question whether Ri f∗ F commutes with base change is controlled by the base change to residue fields of S. Therefore let us look at this situation more closely. Consider for s ∈ S the cartesian diagram Xs

/X

is

fs

 Spec κ(s)

f

g

 / S.

For a bounded complex F of S-flat OX -modules of finite presentation we denote by Fs the restriction F|Xs = i∗s F (no derived inverse image here). Then the functorial homomorphism (23.28.4), which is an isomorphism by Corollary 22.103, becomes the isomorphism ∗ ∼ Rf∗ F ⊗L OS κ(s) = Lg Rf∗ F = Rfs,∗ (Fs ). Taking for i ∈ Z the i-th cohomology on both sides, we obtain a functorial isomorphism (23.28.5)

i H i (Rf∗ F ⊗L OS κ(s)) = H (Xs , Fs ).

Therefore the base change map (23.25.3) takes the form (23.28.6)

β i (κ(s)) : Ri f∗ (F ) ⊗OS κ(s) −→ H i (Xs , Fs ).

We can now apply the results from Sections (23.24) and (23.25). By applying Proposition 23.117 we obtain:

358

23 Cohomology of projective and proper schemes

Theorem 23.139. In the Situation (S) one has the following assertions. (1) The Euler characteristic X χF : S → Z, s 7→ (−1)i dimκ(s) H i (Xs , Fs ) i≥0

is locally constant on S. (2) For each i ∈ Z the function S → Z,

s 7→ dimκ(s) H i (Xs , Fs )

is upper semicontinuous and constructible, i.e., for all n ≥ 0 the subset {s ∈ S; dimκ(s) H i (Xs , Fs ) ≥ n} is closed and constructible in S. Theorem 23.140. (Cohomology and base change) In Situation (S) fix i ∈ Z and a point s ∈ S. (1) The following conditions are equivalent: (i) The map β i (κ(s)) : Ri f∗ F ⊗ κ(s) → H i (Xs , Fs ) is surjective. (ii) There exists an open neighborhood U of s such that the formation of Ri f∗ F |U commutes with base change (Definition 23.138). (2) Assume that β i (κ(s)) is surjective. Then the following conditions are equivalent: (i) The map β i−1 (κ(s)) is surjective (and hence the formation of Ri−1 f∗ F commutes with base change in an open neighborhood of s). (ii) There exists an open neighborhood V of s such that the OV -module Ri f∗ F |V is finite locally free. In this case, the function s 7→ dimκ(s) H i (Xs , Fs ) is locally constant on V . (3) (Grauert’s Theorem) Conversely: Suppose that S is reduced and that the function s 7→ dimκ(s) H i (Xs , Fs ) is locally constant on S. Then Ri f∗ F is a finite locally free OS -module, and the formation of Rj f∗ F commutes with base change for j = i, i − 1. In particular one has for all s ∈ S and j = i, i − 1 functorial isomorphisms of κ(s)-vector spaces Rj f∗ F ⊗ κ(s) → H j (Xs , Fs ). Proof. We apply the general base change results for pseudo-coherent complexes of Section (23.25) to the complex Rf∗ F which is perfect and whose formation commutes with base change by Corollary 23.135. Let us show Assertion (1). Clearly, (ii) implies (i). Conversely, if β i (κ(s)) is surjective, we apply Proposition 23.127 and Remark 23.129 to the perfect complex Rf∗ F . Therefore we find after replacing S be an open neighborhood of s for every morphism g : S ′ → S yielding a cartesian diagram (23.28.1) that g ∗ Ri f∗ F = H i (Lg ∗ Rf∗ F ) = Ri f∗′ (g ′∗ F ), where the second equality holds by Corollary 23.135. Therefore the formation of Ri f∗ F commutes with base change. Part (2) follows from Proposition 23.127. Part (3) follows from Proposition 23.128 using Assertion (1).

359 Corollary 23.141. In Situation (S) fix m ∈ Z. Then the formation of Ri f∗ F commutes with base change for all i ≥ m if and only if Ri f∗ F is finite locally free for all i > m. Proof. We may assume that S is quasi-compact. Then s 7→ dim(Xs ) is bounded (Proposition 14.107) and hence H i (Xs , Fs ) = 0 for i large enough. Therefore we may argue by descending induction on m and can assume that the assertion holds for m + 1. If the formation of Ri f∗ F commutes with base change for i ≥ m, then by induction hypothesis Ri f∗ F is finite locally free for i > m + 1. Moreover, by Theorem 23.140 (2), Rm+1 f∗ F is finite locally free if and only if the formation of Rm f∗ F commutes with base change. Conversely, if Ri f∗ F is finite locally free for all i > m, then by induction hypothesis the formation of Ri f∗ F commutes with base change for all i > m. Again applying Theorem 23.140 (2) shows that also the formation of Rm f∗ F commutes with base change since Rm+1 f∗ F is locally free. Proposition 23.142. Let S be a scheme, let f : X → S be a proper morphism of finite presentation, and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Fix i ∈ Z and set U := { s ∈ S ; H i (Xs , Fs ) = 0 }. Then U is open and constructible, Ri f∗ F |U = 0, and the formation of Rj f∗ F |U commutes with base change for j = i − 1, i. Proof. The first assertion holds because { s ∈ S ; dimκ(s) H i (Xs , Fs ) ≤ 0 } is open and constructible by Theorem 23.139. Moreover, for all s ∈ U the base change map β i (κ(s)) has target H i (Xs , Fs ) = 0 and hence is surjective. Therefore the formation of Ri f∗ F |U commutes with base change by Theorem 23.140 (1). Moreover, Ri f∗ F |U = 0 by Corollary 23.125 applied to the perfect complex E = Rf∗ F . We now can use Theorem 23.140 (2) and hence conclude that the formation of Ri−1 f∗ F |U commutes with base change. Corollary 23.143. Let f : X → S be a proper morphism of finite presentation, and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Fix i ∈ Z. Suppose that there exists an s ∈ S such that H i+1 (Xs , Fs ) = H i−1 (Xs , Fs ) = 0. Then there exists an open neighborhood U of s such that the following assertions hold. (1) Ri+1 f∗ F |U = Ri−1 f∗ F |U = 0. (2) The formation of Rj f∗ F |U commutes with base change for j = i − 2, i − 1, i, i + 1. (3) Ri f∗ F |U is finite locally free. Proof. Proposition 23.142 implies (1) and (2). Then Theorem 23.140 (2) shows that Ri f∗ F is finite locally free in a neighborhood of s. Corollary 23.144. Let f : X → S be a proper morphism of finite presentation, and let F be an OX -module of finite presentation that is flat over S. Suppose that there exists an s ∈ S such that H 1 (Xs , Fs ) = 0. Then there exists an open neighborhood U of s such that R1 f∗ F |U = 0, f∗ F |U is finite locally free and its formation commutes with base change.

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23 Cohomology of projective and proper schemes

Proof. This follows from Corollary 23.143 since there is no cohomology in degree −1 if F is concentrated in degrees ≥ 0. Remark 23.145. If one wants to use Theorem 23.140 and Corollaries 23.143 and 23.144 to prove local freeness of higher direct images and compatibility with base change on all of S (and not only locally in a neighborhood of a point) one sees that all the conditions on the base change maps β i (κ(s)) and β i−1 (κ(s)) and on the cohomology of the fibers have only to be checked for s in a subset T of S such that the only open subset of S containing T is all of S. For instance, if S is quasi-compact, one can choose T to be the set of closed points of S (Lemma 23.4). Corollary 23.146. Let f : X → S be a proper morphism of finite presentation and let F be an OX -module of finite presentation that is flat over S. Let d ≥ 0 be an integer such that dim(Xs ) ≤ d for all s ∈ S. Then Ri f∗ F = 0 for all i > d and the formation of Ri f∗ F commutes with base change for all i ≥ d. See also Corollary 24.44 and the remark following it for a similar stronger result. Proof. The hypothesis implies that H i (Xs , Fs ) = 0 for all s ∈ S and i > d by the vanishing theorem of Grothendieck-Scheiderer 21.57. Hence we conclude by Proposition 23.142. Proposition 23.147. Let f : X → S be a proper morphism of finite presentation and let F be an OX -module of finite presentation that is flat over S. Let b ≥ 0 be an integer and let r : [0, b] → Z≥0 be a map. Let u : T → S be a morphism of schemes yielding a cartesian diagram v /X XT fT

 T

f

u

 / S.

Then there exists a unique subscheme Z of S such that a morphism of schemes u : T → S factors through Z if and only if Ri fT,∗ (v ∗ F ) is locally free of rank r(i) for all i ∈ [0, b] and the formation of Ri fT,∗ (v ∗ F ) commutes with base change. Moreover, the inclusion Z → S is of finite presentation and as sets one has Z = { s ∈ S ; dimκ(s) H i (Xs , Fs ) = r(i) for all i ∈ [0, b]}. Proof. The subscheme Z is clearly unique if it exists. Therefore we may assume that S = Spec R is affine. By Theorem 23.133 and Remark 23.134, E := Rf∗ F is a perfect complex of tor-amplitude in [0, c] for some c ≥ 0, and its formation commutes with base change. Hence, all assertions follow from Proposition 23.130 using that Lu∗ E = RfT,∗ v ∗ F and H i (Xs , Fs ) = H i (E ⊗L OS κ(s)). The same proof combined with Corollary 23.146 also shows the following variant of Proposition 23.147. Remark 23.148. Under the hypotheses in Proposition 23.147, let d be an integer such that dim Xs ≤ d for all s ∈ S, let 0 ≤ a ≤ d be an integer and let r : [a, d] → Z≥0 be a map. Then there exists a unique subscheme Z of S parametrizing, similarly as in Proposition 23.147, the locus where Ri f∗ F is locally free of rank r(i) and its formation commutes with base change for all i ∈ [a, d].

361 Indeed, to apply Proposition 23.130 we have to show that Rf∗ F has tor-amplitude in [a′ , d] for some a′ ≤ d. But Corollary 23.146 shows that Rf∗ F has no cohomology in degrees > d and hence we conclude by Remark 21.165 (2). Remark 23.149. There are the following variants of the assumptions made in (S) and of the results shown above. (1) All the results above using Situation (S) also hold if one assumes that f is only of finite presentation and that F is a bounded complex of OX -modules of finite presentation that are flat and of proper support over S. (2) Let f be flat, proper and of finite presentation and let E ∈ Dqcoh (X) be pseudocoherent. Then Corollary 23.136 shows that Rf∗ E is pseudo-coherent and commutes with derived base change. In this case, for fixed i ∈ Z we say that the formation of Ri f∗ E commutes with base change, if for every cartesian diagram (23.28.1) the functorial morphism (23.28.7)

g ∗ Ri f∗ E −→ Ri f∗′ Lg ′∗ E

is an isomorphism. Note the derived inverse image on the right side which makes this definition different from Definition 23.138. By the same argument as above, this is the case if and only if the base change morphism β : g ∗ Ri f∗ E −→ H i (Lg ∗ Rf∗ E) is an isomorphism. In the special case that g : Spec κ(s) → S is the canonical morphism for a point s ∈ S and is : Xs → X denotes the inclusion of the fiber, the base change map β i (κ(s)) takes the form i ∗ β i (κ(s)) : H i (Rf∗ E ⊗L OS κ(s)) −→ H (Xs , Lis E).

All results from Theorem 23.139 to Corollary 23.146 have an analogue in this setting. One simply has to replace in all assertions Ri f∗ F by Ri f∗ E and Fs by Li∗s E. In addition, one has to assume that E is perfect for Theorem 23.139 (1) as this result needs Rf∗ E to be perfect (Corollary 23.136). There is also a variant of Proposition 23.147. For this one has to assume in addition that E is perfect of tor-amplitude contained in [a, ∞), that b ≥ a is an integer and that the map r is given as map [a, b] → Z≥0 . Then Rf∗ E is also of tor-amplitude contained in [a, ∞) (Remark 23.134). The subscheme Z in Proposition 23.147 then classifies the locus where Ri fT,∗ (Lv ∗ E) is locally free of rank r(i) for i ∈ [a, b].

Hilbert polynomials and flattening stratification We now study how the Hilbert polynomial varies in proper families of finite presentation. It follows easily from previous results that it is locally constant in flat families (Section (23.29)). The converse is also true. For families over a reduced locally noetherian base the fact that the Hilbert polynomial is locally constant implies flatness (Theorem 23.155). In the last section we combine (and generalize) these results by showing that for every proper morphism of finite presentation X → S there exists a stratification of S into

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23 Cohomology of projective and proper schemes

subschemes SΦ that are characterized by the fact that over them the morphism is flat and has a Hilbert polynomial Φ universally (Theorem 23.159). This is the main theorem of this part. (23.29) Local constancy of Hilbert polynomials and of intersection numbers. Let X → S be a morphism of schemes. As usual, we denote for s ∈ S its fiber X×S Spec κ(s) by Xs . For each complex F of OX -modules we denote by Fs its (non-derived) restriction to Xs . Proposition 23.150. Let f : X → S be a proper morphism of finite presentation, and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Let L be an ample line bundle on X. Then the function S → Q[T ] that sends s ∈ S to the Hilbert polynomial of Xs , Fs , and Ls (Definition 23.91) is locally constant. Proof. As F ⊗ L ⊗n is for all n ∈ Z again a bounded complex of OX -modules of finite presentation that are flat over S, the claim follows because the Euler characteristic of Rf∗ (F ⊗ L ⊗n ) is locally constant (Theorem 23.139 (1)). Proposition 23.151. Let f : X → S be a proper morphism of finite presentation, and let F be a bounded complex of OX -modules of finite presentation that are flat over S. Let L1 , . . . , Lt be line bundles on X. (1) The map S → Q[T1 , . . . , Tn ] sending s ∈ S to the Snapper polynomial of Xs , of b Fs ∈ Dcoh (Xs ), and of the line bundles L1,s , . . . , Lt,s (Definition 23.69) is locally constant. (2) Suppose that dim Supp(H i (Fs )) ≤ t for all i ∈ Z and all s ∈ S. Then the function S 7→ Z that sends s ∈ S to the intersection number (L1,s · · · Lt,s · Fs ) (Definition 23.70) is locally constant. The conditions that dim Supp(H i (Fs )) ≤ t for all i ∈ Z can be replaced by the weaker hypothesis that the class of Fs in the Grothendieck group K0′ (Xs ) is contained in K0′ (Xs )t , see Remark 23.45. Proof. The first assertion again follows from Theorem 23.139 (1). And the second assertion follows immediately from the first because the intersection number is a coefficient in the Snapper polynomial. (23.30) Flatness on projective schemes. In this section S denotes a scheme, f : X → S a proper morphism of finite presentation, and L an S-ample line bundle. For every quasi-coherent OX -module F we set F (n) := F ⊗OX L ⊗n for all n ∈ Z. The following remark often allow us to reduce to the noetherian case. Remark 23.152. Suppose that F is an OX -module of finite presentation and that S = Spec R is affine. By noetherian approximation, there exists a finitely generated Z-subalgebra R0 of R, a proper R0 -scheme X0 with X ∼ = X0 ⊗R0 R, an ample line bundle L0 on X0 whose pullback to X is isomorphic to L , and a coherent OX0 -module F0 whose pullback to X is isomorphic to F . If F is flat over S, we can find R0 , X0 , and F0 such that F0 is flat over R0 ([Sta] 05LY).

363 Proposition 23.153. Let F be an OX -module of finite presentation. Then F is flat over S if and only if there exists an integer N ≥ 1 such that f∗ F (n) is finite locally free for all n ≥ N . In this case the formation of f∗ F (n) commutes with arbitrary base change for all n ≥ N . Proof. We may assume that S = Spec R is affine. By Remark 23.152 we may assume that R is noetherian. There exists N ≥ 1 such that H p (X, F (n)) = 0 for all p ≥ 1 and all n ≥ N (Theorem 23.6). Assume that F is flat over S. Then the formation of H p (X, F (n)) commutes with base change for all p ≥ 0 and all n ≥ N (Corollary 23.141). In particular H 1 (Xs , F (n)s ) = 0 for all s ∈ S and n ≥ N . Hence f∗ F (n) = H 0 (X, F )∼ is finite locally free by Corollary 23.144 for n ≥ N and its formation commutes with base change. Conversely, suppose that there exists N ≥ 1 such that f∗ F (n) is finite locally free for n ≥ N . Then this also holds if we replace L by some positive tensor power. Hence we may assume that L is very ample. Then there exists a closed immersion i : X → PdR for some d such that L ∼ = i∗ OPdR (1). Replacing F by i∗ F and X by PdR , we may assume d that X = PR = Proj A with A = R[T0 , . . . , Td ]. By Theorem 13.20 and Proposition 23.12 (3), the L OX -module F is the quasi-coherent module associated with the graded A-module M := n≥N H 0 (X, F (n)). By hypothesis, H 0 (X, F (n)) is a flat R-module for n ≥ N . Hence M is a flat R-module. Therefore its localization MTi and its graded localization M(Ti ) , which is a direct summand of MTi , is flat over R. As F |D+ (Ti ) is the quasi-coherent module corresponding to M(Ti ) , we see that F is flat over R. Lemma 23.154. Let S be a noetherian scheme, let A be a quasi-coherent graded OS algebra such that A1 is of finite type and generates A . Set X := Proj A . Let g : S ′ → S be a morphism of noetherian schemes, and consider the following cartesian diagram (Remark 13.27). h / X ′ := Proj g ∗ A X f′

f

 S′

g

 / S.

Let F be a quasi-coherent OX -module of finite type. Then for large n the base change homomorphism is an isomorphism ∼

g ∗ f∗ F (n) −→ f∗′ h∗ F (n). Proof. As S and S ′ are quasi-compact, we may assume that S = Spec R and S ′ = ˜ Spec R′ are affine. Let A be the graded R-algebra corresponding to A . One has F = M for a finitely generated graded A-module M (Lemma 23.14) and Mn = Γ(X, F (n)) (Proposition 23.13). Moreover, h∗ F is associated to the graded module for large n L M ⊗R R′ = n (Mn ⊗R R′ ) and hence Mn ⊗R R′ = Γ(X ′ , h∗ F (n)) for large n, again by Proposition 23.13. This shows the claim. Theorem 23.155. Let S be a reduced locally noetherian scheme, let f : X → S be a proper morphism, and let L be an S-ample line bundle on X. Let F be a coherent OX -module. Then F is flat over S if and only if the map S → Q[T ] that sends s ∈ S to the Hilbert polynomial ΦXs ,Ls ,Fs of the fiber Xs is locally constant.

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23 Cohomology of projective and proper schemes

Proof. The condition is necessary (without any hypothesis on S) by Proposition 23.150. Hence suppose that s 7→ Φs := ΦXs ,Ls ,Fs is locally constant. We may L assume that S = Spec R is affine. Then X = Proj A for the graded R-algebra A := d≥0 Γ(X, L ⊗d ) (Corollary 13.75). To show that F is flat over S we will show that H 0 (X, F (n)) is a projective R-module for large n (Proposition 23.153). As cohomology commutes with flat base change, we may assume that R is local. Then S is connected and hence s 7→ Φs is constant. Let s0 ∈ Spec R be the closed point. As R is noetherian, there are only finitely many irreducible components of Spec R. Let η1 , . . . , ηm be their generic points. As R is reduced, it suffices to show that for large n the function (*)

s 7→ dimκ(s) H 0 (X, F (n)) ⊗R κ(s)

on Spec R is constant (Corollary 11.19). By semicontinuity of (*) (Corollary 7.31 and the remark following it) it suffices to show that (*) is constant on the finite set {s0 , η1 , . . . , ηm }. But for each s ∈ Spec R we find by Remark 23.92 and Lemma 23.154 an integer Ns ≥ 1 such that Φs (n) = dim H 0 (Xs , F (n)s ) = dim(H 0 (X, F (n)) ⊗R κ(s)) for n ≥ Ns . Therefore the constancy of the Hilbert polynomial shows that (*) is constant on {s0 , η1 , . . . , ηm } for n ≥ max{Ns0 , Nη1 , . . . , Nηm }. (23.31) Flattening stratification by Hilbert polynomials. In this section, f : X → S is a morphism of schemes and F is a quasi-coherent OX -module. For every morphism T → S we set XT := X ×S T and denote by FT the pullback of F to XT . If T = Spec B is affine, we write XB and FB instead of XT and FT . If T = Spec κ(s) → S is the canonical morphism for a point s ∈ S, then we write Xs and Fs instead of Xκ(s) and Fκ(s) . Recall that a polynomial Φ ∈ Q[T ] is called numerical if Φ(n) ∈ Z for all n ∈ Z. We denote the subring of Q[T ] of all numerical polynomials by Q[T ]num . We endow this subring with a total order by defining Φ ≤ Φ′ if Φ(n) ≤ Φ′ (n) for n ≫ 0. In other words, one has a0 + a1 T + a2 T 2 + . . . ≤ b0 + b1 T + b2 T 2 + . . . if and only if there exists r ≥ 0 such that ai = bi for i ≥ r and ar−1 < br−1 (where we set a−1 := b−1 := 0). In the proof of Theorem 23.159 below we will use the following lemma. Lemma 23.156. Let d ≥ 0 and N be integers. Then the map { Φ ∈ Q[T ]num ; deg Φ ≤ d } −→ Zd+1 , Φ 7−→ (Φ(N ), Φ(N + 1), . . . , Φ(N + d)) is an isomorphism of abelian groups. Proof. It is clear that the map is the restriction of the homomorphism { Φ ∈ Q[T ] ; deg Φ ≤ d } → Qd+1 ,

Φ 7→ (Φ(N ), Φ(N + 1), . . . , Φ(N + d))

which is an isomorphism because for every (d + 1)-tuple of rational numbers there exists a unique polynomial of degree ≤ d that takes these values at a given set of d + 1 rational numbers. It remains to show that if Φ(N + i) ∈ Z for i = 0, . . . , d, then Φ is numerical.

365 We argue by induction on d. The case d = 0 is clear. Let d ≥ 1 and set Ψ(n) = Φ(n + 1) − Φ(n). Then deg Ψ ≤ d − 1 and Ψ(n) ∈ Z for N ≤ n ≤ N + d − 1. Hence Ψ(n) ∈ Z for all n ∈ Z by induction hypothesis. Therefore, if Φ(n) ∈ Z, then Φ(n ± 1) ∈ Z. As Φ(N ) ∈ Z, this finishes the proof. Now let S be a scheme, let f : X → S be proper and of finite presentation, and let L be an S-ample line bundle on X. Let F be an OX -module of finite presentation. For every numerical polynomial Φ ∈ Q[T ] define a subset of S by (23.31.1)

SΦ (X, L , F ) := SΦ := { s ∈ S ; ΦXs ,Ls ,Fs = Φ },

where ΦXs ,Ls ,Fs denotes the Hilbert polynomial of Fs on the fiber Xs with respect to Ls . Remark 23.157. As the Euler characteristic and hence the Hilbert polynomial are invariant under change of the base field (Lemma 23.64), for every morphism g : T → S of schemes one has an equality of subsets of T (23.31.2)

g −1 (SΦ (X, L , F )) = SΦ (XT , LT , FT ).

Lemma 23.158. Suppose that S is noetherian. Then S is the set-theoretic disjoint union of finitely many affine subschemes Si such that FSi is flat over Si and such that s 7→ ΦXs ,Ls ,Fs is constant on Si . In particular, there exist only finitely many numerical polynomials Φ such that SΦ is non-empty. Proof. By Proposition 10.86, S is the set-theoretic disjoint union of finitely many affine subschemes Sj′ such that FSj′ is flat over Sj′ and hence the Hilbert polynomial is locally constant on Sj′ (Proposition 23.150). As the Sj′ are noetherian (being locally closed in a noetherian topological space) and in particular quasi-compact, Sj′ is the disjoint union of finitely many open and closed affine subschemes Si on which the Hilbert polynomial is constant. Theorem and Definition 23.159. Let f : X → S be proper of finite presentation, let L be an S-ample line bundle, and let F be an OX -modules of finite presentation. Let Φ ∈ Q[T ] be a numerical polynomial. (1) There exists on SΦ (23.31.1) a unique structure of a locally closed subscheme of S such that a morphism T → S factors through SΦ if and only if FT is flat over T and has Hilbert polynomial Φ in each point of T . Moreover, the inclusion SΦ → S is of finite presentation. S (2) The subspace Φ′ ≥Φ SΦ′ is closed, in particular every point in the closure of SΦ is contained in some SΦ′ for some numerical polynomial Φ′ with Φ′ ≥ Φ. The family of locally closed subschemes (SΦ )Φ is called the flattening stratification of F with respect to (X, L ). If S is quasi-compact, then there are only finitely many Φ such that SΦ ̸= ∅. Moreover, the flattening stratification is compatible with base change g : T → S, i.e., (23.31.2) is an identity of subschemes of T . If X = S and f = idS , then SΦ = ∅ whenever deg Φ > 0 or Φ(0) < 0. Moreover, the Hilbert polynomial is constant of value m ≥ 0 in a point s ∈ S if and only if dimκ(s) (F ⊗κ(s)) = m. Therefore the flattening stratification in this special case coincides with the flattening stratification defined in Section (11.8).

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23 Cohomology of projective and proper schemes

Proof. The uniqueness of the subscheme structure on SΦ is clear by its characterization as a subfunctor of S. This also shows the compatibility with base change once we have shown the existence of the flattening stratification. Therefore we may assume that S = Spec R is affine. By Remark 23.152 and Remark 23.157 we also may assume that R is noetherian. In the noetherian case, any immersion is of finite presentation and hence the same holds for the inclusions SΦ → S is general once we have proved the theorem in the noetherian case. (I). By Lemma 23.158, S is the set-theoretic disjoint union of finitely many affine subschemes Si such that FSi is flat over Si . Let fi : f −1 (Si ) → Si be the restriction of f . By Lemma 23.154 and since there are only finitely many Si , there exists an integer N ′ ≥ 1 such that (*)

f∗ F (n)|Si = fi∗ FSi (n)

for all n ≥ N ′ and for all i. Let N ≥ N ′ such that Rp fi∗ (FSi (n)) = 0 for all p ≥ 1, all n ≥ N , and all i (Theorem 23.6 using again that there are only finitely many Si ). As FSi is flat over Si , we may apply Corollary 23.141 to see that the formation of Rp fi∗ (FSi (n)) commutes for all i with base change for all p ≥ 0 and for all n ≥ N . Combining this with (*) we obtain that for n ≥ N we have the following properties. (A) H p (Xs , F (n)s ) = 0 for p ≥ 1 and for all s ∈ S, (B) f∗ F (n) ⊗ κ(s) = H 0 (Xs , Fs (n)) for all s ∈ S. (II). Let d := sups∈S dim Xs , a finite integer by Proposition 14.107. We set Ej := f∗ F (N + j),

j = 0, . . . , d,

which are coherent OS -modules by Theorem 23.17. Applying the flattening stratification of Section (11.8) to E0 , . . . , Ed and taking scheme-theoretic intersections of all strata, we obtain subschemes We0 ,...,ed for integers e0 , . . . , ed ≥ 0 such that a morphism g : T → S of schemes factors through We0 ,...,ed if and only if g ∗ Ei is locally free of rank ej for all j. Let Φ ∈ Q[T ] be the unique numerical polynomial of degree ≤ d with Φ(N + i) = ej for j = 0, . . . , d (Lemma 23.156) and write WΦ instead of We0 ,...,ed . We claim that the underlying topological space of WΦ is the subspace SΦ . Indeed, for s ∈ S set Φs := ΦXs ,Ls ,Fs . It is a numerical polynomial of degree ≤ d and Φs (n) = dim H 0 (Xs , Fs (n)) for n ≥ N by (A). It is determined by its values Φs (N ), . . . , Φs (N + d). Hence the claim follows from (B). (III). Next we endow SΦ with the correct scheme structure. This may differ from the scheme structure of the WΦ which ensures only that f∗ F (N + j) is locally free of rank Φ(N + j) for j = 0, . . . , d. For arbitrary j ≥ 0 the coherent module f∗ F (N + j) has fibers of rank Φ(N + j) in all points s ∈ WΦ by the claim at the end of Step (II). (ℓ) For ℓ ≥ d define a sequence of closed subschemes WΦ of WΦ with the same topological (ℓ) space as WΦ such that a morphism of schemes g : T → WΦ factors through WΦ if and ∗ only if g (f∗ F (N + j)|WΦ ) is locally free of rank Φ(N + j) for all d ≤ j ≤ ℓ. As WΦ is a (ℓ) (ℓ ) noetherian scheme, there exists an ℓ0 ≥ d such that WΦ = WΦ 0 for all ℓ ≥ ℓ0 . Endow (ℓ0 ) SΦ with the scheme structure of WΦ . (IV). We now show that the subschemes SΦ have the desired universal property. By Lemma 23.154 there exists an integer M ≥ N such that (f∗ F (n))|SΦ = (fSΦ )∗ FSΦ (n) for all n ≥ M . Let g : T → S be a morphism of schemes. Then g factors through SΦ if and only if the Hilbert polynomial of FT is in each point of T equal to Φ and if g ∗ f∗ F (n) is locally free of rank Φ(n) for all n ≥ M . Hence we conclude by Proposition 23.153.

367 (V). It remains to prove Assertion (2). As only finitely many SΦ are non-empty by Lemma 23.158, there exists an m ≥ N such that for any two polynomials Φ and Φ′ with SΦ = ̸ ∅= ̸ SΦ′ one has Φ < Φ′ if and only if Φ(m) < Φ′ (m). Then the non-empty SΦ are the flattening stratification of the coherent OS -module f∗ F (m) and we conclude using that the fiber rank of a coherent OS -module is upper semicontinuous (Corollary 7.31 and the remark following it). For the flattening stratification of a finitely presented module on S there is also a canonical structure of a closed finitely presented subscheme where the module is of rank ≥ r for every fixed r (Section (16.9)). Using similar ideas it is possible to define for every numerical polynomial Φ a natural structure of a closed finitely presented scheme on S ′ Φ ≥Φ SΦ′ which is compatible with ` base change T → S. The morphism of schemes S ′ := Φ SΦ → S is a surjective monomorphism. A morphism T → S factors through S ′ if and only if FT is flat over T . Such a universal flattening exists without assuming the existence of an ample line bundle: Theorem 23.160. Let f : X → S be a proper morphism of finite presentation between schemes, let F be an OX -module of finite presentation. Then there exists a surjective monomorphism of schemes S ′ → S such that a morphism T → S factors through S ′ if and only if FT is flat over T . We refer to [Sta] 05UH for a proof and for further generalizations.

Exercises Exercise 23.1. Let i : Y → X be a closed immersion of locally noetherian schemes, that is a homeomorphism on underlying topological spaces. Show that X is quasi-affine if and only if Y is quasi-affine. Remark : By noetherian approximation one can show that the hypothesis “locally noetherian” is in fact superfluous. Exercise 23.2. Let X be a locally noetherian scheme and let Z ⊆ X be a closed subscheme such that there exists an irreducible component X0 of X such that codim(Z ∩ X0 , X0 ) = 1. Let j : U := X \ Z → X be the inclusion. Show that j∗ OU is not coherent. Exercise 23.3. Let R be an Artinian ring and let X be a proper scheme over Spec R. Show that the abelian category (Coh(X)) of coherent OX -modules satisfies the bi-chain condition (Exercise F.7) and deduce that every coherent OX -module F has a unique (up to order and isomorphism) decomposition F ∼ = F1 ⊕ · · · ⊕ Fr into indecomposable coherent OX -modules Fi . Hint: Exercise F.8. Exercise 23.4. Let k be a field, n ≥ 1 and integer, and let F be a coherent OPnk -module. Show that the following assertions are equivalent. (i) For all i = 1, . . . , n one has H i (Pnk , F (−i)) = 0. (ii) For all i ≥ 1 and all m ≥ −i one has H i (Pnk , F (m)) = 0. (iii) There exists an exact sequence of the form 0 −→ OPnk (−n)rn −→ OPnk (−n + 1)rn−1 −→ · · · −→ OPnk (−1)r1 −→ OPrn0 −→ F −→ 0 k

for some integers ri ≥ 0.

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23 Cohomology of projective and proper schemes

If these conditions hold, show that the canonical map H 0 (Pnk , F ) ⊗ H 0 (Pnk , OPnk (d)) −→ H 0 (Pnk , F (d)) is surjective for all d ≥ 0. Exercise 23.5. Let k be a field and let X → Spec k be a proper smooth morphism. Then the numbers hij (X) := dimk H j (X, ΩiX/k ) are called the Hodge numbers of X. Show that ( 1, if 0 ≤ i = j ≤ n; hij (Pnk ) = 0, otherwise. Remark : See Exercise 23.33 for a generalization. Exercise 23.6. Let X be a qcqs scheme and let L be an ample line bundle ob X. Show that every pseudo-coherent (resp. perfect) complex in D(X) is isomorphic to a bounded above (resp. bounded) complex whose terms are finite direct sums of tensor powers of L . Hint: Use Proposition F.160. Exercise 23.7. Let X be a hypersurface of degree d over a field k, i.e., a closed subscheme of Pnk defined by a homogeneous polynomial of degree d. Show that the dimension of the k-vector space H i (X, OX ) depends only on d and n. Hint: First compute the ideal sheaf of OPnk defining X (a line bundle on Pnk ). Exercise 23.8. Let A be a ring, let I ⊆ A be an ideal, and consider the closed immersion i : Spec A/I → Spec A (which is in particular a proper morphism). (1) Suppose that I is not finitely generated. Show that i is not cohomologically proper. (2) Give an example of a ring A and a finitely generated ideal I (and hence i is proper of finite presentation) such that i is not cohomologically proper. (3) Show that i is cohomologically proper if and only if I is a pseudo-coherent A-module. Exercise 23.9. Let Y = Spec A be an affine scheme, let r ≥ 1, and let f : X := PrY → Y be the projection. Let E ∈ Dqcoh (X) be pseudo-coherent and a ∈ Z. (1) Show that Rf∗ E is pseudo-coherent, i.e., f is cohomologically proper. (2) Suppose that E is acyclic in degrees > a. Then Rf∗ E is acyclic in degrees > a + r. Show that there exists an integer N such that Rf∗ (E ⊗OX OX (n)) is acyclic in degrees > a for all n ≥ N . Exercise 23.10. Show that every morphism of schemes f : X → Y that can be locally on Y factorized into a closed immersion i : X → PrY followed by the structure map PrY → Y with i∗ OX pseudo-coherent is cohomologically proper. Hint: Exercise 23.9. Exercise 23.11. Let R be a noetherian ring. Show that the following assertions are equivalent. (i) R is regular. (ii) For every perfect complex E of R-modules its truncation τ ≥0 E is again perfect. (iii) For every perfect complex E of R-modules its truncation τ ≤0 E is again perfect.

369 Exercise 23.12. Let f : X → Y be a morphism of schemes, suppose that Y is qcqs, and let t ≥ 0 be an integer. Show that f is of tor-dimension ≤ t if and only if H i (Lf ∗ G ) = 0 for i < −t and for every OY -module G of finite presentation. − + Exercise 23.13. Let k be a field and S = Spec k. Show that K0 (Dcoh (S)), K0 (Dcoh (S)), and K0 (Dcoh (S)) are all 0.

Exercise 23.14. Let f : X → Spec k be a proper morphism of noetherian schemes. Show that f∗ : K0′ (X) → K0′ (Y ) = Z is given by the Euler characteristic, where the equality is Exercise 23.24. Exercise 23.15. Let X be a scheme. Let (VB(X)) be the category of locally free OX modules of finite type. Define a short sequence 0 → E ′ → E → E ′′ → 0 of vector bundles to be admissible exact if it is exact as a sequence of OX -modules. Show that this defines on (VB(X)) the structure of an exact category (Exercise F.15). Hint: Exercise F.16 Exercise 23.16. Let A be an exact essentially small category (Exercise F.15). Define K0 (A) as the abelian group with generators [X] for each isomorphism class of objects X in A and relations of the form [X] = [X ′ ] + [X ′′ ] for each admissible exact sequence 0 → X ′ → X → X ′′ → 0 in A. Let R be a ring and let FR be the category of finitely generated free R-modules endowed with its natural exact structure. Show that K0 (FR ) ∼ = Z. Exercise 23.17. Let X be a quasi-compact scheme and let (Li )i∈I be a family of line bundles on X. Show that the following assertions are equivalent. (i) The open subsets Xs for s ∈ Γ(X, Li⊗n ), i ∈ I and n ≥ 1, form a basis of the topology of X. (ii) The open subsets Xs that are in addition affine form a basis of the topology of X. (iii) The open subsets Xs that are in addition affine cover X. (iv) For every quasi-coherent OX -module F of finite type there exist families (ri )i∈I and (ni )i∈I of integers ri ≥ 0 and ni > 0 and a surjective map of OX -modules M ri OX ⊗ Li⊗−ni −→ F . i∈I

(v) For every quasi-coherent OX -module F the evaluation map M Γ(X, F ⊗OX L ⊗n ) ⊗Γ(X,OX ) Li⊗−ni −→ F i,n≥1

is surjective. If (Li )i∈I satisfies these conditions, it is called an ample family. Remark : One can show that every noetherian, semiseparated (Exercise 22.4) scheme X such that OX,x is factorial for all x ∈ X (e.g., if X is regular) admits an ample family of line bundles ([SGA6] O II 2.2.72 ). Exercise 23.18. Let X be a quasi-compact scheme that carries an ample family of line bundles (Exercise 23.17). (1) Show that there exists a finite ample subfamily. 2

The result in [SGA6] O is formulated for separated schemes but the proof carries over to the case where X is only semiseparated.

370

23 Cohomology of projective and proper schemes

(2) Show that X is semiseparated (Exercise 22.4). (3) Let k be a field and let X be the affine line with zero doubled. Denote by U1 , U2 ⊆ X be the two copies isomorphic to A1k . For n ∈ Z define the Cartier divisor Dn given by (U1 , 1) and (U2 , z n ), where z is the coordinate of the second copy of the affine line. Let Ln = OX (Dn ) be the attached line bundle. Show that (L1 , L−1 ) is an ample family of line bundles on X. Remark : Note that there exists no ample line bundle on X since this would imply that X is separated. (4) Let f : Y → X be a quasi-projective morphism of schemes. Show that Y carries an ample family of line bundles. Exercise 23.19. Let X be a quasi-compact scheme with an ample family (Li )i∈I (Exercise 23.17). Show the following assertions. (1) For every quasi-coherent OX -module F , there exists a locally free OX -module E and a surjection E → F . (2) For every quasi-coherent OX -module F of finite type, there exists a finite locally free OX -module E and a surjection E → F . (3) For every surjection G → F of quasi-coherent OX -modules with F of finite type, there exists a finite locally free OX -module E and a map of OX -modules E → G such that the composition E → G → F is surjective Show that in all three assertions one may take E as a direct sum of tensor powers of the Li . Exercise 23.20. Let X be a quasi-compact scheme with an ample family of line bundles (Exercise 23.17). Show that for every perfect complex E on X there exists a strictly perfect complex F and an isomorphism F ∼ = E in D(X). Hint: Use Exercise 23.18 to show that E can be represented by a complex of quasi-coherent OX -modules. Exercise 23.21. Let C be a connected normal scheme of dimension 1. Show that C is semiseparated and deduce that C has the resolution property. Hint: Lemma 15.17, the remark in Exercise 23.17, Exercise 23.19 Exercise 23.22. Let f : X → Y be a morphism of schemes and suppose that Y is regular noetherian of finite dimension d. (1) Show that for every coherent OY -module G there exists a strictly perfect complex F ∼ on Y concentrated in degrees [−d, 0] and an isomorphism F → G in D(Y ). Hint: Use the Remark at the end of Exercise 23.17. (2) Show that f is of tor-dimension ≤ d. Hint: Exercise 23.12. Exercise 23.23. Let X be a scheme and let (VB(X)) be the exact category of finite locally free OX -modules (Exercise 23.15) and let K0V B (X) := K0 (VB(X)) (Exercise 23.16). (1) Show that the multiplication ([E ], [F ]) 7→ [E ⊗OX F ] defines a commutative ring structure on K0V B (X) and that for quasi-compact3 schemes X there is a ring homomorphism K0V B (X) → K0 (X) given by [E ] 7→ [E [0]] which is functorial with respect to pullback f ∗ for morphisms f of quasi-compact schemes. 3

The quasi-compactness assumption is only needed because we defined K0 (X) only for quasi-compact schemes.

371 (2) Let X be a quasi-compact scheme such that there exists an ample family of line bundles on X (Exercise 23.17). Show that the natural map K0V B (X) → K0 (X) is an isomorphism. Hint: Exercise 23.20. (3) Let k be a field and let X be Ank with a double origin, i.e. X is obtained by gluing two copies of Ank along Ank \ {0} (Examples 3.13, 9.10). Let n ≥ 2. Show that K0 (X) = K0′ (X) ∼ = Z ⊕ Z and that K0V B (X) ∼ = Z. Exercise 23.24. Let R be a local Artinian ring. Show that K0′ (R) ∼ = Z given by sending a finitely generated R-module M to its length. Show that the composition K0 (R) → K0′ (R) ∼ = Z is a ring isomorphism. Exercise 23.25. Consider the polynomial ring Q[X1 , . . . , Xt ]. For i ≥ 0 set   X X(X − 1) · · · (X − i + 1) ∈ Q[X]. := i! i

Show that the polynomials of the form Xi11 , . . . , Xitt for ik ≥ 0 form a Q-basis of Q[X1 , . . . , Xt ]. Show that for a polynomial p ∈ Q[X1 , . . . , Xt ] the following conditions are equivalent. (i) For every (n1 , . . . ,P nt ) ∈ Zt the value
p(n1 , .
. . , nt ) is an integer. (ii) If we express p = ik ≥0 ai1 ,...,it Xi11 · · · Xitt , then ai1 ,...,it ∈ Z. If p satisfies these conditions, then p is called numerical . Often a map Zn → Z, that is given by a (necessarily unique) numerical polynomial is also called numerical polynomial. Exercise 23.26. Let f : Z → Z be a map such that n 7→ f (n) − f (n − 1) is a numerical polynomial (Exercise 23.25), then f is a numerical polynomial. Exercise 23.27. Let k be a field, let X be a proper k-scheme, and let L1 , · · · , Lt be b (X) such that [F ] ∈ K0′ (X)t . Show that line bundles on X and let F ∈ Dcoh O X (−1)|I| χ(X, F ⊗L L∨ (L1 · · · Lt · F ) = i ). I⊆{1,...,t}

i∈I

Exercise 23.28. Let k by a field, n ≥ 1 an integer and X ⊆ Pnk a non-empty closed subscheme of dimension d. Show that for all m1 , . . . , md ∈ Z one has (OPnk (m1 ) · · · OPnk (md ) · X) = m1 · · · md deg(X). Exercise 23.29. Let X be a quasi-compact scheme with an ample line bundle. Show Jounanolou’s trick , i.e., show that there exists a vector bundle torsor E → X such that E is an affine scheme. Hint: Use the ample line bundle to construct an affine map s : X → PN Z . Show that the scheme W of rank 1 idempotent matrices in MN +1 (Z) is a vector bundle torsor over PN Z and pull W back to X. Exercise 23.30. Let X be an integral scheme and let E be a finite locally free OX module. Show that there exists an integral scheme X ′ and a birational proper morphism f : X ′ → X such that f ∗ E has a filtration whose graded pieces are invertible OX ′ -modules. Hint: It suffices to show that there exists an f as above such that P(f ∗ E ) has a section over X ′ .

372

23 Cohomology of projective and proper schemes

Exercise 23.31. This problem exhibits the jumping phenomenon in the upper semicontinuity theorem. Let C be a smooth projective connected curve of positive genus (see Chapter 26) over an algebraically closed field k, and x0 a closed point on C. Let D be the divisor ∆C − {x0 } ×k C on C ×k C, where ∆C denotes the divisor on C ×k C given by the image of the diagonal embedding (1C , 1C ) : C → C ×k C. Consider the projection p : C ×k C → C to the second factor, then compute dimk H i (C × {x}, OC×k C (D)|C×{x} ) for x ∈ C(k). Hint: It might be useful to use the fact that, since C has genus > 0, we have H 0 (C, OC (x− x0 )) = 0 for all x ̸= x0 , see Corollary 26.20. Exercise 23.32. Let f : X → S be a morphism of schemes and let Ω•X/S be the de Rham complex. It is a complex of f −1 OS -modules (but the differentials are usually not OX -linear) and we can consider it as an object of D(X, f −1 OS ). Considering f as a morphism of ringed spaces (X, f −1 OS ) → (S, OS ) we can form Rf∗ Ω•X/S ∈ D(S). Forming global sections we also obtain RΓ(X, Ω•X/S ) ∈ D(Γ(S, OS )) and in particular the de Rham cohomology p HdR (X/S) := H p (RΓ(X, Ω•X/S )) ∈ (Γ(S, OS )-Mod).

(1) Let X → S be a morphism of schemes given by a ring homomorphism R → A. Show that in D(R) one has an isomorphism RΓ(X, Ω•X/S ) = Ω•A/R and in particular i (X/S) = H i (Ω•A/R ). HdR (2) Show that the first hypercohomology sequence yields a converging spectral sequence E1pq = Rq f∗ ΩpX/S ⇒ Rf∗ Ω•X/S , called the Hodge spectral sequence. (3) Let f be qcqs. Show that Rf∗ Ω•X/S ∈ Dqcoh (S). (4) Let S be locally noetherian and let f be proper. Show that Rf∗ Ω•X/S ∈ Dcoh (S). (5) Let f be proper and smooth. Show that Rf∗ ΩpX/S for p ≥ 0 and Rf∗ Ω•X/S are perfect objects in D(S) whose formation commutes with arbitrary base change. (6) Show that one can apply Remark 21.130 to the de Rham complex Ω•X/S and that one L p obtains on p∈Z HdR (X/S) the structure of a strictly graded commutative graded Γ(S, OS )-algebra. The multiplication p q p+q ∪ : HdR (X/S) ⊗ HdR (X/S) −→ HdR (X/S)

is called the cup product. Exercise 23.33. Let R be a ring, n ≥ 1 an integer, and set X := PnR . (1) Show that H q (X, ΩpX/R ) is a free R-module of rank ( 1, if 0 ≤ p = q ≤ n; p q rkR H (X, ΩX/R ) = 0, otherwise. i (X/R) is a free R-module (2) Show that the de Rham cohomology (Exercise 23.32) HdR of rank 1 if 0 ≤ i ≤ 2n is even and of rank 0 otherwise.

373 Exercise 23.34. Let k be a field, let n1 , n2 ≥ 1 be integers, let pi : X := Pnk 1 ×k Pnk 2 → Pnk i be the projections. Show that OX (1, 1) := p∗1 OPn1 (1) ⊗ p∗2 OPn2 (1) is an ample line bundle k k on X. What is the Hilbert polynomial of Pnk ×k Pm k with respect to OX (1, 1)? Exercise 23.35. Let X be a finite scheme over a field. Show that the Hilbert polynomial ΦX is a constant polynomial with value dimk Γ(X, OX ).
− 1. Let Exercise 23.36. Let k be a field and let n, d ≥ 1 be integers and set N := n+d d N n → P Z = Pnk embedded into PN via the d-fold Veronese embedding v : P (Exercise 13.8). d k k k Show that the degree of (Z, vd ) is dn . Exercise 23.37. Let S be a scheme and let E ∈ D(S) be a pseudo-coherent complex. (1) Show that E is perfect if and only if for every point s ∈ S the complex E ⊗L κ(s) is perfect. (2) Let f : X → S be a surjective morphism of schemes. Show that E is perfect if and only if Lf ∗ E is perfect. (3) Let S = Spec R be affine and let I ⊆ R be an ideal contained in the Jacobson radical of R. Show that if E ⊗L R R/I is perfect, then E is perfect. Exercise 23.38. Let i : Z → S be a thickening of schemes. Let E ∈ Dqcoh (S) be a complex. Show that E is pseudo-coherent (resp. perfect) if and only if Li∗ E is pseudo-coherent (resp. perfect). Exercise 23.39. Let A be a ring and I ⊆ A be an ideal such that A is I-adically complete. Let E ∈ D(A) be a complex. Show that E is pseudo-coherent (resp. perfect) if and only if E ⊗L A A/I is pseudo-coherent (resp. perfect). Hint: Use Exercise 23.38. Exercise 23.37 (3) might also be useful. Exercise 23.40. Let S be a scheme, let E be a pseudo-coherent complex in D(S), and let b ∈ Z such that H p (E) = 0 for all p > b. Let a ≤ b and let r : [a, b] → Z≥0 , i 7→ ri , be a map. Show that there exists a unique locally closed subscheme j : Z = Zr → S such that a morphism f : T → S factors through Z if and only if for all morphisms g : T ′ → T the OT ′ -modules H i (L(f ◦ g)∗ E) is locally free of rank ri for all i ∈ [a, b]. Show that moreover the following assertions hold. (1) The immersion j : Z → X is of finite presentation. (2) As a set one has (23.31.3)

Z = { s ∈ S ; bi (s) = ri for all i ∈ [a, b]}. ¯

f j (3) If f : T → S factors into T −→ Z −→ X, then H i (Lf ∗ E⊗OT G ) = f¯∗ H i (Lj ∗ E)⊗OT G for all i ∈ [a, b] and for all quasi-coherent OT -modules G .

Exercise 23.41. Let f : X → S be a flat projective morphism of noetherian schemes. Assume that H 0 (Xs , OXs ) = κ(s) for all s, where Xs denotes the scheme-theoretic fiber of f over s. Show that f∗ OX = OS . Hint: As a lemma, prove the following: If φ : A → B is a ring homomorphism such that B is a locally free A-module of rank 1 via φ, then φ is an isomorphism. Remark : The assumption H 0 (Xs , OXs ) = κ(s) holds if all fibers of f are geometrically reduced and geometrically connected (Proposition 12.66). See also Corollary 24.63 for a more general result.

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23 Cohomology of projective and proper schemes

Exercise 23.42. Let k be a field, A := k[X, Y ]/(XY, Y 2 ), and M := A/(Y ) = k[X] as an A-module. (1) Find a partial flat resolution of the A-module M of length at least 4, i.e., an exact sequence Fn → · · · → F2 → F1 → F0 → M → 0 of A-modules with n ≥ 3 and all Fi flat. (2) Now consider the fiber product diagram v

Spec k[Y ]/(Y 2 )

/ Spec A ,

g

 Spec k



f

/ Spec k[X]

u

where u corresponds to the origin of the affine line Spec k[X], and f is given by the canonical inclusion k[X] → A. Using Part (1), show that the canonical map ˜ → Rg∗ Lv ∗ M ˜ Lu∗ Rf∗ M is not an isomorphism, but the composition ˜ → Rg∗ Lv ∗ M ˜ → Rg∗ v ∗ M ˜ Lu∗ Rf∗ M is an isomorphism. Exercise 23.43. Let S be a locally noetherian scheme, and f : X → S a flat proper morphism. We say that f is cohomologically flat (in dimension 0), if for any base change diagram v /X , XT g

 T

u

 /S

f

the canonical map u∗ f∗ OX → g∗ v ∗ OX is an isomorphism. For example, a map f as in Exercise 23.41 is cohomologically flat. (1) Let T → S be a faithfully flat morphism. Show that f is cohomological flat if and only if the base change X ×S T → T is cohomologically flat. (2) Let R be a discrete valuation ring, m its maximal ideal, S = Spec R, and let f : X → S be a proper flat morphism of schemes. Show that f is cohomologically flat if and only if R1 f∗ OX is a free OS -module, in other words, if and only if H 1 (X, OX ) has no torsion. (3) In the setting of Part (2), now assume that H 0 (X, OX ) = R and suppose that f is not cohomologically flat. For n ≥ 1, we set Xn = X ×S Spec(R/mn ). Show that for n sufficiently large, Xn is a flat R/mn -scheme with the property that the module H 0 (Xn , OXn ) of global sections is not flat over R/mn . Remark : For another, explicit example of a flat proper map which is not cohomologically flat, see [Liu] O , Chapter 5, Exercise 3.15.

375 Exercise 23.44. Let k be a field and let X be a proper k-scheme of dimension d. Define the arithmetic genus of X by pa (X) := (−1)d (χ(OX ) − 1). Let i : X → Pnk be a closed embedding and let Φ be the Hilbert polynomial of X with respect to the ample line bundle i∗ OPnk (1). Suppose that X is geometrically connected and geometrically reduced over k. Show that pa (X) =

d X

(−1)d−i dim H i (X, OX ).

i=1

Show the following assertions. (1) pa (Prk ) = 0.
(2) Let H ⊂ Prk be a hypersurface of degree d ≥ 1. Show that pa (H) = d−1 r . (3) Let E be a finite locally free OX -module of rank r + 1. Show that pa (P(E )) = (−1)r pa (X). Exercise 23.45. Let f : X → S be a morphism of schemes. We call an effective Cartier divisor D ⊂ X a relative effective Cartier divisor , if D (considered as a closed subscheme of X) is flat over S. (1) Let f : X → S be a morphism of schemes and let D ⊂ X be a relative effective Cartier divisor. Let S ′ → S be a morphism of schemes. Show that the base change D ⊗S S ′ is a relative effective Cartier divisor in X ×S S ′ over S ′ . (2) Let f : X → S be a flat morphism of finite type between noetherian schemes, and let D ⊂ X be a closed subscheme of X that is flat over S. Show that D is a relative effective Cartier divisor (over S) if and only if for every s ∈ S, the fiber Ds of D over s (as a closed subscheme of the fiber Xs = f −1 (s)) is an effective Cartier divisor of Xs . Show that in this case the line bundle associated with Ds is OX (D)|Xs . Exercise 23.46. Let k be a field and let X be a proper k-scheme. Let A ⊆ Div(X) be the subgroup of the group of Cartier divisors on X generated by all differences D1 − D0 where T is a smooth integral k-scheme of finite type, D ⊂ X × T is a relative effective Cartier divisor over T (Exercise 23.45), x0 , x1 ∈ T (k), and Di is the fiber of D over xi , i = 0, 1. We call divisors D0 , D1 ∈ Div(X) algebraically equivalent , if D1 − D0 ∈ A. The quotient Div(X)/A is called the group of divisors on X up to algebraic equivalence. (1) Let D0 , D1 ∈ Div(X) be linearly equivalent. Prove that D0 and D1 are algebraically equivalent. (2) Let Di,0 , Di,1 ∈ Div(X) be divisors, i = 1, . . . , d = dim(X), such that for every i, D0,i and D1,i are algebraically equivalent. Prove that for the intersection numbers, we have (D0,1 · · · D0,d ) = (D1,1 · · · D1,d ). Hint: : Use Proposition 23.151. (3) Let X be a proper k-scheme of dimension 2, let f : X → C be a surjective flat morphism from X to a connected smooth proper curve C over k, let x0 , x1 ∈ C(k), and consider the divisors Di = f ∗ [xi ] as effective Cartier divisors on X. Show that for every Cartier divisor E on X, (D0 · E) = (D1 · E). Hint: Consider the graph of f .

24

Theorem on formal functions

Content – Derived Completion – The theorem of formal functions – Stein factorization – Algebraization This chapter focuses on the theorem of formal functions and some of its applications. In the simplest form, this theorem says the following. Let S be a locally noetherian affine scheme and let f : X → S be proper. Let J ⊆ OS be a quasi-coherent ideal sheaf, so that we have the closed subschemes Xn := f −1 (V (J n )) ⊆ X, an ascending sequence of closed subschemes which all have the same topological space. Then the limit limn H 0 (Xn , OXn ) (these are the “formal functions”) equals the completion of H 0 (X, OX ) with respect to the topology induced by J . More generally, the theorem covers non-affine base schemes, higher direct images rather than just global sections and arbitrary coherent modules. If one replaces the completion by derived completion, one obtains a version for arbitrary pseudo-coherent objects in D(X) and even a version which does not require the assumption that the base be noetherian. See Theorems 24.28 and Proposition 24.35.

Derived Completion Let A be a ring and let I ⊆ A be an ideal. Then classically the I-adic completion of an ˆ := limn M/I n M . If A is noetherian, then this functor is A-module M is defined by M ˆ is I-adically complete, i.e. the natural exact on finitely generated A-modules M , and M ˆ is an isomorphism (Proposition B.41). ˆ /I n M ˆ → limn M map M But beyond the case of finitely generated modules over noetherian rings this classical completion functor is often very badly behaved. It is the composition of the right exact functor that sends a module M to the projective system · · · → M/I 2 M → M/IM followed by the left exact limit functor. The composition is often not exact, not even ˆ is not necessarily exact in the middle even if A is noetherian (Exercise 24.5). Moreover, M I-adically complete (Exercise 24.6), although this can only happen if I is not finitely generated (Proposition 24.2). Here we will introduce the more sophisticated approach of derived completion – at least if I is finitely generated. The idea is first to construct a “derived version” of the functor M 7→ M/I n M and then to take the derived limit or, if one starts with an object M in D(A), the homotopy limit. To explain this idea in more detail let us assume first that I = (f ) is generated by one element.

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_9

377 fn

Instead of M/f n M we consider the two-term complex M −→ M , concentrated in degrees −1 and 0. There exists a quasi-isomorphism from this complex to M/f n M if f is M -regular. Then the derived completion of M is given by (*)

fn

M ∧ := R lim(M −→ M ) = R HomA (A → Af , M ). n

This is an object in D(A) which is not necessarily concentrated in degree 0. Another possibility for a derived version of M 7→ M/f n M would be to use the functor M 7→ L n ∧ n M ⊗L A A/f . In general, the homotopy limit of M ⊗A A/f might not be M although this is the case if the f -torsion is bounded in A (Exercise 24.7). If I = (f1 , . . . , fr ) is generated by finitely many elements, we first observe that the projective systems (M/I n M )n and (M/(f1n , . . . , frn ))n are equivalent and then replace in the definition (*) of the completion the complex fn

fn

(M −→ M ) = M ⊗A (A −→ A) by M ⊗A K(f1n , . . . , frn )[−r], where fn

fn

1 r K(f1n , . . . , frn )[−r] = (A −→ A) ⊗A · · · ⊗A (A −→ A)

is the Koszul complex of the sequence (f1n , . . . , frn ) shifted into degrees [0, r]. This amounts to replacing A → Af in (*) by the tensor products of the complexes A → Afi . If A is noetherian and M is finitely generated, we will see that M ∧ is the classical completion limn M/I n M . All these constructions can be generalized to consider the derived completion with respect to an ideal sheaf of finite type on a ringed space. Here we will give the construction only if the ideal is globally generated by finitely many elements, which is the only case that we use in the sequel. Let us finish this introduction with a note on derived limits. Let A be a Grothendieck abelian category. If (**)

· · · −→ M2 −→ M1 −→ M0

is a projective system of complexes in C(A), then we can view (**) as a complex of opp projective systems, i.e., as an element of C(AN ), and form the termwise derived limit R limn Mn ∈ D(A). If (**) is a projective system of objects in D(A), i.e., an object in opp opp D(A)N , then it is in general not possible to view it as an element of D(AN ). Instead one takes the homotopy limit, cf. Section (F.54). If (**) is a projective system of complexes in C(A) one has R lim Mn = holim Mn in D(A) by Proposition F.231, where on the right side we consider the image of (Mn )n in opp D(A)N . (24.1) Reminder on completions. Before defining derived completions, we collect some definitions and properties of the classical completion of modules. Let A be a ring, let I ⊆ A be an ideal, and let M be an A-module. The I-adic completion of M is defined as the limit

378

24 Theorem on formal functions ˆ := lim M/I n M. M

(24.1.1)

n

In particular we have the I-adic completion Aˆ = lim A/I n A

(24.1.2)

ˆ as an A-module. ˆ of A. We consider Aˆ as an A-algebra and M An A-module M is called I-adically complete if the map ˆ = lim M/I n M M −→ M n

is an isomorphism. The ring A is called I-adically complete if it is I-adically complete as an A-module. Remark 24.1. Let us link these notions to topological algebra. Given a ring A and an ideal I there exists on A a unique topology making the underlying abelian group (A, +) into a topological group such that (I n )n is a basis of neighborhoods of 0. Moreover, this topology makes A into a topological ring ([BouGT] O III, §6.3, Example 3). Similarly, given an A-module M , there exists on M a unique topology making (M, +) into a topological group such that (I n M )n≥1 is a basis of neighborhoods of 0. This topology makes M into a topological module over the topological ring A. It is called the I-adic topology on A and on M . There is the general notion of the completion of an abelian topological group (with respect to its canonical uniform structure), see [BouGT] O III, §3.4, §3.5. In this case, this is easily described since the I-adic topology on M is by definition first countable: We call a sequence (xn )n of elements xn ∈ M a Cauchy sequence if for all integers m ≥ 0 there exists an N ≥ 0 such that xn − xn′ ∈ I m M for all n, n′ ≥ N . Then the Hausdorff completion of M is given as usual by equivalence classes of Cauchy sequences. In fact, the I-adic topology on M is given by a non-archimedean pseudo-metric, i.e., by a map d : M × M → R≥0 that satisfies the following conditions for all m, m′ , m′′ ∈ M (a) m = m′ ⇒ d(m, m′ ) = 0, (b) d(m, m′ ) = d(m′ , m), (c) d(m, m′′ ) ≤ max{d(m, m′ ), d(m′ , m′′ )}. ′ For instance, d can be defined by d(m, m′ ) = cv(m,m ) , where 0 < c < 1 is a fixed real number and v(m, m′ ) = sup{ n ∈ N ; m − m′ ∈ I n M } with the convention c∞ := 0. With this definition the completion becomes the usual Hausdorff completion of a pseudo-metric space. By [BouGT] O III, §7.3, Cor. 1 of Prop. 2, the Hausdorff completion of A is a metrizable ˆ as defined in (24.1.2), and the Hausdorff complete topological ring isomorphic to A, ˆ , as completion of M is a metrizable complete topological A-module isomorphic to M ˆ , +) into a topological defined in (24.1.1). Its topology is the unique topology making (M group such that the n M = lim I n M/I m M = Ker(M ˆ → M/I n M ) I[ m≥n

form a basis of neighborhoods of 0. The T topological module M is itself Hausdorff (resp. Hausdorff and complete) if and only if n I n M = 0 (resp. it it is I-adically complete in the sense defined above).

379 nM = I nM ˆ will in ˆ and the topology on M However, in general one does not have I[ ˆ ˆ considered as an A-module) nor the I A-adic general be neither the I-adic topology (M ˆ considered as an A-module), ˆ topology (M see Exercise 24.6. However, if I is finitely generated, then the situation is better, as the following result shows.

Proposition 24.2. Let A be a ring, let I ⊆ A be a finitely generated ideal, and let M be n M for all n ≥ 0, in particular the topology on M ˆ is the ˆ = I[ an A-module. Then I n M ˆ I-adic topology and M is I-adically complete. Proof. As I is finitely generated, I n is finitely generated, say I n = (f1 , . . . , fr ). Then X u : M r −→ I n M, (m1 , . . . , mr ) 7→ fi mi , i

is surjective. Hence the induced map on I-adic completions n M = Ker(M ˆ )r → I[ ˆ → M/I n M ), u ˆ : (M

(m ˆ 1, . . . , m ˆ r ) 7→

X

fi m ˆi

i nM . ˆ , we see that I n M ˆ = I[ is surjective (Proposition B.49). Since the image of u ˆ is I n M

(24.2) Derived complete complexes. Let A be a ring. For a complex M of A-modules and f ∈ A we set f

f

T (M, f ) := R lim(M ←− M ←− M ←− . . . ) ∈ D(A). f

f

f

As Af = colim(A −→ A −→ A −→ . . . ) we see that f

f

HomC(A) (Af , M ) = lim(M ←− M ←− M ←− . . . ) and hence obtain an isomorphism in D(A), functorial in M , T (M, f ) ∼ = R HomA (Af , M ).

(24.2.1)

If M is an A-module, we define Y Y δf : M −→ M, N

(xn ) 7→ (xn − f xn+1 ).

N

Then Proposition F.224 shows that H 0 T (M, f ) = HomA (Af , M ) = Ker(δf ), (24.2.2)

H 1 T (M, f ) = Ext1A (Af , M ) = Coker(δf ), H p T (M, f ) = ExtpA (Af , M ) = 0 for all p ̸= 0, 1.

Lemma 24.3. Let A be a ring, M a complex of A-modules and f ∈ A such that T (M, f ) = 0. Then R HomA (E, M ) = 0 for every complex E of Af -modules. Proof. In view of (24.2.1), the claim follows from the following equality. (24.2.3)

R HomA (E, M ) = R HomAf (E, R HomA (Af , M )).

380

24 Theorem on formal functions

To show the equality, let M → I be a K-injective resolution. Then HomA (Af , I) is a K-injective complex of Af -modules. Indeed, let X be an exact complex of Af -modules. Then using (*) and Proposition F.179 one sees that HomK(Af ) (X, HomA (Af , I)) = HomK(Af ) (X ⊗A Af , HomA (Af , I)) = HomK(A) (X, I) = 0, which shows that HomA (Af , I) is K-injective. As E ⊗A Af = E, we find R HomAf (E, R HomA (Af , M )) = R HomAf (E, HomA (Af , I)) = HomAf (E, HomA (Af , I)) = HomAf (E ⊗A Af , I) = R HomA (E, M ). Lemma 24.4. Let A be a ring and let M be a complex of A-modules. Then the set I of f ∈ A such that T (M, f ) = 0 is a radical ideal of A. Proof. Recall that T (M, f ) = R HomA (Af , M ) by (24.2.1). Let f ∈ I and g ∈ A. Then Af g is an Af -module. Hence R HomA (Af g , M ) = 0 by Lemma 24.3 and thus f g ∈ I. Now let f, g ∈ I. As f and g generate the unit ideal in Af +g , there is an exact sequence 0 −→ Af +g −→ Af (f +g) × Ag(f +g) −→ Agf (f +g) −→ 0 ˇ It is the extended alternating Cech complex of the structure sheaf on Spec(Af +g ) attached to the open covering (D(f ), D(g)) which is exact because the higher cohomology groups vanish for the structure sheaf. All modules except possibly Af +g in the sequence are modules over Af or Ag and hence are annihilated by the functor R HomA (−, M ). Hence R HomA (Af +g , M ) = 0. Finally, let f ∈ A and n ≥ 1 with f n ∈ I. Then f ∈ I because Af = Af n . Definition 24.5. Let A be a ring and let I ⊆ A be an ideal. Then a complex M in D(A) is called derived complete with respect to I or derived I-complete if T (M, f ) = 0 for all f ∈ I. The full subcategory of derived I-complete complexes in D(A) is denoted by Dcomp (A, I). As usual we consider A-modules as complexes concentrated in degree 0 and hence have also the notion of derived I-complete A-module. This notion is closely linked to the notion of an I-adically complete module, as we will see in Proposition 24.7 below. Remark 24.6. Let A be a ring, let I ⊆ A be an ideal, and let S ⊆ A be a subset such that the ideal generated by S and the ideal I have the same radical ideal. (1) A complex M in D(A) is derived I-complete if and only if T (M, f ) = 0 for all f ∈ S. This follows from Lemma 24.4. In particular, derived completeness with respect to I for a complex M depends only on the radical of I, i.e., only on the closed subspace V (I) of Spec A. Moreover, for every complex M in D(A) there exists a greatest ideal IM ⊆ A such that M is derived IM -complete. Exercise 24.16 shows that IA is always contained in the Jacobson radical of A. (2) By (24.2.2) an A-module M is derived I-complete if and only if HomA (Af , M ) = Ext1A (Af , M ) = 0 for all f ∈ S.

381 (3) The category Dcomp (A, I) is a triangulated subcategory (Definition F.131) because R HomA (Af , −) is a triangulated functor. Moreover, Dcomp (A, I) is stable under passing to direct summands (i.e., if M ⊕ N are in Dcomp (A, I), then M and N are in Dcomp (A, I)). (4) As the formation of T (M, f ) commutes with products in M , the category Dcomp (A, I) is stable under arbitrary products. (5) By Lemma F.230, R HomA (Af , −) commutes with homotopy limits, so Dcomp (A, I) is stable under homotopy limits in D(A). We now relate derived completeness and adic completeness. Proposition 24.7. Let A be a ring, I ⊆ A an ideal, and let M be an A-module. (1) Suppose that M is I-adically complete, i.e., M → limn M/I n M is an isomorphism. Then M is derived I-complete. (2) Conversely, suppose that I is finitely generated and that M is derived I-complete. Then M → limn M/I n M is surjective. The proof will show that in (2) it suffices to assume that I is finitely generated and that H 1 T (M, f ) = 0, where f runs through a generating system of I. Proof. Part (1).. Assume that M is I-adically complete. By (24.2.2) it suffices to show that HomA (Af , M ) = 0 and Coker(δf ) = 0 for f ∈ I, where Y Y δf : M→ M, (xn )n 7→ (xn − f xn+1 ). N

N

But HomA (Af , M ) = lim HomA (Af , M/I n M ) and HomA (AfQ , M/I n M ) = 0 since f ∈ I. To show that the map δf is surjective, we define for (yn )n ∈ N M elements xn :=

∞ X

f k yn+k .

k=0

The series converges because M is I-adically complete. Then δf ((xn )n ) = (yn )n . Part (2).. Conversely, assume that M is derived I-complete for an ideal generated by elements f1 , . . . , fr . We first show that M → lim M/fin M is surjective for all i = 1, . . . , r. For this we may assume that I = (f ) and T (M, f ) = 0. An element of lim M/f n M n is given by a family (xn )n of elements xn ∈ M such Pn that xn+1 = xn + f yn for some yn ∈ M . Setting y0 := x1 we hence have xn+1 = i=0 f i yi . We look for an x ∈ M such Pn−1 that x ≡ i=0 f i yi mod f n M , for all n. As δf is surjective, we can find zn ∈ M such that yn = zn + f zn+1 for all n ≥ 0. We Pn−1 then have i=0 yi f i = z0 − f n zn , so that we can choose x := z0 . To show that M → lim M/I n M is surjective, note that I rn ⊆ (f1n , . . . , frn ) ⊆ I n implies lim M/I n M = lim M/(f1n , . . . , frn ). An element x in lim M/(f1n , . . . , frn ) is given by a family of elements xn ∈ M such that xn+1 = xn + f1n yn,1 + · · · + frn yn,r for elements yn,i ∈ M . Setting y0,1 := x1 and y0,i := 0, we can represent it as a formal sum x=

r XX

fin yn,i .

n≥0 i=1

lim M/fin M

P n As M → isPsurjective, there exists yi ∈ M mapping to n fi yn,i in n n y , . . . , f lim M/fin M . Then y = maps to x in lim M/(f ). This shows the suri r 1 i jectivity of M → lim M/I n M .

382

24 Theorem on formal functions

Corollary 24.8. Let A be a ring, I ⊊ A be an ideal, and let M be T an A-module. (1) Let I be finitely generated by elements f1 , . . . , fr and suppose n I n M = 0. Then the following assertions are equivalent. (i) The A-module M is I-adically complete. (ii) The A-module is derived I-complete. (iii) One has H 1 T (M, fi ) = 0 for all i = 1, . . . , r. (2) Let A be noetherian and let M be a finitely generated A-module. Suppose that I is contained in the Jacobson radical of A. Then M is I-adically complete if and only if it is derived I-complete. Exercise 24.15 shows that if A is a reduced ring and I is a finitely generated ideal, then A is I-adically complete if and only if A is derived I-adically complete. Proof. The first assertion follows immediately from Proposition 24.7. It implies the second T because one has n I n M = 0 in this situation by Proposition B.42. Proposition 24.9. Let A be a ring, let M in D(A) be a complex and let I ⊆ A be an ideal. Then M is derived I-complete if and only if H q (M ) is derived I-complete for all q ∈ Z. Proof. Let f ∈ I. We have to show that ExtpA (Af , H q (M )) = 0 for all p, q ∈ Z if and only if ExtpA (Af , M ) = 0 for all p ∈ Z. The second hypercohomology spectral sequence for the functor HomA (Af , −) (F.49.4) yields a spectral sequence E2pq = ExtpA (Af , H q (M )) ⇒ Extp+q A (Af , M ). By (24.2.2) we have ExtpA (Af , H q (M )) = 0 for p = ̸ 0, 1. Therefore the spectral sequence degenerates at E2 and the filtration of the limit term has only two steps. Thus we get for all p exact sequences 0 → Ext1A (Af , H q−1 (M )) → ExtpA (Af , M ) → HomA (Af , H q (M )) → 0 which proves the claim. (24.3) Derived completion. Let A be a ring, and let I ⊆ A be an ideal generated by finitely many elements f1 , . . . , fr ∈ I. In other words r [ Spec(A) \ V (I) = D(fi ) i=1

ˇ is quasi-compact. Recall that we defined in Section (22.4) the extended ordered Cech complex Y Y Afi1 fi2 −→ . . . −→ Af1 f2 ···fr , Afi1 −→ (24.3.1) CA (f1 , . . . , fr ) : A −→ i1

where we put A in degree 0.

i1 b, we see that H b (M ⊗ K(f n )) = H b (M ) ⊗A H 0 (K(f n )) = H b (M ) ⊗A A/(f1n , . . . , frn ). Hence the transition maps of the inverse system H b (M ⊗A K(f n )) are all surjective. Therefore R1 lim H b (M ⊗A K(f n )) = 0 by Proposition F.227. We now compare the derived I-completion of a complex M ∈ D(A) to the “naive derived completion” holim(M ⊗L A/I n ). There are several important cases in which this already yields the derived completion. One is that of noetherian rings, see Proposition 24.16 below. Another is the case of ideals generated by one element f provided the f -torsion is bounded. Both results rely on the following remark. Remark 24.15. Let A be a ring and let I ⊆ A be an ideal generated by a finite family f = (f1 , . . . , fr ). By Proposition 24.13 we have M ∧ ∼ = holim(M ⊗L K(f n )). So if the n n pro-objects (K(f ))n and (A/I )n are isomorphic (Remark 24.12 (2)), then it follows that M ∧ ∼ = holim(M ⊗L A/I n ). As I rn ⊆ (f1n , . . . , frn ) ⊆ I n , the pro-objects (A/I n )n and (A/(f1n , . . . , frn ))n are isomorphic. We see that if the pro-objects (K(f n ))n and (A/(f1n , . . . , frn ))n are isomorphic, in which the sequence (f1 , . . . , fr ) is said to be weakly proregular , then for every complex M in D(A) one has an isomorphism (24.3.3)

M∧ ∼ = holim(M ⊗L A/I n ),

which is functorial in M . See also Exercise 24.8 for more results on weakly proregular sequences. This is in particular the case if f is completely intersecting. Then f n is also completely intersecting (Proposition 19.17) and hence K(f n ) ∼ = A/(f1n , . . . , frn ) in D(A) for all n. Proposition 24.16. Let A be a noetherian ring. Then for every complex M in D(A) there is a functorial isomorphism M ∧ ∼ = holim(M ⊗L A/I n ). Proof. As explained in Remark 24.15 it suffices to show that for every n there exists m ≥ n such that K(f m ) → K(f n ) factors through K(f m ) → τ ≥0 (K(f m )) = A/(f1m , . . . , frm ). (I). We first claim that for all p < 0 and for all n ≥ 1 there exists an m ≥ n such that the homomorphism H p (K(f m )) → H p (K(f n )) is zero. Indeed, as A is noetherian, H p (K(f n )) is a finitely generated A-module for all p and n ≥ 1. By Remark 22.15 (1) the transition maps K(f m )p → K(f n )p for m > n and p < 0 are given by multiplying the vectors in the standard basis by elements of the ideal Im−n with Is := (f1s , . . . , frs ) for s ≥ 1.

386

24 Theorem on formal functions

Now we have Im−n ⊆ Int for m = n + tn. The Artin-Rees lemma (Proposition B.40) shows that for large t one has

Int K(f n )p ∩ Ker K(f n )p → K(f n )p+1 ⊆ In Ker K(f n )p → K(f n )p+1 . Putting things together, we obtain

Im K(f m )p → K(f n )p ∩ Ker K(f n )p → K(f n )p+1
⊆ Im−n K(f n )p ∩ Ker K(f n )p → K(f n )p+1
⊆ Int K(f n )p ∩ Ker K(f n )p → K(f n )p+1 ⊆ In Ker(K(f n )p → K(f n )p+1 ). This shows the claim because In annihilates H p (K(f n )) (Remark 19.6). (II). Let n ≥ 1. As H p (K(f m )) = 0 for p ∈ / [−r, 0] and for all m, we find m = n0 ≥ n1 ≥ · · · ≥ nr = n such that H p (K(f ni−1 )) → H p (K(f ni )) is zero for all i and for all p by our claim. Set Ki := K(f ni ). We claim that Ki → Kr = K(f n ) factors through τ ≥−i Ki . Then this claim for i = 0 shows the proposition. This is clear for i = r because τ ≥−r Ki = Ki for all i. Consider now the case i = r − 1 assuming r > 0. There is a distinguished triangle (Proposition F.157) in D(A) +1

H −r (Kr−1 )[r] −→ Kr−1 −→ τ ≥1−r Kr−1 −→ yielding an exact sequence HomD(A) (τ ≥1−r Kr−1 , Kr ) −→ HomD(A) (Kr−1 , Kr ) −→ HomD(A) (H −r (Kr−1 )[r], Kr ). The composition H −r (Kr−1 )[r] → Kr−1 → Kr is zero because it is equal to the composition H −r (Kr−1 )[r] → H −r (Kr )[r] = τ ≤−r Kr → Kr , where the first map vanishes by construction of the Ki . Therefore Kr−1 → Kr factors through τ ≥1−r Kr−1 . Now the same argument applied to τ ≥1−r Kr−2 → τ ≥1−r Kr−1 shows that this morphism factors through τ ≥2−r Kr−2 . Hence Kr−2 → Kr factors through τ ≥2−r Kr−2 . Proceeding by induction shows our claim. Proposition 24.17. Let A be a noetherian ring, let I ⊆ A be an ideal, and let M be an object in D(A) such that H p (M ) is a finitely generated A-module for all p ∈ Z. Then the cohomology modules H p (M ∧ ) of the derived I-completion are the I-adic completions of the cohomology modules H p (M ). In particular we see that if M is as in the proposition and in D[a,b] (A), then M ∧ ∈ D (A). [a,b]

Proof. (I). We first assume that M is in D− (A). Then M is pseudo-coherent by Proposition 22.61. Hence it is isomorphic in D(A) to a bounded above complex F of finite free A-modules by Proposition 21.162. As F is K-flat by Lemma 21.93, we have n n M ⊗L A A/I = F/I F and hence M ∧ = R lim F/I n F = lim F/I n F, where the first equality holds by Proposition 24.16 and Proposition F.231 and the second because the transition maps are surjective (Proposition F.227). As the usual I-adic completion is exact on finite A-modules (Proposition B.39) this shows the assertion.

387 (II). Now let M in D(A) be arbitrary with H p (M ) finitely generated. Fix p ∈ Z and consider a distinguished triangle +1

τ ≤p+1+r M −→ M −→ C −→, where r is the cardinality of some finite set of generators of I. Then C is in D[p+r+1,∞) (A) and hence C ∧ is in D[p+1,∞) (A) by Corollary 24.14. Hence the long exact cohomology sequence of the distinguished triangle (τ ≤p+1+r M )∧ → M ∧ → C ∧ → yields an isomorphism
∼ H p (τ ≤p+1+r M )∧ → H p (M ∧ ) and we can conclude by Step (I). Corollary 24.18. Let A be a noetherian ring, let I ⊆ A be an ideal, and let M be a finite A-module. Then the derived I-completion of M is the usual I-adic completion of M . (24.4) Globalization of derived completion. Most of the above constructions about the derived completion of modules over a ring can be globalized to derived completions of modules over ringed spaces X (or even general “ringed topoi”) with respect to an ideal of finite type. These generalizations are rather straightforward if the ideal is globally generated by finitely many elements, the only case that we will need in the sequel. We fix a ringed space (X, OX ). For U ⊆ X open, f ∈ OX (U ) and F a complex of OU -modules we set f

f

f

T (F , f ) := R lim(F ←− F ←− F ←− . . . ) ∈ D(U ). If F is a complex of OX -modules, we set T (F , f ) := T (F |U , f ). We also denote by OU,f the sheaf of rings on U attached to the presheaf of localizations V 7→ (OX (V ))f . Then for every F in D(U ) one has functorial isomorphisms (24.4.1)

T (F , f ) ∼ = R Hom OU (OU,f , F )

in D(U ). For V ⊆ U open we therefore have by (21.18.9) that (24.4.2)

T (F , f )|V = T (F , f |V ).

Lemma 24.19. Let I ⊆ OX be an ideal and let F be a complex of OX -modules. Let U ⊆ X be open and f ∈ I (U ). Then the following assertions are equivalent. (i) One has T (F , f ) = 0. (ii) For all E in D(OU,f ) one has R Hom OU (E , F |U ) = 0. Proof. By (24.4.1), (ii) implies (i). Let us assume that (i) holds and let E be in D(OU,f ). Choose a K-injective resolution F |U → I in D(U ). The proof of (24.2.3) shows that R Hom OU (E , F |U ) = R Hom OU,f (E , T (F , f )) = 0. Hence R Hom OU (E , F |U ) = 0.

388

24 Theorem on formal functions

Definition 24.20. Let I ⊆ OX be an ideal and let F be a complex of OX -modules. Then F is called derived I -complete if for all U ⊆ X open and f ∈ I (U ) one has T (F , f ) = 0. Denote by Dcomp (X) := Dcomp (X, I ) the full subcategory of D(X) consisting of derived I -complete complexes. Remark 24.21. Let I ⊆ OX be an ideal and let F be a complex of OX -modules. (1) Let (Ui )i be an open covering of X. Then (24.4.2) implies that F is derived I complete if and only if F |Ui is derived I |Ui -complete for all i. (2) If F is derived I -complete and J ⊆ I is a subideal, then F is derived J -complete. Proposition and Definition 24.22. Let I ⊆ OX be an ideal of finite type. Then the inclusion functor Dcomp (X, I ) → D(X) has a left adjoint called derived I -completion. It is denoted by F 7→ F ∧ . One has F ∧ = R Hom OX (C , F ) for a complex C in D(X). For schemes and quasi-coherent ideals we will use the following terminology. Definition 24.23. Let X be a scheme, let I be a quasi-coherent ideal of OX , and let Z be the corresponding closed subscheme of X. Then a complex F in D(X) is also called derived complete along Z if it is derived I -complete and we set Dcomp (X, Z) := Dcomp (X, I ). Moreover, we also call F ∧ the derived completion of F along Z and sometimes write ∧ . F/Z We refer to [Sta] 099F for a proof of Proposition 24.22 in general. We will explain the proof in the special case that the ideal I is globally generated by finitely many sections, where we can use the same argument as before. This hypothesis is for instance satisfied if there exists a morphism f : X → S of schemes, where S = Spec R is affine, and if I is the quasi-coherent ideal of OX defining a closed subscheme of the form f −1 (Z) for some closed subscheme Z of S that is defined by a finitely generated ideal of R. In all applications of derived completion in this book we will be in this situation. Proof. If I is globally generated by sections f1 , . . . , fr , then one can use the same argument as in the proof of Proposition 24.10 using as derived completion the functor (24.4.3)

F 7→ F ∧ := R Hom OX (CX (f1 , . . . , fr ), F ),

where CX (f1 , . . . , fr ) is the complex Y Y OX,fi1 fi2 −→ · · · −→ OX,f1 f2 ···fr OX,fi1 −→ (24.4.4) OX −→ i1

i1 0 and all n ≥ N . For the proof of the formal function theorem only Assertion (1) is applied. Assertion (2) will be used to prove openness of ampleness on proper schemes (Theorem 24.46 below).

395 Proof. To show (1) define a morphism π : Y → X by the cartesian diagram Y g

 Spec B

π

/X f

 / Spec A.

As the formation of the relative Spec commutes with base change (11.2.5), we have Y = Spec B. The B-module F corresponds to a coherent OY -module F ′ via the equivalence between the category of quasi-coherent B-modules and quasi-coherent OY modules (Proposition 12.5). In particular one has π∗ F ′ = F . As g is proper, H p (Y, F ′ ) is a finite B-module (Corollary 23.18). As Spec B → Spec A is affine, π is affine and hence H p (Y, F ′ ) = H p (X, F ) by Corollary 22.6. This proves (1). To show (2) we set M := π ∗ L . As π is affine, M is an ample line bundle on Y by Proposition 13.83. By the projection formula (Proposition 22.81) we find that π∗ (F ′ ⊗ M ⊗n ) = F ⊗ L ⊗n for all integers n and thus H p (Y, F ′ ⊗ M ⊗n ) = H p (X, F ⊗ L ⊗n ) by again using Corollary 22.6. Therefore (2) follows from Serre’s vanishing criterion (Theorem 23.2). Lemma 24.40. Let A be a noetherian ring and let I ⊆ A be an ideal. Let f : X → Spec A be a proper morphism and let F be a coherent OX -module. Fix p ≥ 0. Then the inverse system (H p (X, F /I n F ))n satisfies the Mittag-Leffler condition. Proof. We set Hk := H p (X, F /I k F ) and H := H p (X, F ). We fix n ≥ 1. We claim that for k ≥ n sufficiently large one has Im(Hk → Hn ) ⊆ Im(H → Hn ) which in particular shows that (H p (X, F /I n F ))n satisfies the Mittag-Leffler condition. Let Ck be the cokernel of H → Hk . It suffices to show that the map Ck → Cn is zero for k ≥ n sufficiently large. The exact sequence 0 → I k F → F → F /I k F → 0 yields the long exact cohomology sequence (24.7.1) · · · → H p (X, I k F ) → H p (X, F ) → H p (X, F /I k F ) → H p+1 (X, I k F ) → · · · One obtains the following two descriptions: Ck = Im(H p (X, F /I k F ) → H p+1 (X, I k F )) = Ker(H p+1 (X, I k F ) → H p+1 (X, F )). L L Now C := k≥0 Ck is a module over the A-algebra B := k≥0 I k via multiplication I ⊗ Ck → Ck+1 . Since B is generated over I, the ring B is L A by a generating system of L noetherian. Now C is a B-submodule of k H p+1 (X, I k F ) = H p+1 (X, k I k F ) which ˜ is generated B-module by Proposition 24.39 (1) applied to the f ∗ B-module La finitely k k I F . Therefore C is itself a finitely generated B-module. The first description of Ck shows that Ck is annihilated by I k . Since C is a finitely generated B-module, there exists an integer N such that the ideal I N ⊕ I N +1 ⊕ · · · of B annihilates C. Consider the composition I r ⊗ Cn → Cn+r → Cn . As C is a finitely generated B-module, multiplication I r ⊗ Cn → Cn+r is surjective for large r. Moreover, the composition is zero for r ≥ N . Hence Cn+r → Cn is zero for sufficiently large r.

396

24 Theorem on formal functions

Remark 24.41. Assume that in the situation of Lemma 24.40 the ideal I is a maximal ideal of A. This special case suffices for showing Theorem 24.42 below and its many applications (e.g., Corollary 24.44, Theorem 24.46, Theorem 24.49). In this case one can simply show the Mittag-Leffler condition for (H p (X, F /I n F ))n as follows. Let Xn = X ⊗A A/mn . Then H p (X, F /I n F ) = H p (Xn , F /I n F ) is a finite A/I n -module by Corollary 23.18 and hence is an Artinian A-module since A/I n is Artinian. But any inverse system of Artinian A-modules satisfies the Mittag-Leffler condition. The above lemmas also allow to give a short direct proof of the noetherian version of the theorem of formal functions, Theorem 24.37, without referring to the formalism of derived completion. From this chapter it uses only Proposition 24.39 (1) and the proof of Lemma 24.40. Then it is completed by the following argument. Proof. (Direct proof of the formal function theorem) We use the notation introduced in the proof of Lemma 24.40. Let Ek be the image of H p (X, I k F ) → H p (X, F ). Then (24.7.1) yields a projective system of short exact sequences 0 → H/Ek → Hk → Ck → 0. We have already seen that for all n ≥ 1 there exists a k ≥ n such that Ck → Cn is zero, hence limk Ck = 0. It remains to show that the family (Ek )k defines the I-adic topology on H since then taking limits we see that the I-adic completion of H is limk Hk which is the statement of the theorem of formal functions. We have I k H ⊆ Ek , because Ik Hk = 0. Hence it suffices to show that there exists k0 such that IEk = Ek+1 for all k ≥ k0 , since then Er+k0 = I r Ek0 ⊆ I r H. L L k We have already seen that M := k H p (X, I k F ) is a graded module over B = I which is finitely generated as a B-module by Proposition 24.39 (1). In particular it is a noetherian B-module. Hence the ascending chain of submodules of the form Mn :=

n M


M r p H p (X, I k F ) ⊕ I H (X, I n F ) r≥1

k=0

S

has to stabilize. As n Mn = M we find a k0 such that Mk = M for all k ≥ k0 . This shows that IH p (X, I k F ) = H p (X, I k+1 F ) for k ≥ k0 . As the Ek are defined as images of H p (X, I k F ) we deduce our claim. (24.8) Theorem of formal fibers. Theorem 24.42. Let S be a locally noetherian scheme, let f : X → S be a proper morphism, and let F be a coherent OX -module. Let s ∈ S be a point and let ms ⊂ OS,s be the maximal ideal. Set Xn := X ×S Spec(OS,s /mns ), let in : Xn → X be the projection, and set Fn := i∗n F . Then for all p ≥ 0 one has an isomorphism of ObS,s -modules (24.8.1)

p ∼ (Rp f∗ F )∧ s = lim H (Xn , Fn ). n

Here the left hand side is the ms -adic completion of the stalk (Rp f∗ F )s . Proof. As the natural morphism Spec OS,s → S is flat, we may replace S by Spec OS,s (Proposition 22.90). Then (Rp f∗ F )s = Γ(S, Rp f∗ F ) = H p (X, F ) (Theorem 22.27) and we conclude by the formal function theorem (Theorem 24.37).

397 The theorem is often applied in the following form. Corollary 24.43. In the situation of Theorem 24.42 suppose that limn H p (Xn , Fn ) = 0. Then there exists an open neighborhood V of s such that Rp f∗ (F )|V = 0. p Proof. By Theorem 24.42 we have (Rp f∗ F )∧ s = 0. As f is proper, R f∗ F is a coherent OS module. Therefore the ms -adic topology on the finite OS,s -module (Rp f∗ F )s is Hausdorff (Proposition B.42). Together with the above, this implies (Rp f∗ F )s = 0. Since Rp f∗ F is coherent, the subset { s ∈ S ; (Rp f∗ F )s = 0 } is open in S (Corollary 7.32).

Corollary 24.44. Let S be a scheme, let f : X → S be a proper morphism, and let F be a quasi-coherent OX -module. Let s ∈ S be a point and let d = dim Xs be the dimension of the fiber of f in s. Then for all p > d there exists an open neighborhood V of s such that Rp f∗ F |V = 0. We will prove the result only if S is locally noetherian and F is coherent. The general case is then obtained by approximation of proper morphisms by proper morphisms of finite presentation, by noetherian approximation and by writing every quasi-coherent module as a filtered colimit of modules of finite presentation. We refer to [Sta] 0E7D for the details. The reference also shows that the formation of Rd f∗ F |V commutes with arbitrary base change T → V . Proof. (if S is locally noetherian and F is coherent) The schemes Xn have the same underlying topological space as Xs for all n ≥ 1. Hence by Grothendieck’s vanishing theorem (Theorem 21.57) one has H p (Xn , F ⊗ (OS,s /mns )) = 0 for p > d. Hence we can apply Corollary 24.43. (24.9) Ampleness is open on proper schemes. We will use the following notation. For a morphism f : X → S of schemes, a point s ∈ S and an OX -module F , we denote by Fs the restriction (i.e., the pullback) of F to the schematic fiber Xs = f −1 (s) = X ×S Spec κ(s). Lemma 24.45. Consider the standard set up for inductive limits of schemes (as given in Section (10.13) 1.–3.), and assume that X0 → S0 is of finite presentation. Let L0 be a line bundle on X0 and set L := x∗0 L0 and Lλ := x∗0,λ L0 . Then L is relatively ample over S (resp. very ample over S) if and only if there exists an index λ0 such that Lλ is relatively ample over Sλ (resp. very ample over Sλ ) for all λ ≥ λ0 . Proof. As “ample over S” and “very ample over S” are compatible with base change S ′ → S, it suffices to show that the conditions are necessary. As Xλ → Sλ is of finite type, we may replace L by some suitable power and can assume that L is very ample (Theorem 13.62). Then there exists an immersion i : X → PnS =: P such that i∗ OP (1) = L . By Theorem 10.63, i is the base change of some Sλ -morphism iλ : Xλ → PnSλ =: Pλ . By Proposition 10.75, we can assume that iλ is an immersion after enlarging λ. Finally by applying a gluing argument to Theorem 10.58 (see Exercise 10.32), we may assume that i∗λ OPλ (1) ∼ = Lλ which shows that Lλ is very ample. Theorem 24.46. Let f : X → S be a proper morphism of schemes and let L be an invertible OX -module. Let s ∈ S be a point and assume that Ls is ample on the fiber Xs . Then there exists an open affine neighborhood U ⊆ S of s such that L |f −1 (U ) is ample.

398

24 Theorem on formal functions

We will prove the theorem only if S is locally noetherian. In general one first approximates X locally on S by proper S-schemes of finite presentation and uses noetherian approximation as explained in Chapter 10 and Lemma 24.45. For details see [Sta] 0D2S. Proof. (if S is locally noetherian) We may assume that S is affine. Then by our additional assumption S is the spectrum of a noetherian ring. Since Spec OS,s is the filtered limit of its affine neighborhoods, we may replace S be Spec OS,s by Lemma 24.45. Hence we can assume that S is the spectrum of a local noetherian ring R and s is its closed point corresponding to the maximal ideal m of R. We will show that for every coherent OX -module F there exists an integer N such that H 1 (X, F (n)) = 0 for n ≥ N , where F (n) := F ⊗ L ⊗n . This will prove that L is ample by Lemma 23.5. As s is a closed point, Xs is a closed subscheme of X defined by the ideal sheaf J := mOX . For every L OXs -module G we set G (n) := G ⊗ Ls⊗n . Let i : Xs → X be the inclusion. Set C := j≥0 mj /mj+1 which is a finitely generated R-algebra because ˜ is a quasi-coherent OX -algebra of finite type. Since R is noetherian. Hence C := f ∗ (C) L J C = 0 we have C = i∗ (Cs ), where Cs := i∗ C . We also set M := j≥0 mj F /mj+1 F . Again we have M = i∗ Ms with Ms := i∗ M . As F is coherent, M is a quasi-coherent C -module of finite type. Since Ls is ample, we can apply Proposition 24.39 (2) to the Cs -module Ms and see that there exists an integer N such that H p (Xs , Ms (n)) = 0 for all p > 0 and all n ≥ N . By the projection formula (Proposition 22.81) we have i∗ (Ms (n)) = M (n). As i is affine, we deduce that H p (X, M (n)) = 0 for all p > 0 and all n ≥ N (Corollary 22.6). Therefore we see that (*)

H p (X, mj F /mj+1 F (n)) = 0,

for all

p > 0, j ≥ 0, n ≥ N.

In particular, H 1 (X, F /mF (n)) = 0 for all n ≥ N . Moreover, the long cohomology sequence attached to the short exact sequence 0 −→ mj F /mj+1 F (n) −→ F /mj+1 F (n) −→ F /mj F (n) −→ 0 shows that (*) also implies that H 1 (X, F /mj+1 F (n)) → H 1 (X, F /mj F (n)) is an isomorphism for all j and n ≥ N . In particular limj H 1 (X, F /mj F (n)) = 0 for all n ≥ N . Hence Corollary 24.43 implies that (R1 f∗ F (n))s = H 1 (X, F (n)) is zero for n ≥ N .

Stein factorization As an application of the theorem of formal functions, we come to the Stein factorization of a proper morphism, which is a factorization as a morphism with geometrically connected fibers followed by a finite morphism. Similarly as many other results on proper morphisms in this book, it has an analogue in complex geometry, that was proved several years earlier. (24.10) Stein factorization. Let f : X → S be a qcqs morphism of schemes. Then f∗ OX is a quasi-coherent OS -algebra (Corollary 10.27) and we obtain a factorization of f as

399 (24.10.1)

f′

π

X −→ S ′ = Spec(f∗ OX ) −→ S

with the following properties (Corollary 12.2) (1) The homomorphism OS ′ → f∗′ OX is an isomorphism. (2) The morphism π is affine. If S is affine, one has S ′ = Spec Γ(X, OX ). Remark 24.47. Let f : X → S be a qcqs morphism of schemes and let T → S be an affine morphism. Then every morphism of S-schemes X → T factorizes through X → Spec(f∗ OX ). Indeed, by Proposition 12.1 we find T = Spec A for some quasi-coherent OS -algebra A and hence HomS (X, T ) = HomOS -Alg (A , f∗ OX ) = HomS (Spec(f∗ OX ), T ) by Proposition 11.1. Remark 24.48. Let f : X → S be a qcqs morphism of schemes. The formation of f∗ OX commutes with flat base change (Proposition 12.6). Therefore the factorization (24.10.1) is compatible with base change by a flat morphism S1 → S, i.e., the factorization (24.10.1) of the base change f1 : X1 := X ×S S1 → S1 of f is given by X1′ → S1′ → S1 with S1′ = S ′ ×S S1 . If f is proper, the factorization (24.10.1) is called the Stein factorization. It has the following properties. Theorem 24.49. Let f : X → S be a proper morphism of schemes. Then the factorization (24.10.1) of f has the following properties. (1) The morphism f ′ is proper with geometrically connected fibers. (2) The morphism π is integral with finite discrete fibers. For s ∈ S and s′ ∈ π −1 (s) the field extension κ(s) → κ(s′ ) is finite. (3) If S is locally noetherian, then π is finite. Note that if π is finite, then all assertions of (2) are automatically satisfied. Recall that we assume connected spaces to be non-empty. In particular, f ′ is surjective. We will prove the theorem only in the case that S is locally noetherian. The general case is deduced from this case by nontrivial noetherian approximation (see [Sta] 03H2, 0E0M for details). Proof. (if S is locally noetherian) As π is affine, it is in particular separated. Hence f ′ is proper because f is proper (Proposition 12.58). By the coherence theorem (Theorem 23.17), f∗ OX is a coherent OS -module, hence π is finite. It remains to apply the following corollary to f ′ . Corollary 24.50. Let f : X → S be a proper morphism with OS = f∗ OX . Then f has geometrically connected fibers. Proof. We again restrict to the case that S is locally noetherian. Fix s ∈ S. To show that the fiber Xs = f −1 (s) is geometrically connected, we may replace S by Spec OS,s by Remark 24.48. Hence we may assume that S = Spec R for some local noetherian ring R with maximal ideal m. To see that f is surjective it suffices to see that f is dominant because f is closed. But every morphism f : X → S with OS = f∗ OX is dominant because R → Γ(X, OX ) is injective and thus f cannot factor through any closed subscheme V (a) ⊊ Spec R. To prove that Xs is geometrically connected we proceed in two steps.

400

24 Theorem on formal functions

(I). We first show that Xs is connected which proves that every proper morphism f with OS = f∗ OX has connected fibers. The fiber Xs is closed in X. The theorem of formal fibers 24.42 shows that lim Γ(Xn , OXn ) = R∧ , n

where Xn = V (mn+1 OX ) is the n-th infinitesimal neighborhood of Xs . If Xs was not connected, then there existed a non-trivial idempotent element (en )n ∈ limn Γ(Xn , OXn ) as all Xn have the same underlying topological space as Xs . This contradicts the fact that R∧ is local and hence cannot have a non-trivial idempotent element. (II). To show that Xs is geometrically connected, it suffices to show that for every finite separable extension k of κ(s) the base change Xs ⊗κ(s) k is connected (Proposition 5.53). By Lemma 18.65 there exists a finite flat local R-algebra R′ with residue field k. Let f ′ : X ′ := X ⊗R R′ → S ′ := Spec R′ be the base change of f , and let s′ ∈ S ′ be the closed point. Again flatness and the corresponding fact for f imply that OS ′ = f∗′ OX ′ . As we already proved Step (I), we know that the fiber Xs′ ′ = Xs ⊗κ(s) k is connected. Recall from Remark 5.55 that for a scheme X over a field k the geometric number of connected components is the number of connected components of X ⊗k K, where K is some algebraically closed extension of k. This number is independent of the choice of K. It is finite if X is of finite type. If f : X → S is a morphism of schemes, we denote for s ∈ S by nX/S (s) the geometric number of connected components of the κ(s)-scheme Xs . This defines a map (24.10.2)

nX/S : S → N ∪ {∞}.

Proposition 24.51. Let f : X → S be a proper morphism of schemes and let f′

π

X −→ S ′ −→ S be its Stein factorization. Let s ∈ S. Then nX/S (s) is finite and one has nX/S (s) = nS ′ /S (s) =

X

[κ(s′ ) : κ(s)]sep ,

s′ ∈π −1 (s)

where [L : K]sep denotes the separability degree of a finite field extension K → L. Proof. Let us show the first equality. As in the proof of Corollary 24.50 we can reduce first to the case that S = Spec R for a local ring R and that s ∈ S is the closed point. Then we may assume that κ(s) is algebraically closed again using Lemma 18.65. The morphism g : f −1 (s) → π −1 (s) induced by f ′ is proper and surjective with connected fibers. Hence the first equality follows from the purely topological Lemma 24.52 below. To show the second, essentially elementary, equality, we may assume that S = Spec k. Then S ′ is a finite discrete k-scheme and hence we may assume the underlying topological space of S ′ consists of a single point s′ with residue field k ′ such that [k ′ : k] is finite. ′ As the geometric connected components of S ′ and Sred are the same, we may assume ′ ′ that S = Spec k . Let K be an algebraically closed extension of k. Then the number of connected components of Spec(k ′ ⊗k K) is equal to # Homk (k ′ , K) = [k ′ : k]sep .

401 If X is a topological space, we define an equivalence relation on X by x ∼ x′ if x and x′ are in the same connected component. The quotient space is the space of connected components of X and denoted by π0 (X). We obtain a functor π0 from the category of topological spaces to the category of totally disconnected topological spaces. It is left adjoint to the inclusion functor. If X is a scheme, then π0 (X) denotes the space of connected components of the underlying topological space of X. If X has only finitely many connected components (e.g., if X is noetherian), then π0 (X) is a finite discrete space, in other words, X has only finitely many connected components, and each of them is open and closed in X. Lemma 24.52. Let g : Z → Y be a continuous surjective map of topological spaces such that Y carries the quotient topology of Z (e.g., if g is open or closed). Suppose that all ∼ fibers of g are connected. Then the induced map π0 (Z) −→ π0 (Y ) is a homeomorphism. The homeomorphism is given by sending a connected component Z ′ of Z to g(Z ′ ) and its inverse is given by Y ′ 7→ g −1 (Y ′ ). Proof. For topological spaces X and X ′ let C(X, X ′ ) be the set of continuous maps X → X ′ . Let S be a totally disconnected space. We have functorial bijections C(π0 (Y ), S) = C(Y, S) = { f ∈ C(Z, S) ; f |g−1 (y) is constant for all y ∈ Y } = C(Z, S) = C(π0 (Z), S), where the first and last equality hold by adjointness of π0 and the inclusion functor, the second equality holds because Y carries the quotient topology of Z, and the third ∼ equality holds because all fibers of g are connected. Therefore π0 (g) : π0 (Z) −→ π0 (Y ) is a homeomorphism by Yoneda’s lemma. Remark 24.53. Let f : X → S be a proper morphism and let X → T → S be its Stein factorization. (1) As X → T is closed and surjective with connected fibers, we can apply Lemma 24.52 to see that π0 (X) → π0 (T ) is a homeomorphism. (2) Let s ∈ S. Then we may apply Lemma 24.52 to the surjective proper morphism Xs → Ts induced by X → T to see that π0 (Xs ) = π0 (Ts ) is a homeomorphism. Corollary 24.54. Let R be a local henselian ring, S = Spec R, and let s ∈ S be its closed point. Let X → S be a proper morphism. Then the inclusion Xs → X induces a bijection ∼ of finite discrete spaces π0 (Xs ) −→ π0 (X). ∼

Proof. Let X → T → S be the Stein factorization. The map π0 (Xs ) −→ π0 (X) can be factorized as ∼ ∼ π0 (Xs ) −→ π0 (Ts ) → π0 (T ) −→ π0 (X), where one has the first and the third bijection by Remark 24.53. Moreover, π0 (Ts ) is a finite discrete space by Theorem 24.49 (2). It remains to show that the second map is bijective. If S is noetherian and hence T → S is finite, this follows from the definition of a local henselian ring (Definition 20.1). In general, it follows from Exercise 20.10. Corollary 24.55. Let k be a field and let X → Spec k be a proper k-scheme. Then X is geometrically connected if and only if Γ(X, OX ) is a local ring of dimension 0 whose residue field is a purely inseparable extension of k.

402

24 Theorem on formal functions

Proof. By Proposition 24.51, X is geometrically connected over k if and only if the affine scheme Spec Γ(X, OX ) consists of only one point whose residue field is a purely inseparable extension of k. (24.11) Properties of Stein factorization. Proposition 24.56. Let f : X → S be a proper morphism with geometrically connected fibers and let X → S ′ → S be its Stein factorization. Then S ′ → S is a universal homeomorphism. Proof. As f has non-empty fibers by hypothesis, it is surjective and hence the integral morphism π : S ′ → S is surjective. By Proposition 20.29 it suffices to show that S ′ → S is universally injective. By Proposition 4.35 we have to show that for all s ∈ S the fiber π −1 (s) consists of a single point s′ and that κ(s′ ) is a purely inseparable extension of κ(s). But π −1 (s) is geometrically connected over κ(s) by Proposition 24.51. As it is affine, we can conclude by Corollary 24.55. We can now prove Zariski’s connectedness theorem, which also can be seen as a version of Zariski’s main theorem. Theorem 24.57. (Zariski’s Connectedness Theorem) Let f : X → S be a proper dominant morphism of integral schemes and let η ∈ S be the generic point. Let X → S ′ → S be its Stein factorization. Suppose that S is geometrically unibranch (Definition 20.49, e.g., if S is normal) and that the generic fiber Xη is geometrically connected. Then S ′ → S is a universal homeomorphism and all fibers of f are geometrically connected. Proof. We may assume that S = Spec R is affine. It suffices to show that the integral morphism S ′ → S is a universal homeomorphism by Proposition 24.51. Again it suffices to show that S ′ → S is universally injective by Proposition 20.29. By construction of the Stein factorization we have S ′ = Spec R′ with R′ = Γ(X, OX ). As X is integral, S ′ is also integral. The properties of the Stein factorization imply that R′ is integral over R (Theorem 24.49). The function field of S ′ is the field of fractions of Γ(X, OX ) or also the field of fractions of the localization Γ(Xη , OXη ). The latter is a purely inseparable extension K ′ of κ(η) because Xη is integral and geometrically connected (Corollary 24.55). Let s ∈ S and let A be the localization of the R-algebra Γ(X, OX ) in s. It is integral over the geometrically unibranch ring OS,s . As K ′ = Frac A is a purely inseparable extension of κ(η) = Frac OS,s , the normalization of OS,s in K ′ is a local ring whose residue field is a purely inseparable extension of κ(s) by Proposition 20.51 and Lemma G.32. Since A is integral over OS,s , its normalization equals the normalization of OS,s in K ′ . In particular the fiber of S ′ → S over s consists of a single point and its residue field is purely inseparable over κ(s). Therefore S ′ → S is universally injective (Proposition 4.35). Corollary 24.58. Let f : X → S be a proper dominant morphism of integral schemes and let η ∈ S be the generic point. Assume that S is normal and that the generic fiber of f is geometrically reduced. Then all fibers of f are geometrically connected. Proof. We have Γ(Xη , OXη ) = κ(η) (Proposition 12.66). Therefore S ′ → S is birational and thus an isomorphism because S ′ → S is integral and S is normal. Therefore f∗ OX = OS , and thus the morphism S ′ → S in the Stein factorization X → S ′ → S of f is an isomorphism.

403 Remark 24.59. In the situation of the corollary, if κ(η) is a perfect field, e.g., if κ(η) is a field of characteristic zero, then we may drop the assumption that the generic fiber of f be geometrically reduced. In fact, as X is assumed to be integral and hence reduced, the generic fiber Xη is certainly reduced. If κ(η) is perfect, then Xη is necessarily geometrically reduced (Corollary 5.57). The formation of f∗ OX does not commute with base change in general, but we gave several criteria for this to hold in Section (23.28). The results above show that this always holds “up to universal homeomorphism” in the following sense. Corollary 24.60. Let f : X → S be a morphism of proper schemes, let g : T → S be a morphism of schemes, and let fT : XT := X ×S T → T be the base change of f . Let S ′ = Spec f∗ OX and ST′ := S ′ ×S T = Spec g ∗ f∗ OX be its base change. Then the canonical morphism Spec(fT )∗ OXT → ST′ is a universal homeomorphism. Proof. Let f ′ : X → S ′ be the first morphism of the Stein factorization of f which is proper with geometrically connected fibers by Theorem 24.49. Therefore its base change fT′ : XT → ST′ is proper with geometrically connected fibers. Hence the second morphism of its Stein factorization, which is Spec(fT )∗ OXT → ST′ is a universal homeomorphism by Proposition 24.56. We have the following criterion when the Stein factorization is finite ´etale and commutes with arbitrary base change. Theorem 24.61. Let S be a scheme and let f : X → S be a proper flat morphism of finite presentation. Let s ∈ S be a point such that the fiber Xs is geometrically reduced. Let f′ π X −→ S ′ −→ S be the Stein factorization of f . Then there exists an open neighborhood U of s such that π −1 (U ) → U is finite ´etale and that the formation of the Stein factorization of f −1 (U ) → U commutes with arbitrary base change. The theorem can in particular be applied if f is proper and smooth, and in that case we can take U = S. Proof. We may assume that S = Spec R is affine. (I). As Xs is geometrically reduced, Γ(Xs , OXs ) is a finite ´etale κ(s)-algebra by Example 18.26. (II). We now claim that it suffices to show that β 0 (κ(s)) : Γ(X, OX ) ⊗R κ(s) → Γ(Xs , OXs ) is surjective. Indeed, if β 0 (κ(s)) is surjective, then Theorem 23.140 shows that there exists an open neighborhood V of s such that the formation of f∗ OX |V commutes with base change. Hence Theorem 23.140 (2) shows that f∗ OX |V is a locally free OS -module. We conclude that the Stein factorization of f −1 (V ) → V commutes with arbitrary base change. It follows that (*)

π −1 (s) = Spec Γ(Xs , OXs ).

By Step (I), Γ(Xs , OXs ) is an ´etale κ(s)-algebra. Therefore π is ´etale over an open neighborhood of s because the locus where the fibers of a finite locally free morphism are ´etale is open (Corollary 20.73).

404

24 Theorem on formal functions

(III). To show that β 0 := β 0 (κ(s)) is surjective, we may assume that R is local with closed point s. As cohomology commutes with flat base change, we may pass to a faithfully flat R-algebra and hence can assume that R is strictly henselian. Then π −1 (s) is the scheme theoretic sum of finitely many copies of Spec κ(s) by Step (I) and (*). Passing to connected components of X, we may assume that π −1 (s) = Spec κ(s) by Corollary 24.54. But then β 0 is surjective because the composition β0

R −→ Γ(X, OX ) −→ Γ(X, OX ) ⊗ κ(s) −→ Γ(Xs , OXs ) = κ(s) is surjective. Corollary 24.62. Let f : X → S be a flat proper morphism of finite presentation with geometrically reduced fibers. Then the function nX/S of the geometric number of connected components is locally constant. Proof. Let X → S ′ → S be the Stein factorization. By Proposition 24.51 it suffices to show nS ′ /S is locally constant. But this is clear because S ′ → S is finite ´etale by Theorem 24.61. Corollary 24.63. Let f : X → S be a flat, proper morphism of finite presentation. If f has geometrically connected and geometrically reduced fibers, then OS → f∗ OX is an isomorphism and the formation of f∗ OX is compatible with base change. Note that even if S is noetherian and one omits one of the properties “flat”, “proper”, “geometrically connected fibers” or “geometrically reduced fibers”, then one does not have OS = f∗ OX in general: If one omits “flat”, take a surjective closed immersion that is not an isomorphism. If`one omits “proper”, take A1S . If one omits “geometrically connected fibers”, consider S S → S. If one omits “geometrically reduced fibers” take Spec OS [T ]/(T 2 ) → S or the Frobenius A1k → A1k for a field k of positive characteristic. Proof. Let X → S ′ → S be the Stein factorization. Then S ′ → S is finite ´etale and its formation commutes with arbitrary base change by Theorem 24.61, and it is of rank 1 by Proposition 24.51, hence S ′ = Spec f∗ OX → S is an isomorphism. Remark 24.64. We proved Theorem 24.49 about the main properties of the Stein factorization only in the locally noetherian case. To deduce the result for general base schemes requires a highly nontrivial noetherian approximation argument which is not covered by the techniques explained in Chapter 10. However, for Theorem 24.61 and its Corollaries 24.62 and 24.63 the noetherian approximation is easier and covered by the results and ideas described in Chapter 10. The main reason is that the formation of the Stein factorization commutes with arbitrary base change in this case. Let us go into more detail. Suppose that f : X → S and s ∈ S are as in Theorem 24.61 and we know the theorem for S locally noetherian. To prove it for an arbitrary scheme S, we may assume that S = Spec R is affine. We write R = colimλ Rλ for a filtered inductive system (Rλ )λ of noetherian rings Rλ . Set Sλ = Spec Rλ . As X → S is of finite presentation, there exists a scheme fλ : Xλ → Sλ of finite type such that Xλ ×Sλ S ∼ =X (Theorem 10.66). As the properties “flat” and “proper” are compatible with filtered colimits of rings (Appendix C), we may assume that Xλ → Sλ is proper and flat. Moreover, let sλ ∈ Sλ be the image of s under S → Sλ . Then Xs = (Xλ )sλ ⊗ κ(sλ )κ(s) and therefore the fiber (Xλ )sλ is geometrically reduced because Xs is assumed to be geometrically reduced. Hence we can apply Theorem 24.61 in the noetherian case and

405 we see that there exists an open neighborhood Uλ of sλ such that (fλ )∗ (OXλ )|Uλ is finite locally free and that its formation commutes with arbitrary base change. Applying this to the base change ρλ : S → Sλ we deduce Theorem 24.61 for X → S and s by choosing U = ρ−1 λ (Uλ ). (24.12) The seesaw theorem. In this section, we will prove the seesaw theorem, Theorem 24.66. We can motivate it by the following question. Assume that we have line bundles L and L ′ on an S-scheme f : X → S such that for all s ∈ S the restrictions Ls and Ls′ to the fiber X ×S Spec κ(s) are isomorphic. Can we conclude that L and L ′ are isomorphic? It is easy to see that there are two reasons why this cannot be true: For one thing, the condition on the fibers will not change when we replace S by its underlying reduced subscheme, so we should certainly add the condition that S be reduced. Second, if we set L ′ := L ⊗ f ∗ M for a line bundle M on S, then the restrictions of L and L ′ to the fibers will be isomorphic. So the best we can hope for is that for S reduced, if Ls ∼ = Ls′ for all s ∈ S, then L and ′ L are isomorphic up to tensoring one of them by the pullback of a line bundle from S. It is an immediate consequence of the seesaw theorem that this is true if X is a flat proper S-scheme of finite presentation with geometrically connected and geometrically reduced fibers. The term seesaw stems from the fact that for a line bundle on a product X ×S T one looks at restrictions to fibers X × {t} on the one hand, and to fibers {x} × T on the other hand, thus in a sense going back and forth between X and T . Lemma 24.65. Let k be a field, let X → Spec k be proper and suppose that X is integral. Then a line bundle L on X is trivial if and only if Γ(X, L ) ̸= 0 and Γ(X, L −1 ) ̸= 0. Proof. The condition is clearly necessary. Conversely, let 0 ̸= s ∈ Γ(X, L ) and 0 ̸= t ∈ Γ(X, L −1 ) which we consider as non-zero linear maps s : OX → L and t : L → OX . As X is integral, the composition t ◦ s ∈ Γ(X, OX ) is non-zero. As X is proper over k and integral, Γ(X, OX ) is a finite field extension of k, hence t ◦ s is an isomorphism. Therefore t is an isomorphism as surjective homomorphisms of locally free OX -modules of the same rank are isomorphisms (Corollary 8.12). Theorem 24.66. (Seesaw Theorem) Let f : X → S be a flat, proper morphism of finite presentation with geometrically connected and geometrically reduced fibers, and let E be a finite locally free OX -module. Then there exists a unique subscheme Z of S with the following property. For u : T → S a morphism of schemes we consider the cartesian diagram XT (C)

fT

 T

v

/X

u

 / S.

f

Then u factors through Z if and only if there exists a finite locally free OT -module M with fT∗ M ∼ = v ∗ E . Moreover one has: (1) The immersion Z → S is of finite presentation and as sets one has Z = { s ∈ S ; Es is a free OXs -module}.

406

24 Theorem on formal functions

(2) Suppose that there exists s0 ∈ Z such that H 1 (Xs0 , OXs0 ) = 0. Then there exists an open subscheme U of S such that s0 ∈ U ⊆ Z. (3) Suppose that f has geometrically integral fibers and that E is a line bundle. Then Z is a closed subscheme of S. In Section (27.21) we will introduce the Picard space classifying line bundles on X up to pullback of line bundles from S and show that the condition H 1 (Xs0 , OXs0 ) = 0 means that it has a 0-dimensional tangent space at s0 (Proposition 27.122) and hence is ´etale (Exercise 27.3). This gives a more geometric interpretation of Assertion (2), at least if E is a line bundle. For the proof of the seesaw theorem, we will use the following criterion when a vector bundle is the pullback of a vector bundle. Lemma 24.67. For a ringed space Y we denote by (VB(Y )) the category of locally free OY -modules of finite type. Let f : X → S be a morphism of ringed spaces. For U ⊆ S open consider the functor fU∗ : (VB(U )) → (VB(f −1 (U ))),

M 7→ fU∗ M ,

where fU : f −1 (U ) → U denotes the restriction of f . (1) The functor fU∗ is fully faithful for all open subspaces U ⊆ S if and only if OS = f∗ OX . (2) Suppose that OS = f∗ OX . Let E be a finite locally free OX -module of rank r and let E ∨ be its dual. Then the following assertions are equivalent. (i) The OX -module E is in the essential image of f ∗ : (VB(S)) → (VB(X)). (ii) The OS -module f∗ E is locally free of finite type and the canonical homomorphism f ∗ f∗ E → E is an isomorphism. (iii) The OS -modules f∗ E and f∗ E ∨ are locally free of rank r and the canonical pairing (24.12.1)

f∗ E ⊗OS f∗ E ∨ −→ f∗ (E ⊗OX E ∨ ) −→ f∗ OX = OS

is perfect. If these equivalent conditions are satisfied, then E ∼ = f ∗ f∗ E . Moreover, the formation of f∗ E commutes with base change if f is a morphism of schemes and f∗ OX = OS holds after arbitrary base change. Proof. Let us show (1). The condition that fU∗ is fully faithful for all U ⊆ S open holds if and only if for all open subspaces U ⊆ S and for all OU -modules M1 and M2 the canonical map (*)

Hom OU (M1 , M2 ) −→ (fU )∗ fU∗ Hom OU (M1 , M2 )

of OU -modules is an isomorphism. But if (*) is an isomorphism for U = S and M1 = M2 = OS , then OS = f∗ OX . Conversely, suppose that OS = f∗ OX . Then for U ⊆ S open one also has that OU = (fU )∗ Of −1 (U ) as the formation of f∗ commutes with passing to open subspaces. Let M be any finite locally free OU -module, for instance Hom OU (M1 , M2 ) = M ∨ 1 ⊗ M2 as above. We claim that the canonical map (fU )∗ fU∗ M → M is an isomorphism. Indeed, as this can be checked locally on U , we may assume that M = OUn for some n ≥ 0 and we conclude by OU = (fU )∗ Of −1 (U ) . In particular, (*) is an isomorphism.

407 Next we prove (2). Suppose that E is in the essential image of f ∗ , i.e., there exists a finite locally free OS -module M with E ∼ = f ∗ M , then necessarily M ∼ = f∗ E by the claim above, in particular f∗ E is finite locally free, necessarily of rank r because f ∗ f∗ E = E is of rank r. The same argument shows that f∗ E ∨ is finite locally free of rank r. Moreover, f ∗ (M ∨ ) = E ∨ and hence f∗ (E ∨ ) = M ∨ and (24.12.1) is given by the canonical pairing M ⊗ M ∨ → OS and in particular is perfect. To see that f ∗ f∗ E → E is an isomorphism, n we can work locally on X and hence assume that E = OX and we conclude by the hypothesis f∗ OX = OS . Hence we have seen that (i) implies (ii) and (iii). Suppose that (ii) holds. Then E ∼ = f ∗ M with M := f∗ E . Hence (ii) implies (i). Let us show that (iii) implies (ii). Set M := f∗ E which is locally free of rank r by hypothesis. To see that the canonical homomorphism of locally free modules ψ : f ∗ M = f ∗ f∗ E → E is an isomorphism, we can work locally on S. Therefore, we may assume that r M ∼ → E . It corresponds to a homomorphism = OSr ∼ = f∗ (E ∨ ). Then ψ becomes a map OX r in HomOS (OS , f∗ E ). By hypothesis we find a homomorphism r r φ ∈ HomOS (OSr , f∗ (E ∨ )) = HomOX (OX , E ∨ ) = HomOX (E , OX ) r . Hence φ and ψ are isomorphisms by Corollary 8.12. such that φ ◦ ψ = idOX It remains to show that the formation of f∗ E is compatible with arbitrary base change for scheme morphisms. If E = f ∗ M , then for a cartesian diagram as in Theorem 24.66 (C) one has (fT )∗ (v ∗ E ) = (fT )∗ fT∗ (u∗ M ) = u∗ M = u∗ f∗ E

which shows the claim. Proof. (of Theorem 24.66) The hypotheses imply that f∗ OX = OS , compatibly with base change (Corollary 24.63). All hypotheses and assertions are local on S. The uniqueness of Z is clear. For an integer r ≥ 0 let Xr be the open and closed subscheme where E has a fixed rank r. As all fibers of f are connected and as f is open and closed, Xr = f −1 (Sr ) for the open and closed subscheme Sr := f (Xr ) of S. Hence, we may assume that E is of constant rank r. Now Lemma 24.67 shows that Z is the locus in S, where f∗ E is locally free of rank of r with formations commuting with base change and where the pairing f∗ E ⊗OS f∗ E ∨ → OS of locally free OX -modules (24.12.1) is perfect. The locus Z ′ defined by the first condition is given by an immersion Z → S of finite presentation by Proposition 23.147 applied with b = 0. The locus in Z ′ defined by the second condition is given by the nonvanishing of the r-th exterior power of the homomorphism of locally free rank r modules f∗ (E ∨ )|Z ′ → f∗ (E )∨ |Z ′ which defines an open subscheme Z → Z ′ of finite presentation. Moreover s ∈ S is in Z if and only if Spec κ(s) → S factors through Z. This shows the set-theoretic description of Z in (1). We now show (3). Lemma 24.65 shows that for E = L a line bundle, Z as a set can also be described as Z = { s ∈ S ; dimκ(s) H 0 (Xs , Ls ) > 0 and dimκ(s) H 0 (Xs , Ls−1 ) > 0}, which is a closed subset by semicontinuity (Theorem 23.139).

408

24 Theorem on formal functions

It remains to show (2). As s0 ∈ Z, Es0 = E |Xs0 is a free OXs0 -module, say isomorphic r to OX . Hence by hypothesis we have H 1 (Xs0 , Es0 ) = 0. Thus Corollary 23.144 shows s0 that, after possibly replacing S by an open neighborhood of s0 , f∗ E is finite locally free and that its formation commutes with base change. Hence a basis (e1 , . . . , er ) of H 0 (Xs0 , Es0 ) lifts to a basis of f∗ E again after possibly passing to a smaller neighborhood ∼ of s0 . This basis defines an isomorphism OSr → f∗ E which corresponds to a homomorphism ∗ r r u : f OS = OX → E . As u comes from a basis of H 0 (Xs0 , Es0 ), its restriction to Xs0 is an isomorphism. Therefore u is surjective in an open neighborhood W ⊆ X of Xs0 by Nakayama’s lemma and hence an isomorphism on W , being a surjective map of finite free modules of the same rank. As f is closed, there exists an open neighborhood U ⊆ S of s0 such that f −1 (U ) ⊆ W . Then E |f −1 (U ) is isomorphic to the pullback of OUr and hence the open immersion U → S factors through the subscheme Z by the universal property of Z. Corollary 24.68. Let f : X → S be a flat, proper morphism of finite presentation with geometrically connected and geometrically reduced fibers. Let E be a finite locally free OX -module and let T ⊆ S be a subset such that E |Xs is a free OXs -module and such that H 1 (Xs , OXs ) = 0 for all s ∈ T . Then there exists an open neighborhood U of T in S and a finite locally free OU -modules M such that fU∗ M ∼ = E |f −1 (U ) , where fU : f −1 (U ) → U is the restriction of f . (24.13) Application: Picard group of projective bundles and of products. Proposition 24.69. Let S be a non-empty scheme, let E be a finite locally free OS -module of rank r + 1 with r ≥ 1, let π : P(E ) → S be the structure morphism of the projective bundle defined by E . Then the map µ : Pic(S) × Z −→ Pic(P(E)),

(L , d) 7→ π ∗ L ⊗ OP(E ) (d)

is an isomorphism of groups. ∼ O r+1 and hence P(E ) ∼ Proof. Locally on S one has E = = PrS locally on S. Therefore all S fibers of π are projective spaces of dimension r over a field. So π is proper, flat, of finite presentation with geometrically integral fibers. In particular π∗ OP(E ) = OS . Recall also that we have already seen that Pic(Prk ) = Z for any field k (Example 11.45). Finally, we know by Proposition 22.22 that (*)

H 1 (Prk , OPrk ) = 0.

Set X := P(E ) and write F (d) = F ⊗OX OX (d) for every OX -module F , as usual. Let us show that µ is injective. Let (L , d) ∈ Ker(µ), i.e., π ∗ L (d) is trivial. Its restriction to some fiber Xs ∼ = Prκ(s) is OXs (d) hence d = 0. By Lemma 24.67 (1), π ∗ : Pic(S) → Pic(X) is injective and hence L ∼ = OS . Q If S is the disjoint union of open and closed subschemes Si , then Pic(S) = i Pic(Si ) Q and Pic(P(E )) = i Pic(P(E |Si )). Hence to see that µ is surjective, we may pass to subschemes of S that are open and closed. Let N be a line bundle on X. Choose s ∈ S. Then N |Xs is a line bundle on Prκ(s) and hence isomorphic to OXs (d) for some d ∈ Z. Set M := N (−d) such that M |Xs is trivial. Let Z be the subscheme of S over which M is the pullback of a line bundle on the base, which is an open and closed subscheme of S because of (*) (Theorem 24.66). Hence after replacing S by Z, we see that M ∼ = π ∗ L for ∗ ∼ some line bundle L on S. Hence N = π (L )(d).

409 A similar argument shows the following result. Proposition 24.70. Let k be a field, let T be a proper, geometrically connected, geometrically reduced scheme over k such that H 1 (T, OT ) = 0. Let S be an arbitrary k-scheme. Then there is an isomorphism of groups ∼

µ : Pic(S) × Pic(T ) −→ Pic(S ×k T ),

(L , M ) 7→ p∗1 L ⊗ p∗2 M .

Examples of T satisfying the hypotheses are projective spaces (where the result is a special case of Proposition 24.69), K3-surfaces or, more generally, (strict) Calabi-Yau varieties (Exercise 25.18). On the other hand, in general Pic(S) × Pic(T ) ∼ ̸ Pic(S × T ), = even if S and T are smooth projective curves over a field. Proof. The same argument as in the proof of Proposition 24.69 shows that µ is injective. The surjectivity of µ is also shown in the same way using that p2 : S ×k T → S is flat, proper, of finite presentation with geometrically connected and geometrically reduced fibers and that for all s ∈ S one has 1 1 H 1 (p−1 2 (s), Op−1 (s) ) = H (T ⊗k κ(s), OT ⊗k κ(s) ) = H (T, OT ) ⊗k κ(s) = 0. 2

Remark 24.71. In the situation of Proposition 24.69 any line bundle M on P(E ) is of the form M ∼ = π ∗ (L )(d) for some line bundle L on S and some integer d ∈ Z. By the projection formula one has for all i ≥ 0 Ri π∗ (π ∗ L (d)) = Ri π∗ OX (d) ⊗ L and we calculated Ri π∗ OX (d) in Theorem 22.86. (24.14) The Theorem of the Cube. Lemma 24.72. Let S be a scheme and let f : X → S, fi : Xi → S, i = 1, . . . , m be proper, flat S-schemes of finite presentation with geometrically reduced and geometrically connected fibers. Let gi : Xi → X be morphisms of S-schemes, and let E be a finite r for all i. Let locally free OX -module of rank r such that gi∗ E is trivial, i.e., gi∗ E ∼ = OX i Z = Z(E , f ) ⊆ S be the locus on which E is the pullback of a finite locally free OS -module (Theorem 24.66). Let s ∈ Z be a point such that M (24.14.1) H 1 (Xs , OXs ) → H 1 (Xi,s , OXi,s ) i

is injective. Then there exists an open neighborhood U of s in S such that f∗ E |U ∼ = OUr and such that f ∗ f∗ E → E is an isomorphism over U . In particular, U → S factors through Z → S. We will give the proof only if S is locally noetherian. For a (different) proof for general schemes in case that E is a line bundle (the only case that we will use), see [Sta] 0BF2. Proof. Suppose that S is locally noetherian. Let A be a local Artinian quotient ring of OS,s and write XA and Xi,A instead of X ×S Spec A and Xi ×S Spec A, respectively. By Corollary 24.63 one has

410 (*)

24 Theorem on formal functions H 0 (XA , OXA ) = H 0 (Xi,A , OXi,A ) = A.

(I). We claim that E |XA is trivial. We prove this by induction on the length of A. If A has length 1, then A = κ(s) and E |Xs is trivial since s ∈ Z. If the length of A is l > 1, we can choose 0 ̸= ε ∈ mA such that εmA = 0. Then A0 := A/(ε) has length l − 1 and Ker(A → A0 ) ∼ = κ(s) as OS,s -modules. As E is flat over S we obtain an exact sequence of OX -modules 0 −→ E |Xs −→ E |XA −→ E |XA0 −→ 0. ∼

r By induction hypothesis, there exists an isomorphism OX −→ E |XA0 which we view as a A0 0 tuple of sections σ1 , . . . , σr ∈ H (XA0 , E |XA0 ). It suffices to show that we can lift these secr tions to elements of H 0 (XA , E |XA ), i.e., that their images in H 1 (XsL , E |Xs ) = H 1 (Xs , OX ) s 1 are zero. By hypothesis, it suffices to show that their images in i H (Xi,s , OXi,s ) are zero. As gi∗ E is trivial, by (*) the sections gi∗ (σj ) ∈ H 0 (Xi,A0 , gi∗ E |Xi,A0 ) can be lifted to 0 H (Xi,A , gi∗ E |Xi,A ) and therefore their images in H 1 (Xi,s , OXi,s ) are indeed zero. This shows the claim. (II). Let M := f∗ E , which is a coherent OS -module. Set Xn := XOS,s /mns . By the formal function theorem (Theorem 24.42) we find ∧ r Ms∧ = lim H 0 (Xn , E |Xn ) ∼ ) , = (OS,s n

where the second equality holds by Step (I) using (*). Since OS,s is noetherian, the ∧ homomorphism OS,s → OS,s is faithfully flat, so we obtain that Ms is a free OS,s -module of rank r and the map Ms → H 0 (Xs , E |Xs ) is surjective. Therefore there exists an open neighborhood U of s such that M |U ∼ = OUr and such that f ∗ M → E is surjective over U . ∗ As both f M and E are finite locally free of the same rank over U , we see that f ∗ M → E is an isomorphism over U . The following result, usually called the theorem of the cube, will be important for example in Chapter 27 on abelian schemes. Theorem 24.73. (Theorem of the Cube) Let S be a scheme, let X → S and Y → S be flat proper morphisms of finite presentation with geometrically integral fibers, and let T be an S-scheme. Let L be a line bundle on X ×S Y ×S T . Suppose that there exist sections x ∈ X(S) and y ∈ Y (S) and a point t0 ∈ T such that the restrictions of L to X ×S Y ×S {t0 }, to X ×S {y} ×S T , and to {x} ×S Y ×S T are trivial. Then there exists an open and closed neighborhood W of t0 in T such that L |X×S Y ×S W is trivial. Proof. We apply Lemma 24.72 to the projection f : X ×S Y ×S T → T and to f1 : {x} ×S Y ×S T → T and f2 : X ×S {y} ×S T → T with gi given by the inclusions induced by x and y, respectively. For a point t ∈ T with image s ∈ S the fiber f −1 (t) is given by X ×S Y ×S Spec κ(t) = (Xs ×κ(s) Ys ) ⊗κ(s) κ(t). Let Z → T be the closed subscheme associated to (f, L ) given by Theorem 24.66. It is non-empty since t0 ∈ Z. By Lemma 24.72 it suffices to show that for every t ∈ Z the map (24.14.1) is injective in our situation here. But for all t ∈ T this map is of the form, omitting the respective structure sheaves from the notation, (*)

H 1 (Xs ×κ(s) Ys ) ⊗κ(s) κ(t) −→ (H 1 (Ys ) ⊗κ(s) κ(t)) ⊕ (H 1 (Xs ) ⊗κ(s) κ(t))

411 and a right inverse is given by the base change to κ(t) of the K¨ unneth isomorphism M ∼ H 1 (Ys ) ⊕ H 1 (Xs ) = H p (Ys ) ⊗κ(s) H q (Xs ) −→ H 1 (Xs ×κ(s) Ys ), p+q=1

where the first equality holds since X → S and Y → S have geometrically integral fibers and hence H 0 (Xs ) = H 0 (Ys ) = κ(s). This shows that (*) is even an isomorphism.

Algebraization We will now consider the situation that for an I-adically complete ring A, where I is a finitely generated ideal of A, we are given a compatible family of schemes Xn over A/I n+1 (see Definition 24.82 below for a precise definition). This family is said to be algebraizable if there exists a scheme X over A such that X ⊗A A/I n+1 ∼ = Xn for all n ≥ 0. Moreover if we are given compatible quasi-coherent OXn -modules Fn , we say that (Fn )n is algebraizable if there exists a quasi-coherent OX -module F whose pullback to Xn is isomorphic to Fn . The main results in this part are Grothendieck’s algebraization theorem (Theorem 24.113) and Grothendieck’s existence theorem (Theorem 24.94). For both theorems A has to be noetherian. Then the algebraization theorem states that a family (Xn )n of proper schemes can be algebraized (uniquely up to isomorphism) by a proper scheme over A if the Xn carry a compatible system of ample line bundles. The existence theorem states that if a proper family (Xn )n is algebraizable by a proper scheme X, then one has an equivalence between coherent modules over X and compatible systems (Fn )n of coherent modules over (Xn )n . Notation For any scheme X we will denote by |X| its underlying topological space. Then |·| is a functor from the category of schemes to the category of topological spaces that sends every morphism f of schemes to the underlying continuous map |f |. (24.15) Ideals of definition for constructible closed subsets. Let Y be a qcqs scheme. Recall that a closed subset Z ⊆ |Y | is constructible if and only if its complement |Y | \ Z is quasi-compact (Proposition 10.44). In a noetherian scheme, any closed subset is constructible (Lemma 1.25). In general, the closed constructible subspaces are those that are locally defined by the vanishing of finitely many functions: Lemma 24.74. Let Y be a qcqs scheme, and let Z ⊆ |Y | be a closed subspace. Then the following assertions are equivalent. (i) The complement U := |Y | \ Z is quasi-compact. (ii) There exists a quasi-coherent ideal I of OY of finite type with |V (I )| = Z. (iii) For every quasi-coherent ideal J of OY with |V (J )| = Z there exists a finite type quasi-coherent ideal I ⊆ J such that |V (I )| = Z. Proof. There exists a quasi-coherent ideal J of OX such that |V (J )| = Z by Proposition 3.52. Hence (iii) implies (ii).

412

24 Theorem on formal functions

Let us show that (ii) implies (i). As Y is quasi-compact, we can cover Y by finitely many open affine subschemes V . To show that U is quasi-compact it suffices to show that V ∩ U is quasi-compact for all V . Hence we may assume Y = Spec A is affine and that Z is the underlying topological space of Spec A/I, where I is generated by finitely many elements f1 , . . . , fr . But then U is the union of the quasi-compact principal open subsets D(fi ). Hence it is quasi-compact. It remains to show that (i) implies (iii). If |V (J )| = Z, then J |U = OU is an OU -module of finite type. As Y is qcqs, there exists a finite type quasi-coherent ideal I ⊆ J with I |U = OU (Proposition 10.48). Then |V (J )| ⊆ |V (I )| ⊆ X \ U and hence |V (I )| = Z. From now on Y will denote a qcqs scheme and Z will be a closed constructible subspace. Definition 24.75. An ideal of definition for Z is a quasi-coherent ideal I ⊆ OY of finite type such that |V (I )| = Z. Let I(Z) be the set of ideals of definition for Z and let S(Z) be the set of closed subschemes of Y whose underlying topological space is Z and that are defined by a finite type quasi-coherent ideal. Sending I ∈ I(Z) to its vanishing scheme V (I ) yields an inclusion reversing bijection between I(Z) and S(Z). Both sets are non-empty by Lemma 24.74. Example 24.76. Let Y = Spec A be affine. Then each T ideal of definition for Z corresponds to a finitely generated ideal I ⊆ A such that rad(I) = z∈Z pz . We call such an ideal also an ideal of definition for Z. In general rad(I) is not finitely generated and therefore it is not an ideal of definition. Let I ⊆ A be an ideal of definition for Z and J ⊆ I be a finitely generated ideal. Then J is an ideal of definition for Z if and only if all elements of the image of I in A/J are nilpotent in A/J. If I is an ideal of definition, then I n is an ideal of definition for all n ≥ 1. More generally, we have: Lemma 24.77. Let Y be a qcqs scheme, let Z, Z ′ ⊆ |Y | be closed subsets and suppose that Z is constructible. Let I be an ideal of definition for Z and let I ′ ⊆ OY be any quasi-coherent ideal with |V (I ′ )| = Z ′ . (1) Suppose Z ′ ⊆ Z. Then there exists an n ≥ 1 such that I n ⊆ I ′ . (2) Suppose that |Y | \ Z ′ is quasi-compact and that I ′ is an ideal of definition for Z ′ . Then I I ′ is an ideal of definition for Z ∪ Z ′ , and I + I ′ is an ideal of definition for Z ∩ Z ′ . Proof. We may assume that Y = Spec A is affine, using that Y is quasi-compact for (1). Let I, I ′ ⊆ A be the finitely generated ideals corresponding to I and I ′ , respectively. Then II ′ and I + I ′ are finitely generated. Hence (2) follows because for the radicals of ideals one has rad(II ′ ) = rad(I) ∩ rad(I ′ ) and rad(I + I ′ ) = rad(rad(I) + rad(I ′ )). Let us show (1). As I ⊆ rad(I) ⊆ rad(I ′ ), we find generators a1 , . . . , ar of I and integers i mi ≥ 1 such that am ∈ I ′ . Then I m1 +···+mr ⊆ I ′ . i Proposition 24.78. Let X be a qcqs scheme and let Z be a closed constructible subset. Let F be a quasi-coherent OX -module of finite type. Then Supp(F ) ⊆ Z if and only if there exists an ideal of definition I for Z with I F = 0.

413 Proof. The condition is clearly sufficient. Hence suppose that Supp(F ) ⊆ Z. Let I ′ be the annihilator of F which is a quasi-coherent ideal with |V (I ′ )| = Supp F by Proposition 7.35. By Lemma 24.77 we find an ideal of definition I for Z with I ⊆ I ′ . Then I F = 0 since F is annihilated by its annihilator. (24.16) Formal completion of qcqs schemes along closed subspaces. Let S be a scheme. Recall from Section (4.2) that the Yoneda functor Y 7→ hY with hY (T ) = HomS (T, Y ) yields a fully faithful embedding from the category (Sch/S) of \ of contravariant functors from (Sch/S) to the S-schemes into the category (Sch/S) category of sets. We will usually identify a scheme with the functor it represents. In \ we have the full subcategory of Zariski sheaves (Sch/S) ^ (Sch/S) Zar , i.e., of those functors F : (Sch/S)opp → (Sets) such that for every S-scheme T and for every open covering (Ti )i of T one has
Q Q // i,j F (Ti ×T Tj ) , F (T ) = Eq i F (Ti ) where Eq(−) denotes the equalizer of a pair of parallel arrows. By Proposition 14.76 the ^ Yoneda embedding factorizes through (Sch/S) Zar . If I → (Sch/S), i 7→ Yi , is a diagram in (Sch/S) (in other words, simply a functor from a small category I to (Sch/S); the most important case for us will be I = N), then we write colim Yi I

^ for the colimit of the functors hYi in the category (Sch/S) Zar of Zariski sheaves on ^ (Sch/S). Note however that the embedding (Sch/S) → (Sch/S) Zar does not commute with colimits (except if S is the empty scheme), therefore this is usually not the colimit in the category of schemes, and colimits do not exist in (Sch/S) in general. Hence colimI Yi is a Zariski sheaf on the category of S-schemes, but it is usually not represented by a scheme. Proposition and Definition 24.79. Let S be a scheme, let Y be a qcqs S-scheme and let Z ⊆ |Y | be a closed constructible subset. Fix an ideal of definition J ∈ I(Z) and set Yn := V (J n+1 ) for n ≥ 0. (1) The partially ordered set S(Z) (Definition 24.75) is filtered. The system (Yn )n is a final subset of S(Z). (2) We call the Zariski sheaf (24.16.1)

Yˆ := Y/Z := colim Y ′ = colim Yn ′ n

Y ∈S(Z)

the formal completion of Y along Z. If T is a qcqs S-scheme, then (24.16.2)

Y/Z (T ) = colim Y ′ (T ) = colim Yn (T ). ′ Y ∈S(Z)

n

In particular Y/Z is the unique Zariski sheaf such that (24.16.2) holds for all S-schemes T that are affine. (3) For any S-scheme T one has (24.16.3)

Y/Z (T ) = { f : T → Y ; |f |(|T |) ⊆ Z },

where |f | : |T | → |Y | denotes the underlying continuous map of topological spaces.

414

24 Theorem on formal functions

(4) The functor Y/Z is an fpqc-sheaf. The equality (24.16.2) shows that after restriction to qcqs S-schemes, Y/Z is the colimit in the category of all contravariant functors from the category of qcqs S-schemes to the category of sets. Proof. Assertion (1) follows from Lemma 24.77. It also shows the equality in (24.16.1) and the second equality in (24.16.2). (I). The canonical map ψ : colimn Yn (T ) → Y/Z (T ) is bijective for qcqs schemes T by the same argument as in the proof of Lemma 21.52. This proves parts (1) and (2). (II). Let F be the functor on S-schemes defined by the right side of (24.16.3). We show that F is an fpqc-sheaf. Indeed, let (Ti → T )i be an fpqc cover and let si ∈ F (Ti ) such that the images of si and sj in F (Ti ×T Tj ) are equal for all i and j. As Y is an fpqc-sheaf by Proposition 14.76, there exists a unique s ∈ Y (T ) whose image in F (Ti ) is ` si for all i. Since i Ti → T is surjective one necessarily has s ∈ F (T ). (III). As the underlying topological space of Yn is Z, one has a compatible system of monomorphisms of functor f : Yn → F . As F is a sheaf for the Zariski topology, one obtains an induced morphism α sitting in a commutative diagram /F

α

colimY ′ ∈S(Z) Y ′ &

Y.



It remains to show that α is an isomorphism. As source and target of α are Zariski sheaves, we can do so locally on Y . Therefore we may assume that Y = Spec A is affine. Let I ⊆ A be the radical ideal with V (I) = Z. Clearly α is a monomorphism. To show that α is an epimorphism of Zariski sheaves, it suffices to show that α(T ) is surjective for T = Spec B an affine scheme. Let f : T → Y be an element of F (T ). It is given by a ring homomorphism φ : A → B such that every prime ideal of B contains φ(I), in other words, all elements in φ(I) are nilpotent. Let J := Ker(φ). Then J ⊆ I and the injective homomorphism A/J → B sends all elements of the ideal I/J in A/J to nilpotent elements. Hence all elements of I/J are nilpotent in A/J. Therefore rad(J) = I and the element f ∈ F (T ) is the image of a T -valued point of Spec A/J. For affine schemes we have the following situation. Example 24.80. Let A be a ring, X := Spec A, and let Z ⊆ |X| be a closed subset. Then its complement is quasi-compact if and only if there exists a finitely generated ideal I of A such that Z = | Spec A/I|. We assume this from now on. Every such ideal I is then an ideal of definition for Z. The set of ideals of definitions for Z is again denoted by I(Z). Hence X/Z = colim Spec A/I = colim Spec A/J n+1 , I∈I(Z)

n≥0

where J is a fixed ideal of definition for Z. There is a unique topology on A making A into a topological ring such that the ideals I ∈ I(Z) form a basis of neighborhoods of 0 in A. For this topology (J n+1 )n≥0 is also a basis of neighborhoods of 0. Hence this topology is the J-adic topology. The Z-adic completion of this topological ring is given by

415 Aˆ := lim A/I = lim A/J n+1 , I∈I(Z)

n

where A/I and A/J n+1 are rings endowed with the discrete topology (Remark 24.1). We obtain morphisms of Zariski sheaves Spec(A)/Z −→ Spec Aˆ −→ Spec A. ˆ As J is finitely generated, the topology on Aˆ is the J-adic topology where Jˆ := J Aˆ and ˆ is the ˆ Jˆn+1 (Proposition 24.2). Then the underlying closed subset of V (J) Aˆ = limn A/ ˆ ˆ preimage Z ⊆ Spec A of Z. We set ˆ := (Spec A)/Z = (Spec A) ˆ ˆ. Spf(A) Z This Zariski sheaf is then even an fpqc-sheaf by Proposition 24.79 (4) and it is called the formal spectrum of the topological ring A. The formation of the completion along a closed subspace is functorial in the following sense. Remark 24.81. Let f : X → Y be a morphism of qcqs schemes. Let Z ⊆ |Y | be closed constructible. The morphism f is itself quasi-compact (Remark 10.4) and quasiseparated (Proposition 10.25). If I is an ideal of definition for Z and hence |V (I )| = Z, then J := f −1 (I )OX is the quasi-coherent ideal of OX defining the closed subscheme f −1 (V (I )) whose underlying topological space is f −1 (Z). Hence J is an ideal of definition for f −1 (Z). Let T ⊆ f −1 (Z) ⊆ X be any closed subspace such that |X| \ T is quasi-compact. Then by Lemma 24.77 there exists for every ideal of definition K for T an n ≥ 1 such that J n ⊆ K . In particular, for every X ′ in S(T ) there exists an Y ′ ∈ S(Z) such that f restricts to a morphism of schemes X ′ → Y ′ . Hence f induces a morphism of Zariski sheaves f/Z,T : X/T −→ Y/Z . If T = f −1 (Z), then we often write X/Z instead of X/f −1 (Z) and f/Z instead of f/Z,f −1 (Z) . One obtains a functor (−)/Z of completion along Z from the category of qcqs Y -schemes to the category of Zariski sheaves over Y/Z . (24.17) Adic formal schemes over complete rings. In this section we denote by A a ring and by I ⊆ A a finitely generated ideal such that A = limn A/I n . Set S := Spec A, Z := V (I), and Sn := Spec A/I n+1 such that Z is the underlying topological space of Sn for all n ≥ 0. Definition 24.82. An adic formal scheme X over S/Z consists of a family (Xn → Sn )n of scheme morphisms and for all n ≥ 0 a morphism of Sn+1 -schemes Xn → Xn+1 such that the diagram

(24.17.1)

Xn

/ Xn+1

 Sn

 / Sn+1

416

24 Theorem on formal functions

is cartesian for all n ≥ 0. A morphism of adic formal schemes X → X ′ over S/Z is an inductive system of Sn -morphism un : Xn → Xn′ such that the base change (un+1 )Xn is un for all n ≥ 0. The set of these morphisms is denoted by HomS/Z (X , X ′ ). We obtain the category of adic formal schemes over S/Z . Geometrically, we think of formal schemes as a variants of usual schemes which are allowed to carry “infinitely thick infinitesimal neighborhoods”. It is possible to develop the theory in a more geometric way by setting up a theory of topologically ringed spaces, see for instance [EGAInew] I.10. By Proposition 24.79 (1) to give an adic formal scheme over S/Z is the same as to give a family XS ′ → S ′ of scheme morphisms for all closed subschemes S ′ with Z as underlying topological space together with S-morphisms αS2′ ,S1′ : XS1′ → XS2′ such that αS ′ ,S ′

XS1′  S1′

2

1

/ XS ′ 1  / S2′

is cartesian for all such closed subschemes S1′ and S2′ with S1′ ⊆ S2′ and such that for S1′ ⊆ S2′ ⊆ S3′ one has αS3′ ,S2′ ◦ αS2′ ,S1′ = αS3′ ,S1′ . Hence the notion of an adic formal scheme over S/Z does not depend on the choice of the ideal I but only on the subspace Z. Definition and Proposition 24.83. Let X = (Xn )n and Y = (Yn )n be adic formal schemes over S/Z , and let f = (fn : Xn → Yn )n be a morphism of adic formal schemes X → Y. Let P be one of the properties “separated”, “of finite type”, “proper”, “finite”, “affine”, “closed immersion”, “quasi-compact”, “quasi-separated”. Then f is said to have P if the following equivalent conditions hold. (i) The morphisms fn : Xn → Yn have the property P for all n ≥ 0. (ii) The morphism f0 : X0 → Y0 has the property P. We also say that f is finite locally free if fn is finite locally free for all n. Proof. As Sn → Sn+1 is a universal homeomorphism and a closed immersion for all n, so are Xn → Xn+1 and Yn → Yn+1 for all n ≥ 0 in view of the cartesian diagram (24.17.1). In particular, for the underlying reduced subschemes one has (Xn )red = (Xm )red and (Yn )red = (Ym )red as schemes over (Sn )red = (Sm )red for all n, m ≥ 0. Hence for all properties that depend only on the underlying reduced scheme (i) and (ii) are equivalent. This holds for the properties “quasi-compact” and “quasi-separated” which depend only on the underlying topological spaces. It also holds for the properties “separated” (Proposition 9.13) and “affine” (Lemma 12.38). Now suppose that f0 is of finite type. Then f is quasi-compact. To show that fn is locally of finite type, we may assume that Ym = Spec Bm and Xm = Spec Cm is affine for all m. By induction, we find a surjection of Bn−1 -algebras Bn−1 [t1 , . . . , tr ] → Cn−1 which we can lift to a map of Bn -algebras Bn [t1 , . . . , tr ] → Cn . This is surjective by Nakayama’s lemma because Cn → Cn−1 has nilpotent kernel. Hence Cn is a finitely generated Bn -algebra. Now the equivalence of (i) and (ii) for “proper” follows from Proposition 12.58. We can also deduce the equivalence for “finite” because a morphism of schemes is finite if and only if it is affine and proper (Corollary 12.89).

417 If f0 is a closed immersion, then fn is finite for all n. To see that fn is a closed immersion, we may assume that Ym = Spec Bm is affine for all m. Then Xm = Spec Cm for a finite Bm -algebra for all m. We conclude by Nakayama’s lemma which shows that Bn → Cn is surjective if and only if B0 → C0 is surjective. Let X → S = Spec A be a morphism of schemes. Then the family of schemes Xn := X ×S Sn forms an adic formal scheme over S/Z . Lemma and Definition 24.84. If X → Spec A is qcqs, then we have for all n ≥ 0 a commutative diagram of Zariski sheaves in which all rectangles are cartesian

(24.17.2)

Xn

/ X/Z

/X

 Sn

 / S/Z

 / S.

We call an qcqs adic formal scheme (Xn )n over S/Z algebraizable if it is of the form (X ×S Sn )n for a qcqs scheme X over A. Proof. It suffices to show that the right small rectangle is cartesian on T -valued points, where T is an affine scheme. But in this case one has X/Z (T ) = colimn Xn (T ) and S/Z (T ) = colimn Sn (T ) by (24.16.2). Then it is clear that the diagram is cartesian. This shows in particular that X/Z = colimn Xn and the qcqs adic formal scheme (Xn = X/Z ×S/Z Sn )n determine each other in the algebraizable case. This holds indeed in general (Exercise 24.20), but we will not need this fact. We will view the functor (−)/Z as a functor from qcqs schemes over A to qcqs adic formal schemes over S/Z . If f : X → Y is a morphism of qcqs S-schemes, then f/Z is the family of morphisms fn := f × idSn : Xn → Yn . If A is noetherian, we will study some of its properties in Section (24.23). (24.18) Modules over formal schemes. Consider the following situation. For n ≥ 0 let ιn : Xn → Xn+1 be closed immersions of schemes defined by a nilpotent ideal. In the sequel we will be only interested in one of the following examples. (1) Let X be a qcqs scheme and let Z be a closed constructible subset. Let J be an ideal of definition for Z and set Xn := V (J n+1 ). (2) The family X = (Xn )n is a qcqs adic formal scheme over a S/Z as in Definition 24.82. Definition 24.85. (1) A module over (Xn )n consists of a family of OXn -modules Fn together with isomor∼ phisms of OXn -modules αn : ι∗n Fn+1 −→ Fn for all n ≥ 0. A morphism F → G of modules over (Xn )n is a family of homomorphisms un : Fn → Gn of OXn -modules such that ι∗n (un+1 ) = un for all n ≥ 0. We obtain the category of modules over (Xn )n which we denote by ((Xn )n -Mod). (2) Let P be one of the properties “quasi-coherent”, “of finite type”, “coherent”, “locally free”, “locally free of rank r” for a fixed r ≥ 0. Then a module (Fn , αn )n is said to have P, if the OXn -module Fn has property P for all n.

418

24 Theorem on formal functions

(3) Let F = (Fn )n and G = (Gn )n be modules over (Xn )n . Then we define their tensor product by F ⊗ G := (Fn ⊗OXn Gn )n . (4) An algebra over (Xn )n is a module A over (Xn )n together with a map A ⊗ A → A of modules over (Xn )n that makes An into an OXn -algebra for all n. In particular, the system (OXn )n is an algebra over (Xn )n , which we call the structure sheaf of (Xn )n . Remark and Definition 24.86. Let X be a qcqs scheme, let Z be a closed constructible subset of |X|, let J be an ideal of definition for Z, and set Xn := V (J n+1 ). Then we call a module over (Xn )n also a module over X/Z . If (Fn , αn )n is such a module, then it is often notationally easier to view Fn as an OX -module that is annihilated by J n+1 ∼ and to view αn as an isomorphism Fn+1 /J n+1 Fn+1 → Fn of OX -modules. Often one omits the αn from the notation. By definition one has Hom(X/Z -Mod) ((Fn )n , (Gn )n ) = lim HomOX (Fn , Gn ). n

If F is an OX -module, then F/Z := (F /J n+1 F )n is a module over X/Z , called the formal completion of F along Z. We obtain an additive functor (24.18.1)

(OX -Mod) −→ (X/Z -Mod),

F → F/Z .

Moreover, as the pullback of modules via Xn → X commutes with tensor products we see that (24.18.2)

(F ⊗OX G )/Z = F/Z ⊗ G/Z .

Remark 24.87. By Proposition 24.79 (1) to give a module over X/Z is the same as to give for every ideal of definition I for Z an OX -module FI with I FI = 0 together with ∼ isomorphisms of OX -modules FI /I ′ FI → FI ′ for all ideals of definitions I ⊆ I ′ . Hence the notion of a module over X/Z given in Definition 24.86 does not depend on the choice of the ideal J . Similarly, the notion of a module over a qcqs adic formal scheme X over S/Z as in Definition 24.82 depends only on Z. Next we study modules over (Xn )n in the affine situation. Let A be a ring and let I ⊆ A be an ideal. Let limn (A/I n+1 -Mod) be the category of modules over the projective system (A/I n+1 )n , i.e., of families of A/I n+1 -modules Mn together with isomorphisms ∼ Mn /I n Mn → Mn−1 for all n. The category limn (A/I n+1 -Mod) can be identified with the category of quasi-coherent modules over (Spec A/I n+1 )n defined in Definition 24.85. Consider the functor (24.18.3)

(A-Mod) −→ lim (A/I n+1 -Mod), n

M 7→ (M/I n+1 M ).

If I is finitely generated, it corresponds to the functor of formal completion (24.18.1) for quasi-coherent modules. Proposition 24.88. Let A be a ring and let I ⊆ A be an ideal such that A is I-adically complete, i.e., A ∼ = limn A/I n+1 . (1) Let (Mn )n be in limn (A/I n+1 -Mod) such that M0 is a finitely generated A/I-module. Then Mn is a finitely generated A/I n+1 -module for all n and M := limn Mn is a finitely generated I-adically complete A-module. Moreover, M/I n+1 M ∼ = Mn .

419 (2) The functor (24.18.3) yields an equivalence between the category of finite projective A-modules M and the category of families (Mn )n of finite projective A/I n+1 -modules Mn . For a fixed integer r ≥ 0, M it is of rank r if and only if Mn is of rank r for all n. (3) The functor (24.18.3) yields an equivalence between the category of finitely generated I-adically complete A-modules and the category of families (Mn )n with Mn a finitely generated A/I n+1 -module. The proof will show that a quasi-inverse functor is given by (Mn )n 7→ limn Mn in (2) and (3). Proof. Let us show (1). As M0 is finitely generated, we can choose a surjection of A/Imodules p0 : (A/I)N → M0 . Inductively one constructs a family of A/I n+1 -linear maps pn : (A/I n+1 )N → Mn such that pn ≡ pn−1 (mod I n ). As I/I n+1 is nilpotent in A/I n+1 , the maps pn are surjective by Nakayama’s lemma. In particular, Mn is a finitely generated A/I n+1 -module. Set Kn := Ker(pn ). As Ker((A/I n+1 )N → (A/I n )N ) = (I n /I n+1 )N −→ I n Mn = Ker(Mn → Mn−1 ) is surjective, the snake lemma implies that Kn → Kn−1 is surjective. In particular, the system (Kn )n satisfies the Mittag-Leffler condition (Definition F.225). Hence applying the functor limn we obtain an exact sequence 0 → K −→ AN −→ lim Mn → 0, n

where K := limn Kn by Proposition F.227. Hence limn Mn is a finitely generated A-module. Moreover K → Kn is surjective and therefore M/I n+1 M = Coker(K → (A/I n+1 )N ) = (A/I n+1 )N /Kn = Mn . In particular M = limn M/I n+1 M is I-adically complete. Let us now prove (2). As A = lim A/I n , the functor (24.18.3) is fully faithful on finite free modules and hence on finite projective modules by passage to direct summands. It remains to show that if (Mn )n is a system of finite projective A/I n+1 -modules, then M := limn Mn is a finite projective A-module. Construct pn as above. As Mn is projective, we find sections sn : Mn → (A/I n+1 )N of pn for all n. We claim that we can arrange the sections inductively such that sn−1 ≡ sn (mod I n ) for all n ≥ 1. Indeed, write s¯n := sn (mod I n ) and consider the composition s¯n −sn−1

Mn −→ Mn−1 −−−−−−−−→ Kn−1 . As Mn is projective and Kn → Kn−1 is surjective, we can lift this homomorphism to a homomorphism tn : Mn → Kn and we can replace sn by sn − tn . This proves the claim. Hence, we obtain a compatible system of idempotent endomorphisms en := sn ◦ pn ∈ MN (A/I n+1 ) with Im(en ) = Mn . They define an idempotent endomorphism e ∈ MN (A) with Im(e) = M . Hence M is finite projective. The assertion about the ranks is clear. It remains to show (3). We have already seen in (1) that the functor (24.18.3) has (Mn )n 7→ M := limn Mn as a right inverse (up to isomorphism of functors). It is clear that this is also a left inverse.

