Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform 9783642074721, 3642074723

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Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Foige

A Series of Modern Surveys in Mathematics

Editorial Board

s. Feferman, Stanford

M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollar, Princeton H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris M. Rapoport, Ktiln J.Tits, Paris D. B. Zagier, Bonn Managing Editor R. Remmert, Miinster

Volume 42

Springer-Verlag Berlin Heidelberg GmbH

Reinhardt Kiehl Rainer Weissauer

Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform

Springer

Reinhardt Kiehl Institut fUr Mathematik und Informatik Universitat Mannheim D7,27 68159 Mannheim, Germany e-mail: [email protected] Rainer Weissauer Mathematisches Institut Universitat Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg, Germany e-mai!: [email protected]

Library of Congress Cataloging-in-Publication Data Kiehl, Reinhardt. Weil coniectures, perverse sheaves, and l-adic Fourier transform / Reinhardt Kiehl, Rainer Weissauer. p. cm. - (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, v. 42) Includes bibliographica! references and index. ISBN 978-3-642-07472-1 ISBN 978-3-662-04576-3 (eBook) DOI 10.1007/978-3·662-04576-3 1. Weil coniectures. 2. Homology theory. 3. Sheaf theory. 1. Weissauer, Rainer. II. Title. III. Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 42. QA564 .K5 2001 516.3'52··dc21

2001031426

Mathematics Subject Classification (2000): 14-XX

ISSN 0071-1136 ISBN 978-3-642-07472-1 This work is subiect to copyright. AII rights are reserved, whether the whole or part of the materia! is concerned, specifically the rights of translation, reprinting, reuse of Ulustrations, recitation, broadcasting, reproduction on microfilms or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Sprioger-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

http://www.springer.de © Springer. Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcoverrepriot ofthe hardcover Ist edition 2001

Typeset by the authors using a Springer TEX macro package. Edited by Kurt Mattes, Heidelberg, using the MathTime fonts. Printed on acid·free paper SPIN 10723113 44/3142LK - 5 43210

Preface

The initial motivation for writing this book was given by N. Katz and his review of the book E. Freitag / R. Kiehl, Etale Cohomology and the Weil Conjecture Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag in the bulletin of the AMS. In that review N. Katz remarks that it is especially the generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La Conjecture de Weil II", that had the most relevant applications in recent years. He continues: . .. The book does not discuss Weil II at all, except for a two page summary (IV,S) of some of its main results near the end. Perhaps someday if the authors feel ambitious ... Around that time we gave lectures in the Arbeitsgemeinschaft Mannheim-Heidelberg on Laumon's work, especially on his use of the Fourier transform for etale sheaves and his proof of the Weil conjectures. Therefore, we, that is one of the previous authors and the new author, decided to present these important and beautiful methods of Laumon in the form of this book. Pursuing this plan further the authors immediately felt that Deligne's work on the Weil conjectures was closely related to the sheaf theoretic theory of perverse sheaves. It seemed that only in this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore, it was desirable or even necessary for us to include the theory of middle perverse sheaves (as in asterisque 100) completely as a second main part in this Ergebnisbericht. The l-adic Fourier transform appears as a tool and proves to be a useful technique providing natural and simple proofs. This part of the book was also based on the lectures given in the Arbeitsgemeinschaft Mannheim-Heidelberg. To round things off we present significant applications of these theories. For this purpose we included three chapters on the following topics: the Brylinski-Radon transform including a proof of the Hard Lefschetz Theorem, estimates for exponential sums reviewing the results of Katz and Laumon, and, finally, a chapter on the Springer representations ofWeyl groups of semisimple algebraic groups. In these applications the l-adic Fourier transform always turns out to be of importance. So, looking back, it appears to us that in the course of writing this Ergebnisbericht, we were seemingly attracted by this elegant device, both by its vigour and its beauty.

VI

Preface

The authors want to express thanks to all those who gave valuable comments or encouragement at the various stages of this project. Special thanks go to Dr. 1. Ballmann, Dr. H. Baum, Dr. D. Fulea and Dr. U. Weselmann, to all whom we are indebted for their help during the final preparation of the manuscript. We also heartily thank the staff of Springer-Verlag for their friendly cooperation.

Reinhardt Kiehl Rainer Weissauer

Table of Contents

Introduction. . . . . . . . . . . . . . ..................................... . I.

The General Weil Conjectures (Deligne's Theory of Weights) ...... 1.1 Weil Sheaves ............................................. 1.2 Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.3 The Zariski Closure of Monodromy .......................... 1.4 Real Sheaves ............................................. 1.5 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.6 Weil Conjectures (Curve Case) .............................. 1.7 The Weil Conjectures for a Morphism (General Case) .. . . . . . . . .. 1.8 Some Linear Algebra ...................................... 1.9 Refinements (Local Monodromy) .... . . . . . . . . . . . . . . . . . . . . . . ..

5 5 13 25 33 38 45 52 54 58

II.

The Formalism of Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . .. 11.1 Triangulated Categories .................................... 11.2 Abstract Truncations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.3 The Core of at-Structure ................................... 11.4 The Cohomology Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. b 11.5 The Triangulated Category Dc (X, Qil) .. . . . . . . . . . . . . . . . . . . . . .. 11.6 The Standard t-Structure on Dg(X, 0). . . . . . . . . . . . . . . . . . . . . . . .. 11.7 Relative Duality for Singular Morphisms ...................... 11.8 Duality for Smooth Morphisms .............................. 11.9 Relative Duality for Closed Embeddings ...................... 11.10 Proof of the Biduality Theorem .............................. 11.11 Cycle Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.12 Mixed Complexes .........................................

67 67 74 77 81 86 98 106 112 116 119 123 129

III. Perverse Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111.1 Perverse Sheaves .......................................... 111.2 The Smooth Case .......................................... 111.3 Glueing .................................................. I1I.4 Open Embeddings ......................................... I1I.5 Intermediate Extensions .................................... I1I.6 Affine Maps ..............................................

135 135 137 139 144 147 153

VIII

Table of Contents

III.7 111.8 111.9 111.10 111.11 II1.12 111.13 111.14 IIU5 111.16

Equidimensional Maps ..................................... Fourier Transform Revisited ................................. Key Lemmas on Weights ................................... Gabber's Theorem ......................................... Adjunction Properties ...................................... The Dictionary ............................................ Complements on Fourier Transform .......................... Sections ................................................. Equivariant Perverse Sheaves ................................ Kazhdan-Lusztig Polynomials ...............................

156 159 161 167 169 173 177 181 183 189

IV. Lefschetz Theory and the Brylinski-Radon Transform. . . . . . . . . . .. 203 IY.1 The Radon Transform ...................................... 203 IY.2 Modified Radon Transforms ................................. 207 IV.3 The Universal Chern Class .................................. 215 IV.4 Hard Lefschetz Theorem ................................... 217 IY.5 Supplement: A Spectral Sequence ............................ 221 V.

Trigonometric Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 225 Y.1 Introduction .............................................. 225 V.2 Generalized Kloosterman Sums .............................. 226 V.3 Links with l-adic Cohomology .............................. 229 V.4 Deligne's Estimate ......................................... 230 Y.5 The Swan Conductor ....................................... 231 V.6 The Ogg-Shafarevich-Grothendieck Theorem ................. 236 Y. 7 The Main Lemma ......................................... 237 V.8 The Relative Abhyankar Lemma ............................. 240 Y.9 Proof of the Theorem of Katz ................................ 241 V.1O Uniform Estimates ......................................... 244 Y.11 An Application ............................................ 246 Bibliography for Chapter V ....................................... 248

VI. The Springer Representations ................................ , 249 VI. 1 Springer Representations of Weyl Groups of Semisimple Algebraic Groups ............................. 249 VI.2 The Flag Variety,X;J ....................................... 253 VI.3 The Nilpotent Variety ~/f/' .................................. 256 VI.4 The Lie Algebra in Positive Characteristic ..................... 261 VI.5 Invariant Bilinear Forms on 9 ................................ 263 VI.6 The Normalizer of Lie(B) .................................. 264 VI. 7 Regular Elements of the Lie Algebra 9 ........................ 264 VI. 8 Grothendieck's Simultaneous Resolution of Singularities ........ 266 VI.9 The Galois Group W ....................................... 269 VU 0 The Monodromy Complexes and ' •••••••••••••••••••••••• 272 VI.11 The Perverse Sheaf \11 .•.................•.................• 276

Table of Contents

IX

VI.12 The Orbit Decomposition of \11 .............................. 278 VI.13 Proof of Springer's Theorem ................................ 281 VI.14 A Second Approach ....................................... 286 VI.l5 The Comparison Theorem .................................. 290 VI.16 Regular Orbits ............................................ 295 VI.17 W -actions on the Universal Springer Sheaf .................... 301 VI.18 Finite Fields .............................................. 310 VI.19 Determination of cT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Bibliography for Chapter VI ...................................... 319

Appendix A. B. C. D.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ijrSheaves ............................................... Bertini Theorem for Etale Sheaves ........................... Kummer Extensions ....................................... Finiteness Theorems .......................................

323 323 333 336 338

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 355 Glossary ....................................................... 371 Index .......................................................... 373

Frequently Used Notation

Usually 1 denotes a prime number, '1',/ the ring of I-adic integers, Q/ its quotient field of I-adic numbers. Finite extension fields of Q/ are usually denoted E. Q/ denotes the algebraic closure of Q/ and T : Q/ -+ C denotes a chosen isomorphism between the field Q/ and the field C of complex numbers. For tensor products of Q/-vector spaces we often write ® instead of ®ij[' if confusion is unlikely from the context. Usually K denotes a finite field or an algebraically closed field which has a characteristic different from I. If K is a finite field, k denotes its algebraic closure. A~ will denote the n-dimensional affine space over the base scheme S. Often we write As instead of A1. If the base scheme is fixed, we sometimes also write An instead of A~. Similar Ipm, respectively IP'~, denotes the n-dimensional projective space over the base scheme S. Schemes of finite type over K will be called finitely generated schemes over K. They will be often called algebraic varieties or algebraic manifolds over K (not following the usual notation!). If we write Xo for an algebraic variety, the subscript o will be frequently used to indicate that Xo is an algebraic variety over K. Usually X will then denote the scheme Xo XSpec(K) Spec(k) = Xo ®K k over k, which is obtained by base field extension. A similar notation will be used for morphisms. Furthermore, IXol denotes the set of closed points of the scheme Xo. Also if .~ is a sheaf on Xo, its pullback to X will be denoted ;r;. The Galois group Gal(k/K) acts on the field k (from the left), hence acts on Spec(k) (from the right). Similarly for schemes Xo over Spec(K) we get an induced (right) action of Gal(k/K) on X = Xo XSpec(K) Spec(k) from its action on the second factor. For Y E Gal(k/K) the corresponding automorphism of X will be denoted Yx. For schemes Xo, Yo over Spec(K) the group Gal(k/K) acts on H omSpec(k)(Y, X) via f t-+ JY = yxo f oyyl. For Yo = Spec(K), this defines a right action of Gal(k/K) on the set of k-valued points of X. We refer to these actions as Galois actions. For a closed point x of the scheme Xo, K(X) will denote the residue field of x. This is a finite field. N (x) will denote the number of its elements. A geometric point of X0 over K with values in a separable closed extension field k' of K, which lies over the point x E IXol, will usually be denoted x. Any such x defines an embedding of the field K(X) into the field k: K C K(X) C k. A morphism f : X -+ Y in the category of finitely generated schemes over K is called compactifiable if it can be factorized in the form f = 0 j, where j is

7

XII

Frequently Used Notation

an open immersion and where 7 is a proper morphism. This is always the case for quasiprojective morphisms. For the constant sheaf Q!/ y on a scheme Y we often write oy. We do this in particular in situations where i : Y "-+ X is a closed embedding. Then we often write oy instead ofi*Q/y, if there is no confusion possible. In complicated formulas we sometimes write K (2m) instead of K[2m](m), where [2m] indicates a shift of a complex by 2m to the left, and (m) indicates an m-fold Tate twist. We sometimes write i*[n]K instead of i*(K)[nl, if n has a meaning in terms of the morphism i (a relative dimension). We freely use results of the book [FK]. In fact we will assume in this book most of the main results that were proven in [FK], without further mention of references. This includes, in particular, basic facts of etale cohomology theory, such as the proper base change theorem, the Poincare duality theorem for smooth varieties over a field, and the Grothendieck trace formula, but besides this also some of the more elementary concepts, such as, for instance, the notion of etale sheaves. Because of its importance the reader will find a brief review of the statement of the Grothendieck trace formula at the beginning of the first section of Chap. I. Further information can be found also in the Appendix A of this book.

Introduction

This Ergebnisbericht is directed to a reader who is acquainted with the theory of etale cohomology to the extent, say, of the first two chapters of the book of E. Freitag / R. Kiehl Etale Cohomology and the Weil Conjecture [FK]. In Chap. I of this book the theory of weights of l-adic sheaves and the proof of the generalized Weil conjecture for morphisms are described. Historically one of the first examples for the theory of weights is provided by the Gauss sums. If K = IF'q is a finite field with q elements and if 1/1 : K --+ 1[:* is a nontrivial additive character, the Gauss sums

LX (x)1/I(y . x)

XEK

are the values of the Fourier transform of a nontrivial multiplicative character X --+ 1[:*, where both the character and its Fourier transform are viewed as a function with support in K*. Whereas the character X has absolute value one on its support, its Fourier transform has the absolute value q 1/2 on its support. This is the simplest case, illustrating how the Fourier transform shifts weights. Of course this is rather elementary in the context above. If a nontrivial function f and its Fourier 10n K have, by chance, the same support and constant absolute values equal to c, respectively d, then d = q lf2 . c. This is an immediate consequence of Plancherel's formula K*

XEK

XEK

In Chap. I it is shown that the basic idea of this argument can be used to prove the generalized Weil conjectures in the curve case. To make this work, the method has to be adapted to pertain to less rigid situations. Fortunately the Grothendieck fixed point formula gives all the help required. One derives from it rnixedness and semicontinuouity statements, which finally allow to extend the kind of argument sketched above. The details of the arguments are carried through in the first seven sections of Chap. I. The early sections deal with some of the more elementary properties of weights; here the fundamental estimate for real sheaves proved by using the Rankin trick (Sect. 4) is an important step. The Fourier transform of a complex of l-adic sheaves and its remarkable properties form the content of Sect. 5. In the central Sect. 6 we finally give the proof of the Weil conjecture, essentially following Laumon. This fundamental

2

Introduction

result is the basis for the remaining part of Chap. I, which is then devoted to the proof of the generalized conjecture. In the last section of Chap. I on the local monodromy we already use derived categories and some of the material of the Chap. II. Although we might have postponed this, we preferred to have it included in Chap. I. The main sources for this chapter were La conjecture de Weil II, by Deligne [Del], and Laumon's work: Transformation de Fourier, constants d'equations fonctionelles et conjecture de Wei! Conjecture [Lau]. In Chap. II the technical tools are provided; these are needed in the following chapter on perverse sheaves. In Sects. 3 and 4 of Chap. II abstract truncation structures are defined and their most important properties are studied. In particular, in Sect. 3 we describe the proof of the fact that the core of such a t-structure is an abelian category. In Sect. 7 the first important example of such a t-structure, namely the standard t-structure on the category D~ (X, '1',/), is constructed. The triangulated category D~(X, l/) is not defined as the derived category of an abelian category, but rather as a projective limit of the derived categories D~tf(X, '1',/ In'1',). Its objects, therefore, are not complexes of l-adic sheaves in the usual sense. So the natural simple truncation operators Tn for complexes are not defined a priori. It is therefore necessary to defi~e on D~(X, '1',/) an abstract t-structure - called the standard t-structure - in a rather involved way to get a substitute for the non-existing naive truncation operators of complexes. The core of this triangulated category is isomorphic to the abelian category of l-adic sheaves. In Sects. 7-9, using the absolute Poincare duality for smooth quasiprojective schemes over a field [FK, chap. ILl, theorem 1.13], we develop relative Poincare duality for singular morphisms as far as this is needed in Chap. III. After Chap. I, Chap. III is the second central chapter of this book. In this chapter we place the theory of global weights and the notation of purity of sheaves into the framework of the category of perverse sheaves. It is only in this generality that Deligne's theory of weights obtains its final form. First we develop the complete theory of (middle) perverse sheaves as in asterisque 100 [BBD]. Then this is supplemented by important theorems on the purity and weights of perverse sheaves. One of the main results is Gabber's theorem on the semisimplicity of pure complexes. At this point Fourier analysis again simplifies the picture. It significantly helps to prove the key lemmas in Sect. 10. In Sect. 12 we show how properties of Wei1 sheaf complexes are reflected by properties of certain functions related to these complexes, as already considered in the case of sheaves in Chap. I, Sect. 2. For the proof a variant of Deligne's Fourier transform appears. For this new "Fourier transform" the affine algebraic group An is replaced by another commutative algebraic group, a product En of elliptic curves. In the last section, as a first application of the theory of perverse sheaves, the Kazhdan-Lusztig polynomials - known for the remarkable role they play in the representation theory of groups of Lie type - are constructed using perverse sheaves.

Introduction

3

The following chapters are devoted to applications of the theory. In Chap. IV a new transformation for etale complexes, the Brylinski-Radon transform, is introduced and examined. It is related to Deligne's Fourier transform. In this context we prove the hard Lefschetz theorem. One of the most beautiful applications of Grothendieck's trace formula and of Deligne's theory of weights are non-trivial estimates for trigonometric sums such as the well-known Kloosterman sums, which generalize the Gauss sums. Chapter V tries to convey the basic ideas of Katz and Laumon on the existence of uniform estimates for exponential sums of this type. Such estimates reflect a remarkable quality of the Deligne-Fourier transform. It behaves as if there exists a kind of global Fourier transformation on the affine space over Spec(Z) from which we seemingly get the Fourier transform on the affine spaces over Spec(Z/ pZ) - not uniquely determined - by reducing modulo p for almost all prime numbers p. Chapter VI plays a special role and leads to further areas. It deals with the mysterious representations of Weyl groups of semisimple algebraic groups, discovered by Springer and named after him. Here we venture into the area of algebraic groups and of the representations of finite groups of Lie type. In Sects. 2-9 we report all the necessary background material, e.g. Lie algebras mod p, Grothendieck's simultaneous resolution of singUlarities etc. Proofs are carried out only where this is necessary and possible. The main content of Sects. 10-18 is Brylinski's construction of the Springer representations using perverse sheaves, using Gabber's decomposition theorem and in particular using the Fourier transform. Here it was our intention to present the ideas outlined by Brylinski in [49, § 11 pp. 119-128] in greater detail, to make them available to a larger readership. Some other constructions are discussed as well, and their mutual relations and especially their relations to Springer's original construction are examined. The appendices contain useful supplements. Appendix A represents a bridge between the book of Freitag/Kiehl [FK] and what is required for this Ergebnisbericht. In Freitag/Kiehl's book the theory of Q/-sheaves was developed. Without difficulty it is shown in Appendix A how the essential results on Q/-sheaves - e.g. Poincare duality and Grothendieck's trace formula - carry over to the case of sheaves over a finite extension field E of the field Q/ of l-adic numbers. These are prerequisites for defining the l-adic Fourier transform. The headings of the Appendices B and C certainly speak for themselves. These appendices contain auxiliary results which are needed in Chap. I. For the convenience of the reader in Appendix D we finally present the proof of the finiteness theorems for the direct image functor in the case of non-proper morphisms. For the sake of completeness - it is not needed for this book - the finiteness theorems are proved also for the case of mixed characteristics, and furthermore the corresponding theorems for vanishing cycles and for nearby cycles are included, with short proofs.

I. The General Weil Conjectures (Deligne's Theory of Weights)

1.1 Weil Sheaves Let K be a finite field and k its algebraic closure. Fix a prime number I. The number q of elements of K will always be assumed not to be divisible by the prime number I. The Galois group Gal (k / K) of k over K contains the arithmetic Frobenius element CJ = CJk/K, which acts on k as the automorphism

This arithmetic Frobenius is a topological generator of the pro-cyclic Galois group Gal(k/K). Its inverse element F is called the geometric Frobenius automorphism. The dense cyclic subgroup of the Galois group generated by F is called the Weil group W(k/K) of K. Under the map W(k/K) ::;::: Z

F

r-+l

.z

W(k/K) becomes canonically isomorphic to Z, such that Gal(k/K) ::;::: becomes the pro finite completion of its subgroup W (k / K). More generally for arbitrary schemes X over K one has the morphism CJX/K : X --+ X, which is the identity on the underlying space and is defined by a f-+ a q on the structure sheaf. These morphisms CJX/K are functorial in the category of schemes over K. Let X 0 be a finitely generated scheme Xo over K. Let X be the scheme Xo x SpeC(K) Spec(k) = Xo &h k over k, obtained by base field extension. Similar notation will be used for morphisms. Also if ;~o is a sheaf on Xo, its pullback to X will be denoted ;~ . By its functoriality the morphism CJX/K can be written in the form (idxo x CJk/K) 0 Frx, where Frx : X --+ X

is now a morphism over k. It is called the Frobenius endomorphism of X.The morphism F r x is finite, hence proper. On the other hand the geometric Frobenius element F E W(k/K) C Gal(k/K) acts on the scheme via its Galois action, which is defined by Fx = idxo x FSpec(k). The automorphism Fx is called the Frobenius automorphism of X. It is an automorphism over Spec(K) but not over Spec(k). For schemes Xo, Yo over Spec(K) the Frobenius acts on the k-morphisms HomSpec(k)(Y, X) by conjugation f r-+ fF = Fx 0 f 0 FyI. This defines an action of F on the k-valued

6

I. The General Weil Conjectures (Deligne's Theory of Weights)

points of X. This Galois action of F and the action of Frx on the set of k-valued points coincide. We write X F = X (K) for the set of fixed points of F on the set X (k) of k-valued points of X. Example. For the affine space Xo = .ho, the action of F respectively Frx on .ho(k) k is given by k :3 a f-+ F- 1(a) = (T(a) = a q .

=

Let x E IXol be some closed point of the scheme XO. Let K(X) be the residue field of x and let N (x) denote the number of elements of the finite field K (x). A geometric point of Xo over K with values in a separably closed extension field of k, which lies over the point x E IXo I, will usually be denoted x. Any such x defines an embedding of the field K(X) into the field k: K C K(X) C k. Put d(x)

=

[K(X) : K] .

One of the most important results of the theory of etale sheaves is the Grothendieck trace formula and the corresponding formula for the L-series of a Q/-sheaf .(~ on an algebraic scheme Xo over K. See for instance [FK] or [SGAS].ln [FK] this formula is proven in the context of etale Ql/-sheaves. We also refer the reader to the Appendix A of this book for the precise definition of the notion of Ql/-sheaves. It is also explained there, how the results proved in [FK] extend to the case of Q/-sheaves. Let us now formulate this important result of Grothendieck. Let Xo be an algebraic scheme and let 0J be an etale Ql[-sheaf on Xo. Then there exists a canonical isomorphism * . F rx'y * ( '~) ---+ ~ ,y (~ . F r:t;. The existence of such an isomorphism can be reduced to the fact, that for an etale algebraic space 7T : Go --+ Xo over Xo we have a diagram

X - : W(k/K) ~ Z:: ------ Q/

.

Such a character is completely determined by the image b Frobenius element F. We write

= 1> (F)

of the geometric

for this Wei I sheaf, respectively its pullback from Spec(K) to Xo.

Theorem 1.4 Let Xo be an algebraic scheme over K, and let Xo. Then

.'~

be a Weil sheaf on

(1) Let Xo be normal and geometrically connected over K and let .(~ be an irre-

ducible smooth sheaf of rank n on Xo. Then .~o is an etale sheaf iff the determinant sheaf 1\ n .(~ is an etale sheaf Especially one can always find an elale sheaf.'YQ and an element b E -* Q/ such that .(~ is isomorphic to .:Y{) @

'hb .

(2) In the general case there always exists a chain of subsheaves

o=

;~0(0) C .(~(l) C .. , C .'~o(r) = ,(~

with the following property: Each factor sheaf is of the form 'c, (j)/,!::(j-l) ~ J7(j),o, (/ 'Y() - ,fO 'jt. By additivity the formula extends to all Weil sheaves, using I.1.4. D

1.2 Weights In this and in the following section we fix an isomorphism

between the field Q/ and the field of complex numbers. Definition 2.1 Let Xo be an algebraic scheme over K, real number.

:~

a sheaf on Xo and fJ be a

(1) Choose a k valued geometric point xfor each closed point x E IXol. Let K(X) be the residue field ofx. The Weil group W(k/K(X)) operates on the stalk '(~x' .(~ is said to be r-pure of weight fJ, iffor all points x E IXol and all eigenvalues Ci E Qz of the geometric Frobenii Fx

E

W(k/K(X))

the following holds: N(x) = #K(X) .

(2) The sheaf ;~o is said to be r -mixed, if there exists afinite filtration by subsheaves

o = .,~(o)

c .,~(l) c ... C ~(r)

=

;§o,

such that all factor sheaves .'~o (j) j'.f/o (j -I) are r -pure (of weight say fJ J). (3) The sheaf :f/o is said to be (pointwise) pure of weight fJ, iffor all isomorphisms r : Q/ -+ C the sheaf .'f/o is r-pure of weight fJ. (4) A sheaf .'f/o is mixed, if there exists afinitefiltration of·'f/o such that all successive factor sheaves are pure sheaves.

Remark 2.2 (Twisting) Let '(~o be a sheaf, which is r-pure for all isomorphisms r : Q/ -+ C, but with weights depending on r. Then there exists a pure sheaf .yo in the sense of 1.2.1 (3) and an element b E ijz such that ;~ is twisted of type

1. The General Weil Conjectures (Deligne's Theory of Weights)

14

Later in I.2.8 we will show: Suppose .(~ is i-mixed for all i : Q/ ~ C, then there exists a finite filtration of .~, whose successive factor sheaves are i -pure for all i : Q/ ~ C, hence can be derived from pure sheaves by a twist. We list the following elementary Permanence Properties

(1) Let fo : Xo ---+ Yo be a morphism over K, and ~ be a sheaf on Yo. Then the pullback f ('%) of.~ is i-pure of weight fJ, if 0i is i-pure of weight fJ·1f fo is surjective, then fo(~) is i-pure of weight fJ iff ~ is i-pure of weight fJ. (2) Suppose fo : Xo ---+ Yo is a finite morphism over K, then the direct image fo* (~) of a i -pure sheaf ~ of weight fJ is r -pure of weight fJ. (3) Let Xo be an algebraic scheme over K, 0J a sheaf on Xo and K' a finite extension field of K. Then .(~ is i-pure of weight fJ iff its inverse image on Xo ®K K' is i-pure of weight fJ.

o

Similar properties hold for pure sheaves. From (1 )-(3) one derives permanence properties for i-mixed sheaves in an obvious way. One exception should be noted. The statement "If the inverse image of a sheaf under a surjective morphism is i-mixed, then the sheaf is i-mixed itself"

is true in general only for finite morphisms. Definition 2.3 Let

~

be a sheaf on an algebraic scheme Xo. For fixed

we define

log(lr(a)1 2 )

. ..

w(;§O) = supxEIXol

i

SUPa

(

log N(x)

)'

where a runs through all the eigenvalues of Fx : .~x ---+ ;~x' if .(~ is nontrivial. Here:X always means a geometric point over x. For the trivial sheaf put w(~) = -00.

