The p-adic Simpson Correspondence (AM-193) 9781400881239

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Annals of Mathematics Studies Number 193

The p-adic Simpson Correspondence

Ahmed Abbes Michel Gros Takeshi Tsuji

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2016

c 2016 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Abbes, Ahmed, 1970The p-adic Simpson correspondence / Ahmed Abbes, Michel Gros, Takeshi Tsuji. pages cm. – (Annals of mathematics studies : number 193) Includes bibliographical references and index. ISBN 978-0-691-17028-2 (hardcover : alk. paper) – ISBN 978-0-691-17029-9 (pbk. : alk. paper) 1. Hodge theory, 2. p-adic fields, 3. Arithmetical algebraic geometry. I. Gros, Michel, 1956- II. Tsuji, Takeshi, 1967- III. Title. QA179.A23 2016 512’.2–dc23 2015031778 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX. Printed on acid-free paper. ∞ The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed in the United States of America 1

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In Memoriam Hyodo Osamu

Contents Foreword

ix

Chapter I. Representations of the fundamental group and the torsor of deformations. An overview by Ahmed Abbes and Michel Gros I.1 Introduction I.2 Notation and conventions I.3 Small generalized representations I.4 The torsor of deformations I.5 Faltings ringed topos I.6 Dolbeault modules

1 1 3 5 6 13 19

Chapter II. Representations of the fundamental group and the torsor of deformations. Local study 27 by Ahmed Abbes and Michel Gros II.1 Introduction 27 II.2 Notation and conventions 28 II.3 Results on continuous cohomology of profinite groups 35 II.4 Objects with group actions 50 II.5 Logarithmic geometry lexicon 63 II.6 Faltings’ almost purity theorem 71 II.7 Faltings extension 84 II.8 Galois cohomology 98 II.9 Fontaine p-adic infinitesimal thickenings 110 II.10 Higgs–Tate torsors and algebras 120 II.11 Galois cohomology II 132 II.12 Dolbeault representations 143 II.13 Small representations 153 II.14 Descent of small representations and applications 166 II.15 Hodge–Tate representations 175 Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects 179 by Ahmed Abbes and Michel Gros III.1 Introduction 179 III.2 Notation and conventions 180 III.3 Locally irreducible schemes 184 III.4 Adequate logarithmic schemes 185 III.5 Variations on the Koszul complex 190 III.6 Additive categories up to isogeny 194 III.7 Inverse systems of a topos 203 III.8 Faltings ringed topos 211 vii

viii

CONTENTS

III.9 III.10 III.11 III.12 III.13 III.14 III.15

Faltings topos over a trait Higgs–Tate algebras Cohomological computations Dolbeault modules Dolbeault modules on a small affine scheme Inverse image of a Dolbeault module under an étale morphism Fibered category of Dolbeault modules

Chapter IV. Cohomology of Higgs isocrystals by Takeshi Tsuji IV.1 Introduction IV.2 Higgs envelopes IV.3 Higgs isocrystals and Higgs crystals IV.4 Cohomology of Higgs isocrystals IV.5 Representations of the fundamental group IV.6 Comparison with Faltings cohomology Chapter V. Almost étale coverings by Takeshi Tsuji V.1 Introduction V.2 Almost isomorphisms V.3 Almost finitely generated projective modules V.4 Trace V.5 Rank and determinant V.6 Almost flat modules and almost faithfully flat modules V.7 Almost étale coverings V.8 Almost faithfully flat descent I V.9 Almost faithfully flat descent II V.10 Liftings V.11 Group cohomology of discrete A-G-modules V.12 Galois cohomology Chapter VI. Covanishing topos and generalizations by Ahmed Abbes and Michel Gros VI.1 Introduction VI.2 Notation and conventions VI.3 Oriented products of topos VI.4 Covanishing topos VI.5 Generalized covanishing topos VI.6 Morphisms with values in a generalized covanishing topos VI.7 Ringed total topos VI.8 Ringed covanishing topos VI.9 Finite étale site and topos of a scheme VI.10 Faltings site and topos VI.11 Inverse limit of Faltings topos

222 229 250 266 284 290 299 307 307 313 350 369 383 402 449 449 450 452 453 455 459 461 464 467 471 478 481 485 485 493 494 502 511 527 532 537 542 550 570

Facsimile : A p-adic Simpson correspondence by Gerd Faltings, Advances in Mathematics 198 (2005), 847-862

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Bibliography

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Indexes

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Foreword In 1965, generalizing and clarifying a result of Weil, Narasimhan and Seshadri [56] established a bijective correspondence between the set of equivalence classes of unitary irreducible representations of the fundamental group of a compact Riemann surface X of genus ≥ 2 and the set of isomorphism classes of stable vector bundles of degree zero on X. The correspondence was then extended to all projective and smooth complex varieties by Donaldson [21]. The analogue for arbitrary linear representations is due to Simpson; to obtain a correspondence of the same type as Narasimhan and Seshadri, we need an additional structure on the vector bundle. This led to the notion of a Higgs bundle that was first introduced by Hitchin for algebraic curves. If X is a smooth scheme over a field K, a Higgs bundle on X is a pair (M, θ) consisting of a locally free OX -module of finite type M and an OX -linear morphism θ : M → M ⊗OX Ω1X/K such that θ ∧ θ = 0. Simpson’s main result [67, 68, 69, 70] establishes an equivalence between the category of (complex) finite-dimensional linear representations of the fundamental group of a smooth and projective complex variety and the category of semi-stable Higgs bundles with vanishing Chern classes (cf. [54]). Simpson’s results and the important developments they inspired have led, in recent years, to the search for a p-adic analogue. Looking back, the first examples of such a construction (which did not yet use the terminology of Higgs bundles) can be found in the work of Hyodo [43], who had treated the conceptually important case of p-adic variations of Hodge structures, called Hodge–Tate local systems. At present, the most advanced approach to such a correspondence is due to Faltings [27, 28]. It aims to describe all p-adic representations of the geometric fundamental group of a smooth algebraic variety over a p-adic field in terms of Higgs bundles. The constructions use several tools that he developed to establish the existence of Hodge–Tate decompositions [24], in particular his theory of almost étale extensions [26]. Once completed, this p-adic Simpson correspondence should thus naturally provide the best Hodge–Tate type statements in p-adic Hodge theory. But at present, Faltings’ construction seems satisfying only for curves, and even in that case, many fundamental questions remain open. In this volume, we undertake a systematic development of the p-adic Simpson correspondence started by Faltings following two new approaches, one by the first two authors (A.A. and M.G.), the other by the third author (T.T.). The need to resume and develop Faltings’ construction was felt given the number of results sketched in a rather short and extremely dense article [27]. This correspondence applies to objects more general than p-adic representations of the geometric fundamental group, introduced by Faltings and called generalized representations. We focus mainly on those p-adically close to the trivial representation, qualified by him as small. Though some of our constructions extend beyond this setting, let us make clear right away that we do not, however, discuss the descent techniques that enabled Faltings, in the case of curves, to get rid of this smallness condition.

ix

x

FOREWORD

Independently of the work of Faltings, Deninger and Werner [19, 20, 18] have developed a partial analogue to the theory of Narasimhan and Seshadri for p-adic curves, which should correspond to Higgs bundles with vanishing Higgs field in the p-adic Simpson correspondence. On the other hand, also inspired by the complex case, Ogus and Vologodsky [59] have introduced a correspondence between modules with integrable connection and Higgs modules for varieties in characteristic p. That work, in turn, inspired the first approach developed in this volume for the p-adic Simpson correspondence. Let us, however, note that unlike the complex case where a link between the two variants of the Simpson correspondence is established (by definition) through the Riemann–Hilbert correspondence, the link between the p-adic and the modulo p Simpson correspondences is not, at present, known. Several works exploring this direction are in progress [52, 61, 66]. Let us now give some indications on the structure of this volume. The first approach is presented in Chapters I, II, and III. Chapter I provides an overview of this approach and can also serve as an introduction to the general theme of this volume. In Chapter II, we study the case of an affine scheme of a particular type, qualified also as small by Faltings. We introduce the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then construct a natural equivalence between these two categories. We prove that this approach generalizes simultaneously Faltings’ construction for small generalized representations and Hyodo’s theory of p-adic variations of Hodge–Tate structures. In Chapter III, we address the global aspects of the theory. We introduce the Higgs–Tate algebra, which is the main novelty of this approach compared to that of Faltings, the notion of Dolbeault module that globalizes that of Dolbeault generalized representation, and the companion notion of solvable Higgs bundle. The main result is the equivalence between the category of Dolbeault modules and that of solvable Higgs bundles. We also prove the compatibility of this equivalence with the natural cohomologies. The general construction is obtained from the affine case by a gluing technique relying on the Faltings topos, developed in a more general context in Chapter VI. This first approach, like Faltings’ original one, requires the datum of a deformation of the scheme over a p-adic infinitesimal thickening of order 1 introduced by Fontaine. The second approach, developed in Chapter IV of this volume, avoids this additional datum. For this purpose, we introduce a crystalline type topos, and replace the notion of Higgs bundles by that of Higgs (iso)crystals. The link between these two notions uses Higgs envelopes and calls to mind the link between classical crystals and modules with integrable connections. The main result is the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized representations. We also prove the compatibility of this functor with the natural cohomologies. Finally, we compare the period rings used in the two approaches developed in this volume, showing the compatibility of the two constructions. The last part of the volume, consisting of Chapters V and VI, contains results of wider interest in p-adic Hodge theory. Chapter V provides a concise introduction to Faltings’ theory of almost étale extensions, a tool that has become essential in many questions in arithmetic geometry, even beyond p-adic Hodge theory. The point of view adopted here is closer to Faltings’ original one than the more systematic development given later by Gabber and Ramero. Chapter VI is devoted to the Faltings topos. Though it is the general framework for Faltings’ approach in p-adic Hodge theory, this topos remains relatively unexplored. We present a new approach to it based on a generalization of Deligne’s covanishing topos. Along the way, we correct the original definition of Faltings.

FOREWORD

xi

The reader will find at the end of the volume the facsimile of Faltings’ article, which is reprinted from Advances in Mathematics 198(2), Faltings, Gerd, “A p-adic Simpson Correspondence,” pp. 847–862, Copyright 2005, with permission from Elsevier. We thank the author warmly for having allowed us to reproduce it. Ahmed Abbes, Michel Gros, and Takeshi Tsuji September 2014

The p-adic Simpson Correspondence

CHAPTER I

Representations of the fundamental group and the torsor of deformations. An overview Ahmed Abbes and Michel Gros I.1. Introduction I.1.1. We develop a new approach to the p-adic Simpson correspondence, closely related to Faltings’ original approach [27], and inspired by the work of Ogus and Vologodsky [59] on an analogue in characteristic p of the complex Simpson correspondence. Before giving the details of this approach in Chapters II and III, we give a summary in this introductory chapter. I.1.2. Let K be a complete discrete valuation ring of characteristic 0, with perfect residue field of characteristic p > 0, OK the valuation ring of K, K an algebraic closure of K, and OK the integral closure of OK in K. Let X be a smooth OK -scheme of finite type with integral generic geometric fiber XK , x a geometric point of XK , and X the formal scheme p-adic completion of X ⊗OK OK . In this work, we consider a more general smooth logarithmic situation (cf. II.6.2 and III.4.7). Nevertheless, to simplify the presentation, we restrict ourselves in this introductory chapter to the smooth case in the usual sense. We are looking for a functor from the category of p-adic representations of the geometric fundamental group π1 (XK , x) (that is, the finite-dimensional continuous linear Qp -representations of π1 (XK , x)) to the category of Higgs OX [ p1 ]-bundles (that is, the pairs (M , θ) consisting of a locally projective OX [ p1 ]-module of finite type M and an OX [ p1 ]-linear morphism θ : M → M ⊗OX Ω1X/OK such that θ ∧ θ = 0). Following Faltings’ strategy, which at present has been only partly achieved, this functor should extend to a strictly larger category than that of the p-adic representations of π1 (XK , x), called category of generalized representations of π1 (XK , x). It would then be an equivalence of categories between this new category and the category of Higgs OX [ p1 ]-bundles. The main motivation for the present work is the construction of such an equivalence of categories. When XK is a proper and smooth curve over K, Faltings shows that the Higgs bundles associated with the “true” p-adic representations of π1 (XK , x) are semi-stable of slope zero and expresses the hope that all semi-stable Higgs bundles of slope zero are obtained this way. This statement, which would correspond to the difficult part of Simpson’s result in the complex case, seems out of reach at present. I.1.3. The notion of generalized representations is due to Faltings. They are, in simplified terms, continuous p-adic semi-linear representations of π1 (XK , x) on modules over a certain p-adic ring endowed with a continuous action of π1 (XK , x). Faltings’ approach in [27] to construct a functor H from the category of these generalized representations to the category of Higgs bundles consists of two steps. He first defines H for the generalized representations that are p-adically close to the trivial representation, which he calls small. He carries out this step in arbitrary dimension. In the second step, 1

2

I. AN OVERVIEW

achieved only for curves, he extends the functor H to all generalized representations of π1 (XK , x) by descent. Indeed, every generalized representation becomes small over a finite étale cover of XK . I.1.4. Our new approach, which works in arbitrary dimension, allows us to define the functor H on the category of generalized representations of π1 (XK , x) satisfying an admissibility condition à la Fontaine, called Dolbeault generalized representations. For this purpose, we introduce a family of period rings that we call Higgs–Tate algebras, and that are the main novelty of our approach compared to that of Faltings. We show that the admissibility condition for rational coefficients corresponds to the smallness condition of Faltings; but it is strictly more general for integral coefficients. Note that Faltings’ construction for small rational coefficients is limited to curves and that it presents a number of difficulties that can be avoided with our approach. I.1.5. We proceed in two steps. We first study in Chapter II the case of an affine scheme of a certain type, called also small by Faltings. We then tackle in Chapter III the global aspects of the theory. The general construction is obtained from the affine case using a gluing technique presenting unexpected difficulties. To do this we will use the Faltings topos, a fibered variant of Deligne’s notion of covanishing topos, which we develop in Chapter VI. I.1.6. This introductory chapter offers, in a geometric situation simplified for the clarity of the exposition, a detailed summary of the global steps leading to our main results. Let us take a quick look at its contents. We begin, in I.3, with a short aside on small generalized representations in the affine case, which will be used as intermediary for the study of Dolbeault representations. Section I.4 summarizes the local study conducted in Chapter II. We introduce the notion of generalized Dolbeault representation for a small affine scheme and the companion notion of solvable Higgs module, and then construct a natural equivalence between these two categories. We in fact develop two variants, an integral one and a more subtle rational one. We establish links between these notions and Faltings smallness conditions. We also link this to Hyodo’s theory [43]. The global aspects of the theory developed in Chapter III are summarized in Sections I.5 and I.6. After a short introduction to Faltings’ ringed topos in I.5, we introduce the Higgs–Tate algebras (I.5.13). The notion of Dolbeault module that globalizes that of generalized Dolbeault representation and the companion notion of solvable Higgs bundle are defined in I.6.13. Our main result (I.6.18) is the equivalence of these two categories. For the proof of this result, we need acyclicity statements for the Higgs–Tate algebras that we give in I.6.5 and I.6.8, which also allow us to show the compatibility of this equivalence with the relevant cohomologies on each side (I.6.19). We also study the functoriality of the various introduced properties by étale morphisms (I.6.21), as well as their local character for the étale topology (I.6.22, I.6.23, I.6.24). Finally, we return in this global situation to the logical links (I.6.26, I.6.27, I.6.28), for a Higgs bundle, between smallness (I.6.25) and solvability. At the beginning of Chapters II and III, the reader will find a detailed description of their structure. Chapter VI, which is of separate interest, has its own introduction. Acknowledgments. This work could obviously not have existed without the work of G. Faltings, and first and foremost, that on the p-adic Simpson correspondence [27]. We would like to convey our deep gratitude to him. The genesis of this work immediately followed a workshop held in Rennes in 2008–2009 on his article [27]. We benefited, on that occasion, from the text of O. Brinon’s talk [13] and from the work of T. Tsuji [75] presenting his own approach to the p-adic Simpson correspondence. These two

I.2. NOTATION AND CONVENTIONS

3

texts have been extremely useful to us and we are grateful to their authors for having made them available to us spontaneously. We also thank O. Brinon, G. Faltings, and T. Tsuji for all the exchanges we had with them on questions related to this work, and A. Ogus for the clarifying discussions we had with him on his work with V. Vologodsky [59]. We thank Reinie Erné warmly for translating, with great skill and under tight deadlines, Chapters I–III and VI of this volume, keeping in mind our stylistic preferences. The first author (A.A.) thanks the Centre Émile Borel, the Institut des Hautes Études Scientifiques, and the University of Tokyo for their hospitality. He also thanks those who followed the course he gave on this subject at the University of Tokyo during the fall of 2010 and the winter of 2011, whose questions and remarks have been precious for perfecting this work. The second author (M.G.) thanks the Institut des Hautes Études Scientifiques and the University of Tokyo for their hospitality. Finally, we thank the participants of the summer school Higgs bundles on p-adic curves and representation theory that took place in Mainz in September 2012, during which our main results were presented, for their remarks and their stimulating interest. This work was supported by the ANR program p-adic Hodge theory and beyond (ThéHopaD) ANR-11-BS01-005. I.2. Notation and conventions All rings in this chapter have an identity element; all ring homomorphisms map the identity element to the identity element. We mostly consider commutative rings, and rings are assumed to be commutative unless stated otherwise; in particular, when we take a ringed topos (X, A), the ring A is assumed to be commutative unless stated otherwise. I.2.1. In this introduction, K denotes a complete discrete valuation ring of characteristic 0, with perfect residue field k of characteristic p > 0, OK the valuation ring of K, K an algebraic closure of K, OK the integral closure of OK in K, OC the padic Hausdorff completion of OK , and C the field of fractions of OC . From I.5 on, we will assume that k is algebraically closed. We set S = Spec(OK ), S = Spec(OK ), and ˇ = Spec(O ). We denote by s (resp. η, resp. η) the closed point of S (resp. the generic S C point of S, resp. the generic point of S). For any integer n ≥ 1 and any S-scheme X, we set Sn = Spec(OK /pn OK ), (I.2.1.1)

Xn = X ×S Sn ,

X = X ×S S,

ˇ = X × S. ˇ and X S

c its p-adic Hausdorff completion. For any abelian group M , we denote by M I.2.2. Let G be a profinite group and A a topological ring endowed with a continuous action of G by ring homomorphisms. An A-representation of G consists of an A-module M and an A-semi-linear action of G on M , that is, such that for all g ∈ G, a ∈ A, and m ∈ M , we have g(am) = g(a)g(m). We say that the A-representation is continuous if M is a topological A-module and if the action of G on M is continuous. Let M , N be two A-representations (resp. two continuous A-representations) of G. A morphism from M to N is a G-equivariant and A-linear (resp. G-equivariant, continuous, and A-linear) morphism from M to N . I.2.3. Let (X, A) be a ringed topos and E an A-module. A Higgs A-module with coefficients in E is a pair (M, θ) consisting of an A-module M and an A-linear morphism θ : M → M ⊗A E such that θ ∧ θ = 0 (cf. II.2.8). Following Simpson ([68] p. 24), we call Dolbeault complex of (M, θ) and denote by K• (M, θ) the complex of cochains of A-modules (I.2.3.1)

M → M ⊗A E → M ⊗A ∧2 E . . .

4

I. AN OVERVIEW

deduced from θ (cf. II.2.8.2). I.2.4. Let (X, A) be a ringed topos, B an A-algebra, M a B-module, and λ ∈ Γ(X, A). A λ-connection on M with respect to the extension B/A consists of an A-linear morphism (I.2.4.1)

∇ : M → Ω1B/A ⊗B M

such that for all local sections x of B and s of M , we have (I.2.4.2)

∇(xs) = λd(x) ⊗ s + x∇(s).

It is integrable if ∇ ◦ ∇ = 0 (cf. II.2.10). We will leave the extension B/A out of the terminology when there is no risk of confusion. Let (M, ∇), (M 0 , ∇0 ) be two modules with λ-connections. A morphism from (M, ∇) to (M 0 , ∇0 ) is a B-linear morphism u : M → M 0 such that (id ⊗ u) ◦ ∇ = ∇0 ◦ u. Classically, 1-connections are called connections. Integrable 0-connections are the Higgs B-fields with coefficients in Ω1B/A . Remark I.2.5. Let (X, A) be a ringed topos, B an A-algebra, λ ∈ Γ(X, A), and (M, ∇) a module with λ-connection with respect to the extension B/A. Suppose that there exist ∼ an A-module E and a B-linear isomorphism γ : E ⊗A B → Ω1B/A such that for every local section ω of E, we have d(γ(ω ⊗ 1)) = 0. The λ-connection ∇ is integrable if and only if the morphism θ : M → E ⊗A M induced by ∇ and γ is a Higgs A-field on M with coefficients in E (cf. II.2.12). I.2.6. If C is an additive category, we denote by CQ and call category of objects of C up to isogeny the category with the same objects as C , and such that the set of morphisms between two objects is given by (I.2.6.1)

HomCQ (E, F ) = HomC (E, F ) ⊗Z Q.

The category CQ is none other than the localized category of C with respect to the multiplicative system of the isogenies of C (cf. III.6.1). We denote by (I.2.6.2)

C → CQ ,

M 7→ MQ

the localization functor. If C is an abelian category, the category CQ is abelian and the localization functor (I.2.6.2) is exact. Indeed, CQ identifies canonically with the quotient of C by the thick subcategory of objects of finite exponent (III.6.1.4). I.2.7. Let (X, A) be a ringed topos. We denote by Mod(A) the category of Amodules of X and by ModQ (A), instead of Mod(A)Q , the category of A-modules up to isogeny (I.2.6). The tensor product of A-modules induces a bifunctor (I.2.7.1)

ModQ (A) × ModQ (A) → ModQ (A),

(M, N ) 7→ M ⊗AQ N

making ModQ (A) into a symmetric monoidal category with AQ as unit object. The objects of ModQ (A) will also be called AQ -modules. This terminology is justified by considering AQ as a monoid of ModQ (A). I.2.8. Let (X, A) be a ringed topos and E an A-module. We call Higgs A-isogeny with coefficients in E a quadruple (I.2.8.1)

(M, N, u : M → N, θ : M → N ⊗A E)

consisting of two A-modules M and N and two A-linear morphisms u and θ satisfying the following property: there exist an integer n 6= 0 and an A-linear morphism v : N → M such that v ◦ u = n · idM , u ◦ v = n · idN , and that (M, (v ⊗ idE ) ◦ θ) and (N, θ ◦ v) are Higgs A-modules with coefficients in E (I.2.3). Note that u induces an isogeny of

I.3. SMALL GENERALIZED REPRESENTATIONS

5

Higgs modules from (M, (v ⊗ idE ) ◦ θ) to (N, θ ◦ v) (III.6.1), whence the terminology. Let (M, N, u, θ), (M 0 , N 0 , u0 , θ0 ) be two Higgs A-isogenies with coefficients in E. A morphism from (M, N, u, θ) to (M 0 , N 0 , u0 , θ0 ) consists of two A-linear morphisms α : M → M 0 and β : N → N 0 such that β ◦ u = u0 ◦ α and (β ⊗ idE ) ◦ θ = θ0 ◦ α. We denote by HI(A, E) the category of Higgs A-isogenies with coefficients in E. It is an additive category. We denote by HIQ (A, E) the category of objects of HI(A, E) up to isogeny. I.2.9. Let (X, A) be a ringed topos, B an A-algebra, and λ ∈ Γ(X, A). We call λ-isoconnection with respect to the extension B/A (or simply λ-isoconnection when there is no risk of confusion) a quadruple (I.2.9.1)

(M, N, u : M → N, ∇ : M → Ω1B/A ⊗B N )

where M and N are B-modules, u is an isogeny of B-modules (III.6.1), and ∇ is an A-linear morphism such that for all local sections x of B and t of M , we have (I.2.9.2)

∇(xt) = λd(x) ⊗ u(t) + x∇(t).

For every B-linear morphism v : N → M for which there exists an integer n such that u ◦ v = n · idN and v ◦ u = n · idM , the pairs (M, (id ⊗ v) ◦ ∇) and (N, ∇ ◦ v) are modules with (nλ)-connections (I.2.2), and u is a morphism from (M, (id ⊗ v) ◦ ∇) to (N, ∇ ◦ v). We call the λ-isoconnection (M, N, u, ∇) integrable if there exist a B-linear morphism v : N → M and an integer n 6= 0 such that u ◦ v = n · idN , v ◦ u = n · idM , and that the (nλ)-connections (id ⊗ v) ◦ ∇ on M and ∇ ◦ v on N are integrable. Let (M, N, u, ∇) and (M 0 , N 0 , u0 , ∇0 ) be two λ-isoconnections. A morphism from (M, N, u, ∇) to (M 0 , N 0 , u0 , ∇0 ) consists of two B-linear morphisms α : M → M 0 and β : N → N 0 such that β ◦ u = u0 ◦ α and (id ⊗ β) ◦ ∇ = ∇0 ◦ α. I.3. Small generalized representations I.3.1. In this section, we fix a smooth affine S-scheme X = Spec(R) such that Xη is connected and Xs is nonempty, an integer d ≥ 1, and an étale S-morphism (I.3.1.1)

X → Gdm,S = Spec(OK [T1±1 , . . . , Td±1 ]).

This is the typical example of a Faltings’ small affine scheme. The assumption that Xη is connected is not necessary but allows us to simplify the presentation. The reader will recognize the logarithmic nature of the datum (I.3.1.1). Following [27], we consider in this work a more general smooth logarithmic situation, which turns out to be necessary even for defining the p-adic Simpson correspondence for a proper smooth curve over S. Indeed, in the second step of the descent, we will need to consider finite covers of its generic fiber, which brings us to the case of a semi-stable scheme over S. Nevertheless, to simplify the presentation, we will restrict ourselves in this introduction to the smooth case in the usual sense (cf. II.6.2 for the logarithmic smooth affine case). We denote by ti the image of Ti in R (1 ≤ i ≤ d), and we set (I.3.1.2)

R1 = R ⊗OK OK .

I.3.2. Let y be a geometric point of Xη and (Vi )i∈I a universal cover of Xη at y. We denote by ∆ the geometric fundamental group π1 (Xη , y). For every i ∈ I, we denote by X i = Spec(Ri ) the integral closure of X in Vi , and we set (I.3.2.1)

R = lim Ri . −→ i∈I

6

I. AN OVERVIEW

b In this context, the generalized representations of ∆ are the continuous R-representations b of ∆ with values in projective R-modules of finite type, endowed with their p-adic topolob gies (I.2.2). Such a representation M is called small if M is a free R-module of finite type having a basis made up of elements that are ∆-invariant modulo p2α M for a rational 1 . The main property of the small generalized representations of ∆ is number α > p−1 their good behavior under descent for certain quotients of ∆ isomorphic to Zp (1)d . Let (n) us fix such a quotient ∆∞ by choosing, for every 1 ≤ i ≤ d, a compatible system (ti )n∈N n c1 -representation of ∆∞ similarly. of p th roots of ti in R. We define the notion of small R The functor b (I.3.2.2) M 7→ M ⊗ R c1 R

b c1 -representations of ∆∞ to that of small R-representations from the category of small R of ∆ is then an equivalence of categories (cf. II.14.4). This is a consequence of Faltings’ almost purity theorem (cf. II.6.16; [26] § 2b). c1 -representation of ∆∞ , we can consider the logarithm I.3.3. If (M, ϕ) is a small R of ϕ, which is a homomorphism from ∆∞ to EndR c1 (M ). By fixing a Zp -basis ζ of Zp (1), the latter can be written uniquely as log(ϕ) =

(I.3.3.1)

d X i=1

θi ⊗ χi ⊗ ζ −1 ,

−1

where ζ is the dual basis of Zp (−1), χi is the character of ∆∞ with values in Zp (1) (n) c1 -linear endomorphism of M . that gives its action on the system (ti )n∈N , and θi is an R We immediately see that θ=

(I.3.3.2)

d X i=1

θi ⊗ d log(ti ) ⊗ ζ −1

c1 -field on M with coefficients in Ω1 b is a Higgs R R/OK ⊗R R1 (−1) (I.2.3) (to simplify we 1 will say with coefficients in ΩR/OK (−1)). The resulting correspondence (M, ϕ) 7→ (M, θ) c1 -representations of is in fact an equivalence of categories between the category of small R 1 c ∆∞ and that of small Higgs R1 -modules with coefficients in Ω (−1) (that is, the catR/OK

c1 -modules with coefficients in Ω1 c egory of Higgs R R/OK (−1) whose underlying R1 -module is free of finite type and whose Higgs field is zero modulo p2α for a rational number 1 α > p−1 ). Combining this with the previous descent statement (I.3.2.2), we obtain an b equivalence between the category of small R-representations of ∆ and that of small Higgs 1 c R1 -modules with coefficients in Ω (−1). The disadvantage of this construction is its R/OK

(n)

dependence on the (ti )n∈N (1 ≤ i ≤ d), which excludes any globalization. To remedy this defect, Faltings proposes another equivalent definition that depends on another choice that can be globalized easily. Our approach, which is the object of the remainder of this introduction, was inspired by this construction. I.4. The torsor of deformations I.4.1. In this section, we are given a smooth affine S-scheme X = Spec(R) such that Xη is connected, Xs is nonempty, and that there exist an integer d ≥ 1 and an étale S-morphism X → Gdm,S (but we do not fix such a morphism). We also fix a geometric

I.4. THE TORSOR OF DEFORMATIONS

7

point y of Xη and a universal cover (Vi )i∈I of Xη at y, and we use the notation of I.3.2: ∆ = π1 (Xη , y), R1 = R ⊗OK OK , and R (I.3.2.1). I.4.2. (I.4.2.1)

Recall that Fontaine associates functorially with each Z(p) -algebra A the ring RA = lim A/pA ←−

x7→xp

and a homomorphism θ from the ring W(RA ) of Witt vectors of RA to the p-adic b of A (cf. II.9.3). We set Hausdorff completion A (I.4.2.2)

A2 (A) = W(RA )/ ker(θ)2

b the homomorphism induced by θ. and denote also by θ : A2 (RA ) → A For the remainder of this chapter, we fix a sequence (pn )n∈N of elements of OK such that p0 = p and ppn+1 = pn for every n ≥ 0. We denote by p the element of ROK induced by the sequence (pn )n∈N and set (I.4.2.3)

ξ = [p] − p ∈ W(ROK ),

where [ ] is the multiplicative representative. The sequence (I.4.2.4)

·ξ

θ

0 −→ W(ROK ) −→ W(ROK ) −→ OC −→ 0

is exact (II.9.5). It induces an exact sequence (I.4.2.5)

·ξ

θ

0 −→ OC −→ A2 (OK ) −→ OC −→ 0,

where ·ξ again denotes the morphism deduced from the morphism of multiplication by ξ in A2 (OK ). The ideal ker(θ) of A2 (OK ) has square zero. It is a free OC -module with basis ξ. It will be denoted by ξOC . Note that unlike ξ, this module does not depend on the choice of the sequence (pn )n∈N . We denote by ξ −1 OC the dual OC -module of ξOC . For every OC -module M , we denote the OC -modules M ⊗OC (ξOC ) and M ⊗OC (ξ −1 OC ) simply by ξM and ξ −1 M , respectively. Likewise, we have an exact sequence (II.9.11.2) (I.4.2.6)

·ξ θ b b −→ A2 (R) −→ R −→ 0. 0 −→ R

b The ideal ker(θ) of A2 (R) has square zero. It is a free R-module with basis ξ, canonically b isomorphic to ξ R. The group ∆ acts by functoriality on A (R). 2

b and A (Y ) = We set A2 (S) = Spec(A2 (OK )), Y = Spec(R), Yb = Spec(R), 2 Spec(A2 (R)). ˇ that is, a smooth e of X, I.4.3. From now on, we fix a smooth A2 (S)-deformation X e that fits into a Cartesian diagram A2 (S)-scheme X (I.4.3.1)

ˇ X

/X e

 ˇ S

 / A2 (S)

This additional datum replaces the datum of an étale S-morphism X → Gdm,S ; in fact, such a morphism provides a deformation. We set b b ξ R). (I.4.3.2) T = Hom (Ω1 ⊗ R, b R

R/OK

R

8

I. AN OVERVIEW

b (I.4.2) and denote by T the assob with ξ −1 Ω1R/OK ⊗R R We identify the dual R-module ciated Yb -vector bundle, in other words, b T = Spec(Sym b (ξ −1 Ω1R/OK ⊗R R)).

(I.4.3.3)

R

e the open subscheme of A2 (Y ) defined by U . Let U be an open subscheme of Yb and U We denote by L (U ) the set of morphisms represented by dotted arrows that complete the diagram (I.4.3.4)

U

/U e

 ˇ X

 /X e

 ˇ S

 / A2 (S)

in such a way that it remains commutative. The functor U 7→ L (U ) is a T -torsor for b the Zariski topology of Yb . We denote by F the R-module of affine functions on L (cf. II.4.9). The latter fits into a canonical exact sequence (II.4.9.1) b → F → ξ −1 Ω1 b 0→R R/OK ⊗R R → 0.

(I.4.3.5)

This sequence induces for every integer n ≥ 1 an exact sequence (I.4.3.6)

b → 0. n −1 1 n ΩR/OK ⊗R R) 0 → Symn−1 b (F ) → Sym b (F ) → Sym b (ξ R

R

R

b The R-modules (Symnb (F ))n∈N therefore form a filtered direct system whose direct limit R

(I.4.3.7)

C = lim Symnb (F ) −→

R

n≥0

b is naturally endowed with a structure of R-algebra. By II.4.10, the Yb -scheme (I.4.3.8)

L = Spec(C )

is naturally a principal homogeneous T-bundle on Yb that canonically represents L . b The natural action of ∆ on the scheme A2 (Y ) induces an R-semi-linear action of ∆ on F , such that the morphisms in sequence (I.4.3.5) are ∆-equivariant. From this we b deduce an action of ∆ on C by ring automorphisms, compatible with its action on R, which we call canonical action. These actions are continuous for the p-adic topologies b (II.12.4). The R-algebra C , endowed with the canonical action of ∆, is called the Higgs– b e The R-representation Tate algebra associated with X. F of ∆ is called the Higgs–Tate e extension associated with X. c1 -module with coefficients in ξ −1 Ω1 Let (M, θ) be a small Higgs R R/OK (that −1 1 c1 -module with coefficients in ξ Ω c1 whose underlying R c1 -module is, a Higgs R ⊗R R I.4.4.

R/OK

is free of finite type and whose Higgs field is zero modulo pα for a rational number 1 α > p−1 ) and let ψ ∈ L (Yb ). For every σ ∈ ∆, we denote by σ ψ the section of L (Yb )

I.4. THE TORSOR OF DEFORMATIONS

9

defined by the commutative diagram (I.4.4.1)

σ

AL

/L ] σ

ψ

 Yb

σ

 / Yb

ψ

b R). b The endoThe difference Dσ = ψ − σ ψ is an element of Hom b (ξ −1 Ω1R/OK ⊗R R, R b morphism exp((Dσ ⊗ idM ) ◦ θ) of M ⊗R c1 R is well-defined, in view of the smallness of θ. b b The resulting corresponWe then obtain a small R-representation of ∆ on M ⊗ R. c1 R

c1 -modules dence is in fact an equivalence of categories from the category of small Higgs R b with coefficients in ξ −1 Ω1R/OK to that of small R-representations of ∆. It is essentially a quasi-inverse of the equivalence of categories defined in I.3.3. To avoid the choice of a section ψ of L (Yb ), we can carry out the base change from b to C and use the diagonal embedding of L. In this setting, the previous construcR tion can be interpreted following the classic scheme of correspondences introduced by Fontaine (or even the more classic complex analytic Riemann–Hilbert correspondence) by taking for period ring making the link between generalized representations and Higgs modules a weak p-adic completion C † of C (the completion is made necessary by the exponential). With this ring is naturally associated a notion of admissibility; it is the notion of generalized Dolbeault representation. Before developing this approach, we will say a few words about the ring C that can itself play the role of period ring between the generalized representations and Higgs modules. Indeed, C is an integral model of the Hyodo ring (cf. (I.4.6.1) and II.15.6), which explains the link between our approach and that of Hyodo. b I.4.5. Recall that Faltings has defined a canonical extension of R-representations of π1 (X, y) b → E → Ω1 b 0 → ρ−1 R R/OK ⊗R R(−1) → 0,

(I.4.5.1)

1 that plays an important role in his where ρ is an element of OK of valuation ≥ p−1 approach to p-adic Hodge theory (cf. II.7.22). We show in II.10.19 that there exists a b ∆-equivariant and R-linear morphism 1

p− p−1 F → E

(I.4.5.2)

that fits into a commutative diagram (I.4.5.3)

0

1 / p− p−1 b _ R

1 / p− p−1 F

1 b / p− p−1 ξ −1 Ω1R/OK ⊗R R

/0

−c



0

/ ρ−1 R b

 /E

/ Ω1 R/OK

 b ⊗R R(−1)

/0

1 ∼ b b where c is the isomorphism induced by the canonical isomorphism R(1) → p p−1 ξ R e (II.9.18). The morphism (I.4.5.2) is canonical if we take for X the deformation induced by an étale S-morphism X → Gdm,S . It is important to note that in the logarithmic setting that will be considered in this work, the Faltings extension changes form slightly b is replaced by (πρ)−1 R, b where π is a uniformizer for R. because the factor ρ−1 R

10

I. AN OVERVIEW

I.4.6. Taking Faltings extension E (I.4.5.1) as a starting point, Hyodo [43] defines b an R-algebra CHT using a direct limit analogous to (I.4.3.7). Note that p being invertible in CHT , this is equivalent to beginning with E ⊗Zp Qp , which corresponds to Hyodo’s original definition. The morphism (I.4.5.2) therefore induces a ∆-equivariant isomorphism b of R-algebras 1 ∼ C [ ] → CHT . p

(I.4.6.1)

For every continuous Qp -representation V of Γ = π1 (X, y) and every integer i, Hyodo b 1 ]-module Di (V ) by setting defines the R[ p Di (V ) = (V ⊗Qp CHT (i))Γ .

(I.4.6.2)

The representation V is called Hodge–Tate if it satisfies the following conditions: (i) V is a Qp -vector space of finite dimension, endowed with the p-adic topology. (ii) The canonical morphism (I.4.6.3)

⊕i∈Z Di (V ) ⊗R[ b 1 ] CHT (−i) → V ⊗Qp CHT p

is an isomorphism. b I.4.7. For any rational number r ≥ 0, we denote by F (r) the R-representation of ∆ deduced from F by inverse image under the morphism of multiplication by pr on b so that we have an exact sequence ξ −1 Ω1R/OK ⊗R R, (I.4.7.1)

b → F (r) → ξ −1 Ω1 b 0→R R/OK ⊗R R → 0.

For every integer n ≥ 1, this sequence induces an exact sequence (I.4.7.2)

b → 0. (r) 0 → Symn−1 ) → Symnb (F (r) ) → Symnb (ξ −1 Ω1R/OK ⊗R R) b (F R

R

R

b The R-modules (Symnb (F (r) ))n∈N therefore form a filtered direct system, whose direct R limit (I.4.7.3)

C (r) = lim Snb (F (r) ) −→

n≥0

R

b is naturally endowed with a structure of R-algebra. The action of ∆ on F (r) induces b which we an action on C (r) by ring automorphisms, compatible with its action on R, b call canonical action. The R-algebra C (r) endowed with this action is called the Higgs– e We denote by Cb(r) the p-adic Hausdorff Tate algebra of thickness r associated with X. (r) completion of C that we always assume endowed with the p-adic topology. For all rational numbers r ≥ r0 ≥ 0, we have an injective and ∆-equivariant canonical 0 0 b R-homomorphism αr,r : C (r ) → C (r) . One easily verifies that the induced homomor0 b(r0 ) → Cb(r) is injective. We set phism hr,r α : C (I.4.7.4)

C † = lim Cb(r) , −→

r∈Q>0

b which we identify with a sub-R-algebra of Cb = Cb(0) . The group ∆ acts naturally on C † b and on Cb. by ring automorphisms, in a manner compatible with its actions on R We denote by (I.4.7.5)

dC (r) : C (r) → ξ −1 Ω1R/OK ⊗R C (r)

I.4. THE TORSOR OF DEFORMATIONS

11

b of C (r) and by the universal R-derivation dCb(r) : Cb(r) → ξ −1 Ω1R/OK ⊗R Cb(r)

(I.4.7.6)

its extension to the completions (note that the R-module Ω1R/OK is free of finite type). b The derivations d (r) and d with are ∆-equivariant. They are also Higgs R-fields C ξ −1 Ω1R/OK

Cb(r)

b = d (r) (F (r) ) ⊂ d (r) (C (r) ) (cf. coefficients in because ξ −1 Ω1R/OK ⊗R R C C I.2.5). For all rational numbers r ≥ r0 ≥ 0, we have 0

0

0

pr (id × αr,r ) ◦ dC (r0 ) = pr dC (r) ◦ αr,r .

(I.4.7.7)

b The derivations pr dCb(r) therefore induce an R-derivation dC † : C † → ξ −1 Ω1R/OK ⊗R C † ,

(I.4.7.8)

that is none other than the restriction op dCb to C † . b c1 -module I.4.8. For any R-representation M of ∆, we denote by H(M ) the Higgs R −1 1 with coefficients in ξ ΩR/OK defined by H(M ) = (M ⊗ b C † )∆

(I.4.8.1)

R

c1 -module (N, θ) with coefficients and by the Higgs field induced by dC † . For every Higgs R b −1 1 in ξ Ω , we denote by V(N ) the R-representation of ∆ defined by R/OK

† θtot =0 , V(N ) = (N ⊗R c1 C )

(I.4.8.2)

where θtot = θ ⊗ id + id ⊗ dC † , and by the action of ∆ induced by its canonical action on C † . In order to make the most of these functors we establish acyclicity results for C † ⊗Zp Qp for the Dolbeault cohomology (II.12.3) and for the continuous cohomology of ∆ (II.12.5), slightly generalizing earlier results of Tsuji (cf. IV). b A continuous R-representation M of ∆ is called Dolbeault if it satisfies the following conditions (cf. II.12.11): b (i) M is a projective R-module of finite type, endowed with the p-adic topology; c1 -module of finite type; (ii) H(M ) is a projective R (iii) the canonical C † -linear morphism † † H(M ) ⊗R c1 C → M ⊗ b C

(I.4.8.3)

R

is an isomorphism. c1 -module (N, θ) with coefficients in ξ −1 Ω1 A Higgs R

R/OK

the following conditions (cf. II.12.12): c1 -module of finite type; (i) N is a projective R b (ii) V(N ) is a projective R-module of finite type; (iii) the canonical C † -linear morphism

† V(N ) ⊗ b C † → N ⊗R c1 C

(I.4.8.4)

R

is an isomorphism.

is called solvable if it satisfies

12

I. AN OVERVIEW

One immediately sees that the functors V and H induce equivalences of categories quasib inverse to each other between the category of Dolbeault R-representations of ∆ and that −1 1 c of solvable Higgs R1 -modules with coefficients in ξ ΩR/OK (II.12.15). b We show that small R-representations of ∆ are Dolbeault (II.14.6), that small Higgs c1 -modules are solvable (II.13.20), and that V and H induce equivalences of categories R quasi-inverse to each other between the categories of these objects (II.14.7). We in fact recover the correspondence defined in I.3.3, up to renormalization (cf. II.13.18). b 1 ]-representation of ∆ and solvable I.4.9. We define the notions of Dolbeault R[ p c1 [ 1 ]-module with coefficients in ξ −1 Ω1 Higgs R R/OK by copying the definitions given in p the integral case (cf. II.12.16 and II.12.18). We show that the functors V and H induce equivalences of categories quasi-inverse to each other between the category of Dolbeault b 1 ]-representations of ∆ and that of solvable Higgs R c1 [ 1 ]-modules with coefficients in R[ p p ξ −1 Ω1R/OK (II.12.24). This result is slightly more delicate than its integral analogue (I.4.8). Unlike the integral case, the rational admissibility conditions can be interpreted b 1 ]in terms of divisibility conditions. More precisely, we say that a continuous R[ p representation M of ∆ is small if it satisfies the following conditions: b 1 ]-module of finite type, endowed with a p-adic topology (i) M is a projective R[ p

(II.2.2); b 2 (ii) there exist a rational number α > p−1 and a sub-R-module M ◦ of M of finite type, stable under ∆, generated by a finite number of elements ∆-invariant b 1 ]. modulo pα M ◦ , and that generates M over R[ p c1 [ 1 ]-module (N, θ) with coefficients in ξ −1 Ω1 We say that a Higgs R R/OK is small if it p satisfies the following conditions: c1 [ 1 ]-module of finite type; (i) N is a projective R p

1 c1 -module N ◦ of N of finite (ii) there exist a rational number β > p−1 and a sub-R c1 [ 1 ], such that we have type that generates N over R p

(I.4.9.1)

θ(N ◦ ) ⊂ pβ ξ −1 N ◦ ⊗R Ω1R/OK .

c1 [ 1 ]-module with coefficients in Proposition I.4.10 (cf. II.13.25). A Higgs R p ξ −1 Ω1R/OK is solvable if and only if it is small. b 1 ]-representation of ∆ is small. Proposition I.4.11 (cf. II.13.26). Every Dolbeault R[ p We prove that the converse implication is equivalent to a descent property for small b 1 R[ p ]-representations of ∆ (II.14.8). b 1 ]-representation of ∆ and Proposition I.4.12 (cf. II.12.26). Let M be a Dolbeault R[ p c1 [ 1 ]-module with coefficients in ξ −1 Ω1 (H(M ), θ) the associated Higgs R R/OK . We then p 1 + c have a functorial canonical isomorphism in D (Mod(R1 [ ])) p

(I.4.12.1) C•cont (∆, M )

∼ C•cont (∆, M ) →

•

K (H(M ), θ),

where is the complex of continuous cochains of ∆ with values in M and K• (H(M ), θ) is the Dolbeault complex (I.2.3).

I.5. FALTINGS RINGED TOPOS

13

This statement was proved by Faltings for small representations ([27] § 3) and by Tsuji (IV.5.3.2). It follows from (I.4.6.1) that if V is a Hodge–Tate Qp -representation of Γ, b b 1 ]-representation of ∆; we have a functorial R c1 -linear then V ⊗Zp R is a Dolbeault R[ p isomorphism ∼ b → c (−1), (I.4.13.1) H(V ⊗ R) ⊕ Di (V ) ⊗ R I.4.13.

i∈Z

Zp

b R

1

b is induced by the R-linear b morphisms and the Higgs field on H(V ⊗Zp R) (I.4.13.2)

Di (V ) → Di−1 (V ) ⊗R Ω1R/OK

b 1 ] (cf. II.15.7). Moreover, the deduced from the universal derivation of CHT over R[ p e the deformation induced by an étale isomorphism (I.4.13.1) is canonical if we take for X S-morphism X → Gdm,S . I.4.14. Hyodo ([43] 3.6) has proved that if f : Y → X is a proper and smooth morphism, for every integer m ≥ 0, the sheaf Rm fη∗ (Qp ) is Hodge–Tate of weight between 0 and m; for every 0 ≤ i ≤ m, we have a canonical isomorphism ∼ b (I.4.14.1) Di (Rm fη∗ (Qp )) → (Rm−i fη∗ (Ωi )) ⊗R R, Y /X

and the morphism (I.4.13.2) is induced by the Kodaira–Spencer class of f . It follows that the Higgs bundle associated with Rm fη∗ (Qp ) is equal to the vector bundle (I.4.14.2)

⊕0≤i≤m Rm−i fη∗ (ΩiY /X ),

endowed with the Higgs field θ defined by the Kodaira–Spencer class of f . I.5. Faltings ringed topos I.5.1. We will tackle in Chapter III the global aspects of the theory in a logarithmic setting. However, in order to maintain a simplified presentation, we again restrict ourselves here to the smooth case in the usual sense (cf. III.4.7 for the smooth logarithmic case). In the remainder of this introduction, we suppose that k is algebraically closed and we denote by X a smooth S-scheme of finite type. From I.5.12 on, we will moreover ˇ that we will fix. e of X suppose that there exists a smooth A2 (S)-deformation X I.5.2. The first difficulty we encounter in gluing the local construction described in I.4 is the sheafification of the notion of generalized representation. To do this, we use the Faltings topos, a fibered variant of Deligne’s notion of covanishing topos that we develop in Chapter VI. We denote by E the category of morphisms of schemes V → U over the canonical morphism Xη → X, that is, the commutative diagrams /U (I.5.2.1) V  Xη

 /X

such that the morphism U → X is étale and that the morphism V → Uη is finite étale. ´ /X of étale X-schemes, by the functor It is fibered over the category Et (I.5.2.2)

´ /X , π : E → Et

(V → U ) 7→ U.

´ f/U of finite étale schemes The fiber of π over an étale X-scheme U is the category Et η over Uη , which we endow with the étale topology. We denote by Uη,f´et the topos of

14

I. AN OVERVIEW

´ f/U (cf. VI.9.2). If Uη is connected and if y is a geometric point of sheaves of sets on Et η Uη , denoting by Bπ1 (Uη ,y) the classifying topos of the fundamental group π1 (Uη , y), we have a canonical equivalence of categories (VI.9.8.4) ∼

νy : Uη,f´et → Bπ1 (Uη ,y) .

(I.5.2.3)

We endow E with the covanishing topology generated by the coverings {(Vi → Ui ) → (V → U )}i∈I of the following two types: (v) Ui = U for every i ∈ I, and (Vi → V )i∈I is a covering; (c) (Ui → U )i∈I is a covering and Vi = Ui ×U V for every i ∈ I. e and The resulting covanishing site E is also called Faltings site of X. We denote by E call Faltings topos of X the topos of sheaves of sets on E. We refer to Chapter VI for a e detailed study of this topos. Let us give a practical and simple description of E. Proposition I.5.3 (cf. VI.5.10). Giving a sheaf F on E is equivalent to giving, for every ´ /X , ´ /X , a sheaf FU of Uη,f´et , and for every morphism f : U 0 → U of Et object U of Et a morphism FU → ff´et∗ (FU 0 ), these morphisms being subject to compatibility relations ´ /X , if for any (m, n) ∈ Σ2 , such that for every covering family (fn : Un → U )n∈Σ of Et we set Umn = Um ×U Un and denote by fmn : Umn → U the canonical morphism, the sequence of morphisms of sheaves of Uη,f´et Y Y (I.5.3.1) FU → (fn,η )f´et∗ (FUn ) ⇒ (fmn,η )f´et∗ (FUmn ) (m,n)∈Σ2

n∈Σ

is exact. From now on, we will identify every sheaf F on E with the associated functor {U 7→ ´ f/U of π over U . FU }, the sheaf FU being the restriction of F to the fiber Et η ´ f/X → E is continuous and left exact I.5.4. The canonical injection functor Et η (VI.5.32). It therefore defines a morphism of topos e → Yf´et . β: E

(I.5.4.1) Likewise, the functor (I.5.4.2)

´ /X → E, σ + : Et

U 7→ (Uη → U )

is continuous and left exact (VI.5.32). It therefore defines a morphism of topos (I.5.4.3)

e → X´et . σ: E

I.5.5. Let x be a geometric point of X and X 0 the strict localization of X at x. e 0 the topos of sheaves of sets We denote by E 0 the Faltings site associated with X 0 , by E 0 on E , and by 0 e 0 → Xη,f´ (I.5.5.1) β0 : E et

the canonical morphism (I.5.4.1). We prove in VI.10.27 that the functor β∗0 is exact. This property is crucial for the study of the main sheaves of the Faltings topos considered in this work. The canonical morphism X 0 → X induces, by functoriality, a morphism of topos (VI.10.12) (I.5.5.2) We denote by (I.5.5.3)

the composed functor β∗0 ◦ Φ∗ .

e 0 → E. e Φ: E 0 e → Xη,f´ ϕx : E et

I.5. FALTINGS RINGED TOPOS

15

We denote by Vx the category of x-pointed étale X-schemes, or, equivalently, the ´ /X . For every object (U, ξ : x → U ) of Vx , category of neighborhoods of x in the site Et 0 we denote also by ξ : X → U the X-morphism induced by ξ. We prove in VI.10.37 that e we have a functorial canonical isomorphism for every sheaf F = {U 7→ FU } of E, ∼

ϕx (F ) → lim (ξη )∗f´et (FU ).

(I.5.5.4)

−→

U ∈V◦ x

0

Assume that x is over s. We prove (III.3.7) that X is normal and strictly local (and in particular integral). Let y be a geometric point of Xη0 (which is integral), Bπ1 (Xη0 ,y) the classifying topos of the fundamental group π1 (Xη0 , y), and ∼

0 νy : Xη,f´ et → Bπ1 (Xη0 ,y)

(I.5.5.5)

the fiber functor at y (VI.9.8.4). The composed functor (I.5.5.6)

e E

ϕx

0 / Xη,f´ et

νy

/ Bπ1 (X 0 ,y) η

/ Set,

where the last arrow is the forgetful functor of the action of π1 (Xη0 , y), is a fiber functor e (VI.10.31 and VI.9.9). It corresponds to a point of geometric origin of the topos E, denoted by ρ(y x) (cf. III.8.6). Theorem I.5.6 (cf. VI.10.30). Under the assumptions of I.5.5, for every abelian sheaf e and every integer i ≥ 0, we have a functorial canonical isomorphism (I.5.4.3) F of E (I.5.6.1)

∼

0 Ri σ∗ (F )x → Hi (Xη,f´ et , ϕx (F )).

Corollary I.5.7. We keep the assumptions of I.5.5 and moreover assume that x is over e and for every integer i ≥ 0, we have a canonical s. Then, for every abelian sheaf F of E functorial isomorphism (I.5.7.1)

∼

Ri σ∗ (F )x → Hi (π1 (Xη0 , y), νy (ϕx (F ))).

Proposition I.5.8 (cf. VI.10.32). When x goes through the set of geometric points of X, the family of functors ϕx (I.5.5.3) is conservative. I.5.9. For every object (V → U ) of E, we denote by U U = U ×S S in V and we set (I.5.9.1)

V

the integral closure of

V

B(V → U ) = Γ(U , OU V ).

We thus define a presheaf of rings on E, which turns out to be a sheaf (III.8.16). Note that B is not in general a sheaf for the topology of E originally defined by Faltings in ´ /X ), we denote by B U the restriction ([26] page 214) (cf. III.8.18). For every U ∈ Ob(Et ´ of B to the fiber Etf/Uη of π over U , so that B = {U → B U }. In I.5.10 below, we give an explicit description of this sheaf. For any integer n ≥ 0, we set (I.5.9.2)

Bn

(I.5.9.3)

B U,n

= B/pn B,

= B U /pn B U .

Note that the correspondence {U 7→ B U,n } naturally forms a presheaf on E whose associated sheaf is canonically isomorphic to B n . It is in general difficult, if not impossible, to describe explicitly the restrictions of B n to the fibers of the functor π (I.5.2.2). However, its images by the fiber functors (I.5.5.6) are accessible (III.10.8.5). We denote by ~ : X → X the canonical projection (I.2.1.1) and by (I.5.9.4)

~∗ (OX ) → σ∗ (B)

16

I. AN OVERVIEW

´ /X ) by the canonical homomorphism the homomorphism defined for every U ∈ Ob(Et (I.5.9.5)

Γ(U , OU ) → Γ(U

Uη

, OU Uη ).

e → X´et (I.5.4.3) as a morphism of Unless explicitly stated otherwise, we consider σ : E ringed topos (by B and ~∗ (OX ), respectively). ´ /X , y a geometric point of Uη , and V the conI.5.10. Let U be an object of Et nected component of Uη containing y. We denote by Bπ1 (V,y) the classifying topos of the fundamental group π1 (V, y), by (Vi )i∈I the normalized universal cover of V at y (VI.9.8), and by (I.5.10.1)

∼

νy : Vf´et → Bπ1 (V,y) ,

F 7→ lim F (Vi ) −→

i∈I ◦

the fiber functor at y. For every i ∈ I, (Vi → U ) is naturally an object of E. We can Vi therefore consider the inverse system of schemes (U )i∈I . We set (I.5.10.2)

y

Vi

RU = lim Γ(U , OU Vi ), −→

i∈I ◦

which is a ring of Bπ1 (V,y) . By III.8.15, we have a canonical isomorphism of Bπ1 (V,y) (I.5.10.3)

y

∼

νy (B U |V ) → RU .

I.5.11. Since Xη is a subobject of the final object X of X´et , σ ∗ (Xη ) is a subobject e We denote by of the final object of E. (I.5.11.1)

e/σ∗ (X ) → E e γ: E η

e at σ ∗ (Xη ). We denote by E es the closed subtopos of E e the localization morphism of E ∗ e complement of σ (Xη ), that is, the full subcategory of E made up of the sheaves F such e/σ∗ (X ) , and by that γ ∗ (F ) is a final object of E η es → E e δ: E

(I.5.11.2)

es → E e is the the canonical embedding, that is, the morphism of topos such that δ∗ : E canonical injection functor. There exists a morphism (I.5.11.3)

es → Xs,´et , σs : E

unique up to isomorphism, such that the diagram (I.5.11.4)

es E

σs

/ Xs,´et

σ

 / X´et

ι´et

δ

 e E

where ι : Xs → X is the canonical injection, is commutative up to isomorphism (cf. III.9.8). For every integer n ≥ 1, if we identify the étale topos of Xs and X n (I.2.1.1) (k being algebraically closed), the morphism σs and the homomorphism (I.5.9.4) induce a morphism of ringed topos (III.9.9) (I.5.11.5)

es , B n ) → (Xs,´et , O ). σn : ( E Xn

I.5. FALTINGS RINGED TOPOS

17

I.5.12. For the remainder of this introduction, we assume that there exists a ˇ that we fix (cf. (I.2.1.1) and I.4.2 for the notation): e of X smooth A2 (S)-deformation X (I.5.12.1)

ˇ X

/X e

 ˇ S

 / A2 (S)

´ /X admitting an étale S-morphism to Let Y = Spec(R) be a connected affine object of Et d Gm,S for an integer d ≥ 1 and such that Ys 6= ∅ (in other words, Y satisfies the conditions ˇ For any geometric e the unique étale morphism that lifts Yˇ → X. of I.4.1) and Ye → X 0 point y of Yη , we denote by Yη the connected component of Yη containing y, by ∼

0 νy : Yη,f´ et → Bπ1 (Yη0 ,y)

(I.5.12.2)

b y -extension associated with (Y, Ye ) the fiber functor at y, and by FYy the Higgs–Tate R Y (I.4.3). For every integer n ≥ 0, there exists a B Y,n -module FY,n , where B Y,n = B Y /pn B Y (I.5.9.3), unique up to canonical isomorphism, such that for every geometric point y of y Yη , we have a canonical isomorphism of RY -modules (I.5.10.3) ∼

νy (FY,n |Yη0 ) → FYy /pn FYy .

(I.5.12.3)

The exact sequence (I.4.3.5) induces a canonical exact sequence of B Y,n -modules 0 → B Y,n → FY,n → ξ −1 Ω1X/S (Y ) ⊗OX (Y ) B Y,n → 0.

(I.5.12.4)

(r)

For every rational number r ≥ 0, we denote by FY,n the extension of B Y,n -modules of Yη,f´et deduced from FY,n by inverse image under the morphism of multiplication by pr on ξ −1 Ω1X/S (Y ) ⊗OX (Y ) B Y,n , so that we have a canonical exact sequence of B Y,n modules (r)

0 → B Y,n → FY,n → ξ −1 Ω1X/S (Y ) ⊗OX (Y ) B Y,n → 0.

(I.5.12.5)

This induces, for every integer m ≥ 1, an exact sequence of B Y,n -modules (r)

(r)

m−1 0 → SB (FY,n ) → Sm B Y,n

The B Y,n -modules (Sm B (I.5.12.6)

Y,n

(FY,n ) → Sm B

Y,n

(ξ −1 Ω1X/S (Y ) ⊗OX (Y ) B Y,n ) → 0.

(r)

Y,n

(FY,n ))m∈N therefore form a direct system whose direct limit (r)

CY,n = lim Sm B −→

(r)

Y,n

(FY,n )

m≥0

is naturally endowed with a structure of B Y,n -algebra of Yη,f´et . For all rational numbers r ≥ r0 ≥ 0, we have a canonical B Y,n -linear morphism (I.5.12.7)

0

(r 0 )

(r)

r,r aY,n : FY,n → FY,n 0

that lifts the morphism of multiplication by pr −r on ξ −1 Ω1X/S (Y ) ⊗OX (Y ) B Y,n and that extends the identity on B Y,n (I.5.12.5). It induces a homomorphism of B Y,n -algebras (I.5.12.8)

0

(r)

(r 0 )

r,r αY,n : CY,n → CY,n .

(r) (r) e Note that CY,n and FY,n depend on the choice of the deformation X.

18

I. AN OVERVIEW

´ /X such that We extend the previous definitions to connected affine objects Y of Et (r) (r) Ys = ∅ by setting CY,n = FY,n = 0. I.5.13. Let n be an integer ≥ 0 and r a rational number ≥ 0. The correspondences (r) (r) {Y 7→ FY,n } and {Y 7→ CY,n } naturally form presheaves on the full subcategory of E made up of the objects (V → Y ) such that Y is affine, connected, and admits an étale morphism to Gdm,S for an integer d ≥ 1 (cf. III.10.19). Since this subcategory is clearly e topologically generating in E, we can consider the associated sheaves in E (I.5.13.1)

Fn(r)

(I.5.13.2)

Cn(r)

(r)

= {Y 7→ FY,n }a , (r)

= {Y 7→ CY,n }a .

es (III.10.22). We call Fn(r) the These are in fact a B n -module and a B n -algebra of E (r) Higgs–Tate B n -extension of thickness r and call Cn the Higgs–Tate B n -algebra of e We have a canonical exact sequence of B n -modules thickness r associated with X. (I.5.11.5) (I.5.13.3)

0 → B n → Fn(r) → σn∗ (ξ −1 Ω1X n /S n ) → 0. (r)

(r)

In III.10.29 we describe explicitly the images of Fn and Cn by the fiber functors (I.5.5.3). For all rational numbers r ≥ r0 ≥ 0, the homomorphisms (I.5.12.8) induce a homomorphism of B n -algebras 0

0

αnr,r : Cn(r) → Cn(r ) .

(I.5.13.4)

For all rational numbers r ≥ r0 ≥ r00 ≥ 0, we have 00

0

00

0

αnr,r = αnr ,r ◦ αnr,r .

(I.5.13.5) (r)

We have a canonical Cn -linear isomorphism ∼

Ω1C (r) /B → σn∗ (ξ −1 Ω1X n /S n ) ⊗Bn Cn(r) .

(I.5.13.6)

n

n

(r)

The universal B n -derivation of Cn B n -derivation

corresponds through this isomorphism to the unique

(r) ∗ −1 1 d(r) ΩX n /S n ) ⊗Bn Cn(r) n : Cn → σn (ξ

(I.5.13.7)

(r)

1 that extends the canonical morphism Fn → σn∗ (ξ −1 ΩX ). For all rational numbers n /S n 0 r ≥ r ≥ 0, we have 0

0

0

0

(r ) pr−r (id ⊗ αnr,r ) ◦ d(r) ◦ αnr,r . n = dn

(I.5.13.8)

es (I.5.4) indexed by the ordered set of I.5.14. The inverse systems of objects of E esN◦ . Section III.7 contains useful natural numbers N form a topos that we denote by E ˘ the ring (B e N◦ , by O the results for this type of topos. We denote by B ) of E n+1 n∈N

◦

N −1 1 ring (OX n+1 )n∈N of Xs,´ Ω˘ et , and by ξ

˘ X/S

−1 1 the OX ΩX ˘ -module (ξ

s

n+1 /S n+1

˘ X

◦

N )n∈N of Xs,´ et .

The morphisms (σn+1 )n∈N (I.5.11.5) induce a morphism of ringed topos ˘ → (X N◦ , O ). e N◦ , B) (I.5.14.1) σ ˘ : (E s

s,´ et

˘ X

˘ esN◦ is adic if for all integers m and n We say that a B-module (Mn+1 )n∈N of E such that m ≥ n ≥ 1, the morphism Mm ⊗Bm B n → Mn deduced from the transition morphism Mm → Mn is an isomorphism.

I.6. DOLBEAULT MODULES

19

Let r be a rational number ≥ 0. For all integers m ≥ n ≥ 1, we have a canonical (r) (r) B m -linear morphism Fm → Fn and a canonical homomorphism of B m -algebras (r) (r) Cm → Cn such that the induced morphisms (r) Fm ⊗Bm B n → Fn(r)

(I.5.14.2)

(r) and Cm ⊗Bm B n → Cn(r)

are isomorphisms. These morphisms form compatible systems when m and n vary, so (r) (r) ˘ that (Fn+1 )n∈N and (Cn+1 )n∈N are inverse systems. We call Higgs–Tate B-extension of (r) ˘ (r) N◦ ˘ e e thickness r associated with X, and denote by F , the B-module (Fn+1 )n∈N of Es . ˘ e and denote by C˘(r) , the We call Higgs–Tate B-algebra of thickness r associated with X ◦ (r) ˘ ˘ e N . These are adic B-modules. We have an exact sequence of B-algebra (C ) of E n+1 n∈N

s

˘ B-modules ˘ → F˘ (r) → σ 0→B ˘ ∗ (ξ −1 Ω1˘

(I.5.14.3)

˘ X/S

) → 0. 0

For all rational numbers r ≥ r0 ≥ 0, the homomorphisms (αnr,r )n∈N induce a homomor˘ phism of B-algebras 0 0 α ˘ r,r : C˘(r) → C˘(r ) .

(I.5.14.4)

For all rational numbers r ≥ r0 ≥ r00 ≥ 0, we have 00

0

00

0

α ˘ r,r = α ˘ r ,r ◦ α ˘ r,r .

(I.5.14.5) (r)

The derivations (dn+1 )n∈N define a morphism (I.5.14.6)

d˘(r) : C˘(r) → σ ˘ ∗ (ξ −1 Ω1˘

˘ X/S

) ⊗ ˘ C˘(r) B

˘ of C˘(r) . For all rational numbers that is none other than the universal B-derivation 0 r ≥ r ≥ 0, we have (I.5.14.7)

0

0

0

0

pr−r (id ⊗ α ˘ r,r ) ◦ d˘(r) = d˘(r ) ◦ α ˘ r,r . I.6. Dolbeault modules

I.6.1. We keep the hypotheses and notation of I.5 in this section. We set S = Spf(OC ) and denote by X the formal scheme p-adic completion of X and by ξ −1 Ω1X/S the p-adic completion of the OX -module ξ −1 Ω1X/S = ξ −1 Ω1X/S ⊗OX OX . We denote by (I.6.1.1)

◦

◦

N N u ˘ : (Xs,´ ˘ ) → (Xs,zar , OX ˘) et , OX

the canonical morphism of ringed topos (I.5.14 and III.2.9), by (I.6.1.2)

◦

N λ : (Xs,zar , OX˘ ) → (Xs,zar , OX )

the morphism of ringed topos whose corresponding direct image functor is the inverse limit (III.7.4), and by (I.6.1.3)

˘ → (X , O ) esN◦ , B) > : (E zar X

the composed morphism λ ◦ u ˘◦σ ˘ (I.5.14.1). For modules, we use the notation >−1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation >∗ for the inverse image in the sense of modules; we do likewise for σ ˘. We denote by (I.6.1.4)

˘ δ : ξ −1 Ω1X/S → R1 >∗ (B)

20

I. AN OVERVIEW

the OX -linear morphism of Xs,zar composed of the adjunction morphism (III.11.2.5) σ ∗ (ξ −1 Ω1˘ ξ −1 Ω1X/S → >∗ (˘

(I.6.1.5)

˘ X/S

))

and the boundary map of the long exact sequence of cohomology deduced from the canonical exact sequence ˘ → F˘ → σ (I.6.1.6) 0→B ˘ ∗ (ξ −1 Ω1˘ ˘ ) → 0. X/S

Note that the morphism ˘ ∗ (ξ −1 Ω1˘ >∗ (ξ −1 Ω1X/S ) → σ

(I.6.1.7)

˘ X/S

),

adjoint to (I.6.1.5), is an isomorphism (III.11.2.6). Theorem I.6.2 (cf. III.11.8). There exists a unique isomorphism of graded OX [ p1 ]algebras 1 ∼ ˘ 1] (I.6.2.1) ∧ (ξ −1 Ω1X/S [ ]) → ⊕i≥0 Ri >∗ (B)[ p p whose component in degree one is the morphism δ ⊗Zp Qp (I.6.1.4). This statement is the key step in Faltings’ approach in p-adic Hodge theory. We encounter it here and there in different forms. Its local Galois form (II.8.21) is a consequence of Faltings’ almost purity theorem (II.6.16). The global statement has an integral variant (III.11.3) that follows from the local case by localization (I.5.7). I.6.3. The canonical exact sequence (I.6.1.6) induces, for every integer m ≥ 1, an exact sequence −1 1 ˘ (I.6.3.1) 0 → Symm−1 (F˘ ) → Symm ˘ ∗ (Symm Ω ˘ ˘ )) → 0. ˘ (F ) → σ O ˘ (ξ ˘ B

B

X

X/S

Proposition I.6.4 (cf. III.11.12). Let m be an integer ≥ 1. Then: (i) The morphism 1 ˘ 1 (F˘ ))[ ] → >∗ (Symm (I.6.4.1) >∗ (Symm−1 ˘ (F ))[ ] ˘ B B p p induced by (I.6.3.1) is an isomorphism. (ii) For every integer q ≥ 1, the morphism 1 ˘ 1 (F˘ ))[ ] → Rq >∗ (Symm (I.6.4.2) Rq >∗ (Symm−1 ˘ (F ))[ ] ˘ B B p p induced by (I.6.3.1) is zero. The local Galois variant of this statement is due to Hyodo ([43] 1.2). It is the main ingredient in the definition of Hodge–Tate local systems. Proposition I.6.5 (cf. III.11.18). The canonical homomorphism 1 1 (I.6.5.1) OX [ ] → lim >∗ (C˘(r) )[ ] −→ p p r∈Q >0

is an isomorphism, and for every integer q ≥ 1, (I.6.5.2)

1 lim Rq >∗ (C˘(r) )[ ] = 0. −→ p r∈Q >0

The local Galois variant of this statement (II.12.5) is mainly due to Tsuji (IV.5.3.4).