420

24 Theorem on formal functions

Remark 24.89. The proof shows that all assertions of Proposition 24.88 also hold if one replaces I n by In , where (In )n is a sequence of ideals with In+1 ⊆ In for all n, with In /In+1 ⊆ A/In+1 nilpotent, and with A = limn A/In . If I is a finitely generated ideal in a ring A that is I-adicallyTcomplete, then a finitely generated A-module M is I-adically complete if and only if n I n M = 0 ([Sta] 031B). This condition is automatic if A is noetherian (Corollary B.44). Hence we obtain the following corollary. Corollary 24.90. Let X = Spec A be an affine noetherian scheme, let J ⊆ A be an ideal, and let Z := V (J) be its vanishing locus. Then the category of coherent modules over ˆ X/Z is equivalent to the category of finitely generated A-modules, where Aˆ is the J-adic completion of A. In particular, it is an abelian category. The equivalence is given by sending a coherent module (Fn )n over X/Z to limn Γ(X, Fn ) ˆ with quasi-inverse sending a A-module M to ((M/J n+1 M )∼ )n . Proposition 24.91. Let X be a noetherian scheme, let Z be a closed subspace of X. (1) The category of coherent modules over X/Z is abelian. (2) The functor F 7→ F/Z from the category of coherent OX -modules to the category of coherent modules over X/Z is exact. Proof. Let U = Spec A ⊆ X be open affine. Then the category of modules over U/(U ∩Z) is abelian by Corollary 24.90. Hence kernels and cokernels of morphisms of coherent modules over X/Z exist locally on affine schemes. As these form a basis of the topology, they exist globally by gluing. Moreover, the question whether for a morphism u of coherent modules over X/Z the map Coim(u) → Im(u) is an isomorphism also immediately reduces to the affine situation, where it is answered affirmatively again by Corollary 24.90. This shows (1). The exactness of F 7→ F/Z can be checked locally. Hence we may assume that X = Spec A is affine. Let J ⊆ A be an ideal of definition for Z, and set Aˆ := limn A/J n+1 . Then the functor F 7→ F/Z is identified via the equivalence of categories of Corollary 24.90 ˆ = limn M/J n+1 M which is exact by Proposition B.41. with the functor M 7→ M Remark 24.92. In the situation of Proposition 24.91 let u = (un )n : F → G be a map of coherent modules over X/Z . Then the cokernel of u in the abelian category of coherent modules over X/Z is the coherent module (Coker(un ))n over X/Z . Indeed, to check this we may assume that X = Spec A is affine. Let Aˆ be the I-adic completion for some ideal of definition I for Z. Then u corresponds to a projective system ˆ = limn Mn → limn Pn the corresponding map of A-linear maps un : Mn → Pn . Let u ˆ: M of Aˆ modules. The proof of Proposition 24.91 shows that the cokernel of u is given by the ˆ n+1 A) ˆ n = (Coker(un ))n . system (Coker(ˆ u) ⊗Aˆ A/I ˆ n+1 A) ˆ n gives the kernel of Similarly, in the affine situation the system (Ker(ˆ u) ⊗Aˆ A/I u but it is more difficult to describe it in terms of the kernels of the un . Remark 24.93. Let X be a noetherian scheme, let Z be a closed subspace of X, and let J be an ideal of definition for Z. If (Fn )n is a coherent module over X/Z , define an OX -module F := limn Fn . This module is usually not a coherent OX -module but it determines (Fn )n in the following sense. The canonical map F → Fn factors through F /J n+1 F → Fn . Working locally one deduces from Corollary 24.90 that this is an isomorphism.

421 (24.19) Grothendieck’s existence theorem for coherent modules. In this section let A be a noetherian ring, let I ⊆ A be an ideal, and let Z := V (I) be its vanishing space. We suppose that A is I-adically complete, i.e., the homomorphism A → limn A/I n is an isomorphism. Set Y := Spec A and Yn := Spec A/I n+1 . Let X → Spec A be a proper morphism of schemes. We set Xn := X ×Y Yn and denote by in : Xn → X the canonical closed immersion. The following theorem has some similarity with the GAGA theorems of Section (20.12) and is sometimes called the “formal GAGA” principle: In usual GAGA we compare complex spaces (with holomorphic functions) and schemes (with “polynomial functions”), whereas here we have “formal schemes” (with complete coordinate rings, think of formal power series) and schemes. Theorem 24.94. (Grothendieck’s existence theorem) Let X → Spec A be a proper morphism. Then the functor F 7→ F/Z is an equivalence between the category of coherent OX -modules and the category of coherent modules over X/Z . We will prove Theorem 24.94 in the following sections. Full faithfulness of F 7→ F/Z will be proved in Corollary 24.100. Essential surjectivity of F 7→ F/Z will be proved in Lemma 24.103. Below in Theorem 24.108 we will also see a generalization in which X is only assumed to be separated and of finite type over A and where one obtains an equivalence of coherent modules with proper support. Via the equivalence of Theorem 24.94 finite locally free modules correspond to each other. Proposition 24.95. Let f : X → Spec A be a proper morphism and let r ≥ 0 be an integer. Then the functor F 7→ F/Z yields an equivalence between the category of locally free OX -modules of rank r and the category of locally free modules over X/Z of rank r. Proof. If F is locally free of rank r, then F/Z = (i∗n F )n is clearly locally free of rank r. Conversely, let F be a coherent OX -module such that F/Z is locally free of rank r. We claim that it suffices to show that the stalk Fx is a flat OX,x -module for all x ∈ f −1 (Z). Indeed, then F is locally free of rank r in an open neighborhood of f −1 (Z) (Proposition 7.41), which is necessarily X by Lemma 24.96 below. Now the condition x ∈ f −1 (Z) means that the image of I is contained in the maximal ideal of OX,x . By hypothesis, Fx ⊗OX,x OX,x /I n OX,x is a projective module over OX,x /I n OX,x for all n. Hence Fx is a flat OX,x -module by the local criterion for flatness (Theorem B.51). We used the following lemma. Lemma 24.96. Let f : X → Spec A be a closed morphism and let U be an open neighborhood of f −1 (Z). Then U = X. Proof. As f is closed, f (X \ U ) is a closed subset of Spec A that does not meet Z = V (I). But I is contained in the Jacobson radical by Proposition B.42, i.e. Z contains all closed points of Spec A. Therefore f (X \ U ) = ∅ and hence U = X. There are also derived versions of the existence theorem. It works also in the nonnoetherian context if one imposes a stronger assumption than “proper” on the morphism. We state the following result as an example but refer to [Sta] 0DIG for the proof.

422

24 Theorem on formal functions

Theorem 24.97. Let · · · → An → An−1 → · · · → A0 be a projective system of surjective ring homomorphisms whose kernels consist of nilpotent elements and set A := limn An . Let X → Spec A be a proper, flat morphism of finite presentation. We view Xn := X ⊗A An as a closed subscheme of X for all n and denote by in : Xn → Xn+1 and In : Xn → X the inclusions. Let (Kn , φn )n be a family consisting of pseudo-coherent objects in D(Xn ) and isomorphisms φn : Li∗n Kn+1 → Kn in D(Xn−1 ). Then there exists a pseudo-coherent object in ∼ D(X) and isomorphisms LIn∗ K → Kn for all n that are compatible with the φn . (24.20) Full faithfulness of F 7→ F/Z . If X is a locally noetherian scheme, we denote by Dcoh (X) the full subcategory of D(X) consisting of complexes F such that H p (F ) is coherent for all p ∈ Z. Proposition 24.98. Let A be a noetherian ring and I ⊆ A an ideal with A = limn A/I n+1 . − + Let f : X → Y := Spec A be a proper morphism. Let F ∈ Dcoh (X) and G ∈ Dcoh (X). Then for all p ∈ Z one has functorial isomorphisms (24.20.1)

∼

ExtpOX (F , G ) → ExtpOX (F ∧ , G ∧ )

of finitely generated A-modules, where (−)∧ denotes the derived completion with respect to IOX . Proof. By Corollary 24.31 one has (*)

RΓ(X, R Hom OX (F , G ))∧ = RΓ(X, R Hom OX (F ∧ , G ∧ )).

One has RΓ(X, R Hom OX (F ∧ , G ∧ )) = R HomOX (F ∧ , G ∧ ) by Proposition 21.107 and hence applying H p (−) to the right side of (*) yields Extp (F ∧ , G ∧ ). As f is proper, Extp (F , G ) is a finite A-module (Corollary 23.38). Applying Proposition 24.17 one obtains H p (RΓ(X, R Hom OX (F , G ))∧ ) = Extp (F , G )∧ = Extp (F , G ), where the second equality holds because every finitely generated A-module is I-complete (Proposition B.44). As the derived completion for a coherent module on a noetherian scheme coincides with the classical completion (Corollary 24.34) one obtains the following result. Corollary 24.99. Let A be a noetherian ring, I ⊆ A an ideal with A = limn A/I n+1 . Let f : X → Y := Spec A be a proper morphism. Let F and G be coherent OX -modules. Then one has for all p ∈ Z a functorial isomorphism ∼

ExtpOX (F , G ) → ExtpOX (F ∧ , G ∧ ), where (−)∧ denotes the usual I-adic completion. The special case p = 0 yields the full faithfulness of the functor F 7→ F/Z . Corollary 24.100. Let A be a noetherian ring, I an ideal of A, and let Z = V (I) be the closed subscheme of Y := Spec A defined by I. Let f : X → Y be a proper morphism. Then the functor F 7→ (F /I n+1 F )n from the category of coherent OX -modules to the category of coherent modules over X/Z is fully faithful.

423 Proof. By Corollary 24.99 one has HomOX (F , G ) = HomOX (F ∧ , lim G /I n+1 G ) = lim HomOX (F ∧ , G /I n+1 G ) n

n

= lim HomOX (F ∧ /I n+1 F ∧ , G /I n+1 G ) n

= lim HomOX (F /I n+1 F , G /I n+1 G ), n

where the equality F /I n+1 F ∧ = F /I n+1 F can be checked locally on X and hence holds by Proposition B.41. ∧

(24.21) Essential surjectivity of F 7→ F/Z . We keep the notation from the beginning of Section (24.19), i.e., A is a noetherian ring, I ⊆ A an ideal, and Z := V (I) its vanishing space. We suppose that A is I-adically complete and set Y := Spec A, Yn := Spec A/I n+1 . Let X → Spec A be a separated morphism of schemes of finite type. We set Xn := X ×Y Yn . To prove the essential surjectivity if X → Spec A is proper we proceed somewhat similarly as in the proof that higher direct images of coherent modules are again coherent. In other words, we prove the result first if X is projective and then generalize to the proper case using the lemma of Chow and a d´evissage argument. We say that a coherent module (Fn )n over X/Z is algebraizable if it is isomorphic to a module of the form F/Z for a coherent OX -module F . We then also say that (Fn )n is algebraizable by F . We want to prove that every coherent module over X/Z is algebraizable if X → Spec A is proper. Recall that the coherent modules over X/Z form an abelian category and that F 7→ F/Z is exact (Proposition 24.91). Lemma 24.101. Let X → Spec A be proper. (1) Let u : (Fn )n → (Gn )n be a morphism of coherent modules over X/Z . If (Fn )n and (Gn )n are algebraizable, then Ker(u), Coker(u), and Im(u) are algebraizable. (2) Let 0 → (Fn )n → (Gn )n → (Hn )n → 0 be an exact sequence of modules over X/Z . If (Fn )n and (Hn )n are algebraizable, then (Gn )n is algebraizable. Proof. (1). Let F and G be coherent OX -modules with F/Z = (Fn )n and G/Z = (Gn )n . Then u corresponds to a map of OX -module v : F → G because we have already seen that the functor (−)/Z is fully faithful (Corollary 24.100). As (−)/Z is also exact (Proposition 24.91) we deduce that Ker(u), Coker(u) and Im(u) are algebraizable by Ker(v), Coker(v) and Im(v), respectively. (2). Let F and H be coherent OX -modules with F/Z = (Fn )n and H/Z = (Hn )n . Then (Gn )n defines a class in Ext1 (F ∧ , H ∧ ) by Proposition F.220. Hence we conclude by Corollary 24.99. The following proposition will be the key step for the proof of the essential surjectivity of F 7→ F/Z when X is projective over A. It will be also helpful for the proof of Theorem 24.113 below which gives a criterion when an adic formal scheme is algebraizable. Let X = (Xn )n be a proper adic formal scheme over Y/Z and let L = (Ln )n be a locally free module over X of rank 1 such that L0 is ample. For every coherent module E = (En )n over X and for every m ∈ Z we define the coherent module E (m) = (En (m))n over X by En (m) := En ⊗OXn Ln⊗m . Let OX = (OXn )n be the structure sheaf.

424

24 Theorem on formal functions

Proposition 24.102. Let E be a coherent module over X . Then there exists an integer m0 ≥ 0 such that (1) The map Γ(X , E (m)) := limn Γ(Xn , En (m)) → Γ(X0 , E0 (m)) is surjective for all m ≥ m0 . r (2) For all m ≥ m0 there exists an r ≥ 0 and a map u : OXr → E (m) such that un : OX → n E (m)n is a surjective map of OXn -modules for all n. Proof. Set An := A/I n+1 and denote by f = (fn )n theLmorphism X → Y/Z given n n+1 . As I/I 2 is a by the proper morphisms fn : Xn → Spec An . Let B := n≥0 I /I finitely generated A0 -module, this is a finitely generated A0 -algebra. Set J := IOX and ˜ = L J n /J n+1 which is a quasi-coherent OX -algebra of finite type. B := f0∗ B 0 n We view the OXn -modules En as OX -modules annihilated by J n+1 and set L Fn := Ker(En → En−1 ). As J Fn = 0 we may view Fn as an OX0 -module. Set F := n≥0 Fn which is a quasi-coherent B-module. We claim that it is a finitely generated B-module. Indeed, this may be checked locally on X. Hence we can assume that X = Spec L C is affine. If J ⊆ C is the ideal corresponding to J , then B corresponds to B := n J n /J n+1 . ˆ The module (En )nL over X/Z corresponds to a finite C-module E by Corollary 24.90 and F corresponds to n J n E/J n+1 E which is therefore a finite module over B. This proves the claim. Hence we may apply Proposition 24.39 and see that there exists an integer m0 such that H 1 (X0 , Fn (m)) = 0 for all n and for all m ≥ m0 . Hence the transition maps Γ(X, En (m)) → Γ(X, En−1 (m)) are surjective for all n and for all m ≥ m0 . This implies (1). As L0 is ample, we may also assume after possibly enlarging m0 that E0 (m) is generated by finitely many global sections for all m ≥ m0 . By (1) we may lift these global sections to Γ(X , E (m)) and obtain a map u : OXr → E (m) of modules over X . As u0 is surjective, un is surjective for all n by Nakayama’s lemma. We can now conclude the proof of Grothendieck’s existence theorem, Theorem 24.94. As we already have seen that F 7→ F/Z is fully faithful (Corollary 24.100), it suffices to show the following lemma. Lemma 24.103. Let X → Spec A be a proper morphism. Then every coherent module over X/Z is algebraizable. Proof. Recall that the category of coherent modules of X/Z is abelian (Proposition 24.91) and that a map u = (un )n : E → F of coherent modules over X/Z is surjective if and only if un : En → Fn is surjective for all n (Remark 24.92). We first prove the lemma under the additional assumption that there exists an ample line bundle L on X, i.e., if X → Spec A is projective. As usual we write G (m) for G ⊗ L ⊗m . Let F be a coherent module over X/Z . Then Proposition 24.102 (2) shows that there exist integers m, m′ , r, r ′ ≥ 0 and an exact sequence ′

w

OX/Z (−m′ )r −→ OX/Z (−m)r −→ F −→ 0. Hence F is algebraizable by Lemma 24.101 (1). To prove the lemma for arbitrary proper morphisms X → Spec A we proceed by noetherian induction on X. Hence we can assume that for every closed subscheme T ⊊ X all coherent modules over T/Z are algebraizable. We use Chow’s lemma (Theorem 13.100) by which we can find a surjective projective morphism π : X ′ → X such that X ′ is projective over A and such that π −1 (U ) → U is an isomorphism for some open dense subscheme U of X. Now we proceed as follows.

425 Let F be a coherent module over X/Z . ′ (1) We first construct a coherent module π ∗ F on X/Z , which is algebraizable because we showed the lemma already in the projective case. Next we construct an algebraizable coherent module π∗ π ∗ (F ) on X/Z and a canonical homomorphism π∗ π ∗ (F ) → F . (2) Let K (resp. C ) be its kernel (resp. its cokernel) such that we have an exact sequence (24.21.1)

0 −→ K −→ π∗ π ∗ (F ) −→ F −→ C −→ 0

of coherent modules on X/Z . Since π |π−1 (U ) is an isomorphism, we will see that K and C can already be considered as coherent modules over (X \ U )/Z where X \ U is endowed with a suitable structure of a closed subscheme of X. Hence they are both algebraizable by the induction hypothesis. (3) Then the exact sequence (24.21.1) shows that F is algebraizable by Lemma 24.101. We will carry out these three steps below in Construction 24.104 for step (1), in Lemma 24.105 for step (2), and in Lemma 24.106 for step (3). To construct inverse and direct images for coherent modules over formal completions a conceptual approach would be to develop first a theory of formal schemes as topologically ringed spaces, see [EGAInew] Ch. X, such that formal completions are special cases of formal schemes. As we did not do this, our definitions will be rather ad hoc. They can be justified by showing that these definitions are equivalent to the more natural definitions found in loc. cit. – and because they allow us to complete the proof of Grothendieck’s existence theorem which is the only reason why they are introduced here. Construction 24.104. Let X be a noetherian scheme and let Z ⊆ X be a closed subspace. We fix an ideal of definition J for Z. Let π : X ′ → X be a morphism of noetherian schemes, Z ′ := π −1 (Z) and let J ′ be the ideal of OX ′ generated by π ♯ (π −1 J ). It is an ideal of definition for Z ′ . We set Xn := V (J n+1 ) and Xn′ := V (J ′n+1 ) for all n ≥ 0. Then π −1 (Xn ) = Xn′ . We usually consider a module over X/Z as a family of OX -modules ′ Fn with J n+1 Fn = 0 and Fn /J n Fn ∼ = Fn−1 , similarly for modules over X/Z ′. (1) Let F = (Fn )n be a coherent module over X/Z . Then we define its pullback ′ π ∗ F := (π ∗ Fn )n . This is a coherent module over X/Z ′. If F is algebraizable by a coherent OX -module G , then π ∗ F is algebraizable by π∗ G . ′ (2) Let F ′ = (Fn′ )n be a coherent module over X/Z ′ and suppose that π is proper. Set c′ := limn F ′ which is an OX ′ -module and π∗ F ′ := (π∗ F c′ )/Z . F n ′ c′ is the J ′ -adic comIf F is algebraizable by a coherent OX -module G , then F ∧ ∧ ′ ∧ c = π∗ G = (π∗ G ) by Proposition 24.35. And hence pletion G of G and π∗ F π∗ F ′ = ((π∗ G )∧ )/Z = (π∗ G )/Z is a coherent module over X/Z which is algebraizable by π∗ G . (3) We continue to assume that π is proper. Let F = (Fn )n be a coherent module over X/Z such that π ∗ F is algebraizable. There is a functorial homomorphism of modules over X/Z (24.21.2)

θ : π∗ π ∗ F → F .

Indeed, for each n ≥ 0 we have by definition (π∗ π ∗ F )n = π∗ (lim π ∗ Fk )/J n+1 π∗ (lim π ∗ Fk ) k

k

426

24 Theorem on formal functions and we define θn : (π∗ π ∗ F )n → Fn to be induced by the composition π∗ (lim π ∗ Fk ) = lim π∗ π ∗ Fk → π∗ π ∗ Fn → Fn , k

k

where the first equality holds because π∗ has a left adjoint functor, namely π ∗ , and hence commutes with limits. Lemma 24.105. In the situation of Construction 24.104 (3) suppose that there exists ∼ an open subscheme U ⊆ X such that π induces an isomorphism p−1 (U ) → U . Let (Kn )n (resp. (Cn )n ) be the kernel (resp. cokernel) of the map θ : π∗ π ∗ F → F (24.21.2). Then there exists an ideal of definition A for X \ Z such that A annihilates Cn and Kn for all n. Proof. We may work locally on X and hence assume that X = Spec B is affine. Let J ⊆ B be the ideal corresponding to J . (I). We first assume that B = limn B/J n . In this case we may apply Proposition 24.90 c = limn Fn . Then π ∗ F and see that F is algebraizable by the coherent OX -module F ∗ c ∗ c by Construcis algebraizable by π F and therefore π∗ π F is algebraizable by π∗ π ∗ F tion 24.104 (2). Hence Ker(θ) and Coker(θ) are algebraizable by the kernel K respective c→ F c of coherent OX -modules. As π is cokernel C of the canonical homomorphism π∗ π ∗ F an isomorphism over U , the support of K and C are contained in X \ U hence there exist ideals of definition I and I ′ for X \ U with I K = 0 and I ′ C = 0 (Proposition 24.78). Then A := I I ′ annihilates K and C and a fortiori all Kn and Cn . ˆ = limn B/J n and thus are reduced (II). We now show that we can replace B by B ˜ := Spec B, ˆ let g : X ˜ → X be the to the case that we already proved in Step (I). Set X ˜ ′, π canonical morphism and define X ˜ , and g ′ by the following cartesian diagram ˜′ X

g′

π

π ˜

 ˜ X

/ X′

g

 / X.

Now g is flat because completions are flat for noetherian rings (Proposition B.41). By hypothesis, π ∗ F is algebraizable by a coherent OX ′ -module G and hence g ∗ π∗ π ∗ F is algebraizable by g ∗ π∗ G = π ˜∗ g ′∗ G , where the equality holds because push forward commutes with flat pullback (Proposition 12.6). Hence g ∗ π∗ π ∗ F = π ˜∗ π ˜ ∗ g ∗ F . By (i) we ∗ know therefore that the kernel and cokernel of g θ are annihilated by an ideal of definition A˜ for g −1 (X \ U ). We may assume A˜ to be of the form g ∗ (A ) for some ideal of definition A for X \ U . Indeed, if B is any ideal of definition for X \ U , then g ∗ B is an ideal of OX˜ because g is flat and it is an ideal of definition for g −1 (X \ U ). Then for some n ≥ 1 one has g ∗ (B)n = g ∗ (B n ) ⊆ A˜ by Lemma 24.77 (1). Hence we may replace A˜ by g ∗ (A ) for A := B ⊗n . Let a ⊆ B be the ideal corresponding to A and let Kn (resp. Cn ) be the finitely generated module over B/J n+1 corresponding to Kn and Cn , respectively. Then g ∗ Kn ˆ ˆ ˆ n+1 which can be identified with Kn since corresponds to the B-module Kn ⊗ B/(J B) n+1 n+1 ˆ ˆ ˆ B) = B/J . Hence Kn is annihilated by a because it is annihilated by aB. B/(J Hence Kn is annihilated by A for all n. The same argument shows that Cn is annihilated by A for all n.

427 Lemma 24.106. Let f : X → Y = Spec A be as in the beginning of this section and suppose that f is proper. Let u

u

u

1 2 3 0 −→ F1 −→ F2 −→ F3 −→ F4 −→ 0

be an exact sequence of coherent modules over X/Z . If F1 , F2 , and F4 are algebraizable, then F3 is algebraizable. Proof. By Lemma 24.101 (1), Im(u1 ) is algebraizable. By Lemma 24.101 (2) to the short exact sequence 0 → Im(u1 ) → F3 → F4 → 0, F3 is algebraizable. (24.22) Grothendieck’s existence theorem for coherent modules with proper support. There is also a refined version of Theorem 24.94 where f is not necessarily proper but all modules have proper support. Compare Section (23.5). Let us explain what we mean by this. Let X → Spec A be a separated morphism of finite type. Recall (Definition 23.21) that a closed subspace T of X is said to be proper over Y if the composition T → X → Spec A is proper for some scheme structure on T making T into a closed subscheme of X. Remark and Definition 24.107. Let F = (Fn )n be a quasi-coherent module of finite type over X/Z . By Nakayama’s lemma for nilpotent ideals (Proposition B.3), the homeomorphism Xn → Xn+1 induces an identification between the support of Fn and the support of Fn+1 . This closed subspace of |X0 | = |X1 | = · · · is called the support of F. Let F be a coherent OX -module with proper support over Y . Then F/Z := (i∗n F )n is a coherent module over the formal completion X/Z with proper support over Y . Theorem 24.108. (Grothendieck’s existence theorem) Let f : X → Y := Spec A be a separated morphism of finite type. Then the functor F 7→ F/Z from the category of coherent OX -modules with proper support over Y to the category of coherent modules over X/Z with proper support over Y0 is an equivalence of categories. A proof of this more general version of Grothendieck’s existence theorem can be given along the same lines as the proof of Theorem 24.94 given above, taking into account the following additional remarks. (1) As basic finiteness result one uses that under the hypotheses on X and Y the higher direct images Rp f∗ F are coherent for all coherent OX -modules F whose support is proper over Y (Corollary 23.22). (2) If F and G are coherent OX -modules, then the support of Ext pOX (F , G ) is contained in Supp(F ) ∩ Supp(G ) (Proposition 22.65). Using (1), then one can argue as in the proof of Proposition 24.98 to obtain an isomorphism of finitely generated A-modules p ExtpOX (F , G ) ∼ = ExtOX (F ∧ , G ∧ ).

(3) As in Corollary 24.100 one deduces an isomorphism of finitely generated A-modules (24.22.1)

∼

HomOX (F , G ) −→ Hom(X/Z -Mod) (F/Z , G/Z )

if Supp(F ) ∩ Supp(G ) is proper over A. This proves that the functor F 7→ F/Z in Theorem 24.108 is fully faithful.

428

24 Theorem on formal functions

(4) Then one shows that all coherent modules over X/Z of proper support are algebraizable if X is quasi-projective over A. For this one chooses an open embedding j : X → P into a projective A-scheme P . Let (Fn )n be a coherent module over X/Z whose support is proper over Y . Then (j∗ Fn )n is a coherent module over P/Z . Indeed, each Fn is of the form in,∗ Gn , where in : Tn → X is a closed immersion such that Tn → Y is proper and such that Gn is a coherent OTn -module. Then the immersion j ◦ in is a proper morphism (Proposition 12.58) and hence a closed immersion. Therefore j∗ Fn = (j ◦ in )∗ Gn is coherent. Then (j∗ Fn )n is algebraizable by a coherent OP module H with Supp(G ) = Supp((j∗ Fn )n ) = Supp(F ) and hence F = (j ∗ j∗ Fn )n is algebraizable by j ∗ G which has proper support over Y . (5) Finally, one reduces the general case to the quasi-projective case with the same arguments as in the proper case using Construction 24.104, Lemma 24.105, and Lemma 24.106. (24.23) Algebraization of proper schemes. In this section let A be a noetherian ring and let I ⊆ A be an ideal such that A = limn A/I n . Let S := Spec A and Z := V (I) ⊆ S. Set Sn := Spec A/I n+1 . If X is an adic formal scheme over S/Z , we set Xn := X ×S/Z Sn . For instance, if X = X/Z for a qcqs S-scheme X, then Xn = X ×S Sn (Lemma 24.84). Proposition 24.109. Let f : Y → S be a separated morphism of finite type. Then X 7→ X/Z induces an equivalence between the category of finite Y -schemes X that are proper over S and the category of finite adic formal schemes over Y/Z that are proper over S/Z . Moreover, a finite morphism X → Y is a closed immersion if and only if X/Z → Y/Z is a closed immersion. Proof. As formal completion of modules is compatible with tensor products (24.18.2), Grothendieck’s existence theorem, Theorem 24.108 implies that F 7→ F/Z induces an equivalence between the category of coherent OX -algebras that have proper support over S and the category of coherent algebras over X/Z that have proper support over S/Z . This implies the first assertion by the equivalence of finite schemes and finite quasi-coherent algebras (Corollary 12.2 and Remark 12.10 (3)). Now let ι : T = (Tn )n → X/Z be a closed immersion of adic formal schemes such that Tn is proper over Y for all n. By what we already showed there exists a finite morphism i : T → X of Y -schemes with i/Z = ι such that T is proper over Y and such that T/Z = T . To see that i is a closed immersion we have to show that C := Coker(i♭ : OX → i∗ OT ) is zero. It is a coherent OX -module whose support is proper because T is proper over Y . Now i♭ corresponds under (24.22.1) to the surjective morphism (OXn → i∗ OTn )n of coherent modules over X/Z . As formal completion of coherent modules is exact, C/T = 0. Hence C = 0 because formal completion on coherent modules with proper support is an equivalence of categories (Theorem 24.108). By Proposition 24.95 we deduce the following result. Corollary 24.110. Let f : Y → S be a proper morphism and let r ≥ 1 be an integer. Then X 7→ X/Z induces an equivalence between the category of finite locally free schemes X of rank r over Y and the category of finite locally free adic formal schemes of rank r over Y/Z .

429 Corollary 24.111. Let f : Y → S be a proper morphism. Then X 7→ X ×Y Y0 yields an equivalence from the category of ´etale covers of Y to the category of ´etale covers of Y0 := Y ⊗A A/I. In particular, the inclusion Y0 → Y yields for every geometric point y¯ of Y0 an isomorphism of ´etale fundamental groups ∼

π1 (Y0 , y¯) −→ π1 (Y, y¯) if Y and Y0 are connected (e.g., if S and S0 are connected and f has connected fibers). Proof. Set Yn := Y ⊗A A/I n+1 . We call an adic formal scheme (Xn )n ´etale (resp. finite ´etale) over Y/Z if Xn → Yn is ´etale (resp. finite ´etale) for all n. By Theorem 20.30, there is an equivalence between the category of ´etale Y0 -schemes and the category of adic formal scheme that are ´etale over Y/Z . By Proposition 24.83 this induces an equivalence between the category of ´etale covers of Y0 and the category of adic formal scheme that are finite ´etale over Y/Z . Hence it remains to show that under the equivalence of Corollary 24.110 a Y -scheme X is an ´etale cover if and only if X/Z → Y/Z is finite ´etale. Clearly the condition is necessary. Conversely, if X/Z → Y/Z is finite ´etale, then X → Y is finite locally free. The locus V in Y where X → Y is ´etale is the non-vanishing locus of the discriminant of X over Y , hence it is open in Y (Corollary 20.73). As X → Y is flat, we have V = { y ∈ Y ; Xy is an ´etale κ(y)-scheme} by Corollary 18.45. It contains |f −1 (Z)| = |Yn | because for y ∈ f −1 (Z) the fiber of X in y is the same as the fiber of Xn in y for all n. Hence V = Y by Lemma 24.96. We will now show that on proper schemes X over A the functor X 7→ X/Z is fully faithful, more precisely: Theorem 24.112. Let g : X → S be a proper morphism and let h : Y → S be a separated morphism of finite type. Then f 7→ f/Z is a bijection ∼

HomS (X, Y ) −→ HomS/Z (X/Z , Y/Z ). To prove the theorem we will apply Proposition 24.109 to the graphs of morphisms. Proof. Let (fn )n : X/Z → Y/Z be a morphism given by a family of morphisms fn : Xn → Yn . Let Γfn ⊆ Tn := Xn ×Sn Yn be the graph of fn . As the first projection induces an ∼ isomorphism Γfn → Xn , Γfn is proper over Sn . Set T := X ×S Y . Then (Γfn )n → T/Z is a closed immersion that corresponds by Proposition 24.109 to a closed subscheme Γ ⊆ T that is proper over S. As Γfn → Xn is finite locally free of rank 1, this is also true for Γ → X by Corollary 24.110. Hence Γ → X is an isomorphism and therefore the graph of a morphism F : X → Y such that ΓF ×S Sn = Γfn , i.e., such that F/Z = (fn )n . Conversely, let f : X → Y be a morphism and let F : X → Y be the morphism attached to (fn )n = f/Z as above. Then Γf and ΓF are both closed subschemes of T which are proper over S such that the corresponding closed adic formal subscheme of T/Z is (Γfn )n . Hence Γf = ΓF by Proposition 24.109 and hence f = F .

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24 Theorem on formal functions

Let X be a proper adic formal scheme over S/Z . If X is algebraizable by a proper S-scheme X, then X is unique up to unique isomorphisms inducing the identity on X by Theorem 24.112. In general, there exist proper adic formal schemes over S/Z that are not algebraizable (see for instance [Har2] O Example 3.3). But if X carries an ample line bundle, then this cannot happen, as the next result shows. Theorem 24.113. (Grothendieck’s algebraization theorem) Let X = (Xn )n be a proper adic formal scheme over S/Z . Let L = (Ln )n be a locally free module over X of rank 1 such that the line bundle L0 on X0 is ample. Then X is algebraizable by a proper S-scheme X and there exists a unique (up to isomorphism) line bundle M on X such that M/Z ∼ = L . Moreover, M is ample. Proof. We find an integer m ≥ 1 such that L0⊗m is very ample (Theorem 13.62) and that limn Γ(Xn , Ln⊗m ) → Γ(X0 , L0⊗m ) is surjective (Proposition 24.102). Let t0 , . . . , tr ∈ r+1 Γ(X0 , L0⊗m ) be sections, corresponding to a surjective map of OX0 -modules u0 : OX → 0 L0⊗m , that define a closed embedding i0 : X0 → PrS0 of S0 -schemes with i∗0 OPrS (1) = 0 L0⊗m . As Γ(Xn , Ln ) → Γ(X0 , L0 ) is surjective for all n, we can lift the sections to Γ(Xn , Ln⊗m ) r+1 and obtain a map of OXn -modules un : OX → Ln⊗m that is equal to u0 modulo the n nilpotent ideal generated by I in OXn . Because u0 is surjective, by Nakayama’s lemma un is surjective for all n. The maps un define morphisms of Sn -schemes in : Xn → PrSn such that (in )n is a morphism of adic formal schemes X → (PrSn )n = (PrS )/Z . As i0 is a closed immersion, in is a closed immersion for all n by Proposition 24.83. Hence (in )n is algebraizable by a closed immersion i : X → PrS by Proposition 24.109. Moreover, there exists a unique line bundle M on X such that M/Z ∼ = L by Grothendieck’s existence theorem, Theorem 24.94. As one has M ⊗m = i∗ OPrS (1), M is ample. (24.24) Remarks on the literature. The notion of derived completion seems to be relatively new. Our main source is the Stacks project [Sta] 091N. Precursors and other sources for derived completions can be found in [GrMay] O , [Lu-DAGXII] X Chapter 4 and [BhSc] O X . There are several variants of derived completions, for instance an idealistic variant which is closer to the original approach of Greenlees and May in [GrMay] O . Nice overviews together with comparisons between the different notions of derived completions can be found in [Yek3] O X and [Pos] X . The general compatibility of derived direct image and derived completion (Theorem 24.28) can also be found in the Stacks project [Sta] 0995. The classical theorem of formal functions for proper morphisms between noetherian schemes and for coherent modules (Theorem 24.37) and the Stein factorization theorem in Algebraic Geometry are of course much older. Still one of the best references is [EGAIII] O . This is also presented in a slightly more modern form, nicely explaining the main ideas, in [FGAex] X , Chapter 8, by Illusie. Somewhat weaker statements are also shown in [Har3] O Chapter III. The formal completion of a scheme along a closed subset can either be seen as a topologically ringed space or as a colimit of schemes in the category of Zariski (or, equivalently, fpqc-) sheaves. The first point of view is taken in [EGAInew] and also briefly explained in [FGAex] X Chapter 8. In the noetherian context they can also be seen as special cases of Huber’s adic spaces [Hu] O . The point of view to consider them (maybe even only locally) as special cases of colimits of schemes, as we do here, can also be found in [Yas] O X or in great generality in the Stacks project [Sta] 0AHW. Good references for

431 Grothendieck algebraization theorem are again [EGAIII] O , [FGAex] X Chapter 8, and the Stacks project [Sta] 0898.

Exercises Exercise 24.1. Let A be a ring. What does it mean for a complex M in D(A) to be derived complete with respect to the ideal I = A or with respect to an ideal that contains only nilpotent elements? Exercise 24.2. Let A be a principal ideal domain, p ∈ A a prime element, I = (p) ⊆ A, and let K be the field of fractions of A. Show that the derived I-completion of the A-module K/A is (lim A/pn A)[1]. Hint: Use that K/A is an injective A-module (Proposition G.22). Exercise 24.3. (Derived Lemma of Nakayama) Let A be a ring and let I ⊆ A be a finitely generated ideal. Show the following assertions. (1) Let M be a derived I-complete A module such that M/IM = 0. Then M = 0. (2) Let K ∈ D(A) be a derived I-complete complex such that K ⊗L A A/I = 0. Then K = 0. Hint: Exercise 22.47 Exercise 24.4. Let A be a ring and I ⊆ A an ideal. Suppose that A is derived I-complete (e.g., if A is I-adically complete). Show that every pseudo-coherent complex in D(A) is derived I-complete. ˆ be the Exercise 24.5. Let k be a field, set A := k[[t]] and I := (t) ⊆ k[[t]]. Let M 7→ M functor of t-adic completion on the category of A-modules. This exercise gives an example which shows that this functor is not exact in the middle (and in particular neither left nor right exact). If M is an A-module and m ∈ M we set ordt (m) := sup{ i ∈ N ; m ∈ ti M }. Let P be the A-module of maps f : N → A such that { n ∈ N ; ordt (f (n)) ≤ i } is finite for all i. We endow it with the t-adic topology. (1) Show that P is t-adically complete. (2) Show that there exists a unique A-linear continuous map u : P → P with u(δn ) = tn δn , where δn ∈ P sends m ∈ N to 1 if m = n and to 0 otherwise. Show that u is injective and that its image is not closed in P . (3) Let Q be the cokernel of u such that one has an exact sequence (*)

u

v

0 −→ P −→ P −→ Q −→ 0

ˆ be the t-adic completion of Q. Show that vˆ : P = Pˆ → Q ˆ is surjective and and let Q that its kernel is the closure of the image of u. ˆ but not exact (4) Deduce that the completion of (*) is exact at the left P = Pˆ and at Q in the middle. Exercise 24.6. Let k be a field, let A := k[t1 , t2 , . . . ] be the polynomial ring in countably many variables, and let I := (t1 , t2 , . . . ) be the ideal generated by the variables. Let Aˆ be the I-adic completion of A. (1) Show that Aˆ is isomorphic to the ring k[[t1 , t2 , . . .]] of formal power series (each of which has only a finite number of terms of given degree).

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24 Theorem on formal functions

(2) Let J := Ker(π) where π : Aˆ → k = A/I is the canonical map. Then J is the closure n ˆ Show that J ̸= I Aˆ by considering the formal power series P of I in A. n≥1 tn . ˆ (3) Show that A is not I-adically complete. (4) Show that Aˆ is not J-adically complete if k is a finite field. Exercise 24.7. Let A be a ring and let f ∈ A be an element. Let A[f ] := { a ∈ A ; f a = 0 } be the ideal of f -torsion. Suppose that there exists N ≥ 1 such that A[f n ] = A[f N ] for all n ≥ N . Show that for every object M in D(A) one has n M∧ ∼ = holim(M ⊗L A A/f ).

Remark : See Exercise 24.9 for a generalization. Exercise 24.8. Let A be a ring. A sequence f := (f1 , . . . , fr ) of elements of A is called weakly proregular if for all p < 0 the inverse system of A-modules (H p (K(f n )))n≥1 is pro-zero, i.e., for every n ≥ 1 there exists m ≥ n such that H p (K(f m )) → H p (K(f n )) is zero. An ideal I ⊆ A is called weakly proregular if it is generated by a weakly proregular sequence. (1) Show that if I is weakly proregular, then one has for all M ∈ D(A) a functorial isomorphism M ∧ −→ holim(M ⊗L A/I n ), where the left hand side denotes the derived I-completion of M . (2) Show that every ideal of a noetherian ring is weakly proregular. (3) Show that a sequence consisting of a single element f ∈ A is weakly proregular if and only if there exists an n ≥ 1 such that A[f m ] = A[f n ] := { a ∈ A ; f n a = 0 } for all m ≥ n, cf. Exercise 24.7. (4) Let A → A′ be a faithfully flat ring homomorphism and let I ⊆ A be an ideal. Show that I is weakly proregular if and only if IA′ is weakly proregular. proregular ideal and let J ⊆ A be a finitely generated ideal (5) Let I ⊆ A√be a weakly √ such that I = J. Show that J is weakly proregular. (6) Let I ⊆ A be a weakly pro-regular ideal and consider the A-linear I-adic completion functor ΛI : (A-Mod) → (A-Mod), M 7→ limn M/I n M . Let LΛI : D(A) → D(A) be its left derived functor (note that since ΛI is in general not right exact, we do not have necessarily L0 ΛI = ΛI ). Show that there exists a unique functorial morphism τM : M → LΛI (M ) whose composition with the canonical morphism LΛI (M ) → ΛI (M ) is the canonical map M → ΛI (M ). Show that a complex M ∈ D(A) is derived I-complete if and only if τM is an isomorphism. Remark : See [Yek3] O X. Exercise 24.9. Let X be a scheme and let I ⊆ OX a quasi-coherent ideal. Show that the following assertions are equivalent. (i) There exists an open affine covering (Ui )i of X such that Γ(Ui , I ) is a weakly proregular ideal (Exercise 24.8) of Γ(Ui , OX ) for all i. (ii) For every U ⊆ X open affine Γ(U, I ) is a weakly proregular ideal of Γ(U, OX ). If these conditions are satisfied, we call the corresponding closed subscheme V (I ) a weakly proregular subscheme of X. (1) Show that the underlying topological subspace of a weakly proregular subscheme is closed and constructible and that the property to be weakly proregular depends only on the underlying topological closed subspace.

433 (2) Show that any closed subscheme of a locally noetherian scheme is weakly proregular. (3) Let I ⊆ OX be defining a weakly proregular subscheme. Show that for any F in D(X) the natural map n F ∧ −→ holim(F ⊗L OX OX /I ) n

is an isomorphism, where F

∧

denotes the I -derived completion of F .

Exercise 24.10. Let A be a ring and let I ⊆ A be an ideal. An A-module M is called I-completely flat if TorA i (M, N ) = 0 for all i > 0 and every I-torsion module N . Show that the following assertions are equivalent for an A-module M . (i) M is I-completely flat. k k (ii) For all k ≥ 1 and all i > 0 one has TorA i (M, A/I ) = 0 and M ⊗A A/I is a flat k A/I -module. (iii) For all i > 0 one TorA i (M, A/I) = 0 and M ⊗A A/I is a flat A/I-module. Exercise 24.11. Let A be a ring, let f = (f1 , . . . , fr ) be a finite family of elements of A, let I be the ideal generated by f . Set X := Spec A, U := X \ Z, and let U be the open covering (D(fi ))i of U . Show that there is a functorial isomorphism ∼

• Tot(Cˇalt (U , F • )) −→ RΓ(U, F )

for every F in Dqcoh (X). Hint: Exercise 21.11 Exercise 24.12. Let A be a ring, let I ⊂ A be an ideal generated by a finite family ˇ f = (f1 , . . . , fr ), and set Z = V (I). Let CA (f ) be the associated extended Cech complex (24.3.1). Let DI ∞ −Tors (A) be the full subcategory of D(A) consisting of complexes whose cohomology modules are all I-power torsion (Exercise 22.47). (1) Show that RΓZ : D(A) −→ DI ∞ −Tors (A), E 7→ CA (f ) ⊗L AE defines a triangulated functor and that RΓZ is right adjoint to the inclusion functor DI ∞ -Tors (A) → D(A). ˜ (2) For E in D(A) show that there is a functorial isomorphism RΓZ (E) ∼ = RΓZ (X, E), where the right hand side denotes the cohomology with support in Z (Exercise 21.7). Hint: Exercise 24.11 Exercise 24.13. Let f : X → S be a flat, proper morphism of finite presentation with geometrically integral fibers and let L be a line bundle on X. Suppose that S is reduced and that L |Xs is trivial for every maximal point s of S (i.e., for generic points of all irreducible component of S). Show that there exists a line bundle M on S such that f ∗M ∼ = L. Remark : For a variant without the assumption that f is proper see Lemma 27.70. Exercise 24.14. Let A be a ring and let I ⊆ A be an ideal. Show that the full subcategory of derived I-complete A-modules is a plump subcategory of the category of all A-modules. T Exercise 24.15. Let A be a ring, and let f ∈ A. Let J := n (f n ) ⊆ A. (1) Let M be a derived (f)-complete A-module. Show J Ker(M → limn M/f n M ) = 0. (2) Deduce that if A is derived (f )-complete, then J 2 = 0. (3) Let I ⊆ A be an ideal that can T be generated by r elements and suppose that A is r derived I-complete. Show that ( n I n )2 = 0.

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24 Theorem on formal functions

(4) Let I be a finitely generated ideal of A and assume that A is reduced. Show that A is I-adically complete if and only if A is derived I-adically complete. Exercise 24.16. Let A be a ring and let IA be the greatest ideal I of A such that A is derived I-adically complete. Show that I is contained in the Jacobson radical of A. Hint: Exercise 24.15 Exercise 24.17. Let A be a ring, let I ⊆ A be an ideal. (1) Suppose that A is derived I-complete. Show that (A, I) is a henselian pair. Hint: Reduce to the case that I is generated by one element using Proposition 20.17. (2) Give an example of a ring A and an ideal I such that (A, I) is henselian but A is not derived I-complete. Exercise 24.18. Let f : X → S be a proper morphism and let X → X ′ → S be its Stein factorization. Show that if X is normal, then X ′ is normal. Exercise 24.19. Let S be a reduced scheme and let f : X → S be a flat proper morphism of finite presentation with geometrically integral fibers. Let L be a line bundle on X and let U ⊆ S be an open dense subscheme such that the restriction of L to the fiber Xs is trivial for all s ∈ U . Show that f∗ L is a line bundle and that f ∗ f∗ L ∼ = L. Exercise 24.20. With the notation used in Section (24.17). Show that the following categories are equivalent. (a) The category of qcqs adic formal schemes over S/Z . (b) The category of morphisms of Zariski sheaves X → S/Z that are representable by schemes and qcqs. Hint: If X → S/Z is a qcqs representable morphism, define (Xn → Sn )n by Xn := Sn ×S/Z X. Conversely, if (Xn → Sn )n is a qcqs adic formal schemes over S/Z , then define a morphism X := colimn Xn → S/Z . To show that these functors are quasi-inverse to each other use that every morphism T → S/Z with T qcqs scheme factors through some Sn by Proposition 24.79. Exercise 24.21. Let X be a noetherian integral scheme, and let Y be a separated scheme. Let f and g be morphisms X → Y such that for some non-empty closed subscheme Z ⊆ X, the restrictions f |X/Z and g |X/Z are equal. Show that then f = g.

25

Duality

Content – The right adjoint f × of Rf∗ – Computation of f × in special cases – The functor f ! – Dualizing complexes – Dualizing sheaves – Duality for schemes over fields – Applications to algebraic surfaces To motivate the theme of coherent duality, the topic of this chapter, let us start with the classical Riemann-Roch theorem for a geometrically connected smooth projective curve C over a field k, see Theorem 15.35, Theorem 26.48. One approach to proving it is to first express the Euler characteristic of a line bundle L in terms of the degree of the line bundle and the Euler characteristic of the structure sheaf (which can easily be expressed in terms of the genus of the curve). What remains to be done is to obtain a more manageable expression for the Euler characteristic χ(L ) = dimk H 0 (C, L ) − dimk H 1 (C, L ) of L . It turns out that one can express the dimension of H 1 (C, L ) as the dimension of a space of global sections of some other line bundle on C. More precisely, dimk H 1 (C, L ) = dimk H 0 (C, L ∨ ⊗OC Ω1C/k ). (For C = P1k , we have already seen this in Theorem 22.22.) In this chapter, we will prove and vastly generalize this result. To give an outline of how we will proceed, we first formulate a souped up version of the above statement. Rather than going for an equality of dimensions, it will turn out that there is even a perfect pairing H 0 (C, L ∨ ⊗OC Ω1C/k ) × H 1 (C, L ) → H 1 (C, Ω1C/k ) ∼ = k, or in other words an isomorphism ∼

Homk (H 1 (C, L ), k) → H 0 (C, L ∨ ⊗OC Ω1C/k ). Even better, we can replace this formulation on the level of cohomology groups by a statement about objects of the derived categories of C and k. Denote by f : C → Spec k the structure morphism. Writing H 0 (C, L ∨ ⊗OC Ω1C/k ) = HomOC (L , Ω1C/k ) = HomD(C) (L , Ω1C/k ), the desired statement takes the following form: HomD(k) (Rf∗ L [1], k) ∼ = HomD(C) (L , Ω1C/k ). We will prove below that for any morphism f : X → S between qcqs schemes, © Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_10

436

25 Duality

(1) the functor Rf∗ : Dqcoh (X) → Dqcoh (S) admits a right adjoint f × (Theorem 25.17), (2) for suitable f (e.g., proper, flat, and of finite presentation), we can compute f × K as × Lf ∗ K ⊗L OX f OX (Theorem 25.37), and (3) for f smooth of relative dimension n and proper we have f × OS = ΩnX/S [n] (Theorem 25.58). The above isomorphism is an immediate corollary of this. We will obtain the existence of f × from “abstract” category-theoretic results, namely a version of the Brown representability theorem. See Section (25.33) for a short discussion of some other approaches. While for proper morphisms f the functor f × behaves well and is the “right” functor to consider, it has the defect that it cannot be computed locally on the source. Moreover, even if f is simply the structure morphism of the affine line over a field k, the functor f × yields extremely large and unwieldy modules (Remark 25.45). It is much nicer to work with a functor which for open immersions coincides with the usual pullback functor. Interestingly, one can combine the functors f × for f proper and j ∗ for j an open immersion in the sense + + that one can construct a twisted inverse image functor f ! : Dqcoh (S) → Dqcoh (X) for every separated morphism of finite type between noetherian schemes, such that f ! = f × for f proper, f ! = f ∗ for f an open immersion, and even for every ´etale morphism f , and such that (f ◦ g)! ∼ = g ! ◦ f ! . In view of Nagata’s compactification theorem, these conditions determine the functors f ! uniquely. Giving the construction and some of the properties of f ! (but at times without full proofs) is the second main goal in this chapter. Coming back to the example of a smooth proper curve f : C → Spec k over a field k, we can consider the functor L 7→ L ∨ ⊗L Ω1C/k [1] = R Hom OC (L , f × OSpec k ) (where we b can take any L ∈ Dcoh (C), not just line bundles) as a duality on the bounded derived b • b category Dcoh (C) of coherent sheaves on C. More generally, an object ωX ∈ Dcoh (X), for a • noetherian scheme X, such that the functor R Hom OX (−, ωX ) induces an auto-equivalence b of Dcoh (X) (and that is of finite injective dimension) is called a dualizing complex, a notion we will study in the third part of the chapter, starting with Section (25.14). At the end, in Section (25.26) we will summarize some of the results in the special situation of proper schemes of finite type over a field. The book [Lip2] O X and the Stacks project [Sta] were very helpful when writing this chapter, and their impact on the exposition will be very visible to the reader. For further pointers to the literature see Section (25.33).

The right adjoint f × of Rf∗ Let f : X → S be a morphism of qcqs schemes. We have the pair Lf ∗ : Dqcoh (S) → Dqcoh (X) and Rf∗ : Dqcoh (X) → Dqcoh (S) of adjoint functors, see Section (21.27). In this chapter we will show the existence of a right adjoint f × of Rf∗ and study its properties. In fact, the existence of f × follows immediately from a suitable Brown representability theorem/adjoint functor theorem for triangulated categories. But note that in general, f × is not the derived functor of a functor (OS -Mod) → (OX -Mod).

437 (25.1) Brown representability for triangulated categories. We start out by some general considerations about representability of triangulated functors between triangulated categories that admit arbitrary coproducts, e.g., the derived category of a Grothendieck abelian category (Lemma F.188). Definition 25.1. Let D be a triangulated category which has arbitrary coproducts. (1) A set S of objects in D is said to generate D if whenever A is an object in D with Hom(S, A) = 0 for all S ∈ S, then A = 0. (2) We call an object S ∈ D compact if for every index set I and every family (Ai )i∈I of objects in D, the natural map M M Hom(S, Ai ) −→ Hom(S, Ai ) i∈I

i∈I

is a bijection. (3) We say that the category D is compactly generated, if there exists a generating set S of compact objects in D. Regarding Part (2) of this definition, note that the universal property of the coproduct talks about morphisms going out of a coproduct and is therefore not directly related to this condition. We will use the following general representability result. Theorem 25.2. (Brown representability for triangulated categories) Let D be a triangulated category which has arbitrary coproducts and which is compactly generated. Let H : Dop → (AbGrp) be an additive functor. Then H is representable if and only if H is cohomological (Definition F.136) and for every index set I and every family (Ai )i∈I of objects in D, we have Y M H(Ai ). Ai ) = H( i

i

It is clear that every representable functor H has the properties stated in the theorem. For the converse we refer to [Nee3] O , Theorem 3.1. Corollary 25.3. (Adjoint functor theorem for triangulated categories). Let D be a triangulated category which has arbitrary coproducts and which is compactly generated, and let E be any triangulated category. Let F : D → E be a triangulated functor. Then F has a right adjoint G : E → D if and only if it preserves arbitrary coproducts, i.e., for every L L index set I and every family (Ai )i∈I of objects in D, we have F ( i Ai ) = i F (Ai ). Proof. By general theory (Proposition F.23), the existence of a right adjoint ensures that F preserves arbitrary coproducts. Conversely, assume that F preserves all coproducts. Let E ∈ E, and consider the functor Dopp → (AbGrp),

A 7→ HomE (F (A), E).

This is a cohomological functor and by our assumption on F we have M M Y HomE (F ( Ai ), E) = Hom( F (Ai ), E) = Hom(F (Ai ), E) i

for every family (Ai )i∈I .

i

i

438

25 Duality

Therefore Theorem 25.2 shows that this functor is representable, i.e., there exists D ∈ D such that (*)

HomE (F (A), E) = HomD (A, D),

functorially in A. This means precisely that we can define the functor G that we are looking for by G(E) := D on objects. By functoriality in E of the left hand side of (*) we can extend this definition to morphisms, and one obtains a functor G which is right adjoint to F . We give two further category theoretic lemmas here that we will use later. Lemma 25.4. Let D be a triangulated category and let S be a set of objects generating D. Then a morphism X → Y in D is an isomorphism if and only if for all S in S the induced map HomD (S, X) → HomD (S, Y ) is bijective. Proof. The “only if” part is clear. For the other direction, extend the morphism X → Y to a distinguished triangle X → Y → Z →. Applying the cohomological functors HomD (S, −) to this triangle, we see that our assumption implies HomD (S, Z) = 0 for all S in S. So Z = 0, which means that X → Y is an isomorphism (Remark F.122 (2)). Lemma 25.5. Let D be a triangulated category that is generated by a set S consisting of compact objects and let E be a triangulated category which has arbitrary coproducts. Let F : D → E be a triangulated functor which preserves coproducts, and let G : E → D be its right adjoint. Then the following are equivalent: (i) The functor G respects all coproducts. (ii) For all S ∈ S, F (S) is a compact object of E. Recall that the existence of G follows from Corollary 25.3. Proof. For S a compact object of D, and any family (Ai )i of objects in E, we have M M HomD (S, G( Ai )) = HomE (F (S), Ai ) i

i

←

M

=

M

HomE (F (S), Ai )

i

HomD (S, G(Ai ))

i

= HomD (S,

M

G(Ai )),

i

where we have used the adjunction between F and G (twice) and the compactness of S. If G respects coproducts, the composition is an isomorphism and hence the arrow is an isomorphism which shows that F (S) is compact. Conversely, if F (S) L is compact, L then the arrow is an isomorphism and by Lemma 25.4 this implies that G( i Ai ) ∼ = i G(Ai ), as desired.

439 (25.2) Dqcoh (X) is compactly generated. We will use the general considerations of the previous section to prove the existence of a right adjoint of the functor Rf∗ : Dqcoh (X) → Dqcoh (S) for a morphism f : X → S between qcqs schemes (see Theorem 25.17 in the next section). For this we will study the triangulated category Dqcoh (X) and in particular show the following results for a qcqs scheme X. (1) An object of Dqcoh (X) is compact if and only if it is perfect (Proposition 25.16). (2) If U ⊆ X is a quasi-compact open subscheme and E is a perfect complex on U , then E ⊕ E [1] can be extended to a perfect complex on X (Proposition 25.12, see also Remark 25.13 for a more precise version). (3) Dqcoh (X) is compactly generated (Theorem 25.14). Remark 25.6. Let X be a scheme. Then arbitrary coproducts of objects in Dqcoh (X) exist (in D(X)) and lie in Dqcoh (X). This follows by reduction to the affine case. The following example gives a quick argument that Dqcoh (X) is compactly generated in the special case that X carries an ample line bundle. Example 25.7. Let X be a qcqs scheme and let L be a line bundle on X. (1) Let us show that L is compact as an object of Dqcoh (X), and so are all tensor powers of L , and all shifts of those (this also follows from the more general Lemma 25.8 below). In fact, we can identify HomD(X) (L , A) = H 0 (X, L −1 ⊗ A) for any A ∈ Dqcoh (X), since tensoring by L −1 is exact and HomD(X) (OX , −) is just the global sections functor on X. Since tensor product and the global sections functor on a qcqs scheme commute with direct sums (Corollary 21.56 (2)), this shows the compactness of L . (2) Now suppose that L is an ample line bundle on X. We claim that the set S = {L ⊗m [n]; m, n ∈ Z} generates the category Dqcoh (X). Let A be any non-zero object in Dqcoh (X). The existence of L implies that X is separated (Proposition 13.48). Hence by Theorem 22.35, we may represent A by a complex all of whose entries are quasi-coherent OX -modules. We need to show that there exist m and n and a non-zero morphism L ⊗−m [−n] → A in Dqcoh (X). Such a morphism corresponds to a non-zero morphism OX → A ⊗L L ⊗m [n], i.e. to a non-zero element in H n (X, A ⊗L L ⊗m ). Since A ̸= 0, there exists n such that the quasi-coherent OX -module H n := H n (A) does not vanish. Choose x ∈ X such that Hxn ̸= 0. As L is ample, we find an integer p ≥ 0 and s ∈ Γ(X, L ⊗p ) such that Xs is an open affine neighborhood of x. As H n is quasi-coherent and Xs is affine, Hxn is a localization of H (Xs ). Hence we can find h ∈ H n (Xs ) which is send to a non-zero element in Hxn . Now H n is a quotient of the quasi-coherent module Ker(An → An+1 ). As Xs is affine, we find a ∈ Ker(An → An+1 )(Xs ) = Ker(An (Xs ) → An+1 (Xs )) that maps to h. By Theorem 7.22 there exist m′ , m′′ ≥ 1 such that we can extend ′ ′ ′′ h′ ⊗ s⊗m to a section a′ ∈ Γ(X, An ⊗ L ⊗pm ) and such that a′′ := a′ ⊗ sm maps to n+1 ⊗p(m′ +m′′ ) ′′ zero in Γ(X, A ⊗L ). Then the image of a gives a non-zero class in ′ ′′ H n (X, A ⊗L L ⊗p(m +m ) ).

440

25 Duality

One has a similar assertion (with a similar proof) if X carries an ample family of line bundles (Exercise 25.3). Next we will study the compact objects in Dqcoh (X). We will show that if X is a qcqs scheme, then an object E in Dqcoh (X) is compact if and only if it is perfect. We will now show one direction and see the other direction later (Proposition 25.16). Lemma 25.8. Let X be a qcqs scheme. Let E ∈ D(X) be perfect. Then E is a compact object. Proof. The main point of the proof is to find a suitable stronger statement that we can check locally (and then eventually reduce to the case E = OX ). Therefore, given aL family (Ai )i∈I of objects in Dqcoh (X), we consider the derived Hom sheaf R Hom(E, i Ai ). By the definition of perfect complexes, it is clear that E is an object of Dqcoh (X). First we remark that the natural morphism M M (*) R Hom OX (E, Ai ) → R Hom OX (E, Ai ) i

i

is an isomorphism. Indeed, we can check this locally and hence we may assume that E can be represented by a strictly perfect complex. Then by Corollary 21.141, it suffices to consider the case E = OX . Then (*) is clearly an isomorphism by (21.18.10). Moreover, if B is a complex in D(X), then we have HomD(X) (E, B) = H 0 (R Hom(E, B)) = H 0 (RΓR Hom(E, B))

(**)

= H 0 (X, R Hom(E, B)), where the first equality holds by (F.52.6) and for the second step we use Proposition 21.107. With these remarks, we conclude as follows. HomD(X) (E,

L

Ai )

(∗∗)

=

H 0 (X, R Hom(E,

i

L

Ai ))

i (∗)

=

H 0 (X,

L

R Hom(E, Ai ))

i

=

L

H 0 (X, R Hom(E, Ai ))

i (∗∗)

=

L

HomD(X) (E, Ai ),

i

where we use for the unmarked equality that derived direct image, hence in particular H 0 (X, −), commutes with coproducts since X is qcqs (Lemma 22.82). Our goal is to prove that for a qcqs scheme X the triangulated category Dqcoh (X) is generated by the shifts of a single perfect complex (see Theorem 25.11 below), and in particular is compactly generated by Lemma 25.8. Lemma 25.9. Let V = Spec R be an affine scheme, f1 , . . . , fS r ∈ R, and let j : U ,→ V r be the inclusion of the open quasi-compact subscheme U := i=1 D(fi ). Denote by K the Koszul complex associated with the family f1 , . . . , fr . Let E ∈ Dqcoh (V ). Then the natural morphism E → Rj∗ (E|U ) is an isomorphism if and only if Hom(K[n], E) = 0 for all n ∈ Z.

441 Proof. First note that the final condition, HomD(V ) (K[n], E) = Ext−n OV (K, N ) = 0 for all n ∈ Z, is equivalent to saying that R HomOV (K, E) = 0, see Remark F.217. If E∼ = Rj∗ (E|U ), then in fact we have R HomOV (K, E) ∼ = R HomOV (K, Rj∗ (E|U )) = R HomOU (K|U , E|U ) = 0, since K|U = 0 (Remark 19.6). Now suppose that R HomOV (K, E) = 0. Write I = (f1 , . . . , fr ). Let us first consider the case E|U = 0, so that the cohomology objects of E are I-torsion. By self-duality of the Koszul complex (22.4.6), we get L ∼ 0 = R HomOV (K, E) ∼ = R HomOV (K, R) ⊗L R E = K[−r] ⊗R E.

In the second step, we have used Lemma 21.145 and that K is perfect. Nr fi / R ) by Remark 19.5 (3). By We can write the Koszul complex as K = i=1 ( R induction, this reduces us to proving the following statement: Let R be a ring, f ∈ R, and let E ∈ Dqcoh (Spec R) such that E|D(f ) = 0 and that / R ) ⊗ E = 0. Then E = 0. (R But E|D(f ) = 0 means that the cohomology of E is killed by localizing with respect to f . The second condition implies that multiplication by f is an isomorphism on the cohomology of E. It follows that E = 0. Now we come to the general case. Let Z = V (I) be the closed subscheme defined by the ideal I, and denote by i : Z → V the inclusion. Recall the functor i! defined in Section (21.13), and the distinguished triangle (Proposition 21.62) f

i∗ Ri! E → E → Rj∗ j ∗ E → Since E is in D(V ) (rather than just a complex of abelian sheaves), this is in fact a distinguished triangle in D(V ), cf. Exercise 21.7. Applying R HomOV (K, −) to this triangle, and applying the implication we have shown already to see that R HomOV (K, Rj∗ j ∗ E) = 0, we obtain R HomOV (K, i∗ Ri! E) ∼ = R HomOV (K, E) = 0. Since (i∗ Ri! E)|U = 0, this gives us i∗ Ri! E = 0, and looking at the above exact triangle once more, that E ∼ = Rj∗ j ∗ E. Corollary 25.10. Let X be an affine scheme, and let U ⊆ X be a quasi-compact open subscheme. Let Z = X \ U , and consider the full subcategory DZ = {F ∈ Dqcoh (X); F|U ∼ = 0}

⊆ Dqcoh (X).

There exists a complex P ∈ DZ which is perfect as an object of Dqcoh (X) and such that all its shifts generate DZ , i.e., if F is in DZ and HomDZ (P [i], F ) = 0 for all i, then F = 0. Proof. We identify Dqcoh (X) = D(Γ(X, OX )), and correspondingly work with complexes of modules rather than complexes of sheaves. Since U is assumed to be quasi-compact, Sr there is a finite cover U = i=1 D(fi ), fi ∈ R. We define P as the Koszul complex of the sequence (f1 , . . . , fr ). It is perfect as an object of Dqcoh (X) and lies in DZ . It follows from Lemma 25.9 that P together with all its shifts generates DZ .

442

25 Duality

We will also use the following general result about compactly generated triangulated categories. Theorem 25.11. ([Nee3] O Theorem 2.1) Let D be a compactly generated triangulated category. Let P be a set of compact objects of D which is stable under shifts, and let P be the smallest triangulated subcategory of D that contains P and is closed under taking coproducts. Let T = D/P be the localization of D with respect to the null system P, cf. Section (F.36). (1) The triangulated category P is compactly generated with P as a generating set. An object of P is compact in P if and only if it is compact as an object of D. If P is closed under formation of triangles and under passing to direct summands, then P equals the set of compact objects of P. (2) If P is a generating set for D, then P = D. (3) If F is a compact object in T , then there exists a compact object G in T and a compact object E in D such that F ⊕ G and E are isomorphic in T . Moreover, G may be chosen to be F [1] or any other object such that [F ] + [G] = 0 in K0 (T ) (Definition 23.41). We omit the proof. Its main part are Lemmas 2.2 and 2.3 in [Nee1] O which are stated and proved entirely in the context of triangulated categories. Fun fact: The octahedral axiom is used. Proposition 25.12. Let X be a qcqs scheme, let U ⊆ X be a quasi-compact open subscheme, and let Q ∈ D(U ) be perfect. Then there exists a perfect object P ′ ∈ D(X) ′ . such that Q ⊕ Q[1] ∼ = P|U Proof. First assume that X = Spec(R) is an affine scheme. In this case Dqcoh (X) = D(R) is obviously compactly generated, in fact, every non-zero complex of R-modules admits a non-zero morphism from some R[d]. Let Z = X \ U and consider the full subcategory DZ = {F ∈ Dqcoh (X); F|U ∼ = 0}, of Dqcoh (X). By Corollary 25.10, DZ is generated by objects that are perfect and hence compact (Lemma 25.8) in Dqcoh (X). Now we apply Theorem 25.11 with D = Dqcoh (X), and with P a generating set for DZ consisting of compact objects of D. Then P = DZ and DZ is compactly generated. Furthermore, in this situation, the category T of the theorem can be identified with the localization Dqcoh (X)/DZ . In fact, it is clear that the restriction functor Dqcoh (X) → Dqcoh (U ) annihilates all objects of DZ , so that we obtain a functor Dqcoh (X)/DZ → Dqcoh (U ) by the universal property of the localization. This functor is an equivalence of categories. This follows formally since j ∗ has a fully faithful right adjoint (namely Rj∗ ) by Lemma F.147. Part (3) of the theorem then implies the statement of the lemma, for X affine. Finally, we deduce the general case by a kind of gluing argument. Using the assumption that X is quasi-compact and an induction argument, we reduce to the case that X = U ∪W with U ⊂ X the open subscheme we are given in the statement Srof the lemma, and W an affine open subscheme. In fact, in general we can write X = i=1 Ui with affine open subschemes Ui , and then “extend” in the way specified in the lemma in steps: from U to U ∪ U1 , then to (U ∪ U1 ) ∪ U2 , and so on.

443 So assume that X = U ∪ W with W ⊆ X affine open, and let Q ∈ D(U ) be perfect. By the case we have already discussed, we find PW ∈ D(W ) such that PW |U ∩W is isomorphic to Q|U ∩W ⊕ Q[1]|U ∩W . Write PU := Q ⊕ Q[1]. We obtain perfect complexes PU on U and ∼ PW on W together with an isomorphism PU |U ∩W → PW |U ∩W . By Proposition 21.49 we can glue PU and PW to some object P in D(X) which is then automatically perfect, since this can be checked locally on X. Remark 25.13. Let X be a qcqs scheme. Recall from Section (23.12) that to each perfect complex on X we can associate its class in K0 (X). Let U ⊆ X be an open quasi-compact subscheme. If a perfect complex on U can be extended to a perfect complex on X, then its class in K0 (U ) obviously must lie in the image of the map K0 (X) → K0 (U ) given by restriction of perfect complex. A theorem by Thomason and Trobaugh ([ThTr] O 5.2.2) states that also the converse holds. Granting the theorem, we obtain immediately that a complex of the form Q ⊕ Q[1] for a perfect Q ∈ D(U ) can always be extended to X, because the class of Q ⊕ Q[1] in K0 (U ) is trivial. It is not difficult to give examples where the map K0 (X) → K0 (U ) fails to be surjective. In that case, not every perfect complex on U can be extended to X. See Exercise 25.6. But if X (and hence every open subscheme U ⊆ X) is in addition assumed to be regular and noetherian, then K0 (X) = K0′ (X) equals the Grothendieck group of the category of coherent OX -modules (Proposition 23.55), and since a coherent OU -module can always be extended to X (Corollary 10.49), in this situation the restriction map K0 (X) → K0 (U ) is surjective. An analogue of the theorem of Thomason and Trobaugh for vector bundles does not hold: Even if the map K0 (X) → K0 (U ) is an isomorphism, there may exist vector bundles on U which do not extend to a vector bundle on X. See Exercise 25.7. Theorem 25.14. Let X be a qcqs scheme. Then the category Dqcoh (X) is compactly generated. In fact, there exists a perfect object P ∈ D(X) such that for all E ∈ Dqcoh (X): E=0

⇐⇒

∀m ∈ Z : HomD(X) (P [m], E) = 0.

Proof. To construct P as in the statement of the theorem, we proceed by induction on the number of open affine subschemes required to cover X. So fix an affine open cover Sn X = i=1 Ui . If n = 1, i.e., X is affine, then we can take P = OX . Sn−1 Now assume that n > 1. Let U = i=1 Ui , V = Un = Spec A, so that U and V are qcqs and such that the statement of the theorem holds for U (and for V ) by induction hypothesis. Let Q ∈ D(U ) be a perfect object on U such that Q and all its shifts generate Dqcoh (U ). Denote by jU and jV the inclusions of U and V , respectively, into X. Let P ′ be a perfect complex on X such that P ′ |U ∼ = Q ⊕ Q[1] (Proposition 25.12). Since U ∩ V is quasi-compact, X being quasi-separated, we can write it as a finite Sr union U ∩ V = i=1 D(fi ) of principal subsets of V , fi ∈ A. Let Z = V (f1 , . . . , fr ), a closed subscheme of V with underlying topological space X \ U . Denote by K ∈ D(V ) the Koszul complex of f1 , . . . , fr , and by ι the closed immersion Z ,→ X. The complex K has support contained in Z, so (RjV,∗ K)|U = 0, and (RjV,∗ K)|V = K. We conclude that RjV,∗ K = Rι∗ (K|Z ) is a perfect object of D(X), because this can be checked locally on X. We define a perfect complex P := P ′ ⊕ RjV,∗ K.

444

25 Duality

Let us show that P has the desired property. Consider an object E ∈ Dqcoh (X) with HomD(X) (P [n], E) = 0 for all n. We need to show that E = 0. First note that RjV,∗ K = jV,! K (recall that jV,! is exact) since K is supported on Z which is closed in X. By adjunction of the exact functors jV,! and jV∗ (Proposition 21.4), this gives us Hom(K[n], E|V ) = Hom(RjV,∗ K[n], E) = 0. By Lemma 25.9 this implies that E|V ∼ = Rj∗ E|U ∩V , where j denotes the inclusion U ∩ V ,→ V . Now consider the natural map E → RjU,∗ E|U . Again this is an isomorphism: We can check this locally on X, and restricting to either U (clearly) or to V (by the above) we obtain an isomorphism. Hence adjunction between RjU,∗ and jU∗ yields ′ HomD(U ) (P|U [n], E|U ) = HomD(X) (P ′ [n], E) = 0,

and hence HomD(U ) (Q, E|U ) = 0 for all n, since P ′ |U contains Q as a direct summand. The defining property of Q then gives us E|U = 0, whence E = RjU,∗ E|U = 0, as well. Remark 25.15. Let u : X ′ → X be a quasi-affine morphism of qcqs schemes and let E ∈ Dqcoh (X) be a perfect complex such that the E [n], n ∈ Z, generate Dqcoh (X). Then the perfect complexes Lu∗ E [n], n ∈ Z, generate Dqcoh (X ′ ). Indeed, let F ∈ Dqcoh (X ′ ) be non-zero. Then Ru∗ F = ̸ 0 by Lemma 22.101 and hence there exists an n ∈ Z such that HomD(X ′ ) (Lu∗ E [n], F ) = HomD(X) (E [n], Ru∗ F ) ̸= 0. Proposition 25.16. Let X be a qcqs scheme. An object E ∈ Dqcoh (X) is compact if and only if it is perfect. Proof. We have already seen (Lemma 25.8) that every perfect complex is a compact object. We can reduce the converse to Theorem 25.11 as follows. Let D = Dqcoh (X) in the theorem, and take for P the set of perfect complexes. We know by Theorem 25.14 that P generates D, and so P = D with the notation of Theorem 25.11 by (2) of the theorem. To conclude, it is now enough by Theorem 25.11 (1) to observe that P is closed under formation of triangles (Proposition 21.175) and under passing to direct summands (Proposition 21.176). (25.3) Construction and first properties of f × . Putting things together, we now obtain the existence of an adjoint of Rf∗ . Theorem 25.17. Let f : X → S be a morphism of qcqs schemes. The derived push forward functor Rf∗ : Dqcoh (X) → Dqcoh (S) admits a right adjoint triangulated functor f × : Dqcoh (S) → Dqcoh (X). Proof. It follows from Corollary 25.3, Lemma 22.82 and Theorem 25.14 that Rf∗ has a right adjoint functor. It is triangulated since Rf∗ is triangulated (Lemma F.129).

445 The adjointness between Rf∗ and f × means that we have isomorphisms (25.3.1)

∼

HomD(X) (F, f × G) → HomD(S) (Rf∗ F, G),

functorially in F ∈ Dqcoh (X) and G ∈ Dqcoh (S). In particular, putting F = f × G, the identity of f × G corresponds to a morphism Rf∗ f × G → G by adjunction. We obtain the counit of the adjunction, a morphism of functors (25.3.2)

Trf : Rf∗ f × −→ id,

called the trace map. It yields the adjunction (25.3.1) by u 7→ Trf (G) ◦ Rf∗ u. The first main goal we will set for ourselves is studying f × G and the trace map Trf (G) : Rf∗ f × G → G for all G in Dqcoh (S). Below in Theorem 25.37 we will see that for × many proper morphisms f : X → S one has f × G ∼ = Lf ∗ G ⊗L OX f OS and that one can describe Trf in terms of of Trf (OS ) : Rf∗ f × OS → OS . This reduces our goal to handling the case G = OS . Remark 25.18. Consider a morphism f : X → S between qcqs schemes. To obtain a good theory, for instance to ensure that the functor f × is compatible with base change one needs to impose further assumptions, for instance that f cohomologically proper (e.g., if S is noetherian and f is proper), see Theorem 25.31 below. Moreover, even if S is the spectrum of a field k and X = A1k , f × sends coherent modules not to objects in Dcoh (X), see Remark 25.45 below. Finally, f × cannot be computed locally on the source X. This fails even if f : P1k → Spec k is the structure map of the projective line over a field k, see Exercise 25.4. This is one of the reasons why we will later modify it and construct a functor f ! which combines the good properties of f × for proper morphisms f and of f ∗ for ´etale morphisms f , at least for separated morphisms of finite type between noetherian schemes. See Section (25.11). We will see that this functor has good finiteness properties (Proposition 25.64 and Proposition 25.65) and can be computed locally on source and target (Proposition 25.62). We start by recording some easy properties of f × that follow from properties of Rf∗ formally. Proposition 25.19. Let f : X → Y , g : Y → Z be morphisms of qcqs schemes. Then there is a natural isomorphism f × g× ∼ = (gf )× of functors Dqcoh (Z) → Dqcoh (X). Proof. This follows immediately from the adjunction with derived direct image, and the corresponding property for that (Proposition 21.115). Lemma 25.20. Consider a cartesian diagram X′

u′

/X

u

 /S

f′

 S′

of qcqs schemes.

f

446

25 Duality

(1) The base change morphism Lu∗ ◦ Rf∗ → Rf∗′ ◦ L(u′ )∗ (Definition 21.129) induces, by passing to the right adjoints of all the four functors involved, a morphism (25.3.3)

R(u′ )∗ ◦ (f ′ )× → f × ◦ Ru∗

of functors Dqcoh (S ′ ) → Dqcoh (X). (2) If f and u are tor-independent, then (25.3.3) is an isomorphism of functors. Proof. Part (1) follows formally from the adjunction. For part (2), we only need to observe that under the assumption that f and u are tor-independent, the base change morphism Lu∗ ◦ Rf∗ → Rf∗′ ◦ L(u′ )∗ is an isomorphism of functors Dqcoh (X) → Dqcoh (S ′ ), see Theorem 22.99, so we can apply the previous reasoning to its inverse, as well. (25.4) Variants of the adjunction of Rf∗ and f × for R Hom and R Hom. Lemma 25.21. Let f : X → S be a morphism between qcqs schemes. Consider E ∈ Dqcoh (X) and F ∈ Dqcoh (S). The composition R HomOX (E, f × F ) → R HomOX (Lf ∗ Rf∗ E, f × F ) → R HomS (Rf∗ E, Rf∗ f × F ) → R HomS (Rf∗ E, F ) is an isomorphism. The individual maps arise from the adjunctions between Lf ∗ and Rf∗ , and between Rf∗ and f × . Even though this method to obtain the resulting morphism may seem a bit roundabout, it is reasonable. Note that we cannot (without further justification) say that “applying a functor” (in this case: Rf∗ ) gives a map R HomOX (E, f × F ) → R HomS (Rf∗ E, Rf∗ f × F ). Proof. To check that the given morphism is an isomorphism (in D(S)), it is enough to check that applying H i gives an isomorphism for every i. Now H i (R HomS (A, B)) = HomD(S) (A, B[i]), see Remark F.217. Since all of the above is compatible with shifts, we can replace all the R Hom terms above with usual Hom groups in the respective derived categories. In that case we get back the adjunction map relating Rf∗ and f × . ∼ R HomS (Rf∗ E, F ), i.e., an Next we aim for a R Hom-version of R HomOX (E, f × F ) = isomorphism ? Rf∗ R Hom X (E, f × F ) ∼ = R Hom S (Rf∗ E, F ). We first prove the following version. Lemma 25.22. Let f : X → S be a morphism between qcqs schemes. Let E, G ∈ Dqcoh (S), F ∈ Dqcoh (X). Then the composition HomD(S) (G, Rf∗ R Hom X (E, f × F )) → HomD(S) (G, R Hom S (Rf∗ E, Rf∗ f × F )) → HomD(S) (G, R Hom S (Rf∗ E, F )) is an isomorphism. As in Lemma 25.21 the maps are obtained from the adjunctions between Lf ∗ and Rf∗ , and between Rf∗ and f × .

447 Proof. Using Remark 21.120 we rewrite the left hand side as Hom(G, Rf∗ R Hom OX (E, f × F )) = Hom(Lf ∗ G, R Hom OX (E, f × F )) = Hom(Lf ∗ G ⊗L E, f × F ) = Hom(Rf∗ (Lf ∗ G ⊗L E), F ), and the right hand side as Hom(G, R Hom OS (Rf∗ E, F )) = Hom(G ⊗L Rf∗ E, F ). We can identify these two terms by the derived projection formula Proposition 22.84. (We omit the check that this actually gives rise to the morphism between the two terms that is constructed in the way outlined above.) Note that we cannot invoke the Yoneda lemma here to simply omit the Hom(G, −) in the statement of the lemma and conclude that the respective second entries are isomorphic. The problem is that the R Hom complexes need not lie in Dqcoh (X), even if E and F do. But we have seen in Corollary 22.67 that this is the case if E is pseudo-coherent and F is bounded below. We start by proving a lemma. Lemma 25.23. Let f : X → S be a morphism between qcqs schemes. Then there exists ≥n ≥n−N an N ≥ 0 such that f × maps objects of Dqcoh (S) to Dqcoh (X). In particular, f × maps + + Dqcoh (S) to Dqcoh (X). ≤n Proof. By Proposition 22.29 there exists N ≥ 0 such that for all n and all E ∈ Dqcoh (X), ≤n+N we have Rf∗ E ∈ Dqcoh (S). ≥m Now let F ∈ Dqcoh (S) for some m ∈ Z. We obtain

HomD(X) (E, f × F ) = HomD(S) (Rf∗ E, F ) = 0 0, i.e., that M has positive-dimensional support. If the depth of M is > 0, there exists f ∈ m giving rise to a short exact sequence f·

0 → M −→ M −→ M/f M → 0 and we can proceed by induction, as follows. Consider the long exact sequence of Ext groups attached to this short exact sequence: • • • • · · · → ExtiA (M/f M, ωA ) → ExtiA (M, ωA ) → ExtiA (M, ωA ) → Exti+1 A (M/f M, ωA ) → · · · .

The dimension of the support of M/f M is d − 1 and depth(M/f M ) = depth(M ) − 1, • ) = 0 for i ̸∈ [1 − d, . . . , 1 − depth(M )]. so by induction we know that Exti (M/f M, ωA It follows that for i < −d, and likewise for i > − depth(M ), multiplication by f is • • ), so by Nakayama’s lemma we obtain ExtiA (M, ωA ) = 0 for surjective on ExtiA (M, ωA i∈ / [−d, − depth(M )]. • On the other hand, for i = − depth(M ), we obtain that ExtiA (M, ωA ) ̸= 0, because 1−depth(M ) • (M/f M, ωA ) ̸= 0. ExtA • It remains to show the bound on the dimension of the support of Ext−i A (M, ωA ). But we have an injection −i −i+1 • • • Ext−i (M/f M, ωA ), A (M, ωA )/f ExtA (M, ωA ) ,→ ExtA

whence −i −i+1 • • • dim Supp(Ext−i (M/f M, ωA ) ≤ i − 1. A (M, ωA )/f ExtA (M, ωA )) ≤ dim Supp ExtA

The claim follows from this, because for any finitely generated A-module N and element f ∈ m we have by Proposition G.8

479 dim(N/f N ) ≤ dim(N ) ≤ dim(N/f N ) + 1. If, on the other hand, the depth of M is 0, let N = M [m∞ ] ⊆ M be the union of all m -torsion submodules of M . Since M is finitely generated, N equals the mr -torsion for some sufficiently large r, and hence is a finitely generated A/mr -module, so in particular an A-module of finite length. Furthermore M/N has positive depth, and M and M/N have the same support. By the previous arguments, the result holds for N and for M/N . It then follows for M , as well, by using the long exact sequence of Ext groups attached to the short exact sequence 0 → N → M → M/N → 0. (2). Let p ⊂ A be a minimal prime ideal with dim A/p = dim A. Since Ap has dimension 0, the ideal pAp is nilpotent, say pn Ap = 0. Let B = A/pn . Then Bp = Ap , dim B = • • dim A/p and by Proposition 25.86, ωB := R HomB A (B, ωA ) is a dualizing complex for the ring B. Moreover, we have r

B

p • • L • • (ωB )p = R HomB A (B, ωA ) ⊗A Ap = R HomAp (Bp , (ωA )p ) = (ωA )p .

The identification in the middle follows from Corollary 22.71, the final one from the equality Bp = Ap . Hence we may replace A by B and may therefore assume that p is the unique minimal prime ideal of A and that dim A = dim A/p. Now part (1), applied to M = A, shows • • ))) < dim(A), hence H i (ωA )p = that for − dim(A) < i ≤ 0, we have dim(Supp(H i (ωA i • • H ((ωA )p ) = 0. Since (ωA )p is a dualizing complex for Ap by Lemma 25.78, it must have at least one non-zero cohomology object. In view of part (1), we conclude that • • H − dim(A) (ωA )p = H − dim(A) ((ωA )p ) ̸= 0. • (3). We can apply part (1) for M = A and obtain that ωA is supported in cohomological • degrees contained in [− dim(A), − depth(A)] and that H − depth(A) (ωA ) ̸= 0. The other inequality follows from (2) as among the finitely many minimal prime ideals there exists a minimal prime ideal p with dim(A/p) = dim(A). (25.19) Existence of dualizing complexes. In general, a noetherian scheme X need not have a dualizing complex. One necessary condition is that X is catenary, as we will show below. We will summarize some of the known sufficient conditions for the existence of dualizing complexes in Remark 25.98, see also Theorem 25.99. Definition 25.93. Let X be a locally noetherian topological space. A dimension function on X is a function δ : X → Z such that δ(x) = δ(y) + 1 whenever y is a specialization of x (i.e., y ∈ {x}) but there is no z ∈ {x} \ {x} that specializes to y. We will use this definition in the context of locally noetherian schemes X. Clearly, a dimension function can only exist if X is catenary (Definition 14.102). Dimension functions are essentially unique on connected spaces, since we can “connect” every point to a fixed point by chains of specializations. Lemma 25.94. Let X be a locally noetherian sober topological space, e.g., the topological space of a locally noetherian scheme. If δ and δ ′ are dimension functions in X, then δ − δ ′ is locally constant. Proof. Let us sketch the purely topological proof. Let x0 ∈ X be any point in X. As locally noetherian spaces are locally connected we may replace X by an open connected and noetherian neighborhood. Define c = δ ′ (x0 ) − δ(x0 ).

480

25 Duality

Let ∼ be the equivalence relation generated by specialization, i.e., we define x ∼ y if either x is a specialization of y, or y is a specialization of x. By the properties of dimension functions, we have δ ′ (x) = δ(x) + c for every x in the equivalence class of x0 . Hence it suffices to show that as X is connected, sober, and noetherian, there is only one equivalence class. This is easy using that X has only finitely many irreducible components. Example 25.95. Let X be a noetherian scheme. (1) If X is an irreducible catenary scheme, then x 7→ − dim(OX,x ) is a dimension function on X. (2) If X is catenary and local, i.e., has a unique closed point, then x 7→ dim({x}) is a dimension function on X. (3) If X is a scheme of finite type over a field k, then x 7→ trdegk (κ(x)) = dim({x}) is a dimension function on X. Note however that in general neither of the formulas of the example defines a dimension formula on a (non-irreducible, non-local) finite-dimensional catenary noetherian scheme. This is the reason why in the following proposition the notion of dimension function cannot be easily replaced by a more explicit expression, and is the reason why we introduced this notion here. Proposition 25.96. The function δ : X → Z that maps x ∈ X to the unique integer • δ(x) such that ωX,x [−δ(x)] is a normalized dualizing complex for the local ring OX,x is a dimension function on X. Proof. Let x, y ∈ X be points where y is a specialization of x, and there is no intermediate specialization. We may assume that X = Spec A where A is a local noetherian ring with • dualizing complex ωA and y is the unique closed point of X, because we can replace • X by Spec OX,y with dualizing complex ωX,y . Moreover, we can reduce to the case that x corresponds to the zero ideal in this ring. In fact, say x corresponds to p ⊂ A. A/p • ) is a dualizing complex for A/p. This is again a normalized Then R HomA (A/p, ωA dualizing complex since by Lemma 25.43 (1) the functor R HomB A (B, −) is compatible with composition of ring homomorphisms, so • • )) ∼ ), R HomκA/p (κ, R HomA (A/p, ωA = R HomκA (κ, ωA A/p

where κ denotes the residue class field of A and A/p. Hence, we may assume that A is a noetherian local integral domain of dimension 1 • • = ωA,p is Frac(A) and X = Spec A has generic point x and special point y. Then ωX,x concentrated in a single degree (Corollary 25.85), namely in degree −δ(x). On the • [−δ(y)] is a normalized dualizing complex for the ring A = OX,y , so other hand, ωA • Proposition 25.92 (2) shows that H i (ωA [−δ(y)])p ̸= 0 if (and only if) i = − dim(A) = −1. We obtain that δ(x) = δ(y) + 1, as desired. • Corollary 25.97. Let X be a noetherian scheme which has a dualizing complex ωX . Then X is universally catenary and has finite dimension.

481 Proof. It follows directly from Proposition 25.96 that X is catenary. It is even universally catenary. To check this we need to check that for each x ∈ X and each OX,x -algebra A of • finite type, A is a catenary ring. But ωX,x then is a dualizing complex for OX,x , and we can apply Proposition 25.87 to the morphism Spec A → Spec OX,x to see that A has a dualizing complex. • Since ωX is bounded, by Proposition 25.80 and Proposition 25.92 we obtain a bound on the dimension of the local rings OX,x which is independent of x. Remark 25.98. While Corollary 25.97 shows that a dualizing complex need not exist in general, in many cases dualizing complexes do exist. It is clear that for k a field, the structure sheaf OX = k is a dualizing complex on X = Spec k. It then follows from Proposition 25.87 that every separated k-scheme of finite type admits a dualizing complex. We will show later (Section (25.25)) that on every finite-dimensional noetherian scheme X which is Gorenstein (i.e., all local rings of X are Gorenstein local rings, Definition G.26) the structure sheaf OX is a dualizing complex. Invoking Proposition 25.87 again, we see that every scheme which is separated and of finite type over a Gorenstein scheme has a dualizing complex. For rings (equivalently, for affine schemes) this condition characterizes the class of noetherian rings which have a dualizing complex. This was conjectured by Sharp and proved by Kawasaki. Theorem 25.99. (Sharp’s conjecture [Sharp] O , [Kaw] O ) Let A be a noetherian ring. Then A admits a dualizing complex if and only if A is a quotient of a finite-dimensional Gorenstein ring. (25.20) Relative dualizing complexes. Definition 25.100. Let Y be a noetherian scheme and let f : X → Y be a separated morphism of finite type. Then • := f ! OY ωX/Y is called the relative dualizing complex for f . Remark 25.101. Let f : X → Y be a separated morphism of finite type between noetherian schemes. + • ∈ Dcoh (X) by Proposition 25.64. (1) One has ωX/Y b • (2) If f is of finite tor-dimension (e.g., if f is flat), then ωX/Y ∈ Dcoh (X) and for all + K ∈ Dqcoh (Y ) one has functorial isomorphisms (25.20.1)

• f !K ∼ = Lf ∗ K ⊗L OX ωX/Y

by Proposition 25.65. • is called relative dualizing complex. The following remark shows why ωX/Y

Remark 25.102. Let S be a noetherian scheme that has a dualizing complex ωS• . Let g : X → S and h : Y → S be separated S-schemes of finite type and let f : X → Y be a • morphism of S-schemes of finite tor-dimension. Then ωX := g ! ωS• and ωY• := h! ωS• are dualizing complexes on X and Y , respectively, by Proposition 25.87. Moreover, (25.20.1) shows that

482

25 Duality • ∼ • ωX = Lf ∗ ωY• ⊗L OX ωX/Y .

(25.20.2)

Next we show that for flat morphisms between affine schemes, the formation of the relative dualizing complex is compatible with arbitrary base change. Remark 25.103. Let A be a noetherian ring and let B be an A-algebra of finite type. Let X = Spec B, Y = Spec A, and let f : X → Y denote the corresponding scheme morphism. We fix a closed embedding i : X → AN Y , i.e., a surjection P := A[X1 , . . . , XN ] ↠ B of A-algebras, and denote by g : AN Y → Y the projection. Then f = g ◦ i. • Let ωB/A be the object in D(B) corresponding to f ! OY = i! g ! OY . By Example 25.63 and Proposition 25.44 we find • = R HomB ωB/A P (B, P [N ]).

(25.20.3)

• From now on suppose that B is flat over A. Then the formation of ωB/A is compatible ′ with base change in the following sense. Let A → A be any ring homomorphism and set B ′ := B ⊗A A′ . We claim that there is an isomorphism in D(B ′ ) ′ ∼ • • ⊗L ωB/A B B = ωB ′ /A′ .

(25.20.4)

To show (25.20.4) note that B is perfect as a P -module by Proposition 25.40. Since B is ′ flat over A, we have B ⊗A A′ ∼ = B ⊗L A A . Together with Lemma 25.43 (2), it follows that B B ′ ∼ • L ′ ∼ L ′ ⊗L ωB/A B B = R HomP (B, P [N ]) ⊗B B = R HomP (B, P [N ]) ⊗A A

in D(A′ ). The canonical map R HomB P (B, P [N ]) → P [N ] thus gives us, tensoring with idA′ , a map B L ′ ∼ L ′ L ′ R HomB P (B, P [N ]) ⊗B B = R HomP (B, P [N ]) ⊗A A → P [N ] ⊗A A

in D(A′ ), and hence by adjunction (Lemma 25.43 (1)) the top row in the following commutative diagram. L ′ R HomB P (B, P [N ]) ⊗B B =

 ′ R HomP (B, P [N ]) ⊗L AA

′ ′ L ′ / R HomB P ′ (B , P [N ] ⊗A A )



=

/ R HomP ′ (B ′ , P [N ] ⊗L A′ ) A

where the vertical identifications are identifications in D(A′ ) and the top arrow is a ′ morphism in D(B ′ ) and where P ′ := P ⊗A A′ = P ⊗L A A . We can rewrite the bottom horizontal arrow, using Lemma 25.43 for the term on the right hand side, as ′ L ′ R HomP (B, P [N ]) ⊗L P P → R HomP (B, P [N ] ⊗P P )

in D(P ). This is the canonical arrow as in (21.26.5) and it is an isomorphism since B is a perfect P -module (Lemma 21.145). Hence the top horizontal row in the above diagram is an isomorphism. This gives the isomorphism (25.20.4) by (25.20.3). Next we collect some general properties of the relative dualizing complex.

483 Proposition 25.104. Let f : X → Y be a separated morphism of noetherian schemes of finite type. Let x ∈ X, set y := f (x), and let Xy := f −1 (y) be the scheme-theoretic fiber. • )x = 0 for all i < − dimx Xy . (1) We have H i (ωX/Y • )x = 0 for all i > 0. (2) If f is flat, then we have H i (ωX/Y • (3) If f is flat and all fibers of f have dimension ≤ d, then ωX/Y , considered as an object −1 of D(X, f (OY )), has tor-amplitude in [−d, 0]. Proof. It is enough to consider an affine situation, say X = Spec B, Y = Spec A, φ : A → B is the ring homomorphism defining f , and q and p are the prime ideals in B and A, respectively, corresponding to x and y. Now consider the setting of Remark 25.103, i.e., we choose a surjection P := A[X1 , . . . , XN ] → B of A-algebras and get an isomorphism • • ! ∼ ωB/A = R HomB P (B, P [N ]), where ωB/A in D(B) corresponds to f OY . Let I be the • kernel of P → B and let r ⊂ P be the inverse image of q. Then H i (ωX/Y )x corresponds to i i+N ∼ ∼ H i (R HomB P (B, P [N ])q ) = ExtP (B, P [N ])r = ExtPr (Pr /IPr , Pr ), where we use Corollary 22.71 for the second isomorphism. (1). By the relation between vanishing of Ext and depth (Proposition G.9) the statement of (1) translates to (*)

depth(IPr , Pr ) ≥ N − d = dim(P ⊗A κ(p)) − dim(P/I ⊗A κ(p)),

where d := dimx Xy . As Pr is a flat Ap -algebra, any sequence in Pr that maps to a regular sequence in Pr ⊗A κ(p) is regular (Proposition G.11). Hence to show (*) we may assume that A is a field. Then Pr is a regular local ring, and in particular Cohen-Macaulay. The above claim then follows from Proposition G.14. (2). We need to show that ExtiP (B, P [N ]) = Exti+N (B, P ) = 0 for i > 0. By P Lemma 25.41, B is a perfect P -module of tor-amplitude in [−N, 0]. Hence B is in D(P ) isomorphic to a complex K of finite projective P -modules concentrated in degrees [−N, 0] (Proposition 22.53). In particular it is K-flat and we find Exti+N (B, P ) = P H i+N (HomP (K, P )) = 0 for i > 0. • ⊗L (3). We need to show that for every A-module M , ωB/A A M has cohomology in degrees [−d, 0] only. This statement is compatible with passing to colimits, so it is enough to consider finitely generated A-modules M . To deal with this case, we apply the base change (25.20.4) to the ring homomorphism A → A′ := DA (M ) = A ⊕ M , where the multiplication on A ⊕ M is given by the scalar multiplication by A on M and the rule mm′ = 0 for all m, m′ ∈ M . As M is finitely generated, A′ is noetherian. Applying (1) and (2) to A′ → B ′ := B ⊗A A′ , we see that ωB ′ /A′ has cohomology in ′ degrees [−d, 0] only. By (25.20.4), the same holds for ωB/A ⊗L A A , and hence in particular L for its direct summand ωB/A ⊗A M .

Dualizing sheaves Although the “correct” notion is the notion of dualizing complex, it turns out that the lowest cohomology sheaf of this dualizing complex already contains interesting information which makes it an object interesting in its own right, called a dualizing sheaf.

484

25 Duality

Moreover, we will see that for Cohen-Macaulay schemes, a dualizing complex is concentrated in a single degree and hence a dualizing sheaf determines a dualizing complex in this case. We will also consider a relative situation in which we introduce Cohen-Macaulay morphisms and study the relative dualizing complex. (25.21) Definition and first properties of dualizing sheaves. Let X be a noetherian scheme. We assume that X has a dualizing complex and fix one, • denoted by ωX . • Definition 25.105. With notation as above, set n := min{i; H i (ωX ) ̸= 0} be the maximal • • value of the dimension function attached to ωX (Proposition 25.96). Then ωX := H n (ωX ) is called a dualizing sheaf, or dualizing module, on X.

Analogously we can pass from a dualizing complex for a ring to the associated dualizing module. Similarly as for a dualizing complex, a dualizing sheaf (if it exists) is not unique, but only unique up to tensoring with a line bundle, cf. Proposition 25.84. If U ⊆ X is open, the integer n in the definition above may change when restricting to U such that one does not have ωX |U = ωU . This cannot happen if X is equi-dimensional, i.e., all irreducible components of X have the same dimension: Proposition 25.106 below shows that if X is equi-dimensional and ωX is a dualizing sheaf on X, then Supp(ωX ) = X, and if U ⊆ X is a non-empty open in X, then ωX|U is a dualizing sheaf on U . • Proposition 25.106. Let X be a connected noetherian scheme, let ωX be a dualizing complex and let ωX be the induced dualizing sheaf on X. (1) Let δ be a dimension function on X (which exists by Proposition 25.96). The support of ωX is the union of those irreducible components of X such that δ takes its maximal value on the generic point of the irreducible component. (2) If X is integral, then ωX is torsion-free, i.e., Γ(U, ωX ) is a torsion-free Γ(U, OX )module for every open ∅ ̸= U ⊆ X. (3) The induced coherent OX -module ωX has the property (S2 ), i.e., for all x ∈ X the OX,x -module ωX,x has depth at least min(2, dim(Supp(ωX,x ))) (Definition G.15). • Proof. By Corollary 25.97 we know that d := dim(X) is finite. By shifting ωX appropri• ately, we may assume that d is the largest integer such that H −d (ωX ) ̸= 0, and hence • that ωX = H −d (ωX ). (1). Since any two dimension functions on X differ by a constant (Lemma 25.94), we • may assume that δ is the dimension function given by ωX in the sense of Proposition 25.96. The support of the coherent sheaf ωX is closed (Corollary 7.32). If x ∈ X is a point with δ(x) = d (hence maximal), then Proposition 25.92 applied to the local ring OX,x shows that ωX,x = ̸ 0. Hence all irreducible components that are closures of points where δ is maximal are contained in the support of ωX,x . If, on the other hand, y ∈ X is a point with ωX,y ̸= 0, then we apply Proposition 25.92 to the local ring OX,y and the normalized dualizing complex ωX,y [−δ(y)] for this local ring. Part (3) of the proposition shows that d − δ(y) ≤ dim OX,y . Let x ∈ X correspond to a minimal prime ideal in OX,y such that δ(x) − δ(y) = dim OX,y , i.e., the closure of x in Spec OX,y is an irreducible component of (maximal) dimension dim OX,y . We find that d ≤ δ(x), hence δ(x) = d. So y lies in one of the irreducible components named in the statement of the lemma.

485 (2). As X is irreducible, all dimension functions on X are of the form d′ − dim(OX,x ) for some d′ ∈ Z by Example 25.95 (1) and Lemma 25.94. By our normalization in the • beginning of the proof, the dimension function defined by ωX is given by x 7→ d−dim(OX,x ) (this can be checked at the generic point of X). To see that ωX is torsion-free can be checked on stalks (Proposition B.30). Let x ∈ X • • and set A := OX,x , a noetherian local ring with dualizing complex ωA := ωX,x . As −d • • ωX,x = ̸ 0 by (1), we find ωX,x = H (ωA ) = ωA is the dualizing module for ωA , i.e., its non-zero cohomology object in minimal degree. Hence we have to show that ωA is a • torsion-free A-module. Note that dim(A) ≤ d and that ωA [dim A − d] is normalized by Proposition 25.92 (3). For 0 ̸= s ∈ m, the maximal ideal of A, the quotient B = A/(s) has dimension < dim(A) • • and ωB := R HomB A (B, ωA ) is a dualizing complex on B (Proposition 25.86). We have −d • • H (ωB ) = 0 by Proposition 25.92 (3). Applying R HomA (−, ωA ) to the short exact sequence 0 → A → A → B → 0, where the map A → A is multiplication by s, gives us a • • • • distinguished triangle ωB → ωA → ωA → ωB [1] and the associated long exact cohomology sequence shows that multiplication by s defines an injective homomorphism ωA → ωA . • (3). We can again reduce to the case of a local noetherian ring A. Let ωA be its • normalized dualizing complex and let ωA = H − dim A (ωA ) its dualizing module. Let p ⊂ A be a prime ideal. We have to show that depthAp (ωA )p ≥ inf{2, dimAp (ωA )p }. By (1) we know that the support of ωA is the union of the irreducible components of Spec A of dimension dim A. As Spec A is catenary (Corollary 25.97), this implies that for a prime ideal p of A one has (ωA )p ̸= 0 if and only if dim(A/p) + dim(Ap ) = dim A. In this case, one has • (ωA )p = H − dim(Ap ) ((ωA )p [− dim A/p]). Hence it suffices to show that depthA (ωA ) ≥ inf{2, dimA (ωA )}. We will show this by induction on dim A. The case dim A = 0 is clear. Hence assume that dim A > 0. If depth(A) > 0, we can find a regular element s ∈ m, the maximal ideal of A. Set B := A/(s). Then dim B < dim A. As in the end of the proof of (2) one sees that (i) multiplication by s on ωA is injective, (ii) and one has an injective A-linear map ωA /sωA ,→ ωB . Then (i) shows that depth(ωA ) ≥ 1. If dim A > 1, then dim B > 0 and by applying the induction hypothesis to B we see that ωB has depth > 0. Hence ωA has depth > 1 by (ii) which shows that ωA satisfies (S2 ) in this case. If depth(A) = 0, let I ⊂ A be the ideal of elements annihilated by some power of the maximal ideal m of A and set B := A/I. As A is noetherian, I is annihilated by some • ) to the short fixed power of m and hence is of finite length. Applying R HomA (−, ωA exact sequence 0 → I → A → B → 0 we get by Proposition 25.86 and by derived Matlis duality (Proposition 25.91) a distinguished triangle (*)

+1

• • ωB −→ ωA −→ HomA (I, E)[0] −→,

• where E is a Matlis module for A and ωB is a normalized dualizing complex for B. As dim A > 0, we have depth B ≥ 1 and (*) shows that ωA ∼ = ωB (again, since dim A > 0). As we proved above that ωB satisfies (S2 ), we are done.

Remark 25.107. Note that in Part (1) of the proposition it is not true in general that the support of ωX equals the union of the irreducible components of X of dimension dim X (Exercise 25.11). However, if X is irreducible, or local, or of finite type over a field, then this is in fact true. See Example 25.95.

486

25 Duality

(25.22) Dualizing complexes and sheaves for proper schemes over local rings. Proposition 25.108. Let A be a local noetherian ring with a normalized dualizing complex • ωA . Let X be a proper A-scheme. Denote by f : X → Spec A the structure morphism, and • • let ωX = f ! ωA . (This is a dualizing complex on X by Proposition 25.87.) • (1) For all i ∈ / [− dim(X), . . . , 0], we have H i (ωX ) = 0. i • (2) We have n := min{i; H (ωX ) ̸= 0} = − dim X. Therefore the OX -module ωX = • H − dim(X) (ωX ) is a dualizing module on X. Proof. Let x be a closed point of X. As f is proper, s := f (x) is the closed point of Spec A. • for the local ring OX,x (Proposition 25.80) We first show that the dualizing complex ωX,x is normalized. Consider the commutative diagram Spec κ(x)

ix

g

 Spec κ(s)

/X f

is

 / Spec A.

Using Proposition 25.44, we have κ(x)

• • • • ) = i!x ωX = i!x f ! ωA = g ! i!s ωA = g ! κ(s) R HomOX,x (κ(x), ωX,x

and we need to show that this complex is concentrated in degree 0. But g is a finite κ(x) morphism, so we can identify g ! = R Homκ(s) (κ(x), −) again by Proposition 25.44. Since κ(x) • κ(x) is a free κ(s)-module, we obtain R HomOX,x (κ(x), ωX,x )∼ = Homκ(s) (κ(x), κ(s)). So • we have proved that ωX,x is indeed normalized. • • Hence for all closed points x ∈ X, we find that H i (ωX )x = H i (ωX,x ) = 0 for all i ̸∈ [− dim(OX,x ), − depth(OX,x )] by Proposition 25.92 (3). Since X is proper over A, it is quasi-compact, so every closed subset of X contains a closed point (Lemma 23.4). • Part (1) follows, because the support of H i (ωX ) is closed. • For part (2), in view of part (1) we only need to prove that H − dim(X) (ωX ) ̸= 0. This follows from Proposition 25.92 (3) and a similar reasoning as before, applied to a closed point x ∈ X with dim OX,x = dim X. Remark 25.109. The proof of Proposition 25.108 shows that one has more precisely • H i (ωX ) = 0 for i ∈ / [− dim(X), −δ] with δ := inf x depth(OX,x ) where x runs through the closed points of X. Proposition 25.110. Let A be a local noetherian ring with a normalized dualizing complex • ωA and let ωA be its dualizing sheaf. Let f : X → Spec A be a proper A-scheme with • dualizing sheaf ωX = H − dim X (f ! ωA ) as above. Denote by s ∈ S := Spec A the closed point and by Xs the special fiber of X. Assume that dim X = dim Xs + dim S and write d = dim(Xs ). Then there are isomorphisms, for every quasi-coherent OX -module F , HomOX (F , ωX ) ∼ = HomA (H d (X, F ), ωA ) that are functorial in F .

487 In other words, ωX represents the functor F 7→ HomA (H d (X, F ), ωA ). The condition that dim X = dim Xs + dim S holds for instance if S is irreducible, f is flat, and X is equi-dimensional (see Theorem 14.116 and the following remarks). Proof. We use the adjunction between f ! (which equals f × since f is proper) and Rf∗ and compute • HomOX (F , ωX ) = HomD(X) (τ ≤− dim X F [dim X], τ ≥− dim(X) f ! ωA ) • = HomD(X) (F [dim X], f ! ωA ) • = HomD(A) (Rf∗ F [dim X], ωA ) • = HomD(A) (τ ≤− dim(A) Rf∗ F [dim X], τ ≥− dim A ωA )

= HomA (H d (X, F ), ωA ). In the first and in the final step we use Proposition F.156, where for the final step we use the equality − dim A = d − dim X. For the second step we use Proposition 25.108. For the fourth step we use Proposition 25.92 (3) and Corollary 24.44 for Rf∗ . The latter says that Ri f∗ F = 0 for i > d, so H i (Rf∗ F [dim X]) = 0 for i > − dim X + d = − dim A. (25.23) Cohen-Macaulay schemes. • Again, let X be a Noetherian scheme with a dualizing complex ωX . Recall from Sections (B.14) and (14.28) the notion of Cohen-Macaulay ring and Cohen-Macaulay scheme: A noetherian local ring is called Cohen-Macaulay if the depth, i.e., the maximal length of a regular sequence in its maximal ideal, equals its Krull dimension. A (locally) noetherian scheme is called Cohen-Macaulay, if all its local rings are Cohen-Macaulay. Assuming that X is connected and of finite type over a field, this implies that X is equi-dimensional by Proposition 14.126 (this does not hold for general connected Cohen-Macaulay schemes, see Exercise 25.12). More generally, we call a coherent OX -module F Cohen-Macaulay if for all x ∈ X its stalk Fx is a Cohen-Macaulay OX,x -module (Definition G.13). Proposition 25.92 implies that the Cohen-Macaulay condition is characterized precisely • by the condition that ωX has a single non-vanishing cohomology sheaf:

Proposition 25.111. • (1) Let (A, m) be a local noetherian ring with a normalized dualizing complex ωA . The following are equivalent. (i) The ring A is Cohen-Macaulay. • (ii) The complex ωA is of the form ωA [d] for some A-module ωA ̸= 0 and d ∈ Z. • In this case, d = dim A and ωA is the dualizing module given by ωA . Moreover ωA is a Cohen-Macaulay module of depth d. • . The (2) Let X be a connected noetherian scheme which has a dualizing complex ωX following are equivalent: (i) X is Cohen-Macaulay, • (ii) the complex ωX is of the form ωX [d] for some OX -module ωX and some d ∈ Z. In this case, ωX is a dualizing sheaf on X in the sense of Definition 25.105, a coherent Cohen-Macaulay OX -module with Supp ωX = X.

488

25 Duality

Proof. All assertions of (1) follow immediately from Proposition 25.92 (3) except the last one. To see that ωA is Cohen-Macaulay it suffices to show that depthA (ωA ) = • • d = dim A (Corollary G.10). By Proposition 25.76 we have A = R HomA (ωA , ωA ) = i • • • R HomA (ωA [d], ωA ) = R HomA (ωA , ωA )[−d]. Hence we see that Ext (ωA , ωA ) ̸= 0 if and only if i = −d which shows depthA (ωA ) = d by Proposition 25.92 (1). • Let us prove (2). First note that for x ∈ X the stalk ωX,x is a dualizing complex for the local ring OX,x by Proposition 25.80. Since the Cohen-Macaulay property of X is defined in terms of its local rings, this shows that the implication (ii) ⇒ (i) in (2) follows directly from part (1). To prove that (i) implies (ii), we need to show that the degree where the localizations • ωX,x have their unique cohomology is constant on X. So let x ∈ X. By (i) and part (1), • there exists a unique dx such that H i (ωX,x ) = 0 for all i ̸= dx . We show that x 7→ dx is locally constant. Let U be an affine open neighborhood of x. Then only finitely many • are non-zero. cohomology sheaves of ωX|U • For all i ̸= dx , the stalk of H i (ωX|U ) at x vanishes, so shrinking U further we may i • assume that H (ωX|U ) = 0. Since we have to do this for only finitely many i, the desired result follows. Finally, Supp ωX = X follows from Proposition 25.106 (1). From the argument in the proof of part (2), we also obtain: • Corollary 25.112. Let X be a noetherian scheme which has a dualizing complex ωX . Then the subset U := {x ∈ X; OX,x is Cohen-Macaulay}

is ` open and dense, and U is Cohen-Macaulay. Moreover, one has a decomposition U = d Ud into open and closed subschemes Ud of U where • Ud = { x ∈ U ; H i (ωX,x ) = 0 for all i ̸= d}

Every scheme locally of finite type over a field k has a locally a dualizing complex. Proof. The openness of U and the decomposition of U follow from the proof of Proposition 25.111 (2) and it is clear that U is Cohen-Macaulay. Moreover, since zero-dimensional rings are Cohen-Macaulay, U contains the generic point of every irreducible component. Hence U is dense in X. Remark 25.113. If X satisfies the equivalent conditions in Proposition 25.111 (2) and is • of finite type over a field, then X is equi-dimensional and after shifting the degree of ωX • one can assume that for all x ∈ X, ωX,x [− dim {x}] is a normalized dualizing complex for OX,x . In this case, the integer d of Proposition 25.111 (2) is dim X. (25.24) The relative dualizing complex for Cohen-Macaulay morphisms. We will now transfer the results of Section (25.23) to a relative setting. Definition 25.114. Let f : X → Y be a morphism of schemes whose fibers are locally noetherian. Then f is called Cohen-Macaulay at x ∈ X, if f is flat at x and if the local ring Of −1 (f (x)),x of x in the schematic fiber f −1 (f (x)) is Cohen-Macaulay.

489 The morphism f is called a Cohen-Macaulay morphism if it is Cohen-Macaulay at all points x ∈ X. This is equivalent to asking that f be flat and that all scheme-theoretic fibers of X be Cohen-Macaulay schemes. Note that we did not impose the condition that the fibers are “geometrically CohenMacaulay”, i.e., stay Cohen-Macaulay after base change to any field extension K of κ(f (x)). The reason is that “Cohen-Macaulay” is already a “geometric property”, see Exercise 25.13 for details. Remark 25.115. (1) Complete intersection rings are Cohen-Macaulay. This shows that any syntomic morphism is Cohen-Macaulay. (2) Every quasi-finite flat morphism is Cohen-Macaulay since zero-dimensional rings are automatically Cohen-Macaulay. (3) Let f : X → Y and g : Y → Z be morphisms such that f , g and g ◦ f have locally noetherian fibers and let x ∈ X. (a) If f is Cohen-Macaulay in x and g is Cohen-Macaulay in f (x), then g ◦ f is Cohen-Macaulay in x. (b) If g ◦ f Cohen-Macaulay in x and f is flat in x, then f is Cohen-Macaulay in x and g is Cohen-Macaulay in f (x). Both assertions follow from Proposition B.82. The main goal of this section is to show that if f : X → Y is a separated CohenMacaulay morphism of finite type of noetherian schemes, then the relative dualizing • := f ! OY is concentrated in one degree, see Proposition 25.120 below. If f sheaf ωX/Y is in addition proper, then we obtain a description for f ! F for every F in Dqcoh (Y ) by Corollary 25.38. The next lemma will allow us to reduce the later discussion to a Cohen-Macaulay morphism with zero-dimensional fibers. Lemma 25.116. Let Y be a quasi-separated scheme, let f : X → Y be a morphism of finite presentation. (1) Then for all x ∈ X there exists an open affine neighborhood U of x and a quasi-finite morphism of Y -schemes g : U → AdY of finite presentation. (2) If f is Cohen-Macaulay in x ∈ X, then the pair (U, g) in (1) can be chosen such that g is flat (and hence g is Cohen-Macaulay by Remark 25.115 (2)). If Y is locally noetherian, then Y is automatically quasi-separated and every finite type morphism f : X → Y is of finite presentation. Proof. Let us first remark that any g as in (1) is of finite presentation. This follows from Remark 10.4 as follows. Since Y is quasi-separated and U is quasi-compact, the inclusion U → X is quasi-compact and hence of finite presentation. Therefore f |U is still of finite presentation. Hence g will be of finite presentation (Proposition 10.35). Therefore we can work locally on X and Y and may assume that X = Spec A and Y = Spec R are affine. Let us show (1). For x ∈ X and y = f (x), using Noether normalization we can find a morphism R[X1 , . . . , Xd ] → A from a polynomial ring over R which induces a finite injective ring homomorphism κ(y)[X1 , . . . , Xd ] → A ⊗R κ(y). By Corollary 12.79 (or Corollary 14.113) the locus of points that are isolated in their fiber is open. So replacing X by an open affine subscheme we may assume that the morphism Spec A → AnR is quasi-finite.

490

25 Duality

To show (2) we first remark that because of openness of flatness (Theorem 14.44) we may choose the open neighborhood U of x and g as in (1) such that f is flat. Replacing X by U it is therefore enough to prove the following. Given a flat morphism f : X → Y of affine schemes that is the composition of a quasi-finite morphism g : X → AdY and the projection AdY → Y , we have that g is flat in x if f is Cohen-Macaulay in x (then again by openness of flatness we see that g is flat after possibly replacing X by a smaller open neighborhood of x). In view of the flatness of f , to prove that g is flat, it is enough to show that the κ(y)-morphism gy obtained by the base change Spec κ(y) → Y is flat for all y ∈ Y (by the fiber criterion for flatness, Corollary 14.27). On the other hand, whether f is Cohen-Macaulay can also be checked on the fibers over points y ∈ Y . Therefore, we may assume that Y is the spectrum of a field. Then X is Cohen-Macaulay in x and we can invoke Theorem 14.128 to see that g is flat in x. Remark 25.117. As compositions of Cohen-Macaulay morphisms are again CohenMacaulay (Remark 25.115 (3)), it follows conversely that every f that has locally a factorization as in (2) is automatically Cohen-Macaulay. Corollary 25.118. Let f : X → Y be a morphism locally of finite presentation. Then V := { x ∈ X ; f is Cohen-Macaulay in x} is open in X. If f is flat, then V is dense in every fiber of f . Proof. We may assume that X and Y are affine and hence that f is of finite presentation. Let x ∈ V and choose (U, g) as in Lemma 25.116 (2). Then Remark 25.117 shows that U ⊆ V , in particular V is open X. If f is flat, then V is dense in every fiber of f by Corollary 25.112. Recall that for morphisms f : X → Y locally of finite type the map (25.24.1)

X −→ Z≥0 ,

x 7→ dimx f −1 (f (x))

is upper semicontinuous (Theorem 14.112). For Cohen-Macaulay morphisms we have the following stronger result. Proposition and Definition 25.119. Let f : X → Y be a Cohen-Macaulay morphism locally of finite presentation. Then the map (25.24.1) is locally constant and it is called the relative dimension of the morphism f . Proof. We may assume that Y = Spec R and X = Spec A. Then f is of finite presentation. By Lemma 25.116 we may assume that there exists a d ≥ 0 such that f can be factorized in a flat quasi-finite morphism g : X → AdY followed by the structure map AdY → Y . It suffices to show that dimx f −1 (f (x)) = d for all x ∈ X. For this we can make the base change to κ(f (x)) and hence may assume that Y = Spec k for a field k. Then it suffices to show that dim OX,x = dim OAdY ,g(x) = d for every closed point x ∈ X. But this holds since g is quasi-finite and flat (Corollary 14.97). Hence we see that in the situation of Proposition 25.119, X decomposes into open and closed subschemes Xd , d ≥ 0, such that all non-empty fibers of f |Xd are equi-dimensional of dimension d.

491 If f is smooth, then the notion of relative dimension defined here coincides with the one in the definition of smooth morphism (Definition 6.14). We can now prove a relative version of the fact (cf. Proposition 25.111) that the dualizing complex of a Cohen-Macaulay scheme is concentrated in a single degree. Proposition 25.120. Let Y be a noetherian scheme and let f : X → Y be a separated Cohen-Macaulay morphism of finite type and of relative dimension d. Then there exists a coherent OX -module ωX/Y , flat over Y , such that f ! OY ∼ = ωX/Y [d]. Proof. First recall that the cohomology sheaves of f ! OY are all coherent, as shown by Proposition 25.64. To proceed, we can work locally on X and Y and hence assume that they are both affine and that d is constant. If d = 0, then we are done by Proposition 25.104. If d > 0, we use Lemma 25.116 to factor f , locally on X and Y , as X

g

/ Ad Y

p

/ Y,

where g is Cohen-Macaulay of relative dimension 0, and p is the projection. We then have f ! OY ∼ = g ! p! O Y ∼ = (g ! OAdY )[d] by Example 25.63, and we can use the case d = 0 to conclude. In fact, the property that f ! OY is concentrated in a single degree characterizes CohenMacaulay morphisms. Proposition 25.121. Let Y be a noetherian scheme and let f : X → Y be a flat separated morphism of finite type. Let x ∈ X. The following are equivalent: (i) The morphism f is Cohen-Macaulay at x, (ii) In an open neighborhood of x, the complex f ! OY has a unique non-vanishing cohomology sheaf. Proof. If f is Cohen-Macaulay in x, then f is Cohen-Macaulay in an open neighborhood of x by Corollary 25.118. Then we have already proved the slightly more precise statement of Proposition 25.120 which shows that “(i) ⇒ (ii)” holds. Now let us show that (ii) implies (i). We write y = f (x) and denote by Xy the fiber over y as before. We may work locally and hence assume that X and Y are both affine and that f ! OY = ω[d] for some d ∈ N and some OX -module ω which is coherent by Proposition 25.64. Since f is flat by assumption, it is enough to show that the fiber Xy is Cohen-Macaulay at x. Let i : Xy → X be the inclusion. We will show that, after possibly passing to an open affine neighborhood of x, the pullback i∗ f ! OY = i∗ ω[d] is a dualizing complex. As it is concentrated in one degree, we can then invoke Proposition 25.111 to finish the proof. By Corollary 25.118 we see that f is Cohen-Macaulay on an open subset V of X such that V ∩ Xy is dense in Xy . Since (f |V )! OY = ω |V [d], it follows from Proposition 25.120 that f |V is of constant relative dimension d and hence that all irreducible components of Xy have dimension d. As x 7→ dimx f −1 (f (x)) is upper semicontinuous (Theorem 14.112), we may assume that all fibers of f have dimension ≤ d after possibly shrinking X. By Proposition 25.104 above, ω (placed in degree 0) has tor-amplitude in [0, d] over Y . But a module in degree 0 has tor-amplitude in (−∞, 0] and we conclude that ω is flat over Y .

492

25 Duality

The flatness of ω that we just proved and the flatness of f imply that the natural map Li∗ ω → i∗ ω is an isomorphism, which can be easily checked on stalks. Hence it remains to show that Li∗ f ! OY is a dualizing complex on Xy . As f is flat, we have Li∗ f ! OY ∼ = g ! κ(y) by (25.20.4) and this is a dualizing complex by Proposition 25.87. Remark 25.122. Let S be a noetherian scheme. Let X, Y be separated S-schemes of finite type. Let f : X → Y be a Cohen-Macaulay morphism of S-schemes of constant relative dimension d. We set ωX/Y = H −d (f ! OY ), the unique non-vanishing cohomology object of the • = f ! OY by Proposition 25.120. One calls ωX/Y the relative dualizing complex ωX/Y relative dualizing sheaf for the morphism f . • If S admits a dualizing complex ωS• , we obtain dualizing complexes ωX = g ! ωS• and • ! • ωY = h ωS on X and Y (where g, h denote the structure morphisms of the S-schemes X, Y ) by Proposition 25.87. We then have by (25.20.2) • ∼ ∗ • ωX = f ωY ⊗ L OX ωX/Y [d].

In case X and Y are equi-dimensional, we can pass to the dualizing sheaves and obtain an isomorphism ωX ∼ = f ∗ ωY ⊗OX ωX/Y of coherent OX -modules. (25.25) Gorenstein schemes. Recall the notion of Gorenstein ring, see Definition G.26 and Definition G.29. Every complete intersection ring is Gorenstein (Remark 19.47 (3)). Every Gorenstein ring is Cohen-Macaulay, but not conversely. In fact the notion of being Gorenstein also admits a simple characterization in terms of the dualizing complex. Definition 25.123. Let X be a locally noetherian scheme. We call X a Gorenstein scheme if for every x ∈ X the local ring OX,x is a Gorenstein local noetherian ring. A noetherian ring A is Gorenstein if and only if Spec A is Gorenstein. Proposition 25.124. Let X be a finite-dimensional noetherian scheme. The following are equivalent. (i) The scheme X is Gorenstein. (ii) The structure sheaf OX is a dualizing complex on X. • • and ωX is an invertible object in D(X). (iii) The scheme X has a dualizing complex ωX Proof. Because of Proposition 25.84, (ii) and (iii) are equivalent. Let us prove (i) ⇒ (ii). It follows from the Definition G.26 that OX has injective amplitude in [0, dim X] as a module over itself, because we can check this on the local rings OX,x (Proposition 22.79) and their dimensions are bounded by dim X. By Proposition 25.80, it is then enough to show that for each x, OX,x is a dualizing complex for the local ring OX,x . By Proposition G.26, OX,x has finite injective dimension over itself. It is then clear that the criterion of Proposition 25.76 is satisfied. Let us prove (ii) ⇒ (i). For every x ∈ X, the complex OX,x (placed in degree 0) is a dualizing complex for the local ring OX,x , and in particular has finite injective dimension. Therefore the ring OX,x satisfies condition (i) of Proposition G.26, and therefore OX,x is Gorenstein.

493 Example 25.125. Let S be a finite-dimensional and Gorenstein noetherian scheme. Then we can choose ωS• = OS as a dualizing complex by Proposition 25.124. Let g : P → S be a separated morphism of finite type which is smooth of constant relative dimension N . Let i : X → P be a regular immersion of S-schemes of constant codimension c. Then f = g ◦ i : X → P is an lci-morphism of dimension d := N − c (Proposition 19.42) • and by Corollary 25.88 we find ωX := f ! OS = ωX [d] with (25.25.1)

ω X = i ∗ ΩN P/S ⊗OX

Vc

C∨ i .

N In the important special case P = PN (−N − 1) by Example 17.59. S we have ΩP/S = OPN S

Remark 25.126. There is also a relative version for the Gorenstein property, similarly as we detailed for Cohen-Macaulay morphisms in Section (25.24). One defines a morphism f : X → Y to be Gorenstein if f is flat and all fibers of f are locally noetherian Gorenstein schemes. Every Gorenstein morphism is Cohen-Macaulay and every syntomic morphism is Gorenstein. A relative version of Proposition 25.124 then says that a flat separated morphism f : X → Y of finite type between noetherian schemes is Gorenstein if and only if the • = f ! OY is an invertible object in D(X) ([Sta] 0C08). relative dualizing complex ωX/Y • Hence if f is Gorenstein of constant relative dimension d, then ωX/Y = ωX/Y [d], where ωX/Y is a line bundle.

Duality for schemes over fields In this part, we consider the particular situation of schemes of finite type over a field, and make some of the above, fairly abstract results more explicit. In this context, duality for vector bundles or more generally for coherent sheaves is often called Serre duality, a reference to the work of Serre that was all-important as a starting point for the subject. To apply Serre duality, it is useful to have a concrete description of the dualizing sheaf. For smooth schemes we have already seen (Theorem 25.68) that the dualizing sheaf is the determinant of the K¨ ahler differentials. In Section (25.27) we will also give a description of the dualizing sheaf for normal schemes (Proposition 25.139). We conclude this part by an important application, the Lemma of Zariski-Enriques-Severi. (25.26) Serre duality. We start by collecting some immediate corollaries to what we have already shown. Theorem 25.127. Let X be a proper scheme over a field k. Denote by f : X → Spec k the structure morphism. • Then ωX := f ! OSpec k (= f × OSpec k ) is a dualizing complex on X. It has the following properties: • • b (X) and H i (ωX ) = 0 for all i ∈ / [− dim(X), 0], (1) ωX is in Dcoh − dim X • (ωX ) is a coherent (S2 )-module OX -module whose support is the (2) ωX := H union of the top-dimensional irreducible components of X, and is a dualizing sheaf on X. If X is integral, ωX is torsion-free.

494

25 Duality

(3) (Duality for complexes) For K ∈ Dqcoh (X), there are isomorphisms for all i ∈ Z ∼

• ExtiX (K, ωX ) → H −i (X, K)∨

(25.26.1)

that are functorial in K and compatible with shifts and distinguished triangles, i.e., give rise to an isomorphism of the associated long exact cohomology sequences for • ExtiX (−, ωX ) and H −i (X, −). (Here −∨ = Homk (−, k) denotes the dual k-vector space.) If K ∈ Dcoh (X), then (25.26.1) is an isomorphism of finite-dimensional k-vector spaces. (4) (Duality for perfect complexes or vector bundles) If K is a perfect complex on X (e.g., a finite locally free OX -module), then there are functorial isomorphisms of finite-dimension k-vector spaces for all i (25.26.2)

∼

• ∨ −i ∨ H i (X, ωX ⊗L OX K ) → H (X, K)

compatible with shifts and distinguished triangles. (5) (Duality for quasi-coherent OX -modules) For a quasi-coherent OX -module F , there are isomorphisms ∼ Hom(F , ωX ) → H dim X (X, F )∨ that are functorial in F . Proof. Since f is proper, by definition we have f ! = f × . It follows from Proposition 25.87 • that ωX is a dualizing complex. Parts (1) and (2) follow from Proposition 25.106 with Remark 25.107 and Proposition 25.108. For (3) consider the functorial isomorphisms • R Hom(K, ωX ) = R Hom(K, f × OSpec k ) ∼ = R Hom(RΓ(X, K), k)

given by the adjunction between Rf∗ and f × (in the form of Lemma 25.21) and identifying • • Rf∗ K with RΓ(X, K). We have H i (R Hom(K, ωX )) = ExtiX (K, ωX ). i Let us compute H L (−) of the right hand side. By Example F.153, we can represent RΓ(X, K) ∈ D(k) by i∈Z H i (X, K)[−i]. Since k is injective as a L k-vector space, we find that R Hom(RΓ(X, K), k) is represented by the complex Homk ( i∈Z H i (X, K)[−i], k) whose cohomology in degree i is given by H −i (X, K)∨ . If K ∈ Dcoh (X), then H i (X, K) is a finite-dimensional k-vector space by Corollary 23.25. Part (4) follows from (3) using that for perfect complexes one has, writing RΓ(−) for the functor RΓ(X, −), • • R Hom(K, ωX ) = RΓR Hom(K, ωX ) • ∨ = RΓR Hom(OX , ωX ⊗L OX K ) • ∨ = RΓ(X, ωX ⊗L OX K ),

where for the first identification we use Proposition 21.107, the second identification holds by Proposition 21.148 (3), and the last identification by (21.18.10). Part (5) just restates Proposition 25.110. The formation of dualizing complexes and dualizing sheaves is compatible with change of the base field.

495 Lemma 25.128. Let k ′ /k be a field extension. (1) Let X be a proper k-scheme and denote by X ′ the base change X ⊗k k ′ . Then the • dualizing complex ωX ′ and the OX ′ -module ωX ′ (as in Theorem 25.127) coincide with • the pullbacks of ωX and ωX to X ′ . (2) Let X be a proper k ′ -scheme and suppose that k ′ /k is finite (equivalently: X is a • k ′ -scheme which is proper as a k-scheme). Then ωX and ωX are independent of ′ whether we consider X as a k -scheme, or as a k-scheme. Proof. Part (1) follows directly from Theorem 25.31. Part (2). The equivalence follows from Theorem 12.65, since k ′ ⊆ H 0 (X, OX ). To prove the independence statement, let us denote by f : X → Spec k and g : X → Spec k ′ the structure morphisms, and by h : Spec k ′ → Spec k the morphism corresponding to the ′ inclusion k ⊆ k ′ . Then f ! = g ! ◦ h! . The claim follows since h! k ∼ = R Homkk (k ′ , k) = ′ Homkk (k ′ , k) ∼ = k ′ , where we obtain the first isomorphism from Proposition 25.44. In the remainder of this section, we always equip a k-scheme f : X → Spec k with • its dualizing complex ωX := f ! OSpec k and the corresponding dualizing sheaf ωX . For a (connected) Cohen-Macaulay scheme X with a dualizing complex, the dualizing complex is concentrated in one degree, and hence we can equivalently phrase the above results in terms of the dualizing sheaf by Proposition 25.111 and Proposition 25.124. Corollary 25.129. Let X be a connected proper scheme over a field k. Assume that X is Cohen-Macaulay (hence equi-dimensional, Proposition 14.126). Let d := dim X. With notation as in Theorem 25.127, we have the following assertions. (1) ωX is a coherent Cohen-Macaulay module on X which is a dualizing module and Supp(ωX ) = X. (2) X is Gorenstein (e.g., if X is lci over k) if and only if ωX is a line bundle. (3) (Duality on Cohen-Macaulay schemes for complexes) For K ∈ Dqcoh (X), there are isomorphisms for all i ∼

ExtiX (K, ωX [d]) → H −i (X, K)∨ that are functorial in K and compatible with shifts and distinguished triangles. (4) (Duality on Cohen-Macaulay schemes for OX -modules) For F ∈ QCoh(X), there are isomorphisms for all i ∼

i ∨ Extd−i OX (F , ωX ) → H (X, F )

that are functorial in F . (5) (Duality on Cohen-Macaulay schemes for vector bundles) For every finite locally free OX -module F there are functorial isomorphisms of finite-dimensional k-vector spaces ∼

H d−i (X, ωX ⊗OX F ∨ ) → H i (X, F )∨ . Also recall the following explicit description of ωX if the k-scheme X is smooth or, more generally, can be regularly embedded into a smooth scheme, obtained as a corollary to Theorem 25.68 and Example 25.125. Note that in these cases X is Gorenstein. Corollary 25.130. Let k be a field and let X be a connected separated k-scheme of dimension d.

496

25 Duality

(1) Suppose that X is smooth over k. Then ωX ∼ = ΩdX/k . (2) Suppose that there exists a closed regular immersion i : X → P , where P is a connected smooth and separated k-scheme of dimension N . Then X is equi-dimensional and ωX ∼ = i ∗ ΩN P/k ⊗OX

VN −d

C∨ i ,

where Ci denotes the conormal bundle of i, a finite locally free OX -module of rank N − d. Note that under either of the assumptions X is Cohen-Macaulay, hence we can apply N Corollary 25.129 if X is in addition proper over k. If P = PN (−N − 1) k , then ΩP/k = OPN k by Example 17.59. Sometimes the following generalization of Serre duality is useful. Corollary 25.131. Let X be a proper scheme over a field k. Let K ∈ Dqcoh (X) and E be a perfect complex on X. Then there are functorial isomorphisms, compatible with shifts and distinguished triangles, ∼

−i • ∨ ExtiOX (K, ωX ⊗L OX E ) → ExtOX (E , K) .

i∈Z

For K ∈ Dcoh (X) this is an isomorphism of finite-dimensional vector spaces. Proof. Since E is perfect, we have for all F , G ∈ D(X) (*)

R HomOX (F ⊗L E ∨ , G ) ∼ = R HomOX (F , E ⊗L G ).

by applying RΓ(X, −) to (21.32.3) and using RΓ(X, R Hom(−, −)) = R Hom(−, −) (Proposition 21.107). Therefore we find for all i ∈ Z • • ExtiOX (K, ωX ⊗L E ) = ExtiOX (K, ⊗L E ∨ , ωX ) ∨ = H −i (X, K ⊗L OX E )

= H −i (X, R Hom OX (E , K))∨ = (H −i R HomOX (E , K))∨ ∨ = Ext−i OX (E , K) .

Here the second equality holds by Serre duality in the form of Theorem 25.127 (3) and the third identity by (21.32.2). The last assertion follows from Corollary 23.38. Corollary 25.132. Let X be a connected Cohen-Macaulay proper scheme over k of dimension d and let E be a finite locally free OX -module. Then we have functorial isomorphisms (25.26.3)

∼

Extd−i (K, ωX ⊗OX E ) → Exti (E , K)∨

for all i ∈ Z and for all K ∈ Dqcoh (X). For K ∈ Dcoh (X) this is an isomorphism of finite-dimensional vector spaces. Proof. As X is Cohen-Macaulay and connected, it is equi-dimensional and ω • = ωX [d]. We also have ωX ⊗L OX E = ωX ⊗OX E if E is finite locally free. Hence (25.26.3) is a special case of Corollary 25.131.

497 (25.27) Dualizing sheaves on normal varieties. Using the notion of reflexive OX -module introduced in the following definition, we can describe the dualizing sheaf on a normal variety (over a perfect field) in terms of the sheaf of differentials on its smooth locus. Let X be a ringed space. For an OX -module F , recall that we denote by F ∨ := Hom OX (F , OX ) its dual. Remark 25.133. (1) If X is a scheme and F is of finite presentation, then F ∨ is again quasi-coherent (Proposition 7.29). ∨ (2) If X is a locally noetherian scheme and F is a coherent OX -module, then F is again coherent. Indeed, working locally, by (1) it suffices to show that if M is a finitely generated module over a noetherian ring, then its dual M ∨ = HomA (M, A) is finitely generated. But a surjection Ar → M yields an injective map M ∨ ,→ HomA (Ar , A) = Ar which shows that M ∨ is finitely generated since A is noetherian. Note that we also could have invoked Corollary 22.67. Definition 25.134. Let X be a locally noetherian scheme. A coherent OX -module F is called reflexive, if the natural homomorphism F → F ∨∨ is an isomorphism. Every locally free OX -module of finite rank is reflexive. Of course, this definition would make formally sense for an arbitrary OX -module over an arbitrary scheme X (or even an arbitrary ringed space), but if we want to have in the affine case the compatibility with the corresponding notion for modules, Remark 25.133 shows that additional conditions on X and on F are necessary. Remark 25.135. (1) Let A be a noetherian ring, X = Spec A, and M a finitely generated A-module. Then ˜ is a reflexive OX -module. M is reflexive (Definition G.16) if and only if M (2) Let X be an integral scheme, F and G quasi-coherent OX -modules with F of finite presentation and G torsion-free. Then Hom OX (F , G ) is a torsion-free OX -module which can be checked easily by reduction to the affine case using Remark 25.133 (1). In particular, we see that every reflexive OX -module is torsion-free if X is locally noetherian. Here we will use this notion for normal integral schemes. Then we have the following simple properties. Lemma 25.136. Let X be a normal connected noetherian scheme. (1) If M is a torsion-free coherent OX -module, then there exists an open subscheme U ⊆ X such that codimX (X \ U ) ≥ 2 and such that M|U is a locally free OU -module of finite rank. (2) Let M be a coherent OX -module. For every open subscheme j : U ,→ X such that codimX (X \ U ) ≥ 2 and such that M|U is a locally free OU -module of finite rank, the natural homomorphism M → M ∨∨ factors through j∗ (M|U ) and the resulting homomorphism j∗ (M|U ) → M ∨∨ is an isomorphism. Proof. For (1) note that for every x ∈ X such that the local ring OX,x has dimension 1, this ring is a discrete valuation ring, and hence every torsion-free finite module is free. For (2), notice that our assumptions imply j∗ OU = OX (Theorem 6.45). Thus the sheaf-version of the adjunction between j ∗ and j∗ gives us functorial identifications

498

25 Duality ∨ , OU ) = j∗ M|U , M ∨∨ = Hom OX (M ∨ , j∗ (OU )) = j∗ Hom OU (M|U

and one checks that these are compatible with the natural maps from M to the left and right hand side, respectively. As an immediate consequence we obtain the following characterization of reflexive OX -modules in this case. Corollary 25.137. Let X be a normal connected noetherian scheme. Let M be a coherent OX -module. Let j : U ,→ X be an open subscheme such that codimX (X \ U ) ≥ 2 and such that M|U is a locally free OU -module of finite rank. The following are equivalent. (i) The OX -module M is reflexive. (ii) The natural map ∼ M → j∗ (M|U ) is an isomorphism. The relevance of this notion for the theory of dualizing sheaves comes from the following result. • Proposition 25.138. Let X be a normal connected noetherian scheme, let ωX be a dualizing complex for X and let ωX be its dualizing sheaf. Then ωX is reflexive.

Proof. By Proposition G.18 this follows from Proposition 25.106 (2) and (3). Therefore we obtain the following description of the dualizing sheaf for normal varieties over perfect fields. Proposition 25.139. Let k be a perfect field, and let X be a normal connected k-scheme of finite type. Let n = dim X and let ωX be the dualizing sheaf of X. Denote by j : Xsm ,→ X the open immersion of the smooth locus of X into X. Then ωX ∼ = j∗ (ΩnXsm /k ) ∼ = (ΩnX/k )∨∨ . Proof. Since k is perfect, the smooth locus Xsm of X is the same as the regular locus by Proposition 18.67. It is open and dense in X (Theorem 6.19). Its complement has at least codimension 2 since X is normal (Proposition 6.40). The restriction of ωX to Xsm equals the dualizing sheaf ΩnXsm /k on Xsm (Corollary 25.130). The first isomorphism in the statement follows then from Corollary 25.137 since ωX is reflexive by Proposition 25.138. For the second one, note that the dual of any coherent OX -module is reflexive. In fact, this can be checked locally, and hence follows from Lemma G.17. Hence (ΩnX/k )∨∨ is a reflexive OX -module whose restriction to Xsm is isomorphic to ωX |Xsm . Hence ωX ∼ = (ΩnX/k )∨∨ by Corollary 25.137. (25.28) The Lemma of Enriques-Severi-Zariski. Theorem 25.140. (Lemma of Enriques-Severi-Zariski) Let k be a field and let X be a projective k-scheme of finite type. Let L be an ample line bundle on X. Let F be a coherent OX -module. Assume that depthOX,x (Fx ) ≥ 2 for all closed points x ∈ X (e.g., this holds, if X is normal of dimension ≥ 2 and F is locally free). Then there exists n0 such that H 1 (X, F ⊗ L −n ) = 0 for all n ≥ n0 .

499 If X is Cohen-Macaulay of dimension ≥ 2 and F is locally free, then the theorem follows easily from Serre duality (Corollary 25.129 (5)) and Serre’s criterion for ampleness (Lemma 23.5). In the general case, we first reduce to the case of projective space, and use Corollary 25.129 (4) in place of Corollary 25.129 (5), which allows us to weaken the assumption on F . Proof. Some power L r of L is very ample and gives rise to a closed embedding ι : X → PN k . Then H 1 (X, G ) ∼ = H 1 (PN k , ι∗ G ) for every OX -module G . Applying this to F , F ⊗ L , . . . , F ⊗ L r−1 , we reduce to the case that X = PN (1). Note that the depth k and L = OPN k of (ι∗ G )ι(x) (over OPN ) equals the depth of Gx (over OX,x ) for x ∈ X, and (ι∗ G )y = 0 k ,ι(x) has depth ∞ (by convention) for y ̸∈ ι(X). For the regular scheme X = PN k , we may use Serre duality in the form of Corol−1 lary 25.129 (4) and obtain H 1 (X, F (−n))∨ ∼ ). = ExtN O N (F (−n), ωPN k P

k

Now for n sufficiently large, we have N −1 −1 −1 ExtN )∼ (n)) ∼ (n)), = ExtN = Γ(PN k , Ext O N (F , ωPN O N (F (−n), ωPN O N (F , ωPN k k k P

k

P

P

k

k

where the first equality follows from 21.148 (3) (and holds for all n), and the second equality follows from the local-to-global spectral sequence for Ext groups (Corollary 21.108) together with Serre’s criterion for ampleness (in the form of Lemma 23.5) which implies the vanishing q H p (X, Ext q (F , ωPN (n))) ∼ )(n)) = 0 = H p (X, Ext (F , ωPN k k

for all q, all p > 0 and all sufficiently large n. Note that by Corollary 22.67 (2), the sheaves Ext q (F , ωPN ) are coherent. k

−1 It is thus enough to show that Ext N (n)) = 0. We claim that in fact O N (F , ωPN k P

k

−1 Ext N O N (F , G ) = 0 for all OX -modules G . To see this, it is enough to check that the P

k

stalks of these sheaves vanish, and those we can identify with the corresponding Ext groups over the regular local ring OPN (see Proposition 22.65), which we can compute k ,x using a projective resolution of the OPN -module Fx . By the Auslander-Buchsbaum k ,x formula, Theorem G.12, our assumption on the depth of Fx ensures that F has projective dimension ≤ N − 2, and this yields the desired vanishing. The condition on the depth is satisfied if X is normal of dimension ≥ 2 and F is locally free by Serre’s criterion for normality, see Proposition B.81. A beautiful geometric consequence of the Lemma of Enriques-Severi-Zariski is the following connectedness result. Corollary 25.141. Let k be a field and let X be a normal projective integral k-scheme of finite type, and such that dim X ≥ 2. Let D be an ample effective Cartier divisor on X. Then the support Supp(D) of D is connected. Proof. We may replace D by an arbitrary positive multiple, because that does not change the support. Thus by Theorem 25.140 we may assume that H 1 (X, OX (−D)) = 0. We regard the effective divisor D as a closed subscheme of X. The short exact sequence 0 → OX (−D) → OX → OD → 0

500

25 Duality

gives rise to a surjection H 0 (X, OX ) → H 0 (X, OD ) = H 0 (D, OD ). Since X is integral and proper over k, H 0 (X, OX ) is a domain and finite over k, hence a field, and it follows that H 0 (D, OD ) ∼ = H 0 (X, OX ) is a field, too. In particular, D is connected.

Applications to algebraic surfaces In the remaining sections of this chapter, we will discuss some applications to the theory of algebraic surfaces. For example we will show that every regular proper surface over a field is projective (Theorem 25.151). We also prove the Hodge index theorem (Theorem 25.156) which will be a crucial ingredient for the proof of the Weil conjectures for curves in Section (26.28). (25.29) The theorem of Riemann-Roch for algebraic surfaces. Let k be a field. We will study algebraic surfaces over k in the following sense. Definition 25.142. An (algebraic) surface is a separated k-scheme of finite type that is equi-dimensional of dimension 2. An important tool for the study of proper surfaces is the intersection pairing that we obtain as a special case of the intersection numbers introduced in Chapter 23. Let us recall the definition and the properties important for us in this special case. By a divisor on X we mean a Cartier divisor. If X is regular, then we can identify Cartier divisors and Weil divisors, see Section (11.13). Definition and Proposition 25.143. Let X be a proper algebraic surface over the field k. The intersection pairing Pic(X) × Pic(X) → Z given by (L , M ) 7→ (L · M · X) = χ(OX ) − χ(L −1 ) − χ(M −1 ) + χ(L −1 ⊗ M −1 ), is bilinear and has the following properties: (1) If M = OX (D) for an effective divisor X, then (L · M · X) = (L · D) = deg(L|D ), where we consider D as a closed subscheme of X. (2) If C and D are effective divisors on X without a common irreducible component, and Z = C ∩ D is their schematic intersection, then X (C · D) = dimk (OZ,z ). z∈Z

Proof. The formula for the pairing is a direct consequence of the definitions in Chapter 23, specifically Definition 23.57, Definition 23.70. The bilinearity was shown in Corollary 23.72 ((2)). For part (1), see Proposition 23.58 (3) and Example 23.75 (1), for part (2) see part (2) of that example.

501 Given a proper surface, usually we denote the intersection product of line bundles L and M , and of divisors C and D, respectively, by (L ·M ) and (C ·D). The self-intersection number (L 2 ) := (L · L ), (D2 ) = (D · D) is an important invariant. If D is a very ample divisor on a proper algebraic surface, then we have a simple interpretation of this number: (D2 ) equals the degree of X with respect to the embedding X ⊆ PN k given by D, see Definition 23.80. On the other hand, in general the self-intersection number of a divisor may be negative, see Lemma 25.148 and Theorem 25.156 below. The theorem of Riemann-Roch expresses the Euler characteristic of a line bundle in terms of the Euler characteristic of the structure sheaf and an intersection product involving the canonical divisor. Theorem 25.144. (Theorem of Riemann-Roch for smooth proper algebraic surfaces) Let k be a field and let X be a smooth proper algebraic surface over k. Let K be a canonical divisor on X, i.e., OX (K) ∼ = Ω2X/k . Then for every divisor D on X we have χ(OX (D)) =

1 (D · (D − K)) + χ(OX ). 2

Proof. We compute the intersection product (−D · (D − K)) using the definition: (−D · (D − K)) = χ(OX ) − χ(OX (D)) − χ(OX (K − D)) + χ(OX (K)). By Serre duality (Corollary 25.129) we have χ(OX ) = χ(OX (K)) and χ(OX (D)) = χ(OX (K − D)). Hence we obtain −(D · (D − K)) = 2χ(OX ) − 2χ(OX (D)), as desired. Proposition 25.145. (Adjunction formula) Let k be a field, and let X be a connected smooth proper algebraic surface over k. Let ωX = Ω2X/k be its canonical bundle, and let K be a canonical divisor on X, i.e., OX (K) ∼ = ωX . Then for every effective Cartier divisor C on X (corresponding to a closed curve C ⊆ X) we have (C · (C + K)) = 2g − 2, where g is the genus of the curve C. Proof. We have (C · (C + K)) = deg((OX (C) ⊗OX ωX )|C ) by Proposition 25.143 (1). By Corollary 25.130 (2) and Remark 19.24, (OX (C) ⊗OX ωX )|C is the dualizing sheaf of the curve C over k, which has degree 2g − 2 by Corollary 26.52. If C is smooth, we could alternatively use Proposition 19.32. For surfaces, the Nakai-Moishezon criterion for ampleness, Theorem 23.90, takes the following form. Theorem 25.146. (Ampleness criterion of Nakai-Moishezon for algebraic surfaces) Let k be a field, and let X be an integral proper algebraic surface over k. Let L be a line bundle on X. (1) The line bundle L on X is ample if and only if (L 2 ) > 0 and (L · C) > 0 for every integral curve C ⊂ X.

502

25 Duality

(2) If H 0 (X, L ) ̸= 0 and (L · C) > 0 for every integral curve C ⊂ X, then L is ample. Proof. The first formulation is a direct translation of the general version of the theorem. The second version follows by inspecting the proof of Theorem 23.90, and noting that the condition (L 2 ) > 0 was used only to show that some power of L has non-trivial global sections. (25.30) Resolution of indeterminacies for surfaces. As an illustration of the usefulness of intersection numbers, let us prove the following theorem. Theorem 25.147. Let k be a field, let X be a regular proper algebraic surface over a field k, and let Y be a projective k-scheme. If f : X 99K Y is a rational map, then there ˜ → X which is an isomorphism over the exists a projective surjective morphism π : X ˜ →Y. maximal domain of definition of f and such that f ◦ π extends to a morphism X Furthermore, π can be constructed as a composition of blow-ups of closed points. By saying that the rational map f ◦π extends to a morphism, we mean that its (maximal) ˜ By abuse of terminology, we domain of definition (see Proposition 9.27) is equal to X. then also say that this rational map is a morphism. Compare the much more general statement that we stated, without proof, as Theorem 13.98, and the special case of rational maps to projective schemes (Proposition 13.99) which almost shows the above theorem, except that it does not give that the morphism ˜ → X can be constructed as a sequence of blow-ups of closed points, but on the X other hand works in arbitrary dimension and does not require X to be regular. For the application in Section (25.31), that proposition is actually sufficient. We start with a lemma which clarifies the intersection product on the Picard group of the surface obtained from X by blowing up a closed point in X. (See Sections (13.19) and (19.5).) Lemma 25.148. Let k be a field and let X be a regular proper surface over k. Let x ∈ X ˜ → X denote the blow-up of X in the point x, and let E ⊂ X ˜ be a closed point. Let π : X be the exceptional divisor. (1) For divisors D, D ′ on X we have (π ∗ D, π ∗ D′ ) = (D, D ′ ). (2) For every divisor D on X we have (π ∗ D, E) = 0. (3) We have (E 2 ) = −[κ(x) : k]. (4) The homomorphism ˜ Pic(X) ⊕ Z −→ Pic(X),

(L , n) 7→ π ∗ L ⊗ OX (nE),

is an isomorphism. Proof. Part (1) follows from Proposition 23.77. We may compute the intersection number in Part (2) as deg(π ∗ OX (D)|E ), which vanishes because OX (D) is trivial on a neighborhood of x, whence π ∗ OX (D)|E is trivial. Let us prove Part (3). Under the identification E ∼ = P1κ(x) (Example 13.95), the line bundle OX (−E)|E is identified with OP1κ(x) (1), cf. Remark 13.94. Therefore (E 2 ) = degk (OX (E)|E ) = [κ(x) : k] degκ(x) (OP1κ(x) (−1)) = −[κ(x) : k].

503 Finally, the surjectivity in Part (4) follows easily by interpreting the Picard groups as ˜ is regular by Remark 19.36). groups of Weil divisor classes (Proposition 11.42 using that X But if π ∗ L ⊗ OX (nE) ∼ = OX , then it follows from Part (3) that n = 0, so π ∗ L is trivial. Writing U = X \ {x}, we find that L|U = (π ∗ L )|π−1 (U ) is trivial, and using Proposition 11.42 once again, we conclude that L is trivial. We can now prove the theorem. The idea of the proof is rather simple. Consider a rational map f : X 99K Y from a regular proper surface X over k to a projective k-scheme Y . Since X is regular and Y is proper, the valuative criterion for properness applied to Y shows that f is defined on an open subset U ⊆ X whose complement has codimension at least 2, i.e., consists of finitely many closed points. We expect that blowing up these points should improve the situation, even though the resulting rational map from the blow-up to Y might still be undefined at some of the points of the exceptional divisors. We repeat the process by blowing up further. To conclude the proof, we have to show that this process must eventually stop. Proof. (of Theorem 25.147) We may embed Y in some projective space over k by assumption. Replacing Y we may then assume that Y = Pm k for some m. Then the (maximal) domain of definition of the rational map f : X 99K Pm k is an open subset U of X whose complement consists of (at most) finitely many closed points. In particular, we have Pic(X) ∼ = Pic(U ) by restriction of line bundles (Proposition 11.42), and the morphism U → Pm k given by f gives us a line bundle L on X. We have Γ(X, L ) = Γ(U, L ) because X is normal (locally, sections on U extend uniquely to sections on X by Hartogs’s theorem, Theorem 6.45; since the extension is unique, the result follows by gluing). The points where f is not defined are precisely the points where all global sections of L vanish. Claim 1. We have (L 2 ) ≥ 0. Proof of claim. The property of L that we will use is that its “base locus”, i.e., the common vanishing locus of all its global sections, has dimension 0, or in other words, does not contain a curve. This property is preserved by changing the base field, and by Remark 23.71 this does not change the intersection number, either. We may hence assume that k is infinite. Let D be an effective divisor such that L ∼ = OX (D), and let C1 , . . . , Cr be the integral curves in the support of D. We will show that there exists a divisor D′ which is linearly equivalent to D and such that none of the Ci lies in the support of D′ . This implies (L 2 ) = (D · D′ ) ≥ 0 by Proposition 25.143 (2). Finding such a D′ amounts to finding a section s ∈ Γ(X, L ) which does not vanish along any of the Ci , i.e., s must not lie in the kernel of the evaluation map Γ(X, L ) → L (ηi ), for any i, where ηi denotes the generic point of Ci . For a fixed i, this kernel is a proper k-subvector space of Γ(X, L ), since by assumption Ci is not contained in the support of every divisor linearly equivalent to D. Since k is infinite, Γ(X, L ) does not equal the union of finitely many proper subvector spaces, and the existence of s satisfying the desired property for all i follows. ˜ → X be the blow-up of X in a closed To continue, assume that U = ̸ X and let π : X point X of the reduced closed subscheme X \ U . Denote by E the exceptional divisor. ˜ ˜ 99K Pm gives rise to a line bundle L˜ on X. Similarly as before, the rational map f ◦ π : X k Claim 2. We have (L˜2 ) < (L 2 ).

504

25 Duality

˜ and their open subschemes Proof of claim. Using Proposition 11.42 for X and X, ˜ \ E, we see that L˜ ⊗O (π ∗ L )−1 ∼ X \ {x} and X (qE) for some q ∈ Z. Therefore O = X˜ ˜ X 2 2 2 2 ˜ (L ) = (L ) + q (E ) by Lemma 25.148. The same lemma shows that (E 2 ) < 0, so it is enough to show that q = ̸ 0. But q = 0 would mean that L˜ ∼ = π ∗ L . This is impossible because every global section of L vanishes at x, while all but finitely many points of E lie in the maximal domain of definition of f ◦ π. To finish the proof, note that we can apply Claim 1 to L˜ just as well, so that we obtain 0 ≤ (L˜2 ) < (L 2 ) whenever we blow up a point where f is not defined. By induction, this finishes the proof. In fact, setting X0 := X, letting Xi be the blow-up of Xi−1 in one closed point where the rational map Xi−1 → X0 → Pm is not defined (if such a point exists), and denoting by Li the line bundle on Xi constructed as above, the sequence (Li2 ) is a strictly decreasing sequence of non-negative numbers, so there exists an n ≥ 0 such that the maximal domain of definition of the rational map Xn → Pm k equals Xn . (25.31) Projectivity of regular proper surfaces over a field. In this section, we will prove that every regular proper surface over a field is projective. We start with the following lemma (which we will prove only for surfaces). It will be a key point of the proof of the theorem below and we will employ several of the difficult results that we have proved in the previous chapters. Lemma 25.149. Let k be a field, and let X be an irreducible proper k-scheme with dim X ≥ 2. Let U ⊆ X be non-empty affine open. Then X \ U is connected. Proof. We will prove the statement only for surfaces, similarly as in [Bad] O Thm. 1.28; this is the only case we will use below. For the general case, see [Har2] O Ch. II, Cor. 6.2 (there k is assumed to be algebraically closed, but it is easy to reduce to this situation). First note that whenever f : X ′ → X is a surjective proper morphism of irreducible k-schemes such that f −1 (U ) is affine, we may replace X and U by X ′ and f −1 (U ). Now choose a closed embedding U ⊆ Ank for some n, embed Ank ⊂ Pnk as usual, and denote by Y the schematic closure of U in Pnk . We regard the inclusion U ,→ Y as a rational map X 99K Y . Using Proposition 13.99 (or Theorem 25.147 in case X is regular), we find a proper surjective morphism π : X ′ → X which is an isomorphism over U and such that the composition X ′ → X 99K Y is a morphism X ′ → Y . Replacing X by X ′ , we may hence assume that the birational map X 99K Y is a birational morphism f : X → Y . By construction, U = f −1 (f (U )) and Y \ f (U ) is the support of an ample effective divisor on Y . Replacing X by its normalization (which is finite over X, Corollary 12.52), and U by its inverse image in the normalization of X, we may assume that X is normal. f′ π Let X −→ Y ′ −→ Y be the Stein factorization of f , see Section (24.10). Since X is normal, the same is true for Y ′ (Exercise 24.18). The morphism Y ′ → Y is finite, and thus pullbacks of ample line bundles are ample (Proposition 13.83). Hence Y ′ \f ′ (U ) = π −1 (Y \ f (U )) is again the support of an ample effective divisor, now on Y ′ . By Corollary 25.141, Y ′ \ f ′ (U ) is connected. Using that f ′ has connected fibers by Theorem 24.49, and that the map (f ′ )−1 (Y ′ \ f ′ (U )) → Y ′ \ f ′ (U ) is closed, we find that X \ U = (f ′ )−1 (Y ′ \ f ′ (U )) is connected, as desired. Lemma 25.150. Let X be a noetherian separated regular scheme and let U ⊆ X be an open dense affine subscheme. Then every irreducible component of X \ U has codimension

505 1. In particular, X \ U endowed with its reduced scheme structure is an effective Cartier divisor. The proof will show that instead of supposing that X is regular it suffices to assume that the local rings OX,x are factorial for all x ∈ X \ U . Proof. Let D := X \ U be endowed with the reduced scheme structure. If we have shown that D is a Weil divisor, then it is an effective Cartier divisor since X is regular and hence locally factorial. Hence it suffices to show the first assertion. Let ξ ∈ D be the generic point of an irreducible component. As U is affine and X is separated, the inclusion U → X is affine (Proposition 12.3 3). Hence Uξ := U ×X Spec OX,ξ is affine. As ξ is in the closure of U , Uξ is non-empty. As ξ is a maximal point of X \ U , topologically Uξ is the complement of the special point ξ of Spec OX,ξ . If ξ had codimension > 1 in Spec OX,ξ , then Γ(Uξ , Spec OX,ξ ) = OX,ξ by Hartogs’ theorem (Theorem 6.45) which is absurd since Uξ is affine. Hence dim OX,ξ = 1. Theorem 25.151. Let k be a field and let X be a regular proper surface over k. Then X is projective over k. More precisely, let U ⊂ X be any non-empty affine open subscheme. Then there exists an effective ample divisor on X with support Z = X \ U . Proof. It is clear that the more precise version implies the projectivity of X. Note that since X is regular, the notions of Cartier divisor and Weil divisor coincide. We now proceed in several steps. We may and will assume that X is connected. Clearly the support of an effective ample divisor is of pure codimension 1, and it is connected by Theorem 25.141. We will show first that these two properties do hold for Z as above. (I). By Lemma 25.150, Z is of pure codimension 1, i.e., that every irreducible component of Z is a curve. It follows from Lemma 25.149 that Z is connected. (II). We may replace k by a finite separable extension k ′ (we will use this in Step (III), if k is finite). Indeed, if k ′ is a separable extension, then X ⊗k k ′ is still regular (Proposition 18.52). If U ⊂ X is non-empty open affine, then its inverse image U ′ in X ′ := X ⊗k k ′ is affine. If there exists a scheme structure on D′ := X ′ \ U ′ such that OX ′ (D′ ) is ample, then looking at local equations one sees that NX ′ /X (OX ′ (D′ )) is a line bundle on X of the form OX (D) for some scheme structure on D := X \ U . It is ample by Proposition 13.66 (1). (III). Let Z1 , . . . , Zr be the irreducible components of Z, considered as prime divisors Pr on X. Let us show that there exists D′ = i=1 di Zi , di ≥ 0, such that (D′ · Zi ) ≥ 0 for all i, and (D ′ · Zi ) > 0 for at least one i. Let s ∈ Γ(U, OX ) ⊂ K(X) be non-constant and such that s does not vanish entirely along any of the Zi . We can produce such an s by taking any non-constant section in Γ(U, OX ) and adding a suitable constant to it. If k is finite, then to find such a constant, we might have to enlarge k; this we can do by Step (II). Let D1 be the divisor of zeros of s, and let D2 the divisor of poles, i.e., D1 and D2 are effective divisors with D1 −D2 = div(s). Now s ∈ Γ(U, X ), so the locus of poles of s is disjoint from U , and hence as a Weil PO r divisor, D2 = i=1 di Zi , di ≥ 0. We claim that we can set D′ := D2 . The condition on the intersection numbers can be checked for D1 in place of D2 , because these two divisors are linearly equivalent by construction.

506

25 Duality

Since s does not vanish along any of the Zi , none of them is contained in the support of D1 , so (D1 · Zi ) ≥ 0 for all i. Furthermore, the support of D1 is closed in X and is thus a proper k-scheme, so it cannot be contained in U . Therefore we have D1 ∩ Zi = ̸ ∅ for at least one i, and this implies (D1 · Zi ) > 0. (IV). Let F0 := D′ and F1 := Zi as in Step (III), i.e., (F0 · F1 ) > 0. Define F2 , . . . , Fs such that each Fj , j ≥ 2, is one of the irreducible components of Z, and all of those occur among the Fi (possibly with repetitions), and such that (Fi−1 · Fi ) > 0 for all i = 2, . . . , s. To find such a sequence, we use that Z is connected. It isPthen enough to find positive natural numbers ei , i = 1, . . . , s, such that for s D := i=0 ei Fi we have (D · Zj ) > 0 for all j = 1, . . . , s. In fact, then D is an effective divisor on X with support X \ U . To check that it is ample, we may use the NakaiMoishezon criterion, Theorem 25.146 (2). If C ⊂ X is an integral curve that is different from all the Zi , then C, being proper, cannot be contained in U , and it follows that (C · D) > 0. On the other hand, (D · Zj ) > 0 holds by construction of D. Let us spell out a sufficient condition on the coefficients ei . Fix j ≥ 1 for a moment and let i0 be minimal with Fi0 = Zj . Then (D · Zj ) =

iX 0 −2

ei (Fi · Zj ) + ei0 −1 (Fi0 −1 · Fi0 ) +

i=0

s X

ei (Fi · Fi0 ),

i=i0

where the first sum on the right hand side is non-negative. Since (Fi0 −1 · Fi0 ) > 0 by construction of the Fi , we see that we can ensure that (D · Zj ) > 0 by requiring that Ps ei (Fi · Fi0 ) ei0 −1 > − i=i0 . (Fi0 −1 · Fi0 ) Therefore we can set es := 1 and define the other coefficients by descending induction so that they are positive and satisfy the above inequality for all i0 . Remark 25.152. The projectivity of regular surfaces was first proved by Zariski. The proof above is due to Goodman [Goo] O ; see also [Har2] O . Slightly more generally, one can show that every proper surface X over a field k such that there exists an affine open U ⊂ X containing all non-regular points of X is projective ([Goo] O Cor. to Thm. 2). On the other hand, there exist non-projective proper surfaces and non-projective smooth proper varieties of dimension 3 (and hence of any higher dimension, as well). (25.32) The Hodge index theorem. Definition 25.153. Let k be a field and let X be a proper surface over k. (1) We call divisors D, D′ on X numerically equivalent, if (D · E) = (D′ · E) for every divisor E on X. This also gives us the notion of numerical equivalence for classes of divisors up to linear equivalence. (2) We denote by Num(X) the quotient of Pic(X) by the subgroup consisting of all divisor classes that are numerically equivalent to 0. By the definition of numerical equivalence, the intersection pairing on Pic(X) induces a non-degenerate pairing on Num(X), again called the intersection pairing.