Next we are interested in the radius of convergence for the complex valued Lseries i L(Xo, .~, t) attached to the sheaf .~. We will give an estimate in terms of the weights of the sheaf .~ in its stalks. Lemma 2.4 Let.~ be a sheaf on an algebraic scheme Xo over K with the property w(.%) :::: fJ: For all closed points x E IXol and corresponding geometric points:X over x the eigenvalues a of

are bounded by

1.2 Weights

Ir(a)1 2 S N(x){3

=

15

qd(x l {3

d(x) = [K(X) : K] . Then the L-series r L(Xo, .~, t) is a power series in t r L(Xo, ;~, t) =

n

rdet(l - Fxtd(xl, ;~x)-l

XEIXol

converging for aUltl < q-{31 2- dim (Xol. It has neither poles nor zeros in this region. Proof Without restriction of generality we can assume Xo to be affine, reduced and irreducible. The Noether normalization theorem provides us with a finite morphism from Xo to the affine space of dimension dim(Xo). By a comparison with the affine space this gives us the following rough estimate for the number An = #XO(Kn) of geometric points over K with values in the field Kn of degree n over K

n = 1,2, ..

for some constant C. From our assumptions the traces of

can be estimated by

IrTr(F;)1 2 S r2. qvd(xl{3 ,

where r = maxxEIXol (dimij/~x)' Lemma I.2.4 follows, because this gives the following majorant power series

L r . C . q(dim(Xo)+f3/2ln . t n- 1 00

n=l

for the logarithmic derivative r L'(X

0,

L(Xo,

'6;

.~,

t)

0>, t)

=

L( L 00

n=!

d(x)Tr(F:1d(X l

») .tn-I.

xEIXol,d(xlln

D Lemma 2.5 Let Xo be a smooth irreducible curve over K. Let Uo jo : Uo "-+ Xo be a nonempty open subset of Xo. Let So = Xo \ Uo be the complement ofUo in Xo. Let ;~ be a sheaf on XO, such that its restriction jo(;~) to Uo is smooth and such that

16

I. The General Weil Conjectures (Deligne's Theory of Weights)

Finally assume, that for all points x eigenvalues of

E

IVo I and geometric points x over x

and all

the following weight inequalities hold d(x)

=

Then the corresponding inequalities hold for all points s i.e. for all eigenvalues a of

[K(X) : K] . E

So in the complement,

the following estimate holds

Proof 1) First of all we can replace K by the algebraic closure of K in the function field of Xo. We can therefore assume Xo to be geometrically irreducible. By our assumptions the homomorphism

is injective. We may therefore assume 0J = Jo* (.9'fi) from the beginning. We can also assume Xo to be affine, possibly by removing a point from Vo. Then the Grothendieck trace formula I.1.1 implies that

TL(Xo,·(~,t)

=

TL(Vo,Jo(·(~o),t).

n T(det(l-f,.td(S),;~s)rJ SEISol

Tdet(l- Ft, H}(X,

.~»)

Tdet(l - Ft, Hc2 (X,

;-;»)

2) For x E lUol the sheaf .:YO defines a representation of the Weil group W(Vo, x) on the vectorspace V = .:YOx = .%. Let .yV denote the dual sheaf of .Y and WV the dual of a Qrvectorspace W. Poincare duality on V implies

where Vrr1(U,x) denotes the largest factor space of V with trivial lTI (V, x) action. Here (-1) is a Tate twist by the dual of the cyclotomic character. Therefore the poles on the right hand side of Grothendieck's formula for the L-series, as stated above, are of the form -I q -J) T (a

I.2 Weights

17

for eigenvalues aq of the geometric Frobenius substitution F E W(k/K) acting on VJTj(u,x)(-I), Equivalently a is an eigenvalue of F on VJr1(U,x). The Frobenius homomorphism induces the homomorphism F d(x)

. .

VJrl(U,X) ---+ VJrl(U,X)

on the factorspace VJr1(U,x) of V. Hence ad(x) is also an eigenvalue of Fx : '~x---+ .}«jx. Therefore by the Grothendieck trace formula the L-series r L(Xo, ,(~, t) has no singularities in the region

It I < q-f3/ 2- 1 .

3) The factor rL(Uo, JO'(.}«j), t) of rL(Xo, :~o, t) converges for all It I < q-f3/ 2- 1 and does not have zeros in this region, by Lemma 1.2.4. Furthermore ISol is finite. Hence by 1) and 2) none of the factors r det (l - Fs . td(s) , '(~sr I for S E ISo I has poles for It I < q-f3/ 2- 1 .

This implies the estimates Ir(a)1 2 :::: qd(s)(f3+ 2 ) = N(s)f3+ 2

for all eigenvalues a of Fs : .%, ---+ ,(~,. Here ;~o, is one of the stalks for geometric points over ISO I· 4) Now we apply these estimates also to the sheaves k=1,2, .... If a is an eigenvalue of

then a k is an eigenvalue of

From the following Lemma 1.2.6 we get the injectivity of the homomorphism

This gives for all k = I, 2, 3, ... the upper estimates Ir(a k )12 :::: N(s)kf3+ 2 ,

hence the improved estimate

18

1. The General Wei! Conjectures (Deligne's Theory of Weights)

The claim of the lemma follows from the last inequality in the limit k ---*

00.

D

Lemma 2.6 We use the same notations as in Lemma 1.2.5. Suppose .YO is smooth, given by the representation of the Wei! group W(Xo, x) on the stalk V = :~x = ,'~ for some x E 1UO I. Let Is be the ramification group of X in s E S

Is

CIT! (U,

Then one has

x) C W(Uo, x) .

~ (J*. (,li"'») s = .J'

VI,"

hence

Proof An immediate consequence of the definition of (1* (.07) )-s'

D

We have the following useful Lemma 2.7 Let Xo be a normal, irreducible algebraic scheme over K and ;~ an irreducible smooth sheaf on Xo. Then the restriction of ,'~ to a nonempty open subscheme io: Vo'-+ Xo

of Xo remains irreducible. Proof This is evident, because for any point a E Vo the homomorphism 1

IT!

(Uo, Zi) ---*

IT! (Xo,

1

Zi) ,

and therefore also the homomorphism W(Uo, Zi) ---* W(Xo, Zi) , is surjective.

D

The next theorem reflects a fundamental property of smooth sheaves. Theorem 2.8 (Semicontinuity of Weights) Let ,'~ be a smooth sheafon an algebraic scheme Xo over K and let io: Uo'-+ Xo

be the inclusion of an open dense subscheme of X o. Then the following holds: (I) w(,'§Q) = w(jO'(.'~». (2) If iO'(,'§Q) is i-pure of weight {J. then

.'~

is i-pure of weight {J.

I.2 Weights

19

(3) Let Xo be normal and irreducible, furthermore let .'~ be irreducible and let Jo (:50) be r-mixed. Then .'~ is r-pure. (4) Suppose Xo is connected, Jo(;~) is r-mixed and ,'~ is r-pure of weight {3 at a single point x E IXol, i.e. all eigenvalues a of Fx : ;§Qx ---+ .'~x satisfy Ir(a)1 2 = N(x)f3. Then .'~ is r-pure of weight (3. Proof We first prove (1). For that we may assume, that Xo is irreducible and we can further replace Xo by the normalization of its reduced subscheme XOred. Nothing has to be proved for dim(Xo) = O. The curve case was dealt with in Lemma 1.2.5. The general case dim(Xo) > I on the other hand can be easily reduced to the curve case. For that observe, that every point s E IXol \ IVol can be connected to Vo by a curve Yo c Xo, which has nonempty intersection with Vo. This completes the proof of (1). Claim (2) follows from (I) applied to ;~o and its dual sheaf. Now consider (3): Jo(.'§Q) is r-mixed. Any sheaf on Vo becomes smooth on a suitable open nonempty subscheme. So by shrinking Vo we can assume, that Jo("~o) has a finite filtration by smooth subsheaves, whose successive quotient sheaves are rpure. But Jo(.'§Q) remains irreducible by Lemma 1.2.7, hence is r-pure. But assertion (2) then implies, that also :§o is r-pure. This proves (3). Assertion (4): It is enough to prove the assertion for each irreducible component of Xo, which contains the point x, and then to proceed by iteration. So we can assume Xo to be irreducible and then even to be normal. Furthermore it is enough to proof the claim for all the irreducible constituents of ,(0), which allows us to assume .~ to be irreducible. This implies Jo (:§o) to be irreducible by Lemma 1.2.7. Then JoC;~o) is r-pure by (3) and assertion (2) proves ;~ to be r-pure. The weight necessarily has to be (3. 0

Definition 2.9 Let ,'~ be a sheaf on an algebraic scheme Xo over K. Then there is an open dense subscheme Jo : Vo such that the inverse image

Jo(::~)

"-+

Xo

is smooth on Vo. We define

Remark. This definition is independent of the choice of Vo because of Theorem 1.2.8(1).

Definition 2.10 (Real Sheaves) Let ·'~o be a sheaf on an algebraic variety Xo. ;~ is said to be r-real, iffor all closed points a E IXol and corresponding geometric points a over a, the characteristic polynomial

20

I. The General Wei I Conjectures (Deligne's Theory of Weights)

of the geometric Frobenius

has real coefficients, i,e. rdet(l - Fat, ~a) E lR[t] C C[t].

We skip listing the obvious permanence properties of this notion.

Lemma 2.11 Let .(~ be a sheaf on an algebraic scheme. Assume .(~ to be smooth and r -pure of weight {3. Then .(~ is a direct summand of a smooth, r -real and r -pure sheaf·Yo of weight (3. Proof Consider the dual sheaf ;~ v = .9!I5'0mU~, Q/) of ;~. The element b E Q/ with reb) = qf3 gives the sheaf.Yo = (;~ v ®':hh) EB .'~ wanted. D

2.12 The Functions /~. For the proof of the Weil conjectures an alternative characterization of the maximal weights of a r-mixed sheaf is useful. Let Xo be an algebraic scheme over K and ;~ a sheaf on Xo. Let m be a natural number, we denote by Km the unique extension field of K of degree mover K in k. Let Fm denote the geometric Frobenius of Gal(k/Km). The set XO(Km) = HomSpeC(K) (Spec(Km), Xo) of Km-valued points of Xo is equal to the set of fixed points Xo(k)Fm, where Xo(k) = HomSpeC(K) (Spec(k), Xo) is the set of k-valued geometric points of Xo. For a sheaf .'~ and an integer m we have the complex valued function defined by

f ·~(t) = 4>.?i1ii(t)

fJ .

+ j"~ we get

00

00

n=1

n=1

+ L2Re(j"J70, j"~)ntn-l + L

1If"~II~tn-l .

The power series 4>.~(t) has the precise radius of convergence q-f3- 1 , as already shown. The two other power series on the right have a larger radius of convergence, as follows from the inequalities

1Z

~

12ReU....u , f"~O()nl

%f

.1Z

~ 211f·/ullnllf"~"Olin ~

C2' qn

(w(.)1(j)+W(."1!) 2

+

1)

deduced from Lemma 1.2.13 and w(9'&o) < (w(.0/6)

+ w(.9f3Q»/2 < fJ .

Therefore 4>~(t) and 4>·J70(t) have the same radius of convergence, namely q-f3- 1.

Case 3. The general case: Let Xo be reduced. Then we can find an open affine smooth curve ho : Vo "-+ Xo ,

such that the complement iO : So "-+ Xo

of Vo in Xo is finite and such that the sheaf ho(~) = .0/6 is smooth on Vo. Now consider io : Uo"-+ Xo and the sheaf io(~) on Uo. From 1.2.8(1) we obtain Vo c Uo Wgen(jo(~» = w(.0/6) .

Put,9'6o = io('~)' Then max (w(.0/6) , w(9'&o») = w(~) max(w(~), w(.9f3Q) -

1) =

max(Wgen(jo(~», w(~) -

1) .

I,3 The Zariski Closure of Monodromy

From the exact sequence 0 ---+ hO! (.9i1Q) ---+

3JO ---+ io*(.9'6'o)

25

---+ 0 we obtain

Now the coefficients of these power series are nonnegative, Therefore the radius of convergence of rP'~(t) is the minimum of the corresponding radii of convergence for rP,J{j(t) and rP'~(t), We have already shown, that the radius of convergence for rP'Yifj (t) is q-w(,9'Q)-1 . The radius of convergence for rP'~ (t) is q-w(3fJo). This implies

II~II

= max( w(§6) , w(.9'6'o) -

= max( w gen (jo(3JO)), w(3JO) -

I)

1) . o

1.3 The Zariski Closure of Monodromy Theorem 3.1 Let Xo be a normal absolutely irreducible algebraic scheme over K and let a be a geometric point. Let X : W(Xo, a) ---+

Q/

be a continuous character of the corresponding Weil group. Then the image X (Xl (X, a)) of the geometric fundamental group is a finite group. Equivalently the following holds: A suitable power xm , m :::: I of the character X becomes trivial on the geometric fundamental group Xl (X, a). In particular X = Xl' X2,

where Xl is a character of finite order and X2 factorizes over the quotient group W(k/K) ~ Z ofW(Xo, a).

Corollary. Let ~ be a smooth sheaf of rank 1 on a normal geometrically irreducible algebraic scheme Xo over K. Then 3JO is r-pure. More precisely: There is an element b E Q/ *and an etale sheaf §6 attached to a finite order character of the Weil group of Xo, such that

The r-weight of ~ is log(lr(b)1 2 )/ log(q). Proof of 1.3.1. Without restriction of generality one can make a finite base field extension and replace X by a power X m , m i= O. X being continuous means that we can further assume, that X is E-valued

26

I. The General Weil Conjectures (Deligne's Theory of Weights)

X: W(Xo,a) ~ E*

for a suitable finite field extension E C Qt of Qt. Then one immediately reduces to the case, where the image X(:IT} (X, a)) is a pro-I group. To proceed further, let us first consider the curve case. First Case. Xo is a smooth, projective geometrically irreducible curve. Class field theory for function fields in one variable implies, that the image of lq (X, a) in the abelianized locally compact Weil group of Xo is canonically isomorphic to the divisor class group Pi cO of divisors of degree 0, which are rational over K. It is enough to use the corresponding result for the I-parts of the groups under consideration. To derive it in this special case, one could use the theory of Kummer extensions (see Appendix C). The group Pico is finite. In fact, Pico is isomorphic to the finite group of K-rational points of the Jacobi variety of Xo. Alternatively one may use finiteness of class numbers for global fields. Now consider the higher dimensional case. For this let us make some preparations first. For an open nonempty subscheme Uo of Xo containing the base point a, the map :IT} (U,

a)

~ :IT} (X,

a)

is surjective. Therefore we can assume Xo to be smooth and quasiprojective. Xo is an open dense subscheme of a normal projective scheme Xo over K. SO Xo is contained in some projective space]P>N of dimension N over K. Then there exists an open smooth subscheme Yo of Xo containing Xo

Xo C Yo C Xo, whose complement Ao is of codimension at least 2 in Xo and such that the complement Do = Yo \ Xo of Xo in Yo is a smooth submanifold of pure codimension 1 in yo. After a base field extension the connected components DlO, .. , Dro

of Do can be assumed to be geometrically irreducible, such that each component Dvo has a K-rational point, saya v . The Second Case. Let Xo = Yo and dim(Xo) :::: 2. We choose a sufficiently general linear subspace L of codimension dim (X) - 1 of the projective space ]p>N, such that A n L is empty and such that C = X n L is a smooth irreducible curve. After base field extension we can again assume L to be defined over K. By our choice of L and the assumption X = Y the curve Co is even projective. From Bertini's theorem and Zariski's theorem on connectivity one deduces, that the homomorphism :IT} (C,

a)

~ :IT} (X,

a)

I.3 The Zariski Closure of Monodromy

27

is surjective. We can assume, that the base point is contained in Co. After these preparations the second case follows from the first. The Last Case is the General Case. Fix one of the components Dvo of Do where v = 1, .. , r. Let yJV) be thehenselisation of Yo in the pointa v, i.e. the spectrum of the henselisation of the local ring of Yo in the point avo Then yJV) ®K k = y(v) is the strict henselisation of Y = Yo ®K k in the geometric point av E Yo(k) over avo Let XCv) denote the inverse image of X in y(v), similarly X6V ) the inverse image of Xo in yJv) and D(v) the inverse image of Dv in X(v). For the following considerations we can assume, that the base point a is the image of a geometric point of XCv) also denoted a for the sake of simplicity. The image ofa in X6V ) is also denoted avo The inverse image of W(k/K) C Gal(k/K) under the sUljection ll'1 (X6v) , a) -+ Gal(k/K) is called the Weil group W(X6v ) , a) of X6V ). We have the following natural commutative diagram with exact lines

o ~ ll'l (X (v) , a) ~ W (X6

V) ,

a)

-----?>-

1

1

W (k / K)

-----?>-

0

I

The character X induces characters

o

(v) -

-

Xv : W(X o ,a) -+ Ql

*

-) ;n; * Xv: W(X (v) ,a -+ "'ll

.

By assumption the image X (ll'1 (X, a)) is a pro-I-group. Therefore Xv is tamely ramified along D(v) and factorizes over the I-part J of the tame ramification group along D(v). This group is isomorphic to 7L1 after a choice of generator. For the arithmetic Frobenius element a E W(k/K) and some inverse image a in W(X~), a) we have a ja- l = jq and Xv ( ].)

. - -1) = = Xvo( ].) = Xvo(U- JU

Xv ( ] .)q

for j E J. Hence the character factors over the quotient group 7Lt/ (q - 1)7LI, a finite group with say m elements. Then the m-th power

X: = 1 of Xv is the trivial character. We now repeat this consideration for all the other components of Do. The theorem of the purity of branch points finally implies: X m is unramified in all points of Do and therefore factorizes over the Weil group W(Yo, a).

28

I. The General Wei! Conjectures (Deligne's Theory of Weights)

This reduces the general case to the second case already treated. Theorem 1.3.1 is proved. 0 For a smooth sheaf ~ on Xo of rank n Theorem 1.3.1 and its corollary imply, that its detenninant sheaf, i.e. the highest exterior power /\n ~ of ~, is r-pure being a sheaf of rank 1. Definition 3.2 (Determinant Weights) Let Xo be a geometrically connected normal algebraic scheme over K and ~ be a smooth sheaf on Xo. For each irreducible constituent.90 of ~ we define the determinant weight of ~ with respect to rand .90 to be n

f3 =

f3/n

w(/\ .90) ,

where n is the rank of.90 and f3 is the r -weight of the determinant sheaf /\ n .90 of .90. A smooth sheaf ~ has finitely many such determinant weights (for fixed r ).

Let ~ be a smooth sheaf on a geometrically connected algebraic scheme Xo. Such a sheaf ~ is defined by a representation p of the Wei! group W(Xo, x) on a finite dimensional Qrvectorspace V = ~x = ~. There exists a finite extension field E C QI of Q/ and an E-vectorspace W C V, such that V = W ®E QI and such that p defines a continuous representation p : W(Xo, x) --+ GI(W). The image p(:rq (X, x» of 7q (X, x) in Gl(W) C GI(V) will be called the geometric monodromy group, whereas the image of W(Xo, x) will be called the arithmetic monodromy group of the representation p or alternatively of the smooth sheaf ~. For an extension K' of the base field K of degree m, the point x defines a geometric point of Xo ®K K', again denoted x. The Weil group W(Xo ®K K', x) is canonically isomorphic to the inverse image in W(Xo, x) of the unique subgroup of index m in W(k/K) ~ Z. Similarly if is a geometrically connected etale covering of degree n over Xo, then for a choice of a geometric point x' over x in X the Weil group W(X x') is canonically isomorphic to a subgroup of index n in W(Xo, x) and 7Tl (X', x') is canonically isomorphic to the subgroup W(X x') n 7Tl (X, x) of 7Tl (X, X). This subgroup again has index n in7Tl (X, x). Conversely every subgroup of W (X0, X) with these properties arises from a geometrically connected etale covering of Xo in this way. GL(V) is an algebraic group over QI, naturally defined over the field E by V = W ®E QI. The Zariski closure G geom of p(7Tl (X, x» in GI(V) is a linear algebraic group over QI, which in fact is defined over the field E. Every element g in the Weil group W(Xo, x) nonnalizes this group

Xo

o'

o'

o'

p(g) G geom p(g)-l = G geom .

This means that every element of the cyclic group (0-) ~ W(k/K), where 0- is some fixed element of degree deg( 0-) = 1, defines an automorphism of the algebraic group G geom , which is defined over the field E. Let G be the semidirect product of G geom and the discrete group scheme Z ~ W(k/K) defined over E via this action

1.3 The Zariski Closure of Monodromy

29

G = Ggeom I consider the etale covering

In general this is not a Galois covering. But after a base field extension with the field of qn elements it becomes the Artin-Schreier extension of the affine line A.o ®K Kn over Kn. It then has the covering group Kn. The morphism n ) admits the following decomposition Kn

P6

x~

",n-I

L...v=ox

q"

.

Over Kn this decomposition induces a surjective homomorphism between the covering groups of the maps n ) and Po = P6 1), defined by the trace

P6

Tr:

Kn ~

(J

Given a character true

f-+

K ",n-I

L...v=O

(J

q"

1/t : K ~ Q[ * of K and an element x from K, the following holds 1/tx

0

Tr

= (1/t 0

Tr

t.

By abuse of notation the character

1/t 0 Tr :

Kn ~

-* Q,

will again be denoted 1/t. Contrary to above, 1/Jx is in general for elements x in only a character of Kn. If 1/t is nontrivial, then of course the induced character

Kn

40

I. The General Weil Conjectures (Deligne's Theory of Weights)

is again nontrivial. Therefore the construction of the sheaves'ho (1/r) and'ho (1/rx) is compatible with finite base field extensions. From the Leray spectral sequence obviously follows I

-

I

-

0= He (A, Q/)

1

-

=

H (A, g;h(Q/»

=

He (A, 8'J*(Q/» .

0= H (A, Q/)

I

-

U sing Lemma I.5.1 and base field extension we get Lemma 5.2 For any character 1/r : K -+ -* Q/ and all elements x

E

k one has

Here we use the following convention. If x is an element of Kn for some n, then C;;(1/rx) will denote the pullback of the sheaCC;;o(1/rx) from Ao 0 K Kn to Ao 0 K k. Definition 5.3 (Fourier Transform) Let 1/r : Consider the diagram

-+ -* Q/ be a nontrivial character.

K

Ao x Ao ___m_ _ _~) Ao

m(x, y)

= xy

y~ and define the functor Fourier transform h

-

b

-

T1jr: De(Ao,Q/) -+ Dc(Ao,Q/)

by T1jr(K)

Here K [I]i left.

=

= Rn6! (nJ*(K) 0

m*CC;;O(1/r») [1] .

Ki + I, hence [1} indicates a degree shift for the complex by 1 to the

Remarks to Definition 1.5.3 (l) D~ (Ao, ij/) is the "derived" category of bounded complexes of etale Q/-sheaves

on Ao with constructible cohomology sheaves. For details see Appendix A or Chap. II, in particular the section On the triangulated category eX, Q/). Usually all results obtained for etale sheaves by Fourier transform in the following sections can be extended by Theorem I.IA to cover the case of Weil sheaves as well. (2) For a sheaf K, i.e. a complex concentrated in degree 0, the Fourier transform T1jr (K) is a complex with cohomology concentrated at most in degree -1,0,1. This is due to the shift of degree of the complex to the left by [1], since the direct image complex Rn6! (.7) of a sheaf.7 has cohomology only in degrees 0,1,2.

Di

1.5 Fourier Transform

41

Similar to the later proof of Lemma 1.5.10 one shows

Lemma 5.4 For a

E K

consider the morphism

Ao : Ao ---+ x

Then for a character 1/1

:K

---+

f--+

Ao ax

Q/ the following holds

After suitable base field extensions the base change theorem and this lemma imply

Theorem 5.5 Fora complex Kofrom Dg(Ao, ijz) anda geometric point a

E

Ao(k)

=

k one has Here K denotes the pullback of Ko onto A = Ao rgh k and 'h"(1/Ia) denotes the pullback of the sheaf '£0 (1/1a) defined over a suitable extension field Kn containing a.

Let n be a fixed natural number. In 1.2.12 we have attached functions

to sheaves .(~ on algebraic varieties Xo over K depending on n. This procedure can be extended to complexes Ko E Dg(Xo, ijz) by setting

where .%V (Ko) is the v-th cohomology sheaf of Ko. Letx be an element from Kn and a E k be a solution of the Artin-Schreier equation t qll - t = x. The arithmetic Frobenius substitution of k over Kn maps the solution a to the new solution f3 = a qn . But f3 - a = x. This shows, that the arithmetic Frobenius acts via translation by the element x, also viewed as the corresponding element x in the covering group Kn. The geometric Frobenius element Fx in the covering group Kn is therefore given by the element -x E K II • From this computation and Grothendieck's fixed point formula one derives, using Theorem 1.5.5, the following

Lemma 5.6 Let n :::: I be a natural number and Ko be a complexfrom Dg(Ao, ij{). The functions are related by

42

I. The General Weil Conjectures (DeJigne's Theory of Weights)

=-

jT-£, SI-->-y)-£, SI-->-Y).

The induced homomorphisms of the covering groups KXK--*K

are (a, r) I-->- a , (a, r) I-->- r - a , (a, r) I-->- r .

The character 1f; : K --* Qz * gives via composition with these three homomorphisms three characters 1f;!, 1f;2, 1f;3 : K x K --* ijz *. These three characters correspond to the three smooth sheaves entering into Lemma I.5.1O, namely n I2 *(m*('ho(1f;))) : a*(m*( C;;o(1f;))) :

n 23 *(m*( 'ho (1f;))) :

1f;1 ((a, r)) = 1f;(a)

= 1/1 (r) /1f;(a) 1f;3((a, r)) = 1f;(r)

1f;2 ((a, r))

The assertion of the Lemma I.5.1 0 then boils down to the trivial statement 1f; il1f;3 = 1f;2. 0

1.6 Weil Conjectures (Curve Case) With all the necessary requisites at hand, we are now prepared to prove the fundamental Theorem 6.1 (Deligne) Let Xo be a smooth, geometrically irreducible, projective curve over K. Let

46

1. The General Weil Conjectures (Deligne' s Theory of Weights)

jo: UO"---+ Xo be an open nonempty subscheme. Let j;iQ be a smooth, r -pure sheaf of weight w on Uo. Then the cohomology groups

are r -pure of weight w + i for i = 0, 1, 2. One easily reduces the proof of this theorem to the case of etale sheaves. Therefore assume, that .91) is an etale sheaf.

Lemma 6.2 ([SGA411 dualite, theorem 1.3, p.156) Let X be a smooth curve over afield, j : U"---+ X an open dense subset and.~ a smooth, constructible Q[-sheaf on U. Then

Theorem 1.6.1 and Lemma 1.6.2 are nowadays better understood in the context of the theory of perverse sheaves. We remark, that Theorem 1.6.1 is a special case of more general purity statements proved in Chap. III. The statement of Lemma 1.6.2 is a special case of the general results proved in Chap. III, §5. In fact, j* turns out to be the intermediate extension of.91) in the sense of the theory of perverse sheaves. It should be emphasized, that a proof of 1.6.2 will be given in Chap. III in the context of the theory of perverse sheaves. This proof is self contained and does not make use of the special case formulated in Lemma 1.6.2 above. Therefore we skip the proof of 1.6.2.

Proof of Theorem /.6.1. The case i = 0 is trivial. Because of Lemma 1.6.2 and Poincare duality it is enough to treat the case i = 1 and to prove in that case: For every eigenvalue ex of the Frobenius homomorphism

the following inequality holds

Let us first make some simplifications: Without restriction of generality we can make a base field extension, and one can furthermore replace Uo by a smaller nonempty open subscheme jb : Ub "---+ Uo, since

Especially we can assume Uo to be affine. Then according to the Noether normalization theorem there exists a finite morphism

1.6 Weil Conjectures (Curve Case)

47

Uo --+ Ao to the affine line, which can be extended to a finite morphism

Xo --+ Po of Xo to the projective line Po over K. The direct image of.31) from Uo onto Ao is a smooth sheaf on an open nonempty subset of Ao. Therefore we can assume without restriction of generality, that Xo = Po and that Uo is an open subscheme of the affine line Ao CPo. The complement So of Uo in Po is finite. The factor sheaf ·9'6'0 = Jo* (§Q) / JO! (.915) is concentrated on the finite set So, so HI (pI, .9'6'0) vanishes. Therefore the homomorphism on cohomology with compact support

is surjective. It is therefore enough to prove the corresponding inequalities for the eigenvalues of the Frobenius homomorphism

Making a base field extension and after shrinking of Uo we can assume, that Ao \ Uo contains a K -rational point s, where the sheaf .rjiQ is unrarnified. One can interchange the points sand 00 using a projective linear transformation. We can then assume, that .jlfj is unramified at the point s = 00, i.e. it can be extended as a smooth sheaf to a neighborhood of the point 00. By Remark 1.3.5 the irreducible constituents of .:yo become geometrically irreducible after a finite base field extension. All constituents are then unramified at the point 00. But Hel (U, -) is a half exact functor. We can therefore assume §Q itself to be geometrically irreducible and unramified at the point 00. First we want to deal with the special case, where.31) is a geometrically constant sheaf, i.e. where the pullback .¥ of .31) is a constant sheaf. Then again the direct image Jo* to pI is geometrically constant. This implies the vanishing of the first cohomology H I (pI, J*(.¥)) = 0 . The factor sheaf .9'6'0 = Jo* (.9iiO) / JO! (.31)) is concentrated on the finite complement So = Uo· Therefore the long exact sequence attached to the short exact sequence

pb \

yields a surjection

Tl (10* (.31)) )s =

HO (pI, .9'6')

---+

Hl (U, .¥) .