I.6. DOLBEAULT MODULES

21

˘ ˘ the category of B-modules ˘ esN◦ , by Modad (B) of E We denote by Mod(B) ˘ the full subcategory made up of the adic B-modules ˘ ˘ (resp. Modaft (B)) (resp. the adic BI.6.6.

˘ (resp. Modad (B), ˘ resp. Modaft (B)) ˘ modules of finite type) (I.5.14), and by ModQ (B) Q Q ˘ (resp. Modad (B), ˘ resp. Modaft (B)) ˘ up to isogeny the category of objects of Mod(B) ˘ is abelian and the canonical functors (I.2.6). The category Mod (B) Q

˘ ˘ ad ˘ Modaft Q (B) → ModQ (B) → ModQ (B)

(I.6.6.1)

are fully faithful. We denote by Modcoh (OX ) (resp. Modcoh (OX [ p1 ])) the category of coherent OX -modules (resp. OX [ p1 ]-modules) of Xs,zar and by Modcoh Q (OX ) the category of coherent OX -modules up to isogeny. By III.6.16, the canonical functor 1 Modcoh (OX ) → Modcoh (OX [ ]), p

(I.6.6.2)

F 7→ FQp = F ⊗Zp Qp ,

induces an equivalence of abelian categories 1 ∼ coh (OX [ ]). Modcoh Q (OX ) → Mod p

(I.6.6.3) I.6.7.

For every rational number r ≥ 0, we denote also by d˘(r) : C˘(r) → >∗ (ξ −1 Ω1X/S ) ⊗ ˘ C˘(r)

(I.6.7.1)

B

˘ the B-derivation induced by d˘(r) (I.5.14.6) and the isomorphism (I.6.1.7), that we identify ˘ ˘ with the universal B-derivation of C˘(r) . This is a Higgs B-field with coefficients in >∗ (ξ −1 Ω1 ) (I.2.5). We denote by K• (C˘(r) , pr d˘(r) ) the Dolbeault complex of the Higgs X/S

˘ ˘ B-module (C˘(r) , pr d˘(r) ) (I.2.3) and by K•Q (C˘(r) , pr d˘(r) ) its image in ModQ (B). By 0 r,r 0 (I.5.14.7), for all rational numbers r ≥ r ≥ 0, the homomorphism α ˘ (I.5.14.4) induces a morphism of complexes (I.6.7.2)

0 0 0 0 ν˘r,r : K• (C˘(r) , pr d˘(r) ) → K• (C˘(r ) , pr d˘(r ) ).

Proposition I.6.8 (cf. III.11.24). The canonical morphism (I.6.8.1)

˘ → lim H0 (K• (C˘(r) , pr d˘(r) )) B Q Q −→

r∈Q>0

is an isomorphism, and for every integer q ≥ 1, (I.6.8.2)

lim Hq (K•Q (C˘(r) , pr d˘(r) )) = 0. −→

r∈Q>0

Observe that filtered direct limits are not a priori representable in the category ˘ ModQ (B). I.6.9. The functor >∗ (I.6.1.3) induces an additive and left exact functor that we denote also by (I.6.9.1)

˘ → Mod(O [ 1 ]). >∗ : ModQ (B) X p

For every integer q ≥ 0, we denote by (I.6.9.2)

˘ → Mod(O [ 1 ]) Rq >∗ : ModQ (B) X p

22

I. AN OVERVIEW

the qth right derived functor of >∗ . By (I.6.6.3), the inverse image functor >∗ induces an additive functor that we denote also by 1 ˘ (I.6.9.3) >∗ : Modcoh (OX [ ]) → Modaft Q (B). p ˘ -module G , we have a bifunctorial For every coherent OX [ p1 ]-module F and every B Q canonical homomorphism Hom ˘ (>∗ (F ), G ) → HomOX [ p1 ] (F , >∗ (G ))

(I.6.9.4)

BQ

that is injective (III.12.1.5). I.6.10.

We denote by HI(OX , ξ −1 Ω1X/S ) the category of Higgs OX -isogenies with

coefficients in ξ −1 Ω1X/S (I.2.8) and by HIcoh (OX , ξ −1 Ω1X/S ) the full subcategory made up of the quadruples (M , N , u, θ) such that M and N are coherent OX -modules. These −1 1 are additive categories. We denote by HIQ (OX , ξ −1 Ω1X/S ) (resp. HIcoh ΩX/S )) Q (OX , ξ the category of objects of HI(OX , ξ −1 Ω1X/S ) (resp. HIcoh (OX , ξ −1 Ω1X/S )) up to isogeny (I.2.6). By Higgs OX [ p1 ]-module with coefficients in ξ −1 Ω1X/S , we will mean a Higgs OX [ p1 ]module with coefficients in ξ −1 Ω1X/S [ p1 ] (I.2.3). We denote by HM(OX [ p1 ], ξ −1 Ω1X/S ) the category of such modules and by HMcoh (OX [ p1 ], ξ −1 Ω1X/S ) the full subcategory made up of the Higgs modules whose underlying OX [ p1 ]-module is coherent. The functor HI(OX , ξ −1 Ω1X/S ) → HM(OX [ p1 ], ξ −1 Ω1X/S ) (M , N , u, θ) 7→ (MQp , (id ⊗ u−1 Qp ) ◦ θQp )

(I.6.10.1) induces a functor

1 HIQ (OX , ξ −1 Ω1X/S ) → HM(OX [ ], ξ −1 Ω1X/S ). p By III.6.21, this induces an equivalence of categories 1 ∼ −1 1 ΩX/S ) → HMcoh (OX [ ], ξ −1 Ω1X/S ). (I.6.10.3) HIcoh Q (OX , ξ p (I.6.10.2)

Definition I.6.11. We call Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S any Higgs OX [ p1 ]-module with coefficients in ξ −1 Ω1X/S whose underlying OX [ p1 ]-module is locally projective of finite type (III.2.8). I.6.12. Let r be a rational number ≥ 0. We denote by Ξr the category of integrable ˘ (I.2.9) and by Ξr the category p -isoconnections with respect to the extension C˘(r) /B Q of objects of Ξr up to isogeny (I.2.6). We denote by Sr the functor ˘ → Ξr , M 7→ (C˘(r) ⊗ M , C˘(r) ⊗ M , id, pr d˘(r) ⊗ id). (I.6.12.1) Sr : Mod(B) r

˘ B

˘ B

This induces a functor that we denote also by ˘ → Ξr . (I.6.12.2) Sr : Mod (B) Q

We denote by K (I.6.12.3)

r

Q

the functor

˘ K r : Ξr → Mod(B),

(F , G , u, ∇) 7→ ker(∇).

This induces a functor that we denote also by (I.6.12.4)

˘ K r : ΞrQ → ModQ (B).

I.6. DOLBEAULT MODULES

23

It is clear that (I.6.12.1) is a left adjoint of (I.6.12.3). Consequently, (I.6.12.2) is a left adjoint of (I.6.12.4). If (N , N 0 , v, θ) is a Higgs OX -isogeny with coefficients in ξ −1 Ω1X/S , (I.6.12.5) (C˘(r) ⊗ ˘ >∗ (N ), C˘(r) ⊗ ˘ >∗ (N 0 ), id ⊗ ˘ >∗ (v), id ⊗ >∗ (θ) + pr d˘(r) ⊗ >∗ (v)) B

B

B

is an object of Ξr (III.6.12). We thus obtain a functor (I.6.10) (I.6.12.6)

>r+ : HI(OX , ξ −1 Ω1X/S ) → Ξr .

By (I.6.10.3), this induces a functor that we denote also by (I.6.12.7)

1 >r+ : HMcoh (OX [ ], ξ −1 Ω1X/S ) → ΞrQ . p

Let (F , G , u, ∇) be an object of Ξr . By the projection formula (III.12.4), ∇ induces an OX -linear morphism (I.6.12.8)

>∗ (∇) : >∗ (F ) → ξ −1 Ω1X/S ⊗OX >∗ (G ).

We immediately see that (>∗ (F ), >∗ (G ), >∗ (u), >∗ (∇)) is a Higgs OX -isogeny with coefficients in ξ −1 Ω1X/S . We thus obtain a functor (I.6.12.9)

>r+ : Ξr → HI(OX , ξ −1 Ω1X/S )

that is clearly a right adjoint of (I.6.12.6). The composition of the functors (I.6.12.9) and (I.6.10.1) induces a functor that we denote also by (I.6.12.10)

1 >r+ : ΞrQ → HM(OX [ ], ξ −1 Ω1X/S ). p

˘ Definition I.6.13 (cf. III.12.10). Let M be an object of Modaft Q (B) and N a Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S . (i) Let r > 0 be a rational number. We say that M and N are r-associated if there exists an isomorphism of ΞrQ (I.6.13.1)

∼

α : >r+ (N ) → Sr (M ).

We then also say that the triple (M , N , α) is r-admissible. (ii) We say that M and N are associated if there exists a rational number r > 0 such that M and N are r-associated. Note that for all rational numbers r ≥ r0 > 0, if M and N are r-associated, they are also r0 -associated. ˘ -module any object of the Definition I.6.14 (cf. III.12.11). (i) We call Dolbeault B Q ˘ for which there exists an associated Higgs O [ 1 ]-bundle with coefcategory Modaft ( B) X p Q ficients in ξ −1 Ω1X/S . (ii) We say that a Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S is solvable if it admits an associated Dolbeault module. ˘ the full subcategory of Modaft (B) ˘ made up of the We denote by ModDolb (B) Q Q ˘ -modules and by HMsol (O [ 1 ], ξ −1 Ω1 ) the full subcategory of Dolbeault B Q X p X/S HM(OX [ p1 ], ξ −1 Ω1X/S ) made up of the solvable Higgs OX [ p1 ]-bundles with coefficients in ξ −1 Ω1X/S .

24

I. AN OVERVIEW

˘ -module M and all rational numbers r ≥ r0 ≥ 0, we have a I.6.15. For every B Q canonical morphism of HM(OX [ p1 ], ξ −1 Ω1X/S ) (I.6.15.1)

0

0

>r+ (Sr (M )) → >r+ (Sr (M )).

We thus obtain a filtered direct system (>r+ (Sr (M )))r∈Q≥0 . We denote by H the functor (I.6.15.2)

˘ → HM(O [ 1 ], ξ −1 Ω1 ), H : ModQ (B) X X/S p

M 7→ lim >r+ (Sr (M )). −→

r∈Q>0

For every object N of HM(OX [ p1 ], ξ −1 Ω1X/S ) and all rational numbers r ≥ r0 ≥ 0, ˘ we have a canonical morphism of ModQ (B) (I.6.15.3)

K r (>r+ (N )) → K

r0

0

(>r + (N )).

We thus obtain a filtered direct system (K r (>r+ (N )))r≥0 . Note, however, that filtered ˘ direct limits are not a priori representable in the category Mod (B). Q

˘ -module M , H (M ) (I.6.15.2) Proposition I.6.16 (cf. III.12.18). For every Dolbeault B Q 1 is a solvable Higgs OX [ p ]-bundle associated with M . In particular, H induces a functor that we denote also by ˘ → HMsol (O [ 1 ], ξ −1 Ω1 ), M 7→ H (M ). (B) (I.6.16.1) H : ModDolb X Q X/S p Proposition I.6.17 (cf. III.12.23). We have a functor 1 ˘ (B), (I.6.17.1) V : HMsol (OX [ ], ξ −1 Ω1X/S ) → ModDolb Q p

N 7→ lim K r (>r+ (N )). −→

r∈Q>0

Moreover, for every object N of HMsol (OX [ p1 ], ξ −1 Ω1X/S ), V (N ) is associated with N . Theorem I.6.18 (cf. III.12.26). The functors (I.6.16.1) and (I.6.17.1) (I.6.18.1)

˘ o (B) ModDolb Q

H

/

V

HMsol (OX [ p1 ], ξ −1 Ω1X/S )

are equivalences of categories quasi-inverse to each other. ˘ -module and q ≥ 0 an inteTheorem I.6.19 (cf. III.12.34). Let M be a Dolbeault B Q • ger. We denote by K (H (M )) the Dolbeault complex of the Higgs OX [ p1 ]-bundle H (M ) (I.2.3). We then have a functorial canonical isomorphism of OX [ p1 ]-modules (I.6.9.2) (I.6.19.1)

∼

Rq >∗ (M ) → Hq (K• (H (M ))).

I.6.20. Let g : X 0 → X be an étale morphism. There exists essentially a unique e0 → X e that fits into a Cartesian diagram (I.2.1.1) étale morphism ge : X (I.6.20.1)

ˇ0 X ˇ g

 ˇ X

/X e0 , g e

 /X e

ˇ 0 . We associate with (X 0 , X e 0 is a smooth A2 (S)-deformation of X e 0 ) objects so that X e analogous to those defined earlier for (X, X), which we will denote by the same symbols

I.6. DOLBEAULT MODULES

25

equipped with an exponent 0 . The morphism g defines by functoriality a morphism of ringed topos (III.8.20) e B). e 0 , B 0 ) → (E, Φ : (E

(I.6.20.2)

e B) at σ ∗ (X 0 ). We prove in III.8.21 that Φ identifies with a localization morphism of (E, Furthermore, Φ induces a morphism of ringed topos ˘ ˘ 0 ) → (E esN◦ , B). ˘ : (E es0N◦ , B Φ

(I.6.20.3)

0

We denote by g : X0 → X the extension of g : X → X to the p-adic completions.

Proposition I.6.21 (cf. III.14.9). Under the assumptions of I.6.20, let moreover M ˘ -module and N a solvable Higgs O [ 1 ]-bundle with coefficients in be a Dolbeault B Q X p 0 ˘ −1 1 ∗ ∗ ˘ (M ) is a Dolbeault B -module and g (N ) is a solvable Higgs O 0 [ 1 ]ξ Ω . Then Φ X/S

X

Q

p

bundle with coefficients in ξ −1 Ω1X0 /S . If, moreover, M and N are associated, then ˘ ∗ (M ) and g∗ (N ) are associated. Φ We in fact prove that the diagrams of functors (I.6.21.1)

˘ ModDolb (B) Q ˘∗ Φ



˘ 0) ModDolb (B Q

H

V

/ HMsol (OX [ 1 ], ξ −1 Ω1 ) X/S p

˘ / ModDolb (B) Q

g∗ H0



/ HMsol (OX0 [ 1 ], ξ −1 Ω1 0 ) X /S p

V0



˘∗ Φ

/ ModDolb (B ˘ 0) Q

are commutative up to canonical isomorphisms (III.14.11). I.6.22. (I.6.22.1)

There exists a unique morphism of topos esN◦ → X´et ψ: E

´ /X ), ψ ∗ (U ) is the constant inverse system (σs∗ (Us ))N such that for every U ∈ Ob(Et ´ coh/X the full subcategory of Et ´ /X made up of étale schemes (I.5.11.3). We denote by Et of finite presentation over X. We have a canonical fibered category (I.6.22.2)

˘ → Et ´ coh/X MODQ (B)

˘ ∗ (U )) and the inverse ´ coh/X is the category ModQ (B|ψ whose fiber over an object U of Et ´ coh/X is the restriction functor (I.6.20.2) image functor under a morphism U 0 → U of Et (I.6.22.3)

˘ ∗ (U )) → Mod (B|ψ ˘ ∗ (U 0 )), ModQ (B|ψ Q

M 7→ M |ψ ∗ (U 0 ).

By I.6.21, it induces a fibered category (I.6.22.4)

˘ → Et ´ coh/X MODDolb (B) Q

˘ ∗ (U )). ´ coh/X is the category ModDolb whose fiber over an object U of Et (B|ψ Q ˘ Proposition I.6.23 (cf. III.15.4). Let M be an object of Modaft Q (B) and (Ui )i∈I a ˘ ∗ (U )) ´ coh/X . Then M is Dolbeault if and only if for every i ∈ I, the (B|ψ covering of Et i Q ∗ module M |ψ (Ui ) is Dolbeault. Proposition I.6.24 (cf. III.15.5). The following conditions are equivalent:

26

I. AN OVERVIEW

(i) The fibered category (I.6.22.4) (I.6.24.1)

˘ → Et ´ coh/X MODDolb (B) Q

is a stack ([35] II 1.2.1). ´ coh/X , denoting by U (resp. Ui , for i ∈ I) (ii) For every covering (Ui → U )i∈I of Et the formal p-adic completion of U (resp. U i ), a Higgs OU [ p1 ]-bundle N with coefficients in ξ −1 Ω1U /S is solvable if and only if for every i ∈ I, the Higgs OUi [ p1 ]-bundle N ⊗OU OUi with coefficients in ξ −1 Ω1Ui /S is solvable. Definition I.6.25 (cf. III.15.6). Let (N , θ) be Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S . (i) We say that (N , θ) is small if there exist a coherent sub-OX -module N of N 1 such that that generates it over OX [ p1 ] and a rational number ε > p−1 (I.6.25.1)

θ(N) ⊂ pε ξ −1 Ω1X/S ⊗OX N.

(ii) We say that (N , θ) is locally small if there exists an open covering (Ui )i∈I of Xs such that for every i ∈ I, (N |Ui , θ|Ui ) is small.

Proposition I.6.26 (cf. III.15.8). Every solvable Higgs OX [ p1 ]-bundle (N , θ) with coefficients in ξ −1 Ω1X/S is locally small.

Proposition I.6.27 (cf. III.15.9). Suppose that X is affine and connected, and that it admits an étale S-morphism to Gdm,S for an integer d ≥ 1. Then, every small Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S is solvable. Corollary I.6.28 (cf. III.15.10). Under the conditions of I.6.24, every locally small Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω1X/S is solvable.

CHAPTER II

Representations of the fundamental group and the torsor of deformations. Local study Ahmed Abbes and Michel Gros II.1. Introduction The current chapter is devoted to the construction and study of the p-adic Simpson correspondence, following the general approach summarized in Chapter I, for an affine logarithmic scheme of a certain type (II.6.2). Section II.2 contains the main notation and general conventions, in particular, those related to Higgs modules (II.2.8). Section II.3 contains several results on the continuous cohomology of profinite groups and, in particular, for lack of a good reference, a treatment of the Künneth formula adapted to the situation. Section II.4 recalls and details the existing relations both between torsors for the Zariski topology and principal homogeneous bundles, and between the associated equivariant notions (under an abstract group). Next, in Section II.5 we recall a few notions from logarithmic geometry that will play an important role in this work, in order to fix the notation and give reference points for readers unfamiliar with this theory. In Section II.6, we introduce the logarithmic setting (II.6.2), the rings (II.6.7), and the Galois groups (II.6.10) used throughout this chapter, and then establish some of their properties (II.6.6, II.6.8, and II.6.15). We then recall Faltings’ almost purity theorem (II.6.16) and a number of corollaries (II.6.17)–(II.6.25). Section II.7 is devoted to the Faltings extension; it contains two variants (II.7.17.2) and (II.7.22.2). For the convenience of the reader, in Section II.8, we present, in some detail, the computation of the Galois cohomology due to Faltings, which is central to his approach of p-adic Hodge theory. In Section II.9, we introduce Fontaine infinitesimal p-adic thickenings, and, following Tsuji, endow them with logarithmic structures (II.9.11). The most novel part of the chapter begins at Section II.10, with the introduction of the Higgs–Tate torsor (II.10.3). Theorem II.10.18 establishes an important link between this torsor and the Faltings extension. Section II.11 is devoted to the study of the de Rham (II.11.4) and Galois (II.11.7) cohomology of various algebras associated with this torsor. In Section II.12, we define the main functors (II.12.8.2) and (II.12.9.2) that link the category of generalized representations to that of Higgs modules. In it we also introduce the notions of Dolbeault representation and of solvable Higgs module. We in fact develop two variants, an integral one (II.12.11 and II.12.12) and a more subtle rational one (II.12.16 and II.12.18). We then show that for each variant, these notions lead to two equivalent categories (II.12.15 and II.12.24). Section II.13 is devoted to studying small representations and small Higgs modules following Faltings’ approach in [27]. We also establish links between these notions and the notions of Dolbeault representation and solvable Higgs module (II.13.20, II.13.25, and II.13.26). Section II.14 contains a descent statement for small representations due to Faltings (II.14.4). From this we deduce new links between the different notions of representations and Higgs modules introduced previously (II.14.6, 27

28

II. LOCAL STUDY

II.14.8, and II.14.16). The last section makes the link (II.15.7) with Hyodo’s theory for Hodge–Tate representations. II.2. Notation and conventions All rings in this chapter have an identity element; all ring homomorphisms map the identity element to the identity element. We mostly consider commutative rings, and rings are assumed to be commutative unless stated otherwise; in particular, when we take a ringed topos (X, A), the ring A is assumed to be commutative unless stated otherwise. II.2.1. In this chapter, p denotes a prime number, K a complete discrete valuation ring of characteristic 0, with perfect residue field k of characteristic p, and K an algebraic closure of K. We denote by OK the valuation ring of K, by OK the integral closure of OK in K, by mK the maximal ideal of OK , by k the residue field of OK , and by v the valuation of K normalized so that v(p) = 1. We denote by OC the p-adic Hausdorff completion of OK , by C its field of fractions, and by mC its maximal ideal. We choose a compatible system (βn )n>0 of nth roots of p in OK . For any rational number ε > 0, we set pε = (βn )εn , where n is a positive integer such that εn is an integer. b We denote by GK = Gal(K/K) the Galois group of K over K and by Z(1) and Zp (1) the Z[GK ]-modules (II.2.1.1)

b Z(1)

=

lim µn (OK ), ←−

n≥1

Zp (1)

(II.2.1.2)

=

lim µpn (OK ), ←−

n≥0

where µn (OK ) denotes the subgroup of nth roots of unity in OK . For any Zp [GK ]module M and any integer n, we set M (n) = M ⊗Zp Zp (1)⊗n . b its p-adic Hausdorff completion. For any abelian group A, we denote by A II.2.2. We endow Zp with the p-adic topology and do the same for every adic Zp algebra (that is, every Zp -algebra that is separated and complete for the p-adic topology). Let A be an adic Zp -algebra and i : A → A[ p1 ] the canonical homomorphism. We call padic topology on A[ p1 ] the unique topology compatible with its structure of additive group for which the subgroups i(pn A), for n ∈ N, form a fundamental system of neighborhoods of 0 ([12] III § 1.2 Prop. 1). It makes A[ p1 ] into a topological ring. Let M be an A[ p1 ]module of finite type and M ◦ a sub-A-module of M of finite type that generates it over A[ p1 ]. We call p-adic topology on M the unique topology compatible with its structure of additive group for which the subgroups pn M ◦ , for n ∈ N, form a fundamental system of neighborhoods of 0. This topology does not depend on the choice of M ◦ . Indeed, if M 0 is another sub-A-module of M of finite type that generates it over A[ p1 ], then there exists an m ≥ 0 such that pm M ◦ ⊂ M 0 and pm M 0 ⊂ M ◦ . It is clear that M is a topological A[ p1 ]-module. II.2.3. Let A be a ring and n an integer ≥ 1. We denote by W(A) (resp. Wn (A)) the ring of Witt vectors (resp. of Witt vectors of length n) with respect to p with coefficients in A. We have a ring homomorphism (II.2.3.1)

Φn :

Wn (A) → A, pn−1 pn−2 (x1 , . . . , xn ) 7→ x1 + px2 + · · · + pn−1 xn .

II.2. NOTATION AND CONVENTIONS

29

called the nth ghost component. We also have at our disposal the restriction, the shift, and the Frobenius morphisms R : Wn+1 (A) → Wn (A), V : Wn (A) → Wn+1 (A), F : Wn+1 (A) → Wn (A).

(II.2.3.2) (II.2.3.3) (II.2.3.4)

When A is of characteristic p, F induces an endomorphism of Wn (A) that we denote also by F. II.2.4. For any abelian category A, we denote by D(A) its derived category and by D− (A), D+ (A), and Db (A) the full subcategories of D(A) of complexes with cohomology bounded from above, from below, and from both sides, respectively. Unless mentioned otherwise, complexes in A have a differential of degree +1, the degree being written as an exponent. II.2.5. Let (X, A) be a ringed topos. We denote by Mod(A) or Mod(A, X) the category of A-modules of X. If M is an A-module, we denote by SA (M ) (resp. ∧A (M ), resp. ΓA (M )) the symmetric algebra (resp. the exterior algebra, resp. the divided power algebra) of M ([45] I 4.2.2.6) and for any integer n ≥ 0, by SnA (M ) (resp. ∧nA (M ), resp. ΓnA (M )) its homogeneous part of degree n. We will leave the ring A out of the notation when there is no risk of confusion. Forming these algebras commutes with localizing over an object of X. II.2.6. Let A be a ring, L an A-module, and u : L → A a linear form. For any x ∈ ∧(L), we denote by du (x) the inner product of x and u ([9] III § 11.7 Example p. 161). By (loc. cit. p. 162), we have (II.2.6.1)

du (x1 ∧ · · · ∧ xn ) =

n X (−1)i+1 u(xi )x1 ∧ · · · ∧ xi−1 ∧ xi+1 ∧ · · · ∧ xn i=1

for x1 , . . . , xn ∈ L. The map du : ∧ (L) → ∧(L) is an anti-derivation of degree −1 and square 0 ([9] III § 11.8 Example p. 165). The algebra ∧(L) endowed with the antiderivation du is called the Koszul algebra (or complex) of u. We denote it by KA • (u); we n A therefore have KA n (u) = ∧ L and the differentials of K• (u) are of degree −1 (cf. [10] § 9.1). For any complex of A-modules C, we define a chain complex ([10] § 5.1) (II.2.6.2)

A KA • (u, C) = K• (u) ⊗A C

and a cochain complex (II.2.6.3)

K•A (u, C) = HomgrA (KA • (u), C).

By ([10] § 9.1 Cor. 2 to Prop. 1), if Ann(C) is the annihilator of C, then u(L) + Ann(C) annihilates H∗ (K•A (u, C)) and H∗ (KA • (u, C)). Suppose that L is the direct sum of L1 , . . . , Lr and denote by ui : Li → A the restriction of u to Li . Then the canonical isomorphism ([9] III § 7.7 Prop. 10) (II.2.6.4)

g

∼

⊗1≤i≤r ∧ (Li ) → ∧(L) ∼

A g is an isomorphism of complexes ⊗1≤i≤r KA • (ui ) → K• (u), where the symbol ⊗ denotes the left tensor product (cf. [9] III § 4.7 Remarks p. 49). Since du is an anti-derivation, taking the product in the algebra ∧(L) induces a morphism of complexes

(II.2.6.5)

A A KA • (u) ⊗A K• (u) → K• (u).

30

II. LOCAL STUDY

Supposing L projective of rank n and composing with the canonical morphism KA • (u) → ∧n L[−n], we deduce from this a morphism of complexes A n KA • (u) → HomgrA (K• (u), ∧ L[−n])

(II.2.6.6)

that is bijective ([9] III § 7.8 p. 87). For any complex of A-modules C, we deduce from this an isomorphism of complexes ([10] § 9.1 p. 149) ∼

• n KA • (u, C) → KA (u, C ⊗A ∧ L[−n]).

(II.2.6.7)

Taking the homology, we therefore have, for every integer i, an isomorphism (II.2.6.8)

∼

n−i Hi (KA (K•A (u, C ⊗A ∧n L)). • (u, C)) → H

II.2.7. Let A be a ring, L an A-module, and u : S(L) ⊗A L → S(L) the linear form defined by u(s ⊗ x) = sx for every s ∈ S(L) and x ∈ L. The canonical isomorphism ([9] III § 7.5 Prop. 8) ∼

∧ (S(L) ⊗A L) → S(L) ⊗A ∧(L)

(II.2.7.1)

S(L)

transforms the differential of the Koszul complex K•

(u) (II.2.6) into the map

d : S(L) ⊗A ∧(L) → S(L) ⊗A ∧(L)

(II.2.7.2)

that is defined, for every x1 , . . . , xn , y1 , . . . , ym ∈ L, by (II.2.7.3) m X (−1)i+1 yi x1 . . . xn ⊗ (y1 ∧ · · · ∧ yi−1 ∧ yi+1 ∧ · · · ∧ ym ). d((x1 . . . xn ) ⊗ (y1 ∧ · · · ∧ ym )) = i=1

For any complex C of S(L)-modules, we set S(L)

S(L)

= K• (u, C), = K•S(L) (u, C)

K• (C) K•S(L) (C)

(II.2.7.4) (II.2.7.5)

(since the morphism u is canonical, it can be left out of the notation). Let L0 be an A-module and u0 : S(L0 ) ⊗A L0 → S(L0 ) the linear form such that 0 0 u (s ⊗ x0 ) = s0 x0 for s0 ∈ S(L0 ) and x0 ∈ L0 . The isomorphism (II.2.6.4) induces an isomorphism (II.2.7.6)

∼

(S(L) ⊗A Λ(L))g A (S(L0 ) ⊗A Λ(L0 )) → S(L ⊕ L0 ) ⊗A Λ(L ⊕ L0 ),

where the left exterior tensor product is taken with respect to the canonical co-Cartesian diagram / S(L) A  S(L0 )

 / S(L ⊕ L0 )

It follows from (II.2.7.3) that (II.2.7.6) is an isomorphism of complexes (II.2.7.7)

S(L)

K•

S(L0 )

(u)g A K•

∼

S(L⊕L0 )

(u0 ) → K•

(u ⊕ u0 ).

For any complex C of S(L ⊕ L0 )-modules, this leads to isomorphisms of complexes S(L⊕L0 )

∼

S(L)

(C) → K•

(II.2.7.8)

K•

(II.2.7.9)

K•S(L⊕L0 ) (C)

∼

→

S(L0 )

(K•

(C)),

K•S(L) (K•S(L0 ) (C)).

II.2. NOTATION AND CONVENTIONS

31

Definition II.2.8. Let (X, A) be a ringed topos and E an A-module. (i) We call Higgs A-module with coefficients in E a pair (M, θ) consisting of an A-module M and an A-linear morphism θ : M → M ⊗A E

(II.2.8.1)

such that θ ∧ θ = 0. We also say that θ is a Higgs A-field on M with coefficients in E. (ii) If (M1 , θ1 ) and (M2 , θ2 ) are two Higgs A-modules, a morphism from (M1 , θ1 ) to (M2 , θ2 ) is an A-linear morphism u : M1 → M2 such that (u ⊗ idE ) ◦ θ1 = θ2 ◦ u. The Higgs A-modules with coefficients in E form a category that we denote by HM(A, E). Let us complete the terminology and make a few remarks. II.2.8.2. Let (M, θ) be a Higgs A-module with coefficients in E. For each i ≥ 1, θi : M ⊗A ∧i E → M ⊗A ∧i+1 E

(II.2.8.3)

is the A-linear morphism defined, for all local sections m of M and ω of ∧i E, by θi (m ⊗ ω) = θ(m) ∧ ω. We have θi+1 ◦ θi = 0. Following Simpson ([68] p. 24), the Dolbeault complex of (M, θ), denoted by K• (M, θ), is the cochain complex of A-modules (II.2.8.4)

θ

θ

1 M −→ M ⊗A E −→ M ⊗A ∧2 E . . . ,

where M is placed in degree zero and the differentials are of degree one. II.2.8.5. Let (M, θ) be a Higgs A-module with coefficients in E with M a locally free A-module of finite type. Consider, for an integer i ≥ 1, the composition (II.2.8.6)

∧i M

∧i θ

/ ∧i (M ⊗A E)

/ ∧i M ⊗A Si E,

where the second arrow is the canonical morphism ([45] V 4.5). We call ith characteristic invariant of θ, and denote by λi (θ), the trace of the morphism (II.2.8.6) viewed as a section of Γ(X, Si E). II.2.8.7. Let (M1 , θ1 ) and (M2 , θ2 ) be two Higgs A-modules with coefficients in E. We call total Higgs field on M1 ⊗A M2 the A-linear morphism (II.2.8.8)

θtot : M1 ⊗A M2 → M1 ⊗A M2 ⊗A E

defined by (II.2.8.9)

θtot = θ1 ⊗ idM2 + idM1 ⊗ θ2 .

We call (M1 ⊗A M2 , θtot ) the tensor product of (M1 , θ1 ) and (M2 , θ2 ). II.2.8.10. Suppose that E is locally free of finite type over A; let F = H omA (E, A). Since for every A-module M , the canonical morphism (II.2.8.11)

E ndA (M ) ⊗A E → H omA (M, M ⊗A E)

is an isomorphism, giving a Higgs A-field θ on M is equivalent to giving a structure of S(F )-module on M that is compatible with its structure of A-module. On the other hand, by ([9] § 11.5 Prop. 7), the A-module H omA (∧(F ), A) can be identified with the graded dual algebra of ∧(F ) and we have a canonical isomorphism of graded algebras (II.2.8.12)

∧ (E) → H omA (∧(F ), A).

One verifies that this induces an isomorphism of complexes of A-modules (II.2.7.5) (II.2.8.13)

∼

K• (M, θ) → K•S(F ) (M ).

32

II. LOCAL STUDY

II.2.9. Let f : (X 0 , A0 ) → (X, A) be a morphism of ringed topos, E an A-module, E an A0 -module, and γ : f ∗ (E) → E 0 an A0 -linear morphism. Let (M, θ) be a Higgs A-module with coefficients in E. The composition 0

(II.2.9.1)

θ0 : f ∗ (M )

f ∗ (θ)

/ f ∗ (M ) ⊗A0 f ∗ (E)

id⊗γ

/ f ∗ (M ) ⊗A0 E 0

is then a Higgs A0 -field with coefficients in E 0 . We call the Higgs A0 -module (f ∗ (M ), θ0 ) the inverse image of (M, θ) under (f, γ). II.2.10. Let (X, A) be a ringed topos, B an A-algebra, Ω1B/A the B-module of Kähler differential forms of B over A ([45] II 1.1.2), and ΩB/A = ⊕n∈N ΩnB/A = ∧B (Ω1B/A )

(II.2.10.1)

the exterior algebra of Ω1B/A . Then there exists a unique A-anti-derivation d : ΩB/A → ΩB/A of degree one and square zero that extends the universal A-derivation d : B → Ω1B/A . This follows, for example, from ([10] § 2.10 Prop. 13) by noting that Ω1B/A is the sheaf associated with the presheaf U 7→ Ω1B(U )/A(U ) (U ∈ Ob(X)). Let M be a B-module and λ ∈ Γ(X, A). A λ-connection on M with respect to the extension B/A is an A-linear morphism ∇ : M → Ω1B/A ⊗B M

(II.2.10.2)

such that for all local sections x of B and s of M , we have ∇(xs) = λd(x) ⊗ s + x∇(s).

(II.2.10.3)

We also say that (M, ∇) is a B-module with λ-connection with respect to the extension B/A. We will leave the extension B/A out of the terminology when there is no risk of confusion. The morphism ∇ extends into a unique graded A-linear morphism ∇ : ΩB/A ⊗B M → ΩB/A ⊗B M

(II.2.10.4)

of degree one such that for all local sections ω of ΩiB/A and s of ΩjB/A ⊗B M (i, j ∈ N), we have (II.2.10.5)

∇(ω ∧ s) = λd(ω) ∧ s + (−1)i ω ∧ ∇(s).

Iterating this formula gives (II.2.10.6)

∇ ◦ ∇(ω ∧ s) = ω ∧ ∇ ◦ ∇(s).

We say that ∇ is integrable if ∇ ◦ ∇ = 0. Let (M, ∇) and (M 0 , ∇0 ) be two modules with λ-connections. A morphism from (M, ∇) to (M 0 , ∇0 ) is a B-linear morphism u : M → M 0 such that (id ⊗ u) ◦ ∇ = ∇0 ◦ u. Classically, 1-connections are called connections. Integrable 0-connections are Higgs B-fields with coefficients in Ω1B/A (II.2.8). II.2.11. Let f : (X 0 , A0 ) → (X, A) be a morphism of ringed topos, B an A-algebra, B an A0 -algebra, α : f ∗ (B) → B 0 a homomorphism of A0 -algebras, λ ∈ Γ(X, A), and (M, ∇) a module with λ-connection with respect to the extension B/A. We denote by λ0 the canonical image of λ in Γ(X 0 , A0 ), by d0 : B 0 → Ω1B 0 /A0 the universal A0 -derivation of B 0 , and by 0

(II.2.11.1)

γ : f ∗ (Ω1B/A ) → Ω1B 0 /A0

II.2. NOTATION AND CONVENTIONS

33

the canonical α-linear morphism. We immediately see that f ∗ (∇) is a λ0 -connection on f ∗ (M ) with respect to the extension f ∗ (B)/A0 . It is integrable if ∇ is. Moreover, there exists a unique A0 -linear morphism (II.2.11.2)

∇0 : B 0 ⊗f ∗ (B) f ∗ (M ) → Ω1B 0 /A0 ⊗f ∗ (B) f ∗ (M )

such that for all local sections x0 of B 0 and t of f ∗ (M ), we have (II.2.11.3)

∇0 (x0 ⊗ t) = λ0 d0 (x0 ) ⊗ t + x0 (γ ⊗ id)(f ∗ (∇)(t)).

This is a λ0 -connection on B 0 ⊗f ∗ (B) f ∗ (M ) with respect to the extension B 0 /A0 . It is integrable if ∇ is. II.2.12. Let (X, A) be a ringed topos, B an A-algebra, λ ∈ Γ(X, A), and (M, ∇) a module with λ-connection with respect to the extension B/A. Suppose that there exist ∼ an A-module E and a B-linear isomorphism γ : E ⊗A B → Ω1B/A such that for every local section ω of E, we have d(γ(ω ⊗ 1)) = 0. Let ϑ : M → E ⊗A M be the morphism induced by ∇ and γ. Then the λ-connection ∇ is integrable if and only if ϑ is a Higgs A-field on M with coefficients in E. Indeed, the diagram (II.2.12.1)

E ⊗A M

−id∧ϑ

 Ω1B/A ⊗B M

∇

/ (∧2 E) ⊗A M  / Ω2B/A ⊗B M

where the vertical arrows are the isomorphisms induced by γ, is clearly commutative. II.2.13. Let (X, A) be a ringed topos, B an A-algebra, λ ∈ Γ(X, A), and (M, ∇) a module with integrable λ-connection with respect to the extension B/A. Suppose that ∼ there exist an A-module E and a B-isomorphism γ : E ⊗A B → Ω1B/A such that for every local section ω of E, we have d(γ(ω ⊗1)) = 0 (cf. II.2.12). Let (N, θ) be a Higgs A-module with coefficients in E. There exists a unique A-linear morphism (II.2.13.1)

∇0 : M ⊗A N → Ω1B/A ⊗B M ⊗A N

such that for all local sections x of M and y of N , we have (II.2.13.2)

∇0 (x ⊗ y) = ∇(x) ⊗A y + (γ ⊗B idM ⊗A N )(x ⊗A θ(y)).

This is an integrable λ-connection on M ⊗A N with respect to the extension B/A. II.2.14. Let A be an adic ring, I an ideal of definition of A, and B an adic Aalgebra; that is, an A-algebra B that is separated and complete for the (IB)-adic topology. Recall that the canonical topology on the B-module Ω1B/A is deduced from that b1 of B ([42] 0.20.4.5). We denote by Ω the Hausdorff completion of Ω1 and by B/A

(II.2.14.1)

B/A

b1 d: B → Ω B/A

the universal continuous A-derivation of B. Let M be a B-module that is separated and complete for the (IB)-adic topology, and λ ∈ A. An adic (or I-adic) λ-connection on M with respect to the extension B/A is an A-linear morphism (II.2.14.2)

b1 ⊗ b ∇: M → Ω B/A B M

such that for all x ∈ B and t ∈ M , we have (II.2.14.3)

b + x∇(t). ∇(xt) = λd(x)⊗t

34

II. LOCAL STUDY

We also say that (M, ∇) is a B-module with adic (or I-adic) λ-connection with respect to the extension B/A. For every integer n ≥ 1, we set An = A/I n , Bn = B ⊗A An , and Mn = M ⊗A An . We denote by λn the class of λ in An . Then ∇ induces a (usual) λn -connection with respect to the extension Bn /An : ∇n : Mn → Ω1Bn /An ⊗Bn Mn .

(II.2.14.4)

Moreover, ∇ can be identified with the inverse limit of the morphisms ∇n . We say that ∇ is integrable if ∇n is integrable for every n ≥ 1. Let (M, ∇) and (M 0 , ∇0 ) be two modules with adic λ-connections. A morphism from b ◦ ∇ = ∇0 ◦ u. (M, ∇) to (M 0 , ∇0 ) is a B-linear morphism u : M → M 0 such that (id⊗u) II.2.15. Let A be an adic ring, λ ∈ A, B an adic A-algebra, B 0 an adic B-algebra, and (M, ∇) a B-module with adic λ-connection with respect to the extension B/A. We b 1 0 the universal continuous A-derivation of B 0 and by denote by d0 : B 0 → Ω B /A b1 b1 γ: Ω B/A → ΩB 0 /A

(II.2.15.1)

the canonical morphism. There exists a unique A-linear morphism b1 0 ⊗ b BM → Ω b BM (II.2.15.2) ∇0 : B 0 ⊗ B /A

0

0

such that for all x ∈ B and t ∈ M , we have b = λd0 (x0 )⊗t b + x0 (γ ⊗id b M )(∇(t)). (II.2.15.3) ∇0 (x0 ⊗t)

b B M with respect to the extension B 0 /A. It is This is an adic λ-connection on B 0 ⊗ integrable if ∇ is. II.2.16. Let A be an adic ring, I an ideal of definition of A, λ ∈ A, B an adic A-algebra, and (M, ∇) a B-module with adic λ-connection with respect to the extension B/A. For every integer n ≥ 1, we set An = A/I n and Bn = B ⊗A An . We assume that the following conditions are satisfied: (i) The ideal I is of finite type over A and Ω1B1 /A1 is a B1 -module of finite type. (ii) There exist a free A-module of finite type E and a B-linear isomorphism ∼ b1 γ : E ⊗A B → Ω B/A such that γ(E) ⊂ d(B). b 1 is the (IB)-adic topology and that we have Ω b 1 ⊗A An = Note that the topology on Ω B/A B/A b1 ⊗ b Ω1Bn /An for all n ≥ 1 ([11] III § 2.12 Cor. 1 to Prop. 14). Moreover, we have Ω B/A B M = 1 b Ω ⊗B M = E ⊗A M . Let ϑ : M → E ⊗A M be the morphism induced by ∇ and γ. B/A

Then the adic λ-connection ∇ is integrable if and only if ϑ is a Higgs A-field on M with coefficients in E. Indeed, ϑ is a Higgs A-field on M with coefficients in E if and only if for every n ≥ 1, ϑ ⊗A An is a Higgs An -field on M ⊗A An with coefficients in E ⊗A An . The desired statement now follows from II.2.12. II.2.17. Let A be an adic ring, λ ∈ A, B an adic A-algebra, and (M, ∇) a Bmodule with integrable adic λ-connection with respect to the extension B/A. Suppose that conditions (i) and (ii) of II.2.16 are satisfied. Let (N, θ) be a Higgs A-module with coefficients in E; we endow N with the topology induced by that on A. Taking the limit of II.2.13, we see that there exists a unique A-linear morphism b1 ⊗ b AN → Ω b BM⊗ b AN (II.2.17.1) ∇0 : M ⊗ B/A

such that for all x ∈ M and y ∈ N , we have b A θ(y)). b = ∇(x)⊗ b A y + (γ ⊗B idM ⊗ (II.2.17.2) ∇0 (x⊗y) b A N )(x⊗

b A N with respect to the extension B/A. This is an integrable adic λ-connection on M ⊗

II.3. RESULTS ON CONTINUOUS COHOMOLOGY OF PROFINITE GROUPS

35

II.2.18. Let X be a smooth projective complex variety. A harmonic bundle on X is a triple (M, D, h , i), where M is a complex vector bundle of class C ∞ on X, D is an integrable connection on M , and h , i is a Hermitian metric on M satisfying a condition described below ([54] § 1). There is a unique way to write D as a sum ∇ + α with ∇ a Hermitian connection and α a differential form of degree one with values in EndOX (M ) that is auto-adjoint with respect to h , i. We decompose ∇ and α using their types: (II.2.18.1)

∇ = ∂ + ∂, α = θ + θ∗ ,

where ∂ and ∂ are of type (1, 0) and (0, 1), respectively, and θ is a differential form of type (1, 0) with values in EndOX (M ). The required condition is that the operator D00 = ∂ + θ 2 is integrable (that is, D002 = 0). This condition is equivalent to saying that ∂ = 0, ∂θ = 0, and θ ∧ θ = 0. Thus the operator ∂ defines on M a structure of holomorphic vector bundle and θ is a Higgs field on M with coefficients in Ω1X/C . The Dolbeault complex (II.2.18.2)

D 00

D 00

0 → A0 (M ) −→ A1 (M ) −→ . . .

of (M, θ) is obtained by extending D00 to the differential forms of class C ∞ . By restriction to the holomorphic differential forms, it induces the complex K(M, θ); this explains the terminology. II.3. Results on continuous cohomology of profinite groups II.3.1. Let G be a profinite group and A a topological ring endowed with a continuous action of G by ring homomorphisms. An A-representation of G is an A-module M endowed with an A-semi-linear action of G; that is, an action such that for all g ∈ G, a ∈ A, and m ∈ M , we have g(am) = g(a)g(m). We say that the A-representation is continuous if M is a topological A-module and the action of G on M is continuous. Let M , N be two A-representations (resp. two continuous A-representations) of G. A morphism from M to N is an A-linear and G-equivariant (resp. A-linear, continuous, and G-equivariant) morphism from M to N . We denote by RepA (G) (resp. Repcont A (G)) the category of A-representations (resp. continuous A-representations) of G. If M and N are two A-representations of G, then the A-modules M ⊗A N and HomA (M, N ) are naturally A-representations of G. Suppose that the action of G on A is trivial. The objects of Repcont A (G) are then also called topological A-G-modules. A topological A-G-module whose topology is discrete is called a discrete A-G-module. We denote by Repdisc A (G) the full subcategory of Repcont A (G) made up of discrete A-G-modules. If R is a ring without topology, it is understood that the topological R-G-modules are defined with respect to the discrete topology on R (and the trivial action of G on R). II.3.2. Let A be a ring, G a profinite group, and M an A-module endowed with the discrete topology. The induced A-G-module of M , denoted by IndA,G (M ), is the A-module of continuous maps from G to M , endowed with the action of G defined, for all f ∈ IndA,G (M ) and g ∈ G, by (II.3.2.1)

(g · f )(x) = f (x · g).

It is a discrete A-G-module. We thus define an exact functor (II.3.2.2)

IndA,G : Mod(A) → Repdisc A (G),

M 7→ IndA,G (M )

that is a right adjoint to the forgetful functor for the action of G (V.11.1). It therefore transforms injective A-modules into injective A-G-modules. The category Repdisc A (G) has enough injective objects. An object of Repdisc A (G) is injective if and only if it is a

36

II. LOCAL STUDY

direct summand of an object of the form IndA,G (I), where I is an injective A-module (V.11.2). A discrete A-G-module is called induced if it is isomorphic to the induced A-G-module of an A-module. We denote by Γ(G, −) the left exact functor (II.3.2.3)

Γ(G, −) : Repdisc A (G) → Mod(A),

M 7→ M G ,

and by (II.3.2.4) (II.3.2.5)

+ RΓ(G, −) : D+ (Repdisc A (G)) → D (Mod(A)),

Hq (G, −) : Repdisc A (G) → Mod(A),

(q ≥ 0),

its right derived functors (cf. II.2.4). II.3.3. Let A be a ring, G a profinite group, and H a normal closed subgroup of G. Then the groups H and G/H are profinite. We again denote by Γ(H, −) the left exact functor (II.3.3.1)

disc Γ(H, −) : Repdisc A (G) → RepA (G/H),

M 7→ M H ,

and by RΓ(H, −) and Hq (H, −) (q ≥ 0) its right derived functors. This abuse of notation is justified by the fact that the diagram (II.3.3.2)

D+ (Repdisc A (G))  D+ (Repdisc A (H))

RΓ(H,−)

/ D+ (Repdisc (G/H)) A  / D+ (Mod(A))

RΓ(H,−)

where the vertical arrows are induced by the forgetful functors, is commutative up to canonical isomorphism (V.11.5). Proposition II.3.4 (cf. V.11.7). Let A be a ring, G a profinite group, H a normal closed subgroup of G, and M a discrete A-G-module. Then we have a canonical functorial isomorphism of D+ (Mod(A)) (II.3.4.1)

∼

RΓ(G/H, RΓ(H, M )) → RΓ(G, M ).

Remark II.3.5. Let A be a ring, G a profinite group, H a normal closed subgroup of G, and M a discrete A-G-module. Then for every integer n ≥ 0, the restriction (II.3.5.1)

ρn : Hn (G, M ) → Hn (H, M )

coincides with the composition (II.3.5.2) ∼ un : Hn (G, M ) → Hn (G/H, τ≤n RΓ(H, M )) → Γ(G/H, Hn (H, M )) → Hn (H, M ), where τ≤n RΓ(H, M ) is the canonical filtration of RΓ(H, M ) ([15] 1.4.6), the first arrow is induced by the isomorphism (II.3.4.1), and the other arrows are the canonical morphisms. Indeed, (ρn ) and (un ) are two morphisms of universal ∂-functors that coincide in degree zero. II.3.6. We denote by D the category whose objects are the ordered sets [n] = {0, . . . , n} (for n ∈ N) and whose morphisms are the increasing maps. We keep the notation ∆ for a Galois group that will come up in (II.6.10). For all n ∈ N and i ∈ [n], we denote by din : [n−1] → [n] the increasing injection that forgets i, and by sin : [n+1] → [n] the increasing surjection that repeats i. When there is no risk of confusion, we leave the index n out of the notation din and sin .

II.3. RESULTS ON CONTINUOUS COHOMOLOGY OF PROFINITE GROUPS

37

Let A be an additive category, q an integer ≥ 1, and X a q-cosimplicial object of A ([45] I 1.1). We call diagonal subobject of X, and denote by ∆X, the cosimplicial object of A defined by [n] 7→ X n,...,n . We call cochain complex of X the complex of q-uple e defined for all (n1 , . . . , nq ) ∈ Nq by X e n1 ,...,nq = X n1 ,...,nq and, for 1 ≤ i ≤ q, cochains X by the differential X (II.3.6.1) di = (−1)j X(id, . . . , dj , . . . , id), j

where dj is placed in ith position ([45] I 1.2.2). For every complex of q-uple cochains M R of A , we call associated simple complex the cochain complex M defined, for all n ∈ N, by Z (II.3.6.2) ( M )n = ⊕Pqi=1 ni =n M n1 ,...,nq , P P where the differential d, when restricted to M n1 ,...,nq , is given by j (−1) i0 → M the map defined, for every a ∈ Z>0 , by h(a) = −

(II.3.20.2)

a X i=1

γ a−i · f (γ i ).

2n

This factors through Z/p Z. Composing this with the surjective homomorphism G → Z/p2n Z that sends γ onto the class of 1 gives a continuous map e h : G → M such that e dγ (h) = f . Lemma II.3.21. We keep the assumptions of II.3.20 and moreover denote by ∂M : M → H1 (G, M )

(II.3.21.1)

the boundary map of the exact sequence (II.3.20.1). Then: (i) For every x ∈ M , there exists a unique continuous crossed homomorphism νx : G → M such that νx (γ) = x. (ii) For every x ∈ M , the class of νx in H1 (G, M ) is equal to −∂M (x). (i) Indeed, assume that such a continuous crossed homomorphism νx exists. Then for every integer a ≥ 1, we have (II.3.21.2)

νx (γ a ) = (γ a−1 + · · · + 1) · x. n

Let n be an integer ≥ 0 such that pn x = 0 and γ p · x = x. For every g ∈ G, we have (II.3.21.3)

2n

2n

νx (γ p g) = g · νx (γ p ) + νx (g) = νx (g).

By continuity, νx therefore factors through the canonical homomorphism G → G/γ p On the other hand, the map (II.3.21.4)

Z>0 → M,

2n

Zp

.

a 7→ (γ a−1 + · · · + 1) · x

factors through Z/p2n Z. Composing this with the surjective homomorphism G → Z/p2n Z that sends γ onto the class of 1 gives a continuous crossed homomorphism from G to M . The statement follows. (ii) For every g ∈ G, we have (II.3.20.1) (II.3.21.5)

dγ (νx )(g) = γ · νx (γ −1 g) − νx (g) = −νx (γ) = −x.

Lemma II.3.22. Let G be a profinite group isomorphic to Zp , γ a topological generator of G, A a ring with p-primary torsion, M a discrete A-G-module, and x ∈ M G . Then we have (II.3.22.1)

∂M (x) = ∂A (1) ∪ x,

where ∂M is the boundary map of the exact sequence (II.3.20.1) and ∪ is the cup product. This follows from II.3.21. Indeed, let νx : G → M and µ1 : G → A be the continuous crossed homomorphisms such that νx (γ) = x and µ1 (γ) = 1. For every integer a ≥ 1, we have νx (γ a ) = ax = µ1 (γ a )x by (II.3.21.2). It follows that νx (g) = µ1 (g)x for every g ∈ G. The lemma follows. Proposition II.3.23. Let n be an integer ≥ 1 and for every integer 1 ≤ i ≤ n, Q let Gi be n a profinite group isomorphic to Zp and γi a topological generator of Gi . Set G = i=1 Gi . Let A be a ring and M a discrete A-G-module whose elements are all p-primary torsion. By induction, we define a complex of discrete A-G-modules Ki• , for integers 0 ≤ i ≤ n,

44

II. LOCAL STUDY

as follows. We set K0• = M [0] and for every 1 ≤ i ≤ n, Ki• is the fiber of the morphism • • γi − 1 : Ki−1 → Ki−1 . Then there exists a canonical isomorphism of D+ (Mod(A)) ∼

RΓ(G, M ) → Kn• . Q Qn For every integer 0 ≤ j ≤ n, we set G≤j = 1≤i≤j Gi and G>j = jj ))

(II.3.23.1)

∼

RΓ(G≤j , M ) → Kj• .

(II.3.23.2)

The assertion is immediate for j = 0. Suppose that the assertion has been proved for every integer 0 ≤ j ≤ n − 1. We denote by IndA,Gj+1 (Kj• ) the image of the complex Kj• by the functor IndA,Gj+1 (II.3.2.2). We endow it with the action of G>j defined, for all q ∈ Z, f ∈ IndA,Gj+1 (Kjq ), g = (g0 , g1 ) ∈ G>j = Gj+1 × G>j+1 , and x ∈ Gj+1 , by (g · f )(x) = g1 · f (x · g0 ) ∈ Kjq .

(II.3.23.3)

It immediately follows from II.3.20 that we have an exact sequence of complexes of Repdisc A (G>j ) (II.3.23.4)

/ Kj•

0

εj

/ IndA,Gj+1 (Kj• )

dγj+1

/0,

/ IndA,Gj+1 (Kj• )

where for all q ∈ Z, c ∈ Kjq , f ∈ IndA,Gj+1 (Kjq ) and x ∈ Gj+1 , we have (II.3.23.5)

εqj (c)(x)

(II.3.23.6)

dqγj+1 (f )(x)

= x · c,

−1 = γj+1 f (γj+1 x) − f (x).

By the induction hypothesis, we therefore have a distinguished triangle (II.3.23.7)

/ IndA,Gj+1 (Kj• )

RΓ(G≤j , M )

dγj+1

/ IndA,Gj+1 (Kj• )

+1

/

in D+ (Repdisc A (G>j )). By II.3.2 and (II.3.3.2), for every integer q, the morphism (II.3.23.8)

Kjq → Γ(Gj+1 , IndA,Gj+1 (Kjq )),

u 7→ (x 7→ u)

induces an isomorphism of D+ (Repdisc A (G>j+1 )) (II.3.23.9)

∼

Kjq [0] → RΓ(Gj+1 , IndA,Gj+1 (Kjq )).

By virtue of II.3.4, the triangle (II.3.23.7) therefore induces a distinguished triangle in D+ (Repdisc A (G>j+1 )) (II.3.23.10)

RΓ(G≤j+1 , M )

/ Kj•

γj+1 −1

/ Kj•

+1

/

The proposition follows. Corollary II.3.24. For every integer d ≥ 1, the p-cohomological dimension of the profinite group Zdp is equal to d. For every p-primary torsion discrete Z-Zdp -module M , the associated complex Kd• in II.3.23 is concentrated in the degrees [0, d]. The p-cohomological dimension of the profinite group Zdp is therefore lesser than or equal to d by virtue of II.3.23 and ([65] I Prop. 11). For the trivial discrete Z-Zdp -module Fp , the differentials of the associated complex Kd• are zero and Hd (Zdp , Fp ) is therefore isomorphic to Fp (II.3.23.1); the corollary follows.

II.3. RESULTS ON CONTINUOUS COHOMOLOGY OF PROFINITE GROUPS

45

Corollary II.3.25. Let n be an integer ≥ 1, G a profinite group isomorphic to Znp with Zp -basis e1 , . . . , en , A a topological ring, and M a topological A-G-module. We denote by ϕ : G → AutA (M ) the representation of G over M , by SA (G) the symmetric algebra of the A-module G ⊗Z A, and by M . the SA (G)-module with underlying A-module M such that for every 1 ≤ i ≤ n, the action of ei on M is given by ϕ(ei ) − idM . Suppose that one of the following conditions is satisfied: (i) M is a p-primary torsion discrete A-G-module. (ii) M is endowed with the p-adic topology, for which it is complete and separated. Then we have a canonical isomorphism of D+ (Mod(A)) (II.3.25.1)

∼

C•cont (G, M ) → K•SA (G) (M . )

that is functorial in M , where the complex on the left is defined in (II.3.8) and that on the right is defined in (II.2.7.5). Case (i) follows from II.3.23 and (II.2.7.9). Let us consider case (ii). For every r ≥ 0, we set Mr = M/pr M . By virtue of (II.3.10.5), we have a canonical isomorphism of D+ (Mod(A)) (II.3.25.2)

∼

C•cont (G, M ) → R+ Γ(G, (Mr )r∈N ).

On the other hand, by case (i), we have a compatible system of isomorphisms (II.3.25.3)

∼

RΓ(G, Mr ) → K•SA (G) (M . /pr M . )

of D+ (Mod(A)). Since for every integer n, the inverse system (KnSA (G) (M . /pr M . ))r≥0 satisfies the Mittag–Leffler condition, in view of (II.3.9.4) and (II.3.10.2), we obtain an isomorphism (II.3.25.4)

∼

C•cont (G, M ) → lim K•SA (G) (M . /pr M . ). ←− r≥0

The corollary follows because for every integer n, KnSA (G) (M . ) is complete and separated for the p-adic topology. Remark II.3.26. We keep the assumptions of II.3.25. By II.2.8.10, the SA (G)-module M . corresponds to a Higgs A-field θ on M with coefficients in HomZ (G, A), and the complex K•SA (G) (M . ) can be identified with the Dolbeault complex of (M, θ). II.3.27. Let G be a profinite group isomorphic to Zp , γ a topological generator of G, H a profinite group, A a p-primary torsion ring, and M a discrete A-H-module that we also view as a discrete A-(G × H)-module through the canonical projection G × H → H. We denote by IndA,G (C• (H, M )) the image of the complex of continuous cochains C• (H, M ) (II.3.8) by the functor IndA,G (II.3.2.2). We endow C• (H, M ) with the trivial action of G. By II.3.20, we have an exact sequence of complexes of Repdisc A (G) (II.3.27.1)

d• γ

ε•

0 −→ C• (H, M ) −→ IndA,G (C• (H, M )) −→ IndA,G (C• (H, M )) −→ 0,

where for all q ∈ Z, c ∈ Cq (H, M ), f ∈ IndA,G (Cq (H, M )) and x ∈ G, we have (II.3.27.2)

εq (c)(x)

= x,

(II.3.27.3)

dqγ (f )(x)

= γf (γ −1 x) − f (x).

For every integer q ≥ 0, εq induces an isomorphism of D+ (Mod(A)) (II.3.27.4)

∼

Cq (H, M )[0] → RΓ(G, IndA,G (Cq (H, M ))).

46

II. LOCAL STUDY

By virtue of II.3.4, (II.3.27.1) therefore induces a distinguished triangle in D+ (Mod(A)) / C• (H, M )

RΓ(G × H, M )

(II.3.27.5)

/ C• (H, M )

0

+1

/

where 0 is the zero morphism. From this we deduce, for every integer n ≥ 0, an exact sequence βn

α

n 0 −→ Hn−1 (H, M ) −→ Hn (G × H, M ) −→ Hn (H, M ) −→ 0.

(II.3.27.6)

Moreover, again by II.3.20, we have a canonical exact sequence (II.3.20.1) 0 → A → IndA,G (A) → IndA,G (A) → 0.

(II.3.27.7)

It induces an isomorphism ∼

∂A : A → H1 (G, A).

(II.3.27.8)

Proposition II.3.28. Under the assumptions of II.3.27, with n an integer ≥ 0, we have: (i) βn is the restriction morphism with respect to the canonical injection H → G × H. (ii) For every x ∈ Hn−1 (H, M ), we have αn (x) = ∂A (1) × x,

(II.3.28.1)

where the cross product is defined in (II.3.12.6). We denote by τ≤n C• (H, M ) the canonical filtration of C• (H, M ) ([15] 1.4.6). By II.3.20, we have a commutative diagram of complexes of Repdisc A (G) (II.3.28.2) 0

/ τ≤n C• (H, M ) un

0

 / Hn (H, M )[−n]

ψn

/ IndA,G (τ≤n C• (H, M ))

φn

IndA,G (un )

 / IndA,G (Hn (H, M ))[−n]

/ IndA,G (τ≤n C• (H, M ))

/0

IndA,G (un )

 / IndA,G (Hn (H, M ))[−n]

/0

where un is the canonical morphism and the horizontal lines are the exact sequences defined as in (II.3.27.1). In fact, since the functor IndA,G is exact, the top horizontal line can be deduced from the exact sequence (II.3.27.1) by applying the functor τ≤n . (i) We clearly have βn = Hn (G, IndA,G (un ) ◦ ψn ). In view of (II.3.28.2), we deduce from this that βn = Hn (G, un ). The statement therefore follows from II.3.5. (ii) It follows by induction from II.3.24 that for every integer i ≥ 2, we have (II.3.28.3)

Hn (G, τ≤n−i C• (H, M )) = 0.

Consequently, the canonical morphism (II.3.28.4)

Hn (G, τ≤n−1 C• (H, M )) → H1 (G, Hn−1 (H, M ))

is an isomorphism. In view of (II.3.4.1), we deduce from this a canonical morphism (II.3.28.5)

vn : H1 (G, Hn−1 (H, M )) → Hn (G × H, M ).

Let us moreover consider the commutative diagram (II.3.28.6)

IndA,G (τ≤n−1 C• (H, M ))

δ≤n−1

IndA,G (un−1 )

 IndA,G (Hn−1 (H, M ))[−n + 1]

/ τ≤n−1 C• (H, M )[1] un−1 [1]

δn−1

 / Hn−1 (H, M )[−n + 2]

II.3. RESULTS ON CONTINUOUS COHOMOLOGY OF PROFINITE GROUPS

47

induced by the diagram (II.3.28.2) (for n − 1 instead of n). Let (II.3.28.7)

∂n−1 = Hn−1 (G, δn−1 ) : Hn−1 (H, M ) → H1 (G, Hn−1 (H, M )),

which is in fact an isomorphism. Since (II.3.28.4) is an isomorphism, we have (II.3.28.8)

αn = vn ◦ Hn−1 (G, un−1 [1] ◦ δ≤n−1 ) = vn ◦ ∂n−1 .

By II.3.22, for every x ∈ Hn−1 (H, M ), we have (II.3.28.9)

∂n−1 (x) = ∂A (1) ∪ x.

It therefore suffices to show that the diagram (II.3.28.10)

H1 (G, A) ⊗A Hn−1 (H, M ) TTTT TTTT× TTTT ∪ TTTT  * v n / Hn (G × H, M ) H1 (G, Hn−1 (H, M ))

where the cup product on the left is defined in (II.3.13.3) and the cross product on the right is defined in (II.3.12.6), is commutative. By II.3.25, we have a canonical isomorphism (II.3.28.11)

∼

∂Zp : Zp → H1 (G, Zp )

that is compatible with the isomorphism ∂A (II.3.27.8) through the canonical homomorphism Zp → A. It therefore also suffices to show that the diagram (II.3.28.12)

H1 (G, Zp ) ⊗Zp Hn−1 (H, M ) TTTT TTTT× TTTT ∪ TTTT  * vn 1 n−1 / Hn (G × H, M ) H (G, H (H, M ))

is commutative. By II.3.19, we have a canonical isomorphism of D+ (Mod(A)) (II.3.28.13)

∼

RΓ(G, τ≤n−1 C• (H, M )) → C• (G, Zp ) ⊗Zp τ≤n−1 C• (H, M ),

and in view of (II.3.28.4), the morphism vn can be obtained by applying the functor Hn to the morphism (II.3.28.14)

C• (G, Zp ) ⊗Zp τ≤n−1 C• (H, M ) → C• (G × H, M )

deduced from (II.3.12.5). The commutativity of (II.3.28.12) follows. Remark II.3.29. Proposition II.3.28 gives a “Künneth formula” for the cohomology of products of profinite groups with values in certain discrete modules that we have not been able to find in the literature. We have only treated a simple case, where one of the groups is Zp . This is the only case we will need in this work. For the general case, [48] contains a weaker statement that does not study the compatibility with the cross product. Proposition II.3.30. Let A be a Zp -algebra that is complete and separated for the padic topology and d an integer ≥ 1. Then there exists a unique isomorphism of graded A-algebras (II.3.30.1)

∼

∧ (H1cont (Zdp , A)) → ⊕n≥0 Hncont (Zdp , A),

where the right-hand side is endowed with the cup product, which is the identity in degree one.

48

II. LOCAL STUDY

Let us first consider the case where A is p-primary torsion. We proceed by induction on d. The case d = 1 is immediate in view of II.3.24 and the canonical isomorphism ∼ A → H1cont (Zp , A). Suppose d ≥ 2 and that the statement has been proved for d − 1. Let G = Zp and H = Zd−1 . It follows from II.3.28 and II.3.14 that the cross product p (II.3.12.6) induces an isomorphism of graded A-algebras (II.3.30.2)

∼

(A ⊕ H1 (G, A))g ⊗A (⊕n≥0 Hn (H, A)) → ⊕n≥0 Hn (G × H, A),

where the symbol g ⊗A denotes the left tensor product (cf. [9] III § 4.7 Remarks p. 49). The desired statement now follows using the induction hypothesis. Also note that we ∼ have a canonical isomorphism Ad → H1cont (Zdp , A). Let us now consider the general case. It follows from the above that for every integer n ≥ 0, the inverse system (Hn (Zdp , A/pr A))r∈N satisfies the Mittag–Leffler condition. By virtue of (II.3.10.2), (II.3.10.4), and (II.3.10.5), the canonical morphism (II.3.30.3)

Hncont (Zdp , A) → lim Hn (Zdp , A/pr A) ←− r≥0

is therefore an isomorphism. The proposition then follows from the earlier case. II.3.31. Let G be a profinite group. A G-set is a discrete topological space endowed with a continuous action of G. The G-sets naturally form a category. A G-group M is a group in this category. We associate with it its subgroup of G-invariants H0 (G, M ) = M G and its first cohomology set H1 (G, M ); we refer to ([65] I § 5) for the definition and main properties of this pointed set. II.3.32. Let G be a profinite group, A a ring endowed with the discrete topology and a continuous action of G, M a free A-module of rank r ≥ 1 endowed with the discrete topology, and (e1 , . . . , er ) a basis of M over A. We denote by Matr (A) the A-algebra of square r by r matrices with coefficients in A and by GLr (A) the group of invertible elements of Matr (A). Note that GLr (A) is naturally a G-group. Giving a continuous A-representation ρ of G over M is equivalent to giving, for every g ∈ G, an element Ug of GLr (A) such that the map g 7→ Ug is continuous and that for every g, h ∈ G, we have (II.3.32.1)

Ugh = Ug · g Uh .

The matrix Ug gives the coordinates of the vectors e1 , . . . , er in the basis g(e1 ), . . . , g(er ). Changing the basis (ei )1≤i≤r changes the cocycle g 7→ Ug into a cohomologous cocycle. The map that sends ρ to the class [ρ] of the cocycle g 7→ Ug in H1 (G, GLr (A)) is a bijection from the set of isomorphism classes of continuous A-representations of G over M to the set H1 (G, GLr (A)). The A-representation of G over M that fixes the (ei )1≤i≤r corresponds to the neutral element of H1 (G, GLr (A)). Let a ∈ AG be such that a is nilpotent in A and ρ an A-representation of G over M such that ρ(g)(ei ) − ei ∈ aM for all g ∈ G and 1 ≤ i ≤ r. Then the cocycle g 7→ Ug defined above takes on values in the subgroup idr + aMatr (A) of GLr (A). If we change the basis (ei )1≤i≤r to a basis (e0i )1≤i≤r such that e0i − ei ∈ aM for every 1 ≤ i ≤ r, then the cocycle g 7→ Ug transforms into a cohomologous cocycle in idr + aMatr (A). The map that sends ρ to the class [ρ] of the cocycle g 7→ Ug in H1 (G, idr + aMatr (A)) is a bijection from the set of isomorphism classes of continuous A-representations of G over M that fix ei modulo aM for every 1 ≤ i ≤ r, modulo the isomorphisms that fix ei modulo aM for every 1 ≤ i ≤ r, to the set H1 (G, idr + aMatr (A)).

II.3. RESULTS ON CONTINUOUS COHOMOLOGY OF PROFINITE GROUPS

49

II.3.33. Let G be a profinite group, A a ring endowed with an action of G, a ∈ AG , and m, n, q, r integers ≥ 1 such that q ≥ n ≥ m and m + n ≥ q. Suppose that the action of G on A is continuous for the a-adic topology and that the multiplication by an in A induces an isomorphism ∼

A/aq−n A → an A/aq A.

(II.3.33.1)

The second condition is satisfied, for example, if a is not a zero divisor in A. Consider the canonical exact sequence of G-groups (II.3.33.2) 1 → idr + an Matr (A/aq A) → idr + am Matr (A/aq A) → idr + am Matr (A/an A) → 1.