507 It is known that Num(X) is a free finitely generated abelian group. In fact, it is clearly torsion-free. It follows from the Theorem of N´eron-Severi (also called Theorem of the base) that it is finitely generated. This is a rather difficult result which we will not use below; see [CJLO] X for a modern proof and further references. In particular, Num(X)R is a (finite-dimensional) R-vector space, equipped with a non-degenerate symmetric bilinear form induced by the intersection pairing. Lemma 25.154. Let X be a smooth proper algebraic surface over a field k. Let H be an ample divisor on X. If D is a divisor on X such that (D2 ) > 0, then the following are equivalent: (i) (D · H) > 0, (ii) for all n sufficiently large, the divisor nD is linearly equivalent to an effective divisor, in other words, H 0 (X, OX (nD)) ̸= 0. Proof. (i) ⇒ (ii). We have dim H 2 (X, OX (nD)) = 0 for n large. Indeed, Serre duality implies dim H 2 (X, OX (nD)) = dim H 0 (X, OX (−nD) ⊗ ωX ). Here ωX = Ω2X/k is the dualizing sheaf of X. If n(D ·H) > (ωX ·H), then we have H 0 (X, OX (−nD)⊗ωX ) = 0 because the intersection number of an ample and an effective divisor is positive (Proposition 23.79). It is thus enough to show that χ(OX (nD)) > 0 for all n that are sufficiently large, and this follows from the Riemann-Roch theorem (Theorem 25.144) together with the assumption that (D 2 ) > 0. (ii) ⇒ (i). It is enough to show that n(D · H) = (nD · H) > 0 for n as in (ii). This follows from Proposition 23.79. Corollary 25.155. Let X be a smooth proper algebraic surface over a field k. Let D be a divisor on X such that (D2 ) > 0. If H and H ′ are ample divisors on X, then one has (D · H) > 0 if and only if (D · H ′ ) > 0. Proof. This is an immediate consequence of Lemma 25.154, because the condition that all sufficiently high multiples of D are effective is independent of H and H ′ . With these preparations we can prove the Hodge index theorem. In view of the theorem of N´eron-Severi mentioned above, which ensures that Num(X)R is finite-dimensional, say of dimension n, it says that the “index” or signature of the non-degenerate quadratic space Num(X)R (with the intersection pairing) is (1, n − 1), i.e., Num(X)R is the orthogonal direct sum of a positive definite line and a negative definite subspace of codimension 1. Theorem 25.156. (Hodge index theorem) Let k be a field and let X be a smooth proper surface over k. Let H be an ample divisor on X. If D is a divisor on X with (D · H) = 0 that is not numerically equivalent to 0, then (D2 ) < 0. In other words, the intersection pairing induces a negative definite pairing on the orthogonal complement {H}⊥ = {D ∈ Num(X); (D · H) = 0} of H in Num(X). Proof. Let D be a divisor with (D · H) = 0 and not numerically equivalent to 0. We first exclude the case (D2 ) > 0. Let H ′ = D + nH. For n sufficiently large, H ′ is ample by Proposition 13.50 (2). But then (D · H ′ ) = (D2 ) > 0 and (D · H) = 0 in contradiction to Corollary 25.155.

508

25 Duality

Using this, we can exclude the case (D2 ) = 0 by the following simple argument about non-degenerate bilinear forms. Let E be such that (D · E) ̸= 0, (E · H) = 0. Then ((aD + E)2 ) = 2a(D · E) + (E 2 ) > 0 for suitable a, and ((aD + E) · H) = 0, contradicting the case we have already handled. (25.33) Further references. Already in the fundamental paper [Ser1] O by Serre, at the very beginning of the study of the cohomology of (quasi-)coherent sheaves on algebraic varieties, instances of (Serre) duality were observed, in particular the duality for cohomology of line bundles on projective space (Corollary 22.23); see also Grothendieck’s articles [Gro4] O , [Gro3] O . The theory was further developed by Grothendieck, Verdier, Hartshorne, Deligne, and others. In Hartshorne’s lecture notes [Har1] O of a seminar held by Grothendieck, the functor f ! is constructed by starting from the cases of projective space and of closed immersions, and then showing the necessary compatibilities. See also Conrad’s book [Con] O that provides a thorough discussion of several subtle points that were glossed over in [Har1] O . Alternatively, one can construct a right adjoint f × to Rf∗ “directly”, see [Ver1] O . More recently, the existence of f × was proved more generally by Neeman [Nee3] O , making use of a Brown representability theorem for triangulated categories; this is the approach we O have followed in this chapter. See also [Nee5] O X , [LN] X . × Once one has the functors f , one can construct the twisted inverse image functors f ! “by hand” using Nagata’s compactification theorem; this is the route taken, e. g., in [Sta] and also the one we have followed here. Another approach is to use a category-theoretic result about “pasting functors” as explained in the Appendix by Deligne in [Har1] O , O × cf. also [Lip2] O and f ! in general. X Section 4.8. See also [ILN] X for the relation between f Instead of obtaining a dualizing complex on a (projective) scheme X by embedding it into some projective space, another approach, simpler in some respects, is to consider a finite surjective morphism X ↠ Pd , and to “pullback” the dualizing complex on projective space to obtain one on X. See [Lip1] O for an exposition largely following this approach. It is often (very. . . ) difficult to align the different approaches, and in particular to make identifications obtained from abstract categorical considerations explicit in specific cases. See for instance [Yek2] O , [Sas1] O , [SaTo] O , [Sas2] O X , [NS1] X , [NS2] X , [Lip3] X for further discussions and results in this direction. The theory has been further extended, for instance to better cover non-noetherian schemes and formal schemes, and towards a non-commutative setting. Some pointers in O O O this direction are [AJL] O X , [LNS] , [Yek1] , [BDS] X . If one is willing to accept further assumptions and/or to content oneself with less complete results, the technical machinery of derived categories can be avoided entirely. See for instance [AlKl] O , [Har3] O Chapter III.7, [Liu] O Section 6.4. More recently, Clausen and Scholze have sketched a new approach to duality for coherent sheaves using their theory of condensed mathematics [Scho] X . More details have been given by Mann in his thesis [Man] X . In this setting, the category of schemes is embedded fully faithfully in the category of discrete adic spaces. In this category, every discrete adic space and every morphism of discrete adic spaces has a functorial “compactification” which simplifies the definition of f ! if f is “sufficiently finite”. One even obtains a full six-functor formalism (see Section (21.30)) if one restricts to “sufficiently finite” morphisms1 for the 1

i.e., separated and +-finite type morphisms as defined by Mann, which include all separated morphisms of finite type and all integral morphisms of arbitrary schemes

509 functors f ! and f! . The theory requires some mathematical machinery that is beyond the scope of this book, among other things the theory of ∞-categories. It is an exciting new perspective on the derived category of quasi-coherent OX -modules on a scheme X (and in fact also more general geometric objects such as derived schemes). Regarding the applications to surfaces, the material above is covered in the books by Beauville [Beau] O and B˘adescu [Bad] O , and partly also in [Har3] O Ch. V. Each of these sources provide a lot more material on algebraic surfaces, a beautiful topic which is more involved than the theory of algebraic curves, but still significantly simpler than the study of algebraic varieties of higher dimension. For instance, there are much stronger classification results, proved by Kodaira, Enriques, Mumford and Bombieri, than in higher dimension. Another example is resolution of singularities for surfaces, proved by Zariski and Abhyankar. See Artin’s article in [CoSi] O for an account of the proof and for further references.

Exercises Exercise 25.1. Let X be a scheme. Show that an object in Dqcoh (X) is perfect if and only if it is dualizable (Definition/Remark 21.149). Hint: To show that the condition is sufficient reduce to the case that X is affine. Now show that every dualizable object is compact. Exercise 25.2. Let X be a scheme, let L ∈ Dqcoh (X) be such that there exists M ∈ ∼ Dqcoh (X) with M ⊗L OX L = OX . Show that there exists an open covering (Ui )i of X such that L|Ui ∼ = Li [ni ], where Li is a line bundle on Ui and ni ∈ Z. Hint: One can assume that X = Spec A is affine. Show that L and M are dualizable and hence perfect by Exercise 25.1. Then use Proposition 22.54. Exercise 25.3. Let X be a quasi-compact scheme that carries an ample family of line bundles (Li )i∈I (Exercise 23.17). Show that { Li⊗m [n] ; i ∈ I, m, n ∈ Z } is a set of compact objects that generates Dqcoh (X). Exercise 25.4. Let R be a ring, n ≥ 1, and let f : PnR → S := Spec R be the structure morphism. Let j : U := AnR → PnR be the inclusion. (1) Show that (f × OS )|U ∼ = OU [n]. (2) Let R = k be a field. Show that (f ◦ j)× OS = Homk (k[T1 , . . . , Tn ], k)∼ , considered as a complex of quasi-coherent OU -modules concentrated in degree 0. ˜ be ˜ := R[T ]/(T 2 ) and let γ ∈ R ⊂ R Exercise 25.5. Let R be a noetherian ring, set R neither a unit nor nilpotent. Form the cartesian diagram Spec R[1/γ]

j′

f′

 ˜ Spec R[1/γ]

/ Spec R f

j

 ˜ / Spec R,

where f and f ′ correspond to the R-algebra homomorphisms sending T to 0 and where j ˜ be the complex and j ′ are the inclusions of principal open subschemes. Let N ∈ D(R) 0

0

0

· · · −→ R −→ R −→ R −→ 0 −→ 0 −→ · · ·

510

25 Duality

˜ concentrated in degree ≤ 0, where we consider R as an R-module. We consider f × as ˜ → D(R), and similarly for the functors j ∗ , (f ′ )× and (j ′ )∗ . functor D(R) Q (1) Show that f ! R = i≥0 R[−i] is the complex with R in every non-negative degree Q Q and with zero differentials. Deduce that f ! N = k≥0 i≥0 R[k − i]. T T ˜ ˜ −→ ˜ −→ R −→ 0 of the R-module Hint: Use the projective resolution · · · −→ R R R. ′ ∗ × ′ × ∗ (2) Show that the base change map (j ) f N → (f ) j N is the map YY YY
R[k − i] [1/γ] −→ (R[k − i][1/γ]) k≥0 i≥0

k≥0 i≥0

and that it does not induce an isomorphism after applying H 0 since γ is not a unit and not nilpotent. Exercise 25.6. Let X be a qcqs scheme and let U ⊆ X be a quasi-compact open subset. (1) Suppose that X is affine and let L be a line bundle on U . Suppose there exists a perfect complex E on X such that E |U ∼ = L in D(U ). Show that then there exists a line bundle M on X and an isomorphism M |U ∼ = L of line bundles on U . Hint: Represent E by a strictly perfect complex E • , such that the isomorphism E |U ∼ = N L is given by a quasi-isomorphism E • |U −→ L . Then set M := i∈Z det(E i )⊗i . Remark : This construction is a special and imprecise case of the determinant of a perfect complex, see [KnMu] O . (2) Find an example of an affine scheme X of finite type over a field k, an open subscheme U and of a line bundle L on U that cannot be extended to a perfect complex on X. Hint: E.g., let X be the cone given by z 2 = xy in A3k and U the complement of the origin. Show that Pic(X) = 0, and that Pic(U ) ̸= 0. Exercise 25.7. Let k be a field, let X = Ank and U = Ank \ {0} for n ≥ 3. ∼ (1) Show that restriction of locally free sheaves induces an isomorphism K0 (X) → K0 (U ). (2) Show that a locally free OU -module is isomorphic to the restriction of a locally free OX -module if and only if it is isomorphic to a direct sum of line bundles, or equivalently, if and only if it is free. Hint: Use “Serre’s conjecture” that every finite locally free OX -module is free (see [Lang] O XXI, 3.7). (3) Show that there exists a locally free OU -module which is not isomorphic to the restriction of a locally free OX -module. Hint: Let F be the pullback of Ω1P2 and show that F is not the direct sum of two k line bundles. Remark : See [Ser4] O §5, [Rou] O , Remark 3.13. Exercise 25.8. Let A be a ring and let B be an A-algebra. Let f : Spec B → Spec A be the corresponding morphism of affine schemes. Consider the following assertions. (i) The A-module B is perfect. (ii) Every perfect complex in D(B) is perfect considered as an object in D(A). (iii) Spec B → Spec A is cohomologically proper and of finite tor-dimension. (iv) The functor Rf∗ sends perfect complexes in D(B) = Dqcoh (Spec B) to perfect complexes in Dqcoh (Spec A). (v) The functor f × : Dqcoh (Spec A) → Dqcoh (Spec B) is naturally isomorphic to the functor B D(A) → D(B), K 7→ K ⊗L A R HomA (B, A).

511 Show that (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇒ (v). Remark : Remark 25.45 shows that in fact all these assertions are equivalent. Exercise 25.9. A qcqs morphism f : X → Y of schemes is called universally cohomologically proper 2 if for every morphism g : Y ′ → Y of schemes the base change X ×Y Y ′ → Y ′ of f is cohomologically proper. Show the following assertions. (1) The property “universally cohomologically proper” is stable under composition and under base change. (2) Let f : X → Y be a morphism of schemes and let Y ′ → Y be a faithfully flat surjective morphism of schemes. Suppose that the base change X ×Y Y ′ → Y ′ of f is universally cohomologically proper. Then f is universally cohomologically proper. (3) Every flat cohomologically proper morphism is universally cohomologically proper. Exercise 25.10. Give an example of a finite morphism f : X → S of affine schemes such that f∗ : (OX -Mod) → (OS -Mod) is not exact. Hint: Consider a Dedekind domain R with exactly two maximal ideals (e.g., X = √ Spec Z[ −1](p) for a prime number p ∈ Z such that p ≡ 1 (mod 4)) and let X = {η, s, t} be its spectrum, where s and t denote the closed points. Set U := {s, η} and V := {t, η}. Let K = Frac R and define an OX -module H by H (X) = K × K, H (U ) = H (V ) = K and H ({η}) = 0. Let F be the constant sheaf with value K on X. Show that there is a unique surjective map F → H of OX -modules such that F (U ) → H (U ) and F (V ) → H (V ) are the identity. Let S be the spectrum of a discrete valuation ring and let f : X → S be a finite morphism, which necessarily sends s and t to the closed point of S (e.g., S = Spec Z(p) in the above example). Show that f∗ F → f∗ H is not surjective. Exercise 25.11. Show that there exists a connected noetherian scheme X of dimension 2 which admits a dualizing complex and which has an irreducible component of dimension < dim(X) at whose generic point a dimension function takes its maximal value. Hint: Exercise 14.24 Exercise 25.12. Let k be a field, A := k[x, y], let p := (x, y) and q := (x − 1), and let S −1 A with S := A \ (p ∪ q) be the semilocalization of A in {p, q}. Show that Spec S −1 A is connected and regular (in particular Cohen-Macaulay) but not equi-dimensional. Exercise 25.13. Let k be a field and let X be a locally noetherian scheme over k. Let x ∈ X be a point. Show that the following assertions are equivalent. (i) The local ring OX,x is Cohen-Macaulay. (ii) For every finitely generated extension K ⊇ k and every point x′ ∈ X ⊗k K mapping to x, the local ring OX⊗k K,x′ is Cohen-Macaulay. (iii) There exists a field extension K ⊇ k such that X ⊗k K is locally noetherian and a point x′ ∈ X ⊗k K mapping to x such that OX⊗k K,x′ is Cohen-Macaulay. If X is locally of finite type over k, show that these conditions are also equivalent to (iv) For every field extension K ⊇ k and every point x′ ∈ X ⊗k K mapping to x, the local ring OX⊗k K,x′ is Cohen-Macaulay. Hint: Proposition B.82 Exercise 25.14. Show the following properties of Cohen-Macaulay morphisms. (1) The composition of Cohen-Macaulay morphisms is again Cohen-Macaulay. 2

This is non-standard terminology.

512

25 Duality

(2) Let f : X → Y be a morphism of schemes such that all fibers are locally noetherian schemes, let g : Y ′ → Y be locally of finite type, and let f ′ : X ′ → Y ′ be the base change of f . Show that if f is Cohen-Macaulay, then f ′ is Cohen-Macaulay. Show that the converse holds if g is faithfully flat. Hint: Exercise 25.13. • Exercise 25.15. Let A be a noetherian local ring with normalized dualizing complex ωA , −d • and let d ≥ 0 be an integer. Show that M 7→ ExtA (M, ωA ) is an anti-auto-equivalence of the category of finitely generated Cohen-Macaulay A-modules of depth d.

Exercise 25.16. Let A → B be a finite local homomorphism of noetherian local Cohen-Macaulay rings. Suppose that a dualizing module ωA for A exists. Show that ωB := ExttA (B, ωA ) with t := dim A − dim B is a dualizing module for B. Exercise 25.17. Let k be a field and let X be a proper k-scheme. Use Proposition 25.110 to give another proof that a dualizing sheaf on X is torsion-free (cf. Proposition 25.106 (3)). Exercise 25.18. Let k be a field. A Calabi-Yau variety 3 over k is a smooth proper geometrically connected k-scheme X of dimension d > 0 such that ΩdX/k ∼ = OX and i H (X, OX ) = 0 for all 0 < i < d. A Calabi-Yau variety of dimension 2 is called a K3-surface. (1) Let X be a Calabi-Yau variety of even dimension d. Prove dim H 0 (X, OX ) = dim H d (X, OX ) = 1 and deduce that χ(OX ) = 1 + (−1)d . (2) Show that any smooth geometrically connected hypersurface X of degree d + 2 in Pd+1 is a Calabi-Yau variety of dimension d. k Hint: Consider the exact sequence 0 → OPd+1 (−d − 2) → OPd+1 → OX → 0 and use k k Exercise 17.7. Exercise 25.19. Let X be a connected smooth proper scheme over a field k, let d = dim X and let hij (X) be the Hodge numbers of X (Exercise 23.5). Show that hij (X) = hd−i,d−j (X). Hint: Exercise 7.28. Remark : If k is a field of characteristic 0, then one also has symmetry of the Hodge numbers, i.e. hij (X) = hji (X).

3

There are different (non-equivalent) definitions in the literature. For instance, sometimes the condition that H i (X, OX ) = 0 for all 0 < i < d is omitted. Then the Calabi-Yau varieties as defined here are called strict Calabi-Yau varieties.

26

Curves

Content – Basic notions – The Theorem of Riemann-Roch – Special classes of curves – Vector bundles on curves – Further topics In this chapter, we will study curves over fields. With the results of the preceding chapters, we can prove some of the highlights of the theory, such as the Theorem of Riemann-Roch, without much further effort. The intention of the chapter is not to give a systematic and comprehensive treatment of the theory, but rather to illustrate some important results and classes of particularly interesting curves, for instance elliptic and hyperelliptic curves, while at the same time providing examples of the usefulness of the general machinery of algebraic geometry we have built up so far. See Section (26.31) for further references and pointers to the literature.

Basic notions We start by recalling (and in part, generalizing) a few facts that we have already proved in Volume I or in the preceding chapters. We then introduce the genus of a proper curve. At the end we study singularities of curves and single out some classes of particularly mild singularities. (26.1) Recollections on curves. Let k be a field. In this chapter we will mean by a curve over k the following. Definition 26.1. A curve over the field k is a separated k-scheme C of finite type that is equi-dimensional of dimension 1. In comparison to Definition 15.14 we add the requirement that C be separated. Because a one-dimensional normal noetherian local ring is a discrete valuation ring, a curve C over a field k is regular, if and only if it is normal. If k is perfect, then this is equivalent to C being smooth over k (Proposition 18.67).

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_11

514

26 Curves

Every reduced curve C is birational to a unique normal proper curve over k. In fact, replacing C by the disjoint union of its irreducible components, and handling each of them individually, we may assume that C is integral. Passing to the normalization, we may assume that C is normal, and then the unique normal compactification (Theorem 15.20) is the proper (and even projective, Theorem 26.16) curve that we are looking for. When C is integral, we can obtain it by using that C is quasi-projective (Theorem 15.18), and taking the normalization of the closure of the image of C under some embedding into a projective space. We extract the following definitions from this discussion: Definition 26.2. Let C be a curve over a field k. Let C1 , . . . , Cn be the irreducible components of C endowed with the reduced scheme structure, and for each i let C˜i be the normalization of Ci . (1) The normalization of C is the disjoint union of the curves C˜i , together with the natural morphism to C. (2) The complete (or proper, or projective) normal model of C is the unique proper normal curve that is birational to the underlying reduced curve Cred . If C is integral, then this notion of normalization coincides with the one of Definition 12.42. As in Proposition 12.44 every morphism from a normal scheme X to C such that each irreducible component of C is dominated by an irreducible component of X factors through the normalization of C. In fact, this characterizes the normalization of C. On reduced affine open subschemes, it corresponds to passing to the integral closure of the coordinate ring in its ring of total fractions. The normalization morphism is the composition of the finite surjective morphisms a a C˜i −→ Ci −→ C i

i

hence it is finite surjective. Also recall the equivalence of categories between proper normal integral curves and their function fields, Theorem 15.21. Theorem 26.3. Let k be a field. Mapping an integral curve C to its function field K(C), and a dominant (equivalently: non-constant) morphism between integral curves to the corresponding field extension of their function fields gives rise to a contravariant equivalence between (i) the category of normal proper integral curves over k with non-constant morphisms, and (ii) the category of extension fields K of k such that the extension K/k is finitely generated and has transcendence degree 1. (26.2) Reminder on geometrically connected schemes. As before, let k be a field. We recall the notions of geometrically irreducible, reduced, and connected k-schemes (and particularly curves), see Section (5.13). We say that a k-scheme X has one of the properties irreducible, reduced, connected1 geometrically, if XK has the property for every field extension K/k, or equivalently if XK has the property for some algebraically closed extension field K of k (Definition 5.48, Corollary 5.54). 1

Recall that by definition connected spaces are non-empty.

515 Let X be an integral k-scheme with function field K(X). Then X is geometrically integral if and only if k is algebraically closed in K(X) and the extension K(X)/k is separable (Proposition 5.51). Every smooth k-scheme is reduced, and even geometrically reduced since smoothness is preserved by base change. A scheme X is connected if and only if the ring H 0 (X, OX ) has no non-trivial idempotent elements (equivalently, if this ring does not admit a non-trivial product decomposition, see Exercise 2.17). Now let X be a k-scheme and let Ω be an algebraically closed extension field of k. Then H 0 (XΩ , OXΩ ) = H 0 (X, OX ) ⊗k Ω, so X is geometrically connected if and only if H 0 (X, OX ) ⊗k Ω has no non-trivial idempotents. For instance, this certainly holds whenever H 0 (X, OX ) = k. (Compare also Corollary 24.50 for a relative version of this statement.) A connected k-scheme with a k-rational point is automatically geometrically connected. This follows from the following result (cf. Exercise 5.23) by applying it to Y = Spec k. Lemma 26.4. Let k be a field and let X be a connected k-scheme. Assume that there exists a geometrically connected k-scheme Y and a k-morphism f : Y → X. Then X is geometrically connected. Proof. Let Ω be an algebraic closure of k. We have to show that XΩ = X ⊗k Ω is connected. Consider the surjective projection p : XΩ → X. As Ω is an algebraic extension of k, p is integral and hence closed (Proposition 12.12). By Theorem 14.38 it is also open. Assume that there existed an open and closed subset ∅ ̸= Z ⊊ XΩ . Then p(Z) = X since X is connected. Moreover fΩ−1 (Z) = Y ×X Z is open and closed in YΩ which is by hypothesis connected. As Z → X is surjective, f −1 (Z) → Y is surjective. In particular, f −1 (Z) is non-empty and therefore f −1 (Z) = YΩ . The same argument also shows that f −1 (XΩ \ Z) = YΩ . This is absurd. For X proper over k and reduced, we have the following simple criterion to be geometrically connected. Lemma 26.5. Let k be a field. Let X be a connected proper reduced k-scheme. Then H 0 (X, OX ) is a finite extension field of k. (1) The k-scheme X is geometrically connected if and only if the H 0 (X, OX ) is a purely inseparable extension of k. (2) If X is even geometrically reduced (e.g., if X is smooth over k), then X is geometrically connected if and only if H 0 (X, OX ) = k. Proof. Since X is proper, H 0 (X, OX ) is a finite-dimensional k-vector space (Theorem 12.65, or Corollary 23.18) and hence a finite product of local Artin k-algebras. Now X is connected, so there can be only one factor, and since X is reduced, H 0 (X, OX ) is a reduced ring, hence a field. Now (1) follows from Corollary 24.55. Under the assumptions of (2), we know that H 0 (X, OX ) is a geometrically reduced k-algebra and hence a separable field extension of k (Proposition 18.24). Therefore (2) follows from (1). (26.3) Singularities of curves. Let us compare different forms of singularities for a curve C over a field k. In general, one has the implications

516

26 Curves C smooth over k

(1)

+3 C regular ks

(2)

(3)

(26.3.1)

 C reduced

 C lci

(4)

 C Gorenstein

+3 C normal

(6)

(5)

 +3 C Cohen-Macaulay

Implication (1) holds for any scheme locally of finite type over a field k and it is an equivalence if k is perfect (Proposition 18.67). The Equivalence (2) is particular to curves (Proposition 6.40). The Implications (3), (4), and (5) are general results for local noetherian rings (Remark 19.47). Finally, Implication (6) is again particular to curves and holds by the following lemma. Lemma 26.6. Let C be a noetherian scheme of dimension 1. Then C is Cohen-Macaulay if and only if C has no embedded components. This is true when H 0 (C, OC ) is reduced. If C is reduced, then H 0 (C, OC ) is reduced, but the converse does not hold in general, see Exercise 26.1. Proof. Example B.80 (2) shows the characterization of being Cohen-Macaulay for curves. Now let us assume that x ∈ C is an embedded associated point. As C is a curve, this is a closed point. We will construct a non-zero nilpotent element of H 0 (C, OC ). To achieve this, let U = Spec A be an affine open neighborhood of x. The point x then corresponds to a maximal ideal m ⊂ A of the form m = Ann(a) for some 0 ̸= a ∈ A. Let p1 , . . . , pr be the minimal prime ideals of A. For each i, if s ∈ m \ pi , then sa = 0 implies a ∈ pi ; thus a is nilpotent. We show that one can extend a to a global Srsection of C. In fact, by prime ideal avoidance (Proposition B.2 (2)), there exists s ∈ m \ i=1 pi . Then a|D(s) = 0 and V := D(s) ∪ {m} is the complement in U of finitely many closed points, hence is open in U and therefore in C. We can then glue a|V ∈ H 0 (V, OC ) with 0 ∈ H 0 (C \ {m}, OC ) and obtain a non-zero nilpotent global section of OC , as desired. All other possible implications in (26.3.1) do not hold, as the following examples show. Example 26.7. (1) Exercise 18.19 gives an example of regular, geometrically reduced curve which is not smooth (necessarily over a non-perfect field). (2) Every reduced plane curve over a field k, i.e., every reduced curve which admits an embedding as a locally closed subscheme of P2k , is locally a complete intersection (Proposition 5.31, Corollary 5.42). See also Proposition 26.78 below. This gives us plenty of examples of reduced lci curves which are not regular, e.g., Spec k[X, Y ]/(Y 2 − X 3 ) or Spec k[X, Y ]/(Y 2 − X 2 (X − 1)). It gives us also non-reduced curves which are lci, e.g., Spec k[X, Y ]/(X 2 ). (3) The subring k[T 5 , T 6 , T 7 , T 8 ] of k[T ] is Gorenstein but not lci (Exercise 26.3). (4) Example 26.42 below shows that the union of the three coordinate axes in A3k gives an example of a reduced curve which is not Gorenstein. Compare this to (2) which shows that the union of finitely many lines in A2k is always lci, in particular Gorenstein. (5) Finally, an example of an irreducible curve which is not Cohen Macaulay would be any curve with an embedded component, e.g. Spec k[X, Y ]/(X 2 , XY ).

517 (26.4) Morphisms between curves. Let us recall some results on morphisms between curves, and collect some new ones. First, we have the foundational fact, used for instance in the proof of Theorem 26.3, that every rational map from a normal curve C to a proper k-scheme X extends to a morphism C → X. This is an immediate consequence of the valuative criterion of properness, Theorem 15.9. Remark 26.8. (Morphisms to projective space) Let k be a field and X a k-scheme. N Recall the description of morphisms X → PN k (i.e., of X-valued points of Pk ) from N ∗ (1) is a Sections (8.5), (13.13). For a morphism f : X → Pk , the pullback L := f OPN k globally generated line bundle and the natural basis T0 , . . . , TN ∈ H 0 (PN , O(1)) gives k rise, by pullback, to sections si = f ∗ Ti ∈ H 0 (X, L ) generating L everywhere. Conversely, given a line bundle L on X with global sections s0 , . . . , sN ∈ H 0 (X, L ), not all = 0, we obtain a rational map X 99K PN k which on k-valued points is given by x 7→ (s0 (x) : · · · : sN (x)) (where we choose an isomorphism L (x) ∼ = k and the resulting point in PN k is independent of the choice of isomorphism). This rational map is defined on the open subset U ⊆ X which is the union of the non-vanishing loci of the si . Denoting ∗ by f : U → PN (1) ∼ = L|U . k the corresponding morphism, we have f OPN k We call the subvector space of H 0 (X, L ) generated by the si the corresponding linear system, and say that this linear system is base-point free, if the morphism described above is defined on all of X, i.e., if for every x ∈ X there exists i with si (x) ̸= 0. If s0 , . . . , sN generate H 0 (X, L ), we speak about the complete linear system attached to L . It is base-point free if and only if L is generated by its global sections. As pointed out above, if X = C is a normal curve, then every rational map f : C 99K PN k can be extended to a morphism (but if the rational map comes from a linear system that is not base-point free, then the pullback f˜∗ OPN (1) under the morphism f˜: C → PN k will k be different from the line bundle giving rise to this linear system). Lemma 26.9. Let f : C ′ → C be a non-constant morphism of curves over a field k. Suppose that C ′ is irreducible. (1) The morphism f is quasi-finite. (2) Let C ′ be integral and let C be normal. Then f is flat. (3) Let C ′ be proper over k. Then f is finite. (4) Let C ′ be integral and proper over k and let C be normal, then f is finite locally free. Proof. As f is non-constant and C ′ is an irreducible scheme of dimension 1, the fibers of f must be 0-dimensional, hence f is quasi-finite. Assertion (2) is a special case of Proposition 15.4 (3) which ultimately follows from the fact that a module over a discrete valuation ring is flat if and only if it is torsion-free. Assertion (3) follows from Zariski’s main theorem as follows. As C ′ is proper over k and C is separated over k, the morphism f is proper. It is quasi-finite by (1). Hence it is finite by Zariski’s main theorem (Corollary 12.89). Finally, (4) follows from (2) and (3) (Proposition 12.19). Lemma 26.10. Let k be a field, and let f : X → Y be a dominant morphism between integral k-schemes of finite type with dim(X) = dim(Y ). The following are equivalent (i) The morphism f is generically ´etale, i.e., there exists a non-empty open subscheme U ⊆ X, such that f|U : U → Y is ´etale.

518

26 Curves

(ii) The morphism f is generically unramified, i.e., there exists a non-empty open subscheme U ⊆ X, such that f|U : U → Y is unramified. (iii) The extension K(X)/K(Y ) of function fields induced by f is separable. Because of point (iii), in the literature morphisms with this property are often called separable. Proof. The extension K(X)/K(Y ) is finite since K(X) and K(Y ) are finitely generated field extensions of k with the same transcendence degree dim(X) = dim(Y ) over k. We remark that this shows that the morphism Spec K(X) → Spec K(Y ) is ´etale if and only if it is smooth, if and only if it is unramified (Theorem 18.44), if and only if the extension is separable (e.g., use Proposition 18.24 and Theorem 18.44). It is clear that (i) implies (ii). By the above remark and by Proposition 18.79 we have (iii) ⇒ (i). It remains to show that (ii) implies (iii). But since Spec K(X) is the scheme-theoretic fiber of f over the generic point Spec K(Y ) of Y , this follows from the above remark and the fact that the property unramified is preserved by base change, see Remark 18.23. An important invariant of a finite morphism between integral curves is its degree. (See also Definition 23.76, and (12.6.1) for the case of finite locally free morphisms). Definition 26.11. Let k be a field, and let f : C ′ → C be a non-constant morphism of integral curves over k. Then via f the function field K(C) of C is contained in K(C ′ ), and the degree deg(f ) of f is defined as the degree of this finite field extension. It is clear that the field extension K(C ′ )/K(C) is indeed finite since both fields are finitely generated and have transcendence degree 1 over k. Corollary 26.12. Let k be a field. Let f : C ′ → C be a non-constant morphism between integral curves over k. If C ′ is proper over k, C is normal, and deg(f ) = 1 then f is an isomorphism. Proof. By Lemma 26.9 we know that f is finite locally free. Hence it is an isomorphism if and only if it is of degree 1. The following example shows that the hypotheses in Corollary 26.12 are indeed necessary. Example 26.13. Let C be an integral curve. (1) Let π : C ′ → C be its normalization, which is a finite morphism by Corollary 12.52. Hence K(C) = K(C ′ ) by definition and π is of degree 1. But π is an isomorphism if and only if C is normal. (2) Every open immersion j : C ′ → C has degree 1. If j is not an isomorphism, then C ′ cannot be proper because otherwise j would be also a closed immersion and hence surjective. Remark 26.14. Let k be a field, and let π : C ′ → C be a non-constant morphism of integral curves over k. Then π is quasi-finite by Lemma 26.9. Recall that we defined in Section (12.5) for c′ ∈ C ′ with image c = π(c′ ) positive integers ec′ /c := lg(Oπ−1 (c),c′ ),

fc′′ /c := [κ(c′ ) : κ(c)]insep ,

fc′′′ /c := [κ(c′ ) : κ(c)]sep .

519 Now suppose that π is flat (e.g., if C is normal, Lemma 26.9). Then π is ´etale in a point c′ ∈ C ′ if and only if ec′ /c = fc′′ /c = 1 by Corollary 18.28. In this case, π is ´etale in some open neighborhood of c′ which is dense in C ′ since C ′ is integral, i.e., π is then generically ´etale and there exist only a finite set S ′ ⊆ C ′ of closed points in which π is not ´etale. There exists a natural structure of a closed subscheme on S ′ given by the different of π (Definition 20.74). If C is normal, then S ′ will be a divisor which we will study in Section (26.13) below. The image S := π(S ′ ) is a finite set of closed points of C. Let U := C \ S be its open complement. Then the restriction of π to π −1 (U ) is ´etale. Now suppose that π is finite locally free (e.g., if C ′ is proper over k and C is normal, see Lemma 26.9). Then we have by Proposition 12.21 for every c ∈ C X ec′ /c fc′′ /c fc′′′ /c . deg(π) = c′ ∈π −1 (c)

As all members of the right hand side sum are ≥ 1, we see in particular #π −1 (c) ≤ deg(f ),

c∈C

with equality if and only if Oπ−1 (c),c′ = κ(c) for all c′ ∈ π −1 (c). In this case we can endow S with a natural structure of a closed subscheme given by the discriminant of π (Definition 20.72). (26.5) Quasi-projectivity of Curves. In this section we complete the proof of the theorem, already stated in Volume I, that every separated curve over a field is quasi-projective. We will prove later, using the Theorem of Riemann-Roch, that a separated irreducible curve over a field is either affine or projective (Proposition 26.61). The key ingredient that allows us to reduce the proof of quasi-projectivity in general to the case of reduced curves (which was handled in Chapter 15) is the cohomological characterization of ampleness. Furthermore, we need the following lemma. Lemma 26.15. Let i : X0 → X be a closed immersion of schemes defined by a quasicoherent ideal I ⊂ OX with I 2 = 0 so that we can view I as OX0 -module. Then there exists an exact sequence of abelian groups (26.5.1)

H 1 (X0 , I ) → Pic(X) → Pic(X0 ) → H 2 (X0 , I ).

In particular, Pic(X) → Pic(X0 ) is surjective if H 2 (X0 , I ) = 0. Proof. Consider the short exact sequence of abelian sheaves f 7→1+f

× × 0 −→ I −−−−−→ OX −→ OX −→ 1 0

on the underlying topological space of X0 which is the same as the underlying topological space of X. Then the long exact cohomology sequence yields the exact sequence (26.5.1), × because we can identify H 1 (X, OX ) with Pic(X), and likewise for X0 , see Section (11.7). Theorem 26.16. Let k be a field and let f : C → Spec k be a curve. Then f is quasiprojective. Recall that in this chapters all curves are by definition separated.

520

26 Curves

Proof. Recall that we proved the theorem already if C is reduced (Theorem 15.18). We will reduce to this case. Since we know that Cred is quasi-projective, it admits an ample line bundle L0 . By Lemma 26.15 and since the second cohomology group occurring there vanishes for dimension reasons (Theorem 21.57), we can lift L0 to a line bundle L on C (note that Cred is defined by a nilpotent ideal of OC since C is noetherian). By Lemma 23.7, L is ample. It follows that C is quasi-projective, cf. Section (13.15). See Proposition 26.168 for a result in this direction for relative curves. Also see Proposition 26.61. (26.6) Divisors on curves. In some of the discussions below we want to include curves C (always over a field) that are not necessarily normal. So the notions of Cartier divisors and of Weil divisors may not coincide. By a divisor, we always mean a Cartier divisor (Definition 11.26), i.e., an element of Div(C) = Γ(C, KC× /OC× ), where KC denotes the sheaf of meromorphic functions on C (Sections (11.10) and (11.11)). For every divisor D on a curve C we have the associated line bundle OC (D). If D is represented by (Ui , fi )i with Ui ⊆ C open and fi ∈ KC (Ui ), then OC (D)|Ui = fi−1 OUi . Since every curve C over a field is quasi-projective by Theorem 26.16, the map D 7→ OC (D) yields an isomorphism (26.6.1)

∼

DivCl(C) −→ Pic(C)

of the divisor class group DivCl(C) of Cartier divisors up to linear equivalence with the Picard group Pic(C), i.e., the group of isomorphism classes of line bundles on C, by Corollary 11.30. Recall from Corollary 15.26 that every divisor on a curve can be expressed as the difference of two effective divisors, i.e., those divisors that are represented by (Ui , fi )i with fi ∈ KC (Ui )× ∩ OC (Ui ) (Definition 11.26 (4)). Given a Cartier divisor D on a curve C over a field k, we have the corresponding Weil divisor X nx [x], cyc(D) = x∈C 1

where the sum extends over the set C 1 of closed points of C, nx ∈ Z, and only finitely many summands are ̸= 0. See Section (11.13). The degree of D is then defined as X deg(D) = nx [κ(x) : k]. x∈C 1

See Definition 15.28. If C is proper over k, then two linearly equivalent line bundles have the same degree (Theorem 15.31), i.e., by (26.6.1), deg induces a homomorphism of abelian groups deg : Pic(C) → Z. By Example 23.75 (1), the degree of a line bundle L is the same as the intersection number (L · C).

521 If D is an effective divisor on an arbitrary curve C, then nx = lg(OC,x /(fx )), where fx is a local equation of D at x. In particular, deg(D) ≥ 0. In this case, D defines a closed subscheme, supported on a finite number of closed points of C. It is defined by the invertible quasi-coherent ideal IC (D) := OC (D)⊗−1 . In particular, the closed immersion D → C is regular of codimension 1. Conversely, we can recover the divisor D from this closed subscheme, which we therefore usually also denote by D. We then see that deg(D) = dimk Γ(D, OD ) = dimk Γ(C, OD ). If x ∈ C is a closed point which lies in the normal locus of C, i.e., such that OC,x is normal and hence a discrete valuation ring (Proposition B.71), then we can view [x] as a Cartier divisor, i.e., the cycle [x] is the Weil divisor attached to an (effective) Cartier divisor on C. The degree of this divisor is [κ(x) : k]. The corresponding closed subscheme of C is simply the scheme Spec κ(x) considered as a closed subscheme of C concentrated at x. If C is normal, i.e., all local rings at closed points are discrete valuation rings, then we have a one-to-one correspondence between Cartier divisors and Weil divisors on C, i.e., 1 the group of divisors is the free abelian group Z(C ) . If D = (nx )x∈C 1 is divisor on a connected normal curve C, Remark 11.41 shows that the corresponding line bundle OC (D) has sections over some U ⊆ C open given by Γ(U, OC (D)) = { f ∈ K(C) ; vx (f ) ≥ −nx for all x ∈ C 1 ∩ U }. For suitable morphisms f : C ′ → C we can define a pullback operation on divisors, see Definition 11.49, Proposition 11.50. Here we record the following special cases: Proposition 26.17. Let k be a field and let f : C ′ → C be a morphism of curves over k. Assume that f is flat or that C ′ is reduced and f is non-constant on every irreducible component of C ′ . Then the homomorphism OC → f∗ OC ′ induces a homomorphism f ∗ : Div(C) → Div(C ′ ), and we call f ∗ D the pullback of D under f . For the associated line bundles we have OC ′ (f ∗ D) ∼ = f ∗ OC (D). Remark 26.18. For an effective Cartier divisor, seen as a closed subscheme D ⊂ C locally defined by a regular element t in some Γ(U, OC ), the conditions on f ensure that t is mapped to a regular element in Γ(f −1 (U ), OC ′ ) so that the scheme-theoretic inverse image f −1 (D) is a closed subscheme in C ′ which corresponds to an effective Cartier divisor on C ′ . The hypothesis on f is for instance satisfied if f is the normalization morphism of an arbitrary curve in the sense of Definition 26.2. For a finite morphism f between integral curves, we can compute the degree of the pullback of D as the product of the degree of f and the degree of D. Compare Proposition 15.30, where we have proved this under the assumption that the target of the morphism is normal. Proposition 26.19. Let f : C ′ → C be a finite morphism of integral curves over k, and let D be a divisor on C. Then we have deg(f ∗ D) = deg(f ) deg(D). If C ′ and C are proper over k, then Proposition 26.19 is a special case of Proposition 23.77.

522

26 Curves

Proof. We may assume that D is effective and can work on local rings, i.e., instead of C we consider the spectrum of a local domain A and assume that D corresponds to the closed subscheme A/tA for some regular element t ∈ A. Then f corresponds to a finite ring homomorphism φ : A → B, and we have to show that dimk B/tB = deg(f ) dimk A/tA, where deg(f ) = [Frac(B) : Frac(A)] =: d. Now let b1 , . . . , bd ∈ B be a Frac(A)-basis of Frac(B) and let B ′ ⊆ B be the free A-module generated by b1 , . . . , bn . We then have isomorphisms (B/tB ′ )/(tB/tB ′ ) ∼ = B/tB,

tB/tB ′ ∼ = B/B ′ ,

(B/tB ′ )/(B ′ /tB ′ ) ∼ = B/B ′ ,

where we use for the second isomorphism that t is regular in B by Remark 26.18. Since all occurring terms are finite-dimensional k-vector spaces, we obtain dimk B/tB = dimk B ′ /tB ′ = d dimk A/tA, where for the second equality we use that B ′ is free of rank d over A. Corollary 26.20. Let k be a field and let C be a connected normal proper curve. If there exist x ̸= y ∈ C(k) such that the divisors [x] and [y] are linearly equivalent, then C ∼ = P1k . Proof. Say [x] − [y] = div(f ) for f ∈ K(C). As C is normal, f defines a non-constant morphism f : C → P1 such that f ∗ [0] = [x]. By Proposition 26.19 we find deg(f ) = 1, so f is an isomorphism by Corollary 26.12. Let X be an integral proper scheme over a field k. Then Γ(X, OX ) is a finite k-algebra without zero divisors, and hence a finite field extension of k. Given a divisor D on X, the line bundle OX (D) is by definition contained in the sheaf KX of rational functions on X, so H 0 (X, OX (D)) is a Γ(X, OX )-sub-vector space of K(X). The map H 0 (X, OX (D)) → Div(X),

s 7→ D + div(s),

induces a bijection (26.6.2)

P(H 0 (X, OX (D))∨ ) = (H 0 (X, OX (D)) \ {0})/Γ(C, OC )× ∼

−→ {D′ ⊆ X effective divisor, linearly equivalent to D}

by Proposition 11.34. Here we use that X is integral to see that all non-zero sections of H 0 (X, OX (D)) are regular. Remark 26.21. Let C be a proper curve over a field k. (1) As we have shown in Theorem 15.31, the degree of a principal divisor on X equals 0. (2) Let C be integral and let D be a divisor of deg(D) < 0. Then there cannot exist effective divisors that are linearly equivalent to D. Hence (26.6.2) shows H 0 (C, OC (D)) = 0. (3) Let C be integral and D be a divisor of degree 0. Then OC (D) ∼ = OC if and only if H 0 (C, OC (D)) ̸= 0. In fact, if H 0 (C, OC (D)) ̸= 0, then C is linearly equivalent to an effective divisor of degree 0, i.e., to the trivial divisor, thus OC (D) ∼ = OC (0) = OC .

523 A divisor D on a k-scheme X is called base-point free, if the invertible sheaf OX (D) is generated by its global sections, or equivalently if it defines a morphism from X to the projective space P(H 0 (X, OX (D)). The term base-point free refers to the latter condition – a base point is a point where all global sections of this line bundle vanish and correspondingly the rational map from X to projective space is not defined. A divisor is called ample (resp. very ample) if the line bundle OX (D) is ample (resp. very ample). A divisor is very ample if and only if it is base point free and it defines an immersion of X into projective space. (26.7) Algebraic curves and compact Riemann surfaces. In this section we work over the field C of complex numbers and compare the algebraic situation to the analytic setting, i.e., we have a look at complex manifolds X that have complex dimension 1, or equivalently, real dimension 2. So considered as differentiable manifolds, these objects are surfaces, which justifies the following terminology. Definition 26.22. A Riemann surface is a connected complex manifold of complex dimension 1. Simple examples are C and open subsets of C, and the Riemann sphere P1 (C). More generally, every connected smooth algebraic curve C over C gives rise to a Riemann surface C an by analytification (Section (20.12)). If the curve C is proper over C, then the analytification C an is a compact Riemann surface (cf. Remark 20.59). In the case of curves, we can also argue by using that a proper curve C admits a closed embedding into some projective space. The analytification of this embedding is then a closed embedding of C an into a complex projective space Pn (C). Every smooth algebraic curve C over C can be compactified by adding finitely many closed points, to obtain the smooth projective model. Therefore C an also admits a compactification by adding finitely many points. On the other hand, the open unit disk D = {z ∈ C; |z| < 1} is a typical example of a Riemann surface that is not algebraizable, i.e., that is not of the form C an for an algebraic curve C. Surprisingly, at first, the result that every proper algebraic curve is projective, has an analog in the complex setting: Every compact Riemann surface is projective, i.e., it admits a closed embedding into some complex projective space. This implies that every compact Riemann surface is algebraizable. Therefore, once this fact is established, the theories of smooth proper algebraic curves over C and of compact Riemann surfaces are in a sense the same. However, different methods can of course be used on either side of this correspondence. We emphasize that the case of dimension ≤ 1 is completely exceptional in this respect. Below we sketch one possible path to proving that every compact Riemann surface is algebraizable, based on the following analytic (and difficult) result which we will use as a black box. Theorem 26.23. Let X be a compact Riemann surface. Then the cohomology group H 1 (X, OX ) (with coefficients in the structure sheaf of the complex manifold X, i.e., the sheaf of holomorphic functions on X) is a finite-dimensional C-vector space. For a proof tailored to the setting of Riemann surfaces, see [For] O §14. Of course, in the algebraic setting this finiteness is already known to us, it is a special case of the result that push-forward under proper morphisms preserves coherence (Theorem 23.1). Unlike

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26 Curves

what follows, this statement also holds for proper morphisms between higher-dimensional compact complex manifolds. See Remark 23.19. With this result at hand, there are several routes to showing that X is algebraizable. The most direct one is to show that there exists a non-constant meromorphic function. Definition 26.24. Let X be a Riemann surface. A meromorphic function on X is a morphism X → P1 (C) of Riemann surfaces. Proposition 26.25. Let X be a compact Riemann surface. Then there exists a nonconstant meromorphic function on X. Proof. Let x ∈ X be any point, and let (U, z) be a chart on X with x ∈ U and z(x) = 0, i.e., z : U → C is a holomorphic map identifying U with an open subset of C which maps x ˇ to 0. Let V = X \ {x}. Then U = (U, V ) is an open cover of X, and the Cech cohomology group H 1 (U , OX ) embeds into H 1 (X, OX ) (see the proof of Lemma 21.81). Thus by Theorem 26.23, H 1 (U , OX ) is finite-dimensional. For j ∈ N>0 , consider z −j ∈ Γ(U \ {x}, OX ). Since U \ {x} = U ∩ V , each z −j defines a class in H 1 (U , OX ), and for sufficiently large r, the classes of z 1 , . . . , z −r must be linearly dependent. This means that there exist a1 , . . . , ar ∈ C, not all zero, and f ∈ Γ(U, OX ), g ∈ Γ(V, OX ) with r X aj z −j = g − f. j=1

Pr

Then f + j=1 aj z = g ∈ Γ(V, OX ) extends to a non-constant meromorphic function X → P1 (C) (by mapping x to ∞ ∈ P1 (C)). −j

Corollary 26.26. Every compact Riemann surface X is algebraizable, i.e., is isomorphic as a complex manifold to the analytification C an of a smooth proper curve C over C. Proof. The previous proposition shows that there exists a non-constant morphism f : X → P1 (C) of Riemann surfaces. One concludes by showing that this is necessarily a finite morphism of complex manifolds, which implies that X can be recovered from the coherent OP1 (C) -algebra f∗ OX by an analogue of the relative Spec-construction, and using the GAGA principle for coherent sheaves over P1C in the form of Theorem 20.60, cf. [SGA1] O X Exp. XII, Corollaire 4.6. With the corollary proved, we can apply the various instances of the GAGA principle to the curve C. For a connected proper smooth curve C over C with analytification C an , the (derived) categories of coherent sheaves coincide (Theorem 20.60, Theorem 23.39), line bundles on C correspond to line bundles on C an , and similarly for locally free sheaves of finite rank, divisors, etc. Moreover, the analytification functor on the category of proper C-schemes is fully faithful (Corollary 20.61). As a consequence, later results of this chapter, such as the Theorem of Riemann and Roch, carry over to the context of Riemann surfaces (of course, historically, all these results were first proved for Riemann surfaces). In particular, any line bundle L of sufficiently high degree on a compact Riemann surface X defines an embedding of X into projective space. This fact can also be shown more directly, using analytic methods. One way to do so is to develop Hodge theory (and in particular prove the Hodge decomposition H 1 (X, C) = H 1 (X, OX ) ⊕ H 0 (X, Ω1X ), where Ω1X denotes the sheaf of holomorphic differentials on X). This gives a handle on proving the Kodaira vanishing

525 theorem, a special case being that H 1 (X, L ) = 0 for every (holomorphic) line bundle L on X which satisfies deg(L ⊗ Ω1,∨ X ) > 0. As a consequence, one shows that L with deg(L ) > deg(Ω1X ) + 2 defines a closed embedding X → P(H 0 (X, L )) with arguments analogous to those in the proof of Proposition 26.59 below. See [GrHa] O Ch. 2.1. In addition to complex-analytic methods, in the setting of Riemann surfaces it is natural and useful to study the underlying topological spaces. The following theorem states the classification of compact Riemann surfaces up to homeomorphism. Theorem 26.27. For every compact Riemann surface X there exists a unique natural number g, the genus of X such that X is homeomorphic to a “sphere with g handles”. The topological fundamental group of X is isomorphicQto the quotient of the free group on g −1 generators a1 , . . . , ag , b1 , . . . , bg modulo the relation i=1 ai bi a−1 = 1. The singular i bi 1 cohomology group H (X, Z) is a free Z-module of rank 2g. More concretely, a “sphere with g handles” is the topological space that can be constructed by attaching g “handles” to a sphere. A relatively convenient way to write this down is to construct the space by starting with a regular 2g-gon in the plane R2 and pasting, i.e., identifying some of its edges in a suitable way. See [FaKr] O I.2.5. Alternatively the genus g of a compact Riemann surface is often described as the “number of holes”, in the sense that the sphere has no holes, a torus has one hole, etc. Every Riemann surface of genus 0 is isomorphic to the Riemann sphere P1,an = P1 (C). C For every g > 0 there exist infinitely many non-isomorphic Riemann surfaces of genus g. See Section (26.19) for the case of genus 1. From the comparison theorem for singular cohomology and de Rham cohomology ([GrHa] O Ch. 0, Section 3), the Hodge decomposition and Serre duality, we obtain the following equivalent descriptions of the genus of a compact Riemann surface. The ones that use cohomology of coherent sheaves will allow us to define a useful notion of genus for proper algebraic curves over arbitrary fields. Theorem 26.28. Let X be a compact Riemann surface. Then 1 1 rkZ H 1 (X, Z) = dimC H 1 (X, C) = dimC H 1 (X, OX ) = dimC H 0 (X, Ω1X ). 2 2 We conclude this section by stating the uniformization theorem, another fundamental result in the theory of Riemann surfaces. Theorem 26.29. ˜ of X as a topological space (1) Let X be a Riemann surface. Then the universal cover X can be equipped, in a unique way, with the structure of Riemann surface such that the ˜ → X is a morphism of Riemann surfaces. projection X (2) Every simply connected Riemann surface is isomorphic to precisely one of the following: C, P1 (C), D = {z ∈ C; |z| < 1}. In the theorem, the open unit disk D carries the structure of Riemann surface inherited from the open embedding D ⊂ C. As Riemann surface, D is isomorphic to the complex upper half plane H = {z ∈ C; Im(z) > 0} (which we also view as an open submanifold of C) via the Cayley transform. Part (1) is easy to prove, and in Part (2) it is clear that the three given Riemann surfaces are simply connected and pairwise non-isomorphic. The hard part is to show that there are no other simply connected Riemann surfaces, up to isomorphism. An equivalent formulation is that every simply connected Riemann

526

26 Curves ∼

surface X that is not isomorphic to either C or P1 (C) admits an isomorphism X → D; this statement is also known as the Riemann Mapping Theorem. See [For] O §27. It is then not very hard to show that the only Riemann surface with universal cover P1 (C) is P1 (C), up to isomorphism, and that only C, C× and complex tori C/Λ, where Λ ⊂ C is a lattice (Definition 26.109), have universal cover C. See also Proposition 26.110. For the compact case, one gets the following connection between genus and universal cover. Theorem 26.30. universal cover. (1) We have g = 0 (2) We have g = 1 (3) We have g ≥ 2

˜ be its Let X be a compact Riemann surface of genus g, and let X if and only if if and only if if and only if

˜ X ˜ X ˜ X

∼ = P1 (C). ∼ = C. ∼ D, the open unit disk. =

The division of compact Riemann surfaces into these three classes according to their universal cover has many other manifestations, for instance it reflects the curvature of the Riemann surface in the sense of differential geometry. Also in algebraic geometry, this trichotomy is visible, sometimes in situations that are quite far from complex geometry, compare for instance Section (26.30). Transferring it to algebraic varieties of higher dimension is a first step towards a theory of classification (up to birational maps, say) in that case. Let us say some words about the proof of Theorem 26.30. From the discussion above we have Part (1), and also that a compact Riemann surface with universal cover C has genus 1. It then remains to prove that every compact Riemann surface of genus 1 has universal ˜ → X be cover C. One approach to this is the following: Let X have genus one and let f : X its universal cover. We use that for a canonical divisor K on X, we have ℓ(K) = g(X) = 1 and deg(K) = 2g(X) − 2 = 0, so that K is (linearly equivalent to) the trivial divisor, compare Corollary 26.52 below. Therefore every non-zero holomorphic differential vanishes ˜ from a nowhere on X. Fixing such a differential form ω, integrating f ∗ ω along paths in X ˜ ˜ ˜ fixed base point to any point on X defines a map X → C. Because X is simply connected, the value of the integral does not depend on the choice of path, but only on its end points, so that the map is well-defined. One can show that it is actually a covering map. Since C ˜∼ is simply connected, it follows that it is an isomorphism, so X = C, as we wanted to show. Alternatively, one can consider the curvature of X and its universal cover. Yet another possibility is to show first that X carries a group structure (cf. Theorem 26.98), and then to use the exponential map for the compact complex Lie group X and to show that it induces an isomorphism between X and a quotient of T0 X ∼ = C by a lattice. While it may look somewhat roundabout, this strategy has the advantage of working also in higher dimensions: Every connected compact complex Lie group (i.e., complex manifold which is a group object in the category of complex manifolds) is isomorphic as a complex manifold, to a complex torus Cdim(A) /Λ, for Λ a lattice in Cdim(A) , i.e., a subgroup generated by 2g elements that are R-linearly independent. In particular this applies to the analytification Aan of an abelian variety A over C (but in dimension > 1 most complex tori are not of the form Aan for an algebraic A). See [Mum1] Ch. I. (26.8) The arithmetic genus of a curve. We start by discussing the genus of a proper curve. See the discussion in Section (26.7) for the notion of genus in the theory of compact Riemann surfaces, in particular Theorem 26.27

527 which states that the genus classifies compact Riemann surfaces up to homeomorphism, and Theorem 26.28 which gives expressions for the genus that can immediately be applied to proper algebraic curves over an arbitrary field. For a smooth proper curve C over a field k, we have dimk H 0 (C, ΩC/k ) = dimk H 1 (C, OC ) by duality (Proposition 25.110). In the non-smooth case, it turns out that working with cohomology of the structure sheaf has better technical properties. (But see Definition 26.44.) The following definition is even more flexible (if only slightly). Definition 26.31. Let k be a field, and let C be a proper curve over k. The (arithmetic) genus of C is defined as g(C) = 1 − χk (OC ) = 1 − dimk H 0 (C, OC ) + dimk H 1 (C, OC ). Sometimes the arithmetic genus is denoted by pa (C) instead of g(C). (Compare Definition 26.44.) Note that the genus depends on the choice of base field and therefore should more precisely be denoted by g(C/k), and we will do so, if necessary. It seems, there is no generally accepted definition of the genus for an arbitrary proper curve. Our guiding principle is that the genus should have the following two properties. (1) For smooth connected proper curves over algebraically closed fields (for instance, for those complex algebraic curves corresponding to compact Riemann surfaces, see Section (26.7)) there is a generally accepted definition of genus, and our definition should agree with this in that case. (2) The genus should vary locally constantly in flat proper families of curves (see Section (26.26) below). This leads us to the definition above since the Euler characteristic does vary locally constantly in flat proper families (Theorem 23.139, see also Proposition 26.164). Here, we follow essentially Serre who even advocates (for general projective varieties, see [Ser1] O §80) to define the arithmetic genus simply as χk (OC ). But this would violate the first of our guiding principles. Note that our definition has the consequence that the genus can be negative (and so one has to be careful with arguments that conclude g(C) = 0 from g(C) ≤ 0). For instance, if X = P1k′ for a finite extension k ′ of k, which we can consider as a proper curve of k, then g(P1k′ /k) = 1 − [k ′ : k] (using that H 0 (P1k′ , OP1 ′ ) = k ′ and H 1 (P1k′ , OP1 ′ ) = 0). k k Moreover, our definition behaves somewhat unexpected if the curve is non-connected: for two curves C and C ′ with arithmetic genus g and g ′ , respectively, one has g(C ⨿ C ′ ) = 1 − χk (OC ) − χk (OC ′ ) = g + g ′ − 1. In case that Γ(C, OC ) = k (which holds, e.g., whenever C is proper, geometrically reduced, and geometrically connected, see Lemma 26.5; in particular this is true if C is proper, smooth and geometrically connected over k), one has g(C) = dimk H 1 (C, OC ) ≥ 0. The condition Γ(C, OC ) = k also implies that C is connected. Hence one could have defined the genus only for proper curves satisfying this additional property. But this has the unpleasant effect that there are curves for which the genus is defined but where the genus is not defined for its normalization (see Example 26.87 below).

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26 Curves

Below in Section (26.10), we will also briefly consider the notion of the geometric genus which depends only on a smooth proper model of the curve. The genus is compatible with extension of the base field, as the following lemma shows. Lemma 26.32. Let k be a field, and let C be a proper curve over k. For every extension field K of k, we have g(C/k) = g(C ⊗k K/K). Proof. This follows immediately from the invariance of the Euler characteristic under extension of the base field, see Lemma 23.64. For a non-smooth curve, we will replace the sheaf of differentials, which already played a role above, by the dualizing sheaf of the curve in the following sense. Definition 26.33. Let C be a curve over a field k and denote by f : C → Spec k the structure morphism. (1) Consider k as a coherent sheaf on Spec k, or rather as a complex of coherent sheaves • • concentrated in degree 0. We then write ωC := f ! k and call ωC the dualizing complex of C. • (2) Furthermore, we call ωC := H −1 (ωC ) the dualizing sheaf of C. • Since k (in degree 0) is a dualizing complex for k, Proposition 25.87 shows that ωC is indeed a dualizing complex, and hence ωC is a dualizing sheaf on C. If C is Cohen-Macaulay (which is true in particular, if H 0 (C, OC ) is a reduced ring see • Lemma 26.6), then ωC is concentrated in degree −1 and hence is represented by ωC [1]. See Proposition 25.111. If C is Gorenstein (and in particular Cohen-Macaulay), then ωC is a line bundle by Proposition 25.124. If C is smooth and proper over k, then we have ωC ∼ = Ω1C/k by Theorem 25.58. = Ω1C/k and hence ωC ∼ If a curve C admits a regular closed immersion i : C → X of codimension n into a • proper smooth k-scheme X, then Example 25.125 gives us ωC = ωC [1] with

(26.8.1)

ωC = i∗ Ωn+1 X/k ⊗OC

Vn

C∨ i .

In this case C is a local complete intersection over k and in particular Gorenstein. By Theorem 25.127 (5), we can express the genus in terms of global sections of the dualizing sheaf as follows: Corollary 26.34. Let C be a proper curve over a field k, and denote by ωC its dualizing sheaf as above. If H 0 (C, OC ) = k, then g(C) = dimk H 0 (C, ωC ). Let C be a reduced proper curve over a field k. We want to compare the (arithmetic) genus of a curve and the genus of (the connected components of) its normalization. Let f : C˜ → C denote the normalization morphism. It is an isomorphism over the normal locus Cnorm of C, and the complement C \ Cnorm consists of finitely many closed points of C. We obtain a short exact sequence 0 → OC → f∗ OC˜ → F → 0, where we define F as the cokernel of the homomorphism on the left. This is an OC -module supported on finitely many closed points, and in particular H 1 (C, F ) = 0, so that we obtain an exact sequence

529 M

˜ O ˜) → 0 → H 0 (C, OC ) → H 0 (C, C (26.8.2)

Fx

x∈C\Cnorm

˜ O ˜) → 0 → H 1 (C, OC ) → H 1 (C, C which will be the basis of our comparison. In fact, we get immediately that M Fx . (26.8.3) χ(OC˜ ) = χ(OC ) + χ(F ) = χ(OC ) + dimk x∈C\Cnorm

Proposition 26.35. Let k be a field and let C be a reduced proper curve with normalization f : C˜ → C. Then we have X ˜ + δx , (26.8.4) g(C) = g(C) x∈C 1

where for a closed point x of C we define δx = dimk (O˜C,x /OC,x ), where O˜C,x is the integral closure of OC,x in its total ring of fractions. In particular, for a closed point x ∈ C, we have δx = 0 if and only x lies in the normal locus of C. Hence the sum in (26.8.4) is finite. In Section (26.9) below we will study δx more closely. ˜ + P dimk (Fx ). FurtherProof. As remarked above, we have g(C) = 1 − χ(C) = 1 − χ(C) more, for each closed point x of C we have O˜C,x = (f∗ OC˜ )x because of the compatibility between taking integral closure and localization, and thus δx = dimk (Fx ). From this we obtain the desired formula. (26.9) Ordinary multiple points. A model for a particularly simple curve singularity is the point of intersection of the coordinate axes in the plane, i.e., the singularity of Spec k[T1 , T2 ]/(T1 T2 ) at the origin, or more generally, the point of intersection of the coordinate axes in m-dimensional affine space, that is the singularityQof Spec k[T1 , . . . , Tm ]/(Ti Tj ; i ̸= j) at the origin. The m normalization of this ring is Spec i=1 k[Ti ], the disjoint union of m copies of the affine line over k. Passing to the completion, we get k[[T1 , . . . , Tm ]]/(Ti Tj ; i ̸= j) as the complete local ring at the origin, and k[[T1 ]] × · · · × k[[Tm ]] as its normalization (in its total ring of fractions). We call curve singularities where the complete local ring is isomorphic to k[[T1 , . . . , Tm ]]/(Ti Tj ; i ̸= j) ordinary multiple points, more precisely: Definition 26.36. (1) Let k be an algebraically closed field and let C be a curve over k. A closed point x ∈ C is called an ordinary multiple point, if there exists an isomorphism ∧ ∼ OC,x = k[[T1 , . . . , Tm ]]/(Ti Tj ; i ̸= j)

of k-algebras. In this case m = dimk TC,x is called the multiplicity of x.

530

26 Curves

(2) Let k be a field, let k be an algebraic closure of k and let C be a curve over k. A closed point x ∈ C is called an ordinary multiple point of multiplicity m, if every point of the base change C ⊗k k lying over x is an ordinary multiple point of multiplicity m in the sense of (1). (3) An ordinary multiple point of multiplicity 2 of a curve over a field is also called an ordinary double point (or a node). (4) A curve over a field is called an (at worst) nodal curve, if every closed point is either normal or an ordinary double point. For a plane curve C, the multiplicity of an ordinary multiple point x on C is either 1 or 2, and it is easy to see that it coincides with the multiplicity in the sense of Exercise 6.11. See also [Ful1] O Section 3.2 for a more extensive discussion, but note that there the notion of ordinary multiple point is different form ours. Let C be a reduced curve over an algebraically closed field k. Let f : C˜ → C be the normalization morphism. For x ∈ C a closed point recall the number δx = dimk (f∗ OC˜ )x /OC,x we defined in Proposition 26.35. We denote by (26.9.1)

mx := #f −1 (x)

the number of points of the fiber f −1 (x). We always have (26.9.2)

mx − 1 ≤ δx .

This follows from the following remark in which we study these numbers and their connection in a slightly more general situation. Remark 26.37. Let f : Y → X be a finite birational morphism of noetherian schemes that are equi-dimensional of dimension 1. Then f∗ OY is a finite OX -algebra. We assume that OX → f∗ OY is injective which is automatic if X is reduced since f is birational. The OX -module f∗ OY /OX is zero over the open dense subset of X over which f is an isomorphism. Hence it has support in a finite closed subset T of X. Let x ∈ X be a closed point. Set
δx (f ) := lgOX,x (f∗ OY )x /OX,x . If X is a curve over the field k and x ∈ X(k), then one has δx (f ) = dimk (f∗ OY )x /OX,x . Let us explain why δx (f ) is a finite number. Set M := (f∗ OY )x /OX,x . If x ∈ / T , then M = 0. For x ∈ T , the support of M is {x}. Since we have Supp(M ) = V (Ann(M )) (as sets) for every finitely generated module M (Proposition 7.35) we see that rad(Ann(M )) = m, the maximal ideal of OX,x . Hence there exists r ≥ 1 such that mr M = 0. This shows in particular, that M is an OX,x -module of finite length. Let m be the maximal ideal of OX,x and denote by ( )∧ the m-adic completion. As (f∗ OY )x is a finite OX,x -algebra, Proposition B.41 and Proposition B.46 show that Y ∧ (26.9.3) (f∗ OY )x ⊗OX,x OX,x = (f∗ OY )∧ OˆY,y , x = y∈f −1 (x)

where OˆY,y denotes the my -adic completion of OY,y . As M is annihilated by some power of m, one has M = M ∧ and we obtain by (26.9.3), using that ( )∧ is an exact functor on finitely generated OX,x -modules,

531 (26.9.4)

M = (f∗ OY )x /OX,x ∼ =

Y


∧ , OˆY,y /OX,x

y∈f −1 (x) ∧ where OX,x is embedded diagonally into the product. The fiber f −1 (x) is a finite κ(x)-scheme, i.e., of the form Spec Bx for a finite κ(x)algebra Bx . As (f∗ OY )x ⊗OX,x κ(x) = Bx , we have M ⊗OX,x κ(x) = Bx /κ(x). This shows that

(26.9.5)

#f −1 (x) − 1 ≤ dimκ(x) Bx − 1 ≤ δx (f ).

The first inequality is an equality if and only if Bx is reduced and κ(y) = κ(x) for all y ∈ f −1 (x). The second inequality is an equality if and only if mM = 0. We return to the situation in which C is a curve over an algebraically closed field with normalization f : C˜ → C. One extreme case in (26.9.2) is that mx = 1, i.e., f is injective in a neighborhood of x. As f is surjective, f is then bijective in a neighborhood of x, i.e., C is unibranch in x (Corollary 20.52). Let y ∈ C˜ be the unique point with f (y) = x. Then C is geometrically unibranch if and only if κ(y) is a purely inseparable extension of κ(x), again by Corollary 20.52. Example 26.38. Let k be a field of characteristic different from 2 and 3. The curve C := V (Y 2 − X 3 ) ⊆ A2k is smooth outside the origin x = (0, 0). Its normalization is given 2 3 2 3 by k[X, Y ]/(Y 2 − X 3 ) → k[T ] with P X 7→ nT , Y 7→ T . This identifies k[X, Y ]/(Y − X ) 2 3 with the subring k[T , T ] = { n an T ∈ k[T ] ; a1 = 0 } of k[T ]. The origin is a singularity and one computes that with the above notation mx = 1 and δx = 1. Moreover, the residue extension of the normalization map in the origin is trivial. This shows that C is geometrically unibranch. The other extreme case is that δx = mx − 1. Over an algebraically closed field this equality holds if and only if x is an ordinary multiple point: Proposition 26.39. Let C be a reduced curve over an algebraically closed field k, let f : C˜ → C be its normalization and let x ∈ C be a closed point. Then δx ≥ mx − 1 and the following are equivalent. (i) The point x is an ordinary multiple point. (ii) We have δx = mx − 1. (iii) Writing m = mx , f −1 (x) = {q1 , . . . , qm } and denoting by ∆ ⊆ k m the diagonal and by η : (f∗ OC˜ )x → k m , s 7→ (s(q1 ), . . . , s(qm )), the evaluation map, we have OC,x = {s ∈ (f∗ OC˜ )x ; η(s) ∈ ∆} =: A. There is a similar criterion (with more complicated notation) over non-algebraically closed fields taking into account the possibility that the normalization may have non-trivial residue field extensions in this case. We do not make this explicit. Proof. Let f : C˜ → C denote the normalization morphism. ∧ ∼ (i) ⇒ (ii). By assumption, we have an isomorphism OC,x = k[[T1 , . . . , Tn ]]/(Ti Tj ; i ̸= j) Qn ∧ ∧ ∼ and f∗ (OC˜ )x = i=1 k[[Ti ]] by (26.9.3). The canonical map OC,x → f∗ (OC˜ )∧ x identifies Qn k[[T1 , . . . , Tn ]]/(Ti Tj ; i ̸= j) with the subring of i=1 k[[Ti ]] consisting of tuples of power series (f1 , . . . , fm ) with f1 (0) = · · · = fm (0). Hence mx = n and by (26.9.4)

532

26 Curves δx = dimk (

n Y

k[[Ti ]]/(k[[T1 , . . . , Tn ]]/(Ti Tj ; i ̸= j)) = n − 1.

i=1

(ii) ⇒ (iii). As δx = mx − 1 we have by (26.9.5) and the following remark that (f∗ OC˜ )x ⊗OC,x κ(x) = k m and we can view η as the map (f∗ OC˜ )x → (f∗ OC˜ )x ⊗OC,x κ(x). Hence η induces an isomorphism (f∗ OC˜ )x /B ∼ = k m /∆ ∼ = k m−1 . By assumption, we obtain dimk (f∗ OC˜ )x /A = δx = dimk (f∗ OC˜ )x /OC,x . Since OC,x is contained in A, this implies the equality. (iii) ⇒ (i). As C˜ is regular over the perfect field k, it is smooth over k (Proposition 18.67) ∧ ∼ ˜ and we have OˆC,y ˜ = k[[T ]] for every closed point y ∈ C (Theorem 6.28). Denoting by ( ) the mx -adic completion, (26.9.3) gives ∼ (f∗ OC˜ )∧ x =

m Y

∼ OC,q ˜ i =

i=1

m Y

k[[Ti ]].

i=1

Now assumption (iii) says that we have a short exact sequence 0 → OC,x → (f∗ OC˜ )x → k m /∆ → 0. ∧ ∼ Qm k[[Ti ]] is the subring of tuples We see that the image of OC,x ⊆ (f∗ OC˜ )∧ x = i=1 (f1 , . . . , fm ) of power series with f1 (0) = · · · = fm (0), i.e., of power series with identical absolute terms. As remarked in the beginning of the proof, this subring can be identified with k[[T1 , . . . , Tm ]]/(Ti Tj ; i ̸= j), giving us the desired isomorphism. Example 26.40. The curve V (Y 2 − X 2 (X − 1)) ⊆ A2k over a field k of characteristic ̸= 2 has a node at the origin and is smooth at all other points. In fact, the normalization of k[X, Y ]/(Y 2 − X 2 (X − 1)) is given by k[X, Y ]/(Y 2 − X 2 (X − 1)) → k[T ],

X 7→ T 2 + 1, Y 7→ T (T 2 + 1),

and one has δx = 1. The fiber over the origin (which corresponds to the prime ideal (X, Y )) is the spectrum of k[T ] ⊗k[X,Y ]/(Y 2 −X 2 (X−1)) k ∼ = k[T ]/(T 2 + 1, T (T 2 + 1)) ∼ = k2 , so it consists of two points. Lemma 26.41. Let k be a field and let C be a curve over k that has at most nodal singularities. Then C is a locally complete intersection, and in particular Gorenstein (Definition G.26). Proof. In fact, we can check that the local rings of C are complete intersection rings (Definition 19.46) after base change to an algebraic closure of C (Remark 19.47 (4)), and after passing to complete local rings (Remark 19.47 (2)). But the ring k[[T1 , T2 ]]/(T1 T2 ) is a locally complete intersection. For the final part, use Remark 19.47 (3). Example 26.42. Let k be a field. The curve C = V (XY, Y Z, XZ), that is the union of the coordinate axes in A3k , is not Gorenstein. In fact, using [Mat2] Theorem 18.11, it is enough to exhibit a reducible parameter ideal in the ring A := k[[X, Y, Z]]/(XY, Y Z, XZ), and (X +Y +Z) = (X, X +Y +Z)∩(Y, X +Y +Z) is such an ideal.

533 Alternatively, we can show that Ext1A (k, A) ̸= k as follows. Using a Koszul complex argument, we have Ext1A (k, A) ∼ = HomB (k, B) for B = A/(X + Y + Z) (see the proof of the theorem in [Mat2] mentioned above), and B ∼ = k[[X, Z]]/(XY, X 2 , Y 2 ), so that dimk HomB (k, B) = 2. (The linear maps with 1 7→ X and 1 7→ Y are a basis of this Hom space.) A similar argument shows that an ordinary multiple point of order > 2 is not Gorenstein. (26.10) The geometric genus. For a smooth proper curve C over a field k with H 0 (C, OC ) = k, we have g(C) = 1 − χ(OC ) = dim H 1 (C, OC ) = dimk H 0 (C, Ω1C/k ), as we have discussed above. Clearly for higher-dimensional varieties X over k, say n = dim X, the quantities (−1)n (χ(OX ) − 1) and dimk H 0 (C, ΩnC/k ) will in general be different. To distinguish these two numbers, both of which are interesting invariants of X, one calls (−1)n (χ(OX ) − 1) the arithmetic genus of X and dimk H 0 (X, ΩnX/k ) the geometric genus of X. Returning to the case of curves, if we drop the smoothness assumption, then we can define the arithmetic genus as before, but for the geometric genus it is preferable to modify the definition because the sheaf of differentials on a singular curve is less well behaved. We thus define the geometric genus as the genus of a smooth proper curve birational to X (if necessary, after an extension of the base field). The reason why we might have to extend the base field is the following. While we can always find a proper normal curve birational to the given one, over non-perfect base fields, a normal curve need not be smooth. In order to handle this case appropriately, we use the following lemma. Recall that every reduced curve C over a field k is birationally e which is unique up to isomorphism and which is equivalent to a normal proper curve C, called the normal proper model of the given curve (Definition 26.2). Lemma 26.43. Let k be a field and let C be a curve over k. (1) There exists a finite, purely inseparable field extension k ′ /k such that the normalization of Ck′ and its normal proper model (Ck′ )∼ are smooth over k ′ . (2) If k ′ /k is a field extension such that (Ck′ )∼ is smooth over k ′ , and k ′′ /k ′ is a further field extension, then (Ck′′ )∼ = (Ck′ )∼ ⊗k′ k ′′ , and in particular this is a smooth k ′′ -scheme. Proof. (1) If k ′′ is a perfect field, then every regular k ′′ -scheme is smooth (Proposition 18.67), and hence (Ck′′ )∼ is smooth over k ′′ , since every normal curve is regular. Thus the perfect closure of k in some algebraic closure gives a purely inseparable extension k ′′ of k with (Ck′′ )∼ being smooth over k ′′ . Now the claim follows from a standard limit argument: The algebraic extension k ′′ is the filtered union of its finite subextensions k ′ which are all purely inseparable extensions of k. By Theorem 10.66 and Theorem 10.63 we can find a finite purely inseparable extension k ′ , a curve C1 , and a morphism of k ′ -schemes π : C1 → Ck′ whose base change to k ′′ is the normalization morphism (Ck′′ )∼ → Ck′′ which is by definition a birational map (Ck′′ )∼ → Ck′′ ,red . After possibly enlarging k ′ , one can assume that C1 is smooth over k ′ (Proposition 18.59) and that there exists an open dense subset V ⊆ Ck′ such that π −1 (V ) → V is a closed surjective immersion (Proposition 10.75). This implies that π is the normalization morphism and hence that the normalization of Ck′ is smooth over k ′ . Hence its base change to any field extension K is also smooth over K and in particular normal. Hence a similar approximation argument

534

26 Curves

shows that, after possibly passing to some further finite purely inseparable extension of k ′ , also its proper normal model is smooth. (2) We have ((C ∼ ) ⊗k k ′ )∼ = (C ⊗k k ′ )∼ , both sides locally being given by the integral closure of rings Γ(U, OC ) ⊗k k ′ in the same product of fields. If (Ck′ )∼ is smooth, then so is (Ck′ )∼ ⊗k′ k ′′ , and the claim follows. Definition 26.44. Let k be a field and let C be an irreducible proper curve over k. The geometric genus of C is defined as ˜ Ω1˜ ′ ), pg (C) = dimk′ H 0 (C, C/k where C˜ is a smooth proper model of C (as in Lemma 26.43), defined over a suitable finite purely inseparable extension k ′ of k. The definition of pg (C) is independent of the choice of k ′ by Lemma 26.43 (2). If k ′ is a purely inseparable extension k, then Spec k ′ → Spec k is a universal homeomorphism. Therefore, as C is irreducible, Ck′ and its smooth projective model are again irreducible. Remark 26.45. Let k be a field, and let X be an n-dimensional smooth proper k-scheme with H 0 (X, OX ) = k. Both the geometric genus pg (X) := dimk H 0 (X, ΩnX ) and the arithmetic genus pa (X) := (−1)n (χ(OX ) − 1) are birational invariants in the following sense: If Y has the same properties as required above for X, and f : X 99K Y is a birational map, then pa (X) = pa (Y ) and pg (X) = pg (Y ). (It is essential that X and Y are both smooth, as already the case of curves shows.) For the geometric genus, it is not too hard to show this, using that the domain of definition of the rational map X 99K Y has complement of codimension ≥ 2 and a version of Hartogs’s principle which allows to extend sections (of ΩnX/k ) from an open whose complement has codimension ≥ 1 to all of X. Here it is preferable to work with the space of global sections of a sheaf rather than to consider H n (X, OX ), even if by Serre duality both have the same dimension. In fact, one can show that even dimk H i (X, OX ) is a birational invariant for each i, which implies that both pa (X) and pg (X) are birational invariants. This result is more difficult. While in characteristic 0 there are several methods to prove this and the statement is a “classical fact”, in positive characteristic it is a relatively recent result by Chatzistamatiou and R¨ ulling [ChR¨ u] O X.

The Theorem of Riemann-Roch We now come to the Theorem of Riemann-Roch, a cornerstone of the theory of curves (which also has a counterpart in complex geometry, specifically for Riemann surfaces) and at the same time the foundation of the far-reaching generalizations for higher-dimensional varieties developed by Hirzebruch and by Grothendieck, see Section (23.23), in particular Example 23.115. In the following sections, we see several applications of the theorem. (26.11) The Theorem of Riemann-Roch. We start with the following preliminary form of the Riemann-Roch theorem, cf. Proposition 15.33.

535 Proposition 26.46. Let C be a proper curve over a field k and let L be a line bundle on C. Then we have (26.11.1)

deg(L ) = χ(L ) − χ(OC ) = χ(L ) − 1 + g,

where g is the genus of C. Proof. The second equality holds by definition of the genus. Let us show the first equality. The assertion is clear for L = OC . By Corollary 15.26, every Cartier divisor on C can be written as a difference of effective Cartier divisors. It is then enough to show that for any line bundle L on C and any effective Cartier divisor D, the statement holds for L if and only if it holds for L ⊗ OC (−D). To show this, consider the short exact sequence 0 → OC (−D) → OC → OD → 0. Since the support of OD consists of just finitely many closed points, we have L ⊗OC OD ∼ = OD , so by tensoring by L we obtain a short exact sequence 0 → L ⊗OC OC (−D) → L → OD → 0. Passing to the associated long exact cohomology sequence and using that deg(D) = dim Γ(C, OD ), we get that χ(L ) = χ(L ⊗OC OC (−D)) + deg(D), as desired. Compare Example 23.75 (1). In view of Definition 23.80, we get the following result. Proposition 26.47. Consider a proper curve C over a field k and a very ample divisor D on C. Let ι : C → PN k be the closed embedding corresponding to D (and a choice of basis of H 0 (C, OC (D))). Then the degree of C as a subvariety of PN k equals the degree of the divisor D. Combining Proposition 26.46 with Grothendieck-Serre duality (in the form of Proposition 25.110) we obtain the Riemann-Roch Theorem. Theorem 26.48. (Theorem of Riemann-Roch) Let k be a field and let C be a proper curve over k. Let ωC denote its dualizing sheaf. For every line bundle L on C we have dimk H 0 (C, L ) − dimk H 0 (C, L ∨ ⊗ ωC ) = deg(L ) + 1 − g(C). Proof. By definition we have χ(OC ) = 1 − g(C), so it only remains to show that the left hand side of the equation equals χ(L ), i.e., to showing dimk H 0 (C, L ∨ ⊗ ωC ) = dimk H 1 (C, L ). But that is a direct consequence of Proposition 25.110 using that HomOC (L , ωC ) = HomOC (OC , ωC ⊗ L ∨ ) = H 0 (C, ωC ⊗ L ∨ ). In the Riemann-Roch formula, the dimension of H 0 (C, L ∨ ⊗ ωC ) is usually thought of as a correction term (which vanishes for instance if the degree of L is sufficiently large, see the corollaries below). Replacing Proposition 26.46 by more general versions (e.g., cf. Section (23.23) for smooth curves) and using Serre duality to express the Euler characteristic, one obtains generalizations of Theorem 26.48. See Theorem 26.138 below for a Riemann-Roch version for vector bundles on curves. Under further assumptions, the situation becomes easier to handle (see also the discussion in Section (26.8)):

536

26 Curves

• If the curve C is Gorenstein, then the dualizing sheaf is a line bundle, and hence corresponds to a linear equivalence class of Cartier divisors. Any divisor in this class is called a canonical divisor , and often one (implicitly) fixes one such divisor KC and calls it the canonical divisor on C. • If C is even normal (equivalently, regular and hence locally factorial), then the notions of Cartier divisor and of Weil divisor coincide. • If C is smooth, then it is normal (and over a perfect field the two notions coincide). In this case, the dualizing sheaf is the sheaf Ω1C/k of K¨ ahler differentials on C and hence is very explicit. For Gorenstein curves, we can rewrite the Riemann-Roch theorem in terms of divisors. To do so, we make the following definition. Definition 26.49. Let C be a proper curve over a field k, and let D be a divisor on C. Then we write ℓ(D) := dimk H 0 (C, OC (D)), where OC (D) is the line bundle on C attached to the divisor D. Corollary 26.50. Let C be a proper Gorenstein curve over a field k of genus g = g(C), and let K be a canonical divisor on C. Then for every divisor D on C we have ℓ(D) − ℓ(K − D) = deg(D) + 1 − g. Remark 26.51. Historically, as the name suggests, the theorem goes back to Riemann and his student Roch. Riemann proved what is sometimes called Riemann’s inequality, i.e., using the notation of the previous corollary, ℓ(D) ≥ deg(D)+1−g. Roch proved the formula with the appropriate correction term. They worked in the setting of compact Riemann surfaces and correspondingly used analytic methods. By now, there are far-reaching generalizations, in particular the Hirzebruch-Riemann-Roch and Grothendieck-RiemannRoch theorems. Those theorems work for higher-dimensional varieties, as well, and express the Euler characteristic of coherent sheaves (or even objects in the derived category), or even more generally describe the behavior of the Chern character under push-forward along a proper map. So they may be seen as generalizations of Proposition 26.46. See Section (23.23). Hirzebruch worked in the setting of complex geometry; he used cobordism to prove the theorem. A different approach is to prove it using the Atiyah-Singer index theorem. Grothendieck, on the other hand, worked in the setting of algebraic geometry. For other proofs of Riemann-Roch in the case of curves, see [Ser6], [Kem] O , [ACGH] O App. A. Corollary 26.52. Let C be a proper Gorenstein curve over a field k of genus g = g(C), and let K be a canonical divisor on C. Then deg(K) = 2g − 2

and

ℓ(K) = g − 1 + ℓ(0).

In particular, if H 0 (C, OC ) = k, then ℓ(K) = g. Proof. Using the Riemann-Roch formula for the trivial divisor and for the divisor K, we have ℓ(0) − ℓ(K) = 1 − g, ℓ(K) − ℓ(0) = deg(K) + 1 − g which implies deg(K) = 2g − 2 and ℓ(K) = g − 1 + ℓ(0).

537 Corollary 26.53. Let C be a integral normal proper curve over a field k. If g(C) = 0 and C(k) ̸= ∅, then C ∼ = P1k . Proof. Since C is integral and proper over k, k ′ := H 0 (C, OC ) is a finite field extension. Saying that g(C) = 0 is equivalent to χ(OC ) = 1. But the cohomology groups occurring here are k ′ -vector spaces, so χ(OC ) is a multiple of [k ′ : k]. It follows that H 0 (C, OC ) = k. Let K be a canonical divisor on C. Let x ∈ C be a closed point with residue class field k (which exists by the assumption C(k) ̸= ∅). Then deg(K − [x]) = −3 (Corollary 26.52), so ℓ(K − [x]) = 0 by Remark 26.21 and hence ℓ([x]) = 2 by the Theorem of Riemann-Roch (Corollary 26.50). Thus the line bundle OC ([x]) defines a non-constant rational function f : C 99K P(H 0 (C, OC ([x]))) ∼ = P1k which extends to f since C is normal (Proposition 15.5). Moreover, deg(f ) deg(OP1k (1)) = deg(f ∗ OP1k (1)) = deg([x]) = 1 by Proposition 26.19, hence deg(f ) = 1. By Corollary 26.12, f is an isomorphism. The Brauer-Severi curve attached to a quaternionic skew field (Section (8.11)) is an example of a smooth, proper, geometrically connected curve of genus 0 that has no k-rational point. See Section (26.16) below for further, similar characterizations of the projective line. (26.12) Divisors on curves, continued. Let us collect some further applications of the Theorem of Riemann-Roch to divisors on curves. We start with the following immediate description of line bundles of degree 0. Example 26.54. Let C be a connected proper normal curve over an algebraically closed field Pn k. Then every divisor D of degree 0 is linearly equivalent to a divisor of the form i=1 ([xi ] − [x0 ]) for some n ≥ 0 and some points x0 , x1 , . . . , xn ∈ C(k). Indeed, as C is normal, Cartier divisors are simply formal linear combinations of closed points which are all k-rational since k is algebraically closed. We can choose any x0 ∈ C(k). By the Theorem of Riemann-Roch, we have dim H 0 (C, OC (D + n[x0 ])) ≥ deg(D) + n + 1 − g(C) = n + 1 − g(C) which is > 0 for n ≥ g(C). Since C is connected, Pn D + n[x0 ] is linearly equivalent to an effective divisor, i.e., a divisor of the form i=1 [xi ]. Note that we can take n = g(C). The Riemann-Roch theorem also allows us estimate how the global sections grow if one adds closed points. Let C be a curve over a field. Then we denote by Creg the open subset of points x ∈ C such that OC,x is normal (or, equivalently, regular). If C is generically reduced, this is an open dense subset of C. Proposition 26.55. Let k be a field, let C be proper curve over k and let D be a divisor on C. (1) ℓ(D) ≤ ℓ(D + [x]) ≤ ℓ(D) + [κ(x) : k] for x ∈ Creg closed. (2) Suppose that C is integral Gorenstein with H 0 (C, OC ) = 1. Let g be the genus of C. If deg D ≥ 2g − 2, then H 1 (C, OC (D)) = 0 and ℓ(D) = deg(D) + 1 − g. Proof. For Part (1) note that the Weil divisor [x] is a (Cartier) divisor since x is regular. Then (1) follows by considering the short exact sequence

538

26 Curves 0 → OC (D) → OC (D + [x]) → κ(x) → 0

and the associated long exact cohomology sequence which shows that
ℓ(D + [x]) − ℓ(D) = dimk Ker [κ(x)] → H 1 (C, OC (D)) . Let us show (2). If deg(D) > 2g − 2, then deg(K − D) = 2g − 2 − deg(D) < 0 in view of Corollary 26.52, and hence ℓ(K − D) = 0 (Remark 26.21). Thus the Riemann-Roch formula gives the statement in part (2). Remark 26.56. If C is a proper reduced connected curve over an algebraically closed field k, then H 0 (C, OC ) = k (Lemma 26.5). If C is Gorenstein (e.g., if C is normal), integral, and proper of genus g, then we can apply Proposition 26.55, where we have κ(x) = k since k is algebraically closed. (1) For g > 0, we see that if we have a chain of divisors 0 = D0 < D1 < D2 < · · · < D2g−2 < D2g−1 < . . . with Di = Di−1 + [xi ] for closed points xi ∈ Creg , then ℓ(Di ) ∈ {ℓ(Di−1 ), ℓ(Di−1 ) + 1} for i = 1, . . . , 2g − 1 and #{ 1 ≤ i ≤ 2g − 1 ; ℓ(Di ) = ℓ(Di−1 ) } = g − 1. For i ≥ 2g − 1 one has ℓ(D) = i + 1 − g. (2) If C is normal and g = 0, then C ∼ = P1k by Corollary 26.53 and ℓ(D) = max(0, deg D) for any divisor D on C. For curves, the Nakai-Moishezon criterion (Theorem 23.90) takes the following form, and we give a direct proof here. Proposition 26.57. Let k be a field, let C be a proper curve over k, and let L be a line bundle on C. Then L is an ample line bundle if and only if deg(L |Ci ) > 0 for every irreducible component Ci of C. Proof. Since we can check ampleness of a line bundle on C after restricting it to each of its irreducible components (Corollary 23.9), we may assume that C is integral. Let g be the genus of C. Suppose that L is ample. Then L ⊗n is very ample for large n and in particular generated by its global sections. If deg(L ) < 0, then H 0 (C, L ⊗n ) = 0 for all n ≥ 1 by Remark 26.21 (2), a contradiction. If deg(L ) = 0, then L ⊗n ∼ = OC for large n by Remark 26.21 (3). Hence C would be quasi-affine and thus finite over k since C is proper over k (Corollary 13.82). So we must have deg(L ) > 0. Conversely, suppose that deg(L ) > 0. To show that L is ample, we may always replace L by some positive power (Proposition 13.50). We will use Serre’s ampleness criterion Lemma 23.5 to show that L is ample, i.e., we will show that for every coherent OC -module F we have H 1 (C, F ⊗ L ⊗n ) = 0 for large n. By Theorem 26.16 there exists an ample line bundle M on C. By Riemann-Roch (Theorem 26.48) we have dim H 0 (C, L ⊗m ⊗ M ⊗−1 ) ≥ m deg(L ) − deg(M ) + 1 − g for all m and hence H 0 (C, L ⊗m ⊗ M ⊗−1 ) = HomOC (M , L ⊗m ) ̸= 0, Replacing L by L ⊗m we have for all n ≥ 1 an exact sequence

for m ≫ 0.

539 0 −→ M ⊗n −→ L ⊗n −→ Gn −→ 0 for some skyscraper sheaf Gn . Tensoring with a coherent OC -module F we obtain an exact sequence u F ⊗ M ⊗n −→ F ⊗ L ⊗n −→ F ⊗ Gn −→ 0. As C is a curve, H 2 (C, Ker(u)) = 0 and hence H 1 (C, F ⊗ M ⊗n ) → H 1 (C, Im(u)) is surjective. As Gn has support of dimension 0 so has F ⊗ Gn , hence H 1 (C, F ⊗ Gn ) = 0 and therefore H 1 (C, Im(u)) → H 1 (C, F ⊗ L ⊗n ) is surjective. We deduce that H 1 (C, F ⊗ M ⊗n ) → H 1 (C, F ⊗ L ⊗n ) is surjective. Now we can apply Serre’s ampleness criterion to see that the ampleness of M implies the ampleness of L . We also have the following sufficient criteria for line bundles to be globally generated or very ample. Proposition 26.58. Let k be an algebraically closed field. Let C be a proper normal curve over k. Let D be a divisor on C, and let L = OC (D) be the corresponding line bundle. (1) The following are equivalent: (i) The line bundle L is generated by global sections. (ii) For every closed point x ∈ C, the inclusion Γ(C, O(D − [x])) ⊆ Γ(C, O(D)) is a proper inclusion. (iii) For every closed point x ∈ C, dimk Γ(C, O(D − [x])) = dimk Γ(C, O(D)) − 1. (2) Then the following are equivalent: (i) The line bundle L is very ample. (ii) For all closed points x, y ∈ C (not necessarily different) the inclusions Γ(C, O(D − [x] − [y])) ⊆ Γ(C, O(D − [x])) ⊆ Γ(C, O(D)) are proper inclusions. (iii) For all closed points x, y ∈ C (not necessarily different) dimk Γ(C, O(D − [x] − [y])) = dimk Γ(C, O(D)) − 2. Proof. Part (1). Given a closed point x ∈ C, consider the short exact sequence (*)

0 → L ⊗ O(−[x]) → L → κ(x) → 0.

The line bundle L is generated by global sections if and only if Γ(C, L ) → κ(x) = k is surjective for all x. Passing to global sections in (*) we see that the surjectivity of the above map in turn is equivalent to the condition that the map Γ(C, O(D − [x])) → Γ(C, O(D)) is not surjective, i.e., that (ii) holds. Furthermore, Proposition 26.55 (1) shows that the difference of the dimensions of the two spaces is at most 1, whence (ii) and (iii) are equivalent.

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Part (2). Again Proposition 26.55 (1) shows that (ii) and (iii) are equivalent. It remains to show the equivalence of (i) and (ii). In view of part (1) either assumption implies that L is generated by its global sections and therefore defines a morphism ι : C → PN k for N = dimk Γ(C, L ) − 1. We rephrase the condition that ι is a closed immersion by Theorem 12.94 which implies that this is equivalent to ι being injective on closed points, and inducing injective homomorphisms on the tangent spaces to closed points of C. Now consider closed points x = ̸ y of C. The condition ι(x) ̸= ι(y) means that for some s ∈ Γ(C, L ), s(x) = 0 (in the fiber L (x) of L at x) while s(y) ̸= 0. But these conditions on s can be rephrased as s ∈ Γ(C, L ⊗ O(−[x])) \ Γ(C, L ⊗ O(−[x] − [y])). (Here we view L = OC (D) as a subbundle of KC , compare the definition in Section (11.12), and similarly for the other divisors appearing here.) This gives the equivalence of injectivity on closed points and condition (ii) for all x ̸= y. Now fix a closed point x of C and consider the map Tx C → Tι(x) PN k induced by ι on the tangent spaces. It is injective if and only if its dual homomorphism mPN /m2PN ,ι(x) → k ,ι(x) k 2 ∼ O N mC,x /mC,x is surjective. Choosing an isomorphism O(1)ι(x) = Pk ,ι(x) we obtain an isomorphism Lx ∼ = OC,x and a commutative diagram m PN /m2PN ,ι(x) k ,ι(x) O k

/ mC,x /m2 O C,x

∼ =

{s ∈ Γ(PN , O(1)); s(ι(x)) = 0}

/ {s ∈ Γ(C, L ); s(x) = 0}.

In this diagram, the left vertical map is an isomorphism and the lower horizontal map is onto. We see that the upper horizontal map is onto (which is what we are interested in) if and only if the right vertical map is onto. Since Tx C is one-dimensional, C being smooth over k (Section (26.3)), the map is onto if and only if it is non-zero, i.e., if there exists s ∈ Γ(C, L ) with s ∈ mC,x L but s ̸∈ m2C,x L . Again viewing the line bundles of divisors as subbundles of KC , this condition translates to the existence of an element s ∈ Γ(C, L ⊗ O(−[x])) \ Γ(C, L ⊗ O(−2[x])). This gives the equivalence of (i) and (ii). Proposition 26.59. Let k be a field. Let f : C → Spec k be a proper geometrically connected smooth curve of genus g. Let L be a line bundle on C. (1) If deg(L ) ≥ 2g, then L is generated by its global sections. (2) If deg(L ) ≥ 2g + 1, then L is very ample. Proof. The line bundle L is generated by its global sections if and only if the canonical homomorphism f ∗ f∗ L → L is surjective (Proposition 13.30). This can be done after base change to some field extension of k. Similarly, L is very ample if and only if it is very ample after base change to some field extension (Proposition 14.58). Moreover, neither the genus of the curve nor the degree of L change by an extension of the base field. Hence we may assume that k is algebraically closed. Let us show (1). Say L ∼ = O(D) for a divisor D on C. By Proposition 26.58 (1) it is enough to show that ℓ(D − [x]) = ℓ(D) − 1 for every closed point x ∈ C. By assumption deg(D) and deg(D − [x]) = deg(D) − 1 are both larger than 2g − 2, so that we have ℓ(D − [x]) = deg(D − [x]) + 1 − g(C) = deg(D) − 1 + 1 − g(C) = ℓ(D) − 1

541 by Remark 26.55. The same argument, now using Proposition 26.58 (2), shows (2) if L is a line bundle of degree at least 2g + 1. Remark 26.60. Consider the situation of Proposition 26.59. (1) For g ≥ 2 the converse to (1) or (2) does not hold in general (Exercise 26.9). (2) If g = 0, then L is ample if and only if it is very ample. Indeed by faithfully flat descent (Proposition 14.58) we may assume that k is algebraically closed. Then C∼ = P1k by Corollary 26.53 and the claim follows since a line bundle on a projective space Pnk is ample if and only if it is very ample (if and only if it is isomorphic to OPnk (d) for some d > 0), see Example 13.45 and Remark 13.58, also cf. Section (26.16). (3) If g = 1, then L is very ample if and only if deg(L ) ≥ 3. Indeed, if there existed a very ample line bundle of degree 1 or 2, then this line bundle would yield a closed immersion into P0k or P1k . The first case is obviously absurd and so is the second case because every closed immersion between integral schemes of the same finite dimension is necessarily an isomorphism (Lemma 5.7 (1)). Compare Theorem 27.279 which generalizes this result to abelian varieties. Proposition 26.61. Let C be a separated irreducible curve over a field. Then C is projective over k or C is affine. Proof. We start by some reduction steps based on Chevalley’s criterion that for a finite surjective morphism X → Y between noetherian schemes, Y is affine, if (and only if) X is affine, Theorem 12.39, and the analogous fact for properness, Proposition 12.59. Moreover, it suffices to show the claim after base change to some field extension (Proposition 14.53 and Proposition 14.57). After passing to a purely inseparable perfect field extension, which leaves C irreducible, passing to Cred , and passing to the normalization we may assume that C and its normal proper model C˜ are smooth over k, cf. Lemma 26.43. We know already that C is quasiprojective (Theorem 26.16). Suppose that C is not projective (equivalently, not proper). Let k¯ be an algebraic closure of k and let k ′ be a finite subextension of k¯ such that every connected component of Ck¯ is already defined over k ′ . If one of the connected components C1 of Ck′ was proper over k ′ , then C1 would be proper over k. As Ck′ → C is finite free and in particular open and closed, its restriction to C1 yields a surjective morphism C1 → C of k-schemes. Hence C would be proper. Therefore no connected component of Ck′ is proper and it suffices to show that every such connected component is affine. Hence we may assume that C is even geometrically connected. If C is not projective, then C˜ \ C =: {x1 , . . . , xr } is a non-empty set of closed points ˜ For n sufficiently large, then n([x1 ] + · · · + [xr ]) is a very ample divisor on C˜ by of C. Proposition 26.59. We obtain an embedding of C˜ into projective space, such that C˜ \ C ˜ (topologically) equals the intersection of C˜ with a hyperplane. Its complement in C, namely C, is therefore closed in an affine space, hence affine. (26.13) The formula of Riemann and Hurwitz. Let k be a field, let X and Y be connected smooth curves over k, and let f : X → Y be a generically ´etale morphism. Then f is non-constant and hence quasi-finite and flat by Lemma 26.9. If X is proper over k, then f is finite locally free.

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We have a short exact sequence (26.13.1)

0 → f ∗ ΩY /k → ΩX/k → ΩX/Y → 0,

see Proposition 17.31. For the injectivity on the left, we use that the domain and the target of the map are line bundles on X and that the given homomorphism is non-zero because f is generically ´etale. (We have already proved that the sequence is exact if f is assumed to be smooth, Corollary 18.72; however, this case is not interesting from the point of view of this section. In fact, if f is smooth, then it is necessarily ´etale and ΩX/Y = 0.) We obtain a global section s ∈ Γ(X, ΩX/k ⊗ (f ∗ ΩY /k )∨ ) and call the associated effective divisor on X the ramification divisor of f . Denoting this divisor by R, we have the explicit description X lgOX,x (ΩX/Y,x )[x] R= x∈X 1 1

of R as a Weil divisor, where X denotes the set of closed points of the curve X. Using Theorem 18.74, we see that the support of R consists of the points of X where f is not ´etale. Since f is flat, being ´etale at x is equivalent to being unramified at x. This justifies the term ramification divisor. We arrive at the following definition. Definition 26.62. Let k be a field and let f : X → Y be a generically ´etale morphism of smooth curves over k. The ramification divisor of f is the divisor X lgOX,x (ΩX/Y,x ) · [x] R= x∈X 1

on X. From the above short exact sequence we see immediately how the ramification divisor is related to the canonical divisors on smooth proper curves X and Y . Proposition 26.63. Let k be a field and let f : X → Y be a generically ´etale morphism of smooth proper curves over k. Let KX and KY denote canonical divisors on X and Y , respectively. Then we have the linear equivalence (of divisors on X) KX ∼ f ∗ KY + R. The main result of this section is the Theorem of Riemann and Hurwitz which expresses the degree of the ramification divisor in terms of the genus of X and the genus of Y , and the degree of the morphism f . We will discuss below how to explicitly compute the degree of the ramification divisor in many cases. Theorem 26.64. (Theorem of Riemann-Hurwitz) Let f : X → Y be a generically ´etale morphism of geometrically connected smooth proper curves, and let R be the ramification divisor of f . Then 2g(X) − 2 = deg(f )(2g(Y ) − 2) + deg(R). Proof. Again denoting by KX and KY canonical divisors on X and Y , Proposition 26.63 shows that deg(R) = deg(KX ) − deg(f ∗ KY ) = deg(KX ) − deg(f ) deg(KY ). Since deg(KX ) = 2g(X) − 2 (Corollary 26.52), and similarly for Y , we obtain the formula stated in the theorem.

543 Recall that by definition of R one has X lgOX,x (ΩX/Y,x )[κ(x) : k]. (26.13.2) deg(R) = x∈X 1

Under some further assumptions, we can relate the degree of R to the ramification index we have introduced in Section (12.5); see below for a brief reminder. We will use the following lemma. Lemma 26.65. Let X be a smooth curve over the field k and let x be a closed point of X. If the extension κ(x)/k is separable and s ∈ OX,x is a uniformizer, then ds is a generator of the free OX,x -module ΩX/k,x of rank 1. Proof. The k-algebra A = OX,x is a discrete valuation ring, and ΩX/k,x = ΩA/k is a free A-module of rank 1 because X is smooth over k and x ∈ X is a closed point. By Nakayama’s lemma it is enough to show that ds ̸= 0 ∈ ΩA/k ⊗k κ(x). But since Ωκ(x)/k = 0, the extension κ(x)/k being finite separable, the differential d induces an isomorphism (s)/(s2 ) → ΩA/k ⊗k κ(x) (e.g., use Corollary 18.73). Let f : X → Y be a generically ´etale morphism of geometrically connected smooth proper curves, let x be a closed point of X and let y = f (x). Let Xy = X ×Y Spec κ(y) be the fiber of f over y, a finite k-scheme, and let A be the local ring of Xy at x. Recall from Section (12.5) the ramification index ex/y = lgA (A) and the inertia index fx/y = [κ(x) : κ(y)]. Proposition 26.66. Let f : X → Y be a generically ´etale morphism of geometrically connected smooth proper curves over a field k, and let x be a closed point of X. (1) If f is tamely ramified at x, i.e., the extension κ(x)/κ(f (x)) is separable and char(k) ∤ e, then lg(ΩX/Y,x ) = ex/y − 1. (2) If the assumptions in part (1) are not satisfied, then lg(ΩX/Y,x ) > ex/y − 1. Proof. We prove the proposition under the assumption that κ(x)/k is separable. See [Liu] O Theorem 7.4.16 or [Sta] 0C1F for the general case. Choose uniformizers s and t of the discrete valuation rings OX,x and OY,y . We write e = ex/y . Under the ring homomorphism OY,y → OX,x given by f , t 7→ use for a unit × u ∈ OX,x , and under the natural homomorphism ΩY /k,y ⊗OY,y OX,x → ΩX/k,x , dt ⊗ 1 7→ d(use ) = se du + ese−1 uds = se wds + ese−1 uds, (where we write du = wds for some w ∈ OX,x ). As κ(x)/k is separable by our assumption, ds is a generator of ΩX/k,x by Lemma 26.65. If in addition e = ̸ 0 in κ(x), we obtain the statement in part (1) from this computation and the short exact sequence (26.13.1). On the other hand, if e = 0 in κ(x), then the length of the quotient in this short exact sequence is at least e. By this result and by (26.13.2), in the tamely ramified case and in particular in the case when the base field has characteristic 0, the following more explicit version of the Riemann-Hurwitz formula holds.

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Corollary 26.67. (Theorem of Riemann-Hurwitz, tamely ramified case) Let f : X → Y be a generically ´etale morphism of geometrically connected proper smooth curves over a field k. Assume that for all closed points x of X the extension κ(x)/κ(f (x)) is separable and the ramification index ex/f (x) is invertible in k (both assumptions are satisfied if k is of characteristic 0). Then X (ex/f (x) − 1)[κ(x) : k], 2g(X) − 2 = deg(f )(2g(Y ) − 2) + x∈X 1

where the sum extends over all closed points of X (and only finitely many summands are non-zero). As a useful consequence we spell out the bounds on the number of points where a generically ´etale morphism is ramified that we obtain in this way. Corollary 26.68. Let f : X → Y be a generically ´etale morphism of geometrically connected proper smooth curves. Let ρX be the number of points x ∈ X 1 in which f is ramified and let ρY be the number of points y ∈ Y 1 such that there exists an x ∈ f −1 (y) in which f is ramified. (1) One has ρY ≤ ρX ≤ 2g(X) − 2 − deg(f )(2g(Y ) − 2). (2) Suppose that k is algebraically closed, that all ramification indices ex/f (x) for f are invertible in k, and that f is not an isomorphism. Then 2g(X) − 2 − deg(f )(2g(Y ) − 2) ≤ ρY . deg(f ) − 1 Proof. Clearly, one has ρY ≤ ρX . Hence the first assertion follows from Theorem 26.64 since ρX ≤ deg(R). Let RY be the set of y ∈ Y 1 such that there exists an x ∈ f −1 (y) in which f is ramified. Since k is algebraically closed, all residue extensions of closed points are trivial and one has by Corollary 26.67 X (ex/f (x) − 1) 2g(X) − 2 − deg(f )(2g(Y ) − 2) = x∈X 1

=

X

(

X

(ex/y − 1))

y∈RY x∈f −1 (y)

=

X

(deg(f ) − #f −1 (y))

y∈RY

≤ ρY (deg(f ) − 1), P where for the third equality one uses deg(f ) = x∈f −1 (y) ex/y (12.6.2). Remark 26.69. Let f : X → Y be a generically ´etale morphism of geometrically connected smooth proper curves over a field. Then Corollary 26.68 (1) implies the following assertions. (1) One has g(X) ≥ g(Y ). We will show below (Corollary 26.75) that this holds for any finite morphism between such curves. (2) If g(X) = g(Y ) > 1, then f is ´etale of degree 1 and hence is an isomorphism. Finally, the Riemann-Hurwitz formula also allows to give a new proof that the projective line over an algebraically closed field is “simply connected” in the sense that every ´etale cover splits completely, i.e., is just a disjoint union of copies of the projective line (see Theorem 20.114). More precisely, we have the following result.

545 Corollary 26.70. Let k be a field, let X be a geometrically connected curve and let f : X → P1k be a finite ´etale morphism of k-schemes. Then f is an isomorphism. Proof. As f is finite and ´etale, X is proper and smooth. By assumption, the ramification divisor of f is trivial. Since g(X) ≥ 0, the formula of Riemann-Hurwitz implies that deg(f ) = 1 and Corollary 26.12 shows that f is an isomorphism. (26.14) Purely inseparable morphisms. Let p be a prime number and let S be an Fp -scheme. In Remark/Definition 4.24, we have defined the absolute Frobenius morphism FrobS : S → S, which is the identity on topological spaces and the homomorphism x 7→ xp on (sections of) the structure sheaves. For an S-scheme f : X → S, we let X (p) := X ×S,FrobS S (an S-scheme via projection to the second factor) and obtain the relative Frobenius morphism FX/S := (FrobX , f ) : X → X (p) , a morphism of S-schemes. Note that the S-scheme X (p) arises from the S-scheme X by base change, so it shares with X all properties that are stable under base change, such as being smooth over S, proper over S, etc. If k is a perfect field of characteristic p, then X (p) is isomorphic to X as an abstract scheme (but not in general as a k-scheme). n Iterating the construction, we have the n-fold twist X (p ) := X ×S,FrobnS S and the n n-fold relative Frobenius X → X (p ) . Proposition 26.71. Let k be a field of characteristic p > 0 and let X be a smooth irreducible k-scheme of finite type. The relative Frobenius morphism FX/k : X → X (p) is a finite homeomorphism and has degree pdim(X) . The k-scheme X (p) is integral and we have K(X (p) ) = k · K(X)p . Proof. The absolute Frobenius morphism of any scheme is a universal homeomorphism. It is also integral (by definition of the relative Frobenius or by Proposition 20.29). As a morphism between schemes of finite type over a field, FX/k is also of finite type and hence finite. Now X (p) is the base change of a smooth scheme and in particular it is reduced and therefore integral. The spectrum of K(X) ⊗k k 1/p is the schematic fiber of the generic point of X under the projection X → X (p) . As X (p) is reduced and X → X (p) is a homeomorphism, K(X) ⊗k k 1/p is a field. We conclude that K(X) ⊗k k 1/p = K(X (p) ). Viewing K(X (p) ) as a subfield of K(X) via the relative Frobenius X → X (p) , we obtain the identification K(X (p) ) = Im(K(X) ⊗k k 1/p → K(X)) = kK(X)p . Since X is smooth, ΩX/k is locally free of rank dim X, so ΩK(X)/k is a K(X)-vector space of dimension dim X. By Lemma G.34, we have [K(X) : kK(X)p ] = pdim X . Remark 26.72. Exercise 18.25 shows further properties of the relative Frobenius in a more general situation. In particular, it shows that the relative Frobenius of a smooth scheme is automatically flat (and hence finite locally free). For morphisms of normal curves we also have the following argument. Let k be a field and f : X → Y a surjective morphism of connected curves over k, where X is smooth over k and Y is normal. Then f is flat by Lemma 26.9 and Y is smooth over k by Proposition 18.60 (2).

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Proposition 26.73. Let k be a field of characteristic p > 0 and let f : X → Y be a morphism of connected proper normal curves over k such that the induced field extension K(Y ) ⊆ K(X) is purely inseparable, of degree pn , say. n If X is smooth over k, then there is a unique isomorphism Y ∼ = X (p ) over k which (pn ) identifies f with the n-fold relative Frobenius morphism X → X . Proof. There is a chain of intermediate extensions K(Y ) = K0 ⊂ K1 ⊂ · · · ⊂ Kn = K(X) such that each extension Ki+1 /Ki has degree p. Each Ki is the function field of a proper normal curve over k (Theorem 26.3) and the inclusions give rise to morphisms between these curves. In view of Remark 26.72, they are even smooth. By induction, it is then enough to consider the case n = 1. In this case, we have an inclusion kK(X)p ⊆ K(Y ) (clearly k ⊂ K(Y ), and since K(X)/K(Y ) is purely inseparable of degree p, we can write K(X) = K(Y )[α] for some α ∈ K(X) with αp ∈ K(Y ), so K(X)p ⊆ K(Y ), as well). By Proposition 26.71 we have kK(X)p = K(X (p) ) and [K(X) : kK(X)p ] = p. We thus obtain equalities K(Y ) = kK(X)p = K(X (p) ). Since a proper normal curve is determined by its function field, the proposition follows. Corollary 26.74. Let k be a field of characteristic p > 0. Let f : X → Y be a morphism between connected smooth proper curves over k such that the induced field extension K(Y ) ⊆ K(X) is purely inseparable. Then we have g(X) = g(Y ). Proof. By Proposition 26.73, it is enough to show that g(X (p) ) = g(X). But X (p) is the base change of X/k with respect to the absolute Frobenius morphism of Spec k, hence is still connected, and base change preserves the genus by Lemma 26.32. Proposition 26.75. Let f : X → Y be a finite morphism of geometrically connected smooth proper curves over a field. Then g(X) ≥ g(Y ). Proof. Let L be the separable closure of K(Y ) in K(X), so that the extension K(X)/K(Y ) is split into a purely inseparable extension K(X)/L and a separable extension L/K(Y ). Accordingly, we can factor f as X → X ′ → Y with X → X ′ purely inseparable and X ′ → Y separable, for some connected smooth proper curve X ′ (see Remark 26.72). Then g(X) = g(X ′ ) by Corollary 26.74 and g(X ′ ) ≥ g(Y ) by the formula of RiemannHurwitz, see Remark 26.69. Corollary 26.76. (L¨ uroth’s theorem) Let k be a field. (1) Let X be a normal connected curve over k such that there exists a non-constant morphism f : P1k → X. Then X ∼ = P1k . ̸ k be (2) Let K = k(T ) be the field of rational functions in one variable over k. Let L = an intermediate field of the extension K/k. Then there exists x ∈ K with L = k(x). (Here x is necessarily transcendental over k, so L is isomorphic to the field of rational functions over k in one variable.) Proof. We prove (1). By Lemma 26.9, f is flat. In particular, it is open. It is also closed since P1k is proper over k. Hence f is surjective because X is connected. Therefore X is proper over k because P1k is proper. By Remark 26.72, X is smooth over k. As P1k is geometrically connected, so is X. Then H 0 (X, OX ) = k and hence g(X) ≥ 0. Hence we can apply Proposition 26.75 to see that g(X) = 0. Moreover, X has a k-valued point since this is true for P1k . The claim then follows from Corollary 26.53.

547 As L contains a non-constant rational function f of k(T ), L is not an algebraic extension and hence has transcendence degree 1 over k. As k(T ) is a finitely generated algebraic extension of k(f ) and hence a finite extension of k(f ), L is a finite extension of k(f ) and therefore a finitely generated extension of k. Hence we can use the dictionary between connected normal proper curves and their function fields (Theorem 26.3), to deduce (2) from (1).

Special classes of curves At this point, we come to several important classes of examples. We start out by studying curves which can be embedded in the projective plane P2k . Afterwards we take a look at curves of small genus and at hyperelliptic curves. (26.15) Plane curves. Definition 26.77. A curve C over a field k is called a plane curve if there exists a locally closed immersion C → P2k . We start by collecting some basic properties of plane curves. Proposition 26.78. Let k be a field. (1) If C is a reduced proper plane curve over k, then C is isomorphic to a closed subscheme of P2k of the form V+ (f ) for a homogeneous polynomial f ∈ k[T0 , T1 , T2 ]. (2) For a non-constant homogeneous polynomial f ∈ k[T0 , T1 , T2 ], V+ (f ) is a proper plane curve over k. The embedding V+ (f ) ,→ P2k is a regular immersion of codimension 1 and in particular V+ (f ) is locally a complete intersection. The curve V+ (f ) is integral if and only if f is an irreducible polynomial. Proof. For C integral, Part (1) is Corollary 5.42. For a general reduced curve with irreducible components C1 , . . . , Cn , each Ci is integral and hence of the form V+ (fi ), and then C = V+ (f1 · · · fn ). For part (2), we use Proposition 5.40 (with n = 2 and X = P2k ) to see that V+ (f ) is in fact a curve, i.e., equi-dimensional of dimension 1. The other assertions are easy to check. Not surprisingly, curves defined by a single homogeneous equation in P2k are particularly explicit. Since they are complete intersections in P2k , they are in particular Gorenstein. Hence the canonical sheaf is a line bundle. Proposition 26.79. Let k be a field and let C = V+ (f ) ⊂ P2k be a plane curve, where f ∈ k[T0 , T1 , T2 ] is a non-constant homogeneous polynomial of degree d. (1) We have an isomorphism ωC ∼ = OP2k (d − 3)|C . (2) The degree deg(C) of C as a subvariety of P2k equals d. (3) We have H 0 (C, OC ) = k, C is connected, and the arithmetic genus of C is g(C) =

(d − 1)(d − 2) . 2

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26 Curves

Proof. Part (1). Let i : C → P2k be the embedding. Then its conormal sheaf Ci is i∗ OP2k (−d) and the top exterior power of Ω1P2 /k is OP2k (−3) (Example 17.59). We conclude by (26.8.1). k Part (2). We have d = deg(C) by Example 23.87. Part (3). We denote by i : C ,→ P2k the closed embedding. We have a short exact sequence (13.4.6) 0 → OP2k (−d) → OP2k → ι∗ OC → 0. From the long exact cohomology sequence for this short exact sequence and our knowledge of the cohomology of line bundles on projective space (Theorem 22.22), we obtain isomorphisms ∼

k = H 0 (P2k , OP2k ) → H 0 (C, OC )

and

∼ H 1 (C, OC ) → H 2 (P2k , OP2k (−d)) ∼ = k (d−1)(d−2)/2 .

As H 0 (C, OC ) = k is a field, C is connected and g(C) = dim H 1 (C, OC ) = (d − 1)(d − 2)/2. Remark 26.80. We could also have calculated the genus of a plane connected curve as in Proposition 26.79 as follows. We have ωC ∼ = OP2k (d−3)|C . Now deg OP2k (1)|C = deg(C) = d, by definition of deg(C). Hence ωC has degree d(d − 3). But we know also that ωC has degree 2g − 2 (Corollary 26.52), so g = (d(d − 3) + 2)/2 = (d − 1)(d − 2)/2. Remark 26.81. Since, for example, 2, 4, 5, 7 are not of the form (d−1)(d−2) for any d, 2 we see in particular that a reduced proper plane curve cannot have genus 2 (or genus 4, 5, 7, . . . ). Correspondingly, reduced proper curves of genus 2 cannot be embedded in P2k . On the other hand, we will see below that over an algebraically closed field k every smooth proper connected curve of genus 0 or 1 is a plane curve, see Proposition 26.89 and Proposition 26.95 for more precise assertions. (26.16) Curves of genus 0. In this section, we discuss curves of genus 0. The most prominent example, of course, is the projective line P1k over the base field k. We will give several characterizations of P1k as a genus 0 curve with some additional properties below. Recall that in Corollary 26.53 we have already seen that a connected normal proper curve of genus 0 over k with a rational point is isomorphic to P1k . On the other hand, Brauer-Severi varieties of dimension 1 (Section (8.11)) also have genus 0 by Lemma 26.32. They correspond to central quaternion algebras over k (Theorem 14.95), i.e., every Brauer-Severi curve corresponding to such an algebra that is not isomorphic to the algebra of 2 × 2 matrices M2 (k) is an example of a smooth projective geometrically connected curve of genus 0 that is not isomorphic to P1k . Proposition 26.82. Let k be a field and let X be an integral proper curve of genus g(X) = 0 over k such that H 0 (X, OX ) = k. Let L be a line bundle on X. (1) If deg(L ) = 0, then L ∼ = OX . (2) If deg(L ) > 0, then dimk H 0 (X, L ) = deg(L ) + 1, H 1 (X, L ) = 0 and L is very ample. Proof. Part (1). The Riemann-Roch theorem in the form of Proposition 26.46 shows that dimk H 0 (X, L ) = deg(L ) + 1 − g(X) + dimk H 1 (X, L ) > 0, so we conclude by Remark 26.21 (3).

549 Part (2). As in the proof of part (1), we see that H 0 (X, L ) ̸= 0 and hence L ∼ = OX (D) for an effective Cartier divisor D on X. We have a short exact sequence 0 → OX → L → OD → 0, where the morphism OX → L corresponds to a non-trivial global section of L which defines D. From the associated long exact cohomology sequence and since H 1 (X, OX ) = 0 by our assumption, we conclude that H 1 (X, L ) = 0. Again using Proposition 26.46, this implies dimk H 0 (X, L ) = deg(L ) + 1. The hypothesis that H 0 (X, OX ) = k implies that X is geometrically connected (Section (26.2)). Hence we have already proved that in this situation L is very ample if X is known to be smooth, see Proposition 26.59. Moreover, Proposition 26.57 shows that L is ample in general. We omit the proof that L is very ample in the general case, see Exercise 26.11 or [Sta] 0C6T. If there exists a Cartier divisor (or equivalently, a line bundle) of degree 1 on a genus 0 curve as in the previous proposition, then the curve is necessarily isomorphic to the projective line. Proposition 26.83. Let k be a field and let X be an integral proper curve of genus g(X) = 0 over k such that H 0 (X, OX ) = k. Let L be a line bundle on X with deg(L ) = 1. Then X is isomorphic to the projective line P1k . Proof. By Proposition 26.82, L is very ample and dimk H 0 (X, L ) = 2, so L induces a closed immersion X → P1k which necessarily is an isomorphism. Clearly, a k-rational point inside the normal locus of X gives rise to a divisor, and hence a line bundle, of degree 1. As a corollary we have the following result. Compare Theorem 14.93. Corollary 26.84. Let k be a field and let X be a geometrically integral proper curve of genus g(X) = 0 over k. For any separably closed extension field k ′ of k, X ⊗k k ′ ∼ = P1k′ as ′ k -schemes. In particular a curve is a Brauer-Severi curve if and only if it is geometrically integral, proper, and of genus 0. Proof. The genus is preserved by base change of the field (Lemma 26.32). Hence we may assume that k is separably closed. We have H 0 (X, OX ) = k by Lemma 26.5. It follows from Remark 6.20 and Proposition 6.21 that there exists a k-rational point x ∈ X that lies in the smooth locus. The divisor [x] then has degree 1. Example 26.85. (1) The curve V+ (T02 + T12 + T22 ) ⊆ P2R over R is a smooth proper geometrically connected curve of genus 0 without rational points. It is the Brauer-Severi curve corresponding to the Hamilton quaternions over R.

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(2) Let C be a reduced curve over the field k all of whose irreducible components are isomorphic to P1k . Assume further that the intersections of the irreducible components are transversal, i.e., at most two irreducible components intersect in a point, and the intersection multiplicity is equal to 1. Let Γ denote the dual graph of this configuration, i.e., the graph with vertex set the set of irreducible components of C and where for each intersection point of irreducible components, there is an edge in Γ between the corresponding vertices. If Γ is a tree, then C has genus 0. This follows from Proposition 26.35 by induction on the number of irreducible components. (3) Let k be a field of characteristic ̸= 2 and let d ∈ k be an element which is not a square. Consider C = V+ (T02 − dT12 ) ⊆ P2k . This is√an integral curve with (0 : 0 : 1) as its only rational point. After base change to k( d) it becomes isomorphic to the union of two projective lines intersecting transversally. Thus the base change has √ genus 0 over k( d) by (2), and by Lemma 26.32, X has genus 0 over k. Compare Example 26.87 below. We state some further variants where the condition on the existence of a degree 1 divisor is relaxed a little. Corollary 26.86. Let k be a field and let X be a geometrically integral proper curve of genus g(X) = 0. (1) If X is Gorenstein (e.g., if X is normal) and there exists a line bundle L on X of odd degree, then X ∼ = P1k . (2) If the greatest common divisor of the degrees [κ(x) : k], where x ranges over all closed point of the normal locus Xnorm of X, is equal to 1 (e.g., if the normal locus contains a k-valued point), then X ∼ = P1k . Proof. For Part (1) note that the Gorenstein condition implies that the canonical sheaf of X is a line bundle; it has degree −2 by our assumption on the genus (Corollary 26.52). Since there exists a line bundle of odd degree, there also exists a line bundle of degree 1, and we conclude by Proposition 26.83. Part (2). By assumption, there exist natural numbers a1 , . . . , ar with greatest common divisor 1 and divisors D1 , . . . , Dr on C with deg(Di ) = ai . Then 1 can be expressed as a Z-linear combination of the ai , and correspondingly, we get a divisor of degree 1 as a linear combination of the Di , using the same coefficients. Over finite fields, Condition (2) is satisfied for all geometrically connected smooth proper curves (see Corollary 26.176 below), so we see again that there are no Brauer-Severi curves over a finite field. As we have seen in Corollary 26.84, a geometrically integral proper curve of genus 0 over an algebraically closed field is necessarily smooth. Over other fields, singular curves of genus 0 do exist, though, as the following example shows. Example 26.87. If k is not algebraically closed, then there exist singular proper curves of ˜ be a non-trivial finite field extension, genus 0, as the following construction shows. Let k/k and let ˜ ˜ ]. A0 = {f ∈ k[S]; f (0) ∈ k}, A1 = k[T Mapping S 7→ T −1 yields an isomorphism A0,S → A1,T between the localizations, and using this for gluing the affine schemes Spec A0 and Spec A1 , we obtain an integral proper ˜ In particular, curve C over k with H 0 (C, OC ) = k. Clearly, the normalization of A0 is k[S]. ˜ is not trivial, A0 is not normal. In fact, the point since by assumption the extension k/k

551 x corresponding to the prime ideal (S) ⊂ A0 is not normal. We have δx = 1, since ˜ dimk k[S]/A 0 = 1. The normalization of C is birational to Spec(A1 ) ∼ = A1k˜ and is hence isomorphic to P1k˜ which has genus g(P1k˜ /k) = 1 − 2 + 0 = −1. We thus have g(C) = 0 by Proposition 26.35. Remark 26.88. Let k be a field and let C be a singular integral proper curve of genus 0 with H 0 (C, OC ) = k. Let f : C˜ → C be the normalization morphism. From the exact ˜ O ˜ ) = 0 and that sequence (26.8.2) we obtain that H 1 (C, C ˜ O ˜) = 1 + dimk H 0 (C, C

X

δx ,

x

where the sum is taken over all closedP points of C and δx is defined as in Proposition 26.35. ˜ O ˜ ), the extension Since C is assumed to be singular, δx ̸= 0, so writing k˜ = H 0 (C, C ˜ k/k is a non-trivial field extension. ˜ = 0. One can show that there is precisely one singular point in ˜ k) We see that g(C/ ˜ C, and that there is precisely one point in C˜ lying over it, and that that is a k-rational 1 ∼ ˜ point. It then follows from Corollary 26.53 that C = Pk˜ . More precisely, one can show that C must be as in Example 26.87. See [Sta] 0DJB. Every smooth, and more generally every Gorenstein proper curve of genus 0 is isomorphic to a conic, i.e., a plane curve of degree 2: Proposition 26.89. Let X be a Gorenstein integral proper curve over a field k with g(X) = 0 and H 0 (X, OX ) = k. Then there exists a closed immersion X ,→ P2k of k-schemes, and the image is a plane curve of degree 2 (a conic). ∨ Proof. The canonical sheaf ωX is a line bundle of degree −2, hence its dual ωX is very 1 ∨ ample and the first cohomology H (X, ωX ) vanishes by Proposition 26.82. We therefore ∨ ∨ ∨ have dim H 0 (X, ωX ) = deg(ωX ) + 1 − g(X) + dimk H 1 (X, ωX ) = 3, so this very ample 2 ∨ line bundle gives us a closed immersion X ,→ Pk . Its image has degree deg(ωX ) = 2.

Example 26.90. Let C be a connected proper smooth curve over an algebraically closed field k. (1) If g(C) = 0, and hence C ∼ = P1k , then all divisors of degree 1 are linearly equivalent to each other. (2) Suppose g(C) ≥ 1. For x, x′ ∈ C the divisors [x] and [x′ ] are linearly equivalent if and only if x = x′ , by Corollary 26.20. Remark 26.91. If C is a Gorenstein proper connected curve over a field with H 0 (C, OC ) = k and −KC is ample, then g(C) = 0, because KC must then have negative degree, but deg(KC ) = 2g(C) − 2 can only be negative if g(C) = 0. Remark 26.92. As a consequence of the Theorem of Hasse and Minkowski ([Ser5]) a conic over a finite extension field k of Q (more generally: over any global field) has a k-valued point if and only if it has a kv valued point in the completion kv of every place v of k. For k = Q this means that a conic has a Q-valued point if and only if it has an R-valued point and for every prime number p has a point with values in the field Qp of p-adic numbers. This situation is often described by saying that genus 0 curves satisfy the local-global principle (or the Hasse principle).

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26 Curves

(26.17) Curves of genus 1, elliptic curves. After studying curves of genus 0, we now consider the case of genus 1. We have already briefly touched upon this topic in Section (16.35). While there is only a single geometrically connected smooth proper curve of genus 0 – the projective line, which we understand very well, in genus 1 we see a fascinating picture which is much richer and in parts still quite mysterious. Here we can only deal with the basics, and in particular mostly have to leave out the arithmetic questions and applications connected to this topic, which have been one of the main driving forces in arithmetic geometry and algebraic number theory over the last one or two centuries. See [Sil1] O , [Sil2] O , [CSS] O for further information and additional references. The Riemann-Roch theorem gives us the following information in the case of genus 1 curves. Corollary 26.93. Let k be a field and let C be a geometrically connected smooth proper curve of genus g(C) = 1 over k. (1) The canonical bundle ωC is trivial. (2) Let L be a line bundle on C. If deg(L ) > 0, then ℓ(L ) = deg(L ). Proof. By the Riemann-Roch theorem (Corollary 26.52), ωC has degree 2g(C) − 2 = 0, and ℓ(ωC ) = g(C) = 1. This implies ωC ∼ = OC by Remark 26.21. The second part then also follows from the Riemann-Roch formula, since deg(L ) > 0 implies deg(L ∨ ⊗ ωC ) = deg(L ∨ ) = − deg(L ) < 0 and hence ℓ(L ∨ ⊗ ωC ) = 0. It is an exciting problem to study the structure of the set of k-valued points of a smooth proper curve C of genus 1. Obviously, this problem only has content, if C(k) is non-empty. As we will see, fixing an element of C(k) gives rise to a structure of abelian group on the set C(k), and we thus make the following definition. Definition 26.94. Let k be a field. An elliptic curve over k is a (geometrically) connected smooth proper curve E of genus g(E) = 1 over k together with a point 0 ∈ E(k). The term elliptic curve comes from the relation between elliptic curves over C (or equivalently, compact Riemann surfaces of genus 1, with a fixed base point, i.e., quotients C/Λ of C by a lattice, see Section (26.19)) and elliptic integrals, e.g., the integral to compute the arc-length of an ellipse. Cf. [Sil1] O Ch. VI. We will study elliptic curves over general base schemes in Section (27.27) and reprove some of the results below in this generality. Here we focus on elliptic curves over a field. Note that since an elliptic curve E has a k-valued point, the properties connected and geometrically connected are equivalent (Lemma 26.4). Proposition 26.95. Let k be a field, let E be a connected smooth proper curve over k and let 0 ∈ E(k). (1) The following are equivalent: (i) The genus of E is equal to 1, i.e., E (with the fixed point 0) is an elliptic curve. (ii) The curve E is isomorphic to a smooth projective curve of degree 3 (a “cubic”) in P2k . (2) If the equivalent conditions of part (1) are satisfied, then E is isomorphic to a cubic given by an equation in Weierstraß form, i.e., of the form V+ (Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 − a2 X 2 Z − a4 XZ 2 − a6 Z 3 ),

ai ∈ k,

553 where X, Y, Z denote homogeneous coordinates on P2k . The isomorphism can be chosen so that 0 ∈ E(k) is mapped to the point (0 : 1 : 0). (3) If the characteristic of the field k is different from 2 and 3, and the conditions of part (1) hold, then E is isomorphic to a cubic given by an equation of the form V+ (Y 2 Z − X 3 − aXZ 2 + bZ 3 ),

a, b ∈ k.

Proof. First note that in part (1), the implication (ii) ⇒ (i) follows immediately from the genus-degree-formula for plane curves, Proposition 26.79. (Cf. also Exercise 17.9.) Now assume (i). We find a Weierstraß equation as in (2) for E as follows. We have ℓ(OE (2[0])) = deg(2[0]) = 2, so there exists x ∈ K(E) such that 1, x is a k-basis of H 0 (E, O(2[0])). Similarly, ℓ(OE (3[0])) = 3, so we can extend the family 1, x to a k-basis 1, x, y of H 0 (E, O(3[0])). Then the rational functions 1, x, y, xy, x2 , x3 , y 2 are 7 elements in the 6-dimensional k-vector space H 0 (E, O(6[0])), so they are linearly dependent over k. On the other hand, 1, x, y, xy, x2 are linearly independent, having pairwise different pole orders at 0. Therefore we find a non-trivial linear relation between the above-named elements where the coefficients u of x3 and v of y 2 are both ̸= 0. Replacing x by uvx and y by u2 vy and dividing by u4 v 3 , we may assume that the coefficients of x and y are both = 1. We conclude that there exist a1 , a2 , a3 , a4 , a6 ∈ k with y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 . We set f := Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 − a2 X 2 Z − a4 XZ 2 − a6 Z 3 ∈ k[X, Y, Z]. The image of the morphism E → P2 given by the very ample sheaf O(3[0]) and the basis x, y, 1 of H 0 (E, O(3[0])) is contained in V+ (f ). Since the latter is integral (the polynomial f being irreducible as is easily checked), we obtain an isomorphism E ∼ = V+ (f ), as desired. In particular, (ii) holds. By the choice of x and y, 0 ∈ E(k) is mapped to the point [0 : 1 : 0]. To get part (3) from part (2), we first do the coordinate change replacing Y by Y + 21 (a1 X + a3 ) and then replace X by X + 31 a2 . The numbering of the coefficients ai in the Weierstraß equation reflects the behavior under a coordinate change of the form X 7→ u2 X, Y 7→ u3 Y (u ∈ k × ). Plugging u2 X and u3 Y into a Weierstraß equation as above and dividing by u6 , we obtain a Weierstraß equation with coefficients u−i ai . See Proposition 27.150 for a more general version of the above proposition which handles the case of a general base scheme and yields a Weierstraß equation locally on the base. The following lemma gives an explicit criterion, when the curve defined by a Weierstraß equation is smooth. Lemma 26.96. Let k be a field. (1) For ai ∈ k, the projective plane curve V+ (Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 − a2 X 2 Z − a4 XZ 2 − a6 Z 3 ) is always smooth at the point (0 : 1 : 0). It is smooth over k if and only if ∆ := −b22 b8 − 8b34 − 27b26 + 9b2 b4 b6 ̸= 0, where b2 = a21 +4a2 , b4 = 2a4 +a1 a3 , b6 = a23 +4a6 , b8 = a21 a6 +4a2 a6 −a1 a3 a4 +a2 a23 −a24 .

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26 Curves

(2) Assume that char k ̸= 2. For a, b ∈ k, the projective plane curve V+ (Y 2 Z − X 3 − aXZ 2 + bZ 3 ) is smooth over k if and only if 4a3 + 27b2 ̸= 0, i.e., if and only if the polynomial x3 + ax + b has no multiple zeros in an algebraic closure of k. Proof. Part (1) can be shown by a direct, but somewhat tedious computation which we omit here. See [Sil1] O Section III.1 and Appendix A. In the situation of part (2), a point (x : y : z) ∈ E(k) is smooth if and only if all partial derivatives of f = Y 2 Z −X 3 −aXZ 2 +bZ 3 vanish. We find that y = 0, and may assume z = 1 without loss of generality, since we are working with homogeneous coordinates, and the only point on E with z = 0 is (0 : 1 : 0). Then the condition on x is that x is a zero of X 3 + aX + b (since the chosen point lies on E) and that x is a zero of the derivative of this polynomial with respect to X (which equals ∂f /∂X). From this, the statement of the lemma follows from the formula for the discriminant of a cubic polynomial of this form. Remark 26.97. The quantity ∆ in part (1) for an equation of the particular form of part (2) is −16(4a3 + 27b2 ). In characteristic ̸= 2, the vanishing of one of these expressions is of course equivalent to the vanishing of the other, but using −16(4a3 + 27b2 ) is arguably a better normalization and reflects the fact that a curve of the form V+ (Y 2 Z − X 3 − aXZ 2 + bZ 3 ) with a, b ∈ k is necessarily singular, if char k = 2. The following theorem gives us the above-mentioned group structure. Theorem 26.98. Let k be a field, let E be a connected smooth proper curve over k and let 0 ∈ E(k). The following are equivalent: (i) The genus of E is equal to 1, i.e., E (with the fixed point 0) is an elliptic curve. (ii) The curve E can be equipped with the structure of a group variety over k with neutral element 0 ∈ E(k), i.e., E is an abelian variety over k (Definition 16.53, Chapter 27). Proof. (i) ⇒ (ii). Consider E of genus 1 together with a rational point 0. We then have a bijection ι0 : E(k) → Pic0 (E) = {L ∈ Pic(E); deg(L ) = 0},

x 7→ OE ([x]) ⊗OE OE ([0])∨ .

Indeed, if ι0 (x) = ι0 (y), then [x] and [y] are linearly equivalent. Since g(E) ̸= 0 this implies x = y (Example 26.90). For the surjectivity, let L be a line bundle on E of degree 0. Then L ⊗OE OE ([0]) has degree 1 and hence ℓ(L ⊗OE OE ([0])) = 1 by Corollary 26.93. Therefore, there exists an effective divisor of degree 1, say [x], x ∈ E(k), such that L ⊗OE OE ([0]) ∼ = OE ([x]). Since Pic0 (E) is an abelian group (a subgroup of Pic(E), i.e., the group structure being given by tensor product), this gives us a structure of abelian group on E(k). It is clear that 0 is the neutral element.

555 By applying the same construction to the base change of E to extension fields k ′ of k, we analogously obtain a group structure on each E(k ′ ). It is clear that E(k) is then a subgroup of E(k ′ ). In particular, taking as k ′ an algebraic closure of k, we see that there is at most one morphism E × E → E which on k ′ -valued points gives the multiplication and at most one morphism E → E which on k ′ -valued points gives the map mapping each element to its inverse, for all k ′ /k. Since the group structure is commutative, it is usually denoted as addition and we will follow this convention. We give two ways of showing that indeed the group structure comes from morphisms E × E → E, E → E of k-schemes. The more elementary approach is to embed E into P2k via Proposition 26.95 and to compute explicit formulas describing the addition and the map mapping a point to its negative. See Remark 26.100. This shows directly and very explicitly that these maps are morphisms of projective k-varieties and gives the well-known geometric description of the group law. Cf. Section (16.35). A different approach is to soup up the isomorphism E(k) ∼ = Pic0 (E) so as to obtain a group structure on E(T ) for every k-scheme T , functorially in T . This will give us a structure of group functor on the functor of T -valued points of E, and by the Yoneda lemma the desired morphisms E ×E → E and E → E defining the group variety structure. Compare Section (27.21) where the functor Pic0 is considered in much greater generality. Let us first set up some notation. For a k-scheme T , we abbreviate E ×k T as ET . Similarly, for t ∈ T we denote by Et = E ⊗k κ(t) the fiber of ET over t. Let Pic0 (ET ) be the subgroup of Pic(ET ) consisting of line bundles L such that for each t ∈ T , the pullback Lt of L to the fiber Et = E ×k Spec(κ(t)) has degree deg(Lt ) = 0. (Cf. Proposition 26.166.) Pullback along the projection p : E ×k T → T gives us a map Pic(T ) → Pic(ET ) which is injective since the point 0 gives rise to a section T → E ×k T and whose image p∗ Pic(T ) lies in Pic0 (E ×k T ). Given a morphism x : T → E, we obtain a morphism (x, idT ) : T → E ×k T of T -schemes. This is a regular closed immersion, so its image is an effective Cartier divisor of ET . We denote by OET /T (x) the corresponding line bundle. Cf. Remark 26.163. We denote by 0T : T → E the composition T → Spec k → E, the second arrow being our fixed point 0 ∈ E(k). Claim. The maps E(T ) → Pic0 (E ×k T )/p∗ Pic(T ),

x 7→ OET /T (x) ⊗ OET /T (0T )∨ ,

for k-schemes T are bijective and functorial in T . Proof of claim. It is easy to check that the given line bundles lie in Pic0 (E ×k T ) and that the construction is functorial in T . In particular, whenever T is the spectrum of a field, this bijection induces on E(T ) the same group structure as above. To prove the bijectivity we construct a map in the other direction such that the two maps are inverse to each other. For a line bundle L ∈ Pic0 (E ×k T ), we write L ′ := L ⊗ OET /T (0T ). Then L ′ is a line bundle on ET which is (fiberwise) of degree 1. We denote its restriction to Et by Lt′ . Given L ∈ Pic0 (E ×k T ), we want to show that there exists a unique morphism x : T → E such that for some M ∈ Pic(T ) we have L ∼ = OET /T (x) ⊗ OET /T (0)∨ ⊗ p∗ M , or equivalently

L′ ∼ = OET /T (x) ⊗ p∗ M .

556

26 Curves

For t ∈ T , we have dimκ(t) H 0 (Et , Lt′ ) = deg(Lt′ ) = 1 and dimκ(t) H 1 (Et , Lt′ ) = 0 by the Theorem of Riemann-Roch, Theorem 26.48. By cohomology and base change for the proper morphism p : ET → T , we conclude that R1 p∗ L ′ = 0, and that p∗ L ′ is locally free of rank 1. See Corollary 23.144, Remark 26.165. Since we work up to tensoring with line bundles pulled back from T , we may replace L by L ⊗ (p∗ p∗ L ′ )∨ , and therefore may assume without loss of generality that p∗ L ′ ∼ = OT . In this situation, the section 1 ∈ Γ(T, OT ) = Γ(ET , L ′ ) defines an effective Cartier divisor on ET (with associated line bundle L ′ ) which we view as a closed subscheme Z ⊂ ET . The construction is compatible with base change T ′ → T . In particular, we obtain a Cartier divisor of degree 1 on each fiber Et . Proposition 14.22 shows that Z is flat over T . Thus the morphism Z → T is finite locally free of rank 1, and hence an isomorphism. ∼ We obtain a morphism T → Z → ET , and composing with the projection ET → E we get a morphism x : T → E. This concludes the definition of a map Pic0 (E×k )/p∗ Pic(T ) → E(T ). It is not hard to check that the two maps are inverse to each other. (i) ⇒ (ii). For the converse, we use the fact (Proposition 27.15 in the next chapter) that the existence of a group variety structure on E implies that ωE ∼ = OE . Hence 2g(E) − 2 = deg(ωE ) = 0. Remark 26.99. From the rigidity lemma, Lemma 16.55, we obtain the following consequences. (1) The choice of the point 0 ∈ E(k) determines the group structure uniquely. In fact, suppose that mν : E × E → E are multiplication morphisms and iν : E → E are morphisms giving the inverse of an element, so that an elliptic curve E is a group variety with neutral element 0 : Spec(k) → E for ν = 1, 2. Consider the morphism E × E → E,

(x, y) 7→ m2 (m1 (x, y), i2 (x)),

i.e., we add x to y with respect to the first group law, and then subtract x with respect to the second one. For y = 0, we obtain the constant morphism E = E × {0} → E, x 7→ 0. By Lemma 16.55, the above morphism factors through the second projection, and that implies that m1 = m2 . (2) A scheme morphism between elliptic curves which maps the fixed rational point on the domain to the fixed rational point on the target, is a group morphism. More precisely, we have, as a special case of Corollary 16.56 (1): Let k be a field, let E, E ′ with rational points 0 ∈ E(k), 0′ ∈ E ′ (k) be elliptic curves over k and let f : E → E be a morphism of k-schemes. If f (0) = 0′ , then f is a morphism of group varieties for the group variety structure on E and E ′ , respectively, given by Theorem 26.98. Remark 26.100. As mentioned above, one can write down explicit formulas for the group law on an elliptic curve E which is given as the closed subscheme of P2k defined by a Weierstraß equation. In fact, we can reformulate the definition of the group law as follows. For points P, Q, R ∈ E(k) we have P +Q+R=0

⇔

[P ] + [Q] + [R] ∼ 3[0],

and since the embedding E ⊂ P2k is given by the line bundle OE (3[0]), the condition on the right hand side is equivalent to saying that P , Q and R are the three intersection points (counted with multiplicity) of E with some line in P2k (cf. Theorem 5.61). It follows from the above results, but can also easily be shown directly, that a line in P2k intersects

557 E in either 0, 1 or 3 points of E(k), counted with multiplicity. If one starts with this, then one can define the addition on E by the rule stated above. However, it is then a non-trivial task to show that this structure is associative. For writing out explicit formulas, let us for simplicity restrict to the case of an elliptic curve E of the form V+ (f ) for f = Y 2 Z − X 3 − aXZ 2 − bZ 3 ,

a, b ∈ k.

(This covers the situation where k has characteristic ̸= 2, 3. See [Sil1] O Section III.2 for the general case.) First note that 0 = (0 : 1 : 0) is the only point of E on the “line at infinity” V+ (Z). For the point 0, being the neutral element for the group law, there is nothing to compute. Therefore we can pass to the affine curve E ′ = V (Y 2 − X 3 − aX − b)

⊂ A2k ,

where now X, Y are affine coordinates on A2k . For a point P = (x, y) ∈ E(k), the negative of P with respect to the group law is −P = (x, −y). In fact, it is clear that the points (x : y : 1), (x : −y : 1) and (0 : 1 : 0) are the intersection points of E with the line V+ (X − xZ). It remains to give formulas for the sum of points Pi = (xi , yi ) ∈ E ′ (k), i = 1, 2 where x1 ̸= x2 or P1 = P2 but 2P1 ̸= 0, i.e., y1 ̸= 0. Case 1. Pi = (xi , yi ) ∈ E ′ (k), i = 1, 2, x1 ̸= x2 . We define λ=

y 2 − y1 . x2 − x1

Case 2. Pi = (xi , yi ) ∈ E ′ (k), i = 1, 2, P1 = P2 . We define λ=

3x21 + a . 2y1

With this case-by-case definition of λ, we can state the result in both cases in a uniform way. We have P1 + P2 = (x3 , y3 ) with x3 = λ2 − x 1 − x 2 ,

y3 = λ(x1 − x3 ) − y1 .

Remark 26.101. Let E be an elliptic curve over a field k. The curve E is a group variety, and the group of k-scheme-automorphisms of E contains all translation morphisms x 7→ x + a, a ∈ E(k). This implies that the group of k-scheme-automorphisms acts transitively on E(k). On the other hand, the group of group variety automorphisms of E is finite, as Proposition 26.103 shows. Example 26.102. In contrast to genus 0 curves (Remark 26.92), curves of genus 1 do not satisfy a local-global principle. The following famous example is due to Selmer. Let C be the smooth projective plane cubic curve over Q given by the homogeneous equation 3X 3 + 4Y 3 + 5Z 3 . It is not hard to show that C(R) ̸= ∅ and C(Qp ) ̸= ∅ for every prime number p. Moreover, C(Q) = ∅ (but proving this is more involved). See [Sil1] O Chapter X for a more thorough discussion and further references.

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26 Curves

(26.18) The Legendre family and the j-invariant. Let k be a field of characteristic ̸= 2. Let E be an elliptic curve over k, with affine Weierstraß equation Y 2 = X 3 + a2 X 2 + a4 X + a6 for E (use Proposition 26.95 and that char(k) ̸= 2 to complete the square on the left hand side). The projection map (x, y) 7→ x yields a finite morphism E → P1k of degree 2. In fact, it corresponds to the extension K(P1k ) = k(T ) ⊆ K(E) mapping T to a rational function x on E with a double pole at 0 = (0 : 1 : 0) and no other poles and zeros. Then K(E) ∼ = k(x)[y]/(y 2 − x3 − a2 x2 − a4 x − a6 ), since we can compute the function field on the standard affine chart of E, and hence the extension has degree 2. Since y, as a rational function on E, has a triple pole at 0, we also see that 0 ∈ E(k) is mapped to ∞, and is the only such point. For a k-valued point (1 : x) ∈ P1k (k), the fiber consists of 2 points, if and only if x3 + a2 x2 + a4 x + a6 is ̸= 0 and a square in k. Otherwise it consists of a single point. Hence if k is algebraically closed, then the morphism E → P1k is ramified precisely over ∞ and over the three zeros of the polynomial X 3 + a2 X 2 + a4 X + a6 , seen as elements of k = A1k (k) ⊂ P1k (k). The ramification degree of each ramified point is 2. Since k has characteristic ̸= 2, f is tamely ramified. Compare with the Riemann-Hurwitz formula (Corollary 26.67). From a different point of view, the points P of the form (x, 0) in (the affine part of) E(k) are precisely the points P = ̸ 0 with 2P = 0 with respect to the group law on E. For equations as above with a4 = 0 this follows directly from the formulas in Remark 26.100, but the computation is basically the same if a4 is arbitrary. In particular, the set of these points is preserved by every group variety automorphism of E. This gives us a handle on the automorphism group of the elliptic curve E. Proposition 26.103. Let k be a field of characteristic ̸= 2 and let E be an elliptic curve over k. Let G be the group of automorphisms of E as a group variety, equivalently, the group of automorphisms of E fixing 0 (Remark 26.101). Then G is non-trivial and finite, of order dividing 12. Proof. The group G is non-trivial because multiplication by −1, i.e., mapping each point to its negative, is not the identity automorphism. For an affine point P = (x, y) ∈ E(k), P = −P is equivalent to y = 0, as remarked above, so there are at most 4 such points in E(k) (including the point at infinity). Therefore multiplication by −1 cannot be the identity morphism, because otherwise it would give rise to the identity on E(k ′ ) for every extension field k ′ of k. Compare Proposition 27.186 which shows that multiplication by 2 is a finite morphism, which in turn implies that multiplication by −1 is not the identity. To show that G is finite, we may assume that k is algebraically closed, because passing to an extension field can only enlarge the automorphism group. We may thus assume that E is of the form Eλ for some λ ∈ k. Let f : E → P1k be a morphism of degree 2 ramified over 0, 1, λ and ∞, and assume that 0 ∈ E(k) is mapped to ∞. The points in E where f is ramified are the 2-torsion points, i.e., the points in E[2](k) := {P ∈ E(k); 2P = 0}, hence every automorphism of E permutes these points. Since it fixes 0, it induces a permutation of E[2](k) \ {0}. In this way we obtain a homomorphism Φ : G → S3 from G to the symmetric group S3 . The kernel of Φ consists of those automorphisms of E that fix the four points where f is ramified. Hence every g ∈ Ker(Φ) induces a Galois cover E \ E[2](k) → P1 \ {0, 1, λ, ∞} with Galois group Z/2 (see Section (20.16)). There are precisely 2 such covers, the identity and a second one, namely multiplication by −1.

559 This implies the claim of the proposition. In general, the image of Φ will be a proper subgroup of S3 (it is in bijection with the set of possible expressions for λ in Proposition 26.105 that equal λ). For fields of characteristic = 2, the automorphism group of an elliptic curve is still necessarily finite, but may have order up to 24. Now specifically, for every element λ ∈ k \ {0, 1}, we may consider the affine curve Eλ′ := V (Y 2 − X(X − 1)(X − λ)) ⊂ A2k . Its schematic closure Eλ ⊂ P2k is an elliptic curve over k (Lemma 26.96). As before, we obtain a morphism f : E → P1k extending the map (x, y) 7→ x on the affine part of E. Then f is ramified precisely over 0, 1, λ and ∞, and the ramification degree equals 2 for each of these points. We can view the curves Eλ as the fibers of a smooth proper morphism E := V+ (Y 2 Z − X(X − Z)(X − λZ)) → S := P1k \ {0, 1, ∞}, where E ⊂ P2k ×k S, the Legendre family of elliptic curves. With the terminology of Definition 27.144, E is a (relative) elliptic curve over S = P1k \ {0, 1, ∞}. Proposition 26.104. For k algebraically closed of characteristic ̸= 2, every elliptic curve over k is isomorphic to a curve of the form Eλ . Proof. Given an elliptic curve E with Weierstraß equation Y 2 = X 3 + a2 X 2 + a4 X + a6 , we can factor the polynomial X 3 + a2 X 2 + a4 X + a6 as (X − α)(X − β)(X − γ), say, where α, β, γ ∈ k are pairwise distinct by Lemma 26.96, since k is algebraically closed. After a suitable change of coordinates, we may assume that α = 0, β = 1. Proposition 26.105. Let k be a field of characteristic ̸= 2 and let E, E ′ be elliptic curves over k. Suppose that there are morphisms f : E → P1k and f ′ : E ′ → P1k of degree 2 that are ramified over 0, 1, ∞ and, respectively, λ and λ′ . Then the following are equivalent: (i) The curves E and E ′ are isomorphic. (ii) We have   1 1 λ 1−λ ′ λ ∈ λ, 1 − λ, , , , . λ 1−λ 1−λ λ Proof. First suppose that E and E ′ are isomorphic. We may then just as well assume that E = E ′ . The automorphism group of the curve E acts transitively on E(k); this follows immediately from the existence of the group structure on E. Replacing f and f ′ by the composition with suitable automorphisms of E, we may therefore assume that there f and f ′ are ramified at 0 ∈ E(k) and that f (0) = f ′ (0) = ∞. It follows that (f ′ )∗ O(1) ∼ = OE (2[∞]) ∼ = g ∗ O(1), i.e., f and f ′ are given by the same line bundle on E, but possibly different choices of basis of H 0 (E, OE (2[∞]). A suitable change of basis corresponds to an automorphism of P1k fixing ∞ and mapping the set {0, 1, λ} to {0, 1, λ′ }. Using the explicit description Autk (P1k ) = P GL2 (k) of the automorphism group of the projective line, see Section (11.15), one arrives at the desired conclusion. If, on the other hand, condition (ii) is satisfied, one sees similarly as in the first part of the proof that we may assume λ = λ′ after replacing f ′ by the composition with a suitable automorphism of P1k . But then E and E ′ are both isomorphic to the elliptic curve with affine Weierstraß equation y 2 = x(x − 1)(x − λ).

560

26 Curves

The proposition allows us to make the following definition: Definition 26.106. Let k be an algebraically closed field of characteristic ̸= 2, and let E be an elliptic curve over k. The j-invariant j(E) ∈ k is defined as j(E) = j(λ) := 28

(λ2 − λ + 1)3 , λ2 (λ − 1)2

where λ ∈ k is chosen so that E ∼ = Eλ . One has to check that j(E) is independent of the choice of λ, i.e., that for all the 6 possibilities for λ′ in the previous proposition one obtains the same value of the j-invariant. In fact, the map λ 7→ j(λ) describes a morphism P1k → P1k of degree 6, ramified only over ∞. A different approach is to express j(E) in terms of a Weierstraß equation. For instance, 3 if E is given by the affine equation y 2 = x3 + ax + b, then j(E) = (4a(4a) 3 +27b2 ) . In this way, one can define j(E) for elliptic curves over arbitrary ground fields, not necessarily algebraically closed. The j-invariant classifies elliptic curves up to isomorphism over algebraically closed fields. Theorem 26.107. Let k be a field of characteristic ̸= 2 and let k¯ be an algebraic closure of k. Let E, E ′ be elliptic curves over k. Then j(E) = j(E ′ ) if and only if E ⊗k k¯ and ¯ or equivalently as k-schemes. ¯ E ′ ⊗k k¯ are isomorphic as elliptic curves over k, Proof. The j-invariant does not change under base change of the ground field, so we have j(E) = j(E ′ ) whenever E and E ′ become isomorphic after base change to any extension field of k. ¯ i.e., we will show that Conversely, assume that j(E) = j(E ′ ). We replace k by k, assuming k is algebraically closed, it follows that E and E ′ are isomorphic. Choose ramified covers E → P1k and E ′ → P1k , ramified over 0, 1, ∞ and λ, λ′ , respectively. (Here we need that k is algebraically closed, to be able to apply Proposition 26.104.) We conclude by Proposition 26.105. This theorem also holds if k has characteristic 2 (with an appropriate definition of the j-invariant), and so does the following corollary. Corollary 26.108. Let k be an algebraically closed field. The map E 7→ j(E) induces a bijection between the set of isomorphism classes of elliptic curves over k and the set k. See [Sil1] O Chapter III for further details. (26.19) Elliptic curves over the complex numbers. Over the complex numbers, a smooth proper curve by analytification gives rise to a compact Riemann surface, see Section (26.7). In this section, we discuss the case of genus 1 in this setting in a bit more (but not always full) detail, because it gives some interesting insight into the general theory. So let X be a compact Riemann surface of genus 1. By Theorem 26.30 the universal cover of X is C → X. This yields a very explicit description of X.

561 Definition 26.109. A subgroup Λ ⊂ C is a lattice if it is generated as a Z-module by two R-linearly independent elements. Using the topological interpretation of the genus it is then easy to see that the quotient C/Λ, with the natural structure of Riemann surface, has genus 1. Proposition 26.110. Let X be a compact Riemann surface of genus 1 and let f : C → X be its universal cover. Then Λ := f −1 (f (0)) is a lattice in C and f induces an isomorphism X∼ = C/Λ of Riemann surfaces. Proof. We sketch the key steps of the proof. The Riemann surface X is the quotient of C by the action of the fundamental group π1 (X) of X on its universal covering. The automorphisms of the complex manifold C are the maps z 7→ µz + λ, µ ∈ C× , λ ∈ C. Covering transformations (̸= id) do not have fix points, so for those we must have µ = 1. Therefore we can identify the fundamental group π1 (X) with a subgroup of C. One checks that it is a discrete subgroup, thus a subgroup generated by 0, or 1 or 2 elements of C that are linearly independent over R. Since C and C/Z ∼ = C× are non-compact, the desired statement follows. Corollary 26.111. Let X be a compact Riemann surface of genus 1. Then X can be equipped with the structure of an abelian group so that it becomes a group object in the category of Riemann surfaces (i.e., multiplication and inverse are given by morphisms of complex manifolds). Proof. This is clear by the above, since the quotient of C by a lattice is an abelian group. Corollary 26.112. Let Λ ⊂ C be a lattice, and let X = C/Λ, a compact Riemann surface of genus 1. As an abelian group, X is isomorphic to R/Z × R/Z. In particular, for every n ∈ N>0 , the kernel of multiplication by n on X (the n-torsion on X) is isomorphic, as an abelian group, to (Z/nZ)2 . Morphisms between elliptic curves are easy to understand in terms of lattices. Proposition 26.113. Let Λ1 , Λ2 ⊂ C be lattices. The map {α ∈ C; αΛ1 ⊆ Λ2 } → {f : C/Λ1 → C/Λ2 ; f holomorphic, f (0) = 0} which maps α to the map z 7→ αz, is a bijection. Proof. We omit the proof (that is not very difficult). See [Sil1] O Chapter VI, Theorem 4.1 (a). From the proposition, we obtain the following corollary. This allows, for instance, to analyze the endomorphism ring of an elliptic curve over C in terms of the corresponding lattice. While in most cases, the endomorphism ring is just Z (i.e., every endomorphism has the form “multiplication by n” by some n ∈ Z), some elliptic curves admit more endomorphisms. Those are called elliptic curves with complex multiplication. It turns out that this property is closely related to arithmetic questions. We do not go into this further here, however, and content ourselves with stating the following corollary about homomorphisms between elliptic curves.