SES

Semicontinuity of weights (Theorem 1.2.8) gives us control over the weights of .~ and HO(pl, .~) and therefore implies the estimates wanted.

I. The General Wei! Conjectures (Deligne's Theory of Weights)

48

We can assume now, in addition, that .:)IQ is not geometrically constant. At this point let us collect things together. It is enough to consider the following situation: Let jo : Uo '---+ Ao be an open nonempty subscheme of the affine line and.~ a smooth r -pure sheaf of weight w on Uo. Let p be the corresponding representation of the fundamental group 7Tl (Uo, a) on the finite dimensional Qrvectorspace V. Assume (I) .~ is geometrically irreducible smooth etale Qt-sheaf, i.e. the representation p

L'q (U,ii) is irreducible.

(2) .g;1Q is geometrically nonconstant, i.e. the representation p 11r1 (U,ii) is nontrivial.

(3) §Q is unramified at the point 00, i.e. p factorizes over a representation of 7TI (UoU 00, a) on V.

To prove Theorem 1.6.1 we have seen, that it will now completely suffice to prove in the situation above the Claim 6.3 The eigenvalues ex of the Frobenius homomorphism F : H; (U,.¥) -+ H; (U, .'7) satisfy

The proof of claim 1.6.3 is the main step of the proof of the Weil conjectures in the curve case. For the proof of 1.6.3, following Laumon, we will use the Fourier transform as defined in §5. The Fourier transform T1/J('(~) of the sheaf

will be shown to have the following properties (a) T1/J(~) is a sheaf, i.e. the complex T1/J(.'~) is concentrated in degree O. (b) There are no sections with compact support, i.e. H~(A, T1/J('~» = O. (c) The sheaf TifF(.'§Q) is a r-mixed sheaf. For the moment assume properties a)-c) to be true. Then

T1/J(~)

is a sheaf by (a). Hence we get from Theorem 1.5.5

TifF(~)O = Hl(A, ;~') = Hc1(U, $') . Obviously w = w(~). Therefore the fundamental claim 1.6.3 is equivalent to the statement, that the Fourier transform shifts upper weights at most by I

To prove this recall: For a sheaf '%0 on a smooth curve Yo we have defined the upper weights

1.6 Wei! Conjectures (Curve Case)

w(.YCo)

=

sup sup log(IT(a)e)/ log(N(y)) , yE!YO!

a

where a runs over the eigenvalues of the Frobenii Fy, y (Definition 1.2.3). We also defined the L 2 -norm

11.7toll

= sup {p

49

E

1

Yo 1 for all stalks of.7to

1 lim sup 111'7&011;' q-m(p+dim(Yo))

> O}

m

(Definition 1.2.14). For T-mixed sheaves .~o on Ao with the property

H~(A, .7&")

=0

we have shown in Theorem 1.2.16(2) that

By the properties (a)-(c) stated above both .YCo = .% and .o/Co = Tljr(~o) are Tmixed and satisfy H2(A, .o/&") = O. So the last observation applied to both sheaves allows to restate the Plancherelformula (Theorem 1.5.7)

in terms of the upper weights

This proves the desired inequality, hence claim 1.6.3.

o

In order to complete the proof of claim 1.6.3 and Theorem 1.6.1 we still have to verify the properties (a), (b) and (C) forthe Fouriertransform Tljr( ;~) = Tljr (jO!.¥O)) of the sheaf ;~ = iO!.~.

Property (a). Let x E AO(Kn) show (using Theorem 1.5.5)

= Kn

C A(k)

H~(A, i! (.r) ®ij, ·'h(o/x))

= k be a geometric point. We have to

= H~(U,.r ®ij,(/)(o/x)) = 0

H;(A, i!(.r) ®ij, .Y(o/x)) = H;(U, .Y ®ij, Y(o/x)) = O.

But H2(u,.r ®ij[ '£ (o/x)) vanishes, because U is affine and.r is smooth. For the second assertion we consider the representation p ® 0/x on the vectorspace V attached to the sheaf .~ ®ij[ 'ho(o/x). Poincare duality implies H;(U,.r®;n; '£(o/x)) 'V!.[

= (VP@'I'x IJTI (U-»)(-I)' ,a ,I,

50

I. The General Wei! Conjectures (Deligne's Theory of Weights)

I

-») is the largest factor space of V with trivial action of rrl (U, a) with respect to the representation p®1/Ix. Suppose now Hl(U,.'7 ®Q/5h( 1/Ix» i= O. where (V

,I,

p0'f'X Jl"I

(u

,a

This will lead to a contradiction. According to assumption (1) V is an irreducible rrl (U, a) module. Therefore p ® 1/Ix would have to be the trivial representation of rrl (U, a) on V. For x = 0, hence p ® 1/Ix = p, this contradicts assumption (2); .JiQ namely was supposed not to be geometrically constant. Let x i= O. Then the character 1/1x is nontrivial. On the other hand p is unramified at 00 by assumption (3). p ® 1/Ix geometrically constant, hence unramified at 00, would therefore imply, that the nontrivial character 1/Ix is unramified at 00, i.e. factorizes over rrl(lP'6,a) = Gal(k/K).

This contradicts the geometric irreducibility of the Artin-Schreier covering (§5). Property c) of the Fourier Transform T1/f(JO!(.JiQ». Let b be an element of such that i(b) = qW .

((2)/

Then, using the notation of Definition 1.5.3, the Wei I sheaf

is i-real. (Compare Lemma 1.2.11). But RV rrl(rr 2*(Jo!(.JiQV)) ®Q/ m*(S56(1/I-1» ®Q/Zb)

=R

V

rr!l(rr 2*(j0!(,JiOV)) ®Q/ m*(ZO(1/I-1))) ®Q/ Zb

= 0

for v

i=

1.

Namely, as well as T1/f(JO!(.JiQ», the complex T1/f-1 (JO!(.JiQv)) is represented by a sheaf, i.e. its cohomology is concentrated in degree zero. For that use the same argument as for the proof of property a). Therefore R V rr!I(.9'6'o) = 0

for v

i=

1

.9'6'0 being i-real, the Grothendieck formula for the L-series of a sheaf, applied to the fibers of the map rrl, shows that the Weil sheafR Irr? (.9'60) is again i-real. Then

according to Theorem 1.4.3(1) R Irr? (.%0) is also i-~ixed. Therefore the direct summand T1/f (j0! (.~)) of R Irr (.96'~) is i-mixed.

l

Property b) of the Fourier Transform T1/f (JO! (.JiQ» = .9i'5'0. The Fourier inversion formula 1.5.8 implies

H~(A, .9'6') = .90'-1 (T1/f-1 ('%'0))0 = .%,-1 (T1/f-1

0

T1/f (JO! (.JiQ))) 0

1.6 Wei! Conjectures (Curve Case)

51

This is obvious, because iO! (Jil')) (-1) is a sheaf, i.e a complex with cohomology concentrated in degree O. All three properties of the Fourier transformed sheaf T", CiO! (.96)) have now been verified. So Theorem 1.6.1 has been proved. 0 Corollary 6.4 Let.96 be a Wei! sheaf on an algebraic scheme Xo over K. Assume dim(Xo) ::::: 1. Let.9Q be r-mixed with highest weight equal to 13. Thenfor all integers i (= 0, 1,2) the highest r-weight f3i appearing in H~ (XO, .'7) satisfies f3i ::::: 13

+i

.

Proof We can assume Xo to be reduced. By Theorem 1.4.2 we can also restrict ourselves to the case of an etale sheaf .31). Let us first discuss the case, when X0 is a smooth curve and where.96 is a smooth sheaf. After some base field extension Xo decomposes into geometrically irreducible connected components. We therefore can assume Xo to be geometrically irreducible. The irreducible constituents of.95 are then r -pure by Theorem 1.2.8(3). On the other hand the functor H~ is half exact. So .9Q can be taken to be r-pure of weight 13 without loss of generality. Xo is an open dense subscheme of a smooth projective curve X 0 over K. io : Xo "-+ Xo . Consider the short exact sequence

.90'0 is concentrated on the finite complement So of Xo in Xo. Therefore Hi(X,.%) =0

i

~

1.

According to the sernicontinuity of weights 1.2.5 the r-weights ex of the stalks of

.9'6'0 are controlled by the weight of .9Q ex ::::: 13 . From Theorem 1.6.1 and the long exact cohomology sequence one obtains the desired estimate for cohomology with compact support. The curve case has now been proven. The general case. Everything is trivial in case dim(Xo) = O. So let us assume dim(Xo) > O. Then we can find an open smooth curve

io : Vo

"-+

Xo

52

I. The General Wei! Conjectures (Deligne's Theory of Weights)

with finite complement iO : So 4

(a)

Go = 'Zt,

where we assume

ata- 1 = q. t

t E

Go

for some natural number q i= 1 not divisible by the prime I. Let p be a continuous representation of the locally compact group G on a finite dimensional E -vectorspace V. Here E C Q/ denotes a finite extension field of the field Qt. We define twisted representations V (i) as the tensor product of the representation p with the character 1jr(a i t) = qi of G. Let the situation be as above. Then we have Lemma 8.1 For all g E GO the matrix peg) is quasi-unipotent, i.e. all eigenvalues of this automorphism of V are roots of unity (whose order is bounded in terms of q and dimE(V»).

1.8 Some Linear Algebra

55

Proof Let A be an eigenvalue of peg). Then

= p(g)q

p(U)p(g)p(U)-1

.

Therefore Aq is also an eigenvalue. Since V is finite dimensional, this implies that A is a root of unity. Its order can be bounded in terms ofq and the dimension dimE(V).

o

Notation. Let G~ be the subgroup of elements having unipotent action on V. Let a be an integer such that a . Go C G~. Such an integer a exists by Lemma I.8.1. Let t be a topological generator of the group Go. Since T a = p(t)a is unipotent on V, there exists an integer b :s dimE(V), such that (T a - l)b = O. The nilpotent endomorphism defined by N

= -a- I L(l

- TQ)n In

n",1

is independent of the choice both of a. In particular there exists a uniquely defined nilpotent endomorphism N of V, such that p(u)Np(U)-1 = q . N , and a subgroup G~ of finite index a in Go p (t)

= ex p (t N) = L

= Zt, such that t i N i Ii!

t E G~

holds for all t in G~. If we replace p by the representation p iZI p on the tensor product V iZI V, the corresponding nilpotent endomorphism is N iZI id + id iZI N. Similarly the nilpotent endomorphism for the contragredient representation is _t N (minus the transposed endomorphism).

The u-Decomposition. For a E v E V, which are annihilated by V

= EEla Va. The equation (p(u)

ij/ let Va (p (u)

C V denote the subspace of all vectors for some integer i 2: I. We have

- a -I

- qa-I)i N

r

=

N qi (p(u) - a-I)i implies

For the representation p iZI p on V iZI V we have

since ((p (u) - a-I) iZI p (u) + a-I iZI (p(u) - ,8-1) )2i annihilates v iZI w, provided (p(u) - a-I)i annihilates v and (p(u) - ,8-I)i annihilates w.

56

I. The General Wei! Conjectures (Deligne's Theory of Weights)

The N -Filtration. For a nilpotent endomorphism N on the vectorspace V, there exists a unique finite increasing filtration F. on V with the properties N(Fi(V»)

c

Fi -2(V)

N i : Gr;(V) ~ Gr-i(V)

i?:

o.

Here Gri(V) = Fi(V)IFi-l(V). Existence and uniqueness of such a filtration is proved by induction on the minimal number m such that N m + 1 = o. For m = 0 one endows V with the trivial filtration, i.e. Gri (V) = 0 for i -:F O. In general put Fm(V) = V and F-m-l (V) = O. This is forced, since N V = 0 for v ?: m + 1 has to induce an isomorphism between Grv (V) and Gr -v (V). Therefore Grv(V) = 0 for Ivl ?: m + 1, hence Fm(V) = V .

In order, that N m induces an isomorphism between Grm(V) and Gr_m(V), we have to put Fm-l(V) = Ker(Nm)andF_m(V) = Im(Nm).ButthenFm_l(V)IF_m(V) is annihilated by N m • Hence a corresponding filtration uniquely exists on this quotient by the induction assumption. The inverse image of this filtration in Fm-l (V) defines the desired filtration F.(V) on V. 0 Existence and uniqueness are obtained in the same way in the following general situation: Let A be a noetherian and artinian abelian category, let V E ob(A) and N: V -+ V be a morphism such that N m + 1 = O. Let us return to our original situation. Consider the dual space V*. It is clear that the filtration of the contragredient representation p* on the dual space V* is then given by Fi(V*) = F-i-l(V)~. The representation p respects the filtration F.(V). We obtain an induced representation of G, actually of the factor group GIGo, on Gr.(V) such that N i : Gri(V)(i) ...:+ Gr_i(V). For simplicity from now on assume Go = Go. Definition 8.2 The primitive part Pi (V) of Gr; (V) is defined to be the kernel of the induced homomorphism N: Gri(V) -+ Gri-2(V) . p induces a representation of GIGo = (a) on Pi (V).

Recall the definition of the twisted representations V(i), where a acts with an additional factor qi. Then the following lemma is now more or less evident from the defining properties of the filtration, considered above. Lemma 8.3 (1) Pi(V) = Ofor i > O. (2) Gri(V) ~ EBj::,:liIJ=i(2) p-j(-i(j

+ i»

as GIGo-module.

I.8 Some Linear Algebra

57

(3) Gri(V*) ~ Gr-i(V)* and Gr-i(V) ~ Gri(V)(i) as GIGo-modules. Especially P-i(V*) ~ P-i(V)*(i). (4) Leti::: O. The map N i : Fj(V) --+ F j -2i(V) issurjectiveforallj::: i. (5) For the inducedjiltration on ker(N) C V we have Gri(ker(N)) ~ Pi(V).

One obtains the following picture

P-2(V)(-2) E!1. ..

Gr2(V)

N

Gro(V)

N

P-l(V)(-I)

Grl (V)

Gr_l(V)

Gr_2(V)

N

Po (V)

-

N

P_I (V)

-

N

EB P_2(V)(-I)

EB ...

EB ...

N

EB ...

P-2(V)

EB ...

-

All vertical morphisms are isomorphisms induced by N. Each G ri (V) is isomorphic to the direct sum of the rows of the diagram, i.e. Gro(V) ::::::: Po(V) EB P-2(V)( -1) EB P-4(V)(-2) EB .... Proof ofthe Lemma. Part (1) is obvious from the fact that N i defines an isomorphism between Gri (V) and Gr -i (V) for i ::: O. Part (2): It is enough to consider Gr -i (V) for i ::: O. Proceed by induction with respect to i starting with Gr -m-I (V) = P- m-l(V) = 0 and Gr-m(V) = P-m(V). Obviously Gr_i(V) = P-i(V) EB N i + 1 Gri +2 (V). Apply the induction hypothesis to the right summand, which is isomorphic to Gr -i -2 (V). The statement of part (3) is obvious. Part (4) is proved by passing to the associated graded, where the corresponding assertion is trivial. See the diagram above. (5) asserts, that F-i(V) n ker(N) maps onto P-i(V) = ker(N) : Gr -i (V) --+ Gr -i-2(V). In other words, one has to show: An element v E F-i (V) with Nv E F-i-4(V) can be modified by an element W E F- i -2(V), such that N(v - w) = O. But this follows from property (4). D

58

I. The General Wei! Conjectures (Deligne's Theory of Weights)

Lemma 8.4 For every eigenvalue a of a-Ion Ker(N) in V assume Ir(a)1 < 1. Assume the corresponding statement also for the representations on the tensor products V ® V resp. V* ® V*. Then every eigenvalue a of a-I on Gri(V) has the property

Proof. Using Lemma 1.8.3(1)-(2) it is enough to consider eigenvectors of p(a-l), which lie in some P-i (V), i :::: O. By 1.8.3(5) such a vector can be represented by a vector v E F_i(V) n Ker(N). Projecting v onto Va, we can furthermore assume E

V

Va and N(v) = O.

By 1.8.3(4) one can find a vector w E Fi (V) such that N i (w) = v. Projecting onto Vqia' we can assume Then

i

u = L(-l)j Nj(w) ® Ni-j(w) j=O

is a nonvanishing eigenvector of a-I in (V ® V)qia2. One immediately checks (N ® id

+ id ® N)u

= 0.

The assumptions of Lemma 1.8.4 imply Ir(q i a 2 )1 ::s I, hence Ir(a)1 2 ::s q-i. The corresponding inequality holds for the eigenvalues of a -Ion P-i (V*) P-i(V)*(i). a-1q-i is such an eigenvalue. Therefore Ir(a- 1q-i)1 2 ::s q-i or Ir(a)1 2 :::: q-i. This proves the required equality Ir(a)1 2 = q-i on P-i(V). 0

1.9 Refinements (Local Monodromy) Let Xo be a smooth geometrically irreducible curve over K, S a geometric point of Xo over K with values in k, s the underlying closed point, let

jo : Uo

C-.+

Xo

be the open complement of sin Xo and .~ a r-pure smooth sheaf on U of weight {J. In this situation the weights w(jo*(.g;iQ)s) ::s {J of the direct image of.J'O in the point of degeneration s can be estimated above by {J, according to Lemma 1.2.5. The linear algebra trick of Lemma 1.8.4 of the last section will then imply refined properties for these weights. Let K be the quotient field of the henselization of the local ring of Xo in s, L the quotient field of the strict henselization with respect to sand the separable closure of L. We look at the geometric point

r

I.9 Refinements (Local Monodromy)

59

YJ : Spec(L) --+ Xo .

The Weil group W (L I K) is the inverse image of the Weil group W (k I K) with respect to the natural homomorphism Gal(LI K) --+ Gal(kIK) .

The ramification group I of Xo in sis Gal(LI L). W(LI K) is the semidirect product of Gal(LIL) and the cyclic group W(kIK). The wild ramification group P is the p-Sylow group of Gal(LI L), p being the characteristic of k.1t is a normal subgroup with factor group Gal (L I L) I P, which is the tame ramification group

I tame = I I P . This tame ramification group is canonically isomorphic to

n

ZI(I)

lip,! prime

or uncanonically isomorphic to

n

Zl.

lip,l prime

Its I-primary part is The sheaf ~ is given by a continuous representation of W(Uo, YJ) in the finite dimensional representation space V over «:h. This defines a continuous representation of Gal (L I K) on V. The image of Gal (L I L) in Gl (V) contains a pro-l subgroup of finite index. Passing to a finite covering of Xo, ramified in S, allows to get rid of the wild ramification part and the tame part which is not pro-I, i.e. the representation of Gal(LI L) then factorizes over the pro-l part of the tame ramification group. In that case the representation of W(LI K) factorizes over the representation p of a semidirect product G = Zl

)-

1~

Z

w

~

lid

X [1]

lid[l]

X ~ Y ~ Z ~ X[l].

Corollary 1.6 If(X, Y, Z, u, v, w) is a distinguished triangle and u is an isomorphism, then Z ~ O. Conversely Z ~ 0 implies that u is an isomorphism. Proof The long exact Hom(Z, .)-sequence implies Hom(Z, Z) = 0 and proves the first statement. The second statement again follows from the long exact H omsequence. 0

Corollary 1.7 If(X, Y, Z, u, v, w) is distinguished and w =0, then Y:;:: ZEBX such that u and v correspond to the inclusion resp. projection map. Proof Exercise! First find p with po u = idx, then find i with idy = u 0 p + i 0 v. Then v 0 i = idz and finally poi 0 follow from the injectivity of the dual Hom-sequence at X[l]. 0

So far we have discussed the axioms TRI-3 of a triangulated category and some trivial implications. We now formulate the remaining axiom TR4 for triangulated categories, the so called octaeder axiom. See also [140], especially for an explanation of the name. Axiom TR4a (Composition Law for Mapping Cones). Suppose we are given morphisms u : X ---+ Y and v : Y ---+ Z. For any choice of mapping cones C u , C v and C vou with defining distinguished triangles

Tu = (X, Y, Cu, u, *, *) Tv = (Y, Z, C v , v, *, *) Tvou = eX, z, C vou , v 0 u, *, *) , there exists a distinguished triangle T relating these mapping cones T

=

(C u , C vou , C v , ex, {3, y) ,

which makes the following diagram commute

ILl Triangulated Categories X[l]

Cv[-l]

3 -y[-I] ~

X[l]

1

3a

1

v

Cu

) y

Cv[-l]

71

~

1

C vou

1

) Z

l'

3f3

~

Cv

) Cv

1'" x,

X

The two vertical lines of the diagram are defined by the morphisms of the two distinguished triangles Tu, Tvou, the two horizontal lines by the morphisms of the two rotated distinguished triangles rot (Tv), rot(T). Note, that although we assume the triangles Tu, Tv, Tvou to be chosen fixed, we preferred not to give names to all morphisms. Axiom TR4b. For the full octaeder axiom one adds the further condition of commutativity for the diagram Z[-l]

Z[-l]

j C u " -y[-I] C v [ -1]

Cu "

1

Y "

"

f3[-I]

j C vou [-1]

a[-I] (

Cu[-l]

j u

X (

j-

1'"

Z

Z.

Cu[-l]

The two vertical lines of the diagram are defined by the morphisms of the distinguished triangles rot 2 (Tvou )' rot 2 (Tv ), the two horizontal lines by the morphisms of the distinguished triangles (id, -id, id)* (rot 3 (T)) and rot (Tu). In the axioms of a triangulated category there is a certain redundancy: axiom TR3 is a consequence of axiom TR4a and TR4b. Consider maps f, g, u', u as in

72

II. The Formalism of Derived Categories

TR3. Then axiom TR4a, which is a kind of pushout axiom, applied to X' ~ y' ~ y and axiom TR4b, which is a kind of pullback axiom, applied to X' ..!,. X -.:+ Y together implyTR3.Fork:= gou' = uoJthisdefinesmorphismsa : CUi --+ Ckrespectively fJ' : Ck --+ Cu· Then h = fJ' 0 a defines a morphism (f, g, h) between the triangles (X', Y', CUi) and (X, Y, C u ).

The Derived Categories D(A) The most prominent example for a triangulated category is the derived category D(A) of an abelian category A, as mentioned in the introduction. It is the localization of the category of all complexes K om (A) over A by the class of quasiisomorphisms. A short review on this is in the appendix II of [FK], p. 292 and in chapter I of [140] (Hartshorne). To verify the axioms TRI-4 for derived categories, it is useful to be aware of the fact that the localization functor is not injective on homomorphism groups. In particular homotopic complex maps become equal in the derived category. However this gives some extra freedom. So one can proceed in two steps: first pass to the category K (A), whose morphisms are homotopy classes of complex maps, and then invert quasiisomorphisms. The axioms TR 1-4 can be established already on the first level as properties of complexes up to homotopy. To invert quasiisomorphisms in the category K (A) becomes much more convenient, because the class of quasiisomorphisms is a localizing class, i.e. it allows a calculus of fractions in the category K(A). See [FK] A 11.1. More details can be found in [Ver], [104], [140]. In the same way one can define the derived categories D+(A), D-(A), and Db (A) as the localization of the full subcategory of complexes which are bounded to the left, bounded to the right respectively are bounded. They can be embedded as full subcategories into D(A). In particular for a morphism J in HomD*(A)(K, L) for * E {+, -, b} there exist quasiisomorphisms s, t, such that J 0 s respectively t 0 J are homotopic to complex maps. We remark that the construction of D(A) by localization gives set theoretic problems unless A is a small category or belongs to some given universe, since a priori Hom D(A) (X, Y) is not a set. However, let the category A satisfy the Grothendieck T6hoku axioms ([113]). Then every - may be unbounded - complex K of K om(A) has a right resolution by a so called K-injective complex. Using this fact we can conclude that D(A) is equivalent to the full subcategory of K om (A) given by all K-injective complexes. See [Spa], [Tar). This implies, that HOmD(A) (X, Y) is a set in the case of an abelian Grothendieck category A. See also the following remark for the case of the subcategory D+(A) of D(A). Remark 1.8 If K, L are in D+ (A) and L is injective, i.e. all components LV of the complex L are injective objects of A, then any morphism J in Hom D*(A) (K, L) is represented by a complex map p : K --+ L. The localizing property of quasiisomorphisms, which is formulated in [FK] A II 1(2), allows to reduce the proof of this statement to the case, where J is the inverse of a quasiisomorphism u : L --+ K

11.1 Triangulated Categories

73

(using fractions with left denominators we find u such that u 0 f is a complex map). For the quasiisomorphism u it is enough to show, that there exists a complex map p : K ~ L which is a left inverse of u up to homotopy. Namely then (using calculus of fractions with right denominators) the morphism f and the complex map p L :/

~d

K u

L

id

id

P

or f: K - - - - L ------ Land p: K ---- K ------ L coincide in the derived category, since we have the commutative diagram

L

K

~

id

L

To construct p let C = C u be the cone in the category K+(A). Then we have a distinguished triangle (L, K, C, u, *, *) and C is acyclic and bounded below. Using Corollary 11.1.7 one reduces the construction of a homotopy left inverse p for u to the following fact: If C is acyclic and I is injective (l = L[l]) and both C and I are bounded from below, then any complex map C ~ I is homotopic to the zero map. This homotopy is constructed inductively using the extension property for injective objects. See e.g. [104], p. 180. So for K, L E D+(A) and L injective, the natural map HomK+(A)(K, L) ~ HomD+(A)(K, L)

is surjective. It is also injective, since u 0 f = 0 for f E HomK+(A)(K, L) and a quasiisomorphism u : K' ~ K implies f = O. In fact u* : HomK+(A)(K, L) ~ HomK+(A)(K', L) is injective. Since HomK+(A)(C u , L) vanishes for the acyclic cone C u of u, this follows from the long exact H om-sequence for the triangulated category K+(A). Therefore the Hom-groups in the derived categories D+(A) and Db (A) can be computed in terms of the homotopy category K (A) using injective resolutions in the second variable. This is convenient, provided the abelian category A has enough injective objects. See Verdier [SGA41], p. 299 for further information. Other examples of triangulated categories can be found in [104]. We will be mainly interested in the triangulated categories D(X) = Di(X, ij[) for finitely generated schemes X over a finite or algebraically closed field. These categories are obtained as certain limits of derived categories. For further details on these categories the reader is referred to Appendix A and the corresponding section of this chapter.

Remark 1.9 The diagram of axiom TR4b is formally obtained from the diagram of axiom TR4a by replacing the direction of arrows (and renaming). This implies,

74

II. The Formalism of Derived Categories

that the notion of triangulated category is self dual: If D is a triangulated category, then also the opposite category DOPP, obtained by inverting arrows with the induced translation functor and induced distinguished triangles, is triangulated. Later we will use this in the proof of Corollary IIA.2. Nevertheless we mention, that the only information added by TR4b to TR4a is the commutativity of the middle square of the TR4b diagram. In other words TR4b is, modulo TR4a, equivalent to either one of the two statements: TR4b': (u, idz, f3) : Tvou -7 Tv is a morphism of distinguished triangles. TR4b": The two hidden ways, to go in the diagram of axiom TR4a over the upper right corner from C vou to Y[ I] , anticommute. More precisely, Axiom TR4b" states (-i[-\])

if Tv

=

(*, *, *, v, *, i) and Tvou

=

0

f3

=

-(-u[\])

(*, *, *, v

0

0

k.

u, j, k) are the triangles chosen.

Remark 1.10 The isomorphism class of the distinguished triangle T, whose existence is imposed by axiom TR4, is determined (up to isomorphism) by the three cones Cu, C v, Cvou and the morphism y, according to Corollary ILLS. However -y[ -I] and therefore also y is uniquely determined by the commutativity of the left square of diagram TR4a - as the composite of the given maps C v [ -I] -7 Yand Y -7 C u , appearing in the two distinguished triangles Tv and Tu. Thus the triangles Tu , Tv, Tvou determine T up to isomorphism.

11.2 Abstract Truncations In the derived category D(A) of an abelian category A one has full subcategories D(A):sn and D(A)?:m, consisting complexes with vanishing cohomology in degrees strictly larger than n resp. strictly smaller than m. By the process of truncation, a given complex can be split into two complexes, one of them in D(A):SO and the other in D(A)?: 1. This has an abstract analog in an arbitrary triangulated category, motivated by the theory of D-modules and the RiemannHilbert correspondence. The notion of t-structures was first introduced in [BBD J, inspired by the non obvious t-structures underlying perverse sheaves respectively holonomic D-modules with regular singularities.