Then idr + an Matr (A/aq A) is contained in the center of idr + am Matr (A/aq A). On the other hand, by (II.3.33.1), we have a canonical isomorphism of abelian G-groups ∼

Matr (A/aq−n A) → idr + an Matr (A/aq A).

(II.3.33.3)

By ([65] I prop. 43), we have a canonical exact sequence of pointed sets (II.3.33.4) H1 (G, Matr (A/aq−n A)) −→ H1 (G, idr + am Matr (A/aq A)) −→ ∂

H1 (G, idr + am Matr (A/an A)) −→ H2 (G, Matr (A/aq−n A)).

II.3.34. We keep the assumptions of II.3.33 and let N be a free (A/aq A)-module of rank r with basis f1 , . . . , fr and ρ, ρ0 two continuous representations of G over N (endowed with the discrete topology) such that ρ(g)(fi ) − fi ∈ am N and ρ0 (g)(fi ) − fi ∈ am N for all g ∈ G and 1 ≤ i ≤ r. We denote by g 7→ Vg and g 7→ Vg0 the cocycles of G with values in idr + am Matr (A/aq A) associated with N and N 0 , respectively, by the choice of the basis (fi )1≤i≤d . By virtue of (II.3.33.4) and ([65] I Prop. 42), the following conditions are equivalent: (i) There exists an A-linear automorphism ∼

u : N/an N → N/an N

(II.3.34.1)

such that u ◦ ρ(g) = ρ0 (g) ◦ u for every g ∈ G and u(fi ) − fi ∈ am N for every 1 ≤ i ≤ r. (ii) There exist a cocycle g 7→ Wg of G with values in Matr (A/aq−n A) and a matrix U ∈ idr + am Matr (A/aq A) such that for every g ∈ G, we have (II.3.33.3) Vg0 = U −1 (idr + an Wg )Vg g U .

(II.3.34.2)

If these conditions hold, we will say that ρ0 is deduced from ρ by twisting by the cocycle g 7→ Wg . Two cohomologous cocycles define representations that are isomorphic by an isomorphism that is compatible with (II.3.34.1). We will then also say that ρ0 is deduced from ρ by twisting by the class c ∈ H1 (G, Matr (A/aq−n A)) of the cocycle g 7→ Wg . Let n0 , m0 , q 0 be integers ≥ 1 such that n ≥ n0 ≥ m0 , m ≥ m0 ≥ q − n, and 0 0 q = q − n + n0 . Suppose that multiplication by an in A induces an isomorphism 0

∼

0

0

Then we have a commutative diagram (II.3.34.4) / idr + am Matr (A/aq A) Matr (A/aq−n A) ·an−n

0

A/aq −n A → an A/aq A.

(II.3.34.3)

0

 0 0 Matr (A/aq −n A)

α

 / idr + am0 Matr (A/aq0 A)

/ idr + am Matr (A/an A) β

 / idr + am0 Matr (A/an0 A)

50

II. LOCAL STUDY

where the lines correspond to the exact sequences (II.3.33.2) and α and β are induced by 0 0 the reduction homomorphisms A/aq A → A/aq A and A/an A → A/an A. Consequently, 0 1 q−n if ρ is deduced from ρ by twisting by a class c ∈ H (G, Matr (A/a A)), the repre0 0 sentation over N/aq N induced by ρ0 is obtained from the representation over N/aq N induced by ρ by twisting by the class 0

0

0

an−n · c ∈ H1 (G, Matr (A/aq −n A)).

(II.3.34.5)

II.3.35. We keep the assumptions of II.3.33 and let M be a free (A/an A)-module of rank r with basis e1 , . . . , er and ρ a continuous representation of G over M (endowed with the discrete topology) such that ρ(g)(ei ) − ei ∈ am M for every g ∈ G and every 1 ≤ i ≤ r. We denote by g 7→ Ug the cocycle of G with values in idr + am Matr (A/an A) associated with M through the choice of the basis (ei )1≤i≤d , and by [M ] its class in H1 (G, idr + am Matr (A/an A)). By virtue of (II.3.33.4) and ([65] I prop. 41), the class ∂([M ]) ∈ H2 (G, Matr (A/aq−n A))

(II.3.35.1)

is the obstruction to lifting M to a continuous (A/aq A)-representation of G with underlying (A/aq A)-module that is free of finite type. Let n0 be an integer such that n ≥ n0 ≥ m, q 0 = q −n+n0 and that the multiplication 0 by an in A induces an isomorphism 0

We denote by (II.3.35.3)

0

∼

0

0

A/aq −n A → an A/aq A.

(II.3.35.2)

0

0

0

∂ 0 : H1 (G, idr + am Matr (A/an A)) → H2 (G, EndA (M/aq −n M ))

the boundary map defined as in (II.3.33.4). By functoriality, we have (II.3.35.4)

0

0

∂ 0 ([M/an M ]) = an−n ∂([M ]). II.4. Objects with group actions

II.4.1. Let Sch be the category of schemes. We choose a normalized cleavage of the category of arrows of Sch ([37] VI § 11); in other words, for every morphism of schemes f : X 0 → X, we choose a base change functor (II.4.1.1)

f • : Sch/X → Sch/X 0 ,

Y 7→ Y ×X X 0

such that for every scheme X, f = idX implies f • = idSch/X . For any scheme X, we denote by Xzar the Zariski topos of X and by (II.4.1.2) the canonical functor.

ϕX : Sch/X → Xzar ,

Y 7→ HomX (−, Y )

Remark II.4.2. If f : X 0 → X is a morphism of schemes, the diagram (II.4.2.1)

Sch/X

ϕX

f•

 Sch/X 0

ϕX 0

/ Xzar 

f∗

0 / Xzar

where the horizontal arrows are the canonical functors (II.4.1.2), is not commutative in general. Nevertheless, we have a canonical morphism of functors (II.4.2.2)

f ∗ ◦ ϕX → ϕX 0 ◦ f • .

If, moreover, f is an open immersion, then the morphism (II.4.2.2) is an isomorphism. Indeed, every open subscheme of X 0 is an open subscheme of X and f ∗ is the restriction

II.4. OBJECTS WITH GROUP ACTIONS

51

functor. In this case, the diagram (II.4.2.1) is therefore commutative, up to canonical isomorphism. II.4.3. Let X be a scheme and G an X-group scheme. In this chapter, a principal homogeneous G-bundle over X will mean a (right) G-pseudo-torsor of Sch/X that is locally trivial for the Zariski topology on X ([35] III 1.1.5). We denote by PHB(G/X) the category of principal homogeneous G-bundles over X and by Tors(ϕX (G), Xzar ) the category of (right) ϕX (G)-torsors of Xzar ([35] III 1.4.1). The functor ϕX (II.4.1.2) induces a functor that we denote also by ϕX : PHB(G/X) → Tors(ϕX (G), Xzar ),

(II.4.3.1)

Y 7→ HomX (−, Y ).

Proposition II.4.4. Let X be a scheme and G an X-group scheme. Then: (i) The functor (II.4.3.1) is fully faithful. (ii) If, moreover, X is coherent (that is, quasi-compact and quasi-separated) and G is affine over X, then the functor (II.4.3.1) is an equivalence of categories. (i) Let Y, Z be two objects of PHB(G/X). We denote by HomG (Y, Z) the set of morphisms from Y to Z in PHB(G/X) and by HomϕX (G) (ϕX (Y ), ϕX (Z)) the set of morphisms from ϕX (Y ) to ϕX (Z) in Tors(ϕX (G), Xzar ). Let us show that the map HomG (Y, Z) → HomϕX (G) (ϕX (Y ), ϕX (Z))

(II.4.4.1)

induced by the functor ϕX (II.4.3.1) is bijective. Let (Ui )i∈I be a Zariski open covering of X. For every i ∈ I, we set Gi = G ×X Ui , Yi = Y ×X Ui , and Zi = Z ×X Ui . For every (i, j) ∈ I 2 , we set Uij = Ui ∩ Uj , Gij = G ×X Uij , Yij = Y ×X Uij , and Zij = Z ×X Uij . We then have a commutative diagram of maps of sets (II.4.4.2) / HomϕX (G) (ϕX (Y ), ϕX (Z)) HomG (Y, Z)  Hom Gi (Yi , Zi ) i∈I /

Q

Q

(i,j)∈I 2

  HomGij (Yij , Zij ) /

 Hom (ϕ Xi (Yi ), ϕXi (Zi )) ϕ (G ) i X i∈I i

Q

Q

(i,j)∈I 2

  HomϕXij (Gij ) (ϕXij (Yij ), ϕXij (Zij ))

where the horizontal arrows are induced by the functor (II.4.3.1) and the vertical arrows are defined by restriction (II.4.2). The vertical columns are clearly exact, so we may restrict to the case Y = G. We then identify the source and the target of the map (II.4.4.1) with the set HomX (X, Z), giving the desired statement. (ii) It suffices to show that the functor in question is essentially surjective. Let F be an object of Tors(ϕX (G), Xzar ) and (Ui )i∈I a covering of X by affine open sets such that for every i ∈ I, F |Ui is trivial. Let X 0 = ti∈I Ui , X 00 = X 0 ×X X 0 , G0 = G ×X X 0 , G00 = G ×X X 00 , and denote by f : X 0 → X the canonical morphism and by pr1 , pr2 : X 00 → X 0 the canonical projections. Since X is quasi-compact, we may assume that I is finite. Since X is quasi-separated, f is then quasi-compact. We have an ∼ 0 isomorphism ϕX 0 (G0 ) → f ∗ (F ) of Tors(ϕX 0 (G0 ), Xzar ). The canonical descent datum on f ∗ (F ) induces a descent datum on ϕX 0 (G0 ) with respect to f (as a ϕX 0 (G0 )-torsor of 0 00 Xzar ); that is, an isomorphism of Tors(ϕX 00 (G00 ), Xzar ) (II.4.4.3)

∼

ψ : pr∗1 (ϕX 0 (G0 )) → pr∗2 (ϕX 0 (G0 ))

that satisfies a cocycle relation. Note that pr∗1 (ϕX 0 (G0 )) and pr∗2 (ϕX 0 (G0 )) are canonically isomorphic to ϕX 00 (G00 ) (II.4.2) and that ψ is in general different from the trivial

52

II. LOCAL STUDY

descent datum (induced by ϕX (G)). In view of statement (i), ψ induces a descent datum on G0 with respect to f (as a principal homogeneous G0 -bundle over X 0 ); that is, an isomorphism of PHB(G00 /X 00 ) (II.4.4.4)

∼

ϕ : pr•1 (G0 ) → pr•2 (G0 )

satisfying a cocycle relation. By virtue of ([37] VIII 2.1), there exists a principal homogeneous G-bundle Y over X that corresponds to ϕ. Since the G-torsor ϕX (Y ) of ∼ Xzar corresponds to the descent datum ψ, there exists an isomorphism ϕX (Y ) → F of Tors(ϕX (G), Xzar ), giving the desired statement. II.4.5. Let f : X 0 → X be a morphism of schemes, T an OX -module, and L a T-torsor of Xzar . From now on, for OX -modules, we will use the notation f −1 to denote the inverse image in the sense of abelian sheaves and will keep the notation f ∗ for the inverse image in the sense of modules. The affine inverse image of L under f , denoted by 0 f + (L ), is the f ∗ (T)-torsor of Xzar deduced from the f −1 (T)-torsor f ∗ (L ) by extending its structural group by the canonical homomorphism f −1 (T) → f ∗ (T), (II.4.5.1)

f + (L ) = f ∗ (L ) ∧f

−1

(T)

f ∗ (T);

in other words, the quotient of f ∗ (L ) × f ∗ (T) by the diagonal action of f −1 (T) ([35] III 1.4.6). Let T0 be an OX -module, L 0 a T0 -torsor of Xzar , u : T → T0 an OX -linear morphism, and v : L → L 0 a u-equivariant morphism of Xzar . By ([35] III 1.3.6), there exists a unique f ∗ (u)-equivariant morphism (II.4.5.2)

f + (v) : f + (L ) → f + (L 0 )

that fits into the commutative diagram (II.4.5.3)

f ∗ (L )  f + (L )

f ∗ (v)

f + (v)

/ f ∗ (L 0 )  / f + (L 0 )

where the vertical arrows are the canonical morphisms. The resulting correspondence (T, L ) 7→ (f ∗ (T), f + (L )) is a functor from the category of torsors of Xzar under an 0 OX -module to the category of torsors of Xzar under an OX 0 -module. 00 0 Let g : X → X be a morphism of schemes and h = f ◦ g : X 00 → X. We have a canonical isomorphism of functors ∼

h∗ → g ∗ ◦ f ∗ .

(II.4.5.4)

Since g ∗ commutes with direct limits, we have a canonical isomorphism (II.4.5.5)

∼

−1

g ∗ (f + (L )) → g ∗ (f ∗ (L )) ∧h

(T)

g −1 (f ∗ (T)).

In view of ([35] III 1.3.5), this induces a canonical h∗ (T)-equivariant isomorphism (II.4.5.6)

∼

g + (f + (L )) → h+ (L ).

One immediately verifies that this isomorphism is functorial and that it satisfies a cocycle relation of the type ([37] VI 7.4 B)) for the composition of three morphisms of schemes.

II.4. OBJECTS WITH GROUP ACTIONS

53

II.4.6. Let f : X 0 → X be a morphism of schemes, T a locally free OX -module of finite type, Ω = H omOX (T, OX ) its dual, T = Spec(SOX (Ω)) the corresponding X-bundle, and L a principal homogeneous T-bundle over X. We have a commutative 0 diagram of group morphisms of Xzar f −1 (T)

(II.4.6.1)

 f ∗ (T)

∼

/ f ∗ (ϕX (T))  / ϕX 0 (T ×X X 0 )

∼

where the vertical arrow on the right is the morphism (II.4.2.2) and the other arrows are the canonical morphisms. Moreover, we have an f −1 (T)-equivariant canonical morphism (II.4.2.2) f ∗ (ϕX (L)) → ϕX 0 (L ×X X 0 ).

(II.4.6.2)

By ([35] III 1.3.6), this induces an isomorphim of f ∗ (T)-torsors ∼

f + (ϕX (L)) → ϕX 0 (L ×X X 0 ).

(II.4.6.3)

II.4.7. Let X be a scheme, T a locally free OX -module of finite type, Ω = H omOX (T, OX ) its dual, and L a T-torsor of Xzar . An affine function on L is a morphism f : L → OX of Xzar satisfying the following equivalent conditions: (i) For every open subscheme U of X and every s ∈ L (U ), the map T(U ) → OX (U ),

(II.4.7.1)

t 7→ f (s + t) − f (s)

is OX (U )-linear. (ii) There exists a section ω ∈ Ω(X), called the linear term of f , such that for every open subscheme U of X and all s ∈ L (U ) and t ∈ T(U ), we have f (s + t) = f (s) + ω(t).

(II.4.7.2)

We then also say that the morphism f is affine. Indeed, (ii) clearly implies (i). Conversely, assume that condition (i) is satisfied. Let (Ui )i∈I be an open covering of X such that for every i ∈ I, there exists an si ∈ L (Ui ). Then there exist ωi ∈ Ω(Ui ) such that for every open subscheme U of Ui and every t ∈ T(U ), we have f (si + t) = f (si ) + ωi (t) ∈ OX (U ).

(II.4.7.3)

For every (i, j) ∈ I 2 , let tij ∈ T(Ui ∩ Uj ) be such that si = sj + tij ∈ L (Ui ∩ Uj ). For every open subscheme U of Ui ∩ Uj and every t ∈ T(U ), we have f (si ) + ωi (t) (II.4.7.4)

= f (si + t) = f (sj + tij + t) = f (sj ) + ωj (tij ) + ωj (t) = f (si ) + ωj (t) ∈ OX (U ).

We therefore have ωi |Ui ∩ Uj = ωj |Ui ∩ Uj . Consequently, the sections (ωi )i∈I glue to define a section ω ∈ Ω(X). Let U be an open subscheme X, s ∈ L (U ), t ∈ T(U ). Let us show that f (s + t) = f (s) + ω(t). We may assume that there exists i ∈ I such that U ⊂ Ui . Let t0 ∈ T(U ) be such that s = si + t0 . Then we have (II.4.7.5)

f (s + t) = f (si + t0 + t) = f (si ) + ω(t0 ) + ω(t) = f (s) + ω(t).

Remark II.4.8. Condition II.4.7(ii) corresponds to saying that there exists ω ∈ Ω(X) such that f is T-equivariant when we endow OX with the structure of T-object defined by ω, or more precisely, by the morphism (II.4.8.1)

T × OX → OX ,

(t, x) 7→ ω(t) + x.

54

II. LOCAL STUDY

II.4.9. Let X be a scheme, T a locally free OX -module of finite type, Ω = H omOX (T, OX ) its dual, and L a T-torsor of Xzar . Condition II.4.7(i) is clearly local for the Zariski topology on X, so we denote by F the subsheaf of H omXzar (L , OX ) consisting of affine functions on L ; in other words, for every open subscheme U of X, F (U ) is the set of affine functions on L |U . We call F the sheaf of affine functions on L . It is naturally endowed with a structure of OX -module. We have a canonical OX -linear morphism c : OX → F whose image consists of the constant functions. The “linear term” defines an OX -linear morphism ν : F → Ω. One verifies that the sequence (II.4.9.1)

c

ν

0 −→ OX −→ F −→ Ω −→ 0

is exact; to do this, we may assume that L is trivial. By ([45] I 4.3.1.7), the sequence induces, for every integer n ≥ 1, an exact sequence (II.2.5) (II.4.9.2)

n n 0 → Sn−1 OX (F ) → SOX (F ) → SOX (Ω) → 0.

The OX -modules (SnOX (F ))n∈N therefore form a direct system, whose direct limit (II.4.9.3)

C = lim SnOX (F ) −→

n≥0

is naturally endowed with a structure of OX -algebra. For any integer n ≥ 0, the canonical morphism SnOX (F ) → C is injective. Note that for every OX -algebra B, if u : F → B is an OX -linear morphism such that u ◦ c is the structural homomorphism, there exists a unique homomorphism of OX -algebras C → B that extends u. There exists a unique homomorphism of OX -algebras (II.4.9.4)

µ : C → SOX (Ω) ⊗OX C

such that for every local section x of F , we have (II.4.9.5)

µ(x) = ν(x) ⊗ 1 + 1 ⊗ x.

We denote by T = Spec(SOX (Ω)) the X-vector bundle associated with Ω and set (II.4.9.6)

L = Spec(C ).

We have a canonical isomorphism of groups of Xzar (II.4.9.7)

∼

T → ϕX (T).

For any s ∈ L (X), the morphism ρs : F → OX that sends a local section f of F to f (s) is a section of the exact sequence (II.4.9.1). It extends to a unique homomorphism of OX -algebras %s : C → OX that induces a section σs ∈ L(X). The map s 7→ σs defines a morphism of Xzar (II.4.9.8)

ι : L → ϕX (L).

Proposition II.4.10. Under the assumptions of II.4.9, the morphism T ×X L → L induced by µ (II.4.9.4) makes L into a principal homogeneous T-bundle over X, and the canonical morphism ι : L → ϕX (L) (II.4.9.8) is an isomorphism of T-torsors.

The questions are local, so we may restrict to the case where L is trivial. For s ∈ L (X), let ρs : F → OX be the associated splitting of the exact sequence (II.4.9.1). The morphism λs : Ω → F deduced from idF − c ◦ ρs extends to an isomorphism of OX -algebras (II.4.10.1)

ψ : SOX (Ω) → C

that is compatible with the filtrations (⊕0≤i≤n SiOX (Ω))n∈N and (SnOX (F ))n∈N . It follows from (II.4.9.2) that ψ is an isomorphism. Denote by (II.4.10.2)

δ : SOX (Ω) → SOX (Ω) ⊗OX SOX (Ω)

II.4. OBJECTS WITH GROUP ACTIONS

55

the homomorphism of OX -algebras such that for any local section ω of Ω, we have δ(ω) = ω ⊗ 1 + 1 ⊗ ω. We immediately see that the diagram

δ

/C

ψ

SOX (Ω)

(II.4.10.3)

µ



SOX (Ω) ⊗OX SOX (Ω)

id⊗ψ



/ SO (Ω) ⊗O C X X

is commutative. Consequently, the morphism T ×X L → L induced by µ makes L into a principal homogeneous T-bundle over X, and the morphism Ψ: L → T

(II.4.10.4)

induced by ψ is an isomorphism of T-torsors. Since ρs ◦ λs = 0, we have Ψ(ι(s)) = 0 in T(X). For any t ∈ T(X), we have ρs+t = ρs + t ◦ ν and therefore ρs+t ◦ λs = t ◦ ν ◦ λs = t. It follows that Ψ(ι(s + t)) = t in T(X), and consequently that ι(s + t) = ι(s) + t.

(II.4.10.5)

Hence ι is a morphism of T-torsors and therefore an isomorphism. Definition II.4.11. Under the assumptions of II.4.9, we say that L is the canonical principal homogeneous T-bundle over X that represents L . By II.4.4(i), there exists at most one principal homogeneous T-bundle over X that represents L , up to canonical isomorphism. The construction in II.4.9 gives a canonical one. II.4.12. Let X be a scheme, T and T0 two locally free OX -modules of finite type, L a T-torsor of Xzar , and L 0 a T0 -torsor of Xzar . We set Ω = H omOX (T, OX ), Ω0 = H omOX (T0 , OX ), T = Spec(SOX (Ω)), and T0 = Spec(SOX (Ω0 )). We denote by F the sheaf of affine functions on L (II.4.9), by F 0 the sheaf of affine functions on L 0 , by L the canonical principal homogeneous T-bundle over X that represents L (II.4.11), and by L0 the canonical principal homogeneous T0 -bundle over X that represents L 0 . Let u : T → T0 be an OX -linear morphism, u∨ : Ω0 → Ω the dual morphism of u, and v : L → L 0 a u-equivariant morphism of Xzar . If h : L 0 → OX is an affine function with linear term ω 0 ∈ Ω0 (X), then the composition h0 = h ◦ v : L → OX is affine, with linear term u∨ (ω 0 ). The resulting correspondence h 7→ h0 induces an OX -linear morphism w: F0 → F

(II.4.12.1) that fits into a commutative diagram (II.4.12.2)

0

/ OX

/ F0

/ OX

 /F

w

0

/ Ω0

/0

u∨

 /Ω

/0

where the lines are the canonical exact sequences (II.4.9.1). The morphism u∨ induces a morphism of X-group schemes α : T → T0 . The morphism w induces an α-equivariant X-morphism (II.4.12.3)

β : L → L0 .

56

II. LOCAL STUDY

The diagram / L0

v

L

(II.4.12.4)

ι

 ϕX (L)

ι0

 ϕX (β) / ϕX (L0 )

where ι and ι0 are the canonical isomorphisms (II.4.9.8), is clearly commutative. The correspondence that sends a T-torsor of Xzar to the canonical principal homogeneous T-bundle over X that represents it therefore defines a functor Tors(T, Xzar ) → PHB(T/X),

(II.4.12.5)

L 7→ L.

This is a quasi-inverse of the functor (II.4.3.1) ϕX : PHB(T/X) → Tors(T, Xzar ),

(II.4.12.6)

L 7→ ϕX (L),

by virtue of II.4.10, II.4.4(i), and (II.4.12.4). II.4.13. Let f : X 0 → X be a morphism of schemes, T a locally free OX -module of finite type, and L a T-torsor of Xzar . We set Ω = H omOX (T, OX ) and T = Spec(SOX (Ω)). We denote by F the sheaf of affine functions on L (II.4.7), by F + the sheaf of affine functions on f + (L ) (II.4.5), by L the canonical principal homogeneous T-bundle over X that represents L (II.4.11), and by L+ the canonical principal homogeneous (T ×X X 0 )-bundle over X 0 that represents f + (L ). Let ` : L → OX be an affine morphism, ω ∈ Ω(X) its linear term, and ω 0 = f ∗ (ω) ∈ f ∗ (Ω)(X 0 ). Endowing OX 0 with the structure of f ∗ (T)-object defined by ω 0 (II.4.8), there exists a unique f ∗ (T)-equivariant morphism `0 : f + (L ) → OX 0 that fits into the commutative diagram f ∗ (L )

(II.4.13.1)

f ∗ (`)

 f + (L )

/ f −1 (OX )

`0

 / OX 0

where the vertical arrows are the canonical morphisms ([35] III 1.3.6). The morphism h0 is therefore affine, with linear term ω 0 . The resulting correspondence ` 7→ `0 induces an OX -linear morphism λ] : F → f∗ (F + )

(II.4.13.2)

that fits into a commutative diagram (II.4.13.3)

OX

/F

 f∗ (OX 0 )

 / f∗ (F + )

λ]

/Ω  / f∗ (f ∗ (Ω))

where the other arrows are the canonical morphisms. The adjoint morphism λ : f ∗ (F ) → F +

(II.4.13.4)

therefore fits into a commutative diagram (II.4.13.5)

0

/ OX 0

/ f ∗ (F )

/ OX 0

 / F+

/ f ∗ (Ω)

/0

/ f ∗ (Ω)

/0

λ

0

II.4. OBJECTS WITH GROUP ACTIONS

57

where the lines are the canonical exact sequences (II.4.9.1). Consequently, λ is an isomorphism. We deduce from this an isomorphism of principal homogeneous (T×X X 0 )-bundles ∼

L+ → L ×X X 0 .

(II.4.13.6) The diagram (II.4.13.7)

f ∗ (L ) f ∗ (ι)

a

/ f + (L )

b

 / ϕX 0 (L+ )

ι+



f ∗ (ϕX (L))

where ι and ι+ are the canonical isomorphisms (II.4.9.8), a is the canonical morphism, and b is the morphism induced by (II.4.2.2) and (II.4.13.6), is commutative. Indeed, it suffices to show that the diagram (II.4.13.8)

a]

L

f∗ (ι+ )

ι

 ϕX (L)

/ f∗ (L + )

b]

 / f∗ (ϕX 0 (L+ ))

where a] and b] are adjoints of a and b, is commutative, or that the diagram deduced from this one by evaluating the sheaves on X is commutative. For s ∈ L (X), let ρs : F → OX be the morphism that sends a local section ` of F to `(s). Let us set s0 = a] (s) ∈ L + (X 0 ) and let ρs0 : F + → OX 0 be the morphism that sends a local section `0 of F + to `0 (s0 ). It immediately follows from the definition of the morphism λ] (II.4.13.2) that the diagram (II.4.13.9)

F ρs



λ]

/ f∗ (F + ) f∗ (ρs0 )

OX

 / f∗ (OX 0 )

is commutative. We deduce from this the relation ρs0 ◦ λ = f ∗ (ρs ), which immediately implies the commutativity of the diagram (II.4.13.8). Let g : X 00 → X 0 be a morphism of schemes and h = f ◦ g : X 00 → X. We denote by † F the sheaf of affine functions on h+ (L ). We then have a canonical isomorphism (II.4.13.10)

∼

θ : h∗ (F ) → F † .

In view of (II.4.5.6), we also have a canonical isomorphism (II.4.13.11) One immediately verifies that (II.4.13.12)

∼

λ+ : g ∗ (F + ) → F † . θ = λ+ ◦ g ∗ (λ).

II.4.14. Let f : X 0 → X be a morphism of schemes, T a locally free OX -module of finite type, T0 a locally free OX 0 -module of finite type, L a T-torsor of Xzar , and L 0 a T0 0 torsor of Xzar . Set Ω = H omOX (T, OX ), Ω0 = H omOX 0 (T0 , OX 0 ), T = Spec(SOX (Ω)), 0 and T = Spec(SOX 0 (Ω0 )). We denote by F the sheaf of affine functions on L (II.4.9), by F 0 the sheaf of affine functions on L 0 , by L the canonical principal homogeneous Tbundle over X that represents L (II.4.11), and by L0 the canonical principal homogeneous T0 -bundle over X 0 that represents L 0 . The sheaf f∗ (L 0 ) is naturally an f∗ (T0 )-object of Xzar . Let u : T → f∗ (T0 ) be an OX -linear morphism and v : L → f∗ (L 0 ) a uequivariant morphism. We denote by γ : f −1 (T) → f ∗ (T) the canonical morphism, by

58

II. LOCAL STUDY

u] : f ∗ (T) → T0 the adjoint morphism of u, by u∨ : Ω0 → f ∗ (Ω) the dual morphism of u] , and by v ] : f ∗ (L ) → L 0 the adjoint morphism of v. Since v ] is (u] ◦ γ)-equivariant, it factors uniquely through a u] -equivariant morphism v + : f + (L ) → L 0 .

(II.4.14.1)

By (II.4.12.1) and (II.4.13.4), this induces an OX 0 -linear morphism w : F 0 → f ∗ (F )

(II.4.14.2)

that fits into a commutative diagram (II.4.14.3)

0

/ OX 0

/ F0

/ Ω0

/ OX 0

 / f ∗ (F )

u∨

w

0

/0

 / f ∗ (Ω)

/0

where the lines are the canonical exact sequences (II.4.9.1). The morphism u∨ induces a morphism of X 0 -group schemes (II.4.14.4)

α : T ×X X 0 → T0 .

The morphism w induces an α-equivariant X 0 -morphism (II.4.14.5)

β : L ×X X 0 → L0 .

It follows from (II.4.12.4) and (II.4.13.7) that the diagram (II.4.14.6)

f ∗ (L ) f ∗ (ι)



v]

/ L0 ι0

f ∗ (ϕX (L))

δ

 / ϕX 0 (L0 )

where ι and ι0 are the canonical isomorphisms (II.4.9.8) and δ is the morphism induced by (II.4.2.2) and β, is commutative. II.4.15. We keep the assumptions of II.4.14 and moreover let g : X 00 → X 0 be a morphism of schemes, T00 a locally free OX 00 -module of finite type, L 00 a T00 -torsor of 00 Xzar , and F 00 the sheaf of affine functions on L 00 . Let u0 : T0 → g∗ (T00 ) be an OX 0 -linear morphism and v 0 : L 0 → g∗ (L 00 ) a u0 -equivariant morphism. By II.4.14, the pair (u0 , v 0 ) induces an OX 00 -linear morphism (II.4.15.1)

w0 : F 00 → g ∗ (F 0 ).

Likewise, the pair (f∗ (u0 ) ◦ u, f∗ (v 0 ) ◦ v) induces an OX 00 -linear morphism (II.4.15.2)

t : F 00 → g ∗ (f ∗ (F )).

We then have (II.4.15.3)

t = g ∗ (w) ◦ w0 .

II.4. OBJECTS WITH GROUP ACTIONS

59

This follows from (II.4.13.12) and from a chase in the commutative diagram ∗

(II.4.15.4)

]

0]

g (v ) v / g ∗ (L 0 ) / L 00 g ∗ (f ∗ (L )) ; NNN 7 M w M o M o w MMM NNN oo ww o w M NNN o MMM oo ∗ + ww 0+ NN' M& ww v ooo g (v ) g + (L 0 ) g ∗ (f + (L )) 8 OOO q q OOO q q q OOO qq+ + OO' qqq g (v ) g + (f + (L )),

where v 0] is the adjoint morphism of v 0 , v 0+ is the morphism induced by v 0] , and the unlabeled arrows are the canonical morphisms. II.4.16. (II.4.16.1)

Let C , F be two categories, π: F → C

a fibration ([37] VI 6.1), and ∆ an (abstract) group. For any X ∈ Ob(C ), we denote by FX the fiber of π over X ([37] VI § 4). We denote also by ∆ the groupoid associated with ∆ (that is, the category having only one object, with morphisms class ∆). Let X be an object of C and ϕ a left action of ∆ on X; in other words, ϕ : ∆ → C is a functor that sends the unique object of ∆ to X. For σ ∈ ∆, we (abusively) denote by σ the automorphism ϕ(σ) of X. The base change of π by ϕ ([37] VI § 3) (II.4.16.2)

π∆ : F ∆ → ∆

is a fibration. The category F∆ has the same objects as the fiber FX , but has more morphisms. The Cartesian sections of π∆ are called the ∆-equivariant X-objects of F (or ∆-equivariant objects of FX ). Giving a ∆-equivariant X-object of F is therefore equivalent to giving an object Y of FX and a left action of ∆ on Y (viewed as an object of F ) that is compatible with its action on X by the functor π. The ∆-equivariant X-objects of F naturally form a category, namely the category of Cartesian sections of π∆ ([2] VI 6.10). Let us choose a normalized cleavage of F over C ([37] VI § 7; cf. also [1] 1.1.2); in other words, let us choose, for every morphism f : Z → Y in C , an inverse image functor (II.4.16.3)

f ∗ : FY → FZ ,

such that for any Y ∈ Ob(C ), f = idY implies that f ∗ = idFY . For every pair of composable morphisms (f, g) in C , we have a canonical isomorphism of functors (II.4.16.4)

∼

cg,f : g ∗ f ∗ → (f g)∗

satisfying compatibility relations ([37] VI 7.4). By ([37] VI § 12), giving a ∆-equivariant object of FX is equivalent to giving an object Y of FX and for every σ ∈ ∆, an isomorphism (II.4.16.5)

∼

τσY : Y → σ ∗ (Y )

Y such that τid = idY and for any (σ, σ 0 ) ∈ ∆2 , we have

(II.4.16.6)

Y 0∗ τσσ ∗ τσY ) ◦ τσY0 . 0 = cσ 0 ,σ ◦ (σ

We will leave it to the reader to explicitly describe the morphisms of ∆-equivariant X-objects of F .

60

II. LOCAL STUDY

II.4.17. Let C be a category in which fibered products are representable and X an object of C endowed with a left action of an (abstract) group ∆. We denote also by ∆ the groupoid associated with ∆, and denote by ϕ : ∆ → C the action of ∆ on X. We denote by Fl(C ) the category of arrows of C and by Fl(C ) → C

(II.4.17.1)

the target functor, which is a fibration ([37] VI § 11 a)). We deduce from this, by base change by ϕ ([37] VI § 3), a fibration Fl(C )∆ → ∆

(II.4.17.2)

whose Cartesian sections are called the ∆-equivariant X-objects of C (or ∆-equivariant objects of C/X ). Giving such an object is equivalent to giving an object Y of C/X and a left action of ∆ on Y viewed as an object of C that is compatible with its action on X. We define the category Gr(C ) as follows. The objects of Gr(C ) are pairs consisting of a morphism G → Y of C and a structure of group of C/Y on G, in the sense of ([35] III 1.1.1). We will leave the group structure out of the notation. Let G → Y and G0 → Y 0 be two objects of Gr(C ). A morphism from G0 → Y 0 to G → Y is a commutative diagram of C /G (II.4.17.3) G0  Y0

 /Y

such that the induced morphism G0 → G ×Y Y 0 is a morphism of groups of C/Y 0 . The target functor (II.4.17.4)

Gr(C ) → C ,

(G → Y ) 7→ Y

is a fibration; its fiber over an object Y of C is the category of groups of C/Y . We deduce from it by base change by ϕ a fibration Gr(C )∆ → ∆,

(II.4.17.5)

whose Cartesian sections are called the ∆-equivariant X-groups of C (or ∆-equivariant groups of C/X ). Giving such an object is equivalent to giving a group G of C/X and a left action of ∆ on G viewed as an object in C that is compatible with its action on X and with the group structure on G in a sense that we will not go into. We define the category Op(C ) as follows (cf. [35] III 1.1.6). The objects of Op(C ) are triples consisting of an object G → Y of Gr(C ), a morphism Z → Y of C , and a (right) action m of G on Z over Y ; that is, a Y -morphism m : Z ×Y G → Z subject to the usual algebraic conditions (cf. [35] III 1.1.2). Let (G → Y, Z → Y ) and (G0 → Y 0 , Z 0 → Y 0 ) be two objects of Op(C ). A morphism from (G → Y, Z → Y, m) to (G0 → Y 0 , Z 0 → Y 0 , m0 ) consists of two commutative diagrams of C /G /Z (II.4.17.6) G0 Z0  Y0

 /Y

 Y0

 /Y

such that the diagram (II.4.17.7)

Z 0 ×Y 0 G0 m0

 Z0

/ Z ×Y G ×Y Y 0 

m×Y Y 0

/ Z ×Y Y 0

II.4. OBJECTS WITH GROUP ACTIONS

61

is commutative. The functor (II.4.17.8)

Op(C ) → C ,

(G → Y, Z → Y, m) 7→ Y

is a fibration. Its fiber over an object Y of C is the category of objects with (right) group actions of C/Y . We deduce from this by base change by ϕ a fibration (II.4.17.9)

Op(C )∆ → ∆

whose Cartesian sections are called the ∆-equivariant X-objects with group actions of C (or ∆-equivariant objects with group actions of C/X ). Giving such an object is equivalent to giving a ∆-equivariant group G of C/X , a ∆-equivariant object Y of C/X , and an action m of G on Y that is compatible with the ∆-equivariant structures in a sense we will not go into. We also say that (Y, m) is a ∆-equivariant G-object of C/X . II.4.18. Let X be a scheme endowed with a left action by an (abstract) group ∆. We define the ∆-equivariant objects of Xzar by taking for F in (II.4.16) the cleaved and normalized fibered category (II.4.18.1)

Z → Sch

obtained by associating with any scheme Y the topos Yzar and with any morphism of schemes f : Z → Y the inverse image functor f ∗ : Yzar → Zzar . We define the ∆equivariant groups of Xzar by taking for F in (II.4.16) the cleaved and normalized fibered category (II.4.18.2)

G → Sch

obtained by associating with any scheme Y the category of groups of Yzar and with any morphism of schemes f : Z → Y the inverse image functor f ∗ . We define the ∆equivariant OX -modules of Xzar by taking for F in (II.4.16) the cleaved and normalized fibered category (II.4.18.3)

M → Sch

obtained by associating with any scheme Y the category of OY -modules of Yzar and with any morphism of schemes f : Z → Y the inverse image functor in the sense of modules. Note that since the inverse images of an OX -module under an automorphism of X in the sense of modules and in the sense of abelian groups are equal, every ∆-equivariant OX -module of Xzar is also a ∆-equivariant group. We define the category P as follows. The objects of P are the triples (Y, G, P ), where Y is a scheme, G is a group of Yzar , and P is a (right) G-object of Yzar ([35] III 1.1.2). Let (Y, G, P ) and (Y 0 , G0 , P 0 ) be two objects of P. A morphism from (Y 0 , G0 , P 0 ) to (Y, G, P ) consists of a morphism of schemes f : Y 0 → Y , a group morphism γ : G0 → 0 0 f ∗ (G) of Yzar , and a γ-equivariant morphism δ : P 0 → f ∗ (P ) of Yzar . The functor (II.4.18.4)

P → Sch,

(Y, G, P ) 7→ Y

is a fibration. Taking for F in (II.4.16) the fibered category above, we obtain the notion of ∆-equivariant objects with group actions of Xzar . Giving such an object is equivalent to giving a ∆-equivariant group G of Xzar , a ∆-equivariant object P of Xzar , and an action m of G on P that is compatible with the ∆-equivariant structures in a sense that we will not go into. We also say that P is a ∆-equivariant G-object of Xzar . If, moreover, P is a G-torsor of Xzar , then we also say that it is a ∆-equivariant G-torsor of Xzar . Remarks II.4.19. Let X and X 0 be two schemes endowed with a left action by an (abstract) group ∆, and f : X 0 → X a ∆-equivariant morphism. (i) For every ∆-equivariant object with group actions (G, P ) of Xzar , (f ∗ (G), 0 f ∗ (P )) is naturally a ∆-equivariant object with group actions of Xzar .

62

II. LOCAL STUDY 0 (ii) For every ∆-equivariant object with group action (G0 , P 0 ) of Xzar , (f∗ (G0 ), 0 f∗ (P )) is naturally a ∆-equivariant object with group action of Xzar . Indeed, for every σ ∈ ∆, the base change morphism for the Zariski topos σ ∗ f∗ → f∗ σ ∗ , deduced from the relation f σ = σf ([1] 1.2.2), is an isomorphism. The statement follows using ([1] 1.2.4(i)).

II.4.20. Let X be a scheme endowed with a left action by an (abstract) group ∆ and G a ∆-equivariant X-group scheme. A ∆-equivariant principal homogeneous Gbundle over X is a ∆-equivariant G-object (Y, m) of Sch/X (II.4.17) such that Y is a principal homogeneous G-bundle over X (II.4.3). In view of II.4.2, the functor (II.4.1.2) ϕX : Sch/X → Xzar ,

(II.4.20.1)

Y 7→ HomX (−, Y )

transforms ∆-equivariant X-schemes (resp. ∆-equivariant X-group schemes) (II.4.17) into ∆-equivariant objects (resp. groups) of Xzar (II.4.18). Likewise, we set G = ϕX (G). The functor (II.4.3.1) ϕX : PHB(G/X) → Tors(G, Xzar ),

(II.4.20.2)

Y 7→ ϕX (Y )

transforms ∆-equivariant principal homogeneous G-bundles over X into ∆-equivariant G-torsors of Xzar . Conversely, let Y be a principal homogeneous G-bundle over X and Y = ϕX (Y ). Giving a ∆-equivariant structure on the G-torsor Y determines on Y a unique structure of ∆-equivariant principal homogeneous G-bundle over X. Indeed, for any σ ∈ ∆, let ∼

τσG : G → σ • (G),

(II.4.20.3)

τσY : Y

(II.4.20.4)

∼

→ σ ∗ (Y )

be the isomorphisms induced by the ∆-equivariant structures of G and Y (II.4.16.5), respectively. By definition, τσG is an isomorphism of X-group schemes. By II.4.2, it induces a group isomorphism of Xzar ∼

τσG : G → σ ∗ (G).

(II.4.20.5) G

The isomorphisms (τσ )σ∈∆ make G into a ∆-equivariant group of Xzar . For any σ ∈ ∆, Y G τσ is an isomorphism of G-torsors, where σ ∗ (Y ) is viewed as a G-torsor via τσ . By Y II.4.4(i), τσ is the image by the functor (II.4.3.1) of an isomorphism of PHB(G/X) ∼

τσY : Y → σ • (Y ),

(II.4.20.6)

where σ • (Y ) is seen as a principal homogeneous G-bundle via τσG . The isomorphisms (τσY )σ∈∆ satisfy the compatibility relations (II.4.16.6). They make Y into a ∆-equivariant principal homogeneous G-bundle over X (cf. II.4.16). II.4.21. Let X be a scheme endowed with a left action by an (abstract) group ∆, T a locally free ∆-equivariant OX -module of finite type (II.4.18), and L a ∆-equivariant T-torsor of Xzar . Set Ω = H omOX (T, OX ) and T = Spec(SOX (Ω)). We denote by F the sheaf of affine functions on L (II.4.9) and by L the canonical principal homogeneous T-bundle that represents L (II.4.11). For any σ ∈ ∆, we denote by (II.4.21.1)

∼

τσT : T → σ ∗ (T)

∼

and τσL : L → σ ∗ (L )

the isomorphisms that define the ∆-equivariant structures on T and L , respectively (cf. II.4.16). By II.4.14, the inverses of τσT and τσL induce an OX -linear morphism (II.4.21.2)

τσF : F → σ ∗ (F )

II.5. LOGARITHMIC GEOMETRY LEXICON

that fits into a commutative diagram / OX (II.4.21.3) 0

/F τσF

0

/ OX

 / σ ∗ (F )

/Ω

63

/0

τσΩ

 / σ ∗ (Ω)

/0

where the lines are the canonical exact sequences (II.4.9.1) and τσΩ is the inverse of the isomorphism induced by τσT . It follows that τσF is an isomorphism. By (II.4.15.3), the (τσF )σ∈∆ satisfy compatibility relations (II.4.16.6). They therefore make F into a ∆-equivariant OX -module. Likewise, the (τσΩ )σ∈∆ make Ω into a ∆-equivariant OX module. We deduce from this on T a ∆-equivariant structure of X-group scheme and on L a structure of ∆-equivariant principal homogeneous T-bundle over X (II.4.20). By II.4.10 and (II.4.14.6), we have an isomorphism of objects with ∆-equivariant group actions of Xzar (II.4.21.4)

∼

(T, L ) → (ϕX (T), ϕX (L)).

II.4.22. Let X and X 0 be two schemes endowed with a left action by an (abstract) group ∆, f : X 0 → X a ∆-equivariant morphism, T a locally free ∆-equivariant OX module of finite type, T0 a locally free ∆-equivariant OX 0 -module of finite type, L a 0 ∆-equivariant T-torsor of Xzar , and L 0 a ∆-equivariant T0 -torsor of Xzar . We denote by 0 F the sheaf of affine functions on L (II.4.9) and by F the sheaf of affine functions on L 0 . The pair (f∗ (T0 ), f∗ (L 0 )) is naturally a ∆-equivariant object with group actions of Xzar (II.4.19). Let u : T → f∗ (T0 ) be a ∆-equivariant OX -linear morphism and v : L → f∗ (L 0 ) a ∆-equivariant and u-equivariant morphism. The pair (u, v) induces an OX 0 linear morphism (II.4.14.2) (II.4.22.1)

w : F 0 → f ∗ (F ).

It immediately follows from (II.4.15.3) that w is ∆-equivariant when we endow F and F 0 with the canonical ∆-equivariant structures (II.4.21). Let u] : f ∗ (T) → T0 be the adjoint of u, v ] : f ∗ (L ) → L 0 the adjoint of v, and v + : f + (L ) → L 0 the morphism induced by v ] (II.4.14.1). If u] is an isomorphism, then v + is an isomorphism of f ∗ (T)-torsors and w is an isomorphism by virtue of II.4.13. II.5. Logarithmic geometry lexicon We recall a few notions of logarithmic geometry that will play an important role in this work, in order to fix the notation and to give readers unfamiliar with this theory points of reference. We refer to [50, 51, 33, 58] for the systematic development of the theory. II.5.1. By monoid we will mean a commutative monoid with an identity element. Homomorphisms of monoids are assumed to map the identity element to the identity element. If M is a monoid, then we denote by M gp the associated group, by M × the group of invertible elements of M , by M ] the set of orbits M/M × (which is also the quotient of M by M × in the category of monoids), and by iM : M → M gp the canonical homomorphism. We set M int = iM (M ) and (II.5.1.1)

M sat = {x ∈ M gp |xn ∈ iM (M ) for an integer n ≥ 1}.

We say that a monoid M is integral if the canonical homomorphism iM : M → M gp is injective, that M is fine if it is integral and of finite type, that M is saturated if it is integral and isomorphic to M sat , and that M is toric if it is fine and saturated and if M gp is free over Z.

64

II. LOCAL STUDY

If M is integral, then M ] is integral, and M is saturated if and only if M ] is saturated. A morphism of monoids u : M → N is said to be strict if the induced morphism u] : M ] → N ] is an isomorphism. II.5.2. Let u : M → N be a morphism of integral monoids. We say that u is exact if the diagram (II.5.2.1)

M  M gp

u

/N

ugp

 / N gp

is Cartesian. We say that u is integral if for every integral monoid M 0 and every homomorphism v : M → M 0 , the amalgamated sum M 0 ⊕M N is integral. We say that u is saturated if it is integral and if for every saturated monoid M 0 and every homomorphism v : M → M 0 , the amalgamated sum M 0 ⊕M N is saturated. II.5.3. Let T be a topos. We denote by MonT the category of (commutative and unitary) monoids of T and by AbT the category of abelian groups of T . The canonical injection functor from AbT to MonT admits a right adjoint (II.5.3.1)

MonT → AbT ,

M 7→ M × .

It is immediate that for any U ∈ Ob(T ), the adjunction morphism M × (U ) → M (U ) ∼ induces an isomorphism M × (U ) → M (U )× . We say that a monoid M of T is sharp if M × = 1T . The canonical injection functor from the full subcategory of sharp monoids of T to MonT admits a left adjoint (II.5.3.2)

M 7→ M ] = M /M × .

The canonical injection functor from AbT to MonT admits a left adjoint (II.5.3.3)

MonT → AbT ,

M 7→ M gp .

We say that a monoid M of T is integral if the adjunction morphism M → M gp is a monomorphism. We denote by MonT,int the full subcategory of MonT made up of integral monoids of T . The canonical injection functor from MonT,int to MonT admits a left adjoint (II.5.3.4)

MonT → MonT,int ,

M → M int .

II.5.4. Let C be a site, Ce the topos of sheaves of sets on C (relative to a fixed universe). For every presheaf of monoids P on C , we denote by P gp (resp. P int ) the presheaf of monoids on C that sends U ∈ Ob(C ) to the monoid P(U )gp (resp. P(U )int ) and by P a the sheaf of monoids associated with P. We then have a functorial canonical isomorphism (II.5.4.1)

∼

(P gp )a → (P a )gp .

Since the functor P 7→ P a is exact, it transforms the presheaves of integral monoids on C into integral monoids of Ce. We have a functorial canonical isomorphism (II.5.4.2)

∼

(P int )a → (P a )int .

Consequently, a monoid M of Ce is integral if and only if for every U ∈ Ob(C ), the monoid M (U ) is integral.

II.5. LOGARITHMIC GEOMETRY LEXICON

65

II.5.5. Let T be a topos. We say that a morphism of integral monoids u : M → N of T is exact if the canonical diagram (II.5.5.1)

M  M gp

u

ugp

/N /N



gp

is Cartesian. It amounts to requiring, for every U ∈ Ob(T ), the homomorphism u(U ) : M (U ) → N (U )

to be exact. Indeed, if the diagram (II.5.5.1) is Cartesian, then the same holds for the diagrams obtained by taking its value at any U ∈ Ob(T ). Consequently, the diagram (II.5.5.2)

M (U )  M (U )gp

u

/ N (U )

ugp

 / N (U )gp

is Cartesian because the canonical morphism M (U )gp → M gp (U ) is injective (and the same holds for N ). The converse follows from (II.5.4.1) and from the exactness of the functor P 7→ P a . Let M be an integral monoid of T and n an integer ≥ 1. We say that M is nsaturated if the endomorphism defined by taking the nth power in M is exact. We say that M is saturated if it is n-saturated for every integer n ≥ 1. It amounts to requiring, for every U ∈ Ob(T ), the monoid M (U ) to be saturated. We denote by MonT,sat the full subcategory of MonT made up of saturated monoids of T . The canonical injection functor from MonT,sat to MonT admits a left adjoint (II.5.5.3)

MonT → MonT,sat ,

M 7→ M sat .

II.5.6. Let C be a site, Ce the topos of sheaves of sets on C (relative to a fixed universe). For every presheaf of monoids P on C , we denote by P sat the presheaf of saturated monoids defined, for U ∈ Ob(C ), by U 7→ P(U )sat . Then the functor “associated sheaf of monoids,” P 7→ P a , transforms the presheaves of saturated monoids on C into saturated monoids of Ce, and we have a functorial canonical isomorphism (II.5.6.1)

∼

(P sat )a → (P a )sat .

Consequently, a monoid M of Ce is saturated if and only if for every U ∈ Ob(C ), the monoid M (U ) is saturated. II.5.7. Let f : T 0 → T be a morphism of topos and M a monoid of T . (i) If M is integral (resp. saturated), then the same holds for f ∗ (M ). (ii) We have functorial canonical isomorphisms (II.5.7.1)

f ∗ (M )gp

(II.5.7.2)

f ∗ (M )int

∼

→ f ∗ (M gp ), ∼

→ f ∗ (M int ).

If M is moreover integral, then we have a canonical isomorphism (II.5.7.3)

∼

f ∗ (M )sat → f ∗ (M sat ).

(iii) If u : M → N is an exact morphism of integral monoids of T , then the morphism f ∗ (u) : f ∗ (M ) → f ∗ (N ) is exact.

66

II. LOCAL STUDY

II.5.8. Let T be a topos. (i) If M is an integral (resp. saturated) monoid, then the constant sheaf of monoids MT with value M on T is integral (resp. saturated). (ii) Suppose that T has enough points. A monoid M of T is integral (resp. saturated) if and only if for every point p of T , the monoid Mp is integral (resp. saturated). A morphism of integral monoids u : M → N of T is exact if and only if for every point p of T , the homomorphism up : Mp → Np is exact. (iii) Let M be an integral monoid of T . Then M ] is integral and M is saturated if and only if M ] is saturated. II.5.9. A prelogarithmic structure on a scheme X is a pair (P, β), where P is a sheaf of abelian monoids on the étale site of X and β is a homomorphism from P to the multiplicative monoid OX . A prelogarithmic structure (P, β) is called logarithmic ∼ × × if β induces an isomorphism β −1 (OX ) → OX . The prelogarithmic structures on X naturally form a category that contains the full subcategory of logarithmic structures on X. The canonical injection from the category of logarithmic structures on X to the category of prelogarithmic structures on X admits a left adjoint. It associates with a prelogarithmic structure (P, β) the logarithmic structure (M , α), where M is defined by the co-Cartesian diagram (II.5.9.1)

× β −1 (OX )

/P

 × OX

 /M

We say that (M , α) is the logarithmic structure associated with (P, β). II.5.10. Let f : X → Y be a morphism of schemes. For sheaves of monoids, we use the notation f −1 to denote the inverse image in the sense of sheaves of monoids and keep the notation f ∗ for the inverse image in the sense of logarithmic structures, defined as follows. The inverse image under f of a logarithmic structure (M , α) on Y is the logarithmic structure (f ∗ (M ), β) on X associated with the prelogarithmic structure defined by the composition f −1 (M ) → f −1 (OY ) → OX . It immediately follows from the definition that the canonical homomorphism f −1 (M ] ) → (f ∗ (M ))]

(II.5.10.1) is an isomorphism.

II.5.11. A prelogarithmic (resp. logarithmic) scheme is a triple (X, MX , αX ) consisting of a scheme X and a prelogarithmic (resp. logarithmic) structure (MX , αX ) on X. When there is no risk of confusion, we will leave αX , and even MX , out of the notation. A morphism of prelogarithmic (resp. logarithmic) schemes (X, MX , αX ) → (Y, MY , αY ) is a pair (f, f [ ) consisting of a morphism of schemes f : X → Y and a homomorphism f [ : f −1 (MY ) → MX such that the diagram (II.5.11.1)

f −1 (MY )

f −1 (αY )

f[

 MX

is commutative.

αX

/ f −1 (OY )  / OX

II.5. LOGARITHMIC GEOMETRY LEXICON

67

We say that a logarithmic scheme (X, MX , αX ) is integral (resp. saturated) if MX is integral (resp. saturated). We say that a morphism of logarithmic schemes f : (X, MX , αX ) → (Y, MY , αY ) is strict if (MX , αX ) is the inverse image of (MY , αY ) under f , or, equivalently, if the canonical homomorphism f −1 (MY] ) → MX] is an isomorphism. II.5.12. Let M be a monoid. For every integer n ≥ 1, we (abusively) denote by $n : M → M the Frobenius homomorphism of order n of M (that is, taking the nth power in M in the multiplicative notation). For any ring R, we denote by R[M ] the R-algebra of M and by e : M → R[M ] the canonical homomorphism, where R[M ] is seen as a multiplicative monoid. For any x ∈ M , we will write ex instead of e(x). We denote by AM the scheme Spec(Z[M ]) endowed with the logarithmic structure associated with the prelogarithmic structure defined by e : M → Z[M ]. For any homomorphism of monoids ϑ : M → N , we denote by Aϑ : AN → AM the associated morphism of logarithmic schemes. II.5.13. Let (X, MX ) be a logarithmic scheme, M a monoid, and MX the constant (étale) sheaf of monoids with value M on X. The following data are equivalent: (i) a homomorphism γ : M → Γ(X, MX ); (ii) a homomorphism γ e : MX → MX ; (iii) a morphism of logarithmic schemes γ a : (X, MX ) → AM .

Moreover, the following conditions are equivalent:

(a) MX is associated with the prelogarithmic structure it induces on MX ; (b) the morphism γ a : (X, MX ) → AM is strict.

We then say that (M, γ) is a chart for (X, MX ). We say that the chart (M, γ) is coherent (resp. integral, resp. fine, resp. saturated, resp. toric) if the monoid M is of finite type (resp. integral, resp. fine, resp. saturated, resp. toric). II.5.14. Let f : (X, MX ) → (Y, MY ) be a morphism of logarithmic schemes. A chart for f is a triple ((M, γ), (N, δ), ϑ : N → M ) consisting of a chart (M, γ) for (X, MX ), a chart (N, δ) for (Y, MY ), and a homomorphism ϑ : N → M such that the diagram (II.5.14.1)

N

δ

/ Γ(Y, MY )

ϑ

 M

γ



f[

/ Γ(X, MX )

is commutative, or equivalently, such that the associated diagram of morphisms of logarithmic schemes (II.5.12) (II.5.14.2)

(X, MX )

γa

f

 (Y, MY )

/ AM Aϑ

δ

a

 / AN

is commutative. We say that the chart ((M, γ), (N, δ), ϑ : N → M ) is coherent (resp. fine) if M and N are of finite type (resp. fine).

68

II. LOCAL STUDY

II.5.15. Let (X, MX ) be a logarithmic scheme. We say that (X, MX ) is coherent if every geometric point x of X admits an étale neighborhood U in X such that (U, MX |U ) admits a coherent chart. We say that (X, MX ) is fine if it is coherent and integral. The logarithmic scheme (X, MX ) is fine (resp. fine and saturated) if and only if every geometric point x of X admits an étale neighborhood U in X such that (U, MX |U ) admits a fine (resp. fine and saturated) chart. We say that (X, MX ) is toric if every geometric point x of X admits an étale neighborhood U in X such that (U, MX |U ) admits a toric chart. Lemma II.5.16 ([73] 1.3.2). For every fine and saturated logarithmic scheme (X, MX ), the following conditions are equivalent: (i) There exists a fine and saturated chart γ : P → Γ(X, MX ) for (X, MX ) (II.5.13) such that the composition (II.5.16.1)

× P → Γ(X, MX ) → Γ(X, MX )/Γ(X, OX )

is an isomorphism. (ii) There exists a coherent chart γ : P → Γ(X, MX ) for (X, MX ) such that the composition (II.5.16.1) is surjective. × (iii) The monoid Γ(X, MX )/Γ(X, OX ) is of finite type and the identity of Γ(X, MX ) is a chart for (X, MX ). The implication (i)⇒(ii) is clear. Let us show (ii)⇒(iii). We set Q = Γ(X, MX ) and denote by P and Q the logarithmic structures on X associated with the prelogarithmic structures defined by the homomorphism γ and the identity of Γ(X, MX ), respectively, θ

and by θ : P → Q the homomorphism induced by γ. Since the composition P → Q → M is an isomorphism, it suffices to show that θ is surjective. Let x be a geometric point of X. We denote by β : Q → Mx the canonical homomorphism and by α : P → Mx the composition β ◦ γ. We then have a commutative diagram of homomorphisms of monoids (II.5.16.2)

× P/α−1 (OX,x )

 × Px /OX,x

γ

] θx

/ Q/β −1 (O × ) X,x  / Qx /O × X,x

where γ (resp. θx] ) is induced by γ (resp. θx ) and the vertical arrows are the canonical × × isomorphisms. Since Γ(X, OX ) ⊂ β −1 (OX,x ), γ is surjective. Consequently, θx] is surjective. Hence θx is surjective, giving the statement. Finally, let us show (iii)⇒(i). Let × P = Γ(X, MX )/Γ(X, OX ). Since Γ(X, MX ) is saturated, P is fine and saturated. The torsion subgroup of P gp is therefore contained in P . Since P is sharp, P gp is a free abelian group of finite type. Let δ : P gp → Γ(X, MX )gp be a section of the canonical homomorphism Γ(X, MX )gp → P gp . The restriction of δ to P induces a homomorphism γ : P → Γ(X, MX ) that satisfies the conditions by virtue of ([73] 1.3.1). Lemma II.5.17 ([73] 1.3.3). Let (X, MX ) be a fine and saturated logarithmic scheme whose underlying scheme X is noetherian, x ∈ X, and γ : P → Γ(X, MX ) a fine and saturated chart for (X, MX ). Then there exists a (Zariski) open neighborhood U of x in X such that for every (Zariski) open neighborhood V of x in U , the logarithmic scheme (V, MX |V ) satisfies the equivalent conditions of (II.5.16). We denote by Spec(P ) the set of prime ideals of P (cf. [51] 5.1), by PX the constant (étale) sheaf on X with value P , and by γ e : PX → MX the homomorphism induced by γ.

II.5. LOGARITHMIC GEOMETRY LEXICON

69

For every geometric point y of X, we set (II.5.17.1)

× ey−1 (OX,y py = P − γ ) ⊂ (PX )y = P,

which is a prime ideal of P . For every t ∈ P , we denote by Yt the cosupport of the image × of t in Γ(X, MX /OX ), or equivalently, since MX is integral, the cosupport of the image gp × of t in Γ(X, MX /OX ) ([2] IV 8.5.2). It is a subobject of the final object of the étale topos of X that we identify with an open subset of X ([2] VIII 6.1). We set Xt = X − Yt . One immediately verifies that the fiber of Xt over a geometric point y of X is nonempty if and only if t ∈ py . For every p ∈ Spec(P ), we set Xp = ∩t∈p Xt . The fiber of Xp over a geometric point y of X is nonempty if and only if p ⊂ py . Since X is noetherian and Spec(P ) is finite (cf. [51] 5.5), there exists a (Zariski) open neighborhood U of x in X with the following property. For every p ∈ Spec(P ) such that Up = U ∩ Xp 6= ∅, x belongs to all irreducible components of Up . Since every (Zariski) open neighborhood V of x in U has the same property, it suffices to prove that (U, MX |U ) satisfies the equivalent conditions of II.5.16. Let x be a geometric point of X over x. Since the composition of the canonical homomorphisms (II.5.17.2)

u

v

× × × P → Γ(U, MX )/Γ(U, OX ) → Γ(U, MX /OX ) → MX,x /OX,x

is surjective and the morphism u is injective, it suffices to show that v is injective. Let × a, b ∈ Γ(U, MX /OX ) be such that v(a) = v(b) and y a geometric point of U . Then there exist a geometric point z of U and a specialization map ϕ : z y such that the canonical image z of z in U is a generic point of Upy . We have γ ez = ϕ∗ ◦ γ ey , where ϕ∗ : MX,y → MX,z is the specialization homomorphism associated with ϕ ([2] VIII 7.7). It follows that pz ⊂ py . Since z ∈ Upy , we have py ⊂ pz and therefore py = pz . The commutative diagram (II.5.17.3)

× P/e γy−1 (OX,y )

γ ey

/ MX,y /O × X,y ϕ∗

× P/e γz−1 (OX,z )

γ ez

 / MX,z /O × X,z

where the horizontal arrows are isomorphisms then shows that × × ϕ∗ : MX,y /OX,y → MX,z /OX,z

is an isomorphism. Since x is a specialization of z in U , the canonical images az and bz × are equal. Hence the canonical images ay and by of a and b in of a and b in MX,z /OX,z × MX,y /OX,y are also equal. It follows that a = b, giving the desired result. II.5.18. Let f : (X, MX ) → (Y, MY ) be a morphism of integral logarithmic schemes. We say that f is integral (resp. saturated) if for every geometric point x of X, the homomorphism MY,f (x) → MX,x is integral (resp. saturated), or, equivalently, if the ] ] homomorphism MY,f (x) → MX,x is integral (resp. saturated). II.5.19. A morphism f : X → Y of integral (resp. fine) logarithmic schemes is integral if and only if for every integral (resp. fine) logarithmic scheme Z and every morphism Z → Y , the fibered product Z ×Y X in the category of logarithmic schemes is integral (resp. fine) ([50] 4.3.1).

70

II. LOCAL STUDY

II.5.20. An integral morphism f : X → Y of fine and saturated logarithmic schemes is saturated if and only if for every fine and saturated logarithmic scheme Z and every morphism Z → Y , the fibered product Z ×Y X in the category of logarithmic schemes is fine and saturated ([74] II 2.13 page 24). II.5.21. We set

Let f : (X, MX ) → (Y, MY ) be a morphism of prelogarithmic schemes.

(II.5.21.1)

Ω1(X,MX )/(Y,MY ) =

Ω1X/Y ⊕ (OX ⊗Z MXgp ) F

,

where Ω1X/Y is the OX -module of relative 1-differentials of X over Y and F is the subOX -module locally generated by the sections of the form (i) (d(αX (a)), 0) − (0, αX (a) ⊗ a) for every local section a of MX ; (ii) (0, 1⊗a) for every local section a of the image of the morphism f [ : f −1 (MY ) → MX . We also denote by (II.5.21.2)

d : OX → Ω1(X,MX )/(Y,MY )

the morphism induced by the universal derivation d : OX → Ω1X/Y and we denote by (II.5.21.3)

d log : MX → Ω1(X,MX )/(Y,MY )

the homomorphism defined, for a local section a of MX , by (II.5.21.4)

d log(a) = 1 ⊗ a.

Then the triple (Ω1(X,MX )/(Y,MY ) , d, d log) is universal for logarithmic derivations. We call Ω1(X,MX )/(Y,MY ) the OX -module of logarithmic 1-differentials of (X, MX ) over (Y, MY ) (or of f ). It satisfies the same functoriality properties as the module Ω1X/Y . If MXa (resp. MYa ) denotes the logarithmic structure on X (resp. Y ) associated with MX (resp. MY ), then f induces a morphism f a : (X, MXa ) → (Y, MYa ), and we have a canonical isomorphism (II.5.21.5)

∼

Ω1(X,MX )/(Y,MY ) → Ω1(X,M a )/(Y,M a ) . X

Y

If f is a morphism of coherent logarithmic schemes, then Ω1(X,MX )/(Y,MY ) is a quasicoherent OX -module. If, moreover, the morphism of schemes underlying f is locally of finite presentation, then Ω1(X,MX )/(Y,MY ) is an OX -module of finite presentation. If f is a strict morphism of logarithmic schemes, then the canonical morphism (II.5.21.6)

Ω1X/Y → Ω1(X,MX )/(Y,MY )

is an isomorphism. II.5.22. A morphism of logarithmic schemes f : (X, MX ) → (Y, MY ) is a closed immersion (resp. exact closed immersion) if the underlying morphism of schemes X → Y is a closed immersion and if the morphism f ∗ (MY ) → MX is an epimorphism (resp. an isomorphism).

II.6. FALTINGS’ ALMOST PURITY THEOREM

II.5.23. (II.5.23.1)

71

Consider a commutative diagram of morphisms of fine logarithmic schemes (Z 0 , MZ 0 )

u0 u

j

 (Z, MZ )

g

/ (X, MX ) 8 f

 / (Y, MY )

where Z 0 is a closed subscheme of Z defined by an ideal I of OZ with square zero and j is an exact closed immersion. We denote by Pf (j, u0 ) the set of (Y, MY )-morphisms u : (Z, MZ ) → (X, MX ) such that u0 = u ◦ j. Then Pf (j, u0 ) is a pseudo-torsor under (II.5.23.2)

HomOZ 0 (u0∗ Ω1(X,MX )/(Y,MY ) , I ).

More precisely ([50] 3.9), if Pf (j, u0 ) is nonempty, then every element u ∈ Pf (j, u0 ) uniquely determines an isomorphism (II.5.23.3)

∼

ϕu : Pf (j, u0 ) → HomOZ 0 (u0∗ Ω1(X,MX )/(Y,MY ) , I )

such that for every v ∈ Pf (j, u0 ), every local section a of OX , and every local section b of MX , we have (II.5.23.4) (II.5.23.5)

ϕu (v)(u0∗ (da)) = v ] (v −1 (a)) − u] (u−1 (a)), ϕu (v)(u0∗ (d log b)) = β − 1,

where β is the unique local section of 1 + I ⊂ OZ× ⊂ MZ such that v [ (v −1 (b)) = β · u[ (u−1 (b)). II.5.24. Let f : X → Y be a morphism of fine logarithmic schemes. We say that f is formally smooth (resp. formally étale) if for every logarithmic scheme Y 0 whose underlying scheme is affine, every exact closed immersion with nilpotent ideal j : Y00 → Y 0 , and every morphism Y 0 → Y , the map (II.5.24.1)

HomY (Y 0 , X) → Hom(Y00 , X)

deduced from j is surjective (resp. bijective). We say that f is smooth (resp. étale) if it is formally smooth (resp. formally étale), if the logarithmic schemes X and Y are coherent, and if the morphism of schemes underlying f is locally of finite presentation. II.5.25. Let f : (X, MX ) → (Y, MY ) be a morphism of fine logarithmic schemes and (N, δ) a fine chart for (Y, MY ) (II.5.13). By ([50] 3.5), f is smooth (resp. étale) if and only if locally for the étale topology on X, f admits a fine chart ((M, γ), (N, δ), ϑ : N → M ) (II.5.14) satisfying the following conditions: (i) the kernel and the torsion subgroup of the cokernel (resp. the kernel and the cokernel) of the homomorphism ϑgp : N gp → M gp are finite of invertible order in X; (ii) the induced morphism X → Y ×AN AM (II.5.14.2) is étale in the classical sense. II.6. Faltings’ almost purity theorem II.6.1. We set S = Spec(OK ) and denote by η (resp. s) its generic (resp. closed) point and by η the generic geometric point corresponding to K (II.2.1). We endow S with the logarithmic structure MS defined by its closed point; in other words, MS = j∗ (Oη× ) ∩ OS , where j : η → S is the canonical injection. We fix a uniformizer π of OK and denote by ι : N → Γ(S, MS ) the homomorphism defined by ι(1) = π, which is a chart for (S, MS ).

72

II. LOCAL STUDY

II.6.2. (II.6.2.1)

For the remainder of this chapter, we fix a morphism of logarithmic schemes f : (X, MX ) → (S, MS ),

a toric chart (P, γ) for (X, MX ) (II.5.13), and a homomorphism ϑ : N → P such that the following conditions are satisfied: (C1 ) The scheme X = Spec(R) is affine and connected. (C2 ) The scheme Xs = X ×S s is nonempty. (C3 ) The triple ((P, γ), (N, ι), ϑ) is a chart for f (II.5.14); in other words, the diagram of homomorphisms (II.6.2.2)

PO

γ

/ Γ(X, MX ) O f[

ϑ ι

N

/ Γ(S, MS )

is commutative or, equivalently, the associated diagram of morphisms of logarithmic schemes (II.6.2.3)

(X, MX )

γa

/ AP

f

 (S, MS )

Aϑ

ιa

 / AN

is commutative. (C4 ) The homomorphism ϑ is saturated (II.5.2). (C5 ) The homomorphism ϑgp : Z → P gp is injective, the order of the torsion subgroup of coker(ϑgp ) is prime to p, and the morphism of usual schemes X → S ×AN AP

(II.6.2.4)

deduced from (II.6.2.3) is étale. (C6 ) Let λ = ϑ(1) ∈ P , (II.6.2.5) (II.6.2.6)

L = HomZ (P gp , Z), H(P ) = Hom(P, N).