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Corollary 26.114. Let Λ1 , Λ2 ⊂ C be lattices. (1) Every morphism C/Λ1 → C/Λ2 which maps 0 to 0 is a group homomorphism. (2) The Riemann surfaces C/Λ1 and C/Λ2 are isomorphic as Riemann surfaces if and only if there exists an isomorphism of Riemann surfaces preserving the group structures, if and only if the lattices Λ1 and Λ2 are homothetic (i.e., there exists α ∈ C× with Λ2 = αΛ1 ). We therefore obtain a bijection between the set of homothety classes of lattices in C and the set of isomorphism classes of elliptic curves over C, which on the level of Riemann surfaces is given by mapping the homothety class of a lattice Λ to C/Λ. It is easy to see that every homothety class can be represented by a lattice of the form Z ⊕ Zτ for some ′ τ ∈ H, the upper complex half-plane.  Lattices Z ⊕ Zτ and Z ⊕ Zτ are homothetic if and a b +b only if there exists a matrix ∈ SL2 (Z) such that τ ′ = aτ cτ +d . Summarizing, we c d obtain the following corollary. Corollary 26.115. Let SL2 (Z) act on the complex upper half-plane H by   aτ + b a b . ·τ = c d cτ + d Then the map τ 7→ C/(Z ⊕ Zτ ) induces a bijection between the set SL2 (Z)\H and the set of isomorphism classes of compact Riemann surfaces of genus 1 with a fixed base point, which in turn is in bijection with the set of isomorphism classes of elliptic curves over C. One can show that one can equip SL2 (Z)\H with the structure of Riemann surface such that the projection H → SL2 (Z)\H is a morphism of Riemann surfaces. The theory of the j-invariant yields an isomorphism SL2 (Z)\H ∼ = C of Riemann surfaces. To conclude this section, let us discuss how in the case of genus 1, one can give a more direct proof of the algebraizability of compact Riemann surfaces (Corollary 26.26). From hindsight (say, the Theorem of Riemann-Roch) we know that a non-constant meromorphic function cannot have only a simple pole at a single point, but we expect that a meromorphic function regular outside a single point and with a double pole at that point should exist. Without loss of generality, we will look for such a function with the double pole located at the fixed point 0 of the given Riemann surface X. Via pullback along the universal cover, a function with this property corresponds to a Λ-invariant morphism f : C → P1 (C) (i.e., f (λz) = f (z) for all λ ∈ Λ) with a double pole at all points of Λ, and regular on C \ Λ. The Weierstraß ℘-function  X  1 1 1 − ℘(z) = 2 + z (z − λ)2 λ2 λ∈Λ\{0}

is a function with these properties. We omit the verification that the series converges absolutely and uniformly, locally on C \ Λ. Omitting all the squares in the definition would not give a convergent series; this gives a different perspective on the fact that a meromorphic function on X that is regular outside 0 and has a simple pole at 0 cannot exist. The existence of a non-constant meromorphic function implies that X is algebraizable, as we have seen in the proof of Corollary 26.26. In fact, in the situation at hand, we can be more precise. Namely, the ℘-function and its derivative ℘′ satisfy the equation

563 (℘′ )2 = 4℘3 − g2 ℘ − g3 , where g2 = 60

X λ∈Λ\{0}

1 , λ4

g3 = 140

X λ∈Λ\{0}

1 . λ6

Again, we omit the necessary (and not very difficult) calculations. We obtain that the map X → P2 (C), x 7→ (℘(x) : ℘′ (x) : 1), where 0 7→ (0 : 1 : 0), because ℘ has a pole of order 2, and ℘′ has a pole of order 3 at 0, is a morphism of Riemann surfaces. Its image is contained in the curve V+ (Y 2 Z − 4X 3 + g2 XZ 2 + g3 Z 3 ) by the differential equation for ℘ stated above. Since the algebraic curve V+ (Y 2 Z − 4X 3 + g2 XZ 2 + g3 Z 3 ) is a compact Riemann surface, equality follows. Moreover, the curve V+ (Y 2 Z − 4X 3 + g2 XZ 2 + g3 Z 3 ) is smooth for every lattice Λ, and the above map is an isomorphism from X onto its image, i.e., a closed embedding of X into P2 (C). The equation we find here is almost a Weierstraß equation. The smooth cubic with “distinguished” point (0 : 1 : 0) which it defines is an elliptic curve. This is even a group isomorphism, as follows from Corollary 26.114, or can be shown more directly by proving an “addition theorem” for the ℘-function. Since the group structure on the quotient C/Λ is obvious, this provides an approach to discovering the geometric description of the group law on smooth cubics in a natural way. Summarizing, we have the following result. Theorem 26.116. Let X be a compact Riemann surface of genus 1. There exists a lattice Λ ⊂ C such that X ∼ = C/Λ, and with g2 , g3 defined above (depending on Λ), X is isomorphic to the analytification of the algebraic elliptic curve V+ (Y 2 Z − 4X 3 + g2 XZ 2 + g3 Z 3 ) ⊂ P2C , as Riemann surfaces and as groups. (26.20) Hyperelliptic curves. In this section we study hyperelliptic curves, which share certain properties with elliptic curves, e.g., they admit a generically ´etale morphism to a smooth proper curve of genus 0, and can also be described by an explicit equation that has a form reminiscent of the Weierstraß equation of an elliptic curve. See Proposition 26.122 for a more precise statement. Definition 26.117. Let k be a field. A geometrically connected smooth proper curve C over k is called hyperelliptic, if g(C) ≥ 2 and if there exists a generically ´etale morphism of degree 2 from C onto a smooth proper curve of genus 0. Curves of genus 0 and of genus 1 also admit a generically ´etale morphism of degree 2 to a genus 0 curve, but are of exceptional nature as far as the considerations of this section are concerned, and are therefore excluded in the above definition. In the literature, sometimes curves of genus 1 are also called hyperelliptic. Over an algebraically closed field (or more generally, whenever C has a k-valued point) the morphism in the definition is a generically ´etale morphism of degree 2 from C to a genus 0 curve with a rational point, i.e., to the projective line P1k (Corollary 26.53). Sometimes in the literature a curve is called hyperelliptic, if it admits such a morphism to P1k . For arbitrary fields our definition is more general, and for instance has the advantage that the property of being hyperelliptic descends along extensions of the base field, see Lemma 26.120.

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Lemma 26.118. Let C be a hyperelliptic curve over a field k of genus g = g(C). Assume that there exists a generically ´etale morphism f : C → P1k of degree 2, and denote by L := f ∗ O(1) the corresponding line bundle on C. Then L ⊗(g−1) ∼ = ωC . In particular, the map C → P1k in the definition is unique up to composition with an automorphism of P1k and of C. given by the very ample line bundle Proof. Recall the Veronese embedding P1k ,→ Pg−1 k x1 : · · · : OP1k (g − 1). On homogeneous coordinates, it is given by (x0 : x1 ) 7→ (xg−1 : xg−2 0 0 g−1 g−1 1 x1 ). Cf. Exercise 13.8. The composition C → Pk → Pk of the morphism C → P1k with the Veronese embedding corresponds to the line bundle L ⊗(g−1) . Since the map C → P1k is surjective, and the image of P1k under the Veronese embedding is not contained in any hyperplane, the pullbacks of the standard basis vectors of H 0 (Pg−1 , OPg−1 (1)) are linearly k k

independent in H 0 (C, L ); it follows that ℓ(L ⊗(g−1) ) ≥ g. Since deg(L ⊗(g−1) ) = 2g − 2, we see, using the Theorem of Riemann-Roch, that L ⊗(1−g) ⊗ ωC has degree 0 and has non-trivial global sections, and hence is trivial (Remark 26.21). For the final part, note that by the above we can identify the given map C → P1k with the morphism from C onto the image of “the” morphism C → Pg−1 induced by ωC , up k to isomorphisms of the source and the target. Proposition 26.119. Let C be a hyperelliptic curve over a field k of genus g = g(C). The generically ´etale morphism C → P from C onto a smooth proper curve P of genus 0 is uniquely determined by C. In other words, there exists a unique subfield K of the function field K(C) such that K(C)/K is a quadratic Galois extension and K is the function field of a smooth proper curve of genus 0. The non-trivial Galois automorphism of K(C) over K defines an automorphism C → C of order 2, called the hyperelliptic involution of C. Proof. Let k ′ /k be a finite field extension such that C(k ′ ) ̸= ∅ and hence P (k ′ ) ̸= ∅. This implies that P ⊗k k ′ ∼ = P1k′ . By Lemma 26.118, we have that P ⊗k k ′ is the image of C under the morphism C ⊗k k ′ → Pg−1 given by the canonical sheaf of C ⊗k k ′ and is hence k′ independent of choices. We claim that K(P ) = K(P ) ⊗k k ′ ∩ K(C) ⊆ K(C) ⊗k k ′ . Clearly the left hand side is contained in the right hand side. The other inclusion holds because k is algebraically closed in K(C), C being geometrically integral (Proposition 5.51). It follows that the subfield K(P ) of K(C) depends only on C and not on the choice of morphism from C to a genus 0 curve. The final part follows from the equivalence of categories between normal proper curves and their function fields. On k-valued points, the hyperelliptic involution fixes the points of C that are ramified over P1k . For all other points, the fiber of the image in P1k has two elements, and those are swapped by the involution. As a consequence, we see that the property of being hyperelliptic is preserved by base change and descends along change of the base field. Lemma 26.120. Let k ′ /k be a field extension and let C be a geometrically connected, smooth proper curve over k. Then C is hyperelliptic as a curve over k if and only if C ⊗k k ′ is hyperelliptic as a curve over k ′ .

565 Proof. It is clear that C ⊗k k ′ is hyperelliptic, if C is. For the converse, we argue as follows. By passing to an extension of k ′ we may assume that C(k ′ ) ̸= ∅. Let g be the common genus of C and C ′ (Lemma 26.32) and consider the morphism C → Pg−1 given k by the canonical sheaf of C. Its base change to k ′ gives us a generically ´etale morphism of degree 2 from C ′ onto P1k′ . This implies that the image of C on Pg−1 is a smooth projective curve P of genus 0, k and that the morphism C → P is generically ´etale of degree 2. See [LoKl] O for a discussion of hyperelliptic relative curves. Next we will show that every curve of genus 2 is hyperelliptic. Proposition 26.121. Let k be a field and let C be a geometrically connected smooth proper curve of genus g(C) = 2. Then C admits a generically ´etale morphism of degree 2 to P1k and in particular is hyperelliptic. Proof. The canonical line bundle ωC satisfies deg(ωC ) = 2g(C)−2 = 2 and ℓ(ωC ) = g = 2, and hence defines a morphism C → P1k of degree 2. Since the genus does not change under purely inseparable morphisms and the degree is a prime number, this morphism must be generically ´etale. A hyperelliptic curve which admits a generically ´etale morphism of degree 2 to the projective line can be described by an equation of a particularly simple form (compare the description of an elliptic curve by a Weierstraß equation, Proposition 26.95). The situation is more delicate, however, since no hyperelliptic curve is a plane curve, see Remark 26.81 for the case of genus 2 curves and Exercise 26.15 for the general case. Proposition 26.122. Let C be a hyperelliptic curve over a field k of genus g = g(C). Assume that there exists a generically ´etale morphism f : C → P1k = Proj(k[T0 , T1 ]) of degree 2. (1) There exist polynomials p, q ∈ K[X] satisfying 2g + 1 ≤ max(deg(p), 2 deg(q)) ≤ 2g + 2, such that, for X = TT01 , (a) the affine open subscheme U := f −1 (D+ (T0 )) of C is isomorphic to U ′ := V (Y 2 + q(X)Y − p(X))

⊂ A2k = Spec k[X, Y ],

(b) the affine open subscheme V := f −1 (D+ (T1 )) of C is isomorphic to V ′ := V (Z 2 + X −(g+1) q(X)Z − X −2(g+1) p(X))

⊂ A2k = Spec k[X −1 , Z],

and C is isomorphic to the curve obtained by gluing U ′ and V ′ via Y = X g+1 Z. If char(k) ̸= 2, then we may arrange that q = 0. (2) In terms of this description the ramification divisor of f , seen as a closed subscheme of C, consists of the vanishing locus V (4p + q 2 ) ⊂ U ′ , and, if and only if deg(4p + q 2 ) < 2g + 2, the one point in C \ U . (See the proof for information on the contribution of each of these components to the degree of the ramification divisor.) Proof. We follow [Liu] O Section 7.4.3. Part (1). Consider the short exact sequence 0 → OP1k → f∗ OC → L → 0,

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where L is defined as the cokernel of the map on the left hand side (which is injective since f is dominant). Here f∗ OC is a locally free OP1k -module of rank 2 (it is locally free because f is finite and flat by Lemma 26.9). We claim that L is locally free of rank 1. Because of Corollary 11.19 it is enough to show that all fibers of L are 1-dimensional vector spaces. But for ξ ∈ P1k the fiber of OP1k → f∗ OC is the ring homomorphism κ(ξ) → (f∗ OC )(ξ) which is injective because its domain is a field. So the restrictions (f∗ OC )|D+ (T0 ) and (f∗ OC )|D+ (T1 ) are free over k[X] and k[X −1 ], respectively, and more precisely we can find bases 1, y ∈ Γ(U, OC ) and 1, z ∈ Γ(V, OC ) over k[X] and k[X −1 ], respectively. Over the intersection D+ (T0 ) ∩ D+ (T1 ), we can express y in terms of 1 and z, say y = a + bz,

a ∈ k[X, X −1 ], b ∈ k[X, X −1 ]× .

After a change of basis given by multiplying z by an element of k × , if necessary, we may assume that b = X r for some r. We write a = a+ + a− with a+ ∈ k[X] and a− ∈ k[X −1 ]. Then y − a+ = a− + T r z. If we had r ≤ 0, then this expression would lie in k[T ] ∩ k[T −1 ] = k which is impossible, since y ∈ K(C) \ k(T ). Therefore r ≥ 1. Replacing y by y − a+ and z by z + X −r a− we find bases 1, y of Γ(U, OC ) and 1, z of Γ(V, OC ) such that y = X r z, r ≥ 1, and expressing y 2 with respect to this basis we find y 2 + q(X)y = p(X)

for polynomials p, q ∈ k[X].

We define U ′ and V ′ as in the statement of the proposition. Dividing by X 2r we obtain z2 +

q(X) p(X) z= . Xr X 2r

Now z (considered as an element of K(C)) is integral over k[X −1 ], hence its minimal p(X) −1 polynomial has coefficients in k[X −1 ], i.e., q(X) ]. This implies that deg(q) ≤ X r , X 2r ∈ k[X r, deg(p) ≤ 2r. To get a lower bound on the degrees, we write q− = q/X r , p− = p/X 2r ∈ k[X −1 ] and evaluate the condition that V ′ is smooth (at the point(s) of V (X −1 )). If the characteristic of k is ̸= 2, then this smoothness condition is equivalent to X −1 not being a multiple zero of (4p + q 2 )/X 2r , which in turn is equivalent to deg(4p + q 2 ) ≥ 2r − 1. Altogether we obtain that 2r − 1 ≤ max(deg(p), 2 deg(q)) ≤ 2r in the case of characteristic ̸= 2. If k has characteristic 2, then the smoothness of V ′ at the points of V (X −1 ) is equivalent to ′ the condition that q− (0) ̸= 0 or p− (0)q− (0) ̸= p′− (0)2 . Therefore, 0 is not a zero of q− (so deg(q) ≥ r) or 0 is not a multiple zero of p− (so deg(p) ≥ 2r − 1). Thus we again obtain that 2r − 1 ≤ max(deg(p), 2 deg(q)) ≤ 2r. If k has characteristic ̸= 2, we can use a coordinate change of A2k to complete the square, and thus may arrange q = 0. At this point we have almost finished the proof of part (1). It only remains to show that r = g + 1. We will see this when we analyze the ramification divisor of f in the course of proving part (2). Part (2). To compute the ramification divisor of f|U , we compute the OU ′ -module ΩU ′ /A1k . Write f = Y 2 + q(X)Y − p(X). Then according to Remark 17.35, ΩU ′ /A1k is associated with the Γ(U ′ , OU ′ )-module

567 k[X, Y ]dY /(f dY, df ) ∼ = k[X, Y ]dY /(f dY, (2Y + q)dY ) ∼ = Γ(U ′ , OU ′ )/(2Y + q). If char k ̸= 2, then this is isomorphic to k[X]/(4p + q 2 ). If char k = 2, then it is isomorphic to Γ(U ′ , OU ′ )/(q) ∼ = Γ(U ′ , OU ′ ) ⊗k k[X]/(q). In either case, the ramification divisor of f|U has degree deg(4p + q 2 ). By the analogous computation for V ′ in place of U ′ , we see that the component of the ramification divisor supported on C \ U = V (X −1 ) ⊂ V ′ is (as a closed subscheme of V ) given by
k[X −1 , Z]/(2Z + q− , Z 2 + q− Z − p− ) (X −1 ) , where we again write q− = q/X r , p− = p/X 2r ∈ k[X −1 ]. 2 If the characteristic of k is ̸= 2, then this is isomorphic to (k[X −1 ]/(4p− +q− ))(X −1 ) , and ′ 2 since the smoothness of V implies that 4p− +q− is separable, it is zero- or one-dimensional 2 )(0) ̸= 0, or equivalently, is zero if deg(4p+q 2 ) < 2r over k, depending on whether (4p− +q− (and thus deg(4p + q 2 ) = 2r − 1) and is one-dimensional, if deg(4p + q 2 ) = 2r. In either case, the degree of the ramification divisor for f is 2r. If the characteristic of k is 2, the situation is different in the sense that the ramification divisor of f|U may be trivial and the ramification of f may be concentrated at the single point V (X −1 ) ⊂ V . See Exercise 26.18. To finish the computation in this case, write mq and mp for the multiplicity of X −1 = 0 as a zero of q− and p− (viewed as polynomials in k[X −1 ]), respectively. In other words, mq = r − deg(q), mp = 2r − deg(p). Then the component of the ramification divisor supported at V (X −1 ) is given by
k[X −1 , Z]/(2Z + q− , Z 2 + q− Z − p− ) (X −1 ) ∼ = (k[X −1 , Z]/(X −mq , Z 2 − uX −mp ))(X −1 ) for some unit u ∈ k[X −1 ]× (X −1 ) . Since max(deg(p), 2 deg(q)) ≥ 2r − 1 (by the smoothness of V , see above), mp and mq cannot both be > 1. If mq = 0, then the above ring is 0, so f is unramified at the point(s) of V (X −1 ). Now let mq > 0. If mp = 0, then the k-dimension of the above local ring is 2mq (and its residue class field is k or an inseparable quadratic extension field of k, depending on whether the constant term of u is a square in k, or not). If mq > 0 and mp = 1, then the above local ring is isomorphic to k[Z]/(Z 2mq ) and hence again has dimension 2mq over k. So the outcome is the same in all cases, and putting things together, we see that the ramification divisor of f has degree 2 deg(q) + 2mq = 2r in the characteristic 2 case, too. This in particular proves the description of the support of the ramification divisor stated in part (2). Furthermore, by the Riemann-Hurwitz formula (Corollary 26.67), the ramification divisor R of f has degree 2g + 2. Therefore we have also proved that r = g + 1. The proof can be simplified, if there exists a point 0 ∈ C(k) with ℓ(2[0]) = 2. If k is algebraically closed, then such a point always exists, but not in general (Example 26.126). See Exercise 26.17. Remark 26.123. (1) In the situation of the proposition, since g(C) > 1, the schematic closure in P2k of U ′ (as in (1)) is not smooth. In fact, smoothness would contradict the genus-degree formula Proposition 26.79.

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(2) To remedy the situation and produce an embedding of a curve C as in the proposition into projective space, we can proceed as follows. We first recall the notion of weighted projective space (see also Exercise 13.1), or rather the special case that we are going to use. Consider the polynomial ring A := k[T0 , T1 , Y ] with the grading that attaches degree 1 to T0 and to T1 , and degree g + 1 to Y . We set Pk (1, 1, g + 1) := Proj(k[T0 , T1 , Y ]). According to the way the Proj construction works, Pk (1, 1, g + 1) is covered by the open subschemes D+ (T0 ), D+ (T1 ) and D+ (Y ) which are the hspectra of i the Y homogeneous localizations A(T0 ) , A(T1 ) and A(Y ) . Now A(T0 ) = k TT10 , T g+1 is a 0 polynomial ring in two variables, and the same holds for A(T1 ) . So the corresponding open charts of Pk (1, 1, g + 1) are isomorphic to A2k . hThey cover all of Pk (1, i 1, g + 1) T g+1

T gT

T g+1

except for one point. On the other hand, A(Y ) = k 0Y , 0Y 1 , . . . , 1Y , and the   g+1 g T T T g+1 T local ring at the “origin”, i.e., at the prime ideal 0Y , 0Y 1 , . . . , 1Y is not regular. (If the characteristic of k does not divide g + 1 and k contains all (g + 1)-th roots of unity, then the affine scheme Spec A(Y ) is the quotient of A2k under the action of the cyclic group µg+1 (k) of (g + 1)-th roots of unity given by ζ · (t0 , t1 ) = (ζt0 , ζt1 ), in the sense of Proposition (12.27).) There is a natural morphism A3k \ {0} → Pk (1, 1, g + 1) induced by the inclusions of the homogeneous localizations in the usual localizations. Correspondingly, we obtain a description of the k-valued points Pk (1, 1, g + 1)(k) in terms of homogeneous coordinates, i.e., as equivalence classes (t0 : t1 : y), not all entries = 0, where (t0 : t1 : y) = (t′0 : t′1 : y ′ ) if and only if there exists λ ∈ k × with t′0 = λt0 , t′1 = λt1 , y ′ = λg+1 y. (And of course similarly for extension fields of k.) We have a closed immersion Pk (1, 1, g + 1) ,→ Pg+1 which in terms of homogeneous k g coordinates is given by (t0 : t1 : y) 7→ (tg+1 t : t : · · · : tg+1 : y). In terms of the Proj 0 0 1 1 construction we can describe this embedding as follows. As in Section (13.1), we write L A(g+1) := i Ai(g+1) . Then P(1, 1, g + 1) = Proj A(g+1) (Remark 13.7) and we have a surjective k-algebra homomorphism k[T0 , . . . , Tg+1 , Y ] ↠ A(g+1) , Ti 7→ T0g+1−i T1i , Y 7→ Y , which induces the above morphism. (3) The way in which the affine curves U ′ and V ′ have to be glued in order to obtain the curve C translates precisely to the statement, that we can embed C in the weighted projective plane Pk (1, 1, g + 1). Composing this embedding with the embedding P(1, 1, g + 1) ⊂ Pg+2 , we in particular obtain an embedding C ,→ Pg+2 whose k k ′ ∼ restriction to U = U ⊂ C is given by (x, y) 7→ (1 : x : · · · : xg+1 : y). The image of C in P(1, 1, g + 1) does not contain the unique singular point (0 : 0 : 1) of Pk (1, 1, g + 1), in other words, it is contained in the union of the two charts D+ (T0 ), D+ (T1 ). Let us compute the fiber of f over (0 : 1) ∈ P1k ; set-theoretically, this is the complement C \ U = V \ (V ∩ U ) of U in C. This fiber is the closed subscheme of V ′ P2r Pr given by the vanishing of X −1 . Writing p = i=0 ai X i , q = i=0 bi X i , we obtain f −1 ((0 : 1)) ∼ = k[X −1 , Z]/(Z 2 + X −r q(X)Z − X −2r p(X), X −1 ) ∼ = k[Z]/(Z 2 + br Z − a2r ). This is a finite k-scheme of degree 2 over k. Depending on whether the quadratic equation Z 2 + br Z − a2r = 0 has 0, 1 or 2 solutions in k, it has 0, 1 or 2 points with values in k. If C \ U has no k-valued point, then it consists of one point, and the

569 residue class field of that point is a quadratic extension of k. If the characteristic of k is ̸= 2 and q = 0, there is precisely one solution if and only if p has degree 2g + 1, and in this case this is a point where f is ramified. Remark 26.124. Let us discuss when an affine equation as in Proposition 26.122 defines a hyperelliptic curve. (1) Consider an affine plane curve U = V (Y 2 + q(X)Y − p(X))

⊂ A2k = Spec k[X, Y ]

with p, q ∈ k[X].

The curve U is smooth if and only if • characteristic ̸= 2: 4p + q 2 is separable, • characteristic = 2: q and (q ′ )2 p + (p′ )2 are coprime. Now assume that U is smooth. Let g ∈ N such that deg(p) ∈ {2g + 1, 2g + 2} q p and assume that g > 1 and deg(q) ≤ g + 1. Let V = V (Z 2 + X g+1 )⊂ Z − X 2(g+1) −1 Spec k[X , Z]. We can glue the affine curves U and V according to the identification Y = X g+1 Z and obtain a projective curve C (which we can embed in P(1, 1, g + 1)). Similarly as for U , one obtains a criterion for the smoothness of V . In characteristic ̸= 2, C is smooth if and only if 4p + q 2 is separable of degree ≥ 2g + 1. If C is smooth, then it is a hyperelliptic curve of genus g. Indeed, the projection (x, y) 7→ x defines a morphism U → A1k which extends to a morphism f : C → P1k . As in the proof of Proposition 26.122, one checks that the ramification divisor of f has degree 2g + 2, so that C indeed has genus g. In particular, then C has genus > 0, thus f is generically ´etale of degree 2. (2) If k is algebraically closed of characteristic ̸= 2, then every hyperelliptic curve is isomorphic to the proper normal model of an affine curve of the form V (y 2 − Q2g+1 i=1 (X − ai )) for pairwise different ai ∈ k, with g = g(C), and conversely every such affine curve with g > 1 defines a hyperelliptic curve of genus g. In this case, there is a unique point “at infinity” which is ramified for the hyperelliptic cover. Alternatively, we can work with 2g + 2 pairwise different elements a1 , . . . , a2g+2 ∈ k Q2g+2 and consider the affine curve V (y 2 − i=1 (X − ai )); then we obtain a hyperelliptic curve where the two points at infinity are unramified. (3) In particular, we see that there exist hyperelliptic curves of any genus > 1. Yet another characterization of hyperelliptic curves (over algebraically closed fields of characteristic ̸= 2) is given by the following proposition. Proposition 26.125. Let k be an algebraically closed field of characteristic ̸= 2, and let C be a connected smooth proper curve over k of genus g > 1. The following are equivalent. (a) The curve C is hyperelliptic. (b) If there exists an involution σ : C → C (i.e., an automorphism with σ 2 = id) with precisely 2g + 2 fix points. Proof. If C is hyperelliptic, let f : C → P1k be the covering map and let σ be the hyperelliptic involution. The fix points of σ are precisely the points of C where f is ramified (cf. Proposition 12.21). Since deg(f ) = 2, the ramification index at each ramified points is equal to 2, so the Riemann-Hurwitz formula (in the tamely ramified case, Corollary 26.67) shows that there are 2g + 2 such points.

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Conversely, given an involution with 2g + 2 fix points, the fix field of the corresponding field automorphism K(C) → K(C) is the function field of a connected smooth proper curve C ′ , and the inclusion of function fields defines a morphism C → C ′ of degree 2, which is necessarily generically ´etale since we have excluded characteristic 2. Hence, as before, the fix points of σ are the points where this cover is ramified. The degree of the ramification divisor is 2g + 2, and so g(C ′ ) = 0, once again using the formula of Riemann-Hurwitz. Example 26.126. A hyperelliptic curve C over a field k which admits a generically ´etale morphism of degree 2 to the projective line need not have a k-valued point. For example, consider a smooth proper model of the affine curve given by the equation y 2 = 2x6 − 4 over Q. Proposition 26.127. Let k be a field. Let f ∈ k[X, Y, Z] be a homogeneous polynomial of degree 4 such that C := V+ (f ) ⊂ P2k is a smooth curve. Then C is not hyperelliptic. In particular, there exist curves over k of genus 3 that are not hyperelliptic. ∼ OP2 (1)|C by Proposition 26.79, so ωC is very ample. But this Proof. We have ωC = k implies that C is not hyperelliptic by the discussion in the proof of Lemma 26.118. By the genus-degree formula, Proposition 26.79, C has genus 3. For the final part, it remains to show that over every field there exists a degree 4 homogeneous polynomial f defining a smooth plane curve. If the characteristic of k is ̸= 2, we can take f = X 4 + Y 4 + Z 4 . In characteristic 2 we can take f = X 3 Y + Y 3 Z + Z 4 . Remark 26.128. There exist curves over k of any genus ≥ 3 that are not hyperelliptic. And in a sense (which we do not make precise here) for fixed g ≥ 3 only a small part of all curves of genus g are hyperelliptic. (26.21) Curves of genus > 2. After discussing the classes of curves of genus 0, of genus 1 and of hyperelliptic curves, which include curves of genus 2, we now briefly look at non-hyperelliptic curves of genus > 2. In many respects, these behave differently. Proposition 26.129. Let k be a field and let C be a geometrically connected smooth proper curve over k of genus g. (1) If g ≥ 1, then the canonical bundle ωC is generated by global sections and hence defines a morphism C → Pg−1 . k (2) The following are equivalent: (i) The curve C has genus > 2 and is not hyperelliptic. (ii) The canonical bundle ωC is very ample. Proof. All properties can be checked after base change to an extension field (see the proof of Proposition 26.59, Proposition 14.58, Lemma 26.120), so we may assume that k is algebraically closed. Let K be a fixed canonical divisor on C. (1) By Proposition 26.58, it is enough to show that ℓ(K − [x]) = ℓ(K) − 1 = g − 1 for all closed points x ∈ C. From the Theorem of Riemann-Roch for the divisor K − [x], we obtain ℓ(K − [x]) − ℓ([x]) = 2g − 3 + 1 − g = g − 2,

571 and since g = ̸ 0, ℓ([x]) = 1. In fact, there is no non-constant rational function on C with at most a single pole at x, because the existence of such a function would entail that [x] is linear equivalent to some other divisor on C, necessarily also of degree 1. In view of Example 26.90, this would contradict that C has genus ̸= 0. (2), (i) ⇒ (ii). By part (1) the sheaf ωC is generated by its global sections, so by Proposition 26.58 it is enough to show that ℓ(K − [x] − [y]) = g − 2 for all x, y ∈ C(k). Applying the Theorem of Riemann-Roch to the divisor K − [x] − [y], we find that ℓ(K − [x] − [y]) − ℓ([x] + [y]) = 2g − 4 + 1 − g = g − 3. Furthermore, we have 1 ≤ ℓ([x] + [y]) ≤ 2 (cf. Remark 26.55 (1)). If we had ℓ([x] + [y]) = 2, then the line bundle OC ([x] + [y]) would define a rational map which we could extend to a morphism C → P1k of degree 2. Since g(C) > 0, this morphism cannot be purely inseparable, and thus C had to be hyperelliptic. (ii) ⇒ (i). If ωC is ample, then it has positive degree, and hence g ≥ 2. Since all curves of genus 2 are hyperelliptic by Proposition 26.121, it is enough to show that the canonical bundle of a hyperelliptic curve if not very ample. But we have seen this in the proof of Lemma 26.118 – in fact, the canonical bundle gives rise to a map C → Pg−1 which factors k through the hyperelliptic cover and therefore is not injective. Definition 26.130. Let C be a geometrically connected smooth proper curve over a field k such that ωC is very ample. We call the closed embedding C → Pg−1 that is defined by k ωC the canonical embedding of C.

Vector bundles on curves In this part of the chapter we study vector bundles on smooth proper geometrically connected curves. The main result (Theorem 26.156) is that every vector bundle E has a unique filtration, called the Harder-Narasimhan filtration, E = E0 ⊋ E1 ⊋ ··· ⊋ Em = 0 by subbundles such that E i−1 /E i is semistable of slope λi where λ1 < λ2 < · · · < λm are rational numbers, see Definition 26.148 for the definition of semistability and Definition 26.142 for the definition of the slope of a vector bundle. Moreover, as the second main result (Proposition 26.152) we show that the category of semistable vector bundles of a fixed slope is abelian and every object in that category has a Jordan-H¨ older series whose successive quotients are so-called stable vector bundles (Definition 26.148). As a warm-up we start by giving a new proof of the classification of vector bundles on P1k using cohomological methods. Then we state a version of the theorem of Riemann-Roch for vector bundles on arbitrary proper curves and prove it in the case that the curve is normal. We then introduce semistability and stability of vector bundles and investigate the category of semistable vector bundles of a fixed slope. Finally we prove the existence and uniqueness of the Harder-Narasimhan stratification.

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(26.22) Vector bundles on P1k . Recall that a vector bundle E on a scheme X is a locally free OX -module of finite rank. A vector bundle of constant rank 1 is called a line bundle. Definition 26.131. An OX -submodule F of a vector bundle E is called a subbundle if E /F is a vector bundle. Then F is a vector bundle and it is locally on X a direct summand of E (Proposition 8.10). Now let k be a field. In Section (11.17) we have shown that every vector bundle on P1k is isomorphic to a sum of line bundles. There we used that every vector bundle can be glued from two copies of a free module on each of the two standard charts of P1k in order to reduce the problem to an explicit statement about matrices with entries in k[T, T −1 ]. Here we will give another proof of this theorem. In fact, interpreting Ext1 groups as Yoneda-Ext-groups (Proposition F.220), we will be able to solve the problem using cohomological methods. We start by stating some facts, which we have essentially already seen, about coherent modules on Dedekind schemes. Remark 26.132. Let C be a connected normal locally noetherian scheme of dimension 1. Such a scheme C is automatically integral and regular (Proposition 6.40) and we denote by η ∈ C its generic point. As C is regular of dimension 1, all local rings OC,c for c ∈ C are principal ideal domains. We obtain the following assertions. (1) A coherent OC -module is a vector bundle if and only if it is torsion free since finitely generated torsion free modules over a principal ideal domain are free. (2) For a coherent OC -module F we denote by Ftors its torsion submodule, i.e., Ftors (U ) = F (U )tors = Ker(F (U ) → Fη ) for every ∅ ̸= U ⊆ C open affine. Then F /Ftors is torsion free and hence a vector bundle. (3) Assertion (1) implies that every coherent submodule F of a vector bundle E on C is again a vector bundle. It is not necessarily a subbundle, but it is always contained in a subbundle F ⊆ E of the same rank, namely F := Ker(E → (E /F )/(E /F )tors ), which is called the saturation of F in E . (4) Let E be a vector bundle on C. Then F 7→ Fη yields a bijection between subbundles F of E and K(C)-subvector spaces of Eη . The inverse map is given by attaching to W ⊆ Eη the subbundle F of E with F (U ) = W ∩ E (U ) for every open affine subset U ⊆ C. (5) Let E ̸= 0 be a vector bundle on C. Then there exists an exact sequence of vector bundles 0 −→ E ′ −→ E −→ L −→ 0, where L has rank 1. Indeed, line bundle quotients E → L are parametrized by the projective bundle P(E ), i.e., we have to show that P(E )(C) ̸= ∅. Let U ⊆ C be an open dense subset such that E |U ∼ = OUn . Then there exists a section of P(E ) defined over U . As P(E ) is proper over C and C is normal and of dimension 1, we can extend this section to a section C → P(E ) by Proposition 15.5.

573 (6) Suppose that C admits an ample line bundle2 , e.g., if C is a curve over a field (Theorem 26.16). For every coherent OC -module F there exists a short exact sequence 0 −→ E −1 −→ E 0 −→ F −→ 0, where E −1 and E 0 are vector bundles. Indeed, by our assumption there exist a vector bundle E 0 and a surjection u : E 0 → F (Proposition 22.57). By (3), E −1 := Ker(u) is a vector bundle. Theorem 26.133. Let k be a field. Let E be a locally free sheaf of finite rank r on P1k . Then there exist uniquely determined integers d1 ≥ d2 ≥ · · · ≥ dr such that E ∼ =

r M

OP1k (di ).

i=1

Specializing to the case of locally free modules of rank 1 the theorem gives a new proof of the isomorphism Z ∼ = Pic(P1k ), d 7→ OP1k (d), proved by elementary methods in Example 11.45, see also Proposition 24.69 for a description of Pic(PnS ) for an arbitrary base scheme S and for all n ≥ 1. We cannot expect a generalization of Theorem 26.133 to more general base schemes for vector bundles of rank > 1. For the existence we start with the following lemma which gives us a way of finding a candidate for the integer d1 . Lemma 26.134. Let k be a field, let C be a proper integral curve over k and let L be a very ample line bundle on C. Let E = ̸ 0 be a locally free sheaf of finite rank n on C. As usual, we write E (d) := E ⊗OC L ⊗d for d ∈ Z. There exists an integer d1 such that H 0 (C, E (−d1 )) ̸= 0 and H 0 (C, E (−d)) = 0 for all d > d1 . Proof. Since L is very ample, for d sufficiently negative, E (−d) is generated by its global sections (Corollary 23.2). In particular we then have H 0 (C, E (−d)) ̸= 0. It remains to show that there exists an integer e such that H 0 (C, E (−d)) = 0 for all d ≥ e. For this we apply the remark at the beginning of the proof to the dual E ∨ . Hence we find an integer e such that for all d ≤ e + 1 there exists an N and a surjection OCN ↠ E ∨ (−d) and hence an injection E (d) ,→ OCN . Thus for all d ≤ e, there exists an N such that E (d) injects into OCN (−1) which has no global sections (because deg(L ∨ ) < 0 by Proposition 26.57). It follows that for d ≥ e, H 0 (C, E (−d)) = 0. On the other hand, since H 0 (C, L ) ̸= 0 (L being very ample, hence generated by its global sections), H 0 (C, E (−d)) = 0 implies H 0 (C, E (−d − 1)) = 0. Altogether we obtain the desired result. Proof. (of Theorem 26.133) Consider d1 as in Lemma 26.134 (for C = P1k and L = OP1k (1)) and let s = ̸ 0 be an element of H 0 (P1k , E (−d1 )). We can view s as a non-zero homomorphism OP1k (d1 ) → E . The homomorphism OP1k (d1 ) → E is injective (this is true for the stalks at the generic point, and the sections over any non-empty open embed into this stalk). Denoting by F the cokernel, we obtain a short exact sequence (*) 2

0 → OP1k (d1 ) → E → F → 0.

This hypothesis is in fact superfluous, see Exercise 23.21.

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We claim that F is locally free (necessarily of rank rk(E ) − 1). To show this, by Proposition 8.10, it is enough to show that for every point x ∈ P1k the induced sequence of the fibers is also exact, or in other words that the global section of E (−d1 ) giving rise to the sequence never vanishes. But if this section had a zero at a (closed) point x, then the image of the corresponding map OP1k → E (−d1 ) would be contained in E (−d1 ) ⊗ J ⊂ E (−d1 ) ⊗ OP1k , where J denotes the ideal sheaf defining the closed subscheme {x}. Since J ∼ = OP1k (−m) with m = [κ(x) : k], we would obtain a non-zero global section of E (−d1 − m), a contradiction to the definitionLof d1 . r By induction hypothesis, there is an isomorphism F ∼ = i=2 OP1k (di ) for integers d2 ≥ · · · ≥ dr . Since E (−d1 −1) has no global sections, the same must hold for F (−d1 −1). In fact, tensoring the above short exact sequence (*) with OP1k (−d1 − 1), the long exact cohomology sequence shows that 0 = H 0 (P1k , E (−d1 − 1)) ∼ = H 0 (P1k , F (−d1 − 1)) ∼ =

r M

H 0 (P1k , OP1k (di − d1 − 1))

i=2 0

(P1k , OP1k (−1))

1

(P1k , OP1k (−1))

Here we use that H = H = 0 by Theorem 22.22. We conclude that d1 ≥ d2 ≥ · · · ≥ dr since a line bundle OP1k (di − d1 − 1) has no non-zero global sections if and only if di − d1 − 1 < 0, i.e., di ≤ d1 . But then M M H 1 (P1k , OP1k (d1 − ei )) = 0. Ext1P1 (OP1 , OP1k (d1 − ei )) ∼ Ext1P1 (F , OP1k (d1 )) ∼ = = k

k

i

i

This proves that the short exact sequence (*) splits, and we obtain the desired isomorphism between E and a direct sum of line bundles. Let us show the uniqueness of the decomposition. By the existence part of the theorem we have a decomposition M (*) E ∼ Vd ⊗k O(d) = d∈Z

for finite-dimensional k-vector spaces Vd of which all but finitely many are zero. We have to show that E determines dim(Vd ). For every α ∈ Z let E α be the image of the evaluation map H 0 (P1k , E (−α)) ⊗k O(α) → E . We obtain a decreasing filtration (26.22.1)

· · · ⊇ E −1 ⊇ E 0 ⊇ E 1 ⊇ . . .

of E with E α = E for α ≪ 0 since E (−α) is globally generated for small α and with E α = 0 for α ≫L0 since H 0 (P1k , E (−α)) = 0 for large α by Lemma 26.134. For E as in (*) we have E α ∼ = d≥α Vd ⊗k O(d) which shows dim(Vd ) = rk(E d /E d+1 ). The proof shows that every vector bundle E on P1k has a unique filtration (26.22.1) whose d-th graded step E d /E d+1 is (non-canonically) isomorphic to a direct sum of copies of the line bundle O(d). Moreover, the filtration is split, though the splitting homomorphisms are not unique. This filtration is called the Harder-Narasimhan filtration. When formulated in a suitable way, this statement can be generalized to arbitrary smooth proper geometrically connected curves as we will see in Section (26.25) below. (26.23) The Riemann-Roch theorem for vector bundles on curves. Let X be a scheme and let E be a finite locally free OX -module. Recall that det(E ) := Vrk(E ) (E ) denotes its determinant line bundle (Remark 17.58).

575 Definition 26.135. Let C be a proper curve over a field k and let E be a vector bundle on C. Then deg(E ) := deg(det(E )) is called the degree of E . We then have deg(E ) = χ(C, det(E )) + g − 1 by Proposition 26.46, where g denotes the genus of C. As with the Euler characteristic, the degree depends on k and we also write degk (E ) instead of deg(E ). Remark 26.136. Let C be a proper curve over a field k and let E be a vector bundle on C. (1) Let k ′ be a field extension of k and let E ′ be the pullback of E to C ⊗k k ′ . Then degk′ (E ′ ) = degk E since the degree of line bundles is preserved under passing to field extensions (Remark 15.29) and since forming exterior powers is compatible with pullback (Proposition 7.49). (2) Let L be a line bundle and suppose that E has constant rank r. Then det(E ⊗OC L ) = det(E ) ⊗ L ⊗r . Hence we have (26.23.1)

deg(E ⊗ L ) = deg(E ) + r deg(L ).

See also Exercise 26.21 for a generalization. ∨ (3) Since det(E ∨ ) = det(E )∨ and deg(L ) = − deg(L ) for every line bundle, we find that (26.23.2)

deg(E ∨ ) = − deg(E ).

Proposition 26.137. Let k be a field, let C ′ and C be proper integral curves over k and let f : C ′ → C be a finite dominant morphism. Let E be a vector bundle on C. Then deg(f ∗ E ) = deg(f ) deg(E ) Proof. As forming exterior powers commutes with pullback f ∗ it suffices to show the equality for line bundles. This is a special case of Proposition 23.84. Next, we prove a version of the Riemann-Roch theorem for vector bundles. It is in fact a generalization of the equality χ(L ) = deg(L ) + 1 − g for line bundles L on a curve (Proposition 26.46). Theorem 26.138. Let C be a proper curve over a field k of genus g and let E be a vector bundle on C of constant rank. Then (26.23.3)

χ(C, E ) = deg(E ) + rk(E )(1 − g).

We will give the proof only if C is normal, which is the only case that we will use in the sequel. For the general case see [Sta] 0DJ5. Proof. (if C is normal) We proceed by induction on n := rk(E ). As remarked above, we know the equality already for vector bundles of rank 1. As we assumed that C is normal, we have seen in Remark 26.132 (5) that there exists a short exact sequence 0 → F → E → L → 0 with L a line bundle and F a vector bundle of rank rk(E ) − 1. By induction we know the description of the Euler characteristic already for L and F , we obtain it for E , since the rank, the degree, and the Euler characteristic of vector bundles on C are additive in short exact sequences of vector bundles (for the degree one uses (17.10.5)).

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Now we can use Serre duality as in Theorem 26.48 to express H 1 (C, E ) as global sections of another vector bundle. Corollary 26.139. Let k be a field and let C be a proper curve over k of genus g. Let ωC be the dualizing sheaf. Then for every vector bundle E of constant rank one has dimk H 0 (C, E ) − dimk H 0 (C, E ∨ ⊗ ωC ) = deg(E ) + rk(E )(1 − g). Remark 26.140. Let C be a proper curve of genus g over a field k. Suppose that C is connected to simplify the exposition (for non-connected curves one should regard the rank as a locally constant function on C as usual, and redefine the degree as a locally constant function, by considering each connected component individually). (1) The rank and the degree of vector bundles on C are additive in short exact sequences of vector bundles (for the degree one uses (17.10.5)). Hence by Remark 23.51 we obtain induced maps rk, deg, χ(C, −) : K0 (C) −→ Z. and the Riemann-Roch formula (26.23.3) holds for every perfect complex E on C. (2) Now suppose that C is in addition normal and hence regular. Then every complex in b b Dcoh (C) is perfect and we have K0 (C) = K0 (Dcoh (C)) by Proposition 23.55. Hence b we can define rk(E ) and deg(E ) for every E in Dcoh (C) and obtain (26.23.3) for every b E ∈ Dcoh (C), in particular for every coherent OC -module. These can be described as follows. As C is normal and connected, it is integral. Denote by η the generic point of C. Let F be a coherent OC -module, then it follows from Remark 26.132 (6) that the rank of F is given by rk(F ) = dimK(C) (Fη ), b where K(C) = OC,η field of C. If F is an arbitrary object in Dcoh (C), P is the function i i we have rk(F ) = i∈Z (−1) rk(H (F )) and therefore for every point c ∈ C

(26.23.4)

rk(F ) = χ(F ⊗OC K(C)) = χ(F ⊗L OC κ(c)),

where the second equality holds since the Euler characteristic is constant by Proposition 23.117 (3). The degree is given by Riemann-Roch (26.23.5)

deg(F ) = χ(C, F ) − rk(F )(1 − g),

b F ∈ Dcoh (C).

Example 26.141. Let F be a coherent OC -module with finite support or, equivalently, of rank 0. By Riemann-Roch we have deg(F ) = dim H 0 (C, F ) ≥ 0. Moreover, one has F = ̸ 0 if and only if deg(F ) > 0 since F is supported on an affine subscheme of C. See also Exercise 26.20 for a more concrete description of deg(F ). As deg is additive in exact sequences we obtain the following assertions. ̸ 0 be a coherent OC -module and let E ′ ⊊ E be a coherent submodule of the (1) Let E = same rank. Then E /E ′ has rank 0 and therefore deg(E ′ ) < deg(E ). ̸ 0 be a coherent OC -module and let E ′′ be a nontrivial coherent quotient (2) Let E = module of the same rank. Then Ker(E → E ′′ ) has rank 0 and it follows that deg(E ′′ ) < deg(E ).

577 (26.24) Semistable vector bundles. In this section C will denote a smooth proper geometrically connected curve over a field k. We denote by g its genus. In Theorem 26.133 we have seen that every vector bundle on P1k is the direct sum of line bundles. This result does not generalize to curves of higher genus. But every vector bundle still has a canonical filtration whose graded pieces are of a “simple” form (semistable bundles in the sense of Definition 26.148). This filtration is called the Harder-Narasimhan stratification, see Theorem 26.156 below. The following definition will be central for the construction of the Harder-Narasimhan stratification. Definition 26.142. Let E ̸= 0 be a vector bundle on C. The rational number µ(E ) := deg(E )/ rk(E ) is called the slope of E . We also define the zero bundle to have every rational number as a slope, i.e., µ(0) = λ holds for every rational number λ. b (C) with rk(E ) ̸= 0. By Remark 26.140 (2), the slope µ(E ) is defined for all E in Dcoh

Remark 26.143. Let E ̸= 0 be a vector bundle on C. (1) Let L be a line bundle. Then (26.23.1) shows that (26.24.1)

µ(E ⊗OC L ) = µ(E ) + deg(L ).

(2) By (26.23.2) we have µ(E ∨ ) = −µ(E ). Example 26.144. Let C = P1k . For Ldr ∈ Z one has deg(OC (d)) = d and therefore µ(O(d)) = d. More generally, if E ∼ = i=1 OC (di ) is a vector bundle of rank r over P1k (Theorem 26.133), then deg(E ) = d1 + . . . dr and µ(E ) = (d1 + · · · + dr )/r. Let us study how the slope behaves in short exact sequences. Remark 26.145. Let 0 −→ F ′ −→ F −→ F ′′ −→ 0 be an exact sequence of coherent OC -modules. ̸ 0 and rk(F ′′ ) = 0, then rk(F ′ ) = rk(F ) and Example 26.141 (1) shows (1) If F ′′ = that µ(F ′ ) < µ(F ) if rk(F ) ̸= 0. (2) If F ′ ̸= 0 and rk(F ′ ) = 0, then rk(F ′′ ) = rk(F ) and Example 26.141 (2) shows that µ(F ′′ ) < µ(F ) if rk(F ) ̸= 0. For coherent OC -modules of rank > 0 one has the following result. Lemma 26.146. Let 0 → E ′ → E → E ′′ → 0 be an exact sequence of coherent OC modules of rank > 0. Then always one of the following cases occurs. µ(E ′ ) = µ(E ) = µ(E ′′ ), µ(E ′ ) < µ(E ) < µ(E ′′ ), µ(E ′ ) > µ(E ) > µ(E ′′ ).

or or

Proof. This is completely elementary. Set d′ := deg(E ′ ), r′ := rk(E ′ ), d′′ := deg(E ′′ ), and r′′ := rk(E ′′ ). Then rk(E ) = r′ + r′′ and deg(E ) = d′ + d′′ . If µ(E ′ ) = µ(E ′′ ), then d′ = d′′ r′ /r′′ and hence

578 (*)

26 Curves µ(E ) =

d′′ r′ /r′′ + d′′ d′′ (r′ + r′′ )/r′′ d′′ = = ′′ = µ(E ′′ ). ′ ′′ ′ ′′ r +r r +r r

If µ(E ′ ) < µ(E ′′ ), then d′ < d′′ r′ /r′′ and we then in (*) instead of the first equality we have a < which shows µ(E ) < µ(E ′′ ). Similarly one obtains µ(E ) > µ(E ′ ) in this case, using that d′′ > d′ r′′ /r′ . The argument that µ(E ′ ) > µ(E ′′ ) implies µ(E ′ ) > µ(E ) > µ(E ′′ ) is then the same with opposite inequalities. Remark 26.147. By the Riemann-Roch theorem (in the form of Theorem 26.138) we can describe the slope of a non-zero vector bundle E on C also by (26.24.2)

µ(E ) =

χ(C, E ) + g − 1 ≤ dimk H 0 (C, E ) + g − 1. rk(E )

Hence if 0 ̸= F is any coherent OC -submodule (and hence F is a vector bundle of rank > 0 by Remark 26.132 (3)), then we have µ(F ) ≤ dimk H 0 (C, F ) + g − 1 ≤ dimk H 0 (C, E ) + g − 1, and we see that the slope of submodules of E is bounded. If the slopes of all submodules are bounded by the slope of E , then we call E semistable: Definition and Proposition 26.148. Let E be a vector bundle on C. Then E is called semistable (resp. stable) if the following equivalent conditions are satisfied (resp. if the following equivalent conditions are satisfied and E ̸= 0). (i) For all subbundles 0 ̸= E ′ ⊊ E one has µ(E ′ ) ≤ µ(E ) (resp. µ(E ′ ) < µ(E )). (ii) For all coherent submodules 0 ̸= E ′ ⊊ E one has µ(E ′ ) ≤ µ(E ) (resp. µ(E ′ ) < µ(E )). (iii) For any surjective homomorphism E → E ′′ of vector bundles with E ′′ ̸= 0 one has µ(E ′′ ) ≥ µ(E ) (resp. µ(E ′′ ) > µ(E )). (iv) For any surjective homomorphism E → E ′′ of coherent OC -modules with rk(E ′′ ) > 0 one has µ(E ′′ ) ≥ µ(E ) (resp. µ(E ′′ ) > µ(E )). The zero bundle is semistable but not stable according to our definition. Proof. We may assume that E ̸= 0. We show the assertion for semistability. The proof for stability is the same with strict inequalities instead. The implications of “(i) ⇔ (iii)” and “(ii) ⇒ (iv)” follow from Lemma 26.146 and it is clear that (ii) implies (i) and that (iv) implies (iii). It remains to show that (i) implies (ii). Let 0 ̸= E ′ ⊆ E be a coherent submodule that is not a subbundle and let E ′ be its saturation (Remark 26.132 (3)). Then E ′ is a subbundle of E containing E ′ with rk(E ′ ) = rk(E ′ ). Therefore µ(E ′ ) < µ(E ′ ) ≤ µ(E ) by Remark 26.145. Remark 26.149. Let L be a line bundle on C. (1) Then L is clearly stable since there exist no non-trivial subbundles of L . Its slope is given by µ(L ) = χ(C, L ) + g − 1 by (26.24.2). (2) Let E be a vector bundle on C. As tensoring by L induces a bijection between subbundles of E and of E ⊗ L and since µ(E ′ ⊗ L ) = µ(E ′ ) + deg(L ) for every such subbundle E ′ (26.24.1), we see that E is semistable (resp. stable) of slope λ if and only if E ⊗ L is semistable (resp. stable) of slope λ + deg(L ).

579 (3) Let E be a semistable vector bundle. Then every direct summand E ′ of E can be viewed as a submodule or as quotient of E . Therefore E ′ is again semistable of the same slope as E . Proposition 26.150. Let E and F be semistable vector bundles on C with µ(E ) > µ(F ). Then HomOC (E , F ) = 0. ̸ 0 be its image. As a Proof. Let u : E → F be a non-zero OC -linear map and let G = subsheaf of F this is a vector bundle. Then semistability of E and of F show µ(E ) ≤ µ(G ) ≤ µ(F ). Lemma 26.151. Let E be a semistable vector bundle of slope λ. ̸ 0 be a coherent submodule of E of slope λ. Then F is semistable and a (1) Let F = subbundle of E . (2) Let G be a coherent quotient of E of rank ̸= 0 and of slope λ. Then G is a semistable vector bundle. Proof. We show (1). By Remark 26.132 (1), F is a vector bundle. By (ii) of the Definition 26.148 of semistability, it is clear that F is semistable. Let F be the saturation of F in E (Remark 26.132 (3)). If F = ̸ F , then µ(F ) < µ(F ) ≤ µ(E ) by Remark 26.145 which contradicts that µ(F ) = µ(E ). Hence F = F is a subbundle. Let us show (2). As E is semistable, the vector bundle quotient G /Gtors of E has slope ≥ λ. On the other hand, Remark 26.145 shows that µ(G /Gtors ) ≤ µ(G ) = λ with equality if and only if Gtors = 0. Therefore G is a vector bundle. Moreover, every quotient vector bundle of G is also a quotient vector bundle of E and hence has slope ≥ λ. Therefore G is semistable. For the next statement, recall that we denote by (Coh(C)) the abelian category of coherent OC -modules. Proposition 26.152. Fix λ ∈ Q and denote by (Vectλ (C)) the full subcategory of (Coh(C)) whose objects are the semistable vector bundles of slope λ. (1) Then (Vectλ (C)) is a plump subcategory (Definition F.43) of (Coh(C)). In particular, it is an abelian category. (2) Every object of (Vectλ (C)) has finite length (Definition F.41) and therefore admits a composition series whose graded pieces are simple objects of (Vectλ (C)). The simple objects of (Vectλ (C)) are the stable vector bundles of slope λ. Here, by a simple object in an abelian category we mean an object that has no nontrivial subobjects (see Section (F.8)). Note that in the literature on vector bundles on curves, often a vector bundle E is called simple, if Endk (E ) = k; as pointed out above, this notion differs from ours. Proof. Let us show (1). Let u : E → F be a map of semistable vector bundles of slopes λ. We want to show that Ker(u) and Coker(u), formed in the category of coherent OC modules, are semistable vector bundles of slope λ. We may assume that u ̸= 0. As E and F are both semistable of slope λ we have λ = µ(E ) ≤ µ(Im(u)) ≤ µ(F ) = λ.

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Hence Im(u) is a non-zero submodule of F with the same slope. Therefore Im(u) is a subbundle and semistable by Lemma 26.151. This shows that Coker(u) is a vector bundle. To show that Ker(u) (resp. Coker(u)) is semistable of slope λ, we may assume that Ker(u) ̸= 0 (resp. Coker(u) ̸= 0). Using Lemma 26.146 applied to the exact sequences 0 → Im(u) → F → Coker(u) → 0 and 0 → Ker(u) → E → Im(u) → 0 we see that µ(Ker(u)) = µ(Coker(u)) = λ. Therefore Lemma 26.151 shows that Ker(u) and Coker(u) are semistable. It remains to show that if 0 → E ′ → E → E ′′ → 0 is an exact sequence of vector bundles with E ′ and E ′′ semistable of slope λ, then E is semistable of slope λ. We may assume that E ′ and E ′′ are non-zero. By Lemma 26.146 we know that µ(E ) = λ. Let F ⊆ E be any non-zero subbundle. Let F ′ = F ∩ E ′ and let F ′′ be the image of F in E ′′ . As E ′ and E ′′ are semistable of slope λ, we see that µ(F ′ ), µ(F ′′ ) ≤ λ. Using the exact sequence 0 → F ′ → F → F ′′ → 0 we see by Lemma 26.146 that µ(F ) ≤ λ. We now show (2). We have seen that every proper subobject in (Vectλ (C)) is a subbundle and therefore has strictly smaller rank. This shows that every object of (Vectλ (C)) has finite length. The vector bundles in (Vectλ (C)) that have no non-trivial subobject in (Vectλ (C)) are precisely those for which every subbundle has slope < λ, i.e., the stable vector bundles. Corollary 26.153. Let E be a stable vector bundle on C. Then EndOC (E ) is a finitedimensional skew field over k. If k is algebraically closed, it follows that EndOC (E ) = k. In general it can happen that the skew field is not commutative (Exercise 26.26) or that the center of EndOC (E ) is strictly larger than k (Remark 27.274). Proof. As EndOC (E ) = Γ(C, E ∨ ⊗OC E ), it is a finite-dimensional k-vector space since C is proper over k. By Proposition 26.152, every endomorphism u of E has semistable kernel and cokernel of the same slope as E . As E is stable, the kernel and the cokernel have to be trivial, i.e., u = 0 or u is an automorphism. Even if k is algebraically closed, there exist semistable, non-stable vector bundles E with EndOC (E ) = k (Exercise 26.23). This can only happen if the genus g of C is ≥ 2: If g = 0, then C = P1k and it by Theorem 26.133 a vector bundle E on P1k is stable if and only if it is a line bundle if and only if End(E ) = k. If g = 1, then C is an elliptic curve and it follows from Atiyah’s classification of vector bundles on elliptic curves (Theorem 27.271 below) that a vector bundle E is stable if and only if End(E ) = k (Corollary 27.275). (26.25) Harder-Narasimhan filtration. We continue to denote by k a field and by C a proper smooth geometrically connected curve over k. Let g be the genus of C. In Proposition 26.152 we have seen that the semistable vector bundles of a fixed slope form an abelian category. We will now show that an arbitrary vector bundle always has a filtration whose graded pieces are semistable. This filtration is canonically indexed by the rational numbers. Hence let us first introduce the following general notion. Definition and Remark 26.154. Let E be a vector bundle on C. An R-filtration of E is a decreasing map Fil : R −→ {subbundles of E },

α 7→ Filα (E ),

581 where the set of subbundles is partially ordered by inclusion, such that the following conditions are satisfied. (1) There exist α, β ∈ R such that Filα (E ) = E and Filβ (E ) = 0, i.e., the filtration is exhaustive and separating. T (2) For every α ∈ R one has βα Filβ (E ) and set grα (E ). As each Fil (E ) := Fil (E )/ Fil strict subbundle has a smaller rank, there are only finitely many α with grα Fil (E ) ̸= 0. These are called the jumps of the filtration. If I ⊆ R, we call Fil an I-filtration if grα Fil (E ) = 0 for all α ∈ R \ I. Definition 26.155. Let E be a vector bundle on C. A Harder-Narasimhan filtration or shorter HN filtration of E is a Q-filtration α 7→ HNα (E ) of E such that grα HN (E ) is semistable of slope α for all α ∈ Q. The main result is the following. Theorem 26.156. Every vector bundle E on C has a unique Harder-Narasimhan filtration. Moreover, it is functorial in the following sense: For every OC -linear map u : E → F of vector bundles and for all α ∈ R one has u(HNα (E )) ⊆ HNα (F ). Proof. Let us show the existence of a Harder-Narasimhan filtration such that the maximal jump equals the maximal slope λ of non-zero submodules of E . Such a λ exists since the slope of all non-zero submodules is bounded by Remark 26.147 and the possible denominators are also bounded by rk(E ). We proceed by induction on rk(E ). If E is a line bundle, and more generally if E is semistable of some slope µ, we can set HNα (E ) = E for α ≤ µ and HNα (E ) = 0 for α > µ. If E is not semistable, choose a coherent OC -submodule 0 ̸= F ⊂ E such that F has slope λ and has maximal rank among all submodules that have slope λ. Then F is semistable by construction and it is equal to its saturation F in E (otherwise, µ(F ) > µ(F ) by Remark 26.145 (1)). Hence F is a subbundle of E . Every non-zero submodule of the vector bundle E /F is of the form G /F for some submodule G of E with rk(G ) > rk(F ). Hence µ(G ) < µ(F ) and therefore µ(G /F ) < µ(F ) by Lemma 26.146. By induction hypothesis, E /F has a HN filtration such that max{ α ; grα HN (E /F ) ̸= 0 } < λ. Let p : E → E /F be the canonical map. Setting HNα (E ) := p−1 (HNα (E /F )) for α ≤ λ and HNα (E ) = 0 for α > λ we obtain a HN filtration of E . It remains to show that HN filtrations are functorial in maps u : E → F of vector bundles (then their uniqueness follows by applying functoriality to idE ). Let α ∈ Q with u(HNα (E )) ̸= 0. Let β be maximal such that u(HNα (E )) ⊆ HNβ (F ). Then grβHN (F ) ̸= 0 u π and the composition HNα (E ) −→ HNβ (F ) −→ grβHN (F ) is non-zero since β was maximal. Assume that HNβ (F ) ⊈ HNα (F ). Then α > β. Now grβ (F ) is semistable of slope β and HNα (E ) has a filtration whose graded pieces are semistable of slope ≥ α > β. Therefore π ◦ u induces the zero map on all these graded pieces by Proposition 26.150. Hence π ◦ u = 0, a contradiction. Remark and Definition 26.157. Let r ≥ 0 be an integer. We set Qrdom := { (α1 , . . . , αr ) ∈ Qr ; α1 ≥ · · · ≥ αr }.

582

26 Curves

Here dom stands for dominant and comes from the connection with the theory of linear algebraic groups and their root systems which we do not discuss further here. Let E be a vector bundle of rank r on C. If λ1 > · · · > λm are the jumps in the HN filtration of E i and ri := rk(grλHN (E )), then we call (λ1 , . . . , λ1 , λ2 , . . . , λ2 , . . . , λm , . . . , λm ) ∈ Qrdom | {z } | {z } | {z } r1 times

r2 times

rm times

the Harder-Narasimhan vector of E . One can picture the Harder-Narasimhan vector by the Harder-Narasimhan polygon. This is the piecewise linear polygon connecting by line segments the points (0, 0),

(rk HNλ1 (E ), deg HNλ1 (E )),

...,

(rk HNλm (E ), deg HNλm (E ))

in R2 . It is a concave polygon with break points in Z2 that has slopes λi with multiplicities ri . Lemma 26.158. Let E and F be vector bundles on C. Suppose that the lowest slope of the HN polygon of E is strictly bigger than the smallest slope of the HN polygon of F . Then HomOC (E , F ) = 0. Proof. Indeed, assume that u : E → F is non-zero. Let λ ∈ Q be maximal with u(HNλ (E )) ̸= 0 and let µ ∈ Q be maximal such that u(HNλ (E )) ⊆ HNµ (F ). Then u induces a non-zero map grλHN (E ) → grµHN (F ). This contradicts Proposition 26.150 because λ > µ by assumption. Proposition 26.159. Let k ′ be a field extension of k. Let p : C ⊗k k ′ → C be the projection. Then for every vector bundle E on C and for all α ∈ Q one has p∗ (HNα (E )) ∼ = HNα (p∗ (E )). In particular, E is semistable of slope λ if and only if p∗ E is semistable of slope λ. Proof. Set C ′ := C ⊗k k ′ . Since p : C ′ → C is faithfully flat, p∗ defines an injective map from the set of submodules of E to the set of submodules of p∗ E . Moreover, degree and rank are preserved by p∗ . Hence we see that if there exists a submodule F = ̸ 0 of E with µ(F ) > µ(E ), then p∗ (F ) is a submodule of p∗ (E ) with µ(p∗ F ) > µ(p∗ E ). Hence if p∗ E is semistable of slope λ, then E is semistable of slope λ. It therefore suffices to show that there exists a Q-filtration E • of E such that p∗ (E α ) = HNα (p∗ E ). To do this we may enlarge k ′ since p∗ is injective on the set of submodules of E . There are only finitely many different HNα (p∗ E ) and each of them is of finite presentation. Hence all of them are defined over some finitely generated subfield k ⊆ K ⊆ k ′ . Filtering K by suitable subfields, we can assume that k ′ = k(x) for some x ∈ k ′ and that we are in one of the following cases. (1) The extension k ′ /k is finite separable. (2) The extension k ′ /k is purely inseparable in characteristic p > 0 and xp ∈ k. (3) The extension k ′ /k is purely transcendental. Let K = K(C) and K ′ = K(C ′ ) be the function fields. Set F ′ := HNα (p∗ E ) for some α. To see that F ′ = p∗ (F ) for some subbundle F of E it suffices to show that the K ′ -subspace F ′ := Fη′ of E ′ := p∗ (E )η descends to E := Eη , i.e., that there exists a K-subspace F ⊆ E such that F ⊗K K ′ = F ′ (Remark 26.132 (4)).

583 For (1) we may assume that k ′ is a Galois extension by passing to the normal hull. Let G be the Galois group of k ′ over k. Then p : C ′ → C is a Galois covering and K ′ is a Galois extension of K with Galois group G. For σ ∈ G let σK ′ be the automorphism ∗ ∗ ∗ ∗ of C ′ over C induced by σ. Then σC ′ p E = p E . As σC ′ preserves degree and rank it • ∗ ∗ follows that σC ′ HN (p E ) is again the HN filtration of p∗ E . In particular σ preserves F ′ . Hence by Galois descent (Theorem 14.85), F ′ descends. Consider Case (2). Then K ′ = Kk ′ is a purely inseparable extension of K with (K ′ )p ⊆ K. By Exercise 17.6, the subspace F ′ of p∗ (E )η descends to Eη if and only if for every D ∈ DerK (K ′ ) one has D(F ′ ) ⊆ F ′ . The restriction of D to k ′ is a k-derivation which we again denote by D. Set DE := D ⊗k idE and consider the composition D

E ψ : F ′ −→ p∗ (E ) −→ p∗ E −→ p∗ E /F ′ .

Even though DE is only k-linear but not k ′ -linear, ψ is k ′ -linear and hence OC ′ -linear because we have ψ(af ) = aψ(f ) + D(a)f = aψ(f ) mod F ′ for local sections f of F ′ and a ∈ k ′ . Now Lemma 26.158 implies ψ = 0, which means DE (F ′ ) ⊆ F ′ . Hence we have D(F ′ ) ⊆ F ′ . Case (3) is similar to (1). In this case K ′ = K(x) is purely transcendental. The group G := AutK (K ′ ) of automorphisms of K ′ fixing K contains the group of K-automorphisms induced by translations x 7→ x + a for a ∈ K. Note that K is infinite since it is the function field of a curve. Now every rational function f /g ∈ K(x), with f, g ∈ K[x], such that f (a)/g(a) = f (0)/g(0) for all a ∈ K is constant. Otherwise g(0)f (x) − f (0)g(x) is a non-zero polynomial and there exists a ∈ K which is not a root of this polynomial which would imply that f (x + a)/g(x + a) ̸= f (x)/g(x). It follows that K ′G = K. Every σ ∈ G induces an automorphism on E ′ and F ′ descends if and only if σ(F ′ ) = F ′ for all σ ∈ G ([BouAI] O II, §8.7, Theorem 1). Each σ ∈ G induces an automorphism σC ′ ∗ ∗ ∗ ∗ ′ ′ of C ′ with σC and hence σ(F ′ ) = F ′ . ′ p E = p E such that σC ′ F = F Definition 26.160. A non-zero vector bundle that is not isomorphic to the direct sum of two non-zero vector bundles is called indecomposable. Proposition 26.161. Suppose that C has genus g ≤ 1. Then the HN filtration of any vector bundle E on C is split. In particular, every indecomposable vector bundle is semistable. Proof. We proceed by induction on the rank of E . Let λ be the smallest jump of the HN filtration of E , i.e., the maximal rational number α such that E = HNα (E ). Then 0 ̸= grλHN (E ) is semistable of slope λ. By induction, the HN filtration for E ′ := HNλ+ (E ) L ′ ∼ is split, i.e., E = Ei , where Ei is semistable of slope µ(Ei ) > λ. We have to show that the exact sequence 0 −→ E ′ −→ E −→ grλHN (E ) −→ 0 is split. For this it suffices to show (Remark F.219) that M Ext1OC (grλHN (E ), E ′ ) = Ext1OC (grλHN (E ), Ei ) = 0. i

Now Serre duality in the form of (25.26.3) gives us (*)

Ext1OC (grλHN (E ), Ei )∨ = HomOC (Ei , grλHN (E ) ⊗ ωC ),

584

26 Curves

where ωC = Ω1C/k is the canonical bundle. Then deg(ωC ) = 2g − 2 ≤ 0 by assumption and hence µ(grλHN (E ) ⊗ ωC ) ≤ λ by Remark 26.149 (2). Therefore the right hand side of (*) is 0 by Proposition 26.150. Hence we see that if C is of genus ≤ 1, every vector bundle is the direct sum of semistable vector bundles. If C = P1k , then the non-zero semistable vector bundles of slope λ necessarily have slope λ ∈ Z and are isomorphic to a direct sum of copies of OP1k (λ) by Theorem 26.133. If C has genus 1 and C(k) ̸= ∅, then C is the underlying curve of an elliptic curve. In this case the category of semistable vector bundles of a fixed slope is not semisimple but can still be described quite explicitly by a theorem of Atiyah. We will explain this in Section (27.50) below. It is also known how the Harder-Narasimhan filtration behaves in families, see Theorem 26.167 below.

Further topics We conclude the chapter by mentioning, in part without proof, some further topics in the theory of curves that are particularly important and interesting. (26.26) Relative Curves. In this section we study which of the above notions behave well in families. To this end we introduce the notion of a relative curve. Definition 26.162. Let S be a scheme. An S-scheme f : C → S is called a (relative) curve over S if f is separated, flat and of finite presentation and all fibers of f are equi-dimensional of dimension 1. Remark 26.163. Let C → S be a smooth relative curve. Then every section x : S → C is a regular immersion of codimension 1 (Theorem 19.30). Since C → S is assumed to be separated, x is a closed immersion and hence defines an effective divisor in C, denoted by [x] or just by x. We denote by I (x) ⊆ OC its defining quasi-coherent ideal sheaf, so that x(S) = Spec(OC /I (x)). It is a line bundle and we denote by OC/S ([x]) := I (x)∨ its dual. Proposition 26.164. Let f : C → S be a proper relative curve. Then the map S → Z,

s 7→ g(Cs )

is locally constant. We call this locally constant function the genus of the relative curve C. Proof. By definition of the genus it is enough to show that the Euler characteristic χ(OCs ) is locally constant on S. But we have seen this in Theorem 23.139. Let us specialize some results from cohomology and base change to the case of relative curves.

585 Remark 26.165. Let f : C → S be a proper relative curve. (1) If F is a quasi-coherent OC -module, then Ri f∗ F = 0 for all i ≥ 2 and the formation of R1 f∗ F commutes with base change S ′ → S (by Corollary 24.44 and the remark following it). If F is an OC -module of finite presentation that is flat over S, then R1 f∗ F is locally free if and only if the formation of f∗ F commutes with base change. In this case, f∗ F is locally free (Theorem 23.140). (2) Assume that f∗ OC = OS and that this continues to hold after arbitrary base change S ′ → S. By Corollary 24.63 this holds, if f has geometrically connected and geometrically reduced fibers. Applying part (1) to F = OC , we obtain that R1 f∗ OC is locally free of rank g, where g denotes the genus of the fibers of C. Note that we can consider g as a locally constant function on S by Proposition 26.164. Proposition 26.166. Let f : C → S be a proper relative curve and let L be a line bundle on C. For s ∈ S let Ls be the restriction of L to the schematic fiber Cs of f over s. Then the map deg : S → Z, s 7→ deg(Ls ) is locally constant. Proof. This follows from Theorem 23.139 and Proposition 26.46. For vector bundles on relative smooth proper relative curves, the Harder-Narasimhan polygon is upper semicontinuous. More precisely one has the following result. Let S be a scheme and let f : C → S be a smooth proper relative curve with geometrically connected fibers. Let E be a vector bundle on C. As usual, we denote for s ∈ S the fiber Cs = C ⊗OS κ(s) and by Es the pullback of E to Cs . Then Cs is a proper smooth geometrically connected curve over κ(s) and Es is a vector bundle on Cs . We want to explain how the Harder-Narasimhan filtration of Es depends on s. Since the rank of Es is constant on Cs , the continuous rank function C → Z, where Z is endowed with the discrete topology, factors through a function S → Z,

s 7→ rk(Es )

which is again continuous, i.e., locally constant, since f is closed and surjective and hence S carries the quotient topology of C. As also the map s 7→ deg(Es ) is locally constant (apply Proposition 26.166 to the determinant of E ), it is harmless to assume that s 7→ rk(Es ) and s 7→ deg(Es ) are constant and we denote their respective values by rk(E ) and deg(E ). For every s ∈ S let HNP(Es ) be the HN polygon of Es (Remark 26.157). This is an element of the set P of piecewise linear concave polygons in R2 with integral break points with start point (0, 0) and end point (rk(E ), deg(E )). We endow P with a partial order by defining p ≤ p′ for p, p′ ∈ P if p lies under p′ . Then by [Shatz] O Theorem 3 and Proposition 10 one has the following result3 . Theorem 26.167. The map S → P,

s 7→ HNP(Es )

is constructible and upper semicontinuous, i.e., { s ∈ S ; HNP(Es ) ≥ p } is closed and constructible for every polygon p ∈ P . 3

In loc. cit., this is only shown if S is noetherian but it is easy to reduce to this situation by noetherian approximation using Proposition 26.159 and Theorem 10.57.

586

26 Curves

As a consequence of Proposition 26.57, let us show that smooth proper curves of genus ≥ 2 over any qcqs base scheme are projective. Recall that the genus of a relative curve is locally constant on the base, Proposition 26.164, and that the fiberwise degree of a line bundle on a relative curve is locally constant (Proposition 26.166). Proposition 26.168. Let S be a scheme, and let f : C → S be a proper curve over S with irreducible fibers. (1) If L is a line bundle on C whose (fiberwise) degree is positive, then L is relatively ample over S. (2) Suppose that f is in addition smooth and that the genus g(C) is at least 2. Then the line bundle Ω1C/S is relatively ample for f . In particular, if S is qcqs, then f is projective. Proof. For part (1), we apply Proposition 26.57 and Theorem 24.46. Part (2) follows from part (1). In fact, for s ∈ S, we have (Ω1C/S )s = Ω1Cs /κ(s) (Proposition 17.30). Hence we can apply Corollary 26.52 and (1) to see that the canonical bundle Ω1C/S is relative ample over S. Then f is projective if S is qcqs by Proposition 13.68. The result can be further generalized to certain nodal curves (so-called stable curves), see [DeMu] O Theorem 1.2. One can also show that whenever A is a Dedekind domain and f : C → Spec A is a proper curve where the scheme C is connected and regular, then f is projective ([Lic] O Theorem 2.8). For a study of relative curves of genus 0, see [LoKl] O Section 3. Finally, let us mention the theorem, proved independently by Fontaine and Abrashkin, that there is no smooth proper relative curve over Spec Z of genus > 0. (26.27) Plane curves birational to a given curve. Recall that every smooth proper curve over an infinite field k can be embedded in P3k (Theorem 14.132). Over finite fields this result is not true, in fact there exist finite fields k and smooth curves with more than #P3k (k) rational points. We also have seen that not every (smooth) curve can be embedded in the projective plane. But at least we can find, for every curve a “plane model”, birational to the original curve, with only ordinary double points as singularities. Proposition 26.169. Let k be an infinite field. Let C be a curve over k. Then C is birationally equivalent to a projective plane curve which has at worst ordinary double points as singularities. The basic idea of the proof, similarly as for Theorem 14.132, is to study the “space” of all non-constant morphisms C → P2k , and to show that the locus of morphisms that do not satisfy the criterion above, has smaller dimension than the whole space. However, filling in all the details is quite involved. See [Sam] O , [Har3] O . The following weaker statement is easier to prove, and a map as in that proposition can be obtained quite explicitly in terms of blow-ups. See [Ful1] O Chapter 7. Proposition 26.170. Let k be an algebraically closed field. Let C be a curve over k. Then C is birationally equivalent to a plane curve whose closed points are normal or ordinary multiple points.

587 (26.28) Curves over finite fields and the Weil conjecture. Let k be a finite field of characteristic p, with q elements. For any r ≥ 1, we denote by kr the unique extension field of k of degree r in a fixed algebraic closure k of k. Let C be a geometrically connected smooth projective curve over k of genus g. Denote by Nr = Nr (C) the cardinality of the finite set C(kr ). It is interesting to study the numbers Nr , for instance, to give upper and lower bounds for them. A first naive guess might be that Nr should not be too far off from #P1k (kr ) = q r + 1, and in fact, Corollary 26.175 below gives a bound of |Nr − (q r + 1)| in terms of q, r and the genus of C. The right tool to study this question is the so-called zeta function of the curve C. Not only does it allow to prove this bound on the number of points, but it also exhibits an intriguing connection with number theory, specifically with the Riemann zeta function and the famous Riemann hypothesis. This connection is made by the so-called Weil conjectures, formulated for curves by E. Artin in 1924, and for higher-dimensional varieties over finite fields by A. Weil in 1949. While the case of curves was proved by Weil, the Weil conjectures for higher-dimensional varieties have been one of the major open problems that led to the development of modern algebraic geometry by Grothendieck and his school, in particular to the construction and study of ´etale cohomology. In this case, too, the analogue of the Riemann hypothesis was the main stumbling block, until P. Deligne proved it and thus completed the proof of the Weil conjectures in [Del] O (1974). By now, several approaches (all relying on a suitable cohomology theory in an essential way) are known. See the books [FrKi] O , [KiWe] O for detailed expositions of two of them. Definition 26.171. Let k be a finite field, and let C be a geometrically connected smooth projective curve over k. The zeta function of C is the formal power series ! ∞ X Tr ∈ Q[[T ]]. Z(C/k, T ) = exp Nr r r=1 Remark 26.172. In the sequel, we use some easy facts on formal power series. We O Ch. 4 §4 forodetails. For any ring R, give a brief summary here and refer to [BouAII] nP i a we have the ring R[[T ]] = limi R[T ]/(T i ) = i≥0 i T ; ai ∈ R of formal power series over R which contains the polynomial ring R[T ] as a subring. Its units are those formal power series with absolute term a0 ∈ R× . If f, g ∈ R[[T ]] and g has constant term 0, the substitution f (g(T )) ∈ R[[T ]] is defined. We regard R[[T ]] = limi R[T ]/(T i ) as a topological ring by equipping each R[T ]/(T i ) with the discrete topology and putting the limit topology on R[[T ]]. This gives us the notion of convergence, and in particular the notion of infinite series and infinite products. If k is a field, then k[[T ]] is a discrete valuation nP ring and T is a uniformizer. o Its field of i fractions is k((T )) := Frac(k[[T ]]) = k[[T ]]T = i≥N ai T ; N ∈ Z, ai ∈ k and is called the field of Laurent series over k. It contains the field of rational functions k(T ), and for f, g ∈ k[T ], g ̸= 0, we call the image of f /g in k((T )) the Laurent expansion of f /g at T = 0. A key example is the geometric series X 1 = T i ∈ k[[T ]]. 1−T i≥0

588

26 Curves

We say that a Laurent series which lies in (the image of) k(T ) has a pole of order n at T = a ∈ k, if this is true for the rational function with which it can be identified. Now let k be a field of characteristic 0, or more generally a Q-algebra. We have the exponential series X Ti ∈ k[[T ]] exp(T ) = i! i≥0

and the map f 7→ exp(f ) is an isomorphism between the additive topological group of formal power series with absolute term 0 onto the multiplicative topological group of power series with constant term 1. Its inverse log is called the logarithm, and we have log(1 + T ) =

X

(−1)i−1

i>0

Ti , i

and more generally the same formula holds if T is replaced by any formal power series with constant term 0. Note that the fact that exp and hence log are group homomorphisms says precisely that the functional equations of the classical exp and log are true for them. Since exp and log are isomorphisms of topological groups, the functional equations can also be applied to convergent infinite series and infinite products, respectively. For any f ∈ k[[T ]] we have its formal derivative f ′ (X). If f has constant term 1, then the derivative of log(f ) equals f ′ /f and is called the logarithmic derivative of f . The logarithmic derivative of a product of such series is the sum of their logarithmic derivatives. We can now state the Weil conjectures for the curve C. Theorem 26.173. (Weil conjectures for curves) Let k be a finite field, let q := #k, let C be a geometrically connected smooth proper curve of genus g over k and let Z(C/k, T ) be its zeta function. (1) (rationality) There exists a polynomial P (T ) ∈ Z[T ] of degree 2g such that Z(C/k, T ) =

P (T ) (1 − T )(1 − qT )

in Q((T )).

(2) (functional equation) The zeta function satisfies the following functional equation: Z(C/k,

1 ) = q 1−g T 2−2g Z(C/k, T ) qT

(in Q((T )).

(3) (“Riemann hypothesis” for curves over finite fields) The polynomial P (T ) in Part (1) is of the form 2g Y P (T ) = (1 − αi T ), αi ∈ C, i=1 2

with |αi | = q for all i. We will give the proof in Section (26.29) below. From this theorem we easily obtain the following reformulation. Theorem 26.174. Under the assumptions of Theorem 26.173, there exist algebraic integers αi , i = 1, . . . , 2g, such that

589 (1) for all r, Nr = q r + 1 −

2g X

αir ,

i=1

(2) for all i = 1, . . . , g, αi α2g−i+1 = q, (3) for all i = 1, . . . , 2g, |αi |2 = q. Proof. Being the zeros of the monic polynomial T 2g P (T −1 ) ∈ Z[T ], the αi are algebraic over Q and even integral over Z, i.e., are algebraic integers. Taking the logarithmic derivative of the equality ! Q2g ∞ X Tr i=1 (1 − αi T ) exp Nr = Z(C/k, T ) = r (1 − T )(1 − qT ) r=1 of Theorem 26.173 and comparing coefficients for T r−1 , we see that r

Nr − (q + 1) =

2g X

αir

for all r ≥ 1.

i=1

The polynomial P has integer coefficients, hence for each zero α of P the complex conjugate α is also a zero, and αα = |α|2 = q −1 by Part (3) of the Weil conjectures. This already shows Part (3) and also that we can group the non-real zeros of P in pairs as √ √ required by (2). Now it is enough to show that the multiplicities of q −1 and of − q −1 as zeros of P are even. Since P has even degree 2g, there is an even number of real roots (counted with multiplicity). To conclude, we use that P has positive leading coefficient, as follows from Theorem 26.173 (2) and the observation that P (0) = Z(C/k, 0) = 1. The information about the growth of the numbers Nr contained in the theorem is a key point. We formulate it as the following corollary which follows immediately from Parts (1) and (3) of Theorem 26.174 and the fact that P1 (kr ) = q r + 1. Corollary 26.175. (Weil bound) With notation as in Theorem 26.173, r

|Nr − #P1 (kr )| ≤ 2gq 2

for all r ≥ 1.

Corollary 26.176. Let e denote the greatest common divisor of the numbers [κ(x) : k], where x ranges over the closed points of C. Then e = 1. In other words: There exists a divisor on C which has degree 1. Proof. The bound in the corollary implies that for all r sufficiently large, we have Nr > 0. But for x ∈ C(kr ), the degree deg([x]) = [κ(x) : k] is a divisor of r. Similarly, for y ∈ C(kr+1 ), [κ(y) : k] divides r + 1. Therefore these two degrees have greatest common divisor 1. Remark 26.177. (1) For C as above of genus g = 0, the corollary implies that C ∼ = P1k (and in particular C(k) ̸= ∅) by Corollary 26.86. As such curves are Brauer-Severi curves and hence are classified by quaternion algebras over k, this follows also from Wedderburn’s theorem, that over a finite field k every central simple algebra is isomorphic to Mn (k) for some n ≥ 1.

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26 Curves

(2) If C has genus g = 1, then the Weil bound shows directly that C(k) ̸= ∅. Hence C can be endowed with the structure of an elliptic curve. (3) There exist curves C with C(k) = ∅ (see Exercise 26.19 for an example). We can also write the zeta function of the curve C as an infinite product, a so-called Euler product. For a closed point x ∈ C, we denote by deg([x]) = [κ(x) : k] the degree of the divisor [x]. Proposition 26.178. (Euler product expansion) Let k be a finite field, let C be a geometrically connected smooth proper curve over k and let Z(C/k, T ) be its zeta function. Let C 1 be the set of closed points of C. Then Y (1 − T deg([x]) )−1 . Z(C/k, T ) = x∈C 1

P Proof. For any r, we can write Nr = d|r dMd , where Md denotes the number of closed points x ∈ C with residue class field κ(x) = kd (in other words, with deg([x]) = d). The sum runs over the positive divisors of r. In the ring of formal power series Q[[T ]] we then have ∞ X

∞

XX Tr Tr dMd = r r r=1 r=1 d|r ! ∞ ∞ X X X T dn = Md =− log(1 − T deg([x]) ). n 1 n=1

log(Z(C/k, T )) =

Nr

d=1

x∈C

Applying exp on both sides, we obtain the desired result. Remark 26.179. The classical Riemann zeta function, defined for complex numbers with real part > 1 by X 1 ζ(s) = , Re(s) > 1, ns n≥1

can also be expressed as the Euler product ζ(s) =

Y p

1 , 1 − p−s

Re(s) > 1,

where the product extends over all prime numbers. The zeta function admits an analytic continuation to a meromorphic function on C, again denoted by ζ. Its only pole is s = 1 and it is a simple pole. This meromorphic continuation satisfies a functional equation relating the values ζ(s) and ζ(1 − s). The Riemann hypothesis predicts that all zeros s of ζ with Re(s) > 0 actually have Re(s) = 12 . More generally, it is classical in algebraic number theory that one can define a zeta function for every finite extension field of Q, i.e., for every number field (or, in other words, every global field of characteristic 0 in the sense of algebraic number theory). As is sketched below, the theory extends to global fields of positive characteristic, i.e., to the fields of rational functions of curves over finite fields. The zeta function of a curve that we have defined above is closely related to this construction. Even more generally, one can attach a zeta function to any scheme of finite type over Spec Z; see [Kah1] X , [Kah2] O and the references given there.

591 Let k be a finite field and let C/k be a curve as above. Replacing Q by the function field K(C), we replace prime numbers (i.e., maximal ideals of Spec Z) by closed points of the curve C. In analogy to the Euler product expression above we are led to consider the series Y 1 ζ(C/k, s) = , −s deg([x]) 1 − q 1 x∈C

where now the product extends over the closed points of C, and deg([x]) = [κ(x) : k], so that q deg([x]) = #κ(x). One can show that the product converges for Re(s) > 1. Replacing q −s by a variable T and considering the result as a formal power series, we obtain by Proposition 26.178 the zeta function of C in the form Z(C/k, T ) as above. Denote by P the polynomial in Theorem 26.173. The expression for Z(C/k, T ) in Part (1) of the theorem gives us in particular a meromorphic continuation of ζ(C/k, s) = Z(C/k, q −s ). Furthermore, we have ζ(C/k, s) = 0 if and only if P (q −s ) = 0. In view of Part (3) of the theorem, this is equivalent to q s = αi for some i. Thus the final statement about the absolute values of the αi can be expressed as a statement about the absolute values |q s |, where s is a zero of ζ(C/k, −). Since |q s | = q Re(s) , the statement of the theorem corresponds precisely to the condition that Re(s) = 12 , perfectly reflecting the classical Riemann hypothesis. (26.29) Proof of the Weil conjecture for curves. For the proof of Theorem 26.173 we start with the following lemma that describes how the zeta function of the curve behaves under passing to a finite extension of k. Lemma 26.180. Let C be a geometrically connected, smooth proper curve over the finite field k, and let kr /k be a field extension of degree r. Let Q be an algebraic closure of Q, × and denote by µr ⊂ Q the subgroup of r-th roots of unity. Then in Q[[T ]] we have Y Z(C ⊗k kr /kr , T r ) = Z(C/k, ζT ). ζ∈µr

Proof. We need to understand the fibers of the map C ⊗k kr → C. For x ∈ C a closed point, the fiber over x is isomorphic to the spectrum of κ(x) ⊗k kr . Writing d = [κ(x) : k] and denoting by g the greatest common divisor of d and r, and by ℓ = dr/g their lowest common multiple, we have g κ(x) ⊗k kr ∼ = κ(x) ⊗kg kg ⊗k kr ∼ = κ(x) ⊗kg (kr )g ∼ = kl ,

i.e., the fiber over x consists of g points, each of which has residue class field kl , and hence has degree l/r = d/g over kr . Thus for the left hand side we obtain, using Proposition 26.178, Y Y (1 − T r[κ(x):k]/g )−g . (1 − T r[κ(x):kr ] )−1 = Z(C ⊗k kr /kr , T r ) = x∈C 1

x∈(C⊗k kr )1

Now writing d = [κ(x) : k] as above, we have Y Y (1 − T rd/g )g = (1 − ξT d )g = (1 − ζ d T d ), ξ∈µr/g

which gives the identification we are looking for.

ζ∈µr

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Proof. (of Theorem 26.173) We divide the fairly long proof into several steps. (I) Rationality. We fix the finite field k and the geometrically connected smooth proper curve C over k of genus g. Starting from the Euler product expression of Proposition 26.178, expanding each of the factors (1 − T deg([x]) )−1 as a geometric series, and multiplying out the product, we may express the zeta function as X Y (1 − T deg([x]) )−1 = Z(C/k, T ) = T deg(D) , x∈C 1

D≥0

the sum running over all effective divisors on C. Grouping effective divisors according to linear equivalence, we may rewrite this as a sum over isomorphism classes of line bundles on C. For a line bundle L , write ℓ(L ) = dimk H 0 (C, L ), so that (q ℓ (L ) − 1)/(q − 1) is the cardinality of the linear equivalence class of divisors D with OC (D) ∼ = L . We then obtain X q ℓ(L ) − 1 Z(C/k, T ) = T deg(L ) . q−1 L ∈Pic(C)

Let g be the genus of the curve C. Let e ≥ 1 be the generator of the image Im(deg) ⊆ Z of the degree map deg : Pic(C) → Z. We will see below that e = 1 (cf. Corollary 26.176). Claim. The zeta function Z(C/k, T ) is a rational function, i.e., an element in Q(T ) ⊂ Q((T )), and has a simple pole at T = 1. Proof of claim. The case g = 0 can be handled easily starting from the above expression, since then ℓ(L ) = deg(L ) + 1 for all L of non-negative degree, by the Theorem of Riemann-Roch. From this one easily computes that for C of genus 0, we have Z(C/k, T ) = 1 (1−T e )(1−(qT )e ) . This proves the claim, and once we have shown e = 1, the full Weil conjectures in this case. Therefore we now assume g ≥ 1. The key to expressing the zeta function of C as a rational function is the observation that for line bundles L of sufficiently high degree (namely for degree > 2g − 2), the dimension ℓ(L ) of the space of global sections depends only on the degree d of L , namely ℓ(L ) = d + 1 − g, see Proposition 26.55 (2). We write (q − 1)Z(C/k, T ) as the sum (q − 1)Z(C/k, T ) = f (T ) + g(T ) with X q ℓ(L ) T deg(L ) , f (T ) := L , 0≤deg(L )≤2g−2

a polynomial in Z[T ] of degree ≤ 2g − 2, and X q ℓ(L ) T deg(L ) − g(T ) := =

L , deg(L )≥2g−1

q

T deg(L )

L , deg(L )≥0

L , deg(L )≥2g−1

X

X

deg(L )+1−g

T

deg(L )

−

X

T deg(L ) ,

L , deg(L )≥0

where for the final equality we use the Riemann-Roch theorem as explained before. By definition of e, for d ∈ Z there exists a line bundle of degree d if and only if e | d. In this case, the group Pic0 (C) of (isomorphism classes of) line bundles of degree 0 acts simply transitively on the set of line bundles of degree d. In particular, the above considerations show that h := # Pic0 (C) is finite (since for given d, there are only finitely many effective divisors of degree d, and for d large, every line bundle of degree d is isomorphic to OX (D) for some effective D). So when e | d, there exist h line bundles of degree d on C, up to isomorphism. Let d0 ≥ 0 be minimal such that d0 e ≥ 2g − 1. We obtain

593 



g(T ) = h q 1−g

X

(qT )ed −

d≥d0

X

T ed  = h



d≥0

1 q 1−g (qT )ed0 − e 1 − (qT ) 1 − Te

 .

The right hand side clearly lies in Q(T ) and has a simple pole at T = 1. This proves the claim. Let us show that e = 1. We apply Lemma 26.180 with r = e. The same reasoning as before, applied to the base change C ⊗k ke , shows that Z(C ⊗k ke /ke , T ) has a simple pole at T = 1, whence Z(C ⊗k ke , T e ) also has a pole of order 1 at T = 1. On the other hand, the expression we have found for Z(C/k, T ) is a rationalQ function in T e , so Z(C/k, ζT ) = Z(C/k, T ) for all e-th roots of unity ζ. This shows that ζ∈µe Z(C/k, ζT ) = Z(C/k, T )e has a pole of order e at T = 1, and thus that e = 1. This also gives us d0 = 2g − 1. Putting things together, we obtain (26.29.1)    1−g  2g−1 X q (qT ) 1 1  . q ℓ(L ) T deg(L ) + h − Z(C/k, T ) = q−1 1 − qT 1−T L , 0≤deg(L )≤2g−2

This gives us an expression of the form Z(C/k, T ) =

P (T ) , (1 − T )(1 − qT )

P ∈ Q[T ], deg(P ) ≤ 2g.

Since Z(C/k, T ) has coefficients in Z, the same holds for (1 − T )(1 − qT )Z(C/k, T ) = P . We will see that deg(P ) = 2g in the next step of the proof. (II) Functional equation. While for the rationality it was enough to use a weak form of the Riemann-Roch theorem, we now use its precise statement, including Serre duality on the curve C. We again may exclude the case g = 0. We can check the functional equation for both summands of the expression (26.29.1) separately. For the second summand, this is an easy computation which we leave to the reader. To deal with the first summand, note that the map L 7→ L ∨ ⊗OC Ω1C/k is an involution of the set of isomorphism classes of line bundles with degree in {0, . . . , 2g − 2}. Therefore X X ∨ 1 ∨ 1 q ℓ(L ⊗OC ΩC/k ) T deg(L ⊗OC ΩC/k ) q ℓ(L ) T deg(L ) = L , 0≤deg(L )≤2g−2

L , 0≤deg(L )≤2g−2

=

X

q ℓ(L )−deg(L )−1+g T 2g−2−deg(L ) .

L , 0≤deg(L )≤2g−2

where the evaluation of ℓ(L ∨ ⊗OC Ω1C/k ) follows from the Riemann-Roch theorem, Theorem 26.48, and we use that deg(Ω1C/k ) = 2g −2, see Corollary 26.52. This immediately yields the functional equation for the first summand. Let us summarize what we have seen so far. It is an easy consequence of the functional equation that deg(P ) = 2g. Furthermore, the polynomial P has absolute term P (0) = Q2g Z(C/k, 0) = 1, hence we can factorize it as i=1 (1 − αi T ) with αi ∈ C. Similarly as in √ √ the proof of Theorem 26.174 we see that q and − q both occur with even multiplicity −1 −1 among the αi . By the functional equation, α 7→ q α is an involution of the set of zeros of P . Renumbering the αi if necessary, Z(C/k, T ) can therefore be written in the form Z(C/k, T ) =

2g Y P (T ) with P (T ) = (1 − αi T ) ∈ Z[T ] (1 − T )(1 − qT ) i=1

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26 Curves

and αi ∈ C, αi α2g+1−i = q for i = 1, . . . , g. (III) Riemann hypothesis. It remains to prove the Riemann hypothesis, the most difficult part of the Weil conjectures. The main idea of the proof below, i.e., using the intersection pairing on the surface Ck × Ck , goes back to Weil. Recall that we have fixed an algebraic closure k of k. We start by showing that proving a suitable bound on the numbers Nr = #C(kr ) entails the Riemann hypothesis (compare Corollary 26.175). Lemma 26.181. With notation as above, the following are equivalent: 1 (1) |αi | = q 2 for all i = 1, . . . , 2g, 1 (2) |αi | ≤ q 2 for all i = 1, . . . , 2g, r (3) |Nr − (q r + 1)| ≤ 2gq 2 for all r ≥ 1. Proof. Since we have αi α2g+1−i = q for i = 1, . . . , g, the equivalence (i) ⇔ (ii) is clear. P2g The implication (ii) ⇒ (iii) follows by writing Nr − (q r + 1) = i=0 αir , see the proof of Theorem 26.174. P r 2g Let us show (iii) ⇒ (ii). In view of the above identity, (iii) says that i=1 αir ≤ 2gq 2 1

for all r ≥ 1. The bound (ii) then follows from the following lemma, applied to βi = αi /q 2 , i = 1, . . . , 2g. Note that to apply that lemma, it would even be enough to know that there r exists a constant C > 0 such that |Nr − (q r + 1)| ≤ Cq 2 for all r ≥ 1. Pn Lemma 26.182. Let β1 , . . . , βn ∈ C such that | i=1 β r | is bounded independently of r ≥ 1. Then |βi | ≤ 1 for all i. Proof. To simplify the notation, we assume β1 is non-zero and has the largest absolute P P n value among all βi . We consider the power series r≥1 ( i=1 βir ) tr over the complex numbers. For t ∈ C with |β1 t| < 1 we have ! n n X X X βi t r β i tr = , 1 − βi t i=1 i=1 r≥1

−1

whence the series on the left has radius of convergence |β1 | . Using the Cauchy-Hadamard formula and the assumption, this yields r1 n X r |β1 | = lim sup β ≤ 1, r→∞ i=1

as desired. Alternatively, one can give a more direct proof applying the simultaneous version of Dirichlet’s approximation theorem to the arguments of the βi to show that there exist infinitely many r such that all βir lie in some small sector of the complex plane containing the positive realP axis. This implies (restricting from now on to elements r with the above P property) P that i |βir | ≤ c | i βir | for some constant c > 0 that does not depend on r. But then i |βir | is bounded, and thus so are all |βir |. In the remainder of the proof, we will prove the bound |Nr − (q r + 1)| ≤ 2gq r/2 . It is convenient to pass to k to carry out the argument. To recover the number of kr -rational points of C after the base change from k to k, we use the relative Frobenius morphism FC /k : Ck → (Ck )(q) , see Remark 4.24. By a slight abuse of notation, here and below we k always use the q-th power Frobenius, where q is the number of elements of k. We collect the essential properties that we will use in the following general lemma.

595 Lemma 26.183. Let k be a finite field with q elements, let k be an algebraic closure of k and let σ : k → k, x 7→ xq , be the q-Frobenius homomorphism on k. Consider a k-scheme X0 , and denote by X := X0 ⊗k k its base change to k. (1) We have a canonical identification X (q) = X as k-schemes. Making this identification, we always view the relative Frobenius FX/k as a morphism F : X → X. It agrees with the base change FX0 /k ⊗k k. (2) The map X(k) → X(k) induced by F coincides with the map given by composition with the Frobenius morphism on Spec k (which is induced by σ). In particular, the r inclusion X0 (kr ) ⊂ X(k) gives us an identification of X0 (kr ) with the set X(k)F of fix points under the r-fold composition of F with itself. (3) For every point x ∈ X(k) fixed by F r , the differential (dF r )x : Tx X → Tx X vanishes. The third point of the lemma will be important below; note that for X0 = A1k it is basically equivalent to the fact the polynomial T q as derivative 0 in characteristic p | q. Proof. For Part (1), note that the restriction of σ to k is the identity, so X (q) = X ⊗k,σ k = X0 ⊗k k ⊗k,σ k = X0 ⊗k,σ k = X. It is easy to check that via this identification, F := FX/k = FX0 /k ⊗k k. It should be emphasized, however, that the identification X (q) = X crucially relies on the fact that X arises by base change from a k-scheme, and is the way in which the Frobenius allows us to keep track of the “k-structure on X”. The first statement of Part (2) follows from Part (1) since X(k) = X0 (k) and because the absolute Frobenius is compatible with all morphisms (cf. the outer part of diagram (4.6.2)). The second statement is an immediate consequence of this. It remains to prove (3). We view the composition F r as the relative q r -Frobenius morphism. It is clear that the differential of the absolute Frobenius of X vanishes everywhere. By definition of the relative Frobenius FX/k , the absolute Frobenius of X equals the composition of FX/k with an isomorphism (namely the base change of the absolute Frobenius of the perfect field k). This implies the desired vanishing. (Cf. Exercise 17.13.) Now let X := Ck × Ck , a smooth projective surface over k. Fix a point x ∈ C(k) and consider the following divisors on X: H = Ck × Spec κ(x),

V = Spec κ(x) × Ck ,

∆ the diagonal,

Γr the graph of FCr

k /k

.

We will need to know the following intersection numbers. Lemma 26.184. With notation as above, we have (1) (H · H) = (V · V ) = 0, (2) (H · V ) = (H · ∆) = (V · ∆) = 1, r (3) (Γr · ∆) = #C(k)F = #C(kr ). Proof. Part (1) follows from Proposition 23.77, applied to the projections from X to Ck . Alternatively one could use that the divisor [x] on Ck (where H = C × Spec κ(x)) is linearly equivalent to a divisor on Ck whose support does not contain x. Thus H is linearly equivalent to a divisor whose support is disjoint from H, and the result follows for H, and by an analogous argument for V . For Part (2), it is enough to observe that all the three intersections under consideration are transversal intersections in the single point (x, x).

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26 Curves

In Part (3) the second equality follows from the above discussion, and for the first one, it is enough to check that the intersection is transversal, because C(k)Fr equals the set-theoretic intersection Γr ∩ ∆ (as subsets of Ck ). So let z ∈ C(kr ), considered as a point of C(k) = Ck (k), so that (z, z) ∈ Γr ∩ ∆. We identify the tangent space of X at (z, z) with Tz (Ck )2 . The tangent space to ∆ at this point is simply the diagonal in Tz (Ck )2 . On the other hand, since passing to tangent spaces is functorial and compatible with products, we may identify T(z,z) Γr ⊆ T(z,z) (Ck × Ck ) = Tz (Ck )2 with the graph of the differential (dFr )z . However (dFr )z = 0 by Lemma 26.183. Hence this graph is just the first coordinate axis, and together with the diagonal it spans this vector space. Now consider the quotient Num(X) of Pic(X) by the subgroup of line bundles numerically equivalent to 0 (see Section (25.32)). Let Num(X)R = Num(X) ⊗Z R, a real vector space equipped with the non-degenerate symmetric bilinear form induced by the intersection pairing, and decompose this space as Num(X)R = U ⊥ U ′ , where U := ⟨H, V ⟩R is the two-dimensional subspace generated by H and V , and U ′ is its orthogonal complement. By Lemma 26.184, U ∩ U ′ = 0, so this is indeed a direct sum decomposition, and moreover H + V has self-intersection number ((H + V )2 ) = 2 > 0. It follows from the Hodge index theorem, Theorem 25.32, that the intersection pairing on U ′ is negative definite (either using Sylvester’s theorem for suitable finite-dimensional subspaces, or observing that H + V is ample). Consider the pullback homomorphism Φ = (F × id)∗ : Pic(X) → Pic(X), see Section (11.16). It induces an endomorphism of Num(X)R , again denoted by Φ, which has the following properties. Lemma 26.185. With notation as above, we have the following equalities in Num(X): (1) Φ(H) = H, (2) Φ(V ) = qV , (3) Φr (∆) = Γr , (4) (Φ(D) · Φ(E)) = q(D · E) for all D, E ∈ Num(X)R . Proof. We start with Part (2). Part (1) can be shown in a similar, but simpler way (and holds already in Pic(X)). We write V as the pullback p∗1 [x], where p1 : X → Ck is the projection to the first factor. Then (F × id)∗ p∗1 [x] = p∗1 F ∗ [x]. Now F −1 (x) settheoretically consists of a single point, say y, and Proposition 26.71 shows that deg(F ) = q. It then follows from Corollary 11.51 and Proposition 26.19 that F ∗ [x] = q[y]. It is thus enough to show that p∗1 [x] and p∗1 [y] are numerically equivalent. But this follows from Proposition 23.151 (2). For Part (3), which holds already in Pic(X), by Corollary 11.51 it is enough to observe the general fact that the schematic inverse image of ∆ under the r-fold composition (F × id)r equals the graph Γr , cf. the right hand square in diagram (9.1.4). Part (4) follows from Proposition 23.77. Write ∆ = u + u′ with u ∈ U , u′ ∈ U ′ . Then u = H + V , since (∆ · H) = (∆ · V ) = 1, and using Lemma 26.184 and Lemma 26.185 we can compute #C(kr ) = (Γr · ∆) = (Φr (∆) · ∆) = ((H + q r V + Φr (u′ )) · (H + V + u′ )) = q r + 1 + (Φr (u′ ), u′ ). Thus

597 |#C(kr ) − (q r + 1)| = |(Φr (u′ ), u′ )| p r ≤ |(Φr (u′ ), Φr (u′ ))| · |(u′ , u)| = q 2 |(u′ · u′ )| , where the estimate in the middle follows from the Cauchy-Schwarz inequality on the negative definite space U ′ . In view of the remark at the end of the proof of Lemma 26.181, this would already be enough to finish the proof of the Weil conjectures. But it is also easy to compute (u′ · u′ ) explicitly, so that we can apply Lemma 26.181 in the form it was stated. By definition and Lemma 26.184, we have (u′ · u′ ) = ((∆ − H − V )2 ) = (∆2 ) − 2. Let us show that (∆2 ) = 2 − 2g; this will give us (u′ · u′ ) = −2g and hence the desired result. (Note that if g = 0, then C ∼ = P1k and U ′ = 0.) By the adjunction formula, Proposition 25.145, we have (∆ · (∆ + K)) = 2g − 2, where K is a canonical divisor on X and g is the genus of C. By Corollary 17.32, we may write K = p∗1 KC + p∗2 KC (in Pic(X)) where KC is a canonical divisor on Ck and p1 , p2 : X → Ck are the projections. We have (p∗i KC · ∆) = deg((p∗i K)|∆ ) = deg(KC ) = 2g − 2 for i = 1, 2 (see Example 23.75, Corollary 26.52), so it follows that (∆2 ) = 2 − 2g. This finishes the proof. Several other proofs of the Weil conjectures for curves are known. Weil gave two proofs, one using the Jacobian of the curve C (see Section (27.26) below for the notion of Jacobians), and another one similar to the proof given above, using intersection theory on the surface C ×k C. See [Mil3] X , [Ras] O . A more elementary proof, based on ideas of Stepanov, was given by Bombieri [Bom] O . See also Lorenzini’s book [Lor] O which gives a detailed exposition of this proof, starting from the basic notions of algebraic geometry, and including many examples. (26.30) Curves over number fields. Let k be a number field, i.e., a finite extension field of the field Q of rational numbers. Similarly as for Riemann surfaces (Theorem 26.30) we see the following trichotomy for proper smooth geometrically connected curves over k. Genus 0. A curve (always in the above sense) of genus 0 which has a k-valued point is isomorphic to P1k , and thus has infinitely many rational points. On the other hand, there exist curves of genus 0 without k-valued points. As we have seen above (and is true over every field), a connected smooth proper curve of genus 0 is isomorphic to a smooth conic, i.e., a smooth degree 2 plane curve. For these curves, the Hasse principle holds (see Remark 26.92), i.e., the curve has a k-valued point if (and only if) it has a kv -valued point for every completion of k with respect to an (archimedean or non-archimedean) absolute value v. Genus 1. Curves of genus 1 may have no k-valued points, or finitely many k-valued points, or infinitely many k-rational points. As we have seen, if there is a k-valued point, then the set C(k) can be equipped with the structure of an abelian group, arising from a group scheme structure on C. The Theorem of Mordell (sometimes called the Theorem of Mordell-Weil in view of its generalization to the case of abelian varieties) states that C(k) is a finitely generated abelian group. The famous Conjecture of Birch and Swinnerton-Dyer gives a precise description of the rank of this group in terms of the L-function of the elliptic curve, a holomorphic function attached to C. (See [Sil1] O , [Sil2] O for further details.) The Hasse principle may fail in this situation, see Example 26.102.

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26 Curves

Genus ≥ 2. Curves of genus ≥ 2 have at most finitely many k-valued points. This statement was conjectured by L. Mordell. It is still often called Mordell’s conjecture, although it was proved by G. Faltings ([Fal] O , see also [CoSi] O for further background on the methods used in Faltings’s proof). (26.31) Literature on algebraic curves. We give some further references on the theory of algebraic curves. Both Hartshorne [Har3] O , and Liu [Liu] O discuss curves in quite some detail. Fulton’s book [Ful1] O does not require as much background. In the book [HiSi] O by Hindry and Silverman, there is a comprehensive survey on the theory of algebraic curves. The two large volumes [ACGH] O and [ACG] O by Arbarello, Cornalba, Griffiths and Harris contain a lot of material. Another classic is Mumford’s book [Mum2] O on Curves and their Jacobians. For elliptic curves, the books [Sil1] O , [Sil2] O by Silverman are standard references, with a particular focus on the arithmetic theory. Alternatives are the books [Kna] O by Knapp and [Was] O by Washington. There seems to be no comprehensive account of hyperelliptic curves in a text book, but [Liu] O and Vakil’s book [Vak] X are good to get started in the theory. See also [Loc] O , [Sto] X , [Gross] X . For relative hyperelliptic (and other) curves, also see [LoKl] O .

Exercises Exercise 26.1. Let k be a field, and let C = V+ (T02 ) ⊂ P2k . Show that C is not reduced and that H 0 (C, OC ) = k. Hint: This can be checked by a direct computation. Alternatively use cohomology, cf. the proof of Corollary 25.141. Exercise 26.2. Let k be a field. Show that C = P1k ⊗k k[ε]/(ε2 ) cannot be embedded as a closed subscheme of P2k . Specify a closed immersion C ,→ P3k . Exercise 26.3. Let k be a field. For a commutative monoid S we denote by k[S] the corresponding monoid algebra. From now on let S ⊆ N be an additive submonoid4 so that ( ) X n k[S] = an T ∈ k[T ] ; an = 0 for n ∈ /S . n

Set CS := Spec k[S] which is called the monomial curve of S. (1) ShowPthat there exists a unique finite Pp generating set n1 < n2 < · · · < np of S (i.e., p S = i=1 Nni ) such that if nj = i=1 mi ni with mi ∈ N, then mi = 0 for i ̸= j. (2) Let ρ : L := Np → S be the homomorphism of monoids sending ei to ni with ni defined as in (1) and ei = (0, . . . , 0, 1, 0, . . . , 0) with 1 at the i-th place. Then ρ yields a k-algebra map k[ρ] : k[L] = k[T1 , . . . , Tp ] → k[S], Ti 7→ T ni . The latter is surjective and thus defines a closed immersion CS → Apk . Show that its kernel is generated by fν,µ := T ν − T µ for (ν, µ) ∈ R := { (ν, µ) ∈ L × L ; ρ(ν) = ρ(µ) }. Let r be the minimal number s such that ker(k[ρ]) is generated by s polynomials of the form fν,µ for (ν, µ) ∈ R. 4

Recall that N denotes the natural numbers including 0.

599 (3) Show that r ≥ p − 1. (4) Show that CS is lci if and only if r = p − 1. (5) Let d > 0 be the greatest common divisor of {n1 , . . . , np }. Show that dZ is the subgroup of Z generated by S and that the inclusion k[S] → k[T d ] is the normalization of k[S]. (6) From now on we assume that d = 1. Show that there exists a greatest integer F (S) that is not in S. Show that c(S) := (T F (S)+1 ) is the conductor of k[S], i.e., the ideal { f ∈ k[S] ; f k[T ] ⊆ k[S] } of k[S] and of k[T ]. (7) Show that CS is Gorenstein if and only if there exists an integer F such that S = {m ∈ N ; F − m ∈ / S }. If such an F exists it is necessarily the integer F (S). In the case that p = 2 show that F = (n1 − 1)(n2 − 1) − 1 is such a number, showing that CS is Gorenstein in this case (which also follows from the fact that reduced plane curves are always lci). (8) Suppose that p ≤ 3. Show that r ≤ 3 and that CS is lci if and only if it is Gorenstein. (9) Let a ≥ 1. Show that k[T a , T a+1 , T a+2 ] is Gorenstein (equivalently by (8), lci) if and only if a = 1, 2, or 4. (10) Let S be the monoid generated by {5, 6, 7, 8}. Show that F (S) = 9 and that CS is Gorenstein but not lci. Remark : See [Her] O , [HeKu] O . Exercise 26.4. Let X be a noetherian scheme, let U ⊆ X be an open dense subspace, and set Y := X \ U . Show that for every closed point y ∈ Y there exists an integral one-dimensional subscheme C ⊆ X such that {y} is open in C ∩ Y . Hint: Use that in any noetherian local ring A with dim A ≥ 1 the intersection of all prime ideals p ⊂ A with dim A/p = 1 is the nilradical of A. Exercise 26.5. Let Y be a noetherian scheme, and let f : X → Y be a separated morphism of finite type. (1) Let Z ⊆ X be a closed subscheme, minimal among those for which the restriction of f is not proper. Show that Z is integral and that dim Z = 1. Hint: For the second assertion use a Nagata compactification Z¯ → Y of Z over Y ¯ and Exercise 26.4 for Z ⊆ Z. (2) Show that if f∗ (OX /I ) is a coherent OY -module for every coherent ideal I ⊆ OX , then f is proper. Hint: Argue by contradiction and assume that f is not proper. Use (1) to reduce to the case that X is integral of dimension 1 and that Y is integral of dimension ≤ 1. If dim Y = 0, then one can assume that Y = Spec k for a field k and hence that X is affine. Then f∗ OX is not coherent. If dim Y = 1, then f is quasi-finite and can be factorized in an open dominant immersion j followed by a finite morphism (Corollary 12.85). Show that j∗ OX has to be coherent and deduce that j is an isomorphism. Remark : For this and the previous exercise, see [Lip2] O X (4.3.9) Exercise 26.6. Let Y be a locally noetherian scheme and let f : X → Y be a morphism of schemes. Show that f is proper if and only if f is cohomologically proper, separated, and of finite type. Hint: Exercise 26.5

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26 Curves

Exercise 26.7. Let k be a field, let X be a proper scheme over k, and let L be a line bundle that is generated by its global sections. Show that L is ample if and only if for every closed immersion i : C ⊆ X of an integral curve C, the restriction i∗ L has degree ̸= 0. Exercise 26.8. Let k be a field and let C be a reduced geometrically connected proper curve over k of genus g. Suppose that the characteristic of k is 0 or prime to g − 1. Show that H 0 (C, OC ) = k. Exercise 26.9. Show that in the situation of Proposition 26.59, for g ≥ 2 the converse to (1) or (2) does not hold in general. Exercise 26.10. Let X be a proper scheme over a field k and let L be an ample line bundle on X such that for all n ≥ 0 the L multiplication map H 0 (X, L ) ⊗k H 0 (X, L ⊗n ) → 0 ⊗n+1 H (X, L ) is surjective. Show that n≥0 H 0 (X, L ⊗n ) is a quotient of Sym H 0 (X, L ) and deduce that L is very ample. Exercise 26.11. Let X be an integral proper curve over a field k of genus 0 such that H 0 (X, OX ) = k. Show that every line bundle on X of degree > 0 is very ample. Hint: Exercise 26.10 Exercise 26.12. Let k be a field, and let C = V+ (f ) ⊂ P2k be a projective curve over k, with f ∈ k[X0 , X1 , X2 ] homogeneous of degree d. A point x ∈ C(k) is called an inflexion point, or a flex , for short, if x is a smooth point of C and the intersection multiplicity of C with the tangent line Tx C at x is ≥ 3. (Cf. Definition 16.60.) (1) The Hessian of f is   ∂2f Hf = det ∂Xi ∂Xj i,j=0,1,2 Show that Hf is a homogeneous polynomial which is invariant up to scalars in k × under change of coordinates, i.e., under automorphisms of P2k . (2) Now suppose that char(k) ̸= 2. Show that the flexes of C are the smooth points of C that lie in V+ (Hf ). (3) Suppose that char(k) ̸= 2. Determine all flexes of the Fermat cubic V+ (X03 + X13 + X23 ) (depending on k). Exercise 26.13. Let k be a field. (1) Let E ⊂ P2k be an elliptic curve defined by a Weierstraß equation. Prove that the point (0 : 1 : 0) is a flex (Exercise 26.12) of E. (2) Now let E ⊂ P2k be an elliptic curve whose fixed neutral element 0 is a flex. Denote by E[3] the kernel of the multiplication E → E by 3, i.e., x 7→ x + x + x where addition is the group law on E. Show that E[3](k) is precisely the set of inflexion points in E(k). Exercise 26.14. Let k be a field, and let E be an elliptic curve over k with neutral element 0 ∈ E(k). Let C = E \ {0}, an affine curve over k. (1) Prove that the groups Pic(C) and E(k) are isomorphic. (2) Now let k = C be the field of complex numbers. Denote by Pican (C an ) the complex analytic Picard group of the (non-compact) Riemann surface C an , i.e., the group of isomorphism classes of locally free OC an -modules of rank 1. Prove that Pican (C an ) ∼ = H 2 (C an , Z) = 0. In particular, the natural map Pic(C) → Pican (C an ) is not injective (in contrast to Theorem 20.60).

601 Exercise 26.15. Show that a plane curve is not hyperelliptic. Hint: Use Proposition 26.79 and Proposition 26.129. Exercise 26.16. Let g ∈ N, and let k be a field. Show that there exists a smooth proper curve C of genus g over k. Exercise 26.17. Let k be a field and let f : C → P1 be a generically ´etale morphism of connected smooth proper curves over k with deg(f ) = 2. In particular, C is hyperelliptic. (1) Assume that there exists a point 0 ∈ C(k) with ℓ(2[0]) > 1. Show that there exists a rational function x ∈ K(C) which has a double pole at 0 and no other poles. Show that ℓ(2g[0]) = g + 1, that ℓ((2g + 1)[0]) = g + 2 and that there is an element y ∈ H 0 (C, OC ((2g + 1)[0])) such that 1, x, . . . , xg , y is a basis of this space. Show that, possibly after changing x and y by elements of k × , they satisfy an equation of the form y 2 + q(x)y = p(x) for a monic polynomial p ∈ k[x] of degree 2g + 1 and a polynomial q ∈ k[x] of degree ≤ g. Compare Proposition 26.95. Show that the polynomial Y 2 + q(X)Y − p(X) is irreducible and conclude that C \ {0} is isomorphic to the affine plane curve defined by the above equation. (2) Assume that k is algebraically closed. Show that there exist at most 2g + 2 points p ∈ C(k) with ℓ(2[p]) > 1, and precisely 2g + 2 such points if char k ̸= 2. Hint. Show that the hyperelliptic cover of C is ramified at p. Apply the Theorem of Riemann-Hurwitz to f . Exercise 26.18. Let k be a field of characteristic 2 and let C be the hyperelliptic curve over k defined by the affine equation y 2 − y − x5 = 0. Compute the ramification divisor of the hyperelliptic cover C → P1k . Exercise 26.19. Let k = F3 be the field with 3 elements, and let C be the hyperelliptic curve over k obtained as the smooth proper model of the affine curve given by the equation y 2 = −x6 + x2 − 1. Show that C(k) = ∅. Exercise 26.20. Let k be a field and let C be a proper connected curve over k. Let F be a coherent OC -module of rank 0. Show that X deg(F ) = [κ(c) : k] lgOC,c (Fc ). c

where the sum runs over all closed points c of C. Exercise 26.21. Let C be a proper curve over a field and let E and F be vector bundles of constant rank over C. Show that deg(E ⊗ F ) = rk(E ) deg(F ) + rk(F ) deg(E ) Hint: Use Exercises 23.30 to reduce to the case that E is a line bundle. Exercise 26.22. Let k be a field, let C be a smooth proper geometrically connected curve over k. Let L0 and L1 be line bundles on C of degree d0 and d1 , respectively. Let E be a vector bundle that sits in an extension (*)

0 −→ L0 −→ E −→ L1 −→ 0.

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26 Curves

1 (1) Show that E has slope d0 +d 2 . (2) Suppose that (*) is non-split and that d1 = d0 + 1. Show that E is a stable vector bundle. (3) Suppose that d1 > d0 . Show that dimk Ext1OC (L1 , L0 ) = d1 − d0 − 1 + g and deduce that there exist non-split exact sequences of the form (*) if and only if g ≥ d0 − d1 + 2.

Exercise 26.23. Let k be a field, let C be a smooth proper geometrically connected curve over k of genus g ≥ 2. Le E1 and E2 be two non-isomorphic stable vector bundles of the same slope λ. (1) Show HomOC (E2 , E1 ) = 0 and deduce that dimk (Ext1OC (E2 , E1 )) = rk(E1 ) rk(E2 )(g − 1) > 0. (2) Let 0 → E1 → E → E2 → 0 be a non-trivial extension. Show that E is semistable of slope λ and that E is not stable. Show that EndOC (E ) = k if k is algebraically closed. Exercise 26.24. Let C be a smooth proper geometrically connected curve over a field k. Let E be a vector bundle on C. Show that for every subbundle E ′ of E the point (deg(E ′ ), rk(E ′ )) lies on or below the HN polygon of E . Deduce that the HN polygon is the concave envelope of the points (deg(E ′ ), rk(E ′ )), where E ′ runs through the set of subbundles of E . Exercise 26.25. Let C be a smooth proper geometrically connected curve and let p : C˜ → C be an etale cover. Show that p∗ OC˜ is a semistable vector bundle of slope 0. Exercise 26.26. Let k be a field. Let C be a smooth proper geometrically connected curve over k of genus 0. Suppose that C(k) = ∅. Then C is a Brauer-Severi curve its corresponding central quaternion k-algebra D is a skew field. Let k¯ be an algebraic closure, fix an isomorphism P1k¯ ∼ = C ⊗k k¯ and denote by p : P1k¯ → C the projection. (1) Show that every non-zero vector bundle on C has an integral slope d ∈ Z. (2) Show that for every d ∈ Z there exists a unique (up to isomorphism) stable vector bundle Ed of slope d. If d is even, then Ed is a line bundle with p∗ Ed ∼ = OP1k¯ (d). If d is odd, then Ed is a non-split extension of Ed−1 by Ed+1 and p∗ Ed ∼ = OP1k¯ (d)2 . (3) Show that one has EndOC (Ed ) = D for d odd. Exercise 26.27. Let k be a field and let C be a connected smooth projective curve of genus 3 over C. Assume that C is not hyperelliptic. Show that C is a plane quartic curve, i.e., C ∼ = V+ (f ) ⊂ P2k for a polynomial f ∈ k[X, Y, Z] of degree 4. Exercise 26.28. Let k be a finite field. Prove the Weil conjectures (Theorem 26.173) for C = P1k by computing the zeta function explicitly in this case.

27

Abelian schemes

Content – Preliminaries and general results about group schemes – Definition and basic properties of abelian schemes – The Picard functor – Duality of abelian schemes – Cohomology of line bundles on abelian schemes In this chapter, we take up the topic of abelian varieties that we already touched in Chapter 16, see Sections (16.31) ff. Recall that an abelian variety over a field k is a connected proper smooth group scheme. While we will prove several further results on abelian varieties below, for instance that every abelian variety is projective (Proposition 27.174), the main focus in this chapter is putting this notion in families: An abelian scheme A over a scheme S is a proper smooth group scheme A over S such that all fibers are (geometrically) connected, see Definition 27.89. In particular, every fiber is an abelian variety. This generalization to families of abelian varieties will allow us to illustrate many of the results of the previous chapters in action. An important feature of the theory is a duality on the category of abelian varieties, and more generally of abelian schemes over an arbitrary base scheme: To an abelian scheme A/S we can attach the dual abelian scheme At over S. The dual abelian scheme is defined in terms of its functor of points. Proving that it is in fact representable is quite subtle. See Sections (27.29), (27.39). Here we follow a proof by Deligne that is sketched in [FaCh] O , see also [GrK]. With the dual abelian scheme at our disposal we can prove several results about abelian schemes. We show the Fourier-Mukai equivalence (Theorem 27.243) for abelian schemes over arbitrary base schemes. This result is based on a good understanding of the cohomology of the Poincar´e bundle (Proposition 27.229). Then we study the cohomology of line bundles on abelian schemes and in particular prove the RiemannRoch theorem for abelian varieties (Theorem 27.253) and characterize relative ample line bundles (Theorem 27.264). We also use the Fourier-Mukai equivalence to prove Atiyah’s classification of vector bundles on elliptic curves (Theorem 27.271). We define the notion of polarization (Definition 27.280). Altogether, we then have all the ingredients to write down the moduli functor giving rise to the moduli space of principally polarized abelian varieties with full level n structure, see Section (27.55), an object of great importance in classical algebraic geometry (when considered over C) as much as in arithmetic algebraic geometry (when considered over Q or over Z). Showing the representability of this functor and studying the moduli space in more detail requires methods beyond the scope of this book, however. Nevertheless we hope that this will open the door for the reader to a fascinating topic in this area. In Section (27.54) at the end we will discuss abelian varieties over the complex numbers.

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3_12

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27 Abelian schemes

The complex manifold associated with an abelian variety over C is a complex torus, i.e., a quotient of a complex vector space by a full lattice. This description allows for simpler proofs for several of the results in this chapter in this particular situation, and of course also historically was studied before the general theory.

Preliminaries and general results about group schemes We start the chapter by collecting some results about general group schemes, including a digression on fppf-sheaves and algebraic spaces which is necessary since some of the functors that we define, e.g., quotients of group schemes or Picard schemes, are only representable by algebraic spaces. (27.1) General facts on group schemes. In this section we denote by S a scheme. Let f : G → S be an S-scheme. Recall the notion of a group scheme over S (a scheme G such that for every S-scheme T the T -valued points G(T ) are equipped with a group structure, functorially in T , see Section (4.15)) and the notion of a homomorphism of group schemes over S. A group scheme G is called commutative 1 if the group G(T ) of morphisms T → G of S-schemes is commutative for all S-schemes. If G → S is a group scheme over S and T → S is a morphism of schemes, then the base change GT = G ×S T → T is a group scheme over T . In particular, if T → S is the canonical map Spec κ(s) → S for a point s ∈ S, then the fiber Gs = G ×S Spec κ(s) is a group scheme over the residue field κ(s). We have the following notion of translation. Definition and Remark 27.1. Let f : G → S be a group scheme, let T be an Sscheme and let g ∈ G(T ) = HomS (T, G). Then the left translation tg : GT → GT is the composition (g,id) GT = T ×T GT −−−→ GT ×T GT −→ GT , where the last map is the multiplication of the T -group scheme GT . For any T -scheme T ′ the left translation tg is given on T ′ -valued points by left multiplication by the image of g in G(T ′ ). The left translation by g is an isomorphism of T -schemes whose inverse is given by left translation by g −1 . There is an analogous definition for the right translation 2 by a T -valued point of G. We have the following criterion for a group scheme to be separated which is the analogue of the purely topological fact that a topological group is Hausdorff if and only if its unit point is closed. 1

2

While for an abstract group the terms commutative and abelian are used interchangeably, in the situation at hand we reserve the term abelian for abelian schemes in the sense of Definition 27.89 below. We do not introduce a notation for the right translation since in the sequel we will mainly deal with commutative group schemes.

605 Proposition 27.2. Let G → S be a group scheme. Then G → S is separated if and only if the unit section e : S → G is a closed immersion. Proof. The condition is necessary by Example 9.12. It is sufficient because we have the following cartesian diagram, /S

G ∆G/S

e

 G ×S G

(g,h)7→gh−1

 / G.

Definition 27.3. Let G be an S-group scheme. A subgroup scheme is a subscheme G′ such that the following equivalent assertions hold. (i) The subset G′ (T ) is a subgroup of G(T ) for all S-schemes T . (ii) The multiplication on G induces by restriction a morphism G′ ×S G′ → G′ , the inversion on G induces by restriction a morphism G′ → G′ , and the unit section of G factors through G′ . Such a subgroup scheme is called a normal subgroup scheme if G′ (T ) is a normal subgroup in G(T ) for all T . Example 27.4. For instance, the unit section e : S → G is always an immersion as is any section of a scheme morphism (Example 9.12). Hence it defines a subgroup scheme of G, where S(T ) ⊆ G(T ) is the trivial group consisting of one element. This notion of subgroup scheme differs from the notion defined in Definition 4.45 where we assumed that the immersion H → G is closed. However, we will show below (Proposition 27.11 and Exercise 27.2) that if S is the spectrum of a field, then these two notions coincide. Moreover, almost all subgroup schemes that we will consider in the sequel will be closed subschemes. Let f : G → H be a homomorphism of S-group schemes, and let H ′ be a subgroup scheme of H (not necessarily closed). Then we define the preimage f −1 (H ′ ) := G ×H H ′ which is a subgroup scheme of G. In particular, the kernel Ker(f ) is defined by the cartesian diagram /S

Ker(f ) (27.1.1)

e

 G

 / H,

f

where e : S → H is the unit section of H. For every S-scheme T one has Ker(f )(T ) = Ker(f (T ) : G(T ) → H(T )). In particular Ker(f ) is a normal subgroup scheme of G. If H → S is separated (which is automatic if S = Spec k for a field k by Corollary 27.9 below), then e is a closed immersion (Proposition 27.2) and hence Ker(f ) will be a closed subgroup scheme of G. Remark 27.5. Let G → S be a group scheme. Then there is a commutative diagram of S-schemes (g,h)7→(gh,h) / G ×S G G ×S G m

#

G,

{

pr1

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27 Abelian schemes

where m is the multiplication of the group scheme, pr1 is the first projection, and where the horizontal arrow, given on T -valued points for any S-scheme T , is an isomorphism with inverse (g, h) 7→ (gh−1 , h). Hence if P is any property of scheme morphisms which is stable under base change and under composition with isomorphisms and if G → S has P, then the multiplication m has P. (27.2) Affine Group schemes and Hopf algebras. Let R be a ring and let G = Spec A be an affine group scheme over Spec R. Then multiplication, inversion, and unit of G induce homomorphisms of R-algebras m∗ : A −→ A ⊗R A,

(27.2.1)

i∗ : A → A,

e∗ : A → R,

called comultiplication, antipode, and counit, respectively. The identities in the definition of a group scheme such as associativity of the group law correspond to the fact that (A, m∗ , i∗ , e∗ ) is a (commutative) Hopf algebra in the following sense. Definition 27.6. A commutative Hopf algebra over R is an R-algebra A together with maps as in (27.2.1) such that A m∗

m∗

/ A⊗A

A

m∗ ⊗id

e∗

/R

a⊗b7→i∗ (a)b

 /A

m∗

 A⊗A

  id ⊗m∗ / A⊗A⊗A A⊗A

commute and such that idA = (e∗ ⊗ id) ◦ m∗ . A Hopf algebra A is called cocommutative if m∗ = s ◦ m∗ , where s is the endomorphism of A ⊗ A with s(a ⊗ a′ ) = a′ ⊗ a. A homomorphism of Hopf algebras over R is a homomorphism of R-algebras that respects comultiplication, antipode, and counit. The equivalence between rings and affine schemes then induces an equivalence between the category of affine group schemes over R and the category of commutative Hopf algebras over R. Via this equivalence, commutative affine group schemes correspond to cocommutative commutative Hopf algebras. More generally, if S is a scheme, then one has the obvious notion of a Hopf OS algebra, and an equivalence between the category of relatively affine (commutative) group schemes over S and the category of quasi-coherent (cocommutative) commutative Hopf OS -algebras. Example 27.7. Let R be a ring, and let Ga,R be the additive group over R, i.e., for every R-algebra A we set Ga,R (A) = A, with the group structure given by addition. The Hopf algebra corresponding to Ga,R is the R-algebra R[T ] with comultiplication, counit, and antipode given by m∗ (T ) = T ⊗ 1 + 1 ⊗ T,

e∗ (T ) = 0,

i∗ (T ) = −T.

Example 27.8. Let R be a ring, and let Gm,R be the multiplicative group over R, i.e., for every R-algebra A one has Gm,R (A) = A× . Then the Hopf algebra corresponding to Gm,R is the R-algebra R[T, T −1 ] with comultiplication, counit, and antipode given by m∗ (T ) = T ⊗ T,

e∗ (T ) = 1,

i∗ (T ) = T −1 .

607 (27.3) Generalities about group schemes over a field. Corollary 27.9. Let k be a field and let G be a group scheme over k. Then G is separated. Proof. Let π : G → Spec k be the structure morphism. Let x ∈ G be the image point of the unit section e ∈ G(k). Let U = Spec A ⊆ G be any open affine neighborhood of x. As π ◦ e = idSpec k the homomorphism A → k corresponding to e : Spec k → U has a section and therefore is surjective. This shows that e is a closed immersion. Hence G is separated by Proposition 27.2. The proof shows that, more generally, any group scheme G → S is separated over S if S consists of only one point. One deduces that every group scheme over a discrete base scheme is separated. But there exist group schemes over discrete valuation rings that are not separated (Exercise 27.1). Recall that in Section (16.31) we proved already the following statements about group schemes over a field. Proposition 27.10. Let k be a field, and let G be a group scheme locally of finite type over k. (1) Then G is smooth over k if and only if G is geometrically reduced over k. (2) The group scheme G is geometrically irreducible if and only if it is connected. (3) Let k be perfect. Then Gred is a subgroup scheme of G which is smooth over k. Proposition 27.11. Let k be a field and let G be a group scheme locally of finite type over k. (1) Let U, V ⊆ G be open dense subsets. Then U · V (the image of U ×k V under multiplication) equals the underlying topological space of G. (2) If G is irreducible, then G is quasi-compact. (3) Let H be a k-subgroup scheme of G. Then H is closed in G. The hypothesis that G is locally of finite type over k is in fact superfluous (Exercise 27.2). Proof. We claim that we may assume that k is algebraically closed. Indeed, let k¯ be an algebraic closure of k. As the projection Gk¯ → G is an open morphism (Theorem 14.38), the inverse image of any dense subset of G is dense in Gk¯ . Moreover, Gk¯ → G is surjective, which shows the claim for (1). To see the claim for Assertion (2) we use that if G is irreducible, then Gk¯ is irreducible by Proposition 27.10 (2). Moreover, quasi-compactness can be checked after surjective base change. The claim for (3) holds since being a closed immersion can be checked after faithfully flat base change (Proposition 14.53). Hence we may assume that k is algebraically closed. Then it suffices to show that U (k) · V (k) = G(k). As the inversion is an isomorphism, V (k)−1 is again open and dense in G(k). Then for g ∈ G(k), g(V (k)−1 ) is still open and dense and hence meets U (k). Therefore there exist u ∈ U (k) and v ∈ V (k) such that gv −1 = u. This shows (1). To see (2) let U ⊆ G be an open dense affine subscheme. Then U ×k U is quasi-compact and its image under multiplication map is G by (1). Therefore G is quasi-compact.

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27 Abelian schemes

¯ be the closure of H, For part (3), it suffices to show that H is closed in G. Let H ¯ → Spec k are endowed with the reduced scheme structure. As H → Spec k and H ¯ H ×k H is dense in H ×k H, ¯ universally open (Theorem 14.38) and since H is dense in H, ¯ Therefore H ×k H is dense in H ¯ Let m be ¯ ×k H. ¯ ×k H. ¯ is dense in H and H ×k H ¯ is closed in G ×k G and contains H ×k H. the multiplication morphism. Then m−1 (H) ¯ ¯ ¯ ¯ ¯ Hence m(H ×k H) ⊆ H. As H ×k H is reduced (Corollary 5.57), m induces by restriction ¯ As H ¯ is also stable under inversion, H ¯ is a subgroup scheme. a multiplication on H. ¯ Then (1) implies that H = H · H = H. Definition and Proposition 27.12. Let k be a field and let G be a group scheme locally of finite type over k. Then the image of the unit section is a point in G. Its connected component is denoted by G0 . As G is locally noetherian, G0 is an open and closed subset and we endow it with the scheme structure making G0 into an open subscheme. It is called the identity component of G. The following assertions hold. (1) The subscheme G0 is a quasi-compact normal open and closed subgroup scheme of G which is geometrically irreducible over k. (2) Every homomorphism of group schemes φ : G → H locally of finite type over k induces a homomorphism G0 → H 0 . (3) For every field extension K of k one has (G ⊗k K)0 = G0 ⊗k K. Proof. As G0 is connected and has a rational point, it is geometrically connected by Lemma 26.4. Hence G0 ×k G0 is connected (Proposition 5.53). Therefore the restriction of the multiplication map to G0 ×k G0 factors through G0 . Clearly, the restriction of the inversion map to G0 factors also through G0 . Therefore G0 is an open and closed connected subgroup scheme. It is geometrically irreducible by Proposition 27.10 and quasi-compact by Proposition 27.11 (2). To show that G0 is normal, consider the action of G on itself by conjugation. If G′ is any connected component of G, then the restriction of the action to G′ ×k G0 factors through G0 since G′ ×k G0 is connected. This shows that the conjugation action of all of G preserves G0 , i.e., G0 is a normal subgroup. Assertion (2) is clear, and (3) holds because G0 is geometrically connected. Lemma 27.13. Let k be a field and let G be a k-group scheme locally of finite type. Then every connected component of G is quasi-compact and irreducible and G is equidimensional. Proof. Suppose first that k is algebraically closed. Let C be a connected component of G. As C is locally of finite type over k and k is algebraically closed, there exists g ∈ C(k). ∼ Translation with g induces an isomorphism G0 → C. Therefore C is quasi-compact and irreducible by Proposition 27.12. Moreover, every irreducible component of G is isomorphic to G0 . In particular G is equi-dimensional of dimension dim G0 . Now let k be arbitrary and let k¯ be an algebraic closure of k. Then Spec k¯ → Spec k is universally open (Theorem 14.38) and universally closed because k → k¯ is integral. Therefore the morphism π : Gk¯ → G obtained by base change is open, closed, integral, and surjective. Let C be a connected component of G and let C ′ be a connected component of Gk¯ mapping to C. As π is open and closed, π(C ′ ) = C. As we have already seen that C ′ is quasi-compact and irreducible, so is C. As π is also integral, we have dim C ′ = dim C (Proposition 12.12) which implies that G is equi-dimensional.

609 Over arbitrary fields in general there exist connected components of G that are not geometrically connected and in particular are not isomorphic to G0 (Exercise 16.7). Proposition 27.14. Let k be a field and let f : G → H be a quasi-compact homomorphism of group schemes locally of finite type over k. (1) The subspace f (G) is closed in H. (2) One has dim G = dim f (G) + dim Ker(f ). (3) Suppose that H is smooth over k and that f is surjective. Then f is faithfully flat. Proof. Let k¯ be an algebraic closure of k. Then π : Hk¯ → H is surjective and integral and in particular closed. Moreover for every closed subspace Z of Hk¯ we have dim π(Z) = dim Z by Proposition 12.12. Hence we may assume that k is algebraically closed. We first prove (3). As H is reduced, its identity component H 0 is an integral scheme. Hence by generic flatness (Theorem 10.84) there exists a non-empty open subset V of H 0 such that the restriction f −1 (V ) → V is flat. As S f is a group homomorphism, it is flat over all translates th (V ) for h ∈ H(k). As H = h∈H(k) th (V ), this shows that f is flat. To show (1) and (2) we may replace G by Gred and assume that G is reduced and hence smooth over k (Proposition 27.10 (1)). We show (1). Let C be the closed reduced subscheme of H whose underlying topological space is the closure of f (G). Then C is stable under the inversion of H. The morphism f : G → C is quasi-compact and dominant, hence f × f : G × G → C × C is also quasicompact and dominant. Hence the multiplication of H maps C × C into C. Therefore C is a subgroup scheme of H. Replacing H by C we may assume that f is dominant. By Chevalley’s theorem (Theorem 10.20), f (G) is constructible. As it is dense in H it contains an open dense subset U of H. By Proposition 27.11 (1) we see that H = U · U ⊆ f (G). Therefore f (G) = H. Let us show (2). We may replace H by f (G) with the reduced scheme structure and can therefore assume that f is surjective and that H is smooth over k. Moreover f is then flat by (3). Then f (G0 ) is open (by openness of flat morphisms locally of finite presentation, Theorem 14.35) and closed (by (1)) in H 0 . Hence the restriction f 0 : G0 → H 0 is still surjective. We have dim(G0 ) = dim(G) and dim(H 0 ) = dim(H). Moreover we have Ker(f )0 ⊆ Ker(f 0 ) ⊆ Ker(f ) and hence dim Ker(f 0 ) = dim Ker(f ). Therefore we may assume that G and H are integral schemes. By Remark 14.119 there exists a non-empty open subscheme V of H such that dim(G) = dim f −1 (v)+dim H for all v ∈ V (k). As all fibers have the same dimension by homogeneity this shows (2). (27.4) Differentials of group schemes. Let S be a scheme, let f : G → S be a group scheme, let e : S → G be its unit section which is an immersion (Remark 9.12). Let Ω1G/S be the sheaf of K¨ahler differentials (Section (17.1)). It is an OG -module which is of finite presentation if G → S is locally of finite presentation (Corollary 17.34). Let Ce be the conormal sheaf of the immersion e (Section (17.3)). By (17.4.4) one has (27.4.1)

Ce = e∗ Ω1G/S .

Proposition 27.15. There are isomorphisms of OG -modules (27.4.2)

Ω1G/S ∼ = f ∗ Ce . = f ∗ e∗ Ω1G/S ∼

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27 Abelian schemes

Thus Ω1G/S is a (locally) free OG -module whenever Ce is a (locally) free OS -module. In particular, if S is the spectrum of a field, then Ω1G/S is a free OG -module. Proof. The second isomorphism follows from (27.4.1). To construct the first isomorphism consider the commutative diagram m

G ×S G p1

 G

/G f

f

 / S,

where m is multiplication of the group scheme and pi is the i-th projection. It is easily checked on T -valued points, where T is some S-scheme, that this diagram is cartesian. Hence by Proposition 17.30 we get isomorphisms m∗ Ω1G/S ∼ = Ω1p1 ∼ = p∗2 Ω1G/S . Pulling back by i := (idG , e) : G → G ×S G we obtain an isomorphism of i∗ m∗ Ω1G/S = id∗G Ω1G/S = Ω1G/S and i∗ p∗2 Ω1G/S = f ∗ e∗ Ω1G/S . Applying exterior powers we obtain the following consequence. Corollary 27.16. For all i ≥ 0 one has isomorphisms of OG -modules ΩiG/S ∼ = f∗ = f ∗ e∗ ΩiG/S ∼

(27.4.3)

Vi

OS Ce




.

Remark and Definition 27.17. Let f : G → S be a group scheme locally of finite presentation. Let e : S → G be its unit section with conormal sheaf Ce . We set ∨ ∗ ∗ 1 Lie(G) := Te (G/S) = C ∨ e = (e ΩG/S ) = e TG/S ,

where (−)∨ denotes the OS -dual and TG/S denotes the tangent sheaf. This is the underlying OS -module of the Lie algebra3 of G. As Ω1G/S is of finite presentation (Corollary 17.34), Lie(G) is quasi-coherent by Proposition 7.29. The OS -module Lie(G) determines the tangent sheaf by TG/S = f ∗ Lie(G) (27.4.2). One has the following functoriality properties for Lie(G). Remark 27.18. Let G and H be group schemes locally of finite presentation over S. (1) Let f : G → H be a morphism of S-schemes sending eG to eH . Then by functoriality (17.6.9) one obtains a homomorphism of OS -modules (27.4.4)

Te (f ) : Lie(G) −→ Lie(H).

(2) By (17.6.10) one has (27.4.5) 3

Lie(G ×S H) = Lie(G) × Lie(H).

As we are interested in the sequel mainly in commutative group schemes, we will not give the definition of the Lie bracket on Lie(G) here. It is obtained by pulling back the Lie bracket on TG/S (Remark 17.40) with some caveat as the Lie bracket on TG/S is only OS -linear. See [SGA3] O X Exp. II.

611 (3) The multiplication map m : G ×S G → G induces the addition Te (m) : Lie(G) × Lie(G) = Lie(G ×S G) −→ Lie(G),

(ξ, ζ) 7→ ξ + ζ.

Indeed, write m∗ := Te (m). As m ◦ (idG , e) = idG , we find m∗ (ξ, 0) = ξ. Similarly, m∗ (0, ζ) = ζ. Hence m∗ (ξ, ζ) = m∗ ((ξ, 0) + (0, ζ)) = m∗ (ξ, 0) + m∗ (0, ζ) = ξ + ζ. (4) For a scheme U we set U [ε] := Spec OU [T ]/(T 2 ). Then the structure morphism U [ε] → U has a section that corresponds to T 7→ 0. It is a thickening of order 1. Combining Proposition 17.43 with (3) one sees that one has for every open subscheme U of S an exact sequence of groups (27.4.6)

0 → Γ(U, Lie(G)) → G(U [ε]) → G(U ) → 1.

Every a ∈ Γ(U, OS ) yields an endomorphism of the U -scheme U [ε] by multiplying ε, i.e., the residue class of T , by a. By functoriality we obtain an endomorphism of the functor U 7→ G(U [ε]) which induces a functorial endomorphism of the abelian group Γ(U, Lie(G)). It is not difficult to see that this is simply the scalar multiplication of the Γ(U, OS )-module Lie(G) (cf. Remark 6.8). Remark 27.19. If G : (Sch/S)opp → (AbGrp) is any functor, then one can define a sheaf of abelian groups Lie(G) on S by Γ(U, Lie(G)) := Ker(G(U [ε]) → G(U )) and endow Lie(G) with the structure of an OS -module by the construction of Remark 27.18 (4). Proposition 27.20. Let G be a group scheme locally of finite type over a field k. Then (27.4.7)

dim Lie(G) ≥ dim G.

The following assertions are equivalent. (i) The scheme G is smooth over k. (ii) One has equality in (27.4.7). (iii) The scheme G is smooth in e ∈ G(k) (viewed as a closed point of G). Proof. Inequality 27.4.7 holds since dim Tg (G) ≥ dim G for all g ∈ G(k). Clearly, (i) implies (iii). Moreover, (iii) and (ii) are equivalent by Corollary 18.68. It remains to show that (iii) implies (i). We can check smoothness fpqc-locally and hence may assume that k is algebraically closed. Using that the translation with g ∈ G(k) is an isomorphism we see that G is smooth in all g ∈ G(k). As k is algebraically closed, G(k) is very dense in G (Proposition 3.35). As the set of points in which G is smooth over k is open, this shows that G is smooth in all points. As we can check smoothness of flat morphisms of locally finite presentation on fibers (Theorem 18.56), we obtain by (27.4.2) the following corollary.

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27 Abelian schemes

Corollary 27.21. Let S be a scheme. Let G → S be a group scheme locally of finite presentation and let g ≥ 0 be an integer. Then the following assertions are equivalent. (i) The morphism G → S is smooth of relative dimension g. (ii) The morphism G → S is flat, all fibers are of dimension g, and Lie(G) is a locally free OS -module of rank g. In this case, Ce = Lie(G)∨ is locally free of rank g and hence ΩiG/S is a locally free
OG -module of rank gi . Moreover, one has for the sheaf of global i-forms Rp f∗ ΩiG/S = Rp f∗ OG ⊗OS

(27.4.8)

Vi

OS C e

for all p ≥ 0 and all i ≥ 0. The last assertion follows from the projection formula (Proposition 22.84). The formation of the Lie(G) commutes with base change in the following cases. Proposition 27.22. Let G → S be a group scheme locally of finite presentation, let h : T → S be a scheme morphism, and set GT = G×S T . Suppose that one of the following hypotheses is satisfied. (1) The morphism G → S is smooth. (2) The morphism h : T → S is flat. Then there is a functorial isomorphism Lie(GT ) ∼ = h∗ Lie(G). Proof. Let e : S → G and eT : T → GT be the unit sections, then h∗ Ce ∼ = CeT . Since ∨ ∗ ∨ ∗ Lie(G) = C ∨ e , we obtain a functorial homomorphism α : h (C e ) → (h Ce ) and we have to show that α is an isomorphism. If G → S is smooth, then Ce is finite locally free and we can conclude by Proposition 7.7. Now suppose that T → S is flat. To see that α is an isomorphism, one can work locally on S and T . Hence we may assume that S = Spec A and T = Spec B are affine. Let M be the A-module of finite presentation corresponding to Ce . Then apply Lemma 27.23 with P = B and N = A. Lemma 27.23. Let A be a ring, let M , N , and P be A-modules. Suppose that M is of finite presentation and that P is flat. Then the functorial A-linear map HomA (M, N ) ⊗A P → HomA (M, N ⊗A P ) is an isomorphism. Proof. Consider both sides as functor in M . Choose a resolution L1 → L0 → M → 0 with Li finitely generated free A-module. As P is flat, both functors map this resolution to a short left exact sequence. Hence we may assume that M is finitely generated and free by the five lemma. As both sides commute with direct sums in M , we may even assume that M = A. Then the assertion is clear. (27.5) Singularities of group schemes. Over arbitrary base schemes we easily deduce the following corollary in the presence of flatness.

613 Corollary 27.24. Let S be a scheme and let G → S be a flat group scheme locally of finite presentation with geometrically reduced fibers. Then G → S is smooth. Proof. By Proposition 27.10 (1) all fibers of G → S are smooth, hence G is smooth over S by Theorem 18.56. In characteristic zero flat group schemes locally of finite presentation are smooth. Theorem 27.25. Let S be a scheme of characteristic 0, i.e. S is a Q-scheme. Let f : G → S be an S-group scheme that is flat and locally of finite presentation over S. Then G is smooth over S. Proof. By Corollary 27.24 we may assume that S = Spec k is a field of characteristic zero. By Proposition 27.15 we have Ω1G/k ∼ = f ∗ Ce , where e ∈ G(k) is the unit section. As Ce is a k-vector space, Ω1G/k is a (globally) free OG -module. Hence G is a smooth k-scheme because char(k) = 0 (Proposition 18.69). In positive characteristic there exist plenty of examples of non-smooth group schemes (see for instance Example 18.47). But the structure morphism of a flat group scheme locally of finite presentation is still syntomic (see (19.9)). Proposition 27.26. Let S be a scheme and let G be an S-group scheme that is flat and locally of finite presentation over S. Then G → S is syntomic. As we will not use this result, we will only reduce it to a local statement and then refer to [SGA3] O X for the crucial point. Proof. Indeed, by Proposition 19.52 we may assume that S = Spec k for a field k. If chark = 0, then Theorem 27.25 shows that G is even smooth over k. Hence we may assume that char(k) = p > 0. As “syntomic” is stable under faithfully flat descent, we may also assume that k is algebraically closed. By Proposition 19.50 it suffices to show that OG,g is a complete intersection ring for every closed point g ∈ G(k). But left multiplication by g yields an automorphism of the k-scheme G which induces an ∼ isomorphism OG,g → OG,e , where e ∈ G(k) is the unit section. Hence it suffices to show that OG,e or, equivalently by Remark 19.47 (2), ObG,e is a complete intersection ring. But one can show (e.g., [SGA3] O integers X Exp. VIIB, Corollaire 5.4) that there exist nr p n1 k[[T , . . . , T , . . . , T ]]/(T s ≥ r ≥ 0 and n1 , . . . , nr ≥ 1 such that ObG,e ∼ , . . . , Trp ) = 1 r s 1 which is clearly a complete intersection ring. Proposition 27.27. Let G be a group scheme that is flat and locally of finite presentation over a scheme S. Then the function s 7→ d(s) := dim Gs is locally constant. Proof. Let π : G → S be the structure morphism. We first claim that the function δ : g 7→ dimg (Gπ(g) ) from G to Z is locally constant, i.e., continuous if Z is given the discrete topology. If π is smooth, then this is clear since the map is given by the relative dimension of G over S by (18.10.3). In general, we can use Proposition 27.26 to see that all fibers of G → S are locally complete intersections and in particular Cohen-Macaulay. Therefore δ is locally constant by Proposition 25.119. As all fibers of G are equi-dimensional (Lemma 27.13), δ factors into d ◦ π for some map d : S → Z. As π is open and surjective (Theorem 14.35), d is also continuous.

614

27 Abelian schemes

(27.6) Digression: Sheaves for the fppf topology. Next we would like to study quotients of group schemes. The notion of quotient is much more subtle than that of subgroup schemes. If G is a group scheme over a base scheme S and H ⊆ G is a subgroup scheme one would like to define a homogeneous space G/H, i.e., (ideally) a scheme G/H with a “canonical projection” G → G/H with suitable properties reflecting the quotient property. As G and H can be both considered as functors (Sch/S)opp → (Grp) (Section (4.15)) and H(T ) is a subgroup of G(T ) for every S-scheme T , a first idea would be to consider the functor T 7→ G(T )/H(T ). But this will almost never be a scheme. In fact it will usually not even be a Zariski sheaf (Section (8.3)). Moreover, we know that S-schemes considered as functors (Sch/S)opp → (Sets) (called presheaves in (Sch/S)) satisfy the sheaf criterion for finer “topologies on (Sch/S)” (more precisely, for suitable Grothendieck topologies on (Sch/S)) as explained in Section (14.18). In order to have a chance that the quotient G/H is a scheme one has to sheafify the presheaf T 7→ G(T )/H(T ) for some Grothendieck topology. It turns out that one possible choice of Grothendieck topology is the fppf topology that we have already briefly mentioned in Remark 14.77. Thus we will define G/H as the fppf-sheafification of the presheaf T 7→ G(T )/H(T ) on (Sch/S), see below for precise definitions. Then it is still not at all clear that the fppf-sheaf G/H is representable by an S-scheme. In fact, in general this will not be the case. But most of the fppf-sheaves that we encounter are at least locally for the ´etale topology (Section (20.5)) representable. Such fppf-sheaves are called algebraic spaces. Therefore we will digress in this and the following two sections from the theory of group schemes and explain fppf-sheaves, algebraic spaces and what it means to be a surjective map of fppf-sheaves. The category (Sch/S) is a large category and hence one runs into certain set-theoretic issues if one considers categories of functors with domain (Sch/S) which we largely ignore. These problems can be resolved, see Section (F.1) for a brief discussion and references. For a functor F on (Sch/S), a morphism g : T ′ → T of S-schemes and an element f ∈ F (T ) we think of F (g)(f ) as the pullback of f along g and hence usually denote this element of F (T ′ ) by g ∗ f . We can also view f ∈ F (T ) as a morphism T → F of functors; then g ∗ f is just the composition f ◦ g. Sometimes, specifically if we think of g ∗ f as the restriction of f to an “open” (in the sense of some Grothendieck topology), we simply write f |T ′ := g ∗ f . Given a morphism f : G → H of functors on (Sch/S) and an element g ∈ G(T ) for some S-scheme T , we also write f (g) = f (T )(g) = f ◦ g, where for the final expression we again adopt the point of view that g corresponds to a morphism T → G which can be composed with f . Definition and Lemma 27.28. Let S be a scheme. (1) A presheaf on (Sch/S) is by definition a functor F : (Sch/S)opp → (Sets). (2) A family of scheme morphisms gi : Ti → T is called S an fppf-covering if gi is flat and locally of finite presentation for all i and if T = gi (Ti ). (3) Let F be a presheaf on (Sch/S). Then F is called a sheaf for the fppf-topology or an fppf-sheaf if it satisfies the following equivalent conditions. (i) For every fppf-covering of S-schemes (gi : Ti → T )i the functor F satisfies the sheaf property for this covering. This means, denoting by p1 : Q Ti ×T Tj → Ti and p2 : Ti ×T Tj → Tj the projections, that for every (ai )i ∈ i F (Ti ) such that p∗1 ai = p∗2 aj ∈ F (Ti ×T Tj ) there exists a unique a ∈ F (T ) with gi∗ a = ai .

615 (ii) The functor F is a sheaf for the Zariski topology (i.e., satisfies the sheaf property S for all families (gi : Ti → T )i , where gi are open immersions with gi (Ti ) = T ) and F satisfies the sheaf property for each family consisting of a single faithfully flat finitely presented morphism g : Spec R′ → Spec R of affine S-schemes. Proof. Clearly, (i) implies (ii). Conversely, suppose ` that F satisfies (ii) and let (gi : Ti → T )i be an arbitrary fppf-covering. Set T ′ = i Ti . Then the gi define a morphism g : T ′ → T such that (T ′ → T ) is an fppf-covering and F satisfies the sheaf condition with respect to (Ti → T )i if and only if it satisfies it with respect to the covering consisting of the single morphism g : T ′ → T . Now choose an open affine covering (Uα )α of T . As F is a Zariski sheaf, it suffices to show that F satisfies the sheaf condition for g −1 (Uα ) → Uα for all α. Hence we may assume that T is affine. In this case we may g precompose with a morphism T ′′ → T ′ such that the composition T ′′ → T is an fppf covering and such that T ′′ is affine (Lemma 14.79). But F satisfies the sheaf condition for such fppf coverings by hypothesis. Example 27.29. Let S be a scheme and let X be an S-scheme. Then the functor X : (Sch/S)opp → (Sets) that sends an S-scheme T to X(T ) = HomS (T, X) is an fppfsheaf by faithfully flat descent for morphisms of schemes, Proposition 14.76. As for usual sheaves, one can construct an fppf-sheafification for presheaves, i.e., functors on (Sch/S). We omit the proof, see [Art1] or [Sta] 00W1. Proposition and Definition 27.30. The inclusion functor from the category of fppfsheaves to the category of presheaves on (Sch/S) admits a left adjoint functor F 7→ F # . In other words, for every presheaf F on (Sch/S) there exists an fppf-sheaf F # and a functorial morphism of presheaves ι : F → F # such that for every fppf-sheaf G on (Sch/S) composition with ι yields a bijection Hom(F # , G) ∼ = Hom(F, G). Moreover, we have (1) For all S-schemes T and all elements a, b ∈ F (T ) the images of a and b in F # (T ) are equal if and only if there exists an fppf-covering (gi : Ti → T )i such that gi∗ (a) = gi∗ (b) for all i. (2) For each S-scheme T and for any α ∈ F # (T ) there exists an fppf-covering (gi : Ti → T )i and ai ∈ F (Ti ) such that ι(ai ) = gi∗ (α). The fppf-sheaf F # is called the fppf-sheafification of the presheaf F . Definition 27.31. An fppf-sheaf of groups is a functor (Sch/S)opp → (Grp) whose composition with the forgetful functor (AbGrp) → (Sets) is an fppf-sheaf. A morphism of fppf-sheaves of groups is a morphism of functors (Sch/S)opp → (Grp). An abelian fppf-sheaf is an fppf-sheaf of groups G such G(T ) is an abelian group for all S-schemes T. Remark 27.32. The category of abelian fppf-sheaves on (Sch/S) is an abelian category. For instance a sequence of abelian fppf-sheaves f

0 −→ F −→ G −→ H −→ 0

616

27 Abelian schemes

is exact if and only if 0 → F (T ) → G(T ) → H(T ) is exact for all S-schemes T and if G → H is an epimorphism in the category of fppf-sheaves, i.e., if for all S-schemes T and for all h ∈ H(T ) there exists an fppf-covering (ai : Ti → T )i and gi ∈ G(Ti ) such that f (gi ) = a∗i (h). See Section (27.8) below for more details. (27.7) Digression: Algebraic Spaces. Let S be a scheme. We will encounter fppf-sheaves on (Sch/S) that are not representable by an S-scheme but are at least representable by an algebraic space in the following sense. Our most important example of such a functor is the relative Picard functor PicX/S introduced in Section (27.21) below. For us algebraic spaces will form a technical tool and it is beyond the scope of this book to give a full introduction into this topic. Therefore we will omit some of the proofs and refer to the stacks project [Sta]. Roughly speaking, the notion of algebraic space will allow us to attach a “geometric meaning” to fppf sheaves Z which may not be representable by a scheme but which still are fairly close to a scheme in the sense that there exists a surjective ´etale representable morphism U → Z (and a further technical condition, see below). In the category of fppf sheaves we can then view Z as the quotient of U by some “equivalence relation”, so by definition, the category of algebraic spaces is much more flexible regarding the construction of quotients. In fact, it turns out that many fppf sheaves defined as moduli functors, i.e., whose T -valued points parametrize algebro-geometric objects of some sort, are representable by an algebraic space. Artin gave a list of criteria that ensure representability by an algebraic space in terms of the “deformation theory” of the objects to be parametrized, i.e., their lifting behavior along nil-immersions of spectra of local Artin rings. See [Art3] O , [Art4] O , [BLR] O 8.3, [Sta] 07T0. To introduce algebraic spaces we consider as usually the category of S-schemes as a full subcategory of the category of fppf-sheaves on (Sch/S) via the Yoneda embedding (Section (8.1) and Section (14.18)). Recall from Section (8.2) the notion of representability of a morphism of functors (Sch/S)opp → (Sets) and what it means that such a representable morphism has some property P, e.g., being surjective or being ´etale. Clearly, the composition of two representable morphisms is again representable. If X′ f′

 Y′

/X  /Y

f

is a cartesian diagram of functors (Sch/S)opp → (Sets) and if f is representable, then f ′ is representable. Lemma 27.33. Let Z : (Sch/S)opp → (Sets) be a functor. Then the diagonal ∆Z : Z → Z ×S Z is representable if and only if for every S-scheme U every morphism U → Z is representable. Proof. We write − × − instead of − ×S −. Suppose ∆Z is representable. Let U → Z be a morphism of functors. We have to show that for every scheme V and every morphism V → Z the fiber product U ×Z V is representable by a scheme. This follows from the cartesian diagram

617 / U ×V

U ×Z V  Z

∆Z

 / Z × Z.