Definition 2.1 A t-structure in a triangulated category D consists of two strictly full subcategories D:So and D?:O of D, such that with the definitions D:s n = D:So[ -n] and D?:n = D?:o[ -n] we have (i) Hom(D:So, D?:l) = o. (ii) D:SO C D:Sl and D?:l C D?:O. (iii) For every object E in D there exists a distinguished triangle (A, E, B) with A E D:SO and BE D?:l.

11.2 Abstract Truncations

75

D is said to be bounded with respect to the t-structure, ifevery object of D is contained in some D?'.a and some D~b for certain integers a, b.

Truncation. Suppose we are given at-structure. In the situation ofI1.2.I(iii) we have B E D?'.l and B[-l] E D?'.l[-l] C D?'.l by property 11.2.1 (ii). Therefore the long exact Hom-sequence Il.1.3 attached to the distinguished triangle (A, E, B, u, *, *) together with the vanishing property II.2.1(i) of t-structures implies u* : Hom(X, A)

~

Hom(X, E)

for X E D~O .

This fundamental fact has the striking consequence that u : A --+ E is a universal morphism from D~o to the given object E E D. Every morphism from some X E D~o to E factors in a unique way over the morphism u

X

E D~o

This universal property characterizes the pair (A, u) uniquely up to isomorphism. We therefore write A = T~o(E). The assignment r~o is functorial in E. For every morphism f : E' --+ E, the composite f 0 u' : T~o(E') --+ E factors through u : r~o(E) --+ E by a unique morphism r~o(f). In other words: r~o : D --+ D~o

defines a functor, which is right adjoint to the inclusion functor of D~o into D. The isomorphism u* established above turns out to be the adjunction isomorphism. Similarly, the assignment E r-+ B defines a functor r> 1 : D --+ D?'.l, which is left adjoint to the inclusion of D?'.l C D. We will ass~me, after making some choices, that the functors r~o, r?'.l are fixed from now on. Resume. From the discussion of t-structures so far we see, that the distinguished truncation triangle of property (iii) for t-structures has now become the unique distinguished triangle iii)' The first two morphisms have become adjunction maps. They uniquely determine the third map of the triangle. For this recall Corollary 11.1.5 and the fact, that h = idT~l(E) istheuniquemorphismh: r?'.l(E) --+ r?'.l(E) with the property h o ad?'. 1 = ad?'. 1 because of the adjunction formula Hom(E, y)

=

Hom(r",l (E), Y)

for Y

E

D?'.l .

76

II. The Formalism of Derived Categories

Similarly the isomorphism u* from above gives the adjunction formula Hom(X, ':'Oo(E))

=

for X

Hom(X, E)

Properties of the truncation functors.

E

D:'OO .

Recursively define ':'On for n

E

Z by

or ':'On(X) = (,:'Oo(X[n])[-n]. Then ':'On is right adjoint to the inclusion functor D:'On C D. Therefore, by the obvious inclusion properties of the underlying categones for m:S n . Lemma 2.2 (Orthogonality) For objects E equivalent (i) E is in D:c: n+l . (ii) Hom(D:'On, E) =

E

D the following statements are

o.

Proof One easily reduces ton = O. Then one direction is the statement of property (i) for t-structures. For the converse direction (ii) :::} (i) it is enough to show ':'00 (E) = 0 by Corollary II. 1.6. But ' n, the same can be achieved

r,

Proof u is obtained by honest complex maps u : K e -+ Ie *where Ie is an injective resolution of Use Remark 11.1.8. Naive truncation je = T~n Ie for n - respecting the size of the finite complexes K e and defines honest complex maps v: K e -+ je

r.

r -

i:

r

-+ je

such that i u = v holds in the derived category, i being a quasiisomorphism. The assertion concerning je follows from (6) above. The assertion concerning i e is obtained, if we define i e to be (the perfectification (9) of) the bounded flat complex

i

e

=

Cone(v+ i: K e EB

r

-+ je)[_1].

The maps k and ware the obvious projections. Then k is a quasiisomorphism and up to homotopy of complex maps the following is true vk

See e.g. II 04] p. 161. Therefore uk

= wi .

=w

holds in the derived category.

D

94

II. The Formalism of Derived Categories

If sizes are to be respected, as claimed in the remark, one better replaces the complex i· by the quasi isomorphic complex T 1 of objects Ar and isomorphisms ¢r+1 in the category Dr such that ¢r+1 : Fr+1 (Ar+l) ~ Ar. A morphism 0/ : (Ar, ¢:)r:::.] -+ (Br , ¢!)r:::.1 in D should be given by a system 0/ = (o/r )r> 1 of morphisms o/r : Ar -+ Br in Dr, such that ¢~+ 1 0 Fr+ 1 (0/r+ d = 0/r 0 ¢~~ 1 holds for all r. In such a setting one could ask, whether D carries the structure of a triangulated category. The category D is obviously an additive category. Define the shift operator.r (A) = A[I] by (Ar, ¢r)r:::.1 [I] = (Ar[l], ¢r[l])r:::.l. To define distinguished triangles in D, the obvious attempt would be the following: A triangle (A, B, C, I, g, h) in D is distinguished if and only if the corresponding triangles (Ar, B r , Cr , Ir, gr, h r ) of the system are distinguished triangles in the categories Dr (for all r). However, with this definition it is unfortunately not clear, how to verify for example axiom TR3. Only under the following strong finiteness assumption, it is easy to verify the axioms TRI-TR4 of triangulated categories. Assumption. For all r :::: 1 the homomorphism groups X, Y E Ob(D r )

are finite groups.

II.5 The Triangulated Category Dg(X, Ql)

95

Under this assumption the limit category D is a triangulated category with the definitions as above. The proof is easy and left as an exercise for the reader. Just as an example, let us verify axiom TR3: For given distinguished triangles (X, Y, Z, u, v, w) and (X', Y', Z', u', v', w') and morphisms f : X' --+ X and g : y' --+ Y, such that g 0 u' = u 0 f, we look for amorphism h : Z' --+ Z, such that (J, g, h) extends to a morphism of triangles. Let Er be the set of morphisms hr : Z~ --+ Zr, such that (Jr, gr, h r ) is morphism between the distinguished triangles (Xr, Yr , Zr, u r , V r , w r ) and (X~, Y;, Z~, u~, v~, w~). Then (1) The set Er is nonempty ( axiom TR3 for Dr) (2) The set Er is a finite set ( the finiteness assumption above)

The sets Er obviously define a projective system

By (1) and (2) it follows, that the projective limit E

= l~ Er i= 0. r

Any choice of h E E C Hom D (Z', Z) now gives the required extension. Let us now come back to our original problem. So we specialize to the case of the triangulated categories Dr = D~(X, Or). The transition functors Fr+! : D r+! --+ Dr are given by the tensor product K:+! ~ K:+! ®~r+l Or. The construction above gives the desired limit category D~(X, 0), provided the finiteness assumption made above holds. In order that this finiteness condition holds, we now need the assumptions, made on the base field k. The relevant property is the following: Let k' be a finite separable extension field of k and let G be the Galois group of the separable closure of K' over k'. Then we want, that the Galois cohomology groups H V (G, 71,/171,) are finite groups; here G is assumed to act trivially on the coefficients 71,/171,. Of course this is a strong assumption on the underlying base field k. Let us assume that all the Galois cohomology groups H V (G, 71,/171,) are finite groups. Under this assumption on the field k one can deduce from the finiteness theorems, proved by Deligne ([SGA41]' finitude; see also Appendix D of this book) that for schemes X finitely generated over k and perfect complexes K·,U E Dctj(X,or),thefollowing

Theorem 5.4 (Finiteness Theorem) Assume all Galois cohomology groups HV(G, 71,/171,) are finite groups. Then the homomorphism groups HOmD(X,Or)(K·,

are finite abelian groups.

L·)

96

II. The Fonnalism of Derived Categories

There are other cases, where the analogous finiteness statements are valid. Since these cases are not relevant for this book, we only mention that the similar statement holds for schemes finitely generated over Z, for schemes finitely generated over strictly henselian rings or over discrete valuation rings with finite residue fields. See Mazur [123]. Building on this finiteness theorem we can now define the category Dg(X, 0) as above as the projective limit

in down-to-earth terms. Objects. First of all, an object of

Dg (X, 0) is a collection = K e = (K;)r:::1

K

of complexes K; in D~tf (X, or) together with quasiisomorphisms A.

•

'l'r+l·

Ke,o,L

r+l

'6IO r + 1

~

Or =

Ke

r

in the categories Dg(X, or). The v-th cohomology sheaf of K e is by definition the induced projective system .%v(Ke) = (.'76'V(K;))r:::l. Morphisms. They are given by compatible systems of morphisms for each r: For two objects of Dg(X, 0) represented by projective systems K e = (K;)r:::1 and L e = (L;)r:::l as above put HomD~(X,o)(Ke, U) = l!!EHomDg(X,Or)(K;, L;) . r

In other words, a homomorphism 1/1 : K e -+ L e in HomD~(X,o)(Ke, L e ) is a family 1/1 = (1/Ir)r:::l of morphisms 1/Ir : K; -+ L; in the derived categories Dg(X, or), such that the following diagrams for r = 1, 2 ... commute 1/Ir+l ®~r+1 Or : K:+I ®~r+1 Or

----+

~-J.-¢~1 1/Ir :

K re

L;+1 ®~r+1 Or

~ -J.- ¢;:+I ----+

L re

Note: A morphism 1/1 = (1/Ir)r:::1 between two objects K e = (K;, ¢f)r:::1 and C = (L;, ¢fk~.I is an isomorphism, if and only if all morphisms 1/Ir : K; -+ L; are isomorphisms. As a consequence of the finiteness theorem formulated above, two objectsK e = (K;,¢f)r:::1 andL e = (L;,¢f)r:::1 are isomorphic in the category Dg(X, 0) if and only if the objects Kr and Lr are isomorphic in Dg(X, or) for all r. In other words, the isomorphism class of an object K e = (K;, ¢r )r::: I does not

II.S The Triangulated Category D~(X,ijl)

97

depend on the underlying transition isomorphisms r, i.e. for any sequence of automorphisms ¢r in Hom Db(X 0 ) (K;, K;), there exists a sequence of automorphisms 1/Ir in HomD~(X,or)(K;,cK;)~ such that

In fact, the obstruction to find such 1/Ir is in lim~ AutD~(X,or)(K;, K;). See Jensen [163]. By the finiteness assumption, this lim I-term vanishes. In order to work with this definition of D~(X, 0), it is useful to make some preliminary remarks: 1. By assumption each K; can be replaced by a quasiisomorphic bounded Orfiat complex, whose bounds above and below - its size - may be controlled by the cohomology sheaves of Ki only (Remark (7) and (8)). So all the complexes (K;k:1 can be chosen of uniform size, i.e. uniformly bounded. 2. Next one can use Remark (9) to replace each complex K; by a quasi isomorphic perfect one, without altering the size. 3. So let us suppose that all complexes K; are uniformly bounded perfect complexes. Flatness implies K:+10~r+! Or = K:+ 10 or +! Or. This gives induced transition morphisms These morphisms, induced by r, are only morphisms in the derived category D~(X, Or+I)' One can choose perfect complexes i~, by modifying K e up to quasiisomorphism without changing the size, such that the transition maps are honest complex maps i:+1 --+ i; instead of being just morphisms in the derived category. ~his is an immediate consequence of the Lemma II.S.3. We may therefore assume K; = K;, with "honest" transition maps

With the preparation 1.-3. above one can show

Lemma 5.5 Let K e sheaves

=

(K;)r~1 be an object of D~(X, 0). Then its cohomology

are A-R Jr-adic sheaves.

Remark . .:?11v (K e ) is a Jr-adic sheaf in the relaxed way of speaking (see the comments on the A-R category). Sketch of Proof ofll.5.5. Over an open dense subscheme there exist locally constant sheaves .t, ./J of 0 I-modules independent from r 2: ro such that

98

II. The Fonnalism of Derived Categories

Since .9!{f'v (K;+I

®Or+1

Or) ~ .9!{f'v (K;) ,

.~eV(K-) is smooth on an open dense subscheme U e X, which is shown by induction on r; for all r the sheaves .O/,fjv (K;) IU are locally constant. The sheaves . ~, ./J are constructed from the cohomology sequences of the distinguished triangles (nr K;+I' K;+I' K;+I ®Or+1 Or) as follows: They are obtained as constituents of .O/,fj- (no. Kj) using the isomorphisms

::::r..oV( n r-I . K-) ~ 9!{f'V(nr . K-r+I' ) r -----*. ~

./'[J

that are induced from the n r K r-

-+---

Kr+1

to.

I

multiple of

'6IO r +1

id@n _ Or ~ K r+1

to.

'6IO r+1

°r+1

=

Kr+I'

The left map is the transition map (a quasiisomorphism), the next map is multiplication by id ® n, which after multiplication by n r - I becomes an isomorphism (divisibility of K:+I !). For the details confer [FK], lemma 12.14. By noetherian induction one therefore reduces to the case, where X is the spectrum of a base field. This field may be replaced by a separably closed field. For this case see [FK], lemma 12.5, or the discussion in the next section below. For further properties of these cohomology sheaves see also Appendix A resp. [FK], I § 12. D

Corollary 5.6 .9f(;'V(K-) is a smooth n-adic sheaf on an open dense subscheme U eX.

On one hand, as already observed, the triangulated categories D~t f (X, or) do not have natural (-structures by naive truncation of complexes. Therefore the naive truncations (TsmK;)rC? 1

do not yield an object in the category D~(X, 0). A natural t-structure on D~(X, 0) therefore a priori seems not to be present! On the other hand it is tempting to define a tstructure on D~ (X, 0) by imposing vanishing conditions for the cohomology sheaves (considered in II.S.S). We will see, that this provides the required good substitute.

11.6 The Standard t-Structure on D~ (X, 0) Let the situation be as in the last section. Our aim to define a t-structure on D~ (X, 0), which will be called the standard t-structure. We first deal with the case of a point

Xo

= Spec(k) , k separably closed field.

II.6 The Standard t-Structure on Dg(X, 0)

99

Then objects in Di(X, 0) can be represented by uniformly bounded projective systems (K;)r?:.1 of perfect complexes of finite or-modules with transition maps

(a) Let Ke = lims K; be the inverse limit complex. Since all the K% are finite groups (!) the lim I-terms of such projective systems vanish (Mittag-Leffler condition). Therefore H V (Ke) ~ I~ H V (K;) . s

The modules K: are free, so the transition maps K;+s 0 composition, are quasiisomorphisms. Furthermore

0 ,+\

Os --+ K;, obtained by

is exact. This holds for all 1 :::; r, s. Take the cohomology sequence. The projective limit of these sequences for s --+ 00 is still exact by the finiteness assumptions made. This gives the exact sequences

It will be called the Tor-sequence. Note Toro(M,o/nro) Torf (M, o/nr 0) ;::: M[nr] for an o-module M.

M/nr and

(b) Now dimo I (HV (K;)/n) :::; dimo I (HV(Kj» + dimol (H v+1(Kj», since dimo l ( Hv+I(Ke)[nr]/n) :::; dimol(Hv+I(Ke)/n) and HV(Ke)/n "-+ HV(Kj), H V+ I (Ke) /n "-+ H v+ 1(Kj). Thus the projective limits H V(Ke) are finitely generated o-modules by Remark 11.5.1(3). Hence by the last sentence of Remark 11.5.1(9) one can choose an o-perfectification L e --+ K e of the inverse limit complex K e . In particular HV(Le)---=+Hv(Ke). The composed map U --+ K e --+ K; induces the quasiisomorphisms L; := L e /nrL e ----+ K; for all r (5-1emma)

We may therefore replace (K;)r?:.1 by the quasiisomorphic system

(c) Let A be the abelian category of finitely generated o-modules. Let Db(A) be the derived category of A with bounded complexes. Any object in Db (A) can be represented by a perfect complex of o-modules.

100

II. The Formalism of Derived Categories

Hence there exists a functor Db(A) ~ D~(Spec(k), 0) ,

which maps an o-perfectcomplex L- to the projective system (L- /nrL-)r;"l. Since a perfect complex is now its own projective resolution, any morphism between perfect complexes in Db(A) is induced by a complex map. Therefore isomorphic o-perfect complexes in Db(A) give isomorphic projective systems. This is true also in the converse direction by taking the projective limits. From the preceding discussion in (b) we therefore see, that this functor induces an equivalence of categories Db (A) ;:: Dt(Spec(k),o).

Recall the equivalence of categories ([FK], 12.3) of the abelian category of A-R n-adic sheaves on Spec(k) and A, established by the functor (Fr)r>l f-+ limr Fr. The projective systems N r = (limsHv+l (Kn)[n r ] arising in the -Tor-sequences define null systems; their objects become stable and the transition maps have an additional factor n. So the equivalence Db(A) ;:: Dt(Spec(k), o) is compatible with the cohomology functors defined on both categories (see Lemma 11.5.5).

Exercise. How do the natural truncation operators T-l('~+s 0 0n +s on) E D~tf(X, on) and is up to quasiisomorphism independent from the chosen s ~ t. This means s,s'~t.

Now study for fixed s and varying n the morphisms

in the derived category. The projective system

is an object of Dt(X, 0). Evidently Del(.c~) is concentrated in degrees 0 and -1. We have .o/fjO(Del(:§» = .'§ .7;(J-l (Del(:tf)

=0.

Concerning the Proof. By passage to the stalks the assertions of this lemma are reduced to the corresponding results for modules. Suppose given a projective system of finite or-modules Mr such that

is n-adic: Mr+l/n r M r+! = Mr. Then M = limr Mr is a finitely generated o-module and we have

Without restriction of generality we can assume, that M is a torsion module and even of the special form

II.6 The Standard t-Structure on D~(X, 0)

103

In particular Mr = 0 In! 0 for all r 2: t. On the level r = n + s for s 2: t we can use the following on+s-ftat resolution of the on+s-module Mn+s

Using these complexes the proof of the lemma, in the special case of the projective system of modules (Mrk::l

D

under consideration, is a straight forward exercise. Short Notation. We define D(X, 0)

=

Di(X, 0)

Definition 6.3 (The standard t-structure on Di(X, 0)). D:"O(X, 0)

=

{K e E D(X, 0) 1.76 v (K e )

= 0 'v'v

> O}

D?:.O(X, 0)

=

(K e E D(X, 0) 1.76 v (K e )

= 0 'v'v

< O} .

Theorem 6.4 D:"o(X, 0) and D?:'o(X, 0) define a t-structure on D(X, 0). The core of this standard t-structure is thefull subcategory of D(X, 0) Core(standard) = (K e E D(X, 0) 1.:ytJv(K e ) = 0 v

-I O}

.

The functor Core(standard) ---+ {n - adic sheaves} Ke

H-

,9'60 (K e )

defines an equivalence of categories between the core of the standard t-structure and the abelian category ofn-adic sheaves on X. The lower truncation operator

of this t-structure is the Deligne truncation operator defined in Lemma 6.1. The upper truncation is obtained by completing the natural map

into a distinguished triangle.

Remark. We can and therefore will identify the cohomology objects of the standard t-structure with the "ordinary" n-adic sheaves. We identify sf T?:.n sf T:"n K e = sf T:"n sf T?:.n K e = ,:yen (K e ). This being said we will later omit the index and write

104

II. The Formalism of Derived Categories

We now give a sketch of the proof of Theorem 11.6.4. 1) First Step. We have K e = (K;)

E Dso(X, 0)

~ .3f(jV(K;) = 0 v::: 1, r ::: 1

K e = (K;) E D?:O(X, 0) ~ .9W(K;) = 0 v

:s -2, r

::: 1 and .~-l (K e ) = O.

Considering the stalks, the proof is reduced to the case of the corresponding statement for modules. This case is an easy consequence of the Tor-sequence in subsection (a) at the beginning of this section. 2) Second Step. For K e E DSO(X, 0) and LeE D?:l (X, 0) we have

Claim. HomD(X,o)(K e , L e ) = O. By step I) and the remarks at the beginning we can assume that K e = (K;) and L e = (L;), such that K; is uniformly bounded, constructible and Or-fiat and

v> O.

L; is an injective complex such that L~

=0

v < O.

Then a morphism is given by a homotopy compatible family of complex homomorphisms

hence by a family of sheaf homomorphisms

v=o such that ljIr (K~) C K er(L?

--7

L;) = 3'lJ o(L;) .

However .~IJO(L e) = 0 implies, that the projective system (.~eO(L;))r?: I is a null system. Therefore for some (sufficiently large) natural number t > 1 the sheaf homomorphism

is the zero map for all r ::: 1.

II.6 The Standard t-Structure on D~(X, 0)

105

By definition and flatness

is a quasiisomorphism. By Remark 11.5.1 (6) we also have a quasiisomorphism

The map induced by 1frr+1

is the zero map! We may replace K e , L e by isomorphic objects in the category Di(X, 0), e.g. by the shifted systems

i;+1 = (L e 0 0r +(

Or )r::: 1 .

Hence the underlying morphism of the projective systems is shown to be zero. 3) Third Step. The truncation axiom for the standard t-structure is an immediate consequence of the properties of Deligne's operator r!jel (Lemma 1I.6.1). 4) Last Step. A right inverse Junctor for the functor Core(standard) -+

IT -

adic sheaves

K e r-+ .~O(Ke)

is the Deligne operator DeW~)

(Lemma 11.6.2). It is therefore enough to show the following: Let

be a morphism between objects in the core of the standard t-structure, which induces the zero map 3!eo(h) = 0 .9'eo(h) : ](j'o(K e ) -+ .9'eo(L e)

- in the A-R category. Then we claim h First of all the functor

= 0 in Di(X, 0).

is an exact functor. This follows from the long exact cohomology sequence. Applied for K er (h), Koker (h) it is therefore enough to show: A morphism

106

II. The Fonnalism of Derived Categories

in the abelian category Core(standard), for which

is an isomorphism - in the A-R category - is itself an isomorphism. For this we can assume that the complexes K;, L~ are bounded, or-flat and that the morphism h is represented by complex homomorphisms hr : K; -+ L~. We then have to show, that the morphisms hr are quasiisomorphisms. Passage to the stalks, reduces to the case where X is replaced by a geometric point of X. The corresponding statement in this particular case of modules is an easy consequence of what was proved already. 0 The category

D~(X, E) is deduced from D~(X, 0) by "localization", the category D~(X, Ql/) on the other hand by a direct limit

D~(X, Q/)

= " l~ " D~(X,

E)

EdJ,

(See Appendix A). In an evident way, the distinguished triangles of D~(X, 0) carryover to distinguished triangles and hence structures of triangulated categories for D~(X, E) respectively D~(X, Q/); and similar for the t-structure. The corresponding truncation operators are again denoted sf T:::n, sf r~n or shorter sf

Again for K e

E

r~n

=

r~n

.

D~(D, Ql/) we identify

with a Ql/-sheaf on X. We often write Short Notation. D(X)

=

D~(X, Ql/).

11.7 Relative Duality for Singular Morphisms In the next Chap. III we will consider the theory of perverse sheaves in the sense of [BBD]. One of the cornerstones of this theory is the formalism of Poincare duality in its relative version for arbitrary, i.e. possibly singular morphisms between schemes. It gives rise to functors D and f!, which have remarkable properties.

To begin with, we want to formulate the main results of this duality theory. These are formulated in terms of the derived categories D~(X, Ql/) of bounded complexes of Q/-sheaves with constructible cohomology sheaves on a scheme X. For the purpose of this book it is enough to restrict oneself to the category of finitely generated schemes

II. 7 Relative Duality for Singular Morphisms

107

over a fixed finite field or algebraically closed field. This restriction allows to define the derived category D~(X, Q/) in a simple and naive way. See also Chap. II §5 and Appendix A. The prime I will always be assumed to be different from the characteristic of the residue field. After stating the main results we present the main steps of the proofs. Theorem 7.1 (Deligne [SGA4], expo XVIII) Let J:X---+S be a compactifiable morphism between finitely generated schemes over a finite or algebraically closed base field. Then the Junctor

admits a right adjoint triangulated Junctor

This means, that we have Junctorial isomorphisms

,

Hom(K, J'(L))

--=+ ~

Hom(RJ!(K), L) .

Jorall K E D~(X, Q/) and all L E D~(S, Q/). Thefunctor Hom( -, -) denotes the Junctor oj homomorphisms in the derived categories D~ (-, Q/).

A slightly stronger statement - including the adjointness statements also for all etale schemes S' over S and X' over X - is the following sheafified version: Relative Poincare Duality. There exists a functorial isomorphism

RJ*R.%'om(K, /(L)) ~ RS6'om(RNK), L) .

At this point remember the similar, but comparatively trivial adjunction formula

Hom(f* K, L) ~ Hom(K, RJ*(L))

respectively

108

II. The Fonnalism of Derived Categories

For the definition ofthe complexes R.9{fom(K, L) we refer to [FK], appendix All, p. 300. Now Poincare duality in the singular case is formulated by means of the dualizing complex. Definition 7.2 Let f : X --+ S = Spec(k) be afinitely generated scheme over thejixed base field k - with the restrictions on k made at the beginning. The dualizing complex of X is

The dualizing complex is compatible with base change for the ground field k. One defines the contravariant dualizing functor by Dx(L) =

R.~om(L,

Kx) .

We will often write DL = D(L) = Dx(L), if the scheme X is fixed. Corollary 7.3 (Poincare Duality) Under the assumptions of Theorem II. 7.1 for f : X --+ Sand K E D~(X, Q,) the following holds

Proof. Apply relative Poincare duality for L Kx of the adjoint functor J'.

=

K s and use the functoriality f! K s

Theorem 7.4 (Biduality) (Deligne, SGA4i, Theoreme de finitude) The natural functorial homomorphism K --+ Dx(Dx(K))

is a canonical isomorphism Dx

0

Dx = 1d.

Therefore the dualizing functor defines an anti-equivalence of categories

Hom(K, L) = Hom(Dx(L), Dx(K)) .

= D

11.7 Relative Duality for Singular Morphisms

109

We now collect some formulas, which will be frequently used later. They are immediate or formal consequences of the adjunction Theorem 11.7.1, the biduality Theorem 11.7.4 and the well known elementary tensor identity f*(K rgi K') f*(K) ®L f*(K') and the R.9'6'om-formula R.9i'6'om(K, R.9i'6om(M, N))

= R.9'6'om(K ®L

M, N)

for K, M, N E D~(X, Ql/). These tensor identities are easily verified. The first is reduced to the corresponding statement for the stalks, since tensor products and pull back commute with passage to the stalks. The R.9'6'om-formula is reduced to the corresponding identity for the functors .9'6'om instead of the functors R.9'6'om. The remaining verification is then an obvious fact already on the level of presheaves. However for the reduction to the case of that 9'6'om-functors - which is done by replacing the complex N = N- by an injective resolution r - one has to use that for a complex M E D~(X, Ql/) the complex R9'6'om(M-, r) = .9'6'om(M-, 1-)has flabby, hence R.9'6'om(K, .)-acyclic components! The details of the argument are left as an exercise to the reader. Corollary 7.5 Suppose f:X-+S is a morphism satisfying the assumptions of Theorem lI.7.I. Then the following formulas hold: a) DoD = id b) Do Rf! = Rf* c) Do Rf* = Rf!

d) e) f) g) h)

D or alternatively Do Rf! 0 D = Rf* D or alternatively Do Rf* 0 D = Rf! Do f* = I' 0 D oralternatively Dol' 0 D = Dol' = f* 0 D or alternatively Do f* 0 D = l R.9i'6'om(A, B) = D(A ®L D(B)) Rf!(A ®L f* B) = Rf!A ®L B (Kiinneth type formula) I'R.9i'6'om(A, B) = R.9i'6'om(f*(A), l(B)) 0 0

r

Proof. a) and b) are stated in II. 7.3 and 11.7.4. c) is an immediate consequence of a) and b). d) and e) follow from b) and c) and Hom(K, L) = Hom(Dx(L), Dx(K)) by the adjunction formulas stated in 11.7.1. f) is the R9'6'om-formula if we put B = D(C) and use a), i.e. the biduality D(B) = C. For g) restate the relative Poincare duality theorem using f) in the form Rf*D(A ®L DJ'C) = D(Rf!A ®L DC)

and use DI'C = f* B for DC = B applying e). Now use Do Rf* Statement h) finally is equivalent to

0

D = Rf!.