Note that H(P ) is a fine, saturated, and sharp monoid and that the canonical homomorphism H(P )gp → Hom((P ] )gp , Z) is an isomorphism ([58] I 2.2.3). We assume that there exist h1 , . . . , hr ∈ H(P ) that are Z-linearly independent in L, such that (II.6.2.7)

ker(λ) ∩ H(P ) = {

r X i=1

ai hi | (a1 , . . . , ar ) ∈ Nr },

where we view λ as a homomorphism L → Z. Recall that P gp is a free Z-module of finite type. Note that we have (II.6.2.8)

S ×AN AP = Spec(OK [P ]/(π − eλ )),

where λ = ϑ(1) (cf. II.5.12). We set (II.6.2.9)

Lλ = HomZ (P gp /λZ, Z),

which we identify with the kernel of the homomorphism L → Z given by y 7→ hy, λi (II.6.2.5). Let d be the rank of Lλ . We denote by X ◦ the maximal open subscheme of

II.6. FALTINGS’ ALMOST PURITY THEOREM

73

X where the logarithmic structure MX is trivial. We have X ◦ = X ×AP AP gp , which is an affine open subscheme of Xη . Let α : P → R be the homomorphism induced by the chart (P, γ). Note that for any t ∈ P , α(t) is invertible on X ◦ ; in particular, α(t) 6= 0. Proposition II.6.3. (i) The morphism f is smooth and saturated. (ii) The scheme X is integral, normal, Cohen–Macaulay, and flat over S. (iii) The scheme X ⊗OK OK is normal. (iv) The scheme X ⊗OK k is reduced. (v) The (usual) scheme X ×S η is smooth over η, X ◦ ×S η is the open complement in X ×S η of a divisor D with strict normal crossings, and MX |(X ×S η) is the logarithmic structure on X ×S η defined by D. (i) This immediately follows from II.5.25 and ([74] Chap. II 3.5). (ii) The last three properties follow from (i), ([50] 4.5), and ([51] 8.2 and 4.1); cf. also ([73] 1.5.1). Since X is moreover noetherian and connected (C1 ), it is then integral. (iii) For every finite extension K 0 of K, with the valuation ring OK 0 , endow S 0 = Spec(OK 0 ) with the logarithmic structure MS 0 defined by its closed point, and let f 0 : (X 0 , MX 0 ) → (S 0 , MS 0 ) be the morphism obtained from f by base change in the category of logarithmic schemes by the canonical morphism (S 0 , MS 0 ) → (S, MS ). Then f 0 is smooth and saturated (II.5.20), and consequently X 0 is normal by (ii). Since X 0 = X ×S S 0 , the assertion follows by taking the direct limit over the finite extensions of K contained in K ([39] 0.6.5.12(ii)). (iv) This follows from (i) and ([74] Chap. II 4.2). (v) Let F be the face of P generated by λ; that is, the set of elements x ∈ P such that there exist y ∈ P and n ∈ N such that x + y = nλ ([58] I 1.4.2). We denote by F −1 P the localization of P by F ([58] I 1.4.4). It immediately follows from the universal properties of localizations of monoids and rings that the canonical homomorphism Z[P ] → Z[F −1 P ] induces an isomorphism ∼

Z[P ]λ → Z[F −1 P ].

(II.6.3.1)

Let P/F (resp. Λ) be the cokernel in the category of monoids of the canonical injection F → P (resp. of the homomorphism ϑ : N → P ) (cf. [58] I 1.1.5). We have canonical isomorphisms ∼

∼

Λ] → P/F → (F −1 P )] .

(II.6.3.2) The canonical homomorphism (II.6.3.3)

Hom(P/F, N) → ker(λ) ∩ H(P )

is an isomorphism. As an amalgamated sum of the saturated homomorphism ϑ and the homomorphism N → 0, Λ is saturated (II.5.2). Consequently, P/F is saturated (II.6.3.2). Therefore, by ([58] I 2.2.3), we have a canonical isomorphism (II.6.3.4)

∼

P/F → Hom(Hom(P/F, N), N).

Condition II.6.2(C6 ) then implies that P/F is a free monoid of finite type. Consequently, ∼ there exists a homomorphism P/F → F −1 P that lifts the canonical isomorphism P/F → −1 ] −1 (F P ) (II.6.3.2) so that the induced homomorphism P/F → Z[F P ] is a chart for AF −1 P ([73] 1.3.1). By (II.6.3.1), we deduce from this a chart (II.6.3.5)

(X ×S η, MX |(X ×S η)) → AP/F .

On the other hand, the logarithmic scheme (X, MX ) is regular by virtue of (i) and ([51] 8.2); cf. also ([57] 2.3) and the proof of ([73] 1.5.1). It then follows from ([42] 0.16.3.7

74

II. LOCAL STUDY

and 0.17.1.7) and the definitions ([51] 2.1 and [57] 2.2) that the scheme X ×S η is regular and therefore smooth over η, that X ◦ ×S η is the open complement of a divisor D with strict normal crossings in X ×S η, and that MX |(X ×S η) is the logarithmic structure on X ×S η defined by D. II.6.4. (II.6.4.1)

For every integer n ≥ 1, we set OKn = OK [ζ]/(ζ n − π),

which is a discrete valuation ring. Let Kn be the field of fractions of OKn and πn the class of ζ in OKn , which is a uniformizer of OKn . We set Sn = Spec(OKn ) which we endow with the logarithmic structure MSn defined by its closed point. We denote by τn : (Sn , MSn ) → (S, MS ) the canonical morphism and by ιn : N → Γ(Sn , MSn ) the homomorphism defined by ιn (1) = πn . Note that ιn is a chart for (Sn , MSn ) and that the diagram (II.6.4.2)

(Sn , MSn )

ιa n

τn

 (S, MS )

/ AN A$n

ι

a

 / AN

is Cartesian (cf. II.5.12). For all integers m, n ≥ 1, we have $mn = $m ◦ $n . We deduce from this a canonical morphism (II.6.4.3)

τm,n : (Smn , MSmn ) → (Sn , MSn )

such that τmn = τn ◦ τm,n . For all integers r, m, n ≥ 1, we have τrm,n = τm,n ◦ τr,mn . Hence the logarithmic schemes (Sn , MSn ) for n ≥ 1 form a cofiltered inverse system indexed by the set Z≥1 ordered by the divisibility relation.

II.6.5. (II.6.5.1)

For every integer n ≥ 1, we set (Xn , MXn ) = (X, MX ) ×AP ,A$n AP

and we denote by ρn : (Xn , MXn ) → (X, MX ) the canonical projection (cf. II.5.12). We also (abusively) denote by ρn : Xn → X the morphism of schemes underlying ρn . Then ρn is finite, the scheme Xn is affine with ring (II.6.5.2)

An = R ⊗Z[P ],$n Z[P ],

and the canonical projection (Xn , MXn ) → AP is strict. Since the diagram (II.6.4.2) is Cartesian, there exists a unique morphism (II.6.5.3)

fn : (Xn , MXn ) → (Sn , MSn ),

II.6. FALTINGS’ ALMOST PURITY THEOREM

75

that fits into the commutative diagram (II.6.5.4)

(Xn , MXn ) NNN NNNfn NNN NN' (Sn , MSn )

/ AN

ρn

/ AP | | Aϑ | | || |} |

A$n

 (S, MS ) p7 f pppp p pp  ppp (X, MX )

A$n

 / AN aBB BB Aϑ BB BB  / AP

We denote by Xn◦ the maximal open subscheme of Xn where the logarithmic structure MXn is trivial. We have Xn◦ = Xn ×AP AP gp and fn (Xn◦ ) = Spec(Kn ). For all integers m, n ≥ 1, we have $mn = $m ◦ $n . We deduce from this a canonical morphism (II.6.5.5)

ρm,n : (Xmn , MXmn ) → (Xn , MXn )

such that ρmn = ρn ◦ ρm,n . For all integers r, m, n ≥ 1, we have ρrm,n = ρm,n ◦ ρr,mn . Hence the logarithmic schemes (Xn , MXn ) form a cofiltered inverse system indexed by the set Z≥1 ordered by the divisibility relation. Proposition II.6.6. Let n be an integer ≥ 1. (i) The morphism of usual schemes Xn → Sn ×AN AP deduced from (II.6.5.4) is étale and the morphism of logarithmic schemes fn is smooth and saturated. (ii) The scheme Xn is normal, Cohen–Macaulay, and flat over Sn . (iii) The scheme Xn ⊗OKn OK is normal and has only a finite number of generic points; in particular, Xn ⊗OKn OK is a finite sum of normal integral schemes. (iv) The morphism Xn ⊗OKn Kn → X ⊗OK K deduced from ρn is flat. If X is moreover regular, then ρn is flat. (v) If n is a power of p, then Xn is integral, and the inverse image of any connected component of X ⊗OK OK under the canonical morphism Xn ⊗OKn OK → X ⊗OK OK

is integral. (vi) The squares in the canonical commutative diagram Xn o

(II.6.6.1)

Xn◦

/ AP gp

 X◦



ρn

 Xo

A$n

/ AP gp

are Cartesian. In particular, Xn◦ is a principal homogeneous space over X ◦ for the étale topology, under the group HomZ (P gp , µn (K)). (i) Indeed, the squares of the commutative diagram of morphisms of usual schemes (II.6.6.2)

Xn ρn

 X

/ Sn ×AN AP

/ AP



 / AP

/ S ×AN AP

A$n

76

II. LOCAL STUDY

deduced from (II.6.5.4) are Cartesian. (ii) This follows from (i) using the same proof as for II.6.3(ii). (iii) It follows from (i) using the same argument as for II.6.3(iii) that Xn ⊗OKn OK is normal. On the other hand, since Xn ⊗OKn OK is flat over OK , its generic points are the generic points of the scheme Xn ⊗OKn K, which is noetherian. Consequently, the set of generic points of Xn ⊗OKn OK is finite. The last assertion follows in view of ([39] 0.2.1.6). Indeed, since Xn ⊗OKn OK is normal, two distinct irreducible components of Xn ⊗OKn OK do not meet. (iv) This follows from ([42] 0.17.3.5) because X ⊗OK K is regular by virtue of II.6.3(v), Xn is Cohen–Macaulay by (ii), and ρn is finite. (v) By (II.6.6.2), we have a Cartesian diagram of S-morphisms (II.6.6.3)

Xn

/ Spec(OK [P ]/(πn − eλ )) n γn

ρn

 X

 / Spec(OK [P ]/(π − eλ )),

where γn is induced by the homomorphism $n . By II.6.2(C2 ), there exists x ∈ Xs . Since n is a power of p, the morphism ρn ⊗OK k is a universal homeomorphism. Hence ρ−1 n (x) contains a single point that we denote by xn . On the other hand, since Xn is flat over Sn , every generic point of Xn is a generic point of Xn ⊗OKn Kn ; consequently, its image by ρn is the generic point of X by virtue of (iv). Since ρn is closed, it follows that xn is a specialization of all generic points of Xn . Since Xn is noetherian and normal by (ii), it is integral. The proof of the second assertion is similar to that of the first one. Indeed, one easily deduces from (II.6.6.3) a Cartesian diagram of OK -morphisms (II.6.6.4)

/ Spec(O [P ]/(πn − eλ )) K

Xn ⊗OKn OK βn





αn

/ Spec(O [P ]/(π − eλ )) K

X ⊗OK OK

By (iii), X ⊗OK OK is a finite sum of normal integral schemes; in particular, it has only a finite number of connected components, which are therefore open. Since X is integral and flat over S by II.6.3(ii), GK acts transitively on the set of connected components of X ⊗OK OK . Let Y be a connected component of X ⊗OK OK and Yn = βn−1 (Y ). In view of II.6.2(C2 ), Y ⊗OK k is nonempty. Let y ∈ Y ⊗OK k. Since n is a power of p, the morphism αn ⊗OK k is a universal homeomorphism. Hence βn−1 (y) contains a single point that we denote by yn . On the other hand, every generic point of Yn lies over the unique generic point of Xn and therefore lies over the generic point of X. Consequently, the image of any generic point of Yn by βn is the generic point of Y . Since βn is closed, it follows that yn is a specialization of all the generic points of Yn . On the other hand, Yn is a finite disjoint union of open and closed subschemes that are integral and normal by (iii). It is therefore integral. (vi) Let us show that the canonical diagram (II.6.6.5)

P  P gp

$n

$n

/P  / P gp

II.6. FALTINGS’ ALMOST PURITY THEOREM

77

is co-Cartesian. Indeed, the amalgamated sum P gp ⊕P,$n P is the quotient of P gp ⊕ P by the congruence relation E defined by the set of pairs ((y, x), (y 0 , x0 )) of elements of P gp ⊕P for which there exist z, z 0 ∈ P such that y + z = y 0 + z 0 ∈ P gp and x + nz = x0 + nz 0 ∈ P ([58] I 1.1.5). It therefore suffices to show that E is the congruence relation defined by the homomorphism (II.6.6.6) 0

0

P gp ⊕ P → P gp , 0

(y, x) → x − ny.

If ((y, x), (y , x )) ∈ E, then x − ny = x − ny . Conversely, suppose x = x0 + n(y − y 0 ) ∈ P gp . Since P is integral, there exist z, z 0 ∈ P such that y +z = y 0 +z 0 ∈ P gp ; we therefore have x + nz = x0 + nz 0 ∈ P , which proves the assertion. Consequently, the diagram / AP (II.6.6.7) AP gp A$n

0

A$n



 / AP

AP gp

induced by (II.6.6.5) is Cartesian. Hence the squares in the diagram (II.6.6.1) are Cartesian. The second assertion follows from the first one and the fact that the kernel of the étale isogeny A$n ⊗Z K of AP gp ⊗Z K corresponds to the Z[GK ]-module HomZ (P gp , µn (K)). II.6.7. Let κ be the generic point of X and F the residue field of X at κ (that is, the field of fractions of R). For the remainder of this chapter, we fix a geometric point κ e of X ⊗OK OK over κ, in other words, the spectrum of a separably closed extension F a of F containing K. We denote by F the union of the finite extensions N of F contained in F a such that the integral closure of R in N is étale over X ◦ , and by R the integral closure of R in F . For all integers m, n ≥ 1, the morphism ρm,n : Xmn → Xn is finite and surjective. Consequently, by virtue of ([42] 8.3.8(i)), there exists an X-morphism (II.6.7.1)

κ e → lim Xn , ←−

n≥1

where the inverse limit is indexed by the set Z≥1 ordered by the divisibility relation. For the remainder of this chapter, we fix such a morphism. Since the set of integers n!, for n ≥ 0, is cofinal in Z≥1 for the divisibility relation, this corresponds to fixing a morphism (II.6.7.2)

κ e → lim Xn! , ←−

n≥0

where the inverse limit is indexed by the set N with the usual ordering. By II.6.6(vi), the morphism (II.6.7.1) factors through an X-morphism (II.6.7.3)

Spec(R) → lim Xn . ←−

n≥1

We deduce from this a direct system of R-homomorphisms un : An → R, indexed by the set Z≥1 ordered by the divisibility relation. We denote by Bn the image of un and set (II.6.7.4)

B∞ = lim Bn , −→

n≥1

which we identify with a sub-R-algebra of R. We denote by Hn the field of fractions of Bn and by H∞ the field of fractions of B∞ . On the other hand, the morphism (II.6.7.3) induces a morphism (II.6.7.5)

Spec(OK ) → lim Sn . ←−

n≥1

78

II. LOCAL STUDY

We can therefore extend the un ’s to a direct system of (R ⊗OK OK )-homomorphisms (II.6.7.6)

vn : An ⊗OKn OK → R,

indexed by the set Z≥1 ordered by the divisibility relation. We denote by Rn the image of vn and set (II.6.7.7)

R∞ = lim Rn , −→

n≥1

which we identify with a sub-(R ⊗OK OK )-algebra of R. We denote by Fn the field of fractions of Rn and by F∞ the field of fractions of R∞ . We set (II.6.7.8)

Rp∞ = lim Rpn , −→

n≥0

where the direct limit is indexed by the set N with the usual ordering. We identify Rp∞ with a sub-(R ⊗OK OK )-algebra of R∞ and denote the field of fractions of Rp∞ by Fp∞ . Proposition II.6.8. (i) For every n ≥ 1, Spec(Bn ) is an open connected component of Xn and Spec(Rn ) is an open connected component of Xn ⊗OKn OK . (ii) The rings Bn , Rn (n ≥ 1), B∞ , R∞ , and Rp∞ are normal integral domains. (iii) For every n ≥ 0, we have (II.6.8.1) (II.6.8.2)

Spec(Bpn ) = Xpn , Spec(Rpn ) = (Xpn ⊗OKn OK ) ×(X⊗OK OK ) Spec(R1 ).

(iv) The extensions Fn (n ≥ 1), F∞ , and Fp∞ of F are Galois and we have canonical injective homomorphisms (II.6.2.9) (II.6.8.3) (II.6.8.4)

b Gal(F∞ /F1 ) → Lλ ⊗ Z(1), ∼

Gal(Fp∞ /F1 ) → Lλ ⊗ Zp (1),

where the second is an isomorphism. Moreover, the diagram (II.6.8.5)

Gal(F∞ /F1 )

/ L ⊗ Z(1) b λ

 Gal(Fp∞ /F1 )

 / Lλ ⊗ Zp (1),

where the vertical arrows are the canonical morphisms, is commutative. By II.6.6(vi), for any n ≥ 1, the morphism (II.6.7.1) induces a morphism (II.6.8.6)

κ e → Xn◦ ⊗Kn K.

We denote by κ en its image, which is a generic point of Xn◦ ⊗Kn K. We denote by κn the image of κ en in Xn◦ , which is a generic point of Xn◦ . Note that Hn is the residue field of Xn at κn and that Fn is the residue field of Xn ⊗OKn OK at κ en . (i) Since Xn is noetherian and normal by II.6.6(ii), Spec(Bn ) is the connected component of Xn containing κn , which is obviously open in Xn . In view of II.6.6(iii), Spec(Rn ) is the connected component of Xn ⊗OKn OK containing κ en , which is open in Xn ⊗OKn OK . (ii) These rings are clearly integral domains. For any n ≥ 1, Bn and Rn are normal by virtue of (i) and II.6.6(ii)-(iii). The same therefore holds for B∞ , R∞ , and Rp∞ . (iii) This follows from (i) and II.6.6(v). (iv) It follows from II.6.6(vi) that for every integer n ≥ 1, Hn is a Galois extension of F , whose Galois group is canonically isomorphic to a subgroup of L ⊗Z µn (OK ); more

II.6. FALTINGS’ ALMOST PURITY THEOREM

79

precisely, Gal(Hn /F ) is the decomposition subgroup of κn . By II.6.6(v), if n is a power of p, we have Gal(Hn /F ) ' L ⊗Z µn (OK ).

(II.6.8.7)

Let Mn be the image of the canonical homomorphism F ⊗K Kn → Hn , so that Mn is a Galois extension of F whose Galois group is a subgroup of Gal(Kn /K). We then have a commutative diagram of field extensions / Fn (II.6.8.8) HO n O FO

/ Mn O

/ F1 O

K

/ Kn

/K

where the homomorphisms F ⊗K Kn → Mn , Mn ⊗Kn K → F1 , and Hn ⊗Mn F1 → Fn are surjective. It follows that F1 is a Galois extension of F , with Galois group canonically isomorphic to a subgroup of GK . Consequently, Fn is a Galois extension of F , since it is the composition of the Galois extensions Hn and F1 of F . In particular, Fn is a Galois extension of F1 , with Galois group canonically isomorphic to a subgroup of Gal(Hn /Mn ). It follows from II.6.6(v) that if n is a power of p, we have Gal(Fn /F1 ) ' Gal(Hn /Mn ).

(II.6.8.9)

On the other hand, we have a commutative diagram (II.6.8.10)

Gal(Hn /F )

/ L ⊗Z µn (O ) K

  Gal(Mn /F ) 

 / µn (O ), K

λn

/ Gal(Kn /K)

∼

where λn is the morphism defined by λ ∈ P (II.6.2.5) and the unlabeled arrows are the canonical morphisms. It follows that Gal(Fn /F1 ) is canonically isomorphic to a subgroup of ker(λn ). If n is a power of p, then the isomorphisms (II.6.8.7) and (II.6.8.9) and a chase in the diagram (II.6.8.10) show that Gal(Fn /F1 ) ' ker(λn ).

(II.6.8.11)

The proposition follows by taking the inverse limit over n. Remark II.6.9. Note that condition II.6.2(C6 ) is not used in the proofs of II.6.6 and II.6.8. In Proposition II.6.3, it is only used in the proof of (v). II.6.10. We set Γ = Gal(F /F ), Γ∞ = Gal(F∞ /F ), Γp∞ = Gal(Fp∞ /F ), ∆ = Gal(F /F1 ), ∆∞ = Gal(F∞ /F1 ), ∆p∞ = Gal(Fp∞ /F1 ), Σ = Gal(F /F∞ ), and Σ0 = Gal(F∞ /Fp∞ ). Γ ∆∞

(II.6.10.1)

/ R1 O

RO

∆p∞

/ Rp∞

Σ0

∆

OK

GK

/O K

' / R∞

Σ

'/

5R

80

II. LOCAL STUDY

b By II.6.8(iv), ∆∞ is canonically isomorphic to a subgroup of Lλ ⊗Z Z(1), ∆p∞ is canonically isomorphic to Lλ ⊗Z Zp (1), and Σ0 is a profinite group of order prime to p. We denote by K + the extension of K contained in K such that Gal(K/K + ) is the image of the canonical homomorphism Gal(F1 /F ) → GK . We have K + = K if and only if X ×S η is integral or, equivalently, connected. II.6.11. Let M be a p-primary torsion discrete Z-∆∞ -module. Since the p-cohomological dimension of Σ0 is zero ([65] I Cor. 2 to Prop. 14), for any q ≥ 0, the canonical morphism Hq (∆p∞ , M Σ0 ) → Hq (∆∞ , M )

(II.6.11.1)

is an isomorphism. Consequently, the p-cohomological dimension of ∆∞ is equal to that of ∆p∞ ; that is, to the rank d of Lλ (II.3.24). Remarks II.6.12. Let A be a Zp -algebra that is complete and separated for the p-adic topology and M an A-module that is complete and separated for the p-adic topology. Then: (i) The canonical homomorphisms (II.6.12.1)

HomZp (∆p∞ , M ) → HomZ (∆p∞ , M ) → HomZ (∆∞ , M )

are bijective. Indeed, since the multiplication by p in Σ0 is an isomorphism, for every homomorphism ψ : Σ0 → M , we have ψ(Σ0 ) ⊂ ∩n≥0 pn M = 0. (ii) The canonical morphism (II.6.12.2)

HomZ (∆∞ , A) ⊗A M → HomZ (∆∞ , M )

is bijective. This follows from (i) because ∆p∞ is a free Zp -module of finite type. (iii) Every homomorphism from ∆∞ (resp. ∆p∞ ) to M is continuous when we endow ∆∞ (resp. ∆p∞ ) with the profinite topology and M with the p-adic topology. Indeed, every homomorphism ψ : ∆∞ → M factors through a homomorphism ϕ : ∆p∞ → M by (i), and ϕ is clearly continuous for the p-adic topologies. In particular, when we endow M with the trivial action of ∆p∞ , the canonical morphisms (II.6.12.3) (II.6.12.4)

HomZ (∆p∞ , M ) → H1cont (∆p∞ , M ), HomZ (∆∞ , M ) → H1cont (∆∞ , M ),

are isomorphisms. Lemma II.6.13. For every a ∈ OK , the canonical homomorphism (II.6.10) (II.6.13.1)

Rp∞ /aRp∞ → (R∞ /aR∞ )Σ0

is an isomorphism. Let N be a finite Galois extension of Fp∞ contained in F∞ , A the integral closure of Rp∞ in N , and G = Gal(N/Fp∞ ). Since we have Rp∞ = A ∩ Fp∞ and A = R∞ ∩ N , the canonical homomorphisms Rp∞ /aRp∞ → A/aA → R∞ /aR∞ are injective. On the other hand, since the order of G is prime to p (and therefore invertible in A), the homomorphism (II.6.13.2)

Rp∞ = AG → (A/aA)G

is surjective. The assertion follows. b c1 , R d d Lemma II.6.14. The rings R p∞ , R∞ , and R are OC -flat and the canonical homob c1 → R d d d d morphisms R p∞ , Rp∞ → R∞ , and R∞ → R are injective.

II.6. FALTINGS’ ALMOST PURITY THEOREM

81

Since R1 , Rp∞ , and R∞ are normal by II.6.8(ii), we have, for every n ≥ 0,

pn R1 = (pn Rp∞ ) ∩ R1 ,

pn Rp∞ = (pn R∞ ) ∩ Rp∞ ,

and pn R∞ = (pn R) ∩ R∞ .

b c1 → R d d d d It follows that the canonical homomorphisms R p∞ , Rp∞ → R∞ , and R∞ → R are injective. On the other hand, by ([11] Chap. III § 2.11 Prop. 14 and Cor. 1), for every n ≥ 0, we have c1 /pn R c1 ' R1 /pn R1 . R

(II.6.14.1)

c1 be an element satisfying px = 0 and x the class of x in R c1 /pn R c1 (n ≥ 1). Let x ∈ R n−1 c nc R1 /p R1 . Consequently, Since R1 is OK -flat, it follows from (II.6.14.1) that x ∈ p c1 = {0}. Therefore p is not a zero divisor in R c1 and hence R c1 is OC -flat x ∈ ∩n≥0 pn R b d d ([11] Chap. VI § 3.6 Lem. 1). The same argument shows that R ∞ , Rp∞ , and R are flat over OC . c1 is normal. Proposition II.6.15. The ring R c1 is an OC -algebra that is topologically of finite presentation ([1] First note that R c1 [ 1 ] is an affinoid algebra over C. 1.10.4), and then that R p c1 [ 1 ] is normal. We identify R c1 with the p-adic Hausdorff comLet us show that R p

c1 be the canonical homomorphism. Let q pletion of B = R1 ⊗OK OC and let ϕ : B → R c1 [ 1 ] and p = ϕ−1 (q). By virtue of ([1] 1.12.18), the canonical be a maximal ideal of R p c1 )q induces an isomorphism between the Hausdorff completions homomorphism Bp → (R of these local rings for the topologies defined by the respective maximal ideals. The scheme Spec(B[ p1 ]) endowed with the inverse image of the logarithmic structure MX is smooth over Spec(C) endowed with the trivial logarithmic structure. Consequently, B[ p1 ] is normal by virtue of ([51] 4.1 and 8.2). Since B[ p1 ] is an excellent ring, it follows c1 [ 1 ] for each of its maximal ideals from the above that the Hausdorff completions of R p

c1 [ 1 ] is an excellent ring ([6] 3.3.3), its localizations are normal ([42] 7.8.3(v)). Since R p c1 [ 1 ] is normal at each of its maximal ideals are normal ([42] 7.8.3(v)). Consequently, R p

([42] 7.8.3(iv)). c1 [ 1 ], set For every h ∈ R p |h|sup =

(II.6.15.1)

sup c1 [ 1 ]) x∈Max(R p

|h(x)|,

c1 [ 1 ]) is the maximal spectrum of R c1 [ 1 ] or, equivalently, the set of rigid where Max(R p p c1 ) ([1] 3.3.2). It is a multiplicative seminorm on R c1 [ 1 ] ([7] 6.2.1/1). Let points of Spf(R p c1 [ 1 ] | |h|sup ≤ 1}. Bsup = {h ∈ R p

(II.6.15.2)

c1 is OC -flat (II.6.14) and R c1 [ 1 ] is reduced, we have R c1 ⊂ Bsup and Bsup is the Since R p 1 c1 in R c1 [ ] ([7] 6.3.4/1 and 6.2.2/3). Since R1 ⊗O k is reduced by integral closure of R p K c1 = Bsup by ([7] 6.4.3/4; cf. also [8] 1.1). Hence R c1 is integrally II.6.3(iv) and II.6.8(i), R 1 c closed in R1 [ ] and therefore normal. p

Theorem II.6.16 (Faltings’ almost purity theorem, [26] § 2b). For every finite extension N of H∞ contained in F , the integral closure of B∞ in N is almost étale over B∞ (II.6.7.4).

82

II. LOCAL STUDY

Let us note here that Scholze, in the setting of his theory of perfectoids, gives a generalization of this result ([64] 1.10 and 7.9). Corollary II.6.17. The extension F of F∞ is the union of a filtered direct system of finite Galois subextensions E of F∞ such that the integral closure of R∞ in E is almost étale over R∞ (II.6.7.7). Let N be a finite extension of H∞ contained in F , E the image of the canonical homomorphism N ⊗H∞ F∞ → F , and N (resp. E ) the integral closure of R in N (resp. E). We know (II.6.16) that N is almost étale over B∞ . Hence N ⊗B∞ R∞ is almost étale over R∞ by V.7.4. Consequently, E is almost étale over R∞ by V.7.11 and V.7.4. If N/H∞ is a Galois extension, the same holds for E/F∞ , giving the corollary. Corollary II.6.18. For every subring A of R∞ , the canonical morphism Ω1R∞ /A ⊗R∞ R → Ω1R/A

(II.6.18.1) is an almost isomorphism.

This follows from II.6.17 and ([24] I 2.4(i)). Corollary II.6.19. Let M be an R-module endowed with a continuous R-semi-linear action of Σ for the discrete topology on M . Then Hi (Σ, M ) is almost zero for all i ≥ 1, and the canonical morphism M Σ ⊗R∞ R → M is an almost isomorphism. Let N be a finite Galois extension of F∞ contained in F , D the integral closure of R∞ in N , G = Gal(N/F∞ ), and ΣN = Gal(F /N ). Suppose that D is almost étale over R∞ . Then D is an almost G-torsor over R∞ by V.12.9. Using V.12.5 and V.12.8, we deduce that for every i ≥ 1, Hi (G, M ΣN ) is almost zero, and the canonical morphism M Σ ⊗R∞ D → M ΣN is an almost isomorphism. The corollary follows by taking the direct limit, by virtue of II.6.17. Corollary II.6.20. Let M be an R-module endowed with a continuous R-semi-linear action of ∆ for the discrete topology on M . Then the canonical morphism Hi (∆∞ , M Σ ) → Hi (∆, M )

is an almost isomorphism for every i ≥ 0.

This follows from II.6.19 and from the spectral sequence E1ij = Hi (∆∞ , Hj (Σ, M )) ⇒ Hi+j (∆, M ).

(II.6.20.1)

b Corollary II.6.21. Let (Mn )n∈N be an inverse system of R-representations of Σ (II.3.1) and M its inverse limit. We suppose that for every n ≥ 0, Mn is annihilated by a power of p, that the action of Σ on Mn is continuous for the discrete topology, and that the morphism Mn+1 → Mn is surjective. Then Hicont (Σ, M ) is almost zero for every integer i ≥ 1. Indeed, by (II.3.10.4) and (II.3.10.5), we have (II.6.21.1)

0 → R1 lim Hi−1 (Σ, Mn ) → Hicont (Σ, M ) → lim Hi (Σ, Mn ) → 0. ←−

←−

n

n

For every q ≥ 1, lim Hq (Σ, Mn ) and R1 lim Hq (Σ, Mn ) are almost zero by virtue of II.6.19 ←−

←−

n

n

and ([32] 2.4.2(ii)). For every n ≥ 0, let Cn be the kernel of the surjective morphism Mn+1 → Mn , so that we have an exact sequence (II.6.21.2)

Σ Mn+1

ψn

/ MΣ n

/ H1 (Σ, Cn ).

II.6. FALTINGS’ ALMOST PURITY THEOREM

83

Then coker(ψn ) is almost zero by virtue of II.6.19. It follows that R1 lim MnΣ is almost ←− n

zero by ([32] 2.4.2(iii) and 2.4.3), giving the corollary. Corollary II.6.22. For every a ∈ OK , the canonical homomorphism R∞ /aR∞ → (R/aR)Σ

(II.6.22.1) is an almost isomorphism.

Let N be a finite Galois extension of F∞ contained in F , D the integral closure of R∞ in N , GP= Gal(N/F∞ ), and TrG the R∞ -linear endomorphism of D (or of D/aD) induced by σ∈G σ. Since we have D = R ∩ N and R∞ = D ∩ F∞ , by II.6.8(ii), the homomorphisms R∞ /aR∞ → D/aD → R/aR are injective. Suppose that D is almost étale over R∞ . Then D is an almost G-torsor over R∞ by virtue of V.12.9. Consequently, the quotient (D/aD)G TrG (D/aD) is almost zero by V.12.8. Since TrG (D) ⊂ R∞ , the homomorphism R∞ /aR∞ → (D/aD)G is an almost isomorphism. The corollary follows by taking the direct limit, by II.6.17. bΣ d Corollary II.6.23. The canonical homomorphism R ∞ → R is an almost isomorphism. Indeed, this homomorphism is the inverse limit of the homomorphisms (r ≥ 0) R∞ /pr R∞ → (R/pr R)Σ .

(II.6.23.1)

This can be easily verified or follows from (II.3.10.5). The corollary therefore follows from II.6.22 and ([32] 2.4.2(ii)). Corollary II.6.24. For every nonzero element a of OK and every integer i ≥ 0, the canonical morphisms (II.6.24.1) (II.6.24.2)

Hi (∆p∞ , Rp∞ /aRp∞ ) → Hi (∆, R/aR), Hi (∆∞ , R∞ /aR∞ ) → Hi (∆, R/aR),

are almost isomorphisms.

Indeed, (II.6.24.2) is an almost isomorphism by II.6.20 and II.6.22. On the other hand, the canonical morphism (II.6.24.3)

Hi (∆p∞ , Rp∞ /aRp∞ ) → Hi (∆∞ , R∞ /aR∞ )

is an isomorphism by II.6.13 and (II.6.11.1).

Corollary II.6.25. For every integer i ≥ 0, the canonical morphisms b d ∞) (II.6.25.1) Hi (∆ ∞ , R → Hi (∆, R), cont

p

p

d Hicont (∆∞ , R ∞)

(II.6.25.2) are almost isomorphisms.

cont

b → Hicont (∆, R),

For every integer r ≥ 0, let

(II.6.25.3)

ψr : Hi (∆∞ , R∞ /pr R∞ ) → Hi (∆, R/pr R)

be the canonical homomorphism and Ar (resp. Cr ) its kernel (resp. cokernel). We know that the OK -modules Ar and Cr are almost zero by II.6.24. Hence the OK -modules lim Ar ,

lim Cr ,

r≥0

r≥0

←−

←−

R1 lim Ar , ←− r≥0

R1 lim Cr ←− r≥0

84

II. LOCAL STUDY

are almost zero by ([32] 2.4.2(ii)). It follows that the morphisms lim ψr ←− r≥0

and

R1 lim ψr ←− r≥0

are almost isomorphisms, and the same holds for (II.6.25.2) in view of (II.3.10.4) and (II.3.10.5). The proof that (II.6.25.1) is an almost isomorphism is similar.

II.7. Faltings extension II.7.1. Let K0 be the field of fractions of W(k) (II.2.3) and DK/K0 the different of the extension K/K0 . By ([30] Thm. 1’), there exists a unique GK -equivariant OK -linear morphism (II.7.1.1)

φ : K ⊗Zp Zp (1) → Ω1OK /OK ,

such that for all ζ ∈ Zp (1), a ∈ OK , and r ∈ N, if ζr ∈ µpr (OK ) is the canonical image of ζ, then (II.7.1.2)

φ(p−r a ⊗ ζ) = a · d log(ζr ).

It is surjective with kernel ρ−1 OK (1), where ρ is an element of OK with valuation v(DK/K0 ). II.7.2. (II.7.2.1)

1 p−1

+

The morphism (II.6.7.5) induces an OK -homomorphism lim OKn → OK , −→

n≥1

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation. For every n ≥ 1, we will from now on identify OKn with a sub-OK -algebra of OK ; in m particular, we view πn as an element of OK (II.6.4). We have π1 = π and πmn = πn for all m, n ≥ 1. For every integer n ≥ 1, there exists an element of Ω1O /OK , which we denote by K d log(πn ), such that for every integer m ≥ 1 divisible by p, we have (II.7.2.2)

d log(πn ) =

m dπmn ∈ Ω1OK /OK . πmn

Indeed, m ∈ πmn OKmn , so that m/πmn ∈ OKmn , and for every m0 ≥ 1, we have (II.7.2.3)

m m 0 m0 −1 mm0 dπmn = m πmm0 n dπmm0 n = dπmm0 n ∈ Ω1OK /OK . πmn πmn πmm0 n

Remember that the element d log(πn ) depends not only on πn , but also on the homomorphism (II.7.2.1). For all integers m, n ≥ 1, we have (II.7.2.4) (II.7.2.5)

πn d log(πn ) = dπn , d log(πn ) = md log(πmn ).

Since the canonical morphisms Ω1OK /OK → Ω1O /OK are injective ([30] 2.4 Lem. 4), n K for every integer n ≥ 1, the annihilator of d log(πn ) in Ω1O /OK is nπOK ([30] 2.1 Lem. 1). K

II.7. FALTINGS EXTENSION

85

II.7.3. Consider the direct system of monoids (N(n) )n≥1 , indexed by the set Z≥1 ordered by the divisibility relation, defined by N(n) = N for every n ≥ 1 and whose transition homomorphism λn,mn : N(n) → N(mn) (for m, n ≥ 1) is the Frobenius homomorphism $m of order m of N (II.5.12). We denote by N∞ its direct limit, N∞ = lim N(n) .

(II.7.3.1)

−→

n≥1

For every n ≥ 1, we denote by an : N(n) → OK the homomorphism defined by an (1) = πn . For all integers m, n ≥ 1, the diagram (II.7.3.2)

N(n)

an

/O

K

/O

K

λn,mn

 N(mn)

amn

is commutative. By taking the direct limit, the an ’s therefore define a homomorphism a∞ : N∞ → OK .

(II.7.3.3)

We will denote N(1) (resp. a1 ) simply by N (resp. a). Lemma II.7.4. Let n be an integer ≥ 1. Then: (i) We have canonical isomorphisms (II.7.4.1)

Ω1(O

(II.7.4.2)

Ω1(O

(n) )/(O ,N) K K ,N

K ,N

(n) )/(O

K ,N)

'

Ω1O

K /OK

⊕ OK /nOK

(dπn − πn )OK

,

' OK /(nOK + πn OK ).

(ii) If p divides n, the kernel of the canonical morphism Ω1OK /OK → Ω1(O

(II.7.4.3)

(n) )/(O ,N) K K ,N

is generated by d log(π) (II.7.2.2). (i) We have a commutative diagram with Cartesian square jn / Spec(O [ξ]/(ξ n − π)) Spec(OK ) K RRR RRR RRR RRR R(  Spec(OK )

(II.7.4.4)

/ Spec(O [N(n) ]) K 

 / Spec(OK [N])

where jn is the closed immersion defined by the equation ξ − πn . On the other hand, we have a canonical isomorphism (II.7.4.5)

Ω1(O

(n) ],N(n) )/(O [N],N) K K [N

' Ω1OK /OK ⊗OK OK [N(n) ] ⊕ OK [N(n) ]/nOK [N(n) ].

The isomorphism (II.7.4.1) follows directly from this. The isomorphism (II.7.4.2) is proved using a diagram analogous to (II.7.4.4). (ii) Let ω ∈ Ω1O /OK be such that its image in Ω1(O ,N(n) )/(OK ,N) is zero. By (II.7.4.1), K

K

there exists x ∈ OK such that ω = x(dπn − πn ) ∈ Ω1O /OK ⊕ OK /nOK . Consequently, K xπn ∈ nOK and ω = xdπn ∈ OK d log(π). Conversely, we have (II.7.4.6)

d log(π) = (n/πn )(dπn − πn ) ∈ Ω1OK /OK ⊕ OK /nOK .

Hence the image of d log(π) by the morphism (II.7.4.3) is zero by virtue of (II.7.4.1).

86

II. LOCAL STUDY

Proposition II.7.5. The canonical morphism Ω1OK /OK → Ω1(OK ,N∞ )/(OK ,N)

(II.7.5.1)

is surjective and its kernel is generated by d log(π). In particular, the morphism φ (II.7.1.1) induces a surjective OK -linear morphism (II.7.5.2) with kernel (πρ)−1 OK (1).

K ⊗Zp Zp (1) → Ω1(OK ,N∞ )/(OK ,N)

Note that the second statement is an immediate consequence of the first and II.7.1. Let us prove the first statement. By the universal property of modules of logarithmic 1-differentials, the canonical morphism lim Ω1(O

(II.7.5.3)

K ,N

−→

(n) )/(O

K ,N)

n≥1

→ Ω1(OK ,N∞ )/(OK ,N) ,

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation, is an isomorphism. It therefore follows from II.7.4(ii) that the kernel of the morphism (II.7.5.1) is generated by d log(π). On the other hand, we have a canonical isomorphism lim Ω1(O

(II.7.5.4)

∼

K ,N

−→

(n) )/(O

K ,N)

n≥1

→ Ω1(OK ,N∞ )/(OK ,N) .

By (II.7.4.2), for every integer n ≥ 1, we have  0 1 (II.7.5.5) Ω(O ,N(n) )/(O ,N) = K K kn ⊗OKn OK

if (n, p) = 1, if p|n,

where kn is the residue field of OKn . Observe that for all integers m, n ≥ 1, the canonical morphism Ω1(O

(II.7.5.6)

K ,N

(n) )/(O ,N) K

→ Ω1(O

(nm) )/(O ,N) K ,N K

identifies with m times the canonical morphism kn ⊗OKn OK → kmn ⊗OKmn OK . It follows that Ω1(O ,N∞ )/(O ,N) = 0 and consequently that the morphism (II.7.5.1) is surjective. K

K

Remark II.7.6. For any integer n ≥ 1, the canonical image of the element d log(πn ) ∈ Ω1O /OK (II.7.2.2) in Ω1(O ,N∞ )/(OK ,N) equals the element d log(1(n) ) ∈ Ω1(O ,N(n) )/(OK ,N) , K

K

∼

K

where 1(n) denotes the image of 1 by the canonical isomorphism N → N(n) . Indeed, in Ω1(O ,N(pn) )/(OK ,N) , we have K

(II.7.6.1)

d log(πn ) − d log(1(n) ) =

p (dπpn − πpn d log(1(pn) )) = 0. πpn

Note that the equality d log(πn ) = d log(1(n) ) holds in Ω1(O ,N(mn) )/(OK ,N) for every integer K m ≥ 1 divisible by p (but in general it does not hold when m = 1). Lemma II.7.7. For every flat OK -algebra A, the OK -modules

(II.7.7.1)

Ω1OK /OK ⊗OK A

and

Ω1(OK ,N∞ )/(OK ,N) ⊗OK A

do not have any nonzero mK -torsion. In view of (II.7.1.1) and (II.7.5.2), it suffices to show that the OK -module A[ p1 ]/A does not have any nonzero mK -torsion. Since the mK -torsion is contained in the p-torsion, it therefore suffices to show that the OK -module A/pA does not have any nonzero mK torsion. A computation of valuations show that OK /pOK does not have any nonzero mK -torsion; in other words, the morphism (II.7.7.2)

OK /pOK → ⊕n≥1 OK /pOK ,

x 7→ (p1/n x)n≥1

II.7. FALTINGS EXTENSION

87

is injective. The same then holds for the morphism obtained by extending the scalars from OK to A, giving the statement. II.7.8. Consider the direct system of monoids (P (n) )n≥1 , indexed by the set Z≥1 ordered by the divisibility relation, defined by P (n) = P for every n ≥ 1 and with transition homomorphism in,mn : P (n) → P (mn) (for m, n ≥ 1) given by the Frobenius homomorphism $m of order m of P (II.5.12). We denote by P∞ its direct limit, P∞ = lim P (n) .

(II.7.8.1)

−→

n≥1

For every n ≥ 1, we denote by (II.7.8.2)

∼

P → P (n) ,

t 7→ t(n) ,

the canonical isomorphism. For all m, n ≥ 1 and t ∈ P , we have in,mn (t(n) ) = (t(mn) )m and consequently t(n) = (t(mn) )m ∈ P∞ .

(II.7.8.3)

For every n ≥ 1, we denote by αn : P (n) → Rn the homomorphism induced by the canonical strict morphism (Xn , MXn ) → AP (II.6.5.1). The reader will remember that Rn depends on the choice of the morphism (II.6.7.1). For all integers m, n ≥ 1, the diagram (II.7.8.4)

P (n)

αn

/ Rn

αmn

 / Rmn

in,mn

 P (mn)

is commutative. By taking the direct limit, the αn ’s therefore define a homomorphism α∞ : P∞ → R∞ .

(II.7.8.5)

We also denote by α∞ : P∞ → R the composition of α∞ and the canonical injection R∞ → R. We will simply write P for P (1) . Note that α1 factors through α (II.6.2). Proposition II.7.9. (i) The canonical sequence (II.7.9.1)

0 → Ω1(R,P )/(OK ,N) ⊗R R∞ → Ω1(R∞ ,P∞ )/(OK ,N) → Ω1(R∞ ,P∞ )/(R1 ,P ) → 0

is exact. (ii) For every integer m ≥ 0, the morphism (II.7.9.2)

Hom(p−m Z/Z, Ω1(R∞ ,P∞ )/(R1 ,P ) ) → Ω1(R,P )/(OK ,N) ⊗R (R∞ /pm R∞ )

deduced from (II.7.9.1) using the snake lemma is an isomorphism. (iii) There exists a canonical R∞ -linear isomorphism (II.7.9.3)

1 ∼ (P gp /λZ) ⊗Z (R∞ [ ]/R∞ ) → Ω1(R∞ ,P∞ )/(R1 ,P ) . p

(i) By II.6.8(i), we have a canonical isomorphism (II.7.9.4)

∼

Ω1(R1 ,P )/(OK ,N) → Ω1(R,P )/(OK ,N) ⊗R R1 .

On the other hand, we have a canonical isomorphism (II.7.9.5)

lim Ω1(Rn ,P (n) )/(O −→

n≥1

K ,N)

∼

→ Ω1(R∞ ,P∞ )/(OK ,N) ,

88

II. LOCAL STUDY

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation. It therefore suffices to show that for every n ≥ 1, the canonical morphism Ω1(R1 ,P )/(OK ,N) ⊗R1 Rn → Ω1(Rn ,P (n) )/(O

(II.7.9.6)

K ,N)

is injective. Recall that Spec(Rn ) is an open connected component of Xn ⊗OKn OK (II.6.8), that the canonical morphism X → Spec(OK [P ]/(π − eλ )) is étale (II.6.2.4), and that we have a Cartesian diagram of OK -morphisms (II.6.6.4) (II.7.9.7)

Xn ⊗OKn OK

/ Spec(O [P (n) ]/(π − eλ(n) )) n K

 X ⊗OK OK

 / Spec(O [P ]/(π − eλ )) K

where λ(n) is the image of λ in P (n) by the isomorphism (II.7.8.2). Consider the commutative diagram with Cartesian squares (II.7.9.8) (n) / Spec(O [P (n) ]/(π − enλ(n) )) / Spec(O [P (n) ]) Spec(OK [P (n) ]/(πn − eλ )) K K VVVV VVVV VVVV ia  n VVVV VVV+   / Spec(O [P ]) Spec(OK [P ]/(π − eλ )) K 

 Spec(OK )

 / Spec(O [N]) K

where ian is the morphism induced by the canonical homomorphism in : P → P (n) . We have a canonical isomorphism ∼

Ω1(OK [P ],P )/(OK [N],N) → (P gp /λZ) ⊗Z OK [P ]

(II.7.9.9)

such that for every x ∈ P , the image of d log(x) is the class of x in P gp /λZ. We can therefore identify the morphism Ω1(OK [P ],P )/(OK [N],N) ⊗OK [P ] OK [P (n) ] → Ω1(O

K [P

induced by (II.7.9.10)

ian

with the OK [P

(n)

(n) ],P (n) )/(O

K [N],N)

]-linear morphism

(P gp /λZ) ⊗Z OK [P (n) ] → (P gp /nλZ) ⊗Z OK [P (n) ]

deduced from multiplication by n in P gp . It follows from the diagram (II.7.9.8) and what precedes it that we have a canonical isomorphism (II.7.9.11)

(P gp /nλZ) ⊗Z Rn ∼ 1 → Ω(Rn ,P (n) )/(O ,N) , K λ ⊗ πn Rn

where λ denote the class of λ in P gp /nλZ. We deduce from this a canonical surjective morphism (II.7.9.12)

Ω1(Rn ,P (n) )/(O

K ,N)

→ (P gp /λZ) ⊗Z Rn .

In view of (II.7.9.10), we identify the composition of the morphisms (II.7.9.6) and (II.7.9.12) with the multiplication by n in (P gp /λZ) ⊗Z Rn , which is injective because the torsion subgroup of P gp /λZ has order prime to p and Rn is flat over Zp . Consequently, the morphism (II.7.9.6) is injective.

II.7. FALTINGS EXTENSION

89

(ii) It suffices to show that the multiplication by pm in Ω1(R∞ ,P∞ )/(O ,N) is an isoK morphism. For all integers n, n0 ≥ 1, we have a commutative diagram (II.7.9.13)

/ (P gp /nn0 λZ) ⊗Z Rnn0

(P gp /nλZ) ⊗Z Rn 

un,nn0 Ω1(Rn ,P (n) )/(O ,N) K



/ Ω1

(Rnn0 ,P (nn0 ) )/(OK ,N)

where the vertical arrows are the surjective morphisms deduced from (II.7.9.11), un,nn0 is the canonical morphism, and the top horizontal arrow is induced by the multiplication by n0 in P gp and the canonical homomorphism Rn → Rnn0 . It follows that un,nn0 (Ω1(Rn ,P (n) )/(O

(II.7.9.14)

K ,N)

) ⊂ n0 · Ω1(R

nn0 ,P

(nn0 ) )/(O ,N) K

.

Consequently, the multiplication by pm in Ω1(R∞ ,P∞ )/(O ,N) is surjective. K Let ω ∈ Ω1(R∞ ,P∞ )/(O ,N) be such that pm ω = 0. Then there exists n ≥ 1 such that K ω ∈ Ω1(Rn ,P (n) )/(O ,N) and pm ω = 0 in Ω1(Rn ,P (n) )/(O ,N) . Consider the canonical exact K K sequence (II.7.9.15) Ω1(O

K ,N

h

(n) )/(O ,N) K

n ⊗OK Rn −→ Ω1(Rn ,P (n) )/(O

K ,N)

−→ Ω1(Rn ,P (n) )/(O

K ,N

(n) )

−→ 0.

The Rn -module Ω1(Rn ,P (n) )/(O ,N(n) ) is free of finite type by virtue of II.6.6(i) and II.6.8(i). K Hence it is Zp -flat. Consequently, ω is contained in the image of hn . It therefore follows from (II.7.5.5) and (II.7.5.6) that there exists n0 ≥ 1 such that the image of ω in Ω1(R 0 ,P (nn0 ) )/(O ,N) is zero. Hence ω is zero in Ω1(R∞ ,P∞ )/(O ,N) . K nn K (iii) We have a canonical isomorphism (II.7.9.16)

∼

lim Ω1(Rn ,P (n) )/(R1 ,P ) → Ω1(R∞ ,P∞ )/(R1 ,P ) , −→

n≥1

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation. For every integer n ≥ 1, it follows from the diagram (II.7.9.8) and what precedes it that we have a canonical isomorphism (II.7.9.17)

Mn =

(P gp /nP gp ) ⊗Z Rn ∼ 1 → Ω(Rn ,P (n) )/(R1 ,P ) , e ⊗ πn Rn λ

e is the class of λ in P gp /nP gp . We denote by d log(λ(n) ) the image of λ e ⊗ 1 in where λ Mn , which can be justified by the isomorphism (II.7.9.17). For every integer m ≥ 1, the multiplication by m in P gp and the canonical homomorphism Rn → Rmn induce a morphism Mn → Mmn . The (Mn )n≥1 form a direct system for the divisibility relation, and we have a canonical isomorphism (II.7.9.18)

∼

lim Mn → Ω1(R∞ ,P∞ )/(R1 ,P ) . −→

n≥1

Let n be an integer ≥ 1. In Mpn , we have (II.7.9.19)

d log(λ(n) ) = pd log(λ(pn) ) =

p πpn d log(λ(pn) ) = 0. πpn

Consequently, for every integer m ≥ 1 divisible by p, the composition of the canonical morphisms (II.7.9.20)

(P gp /nP gp ) ⊗Z Rn → Mn → Mmn

90

II. LOCAL STUDY

factors through an Rn -linear morphism (II.7.9.21)

Nn = ((P gp /λZ)/n(P gp /λZ)) ⊗Z Rn → Mmn .

On the other hand, we have a canonical surjective morphism (II.7.9.22)

Mmn → ((P gp /λZ)/mn(P gp /λZ)) ⊗Z Rmn .

The composition of (II.7.9.21) and (II.7.9.22) is induced by the multiplication by m in P gp /λZ and the canonical homomorphism Rn → Rmn . Since the torsion subgroup of P gp /λZ is of order prime to p and Rn is Zp -flat, multiplication by m induces an injective morphism ((P gp /λZ)/n(P gp /λZ)) ⊗Z Rn → ((P gp /λZ)/mn(P gp /λZ)) ⊗Z Rn .

Since Rn is a normal integral domain by II.6.8(ii), the canonical homomorphism Rn /mnRn → Rmn /mnRmn is injective. It follows that the morphism (II.7.9.21) is injective. For all integers m, n ≥ 1, the multiplication by m in P gp /λZ and the canonical homomorphism Rn → Rmn induce a morphism Nn → Nmn . The (Nn )n≥1 form a direct system for the divisibility relation. Taking the direct limit of the morphisms (II.7.9.21), first with respect to the integers m ≥ 1 that are multiples of p, and then with respect to the integers n ≥ 1, gives an isomorphism (II.7.9.23)

∼

lim Nn → Ω1(R∞ ,P∞ )/(R1 ,P ) . −→

n≥1

It is clear that the canonical morphism lim Nps → lim Nn ,

(II.7.9.24)

−→

−→

s≥0

n≥1

where the first direct limit is indexed by the set N with the usual ordering, is an isomorphism. Statement (iii) follows. Corollary II.7.10. The OK -modules Ω1(R∞ ,P∞ )/(R1 ,P ) and Ω1(R∞ ,P∞ )/(R1 ,P ) ⊗R∞ R do not have any nonzero mK -torsion. It suffices to copy the proof of II.7.7, taking into account (II.7.9.3) and the fact that R∞ and R are OK -flat ([11] Chap. VI § 3.6 Lem. 1). Corollary II.7.11. Every element of Ω1(R∞ ,P∞ )/(R,P ) (resp. Ω1(R,P )/(R,P ) ) is annihi∞ lated by a power of p. Consider the exact sequence (II.7.11.1)

Ω1R1 /R ⊗R1 R∞ → Ω1(R∞ ,P∞ )/(R,P ) → Ω1(R∞ ,P∞ )/(R1 ,P ) → 0.

By II.6.8(i), we have a canonical isomorphism (II.7.11.2)

∼

Ω1R1 /R → Ω1OK /OK ⊗OK R1 .

On the other hand, by (II.7.9.3), every element of Ω1(R∞ ,P∞ )/(R1 ,P ) is annihilated by a power of p, and the same holds for Ω1O /OK (II.7.1). Consequently, every element of K Ω1(R∞ ,P∞ )/(R,P ) is annihilated by a power of p. Since the canonical morphism (II.7.11.3)

Ω1(R∞ ,P∞ )/(R,P ) ⊗R∞ R → Ω1(R,P∞ )/(R,P )

is an almost isomorphism by virtue of II.6.18 and ([50] 1.7), it follows that every element of Ω1(R,P )/(R,P ) is annihilated by a power of p. ∞

II.7. FALTINGS EXTENSION

91

Remark II.7.12. The isomorphism (II.7.9.2) can also be deduced from the isomorphism (II.7.9.3) as follows. First note that we have a canonical isomorphism ∼

(P gp /λZ) ⊗Z R → Ω1(R,P )/(OK ,N) .

(II.7.12.1)

For every integer m ≥ 0, the pm -torsion of R∞ [ p1 ]/R∞ is canonically isomorphic to R∞ /pm R∞ . Since the torsion subgroup of P gp /λZ is of order prime to p, (II.7.9.3) induces an isomorphism (II.7.12.2)

∼

(P gp /λZ) ⊗Z (R∞ /pm R∞ ) → Hom(p−m Z/Z, Ω1(R∞ ,P∞ )/(R1 ,P ) ).

It follows from the proof of II.7.9 that this is the inverse of the isomorphism (II.7.9.2). Proposition II.7.13. (i) The kernel of the canonical morphism Ω1OK /OK ⊗OK R∞ → Ω1(R∞ ,P∞ )/(R,P )

(II.7.13.1)

is generated by d log(π) (II.7.2.2). (ii) The sequence of canonical morphisms (II.7.13.2) 0 → Ω1(OK ,N∞ )/(OK ,N) ⊗OK R∞ → Ω1(R∞ ,P∞ )/(R,P ) → Ω1(R∞ ,P∞ )/(R1 ,P ) → 0 is exact and split. (i) We have a canonical isomorphism (II.7.13.3)

∼

lim Ω1(Rn ,P (n) )/(R,P ) → Ω1(R∞ ,P∞ )/(R,P ) , −→

n≥1

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation. It therefore suffices to show that for every integer n ≥ 1 such that p divides n, the kernel of the canonical morphism Ω1OK /OK ⊗OK Rn → Ω1(Rn ,P (n) )/(R,P )

(II.7.13.4)

is generated by d log(π). By functoriality, this morphism factors through the canonical morphism (II.7.13.5)

Ω1(O

K ,N

(n) )/(O ,N) K

⊗OK Rn → Ω1(Rn ,P (n) )/(R,P ) .

It therefore follows from II.7.4(ii) that d log(π) belongs to the kernel of the morphism (II.7.13.4). Conversely, let us show that the kernel of the morphism (II.7.13.4) is contained in Rn d log(π). Recall that Spec(Rn ) is an open connected component of Xn ⊗OKn OK (II.6.8) and that we have a Cartesian diagram of OK -morphisms (II.6.6.4) (II.7.13.6)

Xn ⊗OKn OK

/ Spec(O [P (n) ]/(π − eλ(n) )) n K

 X

 / Spec(OK [P ]/(π − eλ ))

where λ(n) is the image of λ in P (n) by the isomorphism (II.7.8.2). Consider the commutative diagram with Cartesian square (II.7.13.7) (n) / Spec(O [P (n) ]/(π − enλ(n) )) / Spec(O [P (n) ]) Spec(OK [P (n) ]/(πn − eλ )) K K VVVV VVVV VVVV  VVVV VVV+   / Spec(OK [P ]) Spec(OK [P ]/(π − eλ ))

92

II. LOCAL STUDY

We have a canonical isomorphism (II.7.13.8) ∼ Ω1(O [P (n) ],P (n) )/(OK [P ],P ) → (P gp /nP gp ) ⊗Z OK [P (n) ] ⊕ Ω1OK /OK ⊗OK OK [P (n) ]. K

We deduce from this an isomorphism (II.7.13.9)

∼

Ω1(Rn ,P (n) )/(R,P ) →

(P gp /nP gp ) ⊗Z Rn ⊕ Ω1O

K /OK

(λ ⊗ πn − dπn )Rn

⊗OK Rn

,

where λ is the class of λ in P gp /nP gp . Note for the proof of (ii) that for every integer m ≥ 1, the canonical morphism (II.7.13.10)

Ω1(Rn ,P (n) )/(R,P ) → Ω1(Rmn ,P (mn) )/(R,P )

is induced by the multiplication by m in P gp , the identity of Ω1O /OK , and the canonical K homomorphism Rn → Rmn . Let ω ∈ Ω1O /OK ⊗OK Rn be such that its image by the morphism (II.7.13.4) is zero. K By (II.7.13.9), there exists x ∈ Rn such that (II.7.13.11)

ω = x(λ ⊗ πn − dπn ) ∈ (P gp /nP gp ) ⊗Z Rn ⊕ Ω1OK /OK ⊗OK Rn .

Consequently, λ ⊗ πn x = 0 in (P gp /nP gp ) ⊗Z Rn . Since the torsion subgroup of P gp /λZ is of order prime to p and Rn is flat over Zp , the homomorphism (II.7.13.12)

(λZ/nλZ) ⊗Z Rn → (P gp /nP gp ) ⊗Z Rn

is injective. It follows that πn x ∈ nRn . Hence x ∈ (n/πn )Rn , because p divides n and Rn is flat over OK by virtue of II.6.6(ii) and II.6.8(i). It then follows from (II.7.13.11) that ω = −xdπn ∈ Rn d log(π). (ii) By II.6.8(i), we have a canonical isomorphism ∼

Ω1R1 /R → Ω1OK /OK ⊗OK R1 .

(II.7.13.13)

Hence the sequence (II.7.13.2) is exact by (i), II.7.5, and by the canonical exact sequence (II.7.13.14)

Ω1R1 /R ⊗R1 R∞ → Ω1(R∞ ,P∞ )/(R,P ) → Ω1(R∞ ,P∞ )/(R1 ,P ) → 0.

It remains to construct a splitting of (II.7.13.2). We have a canonical isomorphism (II.7.13.15)

∼

lim Ω1(Rn ,P (n) )/(R1 ,P ) → Ω1(R∞ ,P∞ )/(R1 ,P ) , −→

n≥1

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation. Hence, in view of the isomorphism (II.7.13.3), it suffices to construct, for every integer n ≥ 1 such that p divides n, a right inverse of the canonical morphism (II.7.13.16)

Ω1(Rn ,P (n) )/(R,P ) → Ω1(Rn ,P (n) )/(R1 ,P )

such that the family of these right inverses is a morphism of direct systems. It follows from the diagram (II.7.9.8) and what precedes it that we have an isomorphism (II.7.13.17)

∼

Ω1(Rn ,P (n) )/(R1 ,P ) →

(P gp /nP gp ) ⊗Z Rn , λ ⊗ πn Rn

where λ is the class of λ in P gp /nP gp . For every integer m ≥ 1, the canonical morphism (II.7.13.18)

Ω1(Rn ,P (n) )/(R,P ) → Ω1(Rmn ,P (mn) )/(R,P )

II.7. FALTINGS EXTENSION

93

is induced by the multiplication by m in P gp and by the canonical homomorphism Rn → Rmn . In view of (II.7.13.9) and (II.7.13.17), the morphism (II.7.13.16) is induced by the canonical projection (II.7.13.19)

(P gp /nP gp ) ⊗Z Rn ⊕ Ω1OK /OK ⊗OK Rn → (P gp /nP gp ) ⊗Z Rn .

Since the torsion subgroup of P gp /λZ is of order prime to p and Rn is flat over Zp , the Rn -linear morphism (II.7.13.20)

Rn → P gp ⊗Z Rn

defined by λ admits an Rn -linear left inverse u : P gp ⊗Z Rn → Rn . Consider the morphism (II.7.13.21)

vn : P gp ⊗Z Rn → (P gp /nP gp ) ⊗Z Rn ⊕ Ω1OK /OK ⊗OK Rn

defined, for x ∈ P gp ⊗Z Rn with class x in (P gp /nP gp ) ⊗Z Rn , by (II.7.13.22)

vn (x) = x − d log(πn ) ⊗ u(x),

where d log(πn ) is the element of Ω1O /OK defined in (II.7.2.2). In view of (II.7.13.9), vn K induces an Rn -linear morphism (II.7.13.23)

wn : P gp ⊗Z Rn → Ω1(Rn ,P (n) )/(R,P ) .

For every t ∈ P gp , we have wn (nt) = −nu(t)d log(πn ) = −u(t)d log(π) (II.7.2.5). Since p divides n, the image of d log(π) in Ω1(Rn ,P (n) )/(R,P ) is zero by the proof of (i); hence wn (nt) = 0. On the other hand, by virtue of (II.7.2.4), we have (II.7.13.24)

wn (λ ⊗ πn ) = λ ⊗ πn − πn d log(πn ) = λ ⊗ πn − dπn = 0.

Consequently, in view of (II.7.13.17), wn induces an Rn -linear morphism (II.7.13.25)

ωn : Ω1(Rn ,P (n) )/(R1 ,P ) → Ω1(Rn ,P (n) )/(R,P )

that is the right inverse of the canonical morphism (II.7.13.16). Since md log(πmn ) = d log(πn ) for every m ≥ 1, the ωn ’s form a morphism of direct systems for the divisibility relation, completing the proof. II.7.14. (II.7.14.1)

There exists a unique map h , i : Γ∞ × P∞ → µ∞ (OK ) = lim µn (OK ), −→

n≥1

where the direct limit is indexed by the set Z≥1 ordered by the divisibility relation, such that for every g ∈ Γ∞ and every x ∈ P (n) (n ≥ 1), we have hg, xi ∈ µn (OK ) and (II.7.14.2)

g(α∞ (x)) = hg, xi · α∞ (x),

where α∞ is the homomorphism (II.7.8.5). Indeed, R∞ is an integral domain, and we have α∞ (x)n ∈ α(P ) ⊂ R; hence α∞ (x) 6= 0 (II.7.8) and α∞ (x)n is invariant under Γ∞ . For every g ∈ Γ∞ , the map x 7→ hg, xi is a morphism of monoids from P∞ to µ∞ (OK ). For every x ∈ P∞ , the map g 7→ hg, xi is a 1-cocycle; in other words, for all g, g 0 ∈ Γ∞ , we have (II.7.14.3)

hgg 0 , xi = g(hg 0 , xi) · hg, xi.

∼

Let n be an integer ≥ 1. Recall that we have a canonical isomorphism P (n) → P b (II.7.8.2) and that ∆∞ is canonically isomorphic to a subgroup of Lλ ⊗Z Z(1) (II.6.8.3).

94

II. LOCAL STUDY

We therefore have a canonical homomorphism ∆∞ → L ⊗Z µn (OK ). By II.6.6(vi), the diagram (II.7.14.4)

h , i

/ µn (O ) K 4 j j j jj j j j jjjj jjjj  (L ⊗Z µn (OK )) × P ∆∞ × P (n)

where the slanted arrow is induced by the canonical pairing L ⊗Z P gp → Z, is commutative. II.7.15. (II.7.15.1)

There exists a unique map h , i : GK × N∞ → µ∞ (OK )

such that for every g ∈ GK and every x ∈ N(n) (n ≥ 1), we have hg, xi ∈ µn (OK ) and (II.7.15.2)

g(a∞ (x)) = hg, xi · a∞ (x),

where a∞ is the homomorphism (II.7.3.3). Identifying N(n) with N through the canonical ∼ isomorphism N(n) → N, the relation (II.7.15.2) becomes g(πnx ) = hg, xi · πnx . The map (II.7.15.1) has properties analogous to those of (II.7.14.1). Moreover, the diagram (II.7.15.3)

/ Γ∞ × P∞

Γ∞ × N∞  GK × N∞

h , i

h , i

 / µ∞ (O ) K

where the unlabeled arrows are the canonical homomorphisms, is commutative, which justifies using the same notation. II.7.16. Let M∞ be the logarithmic structure on Spec(R∞ ) associated with the prelogarithmic structure defined by (P∞ , α∞ ) (II.7.8). For every g ∈ Γ∞ , denote by τg the automorphism of Spec(R∞ ) induced by g. The logarithmic structure τg∗ (M∞ ) on Spec(R∞ ) is associated with the prelogarithmic structure defined by (P∞ , g ◦ α∞ ). The homomorphism (II.7.16.1)

P∞ → Γ(Spec(R∞ ), M∞ ),

x 7→ hg, xi · x

induces a morphism of logarithmic structures on Spec(R∞ ) (II.7.16.2)

ag : τg∗ (M∞ ) → M∞ .

Likewise, by virtue of (II.7.14.3), the homomorphism (II.7.16.3)

P∞ → Γ(Spec(R∞ ), τg∗ (M∞ )),

x 7→ g(hg −1 , xi) · x

induces a morphism of logarithmic structures on Spec(R∞ ) (II.7.16.4)

bg : M∞ → τg∗ (M∞ ).

We immediately see that ag and bg are isomorphisms inverse to each other (II.7.14.3), and that the map g 7→ (τg−1 , ag−1 ) is a left action of Γ∞ on (Spec(R∞ ), M∞ ). We denote by L the logarithmic structure on Spec(R) inverse image of M∞ . Then the action above lifts to a left action of Γ on (Spec(R), L ). Let N∞ be the logarithmic structure on Spec(OK ) associated with the prelogarithmic structure defined by (N∞ , a∞ ) (II.7.3.3). Likewise, the map (II.7.15.1) defines a left action of GK on the logarithmic scheme (Spec(OK ), N∞ ).

II.7. FALTINGS EXTENSION

95

Let M be the logarithmic structure on Spec(R1 ) associated with the prelogarithmic structure defined by (P, α) (II.7.8). For every g ∈ Gal(F1 /F ), we denote by ug the automorphism of Spec(R1 ) induced by g. The logarithmic structure u∗g (M ) on Spec(R1 ) is associated with the prelogarithmic structure defined by (P, g ◦ α). Since g ◦ α = α, we deduce from this a canonical isomorphism ∼

cg : u∗g (M ) → M .