For the other direction let U → Z × Z be a morphism, where U is an S-scheme. Then by hypothesis, U ×Z U is representable and we conclude by Z ×∆Z ,Z×Z U = U ×∆U ,U ×U (U ×Z U ). Definition 27.34. An algebraic space over S is an fppf-sheaf Z : (Sch/S)opp → (Sets) such that (a) the diagonal Z → Z ×S Z is representable, and (b) there exists an S-scheme U and a surjective ´etale morphism U → Z (automatically representable by Lemma 27.33). A morphism of algebraic spaces is a morphism of functors (Sch/S)opp → (Sets). Clearly, every scheme is an algebraic space. A surjective ´etale morphism U → Z for a scheme U is sometimes called an atlas. Remark 27.35. Let S be a scheme and let X → Z and Y → Z be morphisms of algebraic spaces over S. Then the fiber product X ×S Y of fppf-sheaves on (Sch/S) is again an algebraic space. Lemma 27.36. Let Y be an algebraic space over S and let g : X → Y be a morphism of functors (Sch/S)opp → (Sets). If g is representable, then X is an algebraic space. Proof. Let U be an S-scheme, and let U → X×Y X be a morphism given by h1 , h2 : U → X with g ◦ h1 = g ◦ h2 . Since the diagonal of Y is representable, so is Eq(h1 , h2 ) → U by (9.1.4). Therefore Eq(h1 , h2 ) is representable by a scheme. Then the cartesian diagrams X ×Y X

/Y

 X ×S X

 / Y ×S Y

∆

Eq(h1 , h2 )

/U

 X

 / X ×Y X

(h1 ,h2 )

show first that X ×Y X → X ×S X is representable and then that X → X ×Y X is representable. Therefore their composition, the diagonal of X, is representable. If U is an S-scheme and U → Y is an ´etale surjective morphism, then U ×Y X is a scheme because X → Y is representable. As the properties “´etale” and “surjective” are stable under base change, the projection U ×Y X → X is ´etale and surjective. We are now going to define the underlying topological space of an algebraic space. For a scheme Y we denote by |Y | its underlying topological space. Let X be an algebraic space. Choose a scheme U and an ´etale surjective morphism U → X. Set R := U ×X U which is a scheme by Lemma 27.33. The image of |R| in |U | × |U | is an equivalence relation ([Sta] 03BW) and we define |X| to be the quotient space of |U | with respect to this equivalence relation. By loc. cit. this is well defined. Lemma and Definition 27.37. The above construction of the topological space |X| does not depend on the choice of atlas U → X and |X| is called the underlying topological space of X.

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27 Abelian schemes

Let f : X → Y be a morphism of algebraic spaces. Choose schemes U and V with ´etale surjective morphisms V → Y and U → V ×Y X. Then the composition U → X is surjective ´etale and U → V induces a continuous map |X| → |Y | independent of all choices ([Sta] 03BX). We obtain a functor from the category of algebraic spaces to the category of topological spaces. Definition 27.38. Let X be an algebraic space. (1) Then X is called quasi-compact if there exists a quasi-compact scheme U and an ´etale surjective morphism U → X. Then X is quasi-compact if and only if its underlying topological space |X| is quasi-compact ([Sta] 03E4). (2) Let P be a property of schemes that is local in the ´etale topology, i.e., if S is a scheme and (Si → S)i∈I is an ´etale covering (Definition 20.21), then S has the property P if and only if Si has the property P. Then an algebraic space X is said to have the property P if there exists a scheme U that has the property P and an ´etale surjective morphism U → X. Properties of schemes that are local in the ´etale topology are for instance “locally noetherian”, “reduced”, “normal”, “regular”. Definition 27.39. Let P be a property of morphisms of schemes that is local on source and target for the ´etale topology. Then a morphism f : X → Y of algebraic spaces is said to have the property P if there exists a commutative diagram (*)

U  X

g

f

/V  /Y

with U and V schemes, U → X and V → Y ´etale, and U → X surjective such that g has property P. Examples for properties that are local on source and target for the ´etale topology are “locally of finite presentation”, “locally of finite type”, “smooth”, “´etale”, “unramified”. Remark 27.40. Let P be a property of morphisms of schemes that is local on source and target for the ´etale topology. If the property P for scheme morphisms is stable under composition (resp. stable under base change), so is the property P for morphisms of algebraic spaces. Working with an atlas one also easily deduces the following fact from the analogous property for morphisms of schemes (Proposition 10.35). Remark 27.41. Let X → Y be a morphism of algebraic spaces over S. Suppose that X → S is locally of finite presentation and that Y → S is locally of finite type. Then X → Y is locally of finite presentation. Remark 27.42. By definition, an algebraic space Z over S is smooth over S if there exist a smooth S-scheme U and an ´etale surjective S-morphism U → Z. If Z is locally of finite presentation over S, then Z is smooth over S if and only if and for every thickening T0 → T of affine S-schemes the canonical map Z(T ) → Z(T0 ) is surjective ([Sta] 04AM). If S is locally noetherian, then it suffices to check this condition for every thickening of local Artinian rings Spec R0 → Spec R ([Sta] 0APN) defined by an ideal I ⊆ R with ImR = 0.

619 These are the analogues of Theorem 18.56 and Theorem 18.63 for schemes. We will also need the following properties for morphisms of algebraic spaces which are not local on the source. Note that if X → Y is a morphism of algebraic spaces over a scheme S, then the diagonal ∆X/Y : X → X ×Y X is representable, because the diagonal X → X ×S X is representable, and X ×Y X → X ×S X is a monomorphism. Definition 27.43. Let f : X → Y be a morphism of algebraic spaces. (1) The morphism f is called separated if the representable morphism ∆X/Y is a closed immersion. (2) The morphism f is called quasi-compact if for every affine scheme T and every morphism T → Y the fiber product X ×Y T is quasi-compact. (3) The morphism f is called proper if f is separated, quasi-compact, locally of finite type, and satisfies the following valuative criterion: For every valuation ring R with fraction field K and for every commutative diagram Spec(K)

/X ;

 Spec(R)

 /Y

f

there exists a dotted arrow making the diagram commutative. It is shown in [Sta] 03KU that the dotted arrow is necessarily unique since f is separated. Remark 27.44. (1) Similarly as for scheme morphisms one see that the properties “separated”, “quasicompact”, and “proper” for morphisms of algebraic spaces are stable under composition and under base change. (2) If S is an affine scheme, then a morphism of algebraic spaces X → S is quasi-compact if and only if X is quasi-compact (combine [Sta] 03KG with Remark 10.2). (3) If Y is a locally noetherian algebraic space, then a morphism X → Y is proper as soon as the criterion in Definition 27.43 (3) holds for all discrete valuation rings R ([Sta] 0ARK). Remark 27.45. Let X be an algebraic space over S. Then every section e : S → X of X → S is representable and an immersion since we have a cartesian diagram S

/X

e

e

 X

(id,e)



∆X/S

/ X ×S X.

If X → S is separated, then e is a closed immersion. Conversely, the same argument as in Proposition 27.2 shows that if X is a group algebraic space (Definition 27.46) such that the unit section is a closed immersion, then X is separated over S. In the sequel we will be mainly considering group algebraic spaces in the following sense.

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27 Abelian schemes

Definition 27.46. Let S be a scheme. An fppf-sheaf in groups on (Sch/S) that is an algebraic space is called a group algebraic space The following result allows to avoid the language of algebraic spaces if the base scheme is the spectrum of a field. Theorem 27.47. Let k be a field and let G be a separated group algebraic space over k. Then G is a scheme. We will use the theorem only if G is in addition locally of finite type over k. For the proof in this case one first reduces to the case that k is algebraically closed, using that every group scheme locally of finite type over a field is quasi-projective (see Corollary 27.73) and that for quasi-projective schemes Galois-descent is effective (Theorem 14.86). Then one uses that G contains an open dense subspace U that is represented by a scheme ([Sta] 06NH). By translating U one deduces that G is represented by a scheme. For details of the proof we refer to [Sta] 0B8G which also covers the general case. (27.8) Digression: Fppf-surjective morphisms. In order to have a notion of exact sequences of fppf-sheaves of groups we need to know what it means for a morphism of fppf-sheaves to be surjective. This is defined as usual for sheaves replacing open coverings by fppf-coverings. We continue to denote by S a scheme. Definition 27.48. Let f : X → Y be a morphism of fppf-sheaves over S. Then f is called fppf-surjective if it is an epimorphism in the category of fppf-sheaves, i.e., if for all S-schemes T and for all y ∈ Y (T ) there exists an fppf-covering (gi : Ti → T )i and xi ∈ X(Ti ) such that f (xi ) = gi∗ (y). (See [MaMo] III.7, Cor. 5, for a proof of the equivalence.) Remark 27.49. Let f : X → Y be a morphism of fppf-sheaves over S. (1) By passing to an open affine covering, which is a special case of an fppf-covering, it suffices to check the condition for fppf-surjectivity in Definition 27.48 only for affine schemes T . If f is fppf-surjective and T is affine, then Lemma 14.79 shows that we can find for all y ∈ Y (T ) a surjective flat morphism of finite presentation T ′ → T of affine schemes and x ∈ X(T ′ ) with f (g) = y |T ′ . Therefore, f is fppf-surjective if and only if for every affine S-scheme T = Spec R and for every y ∈ Y (R) there exists a faithfully flat R-algebra R′ of finite presentation such that the image y in Y (R′ ) lies in the image of X(R′ ) → Y (R′ ). (2) If (gi : T `i → T )i is an fppf-covering, then the family consisting of the Q single morphism T ′ := i Ti → T is also an fppf-covering. Moreover, Z(T ′ ) = i Z(Ti ) for every fppf-sheaf Z over S by the sheaf condition for the fppf-covering (Ti → T ′ )i . (3) If f : X → Y is fppf-surjective and S ′ → S is a morphism of schemes, then the base change fS ′ : X ×S S ′ → Y ×S S ′ is fppf-surjective. Lemma 27.50. Suppose that X and Y are algebraic spaces over S and let f : X → Y be a morphism. (1) If f is faithfully flat and locally of finite presentation, then f is fppf-surjective. (2) If f is fppf-surjective, then f has a section fppf-locally and is surjective.

621 Proof. To see (1) choose an atlas V → Y and an atlas U → X ×Y V . Then U → V → Y is faithfully flat and locally of finite presentation. Let y ∈ Y (T ) for some S-scheme T . Then g : T ′ := V ×S T → T is an fppf-covering and T ′ → U → X is a point x′ ∈ X(T ′ ) with f (x′ ) = g ∗ (y). To see (2) we apply Definition 27.48 to T = Y and y = idY . This shows that f has a section fppf-locally. In particular, it is surjective fppf-locally. Hence it is surjective by Proposition 14.50. Example 27.51. Let k be a field. Suppose that f : X → Y is an fppf-surjective morphism of fppf-sheaves over k. Then for y ∈ Y (k) there exists a finite extension K of k such that the image of y in Y (K) is in the image of X(K) → Y (K). Indeed, by hypothesis there exists a k-algebra R′ = ̸ 0 of finite presentation and ′ x ∈ X(R′ ) such that f (x) = g ∗ (y) for the fppf-covering Spec R′ → Spec k. Let m ⊂ R′ be a maximal ideal, then K := R′ /m is a finite extension of k by the Nullstellensatz and h : Spec K → Spec k is still an fppf-covering. If x ∈ X(K) is the image of x′ , then f (x) = h∗ (y). Lemma 27.52. Let f : X → Y and g : Y ′ → Y be morphisms of S-schemes. Let g be fppf-surjective. Let P be a property of scheme morphisms that is stable under base change and can be checked fppf-locally. Then f has property P if and only if its base change f ′ : X ×Y Y ′ → Y ′ has property P. Proof. The conditions is clearly necessary. To show that it is sufficient, we may work locally for the fppf-topology by hypothesis. Hence we may assume that g has a section i. Then f is the base change of f ′ along i, hence f has property P if f ′ has it. Lemma 27.53. If f : X → Y is fppf-surjective and a monomorphism, then f is an isomorphism. Proof. By Lemma 27.52 we may check that f is an isomorphism after making the fppfsurjective base change with f itself. Hence it suffices to show that the second projection X ×Y X → X is an isomorphism. But this morphism has as a section the diagonal ∆f : X → X ×Y X which is an isomorphism because f is a monomorphism. Therefore the projection is an isomorphism. Proposition 27.54. Let f : G → H be a homomorphism of S-group algebraic spaces that are separated and locally of finite presentation over S. Let G be flat over S, let H have geometrically reduced fibers over S, and let f be surjective. Then H is flat over S, f is faithfully flat locally of finite presentation, and in particular f is fppf-surjective. We will give the proof if G and H are S-group schemes and give references to the Stacks project for the necessary modifications in the general case. Proof. First note that f is locally of finite presentation by Proposition 10.35 ([Sta] 06G4 for algebraic spaces). Hence it suffices to show that f is flat. This follows from Proposition 27.14 (3) by the fiber criterion for flatness (Corollary 14.27, [Sta] 05X1 for algebraic spaces) and because a group scheme locally of finite type over a field is smooth if and only if it is geometrically reduced (Proposition 27.10). We can apply Proposition 27.14 (3) by Theorem 27.47 or by proving an analogue for Proposition 27.14 (3) for group algebraic spaces which is not difficult.

622

27 Abelian schemes

(27.9) Quotient spaces and homogeneous spaces. In this section, S always denotes a scheme. Now having the notion of fppf-surjectivity at our disposal, we say that a (finite or infinite) sequence of homomorphisms of S-group schemes f i−1

fi

· · · −→ Gi−1 −→ Gi −→ Gi+1 −→ · · · is exact if for all i the homomorphism f i−1 factors through Ker(f i ) and the induced homomorphism Gi−1 → Ker(f i ) is fppf-surjective. In particular a short sequence of S-group schemes f g 1 −→ G′ −→ G −→ G′′ −→ 1 is exact if and only if 1 → G′ (T ) → G(T ) → G′′ (T ) is exact for all S-schemes T (use Lemma 27.53 to see that G′ → Ker(g) is fppf-surjective if and only if it is an isomorphism) and g is fppf-surjective. If G′ , G, and G′′ are commutative group schemes viewed as abelian fppf-sheaves this notion of exactness coincides with the exactness notion defined in Remark 27.32. Example 27.55. Let S be a scheme. Let Gm be the multiplicative group over S, i.e., Gm (T ) = Γ(T, OT )× for every S-scheme T . This functor is represented by the flat scheme Spec OS [X, X −1 ]. Let n ̸= 0 be an integer. Then by Proposition 27.54 the n-th power homomorphism Gm −→ Gm , Gm (T ) ∋ t 7→ tn ∈ Gm (T ) is fppf-surjective and hence we have an exact sequence of S-group schemes (−)m

1 −→ µn −→ Gm −→ Gm −→ 1, where µn is the group scheme of n-th root of unity, i.e., for every S-scheme T one has µn (T ) = { ζ ∈ Γ(T, OT )× ; ζ n = 1 }. It is represented by Spec OS [X]/(X n − 1). In the following definition, by a G-action G ×S X → X we mean a morphism of fppf-sheaves such that for every S-scheme T the map G(T ) × X(T ) → X(T ) is a group action of the group G(T ) on the set X(T ). Definition 27.56. Let G be a group algebraic space over S that is flat and locally of finite presentation over S, and let X be an algebraic space over S with a G-action G ×S X → X, simply denoted by (g, x) 7→ gx on T -valued points, T an S-scheme. Then X is called a homogeneous G-space if the following conditions are satisfied. (a) The structure morphism X → S is fppf-surjective. (b) The morphism σX : G ×S X → X ×S X, (g, x) 7→ (gx, x) is fppf-surjective. Definition 27.57. Let G be an fppf-sheaf of groups on (Sch/S) and let H ⊆ G be an fppf-sheaf of subgroups. The fppf-sheafification of the presheaf (G/H)′ : (Sch/S)opp −→ (Sets),

T 7→ G(T )/H(T )

is called the quotient of G by H and is denoted by G/H. The canonical map f : G → G/H is fppf-surjective. Even if G and H are group schemes, then G/H is in general not representable by an S-scheme. But it can be shown ([Sta] 06PH) that if G is a group algebraic space over S and H is a subgroup algebraic space of G that is flat and locally of finite presentation over S, then G/H is always an algebraic space and G → G/H is faithfully flat and locally of finite presentation.

623 Remark 27.58. Let G and H be as in Definition 27.57. Then the formation of the presheaf (G/H)′ is compatible with base change S ′ → S. Hence this holds also for G/H, i.e., one has (G/H) ×S S ′ = (G ×S S ′ )/(H ×S S ′ ). Lemma 27.59. Let G be a group algebraic space, let H be a subgroup fppf-sheaf such that G/H is a separated algebraic space. Then G/H with its natural left action is a homogeneous G-space. Moreover, H is a closed subgroup algebraic space of G and one has an isomorphism of algebraic spaces over S (27.9.1)

∼

G ×S H −→ G ×G/H G,

(g, h) 7→ (gh, g).

Proof. As G → S has a section, it is fppf-surjective. As the composition G → G/H → S is fppf-surjective, G/H → S is fppf-surjective. By definition, for all x1 , x2 ∈ (G/H)(T ) there exists an fppf-covering T ′ → T and an element g ∈ G(T ′ ) such that g · (x1 )T ′ = (x2 )T ′ . But this shows that σG/H is fppf-surjective. Therefore G/H is a homogeneous G-space. The image of the unit section e ∈ G(S) is a section x0 ∈ (G/H)(S) which is a closed immersion because G/H → S is separated. Then H = f −1 (x0 ), i.e., H is defined by the cartesian diagram /S

H (27.9.2)

 G

x0

f

 / X.

This shows that H is a closed subgroup algebraic space of G. Let T be an S-scheme. If g ∈ G(T ) such that gT ′ ∈ H(T ′ ) for some fppf-covering T ′ → T , then g ∈ H(T ) as fppf-coverings are fppf-surjective. Hence for g1 , g2 ∈ G(T ) one has (g1 , g2 ) ∈ (G ×G/H G)(T ) = G(T ) ×(G/H)(T ) G(T ) ⇔ g2−1 g1 ∈ H(T ). This shows that (g1 , g2 ) 7→ (g2 , g2−1 g1 ) is a well defined inverse map to (27.9.1). Remark 27.60. Let G and X be as in Definition 27.56 with X a homogeneous G-space. Let S ′ → S be a morphism of schemes. Then X ×S S ′ is a homogeneous (G ×S S ′ )-space (Remark 27.49 (3)). Lemma 27.61. Let G and X be as in Definition 27.56 with X a separated homogeneous G-space. Then locally for the fppf-topology, X is of the form G/H for a closed subgroup algebraic space H of G. Proof. Working locally for the fppf-topology we may assume that there exists a section x0 ∈ X(S) which is a closed immersions because X is separated over S. Define a morphism of S-schemes f : G → X by f (g) = g · x0 (on T -valued points). Then f is G-equivariant, where we endow G with the action by itself via left multiplication. The cartesian diagram G ×S S = G

f

idg ×x0

 G ×S X

/ X = X ×S S idX ×s0

σX

 / X ×S X

shows that f is fppf-surjective since σX is fppf-surjective.

624

27 Abelian schemes

Let H := f −1 (x0 ). Then H is the stabilizer of x0 in G, i.e., for every S-scheme T one has H(T ) = { g ∈ G(T ) ; gx0,T = x0,T }. Therefore, H is a subgroup algebraic space of G. As x0 is a closed immersion, H → G is a closed immersion. As f is fppf-surjective, it induces an morphism of fppf-sheaves G/H → X which is a monomorphism and fppf-surjective. Hence it is an isomorphism by Lemma 27.53. Proposition 27.62. Let G be a group algebraic space that is flat and locally of finite presentation over S and let H ⊆ G be a subgroup fppf-sheaf such that G/H is a separated algebraic space locally of finite presentation over S. (1) Then H is an algebraic space which is flat and locally of finite presentation over S. The quotient G/H is faithfully flat over S, and G → G/H is locally of finite presentation and faithfully flat. (2) The map G → G/H is an H-torsor for the fppf-topology. (3) Let P be a property of morphisms of algebraic spaces that is stable under composition with isomorphisms, stable under base change, and that can be checked locally for the fppf-topology on the target. Then G → G/H has property P if and only if H → S has property P. (4) If G is smooth (resp. smooth with geometrically connected fibers) over S, then G/H is smooth (resp. smooth with geometrically connected fibers) over S. Proof. Let us show (1). The canonical fppf-surjective morphism f : G → G/H is surjective, and it is locally of finite presentation by Proposition 10.35 ([Sta] 06G4 for algebraic spaces). We have already seen that G/H → S is fppf-surjective and in particular surjective. To see that f and G/H → S are flat it suffices to show that all fibers of f are flat by the fiber criterion for flatness (Corollary 14.27, [Sta] 05X1 for algebraic spaces). Hence we can assume that S is the spectrum of a field. Now we can base change by the fppf-surjective morphism f itself (Lemma 27.52) and it suffices to show that projection G ×S H → G is flat by (27.9.1). But this is clear because the base is the spectrum of a field. As f is flat and locally of finite presentation, the cartesian diagram (27.9.2) shows that H → S is flat and locally of finite presentation. Now (2) follows from (27.9.1) since we have seen in (1) that G → G/H is an fppfcovering. To see (3), also follows from (27.9.1): The morphism G → G/H has P if and only if G ×S H = G ×G/H G → G has P if and only if H → S has P since G → S is an fppf-covering. It remains to show (4). As we already know that G/H → S is flat, we can show its smoothness on geometric fibers (Theorem 18.56). Therefore we can assume that S = Spec k for an algebraically closed field k. As G → G/H is surjective, G/H is connected if G is connected, regardless of smoothness. Now suppose that G is smooth. As G → G/H is faithfully flat, G/H is regular (Corollary 14.60) and hence smooth over k (Theorem 6.28). Corollary 27.63. Let G and G′ be group algebraic spaces locally of finite presentation over S. Suppose that G is flat over S and that G′ is separated over S. Let f : G → G′ be an fppf-surjective homomorphism of group algebraic spaces. Let P be a property of morphisms of algebraic spaces that is stable under composition with isomorphisms, stable

625 under base change, and that can be checked locally for the fppf-topology on the target, i.e., after passing to an fppf-covering. Then G′ is flat over S, and f is faithfully flat locally of finite presentation. Moreover, f has property P if and only if H := Ker(f ) → S has property P. ∼

Proof. As f is fppf-surjective, it induces an isomorphism G/ Ker(f ) → G′ . Now we conclude by Proposition 27.62. Corollary 27.64. Let G be an S-group scheme which is flat and locally of finite presentation (resp. smooth) over S and let X be a separated homogeneous G-space locally of finite presentation over S. Then X is faithfully flat (resp. smooth and surjective) over S. Proof. The assertions may be checked fppf-locally and hence we may assume that X = G/H for some subgroup algebraic space H of G (Lemma 27.61). We conclude by Proposition 27.62. Corollary 27.65. Let G be an S-group scheme which is flat and locally of finite presentation over S and let X be a separated homogeneous G-space locally of finite presentation over S. Then the morphisms σX : G ×S X → X ×S X, (g, x) 7→ (gx, x), aX : G ×S X → X, (g, x) 7→ gx are faithfully flat and locally of finite presentation. Proof. As X and G are both locally of finite presentation, both morphisms are locally of finite presentation. It suffices to show that σX is flat. Then it is also faithfully flat because it is fppf-surjective by hypothesis. And aX is the composition of σX followed by the first projection X ×S X → X which is faithfully flat by Corollary 27.64. But σX is the GX -equivariant morphism GX → XX of X-schemes given by the identity x0 := idX ∈ X(X) that is given on T -valued points by g 7→ g · x0 for every X-scheme T . To see that σX is flat we can work fppf-locally and hence assume that σX is the projection GX → GX /H, where H is a subgroup fppf-sheaf by Lemma 27.61. Now we conclude via Proposition 27.62. Corollary 27.66. Let S be a scheme, let G be a smooth S-group scheme with connected fibers, and let f : X → S be a separated homogeneous G-space that is locally of finite presentation over S. Then X → S is smooth and has geometrically irreducible and quasi-compact fibers. Proof. By Corollary 27.64, X is smooth over S. Locally for the fppf topology on S there exists an fppf-surjective morphism G → X which is in particular surjective (Lemma 27.50). As G has geometrically irreducible and quasi-compact fibers (Proposition 27.10 (2) and Proposition 27.11 (2)), X also has geometrically irreducible and quasi-compact fibers. Remark 27.67. The separatedness hypothesis on X in Lemma 27.61, on G/H in Proposition 27.62, on G′ Corollary 27.63, and on X Corollary 27.65 and in Corollary 27.64 can be weakened. It was only used to see that any section is a closed immersion. If one assumes only that the diagonal of these algebraic spaces is an immersion (such an algebraic space is called locally separated in [Sta] 02X5), then any section is an immersion. This is for instance the case if X is a scheme. If the diagonal is an immersion, H will be a subgroup algebraic space of G but not necessarily be closed in G. Otherwise, all assertions and arguments remain the same.

626

27 Abelian schemes

Theorem 27.68. Let S be a scheme, let G be a quasi-projective S-group scheme and let H ⊆ G be a subgroup scheme that is finite locally free over S. Then G/H is representable by a scheme and the projection G → G/H is an H-torsor for the fppf-topology. We will not give a proof for this representability result here but refer to [SGA3] O X Exp. V, 4.1., or to [Sta] 07S7. (27.10) Digression: Homotopy invariance of Picard group. We digress briefly for the following result that we will use in the next section to prove quasi-projectivity of homogeneous spaces. Proposition 27.69. Let S be a noetherian scheme and let U ⊆ AnS be an open subscheme such that U (S) ̸= ∅. Let Y be a normal noetherian integral scheme over S. Let p : Y ×S U → Y be the projection. Then p∗ : Pic(Y ) → Pic(Y ×S U ) is an isomorphism. The proposition in particular shows that the canonical map Pic(Y ) → Pic(AnY ) is bijective for all n ≥ 0 if Y is a normal integral noetherian scheme. Without the normality assumption on Y , the assertion fails, see Exercise 27.6 for details. Proof. Choose u ∈ U (S). Then uY : Y → Y ×S U is a section of p and hence u∗ ◦ p∗ = idPic(Y ) . In particular, p∗ is injective. To show the surjectivity of p∗ we will use Lemma 27.70 below. Clearly U → S is flat. As it has a section, it is faithfully flat. The generic fiber of p is an open non-empty subscheme Uη of Anκ(η) . Now Pic(Anκ(η) ) → Pic(Uη ) is surjective by Corollary 11.43 and Pic(Anκ(η) ) = 0 because polynomial rings over fields are factorial. Hence we may apply Lemma 27.70. Lemma 27.70. Let Y be a normal integral locally noetherian scheme with generic point η, let f : X → Y be a faithfully flat morphism locally of finite type with integral fibers, and let L be a line bundle on X whose restriction to the general fiber Xη of f is trivial. Then there exists a line bundle M on Y such that L ∼ = f ∗M . In particular, f ∗ : Pic(Y ) → Pic(X) is surjective if Pic(Xη ) = 0. We will not prove this result here but refer to [EGAIV] O (21.4.11) or to [Sta] 0BD7. (27.11) Quasi-projectivity of homogeneous spaces. Theorem 27.71. Let S be a normal noetherian scheme, let G be a smooth S-group scheme with connected fibers, and let f : X → S be a homogeneous G-space that is of finite type over S. Suppose that there exists an open subscheme U of X that is quasi-affine over S and that meets every fiber of X → S. Then X is quasi-projective. We will not use the theorem in the sequel and give the proof only if S = Spec k for a field k. For a proof in the general case we refer to [Ray1] O V 3.10. Raynaud shows the following more precise assertion. Let D1 , . . . , Dr be the irreducible components of X \ U that areP of codimension 1 in X and let n1 , . . . , nr > 0 be positive integers. Then the Weil r divisor i=1 ni Di is a Cartier divisor D such that OX (D) is ample.

627 Here we follow the Stacks Project [Sta], where the theorem is shown if X = G. Note that the existence of U is clear if S consists of only one point. In fact, it is shown in [Sta] 0BF7 that every group scheme of finite type over a field is quasi-projective. The method there can be generalized to prove that for every group scheme of finite type every homogeneous space is quasi-projective (see also Exercise 27.7 for a sketch how to deduce this result from Theorem 27.71). Proof. (if S = Spec k for a field k) We will prove the theorem in several steps. Step (I) and (II) will not use any hypothesis on S (not even that S is normal and noetherian). Only from step (III) on we will assume that S is the spectrum of a field. (I). Let W ⊆ G be an open non-empty subscheme that meets every fiber of G → S. Then we will show that W · U = X, i.e., the restriction of the action morphism a : G ×S X → X to W ×S U is surjective. Indeed, the question is local for the fppf-topology. Hence we can assume that there exists a G-equivariant fppf-surjective morphism π : G → X. Set U ′ := π −1 (U ). It suffices to show that W · U ′ = G. This follows from Proposition 27.11 (1) because all fibers of G are irreducible by Proposition 27.10 (2). (II). Next we show that X is separated over S. We will use the valuative criterion Theorem 15.9. As all hypotheses are stable under base change S ′ → S we may assume that S = Spec R for a complete discrete valuation ring R with algebraically closed residue field (Remark 15.11). Let s ∈ S (resp. η ∈ S) be the special (resp. generic) point. Let x1 , x2 ∈ X(S) be sections that agree on η. We have to prove that x1 = x2 . For this we show that there exists g˜ ∈ G(S) such that x1 , x2 ∈ (˜ g U )(S). As U is separated over S, g˜U is also separated and therefore this implies x1 = x2 . To find g˜ consider the morphisms of κ(s)-schemes πi : Gs −→ Xs ,

g 7→ g · (xi )s ,

i = 1, 2.

By step (I) we have G·U = X and therefore U meets the orbit of (xi )s . Hence Ui := πi−1 (U ) is non-empty. As Gs is irreducible and κ(s) is algebraically closed, there exists g˜s ∈ G(s) with g˜s−1 ∈ U1 ∩ U2 . As R is complete and in particular henselian, we can lift g˜s to a section g˜ ∈ G(S) (Theorem 20.12). Then x1 and x2 both factor through g˜U . (III). From now on we assume that S = Spec k for a field k. By Proposition 14.57 one can even assume that k is algebraically closed. We will now construct a family (Dv )v of effective divisors on X. By Corollary 18.57 we find an open dense subscheme W ′ ⊆ G and an ´etale morphism π : W ′ → Ank , where n = dim(G). As π is generically finite, we find a non-empty open subscheme V ⊆ Ank such that the restriction of π to a morphism W := π −1 (V ) → V is finite ´etale (Proposition 12.11 (4)). Let U ⊆ X be any non-empty open affine subscheme. As X is smooth over S and irreducible (Corollary 27.66), X is regular (Corollary 14.60) and U is dense in X. By Lemma 25.150 there exists an effective Cartier divisor E whose underlying topological space is X \ U . For v ∈ V (k) we set X g −1 E, Dv := g∈π −1 (v)(k)

where the sum is the sum of effective Cartier divisor, i.e., if D1 and D2 are effective Cartier divisors, locally of the form V (f1 ) and V (f2 ) for regular sections f1 and f2 of OX , then D1 + D2 is locally of the form V (f1 f2 ).

628

27 Abelian schemes

(IV). We claim that the isomorphism class L of OX (Dv ) is independent of v ∈ V (k). As V is connected, the degree of π over Q V is constant. As π is also ´etale and k is algebraically closed we find π −1 (v) = Spec g∈π−1 (v)(k) k as k-schemes and the cardinality of π −1 (v) does not depend on v. Let E ⊆ G ×S X be the preimage of E under the action morphism a. This is again an effective Cartier divisor because a is flat (Corollary 27.65). Then the fiber Eg of E over a k-valued point g ∈ G(k) is the effective Cartier divisor g −1 E. Let N := OG×S X (E) be the line bundle corresponding to E and let M := Nπ−1 (V )/V (N |V ×S X ) be the norm (Remark 12.25) of its restriction to V ×S X. For v ∈ V (k) the restriction Mv of M to the fiber of V ×S X → V over v is OX (Dv ). In view of Corollary 11.43, V (and likewise the base change to any extension field of k), has trivial Picard group, thus Proposition 27.69 can be applied to q : V ×S X → X. This shows that there exists a line bundle L on X such that M ∼ = q ∗ (L ) and hence Mv ∼ =L for all v ∈ V (k). (V). Finally, we show that L is ample. By Proposition 13.47 it suffices to show that there exist finitely S many sections fi ∈ Γ(X, L ) such that Xfi := Xfi (L ), cf. (7.11.1), is affine and X = Xfi . As L ∼ = OX (Dv ) for all v ∈ V (k), we find fv ∈ Γ(X, L ) such that Xfv = X \ Dv . Since X is separated, finite intersections of open affine subschemes are again affine (Proposition 9.15). Hence \ g −1 U X fv = g∈π −1 (v)(k)

S

is affine. Moreover v∈V (k) Uv = W ·U = X by step (I) of the proof. As X is quasi-compact, S there exist v1 , . . . , vn ∈ V (k) with X = i Xfvi . Corollary 27.72. Let S be a normal scheme, let G be a smooth S-group scheme with connected fibers, and let f : X → S be a homogeneous G-space that is locally of finite type over S. Then there exists an open covering (Si )i of S such that XSi is quasi-projective over Si . In particular, X is separated over S (Remark after Definition 13.60). We will give the proof only if S is in addition locally noetherian. For a proof in the general case by noetherian approximation we refer to [Ray1] O VI 2.5. Proof. (if S is in addition locally noetherian) We may assume that S is affine. Let s ∈ S and let U ⊆ X be any open affine subscheme meeting the fiber Xs . As f is flat (Corollary 27.64), V := f (U ) is an open neighborhood of s in S by Theorem 14.35. Therefore it suffices to see that XV → V is quasi-projective. But by definition, U meets every fiber of XV → V . Moreover, as S is separated, V is separated and therefore U → V is an affine morphism (Proposition 12.3). Hence we can apply Theorem 27.71. Corollary 27.73. Let R be a local normal ring and let G → Spec R be a smooth group scheme with connected fibers. Then every homogeneous G-space of finite type is quasiprojective. In particular, G is quasi-projective.

629 (27.12) The graded Hopf algebra structure on the cohomology ring of an algebraic group. Let G be a quasi-compact group scheme over a field k. We know from Example 21.131 that the cup product induces on M H • (G, OG ) = H p (G, OG ) p≥0

the structure of a strictly graded commutative graded k-algebra. Moreover, the group law m : G ×k G → G induces by functoriality a homomorphism of graded k-algebras g µ : H • (G, OG ) −→ H • (G ×k G, OG×k G ) ∼ = H • (G, OG ) ⊗k H • (G, OG ),

where the isomorphism is given by the K¨ unneth isomorphism (Corollary 22.110). Here − ⊗g − denotes the graded tensor product (Definition 19.3). The unit section Spec k → G defines a counit ε : H • (G, OG ) −→ k and the inverse map G → G defines an antipode S : H • (G, OG ) → H • (G, OG ). Altogether we obtain on H • (G, OG ) the structure of a strictly graded commutative graded Hopf algebra over k in the following sense. Definition 27.74. Let R be a commutative ring. L (1) A graded bialgebra over R is an N-graded R-algebra H • = p≥0 H p together with a comultiplication and a counit µ : H • −→ H • ⊗gR H • ,

ε : H • −→ R,

that are both homomorphisms of graded R-algebras and such that the diagram H•

/ H • ⊗g H • R

µ

µ

 H • ⊗gR H •

µ⊗id

id ⊗µ

/

H • ⊗gR

 H • ⊗gR H •

commutes and such that (ε ⊗ id) ◦ µ = (id ⊗ε) ◦ µ = id : H • → H • , using H • ⊗gR R = H • . (2) A map between graded bialgebras is called a homomorphism of graded bialgebras over R if it is a homomorphism of graded R-algebra that is compatible with comultiplication and counit. (3) A graded bialgebra H • over R is called (strictly) graded commutative if the underlying graded R-algebra is (strictly) graded commutative (Definition 17.49). (4) A graded bialgebra H • is called graded Hopf algebra if there exists an antipode S : H • → H • , i.e., an isomorphism of graded R-modules such that m ◦ (id ⊗S) ◦ µ = i ◦ ε = m ◦ (S ⊗ id) ◦ µ, where m : H • ⊗g H • → H • is the multiplication and i : R → H • is the unit.

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27 Abelian schemes

An antipode on a bialgebra is unique if it exists. Every homomorphism of graded bialgebras between graded Hopf algebras automatically preserves the antipode. Example 27.75. V•Let R be a commutative ring and let M be an R-module. Then its exterior algebra (M ) is endowed with the structure of a strictly graded commutative graded Hopf algebra: We have already seen that it carries the structure of a strictly graded commutative graded R-algebra (Section (17.8)). The diagonal map M → M ⊕ M induces a comultiplication µ:

V•

(M ) −→

V•

the projection (M ) → induces the antipode.

V0

V•

(M ⊕ M ) =

V•

(M ) ⊗gR

V•

(M ),

(M ) = R defines a counit, and the map M → M , m 7→ −m

Example 27.76. Let k be a field. We consider the finite-dimensional monogeneous case. Let H • be a finite strictly graded commutative k-algebra generated by one homogeneous element x ̸= 0 of degree d = deg(x) ≥ 1. Then H • is the quotient of the (commutative) polynomial ring k[x] by a homogeneous ideal, i.e. H • ∼ = k[x]/(xs ) for some s ≥ 2. As H • is strictly graded commutative, we necessarily have s = 2 if d is odd. If d is even and H • can be endowed with the structure of a graded k-bialgebra, then one can show ([MiMo] O 7.8) that necessarily p := char(k) > 0 and that s is a power of p. For two graded bialgebras H1• and H2• we can form the tensor product of graded R-algebras H1• ⊗gR H2• and endow it with the structure of a graded bialgebra by defining as comultiplication the composition µ1 ⊗µ2

H1• ⊗gR H2• −−−−−−→ H1• ⊗gR H1• ⊗gR H2• ⊗gR H2• id ⊗T ⊗id −−−−−−→ (H1• ⊗gR H2• ) ⊗gR (H1• ⊗gR H2• ) where T : H2• ⊗gR H1• −→ H1• ⊗gR H2• is the map of graded R-modules with T (x2 ⊗ x1 ) = (−1)deg(x2 ) deg(x1 ) x1 ⊗ x2 for homogeneous elements xi ∈ Hi• . Example 27.77. Let k be a field. If V is a k-vector space with basis (x1 , . . . , xr ), then there is an isomorphism of graded k-bialgebras

V•

g g (V ) ∼ = k[x1 ]/(x21 ) ⊗k · · · ⊗k k[xr ]/(x2r ),

with deg(xi ) = 1 for all i. We will use the following structure theorem by Borel and Hopf, see [MiMo] O 7.11. Theorem 27.78. Let k be a perfect field and let H • be a finite-dimensional strictly graded commutative graded k-bialgebra with H 0 = k. Then there exist monogeneous k-bialgebras Hi• , i = 1, . . . , r, as in Example 27.76 and an isomorphism of k-bialgebras g g H• ∼ = H1• ⊗k · · · ⊗k Hr• .

Corollary 27.79. Let k and H • be as in Theorem 27.78 and assume that there exists an integerVg ≥ 0 such that H i = 0 for all i > g. Then dimk (H 1 ) ≤ g. If dimk (H 1 ) = g, then • (H 1 ) as graded Hopf algebras over k. H• ∼ =

631 Proof. Write H • = k[x1 ]/(xs11 ) ⊗gk · · · ⊗gk k[xr ](xsrr ) with di = deg(xi ) as in Theorem 27.78. Then dimk (H 1 ) is the number of generators xi such that di = 1. As x1 ⊗ · · · ⊗ xr ∈ H d1 +···+dr is nonzero we find d1 + · · · + dr ≤ g. In particular dimk (H 1 ) ≤ g. We have equality if and only if di = 1 for all i and r = g. If in this case there existed an i such that x2i ̸= 0, say i = 1, then x21 xV 2 · · · xg is a nonzero element of degree g + 1, • contradicting the hypothesis. Hence H • ∼ H 1 by Example 27.77. = Corollary 27.80. Let k be a field and let G be a group scheme of finite type over k such that H • (G, OG ) is a finite-dimensional k-vector space and such that H 0 (G, OG ) = k. Then dim H 1 (G, OG ) ≤ dim G. V• 1 H (G, OG ) → H • (G, OG ) is an If dim H 1 (G, OG ) = dim G, then the canonical map isomorphism. Proof. We can pass to a perfect extension of k because formation of cohomology and of exterior products commute with flat base change. Then we apply Corollary 27.79 with g = dim(G). More generally, if G is any connected group scheme of finite type over a field k, then Brion [Bri] O X has shown that there is an isomorphism of graded Hopf algebras H • (G, OG ) ∼ = Γ(G, OG ) ⊗

V•

(P 1 ),

where P i := { γ ∈ H i (G, OG ) ; µ∗ (γ) = γ ⊗ 1 + 1 ⊗ γ } denotes the subspace of primitive elements of H i (G, OG ). Moreover, one has P i = 0 for i ≥ 2. (27.13) Cartier duality. Let S be a scheme and let π : G → S be a group scheme over S. Then we can define its functor of characters that sends an S-scheme T to the abelian group X ∗ (G)(T ) := HomGrSch/T (GT , Gm,T ) of homomorphisms G ×S T → Gm,T of group schemes over T . Now suppose that π : G → S is a finite locally free commutative group scheme. Then X ∗ (G) is representable by a finite locally free group scheme that can be described as follows. Let A := π∗ OG be the finite locally free commutative cocommutative Hopf algebra corresponding to G (Section (27.2)). Then we can endow the linear dual A ∨ = Hom OS (A , OS ) again with the structure of a finite locally free commutative cocommutative Hopf algebra. Its multiplication and unit is given by the dual of the comultiplication and of the counit, its comultiplication and counit is given by the dual of multiplication and of the unit, and its antipode is given by the dual of the antipode map. We obtain a finite locally free commutative group scheme GD := Spec A ∨ over S. Proposition 27.81. (Cartier duality) Let G be a commutative finite locally free group scheme over S. The functor X ∗ (G) is representable by GD . In the proof we will use the following notion. Let (B, m∗ , i∗ , e∗ ) be a commutative Hopf algebra over a ring R. An element b ∈ B is called group-like if m∗ (b) = b ⊗ b and e∗ (b) = 1. Then one has 1 = e∗ (b) = (i∗ , id)m∗ (b) = (i∗ , id)(b ⊗ b) = i∗ (b)b. Hence b ∈ B × and i∗ (b) = b−1 . Let B gl be the set of group like elements of B. Then B gl is a subgroup of B × .

632

27 Abelian schemes

Proof. The formation of GD commutes with base change T → S. Hence we may assume that S = Spec R is affine and it suffices that one has functorial isomorphism of abelian groups GD (S) ∼ = HomGrSch/S (G, Gm,S ). Let (A, m∗ , i∗ , e∗ ) = Γ(G, OG ) be the Hopf algebra corresponding to G. A character G → Gm,R corresponds to a homomorphism of Hopf algebras R[T, T −1 ] → A and hence by Example 27.8 to a unit a ∈ A× such that m∗ (a) = a ⊗ a, e∗ (a) = 1, and i∗ (a) = a−1 . In other words we have a functorial bijection (27.13.1)

HomGrSch/S (G, Gm,S ) ∼ = Γ(G, OG )gl .

which is easily seen to be an isomorphism of abelian groups. To identify GD (S) = HomR−Alg (A∨ , R) with Agl we remark that every R-linear map ∨ A → R is of the form eva : λ 7→ λ(a) for some a ∈ A because A is finite locally free and hence A ∼ = (A∨ )∨ . Moreover, for a ∈ A one has eva (1) = 1 if and only if e∗ (1) = 1 and eva is a ring homomorphism if and only if m∗ (a) = a ⊗ a. This yields a functorial bijection (27.13.2)

GD (S) ∼ = Agl

which is an isomorphism of groups because the multiplication of the group scheme GD is given by the dual of the multiplication A ⊗ A → A of the R-algebra A. Definition 27.82. Let G be a finite locally free commutative group scheme over a scheme S. The group scheme GD is called the Cartier dual of G. Remark 27.83. Let G be a finite locally free commutative group scheme over a scheme S. ∼ (1) The biduality isomorphism A → (A ∨ )∨ is an isomorphism of Hopf algebras and hence defines an isomorphism of group schemes ∼

(GD )D −→ G. (2) The functor G 7→ X ∗ (G) is contravariant in G. In particular G 7→ GD yields an anti-equivalence of the category of finite locally free commutative group schemes over S with itself. (3) One has rk(GD ) = rk(A ∨ ) = rk(A ) = rk(G). (4) Formation of Cartier dual is compatible with base change, i.e., for every morphism of schemes T → S one has (GT )D ∼ = (GD )T . Remark 27.84. Let R be a ring. The proof of Proposition 27.81 shows the following assertions. (1) For any affine group scheme G over R one has the functorial isomorphism (27.13.1) of abelian groups. (2) Let G = Spec A be a commutative finite locally free group scheme for a finite locally free Hopf algebra A over R. Then (27.13.2), applied to GD , shows that one has a functorial group isomorphism (27.13.3)

G(R) → (A∨ )gl .

633 Example 27.85. Let S be a scheme. Let n ≥ 1 be an integer and let Z/nZS be the constant group scheme corresponding to the abelian group Z/nZ (Example 4.43). Let µn (T ) be the group scheme of n-th roots of unity, i.e., µn (T ) = { a ∈ Γ(T, OT )× ; an = 1 }. In other words, µn is the kernel of the group scheme homomorphism Gm → Gm given (on T -valued points) by z 7→ z n . Then X ∗ (Z/nZS )(T ) = µn (T ). Therefore we see that the group schemes Z/nZS and µn,S are Cartier dual to each other. If Γ(S, OS ) contains a primitive n-th root of unity (e.g., if S is a k-scheme for a field k which contains a primitive n-th root of unity), then a choice of one such yields an isomorphism µn,S ∼ = Z/nZS . If n is invertible in Γ(S, OS ), then such an isomorphism exists at least ´etale-locally on S. On the other hand, if S is a scheme over a field of positive characteristic p dividing n, then µn,S is not ´etale over S, and the structure morphism µp,S → S is a homeomorphism. (27.14) Annihilation of commutative finite locally free group schemes. Let S be a scheme and let G be a commutative finite locally free group scheme over S. We want to show the following result. Proposition 27.86. Suppose that G is of constant rank r ∈ N>0 . Then G is annihilated by r, i.e., g r = 1 for all g ∈ G(T ) and any S-scheme T . To see this we use the following construction. Suppose that S = Spec R is affine. Then G = Spec A for a finite locally free Hopf algebra. Let A∨ be its dual (Section (27.13)). Let φ : R → R′ be a finite locally free R-algebra. By (27.13.3) we have G(R) = (A∨ )gl and G(R′ ) = (A∨ ⊗R R′ )gl . The norm map N : (A∨ ⊗R R′ )× → (A∨ )× (Remark 12.25) then induces a homomorphism of abelian groups NR′ /R : G(R′ ) −→ G(R). The properties of the norm map yield the following assertions. (1) Let R′ be of constant rank r′ over R. If φ∗ : G(R) → G(R′ ) denotes the group homomorphism induced by φ by functoriality, then for all g ∈ G(R) one has (27.14.1)

NR′ /R (φ∗ (g)) = g r

′

(2) If ψ : R′ → R′ is an automorphism of R-algebras, then for all g ′ ∈ G(R′ ) one has (27.14.2)

NR′ /R (ψ ∗ (g ′ )) = NR′ /R (g ′ ).

Proof. [of Proposition 27.86] Since G(T ) = HomT (T, G ×S T ) it suffices toQshow that g r = 1 for any g ∈ G(S). If (Ui )i is an open covering of S, then G(S) → i G(Ui ) is injective. Hence we may assume that S = Spec R is affine. Then G = Spec A for a finite locally free R-algebra φ : R → A of rank r. Let g ∈ G(R). Denote by tg : G → G the translation by g corresponding to an R-algebra automorphism ψg of A. Consider idG ∈ G(A). Then we have the following equality in the group G(R) which shows that g r = 1 NA/R (idG ) = NA/R (ψg∗ (idG )) = NA/R (idG )NA/R (φ∗ (g)) = NA/R (idG )g r . Here the first equality holds by (27.14.2), the second because ψg∗ (idG ) = idG ·φ∗ (g) (multiplication in G(A)), and the third by (27.14.1).

634

27 Abelian schemes

(27.15) Digression: Collection of some properties of schemes over inductive limits of rings. Sometimes we will study schemes over inductive limits of rings, e.g., to reduce to the case of schemes over a noetherian ring or over a local ring. See Chapter 10. Here we collect some further results that we will use in order to give precise references. Very often we will not give proofs. Let (Rλ )λ∈Λ be filtered inductive system of rings, let R := colim Rλ be their colimit, set Sλ := Spec Rλ and S := Spec R. Most often we will apply this to one of the following cases. (1) We write a ring R as the filtered union of its finitely generated Z-subalgebras Rλ . (2) We write the localization R = T −1 A of a ring A with respect to a multiplicative subset T ⊆ A as the filtered colimit of the localizations Af for f ∈ T . Remark 27.87. Let G → S be a group scheme of finite presentation. It follows from Corollary 10.67 and Theorem 10.63 (applied to multiplication, inverse and identity element) that there exists a λ and a group scheme of finite presentation Gλ → Sλ such that G ∼ = Gλ ×Sλ S. We now assume that Λ has a unique minimal element 0 and that we are given an S0 -scheme X0 . Set Xλ := X0 ×S0 Sλ and X := X0 ×S0 S. We now list a number of properties of X that descend to some Xλ which we will use in this chapter. Lemma 27.88. Suppose that X0 → S0 is of finite presentation. Let P be one of the following properties of a morphism of schemes. (1) flat, (2) proper, (3) separated, (4) smooth, (5) the morphism has geometrically reduced fibers, (6) the morphism has geometrically connected fibers, (7) the morphism has equi-dimensional fibers of dimension d, where d ≥ 0 is a fixed integer. Then X → S has property P if and only if there exists a λ such that Xλ → Sλ has property P. Proof. All of the above properties are stable under base change. Therefore if Xλ → Sλ has one of these properties for some λ, then X → S has the same property. So the core of the lemma is the converse, for which we give references. For the property of being “flat” see [EGAIV] O (11.2.6), for the properties “proper” and “separated” see [EGAIV] O (8.10.5), for “smooth” see Proposition 18.59. For all properties of fibers it suffices to show that for all λ the set of points sλ in Sλ such that the fiber of Xλ → Sλ in sλ has the stated property is constructible. Then one can conclude by Proposition 10.56. For constructibility for the fiber properties “geometrically reduced” see [EGAIV] O (9.7.7), for “geometrically connected” see [EGAIV] O (9.7.9), for “equi-dimensional of dimension d” see [EGAIV] O (9.8.5).

635

Definition and basic properties of abelian schemes We now come to the definition of abelian schemes as a smooth proper group scheme with connected fibers, which we think of as families of abelian varieties (Definition 16.53). Using some rigidity results proved in Section (27.17) we then show that an abelian scheme is a commutative group scheme. As further applications of these rigidity results we show that all reduced connected fibers of morphisms to some scheme are translates of abelian varieties (Proposition 27.106) and that to define the group law on an abelian scheme it suffices to find a composition law that has an identity (Proposition 27.109). (27.16) Definition of abelian schemes. We continue to denote by S a scheme. Recall the definition of an abelian variety over a field k. It is defined as a smooth (equivalently by Proposition 27.10 (1), geometrically reduced) proper connected group scheme over k. Moreover, a group scheme over k is an abelian variety if and only if its base change to an algebraic closure (or to any field extension) is an abelian variety since all defining properties are stable under base change and under faithfully flat descent (for “connected group scheme” use Proposition 27.10 (2)). We would now like to define an abelian scheme over a general base scheme as a group scheme that is a “continuously varying family of abelian varieties”. By this we mean a flat group scheme4 X → S with certain addition properties. First of all we add as a finiteness condition “locally of finite presentation” as one would like that everything is generated locally by finitely many elements with finitely many relations. This is something not visible over a field (or a noetherian ring) as then the notions “locally of finite type” and “locally of finite presentation” coincide. For the other properties to add we distinguish between “global properties” and “fiber properties”. For the defining properties of an abelian variety, the global property is “proper” and the fiber properties are “geometrically reduced” and “(geometrically) connected”: Definition 27.89. A group scheme X over S is called an abelian scheme if it satisfies the following properties. (a) The structure morphism X → S is proper, flat and locally of finite presentation. (b) All fibers of X → S are geometrically reduced and connected. Proposition 27.92 below shows that an equivalent definition of an abelian scheme would be a group scheme X over S such that X → S is proper smooth and has connected fibers (which are then automatically geometrically integral). As recalled before, an abelian scheme over a field k is the same as an abelian variety over k. The next two remarks show that the property of being an abelian scheme satisfies the usual permanence properties. Remark 27.90. All the defining properties of abelian schemes are stable under base change, composition and fpqc descent. Hence we see: 4

In fact, usually it is better to work with algebraic spaces, e.g., since descent often behaves better for algebraic spaces than for schemes. Here it does not make a difference because an abelian algebraic space, defined exactly as an abelian scheme below with the word “scheme” replaced by “algebraic space” can be shown to be automatically a scheme (see Theorem 27.211 below).

636

27 Abelian schemes

(1) If X → S is an abelian scheme and T → S is a morphism of schemes, then the base change X ×S T → T is an abelian scheme. (2) Let X → S be a group scheme and let S ′ → S be a faithfully flat quasi-compact morphism such that the base change XS ′ → S ′ is an abelian scheme, then X → S is an abelian scheme. (3) Let X and Y be abelian schemes over S. Then X ×S Y is an abelian scheme over S. Remark 27.91. Let (Rλ )λ∈Λ be a filtered inductive system of rings, let R := colim Rλ be their colimit, set Sλ := Spec Rλ and S := Spec R. Let X → S be an abelian scheme. As X → S is of finite presentation, there exists a λ and an abelian scheme Xλ → Sλ such that X ∼ = Xλ ×Sλ S by Remark 27.87 and Lemma 27.88. The following properties of the structure morphism of an abelian scheme now follow immediately from earlier results. Proposition 27.92. Let f : X → S be an abelian scheme. Then X → S (1) is faithfully flat and quasi-compact, (2) is smooth, (3) is of finite presentation, (4) is universally open, and (5) has geometrically integral fibers. (6) One has OS = f∗ OX and the formation of f∗ OX commutes with base change. Proof. Assertion (1) holds since proper morphisms are quasi-compact and since X → S is surjective because it has a section. Assertion (2) follows from Corollary 27.24, (3) holds because X → S is proper and locally of finite presentation, (4) holds because every flat morphism locally of finite presentation is universally open by Theorem 14.35, (5) follows from Proposition 27.10 (2), and OS = f∗ OX holds universally by Corollary 24.63. Proposition 27.93. Let f : X → S be an abelian scheme of relative dimension g (g a locally constant function on S). Let Ce be the conormal sheaf of the unit section e. Then Ce is a locally free OS -module of rank g and one has isomorphisms

e∗ Ω1X/S

f ∗ Ce ∼ = Ω1X/S , ∼ = Ce ∼ = f ∗ Ω1 , X/S

that are functorial in X. Proof. By Corollary 19.31, e is a regular immersion of codimension g. Hence Ce is locally free of rank g. The first (resp. second) isomorphism holds for arbitrary group schemes (resp. for arbitrary schemes) by Proposition 27.15 (resp. Equation (17.4.4)). To show the last isomorphism it therefore suffices to show that for every locally free OS -module E the canonical homomorphism E → f∗ f ∗ E is an isomorphism. This can be shown locally, hence we may assume that E = OSr for some r ≥ 0. Then the claim follows since f∗ OX = OS (Proposition 27.92 (6)). (27.17) The constancy locus of a morphism of schemes. We continue to denote by S a scheme. Let f : X → S and g : Y → S be S-schemes. An S-morphism h : X → Y is called constant if it factorizes through f , i.e., there exists a section t : S → Y of g such that h = t ◦ f .

637 For a map of schemes T → S we set XT := X ×S T and write hT : XT → YT for h ×S idT . If T = Spec R is affine, we write XR , YR and hR instead of XT , YT , and hT . If T → S is the canonical morphism Spec κ(s) → S we obtain the fibers over s, denoted by Xs , Ys , and hs instead of Xκ(s) , Yκ(s) , and hκ(s) . Finally, we define the locus of points in S in which the fibers of h are constant, (27.17.1)

Const(h) := { s ∈ S ; Xs ̸= ∅ and hs : Xs → Ys is constant}.

Remark 27.94. Suppose that h is constant. (1) Then hT is constant for every scheme map T → S. (2) If f : X → S is faithfully flat and quasi-compact (or, more generally, an epimorphism of schemes), then the section of Y through which h factors is unique. Moreover, the locus, where morphisms induced on fibers are constant is stable under base change in the following sense. Lemma 27.95. In the situation above, let ξ : S ′ → S be a morphism of schemes. Then ξ −1 (Const(h)) = Const(hS ′ ). Proof. Let s′ ∈ S and let s := ξ(s′ ). Then the morphism induced on the fiber in s′ by h′ := hS ′ is the base change of the morphism induced on the fiber in s by h via the field extension κ(s) → κ(s′ ). One has ξ −1 (Const(h)) ⊆ Const(h′ ) since being constant is stable under base change. For the other inclusion, we have to show that if h′s′ is constant, then hs is constant. Now if the fiber of XS ′ in s′ is non-empty, then Xs is non-empty. Hence there exists for every field extension k ′ of κ(s) at most one section t′ : Spec k ′ → Yk′ such that t′ ◦ fk′ = hk′ . Therefore the existence of such a section can be checked fpqc locally by Proposition 14.76. Proposition 27.96. Let f : X → S and g : Y → S be S-schemes and let h : X → Y be an S-morphism. (1) Suppose that f is proper and that OS = f∗ OX . Then Const(h) is open in S and hConst(h) : XConst(h) → YConst(h) is constant. (2) Suppose that X → S is proper, flat, of finite presentation, and has geometrically connected and geometrically reduced fibers and that Y → S is separated and of finite presentation. Then Const(h) is open and closed in S and hConst(h) : XConst(h) → YConst(h) is constant. Under both hypotheses (1) or (2), the morphism f is surjective and hence Const(h) = { s ∈ S ; hs : Xs → Ys is constant} Proof. We may assume S is affine and that Const(h) is nonempty. Let us prove (1). Let s ∈ Const(h). We have to show that there exists an open neighborhood W of s such that hW is constant. Let V ⊆ Y be an open affine neighborhood of the unique point in fs (Xs ) ⊆ Ys ⊆ Y . Then Xs ⊆ h−1 (V ) and hence there exists an open neighborhood W of s such that XW ⊆ h−1 (V ) because f is closed. Hence h restricts to a map of W -schemes h′W : XW → VW . Now V → S is affine as a morphism between affine schemes and hence VW is affine over W . By Remark 24.47, h′W factors through fW,∗ OXW = (f∗ OX )|W which is equal to OW by hypothesis. This shows that hW is constant.

638

27 Abelian schemes

To prove ((2)), we now suppose that X → S is proper, flat, of finite presentation, and has geometrically connected and geometrically reduced fibers. Then OS = f∗ OX by Corollary 24.63. Therefore, we know already that Const(h) is open in S. Let S = Spec R. By Corollary 10.67 there exists a noetherian subring Rλ and a morphism hλ : Xλ → Yλ of finite type Rλ -schemes whose base change to S is h. By Lemma 27.88 we may assume that Xλ → Sλ := Spec Rλ is proper, flat, and has geometrically connected and geometrically reduced fibers and Yλ → Sλ is separated. Moreover, Const(h) is the inverse image under S → Sλ of Const(hλ ) by Lemma 27.95. Hence it remains to show that Const(h) is closed if S is a noetherian scheme. As we already know that Const(h) is open in a noetherian scheme, it is constructible. Hence it suffices to prove that Const(h) is stable under specialization (Lemma 10.17). For this we may base chance to a discrete valuation ring by Proposition 15.7 using that the constancy locus is stable under base change (Lemma 27.95). Hence we can assume that S = Spec R for a discrete valuation ring R. Let s ∈ S be the special point and let η ∈ S be the generic point. We have to show that if η ∈ Const(h), then s ∈ Const(h). Note that h is proper because X → S is proper and Y → S is separated (Proposition 12.58). Consider the schematic image Im(h) of h (Section (10.8)). As h is proper, the underlying topological space of the schematic image of Im(h) is h(X). As images of proper schemes are proper (Proposition 12.59), Im(h) is proper. Therefore replacing Y by Im(h) we can assume that Y → S is proper (and that h is surjective). Let K = κ(η) be the field of fractions of R and let t : Spec K → YK be the section of YK such that hK = t ◦ fK . As Y is proper over R, there exists a unique extension t˜: S → Y of t by the valuative criterion (Theorem 15.9). It remains to show that h = t˜◦ f . Let E = Eq(h, t˜◦ f ) ⊆ X be the equalizer (Definition 9.1). Then E contains the generic fiber Xη . As Y → S is separated, E is a closed subscheme of X (Proposition 9.7). As X is flat over R, it is the smallest closed subscheme of X containing Xη (Proposition 14.14). Therefore E = X. This concludes the proof. Remark 27.97. Under the assumptions in Proposition 27.96 (1) the proof shows that if there exists an s ∈ S such that f (Xs ) is contained in an open affine subscheme of Y , then there exists an open neighborhood W of s such that the restriction XW → YW is constant. If the hypotheses in (2) are satisfied, then W can be chosen open and closed. Corollary 27.98. Let S be a scheme, let X

f

/Y

g

 T

be a diagram of S-schemes. Suppose that g : X → T is proper, flat, of finite presentation, and has geometrically connected and geometrically reduced fibers and that the structure morphism Y → S is separated and of finite presentation. If there exists t ∈ T such that the set-theoretic image f (g −1 (t)) ⊆ Y consists of a single point, then there exists an open and closed neighborhood W of t in T such that f |X×T W factors through g |X×T W . Proof. We apply Proposition 27.96 (2) and Remark 27.97 to the morphism (f, g)S : X → Y ×S T of T -schemes and obtain that there exists an open and closed neighborhood W of t such that the restriction of (f, g)S to XW := X ×T W factors through a section of

639 the projection Y ×S W → W . This section corresponds to a morphism h : W → Y with h ◦ g |XW = f |XW . By the following remark, this corollary generalizes the rigidity lemma, Proposition 16.55, and we sometimes refer to the results in this section as rigidity. Remark 27.99. The condition that Y → S is of finite presentation in Proposition 27.96 (2) and in Corollary 27.98 is only used in the proof to reduce to the noetherian case. Hence it is superfluous if S is locally noetherian. (27.18) Abelian schemes are commutative. In this section S denotes a scheme. Definition 27.100. A pair (F, e) consisting of a presheaf F : (Sch/S)opp → (Sets) and an element e ∈ F (S) is called a pointed presheaf. A morphism of pointed presheaves (F, e) → (F ′ , e′ ) is a morphism α : F → F ′ of functors such that α(S)(e) = e′ . As special cases we obtain the notions of pointed fppf-sheaves, of pointed algebraic spaces, and of pointed schemes and their morphisms. For instance, a pointed S-scheme is a pair (X, e) consisting of an S-scheme X and a section e ∈ X(S) of the structure morphism X → S. A morphism of pointed S-schemes (X, e) → (X ′ , e′ ) is then simply a morphism of S-schemes f : X → X ′ such that f ◦ e = e′ . If F is a presheaf of groups, for instance a group scheme over S, we will consider F always as pointed via the unit section in F (S). Proposition 27.101. Let X and Y be abelian schemes over S and let f : X → Y be a morphism of S-schemes. Then there exists y ∈ Y (S) such that ty ◦ f is a homomorphism of group schemes over S. Here ty : Y → Y denotes the left translation by y (Definition 27.1). Proof. Translating with the inverse of f (eX ) ∈ Y (S), we may assume that f preserves unit sections. Hence it suffices to show the following corollary. Corollary 27.102. Let X and Y be abelian schemes over S and let f : X → Y be a morphism of pointed S-schemes. Then f is a homomorphism of group schemes over S. In the proof we use Corollary 16.56 which gives the result if S is the spectrum of a field. The proof then relies only on Part (1) of Proposition 27.96. A direct proof that does not use the case over a field but that instead uses also Part (2) of Proposition 27.96 is sketched in Exercise 27.8. Proof. Let h : X ×S X → Y be the morphism of S-schemes that is given on T -valued points by (x, x′ ) 7→ f (x)f (x′ )(f (xx′ ))−1 , where we write the group law in X(T ) and in Y (T ) multiplicatively. We have to show that h factors through the unit section eY of Y . If h factors through some section t, then this section is necessarily unique because X ×S X → S is surjective and faithfully flat (Remark 27.94). As the composition of h with (e, idX ) : X → X ×S X factors through eY , we have t = eY . Hence it suffices to show that h is constant. By Proposition 27.96 (1) we can check this on fibers. Hence we conclude by Corollary 16.56. Applying Corollary 27.102 to the inversion of the abelian scheme we deduce:

640

27 Abelian schemes

Corollary 27.103. Abelian schemes are commutative group schemes. From now on we will use additive notions for abelian schemes, e.g., if X → S is an abelian scheme, we write the group law on X(T ) for T some S-scheme additively. The unit section in X(S) will be denote by e or by 0 and will be called the zero section. Corollary 27.104. Let X → S be an S-scheme and let 0 ∈ X(S) be a section. Then there exists at most one structure of an abelian scheme over S on X such that 0 is the zero section. Proof. Let m1 and m2 be two group scheme structures on X with unit section 0 making X into an abelian scheme. Then idX is a homomorphism of group schemes by Proposition 27.102. Hence m1 = m2 . (27.19) Further applications of rigidity. Proposition 27.105. Let X → S be an abelian scheme and let Y → S be a separated S-group scheme of finite presentation. Let f, g : X → Y be homomorphisms of group schemes and let s ∈ S be a point such that the geometric fibers fs¯ and gs¯ agree. Then there exists an open and closed neighborhood W of s such that fW = gW : XW → YW . Proof. One has fs¯ = gs¯ if and only if the fibers fs and gs are equal (Theorem 14.72). Consider h := f g −1 : X → Y . Then hs factors through the unit section of Gs . Hence there exists by Proposition 27.96 an open and closed neighborhood W of s such that hW factors through a section σ : W → YW . Let π : X → S be the structure morphism. Then σ = σ ◦ πW ◦ 0 = hW ◦ 0. As h preserves unit sections, this shows that σ is the unit section of YW and hence that fW = gW . The following amazing result, which we learned from [EGM] X 2.20, essentially states that all k-rational non-empty fibers of morphisms of an abelian variety to some other scheme are translates of abelian varieties after passing to reduced connected components. Proposition 27.106. Let k be a perfect field, let X be an abelian variety over k, let Y be a k-scheme, and let f : X → Y be a morphism of k-schemes. For x ∈ X(k) let Fx := (f −1 (f (x)))0red , where ( )0 denotes the connected component containing x. Then F0 is an abelian subvariety of X and for x ∈ X(k) one has Fx = tx (F0 ). Proof. As f ◦ x is a closed immersion Spec k → Y , the scheme Fx is a closed subscheme of X. It is by definition reduced and hence geometrically reduced because k is perfect (Corollary 5.57). It is connected and has x as a k-valued point. Hence it is geometrically connected (Lemma 26.4). In particular, the formation of Fx is compatible with passing to field extensions of k. We will show that for all x ∈ X(k) we have (*)

Fx = tx (F0 )

Let us first show how (*) implies that F0 is a subgroup scheme. As F0 is a geometrically ¯ is a subgroup reduced k-scheme of finite type over a field, it suffices to show that F0 (k) ¯ ¯ ¯ we of X(k) for some algebraic closure k of k. Hence it remains to show that for y ∈ F0 (k) ¯ ¯ ¯ ¯ have y + F0 (k) = F0 (k). After base change k → k we may assume that k = k. But then ty (F0 ) = Fy = F0 by (*).

641 As F0 is closed in X, it is a proper k-scheme. It is connected and smooth (Proposition 27.10 (3)) and hence an abelian variety. It remains to prove (*). Showing the equality of two closed subschemes can be done fpqc-locally hence we may assume k is algebraically closed. We apply Corollary 27.98 to the projection g : X ×k Fx → Fx , to φ : X ×k Fx → Y given by (y, z) 7→ f (y + z) and to t = 0 ∈ X using that f (0 + z) = f (x) for all z ∈ Fx (k). Hence φ factors through φ¯ : X → Y , i.e., f (y + z) is independent of z ∈ Fx (k) for all y ∈ X(k). In particular f (y − x + Fx (k)) = {f (y)},

for all x, y ∈ X(k).

For y = 0 we obtain −x + Fx (k) ⊆ F0 (k) and for x = 0 we obtain y + F0 (k) ⊆ Fy (k). This proves (*). Corollary 27.107. Let X be an abelian variety over a field k. Let Y be a k-scheme and let f : X → Y be a morphism of k-schemes. Then all non-empty fibers of f are equi-dimensional of the same dimension. Proof. As the dimension of fibers does not change under field extensions (Proposition 5.38), we may assume that k is algebraically closed. We want to show that the function d : X → Z,

x 7→ dimx f −1 (f (x))

is constant. As it is upper semicontinuous by Chevalley’s theorem, Theorem 14.112, it suffices to show that it is constant on the very dense subspace of k-valued points. But Proposition 27.106 shows that for every x ∈ X(k), the underlying reduced subscheme of every connected component of f −1 (f (x)) is isomorphic to the same abelian variety. In particular all irreducible components of f −1 (f (x)) have the same dimension for all x ∈ X(k). (27.20) Constructing abelian schemes. The main result in this section is Proposition 27.109. For the proof we will use the following lemma. Lemma 27.108. Let k be a field, let X and Y be irreducible k-schemes of finite type. Suppose that X is proper over k and that Y is separated over k. Suppose that dim X ≥ dim Y and let f : X → Y be a morphism of k-schemes such that there exists a point y ∈ Y such that dim f −1 (y) = 0. Then f is surjective and dim X = dim Y . Proof. By passing to the underlying reduced subschemes we may assume that X and Y are integral. The morphism f is proper since X is proper and Y is separated. In particular f (X) ⊆ Y is closed. If we show that dim X = dim f (X), then necessarily f (X) = Y since dim X ≥ dim Y . Hence we may assume that f is surjective and it remains to show that dim X = dim Y . By Corollary 14.115 the subset { y ∈ Y ; dim f −1 (y) = 0 } is open. By hypothesis it is non-empty and hence dense because Y is irreducible. Hence dim f −1 (η) = 0 if η is the generic point and we conclude by Proposition 14.109 (2). Proposition 27.109. Let S be a scheme and let X → S be a proper flat morphism of finite presentation with geometrically integral fibers. Suppose that there is given a point e ∈ X(S) and a morphism of S-schemes m : X ×S X → X such that m(x, e) = x = m(e, x) for all x ∈ X(T ), T any S-scheme. Then X → S is an abelian scheme with group law m and unit e.

642

27 Abelian schemes

Lemma 27.110. Let G be a set endowed with an associative composition law G × G → G, (g, h) 7→ gh. For g ∈ G we denote by lg : G → G,

h 7→ hg

and

rg : G → G,

h 7→ gh

the left and right translation. Suppose that rg is surjective for all g ∈ G and that there exists an f ∈ G such that lf is surjective. Then the composition law defines a group structure on G. Proof. As rf is surjective, we find e ∈ G with ef = f . Let g ∈ G. As lf is surjective, we find h ∈ G with f h = g. Then eg = e(f h) = (ef )h = f h = g, hence e is a left unit. For every g ∈ G we also find h ∈ G such that hg = e since rg is surjective. Hence every g has a left inverse. To show that h is a also a right inverse choose h′ ∈ G with h′ h = e using the surjectivity of rh . Then gh = e(gh) = (h′ h)(gh) = h′ ((hg)h) = h′ (eh) = h′ h = e. It remains to show that e is also a right unit. But we find ge = g(hg) = (gh)g = eg = g. Proof. [of Proposition 27.109] If T is an S-scheme and x, y ∈ X(T ) we write x · y instead of m(x, y). Consider the morphism σ : X ×S X −→ X ×S X,

σ(x, y) = (x · y, y).

Consider the following two claims. (1) σ is an isomorphism of S-schemes. This means that for all S-schemes T and for all y ∈ X(T ) the right translation x 7→ x · y is bijective. In particular, there exists for every y ∈ Y (T ) a unique T -valued point y −1 such that y −1 · y = e. As the right translation is functorial in T , so is y 7→ y −1 and we obtain a morphism i : X → X of S-schemes given on T -valued points by y 7→ y −1 with i(e) = e. (2) The morphism of S-schemes (*)

X ×S X ×S X −→ X,

(x, y, z) 7→ i(x · (y · z)) · ((x · y) · z)

is constant. If we have shown both claims, then setting x = y = z = e in (2) it follows that the morphism (*) necessarily factors through the unit section and hence that the composition law is associative. Now we apply Lemma 27.110 to X(T ) for every S-scheme T and see that (X, m, i, e) is an S-group scheme and hence an abelian scheme. Therefore it suffices to show the claims (1) and (2). For claim (1) (resp. (2)) it suffices to do this on fibers by Corollary 18.77 (resp. by Proposition 27.96). Hence we may assume that S = Spec k is a field. By fpqc descent, see Proposition 14.53 (resp. Lemma 27.95), we may in addition assume that k is algebraically closed. Then it suffices to show that m defines a group structure on X(k). We have σ −1 (e, e) = {(e, e)}, therefore Lemma 27.108 shows that σ is surjective. Consider the closed subscheme Z˜ ⊆ X ×X given by Z˜ = { (x, y) ; x·y = e } = σ −1 ({e}× X). The surjectivity of σ implies that the second projection p˜2 : Z˜ → X is surjective. Let Z ⊆ Z˜ be an irreducible component, considered as a closed integral subscheme, with p˜2 (Z) = X. Then Z is proper over k and dim Z ≥ dim X. As p˜−1 2 (e) = {(e, e)}, we have (e, e) ∈ Z. Let p1 : Z → X be the first projection. Then p−1 (e) = (e, e) and Lemma 27.108 1 shows that p1 is surjective.

643 Define f : Z × X × X → X by f ((x, y), z, w) = x · ((y · z) · w). Then we have f (Z × {e} × {e}) = {e} and hence we find by Corollary 27.98 that x · ((y · z) · w) = z · w

(*)

for (x, y) ∈ Z and w, z ∈ X.

Taking w = e in (*) we obtain in particular x · (y · z) = z

(**)

for (x, y) ∈ Z and z ∈ X.

Now fix y ∈ X(k). As we have shown that both projections Z → X are surjective, we find x, x′ ∈ X(k) with (x, y) ∈ Z and (y, x′ ) ∈ Z. Then (**) gives x = x · e = x · (y · x′ ) = x′ . This shows that y has a unique left and right inverse in X(k), which we call y −1 . Finally, we obtain for all x, y, z ∈ X(k) (∗)

(∗∗)

y · (z · w) = y · (x · ((y · z) · w)) = y · (x′ · ((y · z) · w)) = (y · z) · w, which shows that the composition law is associative.

The Picard functor As mentioned in the introduction to this chapter, one of our goals is to introduce the dual abelian scheme of an abelian scheme. The underlying functor will be defined in terms of line bundles on the abelian scheme, and as a preparation, we will study the relative Picard functor for general schemes in the next few sections. Its representability is a difficult result, useful also in many other places in algebraic geometry, which we cannot prove here. We will also briefly digress to define the Jacobian of a family of curves. Finally we will generalize many results of Section (26.17) about elliptic curves to families of elliptic curves parametrized by a scheme. See below for further comments and references. (27.21) The Picard functor. In this section we collect some results on the Picard functor. Let f : X → S be a morphism of schemes and consider the functor (27.21.1)

PX/S : (Sch/S)opp → (AbGrp),

T 7→ Pic(XT ).

Definition 27.111. The (relative) Picard functor PicX/S : (Sch/S)opp → (AbGrp) is the fppf-sheafification of PX/S for the fppf-topology (Definition 27.30). Remark 27.112. There is a map of functors of abelian groups ι : PX/S → PicX/S , where PicX/S is an abelian fppf-sheaf, and the pair (PicX/S , ι) is characterized up to unique isomorphism by the following two properties. (1) For every element L ∈ PicX/S (T ) there exists an fppf-covering (gi : Ti → T )i such that the image of L in PicX/S (Ti ) lies in the image of PX/S (Ti ) → PicX/S (Ti ). (2) A line bundle L ∈ Pic(XT ) is in the kernel of Pic(XT ) → PicX/S (T ) if and only if there exists an fppf-covering (gi : Ti → T )i such that the pullback of L to XTi is trivial.

644

27 Abelian schemes

Indeed, if (Pic′X/S , ι′ ) is a pair satisfying (1) and (2), then by the universal property of the fppf-sheafification there exists a unique homomorphism of abelian fppf-sheaves α : PicX/S → Pic′X/S because Pic′X/S is an abelian sheaf. Then (1) implies that α is surjective and (2) implies that α is injective. Now consider the following two conditions on f . (a) One has OS = f∗ OX compatibly with base change, i.e., for all S-schemes T the homomorphism OT → (fT )∗ OXT is an isomorphism. This is for instance the case if f is flat, proper, of finite presentation and has geometrically reduced and geometrically connected fibers (Corollary 24.63). (b) The morphism f has a section σ : S → X. Both conditions are satisfied if X is an abelian scheme over S. If (b) is satisfied, then the base change σT : T → XT is a section of fT : XT → T for every S-scheme T . In particular, the map fT∗ : Pic(T ) → Pic(XT ) is injective because it has σT∗ as a left inverse. Lemma 27.113. Suppose f satisfies (a) and (b) above. For every S-scheme T we consider Pic(T ) as a subgroup of Pic(XT ). Then we have (27.21.2)

PicX/S (T ) = Pic(XT )/ Pic(T ).

Proof. We will first show that the functor # T 7→ PX/S (T ) := Pic(XT )/ Pic(T ) ∼ = K(T ) := Ker(σT∗ : Pic(XT ) → Pic(T ))

is an fppf-sheaf. We ` Q will show that K is an fppf-sheaf. Let (Ti → T )i be an fppf-covering. As K( i Ti ) = i K(Ti ), it suffices to check the sheaf condition for an fppf-covering consisting of a single morphism h : T ′ → T . Set T ′′ := T ′ ×T T ′ , T ′′′ := T ′ ×T T ′ ×T T ′ , let pi : T ′′ → T ′ and pij : T ′′′ → T ′′ be the projections, and let pi,X : XT ′′ → XT ′ and pij,X : XT ′′′ → XT ′′ be their base changes. Let L ′ be an element of K(T ′ ) such that there exists an isomorphism ∼

φ : p∗1,X L ′ −→ p∗2,X L ′ . ∼

Since L ′ ∈ K(T ′ ), there exists an isomorphism α : OT ′ −→ σT∗ ′ L ′ . Set γ := p∗2 (α−1 ) ◦ σT∗ ′′ (φ) ◦ p∗1 (α) : OT ′′ → OT ′′ . Scaling φ by γ −1 we may assume that γ = 1. Now p∗X,23 φ ◦ p∗X,12 φ differs from p∗X,13 φ by some invertible section s of OXT ′′′ . By our scaling of φ we know that its pullback by σT ′′′ is 1. Hence s = 1 because f∗ OXT ′′′ = OT ′′′ by assumption on f . Hence by faithfully ∼ flat descent there exists a line bundle L on XT and an isomorphism OT → σT∗ L whose pullback to XT ′ is the pair (L ′ , α). This proves that K and hence T 7→ Pic(XT )/ Pic(T ) is an fppf-sheaf. # It remains to show that the map of functors PX/S → PX/S satisfies the properties (1) and (2) of Remark 27.112. This is clear for (1). It remains to show that for every S-scheme a line bundle L in Pic(XT ) is fppf-locally on T trivial if and only if it is of the form fT∗ M for a line bundle M on T . Replacing S by T and X by XT , we may assume that S = T .

645 The condition is clearly sufficient (take as fppf-covering of S a Zariski covering that trivializes M ). Conversely, let (Si → S)i be an fppf covering such that the pullback LSi ∼ of L to XSi is trivial for all i. Choose an isomorphism αi : OXSi → LSi . The pullbacks of αi and αj to XSi ×S Sj differ by an element × gij ∈ Γ(XSi ×S Sj , OX S

i × S Sj

) = Γ(Si ×S Sj , OSi ×S Sj ),

where for the equality we used the Condition (a) on X → S. The gij satisfy the cocycle condition and hence define a descent datum for an line bundle with respect to the fppfcovering (Si → S)i . All such descent data are effective (by Theorem 14.68 combined with Proposition 14.48) and hence we obtain a line bundle M on S such that f ∗ M and L are given by the same descent datum with respect to the fppf-covering (XSi → X)i . Hence L ∼ = f ∗ M again by Theorem 14.68. Example 27.114. Let k be a field and let X be a k-scheme. If X(k) ̸= ∅, then PicX/k (k) = Pic(X). More generally, this holds whenever k is a ring with Pic(k) = 0, e.g., if k is a local ring or a factorial ring. The formation of PicX/S is compatible with base change: Remark 27.115. Let X → S be a morphism of schemes and let S ′ → S be a morphism of schemes. Then PX/S ×S S ′ = PX×S S ′ /S ′ and hence PicX/S ×S S ′ = PicX×S S ′ /S ′ . To study formal smoothness properties of the Picard functor we will use the following lemma. Lemma 27.116. Let f : X → S be flat, proper, of finite presentation with geometrically reduced and geometrically connected fibers such that f has a section. Let T be an affine S-scheme and let i : T0 → T be a nil-immersion defined by a quasi-coherent ideal I with I 2 = 0, allowing to view I as quasi-coherent module on T0 . Then there exists an exact sequence (27.21.3)

PicX/S (T ) −→ PicX/S (T0 ) −→ H 2 (XT0 , I OXT0 ).

Here we set XT := X ×S T and define XT0 similarly. Proof. By Lemma 27.113 we have PicX/S (T ) = Pic(XT )/ Pic(T ) and PicX/S (T0 ) = Pic(XT0 )/ Pic(T0 ). By Lemma 26.15 we have an exact sequence (*)

Pic(XT ) −→ Pic(XT0 ) −→ H 2 (XT0 , I OXT0 ).

The composition Pic(T0 ) → Pic(XT0 ) → H 2 (XT0 , I OXT0 ) factors through H 2 (T0 , I ) which is zero because T0 is affine and I is quasi-coherent. Hence the right map of (*) induces a map PicX/S (T0 ) −→ H 2 (XT0 , I OXT0 ). Now the surjectivity of Pic(XT ) → PicX/S (T ) and Pic(XT0 ) → PicX/S (T0 ) imply the exactness of (27.21.3).

646

27 Abelian schemes

(27.22) Representability of the Picard functor. We will use in the sequel the following representability theorem of Pic X/S . Theorem 27.117. Let S be a scheme, let f : X → S be a flat proper morphism of finite presentation with geometrically reduced and geometrically connected fibers. Then PicX/S is an algebraic space and PicX/S → S is locally of finite presentation. The theorem can be proved by checking Artin’s criteria for representability by an algebraic space, see [Art3] O , [Art4] O , [Sta] 07T0. We will not give the proof here but refer to [Sta] 0D2C and 0DNI. The Seesaw theorem 24.66 just says (for line bundles) that the diagonal of PicX/S is indeed representable by an immersion. More precisely, we have: Proposition 27.118. Under the hypotheses on f in Theorem 27.117, the diagonal PicX/S → PicX/S ×S PicX/S is representable by an immersion of finite presentation. If f has in addition geometrically integral fibers, then the diagonal is represented by a closed immersion of finite presentation and hence PicX/S is separated over S. Proof. Suppose that f has a section. Let T be an S-scheme and let [L1 ], [L2 ] ∈ PicX/S (T ) for L1 , L2 ∈ Pic(XT ). We have to show that the locus on T , where L1 and L2 are isomorphic up to the pullback of a line bundle from T is a subscheme (resp. a closed subscheme if f has geometrically integral fibers). Set M := L1 ⊗ L2⊗−1 . Then we are looking for a subscheme (resp. closed subscheme) Z of T such that a morphism T ′ → T factors through Z if and only if MXT ′ is the pullback of a line bundle on T ′ . This is given by Theorem 24.66. If f does not necessarily have a section, then we choose a faithfully flat quasi-compact base change S ′ → S such that XS ′ has a section (e.g. S ′ = X with the diagonal as a section) and use that the property of being a (closed) immersion of finite presentation is stable under fpqc-descent (see Proposition 14.53). We omit the details. Many more representability results about PicX/S are known. We refer to [FGAex] X §9.4 for a thorough discussion. Here we mention only two further results that give criteria when PicX/S is represented by a scheme. The first result shows that the Picard functor is representable for proper schemes over a field. Theorem 27.119. Let k be a field and let X be a proper k-scheme. Then PicX/k is representable by a separated scheme locally of finite type over k. We will consider in the proof only the case that X is geometrically integral over k (the only case that we will need in the sequel). For the general case we refer to [Mur] O (II.15) (note that once we know that PicX/k is a group scheme locally of finite type over k, then it is automatically separated by Corollary 27.9). Proof. [if X is geometrically integral over k] By Theorem 27.117, PicX/k is an algebraic space locally of finite type over k. As X is geometrically integral over k, it is separated by Proposition 27.118. Therefore PicX/k is representable by Theorem 27.47.

647 The second representability result we state here, again without proof, is the following theorem by Grothendieck [FGA] X . It was one of the first major theorems that Grothendieck proved after introducing the machinery of schemes and the functorial point of view. By relating line bundles to divisors, the representability of the Picard functor is reduced to the representability of the Hilbert functor, see Section (14.32), and thus ultimately to the representability of the Grassmannian functor. Theorem 27.120. Let f : X → S be flat and of finite presentation with geometrically integral fibers. Suppose that there exists an open covering (Ui )i and for all i a closed immersion X |Ui → PnUi (with n depending on i) (e.g., if X → S is projective). Then PicX/S is represented by a separated S-scheme locally of finite presentation. We refer to [FGAex] X 9.4.8 for a proof, where this result is shown if S is locally noetherian. But by the following remark this implies the result in the general case. Remark 27.121. Let P be a property of scheme morphisms and let Q be a property of morphisms from an fppf-sheaf to a scheme. Let P be Zariski local on the target and compatible with inductive limits of rings (Section (10.13)) and let Q be Zariski local on the target and stable under base change. Suppose that one wants to show that for every morphism of schemes X → S of finite presentation that has P the morphism PicX/S → S has property Q. Then it suffices to show this in the case that S = Spec R0 where R0 is a finitely generated Z-algebra. Indeed, since P and Q are both local on the target, we may assume that S = Spec R is affine. We write R as the filtered union of its finitely generated Z-subalgebras Rλ . As X → S is of finite presentation and P is compatible with inductive limits of rings, we find an index λ and an Rλ -scheme Xλ of finite type such that Xλ → Spec Rλ has property P and such that Xλ ⊗Rλ R ∼ = X. Then PicXλ / Spec Rλ → Spec Rλ has property Q by hypothesis. Hence PicX/S → S has property Q because Q is stable under base change and because the formation of PicX/S commutes with base change. For a list of properties of scheme morphisms that are local on the target and compatible with inductive limits of rings we refer to Appendix C. An example for property Q as above is being representable by a scheme (Theorem 8.9 using that fppf-sheaves are in particular sheaves for the Zariski topology). If one already knows that under hypothesis P on X → S the Picard sheaf PicX/S is an algebraic space, then further examples for Q are “separated”, “proper”, or “smooth”. (27.23) The Lie algebra of the Picard functor. Proposition 27.122. Let f : X → S be a proper flat morphism of finite presentation with geometrically reduced and geometrically connected fibers. Then Lie(PicX/S ) ∼ = R 1 f∗ O X . Here we use the definition of the Lie algebra given in Remark 27.19. In the proof we will use that Lie(PicX/S ) is a quasi-coherent OS -module. This holds by Remark 27.17 if PicX/S is a scheme locally of finite presentation and we will use the proposition only in this case. If PicX/S is only an algebraic space locally of finite presentation, then one can show that Lie(PicX/S ) is still quasi-coherent (by the same argument, using that Ω1PicX/S /S is of finite presentation by [Sta] 05ZF).

648

27 Abelian schemes

Proof. For a scheme U we set U [ε] := Spec OU [T ]/(T 2 ). We denote by ε the image of the indeterminate T (in the sections over open subsets). Then the structure morphism U [ε] → U has a section that corresponds to T 7→ 0. It is a thickening of order 1. In particular, U and U [ε] have the same underlying topological space. We have X[ε] = X ×S S[ε] by (11.2.5). (I). We first construct a homomorphism ψ : R1 f∗ OX → Lie(PicX/S ) of OS -modules. Consider the structure sheaf OX[ε] as a sheaf of rings on X. Then for U ⊆ X open, OX[ε] (U ) = Γ(U, OX )[T ]/(T 2 ). We have a short exact sequence of abelian sheaves on X (*)

× × 0 −→ OX −→ OX[ε] −→ OX −→ 1,

where the first map is given on local sections by a 7→ 1 + aε. The sequence is split by × × → OX[ε] given by a 7→ a + 0ε on local sections. By Section (11.7) we have the map OX × 1 Pic(X) = H (X, OX ) and similarly for X[ε]. Let V ⊆ S be open. Taking cohomology over f −1 (V ) = X ×S V we obtain from (*) a split exact sequence (**)

0 −→ H 1 (f −1 (V ), OX ) −→ Pic(X ×S V [ε]) −→ Pic(X ×S V ) −→ 1.

As Pic(XT ) → PicX/S (T ) is functorial in S-schemes T and as we have (***)

Γ(V, Lie(PicX/S )) = Ker(PicX/S (V [ε]) → PicX/S (V )),

we obtain a homomorphism of abelian groups ψV : H 1 (f −1 (V ), OX ) → Γ(V, Lie(PicX/S )), compatible with restriction to open subsets. In view of Proposition 21.27 this construction yields a homomorphism of abelian sheaves ψ : R1 f∗ OX −→ Lie(PicX/S ). Let us show that ψ is OS -linear. For a ∈ Γ(V, OS ) let ma : V [ε] → V [ε] be the endomorphism of V -schemes given by ε 7→ aε. It induces on Γ(V, Lie(PicX/S )) a scalar multiplication which equips it with the structure of a Γ(V, OS )-module. On PicX/S (V [ε]) = × × H 1 (XV , OX[ε] ) it is induced by the endomorphism of OX[ε] given by ε 7→ aε. Hence ma induces the scalar multiplication by a on H 1 (f −1 (V ), OX ). This shows that ψV is Γ(V, OS )-linear and hence that ψ is OS -linear. (II). It remains to show that ψ is an isomorphism. As the formation of R1 f∗ OX and of Lie(PicX/S ) commutes with flat base change (Proposition 22.90 and Proposition 27.22), we can do this after passing to an fppf-covering of S. Hence, we may assume that X → S has a section and hence that PicX/S (T ) = Pic(XT )/ Pic(T ) for every S-scheme T . We may assume that S = Spec R is affine. Because the source and target of ψ are quasi-coherent OS -modules, it suffices to show that the induced R-linear map on global sections ψS : H 1 (X, OX ) −→ Γ(S, Lie(PicX/S )) is an isomorphism. By passing to stalks we may also assume that R is local. Then R[ε] is also local and hence Pic(S) = Pic(S[ε]) = 1. Therefore Pic(XS ) → PicX/S (S) and Pic(XS[ε] ) → PicX/S (S[ε]) are isomorphisms. Hence ψS is an isomorphism by (**) and (***).

649 (27.24) The identity component of the Picard functor. We will be interested in the identity component of PicX/S , so we want to extend this notion, defined for group schemes in Definition 27.12 for group schemes, to group functors. opp → (Grp) Let us explain what we mean by this. Let S be a scheme and let G : (Sch/S) 0 be a functor of groups. For an S-scheme T let G (T ) be the set of g ∈ G(T ) such that for every geometric point t¯ → T there exist a connected t¯-scheme Z, points z, z0 ∈ Z(t¯), and an S-morphism h : Z → G such that h(z) = g(t¯) and h(z0 ) = 1 in G(t¯). This is a subfunctor of G. Clearly, the unit section is contained in G0 (S) and G0 is stable under the inversion ι of G by replacing h by ι◦h. If g, g ′ ∈ G0 (T ), let t¯ → T be a geometric point and let (Z, z, z0 , h) for g and (Z ′ , z ′ , z0′ , h′ ) for g ′ as above. Then gg ′ ∈ G0 (T ) because ˜ for Z ×t¯ Z ′ is connected (Proposition 5.53) and one can take (Z ×t¯ Z ′ , (z, z ′ ), (z0 , z0′ ), h) ′ ˜ gg , where h is the composition h×h′

m

Z ×t¯ Z ′ −→ Z ×S Z ′ −−−−−→ G ×S G −→ G. Definition 27.123. The subgroup functor G0 of G is called the identity component of G. In particular, for every morphism of schemes f : X → S we have the identity component Pic0X/S of the Picard functor. Lemma 27.124. Let k be a field and let G be a k-group scheme locally of finite type. Then the two notions of identity component defined in Definition 27.12 and in Definition 27.123 coincide. ˜ 0 be the Proof. Let G0 be the connected component of the unit section of G and let G 0 subfunctor defined in Definition 27.123. Let T be a k-scheme. Let g ∈ G (T ) and t¯ → T be a geometric point. As G0 is geometrically connected, G0t¯ is connected and we may take Z = G0t¯ , h the canonical map, z := g(t¯) and z0 the unit section of G0t¯ to see that ˜ 0 (T ). Conversely, let g be in G ˜ 0 (T ) which we consider as a morphism g : T → G. If g∈G g did not factor through G0 we would find a geometric point t¯ → T such that g(t¯) ∈ / G0 . But any morphism h : Z → G from a connected scheme Z such that the image of h contains the unit section has to factor through G0 . Therefore g ∈ G0 (T ). opp

Remark 27.125. Let S be a scheme and let G : (Sch/S) → (Grp) be a functor of opp groups such that for all s ∈ S the fiber Gs : (Sch/κ(s)) → (Grp) is representable by a group scheme over κ(s) locally of finite type. (1) Lemma 27.124 shows that for every S-scheme T and g ∈ G(T ) one has g ∈ G0 (T ) if and only if for every geometric point t¯ → T one has g(t¯) ∈ G0t¯ (t¯) with G0t¯ the identity component of the group scheme Gt¯ defined in Definition 27.12. (2) By Proposition 27.12 (3) for every morphism S ′ → S of schemes one then has an isomorphism (G ×S S ′ )0 ∼ = G0 × S S ′ . The hypothesis on G is in particular satisfied if G = PicX/S , where X → S is proper (Theorem 27.119). Lemma 27.126. Let X → S be a proper, smooth morphism of schemes with geometrically connected fibers such that the monomorphism Pic0X/S → PicX/S is representable by an open and closed immersion. Then the morphism Pic0X/S → S of algebraic spaces is proper.

650

27 Abelian schemes

We will see below that the assumptions of this lemma are satisfied for smooth proper relative curves with geometrically connected fibers (Proposition 27.136) and for abelian schemes (Remark 27.210). Proof. In the proof we will use that the formation of Pic0X/S is compatible with base change which holds by Remark 27.125. As f is smooth and has geometrically connected fibers, it has geometrically integral fibers. Hence PicX/S is an algebraic space locally of finite type over S (Theorem 27.117) and PicX/S → S is separated (Proposition 27.118). Hence Pic0X/S → S is a group algebraic space which is separated and locally of finite presentation over S. The fibers Pic0Xs /κ(s) of Pic0X/S → S are group schemes locally of finite type over κ(s) by Theorem 27.119. By definition, they are connected, hence Pic0Xs /κ(s) is quasi-compact by Proposition 27.12 (1). To see that Pic0X/S → S is proper, we may assume that S is affine and that S is noetherian by noetherian approximation (Remark 27.121). As properness can be checked after passing to an fppf-covering, we may assume X → S has a section (e.g., by passing to the fppf-covering X → S). As properness is compatible with filtered inductive limits of rings, we may even assume that S is local. Let us show that Pic0X/S → S satisfies the valuative criterion of Definition 27.43 (3). By Remark 27.44 (3) it suffices to show the criterion for discrete valuation rings. Hence let R be a discrete valuation ring with fraction field K and let Spec R → S be a morphism. Let L ∈ Pic0X/S (K) be represented by a line bundle L on XK . As XR is smooth over the regular ring R, the scheme XR is also regular. Note that XR \ XK is the zero scheme of the uniformizer of R, pulled back to XR . As XR → Spec R is flat, this is a Cartier divisor, and since X is proper, it is non-empty. So XR \ XK is of codimension 1 in XR . Hence Pic(XR ) → Pic(XK ) is surjective by Corollary 11.43 and we can extend L to a line bundle L ′ on XR . As Pic0X/S is open and closed in PicX/S the induced morphism Spec R → PicX/S factors through Pic0X/S . It remains to show that the structure morphism π : Pic0X/S → S is quasi-compact. As S is affine, it suffices to show that the underlying topological space of Pic0X/S is quasi-compact (Remark 27.44 (2)). Let (Ui )i be an open covering. Let s ∈ S be the closed point. As the fiber Pic0Xs /κ(s) is quasi-compact, there exists a finite subset J ⊆ I such S that V := i∈J Ui contains Pic0Xs /κ(s) . It suffices to show that V equals the underlying topological space of Pic0X/S . Let Z be its complement and assume that Z ̸= ∅. As Pic0X/S → S satisfies the valuative criterion above, π(Z) is stable under specialization (Proposition 15.7) and in particular s ∈ π(Z). This is a contradiction since Z does not meet the special fiber. Remark 27.127. The proof shows that if S is locally noetherian and X → S is a proper, smooth morphism of schemes with geometrically connected fibers that has a section, then PicX/S satisfies the valuative criterion of Definition 27.43 (3) for discrete valuation rings R. In fact, the hypothesis “locally noetherian” on S is superfluous and one can show that PicX/S satisfies the valuative criterion for properness for arbitrary valuation rings if one replaces the reference to Corollary 11.43 in the proof of Lemma 27.126 by the argument in the proof of [Sta] 0DNG. As a consequence of the lemma, PicX/S is formally proper in the following sense.

651 Definition 27.128. Let f : Z → S be a morphism locally of finite type between algebraic spaces. Then f is called formally proper if for every valuation ring R with fraction field K and for every commutative diagram Spec(K)

/Z ;

 Spec(R)

 /S

f

there exists a unique dotted arrow making the diagram commutative. Note that this is not a standard notion. Remark 27.129. (1) Every proper morphism is formally proper. (2) The composition of formally proper morphisms is formally proper. (3) In particular, if X → S is formally proper and if Z is a closed subscheme of X, then Z → S is formally proper. (27.25) The Picard functor of curves. In this section S denotes a scheme. Let us first explain what we mean by a curve over S here and in the sequel. Recall from Chapter 26 that a (relative) curve over S is a separated flat morphism f : C → S of finite presentation such that all fibers are equi-dimensional of dimension 1 (Definition 26.162). If in addition C → S is proper and has geometrically reduced and geometrically connected fibers, then PicC/S is representable by an algebraic space (Theorem 27.117). In fact, Theorem 26.16 and Theorem 27.120 show that PicC/S is a scheme; see Theorem 27.137 below for a different approach to proving this. It is even smooth. Proposition 27.130. Let f : C → S be a relative proper curve with geometrically reduced and geometrically connected fibers. Then PicC/S is smooth. Proof. We can check the smoothness locally for the fppf topology. Working Zariski locally we may assume that S is affine. By noetherian approximation, we may assume that S is noetherian (Remark 27.121). By passing to an fppf-covering (e.g., to C → S) we may assume that C → S has a section. By Remark 27.42 it suffices to show that PicC/S (T ) → PicC/S (T0 ) is surjective for every thickening T0 → T of affine S-schemes T0 = Spec R0 and T = Spec R. We may assume that R0 = R/I with I 2 = 0 and that R is a local Artinian ring. By Lemma 27.116 it suffices to show that H 2 (CT0 , F ) = 0, where F is the coherent module I OCT0 . As T0 is affine, we have H 2 (XT0 , F )∼ = R2 fT0 ,∗ F which is zero because the fibers of CT0 → T0 have dimension 1 (Corollary 24.44). Remark 27.131. Let f : C → S be a relative proper curve and let L be a line bundle on C. Then the map s 7→ deg(L |Cs ) is locally constant on S, because deg(L |Cs ) = χ(OCs ) − χ(Ls⊗−1 ) (Section (15.9)) and the Euler characteristic is locally constant (Theorem 23.139).

652

27 Abelian schemes

Let f : C → S be a proper relative curve, let T be an S-scheme, and let L ∈ PicC/S (T ). We are going to define a locally constant map degL : T → Z.

(27.25.1)

For t ∈ T let Lt be the image of L in PicC/S (κ(t)). Fppf-locally, Lt comes from a line bundle on Ct , i.e., we can choose a field extension k of κ(t) and L ∈ Pic(C ⊗ k) whose image in PicC/S (k) is the pullback of Lt . Set degL (t) := degk (L ). This is well defined, commutes with base change, and defines a locally constant function: Lemma 27.132. In the situation above we have the following assertions. (1) The integer degL (t) does not depend on the choices made. (2) If g : T ′ → T is a morphism of S-schemes, then degg∗ L = degL ◦g : T ′ → Z. (3) The map degL is a locally constant. Proof. If k ′ and L ′ ∈ Pic(Ck′ ) is another pair as above, then we find by Remark 27.112 (2) an extension of K of both of k and k ′ such that the images of L and of L ′ in Pic(CK ) are isomorphic, say to M . Therefore we have degk (L ) = degK (M ) = degk′ (L ′ ) by Remark 15.29. The invariance of the degree under base change also shows the second assertion. To show that degL is locally constant, i.e., continuous if we endow Z with the discrete topology, we choose an fppf-covering g : T ′ → T and L ∈ Pic(CT ′ ) whose image in PicC/S (T ′ ) is the inverse image of L. Then degg∗ L is locally constant by Remark 27.131. As g is surjective and open (Theorem 14.35), it is a topological quotient map, therefore degL is locally constant by (2). Definition 27.133. Let C → S be a proper relative curve and let d ∈ Z be an integer. For any S-scheme T we define (d)

PicC/S (T ) = { L ∈ PicC/S (T ) ; degL constant with value d}. By Lemma 27.132 (2) this defines a subfunctor of PicC/S . (d) The inclusion PicC/S → PicC/S is representable by an open and closed immersion by (d)

Lemma 27.132 (3). In particular, PicC/S is an algebraic space if PicC/S is an algebraic space (Lemma 27.36), for instance, if C → S has geometrically reduced and geometrically connected fibers (Theorem 27.117). Remark 27.134. Let C → S be a proper relative curve. The multiplication on PicC/S induces for all integers d, d′ ∈ Z a multiplication morphism (d′ )

(d)

(d+d′ )

PicC/S ×S PicC/S −→ PicC/S , (d)

∼

(−d)

and the inversion yields an isomorphism PicC/S → PicC/S . (0)

In particular, PicC/S becomes an open and closed subgroup functor. Moreover, multiplication makes

(d) PicC/S

(0)

into an PicC/S -torsor for the fppf-topology.

653 (27.26) The Jacobian of a curve. Definition and Remark 27.135. Let f : C → S be a smooth relative curve. Let x ∈ C(S), i.e., x : S → C is a section of f . Then x is a regular immersion (Theorem 19.30) of codimension 1 (Proposition 19.42 (2)). It is a closed immersion because C → S is separated (Example 9.12). Therefore we may view x as an effective Cartier divisor on C (cf. Remark 26.163). Let OC (x) be the corresponding line bundle, i.e., OC (x)⊗−1 ⊆ OC is the ideal defining the closed subscheme corresponding to x. On each fiber of C → S this (1) line bundle is of degree 1. If C → S is proper, we denote its image in PicC/S (S) again by OC (x). Replacing S by an arbitrary S-scheme T and C by CT we obtain a morphism of fppf-sheaves on (Sch/S) (1)

AC : C → PicC/S

(27.26.1)

defined by CT (T ) ∋ x 7→ OCT (x) on T -valued points, called the Abel morphism or Abel-Jacobi morphism. Now let L be a line bundle on C such that L |Cs has degree 1 for all s ∈ S (such a (1) line bundle does not have to exist in general). Its image in PicC/S is again denoted by L . Then we obtain the following variant of the Abel morphism ιL C : C → PicC/S , (0)

(27.26.2)

x 7→ OC (x) ⊗ L ⊗−1 . O (x0 )

If L = OC (x0 ) for some x0 ∈ C(S), then we also write ιxC0 instead of ιCC

.

Proposition 27.136. Let C → S be a smooth proper relative curve with geometrically (0) connected fibers. Then Pic0C/S = PicC/S . (0)

In particular, PicC/S has geometrically connected fibers over S. Proof. By Remark 27.125 we may assume that S = Spec k for some algebraically closed field k. Then both are open subschemes of the scheme PicC/k which is locally of finite type. Hence it suffices to show that (0)

Pic0C/k (k) = PicC/k (k) We have PicC/k (k) = Pic(C) by Example 27.114. Let L ∈ Pic0C/k (k) and let ξ ∈ Pic0C/k be the corresponding closed point. Then L is the fiber of the universal line bundle on PicC/k ×k C at ξ. The fiber of the universal line bundle in 0 ∈ Pic0C/k (k) is the trivial line bundle and hence has degree 0. As the degree is constant on connected schemes (Remark 27.131), L has degree 0. Conversely, P let L ∈ Pic(C) with deg L = 0. Then there exists a divisor D on C of n the form D = i=1 ([xi ] − [x0 ]) for closed points x0 , . . . , xn on C such that OC (D) ∼ =L (Example 26.54). Since C is connected, the morphism ιxC0 : C → PicC/k (27.26.2) factors through Pic0C/k which shows that OC (x − x0 ) ∈ Pic0C/k (k) for all x ∈ C(k). As Pic0C/k (k) is a subgroup of Pic(C), this shows that also L ∈ Pic0C/k (k).

654

27 Abelian schemes

Theorem and Definition 27.137. Let S be a scheme and let C → S be a smooth proper relative curve with geometrically connected fibers. Then Pic0C/S is an abelian scheme over S called the Jacobian of C → S. In the proof we will use Theorem 27.211 below whose proof uses none of the results of this or the next section. Proof. By Proposition 27.136, Pic0C/S → PicC/S is an open and closed immersion. Hence we can apply Lemma 27.126 and see that Pic0C/S is proper over S. Moreover, it is smooth over S by Proposition 27.130. By definition, Pic0C/S → S has geometrically connected fibers. Hence we conclude by Theorem 27.211 below. (d)

(0)

As PicC/S is a PicC/S -torsor for the fppf-topology (Remark 27.134) we obtain the following corollary. Corollary 27.138. Let C → S be a smooth proper relative curve with geometrically (d) connected fibers. Then PicC/S → S is proper, smooth, and has geometrically connected fibers for all d ∈ Z. One can show ([BLR] O 9.4, Prop. 4) that the Jacobian also has an S-ample line bundle which depends functorially on C → S. In particular, Pic0C/S is projective over S if S is (d)

qcqs (Corollary 13.72). This implies that the algebraic spaces PicC/S are schemes for all d ∈ Z by Remark 14.74. Hence PicC/S is a smooth scheme over S. We will use this fact in the proofs of the results of the rest of this section and of the next section. Let C → S be a smooth proper relative curve with geometrically connected fibers and let L be a line bundle on C. Recall that the functions (27.26.3)

gC : S → Z, degL : S → Z,

s 7→ g(Cs ), the genus of C, s 7→ deg(L |Cs ),

are locally constant on S. Corollary 27.139. Let C → S be a smooth proper relative curve with geometrically connected fibers. Then the relative dimension of Pic0C/S over S is equal to gC . Proof. This can be checked on geometric fibers, hence we may assume that S = Spec k for an algebraically closed field k. As Pic0C/k is smooth over k, we have dim Pic0C/k = dim Lie(Pic0C/k ) = dim H 1 (C, OC ) = genus(C), where the second equality holds because Pic0C/k is open in PicC/k and because of Proposition 27.122. Lemma 27.140. Let S be a scheme and let f : C → S be a smooth proper relative curve. Let L be a line bundle on C which is of degree 1 on each fiber. (1) There exists x ∈ C(S) such that L ∼ = OC (x) if and only if there exists a regular section ℓ ∈ Γ(C, L ) such that Coker(ℓ : OC → L ) is locally free of rank 1 over S. (2) Suppose that C → S has geometrically connected fibers and that for all s ∈ S the fiber Cs has genus ≥ 1. If L satisfies the equivalent assertions in (1), then f∗ L is locally free of rank 1, its formation commutes with arbitrary base change S ′ → S, and x is uniquely determined by L .

655 Proof. (1). Let x : S → C be such that L ∼ = OC (x). Let I (x) ⊆ OC be the ideal corresponding to the closed immersion x. Then I (x) ∼ = L ⊗−1 . Tensoring the exact sequence (*)

0 −→ I (x) −→ OC −→ x∗ OS −→ 0

with L yields an exact sequence (**)

0 −→ OC −→ L −→ x∗ (x∗ L ) −→ 0,

where we use that x∗ OS ⊗OC L = x∗ (x∗ L ) by the projection formula, Proposition 22.81. The injective homomorphism OC −→ L corresponds to a regular section ℓ ∈ Γ(C, L ) and its cokernel is locally free of rank 1 over S. Conversely, let ℓ : OC → L be an injection whose cokernel is locally free of rank 1 over S. Tensoring ℓ with I := L ⊗−1 makes I into an invertible ideal of OC that defines a closed subscheme D of C such that D → S is locally free of rank 1, i.e., an isomorphism. This yields the desired section x : S = D → C. (2). By hypothesis there exists x ∈ C(S) and an exact sequence as in (**). As x∗ (x∗ L ) is flat over S, (**) stays exact after base change S ′ → S. Using f∗ OC = OS we obtain by applying f∗ to (**) an exact sequence 0 → OS → f∗ L → x∗ L → R1 f∗ OC → R1 f∗ L → 0. As the formation of all terms, except possibly f∗ L , commute with base change S ′ → S (Corollary 24.63 and Corollary 23.146), the formation of f∗ L also commutes with base change by the five lemma. This implies that f∗ L is finite locally free by Theorem 23.140 (2). To see that f∗ L has rank 1, we may assume that S = Spec k for a field k. Then by Riemann-Roch, Theorem 26.48, one has dimk H 0 (C, L ) − dimk H 1 (X, L ) = 2 − g(C) ≤ 1 which shows that dimk H 0 (C, L ) ≤ 1. But by hypothesis, there exists a regular global section in H 0 (C, L ). Therefore dimk H 0 (C, L ) = 1. To see that x is unique, we may work locally on S. Hence we may assume S = Spec R is affine and that H 0 (C, L ) is a free R-module of rank 1. Choose ℓ as in (1) which we may consider as an R-linear map R → H 0 (C, L ). As ℓ stays injective after arbitrary base change R → R′ , ℓ is necessarily an isomorphism by Proposition 8.10. Hence any two choices of ℓ differ by some unit of R. Therefore the corresponding sections of C, as constructed in (1), are equal. Lemma 27.141. Let S be a scheme, let f : C → S be a smooth proper relative curve with geometrically connected fibers, and let L be a line bundle on C. Suppose that deg(L ) > 2g(C) − 2 (27.26.3). Then R1 f∗ L = 0 and f∗ L is a locally free OS -module of rank deg(L ) + 1 − g(C) whose formation is compatible with base change S ′ → S. Proof. By Remark 26.55 we have H 1 (Cs , L |Cs ) = 0 for all s ∈ S. Hence R1 f∗ L = 0 and f∗ L is finite locally free and its formation commutes with base change by Corollary 23.144. To calculate its rank, we can suppose that S is the spectrum of a field. Then dim H 0 (C, L ) = deg(L ) + 1 − g(C) by the Riemann-Roch theorem.

656

27 Abelian schemes

Proposition 27.142. Let S be a scheme and let C → S be a smooth proper relative curve with geometrically connected fibers such that gC is nowhere 0 on S. Then the Abel (1) morphism ι : C → PicC/S is a closed immersion. Proof. One can check that a morphism between two proper schemes is a closed immersion fiberwise (Proposition 12.93). Hence we may assume that S = Spec k for a field k. As being a closed immersion can be checked after faithfully flat base change (Proposition 14.53), we may assume that k is algebraically closed. In particular C(k) ̸= ∅. As C is proper over (1) k and PicC/k is separated, ι is proper (Proposition 12.58). Hence it suffices to show that ι is injective on k[ε]-valued points (Remark 17.45). But then PicC/k (k[ε]) = Pic(Ck[ε] ) and the injectivity of C(k[ε]) → Pic(Ck[ε] ) follows from Lemma 27.140 (2). Corollary 27.143. Let C → S be as in Proposition 27.142 and let L be a line bundle on C which is on all fibers of degree 1. Then ιL : C → Pic0C/S is a closed immersion. (27.27) Elliptic curves. We continue to denote by S an arbitrary scheme. In this section we will take a look at elliptic curves over S. See Section (26.17) for the case when S is the spectrum of a field. Definition 27.144. An elliptic curve over S is a pair consisting of a morphism of schemes f : E → S together with a section 0 : S → E of f such that (a) the morphism E → S is flat, proper, and of finite presentation and (b) all fibers of E → S are smooth, geometrically connected curves of genus 1. Remark 27.145. Let (f : E → S, 0) be an elliptic curve over S. (1) If T → S is a morphism of schemes, then the base change (ET , 0T ) is an elliptic curve over T . (2) The morphism E → S is smooth because flat morphisms locally of finite presentation with smooth fibers are smooth (Corollary 18.57). (3) By Corollary 24.63 one has f∗ OE = OS , compatibly with base change. Example 27.146. Let E → S be an abelian scheme of dimension 1. Then (E, 0) is an elliptic curve, where 0 ∈ E(S) is the zero section. Indeed, we have to show that all fibers of E have genus 1. But this is clear because we have seen in Proposition 27.15 that the canonical bundle Ω1Es /κ(s) of each fiber Es is trivial. Hence the genus of Es satisfies 2g(Es ) − 2 = 0 by Corollary 26.52. Theorem 27.147. Let (E, 0) be an elliptic curve over S. Then there exists a unique structure of a group scheme over S on E such that 0 is the unit section. This makes E into an abelian scheme of relative dimension 1. Hence we see that elliptic curves are precisely abelian schemes of relative dimension 1. Proof. By definition, any structure of a group scheme on E makes E into an abelian scheme. Therefore the uniqueness assertion holds by Corollary 27.104. To show the existence it suffices to factorize the functor T 7→ E(T ) from (Sch/S) to the category of sets over the category of abelian groups. Hence it suffices to show the following more precise proposition.

657 Proposition 27.148. Let (f : E → S, 0) be an elliptic curve over S. Then the morphism ι0 : E → Pic0E/S ,

x 7→ OE (x) ⊗ OE (0)⊗−1 ,

defined in (27.26.2) is an isomorphism. The proposition shows that every elliptic curve is self-dual, i.e., isomorphic to its dual abelian scheme (see Definition 27.159 and Corollary 27.212 below), and is also isomorphic to its own Jacobian. Proof. By Proposition 14.28 we may assume S = Spec k for a field k. We know already that Pic0E/S is an abelian scheme, and by Proposition 27.122 it has dimension dimk H 1 (E, OE ) = g(E) = 1. So both sides are integral schemes of dimension 1. By Corollary 27.143, ι0 is a closed immersion. Hence it must be an isomorphism. Remark 27.149. The proposition shows that for points P, Q, R ∈ E(T ) one has R = P +Q if and only if there exists a line bundle M on T such that OET (P ) ⊗ OET (Q) ⊗ OET (0T )⊗−1 ∼ = OET (R) ⊗ fT∗ (M ). If Pic(T ) = 0 (e.g., if T is the spectrum of a local ring or of a factorial ring), then the factor fT∗ (M ) can be omitted. Compare Theorem 26.98, Remark 26.100. The following proposition is a relative version of Proposition 26.95. Proposition 27.150. Let (f : E → S, 0) be an elliptic curve. Then there exists an open affine covering (Ui )i of S with Ui = Spec Ri such that for all i one has E ×S Ui ∼ = V+ (wi ) ⊆ P2Ri , where wi ∈ Ri [X, Y, Z] is a cubic homogeneous polynomial of the form (27.27.1)

wi = Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 − a2 X 2 Z + a4 XZ 2 + a6 Z 3 .

An equation as in (27.27.1) is called a Weierstraß equation for E. The proof will show that if S is a scheme with Pic(S) = 0 (e.g., if S = Spec R with R factorial), then such an embedding into P2 and the Weierstraß equation exist globally. Proof. Let L := OE (0) and let ℓ : OE → L be the injective homomorphism given by the zero section (Lemma 27.140). Tensoring with L ⊗d we obtain injective homomorphisms L ⊗d → L ⊗d+1 for all d. By Lemma 27.141, f∗ L ⊗d is a locally free OS -module of rank d for all d ≥ 1 and its formation commutes with base change of S. As the inclusions f∗ L ⊗d → f∗ L ⊗(d+1) stay injective after arbitrary base change, their cokernels are locally free (Proposition 8.10). We obtain a chain of injections O S = f∗ O E ∼ = f∗ L → f∗ L ⊗2 → f∗ L ⊗3 → · · · , where all arrows have a locally free cokernel of rank 1. We view these injections as inclusions.

658

27 Abelian schemes

By working locally on S we can assume that f∗ L → f∗ L ⊗2 and f∗ L ⊗2 → f∗ L ⊗3 have free cokernel. Then we can choose an OS -basis (1, x, y) of f∗ L ⊗3 (i.e., a triple of elements 1, x, y ∈ Γ(E, L ⊗3 ) = Γ(S, f∗ L ⊗3 ) such that the corresponding morphism OS3 → f∗ L ⊗3 is an isomorphism) such that (1, x) is an OS -basis of f∗ L ⊗2 . As the formation of f∗ L ⊗3 commutes with base change, 1, x, y induce a basis of H 0 (Es , Ls⊗3 ) on each fiber Es . As Ls is very ample (Proposition 26.59), the images of sections 1, x, y in H 0 (Es , Ls⊗3 ) generate Ls⊗3 . Hence 1, x, y generate L ⊗3 by Nakayama’s lemma. The associated Smorphism i : E → P2S is fiberwise a closed immersion. Hence it is a closed immersion by Proposition 12.93. It realizes E in P2S as the vanishing scheme of a homogeneous equation of degree 3. To see that one can choose the equation of the special form (27.27.1), one argues as follows. Let x2 be the image x ⊗ x in Γ(E, L ⊗4 ). For each s ∈ S the image of x2 in H 0 (Es , Ls⊗4 ) is non-zero and not contained in H 0 (Es , Ls⊗3 ). Hence the images of 1, x, y, x2 form a basis of H 0 (Es , Ls⊗4 ). Again Proposition 8.10 shows then that (1, x, y, x2 ) is an OS -basis of f∗ L ⊗4 . Similarly, (1, x, y, x2 , xy) is an OS -basis of f∗ L ⊗5 and that (1, x, y, x2 , xy, y 2 ) and (1, x, y, x2 , xy, x3 ) are both OS -bases of f∗ L ⊗6 . Moreover, y 2 − x3 ∼ is a section of f∗ L ⊗5 . This yields E → V+ (w) with w as in (27.27.1) such that E \ 0(S) 2 2 is given in AS ⊆ PS by the dehomogenization of w with respect to the variable Z.

Duality of abelian schemes We now come to the definition of the dual X t of an abelian scheme X as the identity component of the Picard scheme of X. A priori, this is only an algebraic space (but see below). Every line bundle L on X then defines a homomorphism of group schemes φL : X → X t (Theorem of the Square 27.168) which will play a central role in the sequel. The Theorem of the Square will also be instrumental in our proof that abelian varieties over a field are projective (Proposition 27.174), a result that we will generalize to normal base schemes in Theorem 27.291, and in the result that multiplication by non-zero integers is surjective (Proposition 27.186). In case that X is projective over its base scheme, we will show in Theorem 27.198 that X t is representable by a scheme. A key step in the proof is to show that dimk H 1 (X, OX ) = dim(X) for an abelian variety over a field k. An abstract result about Hopf algebras then gives us that the whole cohomology of OX is the exterior algebra of H 1 (X, OX ) (Corollary 27.200). In Theorem 27.203 and its corollaries we will generalize these results to arbitrary base schemes. In fact, for any base scheme S and any abelian scheme X over S, the dual X t is representable by a scheme, see Theorem 27.211. We sketch the proof. The idea is to reduce to the case that S is noetherian and normal by noetherian approximation and by some gluing arguments. In this case the above mentioned results show that X is projective and hence representable by a scheme. We conclude this part by studying the Poincar´e bundle which gives us a proof that ∼ there is a functorial isomorphism X → (X t )t (Theorem 27.222). (27.28) The space of correspondence classes. In this section, S denotes a scheme. Given a morphism X → Y of S-schemes, its graph is a closed subscheme of the product X ×S Y (which projects isomorphically onto X, and

659 this allows us to recover the given morphism from the graph subscheme). Therefore an arbitrary closed subscheme of X ×S Y , sometimes called a correspondence, can be viewed as a generalization of a morphism between X and Y . For our purposes it will be useful to consider divisorial correspondences which we will define (Definition 27.151) as certain equivalence classes of divisors, or rather of line bundles, on the product X ×S Y . It will turn out to be an important tool in our study of the Pic0 functor of an abelian scheme, see Lemma 27.155, Proposition 27.163, Section (27.39). See Remark 27.288 for the case where X and Y are relative curves. Definition 27.151. Let X → S and X ′ → S be S-schemes and let p : X ×S X ′ → X, p′ : X ×S X ′ → X ′ be the projections. Consider the homomorphism of abelian fppf-sheaves on (Sch/S) (27.28.1)

α : PicX/S × PicX ′ /S −→ PicX×S X ′ /S ,

induced by (L , M ) 7→ p∗ L ⊗ p′∗ M . Then CorrS (X, X ′ ) := Coker(α) is called the functor of divisorial correspondence classes between X and X ′ . In other words, CorrS (X, X ′ ) is the fppf-sheaf associated with the presheaf Corr′S (X, X ′ ) : (Sch/S) −→ (AbGrp), T 7−→ Coker(PicX/S (T ) × PicX ′ /S (T ) → PicX×S X ′ /S (T )). In fact, we will only be interested in the case where X and X ′ are abelian schemes over S. In this case, Corr′S (X, X ′ ) is already an fppf-sheaf and hence CorrS (X, X ′ ) = Corr′S (X, X ′ ). More precisely, we have: Remark 27.152. Let f : X → S and f ′ : X ′ → S be morphisms of schemes such that ′ f∗ OX = OS and f∗′ OX = OS compatibly with base change S ′ → S and endowed with sections e ∈ X(S) and e′ ∈ X ′ (S). Let p : X ×S X ′ → X and p′ : X ×S X ′ → X ′ be the projections. The sections e′ ∈ X(S ′ ) and e ∈ X(S) yield by base change sections s : X → X ×S X ′ of p and s′ : X ′ → X ×S X ′ of p′ . (1) We claim that there is a split exact sequence of fppf-sheaves (27.28.2)

α

0 −→ PicX/S × PicX ′ /S −→ PicX×S X ′ /S −→ Corr′S (X, X ′ ) −→ 0.

A splitting is given by (27.28.3)

β := βe,e′ : PicX×S X ′ /S −→ PicX/S × PicX ′ /S

which is induced by N 7→ (s∗ N , (s′ )∗ N ). Indeed, we have to show that β ◦ α = id on T -valued points for an S-scheme T . By base change, we may assume that T = S. Let L ∈ Pic(X) and L ′ ∈ Pic(X ′ ). By symmetry it suffices to show that L and s∗ (p∗ L ) ⊗ s∗ (p′∗ L ′ ) represent the same element in PicX/S (S) = Pic(X)/ Pic(S). But s∗ p∗ L = L and s∗ (p′∗ L ′ ) lies in the image of Pic(S) → Pic(X) because p′ ◦ s factors through the chosen section e′ : S → X ′ .

660

27 Abelian schemes

(2) This shows that Corr′S (X, X ′ ) is isomorphic to the kernel of the homomorphism β of abelian fppf-shaves and hence is already an fppf-sheaf. In particular Corr′S (X, X ′ ) = CorrS (X, X ′ ). Remark 27.153. Let X → S and X ′ → S be S-schemes. As the formation of PicZ/S commutes with base change S ′ → S for every S-scheme Z, so does the formation of CorrS (X, X ′ ), i.e., for all scheme morphisms T → S one has CorrS (X, X ′ ) ×S T ∼ = CorrT (X ×S T, X ′ ×S T ). Notation and Remark 27.154. Let (X, eX ) and (Y, eY ) be pointed presheaves on (Sch/S). Then we denote by Hom0 (X, Y ) the pointed presheaf on (Sch/S) such that Hom0 (X, Y )(T ) is the set of morphisms of pointed presheaves X ×S T → Y ×S T on (Sch/T ) for every S-scheme T . The base point of Hom0 (X, Y ) is given by the morphism X → Y that sends all x ∈ X(T ) to the image of eY in Y (T ). If Y is a presheaf of abelian groups on (Sch/S), then the sum of two pointed morphisms is again pointed and Hom0 (X, Y ) becomes a presheaf of abelian groups. Lemma 27.155. Let (X, e) and (X ′ , e′ ) be as in Remark 27.152. Then there is an isomorphism of abelian fppf-sheaves on (Sch/S), functorial in (X, e) and (X ′ , e′ ), CorrS (X, X ′ ) ∼ = Hom0 (X, PicX ′ /S ). Proof. Let s′ : X ′ → X ×S X ′ be the section of the second projection obtained by base change from e. Via the chosen section e′ we identify (*)

e′∗

T PicX ′ /S (T ) = Ker(Pic(XT′ ) −→ Pic(T )).

To construct the above isomorphism we will work on T -valued points for an S-scheme T . As everything is compatible with base change T → S, we may assume that T = S. By the Yoneda lemma we have PicX ′ /S (X) ∼ = HomS (X, PicX ′ /S ) and this isomorphism is given by sending a line bundle L on X ×S X ′ such that e′∗ X L ∈ Pic(X) is trivial to the morphism u : X → PicX ′ /S that maps x ∈ X(T ) to x∗X ′ LT , where xXT′ : XT′ → XT ×T XT′ is the section of the second projection obtained T from x by base change. Hence L defines a pointed morphism u if and only if s′∗ L is trivial in Pic(X ′ ). Via the identification (*) for T = X we see that Hom0 (X, PicX ′ /S )(S) = Ker(βe,e′ ) = CorrS (X, X ′ )(S) for the homomorphism βe,e′ defined in (27.28.3). Lemma 27.156. Let S be a scheme, let X → S and X ′ → S be flat proper morphisms of finite presentation with geometrically integral fibers and suppose that one can choose e ∈ X(S) and e′ ∈ X ′ (S). Then the morphism βe,e′ : CorrS (X, X ′ ) → PicX×S X ′ /S (27.28.3) is representable by a closed immersion of finite presentation. In particular CorrS (X, X ′ ) is a separated algebraic space locally of finite presentation over S.

661 Proof. By Remark 27.152, CorrS (X, X ′ ) is isomorphic to the kernel of a homomorphism of abelian fppf-sheaves PicX×S X ′ /S → PicX/S × PicX ′ /S which are all separated group algebraic spaces locally of finite presentation by Theorem 27.117. In particular, the unit section of PicX/S × PicX ′ /S is representable by a closed immersion of finite presentation (Remark 27.45 and Remark 27.41). Therefore this holds for βe,e′ by stability under base change (Remark 27.44 (1)). In fact, CorrS (X, X ′ ) is a separated algebraic space locally of finite presentation over S without the hypothesis that there exist sections of X or X ′ : One has to show that all properties can be checked locally for the fppf topology because after passing to the fppf-covering X ×S X ′ → S, both X and X ′ have sections. For the property of being an algebraic space this is a difficult result ([Sta] 0ADV), for the other properties this can be proved using the analogous result for schemes (Proposition 14.53). Proposition 27.157. Let X → S and X ′ → S be as in Lemma 27.156. Then CorrS (X, X ′ ) is a scheme which is separated, unramified, locally of finite presentation, and formally proper (Definition 27.128) over S. Proof. We already know that CorrS (X, X ′ ) is an algebraic space which is separated locally of finite presentation over S. It is formally proper because it is a closed algebraic subspace of PicX×S X ′ /S (Lemma 27.156) which is formally proper by Remark 27.127. Next we show that CorrS (X, X ′ ) → S is unramified. It suffices to show that all fibers are unramified (for schemes locally of finite type over S this holds by Proposition 18.29, and for algebraic spaces this follows easily by choosing an ´etale atlas U → CorrS (X, X ′ ) and to use that by definition, CorrS (X, X ′ ) is unramified over S if and only if U is unramified over S). As the formation of CorrS (X, X ′ ) is compatible with base change, we may assume S = Spec k for a field k which we may even assume to be algebraically closed by Proposition 18.24. To prove that CorrS (X, X ′ ) → S is unramified we claim that it suffices to show that Lie(CorrS (X, X ′ )) = 0. For this we use that CorrS (X, X ′ ) is a scheme by Theorem 27.475 . By Proposition 18.27 and by Remark 17.44 it suffices to show that Tz (CorrS (X, X ′ )) = 0 for all k-valued points z of CorrS (X, X ′ ). By homogeneity it suffices to prove this for the unit section. This shows the claim. To show that Lie(CorrS (X, X ′ )) = 0, consider the split exact sequence of abelian fppf-sheaves (27.28.2). It yields an exact sequence of k-vector spaces Lie α

0 −→ Lie PicX/S × Lie PicX ′ /S −−−→ Lie PicX×S X ′ /S −→ Lie CorrS (X, X ′ ) −→ 0. Now we use that Lie PicX/S = H 1 (X, OX ) (Proposition 27.122). Hence using the K¨ unneth isomorphism (Corollary 22.110) we can identify the term in the middle with H 1 (X × X ′ , OX×X ′ ) =

1 M

H i (X, OX ) ⊗ H 1−i (X ′ , OX ′ ).

i=0

Since H (X, OX ) = H (X = k (Corollary 24.63), altogether we obtain that Lie(CorrS (X, X ′ )) = 0. It remains to show that CorrS (X, X ′ ) is a scheme. As CorrS (X, X ′ ) → S is unramified, it is locally quasi-finite. Moreover it is separated. Therefore CorrS (X, X ′ ) is a scheme by [Sta] 03XX. 0

5

0

′

′ , OX )

Instead of using the difficult Theorem 27.47 one could also argue as in the sequel by checking that all arguments are also valid for algebraic spaces, which is not too difficult.

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27 Abelian schemes

(27.29) Definition of the dual functor to an abelian scheme. Let S be a scheme and let X be an abelian scheme over S with zero section 0 ∈ X(S). Let us collect some notation and recall some results about PicX/S and CorrS (X, X) in this special case. We have the morphisms m, p1 , p2 : X ×S X −→ X, where m is the group law on X and p1 and p2 are the two projections. The zero section yields sections si of pi given by id

×0

X s1 : X = X ×S 0 −−− −−→ X × X, 0×idX s2 : X = 0 ×S X −−−−− → X ×S X.

If T is an S-scheme, we will often also denote the base changes XT ×T XT → XT of m, p1 , p2 (resp. the base changes XT → XT ×T XT of s1 and s2 ) again by m, p1 , p2 (resp. by s1 and s2 ). For every S-scheme T we have PicX/S (T ) = Pic(XT )/ Pic(T ) and PicX/S is a separated group algebraic space locally of finite presentation. Remark 27.158. As a special case of the results of Section (27.28) we have the S-group scheme CorrS (X, X) of divisorial correspondences on X. It is separated, unramified, and locally of finite presentation. By definition it is equipped with an fppf-surjective homomorphism (27.29.1)

PicX×S X/S −→ CorrS (X, X)

of group algebraic spaces over S. Its kernel consists by definition of those classes of line bundles M on X ×S X that are of the form p∗1 L1 ⊗ p∗2 L2 for line bundles L1 and L2 on X. (1) The zero section 0 of X defines a section of (27.29.1) (27.29.2)

CorrS (X, X) −→ PicX×S X/S

that identifies CorrS (X, X)(T ) with those classes of line bundles M on XT ×T XT such that s∗1 M = s∗2 M = OXT in PicX/S (T ). Via this description, (27.29.1) is induced by the map that sends a line bundle M on XT ×T XT to (27.29.3)

M ⊗ p∗1 s∗1 M −1 ⊗ p∗2 s∗2 M −1 .

(2) We have seen in Lemma 27.155 that we can identify (27.29.4)

CorrS (X, X)(T ) = { ψ : XT → PicXT /T ; ψ pointed morphism over T }

for every S-scheme T . Via this description, the proof of Lemma 27.155 shows that (27.29.1) is induced by the map that sends a line bundle M on XT ×T XT to the following pointed morphism. A point x ∈ X(T ′ ), T ′ a T -scheme is mapped to the pullback of the line bundle in (27.29.3) under x ˜ := (x, idXT ′′ ) : XT ′ → XT ′ ×T ′ XT ′ which in PicX/S (T ′ ) equals the line bundle (27.29.5)

x ˜∗ M ⊗ s∗2 M −1 .

The following definitions will be central for the rest of the chapter.

663 Definition 27.159. Let X be an abelian scheme over S with group law m : X ×S X → X. (1) Define the homomorphism φ : PicX/S → CorrS (X, X) of group algebraic spaces over S as the composition m∗

φ : PicX/S −→ PicX×S X/S −→ CorrS (X, X). (2) If L is a line bundle on X, we denote the image of the class of L in PicX/S (S) under the map φ by φL . Via (27.29.4) we usually consider it as a morphism of pointed algebraic spaces (27.29.6)

φL : X → PicX/S .

(3) The dual abelian space of X is defined as X t := Ker(φ). From the above results we deduce easily: Lemma 27.160. The inclusion X t → PicX/S is representable by an open and closed immersion. Therefore X t is a commutative separated group algebraic space over S locally of finite presentation. We will see below (Proposition 27.207) that X t = Pic0X/S , the identity component of PicX/S (Section (27.24)), and that X t is always an abelian scheme over S (Corollary 27.212), which will be called the dual abelian scheme of X. Proof. As CorrS (X, X) → S is unramified and separated, the unit section of CorrS (X, X) is an open and closed immersion (Remark 18.30 and Remark 9.12). This shows that the inclusion j : X t → PicX/S is representable and an open and closed immersion. As PicX/S is a separated algebraic space locally of finite presentation over S, so is X t . We have the following more explicit description of φ. Proposition 27.161. Let L be a line bundle on the abelian scheme X. (1) If one considers CorrS (X, X) as a subgroup of PicX×S X/S via (27.29.2), then the image of the class of L under φ is given by the class of the line bundle φ(L ) = Λ(L ) := m∗ L ⊗ p∗1 L −1 ⊗ p∗2 L −1 ⊗ [0]∗ L on X ×S X, where [0]∗ L is obtained by pulling back L to S via the zero section of X and then by pulling back to X ×S X via the structure morphism. (2) The morphism φL : X → PicX/S is given by X(T ) ∋ x 7→ the class of t∗x LT ⊗ LT−1 for every S-scheme T . Here XT := X ×S T , LT is the pullback of L to XT , and tx is the translation XT → XT by x. The line bundle Λ(L ) is sometimes called the Mumford bundle. Its class in PicX×S X/S (S) is the same as the class of m∗ L ⊗ p∗1 L −1 ⊗ p∗2 L −1 and often one can ignore the factor “⊗[0]∗ L ”, e.g., if Pic(S) = 0, for instance if S = Spec k for a field k. But the definition of Λ(L ) we have given has the advantage that there are canonical isomorphisms Λ(L )|X×0 ∼ = OX ∼ = Λ(L )|0×X .

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27 Abelian schemes

Proof. This follows from Remark 27.158: One obtains the description in (1) (resp. in (2)) by applying (27.29.3) (resp. (27.29.5)) to M := m∗ L . Corollary 27.162. Let f : X → S be an abelian scheme and let L be a line bundle on X. (1) The following assertions are equivalent. (i) The class of L is in X t (S). (ii) For every geometric point s¯ of S the class of Ls¯ is in X t (¯ s). (iii) The Mumford bundle Λ(L ) is isomorphic to the pullback of a line bundle on S. (iv) For all S-schemes T and all x ∈ X(T ) the line bundle t∗x LT ⊗LT−1 is isomorphic to the pullback of a line bundle on T . (2) There exists an open and closed immersion S0 → S such that a morphism of schemes f : T → S factors through S0 if and only if the class of (idX ×f )∗ L is in X t (T ). In particular, if S is connected and there exists a geometric point s¯ → S such that the class of Ls¯ is in X t (¯ s), then the class of L is in X t (S). Proof. The equivalence of (i), (iii), and (iv) holds by Proposition 27.161. The equivalence of (i) and (ii) and part (2) hold since X t → PicX/S is an open and closed immersion. Proposition 27.163. Let X and Y be abelian schemes over S. (1) Every pointed morphism f : X → PicY /S factors through the open subgroup space Y t . In particular CorrS (X, Y ) ∼ = Hom0 (X, Y t ). (2) One has a functorial isomorphism Hom0 (X, Y t ) ∼ = Hom0 (Y, X t ). (3) Let L be a line bundle on X. The pointed morphism φL (27.29.6) factors uniquely through a pointed morphism (27.29.7)

φL : X → X t .

Proof. As Y t → PicY /S is an open and closed immersion and f map the zero section into Y t , (1) holds since X has connected fibers over S. Then the canonical isomorphism CorrS (X, Y ) ∼ = CorrS (Y, X) implies (2). Finally, (3) is an immediate corollary of (1). Next we show that ( )t commutes with base change and is a functor. Remark 27.164. Let X be an abelian scheme over S. As the formation of PicX/S and CorrS (X, X) is compatible with base change, so is the formation of X t , i.e., for every scheme morphism T → S we have a functorial isomorphism (X ×S T )t ∼ = X t ×S T. Lemma and Definition 27.165. Let f : X → Y be a homomorphism of abelian schemes over S. (1) The homomorphism f ∗ : PicY /S → PicX/S induces a homomorphism f t : Y t → Xt called the dual of f .

665 (2) Let L be line bundle on Y . Then the diagram X (27.29.8)

f

φf ∗ L

 Xt o

/Y φL

f

t

 Yt

commutes. of (27.29.8) if we replace the lower horizontal Proof. We first show the commutativity f∗ row of the diagram with PicX/S ←− PicY /S . We check this on T -valued points for an S-scheme T . As all constructions commute with base change, we may assume T = S. Let x ∈ X(S). Using tf (x) ◦ f = f ◦ tx and hence t∗x f ∗ L = f ∗ t∗f (x) L we find (f t ◦ φL ◦ f )(x) = f ∗ (t∗f (x) L ⊗ L −1 ) (27.29.9)

= t∗x f ∗ L ⊗ (f ∗ L )−1 = φf ∗ L (x).

By Proposition 27.163 (3) it now is enough to show that (1) holds. As L ∈ X t (S) if and only if φL = 0, the commutative diagram shows that f ∗ L ∈ Y t (S). Remark 27.166. Let L be a line bundle on X whose class is contained in X t (S) (cf. Corollary 27.162). Then one has m∗ L = p∗1 L ⊗ p∗2 L in Pic(X ×S X)/ Pic(S). Pulling back along a pair (x, y) of T -valued points of X one obtains (x + y)∗ L = x∗ L ⊗ y ∗ L in Pic(X) up to tensoring with a line bundle that is pulled back from S. In particular, if g1 , g2 : Y → X are homomorphisms of abelian schemes, then we have an equality of homomorphisms of commutative group algebraic spaces X t → Y t (27.29.10)

(g1 + g2 )t = g1t + g2t .

Hence, X 7→ X t defines a contravariant additive functor ( )t from the category of abelian schemes over S to the category of abelian algebraic group spaces over S. In the sequel, we will see that this functor defines a contravariant autoduality of the category of abelian schemes over S and that ( )t ◦ ( )t is isomorphic to the identity. Note that the map Hom(X, Y ) −→ Hom(PicY /S , PicX/S ),

f 7→ f ∗

is (except in trivial cases) not a group homomorphism. For instance, we will see in Proposition 27.234 below that the multiplication by an integer [n] induces on the quotient PicX/S /X t the multiplication by n2 .

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27 Abelian schemes

(27.30) The Theorem of the Square. In this section S denotes a scheme. The goal of this section is to show that φL : X → X t (27.29.6) is a homomorphism of group algebraic spaces. If we already knew that X t is an abelian scheme (see Corollary 27.212 below), then we could simply apply Corollary 27.102 as φL preserves unit sections. But the homomorphism φL will be an important tool in the proof that X t is an abelian scheme. Hence we have to argue differently. Proposition 27.167. (Cubical structure on line bundles) Let X be an abelian scheme over S, let L be a line bundle on X, let T be an S-scheme and let x1 , x2 , x3 ∈ X(T ). Then the line bundle cT (L ) := (x1 + x2 + x3 )∗ L ⊗ (x1 + x2 )∗ L −1 ⊗ (x1 + x3 )∗ L −1 ⊗ (x2 + x3 )∗ L −1 ⊗ x∗1 L ⊗ x∗2 L ⊗ x∗3 L ⊗ 0∗ L −1 on T is trivial. Proof. It suffices to treat the universal case T = X ×S X ×S X and xi = pP i : T → A the i-th projection. For I ⊆ {1, . . . , 3} let pI : X ×S X ×S X be the morphism i∈I pi . Then the line bundle cT (L ) is given by O |I|+1 p∗I (L )(−1) . (27.30.1) c(L ) := I⊆{1,2,3}

We claim that the restrictions of c(L ) to {0} ×S X ×S X, to X ×S {0} ×S X, and to X ×S X ×S {0} are trivial. Indeed, by symmetry it suffices to treat the case {0}×S X ×S X. Then c(L )|{0}×S X×S X = m∗ L ⊗ q1∗ L −1 ⊗ q2∗ L −1 ⊗ m∗ L −1 ⊗ (0∗ L )X×S X ⊗ q1∗ L ⊗ q2∗ L ⊗ (0∗ L )−1 X×S X where m, q1 , q2 are the multiplication and the projections X ×S X → X. Clearly this is trivial. We now use the Theorem of the Cube 24.73. By the claim above, we find in every fiber of X → S, considered as the third factor of X ×S X ×S X, a point x ∈ X such that the restriction of c(L ) to X ×S X ×S {x} is trivial. Namely, we can take x to be the image of the unit section of the fiber. Hence the Theorem of the Cube implies that we find for every s ∈ S an open and closed subscheme Us of X that meet the fiber Xs and such that the restriction of c(L ) to X ×S X ×S Us is trivial. As all fibers of X → S are connected one has Xs ⊆ Us and hence (Us )s∈S is a covering of X. This shows that c(L ) is trivial. Theorem of the Square 27.168. Let X be an abelian scheme over S and let L be a line bundle on X. Then the pointed morphism φL : X → X t is a homomorphism of group algebraic spaces over S. Proof. By the description of φL in Proposition 27.161 (2) it suffices to show that for each S-scheme T and for all x, y ∈ X(T ) we have an equality of classes (27.30.2)

t∗x+y LT ⊗ LT = t∗x LT ⊗ t∗y LT

in PicX/S (T ). Let pX : XT = X ×S T → X and pT : XT → T be the projections. Then LT = p∗X L . Define (X ×S T )-valued points of X by x1 := pX , x2 := x ◦ pT , and x3 := y ◦ pT . Then

667 x 1 + x 2 = pX ◦ tx ,

x 1 + x 3 = p X ◦ ty ,

x2 + x3 = (x + y) ◦ pT ,

x1 + x2 + x3 = pX ◦ tx+y .

Now we apply Proposition 27.167 and see that both sides of (27.30.2) are isomorphic line bundles on XT up to tensoring with p∗T (M ), where M := (x + y)∗ L ⊗ x∗ L −1 ⊗ y ∗ L −1 ⊗ [0]∗ L ∈ Pic(T ). Hence we obtain the desired equality in PicX/S (T ) = Pic(XT )/ Pic(T ). Remark 27.169. Let X be an abelian scheme over S and let L and M be line bundles on X. (1) As φ is a group homomorphism, one has φL ⊗M = φL + φM , φL −1 = −φL .

(27.30.3)

(2) Moreover for x ∈ X(S) we have φt∗x L = φL .

(27.30.4)

Indeed, let T be an S-scheme and let y ∈ X(T ). We denote the image of x in X(T ) again by x. Then we have an equality in PicX/S (T ) φt∗x L (y) = t∗x+y LT ⊗ t∗x L −1 = t∗y L ⊗ L −1 = φL (y), where the equality in the middle holds by (27.30.2).P r (3) Let x1 , . . . , xr ∈ X(S) be S-valued points such that i=1 xi = 0 in the abelian group X(S). By induction, (27.30.2) implies the equality r O

t∗xi L = L ⊗r

i=1

in PicX/S (S) = Pic(X)/ Pic(S).

(27.31) The kernel of φL . In this section S denotes a scheme, f : X → S an abelian scheme over S, and L a line bundle on X. We set (27.31.1)

K(L ) := Ker(φL ).

This is a closed subgroup scheme of X. By definition, K(L ) depends only on the class of L in PicX/S (S), i.e. K(L ⊗ f ∗ N ) = K(L ) for a line bundle N on Pic(S). By Proposition 27.161 there are the following descriptions of K(L ). Remark 27.170. For any S-scheme T one has K(L )(T ) = { x ∈ X(T ) ; t∗x L ⊗ L −1 ∼ = fT∗ N for some N ∈ Pic(T )}. Moreover, K(L ) is the maximal closed subscheme of X such that the restriction of the Mumford bundle Λ(L ) to K(L ) ×S X is isomorphic to the pullback of a line bundle on K(L ) (in the sense of the Seesaw Proposition 24.66).

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27 Abelian schemes

Lemma 27.171. Let S be a scheme. (1) The restriction Λ(L )|K(L )×S X is trivial. (2) The restriction L ⊗ [−1]∗ L to K(L ) is isomorphic to [0]∗ L ⊗−2 , in particular it is trivial if Pic(S) = 0. Proof. By Remark 27.170 we know that Λ(L )|K(L )×S X is the pullback of a line bundle N from K(L ). Let s := (id, e) : K(L ) → K(L ) ×S X. Then N = s∗ (Λ(L )|K(L )×S X ) = (L ⊗ L −1 ⊗ [0]∗ (L −1 ) ⊗ [0]∗ (L ))|K(L ) ∼ = OK(L ) . This shows (1). By (1) we know that the restriction of Λ(L ) to K(L ) ×S K(L ) is trivial. Let τ := (id, − id) : X → X ×S X. Then τ ∗ Λ(L ) = [0]∗ L ⊗ L −1 ⊗ [−1]∗ L −1 ⊗ [0]∗ L and the restriction of τ ∗ Λ(L ) to K(L ) is trivial. This implies the claim. Corollary 27.172. Let L be an ample line bundle on X. Then K(L ) is a finite S-scheme. In Theorem 27.198 below we will see that K(L ) → S is finite locally free. Proof. As K(L ) is a closed subscheme of X, it is proper over S. Hence we can check finiteness of K(L ) → S on fibers (Proposition 12.93). Therefore we may assume that S = Spec k for a field k. As L is ample, L ⊗[−1]∗ L is also ample. By Lemma 27.171 we know that the ample line bundle (L ⊗ [−1]∗ L )|K(L ) is trivial. Hence K(L ) → S is quasi-affine (Definition 13.78) and therefore finite by Corollary 13.82. (27.32) Projectivity of abelian varieties. The results of Section (27.11) on general homogeneous spaces show the following general projectivity result. Proposition 27.173. Let S be a local normal noetherian scheme and let X → S be an abelian scheme. Let U ⊆ X be an open affine neighborhood that meets the special fiber of X → S. Let D1 , . . . , Dr be the irreducible components of X \ U that are of codimension 1 in X, considered asPreduced subschemes of X. Let n1 , . . . , nr > 0 be integers. Then the r Weil divisor D := i=1 ni Di is a Cartier divisor and OX (D) is ample. In particular, X → S is projective. Note that the image of U in S is open and contains the special point of S. Therefore U meets every fiber of X → S. We gave a proof of the general statement for homogeneous spaces only if S = Spec k is a field – and even in this case we omitted the proof of the difficult Lemma 27.70. Below in Theorem 27.291 we will prove, more generally, that every abelian scheme over a normal noetherian scheme is projective. We will now give a self-contained proof for the fact that abelian varieties over a field are projective. In fact we will show the following more precise result.

669 Proposition 27.174. Let X be an abelian variety over a field k and let U ⊆ X be a nonempty open affine subscheme. Then every effective divisor D whose underlying topological space is X \ U is ample. In particular, every abelian variety over a field is projective. For instance (X \ U )red is an effective ample divisor by Lemma 25.150. In Proposition 27.265 below we will see that conversely every ample line bundle is of the form OX (D) with D an effective divisor such that X \ D is affine. To prove Proposition 27.174 we will use the following lemma. Lemma 27.175. Let X be an abelian variety over an algebraically closed field k, let D ⊆ X be an effective Cartier divisor, and set L := OX (D). Then L ⊗2 is globally generated. Below in Proposition 27.278 we will prove that the tensor product of any two ample line bundles on an abelian scheme is globally generated. Proof. Fix x ∈ X(k). We have to find an effective divisor E that is linearly equivalent to 2D such that x ∈ / E. Then the complete linear system of L ⊗2 is base point free and hence L ⊗2 is globally generated (Section (13.13)). For y ∈ X(k) we set Ey := ty (D)+t−y (D) which is linearly equivalent to 2D by (27.30.2). Set U := X \ D. Then x∈ / ty (D) ∪ t−y (D) ⇔ x ∈ ty (U ) ∩ t−y (U ) ⇔ y ∈ t−x (U ) ∩ tx ([−1](U )) =: V. As X is irreducible, V ̸= ∅, and for all y ∈ V one has x ∈ / Ey . Proof. [of Proposition 27.174] Set L := OX (D). Let k¯ be an algebraic closure of k. Then the inverse image of D in Xk¯ is still an effective Cartier divisor whose complement is a non-empty open affine subscheme of Xk¯ . To see that L is ample, we may work fpqc-locally (Proposition 14.58). Hence we can assume that k is algebraically closed. ∗ ∼ ⊗2 and By Lemma 27.175 there exists a morphism f : X → PN k such that f OPN (1) = L N −1 −1 N there exists a hyperplane H of Pk such that f (H) = 2D and hence U = f (Pk \ H). As X is proper over k and PN k is separated over k, the morphism f is proper. Hence for each x ∈ U (k) the fiber f −1 (f (x)) is proper and contained in U . As U is affine, f −1 (f (x)) is finite (Corollary 12.89). As all non-empty fibers of f have the same dimension (Corollary 27.107), f is quasi-finite. Hence f is finite, again by Corollary 12.89. Therefore, L ⊗2 is the pullback of an ample line bundle by a finite morphism and hence is itself ample (Proposition 13.83). Hence L is ample (Proposition 13.50).

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27 Abelian schemes

(27.33) Isogenies. In this section, S denotes a scheme. Definition and Proposition 27.176. Let G and H be S-group schemes that are smooth, separated, of finite presentation6 , and with connected fibers over S. Then a homomorphism f : G → H of S-group scheme is called an isogeny if it satisfies the following equivalent conditions. (i) The homomorphism f is quasi-finite, of finite presentation, and flat. (ii) The homomorphism f is surjective and Ker(f ) → S is quasi-finite. (iii) For all s ∈ S one has dim(Gs ) = dim(Hs ) and Ker(fs ) is finite over κ(s). (iv) For all s ∈ S, fs is surjective and finite. ∼ Every isogeny f : G → H induces an isomorphism G/ Ker(f ) → H of group schemes over S. Proof. We recall the following facts that we have already seen. (1) Every surjective homomorphism f : G → H is faithfully flat (Proposition 27.54). Then ∼ f is fppf-surjective and hence induces an isomorphism G/ Ker(f ) → H. (2) Therefore Corollary 27.63 shows that if f is surjective, then f is quasi-finite if and only if Ker(f ) → S is quasi-finite. (3) A scheme over a field is quasi-finite if and only if it is finite (Proposition 5.20). Moreover, f is automatically of finite presentation since G and H are of finite presentation over S (Proposition 10.35). Hence (ii) implies (i). Moreover, (ii) and (iv) are equivalent because of (3). Let us show that (i) implies (iv). As fs is flat, its image is open (Theorem 14.35) and dense because Hs is by hypothesis irreducible. Hence fs is surjective (Proposition 27.14 (1)). As f is quasi-finite, each fiber fs is quasi-finite. Then Ker(fs ) → Spec κ(s) is quasi-finite by (2) and hence finite by (3). Therefore fs is finite by (2). It remains to show that (iv) and (iii) are equivalent. For this we can assume that S = Spec k for a field k. Then (iv) implies (iii) by Proposition 27.14 (2). Conversely, if Ker(f ) is finite over k, then f is finite by (2). Moreover f (G) is a closed subspace of H of the same dimension, again by Proposition 27.14. As H is irreducible, we see f (G) = H. Corollary 27.177. Let X and Y be abelian schemes over S and let f : X → Y be a homomorphism of abelian schemes. (1) If f is an isogeny, then f is surjective and finite locally free and the relative dimensions of X and of Y over S are equal. (2) The homomorphism f is an isogeny if and only if any two of the following assertions hold. (a) f is finite (equivalently, f has finite fibers). (b) f is surjective. (c) The relative dimensions of X and of Y over S are equal. Moreover all these assertions can be checked on fibers over S. The degree of f is defined as the degree of the finite locally free morphism Ker(f ) → S which is a locally constant function on S. 6

Actually finite presentation follows from the other hypotheses: As G and H are smooth and separated over S, to be in addition of finite presentation only means that G → S and H → S are also quasicompact. But any group scheme G over a scheme S whose fibers are locally of finite type and connected is quasi-compact over S by [SGA3] O X Exp. VIB, 3.6.

671 Proof. As X and Y are proper and of finite presentation over S, so is f . Now any proper quasi-finite morphism is finite (Corollary 12.89) therefore f is finite if any only if it has finite fibers. Moreover, a finite flat morphism of finite presentation is finite locally free (Proposition 12.19). Hence Proposition 27.176 shows the first assertion and that any two of the assertions (a), (b), (c) imply that f is an isogeny. Clearly, all these assertions can be checked on fibers. Proposition 27.178. Let f : X → Y be an isogeny of abelian schemes. (1) Let g1 , g2 : Y → Z be homomorphisms of abelian schemes such that g1 ◦ f = g2 ◦ f . Then g1 = g2 . (2) Let h1 , h2 : Z → X be homomorphisms of abelian schemes such that one has f ◦ h1 = f ◦ h2 . Then h1 = h2 . If S is connected, it suffices to assume that (g1 )s¯ ◦ fs¯ = (g2 )s¯ ◦ fs¯ (resp. fs¯ ◦ (h1 )s¯ = fs¯ ◦ (h2 )s¯) in a single geometric point s¯ → S to conclude that g1 = g2 (resp. that h1 = h2 ) by Proposition 27.105. Proof. The first assertion holds since f is an epimorphism (Lemma 27.50). Now let h := h1 − h2 . Then h factors through Ker(f ) by assumption, and we have to show that h = 0. By Proposition 27.105 it suffices to show this on geometric fibers. Hence we may assume that S = Spec k for an algebraically closed field k. As Z is connected and reduced, h : Z → X factors through a connected and reduced subscheme of Ker(f ). Since Ker(f ) is finite, it follows that h factors through its identity section. Proposition 27.179. Let X and Y be abelian varieties over a field k and let f : X → Y b (Y ) one has be an isogeny. Then for every F in Dcoh (27.33.1)

χ(X, f ∗ F ) = deg(f )χ(Y, F ).

Proof. We denote by gr K0 (Z) the graded K-group of a smooth k-scheme Z, so that Z 7→ gr K0 (Z) is endowed with the structure of an additive cohomology theory (Example 23.109): For any morphism g : Z → Z ′ of smooth k-schemes, E 7→ Lf ∗ E induces a ring homomorphism f ∗ : gr K0 (Z ′ ) → gr K0 (Z). For a proper morphism g : Z → Z ′ of smooth k-schemes we denote by g∗ : gr K0 (Z) → gr K0 (Z ′ ) the push forward (23.11.4), which is a homomorphism of abelian groups. Finally, there is a Chern character (23.23.3) ch : K0 (−) → gr K0 (−)Q , which is compatible with pullback. As tangent bundles of abelian varieties are trivial, the relative tangent bundle Tf = [TX/k − f ∗ TY /k ] ∈ K0 (X) is trivial. Hence for every E in Db (X) the GrothendieckRiemann-Roch theorem (23.23.5) yields that ch(Rf∗ E ) = f∗ (ch(E )). Applying this to E = f ∗ F = Lf ∗ F we obtain (1)

(27.33.2)

ch(Rf∗ f ∗ F ) = f∗ ch(f ∗ F ) = f∗ f ∗ (ch(F )) (2) = ch(F )f∗ f ∗ (OY ) = ch(F )f∗ OX (3) = deg(f ) ch(F ).

672

27 Abelian schemes

where (1) holds by the Grothendieck-Riemann-Roch theorem, Theorem 23.112, (2) by the projection formula (23.20.1), and (3) holds by Corollary 23.49 since f∗ OX is a finite locally free OY -module of degree deg(f ). Let h : Y → Spec k be the structure morphism. Multiplying (27.33.2) with td(TY /k )7 and applying h∗ we obtain by Corollary 23.113 the equality of deg(f )h∗ (td(TY /k ) ch(F )) = deg(f )χ(Y, F ) and of h∗ (td(TY /k ) ch(Rf∗ f ∗ F )) = χ(Y, Rf∗ f ∗ F ) = χ(X, f ∗ F ), where the last equality holds by Corollary 23.63. The proof would have also worked verbatim by using the Chow ring (which we did not introduce) as an additive cohomology theory. Remark 27.180. The proof shows that (27.33.1) holds more generally if f : X → Y is a finite locally free morphism of projective smooth schemes over a field k such that the classes of f ∗ Ω1Y /k and of Ω1X/k in K-theory K0 (X) agree. These properties are for instance satisfied if f is finite ´etale because then f ∗ Ω1Y /k ∼ = Ω1X/k (Corollary 18.19). (27.34) The Frobenius isogeny. Let S be a scheme of characteristic p > 0, i.e. pOS = 0. Recall from Definition 4.24 the absolute Frobenius and the relative Frobenius. The absolute Frobenius FrobS : S → S is the identity on underlying topological spaces and raises local sections to their p-th power. If S = Spec R for a ring R, then FrobS corresponds to the usual Frobenius FrobR : R → R that sends a to ap . If X → S is a morphism of schemes, we set X (p) := X ×S,FrobS S. This defines a functor X 7→ X (p) from the category of S-schemes to itself that commutes with fiber products over S. Hence it transforms S-group schemes over S into S-group schemes. Let FX/S : X → X (p) be the relative Frobenius, i.e., the unique morphism of S-schemes that makes the diagram X FrobX FX/S

(27.34.1)

f

!

$ /X

X (p)   S

f (p) □ FrobS

f

 / S,

commutative. The relative Frobenius FX/S is functorial in the S-scheme X. Hence if G is an S-group scheme, FG/S : G 7→ G(p) is a homomorphism of S-group schemes. Example 27.181. Let S be a scheme of characteristic p and let G be an S-group scheme. Suppose that G is defined over Fp , i.e., there exists a group scheme G0 over Fp such that G 0 × Fp S ∼ = G. Then 7

This is not really necessary here because the tangent bundle of the abelian variety Y is trivial and hence td(TY /k ) = 1; it is only included in the proof for Remark 27.180 below.