J'D(A ®L DB) = D(f* A ®L DJ' B)

via formula f). Now d) and a) reduce this to the obvious statement, that r(A ®L DB) = f*A ®L f*DB. 0

110

II. The Formalism of Derived Categories

We remark, that formula g) above was obtained purely formally as a consequence of the duality theorems. However, the reader should be aware of the fact that formula g) has to be proved independently at an early stage of the theory. See [FK], chap. I, proposition 8.14. This formula implies the Kiinneth formulas (see [FKD and it is also one of the ingredients for the proof of the fundamental duality Theorem 11.7.1. This will be explained in the next section. After having stated the duality theorem, a general remark concerning the proofs of the statements of Theorem II. 7.1 and II. 7.4 is in order. These two theorems were formulated in the Q/-adic setting, i.e. for the Q/-adic derived categories. The usual techniques explained in Appendix A (projective limits, localization etc.) actually allow to reduce the statements made for Q/-sheaves, to analogous statements for etale sheaves over finite commutative self injective rings. On that level the restrictions made on the base field - which had the purpose of defining the Qrderived categories - are no longer necessary. So for the rest of this section, which is devoted to the proof of the duality theorems, we can therefore completely ignore Qrsheaves. As already explained, it is enough to consider the corresponding statements for etale sheaves of A -modules. Here A means a finite commutative self injective ring. It is enough to consider the case, where A is a finite factor ring of the ring of integers of a finite extension field E C Q/ of the field Q/ of I-adic numbers. We will only consider etale sheaves of A-modules on noetherian schemes such that the prime I is invertible on A. The proof of the duality Theorem 11.7.1 in the absolute smooth case, i.e. in the case of a smooth morphism

f : X -+ Spec(k) from X to the spectrum of a field k, is contained in [FK]. See also the Appendix A of this book. Without restriction of generality we can assume k to be a separably or even algebraically closed field. Our intention for the following is to show how the general statement of Theorem 11.7.1 - now in the case of A -sheaves - can be deduced from the special case of an absolutely smooth morphism, at least under the slight restriction that the morphism f can be extended to a smooth compactifiable morphism g. This includes - for instance - the case of a quasiprojective morphism f. This is a setting which is sufficiently general for our purposes. Secondly we will prove the biduality Theorem 11.7.4 - always assuming, that the underlying scheme is finitely generated over the base field. We begin by formulating two fundamental finiteness theorems. Both these finiteness theorems are supplements to a finiteness theorem proved in [FK] for proper morphisms. Theorem 7.6 Let

f:X-+S be a morphism between finitely generated schemes over an arbitrary base field. Then the direct image junctor f* has finite cohomological dimension independent from A.

11.7 Relative Duality for Singular Morphisms

111

Proof This statement is an immediate consequence of M. Artin's theorem on the

cohomological dimension of affine algebraic schemes over a separably closed field. See [SGA4], expose X or [FK], chap. I, theorem 9.1. D Theorem 7.7 (Deligne, [SGA4!1, finitude) Let f:X-+S be a morphism between finitely generated schemes over a field and let ;§ be a constructible sheaf on X. Then all higher direct images R V f* (~) are constructible. Proof The proof is similar to the proof, which gives the permanence properties for

mixed sheaves (Theorem 1.9.4 at the end of Chap. I). We leave it to the reader to fill in the details of the argument. See also Appendix D. D Remark. Be aware of the fact, that the argument for the proof of Theorem 11.7.7 uses special cases of the Poincare duality theorem - namely the case of smooth morphisms considered in the next section! The last theorem now has the following consequence Corollary 7.8 (Deligne, [SGA4!1, finitude corollary 1.6) Let X be a finitely generated scheme over a field and let $ , sheaves on X. Then also the Ext-sheaves

;~

be constructible

gxt~(.¥, ~)

are constructible.

Further consequences ofII.7.6-II.7.8 are Corollary 7.9 Let the assumptions be the same as in Il.7.7. ( 1) R.9fJ om ( -, -) defines a bifunctor

D-;(X, A) x D~(X, A) -+ D~(X, A) .

(2) Thefunctors R.9fJomA(-, -), Rf*, Rf! preserve the derived categories Dt( -, A) and Dctj( -, A).

Recall that D;;(X, A), D-;(X, A) denote the full subcategories of Dc(X, A) of complexes of A-module sheaves with constructible cohomology sheaves, which are bounded to the left respectively to the right. Similar Dt(X, A) denotes the full subcategory of bounded complexes, Dctj(X, A) the full subcategory of complexes with finite Tor-dimension. For further details see also Appendix A and Appendix D.

112

II. The Fonnalism of Derived Categories

11.8 Duality for Smooth Morphisms After our preliminary statements made in the last section we now come to the proof of the first reduction step: Under the additional hypothesis, that I : X --+ S is a smooth morphism, we prove Theorem 11.7.1 by reduction to the absolute smooth case, where I is smooth and S is the spectrum of a field. This remaining case is covered in [FK], chap. II, § 1. Recall that all sheaves are now sheaves of A-modules. Let I:X--+S

be a smooth finitely generated compactifiable morphism between noetherian schemes. Further assumptions on I - except that I is invertible in A - will not be necessary. 1) We begin by assuming, that I has constant fiber dimension. All nonempty fibers of I are then equidimensional of dimension say d. Hence for all sheaves .¥ we have v> 2d. For v = 2d one has the functorial trace map

See [FK], chap. II, theorem 1.6. This trace map has obvious permanence properties, which are stated in loco cit. and which will subsequently be used without further mentioning. On the level of complexes the trace map may be viewed as a complex homomorphism R/!(Ax[2d](d» --+ As

due to the vanishing of the higher direct images with compact support for v > 2d, stated above. Recall that 1. the functor I! has finite cohomological dimension 2. R/!(Ax(d» E D~ttCS, A) Now the sought for functor defined to be

i' : D(S, A) --+

D(X, A) is in the first case 1) simply

Definition 8.1 For smooth I put i'(K) = f*(K)[2d](d). The proper base change theorem implies the projection formula [FK], chap. I, proposition 8.14

for any complex K E D(S, A). From this projection formula we obtain a new "trace map" R/!U!(K» --+ K via

II.S Duality for Smooth Morphisms

113

~ K

Rf!(/K))

r

II

Trf®L;dK

Rf!(Ax[2d](d] 0

L

f*(K))

=

Rf!(Ax[2d](d)) 0

L

K

2) Suppose now f is smooth, but does not have constant fiber dimension. Then f decomposes into morphisms of constant fiber dimension

x = Xl

(disjoint decomposition) .

U .... U Xr

The X v are open subschemes of X such that fv = f I X v: X v -+ S has constant fiber dimension d v . We then define for a complex K in D(S, A) I

f'(K)

ffi = \D

I

f~(K)1

Xv .

v

By a term by term addition we again obtain a trace map of the form

All results stated in [FK], chap. II, § 1 remain true. They carryover from the case of constant fiber dimension - considered in loco cit. - to the slightly more general case of nonconstant fiber dimension. The so defined trace map turns out to be the desired adjunction map. The trace map Trf,K defines a duality homomorphism ex: Ham(L, /(K)) -+ Ham (Rf! (L), K) , where Ham ( -, -) denotes the homomorphism functor in the derived categories D( -, -). By definition this homomorphism maps ¢ : L -+ / (K) to TrK,f 0 Rf!(¢)·

3) By a sheafification we also get a duality homomorphism f3: Rf*R,~am(L, /(K)) -+ R.96'am(Rf!(L), K) .

To be precise we demand L E D-(X, A) and K E D+(S, A). 4) Claim. These two duality homomorphisms ex and f3 are isomorphisms. In the second case the map f3 is an isomorphism in the derived category. As already explained in stating Theorem II.7.1, both statements are more or less equivalent. The proof for the absolute case - that is the case of a scheme X over the spectrum S of a field - is contained in [FK], chap. II, § 1. We deduce the general case from this case. See also Verdier, A duality theorem in the etale cohomology of schemes [306]. The proof of the claim 4) is established by noetherian induction with respect to the base S and will occupy the rest of this section. 5) We start with a remark, which allows reduction to the case of constructible sheaves. This will be used for the noetherian induction later. Every sheaf is a factor sheaf of the - possibly infinite - direct sum of all its constructible subsheaves. See

114

II. The Formalism of Derived Categories

[FK] I §4 for a proof. This in particular implies, that every complex L = L· in D-(X, A) has a resolution L E D-(X, A), which has components LV that are direct sums of constructible sheaves. Obviously one can therefore reduce the proof of claim 4) to the particular case, where L is a single sheaf, which is furthermore a direct sum of constructible sheaves. The functors R V I, commute with direct sums. The hyper-ext functors ExtV(.Y, K)

transform direct sums of sheaves in the first variable to direct products. Therefore the Hom-functor Hom(A, B) of the derived category transforms direct sums of sheaves in the first variable into direct products of abelian groups. Hence we will assume in the following, that for the given fixed morphism I: X ---'? S the complex L - in the first variable - is a single constructible sheaf

L=.Y. 6) Let .9t5'" be the v-th cohomology sheaf of the given complex K E D+ (S, A). The first isomorphism claimed is a consequence of corresponding statements where the complex is replaced by certain translates of sheaves Hom(.Y ,/(.~)[v]) ~ Hom(RI!(.Y), .9t5'[v])

'v'v.

Now, since .¥ and all cohomology sheaves of RI! (.Y) are constructible sheaves, both sides of the identity which shall be proved commute with direct limits in the second variable .:;n. This is easily deduced from the corresponding fact for the hyper-ext functors ExtV(A, B). It is therefore enough to prove the theorem for all constructible subsheaves of the cohomology sheaf .~ = .9t5'V(K).

We can and will furthermore assume that S is reduced. 7) If S is the spectrum of a field, then the statement of the claim is contained in [FK]. This is the start of the noetherian induction. By induction assumption the following may be assumed to be proven already: The claim is true for all proper closed subschemes S' of S and the corresponding morphisms

I' = I

®s S':

We consider the cartesian diagram

X'

=

X ®s S' ---+ S' .

II.8 Duality for Smooth Morphisms

115

with closed embeddings 7, i. By the base change theorems and elementary adjunction properties for the complexes L E D-(X, A), M E D+(S', A), K = i*(M) E D+(S, A) we get Hom(L, /CK)) = HomCL, i*(f'!CM))) = Hom(i*CL), /(M))

= HomCRf{(i*CL)), M)

= Hom(i*CRf!(L)), M) = HomCRf!CL), K).

Therefore - by the reasoning above - the claim is true also for f itself and all complexes K E DCS, A), for which all constructible subsheaves ofthe cohomology sheaves have support in a proper sub scheme of S (depending on K). Let F be the function field of one of the irreducible components of S which has a nonempty fiber over the generic point. If such a component does not exist, then the image of f is already contained in a proper closed sub scheme of S and we are done by the induction hypothesis. Else consider the inclusion map j :T

=

Spec(F) -+ S

induced by the generic point of the component. Consider the cartesian diagram

XT

=

j

X Xs T

ji

j!

• X

j

T

• S

Let B be a complex in D+CT, A) and put K

= Rj*(B) .

The smooth base change theorem implies f*Rj*(B) nition of / = f*[2d](d) /K

=

/R(j*B)

= R]*C!*(B)), hence by defi-

= R]*PCB).

Using the elementary adjunction property ofR]* and ]*, we thus obtain Hom(.¥' , /CK))

= Hom(j*(.¥"), FCB)) .

Since T is the spectrum of a field we already know by the induction assumption the adjunction formula HomC]*(.¥"), l(B)) = HomCRj,(j*.¥"), B). This implies Hom(.¥", /CK))

= Hom(Rj,Cj*.7),

B)

= Hom(j*Rf!('¥"),

B) ,

where for the second equality we have used the proper base change theorem Rj, (j* C.¥")) = j*Rf! (.¥") for the functor Rf,. By the elementary adjunction properties ofRj* and j* then Hom(j*Rf!(.¥"), B) and Hom (Rf! C·¥") , Rj*(B)) can be identified. Therefore

116

II. The Fonnalism of Derived Categories

Hom(oY, /(K»

=

Hom (Rf! (.Y) , K) .

We leave it as an exercise to verify the equalities stated above by checking diverse commutative diagrams! 8) Let now K be an arbitrary complex in D+(S, A). Consider the distinguished triangle

which is defined by the adjunction map in the obvious way. By definition j*(P) is a zero object in D+(T, A). In particular, every constructible subsheaf .if/' of a cohomology sheaf of the complex P has vanishing pullback j*(. /V} All such constructible subsheaves therefore have support in some proper closed subscheme of S. The isomorphism claim is therefore true in this particular case by the induction assumption. On the other hand the isomorphism claim for the complex

has been verified in the last step 7). Hence the claim 4) now follows for arbitrary K as above by the 5-lemma. D

11.9 Relative Duality for Closed Embeddings In this section we continue and complete the proof of the relative duality Theorem 11.7.1, which was proved for smooth morphisms in the last section. We also continue the numbering of steps in the proof, which started in the last section. 9) We consider closed embeddings i:Y--+X.

For an etale sheaf

:~

of A-modules on X let

be the sheaf of sections with support in Y. For a sheaf .¥ on Y there are canonical identities

The functor i*

0

ry

is therefore right adjoint to i* = i!. So for injective sheaves :1/ on X also i* (ry (;~» is injective, since the functor i* is an exact functor. In particular, the hyper-ext functors Ext" for complexes L E D-(Y, A) and K E D+(X, A) satisfy Ext"(L, i*Rry(K»

=

Ext"(i*(L), K) .

II. 9 Relative Duality for Closed Embeddings

Therefore Hom(L, i*Rry(K))

=

117

Hom(i*(L), K) .

Again - the homomorphism groups are understood to be the homomorphism groups in the derived category! In other words

Lemma 9.1 The Junctor i! := i*

0

Rry : D+(X, A) --+ D+(Y, A)

is a (partial) right adjoint Junctor Jor the Junctor

Remark. For smooth morphisms dings i, such that

J :X

--+

Sand g : Y

--+

S and closed embed-

we get canonical isomorphisms K E D+(S,A),

an immediate consequence of the relative duality theorem for the smooth case, proved in claim 4)!

The General Case 10) Let be given a smoothly embedable morphism

J, i.e.

J=goi:X--+S,

can be written as the composite of a closed embedding i : X --+ T and a smooth (finitely generated) morphism g : T --+ S. For Rg! and also for Ri! = i! we have already constructed partial right adjoint functors g! and i!. The composite functor

is the desired partial right adjoint functor for

We now restrict ourselves to the category of finitely generated schemes over a fixed base field. Then the functor J* has finite cohomological dimension (see

118

II. The Formalism of Derived Categories

Theorem 11.7.6). This permits to conclude that for a closed embedding i : Y -+ X the functor ry, hence also the functor i! , has finite cohomological dimension. Note that this is a consequence of Theorem 11.7.6, using the distinguished triangle Rry(K) --- K --- Rj*j*(K) - - Rry(K)[I] .

Here j : U = X \ Y ~ X denotes the embedding of the open complement of Y in X. 11) The same distinguished triangle allows to deduce from Theorem 11.7.7 respectively its corollaries 11.7.8 and 11.7.9, that for a complex

also i!(K) E Dctf(Y, A)

holds. More generally for a smoothly embedable morphisms I : X -+ S as in 10) (in the category of finitely generated schemes over a fixed base field) the functor / defined above is a functor which preserves the subcategory of complexes of finite Tor-dimension / : D~tf(S, A) -+ D~tf(X, A) . In other words

I!

is a right adjoint functor for the functor

12) Finally let A' be a different coefficient ring, subject to the assumptions made earlier, and let A -+ A' be a ring homomorphism. Let

11 : D~tf(S, A) -+ D~tf(X, A) 11, : D~tf(S, A') -+ D~tf(X, A') be the adjoint functors, that were constructed for the underlying rings A and A'. It is easy to see from the definitions that

This observation now implies the existence of the functor

under the hypothesis of Theorem 11.7.1 and with all the required properties. This is shown by the limiting argument explained e.g. in Appendix A. Thus Theorem 11.7.1 ~~~

0

1I.1 0 Proof of the Biduality Theorem

119

11.10 Proof of the Biduality Theorem After having proved Theorem 11.7.1 we now turn to the proof of the biduality Theorem 1I.7.4, again for etale sheaves of A-modules. The biduality theorem for Qrsheaves is then deduced from this by a limiting process. We now restrict ourselves to consider only finitely generated schemes over a given base field k and morphisms between such schemes over k. Let

f : X -+ S = Spec(k) be a finitely generated smoothly imbedable compactifiable scheme over the base field k.

Definition 10.1 The complex J'(As) = Kx will be called the A-dualizing complex.

Lemma 10.2 The A-dualizing complex Kx is an object of the category D~tf(X, A). It has the following finiteness property: There is a natural number N such that for all constructible sheaves on X the following holds .9'8V(R.9IJom(.~, Kx))

=0

,

~

for v> N.

Alternatively: Kx has a finite resolution by a complex p., whose components pv are acyclic with respect to allfunctors .9!6'om(.'Y, -), where :5" is a constructible sheaf on X.

Remark. If the base field k is separable closed we prove below, that the A-dualizing complex Kx has finite injective dimension. So there is a number r such that ExtX (:§, Kx) = 0

V A-sheaves .~

v> r.

Equivalently Kx has a bounded injective resolution. Indeed for all A-sheaves have if v > 0 or v < -2· dim (X) . ExtXU~, Kx) = 0

;§-'

we

Without restrictions on the base field the following holds: if v> Oorv < -2 ·dim(X).

Proof ofthe Second Assertion. The construction of the dualizing complex K x is compatible with arbitrary base field extension. The construction of the hyperext-sheaves

120

II. The Formalism of Derived Categories

tfxtX (,'§', Kx) is compatible with finite separable (= etale) base field extension. For an affine projective limit X of schemes the global group ExtX x (.(~, .37) is the direct limits of the corresponding group for the schemes of the projective system, provided the sheaves ,(~ and are constructible ([FK], chap. I, proposition 4.18). The proof remains valid for more general rings A than the ring Z/nZ considered in loco cit. Hence the construction of the sheaves f5 x t X(,(~, K x) is compatible with arbitrary separable base field extension. Without loss of generality we may therefore assume, that the base field k is separable closed. We now prove the claim of the remark. Since A is a self injective ring, for a bounded complex M of A-modules satisfying HV(M) = 0 for all v > r, v < s, the hyperext-groups ExtX (M, A) vanish for all v > -s, v < -r. For a complex K E D-(S, A) we have ExtXs(K, As) = ExtX (reS, K), A). Since the cohomological dimension of the functors f! and rc(X, -) is bounded above by N = 2dim(X), we obtain from Poincare duality for all A -sheaves ,~ on X

.r

ExtX (.(~, Kx) = ExtX (Rf!(~), A) = ExtX (Rrc(X, .(9), A) = 0 D

for all v> 0, v < -N. Lemma 11.10.2 has the following consequence: For any complex L the dual complex D(L) = Dx(L)

E

Di(X, A)

Dx(L) = R.9'6'om(L, Kx) is again contained in Di(X, A). If furthermore L

(Corollary 11.7.9). For a complex L the biduality map

E

E D~tf(X,

A), then also

Di(X, A) there exists a natural functorial homomorphism, L

~

Dx(Dx(L» .

This defines a natural transformation

Id

~

Dx

0

Dx.

Theorem 10.3 The biduality map is a canonical isomorphism. Remark. For our purposes it is enough to restrict to the case of complexes L in D~tf(X, A). Proof We will make use of the formula

for morphisms f:X~

Y.

11.10 Proof of the Biduality Theorem

121

This was obtained in 11.7.3 as a corollary of the relative duality theorem, which we proved in the last two sections. The assertion of the theorem is local on X and S with respect to the etale topology and it is enough to consider the case where L = .7 is an etale sheaf (5-lemma). We may furthermore assume Y to be A-flat (use a resolution). The Smooth Case. Suppose X is smooth over the base field k and .¥ is locally constant. As the statement is of local nature we can assume that .7 is constant, i.e .¥ = .~6x for some finite A-module vl6. Furthermore we may assume that X is equidimensional. Recalling the definition of Kx = J' (A) and the definition of J' in the smooth case, we get for any geometric point x of X

This follows from the definition of R..9'6om ( -, -) and the Remark 11.1.8. Here we used, that A is an injective A-module. We therefore get R..9'6om(vl6x , Kx) = .9(Jom(vlt, A)x[2d](d) .

Hence the biduality theorem is an immediate consequence of the corresponding biduality theorem for finite A-modules

This identity holds for all finite A-modules .~6, since A is by assumption a quasi Frobenius ring (Gorenstein of dimension 0). The General Case. As the statement is of local nature with respect to the etale topology, we may assume that X is affine, hence a closed subscheme of some standard affine space Am over k. However, because of formula (*) above we can actually assume without restriction of generality, that X = Am. Now the plan of proof is induction on m. We want to show that for constructible sheaves Y on Am we have

The case m = 0 is a trivial case of the smooth case, hence already considered. So suppose m > O. We can find an open dense subscheme j:U w. This is trivial for Xo = Spec(K). We have the following elementary properties (1) w(Ko[n]) = w(Ko) + n (2) w(Ko(m)) = w(Ko) - 2m (Tate twist)

and the permanence properties (3) RfoiD~w(Xo)) C D~w(Yo) for a morphism fo : Xo--+ Yo defined over K. (4)

fd(D~UJ(Yo)) C D~UJ(Xo) for a morphism fo : Xo --+ Yo defined over K -

which follow by duality from the Weil conjectures I.9.3 and I.9.4: (5) RfO!(D~UJ(Xo)) C D~w(Yo) for a morphism fo : Xo --+ Yo defined over K. (6) fo(D~:(Yo)) c D~:(Xo) for a morphism fo : Xo --+ Yo defined over K. -

Homomorphisms. In particular, as a consequence KO

=

R.7Com(Ao, Bo[1])

for Ao E D~UJ(Xo) and Bo D(Ao ® D(Bo[1])).

E

E

D~UJ'-w+l (Xo)

D~UJ'(Xo). Recall that R.9rlom(Ao, Bo[1])

132

II. The Formalism of Derived Categories

The Galois group Gal (k / K) is the profinite completion of the cyclic group generated by the Frobenius element F. This implies: Let Ko E Dg(Xo). Then there is an exact sequence

O ---+ H -I (X, K) F ---+ H 0 (Xo, Ko) ---+ H 0 (X, K) F ---+ 0 in terms of coinvariants and invariants under F (Leray spectral sequence). By property (3) above we have for Ko E D~w'-w+1 (Xo) for

J) :::::

Suppose now w' - w > O. Then this forces HO(Xo, Ko)

-I .

= 0, because

Consider Hom Di(Xo)(AO, Bo[1]). To compute this group, we replace Bo by an injective resolution 10. Then HomD7(Xo)(Ao, Bo[1]) is the group HOmK(Xo)(Ao, 10[1]) of complex maps up to homotopy. See Remark 11.1.8. Under our assumption on Bo this is the same as the hypercohomology group HO(Xo, Ko) HO(Xo, R.31'5om(Ao, 10[1])). See Verdier [308] and also [FK], p.300. Furthermore we have

Proposition 12.4 Let Ao

E

D~w(Xo) and Bo _

E

D~_w ,(Xo). Then

vanishes ifw' > w. Proposition 12.5 Let AO E D~w(Xo) and Bo the algebraic closure k of K

E

D~w'(Xo). Then base change to

is the zero map, ifw' > w - I. If K = R.?15'om(Ao, Bo[l]) happens to be in D::':o(Xo), i.e has nontrivial cohomology sheaves only in degrees::::: 0, then H-I(X, K) = O. We then get the stronger

Proposition 12.6 Suppose R.?60m(Ao, Co) E D::':o(Xo) and furthermore assume and Co E D~_w ,(Xo). Then that Ao E D~w(Xo) _

and this group vanishes for w' > w.

II.l2 Mixed Complexes

133

This is a variant of the last proposition, if we put Co = BO[I]. It holds, since the kernel of the base change restriction map of the last proposition is the cohomology group H-l(X, R.%1om(A, B[1]))F and vanishes by assumption. Lemma-Definition 12.7 Let Ko E D~(Xo, Q[) be a mixed respectively T"mixed complex with upper weight w = w(Ko). If Ko =1= 0 is nontrivial, then the following holds w(DKo) :::: -w(Ko) .

In particular; if Ko E D~w (Xo) holds for some w, then w :s w(Ko)· The complex Ko - suppose Ko =1= 0 - is called pure resp. T-pure of weight w equality w(DKo) = -w(Ko) holds, i.e. if Ko E D~w(Xo)

n D~w(Xo)

if

.

It is also convenient to define Ko = 0 to be (T-)pure of weight

-00.

Remark. If Ko is T-pure of weight w, then w necessarily coincides with the upper weight w(Ko)

Warning. In case of sheaves concentrated in degree zero the notion of being T -pure of weight w given in 11.12.7 above does not coincide with the notion of (pointwise) T -pure sheaves in the sense of Definition I I.2.1. In the following we will have to distinguish these two notions. We will therefore always add "point-wise", if the meaning of pure or T -pure is not to be understood in the sense of the definition above. Proof of //,12.7. Put Ao = Ko and Bo[1] = Ko. Then w = w(Ao) = w(Ko) and w' = -w(D(Bo» = -1- w(D(Ko». Now either w' :s w -1, i.e. -w(D(Ko» :s w(Ko). This is the assertion of 11.12.7. Or w' > w - 1. Then Proposition 11.12.5 implies idK E Hom(K, K)F = 0, hence K = O. Then the stalks of the cohomology sheaves in all geometric points vanish. Therefore Ko = 0 contrary to the assumption.

III. Perverse Sheaves

111.1 Perverse Sheaves The theory of perverse sheaves historically emerged from several independent directions. One of them was the theory of intersection cohomology of Goresky-MacPherson, which originally was not defined in terms of sheaf theory but rather using explicit chain complexes. Perhaps stimulated by the Kazhdan-Lusztig conjectures it was Deligne, who gave a reformulation ofthe notion of intersection cohomology within the setting of sheaf theory. In this form intersection cohomology can be defined also for finitely generated schemes over a field of characteristic zero, over a finite field or over the algebraic closure of a finite field. The theory found its final form in the fundamental treatise [BBD], soon after it was realized that perverse sheaves define an abelian category inside D~(ij/). This was suggested by the algebraic theory of 'Xmodules due to Bernstein, where it turned out that the category ofholonomic !./!-modules with regular singularities provide an analog via the Riemann-Hilbert correspondence. In this book we only consider the so called middle perversity. Middle perverse sheaves turn out to be most interesting perverse sheaves, since their notion is self dual with respect to Verdier duality. It is the abelian category of middle perverse sheaves, which also fits perfectly with Deligne's theory of weights. In particular, one has for it the decomposition theorem due to Gabber. The more general perversities, which are defined in [BBD], do not have this particular rigid structure. Their definitions and properties can be dealt with more or less in the same way as for the case of the middle perversity. Although they are important, we therefore do not consider them in this book. For their definition and properties we refer the reader to the original source [BBD].

Notational Remark. Throughout this chapter on perverse sheaves we often write D(X) instead of Dg(X, Q/) (derived categories of a scheme X) and D(K) or DK instead of D x (K) (Verdier dual of a complex K on X) and ® instead of ®L , in order to simplify notation. It seemed to us, that there is little danger of confusion. Let X be a scheme over a base field, which is either a finite field or an algebraically closed field. Let D be now the triangulated category D(X) = Dg(X, Q/). The selfdual (middle) perverse t-structure is defined by

B E PD?:.O(X)

¢=:=}

dim supp(,'7fj-i DB)

.:s i

Vi

E

Z

Vi

E

Z,

The index P refers to truncation with respect to the perverse t-structure. For the moment let us assume, that these definitions induce a (-structure on D(X), The proof of this fact will be postponed to §3. From this definition is then clear, that every object

136

III. Perverse Sheaves

of D(X) is bounded with respect to the perverse t-structure. In particular almost all perverse cohomology groups of an object in D(X) vanish. Lemma-Definition 1.1 The abelian category Perv(X) of perverse sheaves on X is the core Perv(X) = PD~o(X) n PD~o(X) of the category D(X) = D~(X, Qz) with respect to the perverse t-structure. The corresponding cohomological functors are PH V

:

D(X) ---+ Perv(X) .

For any distinguished triangle (A, B, C) in D(X) this gives a long exact sequence ... ~ PH-l(C) ~ PHo(A) ~ PHo(B) ~ PHo(C) ~ PHI (A) ~ ... in Perv(X).