(II.7.16.5)

The map g 7→ (ug−1 , cg−1 ) is a left action of Gal(F1 /F ) on the logarithmic scheme (Spec(R1 ), M ). Let N be the logarithmic structure on Spec(OK ) associated with the prelogarithmic structure defined by (N, a) (II.7.3). Likewise, we define a left action of GK on the logarithmic scheme (Spec(OK ), N ). Note that all morphisms in the commutative diagram (Spec(R), L )

(II.7.16.6)

/ (Spec(R∞ ), M∞ )

/ (Spec(O ), N∞ ) K

 (Spec(R1 ), M )

 / (Spec(O ), N ) K

are Γ-equivariant. II.7.17. (II.7.17.1)

d We denote by E∞ the R ∞ -representation of Γ∞ defined by E∞ = Hom(Qp /Zp , Ω1(R∞ ,P∞ )/(R,P ) ) ⊗Zp Zp (−1),

where the action of Γ∞ comes from its action on Ω1(R∞ ,P∞ )/(R,P ) (II.7.16). In view of II.7.5 and II.7.9(ii), applying the functor Hom(Qp /Zp , −) ⊗Zp Zp (−1) to the split exact d sequence (II.7.13.2) gives an exact sequence of R ∞ -representations of Γ∞ (II.7.17.2)

1 d d 0 → (πρ)−1 R ∞ → E∞ → Ω(R,P )/(OK ,N) ⊗R R∞ (−1) → 0,

−1 d d where we have written (πρ)−1 R OC ⊗OC R ∞ instead of (πρ) ∞ , which is justified because 1 d R is O -flat (II.6.14). Note that Ω is a free R-module of finite type. To ∞ C (R,P )/(OK ,N) simplify the notation, we set

(II.7.17.3)

e1 Ω R/OK

=

Ω1(R,P )/(OK ,N) ,

(II.7.17.4)

ei Ω R/OK

e1 = ∧i Ω R/OK ,

(i ≥ 1).

d The sequences (II.7.17.2) and (II.7.13.2) are split as sequences of R ∞ -modules (without Γ∞ -action). We denote by (II.7.17.5)

1 −1 d e1 c δ: Ω R∞ )(1) R/OK ⊗R R1 → Hcont (∆∞ , (πρ)

the boundary map of the long exact sequence of cohomology obtained by applying the functor Γ(∆∞ , −)(1) to the exact sequence (II.7.17.2). Later, we will show that ∆∞ d c1 (II.8.16). (R =R ∞) II.7.18. by (II.7.18.1)

For every ζ ∈ Zp (1), we denote by d log(ζ) the element of E∞ (1) defined d log(ζ)(p−n ) = d log(ζn ),

where ζn is the canonical image of ζ in µpn (OK ). It is clear that d log(ζ) is the image of d 1 ⊗ ζ by the injection (πρ)−1 R ∞ (1) → E∞ (1) (II.7.17.2).

96

II. LOCAL STUDY

For every t ∈ P , we denote by d log(e t) the element of E∞ (1) defined by n

d log(e t)(p−n ) = d log(t(p ) ),

(II.7.18.2) n

n

where t(p ) is the image of t in P (p ) by the isomorphism (II.7.8.2). This element is e1 d well-defined by virtue of (II.7.8.3). It is clear that the image of d log(e t) in Ω R/OK ⊗R R∞ is d log(t). e −n ) is the canonical image of the element By (II.7.6), for every n ≥ 0, d log(λ)(p 1 1 e ∈ e d log(πpn ) ∈ ΩO /OK (II.7.2.2) in Ω(R∞ ,P∞ )/(R,P ) . In particular, we have d log(λ) K d (πρ)−1 R ∞ (1) ⊂ E∞ (1) (II.7.17.2). The map P → E∞ (1) defined by t 7→ d log(e t) is a homomorphism; it therefore induces a homomorphism that we also denote by (II.7.18.3) P gp → E∞ (1), t 7→ d log(e t). This fits into a commutative diagram (II.7.18.4)

0

/ Zλ

/ P gp

/ P gp /Zλ

/0

0

 / (πρ)−1 R d ∞ (1)

 / E∞ (1)

 /Ω e1 d ⊗ R R∞ R/OK

/0

where the vertical arrow on the right comes from the canonical isomorphism (II.7.12.1). It is useful to also denote by e1 (II.7.18.5) d log : P gp → Ω R/OK

e1 the homomorphism induced by the logarithmic derivation d log : P → Ω R/OK . II.7.19. (II.7.19.1)

Let t ∈ P . We denote by

χ et : Γ∞ → Zp (1)

the map that associates with each g ∈ Γ∞ the element (II.7.19.2)

n

χ et (g) = lim hg, t(p ) i, ←−

n≥0

n

n

n

where t(p ) is the image of t in P (p ) by the isomorphism (II.7.8.2) and hg, t(p ) i ∈ µpn (OK ) is defined in (II.7.14.1). By (II.7.14.3), for all g, g 0 ∈ Γ∞ , we have (II.7.19.3)

χ et (gg 0 ) = g(e χt (g 0 ))e χt (g).

Hence the restriction of χ et to ∆∞ is a character with values in Zp (1); we denote it by χt : ∆∞ → Zp (1). We clearly have χ e0 = 1, and for all t, t0 ∈ P , (II.7.19.4)

χ ett0 = χ et · χ et0 .

Consequently, the map P → Hom(∆∞ , Zp (1)) defined by t 7→ χt is a homomorphism. It therefore induces a homomorphism that we will also denote by (II.7.19.5)

P gp → Hom(∆∞ , Zp (1)),

Since χλ = 1, we deduce from this a homomorphism (II.7.19.6)

t 7→ χt .

P gp /Zλ → Hom(∆∞ , Zp (1)).

By (II.7.14.4), the latter is equal to the composition (II.7.19.7)

P gp /Zλ → Hom(Lλ ⊗Z Zp (1), Zp (1)) → Hom(∆∞ , Zp (1)),

II.7. FALTINGS EXTENSION

97

where the first arrow is induced by the canonical (biduality) morphism (II.6.2.9) and the second arrow by the canonical morphism ν : ∆∞ → Lλ ⊗Z Zp (1) (II.6.8.3). Recall that ν ∼ induces an isomorphism ∆p∞ → Lλ ⊗Z Zp (1) and that we have a canonical isomorphism (II.6.12.1) ∼

HomZp (∆p∞ , Zp (1)) → HomZ (∆∞ , Zp (1)). Since the torsion subgroup of P gp /Zλ is of order prime to p, the homomorphism (II.7.19.6) induces an isomorphism ∼

(P gp /Zλ) ⊗Z Zp → Hom(∆∞ , Zp (1)).

(II.7.19.8)

c1 -linear isomorphism In view of (II.7.12.1) and (II.6.12.2), we deduce from this an R e1 c ∼ c δe: Ω R/OK ⊗R R1 → Hom(∆∞ , R1 (1)).

(II.7.19.9) II.7.20. we have

It immediately follows from the definitions that for all t ∈ P and g ∈ Γ∞ , g(d log(e t)) = d log(e t) + d log(χt (g)).

(II.7.20.1)

Since both sides of the equation are homomorphisms from P to E (1), we have equality for every t ∈ P gp . Consequently, the diagram e δ

e1 c Ω R/OK ⊗R R1

(II.7.20.2)

/ Hom(∆ , R c1 (1)) ∞

δ

 o d H1cont (∆∞ , (πρ)−1 R ∞ (1))

c1 (1)) H1cont (∆∞ , R

where δ is the morphism (II.7.17.5), δe is the morphism (II.7.19.9), and the bottom horc1 → (πρ)−1 R d izontal arrow is induced by the canonical injection R ∞ , is commutative. 1 e Indeed, since ΩR/OK is generated over R by the elements of the form d log(t) for t ∈ P , it suffices to show the commutativity of this diagram for these elements, which follows from (II.7.20.1). II.7.21. (II.7.21.1)

Consider the commutative diagram of canonical morphisms

Ω1(O

K ,N∞ )/(OK ,N)

⊗OK R

u

/ Ω1

(R∞ ,P∞ )/(R,P ) a

Ω1(O

K ,N∞ )/(OK ,N)

⊗OK R

u0

/ Ω1 (R,P

⊗R∞ R



∞ )/(R,P )

/ / Ω1 (R∞ ,P∞ )/(R1 ,P ) ⊗R∞ R b

/ / Ω1 (R,P



∞ )/(R1 ,P )

Since the sequence (II.7.13.2) is exact and split, u is injective. On the other hand, the kernel of a is annihilated by mK by virtue of II.6.18 and (II.5.21.1). Consequently, the kernel of u0 is annihilated by mK . Since Ω1(O ,N∞ )/(OK ,N) ⊗OK R does not have any K nonzero mK -torsion by II.7.7, u0 is injective. Applying the “Tate module” functor Tp (−) = Hom(Qp /Zp , −) to the diagram above, and setting Ξ = Ω1(OK ,N∞ )/(OK ,N) ⊗OK R,

98

II. LOCAL STUDY

we obtain a commutative diagram  Tp (Ξ)  / Tp (Ω1(R∞ ,P∞ )/(R,P ) ⊗R∞ R) (II.7.21.2) Tp (a)

Tp (Ξ) 



/ Tp (Ω1 (R,P

Tp (b)



∞

/ Tp (Ω1 (R∞ ,P∞ )/(R1 ,P ) ⊗R∞ R)

v

v0

) )/(R,P )

/ Tp (Ω1 (R,P

 ∞ )/(R1 ,P )

)

Since the sequence (II.7.13.2) is exact and split, v is surjective. On the other hand, the kernel and cokernel of b are annihilated by mK , and Ω1(R∞ ,P∞ )/(R1 ,P ) ⊗R∞ R does not have any nonzero mK -torsion by virtue of II.7.10. Consequently, b is injective, and therefore Tp (b) is an isomorphism. It follows that v 0 is surjective and that Tp (a) is an isomorphism. It follows from (II.7.5.2) that the canonical morphism b → T (Ω1 Tp (Ω1(OK ,N∞ )/(OK ,N) ) ⊗OC R p (OK ,N∞ )/(OK ,N) ⊗OK R)

(II.7.21.3)

is an isomorphism. Likewise, it follows from (II.7.9.3) that the canonical morphism (II.7.21.4)

b → T (Ω1 Tp (Ω1(R∞ ,P∞ )/(R1 ,P ) ) ⊗R R p d (R∞ ,P∞ )/(R1 ,P ) ⊗R∞ R) ∞

is an isomorphism. Consequently, the canonical morphism b → T (Ω1 Tp (Ω1(R∞ ,P∞ )/(R,P ) ) ⊗R R p d (R∞ ,P∞ )/(R,P ) ⊗R∞ R) ∞

(II.7.21.5) is an isomorphism. II.7.22. (II.7.22.1)

b We denote by E the R-representation of Γ defined by E = Hom(Qp /Zp , Ω1(R,P∞ )/(R,P ) ) ⊗Zp Zp (−1),

where the action of Γ comes from its action on Ω1(R,P

(II.7.16). It follows from b II.7.21 and (II.7.17.2) that we have a canonical exact sequence of R-representations of Γ (II.7.22.2)

∞ )/(R,P )

b b →E →Ω e1 0 → (πρ)−1 R R/OK ⊗R R(−1) → 0,

b called Faltings extension. We have an isomorphism of R-representations of Γ (II.7.22.3)

∼ b→ E∞ ⊗R R E, d ∞

inducing an isomorphism of the extensions (II.7.17.2) and (II.7.22.2). In particular, the b (without Γ-action). sequence (II.7.22.2) is split as a sequence of R-modules II.8. Galois cohomology Proposition II.8.1. Let n be an integer ≥ 0, ν : ∆p∞ → µpn (OK ) a surjective homomorphism, ζ a generator of the group µpn (OK ), a ∈ OK , and q the ideal of OK generated by a and ζ − 1. Let A be an OK -algebra that is complete and separated for the p-adic topology and OK -flat. We denote by A(ν) the topological A-∆p∞ -module A, endowed with the p-adic topology and the action of ∆p∞ defined by the multiplication by ν (II.3.1). (i) If ν = 1 (that is, n = 0), then we have a canonical isomorphism of graded A-algebras (II.8.1.1)

∼

∧ (HomZp (∆p∞ , A/aA)) → H∗cont (∆p∞ , A(ν)/aA(ν)).

(ii) If ν 6= 1 (that is, n 6= 0), then Hicont (∆p∞ , A(ν)/aA(ν)) is a free A/qA-module of finite type for every i ≥ 0 and is zero for every i ≥ rg(L) = d + 1 (II.6.2.5).

II.8. GALOIS COHOMOLOGY

99

(iii) The inverse system (H∗ (∆p∞ , A(ν)/pr A(ν)))r≥0 satisfies the Mittag–Leffler condition uniformly in ν; in other words, if for all integers r0 ≥ r ≥ 0, we denote by 0

hνr,r0 : H∗ (∆p∞ , A(ν)/pr A(ν)) → H∗ (∆p∞ , A(ν)/pr A(ν))

(II.8.1.2)

the canonical morphism, then for every integer r ≥ 1, there exists an integer r0 ≥ r depending on d but not on ν, such that for every integer r00 ≥ r0 , the images of hνr,r0 and hνr,r00 are equal. Fix a Zp -basis e1 , . . . , ed of ∆p∞ and denote by SA (∆p∞ ) the symmetric algebra of the A-module ∆p∞ ⊗Zp A. The ring A is endowed with a structure of SA (∆p∞ )-algebra defined by the homomorphism of A-algebras SA (∆p∞ ) → A that sends ei to ν(ei ) − 1 for 1 ≤ i ≤ d; we denote it by A(ν). . Note that the SA (∆p∞ )-module underlying A(ν). is none other than the SA (∆p∞ )-module associated with A(ν) and denoted by the same symbol in II.3.25. Consider the linear form u : ∆p∞ ⊗Zp A → A

(II.8.1.3)

that sends ei ⊗ 1 to ν(ei ) − 1 for 1 ≤ i ≤ d. It immediately follows from the definitions (II.2.6.3) and (II.2.7.4) that we have a canonical isomorphism ∼

K•SA (∆p∞ ) ((A(ν)/aA(ν)). ) → K•A (u, A/aA).

(II.8.1.4)

By virtue of II.3.25, we deduce from this a canonical isomorphism ∼

C•cont (∆p∞ , A(ν)/aA(ν)) → K•A (u, A/aA)

(II.8.1.5)

of D+ (Mod(A)), where the left-hand side is the complex of continuous cochains of G with values in A(ν)/aA(ν) (II.3.8). (i) This follows from II.3.30 and II.6.12(iii). Note that since the form u (II.8.1.3) is zero, (II.8.1.5) provides an isomorphism of graded A-modules ∼

H∗cont (∆p∞ , A(ν)/aA(ν)) → ∧(HomZp (∆p∞ , A/aA)).

(II.8.1.6)

But it is not clear, a priori, that this is an isomorphism of graded A-algebras. We can, however, deduce it from II.3.28 (which essentially corresponds to the proof of II.3.30). (ii) We denote also by u the linear form ∆p∞ ⊗Zp (A/aA) → (A/aA) deduced from A/aA u (II.8.1.3). By virtue of (II.8.1.5) and (II.2.6.8), it suffices to show that Hi (K• (u)) is a free (A/qA)-module of finite type for every i ≥ 0 and is zero for every i ≥ d + 1. A/aA The second statement is obvious. We also know that Hi (K• (u)) is annihilated by q (II.2.6). For 1 ≤ j ≤ d, set ζj = ν(ej ). We may assume that ζ1 = ζ 6= 1 and that (II.2.1) v(ζ1 − 1) ≤ v(ζ2 − 1) ≤ · · · ≤ v(ζd − 1).

We proceed by induction on d. The statement for d = 1 is an immediate consequence of the flatness of A over OK . Suppose d ≥ 2 and that the assertion holds for d−1. Denote by G the sub-Zp -module of ∆p∞ generated by e1 , . . . , ed−1 and by u0 : G⊗Zp (A/aA) → A/aA the restriction of u to G ⊗Zp (A/aA). By (II.2.6.4), we have a canonical isomorphism A/aA

(II.8.1.7)

K•

A/aA

A/aA

(ζd − 1) ⊗ K•

∼

A/aA

(u0 ) → K•

(u),

where K• (ζd − 1) is the Koszul complex defined by the linear form ζd − 1 in A/aA. By virtue of ([41] 1.1.4.1), for every integer i, we have an exact sequence A/aA

(II.8.1.8) 0 → H0 (K• →

A/aA

(ζd − 1) ⊗ Hi (K•

A/aA Hi (K• (u))

→

(u0 )))

A/aA H1 (K• (ζd

A/aA

− 1) ⊗ Hi−1 (K•

(u0 ))) → 0.

100

II. LOCAL STUDY A/aA

By the induction hypothesis, Hi (K• (u0 )) is a free (A/qA)-module of finite type for every i ≥ 0. Since (ζd − 1) ∈ q, we deduce from this an exact sequence (II.8.1.9)

A/aA

0 → Hi (K•

A/aA

A/aA

(u0 )) → Hi (K•

A/aA

(u)) → Hi−1 (K•

(u0 )) → 0.

Consequently, since Hi (K• (u)) is annihilated by q, it is free of finite type over A/qA. (iii) It follows from (i) that the inverse system (H∗ (∆p∞ , A(ν)/pr A(ν)))r≥0 satisfies the Mittag–Leffler condition when ν = 1. We can therefore restrict ourselves to the characters ν 6= 1. We set Ar = A/pr A and we denote also by u the linear form u ⊗A idAr : ∆p∞ ⊗Zp Ar → Ar (II.8.1.3). By (II.8.1.5) and (II.2.6.8), it suffices to show the r analogous statement for the inverse system (Hi (KA • (u)))r≥1 . We proceed by induction on d. First take d = 1. The canonical homomorphism A

r H1 (K• r+1 (u)) → H1 (KA • (u))

(II.8.1.10)

is clearly bijective, and since v(ζ − 1) ≤ 1, the canonical homomorphism A

r H0 (K• r+1 (u)) → H0 (KA • (u))

(II.8.1.11)

is zero. The assertion therefore holds with r0 = r + 1. Suppose d ≥ 2 and that the assertion holds for d − 1. The assertion for d then easily follows from the exact sequence (II.8.1.9) by taking a = pr for r ≥ 0 (cf. the proof of [41] 0.13.2.1). Corollary II.8.2. Under the assumptions of II.8.1, the canonical homomorphism H∗cont (∆p∞ , A(ν)) → lim H∗ (∆p∞ , A(ν)/pr A(ν))

(II.8.2.1)

←− r≥0

is an isomorphism. Indeed, by (II.3.10.4) and (II.3.10.5), for every i ≥ 0, we have an exact sequence (II.8.2.2)

0 → R1 lim Hi−1 (∆p∞ , A(ν)/pr A(ν)) ←− r≥0

→ Hicont (∆p∞ , A(ν)) → lim Hi (∆p∞ , A(ν)/pr A(ν)) → 0 ←− r≥0

whose left term is zero by virtue of II.8.1(iii) and (II.3.10.2). Remark II.8.3. Under the assumptions of II.8.1, if ν 6= 1, the canonical homomorphism (II.8.3.1)

H∗cont (∆p∞ , A(ν)) ⊗A A/pr A → H∗ (∆p∞ , A(ν)/pr A(ν))

is not, in general, an isomorphism. Proposition II.8.4. Let n be an integer ≥ 1, ν : ∆p∞ → µpn (OK ) a surjective homomorphism, ζ a generator of the group µpn (OK ), a a nonzero element of OK , b = a(ζ − 1)−1 , and α a rational number. Let A be an OK -algebra that is complete and separated for the p-adic topology and OK -flat, N an (A/aA)-module, and M a discrete (A/aA)-∆p∞ module. We denote by N (ν) the discrete A-∆p∞ -module N endowed with the action of ∆p∞ defined by the multiplication by ν. Suppose that the following conditions are satisfied: (i) inf(v(a), α) > v(ζ − 1); (ii) N is flat over OK /aOK ; (iii) M is projective of finite type over A/aA, and is generated by a finite number of elements that are ∆p∞ -invariant modulo pα M . Then for every i ≥ 0, we have (ζ − 1) · Hi (∆p∞ , (M/bM ) ⊗A N (ν)) = 0.

II.8. GALOIS COHOMOLOGY

101

Since M is a direct summand of a free (A/aA)-module of finite type, we can restrict to the case where it is free of finite type over A/aA. It therefore admits an (A/aA)basis consisting of elements that are ∆p∞ -invariant modulo pα M . Let T = M ⊗A N (ν). Fix a Zp -base e1 , . . . , ed of ∆p∞ and denote by SA (∆p∞ ) the symmetric algebra of the A-module ∆p∞ ⊗Zp A. By virtue of II.3.25, we have a canonical isomorphism (II.8.4.1)

∼

C• (∆p∞ , T /bT ) → K•SA (∆p∞ ) ((T /bT ). )

of D+ (Mod(A)). There exists an integer i with 1 ≤ i ≤ d such that ν(ei ) is a generator of µpn (OK ); we may assume that ν(ei ) = ζ. Let ϕ : ∆p∞ → AutA (M ) be the representation of ∆p∞ over M . Then there exists an A-linear endomorphism U of M such that ϕ(ei ) = idM + pα U . Let c = pα (ζ − 1)−1 ∈ mK and V = idT + cU ⊗ (ζ · idN ), which is an A-linear automorphism of T , so that we have (II.8.4.2)

ϕ(ei ) ⊗ (ν(ei ) · idN ) − idT = (ζ − 1)V ∈ EndA (T ).

For every integer 1 ≤ j ≤ d, we have (II.8.4.3)

(ϕ(ej ) ⊗ (ν(ej ) · idN ) − idT ) ◦ V −1

= V −1 ◦ (ϕ(ej ) ⊗ (ν(ej ) · idN ) − idT ) ∈ EndA (T /bT ).

Indeed, since ∆p∞ is abelian, the products of these endomorphisms by ζ − 1 are equal in EndA (T ), which implies the relation (II.8.4.3) because T is flat over OK /aOK . Hence V −1 is an automorphism of the SA (∆p∞ )-module (T /bT ). and it induces, for every q ≥ 0, an SA (∆p∞ )-linear automorphism of Hq (K•SA (∆p∞ ) ((T /bT ). )). Now, this module is annihilated by ei (II.2.6). Consequently, for every x ∈ Hq (K•SA (∆p∞ ) ((T /bT ). )), we have, by virtue of (II.8.4.2), (II.8.4.4)

ei · V −1 (x) = (ζ − 1)x = 0.

II.8.5. We denote by Λ the cokernel in the category of monoids of the homomorphism ϑ : N → P and by q : P → Λ the canonical homomorphism. By ([58] I 1.1.5), Λ is the quotient of P by the congruence relation E consisting of the elements (x, y) ∈ P × P for which there exist a, b ∈ N such that x + aλ = y + bλ. That E is a congruence relation means that it is an equivalence relation and that E is a submonoid of P × P . The group associated with Λ is canonically identified with P gp /Zλ. Since P is integral, Λ is integral; we can therefore identify it with the image of P in P gp /Zλ. Lemma II.8.6. We keep the notation of II.8.5. Then: (i) The monoid Λ is saturated. (ii) For every x ∈ Λ, the set q −1 (x) admits a unique minimal element x e for the preordering on P defined by the monoid structure. (iii) For every x ∈ Λ and every integer n ≥ 0, we have n fx = ne x. (i) Indeed, Λ is the amalgamated sum of the saturated homomorphism ϑ and the homomorphism N → 0. It is therefore saturated (II.5.2). (ii) First note that two arbitrary elements of q −1 (x) are necessarily comparable (II.8.5). Let us show that the set q −1 (x) admits a minimal element x e. This corresponds to saying that for every t ∈ P , there exists n ∈ N such that the element t − nλ of P gp does not belong to P . Indeed, if this is not the case, then for every n ≥ 0, the element α(t)/π n of RK belongs to R, where α : P → R is the homomorphism defined by the chart (P, γ) (II.6.2). Since α(t) 6= 0 and R is a noetherian integral domain, it follows that π is invertible in R, which contradicts condition II.6.2(C2 ). It also follows that −λ does not belong to P . Hence x e is necessarily unique because P is integral and P gp is without torsion.

102

II. LOCAL STUDY

(iii) Since −λ does not belong to P , we have e 0 = 0. We may therefore restrict to the case where n is a prime number. First note that q(f nx) = q(ne x). If n fx 6= ne x, then there exists an m ≥ 1 such that ne x ≥ mλ. Since the homomorphism ϑ is saturated, there then exists m0 ∈ N such that x e ≥ m0 λ and nm0 ≥ m by virtue of ([74] 4.1 p. 11; cf. also [58] 0 I 4.8.13). Since m ≥ 1, the relation x e ≥ m0 λ contradicts the fact that x e is minimal. n

II.8.7. We keep the notation of II.8.5. For every n ≥ 0, we denote by Λ(p ) the n monoid over Λ defined by the pair (Λ, $pn ); in other words, Λ(p ) is the monoid Λ and (pn ) the structural homomorphism Λ → Λ is the Frobenius of order pn of Λ (II.5.12). We n n n naturally identify Λ(p ) with the cokernel of the homomorphism ϑ : N(p ) → P (p ) and n n denote also by q : P (p ) → Λ(p ) the canonical homomorphism. Let n

Pp∞ = lim P (p

)

−→

and

n

Λp∞ = lim Λ(p ) . −→

N

N

We identify Pp∞ with a submonoid of P∞ (II.7.8.1). We denote also by q : Pp∞ → Λp∞ n n the direct limit of the canonical homomorphisms q : P (p ) → Λ(p ) . For every t ∈ Pp∞ , the map (II.8.7.1)

νt : ∆∞ → µp∞ (OK ) = lim µpn (OK ), −→

n≥0

g 7→ hg, ti,

where hg, ti is defined in (II.7.14.1), induces, by virtue of (II.7.14.3), a homomorphism νt : ∆p∞ → µp∞ (OK ).

(II.8.7.2) It is clear that the map (II.8.7.3)

Pp∞ → Hom(∆p∞ , µp∞ (OK )),

t 7→ νt

n

n

is a homomorphism. For every n ≥ 0, denote by λ(p ) the image of λ in P (p ) by the isomorphism (II.7.8.2). Since νλ(pn ) = 1, (II.8.7.3) induces a homomorphism that we denote also by (II.8.7.4)

Λp∞ → Hom(∆p∞ , µp∞ (OK )),

x 7→ νx .

In fact, this induces, for every n ≥ 0, a homomorphism (II.8.7.5)

n

Λ(p

)

→ Hom(∆p∞ , µpn (OK )).

Let (II.8.7.6) (II.8.7.7)

Ξp∞ Ξpn

= Hom(∆p∞ , µp∞ (OK )), = Hom(∆p∞ , µpn (OK )).

We identify Ξpn with a subgroup of Ξp∞ . Lemma II.8.8. Under the assumptions of (II.8.7), let moreover x ∈ Λp∞ , and n be an n integer ≥ 0. Then we have νx (∆p∞ ) ⊂ µpn (OK ) if and only if x ∈ Λ(p ) ⊂ Λp∞ ; in particular, νx = 1 if and only if x ∈ Λ(1) ⊂ Λp∞ .

II.8. GALOIS COHOMOLOGY n+m

Suppose x ∈ Λ(p commutative diagram (II.8.8.1)

Λ(p

n

)

103

for an integer m ≥ 0. By virtue of (II.7.14.4), we have a $pm

)

 P gp /Zλ

·pm

 Hom(∆p∞ , Zp (1))

·pm

/ Λ(pn+m ) JJ JJ JJ JJ JJa  JJ / P gp /Zλ JJ JJ JJ JJ JJ J$  b / Hom(∆p∞ , Zp (1)) / / Hom(∆p∞ , µpn+m (O )) K

where the lower vertical arrows come from the identification of ∆p∞ and Lλ ⊗Z Zp (1) (II.6.10), a is the homomorphism (II.8.7.5), and b is the canonical morphism. The lower square is Cartesian because the torsion subgroup of P gp /Zλ is of order prime to p, and the upper square is Cartesian because Λ is saturated by virtue of II.8.6(i). We have νx (∆p∞ ) ⊂ µpn (OK ) if and only if a(x) is in the image of pm b. The lemma then follows by a chase in the diagram (II.8.8.1). Lemma II.8.9. There exists a canonical decomposition of Rp∞ into a direct sum of R1 -modules of finite presentation that are stable under the action of ∆p∞ , M (ν) (II.8.9.1) Rp∞ = Rp∞ , ν∈Ξp∞ (ν)

such that the action of ∆p∞ on the factor Rp∞ is given by the character ν. Moreover, for every n ≥ 0, we have M (ν) (II.8.9.2) Rpn = Rp∞ . ν∈Ξpn

For every n ≥ 1, set Cn = An ⊗OKn OK (II.6.5.2) and denote by λn : L ⊗Z µn (K) → µn (K)

(II.8.9.3)

the homomorphism defined by λ. We have a canonical isomorphism (II.6.6.4) (II.8.9.4)

∼

(n)

Cn → OK [P (n) ]/(πn − eλ

) ⊗OK [P ]/(π−eλ ) C1 ,

where λ(n) denotes the image of λ in P (n) by the canonical isomorphism (II.7.8.2). The group L ⊗Z µn (K) acts naturally on OK [P (n) ] by homomorphisms of (OK [P ])-algebras. In view of (II.8.9.4), we deduce from this an action of the group ker(λn ) on Cn by homomorphisms of C1 -algebras. Set Cp∞ = lim Cpn ,

(II.8.9.5)

−→

n≥0

where the direct limit is indexed by the set N with the usual ordering. We have canonical isomorphisms (II.6.8.4) (II.8.9.6)

∼

∼

∆p∞ → Lλ ⊗ Zp (1) → lim ker(λpn ). ←−

n≥0

Taking limits gives an action of ∆p∞ on Cp∞ by homomorphisms of C1 -algebras. By virtue of II.6.8(iii), for every integer n ≥ 0, we have a canonical ker(λpn )equivariant isomorphism (II.8.9.7)

∼

Rpn → Cpn ⊗C1 R1 .

104

II. LOCAL STUDY

From this we deduce a ∆p∞ -equivariant isomorphism ∼

Rp∞ → Cp∞ ⊗C1 R1 .

(II.8.9.8)

It therefore suffices to show that there exists a canonical decomposition of Cp∞ into a direct sum of C1 -modules of finite presentation that are stable under the action of ∆p∞ , M (ν) (II.8.9.9) C p∞ = C p∞ , ν∈Ξp∞ (ν)

such that the action of ∆p∞ on the factor Cp∞ is given by the character ν; moreover, for every n ≥ 0, we have M (ν) Cp∞ . (II.8.9.10) C pn = ν∈Ξpn

In view of (II.8.9.4), we can reduce to the case where R = OK [P ]/(π − eλ ). Note that X is no longer necessarily connected after this reduction. Let Λ be the cokernel in the category of monoids of the homomorphism ϑ : N → P and denote by q : P → Λ the canonical homomorphism (cf. II.8.5). We then have M (II.8.9.11) R= OK · α(e x), x∈Λ

where x e is the minimal lift of x in P defined in II.8.6(ii) and α : P → R is the homomorphism induced by the chart (P, γ) (II.6.2). By (II.8.9.4), for every n ≥ 0, we have (pn )

n

Cpn = OK [P (p ) ]/(πpn − eλ

(II.8.9.12) n

Denote by αpn : P (p phism (II.6.5.1)

)

).

→ Cpn the homomorphism induced by the canonical strict mor(Xn , MXn ) → AP .

This notation is compatible with that introduced in II.7.8 and does not lead to any confusion. Taking the direct limit, we obtain a homomorphism (II.8.9.13)

αp∞ : Pp∞ → Cp∞ .

n

n

We take again the notation of II.8.7 and for every x ∈ Λ(p ) , we denote by x e ∈ P (p ) −1 (pn ) defined by its monoid the unique minimal element of q (x) for the preordering on P structure (cf. II.8.6(ii)). We then have M (II.8.9.14) C pn = OK · αpn (e x). x∈Λ(pn ) n

n

By virtue of II.8.6(iii), the maps Λ(p ) → P (p ) , x 7→ x e are compatible. Therefore, by taking the direct limit, they define a map that we denote also by (II.8.9.15)

Λp∞ → Pp∞ ,

x 7→ x e.

We clearly have q(e x) = x. Since Λ is integral and the torsion subgroup of P gp /Zλ is of n n+1 order prime to p, the homomorphisms Λ(p ) → Λ(p ) are injective. Taking the direct limit of the decomposition (II.8.9.14), we obtain M (II.8.9.16) Cp∞ = OK · αp∞ (e x). x∈Λp∞

II.8. GALOIS COHOMOLOGY

105

Each element αp∞ (e x) ∈ Cp∞ is an eigenvector for the action of ∆p∞ . It immediately follows from (II.7.14.4) that the action of ∆p∞ on αp∞ (e x) is given by the character νx . For every ν ∈ Ξp∞ , we then set M (ν) x), OK · αp∞ (e (II.8.9.17) C p∞ = x∈Λp∞ |νx =ν

so that we have C p∞ =

(II.8.9.18)

M

(ν)

C p∞ .

ν∈Ξp∞ (ν)

Since the map Λp∞ → Ξp∞ , x 7→ νx is a homomorphism, we see that Cp∞ is a sub-C1 module of Cp∞ . It follows from II.8.8 and (II.8.9.14) that for every n ≥ 0, we have M (ν) (II.8.9.19) C pn = Cp∞ . ν∈Ξpn (ν)

Since Cpn is of finite presentation over C1 for every n ≥ 0, Cp∞ is of finite presentation over C1 for every ν ∈ Ξp∞ .

Theorem II.8.10. Let a be a nonzero element of OK and ζ a primitive pth root of unity in OK . Then: (i) There exists a unique homomorphism of graded R1 -algebras (II.8.10.1)

∧ (HomZ (∆p∞ , R1 /aR1 )) → H∗ (∆p∞ , Rp∞ /aRp∞ )

whose degree one component is the composition of the canonical morphisms (II.8.10.2)

∼

HomZ (∆p∞ , R1 /aR1 ) → H1 (∆p∞ , R1 /aR1 ) → H1 (∆p∞ , Rp∞ /aRp∞ ).

As a morphism of graded R1 -modules, this admits a canonical left inverse (II.8.10.3)

H∗ (∆p∞ , Rp∞ /aRp∞ ) → ∧(HomZ (∆p∞ , R1 /aR1 )),

whose kernel is annihilated by ζ − 1. (ii) The R1 -module Hi (∆p∞ , Rp∞ /aRp∞ ) is almost of finite presentation for every i ≥ 0 and is zero for every i ≥ rg(L) = d + 1 (II.6.2.5). (iii) The inverse system (H∗ (∆p∞ , Rp∞ /pr Rp∞ ))r≥0 satisfies the Mittag–Leffler condition; more precisely, if for all integers r0 ≥ r ≥ 0, we denote by (II.8.10.4)

0

hr,r0 : H∗ (∆p∞ , Rp∞ /pr Rp∞ ) → H∗ (∆p∞ , Rp∞ /pr Rp∞ )

the canonical morphism, then for every integer r ≥ 1, there exists an integer r0 ≥ r depending only on d and not on the other data in II.6.2, such that for every integer r00 ≥ r0 , the images of hr,r0 and hr,r00 are equal. Indeed, by II.8.9, we have a canonical decomposition of Rp∞ into a direct sum of R1 [∆p∞ ]-modules M (ν) (II.8.10.5) R p∞ = Rp∞ ⊗OK OK (ν), ν∈Ξp∞

where ∆p∞ acts trivially on

(ν) Rp∞

and acts on OK (ν) = OK by the character ν. Since the

(ν)

Rp∞ ’s are OK -flat, by virtue of II.3.15, we have a canonical decomposition into a direct sum of R1 -modules M (ν) (II.8.10.6) H∗ (∆p∞ , Rp∞ /aRp∞ ) = H∗ (∆p∞ , OK (ν)/aOK (ν)) ⊗OK Rp∞ . ν∈Ξp∞

106

II. LOCAL STUDY (1)

(i) We have Rp∞ = R1 (II.8.9.2), so that the component for ν = 1 of the decomposition (II.8.10.6) is the image of the canonical homomorphism of graded R1 -algebras H∗ (∆p∞ , OK /aOK ) ⊗OK R1 → H∗ (∆p∞ , Rp∞ /aRp∞ ).

(II.8.10.7)

Moreover, the canonical homomorphism of graded R1 -algebras (II.8.10.8)

∧ (HomZ (∆p∞ , OK /aOK )) ⊗OK R1 → ∧(HomZ (∆p∞ , R1 /aR1 ))

is an isomorphism. The statement then follows from (II.8.10.6) and II.8.1 (applied with A = OC ). (ii) Let i, n be two integers ≥ 0 and ζn a primitive pn th root of unity. It follows from (II.8.10.6), II.8.1, and II.8.9 that Hi (∆p∞ , Rp∞ /aRp∞ ) is the direct sum of an R1 -module of finite presentation and an R1 -module annihilated by ζn − 1. It is therefore almost of finite presentation over R1 . The second assertion is clear because the p-cohomological dimension of ∆p∞ is equal to d. (iii) This follows from (II.8.10.6) and II.8.1(iii). Corollary II.8.11. For every nonzero element a of OK and every integer i ≥ 0, the kernel and cokernel of the canonical morphism Hi (∆p∞ , R1 /aR1 ) → Hi (∆, R/aR)

(II.8.11.1)

1

are annihilated by mK and p p−1 mK , respectively. Indeed, the canonical morphism Hi (∆p∞ , Rp∞ /aRp∞ ) → Hi (∆, R/aR)

(II.8.11.2)

is an almost isomorphism by II.6.24. On the other hand, it follows from (II.8.10.6), II.8.1 (1) (applied with A = OC ), II.3.15, and the fact that Rp∞ = R1 (II.8.9.2), that the canonical morphism Hi (∆p∞ , R1 /aR1 ) → Hi (∆p∞ , Rp∞ /aRp∞ )

(II.8.11.3)

is injective with cokernel annihilated by ζ − 1, where ζ is a primitive pth root of unity. 1 . The statement follows because v(ζ − 1) = p−1 Corollary II.8.12. The canonical homomorphism ∗ r d H∗cont (∆p∞ , R p∞ ) → lim H (∆p∞ , Rp∞ /p Rp∞ )

(II.8.12.1)

←− r≥0

is an isomorphism. Indeed, by (II.3.10.4) and (II.3.10.5), for every i ≥ 0, we have an exact sequence 0 → R1 lim Hi−1 (∆p∞ , Rp∞ /pr Rp∞ ) ←− r≥0

i r d → Hicont (∆p∞ , R p∞ ) → lim H (∆p∞ , Rp∞ /p Rp∞ ) → 0 ←− r≥0

whose left term is zero by virtue of II.8.10(iii) and (II.3.10.2).

II.8. GALOIS COHOMOLOGY

107

II.8.13. Let a be a nonzero element of OK . Since the torsion subgroup of P gp /Zλ is of order prime to p, we have a canonical isomorphism (II.6.10) (II.8.13.1)

∼

(P gp /Zλ) ⊗Z Zp (−1) → HomZp (∆p∞ , Zp ).

c1 -linear isomorphisms In view of (II.7.12.1), we deduce from this R ∼ e1 Ω (II.8.13.2) R/OK ⊗R (R1 /aR1 )(−1) → HomZ (∆p∞ , R1 /aR1 ), (II.8.13.3)

∼ c e1 c Ω R/OK ⊗R R1 (−1) → HomZ (∆p∞ , R1 ).

We will, from now on, view the targets of these morphisms as cohomology groups (II.6.12). Moreover, the composition of (II.8.13.3) and the canonical isomorphism (II.6.12.1) (II.8.13.4)

∼

c1 ) → HomZ (∆∞ , R c1 ) HomZ (∆p∞ , R

e is none other than the morphism δ(−1) (II.7.19.9), by II.7.19. c1 -algebras Proposition II.8.14. There exists a unique homomorphism of graded R ∗ e1 c d (II.8.14.1) ∧ (Ω R/OK ⊗R R1 (−1)) → Hcont (∆p∞ , Rp∞ ) c1 whose degree one component is induced by (II.8.13.3). As a morphism of graded R modules, it admits a canonical left inverse d e1 c1 (−1)), (II.8.14.2) H∗cont (∆p∞ , R ⊗R R p∞ ) → ∧(Ω R/OK

whose kernel is annihilated by p

1 p−1

.

This follows from II.8.10(i) and II.8.12. ∆p∞ d c1 . Corollary II.8.15. We have (R =R p∞ ) ∆p∞ c1 → (R d Indeed, by II.8.14, the canonical homomorphism R admits a left p∞ ) 1 ∞ ∆ d d c p p−1 . ∞ Since R inverse (R ) → R whose kernel is annihilated by p p∞ is OC -flat 1 p (II.6.14), the left inverse is injective, proving the statement.

Remark II.8.16. By II.6.13, (II.6.11.1), and (II.6.12.1), theorem II.8.10 and its corollaries II.8.11 and II.8.12 still hold when we replace ∆p∞ by ∆∞ and Rp∞ by R∞ . The same then holds for proposition II.8.14 and its corollary II.8.15. We therefore have ∆∞ d c1 . (R =R ∞) Proposition II.8.17. Let a be a nonzero element of OK . (i) There exists a unique homomorphism of graded R1 -algebras e1 (II.8.17.1) ∧ (Ω ⊗R (R1 /aR1 )(−1)) → H∗ (∆, R/aR) R/OK

whose degree one component is induced by (II.8.13.2). It is almost injective and 1 its cokernel is annihilated by p p−1 mK . (ii) The R1 -module Hi (∆, R/aR) is almost of finite presentation for every i ≥ 0 and almost zero for every i ≥ d + 1. (iii) For all integers r0 ≥ r ≥ 0, denote by (II.8.17.2)

0

~r,r0 : H∗ (∆, R/pr R) → H∗ (∆, R/pr R)

the canonical morphism. Then for every integer r ≥ 1, there exists an integer r0 ≥ r depending only on d and not on the other data in II.6.2, such that for every integer r00 ≥ r0 , the images of ~r,r0 and ~r,r00 are almost isomorphic. This follows from II.6.24 and II.8.10.

108

II. LOCAL STUDY

II.8.18.

For every OK -module M , set M [ = HomOK (mK , M ).

(II.8.18.1)

The canonical morphism M → M [ is an almost isomorphism. A morphism of OK modules u : M → N is an almost isomorphism if and only if the associated morphism u[ : M [ → N [ is an isomorphism (V.2.5). Lemma II.8.19. The canonical morphism c1 → (R c1 )[ (II.8.19.1) : R is an isomorphism.

c1 is flat over OC (II.6.14),  is injective. Let us show that  is surjective. Let Since R [ c c1 [ 1 ] is an affinoid algebra u ∈ (R1 ) . For every α ∈ Q>0 , set xα = u(pα ). Recall that R p c1 is the unit ball for the norm | |sup on R c1 [ 1 ] (cf. the proof of II.6.15). over C and that R p

The relations xα+β = pα xβ = pβ xα (α, β ∈ Q>0 ) then imply that |p−α xα |sup ≤ 1.

(II.8.19.2)

c1 and x is independent of α. It is clear that (x) = u. Consequently, x = p−α xα ∈ R

Remark II.8.20. The canonical morphism OC → (OC )[ is an isomorphism. The proof is a very simple variant of that of II.8.19. c1 -algebras Proposition II.8.21. There exists a unique homomorphism of graded R b e1 c (−1)) → H∗ (∆, R) (II.8.21.1) ∧ (Ω ⊗ R R

R/OK

1

cont

c1 whose degree one component is induced by (II.8.13.3). As a morphism of graded R modules, it admits a canonical left inverse b → ∧(Ω e1 c (−1)), (II.8.21.2) H∗ (∆, R) ⊗ R cont

whose kernel is annihilated by p

R/OK

1 p−1

R

1

.

c1 -algebras Indeed, we have a commutative diagram of homomorphisms of graded R (II.8.21.3)

e1 ∧(Ω R/OK

q c1 (−1)) ⊗R R

w v

/ H∗

cont (∆p∞ , Rp∞ )

u

d

/ H∗ (∆, R) b cont

ι

 1 e c1 (−1)))[ (∧(ΩR/OK ⊗R R m

v[

 [ / (H∗ (∆ ∞ , R d p p∞ )) cont

u[

 / (H∗ (∆, R)) b [ cont

w[

where the functor ( )[ is defined in (II.8.18.1), the vertical arrows and u are the canonical morphisms, v is the homomorphism (II.8.14.1), and w is the section (II.8.14.2) of v. Then ι and u[ are isomorphisms (II.8.19 and II.6.25). On the other hand, it follows from II.8.14 1 that the kernel of w[ is annihilated by p p−1 . The proposition follows by a chase in the diagram (II.8.21.3) because u ◦ v is the homomorphism (II.8.21.1). b ∆=R c1 . Corollary II.8.22. (i) We have (R) b is equal to the kernel of the (ii) The p-primary torsion submodule M 1 of H1cont (∆, R) morphism b → H1 (∆, (πρ)−1 R) b (II.8.22.1) ι : H1cont (∆, R) cont

II.8. GALOIS COHOMOLOGY

109

b b ⊂ (πρ)−1 R. deduced from the canonical injection R (iii) The composition (II.8.22.2)

b ι 1 1 −1 b e1 c Ω R), R/OK ⊗R R1 (−1) −→ Hcont (∆, R) −→ Hcont (∆, (πρ)

where the first arrow is induced by (II.8.13.3), is the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (II.7.22.2). (iv) For every integer i ≥ 0, denote by M i the p-primary torsion submodule of b Then there exists a unique isomorphism of graded R i c -algebras H (∆, R). 1

cont

(II.8.22.3)

∧

e1 (Ω R/OK

∼

c1 (−1)) → ⊗R R

b i ⊕i≥0 (Hicont (∆, R)/M )

such that the composition (II.8.22.4)

b 1 1 1 −1 b c e1 Ω R), R/OK ⊗R R1 (−1) → Hcont (∆, R)/M → Hcont (∆, (πρ)

where the first arrow is the degree one component of (II.8.22.3) and the second arrow is induced by ι, is the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (II.7.22.2). b ∆→ b ∆ admits a left inverse (R) c1 → (R) (i) By II.8.21, the canonical homomorphism R 1 b is O -flat (II.6.14), the left inverse is c1 whose kernel is annihilated by p p−1 . Since R R C injective, proving the statement. 1 (ii) It follows from II.8.21 that M 1 is annihilated by p p−1 . On the other hand, the b → (πρ)−1 R b is the composition of the canonical injection R b multiplication by πρ in R ∼ b b sends (πρ)−1 x to x. Since b → and the isomorphism (πρ)−1 R R that, for every x ∈ R, 1 1 v(ρ) ≥ p−1 , we deduce from this that ker(ι) = M . (iii) This follows from the definition of (II.8.13.3), (II.7.20.2), and (II.7.22.3). (iv) This follows from (ii), (iii), and II.8.21.

Proposition II.8.23. Let M be a discrete R1 -∆p∞ -module, a a nonzero element of 1 OK , and α a rational number. Suppose inf(v(a), α) > p−1 and that M is a projective (R1 /aR1 )-module of finite type generated by a finite number of elements that are ∆p∞ 1 invariant modulo pα M . Let b = ap− p−1 . Then for every i ≥ 0, the kernel and cokernel of the canonical morphism (II.8.23.1)

Hi (∆p∞ , M/bM ) → Hi (∆, (M/bM ) ⊗R1 R) 1

are annihilated by mK and p p−1 mK , respectively. Since M is a direct summand of a free (R1 /aR1 )-module of finite type, we have, by II.6.13, (II.8.23.2)

((M/bM ) ⊗R1 R∞ )Σ0 = (M/bM ) ⊗R1 Rp∞ .

Therefore by virtue of (II.6.11.1), the canonical morphism (II.8.23.3)

Hi (∆p∞ , (M/bM ) ⊗R1 Rp∞ ) → Hi (∆∞ , (M/bM ) ⊗R1 R∞ )

is an isomorphism. On the other hand, by II.6.22, the canonical morphism (II.8.23.4)

(M/bM ) ⊗R1 R∞ → ((M/bM ) ⊗R1 R)Σ

is an almost isomorphism. Consequently, by virtue of II.6.20, the canonical morphism (II.8.23.5)

Hi (∆∞ , (M/bM ) ⊗R1 R∞ ) → Hi (∆, (M/bM ) ⊗R1 R)

is an almost isomorphism. It therefore suffices to show that the canonical morphism (II.8.23.6)

Hi (∆p∞ , M/bM ) → Hi (∆p∞ , (M/bM ) ⊗R1 Rp∞ )

110

II. LOCAL STUDY 1

is injective with cokernel annihilated by p p−1 . By II.8.9, we have a canonical decomposition of (M/bM ) ⊗R1 Rp∞ into a direct sum of R1 [∆p∞ ]-modules M (ν) (II.8.23.7) (M/bM ) ⊗R1 Rp∞ = (M/bM ) ⊗R1 Rp∞ ⊗R1 R1 (ν), ν∈Ξp∞

where ∆p∞ acts trivially on (1) R p∞

(ν) Rp∞

and acts on R1 (ν) = R1 by the character ν. Since

= R1 (II.8.9.2), the statement follows from II.8.4 (applied to A = R1 /aR1 and (ν)

(ν)

N = Rp∞ /aRp∞ for ν 6= 1). II.9. Fontaine p-adic infinitesimal thickenings II.9.1. Let us begin by recalling the following construction due to Grothendieck ([36] IV 3.3). Let A be a commutative Z(p) -algebra and n an integer ≥ 1. The ring homomorphism (II.2.3.1) (II.9.1.1)

Φn+1 : Wn+1 (A/pn A) → A/pn A pn pn−1 (x1 , . . . , xn+1 ) 7→ x1 + px2 + · · · + pn xn+1

vanishes on Vn (A/pn A) and therefore induces, by taking the quotient, a ring homomorphism (II.9.1.2)

Φ0n+1 : Wn (A/pn A) → A/pn A pn pn−1 (x1 , . . . , xn ) 7→ x1 + px2 + · · · + pn−1 xpn .

The latter vanishes on (II.9.1.3)

Wn (pA/pn A) = ker(Wn (A/pn A) → Wn (A/pA))

and in turn factors into a ring homomorphism (II.9.1.4)

θn : Wn (A/pA) → A/pn A.

It immediately follows from the definition that the diagram (II.9.1.5)

Wn+1 (A/pA)

θn+1

RF

 Wn (A/pA)

θn

/ A/pn+1 A  / A/pn A

where R is the restriction morphism (II.2.3.2), F is the Frobenius (II.2.3.4), and the unlabeled arrow is the canonical homomorphism, is commutative. For every homomorphism of commutative Z(p) -algebras ϕ : A → B, the diagram (II.9.1.6)

Wn (A/pA) θn

 A/pn A

/ Wn (B/pB) θn

 / B/pn B

where the horizontal arrows are the morphisms induced by ϕ, is commutative. Proposition II.9.2. Let A be a commutative Z(p) -algebra satisfying the following conditions: (i) A is Z(p) -flat. (ii) A is integrally closed in A[ p1 ]. (iii) The absolute Frobenius of A/pA is surjective.

II.9. FONTAINE P -ADIC INFINITESIMAL THICKENINGS

111

(iv) There exist an integer N ≥ 1 and a sequence (pn )0≤n≤N of elements of A such that p0 = p and ppn+1 = pn for every 0 ≤ n ≤ N − 1. For every integer 0 ≤ n ≤ N , we set ξn = [pn ] − p ∈ Wn (A/pA),

(II.9.2.1)

where pn is the class of pn in A/pA and [pn ] is the multiplicative representative of pn . Then for all integers n ≥ 1 and i ≥ 0 such that n + i ≤ N , the sequence (II.9.2.2)

Wn (A/pA)

·Ri (ξn+i )

/ Wn (A/pA)

θn ◦Fi

/ A/pn A

/0

is exact. Indeed, we have FR(ξn+1 ) = ξn and θn (Fi (Ri (ξn+i ))) = θn (ξn ) = 0. To establish the exactness of (II.9.2.2), we proceed by induction on n. Let i be an integer with i+1 0 ≤ i ≤ N − 1 and α, β ∈ A such that αp = pβ. It follows from conditions (i) and (ii) that α ∈ pi+1 A. Consequently, the sequence A/pA

(II.9.2.3)

·pi+1

/ A/pA

Fi+1

/ A/pA

/0

is exact and the statement is true for n = 1. Next, assume that the statement is true for 1 ≤ n ≤ N − 1 (and every 0 ≤ i ≤ N − n) and let us prove it for n + 1. Let i be an integer with 0 ≤ i ≤ N − n − 1. We have a commutative diagram with exact lines (II.9.1.5) (II.9.2.4)

0

/ A/pA

0

 / A/pA

Vn

·Ri (ξi+1 ) V

n

/ Wn+1 (A/pA)

R

·Ri (ξn+i+1)

 / Wn+1 (A/pA)

R

/ Wn (A/pA)

/0

·Ri+1 (ξn+i+1 )

 / Wn (A/pA)

/0

Fi+1

 A/pA

·pn

0

 / pn A/pn+1 A

θn+1 ◦Fi

 / A/pn+1 A

θn ◦Fi+1

 / A/pn A

/0

The induction hypothesis and (II.9.2.3) then imply the statement for n + 1 using the snake lemma. II.9.3. Let A be a commutative Z(p) -algebra. We denote by RA the inverse limit of the inverse system (A/pA)N whose transition morphisms are the iterates of the Frobenius homomorphisms of A/pA: (II.9.3.1)

RA = lim A/pA. ←−

x7→xp

This is a perfect ring of characteristic p. For every integer n ≥ 1, the canonical projection RA → A/pA onto the (n + 1)th component of the inverse system (A/pA)N (that is, the component of index n) induces a homomorphism (II.9.3.2)

νn : W(RA ) → Wn (A/pA).

Since νn = F ◦ R ◦ νn+1 , taking the inverse limit gives a homomorphism (II.9.3.3)

ν : W(RA ) → lim Wn (A/pA), ←−

n≥1

112

II. LOCAL STUDY

where the transition morphisms of the inverse limit are the morphisms FR. One immediately verifies that it is bijective. In view of (II.9.1.5), the homomorphisms θn induce, by taking the inverse limit, a homomorphism (II.9.3.4)

b θ : W(RA ) → A,

b is the p-adic Hausdorff completion of A. We recover the homomorphism defined where A by Fontaine ([29] 2.2). We set (II.9.3.5)

A2 (A) = W(RA )/ ker(θ)2 ,

b for the homomorphism induced by θ (cf. [31] 1.2.2). and write also θ : A2 (A) → A For every homomorphism of commutative Z(p) -algebras ϕ : A → B, the diagram (II.9.3.6)

W(RA ) θ

/ W(RB ) θ

 b A

 /B b

where the horizontal arrows are the morphisms induced by ϕ, is commutative (II.9.1.6). The correspondence A 7→ A2 (A) is therefore functorial.

Remark II.9.4. The canonical projection RW(k) → k onto the first component (that is, the component of index 0) is an isomorphism. It therefore induces an isomorphism ∼ W(RW(k) ) → W(k), which we use to identify these two rings. The homomorphism θ then identifies with the identity endomorphism of W(k). Proposition II.9.5 ([73] A.1.1 and A.2.2). Let A be a commutative Z(p) -algebra satisfying the following conditions: (i) A is Z(p) -flat. (ii) A is integrally closed in A[ p1 ]. (iii) The absolute Frobenius of A/pA is surjective. (iv) There exists a sequence (pn )n≥0 of elements of A such that p0 = p and ppn+1 = pn for every n ≥ 0. We denote by p the element of RA induced by the sequence (pn )n≥0 and set (II.9.5.1)

ξ = [p] − p ∈ W(RA ),

where [p] is the multiplicative representative of p. Then the sequence (II.9.5.2)

·ξ θ b −→ 0 0 −→ W(RA ) −→ W(RA ) −→ A

is exact. Indeed, for every n ≥ 1, if we set (II.9.5.3)

ξn = [pn ] − p ∈ Wn (A/pA),

where pn is the class of pn in A/pA, then the sequence (II.9.5.4)

·ξn

θ

n Wn (A/pA) −→ Wn (A/pA) −→ A/pn A −→ 0

is exact by virtue of II.9.2. Since the homomorphism RF : Wn+1 (A/pA) → Wn (A/pA) is surjective for every n ≥ 0, the sequence (II.9.5.2) is exact in the middle and on the right ([41] 0.13.2.1(i) and 0.13.2.2). If a = (a0 , a1 , a2 , . . . ) ∈ W(RA ) is such that ξa = 0, then (II.9.5.5)

2

(pa0 , pp a1 , pp a2 , . . . ) = (0, ap0 , ap1 , . . . ).

II.9. FONTAINE P -ADIC INFINITESIMAL THICKENINGS

113

To show that ξ is not a zero divisor in W(RA ), it therefore suffices to show that p is not a zero divisor in RA . Let y = (yn )n∈N ∈ RA satisfy py = 0. For every n ≥ 0, let yen be a n lift of yn in A. We have pn yen ∈ pA. Consequently, yen ∈ ppn −1 A because p is not a zero divisor in A. It follows that p p = (e yn+1 yn = yn+1

(II.9.5.6) because p

n+2

−p≥p

n+1

mod pA) = 0

.

II.9.6. Let Y = (Y, MY ) be an affine logarithmic Z(p) -scheme with ring A, M a monoid, and u : M → Γ(Y, MY ) a homomorphism. Consider the inverse system of multiplicative monoids (A)n∈N , where the transition morphisms are all equal to the pth power homomorphism. We denote by Q the fibered product of the diagram of homomorphisms of monoids (II.9.6.1)

M  /A

lim A ←−p

x7→x

where the horizontal arrow is the projection onto the first component (that is, the component of index 0) and the vertical arrow is the composition of u and the canonical homomorphism Γ(Y, MY ) → A. We denote by q the composition (II.9.6.2)

[ ]

Q −→ lim A −→ RA −→ W(RA ), ←−p

x7→x

where the first and second arrows are the canonical homomorphisms (II.9.3.1) and [ ] is the multiplicative representative. It immediately follows from the definitions that the diagram /M Q (II.9.6.3) q

 W(RA )

θ

 /A b

where the unlabeled arrows are the canonical morphisms, is commutative. b with the logarithmic structure M b inverse image of MY We endow Yb = Spec(A) Y and Spec(W(RA )) with the logarithmic structure Q associated with the prelogarithmic structure defined by q (II.9.6.2). By (II.9.6.3), θ induces a morphism (II.9.6.4)

(Yb , MYb ) → (Spec(W(RA )), Q).

The following statement is inspired by ([73] 1.4.2).

Proposition II.9.7. We keep the assumptions of II.9.6, denote by Y ◦ the maximal open subscheme of Y where the logarithmic structure MY is trivial, and suppose that the following conditions are satisfied: (a) A is a normal integral domain. (b) Y ◦ is a nonempty simply connected Q-scheme. (c) M is integral and there exist a fine and saturated monoid M 0 and a homomorphism v : M 0 → M such that the induced homomorphism M 0 → M/M × is an isomorphism. Then: (i) The monoid Q is integral and the group M 0gp is free.

114

II. LOCAL STUDY

(ii) We can complete the diagram (II.9.6.1) into a commutative diagram (II.9.7.1)

/M

v

M0 w

 lim A

 /A

←−

x7→xp

Denote by β : M 0 → Q the induced homomorphism. (iii) The logarithmic structure Q on Spec(W(RA )) is associated with the prelogarithmic structure defined by the composition (II.9.7.2)

β

q

M 0 → Q → W(RA ).

In particular, the logarithmic scheme (Spec(W(RA )), Q) is fine and saturated. (iv) If, moreover, the composition u ◦ v : M 0 → Γ(Y, MY ) is a chart for Y (II.5.13), then the morphism (II.9.6.4) is strict. (i) Since Y ◦ is nonempty, the canonical image of Γ(Y, MY ) in A does not contain 0. It immediately follows that Q is integral. On the other hand, since M 0 is saturated, the torsion subgroup of M 0gp is contained in M 0 . But M 0 is sharp; therefore M 0gp is without torsion. (ii) Let L be the field of fractions of A and L an algebraic closure of L. Consider the inverse system of multiplicative monoids (L)n∈N , where the transition morphisms are all equal to the pth power homomorphism. By (i) and its proof, there exists a homomorphism (II.9.7.3)

M 0gp → lim L ←−

x7→xp

v

that lifts the homomorphism M 0gp → L induced by the composition M 0 → M → A. Let t ∈ Γ(Y, MY ), x its canonical image in A, and y ∈ L such that y p = x. The extension A[z]/z p − x of A is étale over Y ◦ because p is invertible in Y ◦ and x does not vanish at any points of Y ◦ . Since Y ◦ is simply connected, it follows that y ∈ L and consequently that y ∈ A because A is normal. It follows that the restriction of (II.9.7.3) to M 0 induces a homomorphism (II.9.7.4)

w : M 0 → lim A ←−

x7→xp

as desired. (iii) Let G be the inverse image of M × under the canonical homomorphism Q → M . It immediately follows from the definition (II.9.6.1) that G is a subgroup of Q. On the other hand, the composition M 0 → Q/G → M/M × , where the first arrow is deduced from β, is an isomorphism. Consequently, M 0 → Q/G is an isomorphism. The statement then follows from ([73] 1.3.1). (iv) This immediately follows from (iii). II.9.8. For the remainder of this chapter, we fix a sequence (pn )n≥0 of elements of OK such that p0 = p and ppn+1 = pn for every n ≥ 0. We denote by p the element of ROK induced by the sequence (pn )n≥0 and set (II.9.8.1)

ξ = [p] − p ∈ W(ROK ),

where [p] is the multiplicative representative of p. By II.9.5, the sequence (II.9.8.2)

·ξ

θ

0 −→ W(ROK ) −→ W(ROK ) −→ OC −→ 0

II.9. FONTAINE P -ADIC INFINITESIMAL THICKENINGS

115

is exact. It induces an exact sequence ·ξ

θ

0 −→ OC −→ A2 (OK ) −→ OC −→ 0,

(II.9.8.3)

where we have also denoted by ·ξ the morphism induced by the multiplication by ξ in A2 (OK ) (II.9.3.5). The ideal ker(θ) of A2 (OK ) has square zero. It is a free OC -module with basis ξ. We will denote it by ξOC . Note that, unlike ξ, this module does not depend on the choice of the sequence (pn )n≥0 . The Galois group GK acts naturally on W(ROK ) by ring automorphisms, and the homomorphism θ (II.9.8.2) is GK -equivariant. We deduce from this an action of GK on A2 (OK ) by ring automorphisms such that the homomorphism θ (II.9.8.3) is GK equivariant. We set S = Spec(OK )

(II.9.8.4)

ˇ = Spec(O ), and S C

which we endow with the logarithmic structures inverse images of MS (II.6.1), denoted by MS and MSˇ , respectively. The action of GK on OK and OC extends naturally to a ˇ M ), respectively. left action on the logarithmic schemes (S, M ) and (S, S

We set

ˇ S

A2 (S) = Spec(A2 (OK )),

(II.9.8.5)

which we endow with the logarithmic structure MA2 (S) defined as follows. Let QS be the monoid and qS : QS → W(ROK ) the homomorphism defined in II.9.6 (denoted by Q and q) by taking for (Y, MY ) the logarithmic scheme (S, MS ) and for u the canonical homomorphism Γ(S, MS ) → Γ(S, MS ). We denote by MA2 (S) the logarithmic structure on A2 (S) associated with the prelogarithmic structure defined by the homomorphism QS → A2 (OK ) induced by qS . By II.9.7, the logarithmic scheme (A2 (S), MA2 (S) ) is fine and saturated, and θ induces an exact closed immersion ˇ M ) → (A (S), M iS : (S, ˇ 2 A2 (S) ). S

(II.9.8.6)

The Galois group GK has a natural action on the monoid QS and the homomorphism qS : QS → W(ROK ) is GK -equivariant. From this, we deduce a left action of GK on the logarithmic scheme (A2 (S), MA2 (S) ). The morphism iS is GK -equivariant. Remark II.9.9. We denote by ξ −1 OC the dual OC -module of ξOC . For every OC module M , we denote the OC -modules M ⊗OC (ξOC ) and M ⊗OC (ξ −1 OC ) simply by ξM and ξ −1 M , respectively. Note that, unlike ξ, these modules do not depend on the choice of the sequence (pn )n≥0 . It is therefore important to not identify them with M . c1 -algebras A, in the remainder of this chapter, we consider Higgs A-modules For R e1 with coefficients in ξ −1 Ω R/OK ⊗R A (cf. II.2.8). We will abusively say that they have coeffie1 e1 . We will denote the category of these modules by HM(A, ξ −1 Ω ). cients in ξ −1 Ω R/OK

R/OK

Proposition II.9.10. The absolute Frobenius homomorphisms of R∞ /pR∞ and R/pR are surjective. Let us first show that the absolute Frobenius homomorphism of R∞ /pR∞ is surjective. Recall that for every integer n ≥ 1, Spec(Rn ) is a connected component of

116

II. LOCAL STUDY

Xn ⊗OKn OK (II.6.8) and that we have a Cartesian diagram of OK -morphisms (II.6.6.4) (II.9.10.1)

Xn ⊗OKn OK

/ Spec(O [P (n) ]/(π − eλ(n) )) n K

 X ⊗OK OK

 / Spec(O [P ]/(π − eλ )) K

where for every t ∈ P , we have denoted by t(n) its image in P (n) by the canonical isomorphism (II.7.8.2). Since the canonical morphism X → S ×AN AP is étale (II.6.2.4), it suffices to show that the absolute Frobenius homomorphism of the direct limit of Fp -algebras (with respect to the divisibility relation) (n)

lim (OK /pOK )[P (n) ]/(πn − eλ

(II.9.10.2)

−→

)

n≥1

is surjective. This follows from the fact that the absolute Frobenius homomorphism of (n) (pn) OK /pOK is surjective and that for every t ∈ P , we have et = ept in (II.9.10.2). Next, let us show that the absolute Frobenius homomorphism of R/pR is surjective. Let N be a finite extension of F∞ contained in F and D the integral closure of R in N . Suppose that D is almost étale over R∞ . Then D/pD is almost étale over R∞ /pR∞ (V.7.4(3)). Since the absolute Frobenius homomorphism of R∞ /pR∞ is surjective, the absolute Frobenius homomorphism of D/pD is almost surjective by virtue of V.7.9. For every x ∈ D, there exist x0 , y ∈ D such that p1/2 x = x0p + py. Then x00 = p−1/(2p) x0 ∈ D and we have x = x00p + p1/2 y. By the same argument, there exist y 0 , z ∈ D such that y = y 0p + p1/2 z. Consequently, we have x ≡ (x00 + p1/(2p) y 0 )p mod pD. Taking the direct limit, it follows, by virtue of II.6.17, that the absolute Frobenius homomorphism of R/pR is surjective. II.9.11.

By II.9.5 and II.9.10, the sequence ·ξ θ b 0 −→ W(RR ) −→ W(RR ) −→ R −→ 0,

(II.9.11.1)

where ξ is the element (II.9.8.1), is exact. It induces an exact sequence ·ξ θ b b −→ 0 −→ R A2 (R) −→ R −→ 0,

(II.9.11.2)

where we have also denoted by ·ξ the morphism induced by the multiplication by ξ in b with basis ξ, A2 (R). The ideal ker(θ) of A2 (R) has square zero. It is a free R-module b canonically isomorphic to ξ R (cf. II.9.9). The Galois group Γ (II.6.10) acts naturally on W(RR ) by ring automorphisms, and the homomorphism θ (II.9.11.1) is Γ-equivariant. We deduce from this an action of Γ on A2 (R) by ring automorphisms such that the homomorphism θ (II.9.11.2) is Γ-equivariant. We set b (II.9.11.3) Y = Spec(R) and Yb = Spec(R) and endow these with the logarithmic structures inverse images of MX (II.6.2), denoted b induce left actions on the by M and M , respectively. The actions of Γ on R and R Y

b Y

logarithmic schemes (Y, MY ) and (Yb , MYb ), respectively. We set (II.9.11.4)

A2 (Y ) = Spec(A2 (R))

and endow this with the logarithmic structure MA0 2 (Y ) defined as follows. Let QY be the monoid and qY : QY → W(RR ) the homomorphism defined in II.9.6 (denoted by Q and

II.9. FONTAINE P -ADIC INFINITESIMAL THICKENINGS

117

q) by taking for u the canonical homomorphism Γ(X, MX ) → Γ(Y, MY ). We denote by MA0 2 (Y ) the logarithmic structure on A2 (Y ) associated with the prelogarithmic structure defined by the homomorphism QY → A2 (R) induced by qY . The homomorphism θ then induces a morphism (II.9.6.4) (II.9.11.5)

i0Y : (Yb , MYb ) → (A2 (Y ), MA0 2 (Y ) ).

The Galois group Γ acts naturally on the monoid QY , and the homomorphism qY : QY → W(RR ) is Γ-equivariant. We deduce from this a left action of Γ on (A2 (Y ), MA0 2 (Y ) ). The morphism i0Y is Γ-equivariant. Without the assumptions of II.9.7, we would not know whether the logarithmic scheme (A2 (Y ), MA0 2 (Y ) ) is fine and saturated and whether i0Y is an exact closed immersion. That is why we will endow A2 (Y ) with a different logarithmic structure MA2 (Y ) in II.9.12. II.9.12. For every t ∈ P , we denote by e t the element of QY defined by its projections (II.9.6.1) (II.9.12.1)

n

(α∞ (t(p ) ))n∈N ∈ lim R ←− N

n

and γ(t) ∈ Γ(X, MX ),

n

where t(p ) is the image of t in P (p ) by the isomorphism (II.7.8.2) and α∞ is the homomorphism (II.7.8.5). We will see that this notation is compatible with that introduced in (II.7.18.2) and does not cause any confusion. The reader will note that e t depends on the choice of the morphism (II.6.7.1). The resulting map (II.9.12.2)

P → QY ,

t 7→ e t

is a morphism of monoids. We denote by (II.9.12.3)

qeY : P → A2 (R)

the composition (II.9.12.4)

qY

P −→ QY −→ W(RR ) −→ A2 (R).

We endow A2 (Y ) with the logarithmic structure MA2 (Y ) associated with the prelogarithmic structure defined by qeY . The morphism (II.9.12.2) then induces a morphism of logarithmic structures on A2 (Y ) (II.9.11) (II.9.12.5)

MA2 (Y ) → MA0 2 (Y ) .

b is induced by α (cf. II.7.8). Consequently, It is clear that the composition θ ◦ qeY : P → R θ induces an exact closed immersion (II.9.12.6)

iY : (Yb , MYb ) → (A2 (Y ), MA2 (Y ) ),

which factors through i0Y (II.9.11.5). Proposition II.9.13. Suppose that there exists a fine and saturated chart for (X, MX ), M 0 → Γ(X, MX ) (II.5.13), such that the induced homomorphism (II.9.13.1)

× M 0 → Γ(X, MX )/Γ(X, OX )

is an isomorphism. Then the morphism MA2 (Y ) → MA0 2 (Y ) (II.9.12.5) is an isomorphism.