673 G(p) = G ×S,FrobS S = G0 ×Fp S ×S,FrobS S = G0 ×Fp ,FrobFp Spec(Fp ) ×Fp S = G0 ×Fp S = G because the Frobenius on Fp is the identity. In this case, the relative Frobenius is an endomorphism. It is the base change of the absolute Frobenius on G0 , which is an endomorphism of group schemes over Fp . Consider the following examples. (1) G = Ga,S i.e. Ga (T ) = Γ(T, OT ) for every S-scheme T . In this case the relative Frobenius is given by FGa,S /S (T ) : Ga,S (T ) → Ga,S (T ),

x 7→ xp

(2) G = Gm,S , i.e. Gm (T ) = Γ(T, OT )× for every S-scheme T . In this case the relative Frobenius is again given by x 7→ xp . (3) Let H be any abstract group and let G = H S the corresponding constant S-group scheme, i.e., G(T ) consists of the group of locally constant maps T → H. Then the relative Frobenius is the identity. Proposition 27.182. Let p be a prime number and let S be a scheme with pOS = 0. Let G → S be a separated smooth group scheme of finite presentation with connected fibers. Then the relative Frobenius FG : G → G(p) is an isogeny. Its kernel is finite locally free over S of degree s 7→ pdim Gs . Proof. The relative Frobenius FG is an isogeny by Exercise 18.25. In particular, the kernel of FG is separated, quasi-finite, flat and of finite presentation over S. We will use Exercise 20.17 and show that its degree is equal to the function s 7→ pdim Gs which is locally constant by Proposition 27.27. But this follows again from Exercise 18.25 since dim(Gs ) is the relative dimension of G → S in every point of G that is mapped to s. (27.35) Torsion points. In this section S denotes a scheme. Let X be an abelian fppf-sheaf over S. For all n ∈ Z we denote by (27.35.1)

[n] = [n]X : X → X,

X(T ) ∋ x 7→ nx ∈ X(T ),

T an S-scheme

the multiplication by n. It is a homomorphism of abelian fppf-sheaves over S. From now on let X and Y denote abelian schemes over S. Then [n] is a homomorphism 0 of abelian schemes. For n = 0 it is the composition X → S −→ X, where X → S is the structure morphism. For n = 1, −1 it is an isomorphism. For all n ∈ Z we denote by (27.35.2)

X[n] := Ker([n] : X → X)

its kernel, i.e. X[n](T ) = { x ∈ X(T ) ; nx = 0 } for all S-schemes T . This is a closed subgroup scheme of X. Remark 27.183. For L ∈ X t (S) ⊆ Pic(X)/ Pic(S) by (27.29.10) we have the equality [n]∗ L = L ⊗n for all n ∈ Z, i.e. (27.35.3)

[n]tX = [n]X t

674

27 Abelian schemes

Proposition 27.184. Let X be an abelian scheme over S, let L be a line bundle on X, and let n ∈ Z. (1) One has an equality [n]∗ L = L ⊗(n

2

+n)/2

⊗ [−1]∗ L ⊗(n

2

−n)/2

in PicX/S (S), i.e., both sides are isomorphic line bundles on X up to multiplication by a line bundle that is obtained by pullback from a line bundle on S. (2) One has [−1]∗ L ⊗ L −1 ∈ X t (S). (3) One has in PicX/S (S) that 2

[n]∗ L = L ⊗n ⊗ N ,

for some N ∈ X t (S).

Proof. Let us show (1). The equality is clear for n = 0, 1. For general n we use Proposition 27.167 with x1 := [n + 1], x2 := [1] = idX and with x3 := [−1] and get OX ∼ = [n + 1]∗ L ⊗ [n + 2]∗ L −1 ⊗ [n]∗ L −1 ⊗ [0]∗ L −1 ⊗ [n + 1]∗ L ⊗ L ⊗ [−1]∗ L ⊗ [0]∗ L −1 . Hence we get in PicX/S (S) the equality [n + 2]∗ L ⊗ [n + 1]∗ L −1 = ([n + 1]∗ L ⊗ [n]∗ L −1 ) ⊗ L ⊗ [−1]∗ L . Arguing by increasing and by decreasing induction on n we obtain the claim. Next we show (2). For every S-scheme T and every x ∈ X(T ) we have in PicX/S (T ) φ[−1]∗ L (x) = t∗x [−1]∗ LT ⊗ [−1]∗ LT−1 = [−1]∗ (t∗−x LT ⊗ LT−1 ) = t∗−x LT−1 ⊗ LT = φL −1 (−x) = φL (x), where the third equality holds by Remark 27.183 since t∗−x LT ⊗ LT−1 = φL (−x) lies in X t (T ) (Proposition 27.163 (3)). This shows φ[−1]∗ L ⊗L −1 = 0 by Remark 27.169 (1). Now (3) follows because (1) shows that 2

[n]∗ L = L ⊗n ⊗ N ,

with N := (L −1 ⊗ [−1]∗ L )⊗(n

2

−n)/2

.

and N ∈ X (S) by (2). t

Definition and Remark 27.185. Let L be a line bundle on an abelian scheme X over S. Then L is called symmetric if L = [−1]∗ L in PicX/S (S). For symmetric line bundles L , Proposition 27.184 means that in PicX/S (S) one has 2

[n]∗ L = L ⊗n . It is easy to produce symmetric line bundles: If M is any line bundle on X, then M ⊗ [−1]∗ M is symmetric. If M is S-ample, then [−1]∗ M is S-ample since [−1] is an isomorphism. Therefore M ⊗ [−1]∗ M is a symmetric S-ample line bundle. Proposition 27.186. Let X be an abelian scheme over S. For an integer n ̸= 0, the multiplication [n] : X → X is an isogeny. In particular, X[n] is a finite locally free Sgroup scheme. Let g be the relative dimension of X over S, considered as locally constant function on S. Then deg([n]) = n2g .

675 Proof. By Proposition 27.176 we may assume that S = Spec k for a field k and it suffices to show that X[n] is finite and deg([n]) = n2g . As X is projective over k (Proposition 27.174), we may choose a symmetric ample line bundle L on X (Remark 27.185). 2 2 Restricting [n]∗ L ∼ = L ⊗n to X[n] yields that the ample line bundle L ⊗n |X[n] is trivial. Hence X[n] is quasi-affine over k and thus finite over k (Corollary 13.82). It remains to show that deg([n]) = n2g . From Proposition 23.84 we obtain 2

deg([n]) deg(L ) = deg([n]∗ L ) = deg(L n ) = n2g deg(L ), where the last equality holds by Remark 23.85. As L is ample, one has deg(L ) > 0 (Remark 23.82). This shows that deg([n]) = n2g . If n is invertible on S (e.g., if S is a scheme over Q), then a much more elementary argument shows a much stronger result: Proposition 27.187. Let n ∈ Z be invertible on S. Then [n] : X → X is finite ´etale. In particular X[n] is finite ´etale over S. Proof. By Corollary 18.77 we may assume that S = Spec k for a field k that we may assume to be algebraically closed by flat descent. To see that [n] is ´etale it suffices to show by Theorem 18.74 that [n] induces an isomorphism on tangent spaces. By homogeneity, it suffices to show that Lie([n]) : Lie(X) → Lie(X) is an isomorphism. But Lie([n]) is simply the multiplication by n (Remark 27.18 (3)) which is by hypothesis bijective. Since [n] is proper as a morphism between proper schemes, it is finite ´etale. Proposition 27.188. Let X be an abelian scheme over S of relative dimension g. (1) Let n be an integer that is invertible on S. Then there exists a finite ´etale surjective 2g morphism S ′ → S such that X[n]S ′ ∼ = (Z/nZ)S ′ . (2) Let S = Spec k be a field of characteristic p > 0. Then there exists an integer f (X) with 0 ≤ f (X) ≤ g such that for all m ≥ 1 and for every separably closed extension field K of k one has X[pm ](K) = (Z/pm Z)f (X) . The structure of the finite group scheme X[pm ] over an algebraically closed field of characteristic p can be quite complicated and is beyond the scope of this book. We will prove and use in the sequel only (1). For (2) we refer to [Mum1] §15. Proof. [of (1)] By Proposition 27.186 and Proposition 27.187 we know that X[n] is finite ´etale over S of degree n2g . By Proposition 20.69 there exists a finite ´etale surjective morphism S ′ → S such that X[n]S ′ ∼ = GS ′ for some finite abelian group G of order n2g with nG = 0. Moreover, for every divisor d of n, the d-torsion G[d] has the same properties, i.e., it is of order d2g and dG[d] = 0. By the classification of finite abelian groups this implies hat G ∼ = (Z/nZ)2g . In particular, if S = Spec R for a strictly henselian local ring R (e.g., if R is a separably closed field) and if n is prime to the characteristic of the residue field of R, 2g then X[n] ∼ = (Z/nZ)S . Definition and Remark 27.189. Let p be a prime number and let X be an abelian variety over a field k of characteristic p. The integer p-rk(X) := r in Proposition 27.188 (2) is called the p-rank of X.

676

27 Abelian schemes

One can show that for every algebraically closed field of characteristic p and every integer 0 ≤ f ≤ g there exist abelian varieties of dimension g over k whose p-rank is f . An abelian variety of dimension g over a field of characteristic p is called ordinary if its p-rank is g, i.e., maximal possible. Let S be a scheme such that for all s ∈ S one has char(κ(s)) = p. Let X be an abelian scheme over S. Then one can show (Exercise 27.13) that the p-rank function S −→ Z,

s 7→ p- rk(Xs )

is lower semicontinuous and constructible. In particular, if X is of relative dimension g, then the ordinary locus { s ∈ S ; p- rk(Xs ) = g } is open and constructible in S. An elliptic curve whose p-rank is 0 is called supersingular . There is also the notion of a supersingular abelian variety of higher dimension which is more complicated to define. All supersingular abelian varieties have p-rank 0 but the converse does not hold. Proposition 27.190. Let S be a scheme, let n ≥ 1 be an integer, and let f : X → Y be an isogeny of degree n of abelian schemes over S. Then there exists a unique homomorphism g : Y → X such that g ◦ f = [n]X . Moreover, g is an isogeny and f ◦ g = [n]Y . Proof. As Ker(f ) is a finite locally free group scheme of rank n, it is annihilated by n (Proposition 27.86). Hence [n]X factors as [n]X = g ◦ f for some homomorphism g. By Proposition 27.178 (1), g is unique. As [n]X is surjective (Proposition 27.186), g is surjective. As f is an isogeny, X and Y have the same relative dimension over S, hence g is an isogeny (Corollary 27.177). Finally, we have g ◦ [n]Y = [n]X ◦ g = g ◦ (f ◦ g), where the first equality holds since g is a homomorphism of group schemes. Hence f ◦ g = [n]Y by Proposition 27.178 (2). (27.36) Fundamental groups of abelian varieties. In this section we will compute the ´etale fundamental group of an abelian variety. If A is an abelian variety over the complex numbers C, then A(C) is isomorphic as a complex manifold to a complex torus, i.e., a quotient Cg /Λ, where Λ ⊂ Cg is a lattice (a free Z-module of rank 2g with Λ ⊗Z R = Cg ). Thus the universal cover of A(C) in the sense of algebraic topology is Cg → A(C) and the topological fundamental group is π1 (A(C), 0) = Λ (and in particular is abelian). We conclude that the multiplication by n morphisms A(C) → A(C), which are unramified covers with deck transformation group (Z/nZ)2g by Proposition 27.188, form a cofinal system of unramified connected covers. The latter result also makes sense algebraically, and we will show that the ´etale fundamental group of an abelian variety A over a field k is isomorphic to the limit lim A[n](k s ), where k s is a separable closure of A (Proposition 27.194). The crucial point in the proof is Theorem 27.192 which shows that for any finite ´etale surjective k-morphism f : B → A together with a choice of k-valued point e ∈ f −1 (0)(k), there exists a unique structure of abelian variety on B with neutral element e (and hence f is a homomorphism of abelian varieties). Lemma 27.191. Let f : X → S be a smooth proper morphism of connected schemes that has a section σ. Then all fibers of f are geometrically integral.

677 The proof will show that if f is any flat, proper morphism of finite presentation with geometrically reduced fibers between connected schemes and if f has a section, then f has geometrically connected fibers. f′

π

Proof. Let X −→ S ′ −→ S be the Stein factorization of f . As X is connected and f ′ is surjective, S ′ is connected. Then π is finite ´etale by Theorem 24.61. Moreover, τ := f ′ ◦ σ is a section of π and hence τ is an open and closed immersion (Remark 20.66 (5)). As S ′ is connected, τ is necessarily an isomorphism. Therefore f = f ′ has geometrically connected fibers. As f is smooth, all fibers are hence geometrically integral. Theorem 27.192. Let S be a scheme, let X → S be an abelian scheme, let Y → S be an S-scheme with geometrically connected fibers, and let f : Y → X be a finite ´etale surjective S-morphism, and let eY ∈ Y (S) be a section that is mapped by f to the zero section eX of X. Then there exists on Y a unique structure of an abelian scheme over S such that eY is the zero section and such that f is an isogeny. Proof. We use Proposition 27.109. Hence we have to construct a morphism mY : Y ×S Y → Y of S-morphism for which eY is a left and a right unit. f As Y → S is the composition Y −→ X → S, the structure morphism Y → S is smooth and proper. Hence all fibers are geometrically integral. Moreover, Y → S is surjective because it has a section. If the desired structure of an abelian scheme exists on Y , it is necessarily unique by Corollary 27.104. We may therefore work locally on S and hence can assume that S = Spec R is affine. By noetherian approximation we may assume that R is noetherian. Then Spec R is locally connected and hence equals the direct sum of its connected components. Therefore we can assume that S is connected. Then X I and Y I are connected as well for every finite set I, as the structure morphisms are proper surjective with geometrically connected fibers (Lemma 24.52). Let ΓX ⊆ X ×S X ×S X be the graph of the group law on X and let Γ′Y ⊆ Y ×S Y ×S Y be the inverse image of ΓX under f × f × f . As S is connected, there exists a connected component ΓY of Γ′Y containing the image of (eY , eY , eY ). We claim that ΓY is the graph of a morphism mY : Y ×S Y → Y . For I ⊆ {1, 2, 3} denote by pI : ΓX → X I and by qI : ΓY → Y I the projections. To prove the claim we −1 have to show that q12 is an isomorphism. Then we can set mY := q3 ◦ q12 . The morphism q12 has sections s1 over eY (S) × Y and s2 over Y × eY (S) given on points by s1 (eY , y) = (eY , y, y) and s2 (y, eY ) = (y, eY , y). So once we have proved the claim, we have mY (eY , y) = y = mY (y, eY ) and therefore we can apply Proposition 27.109. Moreover, f respects these group laws as by construction f × f × f maps the graph of mY to the graph of mX . To show that q12 is an isomorphism, consider the commutative diagram / ΓX

ΓY q12

 Y ×S Y

f ×f



p12

/ X ×S X.

678

27 Abelian schemes

The horizontal maps are finite ´etale and p12 is an isomorphism. Hence q12 is finite ´etale (Remark 20.66 (3)). As Y ×S Y is connected and finite ´etale morphisms have open and closed images, q12 is surjective. It suffices to show that q12 has degree 1. As the degree is locally constant and as Y ×S Y is connected, it suffices to show that the restriction −1 r : Z := q12 (Y × eY (S)) → Y ×S eY (S) of q12 is an isomorphism. The projection q2 : ΓY → Y is the composition of q12 and of the second projection Y ×S Y , hence it is proper and smooth. It has a section given on points by y 7→ (eY , y, y). Hence we can apply Lemma 27.191 to see that q2 has geometrically integral fibers. In particular Z = q2−1 (eY (S)) is connected (Lemma 24.52). But the ´etale covering r has a section given on points by (y, eY ) 7→ (y, eY , y) which is an open and closed immersion (Remark 20.66 (5)) and hence an isomorphism. Therefore r is an isomorphism. Remark and Definition 27.193. Let k be a field, let k¯ be an algebraic closure of k, ¯ Let X be an abelian variety of dimension g and let k s be the separable closure of k in k. over k. Let n ≥ 1 be an integer. Then G = Gal(k s /k) acts on X[n](k s ) continuously by functoriality. If n is invertible in k, we have X[n](k s ) ∼ = (Z/nZ)2g by Proposition 27.188. If char(k) = p > 0, then X[pm ](k s ) ∼ = (Z/pm Z)f , where 0 ≤ f ≤ g is the p-rank of X (Definition 27.189). By forming the limit over all n (with respect to the divisibility order), we obtain a topological module T (X) := lim X[n](k s ) n

ˆ := limn Z/nZ with a continuous G-action. It is called the total Tate over the ring Z module of X. By the Chinese remainder theorem one has Y T (X) = Tℓ (X), ℓ

where ℓ runs over all prime numbers. Here Tℓ (X) = lim X[ℓm ](k s ) m

is the ℓ-adic Tate module. It is a Zℓ -module endowed with a continuous G-action, where Zℓ = limn Z/ℓn Z denotes the ring of ℓ-adic integers. If ℓ = ̸ char(k), then Tℓ (X) is a free Zℓ -module of rank 2g. If char(k) = p > 0, then Tp (X) is a free Zp -module of rank f . Proposition 27.194. Let k be a separably closed field and let X be an abelian variety over k. Then there is an isomorphism of abelian pro-finite groups π1 (X, 0) ∼ = T (X). Proof. By Proposition 20.105 we may assume that k is algebraically closed. If f : Y → X is an ´etale cover of degree n with Y connected, we can choose 0Y ∈ Y (k) that is mapped to the zero section of X. By Theorem 27.192 we can equip Y with the structure of an abelian variety such that f becomes an isogeny of degree n. By Proposition 27.190 there exists an isogeny g : X → Y such that f ◦ g = [n]X . By Exercise 27.12 we have X[n] = X[n]0 × X[n]red with X[n]red finite constant and X[n](k) = X[n]red (k). Now g(X[n]0 ) ⊆ Ker(f ) is connected and Ker(f ) is constant. Hence X[n]0 ⊆ Ker(g). Therefore we find a factorization

679 f

[n]0 : X/X[n]0 −→ Y −→ X, where the composition [n]0 is induced by the multiplication by n. Its kernel is the finite ´etale group scheme X[n]red , hence [n]0 is finite ´etale (Corollary 27.63). This shows that the multiplication maps [n]0 : X/X[n]0 → X are cofinal in the category of all connected ´etale covers and hence π1 (X, 0) ∼ = lim X[n]red (k) = lim X[n](k) = T (X). n

n

By Theorem 20.115 and Proposition 20.117 we obtain the following corollaries. Corollary 27.195. Let k be a field and let X be an abelian variety over k. Then via functoriality one obtains a split exact sequence of profinite groups 0 −→ T (X) −→ π1 (X, 0) −→ Gal(k s /k) −→ 0. The splitting is given by the zero section (Remark 20.116) and one obtains an isomorphism of profinite groups π1 (X, 0) ∼ = T (X) ⋊ Gal(k s /k), where the action of Gal(k s /k) on T (X) is the action defined in Definition 27.193. From Proposition 20.117 we now immediately obtain the following corollary. Corollary 27.196. Let S be a connected scheme, let s¯ be a geometric point of S and let x ¯ be its image in X under the zero section of X. Then there is an exact sequence of profinite groups T (Xs¯) −→ π1 (X, x ¯) −→ π1 (S, s¯) −→ 1.

(27.37) The dual abelian scheme for projective abelian schemes. Lemma 27.197. Let k be a field and let X be an abelian scheme over k. Let L ∈ X t (k). Then L is non-trivial if and only if H i (X, L ) = 0 for all i ≥ 0. Proof. Since H 0 (X, OX ) = k, the condition is clearly sufficient. Let L be non-trivial and let us show H i (X, L ) = 0 by induction on i. As L ∈ X t (k) we have [−1]∗ L ∼ = L −1 (27.35.3). Hence if H 0 (X, L ) ̸= 0, then also H 0 (X, L −1 ) ̸= 0. This implies that L is trivial by Lemma 24.65. Now let i ≥ 1 and assume by induction that H 0 (X, L ) = H 1 (X, L ) = · · · = H i−1 (X, L ) = 0. Since idX can be factorized as (id,0)

m

X −−−−−→ X × X −→ X we see that H i (X, L ) is embedded into H i (X × X, m∗ L ). As L ∈ X t (k), we have unneth formula (Corollary 22.110) shows m∗ L = p∗1 L ⊗ p∗2 L and therefore the K¨ H i (X × X, m∗ L ) =

i M j=0

H j (X, L ) ⊗ H i−j (X, L ) = 0.

680

27 Abelian schemes

Theorem and Definition 27.198. Let S be a scheme, let X be an abelian scheme over S and let L be an S-ample line bundle on X. Then the following assertions hold. (1) The homomorphism φL : X → X t is finite locally free and surjective. In particular, K(L ) is a finite locally free S-group scheme. (2) The dual abelian space X t is representable by an abelian scheme with dim(X t /S) = dim(X/S) and we have X t = Pic0X/S . We call X t the dual abelian scheme. We have already seen that on an abelian scheme there exists an ample line bundle if S = Spec k for a field k (Proposition 27.174). Proof. Let us first assume that S = Spec k for a field k. In this case we know that X t is a scheme by Theorem 27.47. (I). Suppose in addition that k is algebraically closed. We claim that the group homomorphism φL (k) : X(k) → X t (k) is surjective. Assume there exists N ∈ X t (k) that is not in the image of φL (k). Consider the line bundle M := Λ(L ) ⊗ p∗1 N −1 on X ×k X. For x ∈ X(k) one has M |X×{x} = t∗x L ⊗ L −1 ⊗ N

−1

,

M |{x}×X = t∗x L ⊗ L −1 .

In particular, M |X×{x} ∈ X t (k) is non-trivial for all x ∈ X(k). By Lemma 27.197 we find H j (X, M |X×{x} ) = 0

for all x ∈ X(k) and all j ≥ 0.

Therefore Rj p2,∗ M = 0 for all j ≥ 0 (Proposition 23.142). The Leray spectral sequence H i (X, Rj p2,∗ M ) ⇒ H i+j (X × X, M ) (Corollary 21.45) shows that H j (X × X, M ) = 0 for all j ≥ 0. If x ∈ X(k), then Lemma 27.197 shows that (*)

H j (X, M |{x}×X ) = 0

for all j

⇔

x∈ / K(L )(k).

Therefore Rj p1,∗ M has support in K(L ), which is a finite k-scheme because L is ample (Corollary 27.172). Therefore H i (X, Rj p1,∗ M ) = 0 for i > 0. The Leray spectral sequence for p1 then shows for all j ≥ 0 that H 0 (X, Rj p1,∗ M ) = H j (X × X, M ) = 0 which implies that Rj p1,∗ M = 0 for all j because Rj p1,∗ M is coherent and has support on the affine scheme K(L ). Hence H j (X, M |{x}×X ) = 0 for all j and for all x ∈ X(k). This is a contradiction to (*), since M |{0}×X ∼ = OX . (II). Now let k be an arbitrary field. We show that X t is smooth over k and that φL is finite locally free and surjective. Indeed, by fpqc descent we may assume that k is algebraically closed. We know by Corollary 27.172 that the kernel K(L ) of φL is finite. Therefore dim X = dim X t because (I) shows that φL is surjective. Moreover, we have (27.37.1)

H 1 (X, OX ) = Lie(PicX/S ) = Lie(X t ).

681 Here the first identity holds by Proposition 27.122 and the second identity because X t is open in PicX/S (Lemma 27.160). We deduce (**)

dim X t ≤ dim Lie(X t ) = dim H 1 (X, OX ) ≤ dim(X) = dim X t ,

where the first inequality is (27.4.7) and the second inequality holds by Corollary 27.80. Hence we have equality everywhere in (**). This shows that X t is smooth (Corollary 27.21) and that dim H 1 (X, OX ) = dim(X).

(27.37.2)

As X t is smooth and φL is surjective, φL is faithfully flat (Proposition 27.54). As its kernel is finite, φL is finite and hence finite locally free. From now on let S be an arbitrary scheme. (III). The surjectivity of φL on fibers implies that φL is surjective. As X is flat over S and φL is flat on fibers, φL is flat and X t is flat over S by the fiber criterion for flatness (Corollary 14.27, [Sta] 05X1 for algebraic spaces). As K(L ) is finite (Lemma 27.172), φL is finite (Corollary 27.63) and hence finite locally free (Proposition 12.19). Therefore K(L ) is finite locally free over S. In particular φL is fppf-surjective and therefore identifies X t with the fppf-quotient X/K(L ). Hence X t is a scheme over S (Theorem 27.68) and φL makes X into a K(L )-torsor for the fppf-topology over X t . Moreover, since there exists a surjective map X → X t , the scheme X t has connected fibers over S and is proper over S. Hence X t is an abelian scheme with X t = Pic0X/S by Lemma 27.160. Remark 27.199. Let X and L be as in Theorem 27.198. Then φL : X → X t is an isogeny with kernel K(L ). In particular, φL makes X into a K(L )-torsor for the fppf-topology over X t . Hence Section (14.21) shows that φ∗L induces an equivalence of categories ∼

QCoh(X t ) −→ (K(L )-equivariant quasi-coherent OX -modules). Via this equivalence, finite locally free OX t -modules (resp. line bundles on X t ) correspond to finite locally locally free OX -modules (resp. line bundles on X) with a K(L )-equivariant structure. (27.38) Cohomology of the structure sheaf and of the sheaves of differentials. If X is an abelian variety over a field, we obtain as a direct corollary the cohomology of the structure sheaf on X. Corollary 27.200. Let k be a field and X be an abelian variety of dimension g over k. Then dim H 1 (X, OX ) = g and the canonical homomorphism (21.29.5) of graded kbialgebras V• 1 H (X, OX ) −→ H • (X, OX )
is an isomorphism. In particular, dimk H p (X, OX ) = gp for all p ≥ 0. Proof. We have seen that dim H 1 (X, OX ) = dim X in (27.37.2). Then we can conclude by Corollary 27.80.

682

27 Abelian schemes

Corollary 27.201. Let X be an abelian variety of dimension g over a field k. For all p and q there are functorial isomorphisms H q (X, ΩpX/k ) ∼ =

Vp

Lie(X)∨ ⊗

in particular hp,q (X) := dim H q (X, ΩpX/k ) =

g p


g
q

Vq

Lie(X t ),

.

In Corollary 27.205 below we will generalize this result to abelian schemes over arbitrary base schemes. Proof. Let e : Spec k → X be the zero section and let Ce = e∗ Ω1X/S be the space of invariant differential forms on X, considered as vector space over k. Then one has H q (X, ΩpX/k ) = H q (X, OX ) ⊗

Vp

Ce =

Vp

Lie(X)∨ ⊗

Vq

Lie(X t ),

where the first identity holds by (27.4.8). The second identity holds because the conormal sheaf is the k-linear 27.17) and since we have Vq 1dual of the Lie algebra of X (Definition H (X, OX ) by Corollary 27.204 and H 1 (X, OX ) = Lie(X t ) (27.37.1). H q (X, OX ) = Corollary 27.202. Let X be an abelian variety over a field. Then for n ∈ Z the multiplication [n] on X induces the multiplication by np+q on H q (X, ΩpX ). Proof. The dual of [n] is again the multiplication by n on X t by (27.35.3). They induce multiplication by n on Lie(X) and Lie(X t ) by Remark 27.18 (3). Hence we conclude by Corollary 27.201. We will now generalize Corollary 27.200 and Corollary 27.201 to arbitrary base schemes. Theorem 27.203. Let S be a scheme and let f : X → S be an abelian
scheme of relative dimension g. Then Rp f∗ OX is a locally free OS -module of rank gp whose formation is compatible with arbitrary base change T → S. We will give a proof of the theorem if S is reduced or if 2 is invertible in OS . For a proof in the general case we refer to [BBM] O 2.5.2. commutes Proof. Once we have shown that Rp f∗ OX is locally free and that its formation
with base change, it follows from Corollary 27.200 that its rank is gp . (I). If S is a reduced scheme, then Grauert’s theorem (Theorem 23.140 (3)) implies the result by Corollary 27.200. (II). For non-reduced base schemes S, we assume that 2 is invertible on S. We will reduce below by a standard argument to the case that S = Spec R where R is a local Artinian ring. Hence let us study this case first. Let k be the residue field of R and let m be its maximal ideal. By Theorem 23.133, E := RΓ(X, OX ) is a perfect complex of R-modules whose p L p formation commutes with base change. Hence
we know that H (E ⊗R k) = H (X ⊗R g k, OX⊗R k ) and hence dimk H p (E ⊗L k) = by Corollary 27.200. By Proposition 22.54 R p g we know that we can assume that E is a complex of finite free modules with E p = R(p) and where the differentials dp : E p → E p+1 are given by matrices Dp with entries in m. We have
to show that E is acyclic, i.e. Dp = 0 for all p. Then H p (X, OX ) = E p is free of rank gp and its formation commutes with base change to k and therefore with arbitrary base change by Theorem 23.140.

683 To show that Dp = 0 let n be an integer such that n and n − 1 are invertible in R. Here our additional hypothesis char(k) ̸= 2 ensures that such an integer exists, e.g., n = 2. Multiplication by n on X induces by functoriality an endomorphism of E in D(R). By Lemma 21.134 (3) we can represent this endomorphism by an endomorphism u : E → E of complexes. Moreover we know that up ⊗ 1 on E p ⊗R k = H p (X ⊗R k, OX⊗R k ) is given by multiplication by np (Corollary 27.202). Hence the matrix Up of up is of the form np I +Xp , where I is the identity matrix and where Xp has entries in m. As up+1 ◦ dp = dp ◦ up we find (*)

Dp Xp − Xp+1 Dp = (np+1 − np )Dp = np (n − 1)Dp .

If we had Dp ̸= 0, we would find a minimal r ≥ 1 such that all entries of Dp are in mr , / mr+1 . As n and n − 1 are invertible, the but for some entry dij we would have dij ∈ (i, j)-th entry of the right side of (*) would again be in mr \ mr+1 . But the (i, j)-th entry of the left side is a sum of elements of the form dx, where d is an entry of Dp and hence d ∈ mr and where x is an entry of Xp or Xp+1 and hence x ∈ m. This is a contradiction. (III). The reduction to the local Artinian case is a special case of Remark 14.56. It works without our additional hypothesis that 2 is invertible. Let us spell out the details. We first may assume that S = Spec R is affine. Next we write R as a filtered colimit of noetherian rings Rλ . For sufficiently large λ we find an abelian scheme fλ : Xλ → Spec Rλ such that its base change to Spec R is X (Remark 27.91). If we have shown that Rp fλ∗ OXλ is locally free and its formation commutes with base change, we get the same assertion for X → S. Hence we can assume that R is noetherian. As f is proper, we know that Rp f∗ OX is a coherent OS -module (Theorem 23.17). To show that it is locally free, we may therefore pass to stalks of OS (Proposition 7.27). Moreover, its formation commutes with arbitrary base change if it commutes with base change to residue fields (Theorem 23.140). Therefore we can in addition assume that R is a local ring. Let k be its residue field. It remains to show that H p (X, OX ) is a free R-module and that its formation commutes with base change to k. As the formation of cohomology commutes with flat base change (Corollary 22.91) and because we can check (local) freeness after a faithfully flat base change (Proposition 14.48), we may even assume that R is a complete local noetherian ring by passing to the completion of R. Let m be the maximal ideal of R, set Rn := R/mn and Xn := X ⊗R Rn . Then Xn is an abelian scheme over the local Artinian ring R/mn . Hence we know the theorem already for Xn , i.e., H p (Xn , OXn ) is a free module and H p (Xn , OXn ) ⊗Rn Rn−1 = H p (Xn−1 , OXn−1 ) for all n, in other words, the family (H p (Xn , OXn ))n defines a finite free module over the formal scheme (Spec Rn )n (Definition 24.85). By the Theorem of Formal Functions, Theorem 24.37, we have lim H p (Xn , OXn ) = lim(H p (X, OX ) ⊗R Rn ) n

n

Hence the family (H p (Xn , OXn ))n corresponds to H p (X, OX ) under the equivalence of finite free R-modules and finite free modules over the formal scheme (Spec Rn )n (Proposition 24.88). This shows that H p (X, OX ) is finite free and that H p (X, OX )⊗R Rn = H p (Xn , OXn ). In particular, the formation of H p (X, OX ) commutes with base change to k = R1 .

684

27 Abelian schemes

Corollary 27.204. Let f : X → S be an abelian scheme over a scheme S. The canonical homomorphism (21.29.4) of graded OS -algebras M V• 1 v: R f∗ O X ∼ R p f∗ O X = p≥0

is an isomorphism. Proof. By Theorem 27.203 we know that v is a homomorphism of locally free modules of the same rank. Hence it suffices to show that v is an isomorphism on fibers, i.e., after base change Spec κ(s) → S for s ∈ S (Corollary 8.12). As the formation of Rp f∗ OX commutes with base change, we conclude by Corollary 27.200. Now one shows as in Corollaries 27.201 and 27.202 the following result. Corollary 27.205. Let f : X → S be an abelian scheme over a scheme S of relative dimension g. Let Ce = e∗ Ω1X/S be the conormal sheaf of e (Remark 17.22). Then for all p, q ≥ 0 there are functorial isomorphisms Rp f∗ ΩqX/S = Rp f∗ OX ⊗

Vq

Ce =

Vp

Lie(X t ) ⊗

Vq

Lie(X)∨



of locally free OS -modules of rank gp gq and the formation of Rp f∗ ΩqX/S commutes with base change. Multiplication by an integer n on X induces multiplication by np+q on Rp f∗ ΩqX/S . (27.39) The dual abelian space is a scheme. Let S be a scheme and let f : X → S be an abelian scheme over S. Recall that PicX/S is represented by a separated algebraic group space locally of finite presentation (Theorem 27.117, Proposition 27.118). In PicX/S we have two subgroup functors Pic0X/S and X t . Moreover, we have seen that the inclusion X t → PicX/S is representable by an open and closed immersion (Lemma 27.160). Lemma 27.206. Let k be an algebraically closed field and let X be an abelian variety over k. Then PicX/k is a smooth group scheme over k. Proof. We know that PicX/k is a group scheme locally of finite type (Theorem 27.119) whose identity component X t is smooth (Theorem 27.198). Hence it is smooth (Proposition 27.20). In [Mum1], Sections 12, 13, the existence of the dual abelian variety of an abelian variety X over an algebraically closed field k is proved directly, i.e., without using general representability results for the relative Picard functor. The strategy is to start with an ample line bundle L and to prove that the scheme K(L ) is a finite group scheme. It is relatively simple to construct the quotient X ∨ := X/K(L ) by this finite group scheme, generalizing Proposition 12.27 where the case of a constant finite group scheme is dealt with. Then one has to show that X ∨ , defined in this way, satisfies the universal property of the dual abelian variety, i.e., of Pic0X/k . Proposition 27.207. Let X be an abelian scheme over a scheme S. Then the two subgroup functors Pic0X/S and X t are equal.

685 Proof. By the definition of Pic0X/S (Definition 27.123) and by the description of X t in Corollary 27.162 it suffices to show that Pic0X/S (¯ s) = X t (¯ s) for all geometric points s¯ of S. As the formation of both functors is compatible with base change, we may assume that S = Spec k for an algebraically closed field. Then we can conclude by Theorem 27.198. Proposition 27.208. Let X be an abelian scheme over a scheme S. Then the morphism φ : PicX/S → CorrS (X, X) (Definition 27.159) is smooth. In particular, X t is smooth over S. Proof. Once we know that φ is smooth, it follows that X t is smooth over S because X t is the kernel of φ. To show that φ is smooth, we may assume that S is affine and then by noetherian approximation that S is noetherian. By Remark 27.42 it suffices to show the infinitesimal lifting criterion for nil-immersions i : T0 = Spec R0 → T = Spec R, where R is a locally Artinian ring and where i is defined by an ideal I ⊆ R with Im = 0. Let k be the residue field of R. We may replace X by its base change to T and hence assume that S = T , and we set X0 := X ⊗R R0 and Xk := X ⊗R k. Since Im = 0, we can view I as a finite-dimensional k-vector space. Then H n (X0 , IOX0 ) = H n (Xk , IOXk ) (Corollary 22.6) and we obtain from Lemma 27.116 a commutative diagram / PicX/S (T0 )

PicX/S (T )

φ0

φ

 CorrS (X, X)(T ) _  PicX×S X/S (T )

/ H 2 (Xk , IOX ) k

ρ

 / CorrS (X, X)(T0 ) _  / PicX× X/S (T ) S

φ ¯

 / H 2 (Xk ×k Xk , IOX ×X ), k k

where the upper and the lower horizontal lines are exact and where φ¯ is the map m∗ −p∗1 −p∗2 on cohomology (see Remark 27.158). Given L0 ∈ PicX/S (T0 ) and λ ∈ CorrS (X, X)(T ) with φ0 (L0 ) = ρ(λ) we have to show that there exists L ∈ PicX/S (T ) that maps to L0 and to λ. A diagram chase shows that it suffices to show that ρ and φ¯ are injective. The map ρ is injective since CorrS (X, X) is unramified over S (Proposition 27.157). To see that φ¯ is injective, we note that H 2 (Xk , IOXk ) = H 2 (Xk , OXk ) ⊗k I and similarly for H 2 (Xk ×k Xk , IOXk ×Xk ). Hence it suffices to show the following lemma. Lemma 27.209. Let X be an abelian variety over a field k and let αp : H p (X, OX ) → H p (X ×k X, OX×k X ) be the k-linear map induced by m∗ − p∗1 − p∗2 . Then αp is injective for p ≥ 2. Proof. Let µ : H • (X, OX ) → H • (X, OX ) ⊗k H • (X, OX ) be the comultiplication of the graded bialgebra H • (X, OX ) induced by m∗ (Section (27.12)). Then the kernel of αp p are the primitive elements in H p (X, OX ), i.e., those elements V• 1 x ∈ H (X, OX ) such that m∗ (x) = x ⊗ 1 + 1 ⊗ x. But we have H • (X, OX ) = H (X, OX ) (Corollary 27.200) and the primitive elements of an exterior algebra are homogeneous elements of degree 1. Therefore αp is injective for p ≥ 2. Remark 27.210. Let X be an abelian scheme over a scheme S. We collect what we have seen about PicX/S and X t by now.

686

27 Abelian schemes

(1) The functor PicX/S is an algebraic space locally of finite presentation over S (Theorem 27.117). (2) The Picard space PicX/S is separated (Proposition 27.118). (3) If S = Spec k for a field k, then PicX/S is representable by a group scheme locally of finite type (Theorem 27.47). (4) If S = Spec k for a field k, then PicX/S is representable by a smooth group scheme. (5) We have X t = Pic0X/S (Proposition 27.207) and the inclusion X t → PicX/S is an open and closed immersion (Lemma 27.160). (6) The algebraic space X t is smooth (Proposition 27.208) and proper (Lemma 27.126) over S and has geometrically connected fibers by (5). (7) If X → S is projective, then X t is an abelian scheme (Theorem 27.198). For the representability results (1) and (3) we only gave references. For all other properties we gave proofs using only some standard properties of morphisms of algebraic spaces. We can now invoke the following result. Theorem 27.211. Let S be a scheme and let Y be a group algebraic space such that Y → S is proper smooth with geometrically connected fibers. Then Y is an abelian scheme over S. By Remark 27.210 we obtain the following corollary. Corollary 27.212. Let S be a scheme and let X be an abelian scheme. Then X t is an abelian scheme over S. As we have already indicated above, the abelian scheme X t is called the dual abelian scheme of X. We will only give a sketch of the proof of Theorem 27.211 and refer to [FaCh] O I, 1.9 or to [GrK] for details. Proof. [of Theorem 27.211, Sketch] (I). As Y is by definition an fppf-sheaf the question whether Y is a scheme is Zariski local (Theorem 8.9), in particular we may assume that S is affine. Next we may assume by noetherian approximation that S is of finite type over Z. Moreover if X → Y is a surjective integral morphism of algebraic spaces and X is a scheme, then Y is a scheme ([Sta] 07VT). In particular, an algebraic space Y is a scheme if and only if its underlying reduced algebraic space is a scheme. Therefore we may assume that S is reduced. (II). By noetherian induction one can now assume the base change of Y to any proper closed subscheme of S is a scheme. One uses gluing along closed subschemes (see [Fer] O and [TeTy] O X ) to reduce to the case that S is integral. Here the general idea is the following. Let Z → X be a closed immersion and let Z → T be a finite morphism of algebraic spaces. Then the pushout T ⨿Z X exists in the category of algebraic spaces ([TeTy] O X 6.2.1) and it is a scheme if Z, T , and X are schemes. If T = Spec R, X = Spec A and Z = Spec A/I are affine, then T ⨿Z X is affine and is the spectrum of the fiber product of rings R ×A/I A. Hence gluing S from its irreducible components, viewed as closed integral subschemes, one can assume that S is integral. Next let S˜ → S be the normalization. Then the projection YS˜ → Y is surjective and integral. Hence if YS˜ is a scheme, then Y will be a scheme, again by [Sta] 07VT. Therefore we may assume that S is noetherian and normal.

687 (III). Finally one shows that if S is noetherian and normal, then Y carries an ample line bundle and hence is a projective abelian scheme. If Y is an abelian scheme, this is proved below in Theorem 27.291 but using ampleness criteria whose proof assumed that one knows already that Y is a scheme. One has to check that the essential arguments in the proof of Theorem 27.291 also work without assuming that Y is a scheme. (27.40) Dual homomorphisms. Let S be a scheme, and let X and Y be abelian schemes over S. We denote by Hom(X, Y ) the set of homomorphisms X → Y of abelian schemes over S. Let f ∈ Hom(X, Y ). Then the pullback f ∗ : PicY /S −→ PicX/S induces the dual homomorphism (Definition 27.165) f t : X t −→ Y t . We obtain a map Hom(X, Y ) −→ Hom(Y t , X t ),

f 7→ f t .

By Remark 27.166 this is a homomorphism of abelian groups. Proposition 27.213. Let f : X → Y be an isogeny of abelian schemes over S. (1) Then the dual homomorphism f t : Y t → X t is also an isogeny. (2) Let M be a line bundle on Y . Then the class of M is in Y t (S) if and only if the class of f ∗ M is in X t (S). (3) The kernel of f t is isomorphic to the Cartier dual of Ker(f ). In particular, one has deg(f ) = deg(f t ). Proof. The formation of f t commutes with base change. To show (1) we may by Proposition 27.176 assume that S is the spectrum of a field. As f is an isogeny, we have dim(X t ) = dim(X) = dim(Y ) = dim(Y t ), hence it suffices to show that f t is surjective. As the base is a field, we can choose an ample line bundle L on Y . As f is finite and in particular quasi-affine, f ∗ L is also ample. Hence φf ∗ L is an isogeny and therefore surjective. Hence f t is surjective by Lemma 27.165. Let us show (2). The condition is clearly necessary (Definition 27.165). For the converse we have to show that if φf ∗ L = 0, then φL = 0. By (27.29.9) we have f t ◦ φL ◦ f = 0. As the locus on S, where a morphism of abelian schemes over S is constant, is open and closed in S (Proposition 27.96), we may assume that S = Spec k for a field k. As f is an epimorphism, we know that f t ◦ φL = 0. As f t has finite kernel by (1), φL factors through a finite k-scheme which is geometrically integral because Y is geometrically integral. Hence φL = 0. Now we show (3). Let T be an S-scheme. We have to show that Ker(f t )(T ) ∼ = Ker(f )D (T ) functorially for all S-schemes T . Replacing f : X → Y by its base change to T we may assume that T = S which simplifies the notation. We will construct functorial isomorphisms (a)

Ker(f t )(S) = Ker(f ∗ : PicY /S (S) → PicX/S (S)) (b) = Ker(f ∗ : Pic(Y ) → Pic(X)) (c) = HomS−Gr (K(L ), Gm,S ). We can identify PicY /S (S) with the subgroup of those L ∈ Pic(Y ) such that e∗ L ∼ = OS , where e is the zero section of Y .

688

27 Abelian schemes

Let L ∈ PicY /S (S) be such that f ∗ L is trivial. Then L ∈ Y t (S) by (2) which shows equality (a). Equality (b) follows from the above interpretation of Pic Y /S (S). To see (c) we use that f makes X into a K(L )-torsor over Y (Proposition 27.62) and that therefore to give a line bundle L on Y is the same as to give f ∗ L together with its canonical K(L )-equivariant structure (Section (14.21)). Hence line bundles in Ker(f ∗ : Pic(Y ) → Pic(X)) correspond to K(L )-equivariant structures on OX , i.e., to actions of K(L ) on A1X lifting the action on X by translation. The trivial line bundle OY corresponds to the trivial lifting (x′ ,x,t)7→(x′ +x,t)

K(L ) ×S A1X = K(L ) ×S X ×S A1S −−−−−−−−−−−−→ X ×S A1S = A1X . As f∗ OX = OS holds after any base change, it is easy to check that any other lifting differs × by a homomorphism K(L ) → H, where H is the group functor T 7→ Γ(XT , OX )= T × Γ(T, OT ), i.e. H = Gm,S . This yields (c). Corollary 27.214. Let X be an abelian scheme over a scheme S and let n = ̸ 0 be an integer. Then X[n] and X t [n] are Cartier dual to each other. Proof. This follows from Proposition 27.213 since [n]tX = [n]X t (27.35.3). Corollary 27.215. Let k be a field and let f : X → Y be an isogeny of abelian varieties over k. Then for every line bundle L on X there exist only finitely many isomorphism classes of line bundles M on Y such that f ∗ M ∼ = L. Proof. As f ∗ : Pic(Y ) → Pic(X) is a group homomorphism, it suffices to show that Ker(f ∗ )(k) is finite. By Proposition 27.213 we know that Ker(f ∗ )(k) ⊆ Y t (k) is finite. (27.41) The Poincar´ e bundle. Let S be a scheme and let X be an abelian scheme over S. Let T be an S-scheme and let F be an OX×S T -module. Then we denote by F |0×T the pullback of F under the zero section 0 of the abelian scheme X ×S T → T . By definition we have PicX/S (T ) = Pic(X ×S T )/ Pic(T ) ∼ = { L ∈ Pic(X ×S T ) ; L |0×T ∼ = OT }. ∼

We call an isomorphism ι : L |0×T → OT a rigidification of L . Given two line bundles ∼ L and L ′ with rigidifications ι and ι′ , respectively, any isomorphism α : L → L ′ can be changed by precomposition with an automorphism of L such that the modified isomorphism α satisfies ι′ ◦ α|0×T = ι because Aut(L ) = Γ(X ×S T, OX×S T )× = Γ(T, OT )× . Therefore PicX/S (T ) can also be identified with the group of isomorphism classes of rigidified line bundles (L , ι) on X ×S T . By Corollary 27.162 one has for every S-scheme T (27.41.1)

X t (T ) = { L ∈ PicX/S (T ) ; m∗ L = p∗1 L ⊗ p∗2 L in PicX×S X (T )}

and L ∈ PicX/S (T ) is in X t (T ) if and only if for every geometric point t¯: Spec κ → T one has (idX ×t¯)∗ L ∈ X t (t¯).

689 Definition 27.216. The rigidified line bundle (P, ιP ) on X ×S X t corresponding to idX t ∈ X t (X t ) via (27.41.1) is called the Poincar´e bundle of X. ∼

Hence P is a line bundle on X ×S X t together with an isomorphism ιP : P |0×X t → OX t such that for every S-scheme T and for every (L , ιL ) ∈ X t (T ) there exists a unique morphism f : T → X t of S-schemes such that (idX ×f )∗ (P, ιP ) = (L , ιL ). Remark 27.217. If L is any line bundle on X and φL : X → X t is the corresponding homomorphism, then (idX ×φL )∗ (P) = Λ(L ) in PicX/S (X), where Λ(L ) is the Mumford bundle associated to L . Indeed, by Proposition 27.161 (1) the composition φL

X −→ X t → PicX/S corresponds to the class of Λ(L ). Remark 27.218. Let X → S be an abelian scheme and let P be its Poincar´e bundle. (1) The formation of the Poincar´e bundle is compatible with base change: If T → S is a morphism of schemes, then the Poincar´e bundle of XT → T is the pullback of P under XT ×T (X t )T = X ×S X t ×S T → X ×S X t . (2) The condition that P ∈ PicX/S (X t ) is an X t -valued point of X t means that m∗ P ∼ = p∗1 P ⊗ p∗2 P, where m, p1 , p2 : (X ×S X t ) ×X t (X ×S X t ) −→ X ×S X t are the multiplication and the projections of the abelian scheme X ×S X t → X t . Identifying the left hand side with X ×S X ×S X t we obtain µ∗ P ∼ = p∗13 P ⊗ p∗23 P, where µ : X ×S X ×S X t → X ×S X t is given by (x, y, ξ) 7→ (x + y, ξ). Remark 27.219. Let f : X → Y be a homomorphism of abelian schemes over a scheme. Its dual f t : Y t → X t is an element of X t (Y t ) and hence corresponds to the element L = (idX ×f t )∗ PX ∈ PicX/S (Y t ). As f t is defined by pullback of line bundles via f we find that there is an isomorphism (27.41.2)

(idX ×f t )∗ PX ∼ = (f × idY t )∗ PY

of rigidified line bundles on X ×S Y t with a rigidification along {0} × Y t . (27.42) Biduality. In this section let S be a scheme and let X be an abelian S-scheme. For abelian schemes X and Y over S we denote by Hom(X, Y ) the abelian group of homomorphisms X → Y of group schemes over S. Recall (Remark 27.152 and Proposition 27.163) that one has identifications (27.42.1)

Hom(X, Y t ) = CorrS (X, Y )(S) = Ker(β),

where β is given by β : PicX×S Y /S (S) −→ PicX/S (S) × PicY /S (S), M 7→ (M |X×{0} , M |{0}×Y ).

690

27 Abelian schemes

Observe that CorrS (X, Y ) is symmetric in X and Y , the identification of CorrS (X, Y ) with CorrS (Y, X) being given by pullback along the morphism σX,Y : X ×S Y → Y ×S X which switches the two factors. We obtain an identification Hom(X, Y t ) = Hom(Y, X t ). Now we set Y = X t . Then idX t corresponds to the Poincar´e bundle PX ∈ Ker(β) and also corresponds to a homomorphism of abelian schemes (27.42.2)

κX : X −→ (X t )t .

∗ By definition, we then have an isomorphism (idX t ×κX )∗ PX t ∼ = σX,X t PX of line t bundles on X × X.

Proposition 27.220. (1) For abelian schemes X, Y over S, the isomorphism Hom(Y, X t ) ∼ = Hom(X, Y t ) is t given by f 7→ f ◦ κX . (2) The formation of κX is functorial in X, i.e., for every homomorphism f : X → Y of abelian schemes over S, we have κY ◦ f = f tt ◦ κX . Proof. Any morphism f : Y → X t is determined by the pullback (id ×f )∗ PX of the Poincar´e bundle, and by definition of the identification Hom(Y, X t ) ∼ = Hom(X, Y t ) the ∗ ∗ morphism f corresponds to σX,Y (id ×f ) PX under this bijection. In view of the definition of κX and of (27.41.2), we may compute ∗ ∗ σX,Y (id ×f )∗ PX ∼ = (f × idX )∗ σX,X t PX ∗ ∼ = (f × idX ) (idX t ×κX )∗ PX t ∼ = (idY ×κX )∗ (f × idX tt )∗ PX t

∼ = (idY ×κX )∗ (idY ×f t )∗ PY ∼ = (idY ×(f t ◦ κX ))∗ PY . This proves part (1), and by symmetry also shows that the inverse of this identification is given by g 7→ g t ◦ κY . In particular, applying this to Y = X t and f = id we find that κtX ◦ κX t = idX t . (We will show in Theorem 27.222 that as a consequence κX is an isomorphism.) Now to prove part (2), let f : X → Y be a homomorphism of abelian schemes, and consider f t ∈ Hom(Y t , X t ). In view of part (1) it corresponds to f tt κX ∈ Hom(X, Y tt ). On the other hand, rewriting it as f t = f t κtY κY t = (κY f )t κY t , we see that at the same time it corresponds to κY f . By Theorem 27.222 below, we can identify X tt = X (via κX ) for every abelian scheme X. Then part (1) of the proposition says that f tt = f for every morphism f of abelian schemes. Remark 27.221. Let X be an abelian scheme over a scheme S. As the identifications in (27.42.1) are compatible with base change, we find that κX×S T = κX × idT for every morphism T → S. Theorem 27.222. Let X be an abelian scheme over a scheme S. Then the biduality morphism κX is an isomorphism.

691 Proof. As the formation of κX is compatible with base change and since we can check whether a morphism of S-schemes is an isomorphism on fibers (Corollary 18.77), we may assume that S is the spectrum of a field which we can even assume to be algebraically closed by faithfully flat descent (Proposition 14.53). In the proof of Proposition 27.220 we have already seen that (κX )t ◦ κX t = idX t . Hence κX t has trivial kernel. As a homomorphism between abelian varieties, it is therefore an isogeny of degree 1, i.e., an isomorphism. This shows that (κX )t is an isomorphism. Hence it is enough to show the following lemma. Lemma 27.223. Let f : X → Y be a homomorphism of abelian varieties over an algebraically closed field such that f t is an isomorphism. Then f is an isomorphism. Proof. By Proposition 27.213 it suffices to show that f is an isogeny. One has dim(X) = dim(X t ) = dim(Y t ) = dim(Y ) because f t is an isomorphism. Hence it suffices to show that f is surjective. As f is proper, Y ′ := f (X) is closed in Y , and when endowed with the scheme structure that makes Y ′ into a reduced closed subscheme of Y , it is an abelian subvariety. By the functoriality of the formation of the dual homomorphism we see that f t factorizes as Y t → (Y ′ )t → X t . As f t is an isomorphism, the kernel of Y t → (Y ′ )t is trivial which implies dim(Y ′ )t ≥ dim Y t (Proposition 27.14) and hence dim Y ′ = dim Y . As Y is irreducible, we see that Y ′ = Y . Remark 27.224. Let X be an abelian scheme over a scheme S. Let PX t ∈ Pic(X t ×S (X t )t ) be the Poincar´e bundle of X t . Let σ : X ×S X t → X t ×S X be the canonical morphism that switches the factors. Then by definition of κX we have an isomorphism of rigidified line bundles (27.42.3)

σ ∗ (idX ×κX )∗ PX t ∼ = PX .

Moreover, the homomorphism φPX : X ×S X t → X t × X tt is given on T -valued points by (27.42.4)

φP (x, ξ) = (ξ, κX (x)).

In the sequel we will usually identify X with (X t )t using κX . Remark 27.225. Let S be a scheme and let X be a pointed S-scheme. We say that an abelian scheme A/S together with a morphism α : X → A of pointed S-schemes is an Albanese scheme for X (sometimes denoted by AlbX/S ), if it satisfies the following universal property for morphisms from X into abelian schemes: For every abelian scheme B/S and every morphism f : X → B of pointed S-schemes, there exists a unique homomorphism φ : A → B of abelian schemes such that f = φ ◦ α. It is clear that an Albanese scheme for X is uniquely determined up to unique isomorphism, if it exists. Now assume that Pic0X/S is representable by an abelian scheme. We claim that (Pic0X/S )∨ is an Albanese scheme for X. First note that we have a canonical morphism X → (Pic0X/S )∨ . In fact, consider the universal line bundle on X × Pic0X/S . We can view this as an actual line bundle on X × Pic0X/S , rather than just an equivalence class in Pic0X/S (Pic0X/S ) by requiring that its restrictions to {x0 }×Pic0X/S and to X ×{0} be trivial. (Here x0 ∈ X(S) denotes the fixed point giving X the structure of a pointed scheme.) Since X × Pic0X/S ∼ = Pic0X/S ×X, this line bundle defines a morphism α : X → (Pic0X/S )∨ .

692

27 Abelian schemes

Now consider any morphism f : X → B of pointed S-schemes, where B is an abelian scheme. Pullback of line bundles gives us a morphism B ∨ = Pic0B/S → Pic0X/S , and passing to duals we obtain a morphism φ : AlbX/S → B. It is not hard to check that f = φ ◦ α and that φ is the unique homomorphism with this property. If S is the spectrum of a field k and X is a geometrically connected, smooth and proper pointed k-scheme, then Pic0X/k is representable by a proper k-group scheme (Theorem 27.119). In characteristic 0 it is necessarily reduced and therefore an abelian variety (Theorem 27.25) and the above discussion shows that its dual (Pic0X/k )∨ is an Albanese variety for X. In positive characteristic, Pic0X/k might be non-reduced. If k is perfect, then its underlying reduced scheme is an abelian variety (Proposition 27.10 (3)), and by similar arguments as above one can show that the dual of (Pic0X/k )red is an Albanese variety for X. See [Moc] X Appendix for further details and a more comprehensive discussion.

Cohomology of line bundles on abelian schemes Next we calculate the cohomology of the Poincar´e bundle (Proposition 27.229) and use it to prove the Fourier-Mukai equivalence (Theorem 27.243) for abelian schemes. We then use the Fourier-Mukai equivalence to prove the Riemann-Roch theorem for abelian varieties (Theorem 27.253). For line bundles with non-vanishing Euler characteristic, so-called non-degenerate line bundles, there exists a unique degree in which the cohomology does not vanish. This degree is called the index of the line bundle and it varies locally constant in families. Moreover, a line bundle L is relatively ample if and only if it is non-degenerate and of index 0 (Theorem 27.264). We then use the Fourier-Mukai equivalence to prove Atiyah’s classification of vector bundles on elliptic curves (Theorem 27.271). The characterization of ampleness is used to show that any tensor product of two (resp. three) relatively ample line bundles is globally generated locally on the base scheme (resp. very ample), see Proposition 27.278 and Theorem 27.279. We also introduce the notion of a polarization and show that the property to be a polarization can be checked on a single geometric fiber for a connected base scheme (Corollary 27.285). We conclude the chapter by proving that abelian schemes over normal noetherian base schemes are projective (Theorem 27.291) and by briefly explaining many of the notions in this chapter in the special case of abelian varieties over the complex numbers, see Section (27.54). (27.43) Cohomology of the Poincar´ e bundle. Let π : X → S be an abelian scheme of relative dimension g, let π ′ : X t → S be the structure morphism of the dual abelian scheme, let p : X ×S X t → X and p′ : X ×S X t → X t be the projections, and let ϖ : X ×S X t → S be the structure morphism. Let e : S → X and e′ : S → X t be the zero sections. Hence we have a commutative diagram

693 X ×S X t p′

p

XV

z

$

ϖ π e

π

$  z S

′

XH t

e′

Lemma 27.226. With the above notation the following assertions hold. (1) The Leray spectral sequence for ϖ = π ′ ◦ p′ induces for all i ≥ 0 an isomorphism ∼

π∗′ (Ri p′∗ P) −→ Ri ϖ∗ P.

(*)

̸ g, Rg ϖ∗ P is finite locally free, and the formation (2) One has Ri ϖ∗ P = 0 for all i = of Ri ϖ∗ P commutes with base change for all i. (3) One has Ri p′∗ P = 0 for all i ̸= g. (4) The formation of Rg p′∗ P commutes with arbitrary base change T → X t . We will prove (3) fully only if S is locally noetherian. For the general case we will give a proof using Lemma 27.227 below for whose proof we essentially only give a reference. Proof. We show (1). Let y ∈ X t and s′ := π ′ (y). Then pt,−1 (y) = X ⊗κ(s′ ) κ(y) =: Xy . The line bundle Py = P |Xy is by definition isomorphic to the line bundle in X t (κ(y)) ⊆ Pic(Xy ) corresponding to the morphism Spec κ(y) → X t . In particular, Py ∼ ̸ OXy if and = / e′ (S) ⊂ X t . Hence H i (Xy , Py ) = 0 for all i if y ∈ X t \ e′ (S) by Lemma 27.197. only if y ∈ So Proposition 23.142 shows that one has for all i Ri p′∗ P |(X t \e′ (S)) = 0.

(27.43.1)

Thus for all s ∈ S we have Supp(Ri p′∗ P)s ⊆ {e′ (s)} ⊆ Xst and in particular we see that H n (Xst , (Ri p′∗ P)s ) = 0 for all n > 0. By Proposition 23.142 this shows that Rn π∗′ (Ri p′∗ P) = 0 for all n > 0 and for all i. Now the Leray spectral sequence for ϖ = π ′ ◦ pX t yields the isomorphism (*). To show (2) we may assume that S = Spec k for a field k by Corollary 23.143 and because the formation of the Poincar´e bundle is compatible with base change, and it suffices to show that H i (X ×k X t , P) = 0 for all i ̸= g. Then the isomorphism (*) becomes H 0 (X t , Ri p′∗ P) ∼ = H i (X ×k X t , P). As p′ is proper with fibers of dimension g, we find that Ri p′∗ P = 0

(**)

for all i > g

by Corollary 24.44 and hence H i (X ×k X t , P) = 0 for i > g. We now apply Serre duality to the Poincar´e bundle. Applying (27.35.3) to the abelian scheme X ×k X t → X t , we have ([−1]∗X , idX t )∗ P ∼ = P −1 ⊗ pt,∗ (M ) for some line bundle t −1 M on X . As P and P are rigidified we find by restriction to 0 × X t that M ∼ = OX t . 2g Therefore P ∼ = P −1 . As X ×k X t is a smooth group scheme over k, Ω1X×k X t /k ∼ = OX×k X t ∼ OX×X t . Therefore Serre duality (Corollary 25.129) yields and hence Ω2g = t X×k X /k

H i (X × X t , P) ∼ = H 2g−i (X × X t , P −1 )∨ ∼ = H 2g−i (X × X t , P)∨ .

694

27 Abelian schemes

This shows H i (X ×k X t , P) = 0 also for i < g. Assertion (4) follows as top degree higher direct image always commutes with base change by Corollary 23.146. It remains to show (3). By (**) it suffices to show that Ri p′∗ P = 0 for all i < g. If S is locally noetherian one can argue as follows. Then Ri p′∗ P is a coherent OX t module (Theorem 23.17). Hence its annihilator is a quasi-coherent ideal (Proposition 7.35) which defines a closed subscheme h : Z ,→ X t and Ri p′∗ P = h∗ Fi for the coherent OZ -module Fi := h∗ Ri p′∗ P. The underlying topological space of Z is contained in the closed subscheme defined by the closed immersion e′ by (27.43.1). Hence the restriction of π ′ to Z is affine by Lemma 12.38. As π∗′ (Ri p′∗ P) = Ri ϖ∗ P = 0 for i = ̸ g, this shows that Ri p′∗ P = 0 for i ̸= g by Proposition 12.5. Let us now give a proof of (3) if S is not necessarily locally noetherian. Set E := Rp′∗ P, which is a perfect complex by Corollary 23.136. Let s ∈ S and denote by Ys the fiber in s of an S-scheme Y . By derived cohomology and base change (Theorem 23.133) applied to the cartesian diagram / X ×S X t Xs ×κ(s) Xst p′s

p′

 Xst

 / Xt

is

we find Li∗s E = Rp′s∗ Ps , where Ps is the Poincar´e bundle for Xs . As we proved (3) already if S is the spectrum of a field, we have Ri p′s∗ Ps = 0 for all i < g. Hence Lemma 27.227 below shows that H i (E) = Ri p′∗ P = 0 for all i < g. Lemma 27.227. Let f : X → S be a flat morphism locally of finite presentation and let E be a pseudo-coherent complex on X. For s ∈ S denote by is : Xs := X ×S κ(s) → X the canonical morphism. Then for all n ∈ Z U := { x ∈ X ; H n (Li∗f (x) E)x = 0 } is open and constructible in X and H n (E)|U = 0. Proof. One can assume that S and X are affine. Then E can be represented by a bounded above complex of finite free OX -modules E (Remark 22.46). Then E is K-flat and Li∗f (x) E = i∗f (x) E . Now we can apply [EGAIV] O (12.3.3) (the fact that U is constructible follows from the proof in loc. cit.). Lemma 27.228. With the notation above we have for every quasi-coherent OS -module G and every quasi-coherent OX t -module H functorial isomorphisms (27.43.2) (27.43.3)

∼

HomOS (Rg π∗ OX , G ) −→ Γ(X, π ∗ G ⊗ ΩgX/S ), ∼

HomOX t (Rg p′∗ P, H ) −→ HomOX×X t (P, pt,∗ H ⊗ p∗ ΩgX/S ).

As π : X → S is a smooth group scheme of relative dimension g, ΩgX/S is the pullback of a line bundle on S (Proposition 27.15, Corollary 27.21). Hence we find ∼

HomOS (Rg π∗ OX , G ) −→ Γ(X, π ∗ G ) if Pic(S) = 0, e.g., if S is the spectrum of a field.

695 Proof. We show the first isomorphism. As π is proper and smooth, for every complex G in g Dqcoh (S), one has π × G = Lπ ∗ G ⊗L OX ΩX/S [g] (Theorem 25.58). Hence by Grothendieck duality (Corollary 25.17, Theorem 25.58) we have g HomD(S) (Rπ∗ OX , G) = HomD(X) (OX , Lπ ∗ G ⊗L OX ΩX/S [g]).

Taking G = G [−g] we obtain g HomD(S) (Rπ∗ OX , G [−g]) = HomD(X) (OX , Lπ ∗ G ⊗L OX ΩX/S )

= HomOX (OX , π ∗ G ⊗ ΩgX/S ) = Γ(X, π ∗ G ⊗ ΩgX/S ) using that π and ΩgX/S are flat. Consider the natural map Rπ∗ OX → τ ≥g Rπ∗ OX = Rg π∗ OX [−g], where the equality holds because Rπ∗ OX has cohomology concentrated in degrees ≤ g. This map induces by functoriality an isomorphism ∼

HomOS (Rg π∗ OX , G ) = HomD(S) (Rg π∗ OX [−g], G [−g]) −→ HomD(S) (Rπ∗ OX , G [−g]) because τ ≥g is left adjoint to the inclusion D≥g (S) → D(S). This gives the first isomorphism. For the second isomorphism one argues similarly, using that Rp′∗ P = Rg p′∗ P[−g] by Lemma 27.226 (3) and that ΩgX×S X t /X t = p∗ ΩgX/S by Proposition 17.30. Proposition 27.229. With the notation above let Ce = e∗ Ω1X/S be the conormal bundle of the zero section, which is a locally free OS -module of rank g, and let det(Ce ) be its determinant, i.e., its g-th exterior power. Then the following assertions hold. (1) ( 0, if i ̸= g, i R ϖ∗ P = det(Ce )−1 , if i = g. (2) ( Ri p′∗ P

=

0, e′∗ det(Ce )−1 ,

if i ̸= g, if i = g.

If S = Spec k for a field k (or, more generally, for a ring k such that Pic(k) = 0), then det(Ce )−1 ∼ = OS . Proof. We have already seen in Lemma 27.226 that Ri ϖ∗ P and Ri p′∗ P vanish for i ̸= g. Hence it suffices by (1) of the lemma to show that Rg p′∗ P ∼ = e′∗ det(Ce )−1 . g ′ As the formation of R p∗ P commutes with base change (Lemma 27.226 (4)), we have et,∗ Rg p′∗ P = Rg π∗ ((idX , e′ )∗ P) = Rg π∗ OX by the cartesian diagram X

(idX ,e′ )

p′

π

 S

Therefore we have

/ X ×S X t

e

′

 / X t.

696

27 Abelian schemes HomOX t (Rg p′∗ P, e′∗ det(Ce )−1 ) = HomOS (et,∗ Rg p′∗ P, det(Ce )−1 ) = HomOS (Rg π∗ OX , det(Ce )−1 )

(27.43.4)

= Γ(X, π ∗ det(Ce )−1 ⊗ ΩgX/S ) = Γ(X, OX ) = Γ(S, OS ),

where the third equality holds by Lemma 27.228. Hence there is a natural homomorphism of OX t -modules ξ : Rg p′∗ P → e′∗ det(Ce )−1 corresponding to 1 ∈ Γ(S, OS ). The identifications in (27.43.4) are compatible with base change T → S, hence ξ is compatible with base change T → S. To prove that ξ is an isomorphism, we may work Zariski locally on S. In particular we can assume that S = Spec R is affine. Moreover we may assume that R is noetherian by writing R as a filtered colimit of finitely generated Z-algebras. As in the proof of Lemma 27.226 (3) we find a closed subscheme h : Z → X t , the vanishing locus of the annihilator of Rg p′∗ P, whose underlying topological space is contained in e′ (S) such that h∗ h∗ Rg p′∗ P = Rg p′∗ P. For s ∈ S we have κ(s) = κ(e′ (s)) because e′ is an immersion. By Lemma 27.226 (4) (*)

Rg p′∗ P ⊗OX t κ(s) = H g (Xe′ (s) , Pe′ (s) ) = H g (Xs , OXs ) ∼ = κ(s).

Therefore, the underlying topological space of Z is equal to e′ (S) and Rg p′∗ P is locally generated by one element. We claim that e′ : S → X t factors through the subscheme Z and hence that Z is a nilpotent thickening of the closed subscheme e′ (S). For this let Z ′ ⊆ Z be the maximal subscheme, where Rg p′∗ P is locally free of rank 1 and its formation is compatible with base change (Remark 23.148). As et,∗ Rg p′∗ P = Rg π∗ OX is finite locally free of rank 1 (Theorem 27.203), e′ factors through Z ′ and hence through Z. This shows the claim. As S is affine, Z is affine as a nilpotent thickening by Corollary 12.40. Hence Z = Spec A and there exists a nilpotent ideal I ⊆ A with R = A/I. Moreover, (*) shows that locally on Z and hence locally on S, which has the same underlying topological space, h∗ Rg p′∗ P corresponds to an A-module generated by one element. Therefore we see that g for some ideal J ⊆ A. Then J is contained in the ideal defining Rg p′∗ P = h∗ (A/J) Z ′ in Z and in particular J ⊆ I. If we set ZJ := Spec A/J and XJ := X ×S ZJ , then X × 0 = XI → XJ is a nilpotent thickening. g Further localizing on S we may assume that det(Ce ) ∼ = OS and therefore ΩX/S = ∗ π det(Ce ) ∼ = OX . Then ξ is given by the canonical map OZJ → OZI and it remains to see that I = J. We set PJ := P |XJ . Then PI = P |X×0 ∼ = OXI and we have a commutative diagram HomOXJ (PJ , OXJ ) = HomOX×

SX

 HomOXI (OXI , OXI ) = HomOX×

(P, OXJ )

∼

t

(P, OXI )

∼

t

SX

/ HomO t (OZ , OZ ) = A/J J J X  / HomO t (OZ , OZ ) = A/I, J I X

where the horizontal maps are the isomorphisms (27.43.3). As the right vertical map is surjective, the identity of OXI can be lifted to a map u : PJ → OXJ . As u is an isomorphism modulo the nilpotent ideal I, u is surjective by Nakayama’s lemma and

697 hence an isomorphism as a map between line bundles. This shows that the restriction of P to XJ is trivial. By the universal property of P this implies that ZJ → X t factors through XI = X × 0. Hence I = J. Applying Proposition 27.229 to the Poincar´e bundle of X t and using Remark 27.224 we obtain also the following result. Corollary 27.230. With the notation above we have det(Ce ) ∼ = det(Ce′ ) and ( 0, if i ̸= g, R i p∗ P = e∗ det(Ce )−1 , if i = g,

(27.44) The N´ eron-Severi group of an abelian scheme. Proposition and Definition 27.231. Let k be an algebraically closed field and let X be an abelian variety over k. Let L1 and L2 be line bundles on X. Then the following assertions are equivalent. (i) One has L1 ⊗ L2−1 ∈ X t (k). (ii) There exists a connected k-scheme Z, a line bundle M on X ×k Z and z1 , z2 ∈ Z(k) such that Li = M |X×{zi } for i = 1, 2. Moreover, the k-scheme Z can be chosen to be smooth and of finite type. If these conditions are satisfied, then L1 and L2 are called algebraically equivalent. Proof. Condition (ii) means that L1 and L2 are in the same connected component Z ′ of PicX/k , i.e., it is satisfied if and only if L1 ⊗ L2−1 ∈ Pic0X/k (k) = X t (k). Moreover, one can then choose for Z this connected component Z ′ , which is smooth and of finite type, and for M some representative of the universal line bundle in PicX/S (Z ′ ) = Pic(X ×k Z ′ )/ Pic(Z ′ ) over it. Definition 27.232. Let S be a scheme and let X be an abelian scheme over S. Then the quotient of abelian fppf-groups NS(X) := PicX/S /X t is called the N´eron-Severi group of X. As X t is the identity component of PicX/S , one can view NS(X) as the functor of connected components of PicX/S . Remark 27.233. Let S be a scheme and let X be an abelian scheme over S. Let Hom(X, X t ) be the functor that sends an S-scheme T to the abelian group of homomorphisms of abelian group spaces XT → XTt . This is an fppf-sheaf by Theorem 14.72. It is isomorphic to CorrS (X, X) (Proposition 27.163) and hence a scheme that is separated, unramified, and locally of finite presentation over S (Proposition 27.157). ̸ 0, then (1) For an integer n ∈ Z and f ∈ Hom(X, X t )(T ) we have n · f = f ◦ [n]X . If n = [n]X is an epimorphism and hence nf = 0 implies f = 0. In other words, Hom(X, X t ) is a sheaf of torsion free Z-modules.

698

27 Abelian schemes

(2) The homomorphism of abelian fppf-sheaves φ : PicX/S → Hom(X, X t ), L 7→ φL , by definition has kernel X t . Hence it induces a monomorphism φ¯ : NS(X) ,→ Hom(X, X t ).

(27.44.1)

In particular, NS(X)(T ) is torsion free for all S-schemes T and NS(X) is formally unramified as a functor, i.e. NS(X)(T ) → NS(X)(T0 ) is injective for all closed immersions T0 → T defined by a locally nilpotent quasi-coherent ideal of OT . Let f : X → Y be a homomorphism of abelian schemes. Then f ∗ : PicY /S → PicX/S induces a homomorphism of abelian fppf-sheaves f ∗ : NS(Y ) → NS(X). Proposition 27.234. Let n ∈ Z. The multiplication [n] : X → X induces on NS(X) the multiplication by n2 . Proof. This follows from Proposition 27.184 (3). We can identify Hom(X, X t ) = CorrS (X, X) with a subgroup functor of PicX×S X/S (Remark 27.152). Composing the inclusion Hom(X, X t ) ,→ PicX×S X/S with the map ∆∗ : PicX×S X/S → PicX/S , where ∆ : X → X ×S X is the diagonal, we obtain a map δ : Hom(X, X t ) → PicX/S . Consider the composition (27.44.2)

φ ¯

δ

NS(X) −→ Hom(X, X t ) −→ PicX/S −→ NS(X),

where the last map is the canonical projection. Lemma 27.235. The map (27.44.2) is the multiplication by 2 on NS(X). φ

Proof. Indeed, the composition PicX/S −→ CorrS (X, X) ,→ PicX×S X/S is given by L 7→ m∗ L ⊗ p∗1 L −1 ⊗ p∗2 L −1 by Proposition 27.161 (1). Applying ∆∗ we obtain the class of [2]∗ L ⊗ L ⊗−2 in PicX/S . Since [2]∗ L is algebraically equivalent to L ⊗4 by Proposition 27.234, this shows the claim. (27.45) Fourier-Mukai transforms. In this section we discuss Fourier-Mukai transforms, a construction invented by S. Mukai which gives rise to autoequivalences of categories of the form Dqcoh (X) and in a sense is analogous to the classical Fourier transform (since push-forward can be thought of as an analogue of summation or integration, cf. formula (23.23.6)). In the next section we will show that the Fourier-Mukai transform given by the Poincar´e bundle yields an equivalence Dqcoh (X) ∼ = Dqcoh (X t ) for any abelian scheme X. Let S be a scheme, let X and Y be S-schemes, and denote by p : X ×S Y → X and q : X ×S Y → Y the projections. Definition 27.236. Let K be a complex in D(X ×S Y ). Then the triangulated functors ΦK : D(X) −→ D(Y ), ΨK : D(Y ) −→ D(X),

E 7→ Rq∗ (Lp∗ E ⊗L OX× ∗

F 7→ Rp∗ (Lq F

SY

⊗L OX×S Y

K), K)

699 are called Fourier-Mukai transforms with kernel K. Remark 27.237. Usually, one restricts the Fourier-Mukai transforms to suitable subcategories of D(X). (1) Let X → S and Y → S be qcqs. Then the projections p : X ×S Y → X and q : X ×S Y → Y are also qcqs. Let K ∈ Dqcoh (X ×S Y ). Then the Fourier-Mukai transforms ΦK and ΨK induce functors ΦK : Dqcoh (X) −→ Dqcoh (Y ),

ΨK : Dqcoh (Y ) −→ Dqcoh (X)

by Proposition 22.40 and Theorem 22.31. (2) Let X → S and Y → S be proper flat morphisms of finite presentation. Then p and q have the same properties. Let K ∈ D(X ×S Y ) be perfect. Then ΦK and ΨK restrict to functors ΦK : (Perf(X)) −→ (Perf(Y )),

ΨK : (Perf(Y )) −→ (Perf(X))

by Remark 21.142, Remark 21.143 and Corollary 23.136. Often one considers the case where S = Spec k for a field k and X and Y are smooth proper schemes over k. Then X, Y , and X ×k Y are regular of finite dimension and hence a complex over X, over Y , or over X ×k Y is perfect if and only if its cohomology modules are coherent and non-zero only in finitely many degrees (Proposition 23.55). Hence in this case we obtain functors b b ΦK : Dcoh (X) −→ Dcoh (Y ),

b b ΨK : Dcoh (Y ) −→ Dcoh (X).

The following example shows that derived pushforward, derived pullback and derived tensor product are all special cases of Fourier-Mukai transforms, at least for qcqs morphisms. Example 27.238. Let f : X → S and g : Y → S be qcqs morphisms of schemes. Let h : X → Y be a morphism of S-schemes, let Γ : X → X ×S Y be its graph morphism. Let M be in Dqcoh (X), and set K := RΓ∗ M . Then Γ is qcqs and hence K ∈ Dqcoh (X ×S Y ). The corresponding Fourier-Mukai transforms are given by ΦK : Dqcoh (X) −→ Dqcoh (Y ),

ΦK (F ) = Rh∗ (F ⊗L M ),

ΨK : Dqcoh (Y ) −→ Dqcoh (Y ),

ΨK (G ) = Lh∗ G ⊗L M .

Indeed, for F ∈ Dqcoh (X) and G ∈ Dqcoh (Y ) one has by the projection formula (Proposition 22.84) ΦK (F ) = Rq∗ (Lp∗ F ⊗L RΓ∗ M ) = Rq∗ (RΓ∗ (LΓ∗ Lp∗ F ⊗L M )) = Rq∗ RΓ∗ (F ⊗L M ) = Rh∗ (F ⊗L M ), ΨK (G ) = Rp∗ (Lq ∗ G ⊗L RΓ∗ M ) = Rp∗ (RΓ∗ (LΓ∗ Lq ∗ G ⊗L M )) = Lh∗ G ⊗L M . As special cases we obtain (1) If M = OX , then ΦK = Rh∗ and ΨK = Lh∗ . (2) If h = idX , then Γ : X → X ×S X is the diagonal, and ΦK = ΨK = (− ⊗L M ). Fourier-Mukai transforms for perfect complexes admit a left and a right adjoint by Grothendieck duality for smooth proper morphisms.

700

27 Abelian schemes

Proposition 27.239. Let S be a scheme, let f : X → S and g : Y → S be smooth proper morphisms of relative dimension m and n, respectively, and denote by p : X ×S Y → X and q : X ×S Y → Y the projections. Let K be a perfect complex in D(X ×S Y ) and set KL := K ∨ ⊗L Lq ∗ ΩnY /S [n].

KR := K ∨ ⊗L Lp∗ Ωm X/S [m],

Then ΨKL is a left adjoint and ΨKR is a right adjoint to ΦK : (Perf(X)) → (Perf(Y )). Proof. As f and g are flat, so are p and q and therefore p∗ = Lp∗ and q ∗ = Lq ∗ . For any E in (Perf(X)) and F in (Perf(Y )) one has functorial isomorphisms HomD(X) (ΨKL (F ), E) = HomD(X) (Rq∗ (KL ⊗L q ∗ F ), E) ∼ HomD(X×Y ) (KL ⊗L q ∗ F, p! E) = ∼ = HomD(X×Y ) (KL ⊗L q ∗ F, p∗ E ⊗L ΩnX×Y /X [n]) ∼ HomD(X×Y ) (K ∨ ⊗L q ∗ Ωn [n] ⊗L q ∗ F, p∗ E ⊗L q ∗ Ωn =

Y /S [n])

Y /S

∼ = HomD(X×Y ) (K ⊗ q F, p E) ∼ HomD(X×Y ) (q ∗ F, K ⊗L p∗ E) = ∨

L

∗

∗

∼ = HomD(Y ) (F, Rq∗ (K ⊗L p∗ E)) = HomD(Y ) (F, ΦK (E)). Here the first isomorphism holds by definition of the functor p! = p× as p is proper, the second by Theorem 25.58, the third by compatibility of K¨ahler differentials with base change (apply the n-th exterior power to Proposition 17.30), the fourth because q ∗ ΩnY /S [n] is an invertible object in D(X×S Y ), the fifth by duality of perfect complexes (apply H 0 ◦RΓ to (21.32.3)), and the sixth by adjointness of q ∗ = Lq ∗ and Rq∗ (Proposition 21.126). Remark 27.240. A theorem of Orlov ([Orl] O X 3.2.1) shows the following result. Let X b and Y be smooth projective schemes over a field k (and hence Dcoh (X) = (Perf(X)) b b b and Dcoh (Y ) = (Perf(Y )) by Proposition 23.55). Let F : Dcoh (X) → Dcoh (Y ) be a fully b faithful triangulated functor. Then there exists K ∈ Dcoh (X ×k Y ) such that F ∼ = ΦK , and K is unique up to isomorphism. In loc. cit. this result is formulated with the additional hypothesis that F has a left or a right adjoint functor. This is in fact automatic by a result of Bondal and van den Bergh (e.g., [BBH] O 1.20). We now show how to compute the composition of Fourier-Mukai transforms. Let S be a scheme and let X → S, Y → S, and Z → S be S-schemes. We write X × Y instead of X ×S Y and similarly for other fiber products. Consider the commuting diagram of projections X ×Y ×Z pX

(27.45.1)

X ×Y

pXY

pY Z

x

& Y ×Z prXY Y

 { X

prXY X

pZ

&

Z prY Y

Y

x

Z prY Z

#  Z.

The middle diamond is cartesian. Moreover denote by pXZ : X × Y × Z → X × Z the projection.

701 Definition and Proposition 27.241. For K ∈ D(X × Y ) and M ∈ D(Y × Z) define their convolution by (27.45.2)

K ∗ M := RpXZ,∗ (Lp∗XY K ⊗L Lp∗Y Z M ) ∈ D(X × Z).

Suppose (a) that the morphisms X → S, Y → S, and Z → S are qcqs, YZ (b) that the two projections prXY Y : X ×Y → Y and prY : Y ×Z → Y are tor-independent (e.g., if X → S or Z → S is flat), and (c) that K ∈ Dqcoh (X × Y ) and M ∈ Dqcoh (Y × Z). Then one has ΦM ◦ ΦK = ΦK∗M : Dqcoh (X) −→ Dqcoh (Z) and similarly ΨK ◦ ΨM = ΨK∗M as functors from Dqcoh (Z) to Dqcoh (X). Proof. As Y → S is qcqs, pXZ is qcqs, and hence K ∗ L ∈ Dqcoh (X × Z). Let prXZ X : X× : X × Z → Z be the projections. By symmetry in X and Z it suffices Z → X and prXZ Z to show ΦM ◦ ΦK = ΦK∗M . Let F ∈ Dqcoh (X). Then XY,∗ (ΦM ◦ ΦK )(F ) = ΦM (R prXY (F ) ⊗L K)) Y,∗ (L prX  
XY,∗ Z L prYY Z,∗ R prXY (F ) ⊗L K) ⊗L M = R prYZ,∗ Y,∗ (L prX  
Z ∼ RpY Z,∗ Lp∗XY (L prXY,∗ (F ) ⊗L K) ⊗L M = R prYZ,∗ X  
Z RpY Z,∗ Lp∗X F ⊗L Lp∗XY K ⊗L M = R prYZ,∗ 
Z ∼ RpY Z,∗ Lp∗X F ⊗L Lp∗XY K ⊗L Lp∗Y Z M = R prYZ,∗
= RpZ,∗ Lp∗X F ⊗L Lp∗XY K ⊗L Lp∗Y Z M 
 XZ,∗ ∗ = R prXZ F ) ⊗L Lp∗XY K ⊗L Lp∗Y Z M Z,∗ RpXZ,∗ LpXZ (L prX   XZ,∗ L ∗ L ∗ ∼ L pr F ⊗ Rp (Lp K ⊗ Lp M ) = R prXZ XZ,∗ Z,∗ XY YZ X   XZ,∗ XZ L = R prZ,∗ L prX F ⊗ (K ∗ M )

= ΦL∗M (F ), where the first isomorphism holds since the middle diamond in (27.45.1) is tor-independent (Theorem 22.99), the second isomorphism holds by the projection formula (Proposition 22.84) for pY Z , and the third isomorphism holds by the projection formula for pXZ . Finally, we study when the Fourier-Mukai transform is compatible with base change. Proposition 27.242. Let S ′ → S be a morphism of schemes, set X ′ = X ×S S ′ and Y ′ = Y ×S S ′ , and let π : X ′ −→ X,

ϖ : Y ′ → Y,

π × ϖ : X ′ ×S ′ Y ′ −→ X ×S Y

be the projections. Suppose that X → S and Y → S are qcqs and let K be an object in Dqcoh (X ×S Y ).

702

27 Abelian schemes

(1) If S ′ → S or X → S are flat (or, more generally, if ϖ and q : X ×S Y → Y are tor-independent), then we have for every E ∈ Dqcoh (X) ΦL(π×ϖ)∗ K (Lπ ∗ E) = Lϖ∗ ΦK (E). (2) Dually, if S ′ → S or Y → S are flat (or, more generally, if π and p : X ×S Y → X are tor-independent), then we have for every F ∈ Dqcoh (X) ΦL(π×ϖ)∗ K (Lϖ∗ F ) = Lπ ∗ ΦK (F ). Proof. We show the first assertion. The proof of (2) is similar. Let p′ : X ′ ×S ′ Y ′ → X ′ and q ′ : X ′ ×S ′ Y ′ → Y ′ be the projections. We have ΦL(π×ϖ)∗ K (Lπ ∗ E) = Rq∗′ (Lp′∗ Lπ ∗ E ⊗L L(π × ϖ)∗ K) = Rq∗′ (L(π ⊗ ϖ)∗ Lp∗ E ⊗L L(π × ϖ)∗ K) = Rq∗′ L(π ⊗ ϖ)∗ (Lp∗ E ⊗L K) = Lϖ∗ Rq∗ (Lp∗ E ⊗L K) = Lϖ∗ ΦK (E). Here we used for the fourth equality that the cartesian diagram X ′ ×S ′ Y ′

π×ϖ

q′

 Y′

/ X ×S Y q

ϖ

 /Y

is tor-independent by hypothesis so that we can apply Theorem 22.99. (27.46) Fourier-Mukai equivalence for abelian schemes. Let S be a scheme, let X → S be an abelian scheme over S, and let X t be its dual abelian scheme. Let PX be the Poincar´e bundle on X ×S X t . We obtain Fourier-Mukai transforms SX := ΦPX : Dqcoh (X) → Dqcoh (X t ). We consider PX also as a line bundle on X t ×S X. This line bundle is identified with PX t if we identify X t ×S X with X t ×S X tt via idX t ×κX , where κX is the biduality isomorphism (Remark 27.224). Hence we obtain a Fourier-Mukai transform SX t := ΦPX t : Dqcoh (X t ) → Dqcoh (X). In the sequel we will use the following notation for various standard maps. (a) We denote by f : X → S the structure morphism and by e : S → X its zero section. (b) We write π : X ×S X t → X and π ′ : X ×S X t → X t for the projections. (c) We identify X t ×S X with X ×S X t and call this identification σ. This yields an identification PX with PX t . (d) We denote by p1 , p2 , m : X ×S X → X the first projection, the second projection, and the group law.

703 (e) We identify X ×S X t ×S X with X ×S X ×S X t and for ∅ = ̸ I ⊂ {1, 2, 3} write prI for the I-th projection on X ×S X ×S X t , for instance the projections pr12 to X ×S X and pr3 to X t . (f) We write µ : X ×S X ×S X t → X ×S X t for the morphism (x, y, ξ) 7→ (x + y, ξ) Theorem 27.243. Let S be a scheme and let X → S be an abelian scheme of relative dimension g. One has g SX t ◦ S X ∼ = [−1]∗ ◦ (− ⊗L (ΩX/S )−1 [−g]), g SX ◦ S X t ∼ = [−1]∗ ◦ (− ⊗L (Ω t )−1 [−g]). X /S

In particular, SX yields an equivalence of triangulated categories ∼

SX : Dqcoh (X) −→ Dqcoh (X t ) ∼

inducing an equivalence (Perf(X)) −→ (Perf(X t )). Moreover, these equivalences are compatible with base change S ′ → S. The relative canonical bundle ΩgX/S is a line bundle on X that is obtained via pullback from the line bundle det(Ce ) on S because X → S is a group scheme (Proposition 27.15). In particular, if Pic(S) = 0, then SX t ◦ S X ∼ = [−1]∗ [−g]. Proof. The last assertion follows from Proposition 27.242 since X → S and X t → S are flat. By Proposition 27.241 we may identify SX t ◦ SX = ΦPX ∗PX t . As [−1] is an automorphism of X of order 2, we have [−1]∗ = L[−1]∗ = R[−1]∗ = [−1]∗ as functors D(X) → D(X). Hence by Example 27.238 it suffices to identify PX ∗ PX t with RΓ∗ (ΩgX/S )[−g], where Γ : X → X ×S X is the graph of [−1]. We set P := PX and P˜ := PX t . By Remark 27.218 (2) we have pr∗13 P ⊗ pr23 P = ∗ µ P = Lµ∗ P. This shows the first of the following isomorphisms P ∗ P˜ ∼ = R pr12,∗ (Lµ∗ P) ∼ Lm∗ (Rπ∗ P) = g ∼ = Lm∗ (e∗ det(Ce )−1 ) = RΓ∗ (ΩX/S )[−g].

The second isomorphism follows by flat base change (m is flat by Remark 27.5) using the cartesian diagram µ / X ×S X t X ×S X ×S X t pr12

 X ×S X

π

 / X.

m

The isomorphism Rπ∗ P = e∗ det(Ce )−1 holds by Corollary 27.230. The last isomorphism follows from flat base change from the cartesian diagram X

/S

f

e

Γ

 X ×S X

m

 /X

using that ΩgX/S is the pullback of det(Ce ) by Proposition 27.15.

704

27 Abelian schemes

Proposition 27.244. Let S be a scheme and let f : X → Y be a homomorphism of abelian schemes over S. Then one has an isomorphism SY ◦ Rf∗ ∼ = L(f t )∗ ◦ SX . of functors Dqcoh (X) → Dqcoh (Y t ). Proof. Denote by ρ : X ×S Y t → X and σ : X ×S Y t → Y t the projections. Then for F in Dqcoh (X) one has functorial isomorphisms ′ SY (Rf∗ F ) = RπY,∗ (LπY∗ Rf∗ F ⊗L PY ) ∼ = Rπ ′ (R(f × idY t )∗ Lρ∗ F ⊗L PY ) Y,∗

′ ∼ (R(f × idY t )∗ (Lρ∗ F ⊗L L(f × idY t )∗ PY )) = RπY,∗ ∼ = Rσ∗ (Lρ∗ F ⊗L L(idX ×f t )∗ PX ) ∗ = Rσ∗ L(idX ×f t )∗ (LπX F ⊗L PX ) ∼ = Lf t∗ (Rπ ′ (Lπ ∗ F ⊗L PX )) X∗

X

= Lf t∗ (SX (F )). Here the first and the last isomorphism hold by base change, the second isomorphism by the projection formula, and the third isomorphism by (27.41.2). Remark 27.245. Let X be an abelian scheme over a scheme S of relative dimension g, let T be an S-scheme, let ξ ∈ X t (T ) be a T -valued point and let Mξ be the corresponding rigidified line bundle on X ×S T . (1) Let pT : X ×S T → T and pX : X ×S T → X be the projections. By derived base change one has for every F in Dqcoh (X) a functorial isomorphism Lξ ∗ SX (F ) = Lξ ∗ (Rπ∗′ (Lπ ∗ F ⊗L PX )) ∼ = RpT ∗ L(idX ×ξ)∗ (Lπ ∗ F ⊗L PX )) = RpT ∗ (Lp∗X F ⊗L Mξ ). (2) Let G be in Dqcoh (X t ). Applying (1) to F = SX t (G ) one obtains by Theorem 27.243 a functorial isomorphism g RpT ∗ (Lp∗X SX t (G ) ⊗L Mξ ) ∼ = Lξ ∗ (G [−g] ⊗L (ΩX t /S )−1 ).

(3) If T = S = Spec R is affine with Pic(S) = 0 (e.g., if R is a local ring), then pX = idX , pT : X → Spec R is the structure morphism, and ΩgX t /S ∼ = OX t . Hence (1) and (2) become (27.46.1) (27.46.2)

Lξ ∗ SX (F ) ∼ = RΓ(X, F ⊗L Mξ ), Lξ ∗ G [−g] ∼ = RΓ(X, SX t (G ) ⊗L Mξ ).

Let X be an abelian scheme over a scheme S. For F and G in Dqcoh (X) we define their convolution F ⊚X G := Rm∗ (p∗1 F ⊗L p∗2 G )

705 This is not the convolution defined in (27.45.2) (which is the reason why we use this slightly unconventional symbol to denote it) but rather a construction reminiscent of the convolution of functions on a locally compact group. Then the Fourier-Mukai equivalence ∼ S : Dqcoh (X) −→ Dqcoh (X t ) interchanges tensor product and convolution, similarly as the usual Fourier transform of a convolution of two (suitable) functions is the pointwise product of their Fourier transforms. Proposition 27.246. For F and G in Dqcoh (X) we have a functorial isomorphism S(F ⊚X G ) ∼ = S(F ) ⊗L S(G ). Proof. Then we find S(F ⊚X G ) = Rπ∗′ (PX ⊗L Lπ ∗ Rm∗ (Lp∗1 F ⊗L Lp∗2 G )) ∼ = Rπ∗′ (PX ⊗L Rµ∗ L pr∗12 (Lp∗1 F ⊗L Lp∗2 G )) ∼ = Rπ ′ Rµ∗ (Lµ∗ (PX ) ⊗L L pr∗ F ⊗L L pr∗ G ) ∗

1

2

∼ = Rπ∗′ Rµ∗ (L pr∗13 (PX ) ⊗L L pr∗23 (PX ) ⊗L L pr∗1 F ⊗L L pr∗2 G ) = R pr3,∗ ((L pr∗13 (PX ) ⊗L L pr∗1 F ) ⊗L (L pr∗23 (PX ) ⊗L L pr∗2 G )) ∼ = Rπ ′ (PX ⊗L Lπ ∗ F ) ⊗L Rπ ′ (PX ⊗L Lπ ∗ G ) ∗

∗

= S(F ) ⊗L S(G ). Here the first isomorphism holds by flat base change with the cartesian diagram X ×S X ×S X t

µ

/ X ×S X t

pr12

π

 X ×S X

m

 / X.

The second isomorphism holds by the projection formula, the third by Remark 27.218 (2), and the last isomorphism by the K¨ unneth isomorphism applied to the cartesian diagram X ×S X ×S X t = (X ×S X t ) ×X t (X ×S X t )

X ×S X t

s

+

pr3

π′

+

 s X t.

X ×S X t

π′

(27.47) Riemann-Roch for abelian varieties. ∼

Let f : X → S be an abelian scheme over a scheme S and denote by SX : Dqcoh (X) −→ Dqcoh (X t ) the Fourier-Mukai equivalence.

706

27 Abelian schemes

Definition 27.247. A line bundle L on X is called non-degenerate if the following equivalent (Proposition 27.177) properties are satisfied. (i) φL is an isogeny. (ii) K(L ) is a finite group scheme over S. (iii) K(L ) is a finite locally free group scheme over S. Example 27.248. We have seen in Corollary 27.172 that every ample line bundle on an abelian scheme is non-degenerate. Remark 27.249. Let L be a line bundle on X. The formation of K(L ) commutes with base change. As K(L ) is proper over S, it is finite over S if and only if all fibers of K(L ) → S are finite (Corollary 12.89). Hence one obtains the following assertions. (1) Let g : T → S be a morphism of schemes. If L is non-degenerate, then its pullback to X ×S T is non-degenerate. The converse holds if g is surjective. (2) The line bundle L is non-degenerate if and only if the restriction of L to the fibers Xs for all s ∈ S is a non-degenerate line bundle on the abelian variety Xs → Spec κ(s). Below in Proposition 27.258 we will see that if S is connected and L |Xs is nondegenerate for one s ∈ S, then L is itself non-degenerate. Proposition 27.250. Let X be an abelian scheme and let L be a line bundle on X. Then Lφ∗L SX (L ) ∼ = f ∗ (Rf∗ L ) ⊗ L −1 ⊗ [0]∗ L , where [0]∗ L denotes the line bundle f ∗ e∗ L . If L is non-degenerate, then φL is faithfully flat and hence Lφ∗L = φ∗L . Proof. We use the notation set up in the beginning of Section (27.46). Then one has Lφ∗L (S(L )) = Lφ∗L (Rπ∗′ (PX ⊗ π ∗ L )) ∼ Rp2,∗ L(id, φL )∗ (PX ⊗ π ∗ L ) = = Rp2,∗ (id, φL )∗ (PX ⊗ π ∗ L ) ∼ Rp2,∗ (Λ(L ) ⊗ p∗ L ) = 1 = Rp2,∗ (m∗ L ⊗ p∗2 (L −1 ⊗ f ∗ e∗ L )) ∼ = Rp2,∗ (m∗ L ) ⊗ L −1 ⊗ [0]∗ L ∼ = f ∗ Rf∗ L ⊗ L −1 ⊗ [0]∗ L . Here the first isomorphism is base change via the cartesian diagram X ×S X (id,φL )



X ×S X t

p2

/X

π′

 / X t.

φL

The second isomorphism holds by Remark 27.217. The third isomorphism holds by the projection formula, and the last isomorphism by flat base change using the cartesian diagram

707 m

G ×S G p2

 G

/G f

f

 / S.

In the following we denote the Euler characteristic (Definition 23.59) of a sheaf (or complex) F on a scheme X by χ(X, F ) rather than χ(F ) to avoid confusion. Corollary 27.251. Let X be an abelian variety over a field k and L be a line bundle on X. Then χ(X, Lφ∗L SX (L )) = (−1)g χ(X, L )2 . Proof. By Proposition 27.250 one has Lφ∗L SX (L ) ∼ = f ∗ Rf∗ L ⊗ L −1 . Hence we get χ(X, Lφ∗L SX (L )) = χ(X, f ∗ Rf∗ L ⊗ L −1 ) = χ(Spec k, Rf∗ L ⊗ Rf∗ L −1 ) = χ(Spec k, Rf∗ L )χ(Spec k, Rf∗ L −1 ) = χ(X, L )χ(X, L −1 ). Here the second equality holds by the projection formula (Proposition 23.65) and the third by the multiplicativity of Euler-Poincar´e characteristic (Example 23.60). As the dualizing sheaf on an abelian variety is trivial, we have H i (X, L −1 ) ∼ = H g−i (X, L )∨ by −1 g Serre duality (Corollary 25.129) and hence χ(X, L ) = (−1) χ(X, L ). Lemma 27.252. Let X be an abelian variety over a field k and let E be a vector bundle on X of rank r. Then χ(X t , SX (E )) = (−1)g r. Proof. We apply (27.46.2) to the dual abelian variety of X whose dual we identify with X and to ξ = e the zero section of X. Then we have RΓ(X t , SX (E )) ∼ = Le∗ E [−g] = e∗ E [−g] ∼ = k r [−g] because E is K-flat. Applying Euler characteristics, we obtain the desired equality. Theorem 27.253. (Riemann-Roch Theorem) Let k be a field, let f : X → Spec k be an abelian variety of dimension g over k, and let L be a line bundle on X. (1) One has χ(X, L ) = (L g )/g!, where (L g ) denotes the g-fold self intersection of L defined in Definition 23.70. (2) One has χ(X, L ) ̸= 0 if and only if L is non-degenerate. In this case (27.47.1)

χ(X, L )2 = deg(φL ).

708

27 Abelian schemes

Proof. Let us show (1). We use the Hirzebruch-Riemann-Roch Theorem (Corollary 23.113), where as additive cohomology theory we choose graded K-theory (Example 23.109). As the tangent bundle of X is trivial, its Todd class is 1 and we obtain χ(X, L ) = f∗ (ch(L )). P∞ ])n ])n ∈ grn K0 (X)Q . As We have ch(L ) = n=0 c1 ([L by Remark 23.114 where c1 ([L n! n! the direct image f∗ in graded K-theory is homogeneous of degree −g we find that f∗ c1 ([L ])n = 0 for n ̸= g and hence f∗ (ch(L )) = f∗

χ(c1 ([L ])g ) (L g ) c1 ([L ])g = = g! g! g!

by the definition of the intersection number. Next we show (27.47.1) if L is non-degenerate. Then Lφ∗L = φ∗L and one has deg(φL )χ(X t , SX (L )) = χ(X, Lφ∗L SX (L )) = (−1)g χ(X, L )2 . Here the first equality holds by Proposition 27.179 and the second by Corollary 27.251. As χ(X t , SX (L )) = (−1)g by Lemma 27.252, this shows (27.47.1). It remains to show that χ(X, L ) = 0 if L is not non-degenerate. For this we may assume that k is perfect. As K(L ) is a proper non-finite group scheme over k, K(L )0red is an abelian variety of positive dimension. In particular, there are finite subgroup schemes H of K(L ) of arbitrary large degree, e.g., the n-torsion points of K(L )0red for all n. g φ Then φL factors into X −→ X/H −→ X t and one has by Proposition 27.179 and Corollary 27.251 (−1)g χ(X, L )2 = χ(X, LΦ∗L SX (L )) = deg(g)χ(X/H, Lφ∗ SX (L )). Hence χ(X, L ) is divisible by arbitrary large numbers and hence χ(X, L ) = 0. (27.48) The index of non-degenerate line bundles. Lemma 27.254. Let X be an abelian variety over a field k of dimension g. Let L be a non-degenerate line bundle on X and let Λ(L ) be its Mumford bundle. Write hn (−) for dimk H n (−). Then for all n ≥ 0 one has ( X 0, if n ̸= g; p q −1 n (27.48.1) h (X, L )h (X, L ) = h (X × X, Λ(L )) = deg(φL ), if n = g. p+q=n Proof. By Proposition 27.229 one has Rn π∗′ P = 0 for n ̸= g and Rg π∗′ P = e′∗ k, where e′ is the zero section of X t . As φL is flat and (idX ×φL )∗ (P) = Λ(L ) we obtain by flat base change ( 0, if n ̸= g; n (*) R p2∗ Λ(L ) = ι∗ OK(L ) , if n = g

709 where ι : K(L ) → X is the inclusion. As K(L ) is finite, the initial terms of the Leray spectral sequence E2ij = H i (X, Rj p2∗ Λ(L )) ⇒ H i+j (X × X, Λ(L )) are all zero except for E20g which is of dimension deg(φL ). This shows the second equality of (27.48.1). By definition of the Mumford bundle and by the projection formula we have Rn p2∗ Λ(L ) = Rn p2∗ (m∗ L ⊗ p∗1 L −1 ) ⊗ L −1 ∼ = Rn p2∗ (m∗ L ⊗ p∗1 L −1 ), where the second isomorphism follows since Rn p2∗ Λ(L ) is supported on the finite kscheme K(L ) and any line bundle on a finite k-scheme is trivial. Again using the Leray spectral sequence for p2 we see the first of the following isomorphisms H n (X × X, Λ(L )) ∼ = H n (X × X, m∗ L ⊗ p∗1 L −1 ) ∼ H n (X × X, p∗ L ⊗ p∗ L −1 ) = 1 2 M p ∼ H (X, L ) ⊗ H q (X, L −1 ). = p+q=n

Here one obtains the second isomorphism by observing that σ : X × X → X × X, (x, x′ ) 7→ (x + x′ , x) is an isomorphism with σ ∗ (p∗1 L ⊗ p∗2 L −1 ) = m∗ L ⊗ p∗1 L −1 . This shows the first equality of (27.48.1). The lemma yields an alternative proof of (27.47.1) avoiding Fourier-Mukai transforms since it implies (−1)g χ(X, L )2 = χ(X, L )χ(X, L −1 ) = χ(X × X, Λ(L )) = (−1)g deg(φL ). Proposition 27.255. Let S be a scheme, let f : X → S be an abelian scheme, and let L be a non-degenerate line bundle on X. Then the formation of Rp f∗ L commutes with arbitrary base change T → S, and there exists a decomposition of S into open and closed subschemes a S= Si i≥0

such that Rp f∗ L |Si = 0 for p ̸= i and such that Ri f∗ L |Si is locally free of rank deg(φL )1/2 > 0. Proof. Let us first assume that S = Spec k for a field k. We have to show that there exists a unique integer i such that hi (L ) := dim H i (X, L ) ̸= 0. By (27.48.1) we have for all n≥0 ( X 0, if n ̸= g; p q −1 h (L )h (L ) = deg(φL ), if n = g. p+q=n This is only possible if there exists unique p and q such that hp (L ) ̸= 0 ̸= hq (L −1 ). Moreover, one has necessarily p + q = g. This shows in particular that χ(X, L −1 ) ̸= 0 and hence that L −1 is also non-degenerate by Theorem 27.253. Now let S be arbitrary. We may assume that the relative dimension of X over S is a constant integer g. For i = {0, 1, . . . , g} we set Si := { s ∈ S ; H i (Xs , Ls ) ̸= 0 }.

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27 Abelian schemes

By semicontinuity (Theorem 23.139) we know that Si is closed in S. By what we have already shown, we know that S is the disjoint union of the finitely many Si . Hence every Si is also open in S. Now we can use Corollary 23.143 and Proposition 23.142 to see that Ri f∗ L |Si is locally free, that Rp f∗ L |Si = 0 for p = ̸ i, and that their formations commute with base change. The compute the rank of Ri f∗ L |Si we can look at the fiber in some s ∈ Si . Then one has (dim H i (Xs , Ls ))2 = χ(Xs , Ls )2 = deg(φLs ) by Theorem 27.253. Definition 27.256. In the situation of Proposition 27.255 we call the locally constant function i(L ) : S → N that attaches to s ∈ S the i with s ∈ Si the index of L . If X has relative dimension g over S, then i(L ) takes values in {0, 1, . . . , g}. Remark 27.257. Let X be an abelian scheme of relative dimension g over a scheme S. If L is a non-degenerate line bundle, then the proof of Proposition 27.255 shows that L −1 is also non-degenerate and that i(L −1 ) = g − i(L ). The property of being non-degenerate for a line bundle is also open and closed on the base scheme. Proposition 27.258. Let X be an abelian scheme over a scheme S and let L be a line bundle on X. Let s0 ∈ S such that Ls0 := L |Xs0 is a non-degenerate line bundle on the fiber Xs0 . Then there exists an open and closed neighborhood U of s0 in S such that the restriction of L to X ×S U is non-degenerate of constant index equal to the index of i(Ls0 ). Proof. By Theorem 23.139 the map s 7→ χ(Xs , Ls ) is locally constant. Hence by Theorem 27.253 there exists an open and closed neighborhood V of s0 such that Ls is non-degenerate for all s ∈ V . Hence L |X×S V is non-degenerate (Remark 27.249). By Proposition 27.255 we find in V an open and closed neighborhood U of s0 such that the index of L is constant on U . Corollary 27.259. Let S be a scheme, let f : X → S be an abelian scheme, and let L be a non-degenerate line bundle on X of constant index i(L ). Then there exists a vector bundle E on X t such that SX (L ) ∼ = E [−i(L )]. In particular, i(L ) is the unique integer i such that H i (SX (L )) ̸= 0. Proof. Set i := i(L ). As φL is faithfully flat, it suffices to show that φ∗L S(L ) ∼ = E ′ [−i] ′ for some vector bundle E on X. But by Proposition 27.259 and by Proposition 27.255 one has φ∗L SX (L ) ∼ = f ∗ (Rf∗ L ) ⊗ L −1 ⊗ [0]∗ L = f ∗ (Ri f∗ L [−i]) ⊗ L −1 ⊗ [0]∗ L = (f ∗ (Ri f∗ L ) ⊗ L −1 ⊗ [0]∗ L )[−i] and Ri f∗ L is finite locally free. Proposition 27.260. Let S be a scheme, let f : X → Y be an isogeny of abelian schemes, and let L be a non-degenerate line bundle on Y . Then f ∗ L is non-degenerate and i(f ∗ L ) = i(L ).

711 Proof. One has φf ∗ L = f t ◦ φL ◦ f by Lemma 27.165 and hence φf ∗ L is a composition of three isogenies (Proposition 27.213) and therefore itself an isogeny. Hence f ∗ L is non-degenerate. To compute the index of f ∗ L we may assume that i(L ) is constant. Let E be a vector bundle on X t such that SY (L ) ∼ = E [−i(L )] (Corollary 27.259). As f is flat, Lf ∗ = f ∗ , t and as f is finite and in particular affine, Rf∗t = f∗ . Hence we have SX (f ∗ L ) ∼ = f∗t SY (L ) ∼ = f∗t E [−i(L )] by Proposition 27.244, and hence i(L ) is the unique integer i such that H i (SX (f ∗ L )) ̸= 0. This shows i(f ∗ L ) = i(L ). Lemma 27.261. Let S be a scheme, let X be an abelian scheme over S, and let L and M be line bundles on X such that the class of L ⊗ M −1 is in X t (S). If L is non-degenerate, then M is non-degenerate, and i(L ) = i(M ). Proof. As we have φL = φM , the first assertion is clear. To verify i(L ) = i(M ) we may base change to geometric points of S and hence can assume that S = Spec k for an algebraically closed field k. Then L and M are algebraically equivalent (Definition 27.231) and we find a connected k-scheme Z, k-valued points z, w ∈ Z(k) and a line bundle N on X ×k Z with N |X×{z} ∼ = L and N |X×{w} ∼ = M . By Proposition 27.258, N is a non-degenerate line bundle on the abelian scheme X ×k Z → Z and i(N ) is constant on Z as a function on Z. In particular i(M ) = i(N )(w) = i(N )(z) = i(L ). Proposition 27.262. Let X be an abelian scheme over a scheme S and let L be a nondegenerate line bundle on X. For all n > 0, L ⊗n is non-degenerate, and i(L ⊗n ) = i(L ). We will use below only the fact that there exist arbitrary large integers n such that i(L ⊗n ) = i(L ). Hence we will give the proof only in the case that n = m2 is a square. In this case we easily find an isogeny α : X → X such that one has an identity 2 α∗ (L ) = L ⊗m in NS(X), namely α = [m], see the proof below. In general one can reduce to this case by writing an arbitrary positive integer as a sum of four squares and by passing to X 4 , an idea that is often called Zarhin’s trick, see Exercise 27.16. Proof. [if n = m2 ] Consider the isogeny [m] : X → X. Then i([m]∗ L ) = i(L ) by Proposition 27.260. But by Proposition 27.184 (3) we have in PicX/S (S) 2

[m]∗ L = L ⊗m ⊗ N ,

with N ∈ X t (S).

2

Hence i([m]∗ L ) = i(L ⊗m ) by Lemma 27.261. Remark 27.263. In [Mum1] §16 it is shown that the index of a non-degenerate line bundle L on an abelian variety X can be computed by a Hilbert polynomial as follows. Fix an ample line bundle M on X and let PL ∈ Q[T ] be the Hilbert polynomial of L with respect to M , i.e. PL (n) = χ(X, L ⊗ M ⊗n ). This is a polynomial of degree g = dim(X). Then all complex roots of PL are real and i(L ) is the number of positive roots, counted with multiplicities.

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27 Abelian schemes

(27.49) Characterization of ample line bundles on abelian schemes. Theorem 27.264. Let S be a scheme and let f : X → S be an abelian scheme. Then a line bundle L on X is relatively ample over S if and only if L is non-degenerate and i(L ) = 0. Proof. As ampleness is an open condition on proper schemes (Theorem 24.46) we can check it on fibers. The property of being non-degenerate can also be checked on fibers by Remark 27.249. Hence we can assume that S is the spectrum of a field and it suffices to show the following more precise result. Proposition 27.265. Let k be a field, let X be an abelian variety over k, and let L be an line bundle on X. Then the following assertions are equivalent. (i) L is ample. (ii) L is non-degenerate and H 0 (X, L ) ̸= 0. (iii) There exists an effective divisor D ⊆ X such that L ∼ = OX (D) and X \ D is affine. In this case X \ D is affine for all effective divisors D such that L ∼ = OX (D). Proof. The implication (iii) ⇒ (i) has been shown in Proposition 27.174. Let us show that (i) implies (ii). Let L be ample. Then L is non-degenerate by Corollary 27.172. Moreover, any sufficiently high power L ⊗n is very ample and in particular globally generated. Hence i(L ⊗n ) = 0 and therefore i(L ) = 0 by Proposition 27.262. Now suppose that (ii) holds. As H 0 (X, L ) ̸= 0 there exists an effective divisor D on X such that OX (D) ∼ = L . Then L ⊗2 is globally generated by Lemma 27.175 and ∗ hence defines a morphism g : X → PN (1) ∼ = L ⊗2 . If we show that g is k such that g OPN k ⊗2 finite, then L is ample (Proposition 13.83) and hence L is ample (Proposition 13.50). −1 Moreover for every D as above we find a hyperplane H in PN (H) = 2D k such that g N and hence X \ D is the inverse image of the open affine Pk \ H and therefore X \ D is affine because g is affine. Hence it remains to show that g is finite. As g is proper as a morphism between proper schemes, it suffices to show that g has finite fibers (Corollary 12.89). For this we may assume that k is algebraically closed. Let e be the zero section of X considered as a closed point of X. By Corollary 27.107 all non-empty fibers of g are equi-dimensional of the same dimension. Hence it suffices to show that the reduced connected component F0 of g −1 (g(e)) containing e is contained in the finite scheme K(L ). Let x ∈ F0 (k). Let D be an effective divisor given by some non-zero section s ∈ H 0 (X, L ). Now g ◦ tx = g by Proposition 27.106 and hence s2 ∈ H 0 (X, L ⊗ ) and t∗x s2 have the same zero divisor. It follows that one has an equality of divisors t∗x D = D and in particular x ∈ K(L )(k). Corollary 27.266. Let X be an abelian variety over a field k and let L be an ample line bundle on X. Then H p (X, L ) = 0 for all p > 0 and in particular dim H 0 (X, L ) = χ(X, L ) = (deg φL )1/2 . More generally, by Proposition 27.255 one sees the following result. Corollary 27.267. Let L be a relatively ample line bundle on X. Then Rp f∗ L = 0 for all p ≥ 1 and f∗ L is finite locally free of rank deg(φL )1/2 and its formation is compatible with base change.

713 Using Proposition 27.258 one deduces from Theorem 27.264 that being ample is an open and closed condition on abelian schemes. Corollary 27.268. Let S be a scheme, let f : X → S be an abelian scheme, and let L be a line bundle on X. Let s0 ∈ S be a point, such that L |Xs0 is ample. Then there exists an open and closed neighborhood U of s0 such that L |X×S U is relatively ample over U . A special case of Lemma 27.261 is the following assertion. Corollary 27.269. Let S be a scheme, let X be an abelian scheme over S, and let L and M be line bundles on X such that the class of L ⊗ M −1 is in X t (S). Then M is ample if and only if L is ample. (27.50) Vector bundles on elliptic curves. Let us illustrate many of the previous notions for elliptic curves. As an application we will explain the classification of semistable vector bundles on elliptic curves. Here we follow [Pol] O Chap. 14, see also Bhatt’s lecture notes on abelian varieties [Bha] X . Recall that every vector bundle on an elliptic curve is isomorphic to a direct sum of semistable vector bundles by Theorem 26.156 and Proposition 26.161. We denote by k a field and by E an elliptic curve over k. Let us collect the information that we already know in this special case on line bundles. Let us recall some facts about modules on elliptic curves. Remark 27.270. As elliptic curves have genus 1, the Riemann-Roch theorem yields the following assertions. (1) Let L be a line bundle on E. The line bundle L is non-degenerate if and only if deg(L ) ̸= 0. In this case one has deg(L ) = χ(E, L ),

deg(φL ) = deg(L )2 .

Indeed, the first equality holds by Riemann-Roch (Proposition 26.46) and implies the second equality by (27.47.1). This implies the first assertion by Proposition 27.253 (2). Moreover, L is ample if and only if deg(L ) > 0 (Proposition 26.57). b (E) (2) More generally, by (26.23.5) one has for every F in Dcoh deg(F ) = χ(E, F ).

(27.50.1)

The zero section of E determines a line bundle OE ([0]) on E of degree 1 and by Part (1) of the previous remark, we obtain an isomorphism ∼

φOE ([0]) : E −→ E t ,

x 7→ OE ([x]) ⊗ OE ([0])⊗−1 ,

which we use to identify E and E t , cf. Proposition 27.148. Via this identification, for the Fourier-Mukai transforms we have SE = SE t and we can view the Fourier-Mukai equivalence as an auto-equivalence ∼

b b SE : Dcoh (E) → Dcoh (E)

with

SE (SE (F )) ∼ = [−1]∗ F [−1].

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27 Abelian schemes

We want to classify vector bundles on E using the Harder-Narasimhan stratification introduced in Section (26.25). We have already understood line bundles quite well: If we denote by Picd (E) the set of degree d ∈ Z line bundles on E, then one has E(k) ∼ = Pic1 (E) ∼ = Picd (E), where the first isomorphism is given by x 7→ OE ([x]) (Proposition 27.148) and the second is given by tensoring with OE ([0])⊗d−1 . In the language introduced in Section (26.24) this implies that the sets of isomorphism classes of rank 1 vector bundles of fixed slope (= degree) are in bijection to each other and also in bijection to the set of k-rational points of E. Attaching to x ∈ E(k) the skyscraper sheaf κ(x) = k with support in {x} we also obtain for every d ∈ Z a bijection between isomorphism classes of slope d line bundles and torsion OE -modules F with dimk H 0 (E, F ) = 1. This will be categorified and generalized to arbitrary vector bundles by the theorem of Atiyah below. Recall (Theorem 26.156) that every vector bundle E has a unique and functorial Q-filtration (HNλ (E ))λ such that grλHN (E ) is semistable of slope λ, which is called the Harder-Narasimhan filtration. Moreover, since elliptic curves have genus 1, the HarderNarasimhan filtration is (non-canonically) split (Proposition 26.161). Hence for every vector bundle E on E there exists a unique finite sequence of rational numbers λ1 > λ2 > · · · > λr and unique (up to isomorphism) semistable vector bundles Ei of slope λi such that r M E ∼ Ei , = i=1

where this isomorphism is in general not unique. Now for λ ∈ Q the abelian category (Vectλ (E)) of semistable vector bundles on E of slope λ can be described as follows. Let (Coh(E))tors be the category of coherent OE -modules with finite support. Theorem 27.271. (Atiyah) Let λ ∈ Q. Then there is an equivalence of abelian categories ∼

T : (Vectλ (E)) −→ (Coh(E))tors , such that for every semistable vector bundle E of slope λ we have (27.50.2)

dimk H 0 (E, T (E )) = gcd(deg(E ), rk(E )),

the greatest common divisor of deg(E ) and rk(E ). In fact, we will show that (I) there exists an equivalence between the abelian categories (Vectλ (E)) and (Vect0 (E)) preserving gcd(deg(E ), rk(E )) and that ∼ (II) the shifted Fourier-Mukai equivalence SE [1] induces an equivalence T : (Vect0 (E)) −→ (Coh(E))tors satisfying (27.50.2). We start by recalling the following descriptions of the fibers of the Fourier-Mukai transform, which is a special case of Remark 27.245. Let x ∈ E(k) be a k-rational point with corresponding degree zero line bundle Mx = OE ([x]) ⊗ OE ([0])⊗−1 ∈ Pic0 (E). b (E) we have Then for F in Dcoh

(27.50.3)

L SE (F ) ⊗L OE κ(x) = RΓ(E, F ⊗OE Mx ), L F [−1] ⊗L OE κ(x) = RΓ(E, SE (F ) ⊗OE Mx )

b Recall from Remark 26.140 that we can define degree and rank for objects in Dcoh (E). Let us show how these numbers behave under Fourier-Mukai equivalence.

715 b (E) we have Lemma 27.272. For F in Dcoh

deg(SE (F )) = − rk(F ),

rk(SE (F )) = deg(F ).

Proof. Using (27.50.1), we obtain from (27.50.3) applied to x = 0 ∈ E(k) (and viewing it as a morphism Spec k → E) that (∗)

deg(SE (F )) = χ(E, SE (F )) = χ(L0∗ F [−1]) = −χ(L0∗ F ) = − rk(F ), (∗) rk(SE (F )) = χ(L0∗ SE (F )) = χ(E, F ) = deg(F ), where the equalities (*) hold by (26.23.4). The key step in the proof of Atiyah’s theorem is the following. Proposition 27.273. Let λ ∈ Q. Then the Fourier-Mukai transform induces equivalences ∼

SE [1] : (Vectλ (E)) −→ (Vect−λ−1 (E)), ∼

SE : (Vectλ (E)) −→ (Vect−λ−1 (E)),

if λ < 0 if λ > 0

∼

SE [1] : (Vect0 (E)) −→ (Coh(E))tors ∼

SE : (Coh(E))tors −→ (Vect0 (E)). Proof. Recall that SE ◦ SE [1] ∼ = [−1]∗ by Fourier-Mukai equivalence. Since [−1]∗ preserves degrees of line bundles and one has det([−1]∗ F ) = [−1]∗ det(F ) for every vector bundle F , we see that [−1]∗ preserves the degree of vector bundles. Clearly it also preserves the rank of vector bundles. Hence we have µ([−1]∗ F ) = µ(F ) for every vector bundle F = ̸ 0. Since [−1]∗ also induces a bijection between subbundles of F and of [−1]∗ F , we see that [−1]∗ is an auto-equivalence of (Vectλ (E)) for all λ ∈ Q. Hence to show the equivalences it suffices to show the following assertions. (1) If a vector bundle F on E is semistable of slope λ with λ < 0 (resp. with λ > 0), then SE (F )[1] (resp. SE (F )) is a semistable vector bundle of slope −λ−1 . (2) If F is a vector bundle of slope 0, then SE (F )[1] ∈ (Coh(E))tors . (3) If F is in (Coh(E))tors , then SE (F ) is a semistable vector bundle of slope 0. By Proposition 26.159 and since the Fourier-Mukai equivalence is compatible with base change, we may assume that k is algebraically closed. Let us show (1). Let F be a semistable vector bundle of slope λ ̸= 0. We write F as a finite direct sum of indecomposable vector bundles Fi . Then each summand is again semistable of slope λ (Remark 26.149 (3)) and SE (F )[1] (resp. SE (F )) is the direct sum of the SE (Fi )[1] (resp. SE (Fi )). As finite direct sums of semistable vector bundles of the same slope −λ−1 are again semistable of slope −λ−1 by Proposition 26.152, we may assume that F is indecomposable. As SE is an equivalence, it follows that SE (F ) is also indecomposable. Now suppose that λ < 0. We find that for all L ∈ Pic0 (E) one has H 0 (E, F ⊗ L ) = HomOE (L ⊗−1 , F ) = 0 by Proposition 26.150 because L ⊗−1 is semistable of slope 0. As dim E = 1, we see that H i (E, F ⊗ L ) = 0 for all i = ̸ 1. By (27.50.3) it follows that H i (SE (F ) ⊗L OE κ(x)) = ̸ 1. Therefore SE (F ) is of tor-amplitude in [1, 1] 0 for every x ∈ E(k) and all i = by Proposition 23.126 (here we use that k is algebraically closed), i.e., SE (F )[1] is a vector bundle. Its slope is −λ−1 by Lemma 27.272. It is semistable because SE (F )[1] is indecomposable by Proposition 26.161.

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27 Abelian schemes

If λ > 0, then one can argue similarly. As the canonical bundle ωE on E is trivial, Serre duality (Corollary 25.129) yields, for all L ∈ Pic0 (E), that H 1 (E, F ⊗ L ) = H 0 (E, F ∨ ⊗OE L ∨ )∨ = HomOE (L , E ∨ )∨ = 0, since µ(E ∨ ) = −λ < 0 by Remark 26.143 (2). As above it follows that SE (F ) is a semistable vector bundle of slope −λ−1 . Let us show (2). Let F = ̸ 0 be a semistable vector bundle of slope 0. We will argue by induction on rk(F ). If F is a line bundle of degree 0, then F = My for some y ∈ E(k). By (27.50.3) we have for all x ∈ E(k) with x ̸= −y that SE (My ) ⊗L κ(x) = RΓ(E, Mx+y ) = 0, where the second equality follows from Lemma 27.197. Hence we have SE (My )|E\{−y} = 0, in particular SE (My ) is supported on an affine closed subscheme whose underlying topological space is {−y}. By (27.50.3) with x = 0 for its global sections we obtain RΓ(E, SE (My )) = My [−1] ⊗OE k ∼ = k[−1], which shows that (27.50.4)

SE (My ) = κ(−y)[−1].

In particular SE (My ) ∈ (Coh(E))tors . Now consider F of rank > 1. We claim that there exists a line bundle M of degree 0 and a non-zero map M → F . Otherwise we would have H 0 (E, E ⊗ L ) = HomOE (L ∨ , E ) = 0 for all L ∈ Pic0 (E) and hence as in the argument for λ < 0 one sees that SE (F )[1] is a vector bundle. By Lemma 27.272 we have rk(SE (F )[1]) = − rk(SE (F )) = − deg(F ) = 0. Hence SE (F ) = 0 and therefore F = 0 since SE is an equivalence. This contradiction shows the claim. So let u : M → F be non-zero with M a line bundle of degree 0. Then u is injective since rk(M ) = 1 and we can consider M as a submodule of F . If M was not a subbundle, then its saturation would have slope > 0 (Remark 26.145) which is not possible, since F is semistable of slope 0. Therefore F /M is a vector bundle which is semistable of slope 0 by Proposition 26.152. By induction, SE (M ) and SE (F /M ) are rank 0 coherent sheaves, and hence the same holds for SE (F ). It remains to show (3). Every torsion coherent sheaf F is an iterated extension of structure sheaves of closed points. Using that SE ◦ SE = [−1]∗E [−1], (27.50.4) shows that for every closed point y ∈ E(k) one has S(κ(y)) = [−1]∗ M−y = My . Hence SE (F ) is an iterated extension of degree 0 line bundles and hence SE (F ) ∈ (Vect0 (E)) by Proposition 26.152. Proof. [of Theorem 27.271] Let λ ∈ Q be any rational number. We will show that there is an equivalence of (Vectλ (E)) and (Vect0 (E)) preserving gcd(deg(E ), rk(E )) obtained by repeatedly applying SE [1] and (−) ⊗ OE (0)⊗d for d ∈ Z. First note that the functors SE [1] and SE preserve gcd(deg(E ), rk(E )) by Lemma 27.272. The functor (−) ⊗ OE (0) preserves it since if ab is a rational number, then gcd(a, b) = gcd(a + b, b).

717 Applying SE [1] and (−) ⊗ OE (0)⊗d allows us to replace λ by −λ−1 if λ = ̸ 0 and λ by λ + d. Let us show that any rational number λ can be transformed into 0 by these operations. Let λ ̸= 0. We argue by induction on bλ , where λ = abλλ with aλ and bλ unique coprime integers with bλ ≥ 1. If bλ = 1, λ is an integer and the claim is clear. If bλ > 1 we may add same integer to λ so that −1 < λ < 0. Then −bλ < aλ < 0 and hence b−λ−1 = −aλ < bλ so that we know by induction hypothesis that −λ−1 can be transformed into 0. ∼ It remains to show (27.50.2) for the functor SE [1] : (Vect0 (E)) → (Coh(E))tors , i.e., that dim H 0 (E, SE (F )) = gcd(rk(F ), deg(F )) = rk(F ) for a semistable vector bundle F of slope 0 and hence of degree 0. Since we have SE (SE (F [1])) = [−1]∗ F we find rk(F ) = rk([−1]∗ F ) = deg SE (F )[−1] = χ(E, SE (F )[−1]) = dim H 0 (SE (F )[−1]), where the second equality holds by Lemma 27.272, the third by (27.50.1), and the last uses that SE (F )[−1] is a coherent sheaf with finite support and hence H i (E, SE (F )[−1]) = 0 for i ̸= 0. We obtain the following results about simple and indecomposable vector bundles. Remark 27.274. Every torsion coherent sheaf on E is a finite direct sum of skyscraper sheaves OE,x /(πx )e concentrated in {x}, where x ∈ E is a closed point, πx is a uniformizing element of the discrete valuation ring OE,x , and where e ≥ 1 is an integer. These skyscraper sheaves are indecomposable. The simple objects of (Coh(E))tors are the skyscraper sheaves κ(x) for x ∈ E closed. Fix λ ∈ Q. (1) Via the equivalence of (Vectλ (E)) and (Coh(E))tors , the skyscraper sheaves κ(x) for x ∈ E closed correspond to the simple objects of (Vectλ (E)), which are the stable vector bundles of slope λ by Proposition 26.152. This sets up a bijection ∼

{closed points of E} −→ {stable vector bundles of slope λ},

x 7→ Fλ (x).

The greatest common divisor of the rank and the degree of Fλ (x) is [κ(x) : k]. (2) Let F be an indecomposable vector bundle and let λ be its slope. Then F is semistable by Proposition 26.161. Via the equivalence of (Vectλ (E)) and (Coh(E))tors , it corresponds to the skyscraper sheaf OE,x /(πx )e for a unique closed point x and a unique integer e ≥ 1. Moreover, gcd(rk(F ), deg(F )) = e[κ(x) : k]. Hence we obtain a bijection ∼

{closed points of E} −→ {indecomposable vector bundles of slope λ}. Now OE,x /(πx )e has decomposition series of length e whose graded pieces are all fλ (x) of slope λ isomorphic to κ(x). Therefore the indecomposable vector bundle F corresponding to the closed point x has a decomposition series whose graded pieces are all isomorphic to the stable vector bundle Fλ (x). The length e of this decomposition series is given by fλ (x)), deg(F fλ (x))) gcd(rk(F . e= [κ(x) : k] Corollary 27.275. Let F be a vector bundle on E. Then F is stable if and only if EndOE (F ) is a skew field. In this case EndOE (F ) is a finite commutative field extension of k.

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27 Abelian schemes

Proof. If F is stable, say of slope λ, then F corresponds to the skyscraper sheaf κ(x) for x ∈ E closed via the equivalence T : (Vectλ (E)) ∼ = (Coh(E))tors (Remark 27.274) and EndOE (F ) = EndOE (κ(x)) = κ(x). Conversely, suppose that EndOE (F ) is a skew field. Then F is necessarily indecomposable and therefore semistable (Proposition 26.161). Let λ be its slope. Hence F corresponds via T to an indecomposable coherent torsion sheaf on E, i.e., to a skyscraper sheaf OE,x /mex for some x ∈ E closed and e ≥ 1. Then EndOE (F ) = EndOE (OE,x /mex ) = OE,x /mex which is a skew field if and only if e = 1, i.e., if and only if F is stable (Remark 27.274). (27.51) Very ample line bundles on abelian schemes. In this section we show that the tensor product of any three relatively ample line bundles on an abelian scheme is already very ample. We will need some properties of divisors on an abelian variety X over an algebraically closed field k. Recall that an abelian variety over a field is regular and hence the notions of (Cartier) divisor and of Weil divisor coincide. For x, y ∈ X(k) and a divisor D one has Supp(t∗x D) = t−x (Supp(D)); sometimes we denote this set by Supp(D) − x. We have y ∈ Supp(t∗x D) ⇔ x ∈ Supp(t∗y D). Moreover, if L is a line bundle on X we denote by |L | := P(Γ(X, L )∨ )(k) the corresponding complete linear system, i.e., the projective space of effective Cartier divisors D with OX (D) ∼ = L (Proposition 11.34 and Section (13.13)). Recall that L is globally generated if for all x ∈ X(k) there exists a D ∈ |L | such that x ∈ / Supp(D). Lemma 27.276. Let k be an algebraically closed field, let X be an integral k-scheme of finite type, let D be a Cartier divisor on X and let L = OX (D) be the corresponding line bundle. Let |L | := P(Γ(X, L )∨ )(k) be the linear system given by D. Then there exists an open subscheme U ⊆ P(Γ(X, L )∨ ) such that U (k) is the subset of points of |L | corresponding to reduced divisors. Proof. We will construct a divisor that is “universal” for the complete linear system |L |, ˜ ⊂ X ×k P(Γ(X, L )∨ ) such that for every p ∈ P(Γ(X, L )∨ )(k), i.e., an effective divisor D ˜ over p is the effective divisor in X = X × {p} (considered as a the schematic fiber of D ˜ is proper closed subscheme) corresponding to p. The lemma follows from this because D ∨ O and flat over P(Γ(X, L ) ) and we can apply (11) in Section (E.1), [EGAIV] (12.2.1). ˜ write P = P(Γ(X, L )∨ ) and let M = L ⊠ OP (1) denote the exTo construct D, terior tensor product, i.e., the tensor product of the two pullbacks along the projections. Then Γ(X × P, M ) = Γ(X, L ) ⊗k Γ(X, L )∨ which we can canonically identify with Endk (Γ(X, L )). Denote by s ∈ Γ(X × P, M ) the element corresponding to id ∈ Endk (Γ(X, L )). Then s defines a Cartier divisor in X × P . Fix p˙ ∈ Γ(X, L ) \ {0} ˜ over p is the Cartier and let p ∈ P be the point determined by p. ˙ Then the fiber of D divisor given by the line bundle L and the section obtained as the image of id under the map Endk (Γ(X, L )) = Γ(X, L ) ⊗k Γ(X, L )∨ −→ Γ(X, L ),

t ⊗ λ 7→ λ(p)t, ˙

and this is just p. ˙ The point in |L | corresponding to this divisor is p, as desired.

719 Lemma 27.277. Let k be an algebraically closed field and let X be an abelian variety over k. Let D be an effective divisor on X and let L = OX (D) be the line bundle attached to D. (1) The subset of points of |L | corresponding to reduced effective divisors is a dense open subset. (2) Assume in addition that D is ample. Then for all D′ in an open dense subset of |L | and all 0 ̸= x ∈ X(k), one has t∗x D′ ̸= D′ . Note that in part (2) of the lemma we speak about inequality of divisors (as opposed to the two sides not being linearly equivalent). Proof. Let us show (1). Suppose that D = rE + FPwith E irreducible, r > 1, and F ≥ 0 r ∗ some divisor. Then P rE is linearly equivalent to i=1 txi E for all families of sections xi ∈ X(k) with i xi = 0 (Remark 27.169 (3)). For a suitable choice of the xi the t∗xi E are all distinct and distinct from the other components of D. This shows that the subset in question is non-empty. In view of Lemma 27.276 it is thus open and dense. Next we prove (2). Since t∗x D′ = D′ implies t∗x L ∼ = L and hence x ∈ K(L )(k), we can fix one of these finitely many x. For the finiteness assertion we use Corollary 27.172 and the assumption that L is ample. We will now show that every reduced effective divisor D′ in |L | with t∗x D′ = D′ lies in a union of linear subspaces of strictly smaller dimension. To avoid introducing further notation, let us assume that D is itself reduced and show this claim for D. The union of linear subspaces that we will find will be independent of D, so this finishes the proof. Let G ⊆ K(L ) be the finite group generated by x, considered as a finite ´etale group scheme over k, and consider the finite ´etale isogeny of abelian ¯ := π(Supp D) is a closed subset varieties π : X → X/G of degree d := #G > 1. Then D of pure codimension one in X/G which we consider as a reduced effective divisor. As π is ¯ is again reduced with the same support as D and hence D = π −1 (D) ¯ and ´etale, π −1 (D) ∗ ¯ ¯ D). D L ∼ π O ( As D is ample, is ample as well (Proposition 13.66) and hence one = X/G has dim H 0 (X, L ) = χ(X, L ) ¯ = dχ(X/G, OX/G (D)) ¯ = d dim H 0 (X/G, OX/G (D)) 0 ¯ > dim H (X/G, OX/G (D)). Here the first and the last equality hold by Corollary 27.266, and the second equality ¯ considered as a point in |L |, D lies in by Proposition 27.179. Since D = π −1 (D), ∗ 0 ¯ π H (X/G, OX/G (D)). By Corollary 27.215 there are only finitely many isomorphism classes of line bundles M on X/G such that π ∗ M ∼ = L . The union in H 0 (X, L ) of their global sections is the finite union of lower-dimensional subspaces we wanted to construct. Proposition 27.278. Let S = Spec R be an affine scheme and let f : X → S be an abelian scheme. Let L1 and L2 be ample line bundles on X. Then L1 ⊗ L2 is globally generated. Proof. Let L := L1 ⊗ L2 . In view of Proposition 13.30 we have to show that f ∗ (f∗ L ) → L is surjective. By Nakayama’s lemma we can do this on fibers and hence can assume that S = Spec k for a field k. By faithfully flat descent we can in addition assume that k is algebraically closed.

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Let x ∈ X(k). We have to show that there exists D ∈ |L | such that x ∈ / D. Consider the homomorphism Φ := mX t ◦ (φL1 × φL2 ) : X ×k X −→ X t , (x1 , x2 ) 7−→ t∗x1 (L1 ) ⊗ L1−1 ⊗ t∗x2 (L2 ) ⊗ L2−1 . As L1 and L2 are ample, Φ is surjective and therefore dim Ker(Φ) = g := dim(X). For x0 ∈ X(k) we have Ker(Φ) ∩ ({x0 } × X) ∼ = φ−1 L2 (−φL1 (x0 )) and similarly for Ker(Φ) ∩ (X × {x0 }). Therefore (*)

dim(Ker(Φ) ∩ ({x0 } × X)) = dim(Ker(Φ) ∩ (X × {x0 })) = 0

because φLi is finite for i = 1, 2. As L1 and L2 are ample, we can choose effective divisors Di ∈ |Li | (Proposition 27.265). By (*) we have dim(Ker(Φ) ∩ (t∗x D1 × X)) = dim(Ker(Φ) ∩ (X × t∗x D2 )) = g − 1. Therefore we find (x1 , x2 ) ∈ Ker(Φ)(k) such that (x1 , x2 ) ∈ / (t∗x D1 × X) + (X × t∗x D2 ) and hence x1 ∈ / t∗x D1 and x2 ∈ / t∗x D2 . This shows x∈ / D := t∗x1 D1 + t∗x2 (D2 ). As (x1 , x2 ) are in the kernel of Φ we have t∗x1 L1 ⊗ t∗x2 L2 ∼ = L1 ⊗ L2 = L and hence D ∈ |L |. Theorem 27.279. Let S be a scheme, and let X → S be an abelian scheme. Let L1 , L2 , L3 be relatively ample line bundles on X. Then L1 ⊗ L2 ⊗ L3 is very ample over S. Proof. (I). We may assume that S is affine. Set L := L1 ⊗ L2 ⊗ L3 . We know by Proposition 27.278 that L is globally generated. In particular the corresponding morphism of S-schemes r : X → P(f∗ L ) is defined on all of X. Now f∗ L is a vector bundle on S (Corollary 27.267) and hence r is a morphism of proper S-schemes. We have to show that r is a closed immersion (Proposition 13.56). This can be done on fibers (Proposition 12.93) and hence we may assume that S = Spec k for a field k. By faithfully flat descent we can in addition assume that k is algebraically closed. By Proposition 12.94 it suffices to show that r is injective on k-valued points and on tangent spaces in k-valued points. (II). Let us now show that r is injective on k-valued points. Let y1 , y2 ∈ X(k) such that r(y1 ) = r(y2 ), so that for all D ∈ |L | one has (*)

y1 ∈ Supp(D) ⇔ y2 ∈ Supp(D).

As H 0 (X, L1 ) ̸= 0 (Proposition 27.265), there exists an effective divisor D1 ∈ |L1 |. We choose D1 reduced and as in Lemma 27.277 (2). To prove that y1 = y2 , it then suffices to show that

721 (**)

(Supp D1 − y1 ) ⊆ (Supp D1 − y2 ).

In fact, (**) implies t∗y1 D1 = t∗y2 D1 because D1 is reduced and Lemma 27.277 (2) shows that y1 = y2 . Let x1 be any k-valued point of Supp D1 − y1 . Then L ⊗ t∗x1 L1−1 ∼ = L2 ⊗ L3 ⊗ (L1 ⊗ t∗x1 L1−1 ) is the tensor product of the ample line bundles L2 and L3 ⊗ (L1 ⊗ t∗x1 L1−1 ) (Corollary 27.269) and hence is globally generated by Proposition 27.278. Therefore we find an effective divisor D′ ∈ |L ⊗ t∗x1 L1−1 | such that y2 ∈ / D′ . The divisor tx∗1 D1 + D′ ∈ |L | ′ contains y1 and hence y2 by (*). By our choice of D we must have y2 ∈ Supp(tx∗1 D1 ), i.e. x1 ∈ Supp(t∗y2 D1 ). This shows (**). (III). Assume that r is not injective on tangent spaces. This means that there exist a point x0 ∈ X(k) and a tangent vector 0 ̸= ξ ∈ Tx0 (X) = (mx0 /m2x0 )∨ such that for all effective divisors D ∈ |L | with x0 ∈ Supp(D) the tangent vector ξ is tangential to D at x0 , i.e., if s = 0 is a local equation of D at x0 with differential ds ∈ x∗0 ΩX/k = mx0 /m2x0 at x0 , then ⟨ξ, ds⟩ = 0. Recall that the tangent bundle of X is free. Therefore the tangent vector ξ corresponds to a translation invariant section Σ of the tangent bundle (Section (17.6)) such that Σx0 = ξ. As in Step (II) choose D1 ∈ |L1 | reduced, and let x1 ∈ Supp(t∗x0 D1 ). As before there exists D′ ∈ |L ⊗ t∗x1 L1−1 | such that x0 ∈ / D′ . Then D := D′ + t∗x1 D1 ∈ |L | and hence ξ is tangent to D at x0 by assumption. As x0 ∈ / D′ , ξ = Σx0 must be tangent to t∗x1 D1 at x0 , i.e., Σx0 +x1 is tangent to D1 in x0 + x1 . Since x1 ∈ Supp(D1 ) − x0 was arbitrary, we conclude that Σx is tangent to D1 at every point x ∈ Supp(D1 ). We may consider Σ as a global derivation OX → OX . We claim that it preserves the ideal sheaf defining D1 , in other words, that the following property holds. (T) For U ⊆ X affine open and for all s ∈ OX (U ) such that D1 ∩ U = div(s) one has Σ(s) ∈ (s). Since D1 is reduced, the ideal (s) is a radical ideal, so to show that it contains Σ(s), it is enough to show that Σ(s) lies in every maximal ideal m ⊆ OX (U ) in the support of D1 ∩ U . Since the restriction of Σ to a map m → OX (U ) induces, by tensoring with the residue class field, the map Σx : m/m2 → κ(m), the vanishing of Σx (s) implies that Σ(s) ∈ m, as desired. So when we consider Σ0 as a k[ε]/(ε2 )-valued point of X whose underlying topological point is 0, the translation tΣ : Xk[ε]/(ε2 ) → Xk[ε]/(ε2 ) preserves the divisor D1,k[ε]/(ε2 ) and a fortiori preserves the associated line bundle. This implies Σ0 ∈ T0 (K(L1 )) = Lie(K(L1 )). If the characteristic of k is zero, then all algebraic groups over k are smooth (Theorem 27.25) and the finite group scheme K(L1 ) is ´etale and therefore Lie(K(L1 )) = 0, a contradiction. Hence we can from now on assume that char(k) = p > 0. Let H ⊆ K(L )0 be the smallest group scheme with Σ0 ∈ Lie(H). Consider the isogeny π : X → X/H which is an H-torsor. The scheme-theoretic action of H on X by translation preserves D1 by (T), i.e., the locally free ideal sheaf ID1 of rank 1 defining D1 as a closed subscheme is H-equivariant. By descent along torsors (Section (14.21)) we obtain a locally free ideal sheaf of OX/H that defines a divisor D1′ on X/H such that π ∗ D1′ = D1 . As in the proof of Lemma 27.277 we compute dim H 0 (X, L1 ) = deg(π) dim H 0 (X/H, OX/H (D1′ )) > dim H 0 (X/H, OX/H (D1′ ))

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27 Abelian schemes

and conclude that all global sections of L1 which define reduced divisors lie in a finite set of proper subspaces. This contradicts Lemma 27.277 (1). (27.52) Symmetric homomorphisms and polarizations. Let S be a scheme and let X be an abelian scheme over S. We want to introduce the notion of polarization of an abelian scheme. Over an algebraically closed field, a polarization is an equivalence class of ample line bundles. For an abelian variety X over the complex numbers, we can view a polarization as a positive definite hermitian form on its Lie algebra, i.e., the tangent space at the zero element, with a certain integrality property; see Section (27.54). In general, polarizations can be seen, in some sense, as analogues of symmetric bilinear forms satisfying a positivity condition. More concretely, polarizations are indispensable for the construction of well-behaved moduli spaces of abelian varieties, see Section (27.55). Definition 27.280. A homomorphism λ : X → X t is called (1) symmetric if λ = λt , identifying X with X tt , (2) a polarization if for every geometric point s¯ → S there exists an ample line bundle L on Xs¯ such that λs¯ = φL . A polarization λ is called a principal polarization if λ is an isomorphism. Given a polarization λ, for every geometric point s¯ → S the morphism λs¯ is an isogeny, hence λ is an isogeny. Lemma 27.281. Let λ : X → X t be a homomorphism of abelian schemes. (1) Let s ∈ S be a point such that λs¯ is symmetric for some geometric point s¯ → S with image s. Then there exists an open and closed neighborhood U of s such that λU : X ×S U → X t ×S U is symmetric. (2) If λ is of the form φL for some line bundle L on X, then λ is symmetric. (3) The homomorphism λ is symmetric if and only if there exists a line bundle L on X such that φL = 2λ. Proof. Let us show (1). By hypothesis, λs¯ − λts¯ factors through the zero section of Xs¯t . Hence we also have on fibers λs − λts = 0 since equality of morphisms can be checked after faithfully flat base change. As the constancy locus of λ − λt is open and closed (Proposition 27.96), there exists an open and closed neighborhood U of s such that λU − λtU factors through a section. As λ − λt is a homomorphism of group schemes, that section is necessarily the zero section. Hence λU is symmetric. Assertion (2) holds because the automorphism σ of X ×S X that switches the factors satisfies σ ∗ Λ(L ) ∼ = Λ(L ), where Λ(L ) denotes the Mumford bundle. It remains to show (3). The condition is sufficient by (2). Conversely, for every homomorphism λ : X → X t set L := (idX , λ)∗ PX . If λ is symmetric, we have φL = λ + λt = 2λ, where the first equality follows from Proposition 27.220 and (27.42.4). Corollary 27.282. Polarizations are symmetric. Remark 27.283. One can show that for every line bundle L on X and for every integer n ≥ 2 there exists an fppf-covering (Ui → S)i such that the pullback of L to X ×S Ui is isomorphic to Mi⊗n for line bundles Mi on X ×S Ui ([DePa] O 1.2 and its proof).

723 Hence Lemma 27.281 (3) implies that every symmetric homomorphism φ is fppf-locally of the form φM for some line bundle M . If φ is a polarization, then M is necessarily ample. In fact this holds even locally for the ´etale topology (loc. cit. 1.3). One can use this remark to show the following two results. Proposition 27.284. A homomorphism of abelian schemes λ : X → X t is a polarization if and only if λ is symmetric and (idX , λ)∗ PX is a relative ample line bundle on X over S. Proof. By Lemma 27.281 (1) and since we can check relative ampleness over proper schemes on fibers (Theorem 24.46) and hence on geometric fibers (Proposition 14.58), we may assume that S = Spec k for an algebraically closed field k. We set L := (idX , λ)∗ PX . Let λ be a polarization, i.e., of the form φM for an ample line bundle M on X. Then φM ⊗2 = 2λ = φL . Therefore M ⊗2 and L are algebraically equivalent and hence L is ample. Conversely suppose that L is ample and that λ is symmetric. Then λ = φM for some line bundle M on X by Remark 27.283 and as above M ⊗2 is algebraically equivalent to L . Hence M is ample and λ is a polarization. Corollary 27.285. Let λ : X → X t be a homomorphism of abelian schemes and let s ∈ S be a point such that λs¯ is polarization for some geometric point s¯ → S with image s. Then there exists an open and closed neighborhood U of s such that λU : X ×S U → X t ×S U is a polarization. Proof. By Proposition 27.284 this follows from Lemma 27.281 (1) and from Corollary 27.268. In general, given an abelian scheme there might not exist any polarization. For an abelian variety over a field, a polarization always exists, because every abelian variety is projective. But even if the base field is algebraically closed, a principal polarization need not exist. Remark 27.286. It follows from Proposition 27.148 that every elliptic curve carries a canonical principal polarization. Remark 27.287. Let S be a scheme and let f : C → S be a proper smooth curve with geometrically connected fibers. Then we have the Jacobian JacC = Pic0C/S of C as in Section (27.26). One can show that the abelian scheme JacC carries a principal polarization. See [MFK] O Prop. 6.9 and [Mil2] O for a proof of this fact. If f has a section, then we have the Abel morphism C → JacC , (27.26.2). Pullback of line bundles along this morphism induces a morphism Jac∨ C → JacC , and one can show that this morphism is an isomorphism. The famous Theorem of Torelli states that given curves C, C ′ over a field k such that there exists an isomorphism between their Jacobians which is compatible with their canonical principal polarizations, then C and C ′ are isomorphic. For the case where k is algebraically closed, see [Mil2] O Sections 12, 13 and the references given in the section at the end of that paper. See the appendix by Serre in [Lau] X for a discussion of the result over an arbitrary field.

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27 Abelian schemes

Remark 27.288. Let S be a scheme and let f : X → S, g : Y → S be smooth proper relative curves over S with geometrically connected fibers. Let us also assume that X and Y are pointed S-schemes, i.e., are equipped with a fixed section in X(S) and Y (S), respectively. As in Lemma 27.155, we can identify divisorial correspondences in CorrS (X, Y )(S) with morphisms X → Pic0Y /S of pointed schemes. Using the Albanese property of (Pic0X/S )∨ (Remark 27.225) and the fact that Pic0X/S ∼ = (Pic0X/S )∨ canonically (Remark 27.287), we obtain a morphism Pic0X/S → Pic0Y /S between the Jacobian of X and the Jacobian of Y . Using the same reasoning after base change to any S-scheme T , in this way we obtain isomorphisms ∼

∼

Corr(X, Y ) → Hom0 (X, Pic0Y /S ) → Hom(Pic0X/S , Pic0Y /S ). In fact, Lemma 27.155 implies that the first map is an isomorphism. The natural map X → (Pic0X/S )∨ ∼ = Pic0X/S induces an inverse of the second map. (27.53) Projectivity of abelian schemes over normal base schemes. Recall that in Proposition 27.174 we proved that every abelian variety (i.e., abelian scheme over a field) is projective. Equivalently, in view of the properness, there exists an ample line bundle. In this section we will generalize this result to abelian schemes over any normal noetherian scheme S. Lemma 27.289. Let f : Z → S be a formally proper (Definition 27.128) morphism of schemes. Then the image f (Z) is closed under specialization. Proof. Let s be a point of f (Z) and let t ∈ S be a specialization of s. We have to show that t is in the image of f . We may assume that t ̸= s. Let z ∈ Z with f (z) = s. By Proposition 15.7 there exists a valuation ring R with field of fraction K = κ(z) and a morphism g : Spec R → S such that g(η) = s and g(x) = t, where η (resp. x) is the generic point (resp. special point) of Spec R. As f is formally proper, there exists a morphism g˜ : Spec R → Z with f ◦ g˜ = g. In particular, t lies in the image of f . Lemma 27.290. Let f : Z → S be a separated, unramified, and formally proper (Definition 27.128) morphism of schemes. Suppose that S is locally noetherian and normal. Let v : S 99K Z be a rational section of f . Then v is defined on all of S. Proof. We may assume that S is connected and hence integral. Let U := dom(v) be the domain of definition of v. As Z → S is separated there exists a representative U → Z of v which we again call v (Proposition 9.27). We have to show that U = S. Assume that there exists s ∈ S\U . As v is a section of the unramified morphism f |f −1 (U ) , it is an open immersion v : U → f −1 (U ). Let Y be the closure of v(U ) in X, endowed with the reduced subscheme structure. Then Y is an integral scheme. Let f ′ := f |Y : Y → S. Then f ′ is unramified, separated, formally proper, and birational. The image of f ′ contains the generic point of S and is stable under specialization by Lemma 27.289. Hence f ′ is surjective and its fiber Ys := f −1 (s) ∩ Y in s is non-empty. Let y ∈ Y with f ′ (y) = s. Let V ⊆ Y be an open affine neighborhood of y. Then f ′ |V : V → S is quasi-compact, because S is locally noetherian and in particular quasi-separated. Therefore f ′ |V is quasi-finite (Corollary 18.28), separated, and birational. Hence it is an open immersion by Zariski’s main theorem (Corollary 12.88). But this implies that the section v can be extended to an open neighborhood of s. Contradiction.

725 Theorem 27.291. Let S be a normal noetherian scheme and let X be an abelian scheme over S. Then X → S is projective. Proof. We may assume that S is connected and hence integral. Let η ∈ S be its generic point. By Proposition 27.174 there exists an ample line bundle M on the generic fiber Xη . By Lemma 24.45 there exists then also an open non-empty subscheme U of S and a relatively ample line bundle M ′ on X ×S U whose restriction to Xη is M . The associated homomorphism φM ′ can be considered as a section of CorrS (X, X) over U . By Remark 27.127 we know that PicX×S X/S is formally proper. Since CorrS (X, X) → PicX×S X/S is a closed immersion (Lemma 27.156), the same holds for CorrS (X, X). Hence by Proposition 27.157 we can apply Lemma 27.290 and obtain a section of CorrS (X, X), i.e., a homomorphism λ : X → X t . The morphism λ is generically on S symmetric (Lemma 27.281 (2)), hence it is symmetric (Lemma 27.281 (1)). Therefore there exists a line bundle L on X such that φL = 2λ (Lemma 27.281 (3)). Then the restriction Lη to Xη is algebraically equivalent to M ⊗2 . Therefore Lη is ample. As ampleness is open and closed on abelian schemes (Corollary 27.268), we see that L is relatively ample. Hence X is quasi-projective over S and hence projective because S is qcqs (Corollary 13.72). The normality hypothesis in the theorem cannot be dropped. In [Ray1] O Ch. XII, Raynaud gives an example of an abelian scheme over a local noetherian ring (which is not normal) which does not have an ample line bundle. (27.54) Abelian varieties over the complex numbers. The theory of abelian varieties over the complex numbers forms a vast topic, and our treatment here will necessarily be very sketchy and incomplete. For further details, see the first chapter (and Sections 9 and 24) of Mumford’s book [Mum1], and the comprehensive volume [BiLa] O by Birkenhake and Lange, which has a lot of material on abelian varieties over the complex numbers and also contains very informative introductions to the book and the individual chapters, including further references. Let A be an abelian variety over C. Its analytification Aan (Section (20.12), Section (23.9)) is a complex Lie group, i.e., a group object in the category of complex manifolds. For every complex Lie group G with neutral element e, we have the exponential map exp : Te G → G, a homomorphism of complex Lie groups, with the property that 0 7→ e and that d expe = id. See [BouLie23] O Ch. III §6 no. 4. Since A is proper over C, Aan is compact (Remark 20.59), and we can apply the following proposition. Here, for any finite-dimensional complex vector space V , of dimension g, say, a lattice Λ ⊂ V is a subgroup that is a free Z-module of rank 2g which spans V as an R-vector space; in other words, there exists a Z-basis of Λ which is an R-basis of V . It is then easy to see that the quotient V /Λ carries a unique structure of complex manifold so that the projection V → V /Λ is a morphism of complex manifolds which is locally on V an isomorphism. Proposition 27.292. Let G be a compact complex Lie group with neutral element e. Then the exponential map Te G → G induces an isomorphism Te G/Λ ∼ = G of complex Lie groups, and Λ ⊂ Te G, the kernel of the exponential map, is a lattice in Te G. In particular, as an abstract group G is commutative. Proof. See [Mum1] Section 1.

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27 Abelian schemes

We call a compact complex Lie group of the form V /Λ a complex torus. It is useful to have a closer look at complex tori in general before we come back to the special case of abelian varieties. Let X = V /Λ be a complex torus of dimension g. (When we write this, it is always understood that V is a complex vector space and Λ is a lattice in V .) Viewing X as a real manifold, a choice of basis of Λ yields an identification X ∼ = (R/Z)2g of real Lie groups. In particular, for the underlying topological space we conclude that the canonical projection V → X is the universal cover of X. For the fundamental group of X (with base point 0) we obtain the identification π1 (X) = Λ. Compare Section (27.36). In particular, the fundamental group is abelian and can therefore also be identified with the first singular homology H1 (X, Z) of X. As an abstract group, the group (R/Z)g is divisible, i.e., multiplication by any non-zero integer n is surjective, and the kernel of multiplication by n is isomorphic to ( n1 Z/Z)2g ∼ = (Z/nZ)2g . Compare Proposition 27.186, Proposition 27.187. We also see that the set of all n-torsion points for n ∈ N>0 is dense in X. Let X = V /Λ, X ′ = V ′ /Λ′ be complex tori, and let f : X → X ′ be a morphism of complex manifolds that maps 0 to 0. Then there is a unique lift f˜: V → V ′ of f to the universal covers with f˜(0) = 0. It is clear that f˜ is a morphism of complex manifolds. Then for λ ∈ Λ, the map v 7→ f˜(v + λ) − f˜(v) is continuous on V and takes images in Λ′ , and is thus constant. It follows that f˜(v + λ) = f˜(v) + f˜(λ) for all v ∈ V , λ ∈ Λ. Fixing λ and taking partial derivatives, the constant term f˜(λ) disappears, so all partial derivatives of f˜ are Λ-periodic, and are hence constant by Liouville’s Theorem. This implies that f˜ is a homomorphism of C-vector spaces. Clearly, f˜(Λ) ⊆ Λ′ and the restriction of f˜ to Λ determines f˜ and hence f . In particular, we have proved the following result. (Compare Corollary 16.56 (2) and Remark 27.233.) Proposition 27.293. Let X = V /Λ, X ′ = V ′ /Λ′ be complex tori, where V and V ′ are complex vector spaces of dimension g and g ′ , respectively. Every morphism X → X ′ of complex manifolds which maps 0 to 0 is a homomorphism of complex Lie groups. The above construction defines an embedding Hom(X, X ′ ) → Hom(Λ, Λ′ ) of the group of homomorphisms X → X ′ of complex Lie groups into the group of group homomorphisms between the corresponding lattices. In particular, Hom(X, X ′ ) is a free Z-module of rank ≤ 4gg ′ . Note that the GAGA principle (Corollary 20.61) implies that for abelian varieties A, A′ over C, the group Hom(A, A′ ) of homomorphisms between them coincides with the group of homomorphisms between their analytifications, and hence is a free Z-module of rank ≤ 4 dim(A) dim(A′ ) by the proposition. For a more precise analysis, covering also base fields of positive characteristic, see [Mum1] Section 19. Let us describe the line bundles on a complex torus. First observe that, as in the algebraic situation, there are no non-trivial line bundles on a complex vector space. Lemma 27.294. Let V be a finite-dimensional complex vector space, considered as a complex manifold. Then every line bundle on V is trivial, i.e., every locally free module of rank 1 over the structure sheaf of the complex manifold V is free. Proof. We use the exponential sequence 0 → Z → OV → OV× → 1.

727 ¯ Since H 1 (V, OV ) = 0 by the ∂-Poincar´ e lemma and H 2 (V, Z) = 0 since V is a contractible topological space, it follows from the corresponding long exact cohomology sequence that Pic(V ) = H 1 (V, OV× ) is trivial. Thus if X = V /Λ is a complex torus, then, similarly as in Section (14.21), we can identify line bundles on X and line bundles on V with a Λ-action that is compatible with the action of Λ on V by translations. Since for every line bundle L on the complex manifold X the pullback of L under the projection V → V /Λ is trivial, up to isomorphism a Λ-equivariant line bundle on V is the same thing as a Λ-action on the trivial line bundle OV that lies over the natural action of Λ on V . Such an action is given by a map Λ → Aut(OV ) = Γ(V, OV )× = Γ(V, OV× ) with certain properties (satisfying a cocycle condition), and two such maps give rise to isomorphic line bundles if and only if they differ by a coboundary. One obtains an identification Pic(X) = H 1 (Λ, Γ(V, OV× )). Let us make this more explicit. The exponential sequence for X, × 0 → Z → OX → OX → 1, × gives rise to a map Pic(X) = H 1 (X, OX ) → H 2 (X, Z). The kernel of this map is in × bijection with the group of group homomorphisms Hom(Λ, C× 1 ), where C1 = {z ∈ × C; |z| = 1} is the unit circle, which naturally embeds into Γ(V, OV ). On the other hand, V2 1 V2 1 H (X, Z), and we can view H (X, Z) as the we have an identification H 2 (X, Z) = space of Z-valued alternating forms on H1 (X, Z) = Λ. One can describe the image of × H 1 (X, OX ) in this space as follows.

Lemma 27.295. There are natural group isomorphisms between × (i) the image of the map Pic(X) = H 1 (X, OX ) → H 2 (X, Z), (ii) the set of R-bilinear alternating forms E : V × V → R such that E(iz, iz ′ ) = E(z, z ′ ) for all z, z ′ ∈ V and E(z, z ′ ) ∈ Z for all z, z ′ ∈ Λ, (iii) the set of hermitian forms H such that Im(H(z, z ′ )) ∈ Z for all z, z ′ ∈ Λ. We denote the set of hermitian forms with this property by H . Proof. The map from the set (i) to the set (ii) is given by the restriction of the identification of H 2 (X, Z) with the set of Z-valued alternating forms on Λ discussed above. Given a form E as in (ii), we define H by H(z, z ′ ) = E(iz, z ′ ) + iE(z, z ′ ), and conversely given H, we set E = Im(H). We omit the further computations, see [Mum1] Chapter I or [BiLa] O Chapter 2. The image of L ∈ P ic(X) in H 2 (X, Z) is the singular cohomology version of the first Chern class of L ; one can show that the first Chern class of L vanishes if and only if L is trivial as a topological line bundle. Putting things together, one obtains the following description of the line bundles on a complex torus X = V /Λ. We define P = {(H, α); H ∈ H , α : Λ → C× 1, α(λ + λ′ ) = exp(iπ Im(H(λ, λ′ )))α(λ)α(λ′ ) for all λ, λ′ ∈ Λ}. For H = 0, the condition for α simply says that α is a group homomorphism. We equip this set with the structure of an abelian group by defining (H, α)(H ′ , α′ ) := (H + H ′ , αα′ ).