Warning. In general a perverse sheaf on X is not a sheaf, but only represented by a complex of sheaves on X. In the following the truncation operators Pr and r in the category D(X) will be always understood as truncation operators with respect to the perverse t-structure respectively with respect to the standard t-structure, if not stated otherwise. Remark 1.2 By definition the perverse t-structure is self dual on D(X), i.e. B E Ib~o(X) iff DB E Ib~o(X) .

Let x be an arbitrary point x of X, let i : Y ~ X its closure in X with reduced subscheme structure. Any B E D(Y) becomes a smooth complex on a suitable smooth open neighborhood U of x. For such U the dualizing complex is Ku ~ Qzu[2d(x)](d(x», where d(x) = dim(Y). Hence (Dy(B»'I ~ (B'I)V[2d(x)](d(x» for any geometric point TJ over x. Here M V denotes the dual of M as a complex of Qz-vectorspaces, where stalk complexes are considered as complexes of Q,-vectorspaces. Definition 1.3 For L in D(X) define

i~L

=

(i!L)'I

Since i;(Dx(L» = (i*(Dx(L»'I = (Dy(i!L»'I = il(L)[2d(x)](d(x», another characterization for the perverse t-structure on D(X) is provided by B

E

Ib~o(X) {::::::} .9!6' vi;B = 0 for v > -d(x)

B E Ib~o(X) {::::::} .~vi~B = 0 for v < -d(x) ,

where this should hold for all points x in X.

III.2 The Smooth Case

137

111.2 The Smooth Case Let X be a scheme over some arbitrary base field. Let X denote the scheme obtained from X by base change to the algebraic closure of the base field. The scheme X is called essentially smooth, ifthe reduced scheme Xred is smooth. Under the assumption that the base field is finite or algebraically closed, the scheme X is essentially smooth if and only if the reduced scheme Xred is smooth. Suppose X is essentially smooth, and suppose X is equidimensional of dimension d. Then the dualizing complex on X has the form

It is a complex K x, whose cohomology is concentrated in degree - 2d and such that its cohomology sheaf .9t'f- 2d (Kx) is isomorphic to the smooth sheaf Q, (d). For a smooth sheaf .~ on X - sheaf to be understood in the ordinary sense - the dual sheaf was denoted .(~V ~v = .9J6'om(;§, Q,) .

A sheaf complex K E D(X) = Dg(X, Q,) will be called a smooth complex, if all its cohomology sheaves ,9!6V(K) are smooth sheaves on X.

Proposition 2.1 ( i) Let X be an essentially smooth equidimensional scheme of dimension d and let K E D(X) be a smooth complex on X. Then

in particular .9J6'v (D K) vanishes if and only if.9J6'-v-2d (K) vanishes. (2) Suppose X is irreducible and let K be in D(X). Then there exists an open dense essentially smooth subscheme j:U"-+X

of X, such that j*(K)

is a sheaf complex, which is smooth on U. Proof Use induction on the cohomology degree in which there are non vanishing cohomology sheaves, using the "ordinary" truncation operators r:::;s for the standard tstructure on the triangulated category D(X). For K E D(X) wehavei~s (K)[ -s] = r~sr::::sK. See Chap. 11.6.1 for details on the standard t-structure. For a smooth complex K "I 0 there exists s such that the distinguished triangle

138

III. Perverse Sheaves

defines the smooth sheaf.'i/ = i~s K = i~s i~s K = .9ffJs (K) =1= 0, such that A = i9-1K is a smooth complex with cohomology sheaves .9ffJV(A) = .9ffJV(K) for v < sand .%V (A) = 0 for v ~ s. For smooth Q/-sheaves .~~ and s E Z consider the complex

concentrated in degree s. Then.9'(5v (;~'[ -s D = 0 for v =1= sand.9ffJs (::7[ -s D = :§. Then, for any smooth ij/-sheaves .'7, .'~ on X,

For.'7

= Q/(d) this gives

D(;~[ -sD

= R.%om(.'~[ -s], .¥"[2dD = .9IJom(~, .¥)[2d + s]

.

Therefore, if we apply the functor R.%om ( -, .¥[2dD to the triangle (A, K, .~ [ -s D defined above, the long exact cohomology sequence of its cohomology sheaves .9IJVR.9ffJom(;~[ -s],

.¥[2dD

~ .9IJ vR.9ffJom(K,

.'7[2dD

~

.9IJvR.%om(A, .¥[2dD ~

and the obvious facts CYL'VRCZL' ('r.[ ] .-r--[2d]) _ { 0 v =1= -2d - s ./'Dom.:/ -s ,.¥ .9ffJom(~, .¥) v = -2d - s

./0

and.9IJ VR.9ffJom(A, Y[2dD

ofor v
-2d - s . 0

.?7(5om(~',

Using these identities and the induction hypothesis the proof is now complete.

D

Assume that X is irreducible and essentially smooth. Recall, that this was defined at the beginning of this section. Let B be a complex in D(X) such that all cohomology sheaves .9IJi B of B are smooth sheaves on X. This will be referred to as the "smooth case" for the rest of this section and the next section. A smooth sheaf on X has support of dimension dim(X). All.9ffJi B were assumed to be smooth, therefore we obtain from the last proposition. Remark 2.2 Under the assumptions above, i.e. X essentially smooth, equidimensional and B a smooth complex, we have

1II.3 Glueing

B E PD:'5. o(X) iff .%V B

= 0 for all v >

B E PD~o(X) iff .%V B

= 0 for all v

139

-dim(X) .

By duality < -dim(X) .

o In the smooth case the perverse t-structure therefore behaves like the standard t-structure shifted by dim(X). Let X be essentially smooth and let B be a smooth sheaf complex on X. We get in this case: B E Perv(X) if and only if B = ~[dim(X)]

for a smooth sheaf ;~. In particular, the cohomology sheaves of B are trivial in degrees different from -dim(X). That in general the definitions preceding 111.1.1 define a t-structure on D(X) is not that obvious. The proof will be given in §3. In order to verify the properties (iii) and (i) of a t-structure one has to use stratifications of X and for property (iii) one has to use glueing oft-structures. These techniques allow to reduce everything to the above mentioned simple "smooth case". Details will be explained in the next section.

111.3 Glueing Again in this section we consider finitely generated schemes over a finite field or over an algebraically closed field. Let be given a scheme X and an open sub scheme U j:Uli*F) i' (A, E, B) by definition. From this we get 1) A E T"::'o(X, U): In other words j* A ~ i"::'Oj* E E T"::'O(U)

and 2) BET?:! (X, U): In other words

142

III. Perverse Sheaves

j*B ~ i~lj*E

E

'!B = ~ I.!.'*i?:.l''*F = ~

T?:.l(U)

I

i?:.ll'*F E

T>l(y) -

.

o

This finishes the proof.

Example "i~o' One useful special case is the t-structure, which is obtained from glueing with the degenerate t-structure on T(U), i.e. T(U)~l = O.

In this special case the above construction gives F~E

For the glued t-structure on T(X, U) this determines the upper truncation functor y E ~ . '*E i?:.O = 1* i?:.OI

.

For the corresponding lower truncation operator, denoted i~O' we therefore get a distinguished truncation triangle -

(i~oE, E, i*"i?:.li* E) . Using the gluing construction and noetherian induction, we are now prepared to prove that the perverse truncation structure PD=5.o(X), PD?:'o(X) B B

E

PD=5.o(X)

¢=:::}

E PD~o(X) ¢=:::}

dim supp(9(j-i (B)) :::: i dim supp(.9fIj-i (DB)) :::: i

imposed on D(X), which was introduced in the last section § 1, is indeed at-structure. Lemma 3.1 Let j : U ~ X be an open subscheme of X, and let i : Y closed complement. Thenfor BE D(X) the following holds B E ID=5.o(X) BE ID?:'o(X)

¢=:::} ¢=:::}

~

X be the

j* B E ID=5.o(U) and i* B E ID=5.o(y) /B = j*B E ID?:.o(U) andi!B E ID?:.o(y).

Proof. First observe, that a complex E on X satisfies the semiperversity condition E E PD=5.o(X) iff its restrictions to U and Y satisfy the corresponding conditions

j*(E) E PD=5.o(U) and i*(E) E PD=5.o(y). This is a trivial consequence of the definition and the exactness property of the functor f*. By the commutation rules for the dualizing functor

t'

0

D = D

0

f*

f*

0

D= DOt'

on the category of etale sheaves, we immediately get the assertion of the second equivalence. 0 The compatibility properties 111.3.1 say, that the perverse truncation structure PD=5.o(X), PD?:.o(X) on X is obtained from the perverse t-structures on D(U) and

III.3 Glueing

143

D(Y) by gluing (provided we already know that the Definition 1I1.1.1 defines tstructures on D(V) and D(Y». There is an obvious generalization of Lemma 111.3.1, if X is a finite disjoint union of locally closed subschemes. We do not formulate this.

Proof of Lemma III.I.I. Let X be a scheme. Assume, that X is reduced without restriction of generality. Let j:V'-+X

be a nonempty open essentially smooth subscheme of X and let

its closed complement. Suppose, that all generic points of X are in V. Then by induction on the dimension we can assume, that the perverse truncation structure already defines a t-structure on Y. Put T(Y) = D(Y) together with the perverse t-structure on it. On the other hand let T(V) be the full subcategory of D(V), consisting of complexes with smooth cohomology sheaves. According to Remark 111.2.2 the axioms for a t-structure are valid for the restriction of the perverse truncation structure to T (V), by trivial reasons. Furthermore, the induced t-structure on T(V) coincides with the standard t-structure up to a shift of degree. By gluing we get T(X, V) = {E E D(X)

I j* E has smooth cohomology sheaves on V}

.

By Lemma II1.3.1 the glued t-structure on T(X, V) coincides with the perverse truncation structure, which is obtained by restriction from VeX) to T (X, V). In other words, the perverse truncation structure satisfies the axioms (i)-(iii) of at-structure on T(X, V). Now let E be an arbitrary complex in D(X). Then there always exists an open dense essentially smooth subscheme V '-+ X, such that the restriction of E to V has smooth cohomology sheaves. In other words D(X)

= UeX,

u

T(X, V).

dense open ess. smooth

Therefore axiom (iii) holds for the perverse truncation structure. A similar argument proves axiom (i). For complexes E E JI):"':o(X) and E' E JI)2:1 (X) there exist V and Vi as above, such that E E T(X, V) and E' E T(X, V'). We can replace V, Vi by V" = V n V'. Then Rom(E, E') = 0 can be verified in the full subcategory T(X, V"). Therefore the perverse truncation structure on D(X) satisfies all axioms of at-structure. 0

Truncation of Mixed Perverse Sheaves. In the remaining part of this section let the base field K be a finite field. Let Xo be a an algebraic variety, i.e. a finitely generated scheme over K. A complex Ko E D(Xo) was called i-mixed, if all its cohomology

144

III. Perverse Sheaves

sheaves .76 v (Ko) are i-mixed sheaves in the sense ofll.2.1. The i-mixed complexes define a full triangulated subcategory

of D(Xo). This subcategory is stable under the functors

Again by noetherian induction and the construction of truncated objects via gluing, one obtains Lemma 3.2 Let X0 be a variety over a finite field K. Let Kobe a i -mixed complex Ko E Dmixed(XO). Then the perverse truncation operations preserve the property of being i-mixed

The perverse t-structure on the category D(Xo) therefore induces a t-structure on Dmixed(XO)·

Remark3.3 Suppose given a complex K E D(X). Then there exists a "stratification" of X by finitely many locally closed essentially smooth equidimensional subschemes

such that the restrictions i~ K and i~K are smooth on these subschemes, i.e have smooth cohomology sheaves. X is a disjoint union of its strata Y v . We can additionally assume, that the closure of each stratum is a union of strata. Fixing this stratification we consider the full triangulated subcategory of D(X) of complexes L E D(X) such that i~L

have smooth cohomology sheaves. The restriction of the perverse t-structure to this subcategory is obtained by gluing of shifted standard t-structures (depending on the dimensions; see Remark III.2.2) in the categories of smooth complexes on the strata Yv~

X.

This defines the t-structure in terms of iterates of the gluing method.

111.4 Open Embeddings Let T be an additive, translation preserving functor between triangulated categories A and B with t-structures, which transforms distinguished triangles in distinguished triangles. Such a functor is called

IlI.4 Open Embeddings

145

t-right exact iff T(D:::o(A)) C D:::o(B) and t-Ieft exact iff T(D:o:o)(A) C D:o:o(B). Finally T is called t-exact, if T is t-Ieft and t-right exact.

Lemma 4.1 Let X be afinitely generated scheme over afinite or algebraically closed field. Let j : U ~ X be an open embedding with closed complement i: Y ~ X. Then the functors are t-right exact, are t-exact, are t-left exact for the perverse t-structures on D(X) respectively D(U) and D(Y). Proof The properties are obvious for the functors j*, i *, i! and i* using Lemma III.3.1. In the case of i* recall that j*i* = 0 and i*i* = i!i* = id. For Rj* and j! the assertion follows by adjunction from the t-exactness of j*. Namely

for B E PD:O:o(U). Therefore Rj*B E PD:O:o(X) by Lemma 11.2.2. So Rj* is t-Ieft 0 exact. Then j! is t-right exact by duality. As a consequence of the t-exactness of j* and the t-Ieft exactness of Rj* in the situation of Lemma 111.4.1 above we get for perverse sheaves E E Perv(X) Pfr1Rj*j*(E) =

o.

The distinguished triangle (i*i!E, E, j*j*E) therefore induces the following exact sequence of perverse sheaves in the abelian category Perv(X)

Restriction Sequence

Now let us consider two perverse sheaves A E Perv(Y) and E E Perv(X). We have A = T:::oA and i!E E D:O:o. Furthermore i* is fully faithful. This implies by adjunction and the long exact sequence of the Hom-functor, together with t-structure axiom (i), Hom(i*A, E)

=

Hom(A, i! B)

=

Hom(A, PfI°(i!E))

= Hom(i*A, i*PfI°(i!E)) .

Thus i*Pf{°(i! B) is the largest perverse subobject of the perverse sheaf E, which comes from Perv(Y). More precisely

146

III. Perverse Sheaves

Lemma 4.2 Let i : Y -* X be a closed embedding and B a perverse sheaf on X. Then

is the largest perverse subobject of B, which is isomorphic to an object of the subcategory i*Perv(Y) of Perv(X). As the dual statement,

is isomorphic to the largest perverse quotient of B with that property.

Suppose X to be a finitely generated scheme over a finite or algebraically closed field. In III, §3 we obtained the truncation axiom

This global identity has a local analog for the functor R.9i'5' om ( -, -).

Lemma 4.3 Let B be in PD:SO(X) and let C be in PD?:.O(X). Thenfor the standard t-structure we have R.9llJom(B, C) E D?:.o(X, Q/), i.e.

v
oBo) :5 w imply w(PToBo), w. Therefore d (perversity

imply the conditions of the weight criterion m.lD.I for the perverse sheaf Pj{o Bo. Therefore Theorem III.lO,! shows w(Pj{o Bo) :5 w. This proves the inductive step. Thus the nontrivial direction of the corollary follows. 0 Exercise 10.3 Let j : U '-+ X be an open embedding, let B E Perv(U) be T-pure of weight w. Then Ii = j!*B is the graded component of weight w in the weight filtration of PHORj, B. (Use the diagram of III.5.S). Similar for PHORj*B. The last corollary and its dual imply Corollary 10.4 A T-mixed complex Bo Pj{" Bo are T-pure of weight w

+ v.

E

Dg(Xo, 02/) is T-pure of weight w

iff all

Corollary 10.5 In the situation of III. 10.4 the T-mixed perverse sheaves Pj{" Bo are T-pure of weight w

+ v iff all their simple constituents are T-pure of weight w + v.

Proof §9, Lemma II and its dual for the nontrivial direction

===}.

o

This together with an argument similar to the one used in the proof of §9, Lemma III implies one of the most striking results, namely Gabber's decomposition theorem Theorem 10.6 (Semisimplicity) Let Bo

E Dg(Xo, 02/) be T-pure of weight w. Consider the base change B of Bo to the algebraic closure B E D~(X, 02/), X = Xo X K k. Then B is isomorphic to a finite direct sum of translates A[i], i E Z of simple perverse sheaves A on X

B ~ EBA[i]

A

E

Perv(X) .

Proof Using induction on the (finite) number of nonvanishing perverse cohomology sheaves PH" (Bo) "lOwe first prove B ~ EEl" PH" (B)[ -v]. For Bo "I 0 as in the

theorem consider the distinguished triangle

with respect to the last nonvanishing perverse cohomology sheaf PH n (Bo). By Corollary III.lOA and the induction assumption it is immediately clear that PT

As .

/.~ £'

Similar to Chap. I, §5 we now define the relative Fourier transform vector bundle £ / S

T:/ s

for the

178

III. Perverse Sheaves

by T:IS(K e )

=

RP2{pj(K e ) ®f-L*(5((1/f»)[r].

We often write T: IS = T! or T: IS = T1jf, if the underlying vector bundle is understood from the context. This generalized Fourier transform for vector bundles E / S enjoys a number of properties, whose proofs are easy or similar to the special cases already considered in Chap. I, §5 and Chap. III, §8. Hence no proofs are given for the following theorems. Theorem 13.1 (Inversion formula) Let E / S be a vector bundle of rank rover S. For complexes K e E Di(E, Q1/), the equation

canonically holds.

Theorem 13.2 (Functoriality) For the Fourier transform the following holds:

»

1) Compatibility T:xsT IT (f* K e ) = !*(T:IS (K e for arbitrary base change f: T -+ S. 2) Suppose given a homomorphism u of vector bundles over S and its dual

u:E-+F

u:F-+E "

I

Put d = rank(F) - rank(E). Then for arbitrary complexes K e and C E Dg(F, Q/) the following holds: (a) (b)

T;IS(Ru!K e )

= u'*(TrIS(Ke))[d]

E

b

-

Dc(E, Q1,)

,

T:IS(u*(C» = Ru; T;'IS(C)[d](d) .

The arguments for (1) and (2a) are obvious. (2b) follows from (2a) by the inversion formula. Theorem 13.3 (Direct images) Let E be a vector bundle over S, let B -+ S be a base change map and suppose K e E Dg (B x S E, Q1/). Consider the projections pr : B Xs E -+ E,

,

I

pr : B Xs E -+ E

I

Again this is obvious from the definitions and the proper base change theorem.

III.13 Complements on Fourier Transform

179

Remark. The corresponding relations hold, if Rpr! and Rpri are replaced by Rpr* and Rp

T1/r(OO.E) @

. V; . CZJ(1j!).

Remark. We used, that the restriction of Vs to the zero section is the zero map. Hence

V; (-isotypic components, where t/> runs over the irreducible characters of W 7r*Q/y

=

E9 ~ ®ij/ Vcf>. cf>EW

Here ~ are smooth Q/-sheaves on X and Vcf> are Qrvectorspaces, such that W acts on 7r*Q/y via irreducible representations t/> : W ~ Gl(Vcf». Obviously in all stalks 7r*(Q/)x the group W acts through the regular representation. In particular ~ i= 0 for all t/>. Furthermore 7r*(~) ~ (Q/y t(cf»

for the ranks r (t/» of the smooth sheaves ~. In particular # W = Lcf>E W r (t/» . deg(t/». Since R7r!

#W =

= 7r! = 7r* and 7r! = 7r* the adjunction formula gives

deg(t/>!) . deg(t/>2) . dimij/HomDg(X)(~I'~) .

L cf>1,cf>2

Similar for C = ~ the rank r(t/» = dimij/HomDgCy)(Q/y, 7r*(.~» is equal to dimij/ Hom Dg(X) (EB x ~ ®ij/ Vx , .~) by the adjunction formula. Hence r(t/» = Ldeg(x)· dimij/HomDg(X)(~'~) ~ deg(t/».

x

Now it is easy to see, that the character formula #W = Lcf>EW deg(t/»2 and the inequality r(t/» ~ deg(t/» imply r(t/» = deg(t/» for all t/> E W. Furthermore we obtain, that ~l ~ ~2 are isomorphic as Qrsheaves iff t/>! ~ t/>2 are isomorphic as representations of W. More precisely dimij/HomDgCX)(~I'~) = 1, if t/>! ~ t/>2, and is zero else. In particular the sheaves ~ are irreducible smooth sheaves. e) Let E ~ S be a vector bundle. Assume, that the finite group W acts on E and S so that 7r is equivariant 0" E

W.

186

III. Perverse Sheaves

Then E -+ S is called an equivariant vector bundle iff the fiber mappings (J' : Es ---+ Eu(s) for s E S and (J' E W are linear. If E -+ S is equivariant, then also the dual bundle E' -+ S. The action (J' : E; --+ E~(s) on E', for s E S, (J' E W, is given in terms of the fibers by ((J'(y) , x) = (y, a-I (x)}, where y E E; and x E Eu(s). By the permanence properties of the Fourier transform (Theorem 111.13.2, 111.13.3) there are natural isomorphisms

With their help one constructs a canonical W -action on

for each complex K with W -action. This construction is consistent with the Fourier inversion formula. f) Suppose that (K e , (CPU)UEW) is a W-equivariant complex on X. Furthermore let y : C ~ K e be an isomorphism in the derived category D~(X, Qz). Then (L e , (y-lcpua*(Y))uEw) is a W-equivariant complex on X.

Lemma 15.4 Let f : X -+ S be an equidimensional smooth morphism with geometrically irreducible fibers. Especially, all fibers are nonempty. Suppose a finite group W acts on X over S, i.e. such that f is W -equivariant and such that W acts trivially on S. Let K e E D~(S, Qz) be a complex, such that there is an integer n with Ke[n] E Perv(S). Assume W acts on the pullback f*(K e ) compatible with the action of Won the scheme X. Then there is a representation of Won K e, i.e. there exists a group homomorphism

such that the action ofW on f*(K e ) is obtainedfrom that representation ofW by pullback. Proof. We abbreviate C = f*(K e ) E D~(X, Qz). Then according to 111.7.2 and IIL7.8 of Chap. III we have EndD~(S,ijl) (Ke) = EndD~(X,ijl) (L e), hence AutD~(S,ijl)(Ke) = AutD~(X,ijl)(C) .

For every group element (J' E W the complex the complex L e

(J'

*(L e) is canonically isomorphic to

a(a) : C ~ (J'*(C)

(J'

E

W.

These isomorphisms a «(J') satisfy cocyc1e conditions (J' *(a (r))a «(J') = a «(J' r) for the obvious reasons. Furthermore, for any homomorphism h : K e -+ K e in D~ (S, Qz) the pullback f* (h) = h : L e -+ C of this homomorphism satisfies (*)

(J'*(h) = a(a)

0

h

0

a(a)-l .

1II.15 Equivariant Perverse Sheaves

187

Since the given group action of W on Leis compatible with the action on the scheme X, this gives rise to a family of isomorphisms q;a : a*(Le) --+ U for a E Was defined in Definition IIUS.I. Then c(a) = q;a 0 a(a) is an automorphism of U a

W.

E

The cocycle conditions and the identity (*) imply q;a 0 a(a) 0 q;, 0 a(r) = q;a 0 0 a(r)) 0 a(a) = q;a, 0 aa,' Hence c(ar) = c(a) 0 c(r) for all a, r E W. Since Aut(K e ) = Aut(L e) there is a unique automorphism

a*(q;,

such that j*(co(a)) = c(a). Furthermore c(e) element e of W. It follows, that co(ar) = coCa)

0

= Idu and co(e) = IdK" for the unit

co(r)

a,r E W

defines a representation of W, i.e. a homomorphism co : W --+ Aut(K e ). The pullback of this action on K e is the action of W on L e. This proves the lemma. 0 Now let G be a general group again. Also assume now, that G acts on X from the left. Let m : G x G --+ G be the multiplication and let e : X --+ G x X be the zero section. Let a : G x X --+ X denote the left action on X and let P2 denote the projection P2 : G x X --+ X. A complex K e E Di(X, Q[) is called G-equivariant, if there exists an isomorphism q; in Di (G x X, Q[)

which is rigidified along the zero section e* (q;) = i d K., such that furthermore the cocycle condition (m x idx)*(q;) = P23(q;) 0 (idG x a)*(q;) holds over G x G xX. To be precise, we demand commutativity of the diagram (P2 (;de

0

(idG x a))*(K) = = = = (a

'.,"fe'

0

P23)*(K)

j

1

(a

0

(idG x a))*(K)

(a

0

(m x idx))*(K) (mxidxl*aex)

=

maxa(Zogqlr(ex)1 2 ) ,

a

where the maximum is over all ex such that na f= O. The tensor product induces a ring structure on Z[F] with unit element I given by ex = 1. We also consider the subring R generated by ex = qn/2 for n E Z

Let P*(La naex) = La na be the augmentation. Define D : Z[F] ~ Z[F] by D(L naex) = L naex- I . An element Q in Z[F] is called pure of weight w, if w(Q) :s wand w(D(Q)) :s -w. Any Q with w(Q) :s w can be uniquely written in the form

Q= v

+ q -I /2 . Q'

w(Q')

::s

w,

where v E Z[F] is pure of weight w (integrality of weights). Let G be a semisimple connected algebraic group over the algebraic closure k of the finite field K. We assume G to be of Chevalley type. Suppose the corresponding group Go acts on Xo, such that the induced action of G on X (over the algebraic closure k of K) satisfies

16.1 Assumptions. a) The stabilizers are connected subgroups of G. b) There are only finitely many G-orbits, each containing a K-rational point.

190

III. Perverse Sheaves

Then by Lang's theorem each orbit & = G/G x (for a fixed K-rational point x) satisfies &(K) = G(K)/Gx(K). For the G-orbit (91 in X let d(&) denote the dimension of (9. For a Go-equivariant Weil sheaf §Q on Xo the extension:7" on X is both a G- and Fx-equivariant Qrsheaf on X. Both actions are compatible (with notations as in Chap. III § 15)

a*(F;(:7"))

F;(rp)

~ pi(F;(.¥))

"',F', j a*(:7")

jP"F" rp

.. pi(·¥)

A G- and F x-equivariant Q[-sheaf on X is called a G-equivariant Weil sheaf, if the actions are compatible in this sense. If the Weil sheaf structure F* : F;(:7") ~:7" on:7" is understood from the context, by abuse of notation we also write w(.9T) for the weight w(.9ifi). The Grothendieck Group. In the situation above let Ko(G, X, Fx) denote the Grothendieck group of the abelian category of all G-equivariant Weil sheaves. In the following we will define two standard bases of this Grothendieck group, the T and the I basis. These two bases are related by "triangular" coordinate transformations. In the remaining part of this section this will be applied in two related situations, where the coefficients ofthe coordinate transformation define the KAZDHAN-LUSZTIG polynomials up to some normalization. The T -Basis. For each orbit define a Go-equivariant Weil sheaf on Xo by

extended from fi} to Xo by zero. The corresponding Weil sheaf has upper weight w(Tr:) = -d(fJi} The class of Tr: in the Grothendieck group is denoted Tr:. Under the assumptions a) and b) above Ko(G, &, Fx) ~ Z[F]. The isomorphism is induced by restriction to the fixed K-rational base point x of the orbit (0/. Similarly Lemma 16.2 Under the assumptions above the classes T r: form a Z[F] basis of the Grothendieck group ofGo-equivariant Wei! sheaves on Xo

KO(G, X, Fx) ~

EB Z[F] . Tr: r:

III.16 Kazhdan-Lusztig Polynomials

191

The classes of Qr(co (d(i)), corresponding to the closures 0 of the orbits ~, define another natural basis of the Grothendieck group. However, there is a better choice: Consider the category whose objects are complexes in D~ (X, QI), together with a compatible G-equivariant and Fx-equivariant structure. Morphisms are assumed to be G and F x equivariant. This need not define a triangulated subcategory, but this category is stable under Verdier duality and the other functors. The subcategory, where the complexes in addition are pure complexes of weight zero, is denoted H (G, X, Fx). If we forgetthe Weil sheaf structure, by Gabber's theorem each object I in H (G, X, F x) is a finite sum of translates of G-equivariant perverse sheaves. Since the cohomology sheaves of I are G-equivariant, we can define the class of I

v

in the Grothendieck group Ko(G, X, Fx). The I -Basis. Consider the inclusions

~ . ~ (0 . ~ X . Let w(/,,)

=0

be the intermediate extension of QI" (d(6)) to X, where (m) = [m]('r). In the obvious way this defines a G-equivariant Weil complex on Xo, which is pure of weight zero. Let I" = (-1 )d V )cl (/" ) be its class in the Grothendieck group. Then (*)

for some Q"

I"

E

I" =

L

", e"

Q"

I" (F) . Te ,

Z[F] such that Q",,(F) = 1 and

Lemma 16.3 The weights of the coefficients Q" r i ' ~ 6, ij6' "I (c.

for

I" in Z[F] satisfy w(Qe,,) < 0

Proof I" is pure of weight zero. In particular w(.:;:C;-V(/,,)) ::: -\!. Furthermore - \ ! < -dee') if.3'6'-v (/" )I(C' "10. Since the cohomology sheaves .9'13'-"(/" )I(C' are G-equivariant, this follows from the support condition 1lI.9.3 satisfied by a intersection cohomology sheaf. Since w(T",) = -d(~') and w(.9'6'-"(/,,)le') ::: - \ ! < -d((c'), the lemma immediately results from cl(.?13'-V(/,,))I(C ') E Z[F] . Te. D

The I" form a Z[F]-basis of the Grothendieck group Ko(G. X, Fx). Verdier duality induces a map D on Ko(G, X, Fx), for which this basis is self dual

D(I,,)

= I"

192

III. Perverse Sheaves

Inductively we deduce from (*) - by inverting a triangular matrix - the similar relations (**)

T(C

=

I(C

+

L

S(C 1(' (F) . I(C I

,

(CI~T

for some S(CI(C (F) E Z[F] with upper weight W(S(CI(C) < O. Lemma 16.4 The elements I(C in the Grothendieck group are uniquely determined by self duality and the equations (*) together with the weight conditions of Lemma III. 16.3.