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II. LOCAL STUDY

Indeed, by II.9.7, the logarithmic scheme (A2 (Y ), MA0 2 (Y ) ) is fine and saturated, and the morphism i0Y (II.9.11.5) is an exact closed immersion. Consequently, for every geometric point y of Yb , if we also denote by y the geometric point iY (y) = i0Y (y) of A2 (Y ), the homomorphism (II.9.13.2)

× × MA2 (Y ),y /OA → MA0 2 (Y ),y /OA 2 (Y ),y 2 (Y ),y

induced by (II.9.12.5) is an isomorphism. Since MA0 2 (Y ),y is integral, it follows that the morphism MA2 (Y ),y → MA0 2 (Y ),y (II.9.12.5) is an isomorphism; the proposition follows. Remark II.9.14. Note that, unlike MA0 2 (Y ) , the logarithmic structure MA2 (Y ) depends on the chart (P, γ) for (X, MX ) (II.6.2). Nevertheless, by II.5.17, after replacing X by an affine open covering, if necessary, we may assume that the condition of II.9.13 is satisfied, in which case MA2 (Y ) no longer depends on the chart (P, γ). Remark II.9.15. We denote by π e the element of QS defined by its projections (II.9.6.1) (II.9.15.1)

(πpn )n∈N ∈ lim OK ←− N

and π ∈ Γ(S, MS )

(cf. II.6.4 and II.9.8) and by (II.9.15.2)

qeS : N → A2 (OK )

the composition (II.9.15.3)

q

S N −→ QS −→ W(ROK ) → A2 (OK ),

where the first arrow sends 1 to π e. The logarithmic structure on A2 (S) associated with the prelogarithmic structure defined by qeS is canonically isomorphic to MA2 (S) (II.9.8). This follows from II.9.7 as in the proof of II.9.13. II.9.16. We have a canonical homomorphism QS → QY that fits into a commutative diagram N

/P

 QS

 / QY

ϑ

(II.9.16.1)

qS

 W(ROK )

qY

 / W(R ) R

where ϑ is the homomorphism given in (II.6.2), the left upper vertical arrow sends 1 to π e (II.9.15), and the one on the right is (II.9.12.2). We deduce from this a commutative diagram (II.9.16.2)

(Yb , MYb )  ˇ (S, MSˇ )

iY

iS

/ (A2 (Y ), MA2 (Y ) )  / (A2 (S), M

A2 (S) )

II.9. FONTAINE P -ADIC INFINITESIMAL THICKENINGS

II.9.17.

119

We have a canonical homomorphism × Zp (1) → RR .

(II.9.17.1)

× For every ζ ∈ Zp (1), we denote also by ζ its image in RR . Since θ([ζ] − 1) = 0, we obtain a group homomorphism

Zp (1) → A2 (R),

(II.9.17.2)

ζ 7→ log([ζ]) = [ζ] − 1,

b whose image is contained in ker(θ) = ξ R. Lemma II.9.18. The homomorphism Zp (1) → A2 (R) (II.9.17.2) is Γ-equivariant and 1 b of A (R) and the R-linear b Zp -linear; its image generates the ideal p p−1 ξ R morphism 2 1 b b R(1) → p p−1 ξ R

(II.9.18.1)

b and ζ ∈ Z (1), to x · log([ζ]) is an isomorphism. that sends x ⊗ ζ, where x ∈ R p

The first statement is immediate because Zp (1) and A2 (R) are complete and separated for the p-adic topologies. Let ζ = (ζn )n≥0 ∈ Zp (1) such that ζ1 6= 1 (we have Pp−1 ζ0 = 1). Denote by ζ 0 the canonical image of (ζn+1 )n≥0 in RR and set ω = i=0 [ζ 0 ]i ∈ W(RR ). Then ω is a generator of ker(θ) ([73] A.2.6), and in A2 (R), we have (II.9.18.2) Since v(ζ1 − 1) =

[ζ] − 1 = ω([ζ 0 ] − 1) = ωθ([ζ 0 ] − 1) = ω(ζ1 − 1).

1 p−1 ,

the rest of the assertion follows.

II.9.19. The canonical homomorphism Zp (1) → (RR , ×) (II.9.17.1) and the trivial homomorphism Zp (1) → Γ(X, MX ) (with value 1) induce a homomorphism (II.9.19.1)

Zp (1) → QY .

For all g ∈ Γ and t ∈ P , we have, in QY , (II.9.19.2)

g(e t) = χ et (g) · e t,

where we have (abusively) denoted by χ et : Γ → Zp (1) the map deduced from (II.7.19.1). We deduce from this the following relation in A2 (R): (II.9.19.3)

g(e qY (t)) = [e χt (g)] · qeY (t),

where [e χt (g)] denotes the image of χ et (g) by the composition [ ]

Zp (1) −→ RR −→ W(RR ) −→ A2 (R).

Denote by τg the automorphism of A2 (Y ) induced by the action of g on A2 (R). The logarithmic structure τg∗ (MA2 (Y ) ) on A2 (Y ) is associated with the prelogarithmic structure defined by the composition g ◦ qeY : P → A2 (R) (II.9.12.3). The map (II.9.19.4)

P → Γ(A2 (Y ), MA2 (Y ) ),

t 7→ [e χt (g)] · t

is a morphism of monoids (II.7.19.4). It therefore induces a morphism of logarithmic structures on A2 (Y ) (II.9.19.5)

ag : τg∗ (MA2 (Y ) ) → MA2 (Y ) .

Likewise, by virtue of (II.7.19.3), the morphism of monoids (II.9.19.6)

P → Γ(A2 (Y ), τg∗ (MA2 (Y ) )),

t 7→ [g(e χt (g −1 ))] · t

induces a morphism of logarithmic structures on A2 (Y ) (II.9.19.7)

bg : MA2 (Y ) → τg∗ (MA2 (Y ) ).

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II. LOCAL STUDY

We immediately see that ag and bg are isomorphisms inverse to each other (II.7.19.3), and that the map g 7→ (τg−1 , ag−1 ) is a left action of Γ on the logarithmic scheme (A2 (Y ), MA2 (Y ) ). One immediately verifies that the morphism iY (II.9.12.6) and the canonical morphism (II.9.16.2) (II.9.19.8)

(A2 (Y ), MA2 (Y ) ) → (A2 (S), MA2 (S) )

are Γ-equivariant. Moreover, for every g ∈ Γ, the diagram (II.9.19.9)

τg∗ (MA2 (Y ) )

/ τg∗ (MA0 (Y ) ) 2 a0g

ag

 MA2 (Y )

 / MA0 (Y ) 2

where the horizontal arrows are induced by the homomorphism (II.9.12.5) and a0g is the automorphism of logarithmic structures on A2 (Y ) induced by the action of g on QY , is commutative. II.10. Higgs–Tate torsors and algebras We will use the notation introduced in II.9.8, II.9.11, and II.9.12 for the remainder of this chapter. ˇ = X × S, ˇ and endow these with the logaII.10.1. We set X = X ×S S and X S rithmic structures inverse images of MX (II.6.2), denoted by MX and MXˇ , respectively. We then have canonical isomorphisms (II.10.1.1) (II.10.1.2)

∼

(X, MX ) → (X, MX ) ×(S,MS ) (S, MS ), ∼ ˇ M ) → ˇ M ), (X, (X, MX ) ×(S,MS ) (S, ˇ ˇ X S

where the product is taken indifferently in the category of logarithmic schemes or in that of fine logarithmic schemes. We in particular deduce from this a left action of GK ˇ M ). We have a Γ-equivariant canonical on the logarithmic schemes (X, MX ) and (X, ˇ X morphism (II.9.11.3) (II.10.1.3)

ˇ M ). (Yb , MYb ) → (X, ˇ X

ˇ M ) consists of a smooth morphism A smooth (A2 (S), MA2 (S) )-deformation of (X, ˇ X ˇ M )-isomorphism e of fine logarithmic schemes (X, MXe ) → (A2 (S), MA2 (S) ) and an (S, ˇ S (II.10.1.4)

∼ ˇ M )→ e M e) × (X, (X, ˇ (A2 (S),M X X

A2 (S) )

ˇ M ). (S, ˇ S

ˇ M ) → (S, ˇ M ) is smooth and X ˇ is affine, such a deformation exists and is Since (X, ˇ ˇ X S unique up to isomorphism by virtue of ([50], 3.14). Its automorphism group is isomorphic to (II.10.1.5)

e1 HomOXˇ (Ω ˇ , ξOX ˇ ). R/OK ⊗R OX

1 e1 Recall that we have set Ω R/OK = Ω(R,P )/(OK ,N) (II.7.17.3). e M e ) for the remainder of this section. We fix such a deformation (X, X

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

121

Remark II.10.2. By II.6.8(i), the scheme Spec(R1 ) is an open connected component of ˇ through X. Consequently, Z = Spec(R1 ⊗OK OC ) is an open and closed subscheme of X, which the morphism (II.10.1.3) factors. An (A2 (S), MA2 (S) )-deformation of (Z, MX ˇ |Z) suffices for this chapter. II.10.3.

Set b b e1 T = Hom b (Ω R/OK ⊗R R, ξ R).

(II.10.3.1)

R

b b e1 We identify the dual R-module with ξ −1 Ω R/OK ⊗R R (II.9.9) and denote by S the b (II.2.5) associate symmetric R-algebra b e1 S = ⊕n≥0 S n = S b (ξ −1 Ω R/OK ⊗R R).

(II.10.3.2)

R

b (II.9.11.3), by T e the O b -module Denote by Ybzar the Zariski topos of Yb = Spec(R) Y b associated with T, and by T the Y -vector bundle associated with its dual; in other words, T = Spec(S ).

(II.10.3.3)

e the corresponding open subscheme Let U be a Zariski open subscheme of Yb and U of A2 (Y ) (cf. II.9.11). We denote by L (U ) the set of morphisms represented by dotted arrows that complete the diagram (U, MYb |U )

(II.10.3.4)

iY |U

/ (U e , MA (Y ) |U e) 2

 ˇ M ) (X, ˇ X  ˇ M ) (S, ˇ S

 / (X, e M e) X

iS

  / (A2 (S), M A2 (S) )

in such a way that it remains commutative. By II.5.23, the functor U 7→ L (U ) is a e e M e ). We denote T-torsor of Ybzar . We call it the Higgs–Tate torsor associated with (X, X b by F the R-module of affine functions on L (cf. II.4.9). This fits into a canonical exact sequence (II.4.9.1) b → F → ξ −1 Ω b e1 0→R R/OK ⊗R R → 0.

(II.10.3.5)

By ([45] I 4.3.1.7), for every integer n ≥ 1, this sequence induces an exact sequence (II.2.5) b → 0. n n −1 e 1 ΩR/OK ⊗R R) 0 → Sn−1 b (F ) → S b (F ) → S b (ξ

(II.10.3.6)

R

R

R

b The R-modules (Snb (F ))n∈N therefore form a filtered direct system whose direct limit R

(II.10.3.7)

C = lim Snb (F ) −→

n≥0

R

b is naturally endowed with a structure of R-algebra. By II.4.10, the Yb -scheme (II.10.3.8)

L = Spec(C )

122

II. LOCAL STUDY

is naturally a principal homogeneous T-bundle over Yb that canonically represents L . e M e ). Remember that L , F , C , and L depend on (X, X II.10.4. We endow Yb with the natural left action of ∆; for every g ∈ ∆, the automorphism of Yb defined by g, which we denote also by g, is induced by the automorphism b We view T e as a ∆-equivariant O b -module using the descent datum correg −1 of R. Y c c c e1 sponding to the R1 -module HomR c1 (ΩR/OK ⊗R R1 , ξ R1 ) (cf. II.4.18). For every g ∈ ∆, we therefore have a canonical isomorphism of OYb -modules e e ∼ ∗ e τgT : T → g (T).

(II.10.4.1)

This induces an isomorphism of Yb -group schemes ∼

τgT : T → g • (T),

(II.10.4.2)

where g • denotes the base change functor by the automorphism g of Yb (II.4.1.1). We thus obtain a ∆-equivariant structure on the Yb -group scheme T (cf. II.4.17) and consequently a left action of ∆ on T that is compatible with its action on Yb ; the automorphism of T defined by an element g of ∆ is the composition of τgT and the canonical projection g • (T) → T. We deduce from this an action of ∆ on S by ring automorphisms that is b we call it the canonical action. Concretely, it is induced compatible with its action on R; −1 e 1 c1 ). by the trivial action on S c (ξ Ω ⊗R R R1

R/OK

The left action of ∆ on the logarithmic scheme (A2 (Y ), MA2 (Y ) ) defined in II.9.19 e induces a ∆-equivariant structure on the T-torsor L (cf. II.4.18). In other words, it e T induces, for every g ∈ ∆, a τg -equivariant isomorphism (II.10.4.3)

∼

τgL : L → g ∗ (L ),

these isomorphisms being subject to compatibility relations (II.4.16.6). Indeed, for every Zariski open subscheme U of Yb , we take for (II.10.4.4)

∼

τgL (U ) : L (U ) → L (g(U ))

e be the open subscheme of A2 (Y ) corresponding the isomorphism defined as follows. Let U to U and ψ ∈ L (U ) that we will view as a morphism (II.10.4.5)

e , MA (Y ) |U e ) → (X, e M e ). ψ : (U 2 X

ˇ M ) (II.10.1.3) are ∆-equivariant, so The morphisms iY (II.9.12.6) and (Yb , MYb ) → (X, ˇ X the composition (II.10.4.6)

g −1

ψ

e ), MA (Y ) |g(U e )) −→ (U e , MA (Y ) |U e ) −→ (X, e M e) (g(U 2 2 X

e M e ). It corresponds to the extends the canonical morphism (g(U ), MYb |g(U )) → (X, X e L image of ψ by τg (U ). One immediately verifies that the resulting morphism τgL is a τgT equivariant isomorphism and that these isomorphisms satisfy the required compatibility relations (II.4.16.6). e and L induce a ∆-equivariant structure By II.4.21, the ∆-equivariant structures on T b action of ∆ on on the OYb -module associated with F , or, equivalently, an R-semi-linear F , such that the morphisms in the sequence (II.10.3.5) are ∆-equivariant. We deduce from this a structure of ∆-equivariant principal homogeneous T-bundle over Yb on L (cf. II.4.20). For every g ∈ ∆, we therefore have a τgT -equivariant isomorphism (II.10.4.7)

∼

τgL : L → g • (L).

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

123

This structure determines a left action of ∆ on L that is compatible with its action on Yb ; the automorphism of L defined by an element g of ∆ is the composition of τgL and the canonical projection g • (L) → L. We thus obtain an action of ∆ on C by ring b we call it the canonical action. automorphisms that is compatible with its action on R; Concretely, it is induced by the action of ∆ on F . For every g ∈ ∆, we denote by ∼ L(Yb ) → L(Yb ),

(II.10.4.8)

ψ 7→ g ψ

the composition of the isomorphisms ∼ τgL : L(Yb ) → g • (L)(Yb ),

(II.10.4.9)

∼ g • (L)(Yb ) → L(Yb ),

(II.10.4.10)

ψ 7→ pr ◦ ψ ◦ g −1 ,

where g −1 acts on Yb and pr : g • (L) → L is the canonical projection, so that the diagram (II.10.4.11)

LO

g

/L O g

ψ g

Yb

ψ

/ Yb

is commutative. b Definition II.10.5. The R-algebra C (II.10.3.7), endowed with the canonical action of ∆ b e M e ). The R-representation (II.10.4), is called the Higgs–Tate algebra associated with (X, X e M e ). F (II.10.3.5) of ∆ is called the Higgs–Tate extension associated with (X, X

b Note that under the assumptions of II.9.13, these two R-representations of ∆ do not depend on the choice of the chart (P, γ) (II.6.2), by II.9.13 and (II.9.19.9). II.10.6. (II.10.6.1)

For every u ∈ T = T(Yb ), we denote by tu : T → T

the translation by u. For every ψ ∈ L (Yb ), we denote by tψ the isomorphism of principal homogeneous T-bundles over Yb (II.10.6.2)

∼

tψ : T → L,

v 7→ v + ψ.

The structure of ∆-equivariant principal homogeneous T-bundle over L transfers through tψ to a structure of ∆-equivariant principal homogeneous T-bundle over T. For every g ∈ ∆, we therefore have a τgT -equivariant isomorphism (II.10.6.3)

∼

T τg,ψ : T → g • (T).

This structure determines a left action of ∆ on T that is compatible with its action on T Yb ; the automorphism of T defined by an element g of ∆ is the composition of τg,ψ and • the canonical projection g (T) → T. We deduce from this an action (II.10.6.4)

ϕψ : ∆ → AutR c1 (S )

b of ∆ on S (II.10.3.2) by ring automorphisms that is compatible with its action on R; for every g ∈ ∆, ϕψ (g) is induced by the automorphism of T defined by g −1 . ∼ The isomorphism t∗ψ : C → S induced by tψ is ∆-equivariant if we endow C with the canonical action of ∆ and S with the action ϕψ .

124

II. LOCAL STUDY

One immediately verifies that for every g ∈ ∆, the diagram (II.10.6.5)

τgL

τgT

 g g • (T)

/L

tψ

T •

 / g • (L)

(tg ψ )

is commutative (II.10.4.8). Consequently, the diagram τgT

/ g • (T) TD DD DD DD g• (t(g ψ−ψ) ) T τg,ψ D!  g • (T)

(II.10.6.6)

is also commutative. It follows that ϕψ (g) = t∗(ψ−g ψ) ◦ g,

(II.10.6.7)

where t∗(ψ−g ψ) is the automorphism of S induced by t(ψ−g ψ) and g acts on S by the canonical action. b b e1 The pairing T⊗ b (ξ −1 Ω bS → R/OK ⊗R R) → R extends into a pairing T⊗R R b S , where the elements of T act as R-derivations of S . We thus define a morphism II.10.7.

T → Γ(T, TT/Yb ),

(II.10.7.1)

u 7→ Du ,

where TT/Yb is the tangent bundle of T over Yb . This identifies T with the module of invariant vector fields of T over Yb . It also induces an isomorphism ∼

T ⊗ b OT → TT/Yb .

(II.10.7.2)

R

Denote by Π = ⊕n≥0 Πn the divided power algebra of T (II.2.5). We have a canonical pairing Πn ⊗ b S n+m → S m

(II.10.7.3)

R

that is perfect if m = 0 ([5] A.10). For every u ∈ T = T(Yb ), Du belongs to the divided power ideal of Π, and we can define exp(Du ) as a differential operator of S of infinite order. For every x ∈ S , we have the Taylor formula t∗u (x) = exp(Du )(x),

(II.10.7.4)

where t∗u is the automorphism of S induced by tu (II.10.6.1). II.10.8.

We have a canonical S -linear isomorphism Ω1

(II.10.8.1)

b S /R

∼

e1 → ξ −1 Ω R/OK ⊗R S .

We denote by e1 dS : S → ξ −1 Ω R/OK ⊗R S

(II.10.8.2)

b of S . Explicitly, the morphism (II.10.7.1) is defined as follows: the universal R-derivation b D is the composed R-derivation b b ξ R), e for every u ∈ T = Hom b (Ω1R/OK ⊗R R, u R

(II.10.8.3)

S

dS

e1 / ξ −1 Ω R/OK ⊗R S

(ξ −1 ·u)⊗id

/S .

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

125

We can then prove again, directly, that the differential operator exp(Du ) : S → S

(II.10.8.4)

b (II.10.7.4). Indeed, for every is well-defined and that it is an isomorphism of R-algebras n n−1 n ≥ 0, Du sends S to S (II.10.3.2); it is therefore nilpotent on ⊕0≤i≤n S i . Conseb 1 ]-algebra S [ 1 ]. On the quently, exp(D ) is well-defined as an automorphism of the R[ u

p

other hand, for every x ∈ S 1 ⊂ S , we have (II.10.8.5)

p

exp(Du )(x) = x + (ξ −1 u)(x).

It follows that exp(Du )(S ) ⊂ S and consequently that exp(Du )(S ) = S (because D−u = −Du ). II.10.9.

The isomorphism (II.10.8.1) induces an isomorphism ∼

Ω1

(II.10.9.1)

b C /R

e1 → ξ −1 Ω R/OK ⊗R C .

We denote by e1 dC : C → ξ −1 Ω R/OK ⊗R C

(II.10.9.2)

b of C . For every x ∈ F , dC (x) is the canonical image of x in the universal R-derivation b (II.10.3.5). Consequently, for every g ∈ ∆, the diagram e1 ξ −1 Ω ⊗ R R R/OK C

(II.10.9.3)

dC



e1 ξ −1 Ω R/OK ⊗R C

/C

g

id⊗g

dC



e1 / ξ −1 Ω R/OK ⊗R C

is commutative. For every ψ ∈ L (Yb ), the diagram C

(II.10.9.4)

dC



e1 ξ −1 Ω R/OK ⊗R C

t∗ ψ

id⊗t∗ ψ

/S dS



e1 / ξ −1 Ω R/OK ⊗R S

where t∗ψ is the isomorphism induced by tψ (II.10.6.2), is clearly commutative. It follows that the diagram S

(II.10.9.5)

dS



e1 ξ −1 Ω R/OK ⊗R S

ϕψ (g)

id⊗ϕψ (g)

/S 

dS

e1 / ξ −1 Ω R/OK ⊗R S

is commutative, where ϕψ is the action of ∆ on S defined in (II.10.6.4). e 0 , M e 0 ) be another smooth (A2 (S), M Remark II.10.10. Let (X A2 (S) )-deformation X ˇ b 0 0 0 of (X, MX ˇ ), L the Higgs–Tate torsor, C the Higgs–Tate R-algebra, and F the b e 0 , M ). Then we have an isomorphism of Higgs–Tate R-extension associated with (X (A2 (S), MA2 (S) )-deformations (II.10.10.1)

e0 X

∼ e M e) → e 0 , M e 0 ). u : (X, (X X X

126

II. LOCAL STUDY

∼ e The isomorphism of T-torsors L → L 0 , ψ 7→ u ◦ ψ (II.10.3.4) induces a ∆-equivariant b R-linear isomorphism ∼

F0 → F

(II.10.10.2)

that fits into a commutative diagram (II.10.3.5) (II.10.10.3)

0

/ b R

/ F0

0

/ b R

 /F

b / ξ −1 Ω e1 R/OK ⊗R R

/0

b / ξ −1 Ω e1 R/OK ⊗R R

/0

b We deduce from this a ∆-equivariant R-isomorphism ∼

C0 → C.

(II.10.10.4)

b of the exact II.10.11. Every section ψ ∈ L (Yb ) defines a splitting vψ : F → R sequence (II.10.3.5), namely the morphism that with any affine function ` on L associates b `(ψ). The morphism id − v induces an R-linear morphism F

(II.10.11.1)

ψ

b e1 uψ : ξ −1 Ω R/OK ⊗R R → F

that is none other than the restriction of the isomorphism (t∗ψ )−1 : S → C (II.10.6.2) to S 1. For all ψ, ψ 0 ∈ L (Yb ), the difference ψ − ψ 0 ∈ T(Yb ) = T (II.10.3.1) determines an b R-linear morphism (II.10.11.2)

b b e1 σψ,ψ0 : ξ −1 Ω R/OK ⊗R R → R.

We have (II.10.11.3)

σψ,ψ0 = uψ0 − uψ .

By (II.10.4.11), for all g ∈ ∆ and ψ ∈ L (Yb ), we have (II.10.11.4)

ug ψ = g ◦ uψ ◦ g −1 .

e1 c g Consequently, for every x ∈ ξ −1 Ω R/OK ⊗R R1 , the map g 7→ σψ, ψ (x) is a cocycle of ∆ b with values in R. The induced map (II.10.11.5)

b 1 e1 c ξ −1 Ω R/OK ⊗R R1 → H (∆, R)

is clearly the boundary map of the long exact sequence of cohomology deduced from the exact sequence (II.10.3.5). By (II.10.6.7) and (II.10.7.4), if we denote by Dψ−g ψ the image of ψ − g ψ by the morphism (II.10.7.1), we have (II.10.11.6)

ϕψ (g) = exp(Dψ−g ψ ) ◦ g,

where g denotes the canonical action of g on S (II.10.4). In particular, by (II.10.8.5), 1 e1 for every x ∈ ξ −1 Ω R/OK ⊂ S , we have (II.10.11.7)

ϕψ (g)(x) = x + σψ,g ψ (x).

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

127

Remark II.10.12. We say that an element ψ of L (Yb ) is optimal if for every g ∈ ∆, the morphism ψ − g ψ ∈ T(Yb ) = T factors into (II.10.12.1)

1 ∼ b b → ξ R, b b e1 p−1 ξ R Ω R/OK ⊗R R → R(1) → p

where the second arrow is the isomorphism (II.9.18.1) and the last arrow is the canonical injection. b Denote by S 0 the graded R-algebra (II.2.5) b e1 S 0 = S b (Ω R/OK ⊗R R(−1)).

(II.10.12.2)

R

The canonical isomorphism (II.9.18.1) (II.10.12.3)

1 b b ∼ p−1 e1 e1 Ω ξ −1 Ω R/OK ⊗R R(−1) R/OK ⊗R R → p

b induces an R-homomorphism : S → S 0

(II.10.12.4)

such that  ⊗Zp Qp is an isomorphism. For every ψ ∈ L (Yb ), we also denote by ϕψ the action of ∆ on S 0 [ p1 ] deduced from its action on S defined in (II.10.6.4). This action preserves S 0 if and only if ψ is optimal. Indeed, denote by (II.10.12.5)

1 b b − p−1 e1 R ςψ,g ψ : Ω R/OK ⊗R R(−1) → p

b the R-linear morphism deduced from ψ−g ψ ∈ T(Yb ) = T and the isomorphism (II.9.18.1). 0 e1 For all g ∈ ∆ and x ∈ Ω R/OK (−1) ⊂ S , we then have ϕψ (x) = x + ςψ,g ψ (x).

(II.10.12.6)

1 b ⊂ p− p−1 b for every Consequently, ϕψ preserves S 0 if and only if ςψ,g ψ factors through R R g ∈ ∆, which is equivalent to ψ being optimal.

II.10.13. (II.10.13.1)

By II.6.2(C5 ), there essentially exists a unique étale morphism e0 , M e ) → (A2 (S), M (X A2 (S) ) ×AN AP X0

that fits into a commutative diagram with Cartesian squares (II.10.13.2)

ˇ M ) (X, ˇ X

ˇ M )× A / (S, ˇ AN P S

ˇ M ) / (S, ˇ S

 e0 , M e ) (X X0



/ (A2 (S), M A2 (S) ) ×AN AP

 / (A2 (S), M A2 (S) )

 AP

 / AN

iS

a

where the morphism a is defined by the chart N → Γ(A2 (S), MA2 (S) ), 1 7→ π e (II.9.15). ˇ e0 , M e ) is the smooth (A2 (S), M We say that (X ˇ ) deA2 (S) )-deformation of (X, MX X0 fined by the chart (P, γ). We denote by L0 the Higgs–Tate torsor (II.10.3), by C0 the b b e0 , M e ) Higgs–Tate R-algebra, and by F0 the Higgs–Tate R-extension associated with (X X0

(II.10.5).

128

II. LOCAL STUDY

The diagram (Yb , MYb )

(II.10.13.3)

 ˇ (S, MSˇ )

iY

iS

/ (A2 (Y ), MA2 (Y ) )

b

/ AP

 / (A2 (S), M A2 (S) )

a

 / AN

where the morphism b is defined by the canonical chart P → Γ(A2 (Y ), MA2 (Y ) ) (II.9.12), is commutative by (II.9.16.1). It is clear that the diagram (II.10.13.4)

ˇ M ) / (X, ˇ

(Yb , MYb )

/ (X e0 , M e ) X0 3

X

ψ0

iY

 (A2 (Y ), MA2 (Y ) )

 / (A2 (S), M A2 (S) ) ×AN AP

φ0

(without the dotted arrow), where φ0 is defined by (II.10.13.3), is commutative. We can complete it with a unique dotted arrow ψ0 ∈ L0 (Yb ) in such a way that it remains commutative. We say that ψ0 is the section of L0 (Yb ) defined by the chart (P, γ). The e0 , M e ) and ψ0 depend on the choice of the (II.6.7.1), because reader will note that (X X0 the charts a and b depend on it. Proposition II.10.14. We keep the notation of II.10.13. For all t ∈ P gp and g ∈ ∆, we have (II.10.14.1)

(ψ0 − g ψ0 )(d log(t)) = − log([χt (g)]),

where we view ψ0 − g ψ0 as an element of T(Yb ) = T (II.10.3.1), d log(t) is the canonical e1 image of t in Ω R/OK (II.7.18.5), and log([χt ]) denotes the composition (II.10.14.2)

∆

/ ∆∞

χt

1 / Zp (1) log([ ])/ p p−1 b ξR

/ ξR b ,

where the first and last arrows are the canonical morphisms, χt is the homomorphism (II.7.19.5), and the third arrow is induced by the isomorphism (II.9.18.1). In particular, ψ0 is optimal (II.10.12). b Since the two sides of equation (II.10.14.1) are homomorphisms from P gp to ξ R, −1 we can restrict to the case where t ∈ P . The morphisms φ0 and φ0 ◦ g , where φ0 is the morphism defined in (II.10.13.4) and g −1 acts on (A2 (Y ), MA2 (Y ) ), extend the same morphism (Yb , MYb ) → (A2 (S), MA2 (S) ) ×AN AP .

By the definitions and condition II.6.2(C5 ), the difference φ0 −φ0 ◦g −1 corresponds to the morphism ψ0 −g ψ0 ∈ T. On the other hand, we have g(e t) = [χt (g)]·e t in Γ(A2 (Y ), MA2 (Y ) ) (II.9.19.4). The first statement follows from this in view of II.5.23 and (II.9.17.2). The second statement is an immediate consequence of the first. Lemma II.10.15. We have a commutative diagram (II.10.15.1)

∂

c e1 ξ −1 Ω R/OK ⊗R R1 u



b H1 (∆, ξ −1 R(1))

/ H1 (∆, R) b O w

−v ∼

1 / H1 (∆, p p−1 b R)

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

129

where ∂ is the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (II.10.3.5), u is induced by the morphism (II.8.13.3), v is induced 1 b b → R. by the isomorphism (II.9.18.1), and w is induced by the canonical injection p p−1 R Indeed, it suffices to show that the diagram / H1 (∆, R) b O

∂

c e1 ξ −1 Ω R/OK ⊗R R1

(II.10.15.2)

e ξ −1 δ

w0

 c1 (1)) HomZ (∆∞ , ξ −1 R

−v 0 ∼

1 / Hom (∆ , p p−1 c1 ) R Z ∞

∼

1

where δe is the isomorphism (II.7.19.9), v 0 comes from the isomorphism OC (1) → p p−1 ξOC 1 b is commutative. c1 → R, (II.9.18.1), and w0 is induced by the canonical injection p p−1 R e M e ). We may therefore restrict By II.10.10, ∂ does not depend on the deformation (X, X e M e ) is the deformation defined by the chart (P, γ) (II.10.13). to the case where (X, X Let ψ0 be the section of L0 (Yb ) defined by the chart (P, γ). By II.10.11, for every e1 c1 , ∂(x) is the class of the cocycle g 7→ σψ ,g ψ (x) of ∆ with values x ∈ ξ −1 Ω ⊗R R R/OK

0

0

b The statement therefore follows from II.10.14. in R. Remark II.10.16. It immediately follows from II.10.15 that for every nonzero element a of OK , we have a commutative diagram (II.10.16.1)

∂a

e1 ξ −1 Ω R/OK ⊗R (R1 /aR1 )

/ H1 (∆, R/aR) O

ua

wa

 H1 (∆, ξ −1 R(1)/aξ −1 R(1))

−va ∼

1 1 / H1 (∆, p p−1 R/ap p−1 R)

where ∂a is induced by the boundary map of the long exact sequence of cohomology deduced from the short exact sequence obtained by reducing (II.10.3.5) modulo a, ua is induced by the morphism (II.8.13.2), va is induced by the isomorphism (II.9.18.1), and 1 wa is induced by the canonical injection p p−1 R → R. II.10.17. We denote by ϕ0 = ϕψ0 (II.10.6.4) the action of ∆ on S induced by the section ψ0 ∈ L0 (Yb ) defined by the chart (P, γ) (II.10.13). Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ)⊗Z Zp form a Zp -basis, so that (d log(ti ))1≤i≤d is an R-basis −1 e1 e1 of Ω d log(ti ) ∈ ξ −1 Ω R/OK (II.7.18.5). For every 1 ≤ i ≤ d, set yi = ξ R/OK ⊂ S and denote by χi the composition (II.10.17.1)

∆

/ ∆∞

χti

1 / Zp (1) log([ ])/ p p−1 ξOC ,

where the first arrow is the canonical morphism, χti is the homomorphism (II.7.19.5), and the third arrow is induced by the isomorphism (II.9.18.1). It then follows from (II.10.11.6) and (II.10.14.1) that for every g ∈ ∆, we have (II.10.17.2)

ϕ0 (g) = exp(−

d X

ξ −1

i=1

c1 -algebra Consequently, ϕ0 preserves the sub-R e1 (II.10.17.3) S = S c (ξ −1 Ω R1

∂ ⊗ χi (g)) ◦ g. ∂yi

R/OK

c1 ) ⊗R R

of S , and the induced action of ∆ on S factors through ∆p∞ .

130

II. LOCAL STUDY

Theorem II.10.18. We keep the notation of II.10.13 and denote by (P gp /Zλ)lib the quotient of P gp /Zλ by its torsion submodule. Then giving a right inverse w : (P gp /Zλ)lib → P gp

(II.10.18.1)

b of the canonical morphism P gp → (P gp /Zλ)lib uniquely determines an R-linear and ∆-equivariant morphism 1

βw : p− p−1 F0 → E

(II.10.18.2)

that fits into a commutative diagram (II.10.18.3)

0

1 / p− p−1 b _ R

1 / p− p−1 F0

1 b / p− p−1 e1 ξ −1 Ω R/OK ⊗R R

−c

βw

0

 / (πρ)−1 R b

 /E

/0

/Ω e1

R/OK

 b ⊗R R(−1)

/0

where the lines come from the exact sequences (II.7.22.2) and (II.10.3.5) and c is induced by the isomorphism (II.9.18.1). Since the torsion subgroup of P gp /Zλ is of order prime to p, the isomorphism (II.7.12.1) induces an isomorphism ∼ e1 (P gp /Zλ)lib ⊗Z R → Ω R/OK .

Consequently, the composition of w with the homomorphism P gp → E (1), t 7→ d log(e t) b (II.7.18.3) induces an R-linear morphism that we denote by (II.10.18.4)

b e1 σψ0 : Ω R/OK ⊗R R → E (1).

By (II.7.18.4), σψ0 is a splitting of the exact sequence (II.7.22.2) twisted by Zp (1). For every x ∈ E (1), set (II.10.18.5)

b hψ0 , xi = x − σψ0 (ν(x)) ∈ (πρ)−1 R(1),

b e1 where ν : E (1) → Ω R/OK ⊗R R is the canonical morphism. We define a map (II.10.18.6)

b h , i : L0 (Yb ) × E (1) → (πρ)−1 R(1)

as follows: for every φ ∈ T (II.10.3.1), if we denote by φ the composition (II.10.18.7)

1 b b φ b ∼ − p−1 b e1 Ω R(1) −→ (πρ)−1 R(1), R/OK ⊗R R −→ ξ R −→ p

where the second arrow is induced by the isomorphism (II.9.18.1) and the third arrow is the canonical injection, we have (II.10.18.8)

hψ0 + φ, xi = hψ0 , xi − φ(ν(x)).

b For every ψ ∈ L0 (Yb ), the morphism E (1) → (πρ)−1 R(1), x 7→ hψ, xi is a splitting of the exact sequence (II.7.22.2) twisted by Zp (1). Let us show that for all ψ ∈ L0 (Yb ), x ∈ E (1), and g ∈ ∆, we have (II.10.18.9)

g(hψ, xi) = hg ψ, g(x)i.

b e1 First consider the case ψ = ψ0 . It then suffices to show that for every x ∈ Ω R/OK ⊗R R, we have (II.10.18.10)

σ ψ0 ,g ψ0 (x) = σψ0 (x) − g(σψ0 (g −1 x)),

II.10. HIGGS–TATE TORSORS AND ALGEBRAS

131

where σ ψ0 ,g ψ0 is the morphism defined in (II.10.18.7) from ψ0 − g ψ0 ∈ T (II.10.11). Let t ∈ P , y its canonical image in (P gp /Zλ)lib , and z = w(y) ∈ P gp . Then we have σψ0 (d log(t)) = d log(e z ). On the other hand, by virtue of II.10.14, we have σ ψ0 ,g ψ0 (d log(t)) = (χt (g))−1 , b b where (χt (g))−1 ∈ Zp (1) ⊂ R(1) ⊂ (πρ)−1 R(1). Note that the character log([ ]) disappears from the formula because of the definition (II.10.18.7). Since the image of χt (g) b → E (1) is d log(χ (g)) (II.7.18), it suffices to show by the canonical injection (πρ)−1 R(1) t

the following relation in E (1):

g(d log(e z )) = d log(e z ) + log(χt (g)).

(II.10.18.11)

We can obviously replace t by a power, and therefore assume t − z ∈ Zλ. Since ∆ fixes e the relation (II.10.18.11) follows from (II.7.20.1). d log(λ), In the general case, for all φ ∈ T, x ∈ E (1), and g ∈ ∆, we have = g(hψ0 , xi) − g ◦ φ(ν(x))

g(hψ0 + φ, xi)

= hg ψ0 , g(x)i − g ◦ φ ◦ g −1 (ν(g(x))) = hg ψ0 + g ◦ φ ◦ g −1 , g(x)i = hg (ψ0 + φ), g(x)i,

which concludes the proof of (II.10.18.9). For every x ∈ E (1), the map ψ 7→ hψ, xi is an affine function on L0 with values in b (πρ)−1 R(1) (cf. II.4.7). It is therefore naturally an element of (πρ)−1 F0 (1). The map E (1) → (πρ)−1 F0 (1),

(II.10.18.12)

x 7→ (ψ 7→ hψ, xi)

b is an R-linear and ∆-equivariant morphism by virtue of (II.10.18.9). The induced morphism E → (πρ)−1 F0 fits into a commutative diagram 0

/ (πρ)−1 R b

/E

b /Ω e1 R/OK ⊗R R(−1) 

/0

i1

1 b e1 (πρ)−1 p p−1 Ω R/OK ⊗R R(−1)

/0

−c−1

0

/ (πρ)−1 R b

 / (πρ)−1 F0

 b / (πρ)−1 ξ −1 Ω e1 R/OK ⊗R R

/0

132

II. LOCAL STUDY

where i1 is the canonical injection. Consider the commutative diagram 1 / p− p−1 b R

0

1 / p− p−1 F0

1 b / p− p−1 e1 ξ −1 Ω R/OK ⊗R R

/0

1 b / p− p−1 e1 ξ −1 Ω R/OK ⊗R R

/0

i3

 / (πρ)−1 R b

0

 /H

i2

/ (πρ)−1 R b

0

 b / (πρ)−1 ξ −1 Ω e1 R/OK ⊗R R



/ (πρ)−1 F0

/0

where i2 and i3 are the canonical injections and H is the inverse image of (πρ)−1 F0 ∼ under i2 . Since i2 ◦ c−1 = c−1 ◦ i1 , we deduce from this an isomorphism α : H → E that fits into a commutative diagram /H

/ (πρ)−1 R b

0

1 b / p− p−1 e1 ξ −1 Ω R/OK ⊗R R

−c

α

 /E

/ (πρ)−1 R b

0

/0

/Ω e1 R/OK

 b ⊗R R(−1)

/0 1

1

The morphism p− p−1 F0 → E composed of the canonical injection p− p−1 F0 → H and α then has the desired property. b Corollary II.10.19. There exists a ∆-equivariant R-linear morphism 1

p− p−1 F → E

(II.10.19.1) that fits into a commutative diagram (II.10.19.2)

0

1 / p− p−1 b _ R

1 / p− p−1 F

1 b / p− p−1 e1 ξ −1 Ω R/OK ⊗R R

/0

−c

0

 / (πρ)−1 R b

 /E

/Ω e1 R/OK

 b ⊗R R(−1)

/0

where the lines come from the exact sequences (II.7.22.2) and (II.10.3.5), and c is induced by the isomorphism (II.9.18.1). This follows from II.10.10 and II.10.18. II.11. Galois cohomology II In this section, we slightly generalize the results of T. Tsuji from IV. II.11.1. (II.11.1.1)

b We denote by S the symmetric R-algebra (II.10.3.2) b e1 S = S b (ξ −1 Ω R/OK ⊗R R) R

c its p-adic Hausdorff completion. For every rational number r ≥ 0, we denote and by S b (r) by S the sub-R-algebra of S defined by (II.2.5) (II.11.1.2)

b e1 S (r) = S b (pr ξ −1 Ω R/OK ⊗R R) R

II.11. GALOIS COHOMOLOGY II

133

c(r) its p-adic Hausdorff completion, which we always assume endowed with the and by S c(r) ⊗Z Qp with the p-adic topology (II.2.2). In view of p-adic topology. We endow S p c(r) are OC -flat. For all rational numbers r0 ≥ r ≥ 0, we II.6.14 and its proof, S (r) and S 0 0 have a canonical injective homomorphism αr,r : S (r ) → S (r) . One immediately verifies 0 c(r0 ) → S c(r) is injective. We set that the induced homomorphism hr,r : S α

c(r) , S † = lim S

(II.11.1.3)

−→

r∈Q>0

b c by the direct limit of the homomorof S which we identify with a sub-R-algebra 0,r phisms hα . We have a canonical S (r) -isomorphism Ω1

(II.11.1.4)

b S (r) /R

∼

(r) e1 → ξ −1 Ω . R/OK ⊗R S

We denote by (r) e1 dS (r) : S (r) → ξ −1 Ω R/OK ⊗R S

(II.11.1.5)

b of S (r) and by the universal R-derivation c(r) → ξ −1 Ω c(r) e1 dSc(r) : S R/OK ⊗R S

(II.11.1.6)

e1 its extension to the completions (note that the R-module Ω R/OK is free of finite type). b b (r) −1 e 1 with Since ξ ΩR/OK ⊗R R ⊂ dS (r) (S ), dS (r) and dSc(r) are also Higgs R-fields c(r) , pr d c(r) ) e1 (cf. II.2.12, II.2.16, and II.9.9). We denote by K• (S coefficients in ξ −1 Ω S

R/OK

c(r) , pr d c(r) ) (II.2.8.2) and denote by the Dolbeault complex of the Higgs module (S S • c(r) r e K (S , p dSc(r) ) the augmented Dolbeault complex (II.11.1.7) b → K0 (S c(r) , pr d c(r) ) → K1 (S c(r) , pr d c(r) ) → · · · → Kn (S c(r) , pr d c(r) ) → . . . , R S

S

S

b is placed in degree −1 and the differential R b →S c(r) is the canonical homomorwhere R phism. For all rational numbers r0 ≥ r ≥ 0, we have (II.11.1.8)

0

0

0

pr (id × αr,r ) ◦ dS (r0 ) = pr dS (r) ◦ αr,r . 0

The homomorphism hr,r therefore induces a morphism of complexes α (II.11.1.9)

0 c(r0 ) , pr0 d c(r0 ) ) → K c(r) , pr d c(r) ). e • (S e • (S ν r,r : K S S

b By (II.11.1.8), the derivations pr dSc(r) induce an R-derivation (II.11.1.10)

† e1 dS † : S † → ξ −1 Ω R/OK ⊗R S

that is none other than the restriction of dSc to S † . We denote by K• (S † , dS † ) the b is O -flat (II.6.14), for every Dolbeault complex of the Higgs module (S † , d † ). Since R rational number r ≥ 0, we have (II.11.1.11)

S

C

b ker(dS † ) = ker(dSc(r) ) = R.

Proposition II.11.2. For all rational numbers r0 > r > 0, there exists a rational number α ≥ 0 depending on r and r0 , but not on the data in II.6.2, such that (II.11.2.1)

0 c(r0 ) , pr0 d c(r0 ) ) → K c(r) , pr d c(r) ) e • (S e • (S pα ν r,r : K S S

134

II. LOCAL STUDY

b homotopy. is homotopic to 0 by an R-linear Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ) ⊗Z Zp form a Zp -basis, e1 so that (d log(ti ))1≤i≤d form an R-basis of Ω R/OK (II.7.18.5). For every 1 ≤ i ≤ d, set −1 −1 e 1 ⊂ S . We denote by yi = ξ d log(ti ) ∈ ξ Ω R/OK

b⊗ Q c(r0 ) ⊗Z Qp → R h−1 : S Zp p p

(II.11.2.2)

b the R-linear morphism defined by X

h−1 (

(II.11.2.3)

an

n=(n1 ,...,nd )∈Nd

Y

yini ) = a0 .

1≤i≤d

b For every integer m ≥ 0, there exists a unique R-linear morphism (II.11.2.4)

0

c(r ) ⊗Z Qp → ξ −m Ω c(r) ⊗Z Qp e m+1 ⊗R S em hm : ξ −m−1 Ω p p R/OK ⊗R S R/OK

such that for all 1 ≤ i1 < · · · < im+1 ≤ d, we have Y X yini ⊗ ξ −1 d log(ti1 ) ∧ · · · ∧ ξ −1 d log(tim+1 )) an hm ( n=(n1 ,...,nd )∈Nd

1≤i≤d

X

=

n=(n1 ,...,nd )∈Ji1 −1

Y ni +δii an 1 yi ⊗ ξ −1 d log(ti2 ) ∧ · · · ∧ ξ −1 d log(tim+1 ), n i1 + 1 1≤i≤d

where Ji1 −1 is the subset of Nd consisting of the elements n = (n1 , . . . , nd ) such that n1 = · · · = ni1 −1 = 0. Let α be a rational number such that (II.11.2.5)

α ≥ sup (logp (x + 1) + (x + 1)r − xr0 ), x∈Q≥0

where logp is the logarithm to base p. For every integer m ≥ 0, we clearly have (II.11.2.6)

0

c(r ) ) ⊂ ξ −m Ω c(r) . e m+1 ⊗R S em pα hm (ξ −m−1 Ω R/OK ⊗R S R/OK

One immediately verifies that the morphisms (pα hm )m≥−1 define a homotopy linking 0 0 to the morphism pα ν r,r . Corollary II.11.3. For all rational numbers r0 > r > 0, the canonical morphism (II.11.1.9) (II.11.3.1)

c(r0 ) , pr0 d c(r0 ) ) ⊗Z Qp → K c(r) , pr d c(r) ) ⊗Z Qp e • (S e • (S K p p S S

is homotopic to 0 by a continuous homotopy. b 1 ]. Corollary II.11.4. The complex K• (S † , dS † ) ⊗Zp Qp is a resolution of R[ p Indeed, we have a canonical isomorphism of complexes (II.11.4.1)

∼ c(r) , pr d c(r) ) ⊗Z Qp → lim K• (S K• (S † , dS † ) ⊗Zp Qp . p S −→

r∈Q>0

The corollary therefore follows from II.11.3.

II.11. GALOIS COHOMOLOGY II

II.11.5.

135

e0 , M e ) the smooth (A2 (S), M We denote by (X A2 (S) )-deformation of X0

ˇ M ) defined by the chart (P, γ) (II.10.13.1), by L the associated Higgs–Tate tor(X, ˇ 0 X sor (II.10.3), by ψ0 ∈ L0 (Yb ) the section defined by the same chart (II.10.13.4), and by ϕ0 = ϕψ0 the action of ∆ on S induced by ψ0 (II.10.6.4). By II.11.6 below, for every b rational number r ≥ 0, the sub-R-algebra S (r) of S is stable under ϕ0 . Unless stated (r) (r) † c otherwise, we endow S , S , and S with the action of ∆ induced by ϕ0 . The derivations dS and dSc are ∆-equivariant by (II.10.9.5). The same therefore holds for the derivations dS (r) , dSc(r) , and dS † , in view of (II.11.1.8). b Proposition II.11.6. For every rational number r ≥ 0, the sub-R-algebra S (r) of S is c(r) stable under the action ϕ0 of ∆ on S , and the induced actions of ∆ on S (r) and S are continuous for the p-adic topologies. We take again the assumptions and notation of II.10.17. By (II.10.17.2), for every g ∈ ∆ and every 1 ≤ i ≤ d, we have (II.11.6.1) 1

ϕ0 (g)(yi ) = yi − ξ −1 χi (g).

Since ξ −1 χi (g) ∈ p p−1 OC (II.9.18), S (r) is stable under ϕ0 (g). Let ζ be a generator of Zp (1). There exists ag ∈ Zp such that χi (g) = [ζ ag ] − 1 ∈ A2 (OK ). By linearity, we have log([ζ ag ]) ∈ pvp (ag ) ξOC , and consequently ϕ0 (g)(yi ) − yi ∈ pvp (ag ) S . For every integer n ≥ 0, since the set of g ∈ ∆ such that vp (ag ) ≥ n is an open subgroup of ∆, it follows that the stabilizer of the class of pr yi in S (r) /pn S (r) is open in ∆. The second assertion follows because the action of ∆ on R/pn R is continuous for the discrete topology. Theorem II.11.7 (IV.5.3.4). Let r be a rational number > 0. Then: (i) The canonical morphism c(r) ⊗Z Qp )∆ c1 ⊗Z Qp → (S R p p

(II.11.7.1)

is an isomorphism. (ii) For every integer i ≥ 1,

c(r) ⊗Z Qp ) = 0. lim Hicont (∆, S p

(II.11.7.2)

−→

r∈Q>0

The proof of this theorem will be given in II.11.17. We have pushed it to the end of this section because it requires rather heavy notation. Note that this statement is slightly more general than that of Tsuji (IV.5.3.4). c(r) )∆ Corollary II.11.8. For every rational number r > 0, we have (S † )∆ = (S c = R1 . b → S c(r) is injective. c(r) is OC -flat and the canonical homomorphism R Indeed, S 1 (r) ∆ (r) c ) = S c ∩R c1 [ ]. On the other hand, we have We deduce from II.11.7 that (S p b b b 1 (r) ∆ c c(r) )∆ = R c c1 and S ∩ R[ ] = R and (R) = R1 by II.8.22(i). It follows that (S p

c1 . consequently that (S † )∆ = R II.11.9.

(II.11.9.1)

c1 -algebra of S defined by (II.2.5) We denote by S the sub-R −1 e 1 c1 ). S = SR ΩR/OK ⊗R R c1 (ξ

By II.10.17, the action ϕ0 of ∆ on S preserves S, and the induced action on S factors through ∆p∞ . We also denote by ϕ0 the resulting action of ∆p∞ on S. Copying the

136

II. LOCAL STUDY

proof of II.11.6, we prove that the latter is continuous for the p-adic topology on S. We can also deduce this property directly from II.11.6 by observing that for every integer n ≥ 0, the canonical homomorphism S/pn S → S /pn S is injective (cf. the proof of II.6.14). We set (II.11.9.2)

S∞

(II.11.9.3)

Sp∞

d = S ⊗R c1 R∞ , d = S ⊗R c1 Rp∞ .

The action ϕ0 of ∆p∞ on S induces actions of ∆p∞ on Sp∞ and of ∆∞ on S∞ . (r) d For every rational number r ≥ 0, we denote by S∞ the sub-R ∞ -algebra of S∞ defined by (r) d e1 S∞ = SR (pr ξ −1 Ω d R/OK ⊗R R∞ ), ∞

(II.11.9.4) (r)

(r)

c∞ its p-adic Hausdorff completion. In view of II.6.14 and its proof, S∞ and and by S (r) c S∞ are OC -flat. For every rational number r0 ≥ r, we have a canonical injective ho(r) (r 0 ) momorphism S∞ → S∞ . One immediately verifies that the induced homomorphism 0 (r) (r) (r ) c∞ c∞ is injective. The proof of II.11.6 shows that S∞ is stable under the → S S (r) (r) (0) c∞ are action of ∆∞ on S∞ = S∞ , and the induced actions of ∆∞ on S∞ and S (r) continuous for the p-adic topologies. Unless explicitly stated otherwise, we endow S∞ (r) c∞ and S with these actions and with p-adic topologies. (r) d We denote by Sp∞ the sub-R p∞ -algebra of Sp∞ defined by (r)

e1 d Sp∞ = SRd (pr ξ −1 Ω R/OK ⊗R Rp∞ ), p∞

(II.11.9.5)

(r) c(r) and by S p∞ its p-adic Hausdorff completion. The algebra Sp∞ has properties analogous (r)

(r)

(0)

to those of S∞ . In particular, Sp∞ is stable under the action of ∆p∞ on Sp∞ = Sp∞ , (r) c(r) and the induced actions of ∆p∞ on S ∞ and S ∞ are continuous for the p-adic topologies. p

p

(r) c(r) Unless explicitly stated otherwise, we endow Sp∞ and S p∞ with these actions and with p-adic topologies.

Lemma II.11.10. For every rational number r > 0 and every integer i ≥ 0, the canonical morphism (II.11.10.1)

(r)

c∞ ) → Hi (∆∞ , S c(r) ) Hicont (∆p∞ , S p cont ∞

is an isomorphism, and the canonical morphism (II.11.10.2)

c(r) ) → Hi (∆, S c(r) ) Hicont (∆∞ , S ∞ cont

is an almost isomorphism. Let n be an integer ≥ 0. The canonical homomorphism (II.11.10.3)

(r)

(r)

(r) (r) Σ0 Sp∞ /pn Sp∞ → (S∞ /pn S∞ )

is an isomorphism by virtue of II.6.13. By (II.6.11.1), it follows that the canonical morphism (II.11.10.4)

(r)

(r)

(r) (r) Hi (∆p∞ , Sp∞ /pn Sp∞ ) → Hi (∆∞ , S∞ /pn S∞ )

is an isomorphism. We deduce from this, by (II.3.10.4) and (II.3.10.5), that the morphism (II.11.10.1) is an isomorphism. On the other hand, the canonical homomorphism (II.11.10.5)

(r) (r) S∞ /pn S∞ → (S (r) /pn S (r) )Σ

II.11. GALOIS COHOMOLOGY II

137

is an almost isomorphism by virtue of II.6.22. By II.6.20, it follows that the canonical morphism (r) (r) ψn : Hi (∆∞ , S∞ /pn S∞ ) → Hi (∆, S (r) /pn S (r) )

(II.11.10.6)

is an almost isomorphism. We denote by An (resp. Cn ) the kernel (resp. cokernel) of ψn . Then the OK -modules lim An ,

lim Cn ,

n≥0

n≥0

←−

R1 lim An ,

←−

R1 lim Cn

←−

←−

n≥0

n≥0

are almost zero by virtue of ([32] 2.4.2(ii)). Consequently, the morphisms lim ψn

R1 lim ψn

and

←−

←−

n≥0

n≥0

are almost isomorphisms. We deduce from this, by (II.3.10.4) and (II.3.10.5), that the morphism (II.11.10.2) is an isomorphism. II.11.11. Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ) ⊗Z Zp form a Zp -basis, (χti )1≤i≤d their images in Hom(∆p∞ , Zp (1)) (II.7.19.5), and ζ a Zp -basis of Zp (1). The (χti )1≤i≤d form a Zp -basis of Hom(∆p∞ , Zp (1)) (II.7.19.8). Hence there exists a unique Zp -basis (γi )1≤i≤d of ∆p∞ such that χti (γj ) = δij ζ for all 1 ≤ i, j ≤ d (II.6.10). For every integer 0 ≤ i ≤ d, we denote by i Ξp∞ the subgroup of (II.8.7.7) Ξp∞ = Hom(∆p∞ , µp∞ (OK ))

(II.11.11.1)

consisting of the homomorphisms ν : ∆p∞ → µp∞ (OK ) such that ν(γj ) = 1 for every 1 ≤ j ≤ i. e1 The (d log(ti ))1≤i≤d form an R-basis of Ω R/OK (II.7.18.5). For every 1 ≤ i ≤ d and Pd d −1 e1 ∞ every n = (n1 , . . . , nd ) ∈ N , set yi = ξ d log(ti ) ∈ ξ −1 Ω i=1 ni , R/OK ⊂ Sp , |n| = Qd ni n and y = i=1 yi ∈ Sp∞ . Note that Rp∞ is separated for the p-adic topology and d therefore identifies with a subring of R p∞ ; this follows, for example, from (II.8.9.8), (II.8.9.17), and the fact that Spec(R1 ) is an open subscheme of Spec(C1 ). For every rational number r > 0, every integer 0 ≤ i ≤ d, and every ν ∈ i Ξp∞ , in view of II.8.9 (r) (r) and using the same notation, we denote by i S p∞ (ν) and i S p∞ the sub-R1 -modules of (r)

Sp∞ defined by (II.11.11.2)

(r) i S p∞ (ν)

(II.11.11.3)

(r) i S p∞

=

M

(ν)

pr|n| Rp∞ y n ,

n∈Ji

=

(r) 0 i S p∞ (ν ),

M ν 0 ∈i Ξp∞

where Ji is the subset of Nd consisting of the elements n = (n1 , . . . , nd ) such that n1 = (r) c ∞ the p-adic Hausdorff completion of i S (r) · · · = ni = 0. We denote by i S ∞ . Since the p

p

d p-adic topology on R is induced by the p-adic topology on R p∞ , it easily follows from (r) (r) (II.8.9.1) that the p-adic topology on i S p∞ is induced by the p-adic topology on Sp∞ . (r) c ∞ is the closure of i S (r) c(r) Consequently, i S ∞ in S ∞ . It follows from II.8.9 and (II.11.6.1) p∞

p

p

p

(r)

that for every 1 ≤ j ≤ d and every ν ∈ i Ξp∞ , γj preserves i S p∞ (ν) and therefore also

(r) i S p∞ .

(r)

(r)

If 1 ≤ j ≤ i, γj fixes i S p∞ (ν) and i S p∞ .

Lemma II.11.12. Under the assumptions of II.11.11, let moreover i be an integer such that 1 ≤ i ≤ d, and r, r0 two rational numbers such that r0 > r > 0. Then:

138

II. LOCAL STUDY (r)

(r)

c∞ = S c(r) c c (i) We have 0 S p∞ and d S p∞ = R1 . p (ii) The kernel of the morphism γi − id :

(II.11.12.1)

(r) c (i−1) S p∞

(r)

c∞ → (i−1) S p

(r)

c∞. is equal to i S p (iii) There exists an integer α ≥ 0 depending on r and r0 but not on the data (II.6.2), such that we have (r 0 )

(r)

c ∞ ⊂ (γi − id)((i−1) S c ∞ ). pα · (i−1) S p p

(II.11.12.2)

(r)

(1) c∞ = R c1 . On the other hand, we have (i) Since Rp∞ = R1 (II.8.9.2), we have d S p (r) 0 S p∞

(II.11.12.3)

e1 = SRp∞ (pr ξ −1 Ω R/OK ⊗R Rp∞ ),

(r)

c∞ = S c(r) which implies that 0 S p∞ . p (ii) For every integer n ≥ 0, we clearly have (II.11.12.4)

(r)

(r)

). pn · i S p∞ = i S p∞ ∩ (pn · (i−1) S (r) p∞

It therefore suffices to show that the sequence (II.11.12.5)

/ i S (r) p∞

0

/

(r) γi −id / (r) (i−1) S p∞ (i−1) S p∞

is exact. Let ν ∈ (i−1) Ξp∞ and (II.11.12.6)

z=

X n∈Ji−1

(ν). pr|n| an y n ∈ (i−1) S (r) p∞

We then have (II.11.6.1) (II.11.12.7)

(γi − id)(z) =

X n∈Ji−1

(ν), pr|n| bn y n ∈ (i−1) S (r) p∞

where for every n = (n1 , . . . , nd ) ∈ Ji−1 , (II.11.12.8) bn = (ν(γi ) − 1)an +

X m=(m1 ,...,md )∈Ji−1 (n)

pr(mi −ni )

  mi ν(γi )am wmi −ni , ni

Ji−1 (n) denotes the subset of Ji−1 consisting of the elements m = (m1 , . . . , md ) such 1 that mj = nj for j 6= i and mi > ni , and w = ξ −1 log([ζ]) is an element of valuation p−1 of OC (II.9.18). 1 Suppose that γi (z) = z and ν(γi ) 6= 1. Then v(ν(γi ) − 1) ≤ p−1 and for every n = (n1 , . . . , nd ) ∈ Ji−1 , we have   X mi an = −(ν(γi ) − 1)−1 pr(mi −ni ) ν(γi )am wmi −ni . ni m=(m1 ,...,md )∈Ji−1 (n)

(ν)

It follows that for every α ∈ N and every n ∈ Ji−1 , we have an ∈ prα Rp∞ (this is proved (ν)

by induction on α); hence z = 0 because Rp∞ is separated for the p-adic topology. Suppose that γi (z) = z and ν(γi ) = 1. Then for every n = (n1 , . . . , nd ) ∈ Ji−1 , if we set n0 = (n01 , . . . , n0d ) ∈ Ji−1 (n) with n0i = ni + 1, we have X wmi −ni −1 (ni + 1)!an0 = − pr(mi −ni −1) mi !am . (mi − ni )! 0 m=(m1 ,...,md )∈Ji−1 (n )

II.11. GALOIS COHOMOLOGY II

139

We have wm−1 /m! ∈ OC for every integer m ≥ 1. It follows that for every α ∈ N and (ν) every n = (n1 , . . . , nd ) ∈ Ji−1 such that ni ≥ 1, we have ni !an ∈ prα Rp∞ (this is proved (r)

by induction on α); hence an = 0. Consequently, z ∈ i S p∞ (ν), which concludes the proof of the exactness of the sequence (II.11.12.5). (iii) It suffices to show that there exists an integer α ≥ 0 depending only on r and r0 such that for every ν ∈ (i−1) Ξp∞ , we have (r)

0

c ∞ ). pα ((i−1) S p(r∞) (ν)) ⊂ (γi − id)((i−1) S p

(II.11.12.9)

(r)

c ∞ ), then by p-adic completion, we will deduce Indeed, if we set M = (γi − id)((i−1) S p from this a commutative diagram (r 0 )

/ c MO O

c∞ ) pα ((i−1) S p

(II.11.12.10)

/

(r) c (i−1) S p∞

: uu uu u uu uu γi −id

(r) c (i−1) S p∞

where the vertical arrow is surjective by virtue of ([1] 1.8.5). Suppose ν(γi ) 6= 1. By (II.11.12.7), we have (II.11.12.11) (ν(γi ) − 1)((i−1) S (r) (ν)) ⊂ (γi − id)((i−1) S (r) (ν)) + (ν(γi ) − 1)pr ((i−1) S (r) (ν)). p∞ p∞ p∞ It follows that (r)

c ∞ ). (ν(γi ) − 1)((i−1) S (r) (ν)) ⊂ (γi − id)((i−1) S p∞ p

(II.11.12.12)

We may therefore take α = 1 because v(ν(γi ) − 1) ≤

1 p−1 . (ν)

(ν))∧ Suppose that ν(γi ) = 1, so that ν ∈ i Ξp∞ . Denote by (Rp∞ )∧ and ((i−1) S (r) p∞ (ν)

the p-adic Hausdorff completions of Rp∞ and

(r) (i−1) S p∞ (ν),

c1 -module of (ν))∧ can be identified with a sub-R ((i−1) S (r) p∞ z of

((i−1) S (r) (ν))∧ p∞

respectively. Note that

(r) c (i−1) S p∞ .

Every element

can be written as a series z=

(II.11.12.13)

X

pr|n| an y n ,

n∈Ji−1 (ν)

where an ∈ (Rp∞ )∧ and an tends to 0 when |n| tends to infinity. Consequently, (γi −id)(z) is also given by formula (II.11.12.7). It therefore suffices to show that there exists an integer α ≥ 0 depending only on r and r0 such that for every X 0 0 ) (II.11.12.14) pr |n| bn y n ∈ (i−1) S (r (ν), p∞ n∈Ji−1

the system of linear equations defined, for n = (n1 , . . . , nd ) ∈ Ji−1 , by (II.11.12.15)

0

pα p(r −r)|n| ni !bn =

X m=(m1 ,...,md )∈Ji−1 (n)

pr(mi −ni ) mi !am

wmi −ni , (mi − ni )!

140

II. LOCAL STUDY (ν)

admits a solution am ∈ (Rp∞ )∧ for m ∈ Ji−1 such that am tends to 0 when |m| tends to infinity. For n = (n1 , . . . , nd ) ∈ Ji−1 , set ni

(II.11.12.16)

a0n

= ni !p− p−1 an ,

(II.11.12.17)

b0n

= pα p(r −r)|n| ni !p− p−1 bn ,

ni

0

so that equation (II.11.12.15) becomes (II.11.12.18)

1

1

X

b0n = pr+ p−1 w

p(r+ p−1 )(mi −ni −1) a0m

m=(m1 ,...,md )∈Ji−1 (n)

w(mi −ni −1) . (mi − ni )!

∧ c1 -linear endomorphism Φ of (⊕n∈J Rp(ν) Consider the R defined, for a sequence ∞) i−1 (ν)

(xn )n∈Ji−1 of elements of (Rp∞ )∧ that tends to 0 when |n| tends to infinity, by X X (II.11.12.19) Φ( xn ) = zn , n∈Ji−1

n∈Ji−1

where for n = (n1 , . . . , nd ) ∈ Ji−1 , X (II.11.12.20) zn =

1

p(r+ p−1 )(mi −ni ) xm

m=(m1 ,...,md )∈{n}∪Ji−1 (n)

w(mi −ni ) . (mi − ni + 1)!

1

Since Φ is congruent to the identity modulo pr+ p−1 , it is surjective by virtue of ([1] 1 (ν) 1.8.5). Consequently, for every sequence b0n ∈ pr+ p−1 w(Rp∞ )∧ for n ∈ Ji−1 that tends (ν)

to 0 when |n| tends to infinity, equation (II.11.12.18) admits a solution a0m ∈ (Rp∞ )∧ for 1 m ∈ Ji−1 that tends to 0 when |m| tends to infinity. On the other hand, v(w) = p−1 and there exists an integer α ≥ 0 such that for every n ∈ N, we have n 2 (II.11.12.21) (r0 − r)n + v(n!) − +α≥r+ . p−1 p−1

The desired assertion follows by taking for b0n for n ∈ Ji−1 the elements defined by (II.11.12.17) (which are in fact zero except for finitely many). II.11.13. We keep the assumptions of II.11.11. For every rational number r > 0, (r),• d we define by induction, for every integer 0 ≤ i ≤ d, a complex Ki of continuous R p∞ (r),• (r) (r),• c representations of ∆p∞ by setting K0 = Sp∞ [0] and for every 1 ≤ i ≤ n, Ki is the fiber of the morphism (II.11.13.1)

(r),•

(r),•

γi − id : Ki−1 → Ki−1 .

It follows from II.3.25 and (II.2.7.9) that we have a canonical isomorphism (II.11.13.2)

∼ (r),• c(r) C•cont (∆p∞ , S p∞ ) → Kd

c1 )). For all rational numbers r0 > r > 0, the canonical homomorphism in D+ (Mod(R (r 0 ) (r) (r 0 ),• (r),• c c Sp∞ → Sp∞ induces, for every integer 0 ≤ i ≤ d, a morphism Ki → Ki of d complexes of continuous Rp∞ -representations of ∆p∞ . Proposition II.11.14. Under the assumptions of II.11.11, let moreover i be an integer such that 0 ≤ i ≤ d, and r, r0 two rational numbers such that r0 > r > 0. Then: c1 -linear isomorphism (i) We have a canonical ∆p∞ -equivariant R (II.11.14.1)

(r) c ∼ i S p∞ →

(r),•

H0 (Ki

).

II.11. GALOIS COHOMOLOGY II

141

(ii) There exists an integer αi ≥ 0 depending on r, r0 , and i but not on the data in II.6.2, such that for every integer j ≥ 1, the canonical morphism (r 0 ),•

Hj (Ki

(II.11.14.2) αi

is annihilated by p .

(r),•

) → Hj (Ki

)

We proceed by induction on i. The statement is immediate for i = 0, by II.11.12(i). Suppose i ≥ 1 and that the statement has been proved for i − 1. The distinguished triangle / K(r),• γi −id / K(r),• i−1 i−1

(r),•

(II.11.14.3)

Ki

/

+1

and the induction hypothesis induce an exact sequence (II.11.14.4)

/

/ H0 (K(r),• ) i

0

(r) γi −id c i−1 S p∞

/

(r) c i−1 S p∞

,

which implies statement (i) by virtue of II.11.12(ii). For every integer j ≥ 1, the distinguished triangle (II.11.14.3) induces an exact c1 -modules sequence of R (r)

0 → Cj

(II.11.14.5) (r)

(r),•

→ Hj (Ki

(r),•

(r)

) → Dj

→ 0,

(r)

(r),•

where Cj is a quotient of Hj−1 (Ki−1 ) and Dj is a submodule of Hj (Ki−1 ). Moreover, by the induction hypothesis, we have a canonical isomorphism (r)

(r) ∼

C1

(II.11.14.6)

(r 0 ),•

(r),•

(r ) Cj

→

(r) Cj

(II.11.14.7)

0

and

(r ) Dj

→

(r) Dj

(r 0 ),•

→ Ki−1 and Ki

The canonical morphisms Ki−1 0

(r)

c ∞ /(γi − 1)(i−1 S c ∞ ). → i−1 S p p (r),•

→ Ki

induce morphisms

that fit into a commutative diagram

0

/ C (r0 ) j

/ Hj (K(r0 ),• ) i

/ D(r0 ) j

/0

0

 / C (r) j

 / Hj (K(r),• ) i

 / D(r)

/0

j

where the vertical arrow in the middle is the canonical morphism. Set r00 = (r + r0 )/2. By the induction hypothesis, there exists an integer αi−1 ≥ 0 depending only on r, r0 , and i − 1, such that for every integer j ≥ 1, the morphism (r 0 ) (r 00 ) Dj → Dj is annihilated by pαi−1 . On the other hand, in view of the induction 0 hypothesis and by virtue of (II.11.14.6) and II.11.12(ii), there exists an integer αi−1 ≥0 0 depending only on r, r , and i − 1, such that for every integer j ≥ 1, the morphism 0 (r 00 ) (r) 0 Cj → Cj is annihilated by pαi−1 . Statement (ii) follows by taking αi = αi−1 + αi−1 . Corollary II.11.15. Let r, r0 be two rational numbers such that r0 > r > 0. Then: (i) The canonical homomorphism ∆p∞ c(r) c1 → (S R p∞ )

(II.11.15.1)

is an isomorphism. (ii) There exists an integer α ≥ 0 depending on r, r0 , and d, but not on the data in II.6.2, such that for every integer j ≥ 1, the canonical morphism 0

(II.11.15.2)

c(r∞ ) ) → Hj (∆p∞ , S c(r) Hjcont (∆p∞ , S p p∞ ) cont

is annihilated by pα .