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For (H, α) ∈ P, we denote by L (H, α) the line bundle on X that is given by the Λ-action on the trivial line bundle V × C on V corresponding to    π Λ → H 0 (V, OV× ), λ 7→ z 7→ α(λ) exp πH(z, λ) + H(λ, λ) . 2 Theorem 27.296. (Theorem of Appell-Humbert) Let X = V /Λ be a complex torus and let P be as defined above. The map ∼

P → Pic(X),

(H, α) 7→ L (H, α),

is a group isomorphism. Proof. See [Mum1] Section 2 or [BiLa] O 2.2. A choice of basis of Λ gives us an identification of the set Hom(Λ, C× 1 ) with the real × 2g torus (C× ) . It is not hard to show that Hom(Λ, C ) even carries a natural structure of 1 1 complex torus (the “dual complex torus” of X). Furthermore, this subgroup of Pic(X) “is” the Pic0 of the complex torus in the sense that it consists precisely of those line bundles L on X such that the associated morphism φL as in 27.159 is trivial. See [Mum1] Section 9, [BiLa] O 2.4. Let us next discuss the question under which conditions a complex torus X = V /Λ is algebraizable. The crucial point is to identify a criterion for a line bundle on X being ample. As a first step, one computes the dimension of the space of global sections of line bundles L (H, α) with H positive definite. We denote by det(E) the determinant of any structure matrix of the alternating form E = Im(H) for a basis of the lattice Λ. Since every unit in Z has square 1, it is independent of the choice of basis. Lemma 27.297. Let X be a complex torus, (H, α) ∈ PX , and let E = Im(H) the alternating form attached to H. Assume that H is positive definite. Then dim H 0 (X, L (H, α)) = p det(E). The sections of L (H, α) are called theta functions (with respect to L (H, α)). Proof. For a proof of this by “complex analytic computations”, see [Mum1] Section 3 or [BiLa] O Corollary 3.2.8. For abelian varieties we have given an algebraic proof above, see Corollary 27.266. The understanding of these spaces of global sections is the key point to the proof of the following theorem which gives a characterization of ample line bundles on a complex torus. Theorem 27.298. (Theorem of Lefschetz) Let X = V /Λ be a complex torus, and let L (H, α) be a line bundle on X, where H ∈ H and α ∈ Hom(Λ, C× 1 ). The following are equivalent. (i) The line bundle L (H, α) is ample. (ii) The hermitian form H is positive definite. Proof. The implication (i) ⇒ (ii) is relatively easy, but the converse is more difficult. See [Mum1] Section 3 or [BiLa] O Proposition 4.5.2. One then has the following characterization when a complex torus is algebraizable.

729 Theorem 27.299. Let X = V /Λ be a complex torus. The following are equivalent. (i) The complex torus X is algebraizable, i.e. X = Aan for some variety A over C. (ii) The complex torus X is of the form X = Aan for an abelian variety A over C, compatibly with the group laws on both sides. (iii) The complex torus X is projective, i.e., there is a closed embedding X ,→ (PnC )an of complex spaces, for some n. (iv) There exists a positive definite hermitian form H on V such that the imaginary part Im(H) of H takes values in Z on Λ × Λ. Proof. The implication (i) ⇒ (ii) follows from Corollary 20.61, since X is compact. Since abelian varieties are projective (Proposition 27.174), (ii) ⇒ (iii) is clear. Obviously (iii) implies (i). The equivalence of (iii) and (iv) follows from Theorem 27.298. Also see [Mum1] Section 3 or [BiLa] O Section 4.5. It is an easy consequence that every 1-dimensional complex torus X is algebraizable. In fact, up to isomorphism X can be written in the form C/Z ⊕ τ Z with τ in the upper 1 ′ complex half plane, and then H(z, z ′ ) = Im(τ ) zz is a positive definite hermitian form as in (iv). Compare Sections (26.7) and (26.19). On the other hand, for g > 1, “most” complex tori of dimension g are not algebraizable. By the theorem of Appell-Humbert, the kernel of the map Pic(X) → Hom(X, X ∨ ), L 7→ φL , is Pic0 (X). Accordingly, we call a positive definite hermitian form H as in (iv) of the theorem a polarization of the complex torus X. Then X is algebraizable if and only if it admits a polarization, and in this case a polarization corresponds to a polarization in the sense of Section (27.52) of the corresponding abelian variety. Let X = V /Λ be an algebraizable complex torus and let H be a positive definite hermitian form as in (iv). For a suitable basis e 1 , . . . , eg , f 1 , . . . , fg of the lattice Λ, the 0 D alternating form Im(H) has structure matrix for a uniquely determined −D 0 diagonal matrix D with entries d1 | d2 | · · · | dg . The tuple (d1 , . . . , dg ) is called the type of the polarization H. If φ = φL (H,α) denotes the morphism X → X ∨ of abelian varieties corresponding to thisQpolarization, the di describe the finite ´etale group scheme Ker(φ). g We have Ker(φ) ∼ = i=1 Z/di Z, a product of constant group schemes. In particular, polarizations of type (1, . . . , 1) are precisely the principal polarizations. The elements d1i fi ∈ Λ ⊂ V form a C-basis B of V . We can now express the Z-basis
e1 , . . . , eg , f1 , . . . , fg of Λ in terms of B and we obtain a (g × 2g)-matrix (λij )ij = Z D with Z ∈ Mg (C). Clearly this matrix determines the complex vector space V , the lattice Λ and the form H (up to isomorphism). The condition that H be positive definite hermitian translates to the conditions Z = Z t , and

Im(Z) positive definite,

i.e., the matrix Z is symmetric, and its imaginary part Im(Z) ∈ Mg (R) is (symmetric and) positive definite. To abbreviate the latter condition one often writes Im(Z ) > 0. Conversely, when we fix D, then every matrix Z with these properties defines an abelian variety of dimension g with a polarization of the fixed type. We denote by Hg the set Hg = { Z ∈ Mg (C) ; Z = Z t and Im(Z) positive definite} of all these matrices, the so-called Siegel upper half space. Note that H1 is just the complex upper half plane. We have constructed a map from Hg to the set of isomorphism classes of abelian varieties over C with a polarization of type D.

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For simplicity, let us now restrict to the principally polarized case, i.e., D is the unit matrix. We use the term principally polarized abelian variety for a pair of an abelian variety with a principal polarization. One checks that principally polarized abelian varieties corresponding to matrices Z, Z ′ are isomorphic if and only if there exists a   a b matrix T = ∈ Sp2g (Z), the symplectic group of matrices fixing the alternating c d  0 1g form , such that −1g 0 Z ′ = (aZ + b)(cZ + d)−1 . Here a, b, c, d ∈ Mg (Z) are the “block components” of T . For g = 1 we recover the usual action of the modular group SL2 (Z) on the complex upper half plane, see Corollary 26.115. We obtain a bijection ∼

Sp2g (Z)\Hg → {princ. pol. abelian varieties over C}/ ∼ =. This hints at the question whether the set on the right hand side can be equipped with the structure of a complex space (or even an algebraic variety), since Hg obviously is a complex manifold. In this sense, the above bijection is the starting point for the theory of moduli spaces of abelian varieties, see Section (27.55) for some further remarks in this direction. Note that Hg has dimension g(g + 1)/2, as one easily checks. Since Sp2g (Z) is a discrete group, the moduli space should also have dimension g(g + 1)/2. See Theorem 27.301. Finally, let us briefly discuss the construction of the Jacobian of a smooth projective curve C over C. Let C be a connected smooth projective curve over C. We implicitly identify C with its analytification C an , a compact Riemann surface. See Section (26.7). Let g be the genus of C. Integration of differential 1-forms along closed paths on C induces a pairing Z H1 (C, Z) × H 0 (C, Ω1C/C ) → C, (γ, ω) 7→ ω. γ

The resulting homomorphism H1 (C, Z) → H 0 (C, Ω1C/C )∨ is injective, so that we can view H1 (C, Z) as a lattice in the g-dimensional complex vector space H 0 (C, Ω1C/C )∨ . Thus the quotient J := H 0 (C, Ω1C/C )∨ /H1 (C, Z) is a complex torus. This gives the complex analytic construction of the Jacobian of C. This complex torus is algebraizable, and in fact carries a canonical principal polarization which can be described easily in terms of a suitable basis of H1 (C, Z) = π1 (C)ab consisting of classes of closed paths in C with simple intersection behavior. We do not go into further details at this point but refer to [BiLa] O Chapter 11. P Any divisor D of degree 0 on C can be written as a finite sum i ([pi ] − [qi ]) for points pi , qi on C. For a differential form ω ∈ H 0 (C, ΩC/C ) a choice of path from pi to qi Rq gives us the path integral pii ω. We obtain a linear form in H 0 (C, ΩC/C )∨ whose class in J is independent of the choice of path. This gives us a map Div0 (C) → J. It follows from classical theorems by Jacobi and Abel that this map induces a group isomorphism Pic0 (C) → J. One can then go on and construct a Poincar´e bundle for J and show that J satisfies the universal property of the identity component of the Picard functor of C, i.e., it can be identified with the analytification of the Jacobian of the algebraic curve C.

731 (27.55) Outlook: The moduli space of principally polarized abelian varieties. Constructing a moduli space (or parameter space) of abelian varieties means that we would like to equip the set of (isomorphism classes of) all abelian schemes of relative dimension g with the structure of a scheme. Compare also Section (16.33). The appropriate way to make this precise is to define a functor on the category of schemes and to study whether this functor is representable. Ideally (or: naively), we would like Agnaive (S) to be the set of isomorphism classes of abelian schemes of relative dimension g over S. However, it turns out that defining Agnaive (S) as the set of isomorphism classes of abelian schemes does not lead to a representable functor. The situation over the complex numbers hints at the fact that a better moduli problem (i.e., a better behaved functor) is obtained when we include a polarization. Let us explain the necessity of considering polarized varieties in more detail since this is also tied nicely to other central topics considered in this book. M. Artin gave criteria when a functor is representable at least by an algebraic space (see [Art3] O and [Art4] O or [Sta] 07SZ). More precisely, let Z be a locally noetherian scheme such that OZ,z is a G-ring (Definition 20.46) for every8 z ∈ Z, which holds for instance, if Z is locally of finite type over a field or over a Dedekind ring whose field of fraction has characteristic 0. Let F : (Sch/Z)opp → (Sets) be a functor. Then Artin gave a list of criteria for F that are necessary and sufficient for F to be representable by an algebraic space locally of finite type over Z, see [Sta] 07Y19 . Let us single out one of these criteria. It says that for every complete local noetherian Z-algebra R with maximal ideal m the canonical map F (R) → limn F (R/mn ) is bijective. If for any scheme (= Z-scheme) S we define Agnaive (S) to be the set of isomorphism classes of abelian schemes over S of relative dimension g, then this criterion means that given a formal abelian scheme over Spf(R), i.e., a compatible family X = (Xn )n of abelian schemes Xn over R/mn (see Section (24.17)), we would always find an abelian scheme X over R (necessarily unique by Theorem 24.112) such that Xn ∼ = X ⊗R R/mn for all n. In other words, we need X to be algebraizable. But this does not hold in general. To ensure that X is algebraizable we would like to argue with Grothendieck’s algebraization Theorem 24.113. Hence we need to have a compatible system of ample line bundles Ln on Xn and should add this as a datum to our functor. Instead of specifying an ample bundle we may also specify a polarization since attaching to an ample line bundle L the polarization φL is fppf-surjective (Remark 27.283) and smooth (Proposition 27.208). Therefore it is better for a scheme S to consider Agcoarse (S), the set of isomorphism classes of pairs (X, λ), where X is an abelian scheme of relative dimension g over S and where λ is a polarization of X. Usually one also fixes the degree d of the polarization, coarse coarse i.e., one considers the functor Ag,d , where Ag,d (S) denotes isomorphism classes of pairs (X, λ) as above such that λ has fixed degree d. As the degree is locally constant in families, this defines an open and closed subfunctor of Agcoarse . coarse Alas, Ag,d is still not quite representable. It admits a coarse moduli scheme in the sense of [MFK] O Definition 7.4, see loc. cit. Theorem 7.10. The problem with the coarse representability of Ag,d is the existence of non-trivial automorphisms of polarized abelian schemes, and even of polarized abelian schemes. Let us give a hint why this coarse was representable it would be an fppf-sheaf, in fact even a is problematic. If Ag,d fpqc-sheaf (Proposition 14.76), and in particular for every finite Galois extension k ′ /k we 8 9

It suffices to make the assumption only for those z ∈ Z that are closed in some open affine neighborhood. Artin proved this under somewhat stronger assumptions on Z.

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coarse coarse ′ would have that base change from k to k ′ yields an injective map Ag,d (k) → Ag,d (k ) since Spec k ′ → Spec k is finite and faithfully flat. Hence given a polarized abelian variety (X ′ , λ′ ) over k ′ there should by at most one (up to isomorphism) k-forms of (X ′ , λ′ ), i.e., at most one isomorphism class of a polarized abelian variety (X, λ) over k whose base change to k ′ is (X ′ , λ′ ). But isomorphism classes of k-forms of (X ′ , λ′ ) are in bijection to H 1 (Gal(k ′ /k), Autk′ (X ′ , λ′ )), cf. Theorem 14.90, and we cannot expect this set to be trivial if Autk′ (X ′ , λ′ ) is nontrivial. There are (at least) three ways to deal with this problem. The first one would be to be satisfied with the existence of a coarse moduli space which at least over algebraically closed fields gives a satisfying answer. But this way falls short for many arithmetic applications of the theory. The second and most conceptual way would be to take these automorphisms into account as they are an important feature of the moduli problem. This means, instead of a set-valued functor one considers the functor on the category of schemes that attaches to S the groupoid Ag,d (S), i.e., the category of abelian schemes of relative dimension g over S endowed with a polarization of degree d in which the only morphisms are the isomorphisms of such pairs. As this is not a set-valued functor any more, it cannot be representable by a scheme or an algebraic space. But it is an algebraic stack (see [Sta] 0ELS for the theory of algebraic stacks and [Lan] O X Theorem 1.4.1.11 for the proof that Ag,d is an algebraic stack), which allows to work with it quite similar as one works with schemes (although there are also some significant differences in the theories). Unfortunately, it is beyond the scope of this book to go into more detail here. The third way to solve this problem is to modify the functor by including an auxiliary additional datum which serves as a rigidification. Below, we will add to the datum (X, λ) ∼ of a polarized abelian scheme an isomorphism η : X[N ] → (Z/N Z)2g of group schemes for some fixed N ≥ 3, a so-called full level N structure. Other rigidifications are possible. Serre’s lemma tells us that there are no nontrivial automorphisms of the triple (X, λ, η) (by Proposition 27.105 it suffices to show this if X is defined over an algebraically closed field, then the result can be found in [Ser3] O ). We arrive at the following definition.

Definition 27.300. Let g ≥ 1, d ≥ 1 and N ≥ 3 be natural numbers. The moduli functor of abelian varieties of dimension g with a polarization of degree d and a full level N structure is the functor Ag,d,N : (Sch/Z[

1 opp ]) →(Sets), N T 7→{(X, λ, η); X an abelian scheme of relative dimension g over S, λ a polarization of degree d of X, ∼ ∼. η : X[N ] → (Z/N Z)2g an isomorphism}/ =

In particular, one has the functor Ag,1,N of principally polarized abelian varieties with a full level N structure. One can show that Ag,d,N is representable by a scheme. More precisely, one has the following theorem.

733 Theorem 27.301. The functor Ag,d,N is representable by a smooth quasi-projective Z[ N1 ]-scheme, the moduli space of abelian varieties of dimension g with polarization of degree d and with full level N structure. Its relative dimension is g(g + 1)/2. The proof of the theorem can be approached in several ways, but all of them require methods that we did not cover. Therefore we just give some references. In [MFK] O , a construction using geometric invariant theory is given. In [Lan] O X , the moduli space for Ag,d is first constructed as an algebraic stack using Artin’s criteria from which the existence of the moduli space for Ag,d,N as an algebraic space is easily deduced using Serre’s lemma (Theorem 1.4.1.11 and Corollary 1.4,1.12 of loc. cit.); representability of Ag,d,N as a (quasi-projective) scheme is obtained as a consequence of the theory of the Baily-Borel compactification (Corollary 7.2.3.10). For the theory over the complex numbers, see also Chai’s article in [CoSi] O or [BiLa] O Chapter 8. Note that one could write down the same moduli functor on the category of all schemes, but by Proposition 27.188 (2) an isomorphism η can only exist for abelian schemes over schemes S on which N is invertible, and therefore the fibers over Spec Fp for p | N would be empty. These moduli spaces are important tools in the study of abelian varieties. They are also extremely interesting varieties which have been and are studied extensively. In fact, moduli of abelian varieties are a classical object of study in “classical” algebraic geometry. They are also useful for studying arithmetic questions. (27.56) Literature on abelian varieties and abelian schemes. The “modern classic” on abelian varieties, modern referring to the fact that is is written in the language of schemes, is Mumford’s beautiful book [Mum1], which also contains a chapter on the analytic theory of abelian varieties over C. In [MFK] O , some foundational material on abelian schemes over general base schemes can be found. The manuscript [EGM] X written by Edixhoven, Moonen and van der Geer, with many finished or nearly finished chapters already and a large bibliography, is a comprehensive modern text on abelian varieties and abelian schemes. Other references are Milne’s articles on abelian varieties and on Jacobians of curves in [CoSi] O , with extended bibliographical notes at the end of the article on Jacobians, and his lecture notes [Mil4] X . For the theory over the complex numbers, there is Debarre’s volume [Deb] O , Rosen’s article in [CoSi] O and the comprehensive volume [BiLa] O by Birkenhake and Lange, which is a standard reference. For more specialized topics, we mention Raynaud’s Lecture Notes in Mathematics volume [Ray1] O , in particular for the question when homogeneous spaces are projective, and the book [FaCh] O , where a proof of the representability of the dual abelian scheme over a general base scheme is sketched, which has been worked out in detail by Große-Kl¨onne in [GrK]. For the interplay of the theory of dual abelian varieties and Fourier-Mukai correspondence, good sources are, in addition to Mukai’s original articles [Muk1] O and [Muk2] O , the books by Huybrechts [Huy] O , Polishchuk [Pol] O , and the lecture notes of Bhatt [Bha] X .

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Exercises Exercise 27.1. Let S be a scheme that has a closed point s ∈ S. Let G be obtained by gluing two copies of S along U := S \ {s}. (1) Show that there exists a unique structure of an S-group scheme on G such that G ×S U is the trivial group scheme and such that the fiber Gs is isomorphic to the constant group scheme attached to Z/2Z over κ(s). (2) Assume that {s} is not open in S. Show that G → S is not separated. Exercise 27.2. Show that all assertions of Proposition 27.11 also hold for an arbitrary group scheme over a base scheme S consisting of a single point s. Hint: Argue similarly as in Proposition 27.11 using that Spec κ(s) → S is a universal homeomorphism. To show that g ∈ G lies in the image in (1) make a base change to κ(g). Exercise 27.3. Let G be a group scheme locally of finite type over a field k. Suppose that Lie(G) = 0. Show that G is ´etale over k. Exercise 27.4. Let R be a ring. Show that Pic(R) → Pic(Rred ) is an isomorphism. Exercise 27.5. A ring R is called seminormal if for all b, c ∈ R with b2 = c3 there exists a ∈ R such that a3 = b and a2 = c. (1) Show that every normal domain is seminormal. (2) Show that every seminormal ring is reduced. (3) Let k be a field. Show that k[x, y]/(xy) is seminormal but not normal. Show that k[x, y]/(y 2 − x3 ) is not seminormal. Exercise 27.6. Let R be a ring. Show that the following assertions are equivalent. (i) Rred is seminormal (Exercise 27.5). (ii) Pic(R) → Pic(A1R ) is an isomorphism. (iii) Pic(R) → Pic(AnR ) is an isomorphism for all n ≥ 0. Hint: This is difficult. See [Coq] X for an elementary proof. Note that one can assume that R is reduced by Exercise 27.4. Exercise 27.7. Let k be a field. Let G be a group scheme of finite type. Show that every homogeneous G-space X is quasi-projective. Hint: You might proceed as follows. (1) Show that one can assume that k is algebraically closed and that G and X are connected. (2) Show that G and X are smooth if char(k) = 0 and that one can therefore assume that char(k) > 0. (3) Show that Xred is a homogeneous space for the smooth group scheme Gred and deduce that Xred is quasi-projective. (4) Now use Exercise 22.21. Exercise 27.8. Let S be a scheme, let X and Y be abelian schemes over S, and let f : X → Y be a morphism of S-schemes that preserves the unit section. Consider g : X ×S X → Y , g(x, x′ ) = f (x+x′ )−f (x)−f (x′ ) on T -valued points x, x′ ∈ X(T ). Show that (without assuming that f is a homomorphism of group schemes) that the restriction of g to {0} × X and to X × {0} is constant and deduce that f is a homomorphism of group schemes.

735 Exercise 27.9. Let S be a scheme and let n ≥ 1. Show that PicPnS /S is represented by the constant group scheme attached to the abelian group Z. Exercise 27.10. Let X be an abelian variety over a field k and let f : X → Y be a morphism of k-schemes. Show that every connected component of a fiber of f is geometrically irreducible. Exercise 27.11. Let S be a scheme, and let G and H be smooth separated group schemes of finite presentation with connected fibers over S. Let f : G → H be an isogeny. Then the function deg(f ) : S −→ Z,

s 7→ dimκ(s) H 0 (Ker(fs ), OKer(fs ) )

is called the degree of f . (1) Show that deg(f ) is a lower semicontinuous constructible function, i.e., for all n ∈ Z the set { s ∈ S ; deg(f )(s) ≥ n } is open and constructible in S. (2) Show that f is finite locally free if and only if deg(f ) is locally constant. Hint: Exercise 20.17. Exercise 27.12. Let k be a field and let G be a finite group scheme over k. (1) Show that one has a functorial exact sequence of finite group schemes (*)

1 → G0 → G → G´et → 1,

where G0 is the identity component of G and G´et is ´etale over k. Show that G0 → Spec k is a universal homeomorphism. (2) Now suppose that k is perfect. Show that G´et ∼ = Gred and that the inclusion Gred → G defines a splitting of (*). (3) Call rk´et (G) := dimk Γ(G´et , OG´et ) the ´etale rank of G and rk0 (G) := dimk Γ(G0 , OG0 ) the local rank of G. Show that rk(G) = rk0 (G) + rk´et (G). Exercise 27.13. Let S be a scheme and let G be an S-group scheme such that G → S is separated, flat, quasi-finite, and of finite presentation. Show that the map S → Z, s 7→ rk´et (Gs ) (Exercise 27.12) is constructible and lower semicontinuous. Hint: Exercise 20.18 The following four exercises are taken from [EGM] X . Exercise 27.14. Let k be a field, let X be an abelian variety over k, and let L be a line bundle on X. (1) Show that for n ∈ Z one has [n]∗ L ∼ = OX if and only if L ⊗n ∼ = OX . ̸ −1, 0, 1 one has [n]∗ L ∼ (2) Show that for n ∈ Z with n = = L if and only if L ⊗n−1 ∼ = OX . (3) Let k be algebraically closed. Show that L ∼ = L1 ⊗ L2 , where L1 is symmetric and L2 ∈ Pic0X/k (k). Exercise 27.15. Let S be a scheme and let X be an abelian scheme over S. Let mX : X ×S X → X be the group law and ∆X : X → X ×S X the diagonal. Show that (mX )t = ∆X t and (∆X )t = mX t . Exercise 27.16. Let S be a scheme, let X be an abelian scheme over S, and let L be a line bundle on X. Let Y := X 4 and M := p∗1 L ⊗ p∗2 L ⊗ p∗3 L ⊗ p∗4 L , where pi : Y → X is the i-th projection. Let n ≥ 1 be an integer and write n = a2 + b2 + c2 + d2 for integers a, b, c, d (this is always possible by a theorem of Lagrange, see for instance [Ser5] Cor. 1 in Appendix to Chapter IV). Consider

736

27 Abelian schemes   a −b −c −d  b a −d c   α :=  c d a −b  d −c b a

which we consider as an isogeny α : Y → Y . Show that one has α∗ M = M ⊗n in NS(Y ). Exercise 27.17. Let k be a field and let X be an abelian variety over k. Let L1 and L2 be non-degenerate line bundles on X such that L1 ⊗ L2 is also non-degenerate. Show i(L1 ⊗ L2 ) ≤ i(L1 ) + i(L2 ). Exercise 27.18. Let X be an abelian scheme over a scheme S and let Hom(X, X t )sym be the subgroup functor of Hom(X, X t ) consisting of symmetric homomorphisms. (1) Show that the inclusion Hom(X, X t )sym ,→ Hom(X, X t ) is representable by an open and closed immersion. (2) Show that NS(X) → Hom(X, X t )sym is an isomorphism.

F

Homological Algebra

Content – Addenda to the language of categories – Additive and abelian categories – Complexes in additive and abelian categories – Spectral sequences – Triangulated categories – Sign conventions – Derived categories – Derived functors In this appendix we collect some notions and results from homological algebra that are needed in our exposition. We will not give any proofs but only references except if we were not able to find a suitable textbook reference. Sometimes results immediately follow from the definitions and in this case there are no references.

Addenda to the language of categories In this part we introduce some more general categorical concepts. We begin with a remark on set-theoretic issues that become more serious when working with derived categories. We also introduce general limits and colimits: In [GWI] O , Appendix A, we explained the notion of an inductive and a projective limit of a family of objects indexed by a preordered set. We now generalize this to “families indexed by a category”, more precisely to functors from some small category into our given category. Finally, we recall the notion of adjoint functors and give some some general results about them. (F.1) Set-theoretical remarks. As in the first volume, we will largely ignore set-theoretic questions. However, in the context of homological algebra as we will use it, this is more problematic: The derived category of an abelian category is constructed as a localization of another category (see Section (F.37)), and the question of existence of localization of categories involves more subtle set-theoretic issues than most constructions within algebraic geometry.

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3

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F Homological Algebra

Because this is an important foundational point, we will give some pointers to the literature where possible approaches are discussed. We work within an axiomatic framework for set theory such as ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice). In particular, we have at our disposal a notion of set (which we think of a collection of objects which is “not too large”) and a notion of class (an arbitrary collection of objects). The ZFC axioms specify which set-theoretic constructions can be carried out inside the world of sets (e.g., we can take the power set of a set; but the collection of all sets is not a set itself). However, for our purposes we need more fine-grained control. Defining the collection of morphisms of the localization of a category, we need to express that the size of this collection —while it might excess the size of “small” sets— is not arbitrarily large, i.e., is not an arbitrary class. One way to do so is using the formalism of universes, and then working within a given universe U , which for us will be a “large” infinite set (see [SGA4] O Exp. I for details). Then the U -small sets are those sets which are elements of U . The axioms defining a universe guarantee that the standard set-theoretic constructions can be carried out inside the fixed universe. The category of sets is defined as the category of all U -small sets; in particular its collection of objects is a set (in ZFC), namely the set U . A U -category is a category C whose set of objects is a subset of U and such that for each pair of objects (X, Y ) the set of morphisms HomC (X, Y ) is U -small. Constructions such as the derived category will not in general be possible within the fixed universe: I.e., the Hom sets will not necessarily be U -small. Since they are ZFC sets nevertheless, the construction can be carried out inside a larger universe. So at certain points one may have to pass to a larger universe (but we will allow ourselves to do so without explicitly keeping track of it). For further details, see e.g. [SGA4] O Exp. I, [KaSh] O 1.1. Another way is to restrict to objects (sets, groups, schemes, . . . ) whose size is bounded explicitly in terms of a cardinal number; cf. [Sta] 000H. Comparing this with fixing a universe, we see that given any universe U , the cardinality of sets in U is bounded since U is a set. Conversely, bounding the size of sets is not enough to obtain a set (meaning that the collection of all sets of some bounded cardinality is still a proper class, not a set), so in order to reduce to a set one has to do something more, e.g., to restrict to a set which contains a representative of every isomorphism class. Nevertheless, this approach allows one to avoid the somewhat arbitrary choice of a universe. Finally, we will use only derived categories of Grothendieck abelian categories (see Section (F.12)), or subcategories thereof. We will see, that in this case the derived category is equivalent to a category which lies in the same universe, so that, from hindsight, no set-theoretic issues arise. (F.2) Categories and functors. We will use the standard terminology of elementary category theory (categories, functors, morphisms of functors, isomorphisms, . . . ) as given in [GWI] O , Appendix A. See also [KaSh] O 1.2, 1.3, 1.4. If C is a category, we sometimes write X ∈ C instead of X ∈ Ob(C). All functors are covariant. If we speak of a contravariant functor F from C to a category D, this means that F is a (covariant) functor F : C opp → D. Occasionally, we will use the following notion.

739 Definition F.1. Let C be a category. A subcategory D of C is called strictly full if it is a full subcategory (i.e., HomD (x, y) = HomC (x, y) for all objects x and y in D) and for every object x in D all objects in C that are isomorphic to x are also in D. Definition F.2. Let F : C → D be a functor of categories and let d ∈ D be an object. Then the slice category of F over d, denoted by F/d or C/d , is the category such that (a) an object is a pair (c, σ), where c is an object of C and σ : F (c) → d is a morphism in D, (b) a morphism (c, σ) → (c′ , σ ′ ) is a morphism u : c → c′ such that σ ′ ◦ F (u) = σ. Dually, define the slice category of F under d, denoted by F \d or C \d , as the category with (1) objects being pairs (c, τ : d → F (c)), (2) morphisms (c, τ ) → (c′ , τ ′ ) being morphisms v : c → c′ such that F (v) ◦ τ = τ ′ . We will also use occasionally the following notion. Definition F.3. A functor F : C → D is called conservative if it reflects isomorphisms, i.e., if whenever α is a morphism in C such that F (α) is an isomorphism in D, then α is an isomorphism. Definition F.4. A category C is called a groupoid if every morphism of C is an isomorphism. (F.3) Limits and Colimits. In Appendix A we already explained the notion of a projective limit and an inductive limit. Here we generalize these concepts by replacing partially ordered index sets by index categories. Let I always denote a small category (i.e., a category where the objects form a set, and for any two objects the collection of morphisms between them is a set). Let C be a category and let C I be the category of functors I → C. For every object A of C let cA : I → C be the constant functor with value A, i.e., cA sends every object of I to A and every morphism in I to idA . Every morphism A → B in C induces a morphism of functors cA → cB . We obtain a functor C −→ C I ,

(F.3.1)

A 7→ cA .

Definition F.5. Let X : I → C be a functor. We also call such a functor an I-diagram in C. We write Xi instead of X(i) for an object i ∈ I. (1) Consider the covariant functor colimI X : C → (Sets) sending every object A in C to the set Hom(X, cA ) of morphisms of functors X → cA . If this functor is representable, the representing object of C is called colimit of X and it is denoted by colim X or colim Xi or colim Xi . I

i∈I

i∈I

In other words, colimI X is an object in C together with morphisms si : Xi → colimI X in C for all objects i in I such that (a) for every morphism φ : i → j in I one has si = sj ◦ X(φ),

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F Homological Algebra

(b) for every object Z in C and for all morphisms ti : Xi → Z such that for all morphism ψ : i → j in I one has ti = tj ◦ X(ψ), there exists a unique morphism t : colimI X → Z such that ti = t ◦ si . (2) Dually, a limit of the functor X is an object lim X

or

I

lim Xi i∈I

or

lim Xi ←− i∈I

in C that represents the contravariant functor limI X : I → C, A 7→ Hom(cA , X). If X, X ′ : I → C are functors, any morphism of functors u : X → X ′ induces a morphism colimI (u) : colimI X ′ → colimI X,

resp. limI (u) : limI X → limI X ′ .

If colimI X and colimI X ′ (resp. limI X and limI X ′ ) are both representable, then this morphism corresponds to a unique morphism colim(u) : colim X → colim X ′ , I

I

I

resp. lim(u) : lim X → lim X ′ . I

I

I

Remark F.6. Let C be a category and let I be a small category such that for all functors X : I → C the colimit (resp. the limit) exists. Then we obtain a functor colim : C I −→ C,

(resp.

lim : C I −→ C),

and this functor is left adjoint (resp. right adjoint, Definition F.21) to the functor A 7→ cA (F.3.1). Remark: Projective and inductive limits as limits and colimits F.7. Recall that every preordered set I can be considered as a category, still denoted by I. The objects are the elements of I and for any two elements i, j ∈ I the set of morphisms i → j consists of one element if i ≤ j and is empty otherwise. If X : I opp → C is a projective system in a category C indexed by I, then limI opp X is the projective limit in the sense of Section (A.3). Similarly, if X : I → C is an inductive system in a category C indexed by I, then colimI X is the inductive limit in the sense of Section (A.3). In a given category C only some limits or colimits may exist. Definition F.8. A category in which arbitrary limits (resp. colimits) exist is called complete (resp. cocomplete). A category in which limits (resp. colimits) of all I-diagrams exist for arbitrary finite categories I is called finitely complete (resp. finitely cocomplete). The category of sets is complete and cocomplete: Example: Limits and colimits of sets F.9. Let I be a small category, I := Ob(I), and let X : I → (Sets) be an I-diagram in the category of sets. (1) The limit limI X exists in (Sets) and can be described by Y (F.3.2) lim X = { (xi )i∈I ∈ Xi ; ∀ φ : i → j in I: X(φ)(xi ) = xj }. I

i∈I

For j ∈ I the map pj : limI X → Xj is given by the projection (xi )i∈I 7→ xj .

741 (2) The colimit colimI X exists in (Sets) and can be described by a (F.3.3) colim X = ( Xi )/ ∼, I

i∈I

`

where ( i∈I Xi ) is the disjoint union of the sets Xi and where ∼ is the equivalence relation generated by the relation xi ∼ xj if xi ∈ Xi , xj ∈ Xj and X(φ)(xi ) = xj for some φ : i → j. For j ∈ I the map`sj : Xj → colimI X is given by attaching to xj ∈ Xj the equivalence class of xj ∈ i∈I Xi . Further examples for categories which are complete and cocomplete are (1) the category of topological spaces, (2) the category of groups, (3) the category of left R-modules (R a fixed not necessarily commutative ring), more generally the category of OX -modules ((X, OX ) a ringed space). Remark F.10. Let X : I → C, i 7→ Xi be a diagram in a category C. Then we can rephrase the universal property in the definition of lim Xi and colim Xi as follows. An object limI X in C together with morphisms pi : limI X → Xi for all objects i of I is a limit of X in C if and only if for all objects Y in C the map u7→(pi ◦u)i

HomC (Y, lim X) −−−−−−−−→ lim HomC (Y, Xi ) I

I

is bijective, where the right hand side denotes the limit in the category of sets. Similarly, an object colimI X in C together with morphisms si : Xi → colimI X for all objects i of I is a colimit of X in C if and only if for all objects Y in C the map u7→(u◦si )i

HomC (colim X, Y ) −−−−−−−−→ lim HomC (Xi , Y ) I

I

is bijective. Remark: Double limits F.11. Let I and J be (small) categories and let C be a category such that limits (resp. colimits) of all I-diagrams and all J -diagrams in C exist. Let X : I × J → C, (i, j) 7→ Xij be a diagram in C. Then because of the definition of limits (resp. colimits) via a universal property one obtains that limI×J X (resp. colimI×J X) exists and one has isomorphisms lim Xij ∼ = lim lim Xij ∼ = lim lim Xij i,j

(resp.

i

j

j

i

colim Xij ∼ = colim colim Xij ∼ = colim colim Xij ). i,j

i

j

j

i

Definition F.12. A category I is called filtered if Ob(I) is non-empty and if the following two conditions are satisfied. (a) For all objects i and j in I there exists an object k and morphisms i → k and j → k. (b) For all objects i and j and all morphisms f, g : i → j there exists a morphism h : j → k such that h ◦ f = h ◦ g. We also say that I is cofiltered if the opposite category I opp is filtered. For instance, a partially ordered set I is filtered if and only if the attached category is a filtered category.

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F Homological Algebra

Example: Filtered colimits of sets F.13. Let I be a filtered category and let X : I → (Sets) be an I-diagram in ` the category of sets (we speak of a filtered diagram). In this case one has colimI X = ( i∈I Xi )/∼ where for xi ∈ Xi and xj ∈ Xj one defines xi ∼ xj if there exist morphisms φ : i → k and ψ : j → k such that X(φ)(xi ) = X(ψ)(xj ) (the properties of a filtered category imply that ∼ is an equivalence relation). Definition F.14. A category I is called connected if it is non-empty and for any pair of objects X, Y ∈ I there is a finite sequence of objects (X0 = X, X1 , . . . , Xn−1 , Xn = Y ) such that HomI (Xj−1 , Xj ) or HomI (Xj , Xj−1 ) is non-empty for all j = 1, . . . , n. Definition F.15. A functor α : J → I is called cofinal if the slice category J \i of objects i → α(j) under i is connected for all objects i ∈ I. If α is the inclusion of a subcategory, we also say that J is cofinal in I. Proposition F.16. ([KaSh] O Proposition 2.5.2) Let α : J → I be a cofinal functor of small categories. Then for every diagram X : I → C in a category C there is a functorial isomorphism ∼ colim Xα(j) −→ colim Xi i∈I

j∈J

if either side exists. Remark F.17. Let J be a filtered category. (1) Then J is connected since for all objects X, Y ∈ J there exists an object Z in J and morphisms X → Z and Y → Z. (2) A functor α : J → I is cofinal if and only if the following two conditions hold. (a) For every object i ∈ I there exists j ∈ J and a morphism i → α(j). (b) For all objects i ∈ I and j, j ′ ∈ J there exists ȷ˜ ∈ J and morphisms j → ȷ˜, j ′ → ȷ˜ in J and morphisms i → α(j) and i → α(j ′ ) such that i { α(j ′ )

# α(j ′ )

/ α(˜ ȷ) o

commutes. (F.4) Special cases of limits and colimits. Remark and Definition F.18. As already explained, one has several special cases of limits and colimits. Let X : I → C be an I-diagram in a category C. (1) If Q I is a category with no morphisms except ` the identities, limI X is the product i∈Ob(I) Xi and colimI X is the coproduct i∈Ob(I) Xi . A special case is if I is the empty category. Then there is a unique I-diagram in every category C. Its limit (resp. its colimit), if it exists, is a final object (resp. an initial object) of C. (2) Let I be the category with three objects j, i1 , and i2 and whose only morphisms except the identities are two morphisms i1 → j and i2 → j. We represent I schematically by i1

/jo

i2

743 Then an I-diagram X in a category C is a diagram of morphisms in C of the form X1

f1

/Y o

f2

X2 .

The limit of X, if it exists, is the fiber product of X1 and X2 over Y , denoted by X1 ×Y X2 . Sometimes a fiber product is called a pullback ; the diagram formed by X1 , X2 , Y and their fiber product is called a cartesian diagram. (3) Dually, there is the notion of a pushout in a category C which is the colimit of a diagram X : J → C, where J is represented schematically by •o

•

/ •.

The pushout of a J -diagram X1 ←− Y −→ X2 in C is denoted by X1 ⨿Y X2 . Sometimes a pushout is called an amalgamated sum. (4) Now consider the case i1 = i2 , i.e., I is the category with two objects j and i whose only morphisms except the identities are two morphisms i → j. Hence an I-diagram X is given by a diagram (F.4.1)

Xi

// X j .

u v

In this case its limit is called the kernel of u and v or the equalizer of u and v and denoted by Eq(u, v). The colimit of the diagram (F.4.1) is called the cokernel of u and v or the coequalizer of u and v and denoted by Coeq(u, v). Let F : C → D be a functor between categories. Let X : I → C be a diagram in C such that the limits limI X and limI (F ◦X) exist in C and D, respectively. For every object i in I the morphism limI X → Xi induces by application of F a morphism F (limI X) → F (Xi ). The family of these morphisms corresponds by the universal property of limI (F ◦ X) to a morphism (F.4.2)

F (lim X) −→ lim(F ◦ X). I

I

We say that F commutes with limits if for every diagram X : I → C such that its limit limI X exists in C, the limit of F ◦ X exists in D and the morphism (F.4.2) is an isomorphism. Dually, there is the notion of a functor that commutes with colimits. Proposition F.19. (1) A category C is complete if and only if arbitrary products exist in C and for each pair of parallel arrows u, v : X → Y its equalizer exists. A functor F : C → D commutes with arbitrary limits if and only if it commutes with products and equalizers. (2) A category C is finitely complete if and only if final objects, products of two objects, and for each pair of parallel arrows u, v : X → Y its equalizer exist in C. A functor F : C → D commutes with finite limits if and only if it commutes with final objects, products of two objects, and equalizers. One has dual criteria for a category to be (finitely) cocomplete and for a functor to commute with (finite) colimits.

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F Homological Algebra

Proof. For a small category I let Mor(I) be the set of morphisms in I and for α ∈ Mor(I) let s(α) and t(α) be its source respectively its target. The limit of a diagram X : I → C can be constructed as the kernel of the two morphisms σ, τ :

Q

i∈I

Xi

//

Q

α∈Mor(I)

Xt(α) ,

such that for all α ∈ Mor(I) the composition with the projection is given by prα ◦ σ = X(α) ◦ prXs(α) , prα ◦ τ = prXt(α) . The other assertions are shown similarly. For details see [KaSh] O 2.2.10, 2.2.11. For instance the category of schemes is finitely complete because it has a final object (namely, Spec Z) and fiber products. Remark F.20. Let C be a category. (1) Suppose that finite coproducts and filtered colimits exist in C. Then arbitrary coproducts exist in C because a a Xi = colim Xj i∈I

J ⊆ I finite

j∈J

for any family (Xi )i∈I of objects in C. (2) Similarly, a functor F : C → D commutes with arbitrary coproducts if it commutes with finite coproducts and filtered colimits. There are dual assertions for products and cofiltered limits. (F.5) Adjoint functors. Definition F.21. Let F : C → D and G : D → F be functors. Recall that F is called left adjoint to G or, equivalently, that G is called right adjoint to F if there exist isomorphisms, functorial in A ∈ C and B ∈ D, (F.5.1)

HomC (A, G(B)) ∼ = HomD (F (A), B).

We also say that (F, G) is an adjoint pair. Choosing A = G(B) in (F.5.1), the identity of G corresponds to a morphism of functors ϵ : F ◦ G −→ idD . Choosing B = F (A) one obtains in the same way a morphism of functors η : idC −→ G ◦ F. These morphism ϵ and η are called the adjunction morphisms. Sometimes ϵ is called the counit and η is called the unit of the adjunction. If a functor admits a right (resp. left) adjoint functor, this adjoint functor is unique up to unique isomorphism.

745 Proposition F.22. ([KaSh] O 1.5.6) Let (F, G) be a pair of adjoint functors with unit η and counit ϵ. (1) The functor G is fully faithful if and only if ϵ is an isomorphism. (2) The functor F is fully faithful if and only if η is an isomorphism. (3) The following conditions are equivalent. (i) F is an equivalence of categories. (ii) G is an equivalence of categories. (iii) F and G are fully faithful. In this case, F and G are quasi-inverse to each other and η and ϵ are isomorphisms. Proposition F.23. ([KaSh] O 2.1.10) Let F : C → D be a functor. (1) Suppose that F is right adjoint to some functor G : D → C. Then F commutes with limits. (2) Dually, suppose that F is left adjoint to some functor. Then F commutes with colimits. Definition F.24. Let C be a category that is finitely cocomplete. Then a functor F : C → C ′ is called right exact if it commutes with finite colimits. Similarly we define for a finitely complete category C a functor F : C → C ′ to be left exact if it commutes with finite limits. Compare Proposition F.38 below which states that for additive functors between abelian categories this definition gives the usual notion of exactness.

Additive and abelian categories We now reintroduce additive categories and abelian categories and give some more notions and results about them. A central notion for us will be the notion of a Grothendieck abelian category since all abelian categories that we will derive are Grothendieck categories. (F.6) Additive categories. See also Section (A.4). Proofs of the results below can be found in [KaSh] O Ch. 8. Definition and Proposition F.25. An additive category is a category C such that there exists the structure of an abelian group on the set HomC (X, Y ) for all objects X, Y of C, such that the composition ◦ of morphisms is bilinear, and such that all finite products exist in C. Such a structure is necessarily unique. Note that being additive is really a property of the underlying category not the datum of an additional structure on the category. If C is additive, then there is a natural structure of additive category on the opposite category C opp , as well. Proposition F.26. Let C be an additive category. (1) The category C has a zero object 0 (i.e., an object which is both initial ` and terminal). (2) For any objects X, Y ` in C the product X × Y and the coproduct X Y exist, and the natural morphism X Y → X × Y is an isomorphism. We write X ⊕ Y := X × Y .

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F Homological Algebra

Definition F.27. Let C and D be additive categories. A functor F : C → D is called additive, if it satisfies the following equivalent conditions: (i) For all objects X, Y of C, the map HomC (X, Y ) → HomD (F (X), F ((Y )) is a group homomorphism. (ii) The functor F commutes with finite products. Similarly, we will often consider a situation where all sets of morphisms carry the structure of an R-module for some given ring R, and thus make the following definition. Definition F.28. Let R be a (not necessarily commutative) ring. An additive category C is called R-linear if the abelian group HomC (X, Y ) is endowed with the structure of a left R-module for all objects X and Y such that the composition is R-bilinear. A functor f : C → D between R-linear categories is called R-linear if for all objects the map F : HomC (X, Y ) → HomD (F (X), F (Y )) is R-linear. For instance, the category of left modules over R is Cent(R)-linear, where Cent(R) denotes the center of R. Almost all functors between additive categories that we will consider are additive functors, and sometimes we will assume without further mention that functors between additive categories are additive. Definition F.29. Let C be an additive category. A subcategory C ′ of C is called an additive subcategory if there exists the structure of an additive category on C ′ (necessarily unique by Proposition F.25) such that the inclusion functor C ′ → C is additive. We can rephrase the definition as requiring that HomC ′ (X, Y ) is a subgroup of HomC (X, Y ) for all objects X, Y of C ′ and that C ′ is closed under taking finite products in C (up to isomorphism). Definition: Kernel and Cokernel F.30. Let u : X → Y be a morphism in an additive category. If the equalizer Eq(u, 0) of u and the zero morphism X → Y exists (Definition F.18), it is called kernel of u and denoted by Ker(u). It is endowed with a morphism i : Ker(u) → X such that u ◦ i = 0. A morphism w : Z → X factors through Ker(u) → X if and only if u ◦ w = 0, and in this case the factorization is unique. The morphism u is a monomorphism if and only if its kernel exists and Ker(u) = 0. For any two morphisms u, v : X → Y one has Eq(u, v) = Ker(u − v). Dually, one defines the cokernel of u by Coker(u) := Coeq(u, 0) it it exists. Then u is an epimorphism if and only if its cokernel exists and Coker(u) = 0. Finally we also set Im(u) = Ker(Y → Coker(u)) and Coim(u) := Coker(Ker(u) → X) if they exist. Definition F.31. A sequence 0

/X

f

/Y

g

/Z

/0

of morphisms in an additive category C is called split if there exists an isomorphism Y ∼ = X ⊕ Z such that via this isomorphism, f is identified with the natural embedding and g is identified with the natural projection. Definition F.32. Let C be an additive category. A non-zero object X is called indecomposable if it is not isomorphic to the direct sum of two non-zero objects in C.

747 (F.7) Abelian categories. Definition F.33. An additive category C is called an abelian category, if (a) every morphism has a kernel and a cokernel, and (b) every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism. Remark F.34. Let C be an additive category. By Remark F.30, Condition (a) of Definition F.33 is satisfied if and only if all equalizer and coequalizers exist. As in any additive category finite coproducts and finite products exist (Proposition F.26), we see that Condition (a) means that C is finitely complete and finitely cocomplete (Proposition F.19). Let us assume that this is the case. For every morphism f : X → Y the universal properties of Ker(f ) and Coker(f ) show that there exists a unique morphism f¯: Coim(f ) → Im(f ) making the diagram X  Coim(f )

f

/Y O

f¯

/ Im(f )

commutative. Then Condition (b) of Definition F.33 is satisfied if and only if u ¯ is an isomorphism for every morphism u in C ([KaSh] O 8.3.4). Altogether we see that an additive category is abelian if and only if it is finitely complete and finitely cocomplete and for every morphism f the induced morphism Coim(f ) → Im(f ) is an isomorphism. In particular a morphism f in an abelian category is an isomorphism if and only if Ker(f ) = 0 and Coker(f ) = 0, i.e., if and only if f is a monomorphism and an epimorphism. Example F.35. The category (AbGrp) of abelian groups is the prototypical example of an abelian category. More generally, for any (possibly non-commutative) ring R, the category of left R-modules is an abelian category, as is the category of right R-modules. The category of finitely generated free Z-modules is an additive category which has all kernels and cokernels (kernels are defined as usual using that every submodule of a free Z-module is again free, cokernels are formed by dividing the cokernel within the category of abelian groups by its torsion). But this category is not abelian: the homomorphism Z → Z given by multiplication by 2 is a monomorphism and an epimorphism in this category, but not an isomorphism. Definition F.36. Let C be an abelian category. A (finite or infinite) sequence ···

/ X i−1

di−1

/ Xi

di

/ X i+1

/ ···

in C is called exact at X i if Ker di = Im di−1 . It is called exact if it is exact at every i. A short exact sequence is an exact sequence of the form 0 → X → Y → Z → 0.

748

F Homological Algebra

Proposition F.37. Let C be an abelian category. For a short exact sequence 0

/X

f

/Y

g

/Z

/0

in C the following conditions are equivalent. (i) The sequence is split (Definition F.31). (ii) There exists a section of g, i.e., a morphism s : Z → Y with g ◦ s = idZ . (iii) There exists a retraction of f , i.e., a morphism r : Y → X with r ◦ f = idX . Proof. Clearly, (i) implies (ii) and (iii). Conversely, if s is a section of g (resp. r is a retraction of f ), then s ◦ g (resp. f ◦ r) is an idempotent endomorphism u of Y (i.e., u2 = u) and hence Y = Ker(u) ⊕ Im(u). A functor commutes with finite limits if and only if it commutes with products and with kernels (Proposition F.19), hence: Proposition F.38. An additive functor F : C → D between abelian categories is (1) left exact if and only if for any exact sequence 0→X→Y →Z in C, the sequence 0 → F (X) → F (Y ) → F (Z) is exact. (2) right exact if and only if for any exact sequence X→Y →Z→0 in C, the sequence F (X) → F (Y ) → F (Z) → 0 is exact. It is enough to test left exactness or right exactness on short exact sequences. Recall that we consider contravariant functors C → D as covariant functors C opp → D. Hence for a contravariant functor to be left exact means that the 0 is on the left in the sequence obtained by applying the functor F . For instance the Hom-functor is left exact in each variable (as it preserves limits in the second variable and sends colimits in the first variable to limits). Definition: Exact Functors F.39. A functor F : C → D of abelian categories is called exact if it is left exact and right exact. Equivalently, for every exact sequence ···

/ X i−1

/ Xi

/ X i+1

/ ···

the sequence ··· is exact.

/ F (X i−1 )

/ F (X i )

/ F (X i+1 )

/ ···

749 (F.8) Length and Jordan-H¨ older series in abelian categories. Definition F.40. Let C be an abelian category and let X be an object of C. (1) X is called simple if X ̸= 0 and 0 and X are the only quotient objects of X up to isomorphism. (2) X is called semisimple if it is isomorphic to a finite direct sum of simple objects. (3) X is said to be of finite length if there exists a finite sequence of subobjects 0 = X0 ,→ X1 ,→ · · · ,→ Xn−1 ,→ Xn = X such that Xi /Xi−1 is a simple object. If such a sequence exists, it is called a JordanH¨older sequence or composition series. An object X in an abelian category is of finite length if and only if it is artinian (i.e., it satisfies the descending chain condition on subobjects) and noetherian (i.e., it satisfies the ascending chain condition on subobjects). Proposition and Definition F.41. (Jordan-H¨ older theorem, [Ste] IV, §5) Let X be an object of finite length in an abelian category C. Let 0 = X0 ,→ X1 ,→ · · · ,→ Xn = X

and

′ 0 = X0′ ,→ X1′ ,→ · · · ,→ Xm =X

be two Jordan-H¨ older sequences. Then m = n and there exists a permutation π of ′ ′ ∼ /Xπ(i)−1 {1, . . . , n} such that Xπ(i) = Xi /Xi−1 . The common length n of all JordanH¨ older sequences of X is called the length of X. It is denoted by lgC (X) or simply by lg(X). The length is additive in exact sequences, more precisely: Proposition F.42. Let C be an abelian category and let 0 → X ′ → X → X ′′ → 0 be a short exact sequence in C. Then X is of finite length if and only if X ′ and X ′′ are of finite length. In this case one has lg(X) = lg(X ′ ) + lg(X ′′ ). (F.9) Subcategories of abelian categories. Let C be an abelian category. A full subcategory D of C is additive if and only if D is closed under finite products in C (by Definition F.27). A full additive subcategory D of C is abelian and the inclusion functor is exact if and only if D is closed under kernels and cokernels in C. Definition and Proposition F.43. We call a full subcategory D of an abelian category C plump1 if the following equivalent conditions are satisfied. (i) D is closed under kernels, cokernels, and extensions in C. (ii) D is a full additive subcategory and for every exact sequence X1 → X2 → X3 → X4 → X5 in C with X1 , X2 , X4 , X5 in D also X3 is in D. The plump subcategory D is then itself abelian and the inclusion functor D → C is exact. 1

O Here we follow Lipman [Lip2] O X . Kashiwara and Schapira [KaSh] use the notion of a thick subcategory which is very often defined differently. The Stacks project calls plump subcategories weakly Serre subcategories.

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F Homological Algebra

Definition and Proposition F.44. Let A be an abelian category. A non-empty full subcategory B of A is called Serre subcategory if the following two equivalent conditions hold. (i) For every short exact sequence 0 → X ′ → X → X ′′ → 0 in A one has X ′ , X ′′ ∈ B if and only if X ∈ B. (ii) For every exact sequence Y ′ → Y → Y ′′ in A with Y ′ , Y ′′ ∈ B one has Y ∈ B. Every Serre subcategory B is abelian and the inclusion B → A is an exact functor. Every Serre subcategory is a plump subcategory. The converse does not necessarily hold (Exercise 22.27). Example F.45. Let C be an abelian category. Then the full subcategory of objects of C of finite length (Definition F.40) is a Serre subcategory. (F.10) Five Lemma and Snake Lemma. Lemma F.46. (Four Lemma) Let C be an abelian category. Let / X1

X0 

f0



/ X2

f1



/ Y1

Y0

/ X3

f2



/ Y2

f3

/ Y3

be a commutative diagram in C whose rows are exact. (1) If f 0 is an epimorphism and f 1 and f 3 are monomorphisms, then f 2 is a monomorphism. (2) Dually, if f 3 is a monomorphism and f 0 and f 2 are epimorphisms, then f 1 is an epimorphism. In fact, it suffices that the rows are complexes (see Definition F.68 below) and that the sequences X 1 → X 2 → X 3 and Y 0 → Y 1 → Y 2 are exact. In particular, we obtain the statement of the classical “five lemma”: Corollary F.47. (Five lemma) Let C be an abelian category. Let



/ X2

/ X1

X0 f0



f1



/ Y1

Y0

/ X3

f2

f3



/ Y2

/ X4

/ Y3



f4

/ Y4

be a commutative diagram in C whose rows are exact sequences. If f 0 , f 1 , f 3 , f 4 are isomorphisms, then f 2 is an isomorphism. Lemma F.48. (Snake lemma) Let C be an abelian category. Let A

f

α

0

 /X

u

/B  /Y

g

/C γ

β v

 /Z

/0

751 be a commutative diagram in C whose rows are exact sequences. Then there is an exact sequence Ker α

f′

/ Ker β

g′

/ Ker γ

/ Coker α

δ

u′

/ Coker β

v′

/ Coker γ

where f ′ , g ′ , u′ , v ′ are the maps induced by f , g, u, and v, and where the “boundary map” δ is defined as follows: Set P := B ×C Ker γ and S := Y ⨿X Coker α. Then the projection P → Ker γ is an epimorphism and Coker α → S is injective. The composition P → B → Y → S factors through a morphism δ

P → Ker γ −→ Coker α → S. If there exists an exact fully faithful functor C → (R-Mod) for some not necessarily commutative ring (which is always the case if C is small by a Theorem of Mitchell), then δ can also be described on elements as follows. Let c ∈ Ker γ, let b ∈ B with g(b) = c, let x ∈ X with u(x) = β(b), and define δ(c) as the image of x in Coker α. (F.11) Injective and projective Objects. Definition F.49. (Injective and projective objects) Let C be an abelian category. (1) An object I in C is called injective, if the following equivalent conditions are satisfied. (i) The functor C opp → (AbGrp), X 7→ HomC (X, I), is exact. (ii) Every exact sequence in C of the form 0 → I → X → X ′′ → 0 is split. (2) Dually, an object P in C is called projective, if the following equivalent conditions are satisfied. (i) The functor C → (AbGrp), X 7→ HomC (P, X), is exact. (ii) Every exact sequence in C of the form 0 → X ′ → X → P → 0 is split. The full subcategory of injective (resp. projective) objects of C is an additive subcategory of C. Definition F.50. An abelian category C is said to (1) have enough injectives, if for every object X in C, there exists a monomorphism X → I from X to an injective object I. (2) have enough projectives, if for every object X in C, there exists an epimorphism P → X from a projective object P to X. Remark F.51. Let R be a not necessarily commutative ring. Then every free R-module is projective, and in particular the category (R-Mod) of left R-modules has enough projectives. Categories of abelian sheaves on some space typically do not have enough projectives. The category (R-Mod) also has enough injectives (see Proposition F.62 below). More generally, for every ringed space (X, OX ) the category of OX -modules has enough injectives (combine Proposition 21.7 and Proposition F.62 below).

752

F Homological Algebra

(F.12) Grothendieck abelian categories. One criterion for an abelian category to have enough injectives is to be a Grothendieck category. We first introduce the notion of generators of a category. Definition F.52. Let C be a category. A family of objects (Gi )i∈I for a (small) index Q set I is called a system of generators if the functor i∈I HomC (Gi , −) : C → (Sets) is conservative (Definition F.3). If the family (Gi )i∈I consists of a single object G, then G is called a generator. ` If C admits coproducts and a system of generators (Gi )i , then i∈I Gi is a generator of C. Proposition F.53. ([KaSh] O 5.2.4) Let A be an abelian category which admits all coproducts. For an object G of C, the following conditions are equivalent. (i) G is a generator. (ii) The functor HomC (G, −) : C → (Sets) is faithful. (iii) For any object X in A, there exist a (small) set I and an epimorphism G(I) → X. Definition F.54. An abelian category C is called a Grothendieck category, if (a) all (infinite) coproducts exist in C (and hence C is cocomplete by Proposition F.19), (b) filtered colimits are exact, (c) the category C admits a generator. For any ring R, the category of left (or: right) R-modules is a Grothendieck abelian category. The ring R, considered as an R-module, is a generator. Conversely, the GabrielPopescu theorem implies that every Grothendieck abelian category admits a fully faithful functor to the category of R-modules for a ring R (one takes as R the endomorphism ring of a generator G and the functor is given by X 7→ Hom(G, X)). This functor has an exact left adjoint functor. The category of finite-dimensional vector spaces over a field is an example of an abelian category which does not have all coproducts, and hence is not a Grothendieck abelian category. Remark F.55. Let C be a cocomplete abelian category. Let (Ui )i∈I be a family of objects in C such that for every epimorphism A → B in C with B = ̸ 0 thereLexists i ∈ I and a morphism Ui → A such that composition Ui → B is non-zero. Then Ui is a generator of C. For contravariant functors on Grothendieck categories there is the following easy criterion to be representable. Theorem F.56. ([Sta] 07D7) Let C be a Grothendieck abelian category, and let F : C opp → (Sets) be a functor. The functor F is representable if and only if it commutes with all small limits, i.e., F (colimi Xi ) = limi F (Xi ) for any diagram X : I → C and colimi Xi the colimit in C. Corollary F.57. Let C be a Grothendieck category and let D be a cocomplete category. Then a functor F : C → D has a right adjoint functor if and only if F commutes with small colimits.

753 Proof. The condition is clearly necessary. Now suppose that F commutes with colimits. ˜ X : C opp → (Sets), G ˜ X (Y ) := Let X be an object in D and consider the functor G ˜ X sends colimits in C to limits. Therefore it is representable HomD (F (Y ), X). Then G by Theorem F.56, i.e., there exists an object GX in C and an isomorphism of functors HomD (F (−), X) ∼ = HomC (−, GX ). By the Yoneda lemma, the construction X 7→ GX is functorial in X and therefore defines a right adjoint functor to F . Remark F.58. (1) Every Grothendieck category is cocomplete, as it is finitely cocomplete, as every abelian category, and as arbitrary direct sums exist, because these are filtered colimits of finite direct sums. Now use Proposition F.19. (2) One can also show that all Grothendieck categories are complete ([Bor] 5.2). This can also be easily deduced from Theorem F.56. Remark F.59. Let C be a Grothendieck category and let I be a small category. Then the category C I of functors I → C is a Grothendieck category ([Gro1] O 1.9.2). As Grothendieck categories are complete and cocomplete (Remark F.58), we obtain functors lim : C I → C,

colim : C I → C

As (co)limits commute with each other (see F.11), the functor lim is left exact and the functor colim is right exact. The property to be a Grothendieck category is inherited by so-called Giraud subcategories: Definition F.60. A subcategory D of a Grothendieck abelian category C is called a Giraud subcategory, if the inclusion functor admits a left adjoint and if this left adjoint functor is left exact (and hence exact). For example, the category of sheaves of OX -modules on a ringed space (X, OX ) is a Giraud subcategory of the category of presheaves of OX -modules because sheafification is an exact functor. Proposition F.61. ([Ste] X, §1) Let D be a Giraud subcategory of a Grothendieck abelian category C and let a : C → D be the exact left adjoint to the inclusion. Then D is itself a Grothendieck abelian category. Moreover, an object of D is injective in D if and only if it is injective in C. If G is a generator of C, then a(G) is a generator of D. (F.13) Injective objects in Grothendieck abelian categories and locally noetherian categories. In this section, A denotes a Grothendieck category. First note that A has enough injective objects. Proposition F.62. Let C be a Grothendieck abelian category. Then C has enough injectives. Below, we will state a stronger result (Theorem F.185). We will use the following notions and results to study the category of OX -modules for a locally noetherian scheme X. A reference is [Ste] X, §2–5.

754

F Homological Algebra

Proposition F.63. Let A be a Grothendieck abelian category and let (Gi )i∈I be a family of generators. An object X of A is injective if and only if for all i, every monomorphism ι : U → Gi , and every morphism φ : U → X, there exists φ˜ : Gi → X such that φ˜ ◦ ι = φ. For instance, if A is a ring, the category of A-modules is generated by A. Hence an A-module X is injective if and only if every homomorphism a → X, where a is an ideal of A, can be extended to A. Definition F.64. Let X be an object of A. An injective hull of X is a monomorphism ι : X → I such that I is injective and such that ι−1 (J) := X ×I J = ̸ 0 for every non-zero subobject J of I. Proposition F.65. Every object X of A has an injective hull ι : X → I. If ι′ : X → I ′ is ∼ another injective hull, then there exists an isomorphism γ : I → I ′ (in general not unique) ′ such that ι = γ ◦ ι. Definition and Proposition F.66. Let A be a Grothendieck abelian category. (1) An object X of A is called finitely generated if the following equivalent assertions hold. (i) Whenever X is the filtered colimit of subobjects Xi , then there exists an index i0 such that X = Xi0 . (ii) The functor HomA (X, −) preserves colimits of filtered diagrams I → A where the transition maps are monomorphisms. (2) An object X of A is called noetherian if the following equivalent conditions hold. (i) Every ascending chain of subobjects of X becomes stationary. (ii) Every subobject of X is finitely generated. (3) The category A is called locally finitely generated (resp. locally noetherian) it it has a family of finitely generated (resp. noetherian) generators. Let A be a ring. Then for the category of A-modules the notions of finitely generated and noetherian for A-modules defined above coincide with the usual notions. The category of A-modules is locally finitely generated (A is a finitely generated generator) and it is locally noetherian if and only if A is a noetherian ring. Proposition F.67. Let A be a locally finitely generated Grothendieck abelian category. Then A is locally noetherian if and only if every coproduct of injective objects is injective. In this Lcase, everyLinjective object Z is a coproduct of indecomposable injective objects. If Z = i∈I Xi = j∈J Yj are two such decompositions, then there exists a bijection α : I → J such that Xi ∼ = Yα(i) for all i ∈ I. In particular, we see that a ring A is noetherian if and only if every direct sum of injective A-modules is again injective.

Complexes in additive and abelian categories As a first step towards the definition of the derived category of an abelian category we recall the notion of a complex, the notion of homotopy between two morphisms of complexes, and the notion of a quasi-isomorphism. Then we give various constructions of complexes.

755 (F.14) Categories of Complexes. In this section A always denotes an additive category. Definition F.68. (1) A complex in A is a family (X j )j∈Z of objects and morphisms (dj : X j → X j+1 )j∈Z of A such that dj ◦ dj−1 = 0 for all j. (2) A morphism of complexes f : X → Y is a family of morphisms f j : X j → Y j for j ∈ Z such that the diagram ···

···

d

d

/ X j−1 

d

f j−1

/ Y j−1

d

/ Xj 

d

fj

/ Yj

d

/ X j+1 

d

/ ···

d

/ ···

f j+1

/ Y j+1

commutes. We denote by C(A) the category of complexes in A. (3) The translation functor T : C(A) → C(A) is defined on objects by (X j )j 7→ (X j+1 )j , djT X = −dj+1 and on morphisms by (f j )j 7→ (f j+1 )j . X As in the above diagram, we often write d instead of dj . The translation functor is sometimes called the shift functor . It is an automorphism of the category of complexes. For i ∈ Z we set ( )[i] := T i . In other words, given a complex X = (X j )j , we have (X[i])j := X j+i ,

djX[i] = (−1)i di+j X .

If A admits complex (Xi )i as a graded L view a L L countable direct sums, then we can also object i Xi together with an endomorphism d : X → i i i Xi which shifts degrees by 1 and such that d ◦ d = 0. A complex · · · → X i−1 → X i → X i+1 → · · · is called acyclic at X i , if Ker di = Im di−1 . It is called acyclic if it is exact at every i. Clearly, an acyclic complex is the same as an exact sequence indexed by the integers. The category C(A) of complexes in A is additive. If A is abelian, then C(A) is abelian, too. In particular, we have the following notions: (1) Let X be a complex in A. A subcomplex of X is a subobject of X in C(A). Explicitly, a subcomplex of X is a complex Y such that Y i ⊆ X i for all i, compatibly with the differentials. (2) For every morphism f : X → Y of complexes there exists the notion of a kernel and a cokernel . If A is abelian, these always exist and they are formed componentwise: Ker(f )i = Ker(f i ), Coker(f )i = Coker(f i ), with differentials induced from the differentials of X. Similarly, we have the image Im(f ) of f , Im(f )i = Im(f i ). If Y ⊂ X is a subcomplex, we set X/Y := Coker(Y → X) if this cokernel exists. (3) Suppose that A is an abelian category. A sequence 0→X→Y →Z→0 of complexes is exact in C(A) if and only if it is termwise exact, i.e. 0 → X i → Y i → Z i → 0 is exact for all i ∈ Z. (4) We can form finite direct sums and finite products in C(A) componentwise. (5) More generally, if A admits limits (resp. colimits) indexed by some category, then the same is true for C(A) and these limits (resp. colimits) are formed componentwise. In particular, if A is complete (resp. cocomplete), then C(A) is complete (resp. cocomplete).

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F Homological Algebra

(6) If A is a Grothendieck abelian category, the same is true for C(A). Definition F.69. Let A be an additive category. A sequence 0 → X → Y → Z → 0 of complexes in C(A) is called termwise split if 0 → X i → Y i → Z i → 0 is a split sequence (Definition F.31) for all i ∈ Z. Definition F.70. A complex X = (X j ) is called bounded ( bounded below, bounded above, resp.), if X j = 0 for |j| ≫ 0 (for j ≪ 0, for j ≫ 0, resp.). We denote by Cb (A), C+ (A), C− (A) the full subcategories of C(A) consisting of all complexes which are bounded, bounded below, bounded above. Let M ⊆ Z be a subset. A complex X is said to be concentrated in degree M if X i = 0 / M . We have the full subcategory CM (A) of C(A) of complexes concentrated in for all i ∈ degrees M . We write C≤a (A) := C(−∞,a] (A), etc. All these categories are abelian. Given a family of objects and morphisms as above with an “interval” in Z as index set, we can extend it to a complex by adding zeros. This gives obvious fully faithful exact functors between the categories above. In particular, we consider A as a full subcategory of C(A) (and of C+ (A), . . . ) by sending an object X ∈ A to the complex · · · → 0 → X → 0 → · · · , where X sits in degree 0. Let A be an abelian category. We frequently consider the following functors C(A) → A: πn : (X i )i 7→ X n , Z n : (X i )i 7→ Ker(d : X n → X n+1 ), B n : (X i )i 7→ Im(d : X n−1 → X n ), H n : X 7→ Z n (X)/B n (X). We call H n (X) the n-th cohomology object of X. Note that H n (X) = H 0 (X[n]). As a reminiscence of the origin of these definitions in algebraic topology, sometimes Z n is called the object of cocycles of degree n, and B n the object of coboundaries of degree n. If M ⊆ Z is a subset, then we say that a complex is acyclic in degrees M if H n (X) = 0 for all n ∈ M . Proposition F.71. Let A be an abelian category. Let 0 → X → Y → Z → 0 be a short exact sequence in C(A). For i ∈ Z define a morphism H i (Z) → H i+1 (X) by applying the Snake lemma F.48 to the diagram / Y i / Im(di−1 ) Y

X i / Im(di−1 X ) diX

0

diY

 / Ker(di+1 ) X

 / Ker(di+1 ) Y

/ Z i / Im(di−1 ) Z

/0

diZ

 / Ker(di+1 ). Z

Then the long cohomology sequence . . . −→ H i−1 (Z) −→ H i (X) −→ H i (Y ) −→ H i (Z) −→ H i+1 (X) −→ . . . is exact. Furthermore, we have the following truncation functors: For a complex X with components in an abelian category, we set (F.14.1)

τ ≤n (X) : · · ·

/ X n−2

/ X n−1

/ Ker dn

/0

/ ··· ,

757 and dually (F.14.2)

τ ≥n (X) : · · ·

/0

/ Coker dn−1

/ X n+1

/ X n+2

/ ··· .

Then τ ≤n (X) is a subcomplex of X and τ ≥n X is a quotient complex of X. For the cohomology objects, we have  i  i H (X) if i ≤ n, H (X) if i ≥ n, i ≤n i ≥n H (τ X) = H (τ X) = 0 if i > n, 0 if i < n. We also have the stupid truncation functors defined by (F.14.3)

σ ≥n (X) := (. . . → 0 → 0 → X n → X n+1 → . . .), σ ≤n (X) := (. . . → X n−1 → X n → 0 → 0 → . . .).

Then σ ≥n (X) is a subcomplex of X and σ ≤n (X) is a quotient of X. Definition F.72. (Mapping cone) For a morphism f : X → Y of complexes with components in an additive category A, we define the mapping cone Cf of f by setting   −dX 0 . Cfn = X n+1 ⊕ Y n , dnCf = f dY Below (Remark F.79) it is explained how to characterize the mapping cone by a universal property. The mapping cone encodes kernel and cokernel simultaneously in the following sense. Example F.73. Let A be an abelian category and let f : A → B be a morphism in A which we consider as a morphism of complexes concentrated in degree 0. Then f

Cf = (. . . −→ 0 −→ A −→ B −→ 0 −→ . . .), with B in degree 0. Hence H 0 (Cf ) = Coker(f ),

H −1 (Cf ) = Ker(f ).

Remark F.74. Let A and A′ be additive categories. Let F : A → A′ be an additive functor. Then on complexes F induces an additive functor C(F ) : C(A) → C(A′ ) by sending a complex ((X i ), (di )) to (F (X i ), F (di )) and a morphism (ui ) to (F (ui )). This functor preserves termwise split sequences and mapping cones (i.e., FCu = CF (u) for every morphism u of complexes). (F.15) Homotopy of complexes. Definition F.75. Let A be an additive category. A homotopy between two morphisms f, g : X → Y of complexes of A is a family of morphisms hi : X i → Y i−1 of complexes such that f i − g i = hi+1 ◦ diX + di−1 ◦ hi . Y If such a family (hi )i exists, we call the morphisms f, g : X → Y homotopic.

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F Homological Algebra

Note that h is not assumed to be a morphism of complexes X → Y [−1]. But if u : X → Y [−1] is a morphism of complexes, then h + u is again a homotopy between f and g. More precisely the set of homotopies between f and g is a principal homogeneous space (possibly empty) under the abelian group HomC(A) (X, Y [−1]). Being homotopic is an equivalence relation on the set HomC(A) (X, Y ). As is easily checked, composition of morphisms induces a pairing on the sets of equivalence classes. Hence we can define: Definition F.76. Let A be an additive category, and let C(A) be the category of complexes in A. The homotopy category K(A) is defined by Ob(K(A)) = Ob(C(A)),

HomK(A) (X, Y ) = HomC(A) (X, Y )/ ∼,

where f ∼ g if and only if f and g are homotopic. For ∗ ∈ {b, +, −} or ∗ ∈ { [a, b] ; a, b ∈ Z ∪ {±∞}, a ≤ b } we denote by K ∗ (A) the full subcategory of K(A) consisting of complexes in C ∗ (A). Definition F.77. Let A be an additive category. A morphism of complexes f : X → Y is called a homotopy equivalence if it is an isomorphism in K(A), i.e., there exists a morphism g : Y → X of complexes, such that g ◦ f is homotopic to idX , and f ◦ g is homotopic to idY . If two complexes are isomorphic in K(A), they are also called homotopy equivalent. Remark F.78. The homotopy category K(A) is an additive category. Finite direct sums are formed componentwise. Note that even if A is an abelian category, K(A) is not abelian in general. For instance if A is the abelian category of abelian groups, then the canonical morphism Z → Z/2Z, considered as morphisms of complexes concentrated in degree 0, cannot be factorized as an epimorphism followed by a monomorphism. This can be shown by a direct argument. Alternatively, one can also use that K(A) can be endowed with the structure of a triangulated category (see Section (F.27) below) and that in a triangulated category every epimorphism has a right inverse (see Remark F.122 below). Remark F.79. Let u : X → Y be a morphism in C(A). Then the inclusions Y i → X i+1 ⊕ Y i = Cui define a morphism of complexes ι : Y → Cu . Then the composition X → Y → Cu is homotopic to zero via the homotopy h defined by the inclusion hi : X i → Cui−1 = X i ⊕ Y i−1 . Moreover, the mapping cone Cu has the following universal property of a “homotopy ˜ : v ◦ u ≃ 0 a homotopy, then cokernel”. If v : Y → Z is a morphism of complexes and h there exists a unique morphism of complexes w : Cu → Z such that v = w ◦ ι and ˜ i = wi−1 ◦ hi . Indeed, we define h ˜ i+1 + v i : X i+1 ⊕ Y i → Z i . wi := h A similar argument shows that Cu [−1] has “the universal property of a homotopy kernel”, i.e., by composition one obtains a bijection between HomC(A) (Z, Cu [−1]) and the set of pairs consisting of a morphism of complexes v : Z → X and a homotopy of u ◦ v to 0. Remark F.80. Let F : A → A′ be a (covariant or contravariant) additive functor of additive categories. Then the induced functor C(F ) : C(A) → C(A′ ) further induces a functor K(F ) : K(A) → K(A′ ).

759 Remark F.81. The truncation functors τ ≤n and τ ≥n induce functors K(A) → K(A). Note that this is in general not true for the stupid truncation functors σ ≤n and σ ≥n . (F.16) Quasi-isomorphisms. Definition F.82. Let A be an abelian category. A morphism f : X → Y of complexes is called a quasi-isomorphism (sometimes abbreviated as qis), if the induced morphisms H i (f ) : H i (X) → H i (Y ) on the cohomology are isomorphisms. A complex is exact if and only if there exists a quasi-isomorphism to the zero complex. Note that if f : X → Y is a quasi-isomorphism, there does not necessarily exist a quasi-isomorphism Y → X! For example, let X be given by Z → Z, x → 2x, (in degrees 0, 1, extended by 0), and let Y be given by Z/2 in degree 1. While the natural map X → Y is a qis, there is no non-zero morphism of complexes Y → X at all. The following example will be used very often. Example F.83. Let · · · → X i−1 → X i → X i+1 → . . . be a sequence in an abelian category A. Then this sequence is exact if and only if for every i ∈ Z the morphism of complexes ...

/ X i−1

...

 /0

di−1

/ Xi 

di

/ X i+1

di+1

/0

/ ...

 / X i+2

/ ...

is a quasi-isomorphism. The following lemma, though easy to prove, is a cornerstone of the theory of derived categories: Lemma F.84. Let A be an abelian category. If f, g : X → Y are homotopic morphisms in C(A), then the induced maps on cohomology H i (f ) and H i (g) are equal. In particular, every homotopy equivalence is a quasi-isomorphism. The converse is not true: The above defined quasi-isomorphism 2

(Z −→ Z) → (0 → Z/2Z) is not a homotopy equivalence. Remark F.85. Let A be an abelian category. A morphism f : X → Y in C(A) is a quasi-isomorphism if and only if Cf is an exact complex. (F.17) Double complexes. Let A be an additive category. Definition F.86. A double complex in A is given by a family (X i,j )i,j∈Z of objects in i,j i,j → X i+1,j (horizontal differential) and di,j → X i,j+1 A with morphisms di,j 1 : X 2 : X (vertical differential) for all i, j ∈ Z, such that for all i, j ∈ Z we have ◦ di,j di+1,j 1 = 0, 1

◦ di,j di,j+1 2 = 0, 2

i,j+1 ◦ di,j ◦ di,j di+1,j 2 . 1 = d1 2

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F Homological Algebra

Together with the obvious notion of morphisms of double complexes, we obtain the category C2 (A) of double complexes in A. We visualize a double complex as follows. (F.17.1)

...

. .O .

...

/ X i,j+1 O / X i,j O

...

...

...

/ X i+1,j+1 O

/ ...

di,j+1 1

di,j 2

...

. .O .

di+1,j 2 di,j 1

/ X i+1,j O

/ ...

...

...

Each column and each row of a double complex is a complex and we can view every double complex as complex of vertical or of horizontal complexes. Both point of views yield an equivalence of categories C2 (A) = C(C(A)). In particular C2 (A) is an additive category. It is abelian if A is an abelian category. Definition F.87. Let X be a double complex in A. Suppose that A admits countable direct sums or that for each k there only finitely many non-zero components X i,j with i + j = k. The total complex attached to X is the complex Tot(X) given by M i i,j i,j Tot(X)k = X i,j , d|X i,j = di,j → X i+1,j ⊕ X i,j+1 . 1 ⊕ (−1) d2 : X i+j=k

This definition is not symmetric in (i, j). We could have also chosen to introduce a sign in the horizontal direction. Remark F.88. Clearly the construction of the total complex is functorial yielding a functor Tot from the full subcategory of double complexes X, such that for each k there only finitely many non-zero components X i,j with i + j = k, to the category of complexes C(A). For the rest of the section let us agree to identify C 2 (A) = Cvert (Chor (A)), i.e., we view a double complex X as a complex in vertical direction of horizontal complexes . .O . X •,j+1 O X •,j O ...,

761 We obtain a fully faithful functor ι[j] : C(A) → C 2 (A) by considering every (horizontal) complex as a double complex concentrated in vertical degree −j. By our choice of sign we find Tot ◦ι[j] = idC(A) 2 . Remark F.89. Suppose that u : X → Y is a morphism of double complexes which we view as a morphism of vertical complexes of horizontal complexes. Suppose that (h•,j : X •,j → Y •,j−1 )j is a homotopy of u with 0, then one checks that M (−1)i hi,j : Tot(X)k −→ Tot(Y )k−1 i+j=k

yields a homotopy of Tot(u) with 0. In particular, Tot induces a functor (F.17.2)

Tot : K + (C + (A)) −→ K + (A).

(F.18) The Hom complex. Let A be an additive category. Let X and Y be complexes in C(A). We have the complex of morphisms HomA (X, Y ), Y HomA (X, Y )i = HomA (X k , Y k+i ), k∈Z

di (f ) = dY ◦ f − (−1)i f ◦ dX

for f ∈ Hom(X, Y )i ,

Q Q i.e., if f = (f k )k ∈ k∈Z HomA (X k , Y k+i ), then di f ∈ k∈Z HomA (X k , Y k+i+1 ) is given by (di f )k = dk+i ◦ f k − (−1)i f k+1 ◦ dkX ∈ HomA (X k , Y k+i+1 ). Y Then HomA (X, Y ) is a complex of abelian groups. This construction is functorial in X and in Y and we obtain a functor (F.18.1)

HomA (−, −) : C(A)opp × C(A) −→ C(AbGrp)

One easily checks that for all i ∈ Z one has (F.18.2)

Z i (HomA (X, Y )) = HomC(A) (X, Y [i]), H i (HomA (X, Y )) = HomK(A) (X, Y [i]).

Remark F.90. The functor (F.18.1) preserves homotopy in both variables ([BouA10] O §5, Prop. 3) and hence induces a functor (F.18.3) 2

HomA (−, −) : K(A)opp × K(A) −→ K(AbGrp).

If we had viewed X as a complex in horizontal direction of vertical complexes, we would obtain inclusions ι′ [i] : C(A) → C 2 (A) with Tot ◦ι′ [i] = (−1)i idC(A) .

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F Homological Algebra

(F.19) Tensor product of complexes. Assume that the additive category A has an associative and commutative tensor product. We will not make this precise (which can be done, see for instance [KaSh] O , Chap. 4) because we will need the construction of tensor products of complexes only if A is the category of R-modules for some (commutative) ring R or the category of OX -modules for some ringed space (X, OX ), or some subcategory of these that is stable under the usual tensor product. In this case a tensor product is defined. The tensor product X ⊗ Y of complexes X and Y in C(A) is defined as the total complex of the double complex X i ⊗ Y j with differentials induced by those of X and Y . Explicitly, this means M Xi ⊗ Y j, (X ⊗ Y )n := i+j=n

dn|X i ⊗Y j

:=

diX

⊗ idY j +(−1)i idX i ⊗djY .

The complex X ⊗ Y exists if A admits countable direct sums or if for each n there are only finitely many (i, j) with i + j = n, X i ̸= 0, and Y j ̸= 0 (for instance if X and Y are both bounded below or are both bounded above). For the rest of the section we will always assume that all tensor products of complexes exist. Remark F.91. The tensor product of complexes is functorial in both variables and we obtain a functor − ⊗ − : C(A) × C(A) −→ C(A) which is additive in each component. This functor preserves homotopy in both components ([BouA10] O §4, Prop. 3) and hence induces a functor (F.19.1)

− ⊗ − : K(A) × K(A) −→ K(A)

Remark F.92. Let X, Y , and Z be complexes in C(A). (1) There is an isomorphism, functorial in X and Y (F.19.2)

∼

X ⊗ Y −→ Y ⊗ X ∼

in C(A) which is given on X i ⊗ Y j by (−1)ij τ , where τ : X i ⊗ Y j → Y j ⊗ X i is given by the commutativity of the tensor product. (2) There is an isomorphism, functorial in X, Y , and Z (F.19.3)

∼

(X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z)

in C(A) whose restriction to (X i ⊗ Y j ) ⊗ Z k is given by the associativity isomorphism ∼ (X i ⊗ Y j ) ⊗ Z k → X i ⊗ (Y j ⊗ Z k ) of the tensor product. Let A be a commutative ring and let C(A) := C(A-Mod) be the category of complexes of A-modules. We denote by K(A) its homotopy category. For complexes X and Y of A-modules the Hom complex HomA (X, Y ) is a complex of A-modules and we obtain functors (F.19.4)

HomA (−, −) : C(A)opp × C(A) −→ C(A) HomA (−, −) : K(A)opp × K(A) −→ K(A).

763 Proposition F.93. ([BouA10] O §5, 7) Let A be a commutative ring. There is an isomorphism in C(A) ∼

HomA (X ⊗ Y, Z) −→ HomA (X, HomA (Y, Z)).

(F.19.5)

which is functorial in the complexes X, Y and Z of A-modules. Remark F.94. Applying the functor Z 0 and H 0 to the isomorphism (F.19.5) we obtain by (F.18.2) functorial isomorphisms ∼

(F.19.6)

HomC(A) (X ⊗ Y, Z) −→ HomC(A) (X, HomA (Y, Z)), ∼

HomK(A) (X ⊗ Y, Z) −→ HomK(A) (X, HomA (Y, Z)).

Therefore the functor X 7→ X ⊗ Y from C(A) to C(A) (resp. from K(A) to K(A)) is left adjoint to the functor Z 7→ HomA (Y, Z). One has the following K¨ unneth formula for the tensor product of complexes of modules. Proposition F.95. Let A be a ring, and let E • , F • be complexes of A-modules and suppose that Z n (E) and B n (E) are flat A-modules for all n ∈ Z. (1) For every n ∈ Z there is a functorial exact sequence M H i (E • ) ⊗A H j (F • ) −→ H n (E • ⊗A F • ) 0 −→ i+j=n

−→

M

i j TorA 1 (H (E), H (F )) −→ 0.

i+j=n+1 n

n

If in addition B (E) and B (F ) are projective for all n ∈ Z, then the exact sequence is (non-functorially) split. (2) Suppose that Z n (E), B n (E) and H n (E) are flat A-modules for all n ∈ Z. Then for all n ∈ Z there is a functorial isomorphism M ∼ H i (E • ) ⊗A H j (F • ) −→ H n (E • ⊗A F • ). i+j=n

For the proof we refer to [BouA10] O §4.7.

Spectral sequences We now give a very short introduction to spectral sequences tailored to our purposes. Definitions, constructions, and levels of generality that we will not need are usually not mentioned, even if they are very important in other areas of mathematics (such as topology). In the first three sections we start by defining spectral sequences and explain some information one gets from them such as certain exact sequences. In the next three sections we will explain how one obtains spectral sequences, first from exact couples, then from filtered complexes, and finally from double complexes. Essentially each of these constructions is a special case of the previous construction.

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F Homological Algebra

(F.20) Graded and Filtered objects. Definition F.96. Let A be an additive category. (1) A graded object in A is a family (X i )i∈Z of objects X i ∈ A. (2) Let d ∈ Z. A morphism f : X = (X i ) → Y = (Y i ) of graded objects of degree d is a family (f i )i∈Z of morphism f i : X i → Y i+d . A bi-graded object in A is a graded object in the category of graded objects of A, i.e., it is a family (X p,q )p,q∈Z of objects X p,q ∈ A. For d, e ∈ Z a family of morphisms f p,q : X p,q → Y p+d,q+e is called a morphism of bi-degree (d, e). (Bi-)Graded objects and morphisms of (bi-)graded objects of degree 0 (resp. bi-degree (0, 0)) form an additive category which is abelian if A is abelian (see also Exercise F.10). Definition F.97. Let A be an additive category and let X be an object of A. A descending filtration on X is a family of subobjects X ⊇ · · · ⊇ F p−1 X ⊇ F p X ⊇ F p+1 X ⊇ . . . ′

We call such a filtration finite if there exist p ≤ p′ with F p X = X and F p X = 0. A filtered object of A is an object of A together with a descending filtration. A morphism f : (X, (F p X)) → (Y, (F p Y )) of filtered objects is a morphism f : X → Y such that f |F p X factors through F p Y for all p ∈ Z. Suppose that A is an abelian category. Then we define for a filtered object (F p X)p the associated graded object (grpF (X))p∈Z by grp X := grpF (X) := F p X/F p+1 X. One obtains the category of filtered objects of A. This category is usually not abelian even if A is abelian (Exercise F.11). If A is abelian, (F p X)p 7→ (grpF (X))p∈Z defines a functor from the category of filtered objects in A to the category of graded objects in A. Of course, one has analogous definitions for ascending filtrations. (F.21) Definition of spectral sequences. In this section, A always denotes an abelian category. Definition F.98. Let r0 ∈ Z. (1) A spectral sequence ( of cohomological type, starting at level r0 ) consists of a family (Er , dr )r≥r0 of bi-graded objects Er = (Erp,q )p,q∈Z in A and morphisms dr : Er → Er of bi-degree (r, 1 − r) such that dr ◦ dr = 0 and of isomorphisms of bi-graded objects of bi-degree (0, 0) ∼ Ker(dr )/ Im(dr ) −→ Er+1 for all r ≥ r0 . (2) A morphism of spectral sequences f : (Er , dr )r≥r0 → (Er′ , d′r )r≥r0 is given by a family of morphisms fr : Er → Er′ of bi-degree (0, 0) such that fr ◦ dr = d′r ◦ fr and such that fr+1 is the morphism induced by fr via the given isomorphisms Ker(dr )/ Im(dr ) ∼ = ′ . Er+1 and Ker(d′r )/ Im(d′r ) ∼ = Er+1

765 Remark F.99. Let (Er , dr )r≥r0 be a spectral sequence. (1) Each level Er and dr of a spectral sequence determines Er+1 up to isomorphism but it does not determine dr+1 . (2) If (Er , dr )r≥r0 is a spectral sequence starting at level r0 and if r0′ ≥ r0 is an integer, then (Er , dr )r≥r0′ is a spectral sequence starting at level r0′ . Let (Er , dr )r≥r0 be a spectral sequence. We define inductively bi-graded subobjects 0 = Br0 ⊆ Br0 +1 ⊆ . . . ⊆ Zr0 +1 ⊆ Zr0 = Er0 as follows:

(F.21.1)

Br0 := 0, Zr0 := Er0 , Br+1 /Br := Im(dr : Zr /Br → Zr /Br ), Zr+1 /Br := Ker(dr : Zr /Br → Zr /Br ),

i.e., Br+1 is the unique subobject of Zr containing Br such that Br+1 /Br is the image of dr , and Zr+1 is the unique subobject of Zr containing Br such that Zr+1 /Br is the kernel of dr . Hence for all r ≥ r0 one has an isomorphism of bi-graded objects Zr /Br ∼ = Er and a short exact sequence d

r 0 −→ Zr+1 /Br −→ Zr /Br −→ Br+1 /Br −→ 0. S If there exists a smallest bi-graded subobject r≥r0 Br of Er0 containing allT Br , we call this subobject B∞ . Similarly, if there exists a largest bi-graded subobject r≥r0 Zr contained in all Zr , we call this subobject Z∞ . We obtain a bi-graded object

E∞ := Z∞ /B∞ . From now on we will always assume that B∞ and Z∞ exist. This is for instance the case, if theL abelian category A has countable direct Q sums and countable products. Then B∞ = Im( r Br → Er0 ) and Z∞ = Ker(Er0 → r Er0 /Zr ). If filtered colimits are exact, we also have B∞ = colimr Br . Remark/Definition F.100. A spectral sequence (Er , dr )r≥r0 is said to degenerate at Er if dk = 0 for all k ≥ r. In this case all ∞-terms exist and one has Br = Br+1 = · · · = B∞ ,

Zr = Zr+1 = · · · = Z∞ ,

Er ∼ = Er+1 ∼ = ... ∼ = E∞ .

Definition/Remark F.101. Let (Er , dr )r≥r0 be a spectral sequence in A. We say that the spectral sequence is bounded if for each n ∈ Z there exists only finitely many Erpq0 = ̸ 0 with p + q = n. Suppose that the spectral sequence is bounded. As Erpq is isomorphic to a subquotient of Erpq0 for r ≥ r0 , there exist for all p, q ∈ Z an integer r(p, q) ≥ 0 such that both maps drp−r,q−1+r : Erp−r,q−1+r −→ Erpq ,

pq p+r,q+1−r dpq r : Er → E

are zero for all r ≥ r(p, q). Hence for all r ≥ r(p, q) one has

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F Homological Algebra

pq pq Brpq = Br+1 = · · · = B∞ ,

pq pq Zrpq = Zr+1 = · · · = Z∞ ,

pq pq Erpq = Er+1 = · · · = E∞ .

We say that the spectral sequence is a first (resp. third ) quadrant spectral sequence if Erpq0 = 0 if p < 0 or q < 0 (resp. if p > 0 or q > 0). Such spectral sequences are bounded. Definition F.102. Let (Er , dr )r≥r0 be a bounded spectral sequence in A. Let (H n )n∈Z be a graded object in A endowed with a descending filtration · · · ⊇ F p H n ⊇ F p+1 H n ⊇ . . . We assume that for each n ∈ Z the filtration (F p H n )p is finite (i.e., there exist p ≤ p′ ′ (depending on n) such that F p H n = H n and F p H n = 0). Then we say that the spectral sequence (Er , dr )r≥r0 converges to (H n , F p H n )n,p∈Z and write Erpq0 ⇒ H p+q if we are given for all p, q ∈ Z an isomorphism (F.21.2)

∼

pq −→ grpF H p+q = F p H p+q /F p+1 H p+q . E∞

The filtered graded object (H n , F p H n )n,p is then also called the limit of the spectral sequence. ˜ n ) of spectral sequences with limit term is ˜r , d˜r , F • H A morphism (Er , dr , F • H n ) → (E ˜ ˜ a pair (f, g), where f : (Er , dr ) → (Er , dr ) is a morphism of bounded spectral sequences, ˜ n )n is a morphism of filtered graded objects such that for ˜ n, F •H g : (H n , F • H n )n → (H all p, q ∈ Z the following diagram commutes pq E∞



∼ =

pq f∞

˜ pq E ∞

/ grp H p+q F p+q grp ) F (g

∼ =

 ˜ p+q , / grp H F

where the horizontal morphisms are the isomorphisms (F.21.2). ˜rp,q ) be a morphism of spectral sequences. Suppose Remark F.103. Let f : (Erp,q ) → (E ∼ ˜ p,q →E there exists r0 such that f induces an isomorphism Erp,q r0 for all p and q. Then the 0 p,q ∼ ˜ p,q Five Lemma implies that f induces isomorphisms Es → Es for all s ≥ r0 and all p and q. ˜ n ) respectively and Suppose that both spectral sequences converge to (H n ) and (H that (f, g) is a morphism of spectral sequences with limit term, then g also induces an ∼ ˜ n )n , again by the Five Lemma. isomorphism of graded objects (H n )n → (H (F.22) Exact sequences attached to spectral sequences. Let A be an abelian category. For each spectral sequence (Er , dr )r≥r0 we continue to assume that the ∞-terms B∞ , Z∞ , and E∞ exist (automatically satisfied if A has countable products and countable direct sums).

767 Remark/Definition F.104. Let (Er , dr )r≥r0 be a spectral sequence. pq pq (1) Let (p, q) ∈ Z and suppose that dpq r = 0 for all r ≥ r0 . Then Zr = Er0 for all r ≥ r0 and one obtains a sequence of quotient maps pq pq pq /B∞ = E∞ . Erpq0 = Zrpq0 /Brpq0 ↠ Zrpq0 /Brpq0 +1 ↠ Zrpq0 /Brpq0 +2 ↠ . . . ↠ Z∞

(2) Dually, if the differentials drp−r,q−1+r with target Erpq are all zero, then Brpq = 0 for all r ≥ r0 and one obtains a sequence of monomorphisms pq pq E∞ = Z∞ ,→ . . . ,→ Zrpq0 +1 ,→ Zrpq0 = Erpq0

(3) Suppose that r0 ≥ 2 (i.e., all differentials go right and down) and that Erpq0 = 0 for all q < 0 (i.e., the spectral sequence is concentrated in the first and second quadrant). Then dp,0 r = 0 for all r ≥ r0 and p ∈ Z and we obtain for all p ∈ Z quotient maps p,0 Erp,0 ↠ E∞ . 0

(F.22.1)

p,0 If the spectral sequence converges to (H n )n∈Z , then H p has a filtration with E∞ as p−t,t p,0 last graded component (because E∞ = 0 for t > 0). Thus E∞ is a subobject of H p and the composition with (F.22.1) yields a morphism p,0 Erp,0 ↠ E∞ ,→ H p , 0

(F.22.2)

which is called edge morphism. (4) Dually, suppose that r0 ≥ 1 (i.e., all differentials go right and not up), that Erpq0 = 0 for all p < 0, and that the spectral sequence converges to (H n )n∈Z . Then one obtains for all q ∈ Z an edge morphism as composition 0,q H q ↠ E∞ ,→ Er0,q . 0

(F.22.3)

The hypotheses in (3) and (4) are both satisfied if r0 ≥ 2 and (Er , dr )r≥r0 is a first quadrant spectral sequence. Note that in (3) the same arguments show similar, notationally slightly more complicated results if one supposes that there exists q0 ∈ Z with Erpq0 = 0 for all q < q0 . A similar remark holds for (4). Proposition F.105. (Exact sequence of low degrees, [CaEi] O XV, 5.12) Let (Er , dr )r≥2 be a first quadrant spectral sequence converging against (H n )n . (1) Then the sequence d0,1

e

e

e

2 E22,0 −→ H 2 0 −→ E21,0 −→ H 1 −→ E20,1 −→

(F.22.4)

is exact. Here all arrows labeled with e are edge morphisms. (2) More generally, let n > 0 and suppose that E2p,q = 0 for all 0 < q < n and for all p. Then for all i < n the edge morphisms are isomorphisms ∼

E2i,0 −→ H i and the sequence (F.22.5)

δ 0,n

e

e

e

2 E2n+1,0 −→ H n+1 0 −→ E2n,0 −→ H n −→ E20,n −→

is exact, where the arrows labeled with e are edge morphisms and where δ20,n is the composition ∼

∼

∼

d0,n n+1

∼

∼

n+1,0 0,n → · · · → E2n+1,0 . −−−→ En+1 E20,n → E30,n → · · · → En+1

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F Homological Algebra

(F.23) Spectral sequences associated to exact couples. Let A be an abelian category. In the constructions below we will form certain limits and colimits in A. We always assume that these exist. As we usually work with complete and cocomplete abelian categories, for instance if A is a Grothendieck abelian category (Remark F.58), this will be harmless for us. Definition F.106. An exact couple in A is a diagram in A of the form /A

α

A_ f

E



g

such that Ker(α) = Im(f ), Ker(f ) = Im(g) and Ker(g) = Im(α). A morphism of exact ˜ E, ˜ α ˜ , f˜, g˜) is a pair of morphisms tA : A → A˜ and tE : E → couples s : (A, E, α, f, g) → (A, ˜ ˜ E such that α ˜ ◦ tA = tA ◦ α, f ◦ tE = tA ◦ f , and g˜ ◦ tA = tE ◦ g. Composition of morphisms of exact couples is defined in the obvious way and we obtain the category of exact couples in A. To every exact couple we can attach its derived exact couple as follows. Remark F.107. Let (A, E, α, f, g) be an exact couple in A. Set d := g ◦ f : E → E. As f ◦ g = 0, one has d ◦ d = 0. Set E ′ := Ker(d)/ Im(d),

and

A′ := Im(α) = Ker(g).

Let α′ : A′ → A′ be the map induced by α. As f is zero on Im(d) and sends Ker(d) to Ker(g), it induces a map f ′ : E ′ → A′ . Finally, let g ′ : A′ → E ′ be the map induced by “g ◦ α−1 ” using that g(Ker(α)) = Im(d). Then it is easily checked that (A′ , E ′ , α′ , f ′ , g ′ ) is again an exact couple which is called the derived exact couple of (A, E, α, f, g). We now explain how to attach a spectral sequence to a bigraded exact couple. Remark F.108. Let (A, E, α, f, g) = (A1 , E1 , α1 , f1 , g1 ) be an exact couple and define inductively for r > 1 an exact couple (Ar , Er , αr , fr , gr ) as the derived exact couple of (Ar−1 , Er−1 , αr−1 , fr−1 , gr−1 ). Then we have Ar = Im(αr−1 ), where αs := α ◦ · · · ◦ α (s times) with α0 := idA . Suppose also that (A, E, α, f, g) is equipped with a bigrading such that deg(α) = (−1, 1),

deg(f ) = (1, 0),

deg(g) = (0, 0).

Hence deg(d) = (1, 0), where d := g◦f . All the derived exact couples (Ar , Er , αr , fr , gr ) are endowed with an induced bigrading with deg(αr ) = (−1, 1), deg(fr ) = (1, 0), deg(gr ) = (r − 1, 1 − r), and deg(dr ) = (r, 1 − r), where dr := gr ◦ fr . Hence we obtain a spectral sequence (Er , dr )r≥1 with (F.23.1)

Br = g(Ker(αr−1 )),

Zr = f −1 (Im(αr−1 )),

dr = “g ◦ α−(r−1) ◦ f ”.

The construction of a spectral sequence from an exact couple is functorial. Let us consider the convergence of a spectral sequence attached to an exact couple. We give two examples (dual to each other) of possible criteria.

769 Proposition F.109. Let (A, E, α, f, g) be an exact couple equipped with a bigrading as above and let (Er , dr )r≥1 be the attached spectral sequence. (1) Assume that for all n ∈ Z the map α : As,n−s → As−1,n−s+1 (a) is zero for s ≫ 0 (dependent on n) and (b) is an isomorphism for s ≪ 0 (dependent on n). Then the spectral sequence attached to the exact couple converges to α

α

α

H n := colim(· · · −→ As,n−s −→ As−1,n−s+1 −→ · · · ), s

i.e., if one identifies As,n−s = As−1,n−s+1 = · · · for large s via the isomorphism α, then H n is this object. Its filtration is given by F p H n := Im(Ap,n−p → H n ). (2) Assume that for all n ∈ Z the map α : As,n−s → As−1,n−s+1 (a) is an isomorphism for s ≫ 0 (dependent on n) and (b) is zero for s ≪ 0 (dependent on n). Then the spectral sequence attached to the exact couple converges to α

α

α

H n := lim(· · · −→ As,n−s −→ As−1,n−s+1 −→ · · · ). s

Its filtration is given by F p H n := Ker(H n → Ap,n−p ). Proof. We only show (1), as the proof of (2) is dual. Using (F.23.1), Condition (a) implies that for all p, q ∈ Z there exists a ρ such that (F.23.2)

Zrpq = Ker(f : E pq → Ap+1,q )

for all r ≥ ρ.

pq In particular Zρpq = Zρ+1 = · · · . Similarly, Condition (b) implies that for all p, q ∈ Z there exists ρ such that

(F.23.3)

pq g(Ker(αρ−1 )) ∩ E pq = Bρpq = Bρ+1 = ··· .

We fix ρ = ρp,q such that (F.23.2) and (F.23.3) both hold. Then with ρ = ρp,n−p we have p,n−p p,n−p p,n−p E∞ = Z∞ /B∞ = Ker(f )/g(Ker(αρ−1 ))
p,n−p = Im(g)/g(Ker(αρ−1 ))
p,n−p = Ap,n−p / Ker(αρ−1 ) + Ker(g)
p,n−p = Ap,n−p / Ker(αρ−1 ) + Im(α) .


p,n−p

By enlarging ρ we may assume that α : Ap−t,n−p+t → Ap−t−1,n−p+t+1 is an isomorphism
p,n−p for t ≥ ρ using Condition (b). Then Ap,n−p / Ker(αρ−1 ) is the image of Ap,n−p in n H and hence p,n−p ∼ E∞ = F p H n /F p+1 H n .

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F Homological Algebra

(F.24) Spectral sequences associated to filtered complexes. We continue to denote by A a complete and cocomplete abelian category. In addition we assume (for simplicity) that filtered colimits are exact in A. All these hypotheses are satisfied is A is a Grothendieck abelian category. Definition F.110. A filtered complex X in A is a complex in A endowed with a descending filtration in the category of complexes of A. If X is a filtered complex, each filtration step is a subcomplex F p X of X. The quotient gr X := F p X/F p+1 X is a complex of A. p

Remark F.111. Let X be a filtered complex. We attach an exact couple to X as follows. The exact sequence of complexes 0 −→ F p+1 X −→ F p X −→ grp X −→ 0 yields a long exact cohomology sequence · · · −→ H p+q (F p+1 X) −→ H p+q (F p X) −→ H p+q (grp X) −→ H p+q+1 (F p+1 X) −→ · · · . Combining the morphisms of this sequence gives us a bigraded exact couple /A

α

A_ f

E



g

where E pq := H p+q (grp X)

Apq := H p+q (F p X).

Then α has degree (−1, 1), f has degree (1, 0), and g has degree (0, 0). Hence the construction of Remark F.108 yields a spectral sequence (Er , dr )r≥1 with E1pq = H p+q (grp X). The convergence assumptions for Proposition F.109 (1) are in this case satisfied if for all n∈Z (a) there exists s1 ∈ Z such that H n (F s X) = 0 for all s ≥ s1 and (b) there exists s0 ∈ Z such that H n (F s X) → H n (F s−1 X) is an isomorphism for all s ≤ s0 . This is for instance the case (even uniformly for all n) if there exist s0 ≤ s1 such that F s0 X = X and F s1 X = 0. Then the spectral sequence converges to [ n s n colim H (F X) = H ( F s X) opp s∈Z

s

because filtered colimits are assumed to be exact. Its filtration is given by [
F p H n = Im H n (F p X) → H n ( F s X) . s

771 (F.25) Spectral sequences associated to double complexes. We continue to denote by A a complete and cocomplete abelian category in which filtered colimits are exact, e.g., a Grothendieck abelian category. Let X be a double complex with components in A which we visualize as in (F.17.1). Let Tot(X) be the associated total complex. There are two natural filtrations on Tot(X) given by M M p k p (F.25.1) X i,j , X i,j I F (Tot(X) ) := II F (Tot(X)) := i+j=k j≥p

i+j=k i≥p

By (F.24) we obtain two spectral sequences (I Erp,q (X), I dr )r≥1 and (II Erp,q (X), II dr )r≥1 attached to the double complex. These spectral sequences define for all n ∈ Z two filtrations on H n (Tot(X)) which are denoted by I F and II F . These spectral sequences are functorial in X. Viewing X as a complex in horizontal direction of vertical complexes we can take its cohomology HIp (X) := H p (· · · → X i,• → X i+1,• → . . . ), q which is a (vertical) complex. Similarly we define the cohomology HII (X) of X considered as complex in vertical direction of horizontal complexes. Then one checks that

(F.25.2)

p,q I E1 (X) p,q I d1 p,q I E2 (X)

p,q II E1 (X) p,q II d1 p,q II E2 (X)

= H q (X p,• ), = H q (dp,• 1 ), q = H p (HII (X)),

= H q (X •,p ), = (−1)q H q (d•,p 2 ), = H p (HIq (X)).

Then (F.24) shows: Lemma F.112. Suppose that X is a double complex such that for all n ∈ Z there are only finitely many nonzero X p,q with p + q = n. Then both spectral sequences associated to X are bounded, converge to H ∗ (Tot(X)), and the two filtrations of H n (Tot(X)) given by the spectral sequences are finite. As a trivial application (which one could also see directly) we will see that forming the total complex “preserves quasi-isomorphisms in each direction”: Lemma F.113. Let A be an abelian category. Let X and Y be double complexes such that for all k ∈ Z there are only finitely many nonzero X i,j and Y i,j with i + j = k. Let f : X → Y be a morphism of double complexes which is a quasi-isomorphism of complexes of horizontal complexes, i.e. f

(. . . → X •,j → X •,j+1 → . . .) −→ (. . . → Y •,j → Y •,j+1 → . . .) is a quasi-isomorphism of complexes of complexes. Then Tot(f ) : Tot(X) → Tot(Y ) is a quasi-isomorphism. Proof. By (F.25.2), f induces an isomorphism I E1p,q (X) → I E1p,q (Y ). Hence we conclude by Remark F.103. In the abelian category of modules over a ring there is also the following variant, see [Sta] 09IZ.

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F Homological Algebra

Lemma F.114. Let R be a (not necessarily commutative) ring, let M be a complex of left R-modules and let · · · −→ P2 −→ P1 −→ P0 −→ M −→ 0, be an exact sequence of complexes of left R-modules such that for all i ∈ Z the sequence of left R-modules · · · −→ Ker(diP2 ) −→ Ker(diP1 ) −→ Ker(diP0 ) −→ Ker(diM ) −→ 0 j is exact. Set P i,j := P−i to obtain a double complex P . Then the morphism Tot(P ) → M induced by P0 → M is a quasi-isomorphism.

Triangulated categories Our final goal in this chapter is to introduce derived categories of an abelian category and derived functors. Derived categories will be additive categories endowed with the structure of a triangulated category and derived functors will be triangulated functors between triangulated categories. In this part we these notions. Our main references are [KaSh] O O Ch. 10, [Lip2] O X Ch. 1, and [Nee4] . (F.26) Definition of triangulated categories. Definition F.115. (1) An additive category with translation is a pair (C, T ) consisting of an additive category C and an automorphism T : C → C (which is called the translation functor). (2) A functor of additive categories with translation F : (C, T ) → (C ′ , T ′ ) is an additive ∼ functor F : C → C ′ together with an isomorphism F ◦ T → T ′ ◦ F of functors. As an automorphism a translation functor commutes with finite products, hence T is automatically additive. Although the isomorphism F ◦ T ∼ = T ′ ◦ F is not in general the identity, we will usually neglect it in the discussion, implicitly asserting that it can be chosen suitably. For i ∈ Z the functor T i is defined. We usually write X[i] := T i (X), and similarly for morphisms. Our principal examples of an additive category with translations will be the category C(A) of complexes with components in an additive category A (Definition F.68), its homotopy category K(A) (Definition F.76), and for an abelian category A the derived category D(A) (Definition F.148 below). Definition F.116. (1) Let (C, T ) be an additive category with translation. A triangle in C is a sequence of morphisms X −→ Y −→ Z −→ X[1]. This is sometimes written as

773 Z_

[1]

~

X

/Y

(hence the name triangle). We also often write /Y

X

/Z

/

+1

or even just X → Y → Z. (2) A morphism of triangles is a commutative diagram X α

 X′

/Y

/Z

/ X[1]

 / Y′

 / Z′

 / X[1].

α[1]

We obtain the category of triangles. Definition F.117. A triangulated category is an additive category (C, T ) with translation together with a family of triangles (called distinguished triangles or exact triangles) such that the following conditions are satisfied: (TR0) A triangle which is isomorphic to a distinguished triangle is itself distinguished. idX

For all X ∈ C, the triangle X

/X

/0

/ X[1] is distinguished. f

(TR1) For every morphism f : X → Y in C there exists a distinguished triangle X −→ Y −→ Z −→ X[1]. f

(TR2) (Rotation of triangles) A triangle X if and only if the triangle Y

g

/Z

h

/Y / X[1]

/Z

g

−f [1]

h

/ X[1] is distinguished

/ Y [1] is distinguished.

′

′

′

f g f h / /Y /Z / Y ′ g / Z ′ h / X ′ [1] be (TR3) Let X X[1] and X ′ distinguished triangles, and let α : X → X ′ and β : Y → Y ′ be morphisms with f ′ ◦ α = β ◦ f . Then there exists a morphism γ : Z → Z ′ such that (α, β, γ) is a morphism of distinguished triangles.

(TR4) (Octahedral axiom) Given distinguished triangles X Y X

f

/Y

h

/ Z′

/ X[1]

g

/Z

k

/ X′

/ Y [1]

g◦f

/Z

ℓ

/ Y′

/ X[1],

there exists a distinguished triangle Z′

u

/ Y′

v

/ X′

w

/ Z ′ [1]

774

F Homological Algebra such that the following diagram commutes: /Y

f

X

/ Z′

ℓ

 / Y′

k

 / X′

v

 / X′

g

id

 X

h

g◦f

 /Z

g

 /Z

u

 / Y′

f

u

h

ℓ

 Z′

id

 / X[1]

v

id

 Y

/ X[1]

f [1]

 / Y [1]

h[1]

id w

 / Z ′ [1].

See for instance [KaSh] O 10.1 for a visualization of the octahedron underlying the octahedral axiom. Remark F.118. Using (TR1) – (TR3) one can show ([Hub] X Appendix B) that (TR4) is also equivalent to the following variant of (TR3). (TR4’) Every commutative solid diagram with distinguished triangles as rows X

/Y

f

g

u

X

f′

 / Y′

/Z

h

/ X[1]

h′

/ X[1]

v

g′

 / Z′

can be completed to a commutative diagram with dotted arrow such that the triangle Y

( −g u )

/ Z ⊕Y′

(v,g ′ )

/ Z′

f [1]◦h′

/ Y [1]

is distinguished. Lemma F.119. (Five lemma, [KaSh] O Prop. 10.1.15) Let C be a triangulated category, and let /Y /Z / X[1] X α

 X′

β

 / Y′

γ

 / Z′

α[1]

 / X[1].

be a morphism of distinguished triangles. If two of the morphisms α, β, and γ are isomorphisms, then so is the third. Remark F.120. Let C be a triangulated category. f g h / /Y /Z X[1] is a distinguished triangle, then g ◦ f = 0, as (1) If X follows from axioms (TR0) and (TR3). Then using the rotation axiom (TR2) we also see that h ◦ g and f [1] ◦ h are zero.

775 (2) By the five lemma, for every morphism f : X → Y in C the distinguished triangle f X −→ Y −→ Z −→ X[1] in (TR1) is unique up to isomorphism (but not up to unique isomorphism). (3) In fact, the morphism γ in (TR3) is not required to be unique. However, we have: In the situation of condition (TR3) in the above definition, assume that HomC (Y, X ′ ) = 0 and that HomC (X[1], Y ′ ) = 0. Then γ is unique ([KaSh] O , Prop. 10.1.17). We remark that there are several problems built into the theory of triangulated categories. A triangulated category consists of an additive category together with the datum of an additional structure (a shift functor and the datum of distinguished triangles). Hence to be triangulated is not a property of an additive category. Moreover, by (2) and (3) of the previous remark it is possible to complete a morphism to a distinguished triangle but this usually cannot be done functorially. It is possible to remove these problems by passing to ∞-categories in the sense of Lurie [Lu-HTT] O X . In that theory one can define what it means that an ∞-category is stable and one can view stable ∞-categories as “upgrades” of triangulated categories that result in a more satisfying theory. Unfortunately, this is beyond the scope of this book. Proposition F.121. ([KaSh] O 10.1.19) Let I be a set and let C be a triangulated category which admits direct sums indexed by I. Then direct sums indexed by I commute with the translation functor, and a direct sum of distinguished triangles indexed by I is distinguished. In particular for all X, Y ∈ C, the obvious triangle (F.26.1)

X

/ X ⊕Y

/Y

0

/ X[1]

is a distinguished triangle. Remark F.122. Let C be a triangulated category. 0 (1) Any distinguished triangle of the form X −→ Z −→ Y −→ X[1] is isomorphic to the split triangle (F.26.1): Indeed by a rotated version of (TR3) there exist a morphism of triangles extending idX and idY which is necessarily an isomorphism because of the five lemma Lemma F.119). (2) Let X → Z be a morphism in C. Then (1) shows that X −→ Z −→ 0 −→ X[1] is a distinguished triangle if and only if X → Z is an isomorphism. (3) If u : Z → Y is an epimorphism in C, then u has a right inverse. u v Indeed, complete u to distinguished triangle Z −→ Y −→ X −→ Z[1]. As v ◦ u = 0 (Remark F.120 (1)) and u is an epimorphism, we find v = 0. Now a rotated version of (1) shows the claim. (4) Dually, any monomorphism has a left inverse. (F.27) Triangulated Structures on categories of complexes. Before continuing with the general theory of triangulated categories we explain the most important source of triangulated categories in this book: categories of complexes of an additive (often abelian) category. Throughout this section, let A denote an additive category. We give the homotopy category K(A) the structure of a triangulated category in the following way: The translation functor is given by translation of complexes, X 7→ X[1] (Definition F.68 (3)).

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F Homological Algebra

Given a morphism f : X → Y of complexes, there is an obvious “inclusion” i : Y → Cf and an obvious “projection” p : Cf → X[1]. In this way, f gives rise to a triangle (F.27.1)

X

/Y

f

−p

/ Cf

i

/ X[1].

Note the sign! By definition, an exact (or distinguished) triangle in K(A) is one which is isomorphic to a triangle of this form. One shows that with these definitions the axioms of a triangulated category are satisfied ([Lip2] O X 1.4). The following construction gives the same set of distinguished triangles. Remark F.123. Let A be an additive category. Let 0

/X

f

/Y

g

/Z

/0

be a termwise split sequence (Definition F.69) of complexes of A. Choose morphisms si : Z i → Y i and ri : Y i → X i such that g i ◦ si = id, ri ◦ f i = id and ri ◦ si = 0 for all i ∈ Z and define a triangle f g δ X −→ Y −→ Z −→ X[1], where δ i := ri+1 ◦ diY ◦ si . Its isomorphism class as a triangle in K(A) does not depend on the choice of the si and the ri because every other choice of splittings defines a morphism homotopic to δ. Then the set of triangles in K(A) that are isomorphic to a triangle of this form is the set of distinguished triangles in K(A) defined above. Using that K(A) is triangulated with the original definition of distinguished triangles, we see that indeed every triangle is isomorphic to one of the form Y → Cf → X → Y [1] for a morphism f : X → Y of complexes; one checks that the morphisms are those attached to the termwise split short exact sequence 0 → Y → Cf → X[1] → 0 (for suitable choices of the splittings). To go in the other direction, the key point is to show that in K(A) every morphism is isomorphic to one coming from a termwise split injection of complexes. Cf. [Sta] 014L. The choice of sign in the definition of exact triangles (i.e., defining the final map as −p in (F.27.1)) implies that the boundary maps in the long exact cohomology sequence for the exact triangle attached to a degree-wise split short exact sequence 0 → X • → Y • → Z • → 0 of complexes coincide with those obtained via the snake lemma. (F.28) Triangulated Functors. Definition F.124. (1) A triangulated functor or exact functor F : C → C ′ between triangulated categories C, C ′ is a functor of additive categories with translation which maps distinguished triangles to distinguished triangles. (2) A triangulated functor which is an equivalence of categories is called an equivalence of triangulated categories. (3) A morphism α : F → F ′ of triangulated functors C → C ′ is a morphism of functors such that the following diagram commutes: F ◦T

α◦T

∼ =

∼ =

 T′ ◦ F

/ F′ ◦ T

T

′

 ◦α / ′ T ◦ F ′.

777 Here T and T ′ denote the translation functors of C and C ′ , resp., and the vertical isomorphisms are the ones given by the structure of a functor of categories with translation. A functor between triangulated categories compatible with translation and mapping distinguished triangles to distinguished triangles is automatically additive (use that F preserves the distinguished triangles 0 → 0 → 0 → 0[1] and (F.26.1)) and hence a triangulated functor. Clearly the composition of triangulated functors is again triangulated. If F is an equivalence of triangulated categories in the sense of the above definition, then every quasi-inverse of F is also a triangulated functor. More generally: Proposition F.125. ([KaSh] O Cor. 10.1.16) Let F : C ′ → C be a fully faithful triangulated functor of triangulated categories. Then a triangle of C ′ is distinguished if and only if the image triangle of C is distinguished. Remark F.126. Let A be an additive category and let C ′ be a triangulated category. By Remark F.123 a functor F : K(A) → C ′ of additive categories with translation is triangulated if and only if it satisfies the following equivalent conditions. (i) The functor F preserves mapping cones, i.e., for every morphism f : X → Y of complexes one has F (Cf ) = CF (f ) . (ii) The functor F sends the distinguished triangle given by a termwise split sequence of complexes to a distinguished triangle in C ′ . Example F.127. Let A be an additive category. Fix Z in K(A). The additive functor of complexes X 7→ HomA (Z, X) (the Hom complex defined in Section (F.18)) preserves homotopy and hence induces an additive functor F : K(A) → K(AbGrp),

X 7→ HomA (Z, X)

Then F commutes with the translation functors and preserves termwise split sequences of complexes. Hence it is a triangulated functor. Definition F.128. Let F : C → C ′ and G : C ′ → C be triangulated functors of triangulated categories. Suppose that F is left adjoint to G and let η : idC → G ◦ F and ϵ : F ◦ G → idC ′ be the adjunction morphisms. Then we say that (F, G) is a triangulated adjoint pair if η (or, equivalently by [Lip2] O X (3.3.1), ϵ) is a morphism of triangulated functors. Lemma F.129. ([Nee2] O 3.9) Let C and D be triangulated categories and let F : C → D be a triangulated functor. Suppose that F has a right (resp. left) adjoint G : D → C. Then G has the structure of a triangulated functor such that (F, G) (resp. (G, F )) is a triangulated adjoint pair. Example F.130. Let F : A → B and G : B → A be additive functors of additive categories. Suppose that F is left adjoint to G. Then for all objects A in A and B in B the adjunction isomorphism (F.5.1) is the composition HomA (A, G(B)) −→ HomB (F (A), F (G(B))) −→ HomB (F (A), B), where the first map is induced by F and the second from the adjunction morphism F ◦ G → idB . In particular it is an isomorphism of abelian groups. The functors F and G induce triangulated functors F : K(A) → K(B) and G : K(B) → K(A) and these two functors form a triangulated adjoint pair (F, G).

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F Homological Algebra

(F.29) Triangulated Subcategories. Definition F.131. Let C be a triangulated category. A triangulated subcategory C ′ of a triangulated category C is a full additive subcategory endowed with the structure of a triangulated category such that the restriction of the translation functor T of C induces the translation functor on C ′ and such that the inclusion functor C ′ → C is triangulated. Let C ′ be a full additive subcategory of C that is stable under isomorphism and stable under the translation automorphism and its inverse. Then C ′ carries at most one structure of triangulated category for which the translation is the restriction of that on C and such that the inclusion C ′ → C is a triangulated functor (Proposition F.125). Such a structure of triangulated category exists if and only if for every distinguished triangle X → Y → Z → X[1] of C with X and Y in C ′ one has that Z lies in C ′ (use Remark F.120 (2)). Remark F.132. If C is the homotopy category K(A) for an additive category A, then a full additive isomorphism-stable subcategory C ′ of K(A) is a triangulated subcategory if and only if: (1) For every X in K(A) the complex X is in C ′ if and only if X[1] is in C ′ . (2) The mapping cone of any morphism X → Y of complexes X and Y in C ′ is homotopy equivalent (i.e., isomorphic in K(A)) to a complex in C ′ . The remark shows the full subcategories K b (A), K + (A), and K − (A) of (left/right) bounded complexes are examples of full triangulated subcategories of K(A). (F.30) The opposite triangulated category. Remark F.133. Let C be a triangulated category. We obtain the opposite triangulated category by endowing its opposite category C opp with the structure of a triangulated category as follows. Its translation automorphism is defined by T opp : X 7→ X[−1]. If u

v

w

X −→ Y −→ Z −→ X[1]

(F.30.1)

is a distinguished triangle in C, we obtain a triangle (F.30.2)

v opp

uopp

Z opp −→ Y opp −→ X opp

w[−1]opp

−→

T opp (Z opp )

in C opp and we define the distinguished triangles of C opp to be all triangles of this form. Note that the distinguished triangle (F.30.2) differs by a sign from the triangle we get by applying the canonical contravariant functor (−)opp to the distinguished triangle −w[−1] u v Z[−1] −→ X −→ Y −→ Z obtained from (F.30.1). If D is a triangulated category, then a triangulated contravariant functor C → D is by definition a triangulated (covariant) functor C opp → D. Example F.134. Let A be an additive category and fix Z in K(A). Consider the additive functor HomA (−, Z) : K(A)opp −→ K(AbGrp), Y 7→ HomA (Y, Z), where the left hand side denotes the Hom complex defined in (F.18). To make HomA (−, Z) into a triangulated functor, we define a functorial isomorphism

779 ∼

θY : HomA (Y [−1], Z) −→ HomA (Y, Z)[1] by multiplication by (−1)i−1 in degree i. Then (HomA (−, Z), θ) is a functor of additive categories with translations. As (HomA (−, Z) preserves termwise split sequences it is a triangulated functor. Remark F.135. Let A be an additive category. Then the opposite category Aopp is again additive. We will identify K(Aopp ) and K(A)opp as triangulated categories as follows. We define an isomorphism of categories ∼

Λ : C(A)op → C(Aop ) by setting Λ(X)i := X −i ,

. diΛ(X) := (−1)i d−i−1 X

If a morphism f in C(A) is homotopic to zero via a homotopy (hi )i∈Z , then Λ(f opp ) is homotopic to zero in C(Aop ) via the homotopy ((−1)i hi,opp )i . Hence Λ induces an isomorphism of categories (F.30.3)

∼

Λ : K(A)op → K(Aop ).

We have an isomorphism θ : Λ ◦ [1] ∼ = [−1] ◦ Λ of functors given by multiplication by (−1)i−1 in degree i ∈ Z, i.e., this isomorphism is given by (−1)i−1

i θX : Λ(X[1])i = X 1−i −→ X 1−i = (Λ(X)[−1])i . ∼

Then (Λ, θ) is an isomorphism of triangulated categories K(A)op → K(Aop ), where we endow K(A)op with the structure of a triangulated category as in Remark F.133. If F is a contravariant additive functor of additive categories from A to A′ , we consider F as usual as a (covariant) functor F : Aop → A′ . By identifying K(Aop ) with K(A)op as above, we obtain a functor K(F ) : C(A)op → C(A′ ), i.e. K(F )(X)i = F (X −i ),

diK(F )(X) = (−1)i F (d−i−1 ). X

(F.31) Cohomological Functors. Definition F.136. Let C be a triangulated category and A an abelian category. An additive functor C → A is called cohomological, if for every distinguished triangle X → Y → Z → X[1] in C the sequence F (X) → F (Y ) → F (Z) in A is exact. The rotation axiom (TR2) shows that given a cohomological functor F : C → A, every distinguished triangle X → Y → Z → X[1] gives rise to a long exact sequence · · · → F (Z[−1]) → F (X) → F (Y ) → F (Z) → F (X[1]) → · · · . Example F.137. With notation as in the definition, for every W ∈ C, the functors HomC (W, −) : C → (AbGrp), are cohomological ([KaSh] O 10.1.13).

HomC (−, W ) : C opp → (AbGrp)

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F Homological Algebra

Example F.138. Let A be an abelian category. Then the functor H 0 : K(A) → A,

X 7→ H 0 (X) = Ker(d0X )/ Im(d−1 X )

is a cohomological functor. Indeed, by Remark F.123 we only have to show that if 0 → X → Y → Z → 0 is a termwise split sequence of complexes, then H 0 (X) → H 0 (Y ) → H 0 (Z) is exact and this follows from Proposition F.71. Hence using that H 0 (X[n]) = H n (X), an exact triangle 0 → X → Y → Z → X[1] gives rise to a long exact cohomology sequence · · · H i (X) → H i (Y ) → H i (Z) → H i+1 (X) → · · · .

Sign conventions Unfortunately, in homological algebra, it is unavoidable to have signs come in in certain places (see the definitions of the Hom complex and the total complex of a double complex above, for instance), and some choices have to be made as to where to put which signs. There is (unfortunately, one could say) some freedom here, and almost every consistent choice of sign conventions occurs somewhere in the literature. We will survey our decision about sign conventions here, and give some references pointing to differing normalizations in other places. Below, unless otherwise stated, all complexes are complexes in an abelian category A. Sometimes we implicitly assume that A has infinite products (or that only finitely many terms in the products arising below are non-zero). (F.32) Cones and Exact Triangles. The shift X[1] of a complex X is defined by X[1]i = X i+1 , and the differential is changed by a sign: dnX[1] = −dnX . Similarly, we define the shift X[j] for any integer j. While this is the most common way to set up the shift, some authors do it differently: In [Wei2] O and [BouA10] O , our X[1] is called X[−1]. In [Law] X , the shift defined above is denoted ΣX, while the notation X[1] there is defined without changing the differential; while this convention is more in line with the slogan that a sign arises because certain operators (like d and Σ) anti-commute, it is so rare that we do not follow it.

Given a morphism f : X → Y of complexes, we defined the mapping cone C = Cf by setting C n = X n+1 ⊕ Y n , with differential given by the matrix   −dX 0 . f dY The sign conventions regarding the mapping cone are different in [BouA10] O , [Har1] O , [Wei2] O .

Given a morphism f : X → Y of complexes, there is an obvious “inclusion” i : Y → Cf and an obvious “projection” p : Cf → X[1]. We define a structure of triangulated category on the homotopy category K(A) by saying that distinguished triangles are those isomorphic to a triangle of the form

781 X

f

/Y

i

/ Cf

−p

/ X[1].

Note the sign: The final map is given by −p. O (see With this convention, we follow [Con] O , [SGA4] O , [Sta]. In [Har1] O , [Law] X , [Lip2] O X , [KaSh]

Prop. 12.3.6), [Wei2] O the sign convention differs from ours. Our choice has the advantage that the boundary morphisms in the long exact cohomology sequence agree with the morphisms “obtained from the snake lemma”, see Remark F.151 below for details.

(F.33) Double complexes and the total complex. A double complex is a complex of complexes, i.e., a collection X i,j of objects, i, j ∈ Z with row differentials d1 : X i,j → X i+1,j and column differentials d2 : X i,j → X i,j+1 such that all squares commute. The total complex Tot(X •,• ) of a double complex X •,• is given by M Tot(X •,• )n = X i,j , i+j=n

dn|X i,j = d1 + (−1)i d2 . Often, in the literature a double complex (or bicomplex) is defined by requiring that the squares anti-commute, rather than commute (e.g., in [Con] O , [Wei2] O , [BouA10] O ). In that case, the definition of the differential of the total complex does not involve a sign.

(F.34) Homomorphisms and tensor products. Let X and Y be complexes. We have the complex of morphisms Hom(X, Y ), Y Hom(X, Y )n = HomA (X i , Y i+n ), i

dn (f ) = dX ◦ f − (−1)n f ◦ dX

for f ∈ Hom(X, Y )n .

Usually the differential on the Hom complex is defined in this way, for instance in [BouA10] O taking into account that complexes there are defined with differentials lowering the degree and one has to pass from Xn there to X −n here. But as always, there are exceptions: In [Har1] O and [Wei2] O , the sign differs by (−1)n+1 .

Assume that A has a tensor product. As defined above, the tensor product X ⊗ Y of complexes X and Y is the total complex of the double complex X i ⊗ Y j with differentials induced by those of X and Y . Explicitly, this means M (X ⊗ Y )n = Xi ⊗ Y j, i+j=n

dn|X i ⊗Y j (x

⊗ y) = dX (x) ⊗ y + (−1)i x ⊗ dY (y).

While this is definitively the most common way of defining the tensor product of complexes, in [Har1] O it is done differently. Our convention has the advantages that it fits well with the definition of the total complex of a double complex, and, more importantly, with adjunction between Hom and ⊗ (see Proposition F.93).

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F Homological Algebra

Derived categories The derived category of an abelian category A will be defined as the homotopy category K(A) “localized by the set of quasi-isomorphisms”, i.e., where we invert all quasiisomorphisms. This process of localizing a category will be explained first. (F.35) Localization of Categories. We follow [KaSh] O Ch. 7. The localization of a category C with respect to a set S of morphisms in C is the “minimal” category together with a functor from C such that all elements of S are mapped to isomorphisms. More precisely, we define Definition F.139. Let C be a category, and let S be a set of morphisms in C. A category CS together with a functor Q : C → CS is called a localization of C with respect to S, if for all s ∈ S, the morphism Q(s) is an isomorphism in CS , and if the following universal property holds: For any category D and any functor F : C → D such that for all s ∈ S, F (s) is an isomorphism, there exists a functor G : CS → D and an isomorphism F ∼ = G◦Q of functors. Moreover, G is unique in the following sense: Given any two functors G, G′ : CS → D, composition with Q induces a bijection Hom(G, G′ ) ∼ = Hom(G ◦ Q, G′ ◦ Q) between the respective sets of morphisms of functors. In particular, G as above is uniquely determined up to unique isomorphism, once the isomorphism F ∼ = G ◦ Q is fixed. From the definition, we see immediately that the localization, if it exists, is unique up to equivalence of categories. If one ignores set-theoretic issues, which for instance can be safely done if C is a small category, then the localization CS always exists ([GaZi] O I.1). But the description of morphisms in CS is somewhat awkward making it difficult to work with CS . The localization category becomes much more concrete if S is a (right or left) multiplicative system in the following sense. Definition F.140. Let C be a category, and let S be a set of morphisms in C. We call S a right multiplicative system, if (MS1) every isomorphism in C belongs to S, and for any two morphisms s : X → Y and t : Y → Z in S, the composite t ◦ s belongs to S, (MS2) for every s : X → X ′ in S, and every morphism f : X → Y , there exist morphisms t : Y → Y ′ in S and g : X ′ → Y ′ such that g ◦ s = t ◦ f , and (MS3) Let f, g ∈ HomC (X, Y ) such that there exists s : W → X in S with f ◦ s = g ◦ s. Then there exists a morphism t : Y → Z in S such that t ◦ f = t ◦ g. Analogously, we have the notion of left multiplicative system, where in (MS2) and (MS3) one reverses the arrows. A set of morphisms in C is called multiplicative system if it is a right multiplicative and a left multiplicative system. Here we follow [KaSh] O . Some authors (e.g., [Sta]) call “right” what we call “left”. For the application to the construction of the derived category this difference does not matter because there we will localize by a system that is (right and left) multiplicative. If S is a right (or left) multiplicative system, there is the following explicit construction of a localization of C by S.

783 Remark: Construction of a localization F.141. ([KaSh] O , §7.1) Let S be a right multiplicative system in a category C. Define a category CSr as follows. (a) The objects of CSr are the objects of C. (b) For X, Y ∈ Ob(CSr ) = Ob(C) we set HomCSr (X, Y ) :=

colim

(Y →Y ′ )∈S

HomC (X, Y ′ ),

where the colimit is taken over the category S Y whose objects are morphisms Y → Y ′ in S and where a morphism (s : Y → Y ′ ) −→ (s′ : Y → Y ′′ ) is defined to be a morphism h : Y ′ → Y ′′ in C (not assumed to be in S!) such that h ◦ s = s′ . The axioms (MS1) – (MS3) show that S Y is a filtered category. Hence a morphism X → Y can be also described as an equivalence class s−1 f of a pair (f, s) of morphisms f

s

X −→ Y ′ ←− Y f1

f2

s

s

1 2 in C with s ∈ S. Here we call two such pairs X −→ Y1 ←− Y and X −→ Y2 ←− Y f s ′ equivalent if there exists a third such pair X −→ Y ←− Y and a commutative diagram in C, 8 Y1 `

f1

s1

 / Y′ o O

f

X

f2

&

Y2

s

Y. s2

~

f

−1

s

(c) The composition of two morphisms s f and t−1 g represented by X −→ Y ′ ←− Y g t and Y −→ Z ′ ←− Z is defined by the diagram below with u ∈ S (which exists by (MS2)): X

f

/ Y′ o

s

h

W

/ Z′ o

g

Y ~

t

Z.

u

This makes CSr into a category. Denote by Q : C → CSr the functor that is the identity on objects and sends a morphism f : X → Y in C to id−1 Y f. Theorem F.142. (Existence of localization, [KaSh] O Thm. 7.1.16) Let C be a category, and let S be a right multiplicative system of morphisms in C. Then (CSr , Q) is a localization of C by S. From now on we write CS instead of CSr and if we speak of the localization of C by S we mean the category CSr as above. If T is a left multiplicative system in C we can define analogously a category CTl . For g t instance morphisms in CTl are equivalence classes gt−1 of diagrams X ←− X ′ −→ Y with t ∈ T , i.e., HomCTl (X, Y ) = colim HomC (X ′ , Y ). ′ (X →X)∈T

If S is a (left and right) multiplicative system, then CSr and CSl are equivalent categories and we can write every morphism in CS either as s−1 f or as gt−1 with s, t ∈ S.

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F Homological Algebra

Proposition F.143. ([KaSh] O , 7.1.20, 7.1.22) Let C be a category, let S be a multiplicative system in C, and let Q : C → CS be the localization functor. (1) The following assertions are equivalent. f g h (i) For all morphisms X −→ Y −→ Z −→ W in C with h ◦ g, g ◦ f ∈ S one has 3 g ∈ S. (ii) A morphism f of C lies in S if and only if Q(f ) is an isomorphism in CS . If S satisfies these conditions, we call S saturated. (2) If C is finitely complete (resp. finitely cocomplete), then CS is finitely complete (resp. finitely cocomplete) and Q is left exact (resp. right exact). (3) If C is additive, then CS is additive and Q is an additive functor. Definition F.144. (Localization of functors, [KaSh] O 7.3)) Let C be a category and let S be a set of morphisms in C such that the localization Q : C → CS exists. Let F : C → E be a functor to some category E. (1) The functor F is called right localizable, if there exists a functor RS F : CS → E together with a morphism F → RS F ◦ Q such that for every functor G : CS → E, the composition Hom(RS F, G) → Hom(RS F ◦ Q, G ◦ Q) → Hom(F, G ◦ Q) is bijective. We call the functor RS F the right localization of F . (2) The functor F is called universally right localizable, if for every functor H : E → E ′ , the functor H ◦ F is right localizable, and RS (H ◦ F ) ∼ = H ◦ RS F . Analogous notions and results are available for left localization. (F.36) Localization of triangulated categories. Throughout this section, let C be a triangulated category. We want to localize C by a multiplicative systems in C such that the localization can be endowed with the structure of a triangulated category. We follow [KaSh] O 10.2. A source for such multiplicative systems are null systems in the following sense. Definition F.145. A null system in C is a strictly full (Definition F.1) triangulated subcategory N of C. Equivalently, N is a full subcategory of C such that (a) N is a strictly full subcategory, (b) the zero object 0 lies in N , (c) N is closed under the shift operator T and under its inverse, (d) whenever X → Y → Z → X[1] is a distinguished triangle with X, Z ∈ N , then Y ∈ N. Given a null system N in C, we write N Q := {f : X → Y a morphism in C ; there ex. a dist. triangle X → Y → Z → X[1] with Z ∈ N }. 3

We can formulate this condition for S to be saturated more symmetrically than in [KaSh] O 7.1.19 because we assumed that S is right and left multiplicative.

785 Then one can show that N Q is a left and right multiplicative system. Denote by Q : C → CN Q the localization functor from C to the localization with respect to this multiplicative system. The localization is additive, and the image of T provides it with the structure of a category with translation. We call a triangle in CN Q distinguished if it is isomorphic to the image under Q of a distinguished triangle in C. Note that in general the localization is a “big” category, i.e., a category in which the set of morphisms between two objects does not belong to our chosen universe. Theorem F.146. ([KaSh] O 10.2.3) (1) The construction above endows CN Q with the structure of a triangulated category and Q is a triangulated functor. (2) For X ∈ N one has Q(X) ∼ = 0, and the localization is universal with respect to this property in the following sense: If F : C → C ′ is a triangulated functor of triangulated categories such that F (X) ∼ = 0 for all X ∈ N , then F factors uniquely through Q. One often writes C/N instead of CN Q . By F.141 and using that N Q is (right and left) multiplicative system, there is the following description of Hom-sets in the localization: With notation as above, let X, Y ∈ C/N . Then HomC/N (X, Y ) ∼ = (F.36.1)

∼ = ∼ =

colim

HomC (X, Y ′ )

colim

HomC (X ′ , Y )

(Y →Y ′ )∈N Q (X ′ →X)∈N Q

colim

(X ′ →X)∈N Q,(Y →Y ′ )∈N Q

HomC (X ′ , Y ′ )

Lemma F.147. ([Rou] O Lemma 3.4) Let Q : C → C ′ be a triangulated functor of triangulated categories. Suppose that Q admits a fully faithful right adjoint functor. Then the full subcategory Ker(Q) of objects X in C such that Q(X) = 0 is a null system stable under direct summands in C and Q induces an equivalence of triangulated categories C/ Ker(Q) ∼ = C′.

(F.37) The derived category of an abelian category. In this section, and also in the remainder of this appendix, A will always denote an abelian category, unless something else is stated explicitly. Recall the category K(A) of complexes in A up to homotopy (Definition F.76). Denote by N the full subcategory of K(A) consisting of objects X such that H n (X) ∼ = 0 for all n (i.e., the unique map of complexes from X to the zero complex is a quasi-isomorphism). Then N is a null system in the triangulated category K(A) (Definition F.145). The attached multiplicative system is the set of quasi-isomorphisms. It is saturated (Proposition F.143). Definition F.148. The localization D(A) := K(A)/N is called the derived category of the abelian category A. The objects in the derived category D(A) are complexes of objects in A. Hence one can define properties of objects in D(A) by specifying properties of complexes. But any “good property” of objects in a category should be stable under passing to isomorphic objects and a “good property” for a complex as object in C(A) might not be a “good

786

F Homological Algebra

property” for a complex as object in D(A). For instance, the property of a complex X to be bounded below (i.e., X n = 0 for sufficiently small n) is not a “good property” of X considered as an object in D(A) since there are complexes that are isomorphic to to a bounded below complex in D(A) and that are themselves not bounded below4 . To stress this point, we will occasionally speak of a representative of an object X in D(A) by which we mean a complex, now considered as object in C(A), that is isomorphic to X in D(A). As noted in Section (F.1), the question of existence of the derived category entails some set-theoretic issues: In the framework of universes, the sets of morphisms in D(A) are not necessarily members of the universe to which A belongs. One sometimes expresses this by saying that D(A) is a big category. We will only consider (variants of) derived categories that are subcategories of D(A) for some Grothendieck abelian category A. In this case we will see (Corollary F.186 below) that the sets of morphisms between any two objects in D(A) still form a set in the same universe. This allows us to ignore these set-theoretic issues in this book. We could equivalently define D(A) as the localization of the category C(A) of complexes in A with respect to quasi-isomorphisms. However to obtain the structure of triangulated category on D(A), it is convenient to work with K(A). Moreover, the quasi-isomorphisms do not form a multiplicative system in C(A) which would make the description of morphisms in D(A) using only morphisms of complexes (not up to homotopy) very awkward. By definition there is a localization functor Q : K(A) → D(A), which maps quasi-isomorphisms to isomorphisms. The “universal property” of the triangulated localization yields: Remark F.149. Composition with the localization functor Q : K(A) → D(A) yields for any triangulated category E an isomorphism (not only an equivalence) of the category of triangulated functors D(A) → E to the full subcategory of the category of triangulated functors K(A) → E whose objects are triangulated functors F that transform quasiisomorphisms in K(A) to isomorphisms in E (equivalently F (X) = 0 for every complex X in K(A) with H i (X) = 0 for all i ∈ Z). Let us describe D(A) more explicitly. Remark F.150. As explained in F.141, objects in D(A) are complexes of A. A morphism X → Y in D(A) can be described as an equivalence class s−1 f of morphisms of complexes X

f

/ Y′ o

s

Y,

where s is a quasi-isomorphism. As the quasi-isomorphisms in K(A) form a saturated multiplicative system, a morphism f : X → Y in K(A) is a quasi-isomorphism if and only if Q(f ) is an isomorphism in D(A). Note that usually there exist isomorphisms in D(A) that are not in the image of Q and hence are not given by a quasi-isomorphism of complexes. 4

In this special example it is in fact possible to find a good replacement of the property to bounded below for objects in D(A), see Section (F.38).

787 The structure of a triangulated category on K(A) makes D(A) into a triangulated category in the following way (Theorem F.146): As for K(A) the translation functor is given by translation of complexes: X 7→ X[1], which obviously induces an automorphism of D(A). Furthermore, we define a triangle in D(A) to be distinguished (or exact) if it is isomorphic to the image of a distinguished triangle in K(A) under the localization functor. This makes D(A) into a triangulated category (Theorem F.146) and the localization functor K(A) → D(A) is a triangulated functor. The distinguished triangles in D(A) can be given as follows. By definition, distinguished triangles in D(A) are those triangles that are isomorphic to triangles of the form f

−p

i

X −→ Y −→ Cf −→ X[1]

(F.37.1)

where f : X → Y is a morphism of complexes, Cf is its mapping cone (Definition F.72), i : Y → Cf is the canonical inclusion and p : Cf → X[1] is the canonical projection. By Remark F.123 the distinguished triangles can equivalently be described as those that are isomorphic to a triangle given by a short exact sequence of complexes that is termwise split. In D(A) an arbitrary short exact sequence (F.37.2)

f

g

0 −→ X −→ Y −→ Z −→ 0

of complexes (not necessarily termwise split) also gives a distinguished triangle as follows. The maps χn : Cfn → Y n → Z n obtained by composing the projection and the map g give a surjective morphism χ : Cf → Z of complexes. We claim that this is a quasi-isomorphism. Indeed, the kernel of χ in C(A) is the mapping cone Cf0 of the isomorphism f0 : X → Im(f ) induced by f , hence χ H i (Cf0 ) = 0 for all i and the long exact cohomology sequence of 0 → Cf0 → Cf −→ Z → 0 shows that χ is a quasi-isomorphism. Hence it is an isomorphism in D(A) and we obtain in D(A) a triangle (F.37.3)

X

f

/Y

g

−1

/ Z −p◦χ / X[1]

which is called the triangle associated to the exact sequence (F.37.2). This triangle is isomorphic to the distinguished triangle (F.37.1). Hence it is distinguished. If the sequence (F.37.2) is termwise split, then χ is a homotopy equivalence (i.e., an isomorphism in K(A)) and the triangle (F.37.3) is the image of the distinguished triangle in K(A) defined in Remark F.123. Remark F.151. As the cohomology functors H i : K(A) → A send quasi-isomorphisms to isomorphisms, they factor through functors H i : D(A) → A. By Example F.138, H 0 (in fact any H i ) is a cohomological functor. Hence for any distinguished triangle X → Y → Z → X[1] in D(A) we have the long exact cohomology sequence (F.37.4)

· · · → H i (X) → H i (Y ) → H i (Z) → H i+1 (X) → · · · .

If that distinguished triangle is the one associated to an exact sequence of complexes, the long exact cohomology sequence (F.37.4) is the same as long exact cohomology sequence given by the Snake Lemma in Proposition F.71.

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F Homological Algebra

(1) As a morphism in K(A) is a quasi-isomorphism if and only if its image under Q is an isomorphism, for X ∈ K(A) we have Q(X) ∼ = 0 if and only if H i (X) ∼ = 0 for all i. (2) A morphism f in D(A) is an isomorphism if and only if H i (f ) is an isomorphism for all i ∈ Z. Indeed, represent by f by s−1 g, where g is a morphism of complexes and s is a quasi-isomorphism. If H i (f ) is an isomorphism, then H i (g) is an isomorphism because H i (s) is an isomorphism. Therefore if H i (f ) is an isomorphism, then g is a quasi-isomorphism and hence an isomorphism in D(A). This shows that f is an isomorphism. (3) Note that usually there are non-zero morphisms f in D(A) such that H i (f ) = 0 for all i ∈ Z. In fact, for objects A, B ∈ A, we can view the group HomD(A) (B, A[1]) = Ext1 (B, A) as a Yoneda Ext group (see (F.52) below), and every non-split extension gives rise to a non-trivial morphism f : B → A[1] with H i (f ) = 0 for all i (since for a given i, at most one of the two sides has non-trivial cohomology in degree i). Remark F.152. We can consider every object X of A as a complex X[0] with this object in degree 0, and 0 elsewhere. This defines a functor A → D(A) and it is not difficult to see that this functor is fully faithful (see also Proposition F.156 below for a more general statement). Its essential image consists of objects X in D(A) such that H i (X) = 0 for all i = ̸ 0 (see Proposition F.154 (2) below for a stronger statement). Hence we can (and usually will) consider A as a full subcategory of D(A). If we want to stress the difference between an object X in A and its image in D(A) we will write X[0] for this image. More generally, for i ∈ Z we will write X[−i] for the complex with X in degree i and 0 elsewhere. Example F.153. Let A be an abelian category such that all short exact sequences in A are split (see also Exercise F.29). Then for any complex X ∈ C(A) there exists a L ∼ functorial isomorphism i∈Z H i (X)[−i] → X in D(A) (see also Exercise F.31 for weaker hypotheses that ensure the existence of such an isomorphism which is then not necessarily functorial in X). L Indeed, note that i∈Z H i (X)[−i] denotes the complex 0

0

0

0

· · · −→ H i−1 (X) −→ H i (X) −→ H i+1 (X) −→ · · · , which exists even if A does not admit infinite coproducts. By assumption on A we i i map Ker(d (X). The ui find a section ui : H i (X) → Ker(diX ) of the L canonical L X) → H i i define a homomorphism of complexes u : i∈Z H (X)[−i] → i∈Z Ker(dX )[−i] and the L i inclusions Ker(diX ) → X i define a morphism of complexes v : i∈Z Ker(dX )[−i] → X. The composition M v ◦ u: H i (X)[−i] → X i∈Z

is a quasi-isomorphism and hence in D(A). If one chooses other splittings L an iisomorphismL i i u ˜i defining a morphism u ˜: H (X)[−i] → ˜i factors i∈Z i∈Z Ker(dX )[−i], then u − u i−1 i−1 i−1 through Im(dX ). Composing it with a section of X → Im(dX ), we obtain morphisms hi : H i (X) → X i−1 which defines a homotopy between v ◦ u and v ◦ u ˜. Hence the isomorphism v ◦ u in D(A) is independent of all choices and one checks that it is functorial in X.

789 (F.38) Bounded derived categories. Let K ⊆ K(A) be a triangulated subcategory. Localizing with respect to all quasiisomorphisms in K, we obtain a category DK together with a triangulated localization functor Q = QK : K → DK which is universal for functors from K which map every quasi-isomorphism to an isomorphism. If J ⊆ K is a second triangulated subcategory the universal property of the localization yields a unique triangulated functor DJ → DK

(F.38.1)

making the following diagram commutative J QI

 DJ

/K 

QK

/ DK .

Typical choices for K are the homotopy categories obtained from the categories C b (A), C + (A), C − (A) of (left/right) bounded complexes. We denote their respective homotopy categories by K b (A), K + (A), K − (A), etc. By localization we obtain the bounded derived b b categories DK (A) , etc. For these, we use the more common notation Db (A) := DK (A) , D+ (A), D− (A), etc. See Proposition F.154 below for a description in terms of the vanishing of cohomology objects and for the definition of further variants. Let I ⊆ Z be an interval. The category C I (A) of complexes concentrated in degrees ∈ I is an abelian subcategory of C(A) and we denote by K I (A) the full subcategory of K(A) of complexes concentrated in degrees ∈ I. Note that this is in general not a triangulated subcategory because the mapping cone of a complexes in K I (A) is not necessarily contained in K I (A). Hence it does not make sense to localize these categories as triangulated categories. Instead we denote by DI (A) the full additive subcategory of D(A) consisting of all X with H p (X) = 0 for all p ̸∈ I. We write D≤n (A) := D(−∞,n] (A), D≥n (A) := D[n,∞) (A). This notation is justified by the following result. Proposition F.154. ([KaSh] O Prop. 13.1.12) (1) The triangulated functors Db (A) → D(A), D+ (A) → D(A), D− (A) → D(A) (F.38.1) induce an equivalence of triangulated categories of Db (A) (D+ (A), D− (A), resp.) to the full triangulated subcategories of D(A) of objects X such that H i (X) = 0 for |i| ≫ 0 (i ≪ 0, i ≫ 0, resp.). (2) Let I ⊆ Z be an interval. Then the restriction K I (A) → DI (A) of the localization functor is essentially surjective. Hence we can consider the various categories D∗ (A) with ∗ ∈ {b, +, −, I} as full subcategories of D(A) with obvious inclusions, e.g., D≥a (A) ⊆ D+ (A) for every integer a ∈ Z. Remark F.155. If A is an R-linear category for some ring R (Definition F.28), then the additive category D? (A) is R-linear for ? ∈ {∅, +, −, b, I}. (F.39) Truncation in derived categories. As the truncation functors τ ≤n , τ ≥n preserve quasi-isomorphisms of complexes, they induce truncation functors on the derived categories

790

F Homological Algebra τ ≤n : D(A) → D≤n (A),

τ ≥n : D(A) → D≥n (A).

Proposition F.156. ([KaSh] O Prop. 13.1.12) Let a ∈ Z and X, Y ∈ D(A). Then HomD(A) (τ ≤a X, τ ≥a Y ) ∼ = HomA (H a (X), H a (Y )). In particular HomD(A) (τ ≤a X, τ ≥a+1 Y ) = 0. Note that the situation is not symmetric. In general HomD(A) (τ ≥a+1 X, τ ≤a Y ) ̸= 0. Proposition F.157. (cf. [KaSh] O Prop. 13.1.15) Let X be an object in D(A). There are distinguished triangles τ ≤n X → X → τ ≥n+1 X → τ ≤n−1 X → τ ≤n X → H n (X)[−n] → H n (X)[−n] → τ ≥n X → τ ≥n+1 X → in D(A). Furthermore, we have H n (X)[−n] ∼ = τ ≤n τ ≥n X ∼ = τ ≥n τ ≤n X. Proposition F.158. (cf. [KaSh] O Prop. 13.1.16) The functor τ ≤n : D(A) → D≤n (A) is right adjoint to the inclusion functor D≤n (A) → D(A). Dually, the functor τ ≥n : D(A) → D≥n (A) is left adjoint to the inclusion functor D≥n (A) → D(A). (F.40) Construction of complexes. Definition and Remark F.159. Let A be an abelian category. Let m ∈ Z. Let u : X → Y be a morphism in D(A), and complete to a distinguished triangle X → Y → C →. Then u is called an m-isomorphism if the following equivalent conditions are satisfied. (i) One has H p (C) = 0 for all p ≥ m. (ii) The induced maps H p (u) : H p (X) → H p (Y ) are isomorphisms for p > m and surjective for p = m. A morphism of complexes in C(A) is called m-isomorphism if it is an m-isomorphism considered as a morphism in D(A). The equivalence of the two conditions follows from the long exact cohomology sequence attached to the distinguished triangle. If u is an m-isomorphism, then it is an n-isomorphism for all n ≥ m. A morphism in D(A) (resp. in C(A)) is an isomorphism (resp. a quasi-isomorphism) if it is an misomorphism for all m. The composition of m-isomorphisms is again an m-isomorphism. For every complex X in C(A) the inclusion σ ≥m X → X is an m-isomorphism. Proposition F.160. ([ThTr] O 1.9.5) Let A be an abelian category, let D be a full additive subcategory of A, and let C be a full subcategory of C(A) such that the following hypotheses are satisfied (a) For any complex X in C one has H p (X) = 0 for p large enough and if s : X → Y is a quasi-isomorphism in C(A), then X is in C if and only if Y is in C. (b) One has C b (D) ⊆ C. For every map of complexes u : X → Y with X in C b (D) and Y in C the mapping cone Cu lies in C.

791 (c) Suppose given m ∈ Z, a complex X in C with H p (X) = 0 for p ≥ m, and an epimorphism M → H m−1 (X) in A. Then there exists an object P in D and a map P → M such that the composition P → M → H m−1 (X) is an epimorphism. Then every map u : Z → X in C with Z in C − (D) can be factorized within C as i

u′

u : Z −→ Z ′ −→ X, where Z ′ is in C − (D), u′ is a quasi-isomorphism, and i is a degree-wise split monomorphism. Moreover, if m ∈ Z and u is an m-isomorphism, then one may choose Z ′ as above such that ip : Z p → Z ′p is an isomorphism for all p ≥ m. Often the proposition is applied to Z = 0. Then the assertion is that for every X in C there exists a complex Z ′ in C − (D) and a quasi-isomorphism Z ′ → X.

(F.41) Variants of the derived category. Let A be an abelian category, and let A′ ⊆ A be a plump abelian subcategory (Defini∗ ∗ tion F.43). For ∗ ∈ {∅, b, +, −}, we denote by KA ′ (A) the full subcategory of K (A) of i ′ complexes X such that H (X) ∈ A for all i ∈ Z. These are triangulated subcategories of K(A), and localizing with respect to all quasi-isomorphisms we obtain the category DKA′ (A) as in Section (F.38). Definition F.161. Let A be an abelian category, and let A′ ⊆ A be a plump abelian ∗ subcategory (Definition F.43). For ∗ ∈ {∅, b, +, −}, we denote by DA ′ (A) the full additive subcategory of D∗ (A) consisting of all X such that H i (X) ∈ A′ for all i. KA′ (A) Lemma F.162. ([Lip2] O → D(A) is fully faithful X (1.9.1)) The canonical functor D KA′ (A) ∗ ∗ and yields an equivalence of D with the full subcategory DA ′ (A) of D (A).

This defines triangulated subcategories, and we have natural functors ∗ D∗ (A′ ) −→ DA ′ (A).

Note that even if A and A′ are both Grothendieck abelian categories, these functors D(A′ ) → DA′ (A) are not in general equivalences. We have the following general criterion: Theorem F.163. (cf. [KaSh] O Theorem 13.2.8) Let A′ be a plump abelian subcategory of the abelian category A. Assume that for every monomorphism Y ′ ,→ X in A with Y ′ ∈ A′ there exists a morphism X → Y with Y ∈ A′ such that the composition Y ′ → Y is a monomorphism. Then the natural functors + D+ (A′ ) −→ DA ′ (A),

are equivalences of categories.

b D b (A′ ) −→ DA ′ (A)

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F Homological Algebra

Derived functors We come now to the main topic in this chapter, the definition and construction of the right (resp. left) derived functor RF : D(A) → D(B) (resp. LF : D(A) → D(B)) of an additive functor F : A → B between abelian categories. After stating the abstract definition we have to give criteria for their existence and how to compute them. For this we will define the notion of a right (resp. left) F -acyclic complex in K(A) and see that RF (resp. LF ) can be computed by evaluating the natural extension of F to a functor K(A) → K(B) on these F -acyclic complexes. The full subcategory of D(A) of objects that are isomorphic to right (resp. left) F -acyclic complexes is a triangulated subcategory and one can define RF (resp. LF ) on that subcategory. Hence to have the derived functor RF (resp. LF ) on all of D(A) one has to study the question if each object in D(A) is isomorphic in D(A) to a right (resp. left) F -acyclic complex. In fact, one can define the notion of a K-injective (resp. K-projective) complex (Definition F.179) and these are complexes that are right (resp. left) F -acyclic for every functor F and hence we can define RF (resp. LF ) for all functors F if we know that every object in D(A) is isomorphic to a K-injective (resp. K-projective) complex. For right derived functors this idea just works fine: In this book we are almost always interested in the case that A is a Grothendieck abelian category since we show that the category of OX -modules for an arbitrary ringed space X is a Grothendieck abelian category (Proposition 21.7). In this case Theorem F.185 shows that indeed every object of D(A) is isomorphic to a K-injective complex. Hence the right derived functor RF exists for every functor F . For left derived functors the situation is more complicated in general. If A is the category of A-modules for some ring A, then one can show, using that every A-module has a left resolution by free modules, that every object in D(A) is isomorphic to a K-projective complex (Theorem F.189). Hence in this case there exists the left derivation for every additive functor (A-Mod) → B. But for arbitrary ringed spaces X this does not hold any more, even for X = P1k , k a field (see Exercise 21.5, which also shows that this problem cannot be removed by working only with quasi-coherent modules). Hence in this case one has to show for a given functor F of interest there exist sufficiently many left F -acyclic complexes. This is what we do in Section (21.19) and the following sections to define the left derived functor of the tensor product and of the pullback functor. (F.42) Definition of derived functors. Let A be an abelian category, and let K ⊂ K(A) be a triangulated subcategory (typical examples of K would be K ∗ (A) for ∗ ∈ {∅, +, −, b}). We denote by Q : K → DK the localization functor. Let F : K → E be a triangulated functor from K to some triangulated category E. Definition F.164. We call the functor F : K → E right derivable on K, if there exists a triangulated functor RF : DK → E together with a morphism ζ : F → RF ◦ Q of triangulated functors such that for every triangulated functor G : DK → E the composition Hom(RF, G) → Hom(RF ◦ Q, G ◦ Q) → Hom(F, G ◦ Q), the final map being induced by ζ, is an isomorphism. Such a pair (RF, ζ) is called the right derived functor of F .

793 Note that we could express the existence of RF by saying that F is right localizable as a triangulated functor (i.e., in Definition F.144 we use only triangulated functors G as test functors). See also Remark F.168 below. If the dependence on K needs to be emphasized, we will denote RF by RK F (or by R+ F for K = K + (A), etc.). Dually, we define Definition F.165. We call the functor F : K → E left derivable on K, if there exists a triangulated functor LF : DK → E together with a morphism ξ : LF ◦Q → F of triangulated functors such that for every triangulated functor G : DK → E the composition Hom(G, LF ) → Hom(G ◦ Q, LF ◦ Q) → Hom(G ◦ Q, F ) is an isomorphism. Such a pair (LF, ξ) is called the left derived functor of F . Below we mostly omit the left derived case from the discussion, but of course, all the notions and results below can be dualized. Remark F.166. Let F, G : K → E be triangulated functors and let α : F → G be a morphism of triangulated functors. Suppose that F and G both admit right derived functors (RF, ζ) and (RG, ξ). Then the universal property of (RF, ζ) implies that there exists a unique morphism Rα : RF → RG of triangulated functors such that the diagram F ζ



RF ◦ Q

α

/G

X7→Rα(QX)

 / RG ◦ Q,

ξ

where Q : K → DK is the localization functor, commutes. Instead of saying that F is right derivable, we often say that RF exists. The most common case how the above functor F arises is the following Remark F.167. Let F : A → A′ be an additive functor between abelian categories A, A′ . Then F induces a triangulated functor K(F ) : K(A) → K(A′ ) by applying F to each degree separately, which we sometimes simply denote by F again. Similarly, we can restrict K(F ) to K + (A) or any other triangulated subcategory K of K(A). Denote by Q : K(A) → D(A) and Q′ : K(A′ ) → D(A′ ) the localization functors. We now consider the functor Q′ ◦ K(F ) : K(A) → D(A′ ) (or its restriction K → E, where E is a triangulated subcategory of D(A′ )). If this functor admits a right derived functor in the sense of the above definition, then we say that F : A → A′ has a right derived functor (on K with values in E), and we denote that derived functor by RF (or by RK F ). In this case we set Ri F := H i ◦ RK F : DK → A′ ,

i ∈ Z.