I>

Proof. For another choice I~ we get I~ = I(C + leT R(C I (C (F)I(C I from (**), "'" R(C I(C = O. such that weight w(R(C I(C) < 0 holds. Self duality forces D

Correspondences on .Xl Let Bo be a Borel group of Go defined over K and let To C Bo be a maximal torus in Bo defined over K. By assumption this is a split torus of dimension say r. Let .)3'0 = Gol Bo denote the flag variety. Let k denote the algebraic closure of K and let G, B, T denote the corresponding groups over k. Let W = N(T)IT denote the Weyl group. For a E W let lea) be the length of a (with respect to B). Let ao denote the element of maximal length I (ao) = N where N = dim(.n), or N = dim (B) - r.

G-Orbits and Bruhat Decomposition. Consider X = .n and Y = X x X. The G-orbits yO' on Yare in one to one correspondence with the elements a E W of the Weyl group. We write a' < a if a' i= a and Ya l C yO' (Bruhat ordering). Consider the G-equivariant map Y-+B\GIB':::!:.W, with trivial action on W, defined by (gIB,g2B) f-+ Bg11g2B. Then yO' = {(g B, ga B) Ig E G} is the inverse image of the double coset Ba Bin B \ G lB. Note

where B a = a Ba- 1. In fact Ba = B a n B is the stabilizer of the point (l B, a B) E Yu). Hence dim(Ya ) = dim(G) - r - l(aao) = N + lea). All assumptions of III.16.1 are satisfied for the action of G on Y. Hence we have the complexes Ty" and Iy", and the two bases

of the Grothendieck group at our disposal. Both X and Y have canonical G-actions from the left, such that the two projections pri : Y -+ X are G-equivariant. Then Y Xx Y ':::!:. X x X x X. Puts = prl2, t = pr23 and m = prl3 : Y x x Y -+ Y for the third projection. These maps are G-equivariant.

III.16 Kazhdan-Lusztig Polynomials

J93

Convolution. For sheaf complexes K I , K 2 E Di (Y, QI) on Y define the convolution

product Kl

* K2 =

Rm! (S*(KI) 0

L

t*(K2)}-N) .

By the proper base change theorem (K I * K2) * K3 ;:; KI * (K2 * K3)' If KI, K2 E Di(Y, QI) are G-equivariant, so is Kl * K2 and the associativity isomorphism is an isomorphism of G-equivariant sheaves.

16.5 Some Special Cases 1. Example. Iy"o

= Q/y (dim(Y»). Then hao *hao = huo 0ij, H*(JJ, ijl)(dim(.Yi»)

with class (-I)Nq-N/20:=aEwql(a) IYaa

=

(-I)dim(Y)cl(/y"o)

I Yao

. cl(huo )' Since dim(Y) =

2N is even

= cl(/Yao)' Hence for the longest element 0"0

* I Yao = (-1 )Nq-N /2( ' " q/(a) ~

E

W

. I Yaa .

aEW

2. Example. hi = Q/ YI (N) is a unit element for the convolution product; note YI ;:; X ~ X x X is the diagonal, hence is closed. 3.Example. G = SI(2) and W = {I, s}. Then X = pI and Y = pI xpl. Furthermore YI = pi is the diagonal in Y = pI x pI and y, = Y \ Yl for s -I I is the open 1 complement. Since /Ys = Q/y[2](1) and Tys = QIY, (1) and TYI = QIYI (2) we have

Iy,

= T y, + q-l/2 . T YI

•

In particular the coefficient defined in Lemma III. 16.3 is QYI.Ys(F) = q-I/2. By example I therefore I y, * I y, = - (q I /2 + q -1 /2) . I Ys ' The same computation carries over for any Ty, and T ys defined by a simple reflection s in the case of an arbitrary semisimple group G

Lemma 16.6 Suppose that both K 1, K 2 are translates ofG-equivariant perverse Weil sheaves. Then convolution commutes with Verdier duality D(KI D(K2).

* K2) ;:; D(K» *

For the proof we need

Another Description Consider .

J.L

G x./J~G x

tv JJxjJ

B

][ ./J~./J

v*(K(-N)

=

(7rfL)*(T)

A

K(-N)


d x Y. Now we consider the universal incidence relation H x Y over Y, where (x, h) E ]p>d x pd is in H if and only if the point x is contained in the hyperplane h i:HxYi :EB L[ -2i]( -i) i

--+ Rp*p*(L) .

i=O

Proof The Kiinneth formula 11.7.5 allows to reduce to the case L = Qz. The classes TJi define morphisms Qz[ -2i]( -i) --+ Qz in the derived category, hence induced morphisms Rp3iz[ -2i]( -i) --+ Rp*Qz. Since Qz[ -2i](-i) r::: 2i (Rp*Qz[-2i]( -i», we have a natural morphism L TJi in the derived category, as stated in the lemma. That it actually defines an isomorphism, can be checked on geometric points. This allows to reduce to the case S = Spec(k) and L = Qz, where the statement is well known. D Radon Inversion d We now introduce an analogue of the Radon transform, with the roles of pd and W interchanged: Define Rad v = R1Th1T2'[d - 1] to be the dual Radon transform

We then get the following inversion formula Lemma 1.4 (Radon Inversion Formula) For every K constant complex pz [d] (K) E Pz(D~(Y, Q/» by

E

D~ (pd x Y, Q/) define a

d-2

(K) =

EB .%[d -

2 - 2i]( -i) ,

i=O

where.% = RP2*K. Then the following formula holds (Rad V

0

Rad)(K) ~ K(l - d) EB Pz[d](K) .

Proof We may assume d 2: 2. For simplicity of notation first assume that Y = Spec(K). By the proper base change theorem (Rad V

0

Rad)(K) = Ru*(Q (g/ v*(K»

where u, v : pd x pd --+ pd are the two projections and

Q

is the "operator kernel"

IV.2 Modified Radon Transforms

such that T(

:

207

X = H x Jll>d H -+ ]p>d x ]p>d is the inclusion of the incidence variety

defined by all (x, h, x') in]p>d x JiDd x ]p>d for which x, x' are both in h. The variety X is smooth over Spec(K) (use the projection to JiDd), hence ((Qllh is a pure complex on X of weight O. Although the morphism 11: is not a smooth morphism, it is proper. Hence Q is again a pure complex. By Gabber's Theorem III. 10.6 it has to be a direct sum of irreducible perverse sheaves. These can be computed by Lemma IY.l.3 d-2 Q[2 - 2d] ~ ~*«Ql)uDd)[2 - 2d](1 - d) EB EB(QlhDdxlPd[-2i](-i), i=O

where ~ : ]p>d "-+ ]p>d x]p>d is the diagonal embedding. Note that 11: is a pd-l-fibration over the image of ~ and a ]p>d-2-fibration outside this image. The claim of the lemma now follows from the last formula for the operator kernel Q. The proof for general Y is the same, since now Rad V

0

Rad(K) = Ru*(w*(Q)

fl!l

v*(K)) ,

where u : IP'd x IP'd x Y -+ ]p>d x Y, v : IP'd x IP'd x Y -+ pd x Y and w : ]p>d x JiDd x Y -+ IP'd x JiDd are defined by u(x, x', y) = (x, y), vex, x', y) = (x', y) and w(x,x' , y) = (x,x ' ). 0 Exercise. Give a direct proof for the formula for Q, which does not use the decomposition theorem.

IV.2 Modified Radon Transforms The Brylinski-Radon transform does not preserve the category of perverse sheaves. To phrase it in a different way: For n i= 0 the perverse sheaves Rad n (K) attached to a perverse sheaf K E Per v (]p>d x Y) are nontrivial in general. This is already evident in the case of constant perverse sheaves K. To analyze this in more detail, we define a modified Radon transform. For this we introduce some further notations. Consider the open complement U of the closed subscheme H x Y of IP'd x JiDd x Y

The restriction

q:U-+JiDdxY

208

IV. Lefschetz Theory and the Brylinski-Radon Transfonn

of the projection P2 to U is an affine map; its fibers are affine spaces of dimension d. In particular, Rq! is t-left exact for the perverse t-structures by 111.6.1. This implies Lemma-Definition 2.1 Define the modified Radon transform by

It preserves upper semi-perversity Rad! : PD?:.°(JF'd x y) --+ PD?:.°(lpd x Y). There is a sheaf complex homomorphism Rad(K) --+ Rad!(K) with constant mapping cone defined by.~ = RP2*(K). In particular there is a long exact sequence of perverse sheaves on

J¥d

x Y

... ~ Rad"(K) ~ Radi(K) ~ pj[d] .~" ~ Rad"+I(K) ~ ....

Remark. Similar one defines the modified Radon transform Rad*(K) Rq*(pnd](K)IU) and Rad~(K). Their properties are obviously dual to those of Rad!(K). Proof The adjunction map ad} : id --+ i*i* induces a natural distinguished triangle

hence the distinguished triangle (Rad(K), Rad!(K), Rp2*Pj[d](K». Now use Rp2*pnd]K ~ pnd]RP2*K. D We get as an immediate consequence the following Lemma 2.2 For upper semiperverse complex K

E

PD?:.o(lP'd x Y) we have the

following: (i) For n < 0 the perverse sheaves Radn(K) fit into a commutative diagram

pj[d] .9"6"n-1 -----~-----?>_» Radn(K)

l~

l~

Pffn-I (Rp2Apj[d]K» ---~-_3> Pf/n (Rp2* (i*i* pj[d - I]K» In particular (n < 0)

IY.2 Modified Radon Transforms

209

and for perverse K also Radn(K) : : : : p]'[d] .%'n+l(l)

(n > 0)

are constant perverse sheaves. (ii) For n = 0 one has the following exact sequence of perverse sheaves, called Lefschetz sequence:

Note, that all the perverse sheaves PJ-rRp2*pr(K) : : : : p]'[d] .7P-d

(v E Z)

are constant perverse sheaves. Proof The statements for n < 0 and (ii) for n (i) for n > 0 follows by duality using III.7.2.

= 0 follow from IY.2.1. The statement D

Definition. For perverse sheaves K on lP'd x Y and G pnd] K we define the constant sheaves d"rimn(K) to be the kernels of the restriction morphisms

~rimn(K) = Kernel (PH n(RP2*G) ~ PHnCRP2*i*i*G») Evidently .'-Y'rimn(K)

= 0 for all n

.

< 0 by Lemma IY.2.2.

Remark. Lemma IY.2.2 expresses this basic phenomenon that arises in Lefschetz theory: Statements on PH nRp2* can be reduced to statements on PHnR7T2* by using restriction - i.e. the map induced by the adjunction morphism id ~ i*i*. This is possible except for the two exceptional degrees n = d - I, d. In the formulation used above above it is then the functor Rad?, which controls the "critical" cokernel for n = d - I and the "critical" kernel for ~ = d. Corollary 2.3 For perverse sheaves K and n i- 0 the perverse sheaves Radn(K) are constant on pd x Y. Also all the perverse cohomology sheaves Rad,!(K) are constant perverse sheaves for n i- o. Proof By duality, it is enough to show the first statement for n < O. This case is covered by IY.2.2(i). The second assertion then follows from the long exact sequence stated in IY.2.l. D

Quotient Categories The fact, that for perverse K and n i- 0 the complex Rad n(K) is a constant perverse sheaf, obviously emphasizes the special role played by the functor Rado. It implies

210

IV. Lefschetz Theory and the Brylinski-Radon Transform

that, although Rad O is not an exact functor itself, it nevertheless induces an exact functor on certain abelian quotient categories. To be more precise, consider the abelian quotient categories which are obtained by dividing the abelian categories Per v (JIDd x Y) resp. Per v (JlDd x Y) by the Serre subcategories of constant perverse sheaves pi[d]Perv(Y) resp. Pj[d]Perv(Y). Then Lemma IY.1.2 and IY.2.3 and the long perverse cohomology sequences imply, that the functor Rado induces an exact functor Perv(wft x Y)lpi[d]Perv(Y) -+ Perv(JlDd x Y)lpj[d]Perv(Y) , K -+ KV.

In fact, both functors Rado and Radp induce the same functor on the level of these quotient categories. Remark. The abelian categories of perverse sheaves are noetherian and artinian abelian categories. Let A be such an abelian category and let B be a Serre subcategory. For the obvious reasons the subcategory B will be called the full subcategory of constant objects. The exact quotient functor

n: A

-+ AlB

admits an additive right inverse functor

AlB -+ A, which attaches to each object in the quotient category a reduced representative in A. Let K be an object of A. Then let

denote the maximal constant subsheaf resp. quotient sheaf, the left reduced quotient K I Ks and the right reduced kernel of K -+ K q . K is called reduced, if it has no nontrivial constant subobject or quotient object. Then for arbitrary K in A

is reduced. The subquotient Kred is called the reduced representative of K. In the quotient categories defined above, every object K becomes isomorphic to its subquotient Kred. Two objects K, L in A become isomorphic in the quotient category AlB if and only if their reduced subquotients Kred, Lred are isomorphic in A. For more details see also 11.3.3. The Radon inversion formula IY.I.3 can be restated in the following way: Corollary 2.4 The exact functor Rad o induces an equivalence

IV.2 Modified Radon Transforms

211

of the two abelian categories Perv(pi x Y)/pi[d]Perv(Y) ~ Perv(ffr4 x Y)/pj[d]Perv(Y). Convention. Sl9'pose K E Perv(jpd x Y). Then we will consider K V al~o as an object of Perv(JiDd x Y) - represented by a reduced representative in Perv(lP'd x Y) (unique up to isomorphism). In this sense K V considered as a reduced perverse sheaf depends only on the reduced perverse sheaf of K. In this sense K 1-+ K V gives rise to a 1-1 correspondence between the reduced perverse sheaves on pd x Y and the reduced perverse sheaves on JPd x Y. Extensions Now we consider the Lefschetz sequence

which was defined in Lemma IY.2.2, in more detail. We give another interpretation of this sequence in terms of the functor

It will turn out that (the image of)

pj[d] ,%-1 is the maximal constant perverse subsheaf RadO(K)s of Rado(K). In fact, this follows if we know that Radp(K) is left reduced, i.e. has no constant nontrivial perverse sub sheaf. This will be' shown in Corollary IY.2.7 below. Let us assume this for the moment. Then the left monomorphism in the Lefschetz sequence can be identified with the inclusion monomorphism

So the Lefschetz sequence gives rise to a monomorphism

between left reduced perverse sheaves, whose cokernel is the constant perverse sheaf tYJrimO(K). The reduced Radon transform K V = RadO(K)red is a perverse subsheaf of rRadO(K), so we can also consider the induced exact sequence

where W"(K) is defined to be the cokernel, which is an interesting constant perverse sheaf on JPd x Y attached to K. In fact for d 2: 2, this extension of the constant perverse sheaf W"(K) by the reduced perverse sheaf K V turns out to be the universal

212

IV. Lefschetz Theory and the Bry1inski-Radon Transform

left reduced constant perverse sheaf extensions of K V Csee IY.2.8). By this we mean the following: Every short exact sequence of perverse sheaves

o~ K

V

~

E

------?-

'6

------?-

0

where E is a left reduced perverse sheaf on Jil>d x Y and where '6 is a constant perverse sheaf on Jil>d x Y, is the pullback

o~ K II o~ K

~ RadPCK)

V

V ---~~

I

E

------?-

~'CK) - - - 0

----~~

r

'6

---~~

0

of the extension defined by RadpCK), with respect to a homomorphism of constant perverse sheaves '6 ---+ 9/"CK). In particular, we get from 111.11.5 Corollary 2.5 For d :::: 2 and perverse K we have

Of course we may then also consider RadPCK) as the universal left reduced constant extension of the left reduced perverse sheaf rRadoCK). In fact

o~ K

V

~ rRadoCK) - - - '0. - - - 0,

where '6~ is the cokernel of the natural map from the maximal constant perverse subsheaf to the maximal constant perverse quotient

The perverse sheaf 'Co;-Y'rim- 1CK) is a perverse subsheaf of the constant perverse sheaf ~·(K). For perverse sheaves K on lP'd x Y we can therefore rephrase the Lefschetz sequence in terms of the exact sequences

The Proofs Essentially all statements made in the following will be a consequence of the next Lemma 2.6 (Seesaw-Lemma) For K E JD"=°Cpd x Y) we have (i) RphCRad!(K)) ~ RP2*(K)[-d](-d). E P D"=o(y).

(ii) Rph(Rad!(K))

IY.2 Modified Radon Transforms

213

Proof The implication (i)

=} (ii) is the fact that RP2*[ -d] maps JD:O:o(pd x Y) to JD:O:o(y). See II1.7.1. For (i) observe ihp2 = P2Pl and PPf2 = P21fJ. Hence

1

1

"adj"

(RPl*RP2*)i*i* Pj(K)

"adj"

=

I

(RP2*Rph)i*i* Pj(K)

I

The left side appears in the distinguished triangle, that is obtained by applying RPl* to the first morphism of the distinguished triangle

- see also the proof of IY.2.1. We will see, that this direct image triangle splits. To understand the relevant morphism "ad}" look at the right side of the upper diagram. In fact, the vertical map is a projection onto a direct summand. To be more precise, by Lemma IY.1.3 the right vertical arrow induces an isomorphism - even before applying RP2* - if we restrict to the direct summand EB1~6 r'; : EB1~6 K[ -2i]( -i) C~_ _ _~:O Rpl*Pj K

1

1

ad}

EB1~6 iji : EB1~6 K[-2i](-i) and it is zero on the image of TJd. To show this vanishing property, use truncation Pr O. Using these formulas we can compare both sides of the formulas 5.4. By the semisimplicity of the category of pure perverse sheaves of fixed weight we can cancel terms, which coincide. Furthermore the pullback pi [d].%-d is the maximal constant perverse subsheaf of the perverse sheaf K. It therefore vanishes, if K is reduced. Furthermore pnd].%-l is the maximal constant perverse subsheaf of Rado(K). See IV.3.5. Now 5.4(i) gives the next theorem by a straight forward calculation using these facts together with the parity law IY.I.2. Notation. Let K V denote the reduced Radon transform Rado(K)red of K. Define the groups (.%Vr = PfrRph(K v ) . Radvm(Kv) = pndl(~vt

for arbitrary n and for m

i= O. Again (~v)m

~ (.%v)m-l for m < 0 etc.

Theorem 5.5 Suppose d ~ 2. Let Ko be a reduced pure perverse sheaf on

pg x Yo.

IV.5 Supplement: A Spectral Sequence

223

(a) We get (.%,v)-l :;::: {.%,O(2;d) d == 0(2) .%1(3;d) d ¢ 0(2)

(b) Furthermorefor 0< m

(c) Finally for 2 - d

:s m

:s d -

2, hence 2 - d

< 0, hence

°

< m

:s m + I -

+d -

I

:s d -

d < 0, we have

2, we have

These formulas express the perverse cohomology sheaves (%v)n of K V in terms of the perverse cohomology sheaves of K for all n f= 0. In the remaining part of this section let us recollect some of the information gathered in the proof of the Hard Lefschetz theorem, and formulate this in terms of the Radon transform. We would like to emphasize that - for pure perverse sheaves Ko as above this gives a rather satisfactory way to express the perverse cohomology sheaves PH n (Rp2K) in terms of the "simpler" perverse cohomology sheaves Rad n+ 1(K). Of course it is enough to consider the cases n :s d using duality; and as already remarked in Lemma IY.2.2 it is also enough to restrict to the two cases n = d - I and n = d, which are linked by the Lefschetz sequence. In fact, for pure perverse sheaves Ko the Lefschetz sequence may be completely expressed in terms of the perverse sheaves Rado(K) and Rad1(K) as follows Corollary 5.6 Suppose d :::: 2 and let Ko be a pure perverse sheaf on ~ x Yo of weight w. Then the Lefschetz sequence is obtained by composing the following short exact sequences: (1) The inclusion of the maximal constant perverse subsheaf of Rado(K) into

Rado(K)

Since Rado (Ko) is a pure perverse sheaf of weight w +d -1, this sequence splits by Gabber's theorem. The quotient is the reduced perverse sheaf Rado(K)red = KV. (2) The second short exact sequence is the universal left reduced constant perverse extension 0--+ K V --+ Radp(K) --+ CP'(K) --+ 0, where now Ko is pure of weight w + d - I and CP'(Ko) is pure of weight w + d.

224

IV. Lefschetz Theory and the Brylinski-Radon Transform

(3) The last short exact sequence is

It splits, since all these perverse sheaves arise from pure perverse sheaves of weight w + d. Furthermore the sheaf Rad!l (Ko) is pure of weight w + d + 1. Proof. The assertions (1) respectively (2) were already obtained in step 4) of the proof of the Hard Lefschetz theorem, in the first respectively second argument given. For the sequence (3) surjectivity on the right follows from Corollary IV.4.2 and IY.3.4. Injectivity on the left follows from 't? ofYJrim- 1(K) = 0, which was also obtained in step 4) of the proof of the Hard Lefschetz theorem. 0 For pure perverse sheaves Ko the corollary therefore implies, that Rado(K) uniquely determines the maximal constant subsheaf pi[d](PH d - 1RP2*K), uniquely determines K V and therefore uniquely determines cp'·(K). Hence pi[d](PHd RP2*K) ~ W(K) EB Rad1(K) is uniquely determined by Rado(K) and Rad 1(K). Therefore all the perverse sheaves VEZ

are uniquely determined by the perverse sheaves Radn(K) for n

E

Z.

v.

Trigonometric Sums

V.I Introduction This chapter has its own bibliography. One of the most beautiful applications of Grothendieck's trace formula and Deligne's theory of weights (La conjecture de Weil II [Del]) are certain non trivial estimates for trigonometric sums. These trigonometric sums are exponential sums, which for instance generalize the well known KLOOSTERMAN sums. It is the DELIGNE Fourier transform, which provides the link between the trigonometric sums and the etale cohomology theory. Hence, one of the essential tools is the ARTIN-SCHREIER sheaf. Let I be a fixed prime. For every prime p i=- I let us choose a non-trivial character

The Fourier transform defined by the character 1/1p on the affine line Arrp over the finite field IF p is denoted

KATZ and LAuMaN have shown, that these Fourier transforms for the various primes have the remarkable property of being uniform: Let

be a sheaf complex on the affine space Az over Z and let K p E Dt (Arr p' Qz) be the pull back of K to the affine spaces Arrp over IF p' Then, for almost all primes p, each of the Fourier transforms T1/1p (K p) has similar properties, from a topological point of view. As a collection they behave, as if there actually exists a global Fourier transform b T(K) E Dc(Az,Qz) - a Fourier transform over Z - with the property

for almost all primes p ([Ka-L], theorem 4.1, corollary 4.2).

226

V. Trigonometric Sums

This property of the Deligne-Fourier transform of being uniform is truly remarkable. KATZ and LA UMON deduced from this fact uniform estimates for trigonometric sums, which therefore hold for almost all primes p. In this book, our intention is merely to present some ofthe most striking results in this area mainly due to DELIGNE, KATZ, and LAUMON. Our emphasis will be to outline the underlying ideas of methods and proofs. We do not intend to give the strongest results obtainable. In particular we concentrate on some of the simple cases, in order to give an idea of the arguments involved. In the following, we choose a prime I and an isomorphism T:QI~ce

fixed from now on. We consider characters

of the additive groups of the underlying finite fields. If necessary, we will view them via T as complex valued characters IFq ~

ce*.

We therefore often write 11/1 (x) 1instead of 1T 1/1 (x) I. For a point x of a scheme, :x will denote a geometric point above x. Let R be a ring and S be a scheme. We then let AR resp. As denote the affine line over Spec(R) respectively S.

V.2 Generalized Kloosterman Sums Let q be a power of a prime p. Let 1/1 be a nontrivial character of the additive group of the finite field IFq with q elements. Let a E IF~ be an element of the multiplicative group IF~. Fix a natural number m ::: 1. Then Generalized Kloosterman Sums are defined as the trigonometric sums Kloosm(q,

1/1, a)

L

=

1/I(Xj

Xl"" ,Xm ElF~ X! ... xm=a

Example.

L

e 2;i (Xl +... +X m ),

X!, .... xmElF'p

Xl···xm=a

For this sum a trivial estimate is

+ ... + xm) E ce.

V.2 Generalized Kloosterman Sums

227

If we vary a, the Kloosterman sums Kloosm(q, 1/1, a) define a function on the multiplicative group lF~, briefly called Kloos m(a), in the following. We may then calculate the Mellin transform ~m (X) of the Kloosterman sum, i.e. the Fourier transform with respect to the multiplicative group lF~ of the field IFq. Then llism (X) is a function on the set of all characters X of lF~. To determine it we may use the classical Gauss sums g(X) = L 1/1 (x)X (x). xEF;

Then the following formulas are easily verified g(X)m = L

x(a) . Kloosm(a) = llism(x)

aEIF;

and

1,,-

Kloosm(a) = - - ~x(a)g(X)m. q-l

x

On the other hand the PLANCHEREL formula applied for the character X gives X==1

else. This in turn implies X

==

1

else.

To obtain an estimate for Kloos m, we can again apply the PLANCHEREL formula L aEIFZ

IKloosm(a)1 2 = _1_ L q-l

IlliSm (x)1 2 =

X

_1_(1

q-l

+ (q -

2)qm) =

m-l

=qm _ Lqv.

v=o Hence IKloosm(q,

1/1, a)11 :::: q!!f .

For m > 2 this estimate is better than the trivial estimate. A much better bound - in fact the best possible one - was found by DELIGNE using cohomological methods. We will give a proof for this estimate later at the end of this chapter, and now merely state the result

Theorem 2.1 (Deligne ([De]) IKloosm(q, always holds.

m-l

1/1, a)1 :::: m . q-2-

m-l

= m . #(lF q )-2-

228

V. Trigonometric Sums

For any natural number a

f::. 0 and fixed m Kloos m(p,

1/1, a mod p)

can be viewed as a function depending on the primes p (not dividing a) and the corresponding character 1/1 of F p' The estimate of Deligne stated above is uniform in p, i.e. it is of the form IKloosm(q,

1/1, a mod p)1

SA. qW

(q

=

p) ,

such that the constants A and w do not depend on the chosen prime p and the chosen character 1/1 . In [Kal], Katz examined a more general class of trigonometric sums, for which he was also able to give similar uniform estimate. To explain this result, let h : X ----+ Az

be a finitely generated scheme over the affine line Az above Z. For every prime p, every power q of it, and every non-trivial character 1/1 of F q , one defines the following generalized Trigonometric Sum

L

1/I(h(x)).

xEX(IF q )

Let N be the supremum of the dimensions of the fibers of the complexified morphism h®C:X®C----+Ac·

Theorem 2.2 (Katz) ([Kal], Theorem 2.3.1) Under the hypotheses above, there exists a constant A not depending on q and 1/1, such that for almost all primes p, any power q = pr (r E Z, r ~ 1) of it, andfor all nontrivial characters 1/1 oflF q the following estimate holds (i)

L

1/I(h(x))

:s A .

qN+!

XEX (IFq)

lfmoreover the generic fiber XI) of the morphism h is irreducible, or ifdim XI) < N holds, then the following stronger estimate holds (ii)

L

1/I(h(x))

:s A . qN .