142

II. LOCAL STUDY

This follows from (II.11.13.2) and II.11.14. Corollary II.11.16. Let r, r0 be two rational numbers such that r0 > r > 0. Then: (i) The canonical homomorphism c(r) )∆ c1 → (S R

(II.11.16.1)

is an almost isomorphism. (ii) There exists an integer α ≥ 0 depending on r, r0 , and d but not on the data in II.6.2, such that for every integer j ≥ 1, the canonical morphism c(r0 ) ) → Hj (∆, S c(r) ) (II.11.16.2) Hjcont (∆, S cont is annihilated by pα .

This follows from II.11.10 and II.11.15. II.11.17. We can now prove theorem II.11.7. Since ∆ is compact, for every rational number r > 0, the canonical morphism (II.3.8) c(r) ) ⊗Z Qp → C• (∆, S c(r) ⊗Z Qp ) C•cont (∆, S cont p p

(II.11.17.1)

is an isomorphism. Theorem II.11.7 therefore follows from II.11.16.

Proposition II.11.18. Let r, r0 be two rational numbers such that r0 > r > 0. Then: (i) For every integer n ≥ 1, the canonical morphism R1 /pn R1 → (S (r) /pn S (r) )∆

(II.11.18.1)

(r)

is almost injective. We denote by Hn its cokernel. (ii) There exists an integer α ≥ 0 depending on r, r0 , and d but not on the data in (r) (r 0 ) II.6.2, such that for every integer n ≥ 1, the canonical morphism Hn → Hn α is annihilated by p . (iii) There exists an integer β ≥ 0 depending on r, r0 , and d but not on the data in II.6.2, such that for all integers n, q ≥ 1, the canonical morphism 0

0

Hq (∆, S (r ) /pn S (r ) ) → Hq (∆, S (r) /pn S (r) )

(II.11.18.2)

is annihilated by pβ .

(i) This follows from II.11.16(i) and from the long exact sequence of cohomology b of ∆ associated with the short exact sequence of R-representations n

·p c(r) −→ c(r) −→ S c(r) /pn S c(r) −→ 0. 0 −→ S S

(II.11.18.3)

c1 -linear morphism We also deduce from this an almost injective R c(r) ). (II.11.18.4) H (r) → H1 (∆, S n

cont

(ii) This follows from (II.11.18.4) and II.11.16(ii). (iii) For all integers n, q ≥ 1, the long exact sequence of cohomology deduced from c1 -modules (II.11.18.3) gives an exact sequence of R (II.11.18.5) c(r) )/pn Hq (∆, S c(r) ) → Hq (∆, S (r) /pn S (r) ) → T (r),q → 0, 0 → Hqcont (∆, S cont n (r),q

c(r) ). Let r00 = (r + r0 )/2. By where Tn is a pn -torsion submodule of Hq+1 cont (∆, S 0 II.11.16(ii), there exists an integer β > 0 depending only on r, r0 , and d, such that for every integer q ≥ 1, the canonical morphisms c(r0 ) ) → Hq (∆, S c(r00 ) ) and Hq (∆, S c(r00 ) ) → Hq (∆, S c(r) ) Hqcont (∆, S cont cont cont 0

are annihilated by pβ . Statement (iii) follows by taking β = 2β 0 .

II.12. DOLBEAULT REPRESENTATIONS

143

II.12. Dolbeault representations For the remainder of this chapter, we fix a smooth (A2 (S), MA2 (S) )ˇ M ) (II.10.1) and denote by F the associated Higgs–Tate e M e ) of (X, deformation (X, ˇ X X b b R-extension and by C the associated Higgs–Tate R-algebra (II.10.5). We denote by Cb II.12.1.

the p-adic Hausdorff completion of C . For every rational number r ≥ 0, we denote by b of ∆ deduced from F by taking the inverse image under the F (r) the R-representation b e1 multiplication by pr on ξ −1 Ω R/OK ⊗R R, so that we have a locally split exact sequence b of R-modules u(r) −1 e 1 b −→ 0. b −→ F (r) −→ 0 −→ R ξ ΩR/OK ⊗R R

(II.12.1.1)

By ([45] I 4.3.1.7), this sequence induces, for every integer n ≥ 1, an exact sequence (II.2.5) b (r) e1 ) → Snb (F (r) ) → Snb (ξ −1 Ω 0 → Sn−1 R/OK ⊗R R) → 0. b (F

(II.12.1.2)

R

R

R

b The R-modules (Snb (F (r) ))n∈N therefore form a filtered direct system whose direct limit R

C (r) = lim Snb (F (r) )

(II.12.1.3)

−→

n≥0

R

b is naturally endowed with a structure of R-algebra. There exists a unique homomorphism b of R-algebras µ(r) : C (r) → C (r) ⊗ b S (r) ,

(II.12.1.4) where S

(r)

R

b defined in (II.11.1.2), such that for every x ∈ F (r) , we have is the R-algebra

(II.12.1.5)

µ(r) (x) = x ⊗ 1 + 1 ⊗ (pr · u(r) (x)).

This makes Spec(C (r) ) into a principal homogeneous Spec(S (r) )-bundle over Yb (cf. the proof of II.4.10). The action of ∆ on F (r) induces an action on C (r) by ring automorphisms that b we call it the canonical action. The R-algebra b is compatible with its action on R; (r) C endowed with this action is called the Higgs–Tate algebra of thickness r associe M e ). We denote by Cb(r) the p-adic Hausdorff completion of C (r) , which ated with (X, X we always assume endowed with the p-adic topology. We endow Cb(r) ⊗Zp Qp with the p-adic topology (II.2.2). In view of II.6.14 and its proof, C (r) and Cb(r) are OC -flat. For all rational numbers r0 ≥ r ≥ 0, we have an injective and ∆-equivariant canoni0 0 b cal R-homomorphism αr,r : C (r ) → C (r) . One immediately verifies that the induced 0 r,r 0 (r ) homomorphism hα : Cb → Cb(r) is injective. Set (II.12.1.6)

C † = lim Cb(r) , −→

r∈Q>0

b which we identify with a sub-R-algebra of Cb = Cb(0) by the direct limit of the homomor0,r phisms (hα )r∈Q>0 . The action of ∆ on the rings (Cb(r) )r∈Q>0 induces an action on C † b and on Cb. by ring automorphisms that is compatible with its actions on R We denote by (II.12.1.7)

(r) e1 dC (r) : C (r) → ξ −1 Ω R/OK ⊗R C

144

II. LOCAL STUDY

b of C (r) and by the universal R-derivation b(r) e1 dCb(r) : Cb(r) → ξ −1 Ω R/OK ⊗R C

(II.12.1.8)

e1 its extension to the completions (note that the R-module Ω R/OK is free of finite type). We immediately see that, as for dC (II.10.9.3), the derivations dC (r) and dCb(r) are ∆-equivarb e1 iant. Furthermore, d (r) and d are also Higgs R-fields with coefficients in ξ −1 Ω , C

because

Cb(r)

R/OK

e1 ξ −1 Ω R/OK • b(r)

b = d (r) (F (r) ) ⊂ d (r) (C (r) ) (cf. II.2.12 and II.2.16). We ⊗R R C C , pr d b(r) ) the Dolbeault complex of (Cb(r) , pr d b(r) ) (II.2.8.2) and by

denote by K (C C C e • (Cb(r) , pr d b(r) ) the augmented Dolbeault complex K C (II.12.1.9) b → K0 (Cb(r) , pr d 1 b(r) r R , p d b(r) ) → · · · → Kn (Cb(r) , pr d b(r) ) → K (C C

Cb(r) )

C

→ ...,

b → Cb(r) is the canonical homomorb is placed in degree −1 and the differential R where R phism. For all rational numbers r0 ≥ r ≥ 0, we have 0

0

0

pr (id × αr,r ) ◦ dC (r0 ) = pr dC (r) ◦ αr,r .

(II.12.1.10) 0

induces a morphism of complexes Consequently, hr,r α 0 e • (Cb(r0 ) , pr0 d b(r0 ) ) → K e • (Cb(r) , pr d b(r) ). ν r,r : K C C

(II.12.1.11)

b By (II.12.1.10), the derivations pr dCb(r) induce an R-derivation † e1 dC † : C † → ξ −1 Ω R/OK ⊗R C

(II.12.1.12)

b that is none other than the restriction of dCb to C † . This is also a Higgs R-field with • † † −1 e 1 coefficients in ξ ΩR/OK . We denote by K (C , dC † ) the Dolbeault complex of (C , dC † ). b is O -flat (II.6.14), for every rational number r ≥ 0, we have Since R C

b ker(dC † ) = ker(dCb(r) ) = R.

(II.12.1.13)

ˇ M ) defined by e M e ) is the deformation (X e0 , M e ) of (X, II.12.2. When (X, ˇ X X0 X the chart (P, γ) (II.10.13.1), we add an index 0 to the objects defined in (II.12.1): C0 , (r) C0 ... The section ψ0 ∈ L0 (Yb ) defined by the chart (P, γ) (II.10.13) then induces an b isomorphism of R-algebras ∼

S → C0 ,

(II.12.2.1)

b where S is the R-algebra defined in (II.10.3.2). It is ∆-equivariant when we endow S with the action ϕ0 = ϕψ0 of ∆ induced by ψ0 (II.10.6.4). It is moreover compatible with the universal derivations, by (II.10.9.4). For every rational number r ≥ 0, ψ0 induces an (r) b b and consequently an isomorphism of R-algebras b R-homomorphism C →R 0

(II.12.2.2)

∼

(r)

S (r) → C0 ,

II.12. DOLBEAULT REPRESENTATIONS

145

b defined in (II.11.1.2). For all rational numbers r0 ≥ r ≥ 0, where S (r) is the R-algebra the diagram 0

S (r )

(II.12.2.3)

S



(r)

∼

∼

/ C (r0 ) 0  / C (r) 0

where the vertical arrows are the canonical homomorphisms, is commutative. It follows that the isomorphism (II.12.2.2) is ∆-equivariant when we endow S (r) with the action of ∆ induced by ϕ0 (II.11.6). One also immediately verifies that the diagram (II.11.1.5) S (r)

(II.12.2.4)

dS (r)

/ C (r) 0

∼

d

 

(r) e1 ξ −1 Ω R/OK ⊗R S

(r) C0

(r)

e1 ξ −1 Ω R/OK ⊗R C0

is commutative. For future reference, we rewrite in the following five propositions the main results of Section II.11 concerning the algebra S in terms of the algebra C , for a general deformaˇ M ), taking into account II.12.2 and II.10.10. e M e ) of (X, tion (X, ˇ X X Proposition II.12.3. (i) For all rational numbers r0 > r > 0, there exists a rational number α ≥ 0 depending on r and r0 , but not on the data in (II.6.2), such that 0 e • (Cb(r0 ) , pr0 d b(r0 ) ) → K e • (Cb(r) , pr d b(r) ), (II.12.3.1) pα ν r,r : K C

C

0

b howhere ν r,r is the morphism (II.12.1.11), is homotopic to 0 by an R-linear motopy. (ii) For all rational numbers r0 > r > 0, the canonical morphism (II.12.3.2)

0 e • (Cb(r0 ) , pr0 d b(r0 ) ) ⊗Z Qp → K e • (Cb(r) , pr d b(r) ) ⊗Z Qp ν r,r ⊗Zp Qp : K p p C C

is homotopic to 0 by a continuous homotopy. b 1 ]. (iii) The complex K• (C † , dC † ) ⊗Zp Qp is a resolution of R[ p This follows from II.11.2, II.11.3, and II.11.4.

Proposition II.12.4. For every rational number r ≥ 0, the actions of ∆ on C (r) and Cb(r) are continuous for the p-adic topologies. This follows from II.11.6. Proposition II.12.5. Let r be a rational number > 0. Then: (i) The canonical morphism (II.12.5.1)

c1 ⊗Z Qp → (Cb(r) ⊗Z Qp )∆ R p p

is an isomorphism. (ii) For every integer i ≥ 1, we have (II.12.5.2)

lim Hicont (∆, Cb(r) ⊗Zp Qp ) = 0. −→

r∈Q>0

This follows from II.11.7.

146

II. LOCAL STUDY

c1 . Corollary II.12.6. For every rational number r > 0, we have (C † )∆ = (Cb(r) )∆ = R This follows from II.11.8 (or from II.12.5). Proposition II.12.7. Let r, r0 be two rational numbers such that r0 > r > 0. Then: (i) For every integer n ≥ 1, the canonical homomorphism R1 /pn R1 → (C (r) /pn C (r) )∆

(II.12.7.1)

(r)

is almost injective. We denote by Hn its cokernel. (ii) There exists an integer α ≥ 0 depending on r, r0 , and d but not on the data in (r 0 ) (r) II.6.2, such that for every integer n ≥ 1, the canonical morphism Hn → Hn is annihilated by pα . (iii) There exists an integer β ≥ 0 depending on r, r0 , and d but not on the data in II.6.2, such that for all integers n, q ≥ 1, the canonical morphism (II.12.7.2)

0

0

Hq (∆, C (r ) /pn C (r ) ) → Hq (∆, C (r) /pn C (r) )

is annihilated by pβ .

This follows from II.11.18. II.12.8. defined by (II.12.8.1)

b c1 -module M of ∆, we denote by H(M ) the R For every R-representation H(M ) = (M ⊗ b C † )∆ . R

e1 c1 -field with coefficients in ξ −1 Ω We endow it with the Higgs R R/OK induced by dC † (II.12.1.12) (cf. II.9.9). We thus define a functor (II.12.8.2)

c1 , ξ −1 Ω e1 H : Rep b (∆) → HM(R R/OK ). R

e1 c1 -module (N, θ) with coefficients in ξ −1 Ω For every Higgs R R/OK (II.9.9), b we denote by V(N ) the R-module defined by II.12.9.

(II.12.9.1)

† θtot =0 , V(N ) = (N ⊗R c1 C )

c1 -field on N ⊗ c C † (II.2.8.8). We where θtot = θ ⊗ id + id ⊗ dC † is the total Higgs R R1 b endow it with the R-semi-linear action of ∆ induced by its natural action on C † . We thus define a functor c1 , ξ −1 Ω e1 (II.12.9.2) V : HM(R ) → Rep b (∆). R/OK

R

Remarks II.12.10. (i) It follows from II.10.10 that the functors H and V do not depend e M e ), up to noncanonical isomorphism. on the choice of the deformation (X, X b (ii) For every R-representation M of ∆, the canonical morphism (II.12.10.1)

b 1 c 1 H(M ) ⊗R b R[ ]) c1 R1 [ ] → H(M ⊗R p p

is an isomorphism. c1 -module (N, θ) with coefficients in ξ −1 Ω e1 (iii) For every Higgs R R/OK , the canonical morphism b 1 ] → V(N ⊗ R c1 [ 1 ]) (II.12.10.2) V(N ) ⊗ b R[ c1 R R p p is an isomorphism.

II.12. DOLBEAULT REPRESENTATIONS

147

b M of ∆ is Dolbeault if Definition II.12.11. We say that a continuous R-representation the following conditions are satisfied: b (i) M is a projective R-module of finite type, endowed with the p-adic topology. c1 -module of finite type. (ii) H(M ) is a projective R (iii) The canonical C † -linear morphism † † H(M ) ⊗R c1 C → M ⊗ b C

(II.12.11.1)

R

is an isomorphism. e M e ) (II.10.10 and II.12.10(i)). This notion does not depend on the choice of (X, X c1 -module (N, θ) with coefficients in ξ −1 Ω e1 Definition II.12.12. A Higgs R R/OK is called solvable if the following conditions are satisfied: c1 -module of finite type. (i) N is a projective R b (ii) V(N ) is a projective R-module of finite type. (iii) The canonical C † -linear morphism

† V(N ) ⊗ b C † → N ⊗R c1 C

(II.12.12.1)

R

is an isomorphism. e M e ) (II.10.10 and II.12.10(i)). This notion does not depend on the choice of (X, X b c1 of ∆. Then the Higgs R Lemma II.12.13. Let M be a Dolbeault R-representation b module H(M ) is solvable, and we have a functorial and ∆-equivariant canonical Risomorphism (II.12.13.1)

∼

V(H(M )) → M.

Indeed, the canonical C † -linear morphism (II.12.13.2)

† † H(M ) ⊗R c1 C → M ⊗ b C R

b e1 is a ∆-equivariant isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK , where † C is endowed with the Higgs field dC † (II.12.1.12), H(M ) is endowed with the trivial b action of ∆, and M is endowed with the zero Higgs field (cf. II.2.13). Since M is R-flat ∼ b we deduce from this a ∆-equivariant R-isomorphism b and ker(dC † ) = R, V(H(M )) → M . † The canonical C -linear morphism

(II.12.13.3)

V(H(M )) ⊗ b C † → H(M ) ⊗ b C † R

R

can then be identified with the inverse of the isomorphism (II.12.13.2), which shows that H(M ) is solvable. c1 -module with coefficients in Lemma II.12.14. Let (N, θ) be a solvable Higgs R b e1 ξ −1 Ω R/OK . Then the R-representation V(N ) of ∆ is Dolbeault, and we have a funcc1 -modules torial canonical isomorphism of Higgs R (II.12.14.1)

∼

H(V(N )) → N.

For all rational numbers r0 ≥ r ≥ 0, the canonical morphism b(r0 ) → N ⊗ c Cb(r) N ⊗R c1 C R1

148

II. LOCAL STUDY

b c1 -flat. Since V(N ) is an R-module of finite type, there exists is injective because N is R a rational number r > 0 such that we have (r) (II.12.14.2) V(N ) = (N ⊗ c Cb(r) )θtot =0 , R1

(r) θtot

c1 -field on N ⊗ c Cb(r) . On the = θ ⊗ id + p id ⊗ dCb(r) is the total Higgs R R1 c1 -flat, ξ −1 Ω e1 ⊗ N ⊗ Cb(r) is OC -flat. other hand, since Cb(r) is OC -flat and N is R R c R/OK R1 b(r) such that θ(r) (pn x) = 0, we Consequently, for every n ≥ 0 and every x ∈ N ⊗R c1 C tot where

r

(r)

have θtot (x) = 0. It follows that (II.12.14.3)

b(r) ) ∩ V(N ) = pn V(N ). (pn N ⊗R c1 C

b(r) . It then follows Hence the p-adic topology on V(N ) is induced by that on N ⊗R c1 C from II.12.4 that the action of ∆ on V(N ) is continuous for the p-adic topology. The canonical C † -linear morphism † V(N ) ⊗ b C † → N ⊗R c1 C

(II.12.14.4)

R

b e1 is a ∆-equivariant isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK , where † C is endowed with the Higgs field dC † , V(N ) is endowed with the zero Higgs field, and N is endowed with the trivial action of ∆ (cf. II.2.13). Since N is a direct summand of a c1 -module of finite type, we have(N ⊗ c C † )∆ = N (II.12.6). We deduce from this free R R1 ∼ c1 -modules H(V(N )) → an isomorphism of Higgs R N . The canonical C † -linear morphism (II.12.14.5)

† † H(V(M )) ⊗R c1 C → V(M ) ⊗R c1 C

then identifies with the inverse of the isomorphism (II.12.14.4), which shows that V(N ) is Dolbeault. Proposition II.12.15. The functors V and H induce equivalences of categories quasib inverse to each other between the category of Dolbeault R-representations of ∆ and that −1 e 1 c of solvable Higgs R1 -modules with coefficients in ξ ΩR/OK . This follows from II.12.13 and II.12.14. b 1 ]-representation M of ∆ is Dolbeault Definition II.12.16. We say that a continuous R[ p if the following conditions are satisfied: b 1 ]-module of finite type, endowed with the p-adic topology (i) M is a projective R[ p

(II.2.2). c1 [ 1 ]-module of finite type (II.12.8.1). (ii) H(M ) is a projective R p (iii) The canonical morphism (II.12.16.1)

† † H(M ) ⊗R c1 C → M ⊗ b C R

is an isomorphism. e M e ) (II.10.10 and II.12.10(i)). This notion does not depend on the choice of (X, X b 1 ]-representation M of ∆ is Dolbeault if and only Remark II.12.17. A continuous R[ p if it satisfies conditions (i) and (ii) of II.12.16, as well as the following condition: (iii’) There exists a rational number r > 0 such that H(M ) is contained in M ⊗ b Cb(r) R

(II.12.1.3) and the canonical morphism

(II.12.17.1)

b(r) → M ⊗ b Cb(r) H(M ) ⊗R c1 C R

II.12. DOLBEAULT REPRESENTATIONS

149

is an isomorphism. Indeed, for any rational number r > 0, the canonical morphism M ⊗ b Cb(r) → M ⊗ b C † R R b Condition (iii’) clearly implies condition II.12.16(iii). is injective because M is R-flat. c1 [ 1 ]-module Conversely, suppose that condition II.12.16(iii) holds. Since H(M ) is an R p of finite type, there exists a rational number r > 0 such that H(M ) is contained in b 1 ], after taking a smaller r, if necessary, M ⊗ b Cb(r) . Since M is of finite type over R[ p R we may assume that the morphism (II.12.17.1) is surjective. Moreover, since H(M ) is c1 -flat, for every rational number r > 0, the canonical morphism R (II.12.17.2) H(M ) ⊗ c Cb(r) → H(M ) ⊗ c C † R1

R1

is injective. The morphism (II.12.17.1) is therefore injective by virtue of II.12.16(iii). c1 [ 1 ]-module (N, θ) with coefficients in Definition II.12.18. We say that a Higgs R p e1 is solvable if the following conditions are satisfied: ξ −1 Ω R/OK c1 [ 1 ]-module of finite type. (i) N is a projective R p b 1 ]-module of finite type. (ii) V(N ) is a projective R[ p

(iii) The canonical morphism † V(N ) ⊗ b C † → N ⊗R c1 C

(II.12.18.1)

R

is an isomorphism. e M e ) (II.10.10 and II.12.10(i)). This notion does not depend on the choice of (X, X c1 [ 1 ]-module (N, θ) with coefficients in ξ −1 Ω e1 Remark II.12.19. A Higgs R R/OK is solvp able if and only if it satisfies conditions (i) and (ii) of II.12.18, as well as the following condition: b(r) (iii’) There exists a rational number r > 0 such that V(N ) is contained in N ⊗R c1 C and the canonical morphism (II.12.19.1) V(N ) ⊗ b Cb(r) → N ⊗ c Cb(r) R1

R

is an isomorphism. The proof, similar to that of II.12.17, is left to the reader. b 1 ]-representation M of ∆, the Higgs R c1 [ 1 ]Lemma II.12.20. For every Dolbeault R[ p p b module H(M ) is solvable, and we have a functorial and ∆-equivariant canonical R[ 1 ]p

isomorphism ∼

V(H(M )) → M.

(II.12.20.1)

Indeed, the canonical C † -linear morphism

† † H(M ) ⊗R c1 C → M ⊗ b C

(II.12.20.2)

R

b e1 is a ∆-equivariant C -isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK , † where C is endowed with the Higgs field dC † (II.12.1.12), H(M ) is endowed with the trivial action of ∆, and M is endowed with the zero Higgs field (cf. II.2.13). Since b b we deduce from this a ∆-equivariant R[ b 1 ]-isomorphism M is R-flat and ker(d † ) = R, †

∼

C

p

V(H(M )) → M . The canonical C † -linear morphism (II.12.20.3)

V(H(M )) ⊗ b C † → H(M ) ⊗ b C † R

R

150

II. LOCAL STUDY

can then be identified with the inverse of the isomorphism (II.12.20.2), which shows that H(M ) is solvable. b b 1 ]-module of finite type, T a sub-R-module of Lemma II.12.21. Let V be a projective R[ p (r) b finite type of V such that V = T ⊗Z Qp , and r a rational number ≥ 0. Let M = V ⊗ b C p

R

and denote by M the canonical image of T ⊗ b Cb(r) in M . Then the canonical morphism R V → M is injective and there exists an integer m ≥ 0 such that T ⊂ V ∩ M ◦ ⊂ p−m T . ◦

b 1 ]-module of finite type V 0 and an R[ b 1 ]-isomorphism We choose a projective R[ p p ∼ b 1 n b 0 ϕ : V ⊕ V 0 → (R[ ]) , where n is an integer ≥ 1. Let T be a subR-module of finite type p (r) 0◦ 0 0 0 0 0 b and denote by M the canonical of V such that V = T ⊗Z Qp . Let M = V ⊗ b C p

R

image of T 0 ⊗ b Cb(r) in M 0 . The isomorphism ϕ induces a (Cb(r) ⊗Zp Qp )-isomorphism R ∼ b → Cb(r) is inφ : M ⊕ M 0 → (Cb(r) ⊗ Q )n . Since the structural homomorphism R Zp

p

jective (II.12.1.13), the canonical morphism V → M is injective. On the other hand, b n ⊂ ϕ(T ⊕ T 0 ) ⊂ p−j R b n and therethere exists an integer j ≥ 0 such that we have pj R fore pj (Cb(r) )n ⊂ φ(M ◦ ⊕ M 0◦ ) ⊂ p−j (Cb(r) )n . It immediately follows from II.10.10 and b 1 ] ∩ Cb(r) = R. b Consequently, (II.12.2.2) that R[ p

n

b ⊂ p−2j ϕ(T ⊕ T 0 ), ϕ(T ⊕ T 0 ) ⊂ ϕ((V ∩ M ◦ ) ⊕ (V 0 ∩ M 0◦ )) ⊂ p−j R

(II.12.21.1)

giving the statement of the lemma. c1 [ 1 ]-module (N, θ) with coefficients in Lemma II.12.22. For every solvable Higgs R p b 1 ]-representation V(N ) of ∆ is Dolbeault, and we have a functorial e1 R[ ξ −1 Ω , the R/OK p c1 [ 1 ]-modules canonical isomorphism of Higgs R p

∼

H(V(N )) → N.

(II.12.22.1)

Let us first show that V(N ) is a continuous representation of ∆ for the p-adic topology (II.2.2). By II.12.19, there exists a rational number r > 0 such that the canonical Cb(r) linear morphism b(r) φ : V(N ) ⊗ b Cb(r) → N ⊗R c1 C

(II.12.22.2)

R

b of finite type of V(N ) such is a ∆-equivariant isomorphism. Let T be a sub-R-module that V(N ) = T ⊗Zp Qp . Let M = V(N ) ⊗ b Cb(r) and denote by M ◦ the canonical R c1 -module of finite type of N such that image of T ⊗ b Cb(r) in M . Let N ◦ be a sub-R R N = N ◦ ⊗Z Qp . Let N = N ⊗ c Cb(r) and denote by N ◦ the canonical image of p

R1

b N ⊗ b Cb(r) in N . Since M ◦ and N ◦ are R-modules of finite type, there exists an R n ◦ ◦ −n ◦ integer n ≥ 0 such that p N ⊂ φ(M ) ⊂ p N . On the other hand, by II.12.21, the canonical morphism V(N ) → M is injective and there exists an integer m ≥ 0 such that T ⊂ V(N ) ∩ M ◦ ⊂ p−m T . It follows that ◦

(II.12.22.3)

pn T ⊂ V(N ) ∩ φ−1 (N ◦ ) ⊂ p−m−n T.

Let x ∈ T and ν be an integer ≥ 0. By (II.12.22.3), there exist an integer b ≥ 1, P y1 , . . . , yb ∈ N ◦ , and α1 , . . . , αb ∈ Cb(r) such that pn φ(x) = 1≤i≤b αi yi . By virtue of

II.12. DOLBEAULT REPRESENTATIONS

151

II.12.4, there exists an open subgroup ∆x,ν of ∆ such that for every g ∈ ∆x,ν and every 1 ≤ i ≤ b, we have g(αi ) − αi ∈ pm+2n+ν Cb(r) . It follows that

g(x) − x ∈ V(N ) ∩ (pm+n+ν φ−1 (N ◦ )) ⊂ pν T. b we deduce from this that there exists an open subgroup Since T is of finite type over R, P 0 ∆ of ∆ such that for every g ∈ ∆0 , we have g(T ) ⊂ T . Consequently, T 0 = g∈∆ g(T ) b is a sub-R-module of finite type of V(N ), stable under ∆. Replacing T by T 0 , we reduce (II.12.22.4)

to the case where T is stable under the action of ∆. It then follows from (II.12.22.4) that the action of ∆ on T is continuous for the p-adic topology, and the same therefore holds for the action of ∆ on V(N ). The canonical C † -linear morphism † V(N ) ⊗ b C † → N ⊗R c1 C

(II.12.22.5)

R

b e1 is a ∆-equivariant C -isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK , b where C † is endowed with the Higgs R-field dC † , N is endowed with the trivial action b of ∆, and V(N ) is endowed with the zero Higgs R-field (cf. II.2.13). Since N is a direct †

c1 [ 1 ]-module of finite type, we have (N ⊗ c C † )∆ = N (II.12.6). We summand of a free R p R1 ∼ c1 [ 1 ]-modules H(V(N )) → deduce from this an isomorphism of Higgs R N . The canonical p

C † -linear morphism

† † H(V(N )) ⊗R c1 C → V(N ) ⊗R c1 C

(II.12.22.6)

is then identified with the inverse of the isomorphism (II.12.22.5), which shows that V(N ) b 1 ]-representation of ∆. is a Dolbeault R[ p

Remark II.12.23. The proof of II.12.22 given above is due to Tsuji. It is simpler and more elegant than the proof we had given in an earlier version of this text. Proposition II.12.24. The functors H and V induce equivalences of categories quasib 1 ]-representations of ∆ and inverse to each other between the category of Dolbeault R[ p e1 c1 [ 1 ]-modules with coefficients in ξ −1 Ω . that of the solvable Higgs R R/OK

p

This follows from II.12.20 and II.12.22. b 1 ]-representation of ∆, (H(M ), θ) the associated II.12.25. Let M be a Dolbeault R[ p

c1 [ 1 ]-module with coefficients in ξ −1 Ω e1 Higgs R R/OK (II.12.8.2), and θtot = θ ⊗id+id⊗dC † p † c1 -field over H(M ) ⊗ c C . It immediately follows from (II.12.16.1) that the total Higgs R R1

b we have a functorial canonical isomorphism of complexes of R-representations ∼

† • † K• (H(M ) ⊗R c1 C , θtot ) → M ⊗ b K (C , dC † ),

(II.12.25.1)

R

where K• (−, −) denotes the Dolbeault complex (II.2.8.4). By II.12.6, for every i ≥ 0, ei c 1 since H(M ) ⊗R Ω R/OK is a direct summand of a free R1 [ p ]-module of finite type, we have (II.12.25.2)

† ∆ −i ei ei (ξ −i H(M ) ⊗R c1 C ⊗R ΩR/OK ) = ξ H(M ) ⊗R ΩR/OK .

c1 [ 1 ]-modules We deduce from this a functorial canonical isomorphism of complexes of R p ∼

K• (H(M ), θ) → (M ⊗ b K• (C † , dC † ))∆ ,

(II.12.25.3)

R

∆

where the functor (−) on the right-hand side is defined component-wise. This result can be refined as follows.

152

II. LOCAL STUDY

b 1 ]-representation Proposition II.12.26 ([27] § 3, IV.5.3.2). Let M be a Dolbeault R[ p e1 c1 [ 1 ]-module with coefficients in ξ −1 Ω of ∆ and (H(M ), θ) the associated Higgs R R/OK p c1 [ 1 ])) (II.12.8.2). Then we have a functorial canonical isomorphism in D+ (Mod(R p

∼ C•cont (∆, M ) →

(II.12.26.1)

•

K (H(M ), θ),

where C•cont (∆, M ) is the complex of continuous cochains of ∆ with values in M (II.3.8) and K• (H(M ), θ) is the Dolbeault complex (II.2.8.4). Indeed, for every integer i ≥ 0 and all rational numbers r0 > r > 0, the canonical morphism e • (Cb(r0 ) , pr0 d b(r0 ) )) → Ci (∆, M ⊗ b K e • (Cb(r) , pr d b(r) )) (II.12.26.2) Ci (∆, M ⊗ b K cont

C

R

cont

R

C

is homotopic to 0, by virtue of II.12.3(ii). Consequently, the complex e • (Cb(r) , pr d b(r) )) lim Cicont (∆, M ⊗ b K C

(II.12.26.3)

R

−→

r∈Q>0

is acyclic. The first spectral sequence of the bicomplex ([41] § 0 (11.3.2.2)) e • (Cb(r) , pr d b(r) )) lim C•cont (∆, M ⊗ b K C

(II.12.26.4)

R

−→

r∈Q>0

then implies that the associated simple complex (II.3.6.2) Z e • (Cb(r) , pr d b(r) )) (II.12.26.5) lim C•cont (∆, M ⊗ b K C R

−→

r∈Q>0

b → K• (Cb(r) , pr d is acyclic. The canonical morphisms R[0] Cb(r) ) therefore induce a quasiisomorphism Z • (II.12.26.6) Ccont (∆, M ) → lim C•cont (∆, M ⊗ b K• (Cb(r) , pr dCb(r) )). R

−→

r∈Q>0

By II.12.17, there exists a rational number r0 > 0 such that H(M ) is contained in M ⊗ b Cb(r0 ) and that for every rational number 0 < r ≤ r0 , the canonical morphism R

b(r) → M ⊗ b Cb(r) H(M ) ⊗R c1 C

(II.12.26.7)

R

(r) θtot

c1 -field over is bijective. If we write = θ ⊗ id + pr id ⊗ dCb(r) for the total Higgs R (r) b H(M ) ⊗R , then we deduce from this an isomorphism c1 C ∼ b(r) , θ(r) ) → K• (H(M ) ⊗R M ⊗ b K• (Cb(r) , pr dCb(r) ). c1 C tot

(II.12.26.8)

R

For every integer i ≥ 0, the canonical morphism (II.12.26.9) H(M ) ⊗ c Ci (∆, Cb(r) ) → Ci R1

cont

cont (∆, H(M )

b(r) ) ⊗R c1 C

c1 [ 1 ]-module is an isomorphism. Indeed, we can reduce to the case where H(M ) is a free R p of finite type, in which case the assertion follows from the compactness of ∆. On the c1 -field δ i,(r) on Ci (∆, Cb(r) ) with other hand, the derivation dCb(r) induces a Higgs R cont −1 e 1 coefficients in ξ ΩR/OK . The differentials of the complex C•cont (∆, Cb(r) ) are morphisms i,(r) c1 -field of Higgs modules. If we denote by ϑtot = θ ⊗ id + pr id ⊗ δ i,(r) the total Higgs R on H(M ) ⊗ c Ci (∆, Cb(r) ), then the canonical morphism R1

cont

i b(r) ), ϑi,(r) ) → Ci (∆, K• (H(M ) ⊗ c Cb(r) , θ(r) )) (II.12.26.10) K• (H(M ) ⊗R c1 Ccont (∆, C tot tot cont R1

II.13. SMALL REPRESENTATIONS

153

is an isomorphism (II.12.26.9). It follows from II.12.5, using the first spectral sequence of bicomplexes, that the canonical morphism Z • b(r) ), ϑ•,(r) ) (II.12.26.11) K• (H(M ), θ) → lim K• (H(M ) ⊗R c1 Ccont (∆, C tot −→

r∈Q>0

is a quasi-isomorphism. The proposition follows. II.13. Small representations We keep the assumptions and notation of II.12 in this section. Definition II.13.1. Let G be a topological group, A an OC -algebra that is complete and separated for the p-adic topology, endowed with a continuous action of G (by homomorphisms of OC -algebras), α a rational number > 0, and M a continuous A-representation of G, endowed with the p-adic topology. (i) We say that M is α-quasi-small if the A-module M is complete and separated for the p-adic topology, and is generated by a finite number of elements that are G-invariant modulo pα M . (ii) We say that M is α-small if M is a free A-module of finite type having a basis over A consisting of elements that are G-invariant modulo pα M . (iii) We say that M is quasi-small (resp. small) if it is α0 -quasi-small (resp. α0 -small) 2 . for a rational number α0 > p−1 cont (G) (resp. Repqsf We denote by Repα-qsf A (G)) the full subcategory of RepA (G) A made up of the α-quasi-small (resp. quasi-small) A-representations of G whose underlying A-module is OC -flat, and by Repα-small (G) (resp. Repsmall (G)) the full subcategory of A A cont RepA (G) made up of the α-small (resp. small) A-representations of G.

If the action of G on A is trivial, then an A-representation M of G is α-small if and only if it is α-quasi-small and M is a free A-module of finite type. Definition II.13.2. Let G be a topological group, A an OC -algebra that is complete and separated for the p-adic topology, endowed with a continuous action of G (by homomorphisms of OC -algebras). We endow A[ p1 ] with the p-adic topology (II.2.2). We say that a continuous A[ p1 ]-representation M of G is small if the following conditions are satisfied: (i) M is a projective A[ p1 ]-module of finite type, endowed with the p-adic topology (II.2.2). 2 and a sub-A-module of finite type M ◦ (ii) There exist a rational number α > p−1 of M that is stable under G, generated by a finite number of elements that are G-invariant modulo pα M ◦ , and that generates M over A[ p1 ]. cont We denote by Repsmall 1 (G) the full subcategory of Rep 1 (G) made up of the small ] A[ p ] A[ p A-representations of G.

Note that unlike in the integral case (II.13.1), we do not require M to be free over A[ p1 ]. Remarks II.13.3. Let G be a topological group and A an OC -algebra that is separated and complete for the p-adic topology, endowed with a continuous action of G (by homomorphisms of OC -algebras). (i) Let M be a projective A[ p1 ]-module of finite type and M ◦ a sub-A-module of finite type of M . Then M ◦ is complete and separated for the p-adic topology.

154

II. LOCAL STUDY

Indeed, M ◦ is complete by virtue of ([11] Chap. III § 2.12 Cor. 1 to Prop. 16). On the other hand, after adding a direct summand to M , if necessary, we may assume that it is free of finite type over A[ p1 ]. Consequently, there exists an integer m ≥ 0 such that pm M ◦ is contained in a free A-module of finite type N . Hence ∩n≥0 pn M ◦ ⊂ ∩n≥0 pn N = 0. (ii) Let M be a small A[ p1 ]-representation of G and M ◦ a sub-A-module of finite type of M satisfying condition II.13.2(ii). It then follows from (i) that M ◦ is a quasi-small A-representation of G. c1 -module with Definition II.13.4. Let β be a rational number > 0 and (N, θ) a Higgs R −1 e 1 coefficients in ξ ΩR/OK (II.9.9). c1 and if θ is a (i) We say that (N, θ) is β-quasi-small if N is of finite type over R e1 multiple of pβ in ξ −1 EndR c1 (N ) ⊗R ΩR/OK (II.2.8.10). We then also say that c1 -field θ is β-quasi-small. the Higgs R (ii) We say that (N, θ) is β-small if it is β-quasi-small and if N is free of finite type c1 . We then also say that the Higgs R c1 -field θ is β-small. over R (iii) We say that (N, θ) is quasi-small (resp. small) if it is β 0 -quasi-small (resp. β 0 1 c1 -field small) for a rational number β 0 > p−1 . We then also say that the Higgs R θ is quasi-small (resp. small). e1 e1 c1 , ξ −1 Ω c1 , ξ −1 Ω ) ) the full subcategory of HM(R We denote by HMβ-qsf (R R/OK

R/OK

c1 -modules whose underlying R c1 -module is (II.9.9) made up of the β-quasi-small Higgs R β-small c −1 e 1 e1 c OC -flat and by HM (R1 , ξ ΩR/OK ) the full subcategory of HM(R1 , ξ −1 Ω R/OK ) c made up of the β-small Higgs R1 -modules. c1 [ 1 ]-module (N, θ) with coefficients in ξ −1 Ω e1 Definition II.13.5. A Higgs R R/OK is said p to be small if the following conditions are satisfied: c1 [ 1 ]-module of finite type. (i) N is a projective R p

1 c1 -module of finite type N ◦ and a sub-R (ii) There exist a rational number β > p−1 1 c1 [ ], such that we have of N that generates it over R p

(II.13.5.1)

e1 θ(N ◦ ) ⊂ pβ ξ −1 N ◦ ⊗R Ω R/OK .

c1 [ 1 ], ξ −1 Ω e1 c 1 −1 Ω e1 We denote by HMsmall (R R/OK ) the full subcategory of HM(R1 [ p ], ξ R/OK ) p 1 c1 [ ]-modules (cf. II.9.9). made up of the small Higgs R p

Note that unlike in the integral case (II.13.4), we do not require N to be free over 1 c R1 [ p ]. Lemma II.13.6. Let A be an integral domain with field of fractions L, B a subring of L containing A, N a flat B-module of finite type, u1 , . . . , u` B-linear endomorphisms of N that commute pairwise and such that for every 1 ≤ i ≤ `, the characteristic polynomial of the endomorphism ui ⊗ id of N ⊗B L has coefficients in A. Then N is generated over B by a sub-A-module of finite type M such that ui (M ) ⊂ M for every 1 ≤ i ≤ `. We proceed by induction on the number of endomorphisms. Suppose that the assertion has been proved for ` − 1 and let us show it for `. Let M 0 be a sub-A-module of finite type of N that generates it over B and satisfies ui (M 0 ) ⊂ M 0 for every 1 ≤ i ≤ ` − 1. Let P (X) = X n + a1 X n−1 + · · · + an ∈ A[X] be the characteristic polynomial of the endomorphism u` ⊗ id of N ⊗B L. Since N is B-flat, we can identify it with a sub-B-module

II.13. SMALL REPRESENTATIONS

155

of N ⊗B L. Consequently, the endomorphism P (u` ) of N is zero and the sub-A-module Pn−1 M = i=0 ui` (M 0 ) of N has the desired property. e1 c1 [ 1 ]-module with coefficients in ξ −1 Ω Lemma II.13.7. Let (N, θ) be a Higgs R R/OK such p that the following conditions are satisfied: c1 [ 1 ]-module of finite type. (i) N is a projective R p

1 p−1 such that for every i ≥ 1, the c e1 to piβ ξ −i SiR (Ω R/OK ) ⊗R R1 (II.2.8.5).

(ii) There exists a rational number β > characteristic invariant of θ belongs Then (N, θ) is small.

ith

b1 ) is OC -flat, its generic points are the generic points of Spec(R b1 [ 1 ]). Since Spec(R p b1 ) has only a finite number of generic points. On Since the latter is noetherian, Spec(R b1 is normal by virtue of II.6.15. It is therefore isomorphic to a finite the otherQhand, R n product j=1 Aj of normal integral domains. By II.13.6, for every 1 ≤ j ≤ n, there exists a sub-Aj -module of finite type Nj◦ of N ⊗Rb1 Aj that generates it over Aj [ p1 ], such e1 that if we denote by θj the Higgs Aj [ 1 ]-field over N ⊗ b Aj with coefficients in ξ −1 Ω R1

p

R/OK

induced by θ, we have (II.13.7.1)

e1 θj (Nj◦ ) ⊂ pβ ξ −1 Nj◦ ⊗R Ω R/OK .

The lemma follows. II.13.8. Recall that on C, the logarithmic function log(x) converges when x ∈ 1 . For every 1 + mC , and the exponential function exp(x) converges when v(x) > p−1 1 x ∈ C such that v(x) > p−1 , we have exp(x) ≡ 1 + x mod (xmC ), log(1 + x) ≡ x mod (xmC ),

exp(log(1 + x)) = 1 + x and log(exp(x)) = x.

c1 -module of finite type. By ([1] 1.10.2), M is complete II.13.9. Let M be an R and separated for the p-adic topology, and the same therefore holds for EndR c1 (M ). (M ) is O -flat and for every rational Suppose, moreover, that M is OC -flat. Then EndR C c1 α number α ≥ 0, the canonical homomorphism pα EndR c1 (M ) → HomR c1 (M, p M ) is an c1 -linear endomorphism of M and α a rational number > 0. isomorphism. Let u be an R If u induces an automorphism of M/pα M , then u is an automorphism of M . Indeed, u is surjective by Nakayama’s lemma. If x ∈ M is such that u(x) = 0, then there exists y ∈ M such that x = pα y; since M is OC -flat, we have u(y) = 0; it follows that x ∈ ∩n≥0 pnα M = 0, giving the assertion. We can therefore identify id + pα EndR c1 (M ) 1 α with a subgroup of AutR c1 (M ). Suppose α > p−1 and v = u − id ∈ p EndR c1 (M ). For every x ∈ M , the series X 1 (II.13.9.1) exp(v)(x) = v n (x), n! n≥0

(II.13.9.2)

log(u)(x)

=

X (−1)n v n+1 (x), n+1

n≥0

c1 -endomorphisms exp(v) and log(u) of M . Morethen converge in M , and define two R α α over, we have exp(v) ∈ id+p EndR c1 (M ) and log(u) ∈ p EndR c1 (M ), and exp(log(u)) = u and log(exp(v)) = v.

156

>

II. LOCAL STUDY

c1 -module of finite type, α a rational number II.13.10. Let M be an OC -flat R 1 and β = α − p−1 . We denote by ΨM the composed isomorphism

1 p−1 ,

∼ / β −1 c (II.13.10.1) HomZ (∆∞ , pα EndR p ξ EndR c1 (M )) c1 (M ) ⊗R c1 HomZ (∆∞ , R1 (1)) WWWWW WWWWW WWWWW e id⊗δ WWWWW ΨM W+  e1 pβ ξ −1 End c (M ) ⊗R Ω R/OK

R1

where δe is the isomorphism (II.7.19.9) and the horizontal isomorphism comes from 1 ∼ (II.6.12.2) and from the canonical isomorphism OC (1) → p p−1 ξOC (II.9.18.1). Note c1 -module End c (M ) is complete and separated for the p-adic topology and that the R R1 OC -flat (II.13.9). We can write ΨM explicitly as follows. Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ) ⊗Z Zp form a Zp -basis, (χti )1≤i≤d their images in e1 HomZ (∆∞ , Zp (1)) (II.7.19.5), and (d log(ti ))1≤i≤d their images in Ω R/OK (II.7.18.5). The 1 e (d log(ti ))1≤i≤d then form an R-basis of Ω (II.7.12.1) and the (χt )1≤i≤d form a Zp R/OK

i

basis of HomZ (∆∞ , Zp (1)) (II.7.19.8). For every 1 ≤ i ≤ d, denote by χi the composition (II.13.10.2)

∆∞

χti

1 / Zp (1) log([ ])/ p p−1 ξOC ,

where the second arrow is induced by the isomorphism (II.9.18.1). In view of (II.6.12.2), β for every homomorphism φ : ∆∞ → pα EndR c1 (M ) (1 ≤ c1 (M ), there exist φi ∈ p EndR i ≤ d) such that φ=

(II.13.10.3)

d X i=1

ξ −1 φi ⊗ χi .

Concretely, if ζ is a Zp -basis of Zp (1), there exist elements γ1 , . . . , γd ∈ ∆∞ such that for all 1 ≤ i, j ≤ d, we have χti (γj ) = δij ζ. For every 1 ≤ i ≤ d, we therefore have ξ −1 log([ζ])φi = φ(γi ).

(II.13.10.4)

Recall that ξ −1 log([ζ]) is an element of OC with valuation d log(ti ) for every 1 ≤ i ≤ d, we have ΨM (φ) =

(II.13.10.5)

d X i=1

1 p−1

e t )= (II.9.18). Since δ(χ i

ξ −1 φi ⊗ d log(ti ).

c1 -representation of ∆∞ over M . Since ∆∞ acts trivially Let ϕ be an α-quasi-small R c on R1 , ϕ is a homomorphism ϕ : ∆∞ → AutR c1 (M )

(II.13.10.6)

whose image is contained in the subgroup id + pα EndR c1 (M ) of AutR c1 (M ). Since ∆∞ is abelian, we can define the homomorphism (II.13.9) log(ϕ) : ∆∞ → pα EndR c1 (M ).

(II.13.10.7)

Moreover, it follows from (II.13.10.4) and (II.13.10.5) that ΨM (log(ϕ))∧ΨM (log(ϕ)) = 0; c1 -field on M with coefficients in in other words, ΨM (log(ϕ)) is a β-quasi-small Higgs R −1 e 1 ξ ΩR/OK . We thus obtain a functor β-qsf c −1 e 1 (II.13.10.8) Repα-qsf (R1 , ξ ΩR/OK ), c (∆∞ ) → HM R1

(M, ϕ) 7→ (M, ΨM (log(ϕ))).

II.13. SMALL REPRESENTATIONS

157

c1 -field over M with coefficients in ξ −1 Ω e1 Let θ be a β-quasi-small Higgs R R/OK . In view of (II.13.10.3) and (II.13.10.4), since θ ∧ θ = 0, the image of the homomorphism α Ψ−1 c1 (M ) consists of endomorphisms of M that commute pairwise. M (θ) : ∆∞ → p EndR We can therefore define the homomorphism (II.13.9) exp(Ψ−1 c1 (M ), M (θ)) : ∆∞ → AutR

(II.13.10.9)

c1 -representation of ∆∞ over M . We thus define a which is clearly an α-quasi-small R functor (II.13.10.10) α-qsf c1 , ξ −1 Ω e1 (M, θ) 7→ (M, exp(Ψ−1 HMβ-qsf (R R/OK ) → Rep c (∆∞ ), M (θ))). R1

1 and β = Proposition II.13.11 (Faltings, [27]). For all rational numbers α > p−1 1 α − p−1 , the functors (II.13.10.8) and (II.13.10.10) are equivalences of categories quasiinverse to each other. They induce equivalences of categories ∼

(II.13.11.1)

c1 , ξ −1 Ω e1 Repα-small (∆∞ ) → HMβ-small (R c R/OK ), R

(II.13.11.2)

α-small c1 , ξ −1 Ω e1 (∆∞ ) HMβ-small (R c R/OK ) → RepR

1

∼

1

quasi-inverse to each other. The first assertion is an immediate consequence of the definition of the functors c1 is OC -flat (II.6.14). (II.13.10.8) and (II.13.10.10). The second assertion follows because R Remark II.13.12. Under the assumptions of II.13.10, if (M, ϕ) is an object of the β-qsf c −1 e 1 category Repα-qsf (R1 , ξ ΩR/OK ) c1 (∆∞ ) and (M, θ) is an object of the category HM R that correspond via the functors (II.13.10.8) and (II.13.10.10), then ϕ and θ are linked by the following formulas: ! d X −1 (II.13.12.1) ϕ = exp ξ θ i ⊗ χi , i=1

(II.13.12.2)

θ

=

d X i=1

Since the χi factor through ∆

p∞

ξ −1 θi ⊗ d log(ti ).

(II.7.19), ϕ factors through ∆p∞ .

c1 -algebra II.13.13. For every rational number r ≥ 0, we denote by S(r) the sub-R (r) of S (II.11.1.2) defined by (II.13.13.1)

r −1 e 1 c1 ), S(r) = SR ΩR/OK ⊗R R c1 (p ξ

b (r) its p-adic Hausdorff completion, which we endow with the p-adic topology. and by S b (r) is R c1 -flat by virtue of ([1] 1.12.4) and therefore OC -flat (II.6.14). We Note that S (0) b =S b (0) . For all rational numbers r0 ≥ r ≥ 0, we have a canonical set S = S and S 0 0 injective homomorphism ar,r : S(r ) → S(r) . One immediately verifies that the induced 0 0 b (r ) → S b (r) is injective. We have a canonical S(r) -isomorphism homomorphism b ar,r : S (II.13.13.2)

∼

−1 e 1 Ω1S(r) /R ΩR/OK ⊗R S(r) . c →ξ 1

We denote by (II.13.13.3)

(r) e1 dS(r) : S(r) → ξ −1 Ω R/OK ⊗R S

158

II. LOCAL STUDY

c1 -derivation of S(r) , which is also a Higgs R c1 -field with coefficients in the universal R −1 e 1 −1 e 1 (r) c1 ⊂ dS(r) (S ). We denote by ξ ΩR/OK because ξ ΩR/OK ⊗R R (II.13.13.4)

b (r) → ξ −1 Ω b (r) e1 dS b (r) : S R/OK ⊗R S

its extension to the completions. For all rational numbers r0 ≥ r ≥ 0, we have (II.13.13.5)

0

0

0

pr −r (id × ar,r ) ◦ dS(r0 ) = dS(r) ◦ ar,r .

ˇ e0 , M e ) the smooth (A2 (S), M We denote by (X ˇ ) deA2 (S) )-deformation of (X, MX X0 fined by the chart (P, γ) (II.10.13.1), by L0 the associated Higgs–Tate torsor (II.10.3), by ψ0 ∈ L0 (Yb ) the section defined by the same chart (II.10.13.4), and by ϕ0 = ϕψ0 the action of ∆ on S induced by ψ0 (II.10.6.4). By II.10.17, ϕ0 preserves S and the induced action of ∆ on S factors through ∆p∞ . We also denote the resulting action of ∆p∞ on S by ϕ0 . Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ) ⊗Z Zp form a Zp -basis, (χti )1≤i≤d their images in HomZ (∆p∞ , Zp (1)) (II.7.19.5), and (d log(ti ))1≤i≤d their im−1 e1 e1 ages in Ω d log(ti ) ∈ ξ −1 Ω R/OK ⊂ S, R/OK (II.7.18.5). For every 1 ≤ i ≤ d, we set yi = ξ and denote by χi the composition (II.13.13.6)

∆p∞

χti

1 / Zp (1) log([ ])/ p p−1 ξOC ,

where the second arrow is induced by the isomorphism (II.9.18.1). By (II.10.17.2), for every g ∈ ∆p∞ , we have (II.13.13.7)

ϕ0 (g) = exp(−

d X i=1

ξ −1

∂ ⊗ χi (g)). ∂yi

It follows that ϕ0 is continuous for the p-adic topology on S (cf. II.11.9). For every rational number r ≥ 0, ϕ0 preserves S(r) , and the induced actions of ∆p∞ on S(r) b (r) are continuous for the p-adic topologies (cf. II.11.6). Unless explicitly stated and S b (r) with these actions. It immediately follows from otherwise, we endow S(r) and S (II.13.13.7) that dS(r) and dS b (r) are ∆p∞ -equivariant. II.13.14. We keep the notation of II.13.13 and let moreover β, r be two rational 1 c1 -module numbers such that β > r + p−1 and r ≥ 0, and (N, θ) a β-quasi-small Higgs R −1 e 1 with coefficients in ξ Ω (II.13.4) such that N is OC -flat. There is a unique way to R/OK

write θ=

(II.13.14.1)

d X i=1

θi ⊗ yi ,

where the θi are endomorphisms of N that belong to pβ EndR c1 (N ) and commute pairwise. Pd Qd Qd d For every n = (n1 , . . . , nd ) ∈ N , let |n| = i=1 ni , n! = i=1 ni !, θn = i=1 θini ∈ Q d ni n b (r) is complete and separated for EndR c1 (N ), and y = c1 S i=1 yi ∈ S. Note that N ⊗R b (r) is R c1 -flat ([1] 1.12.4). the p-adic topology ([1] 1.10.2), and that it is OC -flat because S (r) b Consequently, for every z ∈ N ⊗ c S , the series R1

(II.13.14.2)

X 1 (θn ⊗ y n )(z) n! d

n∈N

II.13. SMALL REPRESENTATIONS

159

b (r) , which b (r) , and defines an S b (r) -linear endomorphism of N ⊗ c S converges in N ⊗R c1 S R1 we denote by (II.13.14.3)

b (r) → N ⊗ c S b (r) . expr (θ) : N ⊗R c1 S R1

For every rational number r0 such that 0 ≤ r0 ≤ r, the diagram (II.13.14.4)

expr (θ)

b (r) N ⊗R c1 S id⊗ar

/ N ⊗c S b (r) R1

0 ,r

id⊗ar

 b (r0 ) N ⊗R c1 S

expr0 (θ)

0 ,r

 / N ⊗c S b (r0 ) R1

is commutative. We may therefore leave the index r out of the notation expr (θ) without risk of confusion. 1 and Proposition II.13.15. Let β, r be two rational numbers such that β > r + p−1 c c r ≥ 0, N an R1 -module of finite type that is OC -flat, θ a β-quasi-small Higgs R1 -field e1 c1 -representation of ∆∞ over N with coefficients in ξ −1 Ω , and ϕ the quasi-small R R/OK

b (r) b (r) -module N ⊗ c S over N associated with θ by the functor (II.13.10.10). Then the S R1 is complete and separated for the p-adic topology, and the endomorphism (II.13.14.3) (II.13.15.1)

b (r) → N ⊗ c S b (r) expr (θ) : N ⊗R c1 S R1

is a ∆∞ -equivariant isomorphism of modules with integrable p-adic pr -connections with b (r) /R b (r) is endowed with the action of ∆∞ c1 (II.2.14), where S respect to the extension S induced by ϕ0 (II.13.13.7) and with the p-adic pr -connection pr dS b (r) (II.13.13.4), the module N on the left-hand side is endowed with the trivial action of ∆∞ and the Higgs c1 -field θ, and the module N on the right-hand side is endowed with the action ϕ of R c1 -field (cf. II.2.17). In particular, expr (θ) is an isomorphism ∆∞ and the zero Higgs R e1 c1 -modules with coefficients in ξ −1 Ω of Higgs R R/OK (II.2.16). b (r) is complete and separated for the p-adic topology by First note that N ⊗R c1 S c1 -generators of N , and for every 1 ≤ ` ≤ d, virtue of ([1] 1.10.2(ii)). Let x1 , . . . , xn be R ` c1 such that for every let A` = (mij )1≤i,j≤n be an n × n matrix with coefficients in R 1 ≤ j ≤ n, we have n X (II.13.15.2) θ` (xj ) = pβ m`ij xi . i=1

Note that the matrices A` do not commute pairwise in general (but they do commute if c1 with basis x1 , . . . , xn ). For every n = (n1 , . . . , nd ) ∈ Nd , we set N is free over R (II.13.15.3)

c1 ). An = An1 1 · An2 2 · · · And d ∈ Matn (R

The series E=

(II.13.15.4)

X pβ|n| An ⊗ y n n! d

n∈N

b (r) . For all (a1 , . . . , an ) ∈ (S b (r) )n , we then defines an n × n matrix with coefficients in S have n n X X b (r) , expr (θ)( xi ⊗ ai ) = xi ⊗ bi ∈ N ⊗R c1 S i=1

i=1

160

II. LOCAL STUDY

where (b1 , . . . , bn ) = E · (a1 , . . . , an ). b (r) , the determinant of E is invertible Since the matrix E −id has coefficients in pβ−r S b (r) . If E −1 = (fij )1≤i,j≤n is the inverse matrix of E in GLn (S b (r) ), then for every in S 1 ≤ i ≤ d, we have d X expr (θ)( xj ⊗ fji ) = xi ⊗ 1.

(II.13.15.5)

j=1

Hence expr (θ) is surjective. b (r) be such that exp (θ)(x) = 0. Since E ≡ id mod (pβ−r ), there Let x ∈ N ⊗R c1 S r (r) β−r b b (r) is OC -flat (II.13.14), we exists y ∈ N ⊗R such that x = p y. Since N ⊗R c1 S c1 S n(β−r) b (r) ) and consequently have exp (θ)(y) = 0. It follows that x ∈ ∩n≥0 p (N ⊗ c S r

R1

b (r) is separated for the p-adic topology ([1] 1.10.2). Hence that x = 0 because N ⊗R c1 S expr (θ) is bijective. It immediately follows from the definition (II.13.14.2) that the diagram expr (θ)

b (r) N ⊗R c1 S

(II.13.15.6)

/ N ⊗c S b (r) R1 pr id⊗dS c(r)

∇(r)

 b (r) e1 ⊗ N ⊗R ξ −1 Ω R c1 S R/OK

expr (θ)⊗id

 b (r) e1 / ξ −1 Ω ⊗ N ⊗R R c1 S R/OK

n where ∇(r) = θ ⊗ id + pr id ⊗ dS b (r) (II.2.17.2), is commutative modulo p for every n ≥ 1. It is therefore commutative; in other words, expr (θ) is a morphism of modules with p-adic b (r) /R c1 . pr -connections with respect to the extension S By II.13.12 and (II.13.13.7), for every g ∈ ∆∞ , we have

kϕ(g)

(II.13.15.7)

=

d X exp( ξ −1 θi ⊗ χi (g)), i=1

ϕ0 (g)

(II.13.15.8)

=

exp(−

d X

ξ −1

i=1

∂ ⊗ χi (g)). ∂yi

∂ exp(−ξ −1 ∂y i

b (r) , we c1 -automorphism of S ⊗ χi (g)) as an R On the other hand, if we view have ∂ (idN ⊗ exp(−ξ −1 ⊗ χi (g))) ◦ expr (θ) ∂yi −1 ∂ = (exp(−ξ −1 θi ⊗ χi (g)) ⊗ idS ⊗ χi (g))). b (r) ) ◦ expr (θ) ◦ (idN ⊗ exp(−ξ ∂yi Indeed, it suffices to verify this equation modulo pn for every n ≥ 1, which follows ∂ formally from the definition (II.13.14.2) (we use first that exp(−ξ −1 ∂y ⊗ χi (g)) is a i (r) c homomorphism of the R1 -algebra S ). Since the endomorphisms θi commute pairwise, it follows that (II.13.15.9)

d d X X ∂ (exp( ξ −1 θi ⊗ χi (g)) ⊗ exp(− ξ −1 ⊗ χi (g))) ◦ expr (θ) ∂yi i=1 i=1

=

expr (θ) ◦ (idN ⊗ exp(−

d X i=1

ξ −1

∂ ⊗ χi (g))). ∂yi

II.13. SMALL REPRESENTATIONS

161

Consequently, the morphism expr (θ) is ∆∞ -equivariant. Corollary II.13.16. Under the assumptions of II.13.15, we have a functorial and ∆b e1 equivariant Cb(r) -isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK (II.13.16.1)

∼ b(r) , b(r) → N ⊗R N ⊗R c1 C c1 C

b where Cb(r) is endowed with the canonical action of ∆ and the Higgs R-field pr dCb(r) (II.12.1.8), the module N on the left-hand side is endowed with the trivial action of ∆∞ c1 -field θ, and the module N on the right-hand side is endowed with the and the Higgs R c1 -field. If, moreover, the deformation (X, e M e ) is action ϕ of ∆∞ and the zero Higgs R X defined by the chart (P, γ) (II.10.13.1), then the isomorphism is canonical. c(r) -isomorphism Indeed, by II.13.15, expr (θ) induces a functorial and ∆-equivariant S b −1 e 1 of Higgs R-modules with coefficients in ξ ΩR/OK (II.11.1) (II.13.16.2)

∼ c(r) , c(r) → N ⊗R N ⊗R c1 S c1 S

b c(r) is endowed with the action of ∆ induced by ϕ0 (II.11.5) and the Higgs R-field where S r p dSc(r) (II.11.1.6), the module N on the left-hand side is endowed with the trivial action c1 -field θ, and the module N on the right-hand side is endowed of ∆∞ and the Higgs R c1 -field. The corollary follows in view of with the action ϕ of ∆∞ and the zero Higgs R II.10.10 and II.12.2. Remark II.13.17. Under the assumptions of II.13.15, by virtue of II.2.15, expr (θ) induces a functorial and ∆-equivariant Cb(r) -isomorphism of modules with integrable pb adic pr -connections with respect to the extension Cb(r) /R, ∼ b c Cb(r) , b c Cb(r) → N⊗ (II.13.17.1) N⊗ R1

R1

where Cb(r) is endowed with the canonical action of ∆ and the p-adic pr -connection pr dCb(r) (II.12.1.8), the module N on the left-hand side is endowed with the trivial action c1 -field θ, and the module N on the right-hand side is endowed of ∆∞ and the Higgs R c1 -field (cf. II.2.17). with the action ϕ of ∆∞ and the zero Higgs R Corollary II.13.18. Under the assumptions of II.13.15, we have a functorial and ∆b e1 equivariant C † -isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK (II.13.18.1)

∼

† † N ⊗R c1 C → N ⊗R c1 C ,

b Higgs dC † where C † is endowed with the canonical action of ∆ and the Higgs R-field (II.12.1.12), the module N on the left-hand side is endowed with the trivial action of ∆∞ c1 -field θ, and the module N on the right-hand side is endowed with the action and the R c1 -field (II.2.8.7). If, moreover, the deformation (X, e M e) ϕ of ∆∞ and the zero Higgs R X is defined by the chart (P, γ) (II.10.13.1), then the isomorphism is canonical. This follows from II.13.16. c1 -module of finite type, θ a quasi-small Corollary II.13.19. Let N be a projective R c1 -field over N with coefficients in ξ −1 Ω e1 c Higgs R R/OK , and ϕ the quasi-small R1 -representation of ∆∞ over N associated with θ by the functor (II.13.10.10). Then we have a c1 -modules with coefficients in ξ −1 Ω e1 functorial isomorphism of Higgs R R/OK (II.13.19.1)

b ∼ H((N, ϕ) ⊗R c1 R) → (N, θ),

162

II. LOCAL STUDY

b of where H is the functor (II.12.8.2), and a functorial isomorphism of R-representations ∆ ∼ b (II.13.19.2) V(N, θ) → (N, ϕ) ⊗ R, c1 R

e M e ) is defined by where V is the functor (II.12.9.2). If, moreover, the deformation (X, X the chart (P, γ) (II.10.13.1), then the isomorphisms are canonical. b and (C † )∆ = R c1 (II.12.6). This follows from II.13.18 and the fact that ker(dC † ) = R c1 -module with coefficients in ξ −1 Ω e1 Corollary II.13.20. Every small Higgs R R/OK is b solvable, and its image by V is a small R-representation of ∆. This follows from II.13.18 and II.13.19. This statement will be strengthened in II.14.7. c1 [ 1 ]-module with coefficients in ξ −1 Ω e1 Let (N, θ) be a small Higgs R R/OK p 1 c1 -module of finite type of N that (II.13.5), β a rational number > p−1 , N ◦ a sub-R c1 [ 1 ], such that we have generates it over R p II.13.21.

e1 θ(N ◦ ) ⊂ pβ ξ −1 N ◦ ⊗R Ω R/OK ,

(II.13.21.1)

e1 c1 -field on N ◦ with coefficients in ξ −1 Ω and θ◦ the Higgs R R/OK induced by θ. Then ◦ ◦ c1 -module (II.13.4). We denote by ϕ◦ the quasi-small (N , θ ) is a quasi-small Higgs R c1 -representation of ∆∞ over N ◦ associated with θ◦ by the functor (II.13.10.10) and R c1 [ 1 ]-representation of ∆∞ over N deduced from ϕ◦ . Let us show that by ϕ the small R p

ϕ does not depend on the choice of N ◦ , and that the correspondence (N, θ) 7→ (N, ϕ) defines a functor (II.13.2) small c1 [ 1 ], ξ −1 Ω e1 HMsmall (R c1 [ 1 ] (∆∞ ). R/OK ) → RepR p p

(II.13.21.2)

c1 [ 1 ]-modules with Indeed, let u : (N, θ) → (N1 , θ1 ) be a morphism of small Higgs R p ◦ e1 c1 -module of finite type of N1 that generates it coefficients in ξ −1 Ω and N a subR 1 R/OK c1 [ 1 ], such that we have u(N ◦ ) ⊂ N ◦ and over R 1

p

e1 θ1 (N1◦ ) ⊂ pβ ξ −1 N1◦ ⊗R Ω R/OK .

(II.13.21.3)

c1 -field on N ◦ with coefficients in ξ −1 Ω e1 We denote by θ1◦ the Higgs R 1 R/OK induced by θ1 , by ◦ ◦ ◦ c ϕ1 the R1 -representation of ∆∞ over N1 associated with θ1 by the functor (II.13.10.10), c1 [ 1 ]-representation of ∆∞ over N1 induced by ϕ◦ . Since the morand by ϕ1 the R 1

p

phism (N ◦ , ϕ◦ ) → (N1◦ , ϕ◦1 ) induced by u is clearly ∆∞ -equivariant, the same holds for u : (N, ϕ) → (N1 , ϕ1 ). Let us show that ϕ does not depend on the choice of N ◦ . Let γ be a rational number 1 c1 -module of finite type of N that generates it over R c1 [ 1 ], such > p−1 and N ? a sub-R p that we have e1 (II.13.21.4) θ(N ? ) ⊂ pγ ξ −1 N ? ⊗R Ω . R/OK

Replacing β and γ by the least of the two and N by N ◦ + N ? , we may assume that β = γ and N ◦ ⊂ N ? . Applying the above to the identity of N , we deduce that ϕ does not depend on the choice of N ◦ . ?

II.13. SMALL REPRESENTATIONS

163

c1 [ 1 ]Let us show that for every morphism v : (N, θ) → (N 0 , θ0 ) of small Higgs R p 0 c1 [ 1 ]-representation of ∆∞ e1 , if ϕ is the small R modules with coefficients in ξ −1 Ω R/OK p over N 0 associated with θ0 , then v : (N, ϕ) → (N 0 , ϕ0 ) is ∆∞ -equivariant. Let β 0 be a 1 c1 -module of finite type of N 0 that generates it rational number > p−1 and N 0◦ a sub-R c1 [ 1 ], such that we have over R p

0 e1 θ0 (N 0◦ ) ⊂ pβ ξ −1 N 0◦ ⊗R Ω R/OK .