As RF is a triangulated functor, the composition with the cohomological functor H 0 is again a cohomological functor R0 F : DK → A′ and for every distinguished triangle X → Y → Z → X[1] in K we obtain a long exact sequence

794

F Homological Algebra . . . −→ Ri−1 F (Z) −→ Ri F (X) −→ Ri F (Y ) −→ Ri F (Z) −→ Ri+1 F (X) −→ . . .

This applies in particular to a short exact sequence 0 → X → Y → Z → 0 of complexes of A with X, Y, Z in K. Remark F.168. Note that in the literature there are different definitions of derived functors. Here we use Verdier’s original definition following [Lip2] O X. A stronger version of derived functors has been defined by Deligne in [SGA4] O XVII, 1.7. This is also the approach followed in [KaSh] O . It is based on the fact that the localization KS of a category K by a right multiplicative system S can be embedded fully faithfully into the category Ind(K) of filtered inductive systems of K. If ιK : K → Ind(K) is the fully faithful canonical functor and Q : K → KS is the localization functor, one obtains a diagram of functors K

/ KS

Q

ιK

α

#  Ind(K),

which is not commutative. But there exists a natural morphism ιK −→ α ◦ Q.

(F.42.1)

Let us suppose that K is a triangulated subcategory of K(A) for an abelian category A and that KS = DK is the localization by the system S of quasi-isomorphisms in K. Then Ind(K) is also triangulated, all functors above are triangulated and the morphism (F.42.1) is a morphism of triangulated functors. Let F : K → E be a triangulated functor of triangulated categories. Rather than requiring the existence of a right localization as a triangulated functor, in [KaSh] O the existence of a universal right localization is required (Def. F.144). In particular, in the universal property, non-triangulated test functors are also allowed. Moreover, the derived functor DK → E in the sense of [KaSh] O exists if and only if the composition α

Ind(F )

DK −→ Ind(K) −→ Ind(E) factors over the fully faithful functor ιE : E → Ind(E). This factorization yields a triangulated functor R : DK → E and a morphism of triangulated functors ζ : F → R ◦ Q induced by (F.42.1). The pair (R, ζ) is a right derived functor in the sense of [KaSh] O . In particular, (R = RF, ζ) is a right derived functor in the sense of Definition F.164 above. It is not clear (to us), whether a derived functor as in Definition F.164 is necessarily a derived functor in the sense of [KaSh] O . In practice this distinction is not important because the results for the existence of derived functors in [KaSh] O show that all techniques we use to construct derived functors (which all rely on Theorem F.173 below) also ensure the existence of the right derived functor in their stronger sense. Compare also Corollary F.174 (2) below which shows that in the situation of Theorem F.173, F is “universally right localizable as a triangulated functor”. (F.43) Construction of derived functors. We now give some existence results for derived functors on the unbounded derived category. We follow [Lip2] O X 2.2. For comparisons with more classical situations see Section (F.48).

795 Classically derived functors are constructed via acyclic resolutions (or injective resolutions, which are acyclic resolutions for all left exact functors). Since unbounded complexes of acyclic (or injective) objects do not have the same good properties as bounded below such complexes, one cannot work with resolutions by complexes of acyclic objects (or with injective resolutions). But it turns out that there is a good generalization, namely the notion of right F -acyclic complexes (or of K-injective complexes). Throughout, let K be a triangulated subcategory of K(A), let E be a triangulated category, and let F: K→E be a triangulated functor. Let Q : K → DK be the localization functor. Usually we will apply the following results in the following “classical” situation: (CL) F is the functor K(A) → D(B) induced by an additive functor A → B of abelian categories, again denoted by F (Remark F.167). There is one easy example where the existence of right derived functors follows immediately from the definition. Example F.169. Recall (Remark F.149) that the following assertions for a triangulated functor F : K → E are equivalent. (i) There exists a (necessarily unique) triangulated functor F¯ : DK → E such that F = F¯ ◦ Q. (ii) The functor F maps quasi-isomorphisms in K to isomorphisms in E. (iii) The functor F maps exact complexes in K to 0 in E. In this case, the functor F¯ is the right derived functor of F . It is also the left derived functor of F and we usually simply write F instead of RF or LF in this case. In Situation (CL), the functor F : K(A) → D(B) maps quasi-isomorphisms to isomorphisms if and only if it is induced by an exact functor A → B. Indeed, if A → B is exact, then the induced functor sends quasi-isomorphisms in K(A) to quasi-isomorphisms in K(B) and hence to isomorphisms in D(B). Conversely, F sends quasi-isomorphisms in K(A) to isomorphisms in D(B) if and only if the functor K(A) → K(B) preserves quasi-isomorphisms. As a complex is exact if and only if there exists a quasi-isomorphism to 0, this shows that the functor F : A → B is exact. The idea to construct derived functors more generally is to reduce to the situation in Example F.169. For this we will use two ingredients: (I) To construct a (preferably large) triangulated subcategory J of K such that F |J maps quasi-isomorphisms to isomorphisms. (II) A recipe to extend derived functors on a triangulated subcategory (such as J as in (I)) to larger triangulated subcategories. We start with Step (II), i.e., we consider the following situation. Let J ⊆ K ⊆ K(A) be triangulated subcategories yielding a commutative diagram J

j

QI

 DJ

ȷ¯

/K 

QK

/ DK ,

where j is the inclusion functor and ȷ¯ is the induced functor.

796

F Homological Algebra

Proposition F.170. ([Lip2] O X (1.7.2) and (2.2.3)) Suppose that for every object X in K there exists a quasi-isomorphism φX : X → AX , where AX is an object of J . ∼ (1) Then ȷ¯ is an equivalence of triangulated categories DJ −→ DK and there exists a K J quasi-inverse functor ρ : D → D with ρ(X) = AX and such that the φX yield ∼ ∼ isomorphisms of triangulated functors idDJ → ρ ◦ ȷ¯ and idDK → ȷ¯ ◦ ρ. (2) Suppose that the restriction of F : K → E to J has a right derived functor RJ F : DJ → E,

ζJ : F |J → RJ F ◦ QJ .

Then F has a right derived functor (RF, ζ) with RF = RJ F ◦ ρ and ζ(X) for X in K the composition F (X)

F (φX )

ζJ (AX )

/ F (AX )

/ RF J (AX ) = RF (X).

Now we come to step (I). The key definition is the following. Definition F.171. We call an object X ∈ K right F -acyclic, if for every quasiisomorphism X → Y in K, there exists a quasi-isomorphism Y → Z such that the map F (X) → F (Z) obtained by applying F to the composition is an isomorphism. (Reversing the directions of the arrows, we obtain the notion of left F -acyclic object.) Then F transforms quasi-isomorphisms between F -acyclic complexes to isomorphisms, more precisely: Lemma F.172. ([Lip2] O X (2.2.5)) The full subcategory (F -acycl) of K consisting of right F -acyclic objects is a triangulated subcategory, and the functor DF -acycl → DK induced by the inclusion is fully faithful. Moreover the restriction of F to (F -acycl) transforms quasi-isomorphisms into isomorphisms. Combining Example F.169, Proposition F.170, and Lemma F.172 we see that if there are enough right F -acyclic objects in the sense of the following proposition, then the right derived functor of F exists: Theorem F.173. Let K ⊆ K(A) be a triangulated subcategory and let F : K → E be a triangulated functor of triangulated categories. Assume that for every object X of K there exists a quasi-isomorphism φX : X → AX , where AX is right F -acyclic. Then F has a right derived functor (RF : DK → E, ζ : F → RF ◦Q) with RF (A) = F (A) for A ∈ Ob(D F -acycl ) = Ob(F -acycl). In particular, RF (X) = F (AX ),

for X ∈ Ob(D K ) = Ob(K).

The morphism ζ : F → RF ◦ Q is given by F (φX ) : F (X) → F (AX ) = RF (X). We have the following complement to the theorem.

797 Corollary F.174. ([Lip2] O X (2.2.6)) Suppose the hypotheses of Theorem F.173 are satisfied. (1) A complex X in K is right F -acyclic if and only if ζ(X) : F (X) → RF (X) is an isomorphism. (2) Let G : E → E ′ be any triangulated functor. Then (G ◦ RF, G(ζ)) is a right derived functor of G ◦ F . Remark F.175. In practice, we will use Corollary F.174 (2) as follows. The triangulated categories E and E ′ will be a triangulated subcategories of D(B) and D(B ′ ), respectively, for some abelian categories B and B ′ . The functor F : K → E will be induced by some additive functor F : A → B. And the functor G will be the functor induced by an exact functor G : B → B ′ (Example F.169). Then (F.43.1)

R(G ◦ F ) = G ◦ RF.

In addition, we have the following compatibility with composition of functors: ′ Proposition F.176. (cf. [Lip2] O X Cor. 2.2.7) Let A, A be abelian categories, K ⊂ ′ ′ ′ K(A) and K ⊂ K(A ) triangulated subcategories, and Q : K → DK , Q′ : K′ → DK the localization functors. Let E be a triangulated category. Let G : K′ → E be a triangulated functor which has a right derived functor RG. Let F : K → K′ be a triangulated functor. Assume that there exists a family of quasiisomorphisms X → AX , X ∈ K, such that AX is right (Q′ ◦ F )-acyclic, and F (AX ) is right G-acyclic. Then Q′ ◦ F has a right derived functor RF and G ◦ F has a right derived functor ∼ R(GF ). There is a unique isomorphism α : R(GF ) → RG ◦ RF of triangulated functors K D → E such that the following diagram of morphisms of functors commutes:

GF

/ R(GF ) ◦ Q

 RG ◦ Q′ ◦ F

 / RG ◦ RF ◦ Q.

α◦Q

Remark F.177. Let A, A′ , K ⊂ K(A), K′ ⊂ K(A′ ), and E be as in Proposition F.176. Let F : K → K′ and G : K′ → E be triangulated functors such that G has a right derived functor RG. Suppose that F preserves quasi-isomorphisms (for instance, if F is induced by an exact functor A → A′ ). Then RF = F (omitting localization functors) and every X in K is right F -acyclic. Hence if there exists a family of quasi-isomorphisms X → AX , X ∈ K, such that F (AX ) is right G-acyclic, then R(G ◦ F ) = RG ◦ F. We can apply Remark F.177 to inclusions of triangulated subcategories: Corollary F.178. Let K′ ⊆ K be triangulated subcategories of K(A) and let j : K′ → K be the inclusion. Let F : K → E be a triangulated functor such that (RF : DK → E, ζ) exists. Suppose that for every complex X in K′ there exists a quasi-isomorphism X → AX in K′ such that AX is right F -acyclic (as object of K).

798

F Homological Algebra ′

Then the right derived functor (RK F, ζ ′ ) of F |K′ exists and there exists a unique isomorphism ′ ∼ α : RK F −→ RF ◦ j such that for all X in K′ one has α(QX) ◦ ζ ′ (X) = ζ(j(X)). (F.44) K-injective resolutions. In many cases, we can find enough objects which are right acyclic for all triangulated functors on K simultaneously. The key notion is the notion of K-injective objects introduced by Spaltenstein. Dually we have the notion of K-projective objects. In the realm of algebraic geometry, such objects are however less abundant, so that in practice this notion turns out to be less relevant, and finding left derived functors often has to be done in different ways. In algebraic geometry, the notion of K-flat complexes, which are left acyclic for the tensor product, is the more important one. Definition and Proposition F.179. ([Lip2] O X Prop. (2.3.8)) Let A be an abelian category, and let K be a triangulated subcategory of K(A). An object I ∈ K is called K-injective (or homotopically injective, or q-injective) in K, if the following equivalent conditions are satisfied. (1) I is right F -acyclic for every triangulated functor F : K → E. (2) One has HomK (X, I) = 0 for every exact complex X in K. (3) For every morphism f : X → I and quasi-isomorphism s : X → Y (both in K), there exists g : Y → I in K with gs = f . (This g is necessarily unique.) (4) Every quasi-isomorphism s : I → Y in K has a left inverse. (5) Every quasi-isomorphism s : I → Y in K is a monomorphism. (6) The triangulated functor HomA (−, I) : Kop → K(AbGrp) (Example F.127) preserves the property quasi-isomorphism. (7) For every X ∈ K, the natural map (F.44.1)

HomK (X, I) ∼ = HomDK (X, I)

is a bijection. One also defines dually the notion of an object P ∈ K that is called K-projective (or homotopically projective, or q-projective). If an object I in K is K-injective (resp. K-projective) in K(A), then it is also K-injective (resp. K-projective) in K. Remark F.180. Property (2) of Proposition F.179 shows that if I is K-injective and exact, then idI = 0 in K(A), i.e., I is homotopy equivalent to 0. Remark F.181. Let A be an abelian category. The connection between K-injective complexes and complexes of injective objects is the following ([Lip2] O X (2.3.3) and (2.3.4), see also Exercise F.20). (1) An object A of A is injective if and only if it is K-injective in K(A) (or, equivalently, in K b (A)) if considered as a complex concentrated in degree 0. (2) Every bounded below complex of injective objects of A is K-injective in K(A). Conversely, if A has enough injective objects, then any complex I in K + (A) which is K + (A)-injective is isomorphic in K + (A) to a bounded below complex of injective objects in A.

799 Example F.182. It is not true that every complex consisting of injective objects is K-injective. Consider the category of modules over the ring Z/4Z, and the complex which in each degree has a free Z/4Z-module of rank 1, and with all differentials given by multiplication by 2. This is an exact complex of injective objects, but it is not K-injective. Because in that case it would necessarily be homotopic to 0 by Remark F.180, however tensoring by − ⊗Z/4 Z/2Z we obtain a complex which is not exact. Note that the same complex also is an example of a complex consisting of projective objects which is not K-projective. Corollary F.183. Let F : A → B be an additive functor of abelian categories. Suppose that F has an exact left adjoint functor G. Then F sends injective objects to injective objects and the induced functor F : K(A) → K(B) sends K-injective complexes to K-injective complexes. Proof. We use (2) of Proposition F.179. Let I in K(A) be a K-injective complex and let Y be an exact complex in K(B). Then by Example F.130 we have HomK(B) (Y, F (I)) = HomK(A) (G(Y ), I) = 0 because G(Y ) is still exact. As an object is injective if and only if it is K-injective as a complex concentrated in degree 0, the first assertion also follows. Definition F.184. Let A be an abelian category, and let K ⊂ K(A) be a triangulated subcategory. We say that K has enough K-injective objects, if for every X in K there exists a quasi-isomorphism X → IX where IX is a K-injective object of K. We then call such a quasi-isomorphism X → IX a K-injective resolution of X. By Proposition F.179 (3), a K-injective resolution is unique up to unique isomorphism in K(A). Grothendieck abelian categories (Definition F.54) have enough K-injective objects. More precisely: Theorem F.185. ([Sta] 079P) Let A be a Grothendieck abelian category. Then for any complex X in K(A) there exists a quasi-isomorphism and monomorphism of complexes u : X → IX such that IX is a K-injective complex in K(A) and such that each component of I is an injective object of A. Moreover, this construction is functorial in X. If there exists n ∈ Z with H i (X) = 0 for i < n, then one can choose I such that I i = 0 for all i < n. Corollary F.186. Let A be a Grothendieck abelian category, let I be the full subcategory of A of injective objects, and let KK−inj (A) ⊆ K(A) be the homotopy category of K-injective + complexes. Then K + (I) = KK−inj (A) and the natural functors (F.44.2)

KK−inj (A) → D(A),

K + (I) → D+ (A)

are triangulated equivalences. In particular we see that for Grothendieck abelian categories A, the derived category D(A) is actually a category with small Hom sets. For us the main application of the results above is the following corollary.

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F Homological Algebra

Corollary F.187. Let A be a Grothendieck abelian category. Let F : K(A) → E be a triangulated functor to a triangulated category E. (1) Then F admits a right derived functor RF : D(A) → E, and RF (X) = F (IX ), where X → IX is a quasi-isomorphism to a K-injective complex. (2) The restriction of F to K + (A) also admits a right derived functor R+ F : D+ (A) → E, and R+ F is the restriction of RF to D+ (A). (3) If G : E → E ′ is any triangulated functor, then G ◦ RF is the right derived functor of G ◦ F. Furthermore, Proposition F.176 on the derived functor of a composition of functors specializes to the situation at hand (using K-injective resolutions) in the obvious way. Lemma F.188. ([Sta] 07D9) Let A be a Grothendieck abelian category. Then the derived category D(A) has arbitrary products and coproducts. Coproducts are obtained by taking termwise direct sums of any representing complexes. Products are obtained by taking termwise products of K-injective representing complexes. A Grothendieck category usually has not K-projective resolutions. But there is one important special case in which this is the case. Theorem F.189. ([Spa] O Theorem C) Let A be a ring. Then for every complex X of A-modules there exists a K-projective complex of A-modules P and a quasi-isomorphism P → X. In fact, the localization functor Q : K(A) → D(A) has a fully faithful left adjoint functor P and for X in K(A) the counit of the adjunction P (Q(X)) → X defines a functorial K-projective resolution of X ([Gil] O X ). (F.45) Adjointness of derived functors. In this section, let F : A → B and G : B → A be additive functors of abelian categories such that G is left adjoint to F . Remark F.190. Let X in C(A) and Y in C(B) be complexes. Then the adjunction iso∼ morphisms HomB (Y i , F (X j )) → HomA (G(Y i ), X j ) for all i, j ∈ Z yield an isomorphism of complexes of abelian groups (F.45.1)

∼

HomB (Y, F (X)) → HomA (G(Y ), X)

which is functorial in X and Y . Applying Z 0 (−) and H 0 (−) to this isomorphism one obtains functorial isomorphisms of abelian groups ∼

(F.45.2)

HomC(B) (Y, F (X)) → HomC(A) (G(Y ), X), ∼

HomK(B) (Y, F (X)) → HomK(A) (G(Y ), X).

Now suppose that for every complex X in K(A) there exists a quasi-isomorphism X → IX with IX right F -acyclic and that for every complex Y in K(B) there exists a quasi-isomorphism PY → Y with PY left G-acyclic. Then the right derived functor RF : D(A) → D(B) and the left derived functor LG : D(B) → D(A) exist.

801 5 Proposition F.191. ([Lip2] O X (3.2.1), (3.2.2) ) The pair (LG, RF ) is a triangulated adjoint pair (Definition F.128). More precisely, there exists for X ∈ D(A) and Y ∈ D(B) a unique functorial isomorphism ∼

ρ : HomD(B) (Y, RF (X)) −→ HomD(A) (LG(Y ), X) such that the following diagram is commutative HomK(B) (Y, F (X))

ν

/ HomD(B) (Y, F (X))

∼ = ρ

(F.45.2) ∼ =

 HomK(A) (G(Y ), X)

/ HomD(B) (Y, RF (X))

µ

/ HomD(A) (G(Y ), X)

 / HomD(A) (LG(Y ), X).

Moreover, if X is K-injective and Y is left G-acyclic (resp. if X is right F -acyclic and Y is K-projective), then ν (resp. µ) is an isomorphism. (F.46) Bounded Functors. Let A and B be abelian categories and let A′ be a plump subcategory of A. Recall that ∗ we defined for ∗ ∈ {b, +, −, ∅} the triangulated subcategory DA ′ (A) of D(A) consisting i ′ of complexes X such that H (X) ∈ A for all i ∈ Z and that H i (X) = 0 for |i| ≫ 0 ∗ (resp. i ≪ 0, resp. i ≫ 0) if ∗ = b (resp. ∗ = +, resp. ∗ = −). Let F : DA ′ (A) → D(B) be a triangulated functor. Typically, F could be the right or left derived functor of some ∗ functor KA ′ (A) → D(B). ∗ As DA′ (A) is a triangulated subcategory stable under the truncation functors τ ≤n and τ ≥n for all n ∈ Z, we find (cf. [Lip2] O X (1.11.2)) (F.46.1) ≤n ≤n+d { d ; F (DA (B) for all (or one) n ∈ Z } ′ (A)) ⊆ D ∼

= { d ; H i F (X) −→ H i F (τ ≥n X) for all X, all (or one) n ∈ Z and all i ≥ n + d }, (F.46.2) ≥n ≥n−d (B) for all (or one) n ∈ Z } { d ; F (DA ′ (A)) ⊆ D ∼

= { d ; H i F (τ ≤n X) −→ H i F (X) for all X, all (or one) n ∈ Z and all i ≤ n − d }. More precisely, in the descriptions given second, the condition is to be understood as saying that the map induced by the exact triangle τ ≤n X → X → τ ≥n+1 X → (Proposition F.157) is an isomorphism. Let dim+ (F ) (resp. dim− (F )) be the infimum of the set (F.46.1) (resp. of (F.46.2)). We say that F is cohomologically bounded above (resp. cohomologically bounded below ) if dim+ F < ∞ (resp. if dim− F < ∞). The functor F is called cohomologically bounded if it is cohomological bounded above and below. If F is the (right or left) derived functor of some functor F ′ , we also write dim+ (F ′ ) instead of dim+ (F ), similarly for the other definitions. 5

The proof in loc. cit. is given only for special functors. In the general case used here the proof is verbatim the same.

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F Homological Algebra

Usually, if F is the right (resp. left) derived functor on D+ (A) (resp. on D− (A)) of some additive non-zero functor A → B, then we will have dim− (F ) = 0 (resp. dim+ (F ) = 0), see Remark F.200 (2) below for details. Hence for right (resp. left) derived functors dim+ F (resp. dim− F ) is the interesting invariant. The cohomological dimension is usually determined as follows. + Lemma F.192. ([Lip2] O X (1.11.2)) Let F : DA′ (A) → D(B) be a triangulated functor. Then dim+ F = inf{ d ∈ Z ; H i F (A) = 0 for all i > d and A in A′ }. − Similarly, if L : DA ′ (A) → D(B) is a triangulated functor, then

dim− L = inf{ d ∈ Z ; H i F (A) = 0 for all i < −d and A in A′ }. ∗ Proposition F.193. ([Lip2] O X (1.11.3)) Let F, G : DA′ (A) → D(B) be triangulated functors and suppose that one of the following conditions hold. (a) ∗ = b. (b) ∗ = + and both F and G are cohomologically bounded below. (c) ∗ = − and both F and G are cohomologically bounded above. (d) ∗ = ∅ and both F and G are cohomologically bounded. Let η : F → G a morphism of triangulated functors. Then η is an isomorphism if and only if η(A) is an isomorphism for every object A of A′ . Moreover, suppose that Condition (b) (resp. Condition (c)) holds, and let I (resp. P ) be a set of objects of A′ such that every object of A′ admits a monomorphism into an object of I (resp. is the target of an epimorphism out of an object of P ). Then η is an isomorphism if η(A) is an isomorphism for every object A of I (resp. of P ).

(F.47) Construction of resolutions. In this section we collect lemmas which allow us to construct resolutions. We will only consider the case of right resolutions of a complex X, i.e., of quasi-isomorphisms X → IX , where IX lies in a special class of complexes (except for Corollary F.197). The case of left resolutions is entirely dual. We will always denote by A an abelian category. The main technical tool to construct right resolutions of bounded below complexes that have components of a prescribed type is the following lemma. Lemma F.194. Let I be a class of objects of A containing 0 such that every object A of A admits a monomorphism into an object I 0 (A) in I. We consider I as a full subcategory. (1) Let n ∈ Z. Every complex X ∈ C(A) with H i (X) = 0 for all i < n admits a quasii isomorphism to a bounded below complex IX of objects in I with IX = 0 for all i < n. (2) Suppose that for all I, J ∈ I one also has I ⊕ J ∈ I and that A 7→ I 0 (A) is an additive functor A → I. Then X 7→ IX can in addition be chosen to be functorial in X, a monomorphism in each degree, and such that IX[k] = IX [k] for all k ∈ Z. We will only prove (and use) the second assertion. For the first one we refer to [Sta] 05T6.

803 Proof. By hypothesis, the quotient map X → τ ≥n X is a quasi-isomorphism. Replacing X by τ ≥n X, we may assume that X i = 0 for i < n. Let us first assume that X = A is concentrated in degree 0. By hypothesis we are given a functorial monomorphism u : A ,→ I 0 (A) with I 0 (A) in I. We define an exact sequence 0 → A → I 0 (A) → I 1 (A) → I 2 (A) → . . . inductively by I 1 (A) := I 0 (Coker(u)) and I i (A) := I 0 (Coker(I i−2 (A) → I i−1 (A))) and i hence obtain a quasi-isomorphism of complexes A → IA with IA := I i (A) for i ≥ 0 and i 0 IA := 0 for i < 0. As A 7→ I (A) is functorial, A 7→ IA is functorial in A. Now let X be an arbitrary complex with X i = 0 for i < n. By the functoriality of A 7→ IA we obtain a complex of complexes . . . → 0 → IX n → IX n+1 → . . . , j j i.e., a double complex I˜X := (IX i )i,j . As X is bounded below and IA = 0 for all j < 0, the total complex IX attached to this double complex exists. If we consider X as a double complex by X i,0 := X i and X i,j := 0 for j = ̸ 0, the 0 ˜ morphisms X i → IX i yield a morphism αX : X → IX of double complexes which is a quasi-isomorphism in vertical direction. Hence it induces a quasi-isomorphism X = Tot(X) → Tot(I˜X ) = IX by Lemma F.113. This quasi-isomorphism is functorial in X, a monomorphism in each degree and compatible with shift.

To construct resolutions of unbounded complexes one can combine Lemma F.194 with the following lemma. Lemma F.195. ([Spa] O 3.7) Let I be a class of bounded below complexes in C(A) satisfying the following properties. (a) If X ′ → X → X ′′ is a termwise split sequence of complexes in C(A) such that X ′ and X ′′ are in I, then X is in I. (b) For every bounded below complex Y in C(A) there exists a quasi-isomorphism Y → IY with IY in I. Let X be a complex. Then there exists a commutative diagram of complexes τ ≥0 X o (F.47.1) 

τ ≥−1 X o

f0

I0 o



f1

I1 o

τ ≥−2 X o 

...

f2

I2 o

...

such that (1) all complexes In are in I, (2) for all n ≥ 1, the morphism In → In−1 is a termwise split epimorphism whose kernel is a complex in I, (3) The vertical morphisms fn are quasi-isomorphisms for all n ≥ 0. Remark F.196. If IY depends functorially on Y , then we can obtain a diagram (F.47.1) satisfying (1) and (3) and depending functorially on X by simply setting In := Iτ ≥−n X . From the commutative diagram (F.47.1) one obtains a morphism of complexes (F.47.2)

X = lim τ ≥−n X −→ lim In n

n

804

F Homological Algebra

if these limits exist. As the morphisms fn are quasi-isomorphisms, this morphism is a quasi-isomorphism if limits indexed by N are exact. Unfortunately this is almost never the case for those abelian categories that we are interested in, namely the abelian category of modules over a ring or a sheaf of rings. But the dual statement that colimits indexed by N (or any filtered colimits) are exact is true in these categories. Hence we deduce from the dual versions of Lemma F.194 and Remark F.196 the following corollary. Corollary F.197. Let A be an abelian category such that colimits indexed by the totally ordered set N exist and are exact (e.g., if A is a Grothendieck abelian category). Let P be a full additive subcategory of A such that for every object A in A there is an epimorphism PA → A with PA in P which is functorial in A. Then for every complex X in C(A) there exists a quasi-isomorphism PX → X, functorial in X and compatible with shift, where PX is a colimit indexed by N of bounded above complexes whose components are in P. (F.48) Derived functors on D+ (A) and higher derived functors of left exact functors. We will now translate the theory to a more “classical” language by linking derived functors on D+ (A) to higher derived functors as (universal) δ-functors (see Definition F.201 below). We will also explain how to obtain quasi-isomorphisms to acyclic complexes via resolutions of complexes of acyclic objects, e.g., injective resolutions. More precisely, let us consider an additive functor F : A −→ B between abelian categories A and B. The induced triangulated functors K + (A) → D+ (B) and K − (A) → D− (B) are again denoted by F . We are interested in right derived functors R+ F : D+ (A) → D+ (B) and left derived functors L− F : D− (A) → D− (B). We will consider only the case R+ F . The case L− F is entirely dual. We will consider the following assumption on A and F that will be usually satisfied in our applications. Condition (Ac) F.198. For every complex X in K + (A) there exists a quasi-isomorphism X → CX , where CX is a right F -acyclic complex in K + (A). Moreover if n ∈ Z is an i integer such that X i = 0 for all i < n, then we may assume that CX = 0 for all i < n. Then the derived functor R+ F : D+ (A) → D+ (B) exists and R+ F (X) = F (CX ) by Theorem F.173. Below we will give a criterion for (Ac) to hold via acyclic resolutions. A special case of this criterion and a principal example, where Condition (Ac) is satisfied, is the existence of enough injective objects in A: Example F.199. Assume that A has enough injective objects. We claim that Condition (Ac) is satisfied for every additive functor F : A → B to an abelian category B. Indeed, by Lemma F.194 every complex X in K + (A) admits a quasi-isomorphism to a i bounded below complex IX of injective objects with IX = 0 for all i < n if X i = 0 for all + i < n, some n ∈ Z. As IX is K-injective in K (A) (Remark F.181 (2)), it is right acyclic for every functor (Proposition F.179) und in particular right F -acyclic.

805 Therefore the right derived functor R+ F : D+ (A) → D+ (A′ ) exists, and R+ F (X) ∼ = F (IX ) for every X ∈ D+ (A). We collect some properties of the functor R+ F : D+ (A) → D+ (B) under the assumption that Condition (Ac) holds. Recall that in this case we have the right derived functors Ri F := H i ◦R+ F : D+ (A) → B for i ∈ Z, which we often restrict to functors Ri F : A → B and call them higher derived functors of F . Remark F.200. Let F : A → B be an additive functor. Suppose that Condition (Ac) holds. (1) Let X be in D+ (A) and let n ∈ Z such that H i (X) = 0 for all i < n. Then Ri F (X) = 0 for all i < n. Indeed, by assumption X → τ ≥n X is a quasi-isomorphism and by (Ac) we can choose a quasi-isomorphism τ ≥n X → Cτ ≥n X to a right F -acyclic complex whose components are 0 in degrees < n. Then RF (X) = RF (Cτ ≥n X ) = F (Cτ ≥n X ) and hence Ri F (X) = 0 for all i < n. (2) In particular, for every object A in A, we find that Ri F (A) = 0 for all i < 0. In other words, dim− (RF ) ≤ 0 (and = 0 if F = ̸ 0). Moreover, from the long exact cohomology sequence attached to a short exact sequence in A (Remark F.167) it follows that R0 F is always left exact. (3) The morphism F → RF of functors induces a morphism F → R0 F of functors A → B, and the latter is an isomorphism if and only if F is left exact. Indeed, by (2) the condition is clearly necessary. Conversely, if F is left exact and A is an object of A, then the quasi-isomorphism A → CA is the same as an exact 0 1 0 1 sequence 0 → A → CA → CA → . . . and 0 → F (A) → F (CA ) → F (CA ) is exact. Therefore F (A) = H 0 (F (CA )) = R0 (A). Considering Ri F as functors A → B, we obtain by Remark F.167 a δ-functor (Ri F )i≥0 in the following sense. Definition F.201. Let A, B be abelian categories. (1) A δ-functor from A to B is a collection (T i )i≥0 of functors A → B together with “connecting” morphisms δ i : T i (A′′ ) → T i+1 (A′ ) for each short exact sequence 0 → A′ → A → A′′ → 0 in A, for all i ≥ 0, such that (a) For each short exact sequence 0 → A′ → A → A′′ → 0 in A, the sequence 0 → T 0 (A′ ) → T 0 (A) → T 0 (A′′ ) → T 1 (A′ ) → · · · · · · → T i (A′ ) → T i (A) → T i (A′′ ) → T i+1 (A′ ) → · · · is exact, and (b) for every commutative diagram with exact rows in A 0

/ A′

/A

/ A′′

/0

0

 / B′

 /B

 / B ′′

/0

the connecting morphisms δ i yield a commutative diagram

806

F Homological Algebra T i (A′′ )  T i (B ′′ )

δi

/ T i+1 (A′ )

δi

 / T i+1 (B ′ ).

(2) A δ-functor T = (T i )i from A to B is called universal, if for every δ-functor U = (U i )i from A to B and every morphism f 0 : T 0 → U 0 of functors, there exist unique morphisms f i : T i → U i of functors, for all i > 0, such that for each short exact sequence 0 → A′ → A → A′′ → 0 in A, the diagram T i (A′′ )

δi

f i (A′′ )

 U i (A′′ )

δ

i

/ T i+1 (A′ ) 

f i+1 (A′ )

/ U i+1 (A′ ).

is commutative. There is the following criterion for a δ-functor to be universal. Proposition F.202. ([Gro1] O 2.2.1) Let (T i )i≥0 be a δ-functor from A to B. Suppose that for every i > 0 and any object A of A there exists a monomorphism u : A → C such that T i (u) = 0. Then (Ti )i is a universal δ-functor. One calls δ-functors satisfying the hypothesis of the proposition effaceable. In order to apply the proposition to a δ-functor of the form (Ri F )i≥0 , one can often use monomorphisms to right F -acyclic objects C, where an object C of A is called right F -acyclic if it is right F -acyclic as complex concentrated in degree 0 for the induced functor K + (A) → D+ (B). Then Ri F (C) = 0 for i > 0, and a fortiori any morphism with target Ri F (C) must vanish. Example F.203. An injective object of A is K-injective as a complex concentrated in degree 0 (Remark F.181 (1)) and hence is right F -acyclic for every additive functor F (Proposition F.179 (1)). The following lemma shows in particular that one obtains the “usual” definition of F -acyclic objects if F is a left exact functor A → B and A “has enough acyclic objects”. Proposition F.204. ([Lip2] O X (2.7.4)) Suppose that F : A → B is left exact and suppose that every object of A admits a monomorphism into a right F -acyclic object of A. Then every bounded below complex of right F -acyclic objects is a right F -acyclic complex, and for every X in K + (A) and every n ∈ Z such that H i (X) = 0 for all i < n there exists a quasi-isomorphism X → CX , where CX is a complex with right F -acyclic components i and CX = 0 for all i < n. In particular, Condition (Ac) holds. Moreover, the following conditions for an object A of A are equivalent. (i) A is right F -acyclic. (ii) Ri F (A) = 0 for all i > 0. (iii) The morphism F (A) → RF (A) is an isomorphism. Combining Proposition F.204, Remark F.200 (3) and Proposition F.202 we obtain:

807 Corollary F.205. Suppose that F : A → B is left exact and that every object of A admits a monomorphism into a right F -acyclic object of A. Then F = R0 F and (Ri F )i≥0 is a universal δ-functor. Remark F.206. Suppose that in the situation of the corollary, (T i )≥0 is a δ-functor with T 0 = F and such that T i (C) = 0 for every right F -acyclic object C and all i > 0. Then the δ-functor (T i )≥0 is universal by Proposition F.202 and hence there is a unique ∼ isomorphisms of δ-functors (T i )i≥0 → (Ri F )i≥0 that is the identity in degree 0. A special case is if A has enough injective objects and T i (I) = 0 for every injective object I of A and all i > 0. The following proposition gives a recipe to simultaneously find acyclic objects and to check that Condition (Ac) holds. Proposition F.207. ([Lip2] O X (2.7.2) and its proof) Let F : A → B be an additive functor of abelian categories. Let I be a class of objects of A such that (a) For every object A of A there exists a monomorphism A → I with I in I. (b) For all I and J in I one has I ⊕ J in I. (c) If 0 → A → B → C → 0 is an exact sequence in A with A and B in I, then C ∈ I and the exact sequence 0 → F (A) → F (B) → F (C) → 0 is exact. Then the following assertions hold. (1) The full subcategory K + (I) of K + (A) of complexes whose components are all in I is a triangulated subcategory of K + (A) and all complexes in K + (I) are right F -acyclic. In particular, every object of I is right F -acyclic. (2) For every X in K + (A) and every n ∈ Z such that H i (X) = 0 for all i < n there i exists a quasi-isomorphism X → IX with IX ∈ K + (I) and IX = 0 for all i < n. In particular, Condition (Ac) holds, there exists a right derived functor R+ F of F , and dim− R+ F = 0 if F ̸= 0. Note that (a) implies that I is non-empty and that (c) with A = B ∈ I shows that 0 ∈ I. Remark F.206 above gives a criterion to identify the higher derived functors Ri F = i H ◦ R+ F . This is of course less precise than identifying the derived functor R+ F itself. For this we can use Proposition F.193 and obtain: Corollary F.208. Let F : A → B be a left exact functor of abelian categories. Let I be a class of right F -acyclic objects in A such that every object of A admits a monomorphism into an object of I (e.g., if A has enough injective objects we can choose as I the class of all injective objects of A). Consider F also as functor K + (A) → D(B). Let Q : K + (A) → D+ (A) be the localization functor. Let G : D+ (A) → D(B) be a triangulated functor such that G(D≥0 (A)) ⊆ D≥0 (B). Let ξ : F → G ◦ Q be a morphism of triangulated functors such that ξ(I) : F (I) → G(I) is an isomorphism for every object I of I. Then the morphism RF → G corresponding to ξ is an isomorphism. (F.49) Hypercohomology spectral sequences. The spectral sequence attached to a filtered complex, Section (F.24), can be generalized as follows. Let T be a triangulated category and let R0 : T → B be a cohomological functor with values in some Grothendieck abelian category B. For all n ∈ Z set Rn := R0 ◦ S n , where S is the functor X 7→ X[1]. Let

808

F Homological Algebra ip

· · · −→ X p+1 −→ X p −→ · · · ,

p∈Z

be a Zopp -diagram in T . We choose for all p ∈ Z a distinguished triangle ip

+1

X p+1 −→ X p −→ Xpp+1 −→ . As in Remark F.111, from the exact sequence · · · → Rp+q (X p+1 ) → Rp+q (X p ) → Rp+q (Xpp+1 ) → Rp+q+1 (X p+1 ) → · · · we obtain a spectral sequence (Er , dr )r≥1 with E1pq = Rp+q (Xpp+1 ). Now suppose that for all n ∈ Z one has Rn (X p ) = 0 for p ≫ 0 (depending on n) and that ip induces an isomorphism Rn (X p+1 ) → Rn (X p ) for p ≪ 0 (again depending on n). We set X := colimp∈Zopp X p , assuming that this colimit exists. Then the spectral sequence converges, (F.49.1)

E1pq = Rp+q (Xpp+1 ) ⇒ Rn (X).

The filtration of the limit term is given by F p Rn (X) = Im(Rn (X p ) → Rn (X)). In practice, one often has that ip is an isomorphism for p ≪ 0, and then the existence of X is automatic. Example F.209. Let T : A → B be an additive functor of Grothendieck abelian categories. Let X be a complex with coefficients in A which is endowed with a descending filtration (F p X)p such that for all n ∈ Z there exist p0 , p1 with F p X n = 0 for p ≥ p0 and F p X = X for p ≤ p1 . Applying the above construction to R0 := H 0 ◦ RT = R0 T we obtain a convergent spectral sequence (F.49.2)

E1pq = Rp+q T (grp (X)) ⇒ Rn T (X).

We consider two special cases that are sometimes called spectral sequences of hypercohomology. (1) First, we consider the “naive filtration” given by the subcomplexes σ ≥p X (F.14.3). In this case the spectral sequence takes the form (F.49.3)

E1pq = Rp+q T (X p [−p]) = Rq T (X p ) ⇒ Rn T (X).

(2) Consider the ascending filtration by the truncated complexes τ ≤n X (F.14.1). To obtain a descending filtration we set F p X := τ ≤−p X. Then grp X = H −p (X)[p] and we obtain a convergent spectral sequence E1pq = R2p+q T (H −p X) ⇒ Rp+q T (X). −q,p+2q we get the convergent spectral sequence Replacing Erpq by Er+1 (F.49.4)

E2pq = Rp T (H q (X)) ⇒ Rp+q T (X).

As both truncations σ ≥p and τ ≤−p are functorial, the spectral sequences (F.49.3) and (F.49.4) are both functorial in X. Another special case is obtained from the filtrations induced by a double complex (Section (F.25)).

809 Example F.210. Let T : A → B be an additive functor of Grothendieck abelian categories and let X be a double complex with components in A. Consider the two filtrations (F.25.1) yielding graded complexes dp,k−p

I

grp (Tot(X)) :

2 · · · −→ X p,k−p −−− −−→ X p,k+1−p −→ · · · ,

II

grp (Tot(X)) :

1 · · · −→ X k−p,p −−− −−→ X k+1−p,p −→ · · · ,

dk−p,p

where in both cases the entries in degree k and k + 1 are described. Now suppose that for all n there exist only finitely many non-zero X i,j with i + j = n. Then we obtain as special cases of (F.49.2) two convergent spectral sequences (F.49.5)

pq I E1 pq II E1

= Rp+q T (X p,•−p ) = Rq T (X p,• ) ⇒ Rn T (Tot(X)), = Rp+q T (X •−p,p ) = Rq T (X •,p ) ⇒ Rn T (Tot(X)).

As the differential in I E1pq (resp. (F.49.6)

pq II E2

pq II E1 )

= Rp+q T (X

pq I E2 •−p,p

is given by d2 (resp. d1 ) we have = Rq T (X p,• ) ⇒ Rn T (Tot(X)),

) = Rq T (X •,p ) ⇒ Rn T (Tot(X)).

(F.50) Grothendieck spectral sequence. We apply Proposition F.176 to the setting considered in the previous section. Let F : A → A′ , F ′ : A′ → A′′ be additive functors between abelian categories. Proposition F.211. Suppose that there exists a class I of objects in A containing 0 such that the following assumptions are satisfied. (a) Every object of I is right F -acyclic, and every object of A admits a monomorphism to an object of I. (b) Every object of A′ admits a monomorphism to a right F ′ -acyclic object of A′ . (c) The functor F maps objects in I to right F ′ -acyclic objects of A′ . Then the canonical morphism R+ (F ′ ◦ F ) → R+ F ′ ◦ R+ F of functors D+ (A) → D+ (A′′ ) is an isomorphism. If A and A′ have enough injective objects, then one can choose I as the class of all injective objects of A and (b) is satisfied for all functors F ′ . Proof. By Lemma F.194 and Proposition F.204, Assumption (a) implies that Condition (Ac) is satisfied for A and F and that we can choose the quasi-isomorphisms X → CX such that all components of CX are in I. By (c), F (CX ) is a bounded below complex of right F ′ -acyclic objects and therefore F (CX ) is right G′ -acyclic objects by Assumption (b) via Proposition F.204. Therefore we can apply Proposition F.176. Proposition F.212. (Grothendieck spectral sequence, [Sta] 015N) Suppose that there exists a full additive subcategory I of A such that the assumptions (a), (b), and (c) of Proposition F.211 are satisfied. Then for all X in D+ (A) there exists a converging spectral sequence E2p,q = Rp F ′ (Rq F (X)) ⇒ Rp+q (F ′ ◦ F )(X),

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which is functorial in X. Moreover, if n ∈ Z is such that H i (X) = 0 for all i < n, then E2p,q = 0 whenever q < n or p < 0. (F.51) Derived bi-functors. Let K1 , K2 , and E be triangulated categories with respective translation functors T1 , T2 , and T . We will define a triangulated bi-functor K1 × K2 → E to be a functor that is “triangulated in each variable”. Here one has to give some care to signs and compatibility with translations functors. Given a functor F : K1 × K2 → E, for each X in K1 and for each Y in K2 we have the partial functors (F.51.1)

FX : K2 → E, FY : K1 → E,

FX (B) = F (X, B), FY (A) = F (A, Y ).

Definition F.213. A triangulated bi-functor K1 × K2 → E is a triple (F, θ1 , θ2 ), where F : K1 × K2 → E is a functor and ∼

θ1 : F ◦ (T1 × idK2 ) −→ T ◦ F,

∼

θ2 : F ◦ (idK1 ×T2 ) −→ T ◦ F

are isomorphisms of functors such that (a) For each Y in K2 and for each X in K1 the pairs (FY , θ1 ) and (FX , θ2 ) are triangulated functors. (b) The composed functorial isomorphisms F (T1 × T2 ) = F (T1 × id)(id ×T2 )

F (T1 × T2 ) = F (id ×T2 )(T1 × id)

via θ1

/ T F (id ×T2 )

via θ2

/ TTF

via θ2

/ T F (T1 × id)

via θ1

/ TTF

are negatives of each other. We also have the obvious notion of a morphism of triangulated bi-functors. Example F.214. Let A be an additive category with a tensor product (see Section (F.19)). Suppose that A admits countable direct sums. Then forming the tensor complex yields a bi-functor ⊗ : C(A) × C(A) → C(A). It preserves homotopy in each variable and hence induces a bi-functor (F.51.2)

⊗ : K(A) × K(A) −→ K(A).

It preserves termwise split sequences in each component. For a fixed complex Y in K(A) ∼ we define θ1 : X[1] ⊗ Y → (X ⊗ Y )[1] to be the identity. For a fixed complex X in K(A) ∼ we define θ2 : X ⊗ Y [1] → (X ⊗ Y )[1] to be the multiplication by (−1)i if restricted to X i ⊗ Y j+1 . Then (− ⊗ −, θ1 , θ2 ) is a triangulated bi-functor. Now suppose that A1 and A2 are abelian categories and that Ki is a triangulated subcategory of K(Ai ), i = 1, 2. Given a triangulated bi-functor (F : K1 × K2 → E, θ1 , θ2 ) as above we consider the following two conditions. (ACI) For all X in K1 there exists a quasi-isomorphism X → IX such that

811 (a) the complex IX is right FY -acyclic for all Y in K2 and (b) the partial functor FIX : K2 → E sends quasi-isomorphisms to isomorphisms. (ACII) For all Y in K2 there exists a quasi-isomorphism Y → IY such that (a) the complex IY is right FX -acyclic for all X in K1 and (b) the partial functor FIY : K1 → E sends quasi-isomorphisms to isomorphisms. Assume that Condition (ACI) is satisfied. Then for fixed Y in K2 the right derived functor R(FY ) exists. Moreover for X in K1 fixed, the functor Y 7→ R(FY )(X) = FY (IX ) = FIX (Y ) sends quasi-isomorphisms to isomorphisms and hence factors through K2 → DK2 . Hence we obtain a triangulated bi-functor RI F : DK1 × DK2 −→ E,

RI F (X, Y ) = F (IX , Y ).

Similarly, if Condition (ACII) holds, one obtains a triangulated bi-functor RII F . If both of the above conditions hold, then RI F (X, Y ) = F (IX , Y ) ∼ = F (IX , IY ) = RII F (IX , Y ) ∼ = RII F (X, Y ), and thus RI F = RII F . Hence if at least one of the Conditions (ACI) or (ACII) hold, then we simply write RF := RI F or RF := RII F and call RF the right derived bi-functor of F. (F.52) The derived Hom functor and Ext Groups. Let A be an abelian category. We consider the bi-functor HomA (−, −) : K(A)opp × K(A) −→ K(AbGrp), where for X and Y in K(A) we denote by HomA (X, Y ) the Hom complex defined in (F.18). This is a triangulated bi-functor ([Lip2] O X (2.4.3)). Let K ⊆ K(A) be a triangulated subcategory with enough K-injective objects. In practice, we will usually have K = K(A). Then the restriction of HomA (−, −) to K(A)opp × K satisfies Condition (ACII) of Section (F.51) and hence we obtain by Section (F.51) the derived bi-functor (F.52.1)

R HomA : D(A)opp × DK → D(AbGrp)

which is a triangulated bi-functor. It is called the derived Hom functor . Moreover, the choice of K-injective resolutions Y → IY for all Y in K induces a morphism of triangulated bi-functors (F.52.2)

η : HomA (X, Y ) −→ HomA (X, IY ) = R HomA (X, Y ),

where as usual we omit the localization functors from the notation. If every X in K(A) has a K-projective resolution PX → X (i.e., a K-injective resolution X opp → (PX )opp in K(A)opp ), then we can also calculate R HomA (X, Y ) = HomA (PX , Y ). This is for instance the case if A is the category of modules over a ring by Theorem F.189. Definition F.215. For i ∈ Z and X and Y in D(A) the i-th extension group (or: Ext group) of X by Y is the group ExtiA (X, Y ) := HomD(A) (X, Y [i]) = HomD(A) (X[−i], Y ).

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F Homological Algebra

In particular for all objects A and B in A the group ExtiA (A, B) is defined by considering A and B as objects in D(A) as usual. If A is an R-linear category (Definition F.28) for some ring R, then the Ext groups are R-modules. Remark F.216. Let X and Y be complexes in D(A). (1) Let D′ be a triangulated subcategory of D(A), such as D′ = D∗ (A) for ∗ ∈ {b, +, −} ∗ ′ or, more generally, D′ = DA ′ (A) for some plump subcategory A (Definition F.161). i ′ If X and Y are both in D , then ExtA (X, Y ) = HomD′ (X, Y [i]). (2) As HomD(A) (·, ·) is a cohomological functor in each variable (Example F.137), any two distinguished triangles X ′ → X → X ′′ → X[1] and Y ′ → Y → Y ′′ → Y [1] in D(A) yield long exact sequences of abelian groups (F.52.3) i i i ′′ ′ ′′ . . . −→ Exti−1 A (X, Y ) −→ ExtA (X, Y ) −→ ExtA (X, Y ) −→ ExtA (X, Y ) −→ . . . , i i i ′ ′′ ′ . . . −→ Exti−1 A (X , Y ) −→ ExtA (X , Y ) −→ ExtA (X, Y ) −→ ExtA (X , Y ) −→ . . .

(3) Every quasi-isomorphism Y → IY to some K-injective complex I by Proposition F.179 (7) for all i yields an isomorphism (F.52.4)

∼

ExtiA (X, Y ) → HomK(A) (X, IY [i]).

Similarly for a quasi-isomorphism PX → X with PX a K-projective complex. The relation between Ext groups and RHom is as follows. Remark F.217. Suppose that K has enough K-injective objects (e.g., if K = K(A) for a Grothendieck abelian category A, which will be the only case that is used by us). For all i ∈ Z there are isomorphisms, functorial in X ∈ D(A)opp and Y ∈ DK , (F.52.5)

∼

H i (R HomA (X, Y )) → ExtiA (X, Y )

and in particular (F.52.6)

∼

H 0 (R HomA (X, Y )) → HomD(A) (X, Y ).

Indeed, for Y in K we can choose a quasi-isomorphism Y → IY with IY a K-injective complex. Then (F.52.5) is given as the composition ∼

H i (R HomA (X, Y )) −→ H 0 (R HomA (X, Y [i])) ∼

−→ H 0 HomA (X, IY [i]) ∼

−→ HomK(A) (X, IY [i]) ∼

−→ ExtiA (X, Y ), where the third isomorphism is (F.18.2) and the last isomorphism is (F.52.4). Remark F.218. Let b, c ∈ Z. Suppose X ∈ D(−∞,b] (A) and Y ∈ D[c,∞) (A). Then we can represent X (resp. Y ) by a complex concentrated in degrees ≤ b (resp. in degrees ≥ c) (Proposition F.154). For these representatives the complex HomA (X, Y ) is concentrated in degrees [c − b, ∞). Now suppose that K(A) has enough K-injective objects. Then we can choose a quasiisomorphism Y → IY with IY a K-injective complex concentrated in degrees [c, ∞). Therefore we find R HomA (X, Y ) ∈ D[c−b,∞) (A), i.e.

813 ExtiA (X, Y ) = 0

for all i < c − b.

Remark F.219. For objects A and B in the abelian category A we can describe ExtiA (B, A) also as follows. An extension of B by A of degree i ≥ 1 is an exact sequence in A of the form (F.52.7)

0 −→ A −→ Z −i+1 −→ Z −i+2 −→ . . . −→ Z 0 → B −→ 0.

E:

An extension of B by A of degree 1 is just a short exact sequence 0 → A → Z 0 → B → 0. Such an extension is called split, if this short exact sequence splits. Under the bijection constructed below, the split extension corresponds to the zero element of the group Ext1A (B, A). ˜ of B by A of the same degree i is a ˜ of extensions E and E A morphism E → E commutative diagram E:

0

/A

/ Z −i+1

/ Z −i+2

/ ...

/ Z0

/B

/0

˜: E

0

/A

 / Z˜ −i+1

 / Z˜ −i+2

/ ...

 / Z˜ 0

/B

/ 0.

Two extensions E and E ′ of B by A of the same degree i are called equivalent if there ˜ of B by A of degree i and morphisms E ˜ → E and E ˜ → E ′ . It will exists an extension E follow from Proposition F.220 below that this is indeed an equivalence relation. For an extension E as in (F.52.7) let σE be the complex . . . → 0 → A → Z −i+1 → . . . → Z 0 → 0 → . . . with Z j in degree j. We define δ(E) ∈ ExtiA (B, A) as the morphism −f s−1 : B → A[i] in D(A), where s is the quasi-isomorphism s : σE −→ B = B[0] and f : σE → A[i] the projection. In other words, δ(E) is the morphism in the distinguished triangle δ(E)

A[i − 1] −→ Z −→ B[0] −→ A[i − 1][1] = A[i] associated to the exact sequence of complexes 0 → A[i − 1] → Z → B[0] → 0 (F.37.3). Proposition F.220. ([Ver3] O III.3) Two extensions E and E ′ are equivalent if and only if δ(E) = δ(E ′ ) and one obtains a bijective map E 7→ δ(E) between the set of equivalence classes of extensions of B by A of degree i and the set ExtiA (B, A). (F.53) Injective dimension. Let A be a Grothendieck abelian category. Definition F.221. Let a ≤ b be integers and set d := b − a. An object X in D(A) is said to have injective amplitude in [a, b] if the functor R HomA (−, X) maps A into [a,b] D(AbGrp) , i.e., ExtiA (F, X) = 0 for all F ∈ A and i ̸∈ [a, b]. In this case, one also says that X has injective dimension ≤ d. We say that X has finite injective dimension if it is of injective dimension ≤ d for some d. Proposition F.222. For an object X in D(A) and integers a ≤ b, the following are equivalent:

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(i) There exists a complex I • of injective objects I i of A with I i = 0 for i ∈ / [a, b] such that X and I • are isomorphic in D(A). (ii) The complex X has injective amplitude in [a, b]. Proof. (i) ⇒ (ii). We may assume that X is a complex of injective objects concentrated in degrees [a, b]. Then X is K-injective (Remark F.181) and hence R Hom(F, X) = Hom(F, X) which shows H i (R Hom(F, X)) = 0 for i ∈ / [a, b]. (ii) ⇒ (i). We first show the following claim. Fix i ∈ Z. If H i (R Hom(F, X)) = 0 for every F ∈ A, then H i (X) = 0. To prove the claim we may replace X by a K-injective complex. Then R Hom(F, X) is represented by the complex HomA (F, X) and hence H i (HomA (F, X)) = 0. We obtain a commutative diagram B i (HomA (F, X))  HomA (F, B i (X))

∼

/ Z i (HomA (F, X)) ∼ =

 / HomA (F, Z i (X)),

where the right vertical arrow is an isomorphism since Hom is left exact. If H i (X) ̸= 0, then B i (X) → Z i (X) is a strict inclusion. Hence we find an object F in A (e.g., F = Z i (X)) such that HomA (F, B i (X)) → HomA (F, Z i (X)) is not surjective. This gives a contradiction. The claim above implies that X ∈ D[a,b] (A). Therefore we find a quasi-isomorphism X → J for a complex I with J i injective for all i ∈ Z and J i = 0 for i < a. Moreover I := τ ≤b J → J is a quasi-isomorphism. Then I is concentrated in degrees [a, b] and it is isomorphic to X in D(A). Moreover I i = J i is injective for all i = a, . . . , b − 1. It remains to show that I b is injective. Set K := σ ≤b−1 I which is a complex of injective modules concentrated in degrees +1 [a, b − 1]. Then we obtain a distinguished triangle I b [−b] −→ I −→ K −→ in D(A) and hence for every F in A an exact sequence ExtbA (F, K) −→ Ext1A (F, I b ) −→ Extb+1 A (F, I). By assumption, the term on the right vanishes and by the implication “(i) ⇒ (ii)” the term on the left vanishes. Therefore Ext1A (F, I b ) = 0 for every F in A. But this shows that HomA (−, I b ) : A → (AbGrp) is exact and hence that I b is injective in A. Example F.223. Let A be a ring and let A be the category of A-modules. Suppose that X is a complex in D(A) such that ExtiA (A/a, X) = 0 for all ideals a ⊆ A and all i ∈ / [a, b]. Then X is of injective amplitude in [a, b]. Indeed, one has ExtiA (A, X) = H i (X) and hence X ∈ D[a,b] (A). Then the proof of “(ii) ⇒ (i)” in Proposition F.222 shows that X is isomorphic in D(A) to a complex of injective A-modules concentrated in degrees [a, b], using Proposition G.21. (F.54) Derived limits and homotopy limits. Let A be a Grothendieck abelian category and let I be a small category. As A is complete (Remark F.58), we obtain a left exact functor limI : AI → A. Moreover AI is again a Grothendieck category (Remark F.59). Hence the derived functor R lim := R lim : D(AI ) → D(A) I

815 exists (Corollary F.187). As usual we set Rp lim(X) := H p (R lim(X)) for X ∈ D(AI ). Sometimes one writes limp (X) instead of Rp lim(X) but we will not use this notation. Mostly we will be interested in the special case, where I is the category given by the partially ordered set Nopp , i.e., elements of AI are diagrams in A of the form X0 ← X1 ← X2 ← . . . . Lemma F.224. ([Nee4] O A.3.6) Suppose that in A countable products are exact. Let u2 u1 · · · → E2 −→ E1 −→ E0 be an Nopp -diagram in A and define Y Y δ: En → En , (en )n 7→ (en − un+1 (en+1 ))n . n

n δ

Then the complex n E n −→ In particular one has Q

Q

n

En (concentrated in degree 0 and 1) represents R lim En .

lim En = Ker(δ), n

(F.54.1)

R1 lim En = Coker(δ), n

p

for p ̸= 0, 1.

R lim En = 0 n

Definition F.225. Let A be an abelian category. A diagram X : Nopp → A is said to satisfy the Mittag-Leffler condition or is ML if for every n ∈ N there exists c = c(n) ≥ n such that Im(Xk → Xn ) = Im(Xc → Xn ) for all k ≥ c. Clearly any Nopp -diagram X in which all transition maps Xn → Xn−1 are epimorphisms satisfies the Mittag-Leffler condition. Remark F.226. Let A be an abelian category. Then for every diagram X : Nopp → A there exists a monomorphism X → M into an Nopp -diagram M that satisfies the MittagLeffler condition. Indeed, for n ∈ N set Mn := Xn ⊕ Xn−1 ⊕ · · · ⊕ X0 with transition maps given by the projections and define Xn → Mn by (idXn , un , . . . , u1 ◦ · · · ◦ un ), where un : Xn → Xn−1 are the transition maps. For modules over a ring the Mittag-Leffler condition ensures that the given diagram is right-lim-acyclic. This follows via Proposition F.207 from the following results. Proposition F.227. ([Wei2] O 3.5) Let R be a ring (not necessarily commutative). Let 0 −→ (En )n −→ (Fn )n −→ (Gn )n −→ 0 be an exact sequence of Nopp -diagrams of R-modules. (1) If (Fn ) is ML, then (Gn ) is ML. (2) If (En ) is ML, then 0 −→ lim En −→ lim Fn −→ lim Gn −→ 0 n

n

is exact. (3) If (En ) is ML, then Rp lim En = 0 for all p ̸= 0.

n

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Analogous statements do not hold in arbitrary complete abelian categories, even if arbitrary products are exact. Neeman constructed in [Nee4] O A.5 an abelian category with exact products and a ML system (En )n in it such that R1 lim En = ̸ 0. Roos gives in [Roo] O 1.18 for all integers m ≥ 1 an example of a quasi-affine noetherian regular scheme U (the complement of the special point in the spectrum of a regular local noetherian ring) and a projective system of coherent OU -modules Fn with surjective transition maps (in particular (Fn )n is a ML system) such that Rp limn Fn ̸= 0 if and only if p ̸= 0, m. For inverse systems in triangulated categories there is the notion of a homotopy limit which is closely related to the derived limit if the triangulated category is the derived category of a Grothendieck abelian category. Definition F.228. Let (Kn , fn : Kn → Kn−1 )n≥0 be an inverse system in a triangulated opp category D, i.e., it is an object of DN . We call anQ object K ∈ D a homotopy limit of this system, and write K = holim Kn , if the product n K n exists in D, and there exists an exact triangle Y Y +1 K −→ Kn −→ Kn −→ n∈N

where the morphism (xn − fn+1 (xn+1 ))n ”.

Q

Kn →

Q

n∈N

Kn is given by “identity minus shift”, i.e., “(xn )n 7→

Note that the homotopy limit, if it exists, is determined uniquely up to isomorphism, but not up to unique isomorphism. In particular, holim is not a functor. Nevertheless it has a weak functoriality property in the following sense. Remark F.229. Let α : (Kn )n −→ (Ln )n be a map of inverse systems in a triangulated category D, and assume that holim Kn and holim Ln exist. By a rotated version of (TR3) there exists a morphism η : holim Kn → holim Ln yielding a morphism of distinguished triangles +1 / (holim Kn )[1] / Q Kn / Q Kn holim Kn η

Q

 holim Ln

αn

Q / Ln

Q

αn

Q / Ln

η[1]

 / (holim Ln )[1].

+1

Moreover, if α is an isomorphism, then η is an isomorphism by the five lemma, Lemma F.119. If the triangulated category D is of the form D(A) for a Grothendieck abelian category A, then arbitrary products and hence homotopy limits exist in D(A) (Lemma F.188). Lemma F.230. ([Sta] 08U1) Let F : A → B be an additive functor from a Grothendieck abelian category A to an abelian category B. By F.187, there exists a derived functor RF of F . Assume that RF commutes with countable products of objects of D(A). Then for every inverse system (En )n≥0 , En ∈ A we have RF (holim En ) ∼ = holim(RF En ) in D(B). Note that the hypotheses on RF is satisfied in the following two cases. (1) The abelian category B has exact countable products and F commutes with countable products (see the proof of [Sta] 08U1).

817 (2) The functor F has a left adjoint functor G, and G has a left derived functor LG. Then RF is right adjoint to LG (Proposition F.191) and in particular commutes with arbitrary products. The next proposition relates the homotopy limit construction to the derived functors R lim. We continue to denote by A a Grothendieck category. Then the category of inverse opp opp systems AN is also a Grothendieck category (Remark F.59). Every complex in C(AN ) can be considered as an inverse system (Xn )n of complexes in C(A) and a morphism (Xn )n → (Yn )n between such complexes is a quasi-isomorphism if and only if Xn → Yn is a quasi-isomorphism for all n. Hence we obtain a functor D(AN

(F.54.2)

opp

) → D(A)N

opp

which however usually is not faithful. To each object of the left hand side we can attach its derived limit in D(A) and to each object of the right hand side we can attach a homotopy limit. opp

Proposition F.231. Let X = (Xn )n be an object in D(AN ) viewed as an inverse opp system of complexes Xn . Then R lim X is a homotopy limit of (Xn )n in D(A)N . Proof. We start with three preliminary remarks. opp (I). The functor em : AN → A, (Yn )n 7→ Ym has an exact left adjoint, namely the functor im that sends Z in A to the inverse system id

id

id

· · · −→ 0 −→ 0 −→ Z −→ Z −→ · · · −→ Z, where the first entry of Z sits in degree m. In particular, em preserves injective objects opp and the functor K(AN ) → K(A) induced by em preserves K-injective complexes (Corollary F.183). opp (II). Let J = (Jn )n be an inverse system in AN that is injective as object of the opp abelian category AN . We claim that Jn → Jn−1 is a split epimorphism for all n. Indeed, for an object Z in A consider the monomorphism of inverse systems in−1 (Z) → in (Z). As J is injective we obtain a surjective map Hom(Z, Jn ) = Hom(in (Z), J) −→ Hom(in−1 (Z), J) = Hom(Z, Jn−1 ). In particular, setting Z = Jn−1 we find a preimage of idJn−1 , which is a right inverse of Jn → Jn−1 . (III). Let (Jn )n be an inverse system in A such that Jn → Jn−1 is a split epimorphism for all n. Then Y Y 0 → lim Jn −→ Jn −→ Jn −→ 0 n

n

n

is a split exact sequence. Indeed, by the Yoneda lemma we may assume that A is the category of abelian groups. Then we can use Proposition F.224. We now come to the proof of Proposition F.231. To calculate R lim X we choose a opp quasi-isomorphism X → I, where I = (In )n in K(AN ) is K-injective and all terms are Nopp injective objects in A (Theorem F.185). Then R lim X = lim In , where the limit is taken termwise. By Step (I), all complexes In are K-injective and consist of injective objects as well. Moreover, we have a termwise split exact sequence of complexes Y Y 0 −→ lim In −→ In −→ In −→ 0

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by Steps (II) and (III). By Lemma F.188, the products in the exact sequence represent the product in the derived category because In is K-injective. Therefore R lim X is a homotopy limit by Definition F.228. opp

opp

Remark F.232. Let A be an abelian category. The functor D(AN ) → D(A)N (F.54.2) is essentially surjective (use [Sta] 091I whose proof verbatim carries over to this ˜ n )n situation). Hence for every inverse system (Xn )n in D(A) we can choose an object (X Nopp in D(A ) that gives rise to (Xn )n , which however in general will not be unique up to isomorphism. Now suppose that A is a Grothendieck category. Then Proposition F.231 shows that ˜ n is a homotopy limit of (Xn )n . Hence the isomorphism class of holim Xn does R lim X ˜ n )n . not depend on the isomorphism class of (X In the category of modules over a ring many aspects of derived and homotopy limits become easier, for instance because arbitrary products are exact and Proposition F.227 holds. Another instance is the following result. Lemma F.233. ([Sta] 07KZ) Let A be a ring and let (En )n≥0 be an inverse system of objects of D(A-Mod). For every m ∈ Z, we have a short exact sequence of A-modules 0 → R1 lim H m−1 (En ) → H m (holim En ) → lim H m (En ) → 0.

(F.55) Homotopy colimits. Definition F.234. Let D be a triangulated category and let f0

f1

K0 −→ K1 −→ K2 −→ . . . be an N-diagram in D. One says L that an object K in D is a homotopy colimit of the system (Kn )n if the direct sum n Kn exists and there is a distinguished triangle M M ε +1 Kn −→ K −→, Kn −→ where ε is the map given by idKn −fn . In this case we write K = hocolim Kn . If the homotopy colimit exists, then it is unique up to non-unique isomorphism. In particular hocolim is not a functor. As any map in DLcan be completed to a distinguished triangle, the homotopy colimit hocolim Kn exists if Kn exists. This is for instance the case in the derived category of any Grothendieck abelian category (Lemma F.188). If A is an abelian category that has exact countable direct sums (e.g., if A is a Grothendieck abelian category), then coproducts in D(A) exist and can be computed by taking termwise direct sums of any representing complexes. This implies the following lemma, see [Sta] 093W. Lemma F.235. Let A be an abelian category that has exact countable direct sums. Let (Xn )n be an N-diagram of complexes in C(A). Then the termwise colimit colim Xn is a homotopy colimit in D(A).

819

Exercises Exercise F.1. For a group G let BG be the category with a single object ∗ and with EndBG (∗) = G, where the composition is given by the multiplication in G. Let G be a groupoid, let x be an object in G and set G := AutG (x). Show that the inclusion BG → G is an equivalence if and only if G is connected. Exercise F.2. Let A be a ring and let S ⊆ A be a multiplicative subset. Consider S as the set of objects in a category, again denoted by S, where for s, t ∈ S one sets HomS (s, t) := { u ∈ S ; us = t }. Composition in S is given by multiplication. (1) Show that S is a filtered category. (2) Let M be an A-module. Define a functor M : S → (A-Mod) as follows. On objects, for s ∈ S set M (s) = M , and for a morphism u : s → t let M (u) : M → M be the ∼ multiplication by u. Show that one has a functorial isomorphism colimS M → S −1 M . Exercise F.3. Let C be an abelian category. Let u : X → Y be an epimorphism in C. Show that the following conditions are equivalent. (i) For any morphism g : Z → X one has that u ◦ g is an epimorphism if and only if g is an epimorphism. (ii) For all subobjects U of X with U + Ker(u) = X one has U = X. If these conditions are satisfied, u is called an essential epimorphism. Show the following assertions. (1) Let u and v be composable epimorphisms. Then v ◦ u is essential if and only if u and v are essential. Lr (2) If ui : Xi → Yi for i = 1, . . . , r are essential epimorphisms, then i=1 ui is an essential epimorphism. Exercise F.4. Let C be a category and let X be an object of C. An object Y of C i r is called retract of X if there exist morphisms Y −→ X −→ Y such that r ◦ i = idY . An endomorphism u of an object of C is called idempotent if u2 = u. An idempotent i r endomorphism u of an object X is called split if there exists a retract Y −→ X −→ Y of X such that u = i ◦ r. If every idempotent in C admits a splitting, then C is said to idempotent complete. (1) Show that if the splitting of an idempotent u of an object X is given by a retract i r Y −→ X −→ Y , then there is a unique commutative diagram whose vertical maps are isomorphisms Eq(u, idX ) O

/X

/ Coeq(u, idX )

/X

 / Y.

∼ =

∼ =

Y

i

r

Therefore a splitting of u is unique up to unique isomorphism it it exists. (2) Show that every small category C has an idempotent completion, i.e., there exists a fully faithful functor i : C → C ′ , where C ′ is a small idempotent complete category such that every object in C ′ is the retract of an object of C under i. Hint: Consider the Yoneda embedding C → (PSh(C)), where (PSh(C)) denotes the category of functors C opp → (Sets), and define C ′ as the category of retracts in (PSh(C)) of the representable functors.

820

F Homological Algebra

(3) Show that an idempotent completion i : C → C ′ has the following universal property and hence we can speak of the idempotent completion of C. If D is a small idempotent complete category, then composition with i induces an equivalence of the category of functors C ′ → D with the category of functors C → D. (4) Show that the idempotent completion of an additive category is an additive category. Remark : If C is the category whose objects are the open sets of Rn for some n ≥ 0 and whose morphisms are infinitely often differentiable maps between such open subsets, then its idempotent completion is the category of (Hausdorff, second countable) smooth manifolds, see https://ncatlab.org/nlab/show/Karoubi+envelope. Exercise F.5. Let C be an additive category. Let X be an object of C. (1) Show that if the ring HomC (X, X) is local, then X is indecomposable. Here we call a not necessarily commutative ring local if the sum of any two non-units is again a non-unit. (2) Let X = X1 ⊕ · · · ⊕ Xn = Y1 ⊕ · · · ⊕ Ym be two decompositions into objects with local endomorphism ring. Show that m = n and that there exists a permutation σ such that Xi ∼ = Yσ(i) for all i = 1, . . . , n. (3) The converse in (1) does not hold in general: Let C be the category of abelian groups. Show that Z is indecomposable in C but that the endomorphism ring of Z is not local. Exercise F.6. An additive category is called Krull-Schmidt category if every object decomposes into a finite direct sum of objects having local endomorphism rings (see Exercise F.5). Show that every Krull-Schmidt category is idempotent complete (Exercise F.4). Exercise F.7. Let C be an essentially small abelian category. A bi-chain in C is a sequence of triples (Xn , in , pn )n≥0 , where Xn is an object in C, in : Xn → Xn−1 is a monomorphism, and pn : Xn−1 → Xn is an epimorphism. Such a bi-chain is said to terminate if there exists an integer N ≥ 1 such that in and pn are isomorphisms for all n ≥ N . If every bi-chain in C terminates, one says that C satisfies the bi-chain condition. (1) Show that an abelian category in which every object is of finite length satisfies the bi-chain condition. (2) Show that a bi-chain (Xn , in , pn )n≥0 in C terminates if and only if the descending chain End(X0 ) ⊇ · · · ⊇ End(Xn−1 ) ⊇ End(Xn ) ⊇ · · · becomes stationary. Here End(Xn−1 ) → End(Xn ) is the injective map u 7→ pn ◦ u ◦ in . Remark : See Exercise 23.3 for an example of an abelian category that satisfies the bi-chain condition. Exercise F.8. Let C be an abelian category that satisfies the bi-chain condition (Exercise F.7). (1) Show that C is a Krull-Schmidt category (Exercise F.6). (2) Show that an object of C is indecomposable if and only if its endomorphism ring is local and deduce that every object in C admits a decomposition into a finite direct sum of indecomposable objects having local endomorphism rings. (3) Let u be an endomorphism of an object of X. Show that one has a Fitting decomposition with respect to u, i.e., that for sufficiently large integers r one has X∼ = Im(ur ) ⊕ Ker(ur ). Exercise F.9. Let A be an abelian category and fix n ∈ Z. (1) Show that the functor A → C(A), M 7→ M [−n], is left adjoint to X 7→ Ker(dn : X n → X n+1 ) and right adjoint to X 7→ Coker(dn−1 ).

821 (2) Let Dn : A → C(A) be the functor that sends M to the complex id

· · · −→ 0 −→ M −−−M −→ M −→ 0 −→ . . . with M sitting in degrees n and n + 1. Show that Dn is left adjoint to the functor X 7→ X n and right adjoint to X 7→ X n+1 . Exercise F.10. Let I be a small category and for a category A consider the functor category AI . Show that if A is additive (resp. abelian), then AI is additive (resp. abelian). Remark : Note that if I is the set Z, considered as a discrete category, then AZ is the category of Z-graded objects in A with morphisms of degree 0. Exercise F.11. Let k be a field. Show that the category of filtered k-vector spaces is not abelian. Exercise F.12. Let A be an abelian category and let B ⊆ A be a full abelian subcategory. Suppose that for every epimorphism X → Y in A with Y in B there exists a morphism Y ′ → X with Y ′ in B such that the composition Y ′ → X → Y is an epimorphism. Show that B is a plump subcategory of A. Exercise F.13. Let F : A → C be an exact functor of abelian categories. Then the full subcategory B of objects X ∈ A with F (X) = 0 is called the kernel of F . Show that B is a Serre subcategory of A. Exercise F.14. Let A be an abelian category that contains a small subcategory A0 such that the inclusion A0 → A is an equivalence of categories. Let B ⊆ A be a Serre subcategory. Show that there exists an abelian category A/B and an exact functor π : A → A/B characterized by the following universal property. For any exact functor F : A → C of abelian categories such that B is contained in the kernel of F (Exercise F.13) there exists a factorization F = F¯ ◦ π for a unique exact functor F¯ : A/B → C. Moreover, π is essentially surjective and its kernel is B. Hint: Show that the arrows u in A such that Ker(u), Coker(u) ∈ B form a saturated multiplicative system S and set A/B = S −1 A. Exercise F.15. Let A be an additive category. A short exact sequence in A is a sequence of the form p i 0 −→ X ′ −→ X −→ X ′′ −→ 0 such that i is a kernel of p and p is a cokernel of i. An exact structure on A is a class E of short exact sequences in A, called admissible, that is closed under isomorphisms and that satisfies the axioms (A)0 - (C)opp below. Here we call a morphism i in A an admissible i monic if there exists an admissible exact sequence 0 → X ′ −→ X → X ′′ −→ 0. Dually, one defines the notion of an admissible epic. (A)0 The identity of the zero object is an admissible epic. (B) The composition of two admissible epics is an admissible epic. (C) The pullback of an admissible epic along an arbitrary morphism exists and yields an admissible epic. (C)opp The pushout of an admissible monic along an arbitrary morphism exists and yields an admissible monic.

822

F Homological Algebra

An exact category is a pair (A, E ) consisting of an additive category and an exact structure. (1) Show that (A)0 , (C), and (C)opp imply the following property. (A) For all objects X in A the identity idX is an admissible monic and an admissible epic. (2) Show that in an exact category also the following property holds. (B)opp The composition of two admissible monics is an admissible monic. Deduce that E is an exact structure for A if and only if E opp is an exact structure for Aopp . (3) Show that all isomorphisms in A are admissible epics and admissible monics. (4) Show that every split exact sequence 0 → X → X ⊕ Y → Y → 0 is admissible. Remark : For (much) more on exact categories see [B¨ uh] O X. Exercise F.16. Let B be an abelian category and let A be a full subcategory stable under extensions in B. Show that A together with the class of sequences 0 → X ′ → X → X ′′ → 0 that are exact in B is an exact category (Exercise F.15). Remark : Conversely, one can show that if (A, E ) is an exact category such that A is equivalent to a small category, then there exists a fully faithful functor y : A → B to an abelian category B such that the essential image of y is stable under extensions and such that a short sequence in A is exact if and only if its image under y is a short exact sequence in B ([B¨ uh] O X App. A). Exercise F.17. Let A be a Grothendieck abelian category and let 0 → X ′ → X → X ′′ → 0 be a short exact sequence in A. (1) Show that if X is finitely generated, then X ′′ is finitely generated. Show that if X ′ and X ′′ are finitely generated, then X is finitely generated. (See Definition F.66.) (2) Show that X is noetherian if and only if X ′ and X ′′ are noetherian. Exercise F.18. Let A be a ring, let E • , F • be complexes of A-modules and suppose that E is bounded above and that E n and H n (E) are flat A-modules for all n ∈ Z. Show that for all n ∈ Z there is a functorial isomorphism M ∼ H i (E • ) ⊗A H j (F • ) −→ H n (E • ⊗A F • ). i+j=n

Exercise F.19. Let T be a triangulated category and let U be a non-empty full additive subcategory of T such that if E → F → G → is a distinguished triangle in T and two of its terms are in U , then the third is in U . Show that U is stable under shifts and that if E is an object in U , then every object of T isomorphic to E is also in U . Deduce that U inherits from T the structure of a triangulated category. Exercise F.20. Let A be a Grothendieck abelian category. (1) Show that a complex I in C(A) is K-injective if and only if for every exact complex E the Hom complex HomA (E, I) is exact. (2) Show that every K-injective complex is isomorphic in K(A) to a K-injective complex K whose components K n are injective objects of A for all n ∈ Z. (3) Show that a complex I in C(A) is K-injective if and only if there exist complexes C1 , C2 in C(A) that are isomorphic to 0 in K(A) and a K-injective complex with injective components K such that I ⊕ C1 ∼ = K ⊕ C2 in C(A). Remark : See [Gil] O X §6, and the Introduction.

823 Exercise F.21. Let A be a Grothendieck category. Let Q : K(A) → D(A) be the canonical functor. Show that there exists a fully faithful right adjoint functor I : D(A) → K(A) whose essential image consists of the full subcategory of K-injective complexes. Show that for each complex X in K(A) the unit X → I(Q(X)) of the adjunction is a K-injective resolution of X. Exercise F.22. Let A be aQnot necessarily commutative ring and let (Ei )i be a family of left A-modules. Show that i Ei is injective if and only if Ei is an injective A-module for all i. Exercise F.23. Let A be a not necessarily commutative ring and let M be a left Amodule. Show that M is an injective A-module if and only if for all left ideals a ⊆ A every A-linear map u : a → A can be extended to an A-linear map A → E. Remark : See also Section (G.8). Exercise F.24. Let A be an integral domain. (1) Show that every injective A-module is divisible (i.e., for all 0 ̸= a ∈ A scalar multiplication by a on E is surjective). (2) Show that every torsion free and divisible module is injective. Deduce that the field of fractions of A is an injective A-module. (3) Suppose that A is a principal ideal domain. Show that an A-module is injective if and only if it is divisible. Hint: Exercise F.23 Exercise F.25. Let A be a not necessarily commutative ring. (1) Show that the left A-module E := HomZ (A, Q/Z) is injective and that for every left A-module M and for all 0 ̸= x ∈ M there exists an A-linear map u : M → E such that u(x) ̸= 0. (2) Let M be a left A-module. Show that I(M ) := E HomA (M,E) is an injective A-module and that the map eM : M → I(M ) that sends m ∈ M to (u(m))u∈HomA (M,E) is A-linear, injective and functorial in M . (3) Show that eM is an injective hull of M . Hint: Exercise F.24 Exercise F.26. Let A be a not necessarily commutative ring. Show that the following assertions are equivalent. (i) A is left noetherian, i.e., every left ideal of A is finitely generated. (ii) Every colimit of a small filtered diagram of injective left A-modules is again injective. L (iii) For all countable families (En )n∈N of injective left A-modules the direct sum n En is an injective A-module. Hint: To show that (i) implies (ii) use Exercise F.23, that every ideal a of A is of finite presentation, and that hence HomA (a, −) commutes with filtered colimits. To show S that (iii) implies (i) let (an )n be an ascending chain of ideals, let a := n an , and let a/an → InL be an injective map of A-modules L into an injective A-module. The induced map a → n In can be extended to u : A → n In . Consider u(1). (Here by ideal we mean left ideal, and similarly for modules.)

824

F Homological Algebra

Exercise F.27. Let A be a noetherian ring and let S ⊆ A be a multiplicative subset. Let E be an injective A-module. Show that S −1 E is an injective A-module and an injective S −1 A-module. Hint: Exercise F.26 and Exercise F.2. Exercise F.28. Let A be an abelian category and let p ≥ 0. Suppose that ExtpA (X, Y ) = 0 for all objects X, Y ∈ A. Show that ExtiA (X, Y ) = 0 for all i ≥ p and all objects X, Y ∈ A. Exercise F.29. Let A be an abelian category. Show that the following assertions are equivalent. (i) Every short exact sequence in A splits. (ii) For all X, Y ∈ A one has Ext1A (X, Y ) = 0. (iii) For all X, Y ∈ A and for all k ≥ 1 one has ExtkA (X, Y ) = 0. An abelian category satisfying these equivalent condition is called semisimple. Hint: Exercise F.28. Exercise F.30. Let A be an abelian category. Show that for an object X in Db (A) the following assertions are equivalent. (i) X is isomorphic in Db (A) to a complex whose differentials are 0. (ii) There morphisms up : H p (X)[−p] → X, p ∈ Z, such that the resulting morphism L exist p u: H (X)[−p] → X satisfies H p (u) = idH p (X) for all p ∈ Z. p L p b (iii) There exists an isomorphism u : p H (X)[−p] → X in D (A). If X satisfies these equivalent conditions, X is called decomposable. Exercise F.31. Let A be an abelian category. (1) Let X be an object in Db (A) such that Extp (H i (X), H j (X)) = 0 for all p ≥ 2 and for all i > j. Show that X is decomposable (Exercise F.30). (2) Suppose that Ext2A (X, Y ) = 0 for all objects X, Y ∈ A. Show that every complex in Db (A) is decomposable. Hint: Exercise F.28 (3) Suppose that Ext2A (X, Y ) = 0 for all objects X, Y ∈ A and that A has exact coproducts. Show that for any complex X in C(A) there exists an isomorphism L ∼ i i∈Z H (X)[−i] → X in D(A). Exercise F.32. Let A be a Dedekind domain. (1) Show that every A-module has injective dimension ≤ 1. Hint: Use Proposition G.21 and that every ideal of A is a projective A-module. (2) Show that every complex in Db (A) is decomposable (Exercise F.30). Hint: Exercise F.31. Exercise F.33. Let A be a ring, M an A-module and let d ≥ 0 be an integer. Show that M has injective dimension ≤ d if and only if Extd+1 A (A/a, M ) = 0 for every ideal a of A. Hint: Proposition G.21. Exercise F.34. Let A be a noetherian ring and let M be an A-module. (1) Show that the injective dimension of the A-module M is the supremum of the injective dimensions of the Ap -modules Mp where p runs through all prime ideals (equivalently, all maximal ideals) of A. Hint: Exercise F.33. (2) Let S ⊆ A be a multiplicative subset. Show that the injective dimension of the S −1 A-module S −1 M is at most the injective dimension of the A-module M . Deduce that if M is an injective A-module, then S −1 M is an injective S −1 A-module.

825 Exercise F.35. Let k be a field and let R := k[X]/(X 2 ). Show that R is an injective R-module. Show that the R-module k does not have finite injective dimension. Exercise F.36. Let A be a Grothendieck abelian category and let X ∈ Db (A). Show that if H p (X) has finite injective dimension for all p, then X has finite injective dimension. Exercise F.37. Let R be a ring and let E and F be bounded above complexes of projective R-modules. Show that every isomorphism E → F in D(R) is induced by a homotopy equivalence E → F of complexes of R-modules. Exercise F.38. Endow N with a topology by declaring a subset U ⊆ N to be open if n ∈ U and m < n imply m ∈ U (i.e., it is the topological space attached to the partially ordered set Nopp ). Let A be a Grothendieck abelian category. (1) Show that there is an equivalence F 7→ E(F ) between the category of abelian sheaves F on N and the category of Nopp -diagrams in A such that Γ(N, F ) = limn E(F ). (2) Deduce that Rp lim E(F ) = H p (N, F ) for all p ≥ 0. (3) Generalize all the above assertions to the situation where one replaces N by the partially ordered set of ordinals that are smaller than some fixed ordinal. Exercise F.39. Let A be an abelian category with countable products and enough injective objects and let X be a complex with components in A. Let (In )n be a system of bounded below complexes with injective components that forms a resolution of (τ ≥−n X)n as in Lemma F.195. (1) Show that I := lim In exists, is K-injective, and that I = R limn In in D(A). (2) Show that X = lim τ ≥−n X → limn In = I is an isomorphism in D(A) if and only if X → R lim τ ≥−n X is an isomorphism. (3) Show that the equivalent conditions in (2) are satisfied if countable products are exact in A.

G

Commutative Algebra II

In this appendix, we collect some results from commutative algebra that were not stated in Appendix B.

On regular and Cohen-Macaulay rings

(G.1) More on flatness. We first give some results on flat modules (also see Section (B.4)). Proposition G.1. ([Mat2] Theorem 7.10) Let A be a local ring and let M be a finitely generated flat A-module. Then M is a free A-module. Proposition G.2. ([Mat2] Theorem 22.5, [Sta] 046Y) Let φ : R → A be a homomorphism of rings, and let M and N two A-modules of finite type. Let u : M → N be an A-linear homomorphism. Assume that N is flat over R and that one of the following conditions are satisfied. (a) R and A are noetherian. (b) A is an R-algebra of finite presentation, and M and N are A-modules of finite presentation. Then the following assertions are equivalent. (i) u is injective and Coker(u) is a flat R-module. (ii) For every maximal ideal m of A the morphism u ⊗ idκ(p) : M ⊗R κ(p) → N ⊗R κ(p) is injective, where p := φ−1 (m) ⊂ R. Theorem G.3. (Lazard’s theorem, [BouA10] O §1.6, Th´eor`eme 1) Let A be a ring and let M be an A-module. Then the following assertions are equivalent. (i) M is a flat A-module. (ii) M is a filtered colimit of finitely generated free A-modules. (iii) For every A-module N of finite presentation, the canonical map HomA (N, A) ⊗A M −→ HomA (N, M ) is surjective.

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3

827 (G.2) Regularity and global dimension. Proposition G.4. ([BouAC10] O §4 no. 2) Let R be a local noetherian ring with residue field k. Then the following assertions are equivalent. (i) R is regular. (ii) There exists an integer d ≥ 0 such that TorR d+1 (k, k) = 0. (iii) There exists an integer e ≥ 0 such that TorR i (M, N ) = 0 for all finitely generated R-modules M and N and for all i > e. If these equivalent conditions are satisfied, the minimal integers d and e satisfying (ii) resp. (iii) are both equal to dim R. Proposition G.5. ([BouAC10] O §4, no. 2.) Let R be a regular noetherian ring. Then for every finitely generated R-module M there exists an exact sequence 0 −→ F−n −→ F−n+1 −→ · · · −→ F0 −→ M −→ 0 where the Fi are finitely generated free R-modules and where n ≤ dim R. (G.3) Dimension of modules. Definition G.6. Let A be a ring and let M be a finitely generated A-module. Then dimA (M ) := dim(Supp M ) is called the dimension of M . Recall that Supp M ⊆ Spec A is a closed subset since M is finitely generated. Then dimA (M ) = −∞ if and only if M = 0. Proposition G.7. ([BouAC89] O VIII 1.4, Prop. 9) Let A be a ring and let 0 → M ′ → M → M ′′ → 0 be a short exact sequence of finitely generated A-modules. Then dim(M ) = sup{dim(M ′ ), dim(M ′′ )}. Proposition G.8. ([BouAC89] O VIII 3.1, Prop. 2) Let A be a noetherian ring, let M be a finitely generated A-module, and let a ⊂ A be an ideal that is contained in the Jacobson radical of A and which is generated by r elements. Then dim M − r ≤ dim M/aM ≤ dim M. Note that loc. cit. is formulated only for M = A but the proof works verbatim for an arbitrary finitely generated A-module. (G.4) Depth. Let A be a local noetherian ring with maximal ideal m and residue field κ. Recall that the depth of a finitely generated A-module M ̸= 0 is defined as the maximal r ≥ 0 such that there exists an M -regular sequence with r elements contained in m. It is denoted by depthA (M ). More generally, let A be a noetherian ring (not necessarily local), let I ⊆ A be an ideal and let M be a finitely generated A-module with M ̸= IM . Then the length of a maximal M -regular sequence in I is well determined and it is called the I-depth of M and denoted by depthA (I, M ). If M = IM we define the I-depth of M to be ∞.

828

G Commutative Algebra II

Proposition G.9. ([BouAC10] O 1.4, Th´eor`eme 2) Let A be a noetherian, I ⊆ A an ideal and let M be a finitely generated A-module. Then depthA (I, M ) = inf{ i ≥ 0 ; ExtiA (A/I, M ) ̸= 0 }. Corollary G.10. ([BouAC10] O 1.4, Cor. 2 de Th´eor`eme 2) Let A be a local noetherian ring and M ̸= 0 be a finitely generated A-module. Then one has depthA (M ) ≤ dim(Supp(M )) < ∞. Proposition G.11. ([BouAC10] O 1.7, Prop. 10, Cor. de Prop. 11, see also Lemma 14.23) Let A → B be a local homomorphism of local noetherian rings and let k be the residue field of A. (1) Let f = (f1 , . . . , fr ) be a sequence of elements in the maximal ideal of B. Then A → B is flat and the image of f is B ⊗A k is regular if and only if B/f B is a flat A-algebra and f is regular. depth(B) = depth(A) + depth(B ⊗A k). Dually to the notion of injective dimension, Section (F.53), we have the notion of projective dimension. For a ring A, an A-module M has projective dimension d, if d is the minimal natural number such that there exists a projective resolution Pd → · · · → P1 → M, i.e., an exact sequence with all Pi projective A-modules. Theorem G.12. (Auslander-Buchsbaum formula, [Mat2] Theorem 19.1) Let A be a local noetherian ring and let M ̸= 0 be a finitely generated A-module of finite projective dimension projdim(M ). Then projdim(M ) + depth(M ) = depth(A).

(G.5) Cohen-Macaulay modules. Definition and Proposition G.13. ([BouAC10] O 2.1, Cor. de Prop. 1) Let A be a noetherian ring. A finitely generated A-module M is called Cohen-Macaulay, if the following equivalent conditions are satisfied. (i) For every maximal ideal m of A with Mm ̸= 0 one has depthAm (Mm ) = dimAm (Mm ). (ii) For every prime ideal p of A with Mp ̸= 0 one has depthAp (Mp ) = dimAp (Mp ). Proposition G.14. ([BouAC10] O 2.2, Cor. de Prop. 2) Let A be a local noetherian ring and let M = ̸ 0 be a finitely generated Cohen-Macaulay module. Then Supp M is equi-dimensional and for every ideal I of A one has depthA (I, M ) = dim M − dim M/IM. Definition G.15. Let A be a noetherian ring, let M be a finitely generated A-module, and let k ≥ 0 be an integer. Then M is said to have the property (Sk ) if one has depthAp (Mp ) ≥ inf{k, dimAp (Mp )} for every prime ideal p of A. Every finitely generated A-module satisfies (S0 ). A finitely generated A-module is Cohen-Macaulay if and only if it satisfies (Sk ) for all k ≥ 0.

829 (G.6) Reflexive modules. Let A be a ring. For an A-module M , we denote by M ∨ := HomA (M, A) its A-dual. We then have the natural homomorphism M → M ∨∨ from M into its double dual. Definition G.16. Let A be a ring. An A-module M is called reflexive, if the natural map M → M ∨∨ is an isomorphism. Clearly locally free A-modules M of finite rank are reflexive (because this can be checked locally on Spec A and is clear for finite free modules). But in general, being reflexive is a weaker notion than being finite locally free. Every reflexive module (or, more generally, the dual of any module) over an integral domain is torsion-free. Lemma G.17. ([Sta] 0AV3) Let A be a noetherian domain and let M be a finitely generated A-module. Then the dual M ∨ is reflexive. Reflexivity is a particularly useful notion for normal domains. Proposition G.18. Let A be a normal noetherian domain, X = Spec A, and let M be a finite A-module. The following are equivalent. (i) The module M is reflexive. (ii) The module M is torsion-free and has the S property (S2 ) (Definition G.15). (iii) The module M is torsion-free and M = p∈Spec A, ht(p)=1 Mp (inside M ⊗A Frac(A)). (iv) The module M is torsion-free, and whenever M ′ is a torsion-free A-module with M ⊆ M ′ and codimSpec A (Supp(M ′ /M )) ≥ 2, then M = M ′ . Proof. The equivalence of (i), (ii) and (iii) is proved in [Sta] 0AVB. Given (iii) and M ′ as in (iv), we have M ⊆ M ′ ⊆ M ⊗ K and Mp = Mp′ for all p of height 1, and hence M = M ′ using the description in (iii). For implication (iv) ⇒ (i) note that there exists an open U ⊆ Spec(A) whose complement has codimension ≥ 2 such that the restriction of M to U is locally free (the local rings Ap for p of height 1 are discrete valuation rings, and since M is torsion-free, Mp is free). Therefore we can take M ′ = M ∨∨ in (iv) and obtain (i). (G.7) Quotients by ideals generated by regular sequences. Proposition G.19. ([BouAC89] O VIII, §5.3) Let A be a local noetherian ring, I ⊊ A an ideal. Let A/I be regular. Then A is regular if and only if I is generated by a regular sequence. Proposition G.20. ([BouAC10] O §2.3, Prop. 4) Let A be a local noetherian ring, f = (f1 , . . . , fc ) a sequence in the maximal ideal of A, and let I be the ideal generated by f . Assume that dim A/I = dim A − c. Then A is is Cohen-Macaulay if and only if f is a regular sequence and A/I is Cohen-Macaulay.

On injective modules and Gorenstein rings In the following sections, A will denote a ring.

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(G.8) On injective modules. Recall (Definition F.49) that an A-module M is injective if and only if the functor HomA (−, M ) : (A-Mod)opp → (A-Mod) is exact. Proposition G.21. ([BouA10] O §1.7, Prop. 10, §5.5, Prop. 11) Let A be a ring and let M be an A-module. Then the following assertions are equivalent. (i) M is injective. (ii) For every ideal a of A every A-linear map a → M can be extended to A. (iii) For every A-module N one has ExtiA (N, M ) = 0 for all i > 0. (iv) For every ideal a of A one has Ext1A (A/a, M ) = 0. Proposition G.22. ([BouA10] O §1.7, Cor. 2 de Prop. 10) Let A be a principal ideal domain. Then an A-module M is an injective A-module if and only if it is divisible (i.e., for all 0 ̸= a ∈ A the multiplication M → M by a is surjective). Proposition G.23. ([Mat2] §18, Lemma 5) Let A be a noetherian ring, let S ⊆ A be a multiplicative subset, and let I be an injective A-module. Then S −1 I is an injective S −1 A-module. Proof. Let b ⊆ S −1 A be an ideal. Then b = S −1 a for some ideal a of A. By Proposition G.21 it suffices to show that HomS −1 A (S −1 A, S −1 I) = S −1 HomA (A, I) −→ HomS −1 A (S −1 a, S −1 I) = S −1 HomA (a, I) is surjective (for the first equality use that a is of finite presentation since A is noetherian). This is clear as I is injective. For different ideas of proofs see Exercise F.27 and Exercise F.34. (G.9) Matlis duality. In this section, A denotes a noetherian local ring with maximal ideal m and residue class field κ. Every A-module M has an injective hull (Definition F.64). By definition, it is an embedding M ,→ E of A-modules such that E is an injective A-module such that N ∩ M ̸= 0 for every non-zero submodule N of E. An injective hull is determined up to isomorphism (but not up to unique isomorphism), see Proposition F.65. An injective hull of the A-module κ is called a Matlis module for A. Proposition G.24. (Matlis duality, [BouAC10] O §8.3, Th´eor`eme 2) Let E be a Matlis module for A. The functor DA : N 7→ HomA (N, E) is an anti-equivalence of the category of finite length A-modules with itself which is its own inverse. For every A-module M of finite length one has lgA (DA (M )) = lgA (M ). Denote by (A-Mod)fl be the full subcategory of the category of A-modules consisting of the A-modules of finite length, i.e., of finitely generated A-modules that are annihilated by some power of m. This is a plump subcategory (Definition F.43), in particular it is an abelian category.

831 Let T : (A-Mod)fl → (A-Mod) be an exact A-linear contravariant functor such that T (κ) is an A-module of length 1. Then for every M in (A-Mod)fl , T (M ) is also of finite length with lgA T (M ) = lgA M . For n ≥ 0 set En := T (A/mn ). Then for m ≥ n, the canonical maps A/mm → A/mn induce injective A-linear maps En → Em . Set E := colimn En . Let M be an A-module of finite length and choose an n ≥ 1 such that mn M = 0. For x ∈ M , the A-linear map A/mn → M that sends 1 to x induces an A-linear map αx,M : T (M ) → En ,→ E and the map θM : T (M ) −→ HomA (M, E),

λ 7→ αx,M (λ)

is independent of the choice of n and A-linear. It defines an A-linear map of A-linear opp functors (A-Mod)fl → (A-Mod)fl (G.9.1)

θ : T (−) −→ HomA (−, E).

Proposition G.25. ([BouAC10] O §8.3, Th´eor`eme 3) The A-module E is a Matlis module and θ is an isomorphism of functors. (G.10) Gorenstein rings. Proposition/Definition G.26. (cf. [Mat2] Theorem 18.1) Let A be a noetherian local ring of dimension n. Let κ be the residue class field of A. The following are equivalent. (i) A has finite injective dimension as an A-module, i.e., A is isomorphic to a bounded complex of injective A-modules in D(A). (ii) ExtiA (κ, A) = 0 for all i ̸= n, and ExtnA (κ, A) ∼ = κ. (iii) There exists i > n, such that ExtiA (κ, A) = 0. (iv) The ring A is Cohen-Macaulay and ExtnA (κ, A) ∼ = κ. We say that the local ring A is a Gorenstein ring, if A satisfies the above conditions. It has then injective amplitude contained in [0, n]. As every local Artinian ring is zero-dimensional and hence Cohen-Macaulay, we have the following corollary. Corollary G.27. Let A be a local Artinian ring with residue field κ. Then the following assertions are equivalent. (i) A is Gorenstein. (ii) The A-module A is injective. (iii) The κ-vector space HomA (κ, A) is one-dimensional. Proposition G.28. ([Mat2] Theorem 18.2) Let A be a Gorenstein local ring. For every prime ideal p ⊂ A, the localization Ap is Gorenstein. Because of this proposition, we can define the notion of Gorenstein ring as follows. Definition G.29. Let A be a noetherian ring. We say that A is a Gorenstein ring, if for every maximal ideal p ⊂ A (equivalently: for every prime ideal p ⊂ A) the localization Ap is a Gorenstein local ring in the sense of Definition G.26.

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Proposition G.30. ([BouAC10] O §3.8, Prop. 12) Let A → B be a flat local homomorphism of local noetherian rings. Let κA be the residue field of A. Then the following assertions are equivalent. (i) B is a Gorenstein ring. (ii) A and B ⊗A κA are Gorenstein rings. Corollary G.31. ([BouAC10] O §3.8, Cor. 2 de Prop. 12) Let A be a noetherian ring and let I ⊆ A be an ideal that is contained in the Jacobson radical of A. Then A is Gorenstein if and only if the I-adic completion of A is Gorenstein. (G.11) Addenda on separable and inseparable field extensions. Lemma G.32. ([BouAC] O V, §2.3, Lemma 4, Remark (1)) Let A be a normal domain, K its field of fractions, let p ≥ 1 be its characteristic exponent. Let K ′ be a purely inseparable extension of K and let A′ be the integral closure of A in K ′ . Then m

A′ = { x′ ∈ K ′ ; ∃ m ≥ 1 : xp ∈ A } and for every prime ideal p of A there exists a unique prime ideal p′ of A′ lying over p. One has m p′ = { x′ ∈ K ′ ; ∃ m ≥ 1 : xp ∈ p }. We also need the following variant of Proposition B.97. Proposition G.33. ([Sta] 030W; cf. [BouAII] O 15.4 Cor. 1) Let K/k be a field extension of fields with positive characteristic p. The following are equivalent. (i) The extension K/k is separable. √ (ii) The ring K ⊗k k 1/p is reduced, where k 1/p = k( p a, a ∈ k) is the extension of k obtained by adjoining all p-th roots of elements of k in some algebraic closure of k. Lemma G.34. ([Mat2] Theorem 26.5; [Sta] 07P2.) Let k be a field of characteristic p > 0 and let K/k be a finitely generated field extension. Let x1 , . . . , xn ∈ K. The following are equivalent. (i) The elements x1 , . . . , xn are a differential basis of K over k, i.e., dx1 , . . . , dxn are a K-basis of ΩK/k . (ii) The elements x1 , . . . , xn are a p-basis of K over k, i.e., the field K equals the compositum kK p (x1 , . . . , xn ) and [K : kK p ] = pn . In particular, we have pdimK ΩK/k = [K : kK p ].

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modules

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sheaves,

Detailed List of Contents 1

Introduction 17 Differentials Differentials for rings and extensions of algebras . . . Derivations and K¨ ahler differentials for rings . . . . . . Extensions of algebras by modules . . . . . . . . . . . . Differentials for sheaves on schemes . . . . . . . . . . . Conormal sheaf of an immersion . . . . . . . . . . . . . Derivations and K¨ ahler differentials for schemes . . . . . Fundamental exact sequences for K¨ ahler differentials . . Tangent bundles . . . . . . . . . . . . . . . . . . . . . . Differentials of Grassmannians and of projective bundles The de Rham complex . . . . . . . . . . . . . . . . . . . . The exterior algebra . . . . . . . . . . . . . . . . . . . . Differential graded algebras . . . . . . . . . . . . . . . . The de Rham complex . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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´ 18 Etale and smooth morphisms Formally unramified, formally smooth and formally ´ etale morphisms Definition of formally unramified, formally smooth and formally ´etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formally unramified morphisms and differentials . . . . . . . . . . . . . Gluing local lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formally smooth resp. formally ´etale morphisms and differentials . . . . Unramified and ´ etale morphisms . . . . . . . . . . . . . . . . . . . . . . Unramified morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . ´ Etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local description of ´etale morphisms . . . . . . . . . . . . . . . . . . . . Characterization of ´etale morphisms . . . . . . . . . . . . . . . . . . . . Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrically regular schemes . . . . . . . . . . . . . . . . . . . . . . . . Characterization of smooth morphisms . . . . . . . . . . . . . . . . . . . Characterizations of smooth morphisms in the noetherian case . . . . . Smooth schemes over a field . . . . . . . . . . . . . . . . . . . . . . . . . Smooth morphisms and differentials . . . . . . . . . . . . . . . . . . . . Smooth and ´etale morphisms between smooth schemes . . . . . . . . . . Open immersions and ´etale morphisms . . . . . . . . . . . . . . . . . . . Fibre criterion for smooth and ´etale morphisms . . . . . . . . . . . . . . Generic Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3

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847 19 Local complete intersections The Koszul complex and completely intersecting immersions Koszul complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regular and completely intersecting sequences . . . . . . . . . . . Regular and completely intersecting immersions . . . . . . . . . . Regular immersions of flat and of smooth schemes . . . . . . . . Blow-up of regularly immersed smooth subschemes . . . . . . . . Local complete intersection and syntomic morphisms . . . . . Local complete intersection morphisms . . . . . . . . . . . . . . . Complete intersection rings . . . . . . . . . . . . . . . . . . . . . Local complete intersection morphisms over a field . . . . . . . . Syntomic morphisms . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ´tale topology 20 The e Henselian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of henselian rings . . . . . . . . . . . . . . . . . . . Sections of smooth morphisms . . . . . . . . . . . . . . . . . . Sections of ´etale and smooth schemes over henselian rings . . Henselian pairs . . . . . . . . . . . . . . . . . . . . . . . . . . The ´ etale topology . . . . . . . . . . . . . . . . . . . . . . . . . ´ Etale topology . . . . . . . . . . . . . . . . . . . . . . . . . . Lifting of ´etale schemes . . . . . . . . . . . . . . . . . . . . . Sheaves in the ´etale topology . . . . . . . . . . . . . . . . . . Points and stalks in the ´etale topology . . . . . . . . . . . . . Stalks of the structure sheaf: (strict) henselization . . . . . . Unibranch schemes . . . . . . . . . . . . . . . . . . . . . . . . Artin approximation . . . . . . . . . . . . . . . . . . . . . . . Analytification of schemes over C . . . . . . . . . . . . . . . . The ´ etale fundamental group of a scheme . . . . . . . . . . . ´ Etale covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lifting of ´etale covers . . . . . . . . . . . . . . . . . . . . . . Fibers of ´etale covers and the fundamental groupoid . . . . . Galois covers . . . . . . . . . . . . . . . . . . . . . . . . . . . Profinite groups and the topology on the automorphism group The ´etale fundamental group . . . . . . . . . . . . . . . . . . Functoriality of the fundamental group . . . . . . . . . . . . . Fundamental groups of fields . . . . . . . . . . . . . . . . . . Examples from number theory . . . . . . . . . . . . . . . . . The fundamental group of P1 . . . . . . . . . . . . . . . . . . Algebraic and analytic fundamental group . . . . . . . . . . . The fundamental exact sequence of fundamental groups . . . Specialization of fundamental groups . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21 Cohomology of OX -modules Categories of abelian sheaves and of OX -modules . . . . . . Sheaves of sections with proper support . . . . . . . . . . . . Categories of sheaves on a ringed space . . . . . . . . . . . . . Restriction to open subsets . . . . . . . . . . . . . . . . . . . Cohomology and derived direct image . . . . . . . . . . . . . Cohomology as derived functor . . . . . . . . . . . . . . . . . Cohomology and restriction to open subspaces . . . . . . . . Higher direct images as derived functors . . . . . . . . . . . . Flasque Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of flasque resolutions . . . . . . . . . . . . . . . Mayer-Vietoris sequences and gluing of complexes . . . . . . . Cohomology groups — a first example . . . . . . . . . . . . . Compatibility with colimits . . . . . . . . . . . . . . . . . . . The Grothendieck-Scheiderer vanishing theorem . . . . . . . . The local cohomology triangle . . . . . . . . . . . . . . . . . . ˇ Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . ˇ The Cech complex . . . . . . . . . . . . . . . . . . . . . . . . ˇ The alternating and the ordered Cech complex . . . . . . . . ˇ Passing to refinements for Cech cohomology . . . . . . . . . . ˇ Cech cohomology versus cohomology . . . . . . . . . . . . . . Derived inverse image, Hom sheaves, and tensor products Derived functor of Hom and Hom . . . . . . . . . . . . . . . K-flat complexes . . . . . . . . . . . . . . . . . . . . . . . . . Derived tensor product and Tor sheaves . . . . . . . . . . . . Ext sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derived functor of the inverse image functor . . . . . . . . . . Derived direct image and composition . . . . . . . . . . . . . Relations between derived functors . . . . . . . . . . . . . . . Derived inverse images and ⊗L . . . . . . . . . . . . . . . . . Adjointness and derived functors . . . . . . . . . . . . . . . . Relations between R Hom and ⊗L . . . . . . . . . . . . . . . Relations between Rf∗ and Lf ∗ . . . . . . . . . . . . . . . . . The base change morphism . . . . . . . . . . . . . . . . . . . Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . . Formalism of six functors . . . . . . . . . . . . . . . . . . . . Perfect and pseudo-coherent complexes . . . . . . . . . . . . Perfect complexes . . . . . . . . . . . . . . . . . . . . . . . . . The dual of a perfect complex . . . . . . . . . . . . . . . . . . Pseudo-coherent complexes . . . . . . . . . . . . . . . . . . . Pseudo-coherent complexes over a ring . . . . . . . . . . . . . Tor-amplitude and tor-dimension . . . . . . . . . . . . . . . . Pseudo-coherent complexes of finite tor-dimension are perfect Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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849 22 Cohomology of quasi-coherent modules ˇ Cohomology of quasi-coherent modules and Cech cohomology . . . . Quasi-coherent cohomology of affine schemes . . . . . . . . . . . . . . . ˇ Cohomology versus Cech Cohomology . . . . . . . . . . . . . . . . . . . Elementary examples: P1 and A2 \ {0} . . . . . . . . . . . . . . . . . . . ˇ The extended ordered Cech complex and Koszul complexes . . . . . . . Example: Cohomology of quasi-affine schemes . . . . . . . . . . . . . . . Example: Cohomology of twisted line bundles on projective spectra and on projective space . . . . . . . . . . . . . . . . . . . . . . . . . . Derived categories of quasi-coherent modules . . . . . . . . . . . . . . Homotopy Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasi-coherence of higher direct images . . . . . . . . . . . . . . . . . . The category of quasi-coherent OX -modules and its derived category . . Comparison of Dqcoh (X) and D(QCoh(X)) . . . . . . . . . . . . . . . . Derived tensor product and pullback of quasi-coherent complexes . . . . Derived categories of coherent modules on noetherian schemes . . . . . . Finiteness properties of complexes on schemes . . . . . . . . . . . . . . Perfect and pseudo-coherent complexes on schemes . . . . . . . . . . . . The resolution property and representing perfect complexes . . . . . . . Pseudo-coherent complexes on noetherian schemes . . . . . . . . . . . . Derived tensor products of pseudo-coherent and perfect complexes . . . Derived Hom and Ext on schemes . . . . . . . . . . . . . . . . . . . . . Injective modules on locally noetherian schemes . . . . . . . . . . . . . . Projection formula, base change and the K¨ unneth formula . . . . . . The projection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: Cohomology of projective bundles . . . . . . . . . . . . . . . . Special case of base change: Non-derived flat base change . . . . . . . . Tor-independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Base change and K¨ unneth formula: Main Theorem . . . . . . . . . . . . Tor-independence implies base change . . . . . . . . . . . . . . . . . . . K¨ unneth isomorphism implies tor-independence . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Cohomology of projective and proper schemes Cohomology of projective schemes . . . . . . . . . . . . . . . . . . . . . Coherence of direct images under projective morphisms and Serre’s vanishing criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serre’s ampleness criterion . . . . . . . . . . . . . . . . . . . . . . . . . . Coherent modules on projective spectra revisited . . . . . . . . . . . . . Coherence of higher direct images for proper morphisms . . . . . . . Finiteness of higher direct images under proper morphisms . . . . . . . Finiteness of cohomology of modules with proper support . . . . . . . . Derived image of pseudo-coherent complexes . . . . . . . . . . . . . . . . Morphisms of finite tor-dimension . . . . . . . . . . . . . . . . . . . . . . Derived image of perfect complexes . . . . . . . . . . . . . . . . . . . . . GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical intersection theory, Euler characteristic, and Hilbert polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

232 233 233 236 236 238 239 240 243 243 245 248 250 253 254 255 255 260 261 262 262 266 269 269 274 276 278 280 282 284 288 297 298 298 299 303 306 306 308 308 309 311 313 314

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Grothendieck group of abelian categories and triangulated categories . . Grothendieck group of noetherian schemes . . . . . . . . . . . . . . . . . K-groups of quasi-compact schemes . . . . . . . . . . . . . . . . . . . . . Chern classes of line bundles on noetherian schemes . . . . . . . . . . . Euler characteristic of schemes over a field . . . . . . . . . . . . . . . . . Numerical intersection number for proper schemes over a field . . . . . . Asymptotic Riemann Roch theorem . . . . . . . . . . . . . . . . . . . . The degree of a closed subscheme . . . . . . . . . . . . . . . . . . . . . . The Nakai-Moishezon criterion for ampleness . . . . . . . . . . . . . . . Hilbert polynomials of proper schemes over a field . . . . . . . . . . . . The Grothendieck-Riemann-Roch theorem . . . . . . . . . . . . . . . . Cohomology theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chern classes of vector bundles . . . . . . . . . . . . . . . . . . . . . . . Examples of Cohomology theories . . . . . . . . . . . . . . . . . . . . . . The Grothendieck-Riemann-Roch theorem for additive cohomology theories Cohomology and base change . . . . . . . . . . . . . . . . . . . . . . . . . Semicontinuity of Betti numbers for pseudo-coherent complexes . . . . . Base change for pseudo-coherent complexes . . . . . . . . . . . . . . . . Subschemes classifying properties of perfect complexes . . . . . . . . . . Criteria when direct images are perfect and commute with base change . Semicontinuity theorems and base change of higher direct images . . . . Hilbert polynomials and flattening stratification . . . . . . . . . . . . Local constancy of Hilbert polynomials and of intersection numbers . . Flatness on projective schemes . . . . . . . . . . . . . . . . . . . . . . . Flattening stratification by Hilbert polynomials . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Theorem on formal functions Derived Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reminder on completions . . . . . . . . . . . . . . . . . . . . . . . . . . Derived complete complexes . . . . . . . . . . . . . . . . . . . . . . . . . Derived completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Globalization of derived completion . . . . . . . . . . . . . . . . . . . . . The theorem of formal functions . . . . . . . . . . . . . . . . . . . . . . . Theorem of formal functions, derived version . . . . . . . . . . . . . . . Theorem of formal functions for locally noetherian schemes . . . . . . . Mittag-Leffler condition for cohomology and direct proof of the Theorem of formal functions . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem of formal fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . Ampleness is open on proper schemes . . . . . . . . . . . . . . . . . . . Stein factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stein factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Stein factorization . . . . . . . . . . . . . . . . . . . . . . . The seesaw theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application: Picard group of projective bundles and of products . . . . . The Theorem of the Cube . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideals of definition for constructible closed subsets . . . . . . . . . . . .

314 315 317 319 320 322 326 327 329 331 332 333 335 337 339 342 343 345 352 353 356 361 362 362 364 367 376 376 377 379 382 387 390 390 392 394 396 397 398 398 402 405 408 409 411 411

851 Formal completion of qcqs schemes along closed subspaces . . . . . . . . Adic formal schemes over complete rings . . . . . . . . . . . . . . . . . . Modules over formal schemes . . . . . . . . . . . . . . . . . . . . . . . . Grothendieck’s existence theorem for coherent modules . . . . . . . . . . Full faithfulness of F 7→ F/Z . . . . . . . . . . . . . . . . . . . . . . . . Essential surjectivity of F 7→ F/Z . . . . . . . . . . . . . . . . . . . . . Grothendieck’s existence theorem for coherent modules with proper support Algebraization of proper schemes . . . . . . . . . . . . . . . . . . . . . . Remarks on the literature . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Duality The right adjoint f × of Rf∗ . . . . . . . . . . . . . . . . . . . . . . . . Brown representability for triangulated categories . . . . . . . . . . . Dqcoh (X) is compactly generated . . . . . . . . . . . . . . . . . . . . Construction and first properties of f × . . . . . . . . . . . . . . . . . Variants of the adjunction of Rf∗ and f × for R Hom and R Hom . . f × and base change . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of f × in terms of f × OS . . . . . . . . . . . . . . . . . . Computation of f × in special cases . . . . . . . . . . . . . . . . . . . The functor f × for morphisms f between affine schemes . . . . . . . The functor i× for a closed immersion i . . . . . . . . . . . . . . . . The example of projective space . . . . . . . . . . . . . . . . . . . . Computation of f × OS for f smooth and proper . . . . . . . . . . . . The functor f ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of the functor f ! . . . . . . . . . . . . . . . . . . . . . . Properties of the functor f ! . . . . . . . . . . . . . . . . . . . . . . . The functor f ! for smooth morphisms f . . . . . . . . . . . . . . . . Dualizing complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dualizing complex . . . . . . . . . . . . . . . . . . . . . . . . . . Local nature of dualizing complexes . . . . . . . . . . . . . . . . . . Uniqueness of dualizing complexes . . . . . . . . . . . . . . . . . . . Dualizing complexes and f ! . . . . . . . . . . . . . . . . . . . . . . . Dualizing complexes over local rings . . . . . . . . . . . . . . . . . . Existence of dualizing complexes . . . . . . . . . . . . . . . . . . . . Relative dualizing complexes . . . . . . . . . . . . . . . . . . . . . . Dualizing sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition and first properties of dualizing sheaves . . . . . . . . . . Dualizing complexes and sheaves for proper schemes over local rings Cohen-Macaulay schemes . . . . . . . . . . . . . . . . . . . . . . . . The relative dualizing complex for Cohen-Macaulay morphisms . . . Gorenstein schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality for schemes over fields . . . . . . . . . . . . . . . . . . . . . . Serre duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dualizing sheaves on normal varieties . . . . . . . . . . . . . . . . . . The Lemma of Enriques-Severi-Zariski . . . . . . . . . . . . . . . . . Applications to algebraic surfaces . . . . . . . . . . . . . . . . . . . . The theorem of Riemann-Roch for algebraic surfaces . . . . . . . . .

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852 Resolution of indeterminacies for surfaces . . . . . Projectivity of regular proper surfaces over a field . The Hodge index theorem . . . . . . . . . . . . . . Further references . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

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502 504 506 508 509

26 Curves Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . Recollections on curves . . . . . . . . . . . . . . . . . . . . Reminder on geometrically connected schemes . . . . . . . Singularities of curves . . . . . . . . . . . . . . . . . . . . Morphisms between curves . . . . . . . . . . . . . . . . . . Quasi-projectivity of Curves . . . . . . . . . . . . . . . . . Divisors on curves . . . . . . . . . . . . . . . . . . . . . . Algebraic curves and compact Riemann surfaces . . . . . The arithmetic genus of a curve . . . . . . . . . . . . . . . Ordinary multiple points . . . . . . . . . . . . . . . . . . . The geometric genus . . . . . . . . . . . . . . . . . . . . . The Theorem of Riemann-Roch . . . . . . . . . . . . . . . The Theorem of Riemann-Roch . . . . . . . . . . . . . . . Divisors on curves, continued . . . . . . . . . . . . . . . . The formula of Riemann and Hurwitz . . . . . . . . . . . Purely inseparable morphisms . . . . . . . . . . . . . . . . Special classes of curves . . . . . . . . . . . . . . . . . . . . Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . Curves of genus 0 . . . . . . . . . . . . . . . . . . . . . . . Curves of genus 1, elliptic curves . . . . . . . . . . . . . . The Legendre family and the j-invariant . . . . . . . . . . Elliptic curves over the complex numbers . . . . . . . . . Hyperelliptic curves . . . . . . . . . . . . . . . . . . . . . Curves of genus > 2 . . . . . . . . . . . . . . . . . . . . . Vector bundles on curves . . . . . . . . . . . . . . . . . . . Vector bundles on P1k . . . . . . . . . . . . . . . . . . . . . The Riemann-Roch theorem for vector bundles on curves Semistable vector bundles . . . . . . . . . . . . . . . . . . Harder-Narasimhan filtration . . . . . . . . . . . . . . . . Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Curves . . . . . . . . . . . . . . . . . . . . . . . . Plane curves birational to a given curve . . . . . . . . . . Curves over finite fields and the Weil conjecture . . . . . . Proof of the Weil conjecture for curves . . . . . . . . . . . Curves over number fields . . . . . . . . . . . . . . . . . . Literature on algebraic curves . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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513 513 513 514 515 517 519 520 523 526 529 533 534 534 537 541 545 547 547 548 552 558 560 563 570 571 572 574 577 580 584 584 586 587 591 597 598 598

853 27 Abelian schemes Preliminaries and general results about group schemes . . . . . . . . General facts on group schemes . . . . . . . . . . . . . . . . . . . . . . . Affine Group schemes and Hopf algebras . . . . . . . . . . . . . . . . . . Generalities about group schemes over a field . . . . . . . . . . . . . . . Differentials of group schemes . . . . . . . . . . . . . . . . . . . . . . . . Singularities of group schemes . . . . . . . . . . . . . . . . . . . . . . . . Digression: Sheaves for the fppf topology . . . . . . . . . . . . . . . . . . Digression: Algebraic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . Digression: Fppf-surjective morphisms . . . . . . . . . . . . . . . . . . . Quotient spaces and homogeneous spaces . . . . . . . . . . . . . . . . . Digression: Homotopy invariance of Picard group . . . . . . . . . . . . . Quasi-projectivity of homogeneous spaces . . . . . . . . . . . . . . . . . The graded Hopf algebra structure on the cohomology ring of an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cartier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annihilation of commutative finite locally free group schemes . . . . . . Digression: Collection of some properties of schemes over inductive limits of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition and basic properties of abelian schemes . . . . . . . . . . . Definition of abelian schemes . . . . . . . . . . . . . . . . . . . . . . . . The constancy locus of a morphism of schemes . . . . . . . . . . . . . . Abelian schemes are commutative . . . . . . . . . . . . . . . . . . . . . . Further applications of rigidity . . . . . . . . . . . . . . . . . . . . . . . Constructing abelian schemes . . . . . . . . . . . . . . . . . . . . . . . . The Picard functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Picard functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representability of the Picard functor . . . . . . . . . . . . . . . . . . . The Lie algebra of the Picard functor . . . . . . . . . . . . . . . . . . . The identity component of the Picard functor . . . . . . . . . . . . . . . The Picard functor of curves . . . . . . . . . . . . . . . . . . . . . . . . The Jacobian of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality of abelian schemes . . . . . . . . . . . . . . . . . . . . . . . . . . The space of correspondence classes . . . . . . . . . . . . . . . . . . . . Definition of the dual functor to an abelian scheme . . . . . . . . . . . . The Theorem of the Square . . . . . . . . . . . . . . . . . . . . . . . . . The kernel of φL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projectivity of abelian varieties . . . . . . . . . . . . . . . . . . . . . . . Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Frobenius isogeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental groups of abelian varieties . . . . . . . . . . . . . . . . . . The dual abelian scheme for projective abelian schemes . . . . . . . . . Cohomology of the structure sheaf and of the sheaves of differentials . . The dual abelian space is a scheme . . . . . . . . . . . . . . . . . . . . . Dual homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Poincar´e bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

603 604 604 606 607 609 612 614 616 620 622 626 626 629 631 633 634 635 635 636 639 640 641 643 643 646 647 649 651 653 656 658 658 662 666 667 668 670 672 673 676 679 681 684 687 688

854

Detailed List of Contents

Biduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cohomology of line bundles on abelian schemes . . . . . . . . . . . Cohomology of the Poincar´e bundle . . . . . . . . . . . . . . . . . . The N´eron-Severi group of an abelian scheme . . . . . . . . . . . . . Fourier-Mukai transforms . . . . . . . . . . . . . . . . . . . . . . . . Fourier-Mukai equivalence for abelian schemes . . . . . . . . . . . . . Riemann-Roch for abelian varieties . . . . . . . . . . . . . . . . . . . The index of non-degenerate line bundles . . . . . . . . . . . . . . . Characterization of ample line bundles on abelian schemes . . . . . . Vector bundles on elliptic curves . . . . . . . . . . . . . . . . . . . . Very ample line bundles on abelian schemes . . . . . . . . . . . . . . Symmetric homomorphisms and polarizations . . . . . . . . . . . . . Projectivity of abelian schemes over normal base schemes . . . . . . Abelian varieties over the complex numbers . . . . . . . . . . . . . . Outlook: The moduli space of principally polarized abelian varieties Literature on abelian varieties and abelian schemes . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F Homological Algebra Addenda to the language of categories . . . . . . . . . Set-theoretical remarks . . . . . . . . . . . . . . . . . . Categories and functors . . . . . . . . . . . . . . . . . Limits and Colimits . . . . . . . . . . . . . . . . . . . Special cases of limits and colimits . . . . . . . . . . . Adjoint functors . . . . . . . . . . . . . . . . . . . . . Additive and abelian categories . . . . . . . . . . . . . Additive categories . . . . . . . . . . . . . . . . . . . . Abelian categories . . . . . . . . . . . . . . . . . . . . Length and Jordan-H¨ older series in abelian categories Subcategories of abelian categories . . . . . . . . . . . Five Lemma and Snake Lemma . . . . . . . . . . . . . Injective and projective Objects . . . . . . . . . . . . . Grothendieck abelian categories . . . . . . . . . . . . . Injective objects in Grothendieck abelian categories and categories . . . . . . . . . . . . . . . . . . . . . Complexes in additive and abelian categories . . . . Categories of Complexes . . . . . . . . . . . . . . . . . Homotopy of complexes . . . . . . . . . . . . . . . . . Quasi-isomorphisms . . . . . . . . . . . . . . . . . . . Double complexes . . . . . . . . . . . . . . . . . . . . . The Hom complex . . . . . . . . . . . . . . . . . . . . Tensor product of complexes . . . . . . . . . . . . . . Spectral sequences . . . . . . . . . . . . . . . . . . . . . Graded and Filtered objects . . . . . . . . . . . . . . . Definition of spectral sequences . . . . . . . . . . . . . Exact sequences attached to spectral sequences . . . . Spectral sequences associated to exact couples . . . . . Spectral sequences associated to filtered complexes . .

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689 692 692 697 698 702 705 708 712 713 718 722 724 725 731 733 734 737 737 737 738 739 742 744 745 745 747 749 749 750 751 752 753 754 755 757 759 759 761 762 763 764 764 766 768 770

855 Spectral sequences associated to double complexes . . . . . . . . . . . . 771 Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 Definition of triangulated categories . . . . . . . . . . . . . . . . . . . . 772 Triangulated Structures on categories of complexes . . . . . . . . . . . . 775 Triangulated Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 Triangulated Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . 778 The opposite triangulated category . . . . . . . . . . . . . . . . . . . . . 778 Cohomological Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 Sign conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 Cones and Exact Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 780 Double complexes and the total complex . . . . . . . . . . . . . . . . . . 781 Homomorphisms and tensor products . . . . . . . . . . . . . . . . . . . . 781 Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Localization of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 782 Localization of triangulated categories . . . . . . . . . . . . . . . . . . . 784 The derived category of an abelian category . . . . . . . . . . . . . . . . 785 Bounded derived categories . . . . . . . . . . . . . . . . . . . . . . . . . 789 Truncation in derived categories . . . . . . . . . . . . . . . . . . . . . . 789 Construction of complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 790 Variants of the derived category . . . . . . . . . . . . . . . . . . . . . . . 791 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 Definition of derived functors . . . . . . . . . . . . . . . . . . . . . . . . 792 Construction of derived functors . . . . . . . . . . . . . . . . . . . . . . 794 K-injective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 Adjointness of derived functors . . . . . . . . . . . . . . . . . . . . . . . 800 Bounded Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 Construction of resolutions . . . . . . . . . . . . . . . . . . . . . . . . . 802 Derived functors on D+ (A) and higher derived functors of left exact functors 804 Hypercohomology spectral sequences . . . . . . . . . . . . . . . . . . . . 807 Grothendieck spectral sequence . . . . . . . . . . . . . . . . . . . . . . . 809 Derived bi-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 The derived Hom functor and Ext Groups . . . . . . . . . . . . . . . . . 811 Injective dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 Derived limits and homotopy limits . . . . . . . . . . . . . . . . . . . . . 814 Homotopy colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 G Commutative Algebra II On regular and Cohen-Macaulay rings . . . . . . . More on flatness . . . . . . . . . . . . . . . . . . . Regularity and global dimension . . . . . . . . . . Dimension of modules . . . . . . . . . . . . . . . . Depth . . . . . . . . . . . . . . . . . . . . . . . . . Cohen-Macaulay modules . . . . . . . . . . . . . . Reflexive modules . . . . . . . . . . . . . . . . . . . Quotients by ideals generated by regular sequences On injective modules and Gorenstein rings . . . . On injective modules . . . . . . . . . . . . . . . . .

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826 826 826 827 827 827 828 829 829 829 830

856

Detailed List of Contents Matlis duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gorenstein rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addenda on separable and inseparable field extensions . . . . . . . . . .

830 831 832

Bibliography

833

Detailed List of Contents

846

Index of Symbols

857

Index

860

Index of Symbols exterior algebra, 24

D ∗ (R)

−∪−

cup product, 207

D ∗ (X) bounded versions of D(X), 153

−⊗−

tensor product of complexes, 762

DC (N ) augmented extension by square zero ideal, 7

V

R

M

− ⊗L − derived tensor product, 193 − ⨿− − pushout in a category, 743 (Ab(X)) category of abelian sheaves on X, 153 Ag,d,N

moduli space of principally polarized abelian varieties, 733

C/N

localization defined by null system, 785

c1 (L )

first Chern class of line bundle, 319

cA (E )

Chern polynomial, 336

cA i (E )

i-th Chern class, 336

bounded versions of D(R), 153

Dcoh (X) derived category of complexes with coherent cohomology objects, 254 Dcomp (A, I) derived I-complete complexes, 380 Dcomp (X) derived complete complexes, 388 Dcomp (X, Z) derived complete complexes, 388 Dcomp (X, I ) derived complete complexes, 388 deg(L ) degree of line bundle, 327 degL (Z) degree of closed subscheme w.r.t. line bundle, 327

ˇ ˇ n (U , F ) alternating Cech C complex, 180 alt

deg(Z) degree of closed subscheme, 327

ˇ n (U , F ) ordered Cech ˇ C complex, 180 ord

DerR (A, M ) R-derivations from A to M , 5

C ∗ (R)

bounded versions of C(R), 153

DerS (OX , F ) S-derivations from OX to F , 14

C ∗ (X)

bounded versions of C(X), 153

DK

C/d

slice category over d, 739

C\d

slice category under d, 739

Dqcoh (X) derived category of complexes with quasi-coherent cohomology objects, 243

Cf•

mapping cone of f , 757

chA

Chern character, 340

χ(F )

Euler characteristic, 320

Ci

conormal sheaf, 11

Clopen(X) set of open and closed subschemes of X, 96 (Coh(X)) category of coherent OX -modules, 254 colimI

colimit, 739

CorrS (X, X ′ ) divisorial correspondences, 659

localization of D w.r.t. K, 789

D(R)

derived category of (R-Mod), 153

D(X)

derived category of (OX -Mod), 153

dX/S

different of X over S, 121

DX/S

discriminant of X over S, 120

εi (A)

dimension of Koszul homology for local ring, 84

ExR (A, M ) Extensions of A by M , 9 ExtiA (X, Y ) i-th Ext group, 811 Ext i (F , G ) Ext sheaf, 195

C OX/S scheme of open and closed subschemes, 96

f an

analytification of f , 114

Fs¯

fiber functor, 124

C(R)

F#

sheafification of presheaf F , 615

f!

twisted inverse image functor, 464

ft

dual homomorphism of abelian schemes, 664

f×

right adjoint of Rf∗ , 444

F∧

derived completion along closed subscheme, 388

∧ F/Z

derived completion along closed subscheme, 388

G0

identity component of group scheme, 608

category of complexes in (R-Mod), 153

CS localization of category C w.r.t. S, 782 ˇ ˇ C(U , F ) Cech complex with respect to covering, 179 category of complexes of OX -modules, 153 ˇ ˇ C(X, F ) Cech complex, 182 C(X)

CY /X

conormal sheaf, 11

(D1 · · · Dt · F ) intersection number, 323 D(A)

derived category of abelian category A, 785

∗ (A) subcategory of D ∗ (A), 791 DA ′

ΓZ (X, F ) sections with support in Z, 177 g(C)

(arithmetic) genus of curve C, 527

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3

858

Index of Symbols Ni

normal bundle, 75

NQ

multiplicative system attached to null system, 784

(G-sets) Category of finite G-sets, 129 ˇ n (X, F ) Cech ˇ H cohomology group, 182

NS(X)

N´ eron-Severi group, 697

Ω1X/S

sheaf of K¨ ahler differentials, 13

Hi (f )

• ωA

dualizing complex (for a ring), 471

H i (X, F ) i-th cohomology group of X with coefficients in F , 159

ωX

dualizing sheaf, 484

H n (X) n-th cohomology object of complex X, 756

• ωX

dualizing complex, 469

Ω•X/S

de Rham complex, 27

n (X, F ) cohomology with support in Z, 178 HZ

OU ⊆X

proper direct image of OU , 155

hocolim Kn homotopy colimit, 818

(OX -Mod) category of OX -modules on ringed space X, 153

gr•F (X) associated graded object to filtration F on X, 764 Gs¯

category of Galois covers, 127

homology of Koszul complex, 69

holim Kn homotopy limit, 816 HomB A (B, −) Hom-functor to (B-Mod), 456 HomA (X, Y ) Hom complex, 761 i(L )

index of line bundle on abelian scheme, 710

Is¯

category of pointed ´ etale covers, 125

I(Z)

set of ideals of definition, 412

j(E)

j-invariant of elliptic curve, 560

j!

proper direct image, 154

pg (C)

geometric genus of curve C, 534

ΦK

Fourier-Mukai transform, 699

φL

morphism to dual abelian scheme attached to L , 680

ΦX,L ,F Hilbert polynomial, 331 π1 (X, x ¯) algebraic fundamental group, 130 π1 (X, x) topological fundamental group, 115 (d)

K0′ (X) Grothendieck group of scheme, 315 K0 (A)

Grothendieck group of abelian category, 314

K0 (T )

Grothendieck group of triangulated category, 314

K0 (X) (zero-th) K-group of scheme X, 317 K(A)

homotopy category, 758

K ∗ (A) bounded homotopy categories, 758 K ∗ (R)

bounded versions of K(R), 153

K ∗ (X) bounded versions of K(X), 153

PicC/S subfunctor of PicC/S of line bundles of degree d, 652 PicX/S (relative) Picard functor, 643 Π(X)

algebraic fundamental groupoid, 124

Π(X)

topological fundamental groupoid, 115

(P-OX -Mod) category of presheaves of OX -modules on X, 153 (PSh(X)) category of presheaves of sets on X, 153 ΨK

Fourier-Mukai transform, 699

QCoh(X) category of quasi-coherent OX -modules, 248

K• (f )

Koszul complex, 69

RF

right derived functor of F , 792

K(L )

kernel of φL , 667

Rf∗

derived direct image, 162

K(R)

category of complexes in (R-Mod) up to homotopy, 153

Rh

henselization of R, 108

Rhs

strict henselization, 108

K(X)

category of complexes of OX -modules up to homotopy, 153

(L1 · · · Lt · F ) intersection number, 323 (L1 · · · Lt · Z) intersection number, 323 ℓ(D)

Dimension of H 0 (C, OC (D)), 536

R HomB A (B, −) R Hom-functor to D(B), 456 O

R Hom OZ (OZ , −) Hom-functor to (OZ -Mod), X 458 R i f∗

higher direct image sheaf, 162

left derived functor of F , 793

σ ≤n ,

Lf ∗

derived inverse image, 197

(Sh(X)) category of sheaves of sets on X, 153

limI X ˆ M

limit, 740

Smk

category of smooth quasi-projective kschemes, 333

(classical) completion of M , 377

M∧

SX

derived completion of M , 383

Fourier-Mukai transform w.r.t. Poincar´ e bundle, 702

LF

σ ≥n

stupid truncation functors, 757

859 S(Z)

set of closed subschemes with support Z, 412

τ ≤n , τ ≥n truncation functors, 756 τX/S

trace form, 119

tdA

Todd class, 340

Tg

tangent sheaf/bundle, 20

Tor n (F , G ) Tor sheaf, 193 trX/S

trace homomorphism, 119

TX/S

tangent sheaf/bundle, 20

X an

analytification of X, 113

Xt

dual abelian scheme, 680

Y/Z

formal completion of Y along Z, 413

Z(C/k, T ) Zeta function of curve over finite field, 587

Index Abel morphism, 653 Abel-Jacobi morphism, 653 abelian category, 747 — semisimple, 824 abelian fppf-sheaf, 615 abelian scheme, 635 — dual, 663, 680, 686 — isogeny, 670 — Poincar´e bundle, 689 — polarization, 722 abelian variety, 554, 635 acyclic complex, 755, 756 acyclic object, 796 additive category, 745 — with translation, 772 additive cohomology theory, 334 additive functor, 746 additive subcategory, 746 adic formal scheme, 415 adjoint functor, 744 Adjoint functor theorem, 437 adjoint pair of functors, 744 adjunction — counit, 744 — morphisms, 744 — of triangulated functors, 777 — unit, 744 adjunction formula, 501 admissible exact short exact sequence, 369 admissible morphism, 821 Albanese scheme, 691 algebra — differential graded, 25 — exterior, 24 — graded commutative, 24 — separable, 39 — standard ´etale, 43 — strictly graded commutative, 24 algebra over formal scheme, 418 algebraic fundamental group, 130 algebraic space, 617 — group –, 620 — properties, 618 — quasi-compact, 618 — underlying topological space, 617 algebraically equivalent, 375, 697 algebraizable module, 423 algebraizable scheme, 417 ˇ alternating Cech complex, 180 alternating map, 88 amalgamated sum, 743 ample family of line bundles, 369

analytic space — complex, 113 — standard, 113 analytification, 113 antipode, 606 approximation property, 112 arithmetic genus, 527 arithmetic genus of a variety, 375 Artin-Schreier cover, 149 associated graded object, 764 atlas of algebraic space, 617 Auslander-Buchsbaum formula, 828 Base change for f × , 450 base change morphism, 206 Betti function, 344 bi-chain, 820 — terminate, 820 bi-chain condition, 820 bi-graded object, 764 bialgebra — graded, 629 bounded above complex, 756 bounded below complex, 756 bounded complex, 756 bounded spectral sequence, 765 branch locus, 121 Brown representability theorem, 437 Calabi-Yau variety, 512 canonical bundle, 52 canonical divisor, 52, 536 canonical embedding, 571 cartesian diagram, 743 Cartier divisor — relative effective, 375 Cartier dual, 632 category — abelian, 747 — abelian semisimple, 824 — additive, 745 — additive sub–, 746 — cocomplete, 740 — cofiltered, 741 — complete, 740 — connected, 742 — derived, 785 — enough K-injective objects, 799 — essentially small, 314 — exact, 822 — filtered, 741 — finitely cocomplete, 740

© Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature 2023 U. Görtz und T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes, Springer Studium Mathematik – Master, https://doi.org/10.1007/978-3-658-43031-3

861 — finitely complete, 740 — Giraud sub–, 753 — Grothendieck, 752 — homotopy, 758 — idempotent complete, 819 — Krull-Schmidt, 820 — localization, 782 — locally finitely generated, 754 — locally noetherian, 754 — opposite triangulated, 778 — plump sub–, 749 — R-linear, 746 — Serre sub–, 750 — triangulated, 773 — triangulated sub–, 778 — with enough injectives, 751 — with enough projectives, 751 category of complexes in ab. cat., 755 ˇ Cech cohomology for a complex of OX -modules, 182 ˇ Cech cohomology group, 182 ˇ Cech complex, 179, 182 ˇ Cech complex for a complex of OX -modules, 182 Chern character, 340 Chern class — of line bundle, 319 i-th Chern class, 336 Chern class of line bundle, 333 Chern polynomial, 336 Chern roots, 337 Chow’s theorem, 115 coboundaries, 756 cocommutative Hopf algebra, 606 cocomplete category, 740 cocycles, 756 codimension — of quasi-regular immersion, 75 coequalizer, 743 cofiltered category, 741 cofinal functor, 742 cofinal subcategory, 742 Cohen-Macaulay — coherent module, 487 — module, 828 — morphism, 489 coherator, 248 coherent — module, 292 — ring, 292 — scheme, 293 cohomological functor, 779 cohomologically bounded functor, 801 cohomologically flat morphism, 374 cohomologically proper, 311

cohomology object of complex, 756 cohomology of OX -module, 159 cohomology theory, 333 — additive, 334 — graded, 334 — morphism, 334 — multiplicative, 334 cohomology with support, 227 cokernel — in additive category, 746 — of morphism of complexes, 755 colimit, 739 commutative group scheme, 604 commutative Hopf algebra, 606 compact object, 437 compactification of a scheme, 464 compactly generated triangulated category, 437 complete category, 740 complete intersection ring, 84 complete normal model, 514 completely flat module, 433 completely intersecting, 274 completely intersecting immersion, 74 completely intersecting sequence, 71 completion — derived, 383 complex, 755 — acyclic, 755, 756 — bounded, 756 — bounded above, 756 — bounded below, 756 — category of -es, 755 — cohomology object, 756 — decomposable, 824 — derived complete, 380 — double, 759 — dual, 214 — dualizable, 215 — exact sequence, 755 — filtered, 770 — finite tor-dimension, 221 — homotopy equivalent, 758 — injective amplitude, 813 — injective dimension, 813 — K-flat, 191 — K-injective, 798 — K-projective, 798 — left F -acyclic, 796 — locally finite tor-dimension, 221 — m-pseudo-coherent, 216 — morphism of, 755 — of finite injective dimension, 813 — perfect, 211, 212, 255, 311 — pseudo-coherent, 216, 255, 308 — right F -acyclic, 796

862 — strictly m-pseudo-coherent, 230 — strictly perfect, 211 — strictly pseudo-coherent, 230 — sub–, 755 — termwise split sequence, 756 — tor-amplitude of, 221 — tor-dimension, 221 — total, 760 complex analytic space, 113 complex torus, 726 composition series, 749 comultiplication, 606 connected category, 742 connection, 30 conormal bundle, 75 conormal sheaf, 11, 64 conservative functor, 739 constant morphism, 636 content, 142 continuous group action, 129 convergent power series, 145 convergent spectral sequence, 766 convolution, 704 coregular sequence, 88 counit, 606 — of adjunction, 744 Cousin problem, 170 cup product, 207 — relative, 206 cup product in de Rham cohomology, 372 curvature of a connection, 30 curve, 513 — elliptic, 552, 656 — Gorenstein, 532 — hyperelliptic, 563 — Jacobian, 654 — monomial, 598 — plane –, 547 — relative, 584, 651 — zeta-function, 587 cuspidal singularity, 146 de Rham cohomology, 372 de Rham complex, 27 decomposable complex, 824 degenerate spectral sequence, 765 degree — of closed subscheme, 327 — of isogeny, 670 — of morphism, 518 — of proper morphism, 325 degree of a vector bundle, 575 degree of an ´etale cover, 118 degree of an isogeny, 735 δ-functor, 805

Index depth, 827 derivation, 5, 14 — graded, 25 derived category, 785 derived complete — along a closed subscheme, 388 — complex, 380 — sheaf, 388 derived completion, 383, 388 — along a closed subscheme, 388 derived direct image, 162 derived exact couple, 768 derived Hom functor, 811 derived inner Hom, 189 Derived Lemma of Nakayama, 431 derived tensor product, 193 descending filtration, 764 determinant line bundle, 28, 574 dga, 25 diagram — filtered, 742 — tor-independent, 278 I-diagram in category, 739 different, 121 differential form, 27 differential graded algebra, 25 differential p-forms, 25 dimension — projective, 828 dimension function, 479 dimension of a module, 827 discriminant, 120 distinguished triangle, 773 divisible module, 823 divisor — algebraically equivalent, 375 — canonical, 536 — numerically equivalent, 506 divisorial correspondence — functor of –, 659 double complex, 759 dual abelian scheme, 680, 686 dual abelian space, 663 dual complex, 214 dual homomorphism of abelian schemes, 664 dual Koszul complex, 238 dualizable complex, 215 dualizable module, 230 dualizing complex, 469, 471, 528 — normalized, 477 — relative, 481 dualizing module, 484 dualizing sheaf, 484, 528 — relative, 492

863 edge morphism in spectral sequence, 767 effaceable δ-functor, 806 elliptic curve, 552, 656 — j-invariant, 560 enough injectives, 751 enough K-injective objects, 799 enough projectives, 751 epimorphism — essential, 819 equalizer, 743 equivalence of extensions, 813 equivalence of triangulated categories, 776 essential epimorphism, 819 essentially small category, 314 ´etale, 42 ´etale cover, 117 — Galois, 126 — pointed, 125 ´etale covering, 101 — refinement, 101 ´etale fundamental group, 130 ´etale neighborhood, 106 etale topology — presheaf, 104 — sheaf, 104 Euler characteristic, 320, 344 Euler product, 590 Euler sequence, 23 exact category, 822 exact couple in abelian category, 768 — morphism, 768 exact functor, 748, 776 exact sequence, 747 — of complexes, 755 exact sequence of group schemes, 622 exact structure on an additive category, 821 exact triangle, 773 exponential map, 725 Ext group, 811 ˇ extended ordered Cech complex, 238 extension — equivalence, 9 — of algebra by module, 8 — of objects in abelian category, 813 — trivial, 9 extension by zero, 155 exterior algebra, 24 Fermat cubic, 600 fiber functor, 124 fiber product, 743 filtered category, 741 filtered complex, 770 filtered diagram, 742 filtered object, 764

— associated graded, 764 — morphism, 764 filtration — descending, 764 — finite, 764 filtration of vector bundle, 580 final object, 742 finite injective dimension, 813 finite length object, 749 finite tor-dimension, 221 finitely cocomplete category, 740 finitely complete category, 740 finitely generated object, 754 first Chern class — of line bundle, 319 Fitting decomposition, 820 flabby sheaf, 163 flasque sheaf, 163 flat dimension of OX -module, 231 flattening stratification, 365 flex, 600 formal completion along a closed subscheme, 413, 418 formal scheme, 415 — structure sheaf, 418 formal spectrum, 415 formally ´etale, 32 formally proper, 651 formally smooth, 32 formally unramified, 32 Fourier-Mukai transform, 699 — convolution, 701 fppf-covering, 614 fppf-quotient of a group scheme, 622 fppf-sheaf, 614 — abelian, 615 — of groups, 615 fppf-sheafification, 615 fppf-surjective, 620 Frobenius morphism, 545, 672 functor — additive, 746 — adjoint, 744 — cofinal, 742 — cohomological, 779 — cohomologically bounded, 801 — commuting with colimit, 743 — commuting with limit, 743 — conservative, 739 — δ-functor, 805 — exact, 748, 776 — left derivable, 793 — left derived, 793 — left exact, 745 — localizable, 784

864 — locally of finite presentation, 54 — of additive categories with translation, 772 — right derivable, 792 — right derived, 792 — right exact, 745 — right F -acyclic, 796 — shift, 755 — translation, 755 — triangulated, 776 — triangulated bi–, 810 — universally localizable, 784 functor of divisorial correspondences, 659 fundamental class, 334 fundamental group — algebraic, 130 — ´etale, 130 — of abelian variety, 676 fundamental groupoid, 124 G-ring, 109 GAGA principle, 114 Galois cover, 126 generating set of triangulated category, 437 generators of a category, 752 generically ´etale, 517 genus, 527 — arithmetic, 375 — geometric, 534 genus of a relative curve, 584 genus of Riemann surface, 525 genus-degree formula, 547 geometric genus, 534 geometric point, 106 geometrically regular, 48 geometrically unibranch — local ring, 110 — scheme, 110 Giraud subcategory, 753 Godement resolution, 164 Gorenstein morphism, 493 Gorenstein ring, 474, 831 Gorenstein scheme, 492 graded bialgebra, 629 — graded commutative, 629 — homomorphism, 629 — strictly graded commutative, 629 graded cohomology theory, 334 graded commutative algebra, 24 graded commutative dga, 25 graded derivation, 25 graded Hopf algebra, 629 Graded module — Property(TF), 304 — Property (TN), 304 graded object, 764

Index — morphism, 764 graded tensor product, 69 Grothendieck category, 752 Grothendieck group — of abelian category, 314 — of scheme, 315 — of triangulated category, 314 Grothendieck-Riemann-Roch theorem, 341 group algebraic space, 620 group scheme, 604 — abelian scheme, 635 — Cartier dual, 632 — commutative, 604 — identity component, 608 group-like element, 631 groupoid, 739 Harder Narasimhan polygon, 582 Harder Narasimhan vector, 582 Harder-Narasimhan filtration, 571, 581 Harder-Narasimhan filtration for P1k , 574 Hasse principle, 551 henselian local ring, 92 henselian pair, 98 henselization, 108 henselization of pairs, 148 higher derived functor, 805 higher direct image sheaves, 162 Hilbert function, 332 Hilbert polynomial, 331 Hirzebruch-Riemann-Roch formula, 341 HN filtration, 581 Hodge decomposition, 524 Hodge numbers, 368 Hodge spectral sequence, 372 homogeneous G-space, 622 homomorphism — of graded bialgebras, 629 — of Hopf algebras, 606 homotopic morphisms of complexes, 757 homotopically injective, 798 homotopically projective, 798 homotopy category, 758 homotopy colimit, 818 homotopy equivalence, 758 homotopy equivalent complexes, 758 homotopy limit, 816 homotopy of morphism of complexes, 757 Hopf algebra — cocommutative, 606 — commutative, 606 — graded, 629 — homomorphism, 606 hypercohomology, 159 hypercohomology spectral sequence, 168, 808

865 hyperelliptic curve, 563 hyperelliptic involution, 564 I-adic completion of a module, 377 I-adic topology, 378 I-adically complete, 378 ideal of definition, 412 idempotent, 819 — split, 819 idempotent complete category, 819 idempotent completion, 819 identity component, 608 — of group functor, 649 image — of morphism of complexes, 755 imperfection module, 64 indecomposable object, 746 indecomposable vector bundle, 583 index — of line bundle, 710 inertia subgroup, 146 infinitesimal lifting criterion, 52 infinitesimal neighborhood, 11 inflexion point, 600 initial object, 742 injective — amplitude, 813 — dimension, 813 — hull, 754 — object, 751 injective module, 830 intersection number, 323 intersection pairing — on proper surface, 500, 506 invertible object in D(X), 473 isogeny, 670

— of an exact functor, 821 — of morphism of complexes, 755 Koszul complex, 69, 89 — dual, 238 Krull-Schmidt category, 820 K¨ unneth map, 280

j-invariant, 560 Jacobi criterion, 49 Jacobian of curve, 654 Jordan-H¨ older sequence, 749 Jordan-H¨ older theorem, 749 Jounanolou’s trick, 371 jumps of a filtration, 581

lattice, 561 Laurent expansion, 587 Laurent series, 587 lci morphism, 81 — relative dimension, 81 left adjoint, 744 left complete, 243 left derivable functor, 793 left derived functor, 793 left exact functor, 745 left F -acyclic, 796 left multiplicative system, 782 Legendre family of elliptic curves, 559 Leibniz rule, 5 length of an object, 749 Leray spectral sequence, 168 limit, 740 limit of spectral sequence, 766 line bundle, 572 — algebraically equivalent, 697 — index, 710 — non-degenerate, 706 local cohomology triangle, 177 local for ´etale topology, 102 local non-commutative ring, 820 local property for the ´etale topology, 101 local-global principle, 551 localizable functor, 784 localization functor, 786 localization of category, 782 locally complete intersection morphism, 81 locally finite tor-dimension, 221 locally finitely generated category, 754 locally noetherian category, 754 locally of finite presentation functor, 54 logarithmic derivative, 588 L¨ uroth’s theorem, 546

K-flat complex, 191 (zero-th) K-group of scheme, 317 K-injective, 798 K-injective resolution, 799 K-projective, 798 K3 surface, 512 K¨ ahler differentials, 5, 13 kernel — in additive category, 746 — of a map of group schemes, 605

m-isomorphism, 790 m-pseudo-coherent complex, 216 m-pseudo-coherent module, 216 map of open coverings, 181 mapping cone, 757 Matlis duality, 477, 830 Matlis module, 830 maximal prime-to-p-quotient, 141 meromorphic function — on Riemann surface, 524

866 Mittag-Leffler condition, 815 ML condition, 815 module — Cohen-Macaulay, 828 — coherent, 292 — completely flat, 433 — dimension, 827 — dualizable, 230 — m-pseudo-coherent, 216 — Matlis, 830 — over formal scheme, 417 — pseudo-coherent, 216 — reflexive, 497, 829 moduli scheme of abelian varieties, 733 monomial curve, 598 morphism — adjunction, 744 — Cohen-Macaulay, 489 — cohomologically flat, 374 — cohomologically proper, 311 — completely intersecting immersion, 74 — constant, 636 — ´etale, 42 — formally ´etale, 32 — formally smooth, 32 — formally unramified, 32 — fppf-surjective, 620 — generically ´etale, 517 — Gorenstein, 493 — lci, 81 — locally complete intersection, 81 — of cohomology theories, 334 — of complex analytic spaces, 113 — of double complexes, 760 — of exact couples, 768 — of extensions, 813 — of filtered objects, 764 — of formal schemes, 416 — of graded objects, 764 — of pointed S-schemes, 639 — of pointed schemes, 132 — of spectral sequences, 764 — of spectral sequences with limit term, 766 — of triangles, 773 — of triangulated bi-functors, 810 — pseudo-coherent, 312 — quasi-regular immersion, 74 — regular immersion, 74 — semiseparated, 288 — separable, 518 — smooth, 49 — — infinitesimal lifting criterion, 52 — smooth in a point, 49 — syntomic, 86 — transversal, 67

Index — universally cohomologically proper, 511 — unramified, 39 — weakly ´etale, 100 — weakly smooth, 100 — weakly unramified, 100 morphism of algebraic spaces, 617 — formally proper, 651 — proper, 619 — quasi-compact, 619 — separated, 619 morphism of complexes, 755 — cokernel, 755 — homotopic, 757 — image, 755 — kernel, 755 multiplicative cohomology theory, 334 multiplicative system, 782 multiplicity of point on a curve, 529 Mumford bundle, 663 Nakai-Moishezon criterion, 329 Nakayama’s lemma — derived, 431 N´eron-Severi group, 697 nil-immersion, 32 nodal curve, 530 node, 530 noetherian object, 754 non-degenerate line bundle, 706 normal bundle, 75 normalization — of curve, 514 null system, 784 number of geometric points, 145 numerical polynomial, 323, 364, 371 numerically equivalent, 506 opposite triangulated category, 778 order of thickening, 32 ˇ ordered Cech complex, 180 ordinary abelian variety, 676 ordinary double point, 530 ordinary multiple point, 529 p-rank of abelian variety, 675 π1 -proper, 141 path between geometric points, 131 perfect complex, 211, 212, 255, 311 Picard functor — of elliptic curve, 554 — relative, 643 plane curve, 547 plump subcategory, 749 Poincar´e bundle, 689 pointed ´etale cover, 125

867

q-injective, 798 q-projective, 798 qis, 759 quasi-flasque sheaves, 165 quasi-isomorphism, 759 quasi-regular immersion, 74 — codimension, 75 quotient by a finite group, 126 quotient of a group scheme, 622

relative cup product, 206 relative curve, 584, 651 relative dimension — of CM morphism, 490 — of lci morphism, 81 — of smooth morphism, 49 relative dualizing complex, 481 relative dualizing sheaf, 492 relative Picard-functor, 643 representative of an object in a derived category, 786 resolution property, 260 retract, 819 Riemann existence theorem, 123 Riemann Mapping Theorem, 526 Riemann surface, 523 — of genus 1, 560 Riemann-Roch theorem — for abelian varieties, 707 — for curves, 535 — for surfaces, 501 right adjoint, 744 right derivable functor, 792 right derived bi-functor, 811 right derived functor, 792 right exact functor, 745 right F -acyclic, 796 right F -acyclic object, 806 right localizable functor, 784 right multiplicative system, 782 rigidification of a line bundle, 688 ring — approximation property, 112 — coherent, 292 — complete intersection, 84 — G-ring, 109 — geometrically unibranch, 110 — Gorenstein, 474, 831 — local henselian, 92 — seminormal, 734 — strictly henselian local, 107 — unibranch, 110 — universally coherent, 294 ring homomorphism — small, 55

ramification divisor, 542 refinement of coverings, 181 refinement of ´etale covering, 101 reflexive — module, 829 — OX -module, 497 regular immersion, 74 regular sequence, 71 relative Cartier divisor, 375

saturated multiplicative system, 784 saturation of a submodule, 572 scheme — coherent, 293 — geometrically regular, 48 — geometrically unibranch, 110 — Gorenstein, 492 — pointed, 639 — semiseparated, 236, 288

pointed presheaf, 639 — morphism, 639 pointed S-scheme, 639 — morphism, 639 pointed scheme, 132 polarization, 722 — principal, 722 polynomial — Hilbert, 331 — numerical, 323, 364, 371 — Snapper, 323 I-power torsion module, 296 presheaf — for ´etale topology, 104 — pointed, 639 presheaf (on category), 614 presheaves of OX -modules, 153 primitive polynomial, 92, 142 principally polarized abelian variety, 730 profinite completion, 138, 149 profinite group, 128 projection formula, 269, 333 — derived, 272 projective dimension, 828 projective object, 751 proper base change theorem, 226 proper continuous map, 225 proper direct image, 154 proper normal model, 514 proper support, 308 pseudo-coherent complex, 216, 255, 308 pseudo-coherent module, 216 pseudo-coherent morphism of schemes, 312 pullback, 743 pushout in a category, 743

868 — unibranch, 110 — universally coherent, 294 scheme of connected components, 149 section conjecture, 139 sections with support, 176 seminormal ring, 734 semiseparated morphism of schemes, 288 semiseparated scheme, 236, 288 semisimple abelian category, 824 semisimple object, 749 semistable vector bundle, 578 separable algebra, 39 separable morphism, 518 separated continuous map, 225 sequence — completely intersecting, 71 — exact, 747 — regular, 71 — short exact, 747 — weakly regular, 71 Serre duality, 493, 495, 496 Serre subcategory, 750 Sharp’s conjecture, 481 sheaf — derived complete, 388 — extension by zero, 155 — flabby, 163 — flasque, 163 — for ´etale topology, 104 — for fppf-topology, 614 sheafification, 615 shift functor, 755 short exact sequence, 747 Siegel upper half plane, 729 simple object, 749 Six Functor Formalism, 208 slice category, 739 slope of a vector bundle, 577 small ring homomorphism, 55 smooth morphism of schemes — in a point, 49 Snapper polynomial, 323 specialization map, 141 spectral sequence, 764 — bounded, 765 — convergent, 766 — degenerate, 765 — edge morphism, 767 — hypercohomology, 808 — limit, 766 — morphism, 764 spectral sequence with limit term — morphism of —, 766 spectral space, 174 split ´etale cover, 117

Index split extension, 813 split idempotent, 819 split sequence, 746 splitting principle, 336 stable vector bundle, 578 stalk — of ´etale presheaf, 106 standard analytic space, 113 standard ´etale algebra, 43 Stein factorization, 399 strict henselization, 108 strictly full subcategory, 739 strictly graded commutative algebra, 24 strictly graded commutative dga, 25 strictly henselian local ring, 107 strictly m-pseudo-coherent complex, 230 strictly perfect complex, 211 strictly pseudo-coherent complex, 230 structure sheaf — of formal scheme, 418 stupid truncation functor, 757 subbundle of a vector bundle, 572 subcategory — cofinal, 742 — strictly full, 739 subcomplex, 755 subgroup scheme, 605 — normal, 605 supersingular elliptic curve, 676 surface, 500 symmetric — morphism to dual abelian scheme, 722 symmetric line bundle, 674 syntomic morphism, 86 tangent bundle, 20 — virtual, 339 tangent sheaf, 20 tangent space, 22 Tate module, 678 tensor product — grading, 69 tensor product of complexes, 762 termwise split sequence, 756 (TF) Property, 304 Theorem of Riemann-Hurwitz, 542, 544 Theorem of Riemann-Roch, 535 — for abelian varieties, 707 — for surfaces, 501 Theorem of the cube, 410, 666 Theorem of the square, 666 theta function, 728 thickening, 32 — order, 32 — universal, 64

869 (TN) Property, 304 (TN)-injective, 304 (TN)-isomorphism, 304 (TN)-surjective, 304 Todd class, 340 topological space — spectral, 174 Tor sheaf, 193 tor-amplitude, 221 tor-dimension, 221 tor-dimension of OX -module, 231 tor-independent, 279 tor-independent morphisms, 278 torsion submodule, 572 total complex, 760 trace form, 119 trace homomorphism, 119 trace map, 445 translation, 604 translation functor, 755, 772 transversal morphism, 67 triangle, 772 — distinguished, 773 — exact, 773 — morphism of –s, 773 triangle associated to an exact sequence, 787 triangulated adjoint pair, 777 triangulated bi-functor, 810 triangulated category, 773 — compactly generated, 437 — equivalence, 776 — generating set, 437 triangulated functor, 776 triangulated subcategory, 778 truncation functor, 756 — stupid, 757 twisted inverse image functor, 464 unibranch — local ring, 110 — scheme, 110 unit — of adjunction, 744 universal δ-functor, 806 universal thickening, 64 universally coherent ring, 294 universally coherent scheme, 294 universally cohomologically proper morphism of schemes, 511 universally localizable functor, 784 unramified, 39 vector bundle, 572 virtual tangent bundle, 339

weakly ´etale, 65 weakly ´etale morphism, 100 weakly proregular ideal, 432 weakly proregular sequence, 385, 432 weakly proregular subscheme, 432 weakly regular sequence, 71 weakly smooth morphism, 100 weakly unramified morphism, 100 Weierstraß equation, 552, 657 Weil conjectures for curves, 588 Z-adic completion, 414 Zariski pair, 148 Zariski’s Connectedness Theorem, 402 zeta function of curve, 587