XEX (IFq)

Proof We give a proof of this theorem in Section nine.

Corollary 2.3 There exists a constant A' such that for any prime p

D

Y.3 Links with /-adic Cohomology

229

respectively under the stronger assumption

L

:s A' . pN

1/I(h(x»

xEX(lF p )

holds.

z

Remark. Let A be the m-dimensional affine space over :2:, a i= 0 an integer, and X ~ A the closed subscheme of A defined by the equation Xl ..... Xm = a. For the choice of h: X ~Az

z

z

X

= (Xl, ...

,Xm )

~ Xl

+ ... + Xm

the corresponding trigonometric sums give back the generalized Kloosterman sums Kloosm(p,

1/1, a

mod p).

Suppose m ~ 2. Then in this case the fiber dimensions of hare :s m - 2. For m ~ 3, the generic fiber X 1) is irreducible. Corollary V2.3 therefore implies the following estimate

1/1, a IKloosm(p, 1/1, a IKlooS2(P,

mod p)1 mod p)1

:s A . p'l., :s A' . pm-2. I

I

in the case m = 2 respectively m ~ 3, which is valid for all primes p. However, this estimate coincides only in the two cases m = 2,3 with the strong estimate of DELlGNE, at least up to a proper choice of the constant A'!

In the next sections we want to present the proof of Theorem V2.2, essentially following KATZ. This requires some preparations.

V.3 Links with l-adic Cohomology In the following fix a prime p i= I and a power q of p. We also fix a finitely generated morphism

Q;

T

For a nontrivial character 1/1 : IF q ~ ~ C* we use the trace TrlFqn /lFq = Trn : IFqn --+ IFq of the corresponding field extension of degree n to define

230

V. Trigonometric Sums

which is a non-trivial character of IFqn. Let'h ('I/},-l) be the Artin-Schreier sheaf on Ao attached to the character 1jJ-l, as explained in Chap. I, §5. Let Y be a point of Ao(lF q) = IFq. The geometric Frobenius element Fy acts on the stalk'h (1jJ -I )y by multiplication with the character value 1jJ(y). More generally, for Y E Ao(lFqn), the Frobenius element Fy acts on the stalk Z (1jJ -I) y ~ Q/ by multiplication with 1jJ(TrIFqnIIF(Y))' See Chap. I, §5. Let x be a point in X(lFqn). Then the action of Frobenius Fx: : h*('25 (1jJ-l)h ---+ h*(':h( 1jJ-I) h on the stalk of the pullback of Y; (1jJ-I) on X at the point x is given by

Fx:: a

f-+

1jJ (TrIFq n IIF q (h (x))) . a .

For every integer n :::: 1, we now define the trigonometric sum fn(X, h, 1jJ)

=

1jJ (TrIFq n IIF q (h (x))) .

Let F be the endomorphism ofRfc(X Q9lFq , h*('h(1jJ-l))) induced by the Frobenius morphism of X over IFq. Then we obtain from Grothendieck's trace formula

v

Let the elements a VJL denote the eigenvalues of F on H~(X Q9lFq , h*('h(1jJ-l))). Then fn(X, h, 1jJ) = L(-I)\la~JL . \I.JL For an upper bound

Wv

of the weights of H~(X Q9lFq , h*('£(1jJ-l))), we obtain

I

la\lJL I :::: q 2wV. This implies the following estimate for the trigonometric sums fn Theorem 3.1 IIn (X, h, 1jJ) I ::::

L q ~nwv

. dim HL~ (X Q9lFq , h* ('h( 1jJ-I)))

II

Remark 3.2 Let K

= Rh!(Q/). Then

Rfc(X Q9lFq , h*C'h(1jJ-I))

=

Rfc(Ao Q9lFq , K 129 '/:(1jJ-l)) .

V.4 Deligne's Estimate Let Ao be the m-dimensional affine space over IFq. For a E lFZ and X := {x = (Xl, ... , Xm)

E

Ao : xl· ... · xm = a}

Y.S The Swan Conductor

231

and the morphism h : X -----+ Ao defined by h :x

=

(X1, ... , xm) r-+ XI

+ ... + Xm

we get Kloosm(qn, 1jJ 0 TrlF q nllFq , a) = fn(X, h, 1jJ). Deligne proves in [De] theorem 7.4

Theorem 4.1 Suppose I does not divide q. Then for all nontrivial characters 1jJ IFq --+ -* Ql the following holds:

and

The sheaf h* ('£ (1jJ-I)) is pointwise of weight zero for all its stalks. Therefore the generalized Weil conjectures imply, that m - I is an upper bound for the weights of H;:,-I (X 0lFq , h*('h(1jJ-I))). This implies Deligne's estimate

Remark. Deligne even proved, that

is pure of weight m - 1. This implies, that Deligne's estimate is best possible.

v.s

The Swan Conductor

This section is a report on [Sel], chap. IV, V, [Se2], §1O, or [Lau2]. Let 0 be a strict Henselian discrete valuation ring with algebraically closed residue class field of characteristic p > 0 and let K be the quotient field of o. Let [ be the Galois group of the separable closure K of K. Then the group [ is a pure ramification group. It has the wild ramification group P as normal subgroup. P is closed in [; P is the profinite p-Sylow group of [. The quotient group [t = [tame = [ / P is called the group of tame ramification. There is a natural isomorphism [tame

=

TI

Zl(l)

(l,p)=1

Zl(l)

= 1~{Lln

,

n

where

{LIn ~ 0*

is the group of In -th roots of unity.

232

V. Trigonometric Sums

Let I -I p be a fixed prime and lB' be a finite field of characteristic I -I p. Consider continuous representations of Ion finite dimensional vector spaces Mover lB'. Then M is an I-module. The term "continuous" means, that an open normal subgroup H of I acts trivially on M. H is an open and closed subgroup of finite index in I. The action of I on M factorizes over the finite quotient group G = 1/ H. M is a G-module.

Definition 5.1 We say that I acts tamely on M if P acts trivially on M. Then M is called a tame or tamely ramified I -module. M then is a module over the tame ramification group It. Let H be an arbitrary closed normal subgroup of I which is of finite index in I such that H n P acts trivially on M. Then the factor group PH / H = P / P n H acts on M. Let L = KH be the finite Galois extension of K corresponding to the Galois group G = 1/ H. Let 0' be the valuation ring of L, n E 0' a primitive element of the maximal ideal of 0'. Let v(·) be the additive valuation of L normalized such that v(n) = 1. Now consider the higher ramification groups of LIK: Gi

= {a = {a

E

G : v(a(n) - n) :::: i

E

G : v(a(x) - x) :::: i

+

I}

+ I \/x

EO'},

Go = G ;2 Gj ;2 G2 ;2 ... ;2 G r = {I}. G j is the wild ramification group of L IK. In other words

Gj=PH/H=p/Hnp. G I acts on M. M is tamely ramified if and only if the finite group G j acts trivially onM.

Let Pi be the preimage of Gi in P. Pi is a normal subgroup of I. The decreasing sequence of groups P = PI ;2 P2 ;2 P3 ;: .... depends on the choice of H. P being a pro-p-group, the action of P and its subgroups on M is semisimple, because the characteristic of the field lB' is different from p by assumption.

Theorem 5.2 (see [SeI], IV §2 and VI §2, [Se2], §I9) ( 1) The rational number

I - ~ dimJIi'(M/MGi) - ~ [Go: G;J is independent of the choice of H.

Y.5 The Swan Conductor

233

(2) The number

is an integer. Remark on (1). For the proof of independence one needs another numbering of the higher ramification groups, the so called upper numbering. This upper numbering defines a filtration of Go by the family of ramification groups G X where x > 0 is in JR. See [Se1], chap. IV, §3. This allows to rewrite the formula 1 '" dimlF(M/MGi) L... [Go: G·] i:::: 1

I

into a formula involving the upper indices x and the groups G X • See also [Ka2] , Chap. IV 1. For a normal subgroup reG the image of G X in G / r is (G / r)x by the theorem of Herbrand. See [Sel], chap. IV, §3, proposition 14 respectively lemma 5. The claimed independence is a consequence of this fact. Concerning (2). If H is chosen small enough we can suppose M to be a G-module. If G is abelian, the integrality property follows from the Hasse-Arf Theorem. See

[Sel], chap. IV, § 1, p.84. For representations of G on a finite dimensional vector space V over a field of characteristic 0, the integrality property for V is deduced from the abelian case via Brauer's Theorem ([Se2], theorem 24). For details see [Sell, chap. VI, §2. Let DE be a complete discrete valuation ring with quotient field E of characteristic 0 and residue field IF. Then there exists a virtual DE-module Vo which is free as a virtual DE-module, such that IF ®o E Vo and M have the same image in the Grothendieck group of finite IF[G]-modules. This is also a consequence of Brauer's Theorem. See [Se2], theorem 33 resp. theorem 42. Put V = E ®o E Vo. Since the characteristic I of E does not divide the order of the wild ramification group G 1, the functor of r -invariants is exact in the category of IF[ G]-modules for any subgroup r of G 1. Hence the following holds true i ~ 1.

Hence the desired integrality property for M follows from the corresponding statement for V in the characteristic 0 case. For a related topic - the construction of the ARTIN and the SWAN representations - see [Se2] 19.1 and 19.2. Definition 5.3 The integer

Swan(M) :=

L

i::::1

1

[Go: Gil

dim(M/MGi)

234

V. Trigonometric Sums

=L .

1

1:0:1

[/:H·P;]

dim(MIMPi)

is called the Swan conductor of M.

Remark. M is tamely ramified if and only if Swan(M) is zero. Due to the semisimple action of P and its subgroups, the invariant Swan(M) is additive for short exact sequences of I modules. Now consider a continuous representation of I on a finite dimensional vector space V over ijl for I f= p. Continuous means, that there exists a continuous representation of I On a finite dimensional vector space Vo over a finite extension field E C Ql of Ql such that V = Ql @E VO (as representation modules). An open and closed normal subgroup of finite index in P acts trivially on V, because P is a pro-p-group and because I f= p. Using the fact that I is compact, one knows that there exists a finite free 0 E-module

FO

~

with

FO@oE E

Vo

=

Vo

and I Fo S; FO. Here 0 E is the valuation ring of E. Since P is a pro- p-group and the action of P on Fo factorizes over a finite quotient group, P and all its subgroups act semisimply on Fo. We choose an open normal subgroup H of I with finite index, such that I n P acts trivially on V. Consider the higher ramification subgroups of G = I I H G

= Go 2

GI

=

PIP n H

2 G2 2 ... 2 G v = {J}

and the preimages in P P = PI 2 P2 2 P3 2 ....

Let IF be the residue field of 0 E. Then put

Then Fcii is a direct summand of Fo and we have

FPi,o..

o

VPi

In) -

'6'0 E '\£1 -

I acts continuously on the finite vector space M over the finite field IF. Therefore

L i>1

1

[I : H . p.] 1

dim;n; (VIVPi) = Swan(M). 2N ora> I we have E b

.'£(1/1) is :s b. Hence the weight of E b is :s

of the theorem. Now we assume the generic fiber of this mapping to be either geometrically irreducible, or the dimension of that fiber to be < N. By a shrinking of S, we may furthermore assume: (e' ) For every morphism t : Spec(IF'q)

~

S

the dimensions of all fibers of the induced morphisms

are:s N. The general fiber is geometrically irreducible or its dimension is < N.

Claim. The weight is always :s 2N for b :s 2N, a :s 1. By the given assumptions, we only have to show that the weight of Ei ,2N is :s 2N. If the dimension of the general fiber of h t is less than N then R 2N h t l(02t) is concentrated on finitely many points and Ei,2N is zero. Let the general geometric fiber of h t be irreducible of dimension N. We now make use of the trace morphism. It is available for flat finitely generated compactifiable morphisms f : X ~ S between noetherian schemes of fiber dimension :s d. It is a canonical morphism of sheaves

with certain natural properties. In particular, if all fibers of the morphism fare geometrically irreducible of dimension d, this trace map is an isomorphism. See

Y.9 Proof of the Theorem of Katz

243

Chap. II, §8 and III, §7 or [SGA4], expose XVIII, p. 553 for further details. Let us therefore - for the moment - assume, that the morphism h t : X t -+ At is flat. Under this additional assumption, we have the trace map R

2N

-

-

ht!(I!J,) -+ !f1t(-N)

at our disposition. By our assumptions, this trace map is an isomorphism at the generic point of At. Consider the exact sequences of sheaves, which is obtained after tensoring wi thY; ( lj! )

o -+ .7~· -+ R2N h t !(Q,)@'£(lj!) -+.Yl -+ 0 0-+ jJ -+ !Qz( -N) @.':£(lj!) -+ () -+ 0

for some sheaf .YJ. The sheaves .76' and Q are skyscraper sheaves concentrated on finitely many points and the weight of Q obviously satisfies w(Q) ::: 2N. Now H:(A t , !Q,( -N) @ (/:'(lj!)) = 0 as shown in Chap. I, §5. Thus H} (At, R 2N h t ,(!Q,) @yj(lj!)) ::::::: H(I(A t , ./J) : : : : H(O(A t , Q).

Hence the weight of Ei,2N is ::: 2N. Let us finally remove our temporary assumption, that h t was a flat morphism. For this we have to reduce the case of a general morphism h t to the case of a flat morphism ht . Without restriction of generality we can replace X t by (X t )red. Whence we can assume, that X t is reduced. Let At = Spec(1Fq [z)), let J C Ox, be the ideal of1Fq [z] torsion elements of the structure sheaf Ox, of X t and let Xt -+ X t be the closed sub scheme defined by the ideal sheaf J

Then, of course, ht : Xt -+ At is a flat morphism. The closed embedding Xt c...,>- X t is an isomorphism over the generic point of At. Hence it is still an isomorphism, if we discard a certain finite number of points of At. But this implies, that the kernel and the cokernel of the map R

2N-

-

c·

ht!(!Qz)@.'h(lj!) -+ R

2N

-

c

ht!(!Q,)@'/j(lj!)

are concentrated on a finite number of closed points. Hence the morphism H;(A t , R 2N ht !(!Q,)

0.'£ (lj!)) -+

H;(A t , R 2N h/!(!Q,) @ ..cZ/(lj!))

is surjective! Now we can easily deduce the desired results from the case of the flat morphism ht , that was already considered above. Now let R be Z. The uniform estimates of weight and dimension derived so far will suffice for the following arguments. In particular, together with Theorem V3.1 they imply the estimates for the trigonometric sums, that were stated in Theorem V2.2. D

244

V. Trigonometric Sums

v.tO Uniform Estimates In this section we give a review of the results of Katz and Laumon [Ka-L] on uniform estimates. As before, let R ~ C be a finitely generated ring, I a prime that is invertible in R, and S = Spec(R). Let Am be the m-dimensional affine space over S and let K be a complex in D~(Am, 1Ql/). For a morphism t : Spec(lF q ) -+ S

let K t be the pullback of K to the affine space of dimension mover IFq

A'; = Am Xs Spec(lFq ). Theorem 10.1 There exists an open nonempty subscheme Sf

~ S and an open nonempty subscheme V ~ Am with the following properties: Let t : Spec(lFq) -+ Sf be a morphism and 1/1 : IFq -+ 1Ql; a non-trivial character. We consider the Fourier transform (see III.S.1)

T1jr : D~(A';, IQlz) -+ D~(A';, IQlz). Then V t = V x Spec(lFq) ~ A'; is nonempty. All cohomology sheaves .96'v (T1jr (K t )) are smooth on V t . The ranks of.9f(jv (T1jr (Kt))!Vt can be estimated independently of the choice oft and 1/1. The number

does not depend on the choice of t and

1/1.

Proof See [Ka-L], theorem 4.1 and corollary 4.2.

D

Remark 10.2 One can choose V to be homogeneous. This means, there exists a homogeneous nonvanishing polynomial F (Z 1, ... , Zm) with coefficients in R so that V is the complement in Am of the subvariety of codimension one defined by the zero locus of F. This property comes from the fact that the theorem also holds for all characters 1/Ia(x) = 1/I(ax) , Va ElF; of lF q .

Remark 10.3 Let K t be a perverse sheaf. Then the Fourier transform T1jr(K t ) is perverse again. Therefore the restriction to V t vanishes

if V respectively F was suitably chosen. Suppose K t to be pure of weight w. Then T1jr (K t ) is pure of weight w + m.

V10 Uniform Estimates

245

Let X be a smooth affine scheme over S of relative dimension d

= d(X/S)

over S and consider an S-morphism

which we assume to be finite. In addition to these fixed data, we suppose that the following additional data are given: an S-morphism into the multiplicative group G m g : X -----+ Gm.s ,

over S, a specialization morphism t : Spec(lFq) -----+ S ,

as above, furthermore a non-trivial character 1/1 : IFq --+ Q7, an arbitrary character X : lF~ --+ an m-tuple a = (al, ... , am) E IF;. These extra data will play the role of variables in the following. For each six-tuple (g, IFq' t, a, 1/1, X) of these variables we define a trigonometric sum:

Q7,

f(g, lFq , t, a,

1/1,

X)

=

L

1/1

xEX,(lFq )

(ta; . h;(X)) X(g(x)). 1=1

Here Xl = X xs Spec(lFq ). Theorem 10.4 ([Ka-LJ, theorem 5.2) There exists a homogeneous nonvanishing polynomial F(ZI, ... ,Zm) E R[ZI, ... , Zm], a non-zero open subscheme S' I)

~

Ql/ [W].

Duality. For the moment suppose, that the characteristic p of the base field k is large enough. Then A(u) is one of the groups (Z/2ZY, S3, S4, Ss. The irreducible characters X of these groups are self dual X * ~ X. Furthermore, by the definition of the Brylinski-Springer correspondence T1jr (i, /'* \fJ ~ ('Yx )) ~ cJ>4, for ¢ = ¢~.x' Furthermore D(cJ>4,) ~ cJ>4,*(n) by VLlO.6 and DT1jr(K) ~ T1jr-,D(K)(n) by III. 12.2. This implies cJ>4,*(n) ~ DcJ>4, ~ DT1jr(i, / '*\fJ~(.¥X)) ~ T1jr-ID(i.I'*\fJ~(.o/x))(n) ~ T1jr-,i, / '*\fJ~(.J dime. Iii) = m - 2 . d(~). Therefore bq,®E::: 1/2· dime· Iii) = N - d(u) for ¢ = Sw 0 ¢u,x or d(u) ::: aq,

Equality holds iff '¥x is constant, i.e. iff X As a special case one obtains

=

1 is the trivial character of A(u).

Corollary 15.4 Let p be a very good prime. Then v =I- 2m

Recall, that. I/" is affine and of dimension m. Thus. V behaves like affine space of dimension m. On the other hand .'V· has singularities. To illustrate this consider the simplest case G = SI(2). Then. V = Spec k[a, b, e)/(a 2 + be), which has a singularity at the origin. Note 16 ;:::: ,/1/" via g t-+ g - 1. A matrix with entries a, b, e, d and trace a + d = 0 is nilpotent iff a 2 + be = O. For very good primes p the computation of the cohomology of .J' strengthens Steinberg's theorem

Theorem 15.5 (Steinberg) The number of unipotent elements in G (18' q) equals qm, the square of the number of elements in a p-Sylow group (if G is defined over the finite field 18'q).

VI.16 Regular Orbits

295

VI.16 Regular Orbits The method used in the previous section can be carried over to study more general situations as well. In particular it is of interest to have information on the Fourier transform of (co )adjoint orbits. This is the starting point of Springer's fundamental papers [S6] and [S2], and it appears in [S6j in disguised form as a problem on trigonometric sums. The case already studied in the last sections was the case of nilpotent orbits. The case of regular orbits, studied in Springer's papers, is the other extreme.

General Reference. [Bry2, § 11] Let p be a very good prime for G. Let T be a maximal torus of G and B a Borel subgroup of G containing T. We fix a nondegenerate G-equivariant symmetric bilinear form K on g. From our assumptions on p the existence of K is guaranteed by VI.5.1. For any y E t* define a morphism Vy : X --+ A on X = G x B b by (y E t*, bEt)

where b is the image of bin t

= blu. The map Vy X

q

"j A

is well defined. We get the diagram

) 9

,'h (1/1)

where q : X --+ 9 is defined by the Grothendieck simultaneous resolution of singularities, and Vy : X --+ A is defined as above. Let K = IF q be a finite field contained in k. Let 1/1 be a nontrivial character of the additive group of K. Let Y; (1/1) be the corresponding Artin-Schreier sheaf on A.

Definition 16.1 For y

E

t* define the Springer sheaf .Y(1/I, y)

Obviously D.'/(1/I, y) ~ .'/(1/1-1, y)(n). The Springer sheaves are perverse sheaves on g. Namely q is a small proper (1/1)) is a smooth sheaf on X. If G, Band T and y morphism, X is smooth and are defined over K, then we obtain a canonical Weil sheaf structure on the Springer sheaf.

v; ((/

Remark 16.2 The morphism Vy is trivial on the closed subset Y = G x B u of X = G x B b. Therefore, since Y = q -1 (. ·1/ ') red, we have the following consequence of the proper base change theorem: Let y E t* and let .'/(1/1, y) E Perv(g) be the

296

VI. The Springer Representations

corresponding Springer sheaf. Then the restriction of .Y' (ljI, y) to the subvariety i. ~. : .1//' -+ 9 is isomorphic to the perverse sheaf \}! [r ]( - N) on ./V. For the definition of \}! see VI.II.2. In particular, the restriction of .'/ (ljI, y) to ./V does not depend on ljI and y. There are natural maps res: :7(ljI, y) ~

i.

/,*(*,.(.'7(ljI, y)) ~

Since \}! ~ D(\}!) = D(i.*,.[-r](N).Y(ljI, y)) implies (using the ljI-independence)

i.

~ ·*\}.

= i.I /.[r](-N).Y(ljI-l, y)(n) this

i.! / .. '/'( ljI, y )[2r](r) ~ i.*, ..':/( ljI, y) .

Now we consider regular (co )adjoint orbits 6 of G. For this we use the fixed G-equivariant bilinear form K as in Remark V1.10.7 to identify t*

~

t

~

9

~

g* .

Then by definition a regular orbit 0' of Gin g* contains an element yin t;eg. Since in fact t* is not contained in g*, we have to proceed as follows to give sense to this statement: y E t* is called regular, if y corresponds to a regular element Y E treg under the isomorphism t* ~ t. Then t;eg ~ treg and (7; is the image of the orbit G (y) c 9 under the isomorphism 9 ~ g*. The orbit is closed in g* and has dimension m. This is a well known consequence of the assumption, that y is regular. See Lemma VI.l6.5 below and its proof. Let iro : (7 ~ g* be the inclusion map and let

ry

Oro

= iro * (Qi[ro)

be the constant sheaf on this orbit viewed as a sheaf on g*, where (m) Since (7.~ with the reduced sub scheme structure is smooth

=

[2N](N).

Oro (m) E Perv(g*) .

The next proposition compares the Springer sheaves with the Deligne-Fourier transform of regular orbits. Formulas of that type first arose in the work of HARISHCHANDRA on the representations of real reductive Lie groups.

Proposition 16.3 For regular elements y E t;eg the Fourier transform T 1/1 (oro, (m) ) of the perverse sheaf oro, (m) on g*, defined by the m-dimensional smooth regular (co)adjoint orbit C: y , and the Springer sheaf .'/(ljI, y) are isomorphic perverse sheaves on 9 hy : .'/(ljI, y)

Y E t~eg

.

Remark 16.4 Even for nonregular y E t* both sides remain defined. This is obvious for the left side. The right side can be defined by some extension of the constant sheaf

VI.16 Regular Orbits

297

to the closure of an orbit. However one can not expect to obtain an isomorphism any longer. E.g. for y = 0 we see that T"A8ro(m)) = 8g [n](m) is not even a perverse sheaf, whereas the left side specializes to the perverse sheaf Y(1/!,O) = •

Now we come to the proof of the proposition Proof Step 1. y E t~eg has stabilizer G y = T, the orbit (0~ is a closed orbit in g*. With respect to the diagram of the Grothendieck simultaneous resolution of singularities (pl)-I (y)c

.. X;eg

i'

1

{y}C

.. t~eg

q'

8'

* .. greg

0(

) (0;'

i"

.. t~eg/W

the following holds

Lemma 16.5 For regular y

E

t~eg the morphisms q' : (pi) -I (y) -+

(0y

is an

isomorphism, hence

Proof The morphism q' : X;eg -+ g~eg is a finite etale Galois covering map (Theorem V1.9.1 respectively its coadjoint version). Since pi is smooth, its fiber F = (pl)-I (y) is a smooth closed subscheme of X' (of "constant" dimension). Indeed Fe X;eg. The orbit ~y = G(y) is smooth and * (i3iY -+ c-!y' (q ')-l (c-'»Y = X'reg x greg

is a finite etale covering map of smooth schemes. The smooth scheme F is contained in the smooth scheme (q') -I (~y) and has the same "constant" dimension. Therefore F is open and closed, as a subscheme of (ql)-l (f'iy). Therefore

q': F -+ (i3~ is a finite etale covering map onto the irreducible scheme fC'Y, so it is enough to show injectivity of this morphism in the set theoretic sense: Use the isomorphism G / T x t~eg -+ X~eg (as in the proof of Theorem VI.9.l). If gl T x Y = g2T x y, then gl = g2t with t E T(k) by the regularity of y. Hence q' is injective on F = (pi) -I (y). This completes the proof of the lemma.

298

VI. The Springer Representations

Noteworthy the closed subset F = (p') -1 (y) of X' maps to a closed subset of g*, since q' : X' --+ g* is a proper morphism. Hence by the proof above the regular orbits (U); = q' (F) are closed in g* - a fact already mentioned and used earlier. 0 Step 2. The vector bundle X' = G x B b' over .71 is the middle term of an exact sequence of vector bundles over $.

y,C

j

Sy

.

X'

9

X

t* .

The CJd !*-extension of .9"'( 1/r) Igreg X t: eg is .9'" (1/r) Igreg x t*, since the latter is a smooth sheaf by Theorem VI.9.1. The (12) !*-extension of.)I' (1/r) Igreg x t* is .Y( 1/r), since q x id is a small map with top stratum greg (Lemma VI.lO.3(a)) and since X x t* and v*U;.l;(1/r)) are smooth. The claim now follows, since intermediate extensions are functorial: J!* = 12!* 0 iI!*. D

Notation. For K E Perv(g) we write K /t* E Perv(g x t*) for the pullback of K[r] under the projection 9 x t* -+ g. With this notation we get similar as in VI. 16.2

where r

= dim(t)

and 2N

+ r = n = dim(g), and where i.Fxt*: .Y·xt*valued character. For maximal tori T, T' in G put N(T, T') = {g E G I geT) = T'}. Recall that the Grothendieck simultaneous resolution Xo -+ go defines the Springer fibers .7Jx . For a maximal torus To define w = weT, g) E WeT) as above, such that T = g-ITg. Lett denote the Lie algebra of T. For y E (t~eg)F putS' = Adg(y) E g.

314

VI. The Springer Representations

Recall y E ({?(T)y)F C (g*)F ~ (g)F. Recall n = dim(g) = m where r = dim(t). Then - as an application of the previous results - we obtain

+r

= 2N

+ r,

Theorem 18.2 (Springer) ([S6], theorem 3.15 and 4.4) For y E (treg)*F andfor x E gF the following values (a) and (b) are equal

(a) Thefunction fLO(x, y) definedfor the Wei! complex Lo = .7(To, ljI) (coadjoint construction) or by transport of structure r(trace(wF, SeT, ljI)(x,y)) , where w acts by the Brylinski action. (b) Thefunction fKO(x, y) definedforthe Wei! complex Ko = T1/J(M(To», where M(To) = (J6)1*(8" (To) (m}[r D. Explicitly

L

q-N.

ljI-l((x,z}).

ZEVy)F

Furthermore: (c)

If x

E 9 F is regular,

then (a) is also equal to

WEN(T',T)F /TF

This value is zero unless the centralizer T~ ofx in GO is conjugate to TO under G F. (c') If x E JV F C gF is nilpotent, then (a) is also equal to L(-I)V. r trace(wF, H~(,'7ix, : D~(X, E) ~ D~(Xs, E, r) , the nearby cycles resp. vanishing cycles functors with coefficients in E. By taking the direct limit over all E such that Q/ c E C Q/ we obtain b

-

b-

R'II, R4>: Dc(X, Q/) ~ Dc (X S , Q/, r).

To fill in the details of the arguments is left as an exercise for the reader.

0

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