(II.13.21.5)

Replacing β and β 0 by the least of the two and N 0◦ by v(N ◦ ) + N 0◦ , we may assume that β = β 0 and v(N ◦ ) ⊂ N 0◦ . We then conclude as above that v : (N, ϕ) → (N 0 , ϕ0 ) is ∆∞ -equivariant. II.13.22.

c1 [ 1 ]-representation of ∆∞ (II.13.2), α a rational Let (N, ϕ) be a small R p c1 -module of finite type of N that is stable under ∆∞ and , and N ◦ a sub-R

2 number > p−1 generated by a finite number of elements that are ∆∞ -invariant modulo pα N ◦ , and that c1 [ 1 ]. By II.13.3(ii), the R c1 -representation ϕ◦ of ∆∞ over N ◦ induced generates N over R p c1 -field on N ◦ with coefficients by ϕ is quasi-small. Denote by θ◦ the quasi-small Higgs R ◦ −1 e 1 associated with ϕ by the functor (II.13.10.8) and by θ the small Higgs in ξ Ω R/OK

c1 [ 1 ]-field on N induced by θ◦ . Proceeding as in II.13.21, one shows that (N, θ) does R p not depend on the choice of N ◦ and that the correspondence (N, ϕ) 7→ (N, θ) defines a functor (II.13.5) small c 1 e1 Repsmall (R1 [ ], ξ −1 Ω c1 [ 1 ] (∆∞ ) → HM R/OK ). R p p

(II.13.22.1)

It immediately follows from II.13.10 that the functors (II.13.21.2) and (II.13.22.1) are quasi-inverse to each other. c1 [ 1 ]-module with coefficients in Proposition II.13.23. Let (N, θ) be a small Higgs R p e1 c1 [ 1 ]-representation of ∆∞ over N associated with ξ −1 Ω (II.13.5), and ϕ the small R R/OK p θ by the functor (II.13.21.2). Then: b (i) We have a functorial ∆-equivariant C † -isomorphism of Higgs R-modules with e1 coefficients in ξ −1 Ω R/OK , ∼

† † N ⊗R c1 C → N ⊗R c1 C ,

(II.13.23.1)

b where C † is endowed with the canonical action of ∆ and the Higgs R-field dC † (II.12.1.12), the module N on the left-hand side is endowed with the trivial c1 -field θ, and the module N on the right-hand side action of ∆∞ and the Higgs R c1 -field. If, moreover, is endowed with the action ϕ of ∆∞ and the zero Higgs R e the deformation (X, MXe ) is defined by the chart (P, γ) (II.10.13.1), then the isomorphism is canonical. c1 [ 1 ]-module (N, θ) is solvable and we have a functorial ∆-equivar(ii) The Higgs R p b 1 iant R[ ]-isomorphism p

(II.13.23.2)

∼ b V(N ) → (N, ϕ) ⊗R c1 R,

e M e ) is where V is the functor (II.12.9.2). If, moreover, the deformation (X, X defined by the chart (P, γ) (II.10.13.1), then the isomorphism is canonical.

164

II. LOCAL STUDY

b 1 ]-representation V(N ) of ∆ is small and Dolbeault, and we have a (iii) The R[ p c1 [ 1 ]-modules functorial isomorphism of Higgs R p ∼

H(V(N )) → (N, θ),

(II.13.23.3)

e M e ) is where H is the functor (II.12.8.2). If, moreover, the deformation (X, X defined by the chart (P, γ) (II.10.13.1), then the isomorphism is canonical. The isomorphism (II.13.23.1) follows from II.13.18; the functoriality is proved as b and in II.13.21. The other assertions follow in view of the fact that ker(dC † ) = R † ∆ c1 (II.12.6). (C ) = R c1 [ 1 ]-module of finite type, θ a Proposition II.13.24 (IV.5.3.10). Let N be a projective R p b 1 ]-module, r a rational c [ 1 ]-field over N with coefficients in ξ −1 Ω e1 Higgs R , M an R[ 1 p

R/OK

p

number > 0, and ∼ b(r) → M ⊗ b Cb(r) N ⊗R c1 C

(II.13.24.1)

R

b b(r) e1 with coefficients in ξ −1 Ω a C -linear isomorphism of Higgs R-modules R/OK , where C is endowed with the Higgs field pr dCb(r) (II.12.1.8) and M is endowed with the zero Higgs c1 [ 1 ]-module with coefficients in ξ −1 Ω e1 field. Then (N, θ) is a small Higgs R (II.13.5). b(r)

R/OK

p

c(r) Indeed, in view of II.10.10 and II.12.2, the isomorphism (II.13.24.1) induces an S b e1 , linear isomorphism of Higgs R-modules with coefficients in ξ −1 Ω R/OK

c(r) ∼

N ⊗R c1 S

(II.13.24.2)

c(r) , → M ⊗b S R

c(r) is endowed with the Higgs field pr d c(r) (II.11.1.6) and M is endowed with where S S b the zero Higgs field. This induces an R-linear isomorphism b ∼ N ⊗R c1 R → M.

(II.13.24.3)

Let t1 , . . . , td ∈ P gp be such that their images in (P gp /Zλ) ⊗Z Zp form a Zp -basis, e1 so that (d log(ti ))1≤i≤d is an R-basis of Ω R/OK (II.7.12.1). For every 1 ≤ i ≤ d, we −1 −1 e 1 set yi = ξ d log(ti ) ∈ ξ ΩR/OK ⊂ S . For every n = (n1 , . . . , nd ) ∈ Nd , we set Pd Qd b 1 ] with the p-adic topology ni n |n| = ∈ S . If we endow R[ i=1 ni and y = i=1 yi p b 1 (r) 1 c (II.2.2), the R[ ]-algebra S [ ] is canonically identified with p

p

{

(II.13.24.4)

b 1 ][[y , . . . , y ]] | an y n ∈ R[ 1 d p d

X

n∈N

lim

|n|→+∞

p−r|n| an = 0}.

Note that under the conventions of (II.11.1.6), we have dSc(r) (pr yi ) = 1 ⊗ yi for every 1 ≤ i ≤ d. b 1 ][[y , . . . , y ]]-module N ⊗ b 1 The R[ 1 d c1 [ 1 ] R[ p ][[y1 , . . . , yd ]] is complete and separated p R p

for the (y1 , . . . , yd )-adic topology. Indeed, it is complete by virtue of ([11] Chap. III § 2.12 Cor. 1 to Prop. 16) and is separated because it is a direct summand of a free module P b of finite type. Consequently, every formal series n∈Nd xn ⊗ y n , where xn ∈ N ⊗R c1 R, b 1 ][[y , . . . , y ]]. Conversely, every element x of converges in N ⊗ R[ c1 [ 1 ] R p

p

1

d

b 1 N ⊗R c1 [ 1 ] R[ ][[y1 , . . . , yd ]] p p

II.13. SMALL REPRESENTATIONS

165

can be written in a unique way as X

x=

(II.13.24.5)

n∈Nd

xn ⊗ y n ,

b b 1 b where xn ∈ N ⊗R c1 R. If we endow the R[ p ]-module N ⊗R c1 R with the p-adic topology b 1 ]-module c(r) [ 1 ]-module N ⊗ c S c(r) is identified with the R[ (II.2.2), then the S p p R1 (II.13.24.6)

{

X n∈Nd

b xn ⊗ y n | xn ∈ N ⊗R c1 R,

lim

|n|→+∞

p−r|n| xn = 0}.

c(r) . To prove equality, we can reduce to the Indeed, the latter clearly contains N ⊗R c1 S c1 [ 1 ], in which case the assertion is obvious case where N is free of finite type over R p (II.13.24.4). Let x ∈ N , z ∈ M the image of x ⊗ 1 by the isomorphism (II.13.24.3), and X c(r) xn ⊗ y n ∈ N ⊗R (II.13.24.7) c1 S n∈Nd

c(r) by the inverse of the isomorphism (II.13.24.2), where the image of z ⊗ 1 ∈ M ⊗ b S R b We clearly have x = x. Let us write x ∈ N ⊗ R. n

0

c1 R

θ=

(II.13.24.8)

d X i=1

θi ⊗ yi ,

c1 [ 1 ]-endomorphisms of N that commute pairwise. Since the section where the θi are R p (II.13.24.7) is annihilated by θ ⊗ 1 + 1 ⊗ pr dSc(r) , for every n = (n1 , . . . , nd ) ∈ Nd and every 1 ≤ i ≤ d, we have θi (xn ) + (ni + 1)xn+1i = 0,

(II.13.24.9)

where 1i is the element of Nd whose components are all zero except for the ith, which is 1. It follows that Y 1 (II.13.24.10) xn = (−1)|n| ( θni )(x). ni ! i 1≤i≤d

Consequently, p−r|n| ( (II.13.24.6).

1 ni 1≤i≤d ni ! θi )(x)

Q

b tends to 0 in N ⊗R c1 R when |n| tends to infinity

b The p-adic topology on N is induced by the p-adic topology on N ⊗R c1 R. Indeed, c1 [ 1 ], we can reduce to the case where N is since N is projective of finite type over R p c1 [ 1 ]. For every n ≥ 1, since the free of finite type, or even to the case where N = R p

b nR b is injective (cf. the proof of II.6.14), we c1 /pn R c1 → R/p canonical homomorphism R b ∩R c1 [ 1 ] = pn R c1 , giving the assertion. have pn R p Q It follows from the above that for every x ∈ N , p−r|n| ( 1≤i≤d n1i ! θini )(x) tends to 0 in N when |n| tends to infinity. c1 -module of finite type of N that generates it over R c1 [ 1 ], and ε a Let N0 be a sub-R p 1 1 rational number such that p−1 < ε < r + p−1 . Since the sequence p(r−ε)n n! tends to 0

166

II. LOCAL STUDY

Q in OC when n tends to infinity, for every x ∈ N , p−ε|n| ( 1≤i≤d θini )(x) tends to 0 in N c1 -module when |n| tends to infinity. We may therefore consider the sub-R X Y p−ε|n| ( (II.13.24.11) N◦ = θini )(N0 ) n∈Nd

1≤i≤d

c1 and generates N over R c1 [ 1 ]. Since we have of N . It is of finite type over R p (II.13.24.12)

e1 θ(N ◦ ) ⊂ pε ξ −1 N ◦ ⊗R Ω R/OK ,

c1 [ 1 ]-module with coefficients in ξ −1 Ω e1 (N, θ) is a small Higgs R R/OK . p e1 c1 [ 1 ]-module with coefficients in ξ −1 Ω Corollary II.13.25. A Higgs R R/OK is solvable p (II.12.18) if and only if it is small (II.13.5). This follows from II.13.23(ii) and II.13.24. b 1 ]-representation of ∆ (II.12.16) is small (II.13.2). Corollary II.13.26. Any Dolbeault R[ p This follows from II.12.24, II.13.25, and II.13.23(iii). We will strengthen this statement, under certain assumptions, in II.14.8. II.14. Descent of small representations and applications We keep the assumptions and notation of II.12 in this section. Proposition II.14.1. Let a be a nonzero element of OK , α a rational number > M1 and M2 two α-small (R1 /aR1 )-representations of ∆p∞ (II.13.1), and (II.14.1.1)

u : M1 ⊗R1 R → M2 ⊗R1 R

1 p−1 ,

1

1 a ∆-equivariant R-linear morphism. Suppose that v(a) > p−1 + α and set b = ap−α− p−1 . Then there exists a unique ∆p∞ -equivariant R1 -linear morphism

(II.14.1.2)

u : M1 /bM1 → M2 /bM2

such that u ⊗R1 R ≡ u mod bM2 ⊗R1 R. Denote by M the discrete R1 -∆p∞ -module HomR1 (M1 , M2 ), which is an α-small (R1 /aR1 )-representation of ∆p∞ . Since R1 is normal by II.6.8(ii), we have pα bR1 = (pα bR) ∩ R1 . Consequently, the canonical morphism (II.14.1.3)

H0 (∆p∞ , M/pα bM ) → H0 (∆, (M/pα bM ) ⊗R1 R) 1

is injective and its cokernel is annihilated by p p−1 mK by virtue of II.8.23. Hence there exists a ∆p∞ -equivariant R1 -linear morphism (II.14.1.4)

v : M1 /pα bM1 → M2 /pα bM2

such that v ⊗R1 R ≡ pα u mod pα bM2 ⊗R1 R. Since pα R1 = (pα R) ∩ R1 and since R1 is OK -flat, representing v by a matrix with coefficients in R1 /pα bR1 , we see that there exists a unique ∆p∞ -equivariant R1 -linear morphism (II.14.1.5)

u : M1 /bM1 → M2 /bM2

such that v = pα u. Using once again that R is OK -flat, we deduce that u ⊗R1 R ≡ u mod bM2 ⊗R1 R. The uniqueness of u follows from the fact that the canonical homomorphism R1 /bR1 → R/bR is injective (cf. the proof of II.6.14).

II.14. DESCENT OF SMALL REPRESENTATIONS AND APPLICATIONS

167

1 Proposition II.14.2. Let α be rational number > p−1 , a a nonzero element of OK such that v(a) > α, and M a free (R/aR)-module of rank r ≥ 1, endowed with the discrete topology and a continuous R-semi-linear action of ∆ such that M admits a basis e1 , . . . , er consisting of elements that are ∆-invariant modulo p2α M . Then there exist a discrete R1 -∆p∞ -module N whose underlying module is free of rank r over R1 /ap−α R1 , with a basis f1 , . . . , fr consisting of elements that are ∆p∞ -invariant modulo pα N , and a ∆-equivariant R-linear isomorphism ∼

N ⊗R1 R → M/ap−α M

(II.14.2.1)

that sends fj ⊗ 1 mod pα N ⊗R1 R to ej mod pα M for every 1 ≤ j ≤ r. The proposition is obvious if v(a) ≤ 3α, in which case we take N = (R1 /ap−α R1 )r endowed with the trivial representation of ∆p∞ and the canonical basis. Suppose, therefore, v(a) > 3α. Let n be an integer ≥ 1 such that ε=

(II.14.2.2)

1 1 v(a) − 3α < (α − ). n 3 p−1

Let us show by a finite induction that for every 0 ≤ i ≤ n, the proposition holds for the R-representation M/p3α+iε M ; in other words, that there exist a discrete R1 -∆p∞ -module (i) (i) Ni whose underlying module is free of rank r over R1 /p2α+iε R1 , with a basis f1 , . . . , fr α consisting of elements that are ∆p∞ -invariant modulo p Ni , and a ∆-equivariant R-linear isomorphism ∼

Ni ⊗R1 R → M/p2α+iε M

(II.14.2.3) (i)

that sends fj ⊗ 1 mod pα Ni ⊗R1 R to ej mod pα M for every 1 ≤ j ≤ r. The representation N = Nn will then answer the question because 2α + nε = v(a) − α. We take N0 = (R1 /p2α R1 )r endowed with the trivial representation of ∆p∞ and the canonical basis. Suppose that Ni has been constructed for 0 ≤ i < n and let us construct Ni+1 . By II.3.35, the obstruction to lifting Ni to a discrete (R1 /p3α+iε R1 )-∆p∞ -module whose underlying module is free of finite type over R1 /p3α+iε R1 is an element o of H2 (∆p∞ , Matr (R1 /pα R1 )). On the other hand, the morphism H2 (∆p∞ , Matr (R1 /pα R1 )) → H2 (∆, Matr (R/pα R))

(II.14.2.4)

is almost injective by virtue of II.8.11. The image of o in H2 (∆, Matr (R/pα R)) is zero because the representation M/p2α+iε M lifts to M/p3α+iε M ; hence pε o = 0. Consequently, by virtue of (II.3.35.4), Ni /p2α+(i−1)ε Ni lifts to a discrete (R1 /p3α+(i−1)ε R1 )0 ∆p∞ -module Ni+1 whose underlying module is free of finite type over R1 /p3α+(i−1)ε R1 . 0 By II.3.34, the lift Ni+1 ⊗R1 R of M/p2α+(i−1)ε M is deduced from M/p3α+(i−1)ε M by twisting by an element c of H1 (∆, Matr (R/pα R)). By virtue of II.8.11, the cokernel of the canonical morphism H1 (∆p∞ , Matr (R1 /pα R1 )) → H1 (∆, Matr (R/pα R))

(II.14.2.5) 1

is annihilated by p p−1 +ε . Since α >

1 p−1

+ 3ε, pα−2ε c is the image of an element

c0 ∈ H1 (∆p∞ , Matr (R1 /pα R1 )). 0 0 Twisting Ni+1 /p2α+(i+1)ε Ni+1 by −c0 , we obtain a discrete (R1 /p2α+(i+1)ε R1 )-∆p∞ module Ni+1 whose underlying module is free of finite type over R1 /p2α+(i+1)ε R1 , which lifts Ni /pα+(i+1)ε Ni . Note that 2α + (i − 1)ε > α + (i + 1)ε. Then Ni+1 has the required properties by the last assertion of II.3.34.

168

II. LOCAL STUDY

Proposition II.14.3. Let a be a nonzero element of OK , α and β two rational numbers 1 such that v(a) > α > β > p−1 , and M a free (R/aR)-module of rank r ≥ 1, endowed with the discrete topology and a continuous R-semi-linear action of ∆ such that M admits a basis consisting of elements that are ∆-invariant modulo p2α M . Then there exist a discrete R1 -∆p∞ -module N whose underlying module is free of rank r over R1 /ap−α R1 , having a basis consisting of elements that are ∆p∞ -invariant modulo p2β N , and a ∆equivariant R-linear isomorphism ∼

N ⊗R1 R → M/ap−α M.

(II.14.3.1)

The proposition is obvious if v(a) ≤ 3α, in which case we take N = (R1 /ap−α R1 )r endowed with the trivial representation of ∆p∞ . Suppose, therefore, v(a) > 3α. Let n be an integer ≥ 1 such that

1 1 v(a) − 3α < inf( (α − ), α − β). n 3 p−1 Let us show by a finite induction that for every 0 ≤ i ≤ n, the proposition holds for the R-representation M/p3α+iε M ; in other words, that there exists a discrete R1 -∆p∞ module Ni whose underlying module is free of rank r over R1 /p2α+iε R1 , having a basis consisting of elements that are ∆p∞ -invariant modulo p2(α−ε) Ni and a ∆-equivariant R-linear isomorphism (II.14.3.2)

(II.14.3.3)

ε=

∼

Ni ⊗R1 R → M/p2α+iε M.

The representation N = Nn will then answer the question because 2α + nε = v(a) − α and α − ε > β. We take N0 = (R1 /p2α R1 )r endowed with the trivial representation of ∆p∞ and the canonical basis. Suppose that Ni has been constructed for 0 ≤ i < n and let us construct Ni+1 . By II.3.35 (applied with aq = p3α+iε , an = p2α+iε , and am = p2(α−ε) ), the obstruction to lifting Ni to a discrete (R1 /p3α+iε R1 )-∆p∞ -module whose underlying module is free of finite type over R1 /p3α+iε R1 is an element o of H2 (∆p∞ , Matr (R1 /pα R1 )). On the other hand, the morphism (II.14.3.4)

H2 (∆p∞ , Matr (R1 /pα R1 )) → H2 (∆, Matr (R/pα R))

is almost injective by virtue of II.8.11. The image of o in H2 (∆, Matr (R/pα R)) is zero because the representation M/p2α+iε M lifts to M/p3α+iε M ; hence pε o = 0. Consequently, 0 by (II.3.35.4), Ni /p2α+(i−1)ε Ni lifts to a discrete (R1 /p3α+(i−1)ε R1 )-∆p∞ -module Ni+1 3α+(i−1)ε whose underlying module is free of finite type over R1 /p R1 . By II.3.34, the lift 0 Ni+1 ⊗R1 R of M/p2α+(i−1)ε M is deduced from the lift M/p3α+(i−1)ε M by twisting by an element c of H1 (∆, Matr (R/pα R)). By virtue of II.6.24, the cokernel of the canonical morphism (II.14.3.5)

H1 (∆p∞ , Matr (Rp∞ /pα Rp∞ )) → H1 (∆, Matr (R/pα R))

is annihilated by pε . Hence pε c is the image of an element

c0 ∈ H1 (∆p∞ , Matr (Rp∞ /pα Rp∞ )). 1

By the proof of II.8.10(i), we have c0 = c01 + c02 , where p p−1 c02 = 0 and c01 is the image of an element c1 ∈ H1 (∆p∞ , Matr (R1 /pα R1 )). 0 0 Twisting Ni+1 /p3α+(i−2)ε Ni+1 by −c1 , we obtain a discrete (R1 /p3α+(i−2)ε R1 )-∆p∞ 00 module Ni+1 whose underlying module is free of finite type over R1 /p3α+(i−2)ε R1 and 00 that lifts Ni /p2α+(i−2)ε Ni . By the last assertion of II.3.34, the lift Ni+1 ⊗R1 R of

II.14. DESCENT OF SMALL REPRESENTATIONS AND APPLICATIONS

169

M/p2α+(i−2)ε M is deduced from the lift M/p3α+(i−2)ε M by twisting by the image c2 1 of c02 in H1 (∆, Matr (R/pα R)). Since p p−1 c2 = 0, we deduce from this, again by the last assertion of II.3.34, that there exists a ∆-equivariant R-linear isomorphism Since α > properties.

∼

1

(II.14.3.6) 1 p−1

1

00 00 (Ni+1 /p3α− p−1 +(i−2)ε Ni+1 ) ⊗R1 R → M/p3α− p−1 +(i−2)ε M.

00 00 + 3ε, the representation Ni+1 = Ni+1 /p2α+(i+1)ε Ni+1 has the desired

Proposition II.14.4 (Faltings, [27]). The functor (II.13.1) (II.14.4.1)

small (∆), Repsmall c (∆∞ ) → Rep b R R

1

b M 7→ M ⊗R c1 R

is an equivalence of categories. Indeed, since the canonical functor (II.14.4.2)

small Repsmall c (∆p∞ ) → RepR c (∆∞ ) R 1

1

is an equivalence of categories (II.13.12), it suffices to show that the functor b (II.14.4.3) Repsmall (∆ ∞ ) → Repsmall (∆), M 7→ M ⊗ R c1 R

p

c1 R

b R

is an equivalence of categories. This functor is fully faithful by virtue of II.14.1. Let us show that it is essentially surjective. Let α, β be two rational numbers such that b 1 and M a (2α)-small R-representation of rank r ≥ 1. By II.14.3, for every α > β > p−1 integer n > α, there exist a discrete R1 -∆p∞ -module Nn whose underlying module is free of rank r over R1 /pn−α R1 , with a basis consisting of elements that are ∆p∞ -invariant modulo p2β N , and a ∆p∞ -equivariant R-linear isomorphism (II.14.4.4)

∼

Nn ⊗R1 R → M/pn−α M.

By virtue of II.14.1, for all integers n ≥ m > α, there exists a unique ∆p∞ -equivariant R1 -linear isomorphism (II.14.4.5)

∼

Nn /pm−α Nn → Nm

that is compatible with the isomorphisms (II.14.4.4). Consequently, the R1 -modules (Nn )n>α form an inverse system, and if N is its inverse limit, then we have a ∆b isomorphism equivariant R-linear (II.14.4.6)

b ∼ N ⊗R c1 R → M.

Remarks II.14.5. (i) Though similar, the proofs of Propositions II.14.2 and II.14.3 0 differ significantly in their last steps that consist of defining Ni+1 from Ni+1 . Note that as far as the descent of the basis is concerned, the proof of II.14.3 does not give better results than that of II.14.2, which is why we left this part out in statement II.14.3. Proposition II.14.2 is due to Faltings ([27] Lemma 1). (ii) The descent statement established by Faltings in [27] is in fact slightly weaker than II.14.4. b Corollary II.14.6. Every small R-representation of ∆ is Dolbeault, and its image by H c e1 (II.12.8.2) is a small Higgs R1 -module with coefficients in ξ −1 Ω . R/OK

This follows from II.14.4, II.13.11, II.13.18, and II.13.19. Corollary II.14.7. The functors H (II.12.8.2) and V (II.12.9.2) induce equivalences of b categories quasi-inverse to each other between the category of small R-representations of −1 e 1 c ∆ and that of small Higgs R1 -modules with coefficients in ξ Ω . R/OK

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II. LOCAL STUDY

This follows from II.12.15, II.13.20 and II.14.6. Proposition II.14.8. The following statements are equivalent: b 1 ]-representation M of ∆ (II.13.2), there exist a small Higgs (i) For every small R[ p c1 [ 1 ]-module N with coefficients in ξ −1 Ω e1 R (II.13.5) and a ∆-equivariant R/OK

p

b 1 ]-isomorphism R[ p ∼

M → V(N ).

(II.14.8.1)

b 1 ]-representation of ∆ is Dolbeault (II.12.16). (ii) Every small R[ p b 1 ]-representation of ∆ is Dolbeault if and only if it is small. (iii) An R[ p

(iv) The functor (II.13.2) small Repsmall c [ 1 ] (∆∞ ) → Rep b 1 (∆), R

(II.14.8.2)

R[ p ]

1 p

b M 7→ M ⊗R c1 R

is an equivalence of categories. Indeed, the implication (i)⇒(ii) follows from II.13.23(iii), the implication (ii)⇒(iii) is a consequence of II.13.26, and the implication (iii)⇒(i) follows from II.12.15 and II.13.25. b 1 ]-representation M of ∆, the Higgs Next, let us show (i)⇒(iv). For every small R[ p

c1 [ 1 ]-module H(M ) is small by (i) and II.13.23(iii). The functor (II.13.21.2) therefore R p c1 [ 1 ]-representation ϕ of ∆∞ on H(M ). The correspondence associates with it a small R p

M 7→ (H(M ), ϕ) defines a functor

small Repsmall b 1 (∆) → RepR c [ 1 ] (∆∞ ).

(II.14.8.3)

R[ p ]

1 p

Let us show that the functors (II.14.8.2) and (II.14.8.3) are quasi-inverse to each other. b b 1 ]-representation M of ∆, we have functorial ∆-equivariant R-isomorFor every small R[ p

phisms (II.13.23.2) ∼ b ∼ H(M ) ⊗R c1 R → V(H(M )) → M.

(II.14.8.4)

c1 [ 1 ]-representation of ∆∞ , and θ the small Higgs R c1 [ 1 ]Moreover, let (N, ϕ) be a small R p p e1 associated with ϕ by the functor (II.13.22.1). field over N with coefficients in ξ −1 Ω R/OK

c1 -modules By II.13.23, we have a functorial isomorphism of Higgs R ∼ b → (II.14.8.5) H((N, ϕ) ⊗ R) (N, θ). c1 R

Applying the functor (II.13.21.2), quasi-inverse of the functor (II.13.22.1), we obtain a c1 -isomorphism functorial ∆∞ -equivariant R ∼ b → (N, ϕ). (II.14.8.6) H((N, ϕ) ⊗ R) c1 R

The functors (II.14.8.2) and (II.14.8.3) are therefore quasi-inverse to each other. b 1 ]-representation of ∆. By (iv), Finally, let us show (iv)⇒(i). Let M be a small R[ p b 1 c there exist a small R [ ]-representation (N, ϕ) of ∆ and a ∆-equivariant R-isomor1 p

∞

phism (II.14.8.7)

∼ b M → (N, ϕ) ⊗R c1 R.

e1 c1 [ 1 ]-field on N with coefficients in ξ −1 Ω Let θ be the small Higgs R R/OK associated with p ϕ by the functor (II.13.22.1). By virtue of II.13.23(ii), since the functors (II.13.21.2) and

II.14. DESCENT OF SMALL REPRESENTATIONS AND APPLICATIONS

171

b 1 ]-isomorphism (II.13.22.1) are quasi-inverse to each other, we have a ∆-equivariant R[ p ∼

M → V(N, θ), proving the proposition. Corollary II.14.9. Assume that the equivalent statements of II.14.8 hold. Then for b 1 ]-representation Hom b 1 b 1 ]-representation M of ∆, the R[ every small R[ b 1 (M, R[ p ]) of p p R[ p ]

∆ is small. b c1 [ 1 ]-representation N of ∆∞ and a ∆-equivariant RIndeed, there exist a small R p ∼ b b isomorphism M → N ⊗R c1 R. We deduce from this a ∆-equivariant R-isomorphism (II.14.9.1)

Hom b

1 R[ p ]

∼ b 1 ]) → b c 1 (M, R[ HomR c1 [ 1 ] (N, R1 [ ]) ⊗R c1 R. p p p

2 c1 -module of finite type of N stable Let α be a rational number > p−1 and N ◦ a sub-R under ∆∞ and generated by a finite number of elements that are ∆∞ -invariant modulo c1 [ 1 ]. By virtue of ([1] 10.10.2(iii)), N ◦ is a coherent pα N ◦ , and that generates N over R p c1 ), and the canonical morphism c1 -module. The same therefore holds for Hom c (N ◦ , R R R1

(II.14.9.2)

◦ c c 1 c 1 HomR c1 R1 [ ] → HomR c1 (N , R1 ) ⊗R c1 [ 1 ] (N, R1 [ ]) p p p

c1 is O -flat, for every rational number is an isomorphism. On the other hand, since R K β > 0, the canonical morphism (II.14.9.3)

◦ c β ◦ β ◦ β HomR c1 (N , R1 ) ⊗OK OK /p OK → HomR1 (N /p N , R1 /p R1 )

◦ c is injective. It follows that the representation of ∆∞ over HomR c1 (N , R1 ) is continuous for the p-adic topology, and is quasi-small, giving the corollary.

Corollary II.14.10. Assume that the equivalent statements of II.14.8 hold. Then for b 1 ]-representations M and M 0 of ∆, and every ∆-equivariant surjective R[ b 1 ]all small R[ p p b 1 0 linear morphism u : M → M , the R[ ]-representation of ∆ over the kernel of u is small. p

This follows from II.14.9. Lemma II.14.11. Let α, ε be two rational numbers such that 0 < ε < α and M an b α-quasi-small R-representation of ∆. Then: (i) The R∞ -module (M/pα M )Σ is almost of finite type. b (ii) The R-module pε M is generated by a finite number of elements that are, on the one hand, ∆-invariant modulo pα M and, on the other hand, Σ-invariant. b → M is almost surjective. (iii) The canonical morphism M Σ ⊗R R d ∞ b that are ∆-invariant modulo pα M . Let x1 , . . . , xd be generators of M over R α d (i) We denote by u : (R/p R) → M/pα M the ∆-equivariant surjective R-linear morphism defined by the (xi )1≤i≤d , and by C its kernel. Then H1 (Σ, C) is almost zero by virtue of II.6.19. On the other hand, the canonical morphism R∞ /pα R∞ → (R/pα R)Σ

is an almost isomorphism (II.6.22). Consequently, the R∞ -linear morphism (II.14.11.1)

(R∞ /pα R∞ )d → (M/pα M )Σ

defined by the (xi )1≤i≤d is almost surjective.

172

II. LOCAL STUDY

b pα M is complete and separated for the p-adic topology; it is (ii) The R-module complete by virtue of ([11] Chap. III § 2.12 Cor. 1 to Prop. 16) and is separated as a submodule of M . Therefore H1cont (Σ, pα M ) is almost zero by virtue of II.6.21. Consequently, the canonical morphism M Σ → (M/pα M )Σ

(II.14.11.2)

is almost surjective. The classes of the elements pε x1 , . . . , pε xn in (M/pα M )Σ lift to elements x01 , . . . , x0d ∈ M Σ . For every 1 ≤ i ≤ d, we have x0i ∈ M Σ ∩ (pε M ) = (pε M )Σ .

(II.14.11.3)

b On the other hand, by Nakayama’s lemma, the (x0i )1≤i≤d generate pε M over R. (iii) This follows from (ii). b Lemma II.14.12. Let M be an R-module that is complete and separated for the p-adic b topology, endowed with a continuous R-semi-linear action of Σ, and x1 , . . . , xd elements d b Then the R d -linear morphism R d → M Σ defined by of M Σ that generate M over R. ∞

∞

the (xi )1≤i≤d , is almost surjective.

bd → M b morphism R Denote by C the kernel of the Σ-equivariant surjective R-linear b d for the p-adic topology. defined by the (x ) . Since M is separated, C is closed in R i 1≤i≤d

It is therefore complete and separated for the topology induced by the p-adic topology on b d ; in other words, C is isomorphic to the inverse limit of the inverse system of R-modules b R d

b )) 1 (C/(C ∩ pn R n∈N . Consequently, Hcont (Σ, C) is almost zero by virtue of II.6.21. On bΣ d the other hand, the canonical homomorphism R ∞ → R is an almost isomorphism by II.6.23, giving the lemma. d Lemma II.14.13. Let α be a rational number > 0 and M an α-quasi-small R ∞Σ 0 d ∞ endowed representation of ∆∞ such that M is Zp -flat. Then the R -module M p d with the induced action of ∆p∞ , is an α-quasi-small R p∞ -representation of ∆p∞ , and the canonical morphism d M Σ0 ⊗Rd R ∞ →M p∞

(II.14.13.1) is surjective.

By (II.3.10.4) and (II.3.10.5), we have (II.14.13.2)

0 → R1 lim (M/pn M )Σ0 → H1cont (Σ0 , M ) → lim H1 (Σ0 , M/pn M ) → 0. ←−

←−

n

n

On the other hand, since Σ0 is a profinite group of order prime to p, H1 (Σ0 , M/pn M ) = 0 for every n ≥ 0, and the inverse system ((M/pn M )Σ0 )n≥0 satisfies the Mittag–Leffler condition. It follows that H1cont (Σ0 , M ) = 0. Likewise, we have H1cont (Σ0 , pα M ) = 0 because pα M is complete and separated for the p-adic topology. Consequently, the canonical morphism (II.14.13.3)

M Σ0 → (M/pα M )Σ0

is surjective. Let x1 , . . . , xd be generators of M over R∞ that are ∆∞ -invariant modulo pα M . Their classes in (M/pα M )Σ0 lift to elements x01 , . . . , x0d of M Σ0 that generate M over

II.14. DESCENT OF SMALL REPRESENTATIONS AND APPLICATIONS

173

Σ0 d d R ⊗Rd R ∞ . Therefore the canonical morphism M ∞ → M is surjective. Copying the p∞ proof of II.14.12, we show that the morphism

(II.14.13.4)

d

Σ0 d R p∞ → M

d defined by the (x0i )1≤i≤d is surjective. Consequently, M Σ0 is an R p∞ -module of finite type. It is therefore complete and separated for the p-adic topology; it is complete by virtue of ([11] Chap. III § 2.12 Cor. 1 to Prop. 16) and is separated as a submodule of M . On the other hand, since M is Zp -flat, the p-adic topology on M Σ0 is clearly induced by the p-adic topology on M . Hence the representation of ∆p∞ over M Σ0 induced by that of ∆ is continuous for the p-adic topology. For every 1 ≤ i ≤ d and every g ∈ ∆p∞ , we have g(x0i ) − x0i ∈ M Σ0 ∩ pα M = pα M Σ0 . It follows that M Σ0 is an α-quasi-small d R p∞ -representation of ∆p∞ , giving the lemma. Proposition II.14.14. Let α, ε be two rational numbers such that 0 < 2ε < α and b of ∆ such that M is Zp -flat. Then there exist M an α-quasi-small R-representation d ∞ an (α − 2ε)-quasi-small R -representation M 0 of ∆p∞ and a ∆-equivariant surjective p b R-linear morphism (II.14.14.1)

b → pε M. R M 0 ⊗Rd p∞

b that are, on the one By II.14.11(ii), there exist generators x1 , . . . , xd of pε M over R α hand, ∆-invariant modulo p M and, on the other hand, Σ-invariant. For every 1 ≤ i ≤ d and every g ∈ ∆, g(xi ) − xi ∈ pα M ∩ M Σ = pα M Σ . Hence by virtue of II.14.12 (applied d b d to the (α − ε)-quasi-small R-representation pε M of Σ), there exist (a ) ∈ R ij 1≤j≤d

∞

such that

(II.14.14.2)

g(xi ) − xi =

d X

pα−2ε aij xj .

j=1

ε d Denote by M1 the sub-R ∞ -module of p M generated by x1 , . . . , xd . By (II.14.14.2), M1 is stable under the action of ∆. Since M1 ⊂ M Σ , the induced action of ∆ on M1 factors through ∆∞ . d The R ∞ -module M1 is complete and separated for the p-adic topology; it is complete by virtue of ([11] Chap. III § 2.12 Cor. 1 to Prop. 16) and is separated as a submodule of M . For every integer n ≥ 2ε, we have

(II.14.14.3)

pn M1 ⊂ M1 ∩ pn M ⊂ pn M Σ ⊂ pn−2ε M1 ,

where the last inclusion is a consequence of II.14.12. Therefore the p-adic topology on M1 is induced by the p-adic topology on M . In view of (II.14.14.2), it follows that M1 d is an (α − 2ε)-quasi-small R ∞ -representation of ∆∞ . Since the canonical morphism (II.14.14.4)

b → pε M R M1 ⊗R d ∞

is clearly surjective, the proposition follows from II.14.13 applied to M1 . b 1 ]-representation of ∆. Proposition II.14.15. Suppose d = 1, and let M be a small R[ p c1 -representation M 0 of ∆∞ and a ∆-equivariant surjective Then there exist a small R b 1 ]-linear morphism R[ p

(II.14.15.1)

b 1 M 0 ⊗R c1 R[ ] → M. p

174

II. LOCAL STUDY

b M ◦ of ∆ such that M ◦ Indeed, by II.13.3, there exist a quasi-small R-representation b 1 is Zp -flat and a ∆-equivariant R[ p ]-linear isomorphism ∼ b 1] → M ◦ ⊗ b R[ M. R p

(II.14.15.2)

By virtue of II.14.14, after replacing M ◦ by pε M ◦ , if necessary, for a rational number d ε > 0, we may assume that there exist a quasi-small R p∞ -representation N of ∆p∞ and b a ∆-equivariant surjective R-linear morphism (II.14.15.3)

b → M ◦. N ⊗Rd R p∞

There exist a rational number α >

1 p−1

d and generators x1 , . . . , xd of N over R p∞ that are 2α 0 d ∆p∞ -invariant modulo p N . We denote by N the free Rp∞ -module with basis e1 , . . . , ed d and by σ : N 0 → N the R p∞ -linear morphism that sends ei to xi for every 1 ≤ i ≤ d. Denote by 0 ϕ0 : ∆p∞ → AutR c1 (N )

(II.14.15.4)

d the trivial R p∞ -representation of ∆p∞ with respect to the basis e1 , . . . , ed . We choose 0 d a Zp -basis γ of ∆p∞ . There exists an R p∞ -linear automorphism u of N such that for 2α 0 every 1 ≤ i ≤ d, we have u(ei ) − ei ∈ p N and σ(u(ei )) = γ(xi ).

(II.14.15.5)

−1 d For every g ∈ ∆p∞ , denote by g u the R ) of p∞ -linear automorphism ϕ0 (g) ◦ u ◦ ϕ0 (g 0 0 d ∞ N . If A ∈ GLd (R ) is the matrix of u with respect to the basis e , . . . , e of N , then p 1 d g(A) is the matrix of g u with respect to the same basis. Let r be an integer ≥ 0 and Ar the class of A in GLd (Rp∞ /pr Rp∞ ). There exists an integer m such that Ar belongs to m GLd (Rpm /pr Rpm ), so that γ p (Ar ) = Ar . Hence for every n ≥ 0, we have

(II.14.15.6)

Ar γ(Ar )γ 2 (Ar ) . . . γ p

m+n

−1

(Ar ) = (Ar γ(Ar )γ 2 (Ar ) . . . γ p

m

−1

n

(Ar ))p .

d Since A ≡ id mod (p2α R p∞ ), for n sufficiently large, the product (II.14.15.6) is equal to the identity of GLd (Rp∞ /pr Rp∞ ). Consequently, for every y ∈ N 0 , the sequence of elements of N 0 (II.14.15.7)

n

n 7→ (u ◦ ϕ0 (γ))p (y) = u ◦ (γ u) ◦ · · · ◦ (γ

pn −1

n

u) ◦ ϕ0 (γ p )(y)

converges to y for the p-adic topology. It follows that the homomorphism (II.14.15.8)

0 Z → AutR c1 (N ),

n 7→ (u ◦ ϕ0 (γ))n

0 d extends to an R p∞ -representation ϕ of ∆p∞ over N , where we identify Z with a subgroup n of ∆p∞ by the injection n 7→ γ . It is clear that ϕ is a continuous representation for the p-adic topology of N 0 , and is even small. Moreover, since ϕ(γ) = u◦ϕ0 (γ), the morphism σ is ∆p∞ -equivariant (II.14.15.5). The proposition then follows from II.14.4.

b 1 ] is faithfully flat over R c1 [ 1 ] (cf. Proposition II.14.16. Suppose d = 1 and that R[ p p II.14.17(i)). Then the statements of II.14.8 are equivalent to the following statement: b 1 ]-representations M and M 0 of ∆, and every ∆-equivariant (?) For all small R[ p

b 1 ]-representation of ∆ on b 1 ]-linear morphism u : M 0 → M , the R[ surjective R[ p p the kernel of u is small.

II.15. HODGE–TATE REPRESENTATIONS

175

By II.14.8 and II.14.10, it suffices to show that (?) implies II.14.8(i). Let M be a b 1 ]-representation of ∆. By II.14.15, there exist a small R-representation b M 0 of small R[ p b 1 ]-linear morphism ∆ and a ∆-equivariant surjective R[ p

b 1 ] → M. u : M 0 ⊗ b R[ R p

(II.14.16.1)

b 1 ]-representation of ∆ on the kernel of u is small. Applying By assumption, the R[ p b M 00 of ∆ and an exact sequence II.14.15 once again, we obtain a small R-representation b 1 of ∆-equivariant R[ ]-linear morphisms p

(II.14.16.2)

v u b 1 ] −→ b 1 ] −→ M 0 ⊗ b R[ M −→ 0. M 00 ⊗ b R[ R R p p

Replacing M 00 by pn M 00 for an integer n ≥ 0, we may assume that v(M 00 ) ⊂ M 0 . c1 [ 1 ]-module with coefficients in ξ −1 Ω e1 We denote by (N, θ) the Higgs R R/OK that is the p cokernel of the morphism (II.14.16.3)

b 1 ]) → H(M 0 ⊗ R[ b 1 ]). H(v) : H(M 00 ⊗ b R[ b R R p p

b c1 -module M 0 of ∆ is Dolbeault, the Higgs R By virtue of II.14.6, the R-representation b 0 H(M ) is small and solvable, and we have a functorial and ∆-equivariant canonical Risomorphism (II.12.13.1) ∼

V(H(M 0 )) → M 0 .

(II.14.16.4)

b Consequently, by (II.13.19.2), we have a functorial R-isomorphism (II.14.16.5)

b ∼ 0 H(M 0 ) ⊗R c1 R → M .

We deduce from this an isomorphism b ∼ N ⊗R c1 R → M.

(II.14.16.6)

b 1 ] is faithfully flat over R c1 [ 1 ], N is projective of finite type over R c1 [ 1 ]. ConseSince R[ p p p b 1 ]), the Higgs R c [ 1 ]-module (N, θ) is small. Hence quently, as a quotient of H(M 0 ⊗ R[ b R

p

1 p

b 1 ]-isomorphism by virtue of II.13.23(ii) and (II.14.16.4), we have a ∆-equivariant R[ p ∼

M → V(N ), giving statement II.14.8(i). b 1 ] to be always faithfully flat over R c1 [ 1 ]. This Remarks II.14.17. (i) We expect R[ p p statement has been established by Tsuji if MX is defined by a divisor with strict normal crossings on X. (ii) Statements II.14.15 and II.14.16 are directly inspired by the approach of Faltings ([27] Theorem 3, p. 852). (iii) We expect the equivalent statements of II.14.8 to hold for every d. II.15. Hodge–Tate representations II.15.1. (II.15.1.1)

Consider the exact sequence (II.7.22.2) b →E →Ω b e1 0 → (πρ)−1 R R/OK ⊗R R(−1) → 0

176

II. LOCAL STUDY

b For every n ≥ 0, let (II.2.5) and denote also by 1 ∈ E the image of 1 ∈ (πρ)−1 R. ιn : Sn (E ) → Sn+1 (E )

(II.15.1.2)

b be the R-linear morphism defined as follows. For n = 0, ι0 is the composition of the b → (πρ)−1 R b → E . For n ≥ 1 and for all x , . . . , x ∈ E , we have canonical injections R 1 n ιn ([x1 ⊗ · · · ⊗ xn ]) = [1 ⊗ x1 ⊗ · · · ⊗ xn ].

(II.15.1.3)

The Sn (E )’s then form a direct system. We set CHT = lim Sn (E ).

(II.15.1.4)

−→

n≥0

The multiplication morphisms Sm (E )⊗ b Sn (E ) → Sm+n (E ) (m, n ≥ 0) induce a structure R b b on C . Note that C is naturally an R-representation of Γ. The resulting of R-algebra HT

HT

representation is called the Hyodo ring ([43] § 1). Remark II.15.2. Note that p is invertible in CHT , so that CHT ' CHT [ p1 ]. Indeed, we b ⊂E ⊂C . have 1 = πρ · (πρ)−1 , where (πρ)−1 ∈ (πρ)−1 R HT

II.15.3. morphism (II.15.3.1)

b For all integers i ≥ 0 and n ≥ 0, we define a Γ-equivariant R-linear n−1 ei e i+1 (−i − 1) κi,n : Sn (E ) ⊗R Ω (E ) ⊗R Ω R/OK (−i) → S R/OK

by setting κi,0 = 0, and if n ≥ 1, (II.15.3.2) κi,n ([x1 ⊗· · ·⊗xn ]⊗ω) =

X

[x1 ⊗· · ·⊗xi−1 ⊗xi+1 ⊗· · ·⊗xn ]⊗(u(xi )∧ω),

1≤i≤n

b en e1 where x1 , . . . , xn ∈ E , ω ∈ Ω R/OK (−n), and u : E → ΩR/OK ⊗R R(−1) is the canonical morphism (II.15.1.1). One immediately verifies that we have κi+1,n−1 ◦ κi,n = 0 and that the diagram κi,n

ei Sn (E ) ⊗R Ω R/OK (−i) ιn ⊗id



ei Sn+1 (E ) ⊗R Ω R/OK (−i)

e i+1 (−i − 1) / Sn−1 (E ) ⊗R Ω R/OK

κi,n+1



ιn−1 ⊗id

e i+1 (−i − 1) / Sn (E ) ⊗R Ω R/OK

b 1 ]-linear morphism is commutative. Taking the direct limit, we obtain a Γ-equivariant R[ p (II.15.3.3)

ei e i+1 κi : CHT ⊗R Ω R/OK (−i) → CHT ⊗R ΩR/OK (−i − 1).

These morphisms define a complex K ( (II.15.3.4)

Ki=

•

b 1 ]-modules by setting of R[ p

ei CHT ⊗R Ω R/OK (−i) 0

if i ≥ 0, if i < 0.

II.15. HODGE–TATE REPRESENTATIONS

177

II.15.4. Let V be a Qp -representation of Γ. For every integer i, we denote by b 1 ]-module defined by Di (V ) the R[ p Di (V ) = (V ⊗Qp CHT (i))Γ .

(II.15.4.1)

b 1 ]-linear Taking the Γ-invariants of the complex V ⊗Qp K • (i) (II.15.3.4), we obtain an R[ p i complex, denoted by D (V ), i−2 e1 e2 Di (V ) → Di−1 (V ) ⊗R Ω (V ) ⊗R Ω R/OK → D R/OK → . . . ,

(II.15.4.2)

b 1 ]-module ⊕i∈Z Di (V )(−i) with where Di (V ) is placed in degree zero. We endow the R[ p e1 the Higgs field with coefficients in Ω (−1) induced by κ0 (II.15.3.3). The Dolbeault R/OK i complex of the resulting Higgs module is ⊕i∈Z D (V )(−i) (II.2.8.4). Definition II.15.5 ([43] 2.1). We say that a continuous Qp -representation V of Γ is Hodge–Tate if the following conditions are satisfied: (i) V is a Qp -vector space of finite dimension, endowed with the p-adic topology (II.2.2). (ii) The canonical morphism ⊕i∈Z Di (V ) ⊗R[ b 1 ] CHT (−i) → V ⊗Qp CHT

(II.15.5.1)

p

is an isomorphism. We can make the following remarks. II.15.5.1. Hyodo shows in loc. cit. that for every continuous Qp -representation V of Γ of finite dimension, the morphism (II.15.5.1) is injective. b 1 ]-modules Di (V ) II.15.5.2. For every Hodge–Tate Qp -representation V of Γ, the R[ p (i ∈ Z) are locally free of finite type and are zero except for a finite number of integers i, called the Hodge–Tate weights of V (cf. [14] 4.2.7). e0 , M e ) be the smooth (A2 (S), M Proposition II.15.6. Let (X A2 (S) )-deformation of X0 ˇ gp gp (X, MX ˇ ) defined by the chart (P, γ) (II.10.13.1), (P /Zλ)lib the quotient of P /Zλ by its torsion submodule, and w : (P gp /Zλ)lib → P gp

(II.15.6.1)

a right inverse of the canonical morphism P gp → (P gp /Zλ)lib . We denote by C0 the b b Higgs–Tate R-algebra (II.10.5) and by F the Higgs–Tate R-extension of (II.10.5) asso0

b 1 ]-algebras e0 , M e ). Then there exists a canonical isomorphism of R[ ciated with (X X0 p 1 βew : C0 [ ] → CHT p

(II.15.6.2) 1

such that for every x ∈ p− p−1 F0 ⊂ C0 [ p1 ], we have βew (x) = βw (x) ∈ E ⊂ CHT ,

(II.15.6.3)

where βw is the morphism (II.10.18.2). Moreover, βew is ∆-equivariant and the diagram (II.15.6.4) ew β

C0 dC0



e1 ξ −1 C0 ⊗R Ω R/OK

/ CHT κ0

v ∼

 ew ⊗id 1 β e1 / CHT ⊗R Ω / p− p−1 e1 C0 ⊗R Ω (−1) R/OK (−1) R/OK

178

II. LOCAL STUDY

where v is the isomorphism induced by (II.9.18.1), is commutative. This follows from (II.10.3.7) and II.10.18. Proposition II.15.7. We keep the assumptions of II.15.6 and let V be a Hodge–Tate b θ) the Higgs module with coefficients in ξ −1 Ω e1 Qp -representation of Γ, (H0 (V ⊗Zp R), R/OK b b 1 associated with the R[ ]-representation V ⊗ R of ∆ by the functor (II.12.8.2) with reZp

p

b with coeffie0 , M e ), and θ0 the Higgs field on H0 (V ⊗Z R) spect to the deformation (X p X0 1 ∼ b b (II.9.18.1). e1 p−1 ξ R cients in Ω R/OK (−1) deduced from θ and the isomorphism R(1) → p b is a Dolbeault R[ b 1 ]-representation of ∆, and we have a functorial canonical Then V ⊗ R Zp

p

c1 [ 1 ]-isomorphism of Higgs R c1 [ 1 ]-modules with coefficients in Ω e1 R R/OK (−1) p p (II.15.7.1)

∼ b c1 (−i) → ⊕i∈Z Di (V ) ⊗Rb R H0 (V ⊗Zp R),

where the left-hand side is endowed with the Higgs field induced by κ0 (II.15.3.3) and the right-hand side is endowed with θ0 . b 1 ]-representation of ∆. If we endow Di (V ) b is a continuous R[ First, note that V ⊗Zp R p b b with the trivial Higgs field, and ⊕ Di (V )⊗ R(−i) with the trivial action of Γ, V ⊗ R i∈Z

Zp

and CHT with the Higgs fields induced by κ0 , then the canonical morphism (II.15.7.2)

b R

⊕i∈Z Di (V ) ⊗R[ b 1 ] CHT (−i) → V ⊗Qp CHT p

b 1 ]-modules with coefficients in is a Γ-equivariant CHT -linear isomorphism of Higgs R[ p e1 Ω (−1). In view of II.15.6, we deduce from this a ∆-equivariant C0† -linear isomorR/OK b 1 ]-modules with coefficients in Ω e1 phism of Higgs R[ R/OK (−1) p (II.15.7.3)

∼

⊕i∈Z Di (V ) ⊗Rb C0† (−i) → V ⊗Zp C0† ,

where C0† [ p1 ] is endowed with the Higgs field induced by dC † and by the isomorphism 0 1 ∼ b (II.9.18.1). Since Di (V ) is a direct summand of a free R[ b b 1 ]-module R(1) → p p−1 ξ R p c1 by virtue of II.12.6. of finite type (II.15.5.2), we have (Di (V ) ⊗Rb C0† )∆ = Di (V ) ⊗Rb R Consequently, by taking, in (II.15.7.3), the invariants under ∆, we obtain an isomorphism c1 [ 1 ]-modules with coefficients in Ω e1 of Higgs R R/OK (−1) p (II.15.7.4)

∼ b c1 (−i) → ⊕i∈Z Di (V ) ⊗Rb R (V ⊗Zp C0† )∆ = H0 (V ⊗Zp R).

Moreover, the canonical morphism (II.15.7.5)

b ⊗ C† → V ⊗ C† H0 (V ⊗Zp R) Zp 0 c1 0 R

b is a Dolbeault R[ b 1 ]-represenidentifies with the isomorphism (II.15.7.3). Hence V ⊗Zp R p tation of ∆.

CHAPTER III

Representations of the fundamental group and the torsor of deformations. Global aspects Ahmed Abbes and Michel Gros III.1. Introduction In this chapter we continue the construction and study of the p-adic Simpson correspondence started in Chapter II, following the general approach summarized in Chapter I. After fixing the notation and general conventions in III.2, we develop, in Sections III.3 to III.7, preliminaries that are useful for later on. Section III.3 contains results and complements on the notion of locally irreducible schemes. In III.4, we fix the logarithmic geometry setting of our constructions. The interlude III.5 contains a number of results on the Koszul complex used in different places in this book. Section III.6 develops the formalism of additive categories up to isogeny. Section III.7 is devoted to the study of the inverse systems of a ringed topos, in particular to the notion of adic modules and to the finiteness conditions adapted to this setting. Let K be a complete discrete valuation ring of characteristic 0, with algebraically closed residue field of characteristic p and let K be an algebraic closure of K. We denote by OK the valuation ring of K, by OK the integral closure of OK in K, and by OC the p-adic Hausdorff completion of OK . We set S = Spec(OK ) and endow it with the logarithmic structure MS defined by its closed point. We consider in this chapter a logarithmic scheme (X, MX ) smooth and saturated over (S, MS ), satisfying a local condition (III.4.7) corresponding to the assumptions made in the first part (II.6.2). We denote by X ◦ the maximal open subscheme of X where the logarithmic structure MX is trivial. The topological setting in which the p-adic Simpson correspondence takes place e associated with the canonical morphism X ◦ ⊗K K → X, is that of the Faltings topos E whose detailed study has been undertaken independently in Chapter VI. In III.8 we es special fiber of endow it with a ring B. We then introduce, in III.9, the topos E ◦ ˘ N e and the ringed topos (E e , B) p-adic formal completion of (E, e B), where our main E s constructions will take place. As sketched in the general introduction in Chapter I, the p-adic Simpson correspondence depends on a smooth (logarithmic) deformation of X ⊗OK OC over the universal p-adic infinitesimal thickening of OC of order ≤ 1, introduced by Fontaine (II.9.8). In the remainder of this introduction, we assume that there exists such a deformation, which we fix. In Section III.10, we define, for every rational number r ≥ 0, the Higgs–Tate ˘ B-algebra of thickness r, denoted by C˘(r) . These algebras form a direct system: for 0 all rational numbers r ≥ r0 ≥ 0, we have a canonical homomorphism C˘(r) → C˘(r ) . These are sheaf-theoretic analogues of the Higgs–Tate algebras introduced in the first part (II.12.1). They are naturally endowed with Higgs fields. Section III.11 contains two acyclicity results that are fundamental for what comes after. We prove (III.11.24) that ˘ up the direct limit of the Dolbeault complexes of C˘(r) , for r ∈ Q , is a resolution of B, >0

179

180

III. GLOBAL ASPECTS

to isogeny. On the other hand, if we denote by X the formal scheme p-adic completion of X ⊗OK OK , then we have a canonical morphism of ringed topos ˘ → (X , O ). esN◦ , B) > : (E zar X We prove (III.11.18) that the canonical homomorphism 1 1 OX [ ] → lim >∗ (C˘(r) )[ ] −→ p p r∈Q >0

is an isomorphism and that for every integer q ≥ 1,

1 lim Rq >∗ (C˘(r) )[ ] = 0. −→ p r∈Q >0

Section III.12 is devoted to the construction of the p-adic Simpson correspondence. We introduce the notions of Dolbeault modules and solvable Higgs bundles (III.12.11) and we show (III.12.26) that they lead to two equivalent categories. We in fact construct two explicit equivalences of categories that are quasi-inverse to each other. We also prove the compatibility of this correspondence with the relevant cohomologies on each side (III.12.34). In III.13 we establish a link with the constructions developed in the first part of Chapter II for affine schemes of a certain type, called small by Faltings. In III.14 we study the functoriality of the p-adic Simpson correspondence for étale morphisms. Section III.15 is devoted to studying the fibered category of Dolbeault modules over the restricted étale site of X. We show the local character for the étale topology on X of the Dolbeault property for modules (III.15.4). The analogous assertion for the solvability property of Higgs bundles is equivalent to the fact that the fibered category of Dolbeault modules is a stack (III.15.5). We say that a Higgs bundle is small if its Higgs field satisfies a certain divisibility condition with respect to a lattice, and that it is locally small if this condition holds locally (III.15.6). We show that every solvable Higgs bundle is locally small (III.15.8). Conversely, if the Dolbeault modules form a stack, then every locally small Higgs bundle is solvable (III.15.10). For a small affine scheme, we show unconditionally that every small Higgs bundle is solvable (III.15.9). III.2. Notation and conventions All rings in this chapter have an identity element; all ring homomorphisms map the identity element to the identity element. We mostly consider commutative rings, and rings are assumed to be commutative unless stated otherwise; in particular, when we take a ringed topos (X, A), the ring A is assumed to be commutative unless stated otherwise. III.2.1. In this chapter, p denotes a prime number, K a complete discrete valuation ring of characteristic 0, with algebraically closed residue field k of characteristic p, and K an algebraic closure of K. We denote by OK the valuation ring of K, by OK the integral closure of OK in K, by mK the maximal ideal of OK , and by v the valuation of K normalized so that v(p) = 1. We denote by OC the p-adic Hausdorff completion of OK , by C its field of fractions, and by mC its maximal ideal. Unless stated otherwise, we view OC as an adic ring, endowed with the p-adic topology ([1] 1.8.7); it is a 1-valuative ring ([1] 1.9.9). We choose a compatible system (βn )n>0 of nth roots of p in OK . For any rational number ε > 0, we set pε = (βn )εn , where n is an integer > 0 such that εn is an integer. b its p-adic Hausdorff completion. For any abelian group A, we denote by A

III.2. NOTATION AND CONVENTIONS

181

ˇ = Spec(O ). We denote by s (resp. η, We set S = Spec(OK ), S = Spec(OK ), and S C resp. η) the closed point of S (resp. generic point of S, resp. generic point of S). For any integer n ≥ 1, we set Sn = Spec(OK /pn OK ). For any S-scheme X, we set ˇ = X × S, ˇ (III.2.1.1) X = X ×S S, X and Xn = X ×S Sn . S

We endow S with the logarithmic structure MS defined by its closed point; in other words, MS = j∗ (Oη× ) ∩ OS , where j : η → S is the canonical injection (cf. II.5.9). Note that a homomorphism of monoids ι : N → Γ(S, MS ) is a chart for (S, MS ) (II.5.13) if and only if ι(1) is a uniformizer of OK . ˇ with the logarithmic structures M and M inverse images of We endow S and S ˇ S S MS (cf. II.5.10). We denote by S = Spf(OC ) the formal scheme p-adic completion of S or, equivaˇ lently, of S. III.2.2. A, the ring

Recall (II.9.3) that Fontaine associates functorially with every Z(p) -algebra RA = lim A/pA

(III.2.2.1)

←−

x7→xp

and a homomorphism θ from the ring W(RA ) of Witt vectors of RA to the p-adic b of A. We set Hausdorff completion A A2 (A) = W(RA )/ ker(θ)2

(III.2.2.2)

b the homomorphism induced by θ. and denote also by θ : A2 (A) → A III.2.3. In this chapter, we fix a sequence (pn )n≥0 of elements of OK such that p0 = p and ppn+1 = pn for every n ≥ 0. We denote by p the element of ROK (III.2.2.1) induced by the sequence (pn )n≥0 and set ξ = [p] − p ∈ W(ROK ),

(III.2.3.1)

where [p] is the multiplicative representative of p. The homomorphism θ : W(ROK ) → OC is surjective and its kernel is generated by ξ, which is not a zero divisor in W(ROK ) (II.9.5). We therefore have an exact sequence (III.2.3.2)

·ξ

θ

0 −→ OC −→ A2 (OK ) −→ OK −→ 0,

where we have denoted also by ·ξ the morphism induced by the multiplication by ξ in A2 (R). The ideal ker(θ) of A2 (OK ) has square zero. It is a free OC -module with basis ξ. We will denote it by ξOC . Note that, unlike ξ, this module does not depend on the choice of the sequence (pn )n≥0 . We denote by ξ −1 OC the dual OC -module of ξOC . For any OC -module M , we denote the OC -modules M ⊗OC (ξOC ) and M ⊗OC (ξ −1 OC ) simply by ξM and ξ −1 M , respectively. Note that, unlike ξ, these modules do not depend on the choice of the sequence (pn )n≥0 . It is therefore important to not identify them with M . We set A2 (S) = Spec(A2 (OK ))

(III.2.3.3)

which we endow with the logarithmic structure MA2 (S) defined in (II.9.8). The logarithmic scheme (A2 (S), MA2 (S) ) is then fine and saturated, and θ induces an exact closed immersion ˇ M ) → (A (S), M (III.2.3.4) i : (S, ). S

ˇ S

2

A2 (S)

182

III. GLOBAL ASPECTS

III.2.4. For any abelian category A, we denote by D(A) its derived category and by D− (A), D+ (A), and Db (A) the full subcategories of D(A) of complexes with cohomology bounded from above, from below, and from both sides, respectively. Unless mentioned otherwise, complexes in A have a differential of degree +1, the degree being written as an exponent. III.2.5. For this entire chapter, we fix a universe U with an element of infinite cardinality. We call category of U-sets, and denote by Ens, the category of sets that are in U. Unless stated otherwise, the schemes in this chapter are assumed to be elements of the universe U. We denote by Sch the category of schemes that are elements of U. III.2.6. Following the conventions of ([2] VI), we use the adjective coherent as a synonym for quasi-compact and quasi-separated. III.2.7. Let (X, A) be a ringed topos. We denote the category of A-modules of X by Mod(A) or Mod(A, X). If M is an A-module, then SA (M ) (resp. ∧A (M ), resp. ΓA (M )) is the symmetric algebra (resp. the exterior algebra, resp. the divided power algebra) of M ([45] I 4.2.2.6). For any integer n ≥ 0, we denote the homogeneous part of degree n by SnA (M ) (resp. ∧nA (M ), resp. ΓnA (M )). We will leave the ring A out of the notation when there is no risk of confusion. Forming these algebras commutes with localizing over an object of X. Definition III.2.8 ([4] I 1.3.1). Let (X, A) be a ringed topos. We say that an Amodule M of X is locally projective of finite type if the following equivalent conditions are satisfied: (i) M is of finite type and the functor H omA (M, ·) is exact; (ii) M is of finite type and every epimorphism of A-modules N → M admits locally a section; (iii) M is locally a direct summand of a free A-module of finite type. When X has enough points, and for every point x of X, the stalk of A at x is a local ring, the locally projective A-modules of finite type are locally free A-modules of finite type ([4] I 2.15.1). ´ /X (resp. X´et ) the étale site (resp. III.2.9. For any scheme X, we denote by Et ´ f/X the finite étale site of X, that is, the full subcategory topos) of X. We denote by Et ´ of Et/X made up of the finite étale schemes over X, endowed with the topology induced ´ /X , and we denote by Xf´et the finite étale topos of X, that is, the topos of by that of Et ´ f/X (cf. VI.9.2). The canonical injection Et ´ f/X → Et ´ /X induces sheaves of U-sets on Et a morphism of topos (III.2.9.1)

ρX : X´et → Xf´et .

We denote by Xzar the Zariski topos of X and by (III.2.9.2)

uX : X´et → Xzar

the canonical morphism ([2] VII 4.2.2). If F is a quasi-coherent OX -module of Xzar , we denote also by F the sheaf on X´et defined, for every étale X-scheme X 0 , by ([2] VII 2 c)) (III.2.9.3)

F (X 0 ) = Γ(X 0 , F ⊗OX OX 0 ).

This abuse of notation does not lead to any confusion. We have a canonical isomorphism (III.2.9.4)

∼

uX∗ (F ) → F.

III.2. NOTATION AND CONVENTIONS

183

We therefore view uX as a morphism from the ringed topos (X´et , OX ) to the ringed topos (Xzar , OX ). For modules, we use the notation u−1 X to denote the inverse image in the sense of abelian sheaves, and we keep the notation u∗X for the inverse image in the sense of modules. The isomorphism (III.2.9.4) induces by adjunction a morphism (III.2.9.5)

u∗X (F ) → F.

This is an isomorphism if F is an OX -module of finite presentation. Indeed, since the question is local, we may restrict to the case where there exists an exact sequence of OX m n modules OX → OX → F → 0 of Xzar . This induces an exact sequence of OX -modules m n OX → OX → F → 0 of X´et . The assertion follows in view of the right exactness of the functor u∗X . III.2.10.

Let X be a connected scheme and x a geometric point of X. We denote

by (III.2.10.1)

´ f/X → Ens ωx : Et

the fiber functor at x, which associates with each finite étale cover Y of X the set of geometric points of Y over x, by π1 (X, x) the fundamental group of X at x (that is, the automorphism group of the functor ωx ), and by Bπ1 (X,x) the classifying topos of the profinite group π1 (X, x), that is, the category of discrete U-sets endowed with a continuous left action of π1 (X, x) ([2] IV 2.7). Then ωx induces a fully faithful functor (III.2.10.2)

´ µ+ x : Etf/X → Bπ1 (X,x)

whose essential image is the full subcategory of Bπ1 (X,x) made up of finite sets ([37] ´ f/X V § 4 and § 7). Let (Xi )i∈I be an inverse system on a filtered ordered set I in Et that prorepresents ωx , normalized by the fact that the transition morphisms Xi → Xj ´ f/X is equivalent (i ≥ j) are epimorphisms and that every epimorphism Xi → X 0 of Et to a suitable epimorphism Xi → Xj (j ≤ i). Such a pro-object is essentially unique. It is called the normalized universal cover of X at x or the normalized fundamental pro-object ´ f/X at x. Note that the set I is U-small. The functor of Et (III.2.10.3)

νx : Xf´et → Bπ1 (X,x) ,

F 7→ lim F (Xi ) −→ i∈I

is an equivalence of categories that extends the functor µx+ (cf. VI.9.8). We call it the fiber functor of Xf´et at x. III.2.11. We keep the assumptions of III.2.10 and let moreover R be a ring of Xf´et . Set Rx = νx (R), which is a ring endowed with the discrete topology and a continuous action of π1 (X, x) by ring homomorphisms. We denote by Repdisc Ry (π1 (Y, y)) the category of continuous Ry -representations of π1 (Y, y) for which the topology is discrete (II.3.1). By restricting the functor νx to R-modules, we obtain an equivalence of categories that we denote also by (III.2.11.1)

∼

νx : Mod(R) → Repdisc Rx (π1 (X, x)).

An R-module M of Xf´et is of finite type (resp. locally projective of finite type (III.2.8)) if and only if the Rx -module underlying νx (M ) is of finite type (resp. projective of finite type). Indeed, the condition is necessary by virtue of VI.9.9. Let us show that it is sufficient. First suppose that the Rx -module νx (M ) is of finite type. Let N be a free Rx module with basis e1 , . . . , ed and u : N → νx (M ) an Rx -linear epimorphism. Since the desired assertion is local for Xf´et , after replacing X by a finite étale cover, if necessary, we may assume that π1 (X, x) fixes the elements u(e1 ), . . . , u(ed ) of νx (M ). Endowing N with the unique Rx -representation of π1 (X, x) that fixes e1 , . . . , ed , the homomorphism

184

III. GLOBAL ASPECTS

u is then π1 (X, x)-equivariant. Consequently, M is an R-module of finite type. Suppose, moreover, that the Rx -module νx (M ) is projective of finite type. Let v : νx (M ) → N be an Rx -linear splitting of u. After once more replacing X by an étale cover, if necessary, we may assume that π1 (X, x) fixes the elements v(u(e1 )), . . . , v(u(ed )) of N . Consequently, v is π1 (X, x)-equivariant. It follows that M is the direct summand of a free R-module of finite type, giving the assertion. III.3. Locally irreducible schemes III.3.1. Let X be a scheme whose set of irreducible components is locally finite. Recall that the following conditions are equivalent ([39] 0.2.1.6): (i) The irreducible components of X are open. (ii) The irreducible components of X are identical to its connected components. (iii) The connected components of X are irreducible. (iv) Two distinct irreducible components of X do not meet. The scheme X is then the sum of the schemes induced on its irreducible components. When these conditions are satisfied, we say that X is locally irreducible. This notion is clearly local on X; that is, if (Xi )i∈I is an open covering of X, then X is locally irreducible if and only if for every i ∈ I, the same holds for Xi . Remarks III.3.2. (i) The set of irreducible components of a locally noetherian scheme is locally finite ([39] 0.2.2.2). (ii) A normal scheme is locally irreducible if and only if the set of its irreducible components is locally finite. Indeed, every normal scheme clearly satisfies condition III.3.1(iv). (iii) A locally irreducible scheme X is étale-locally connected; that is, for every étale morphism X 0 → X, every connected component of X 0 is an open set in X 0 (VI.9.7.3). Lemma III.3.3. Let X be a normal and locally irreducible scheme and f : Y → X an étale morphism. Then Y is normal and locally irreducible. Indeed, Y is normal by virtue of ([62] VII Proposition 2). It therefore suffices to show that the set of its irreducible components is locally finite, by III.3.2(ii). Since the question is local on X and Y , we may restrict ourselves to the case where they are affine, so that f is quasi-compact and consequently quasi-finite. The desired assertion then follows from ([42] 2.3.6(iii)). Lemma III.3.4. Let X be a normal scheme and j : U → X a dense and quasi-compact open immersion. Then X is locally irreducible if and only if the same holds for U . Indeed, if X is locally irreducible, then so is U . Conversely, suppose that U is locally irreducible and let us show that the same holds for X. We may clearly restrict to the case where X is quasi-compact. Consequently, U is quasi-compact and therefore has only a finite number of irreducible components, by III.3.1(i). The same therefore holds for X because X and U have the same generic points. Since X is normal, it is locally irreducible by virtue of III.3.2(ii). Lemma III.3.5. Let A be a henselian local ring and B an A-algebra that is an integral domain and integral over A. Then B is a henselian local ring. If, moreover, A is strictly local, then the same holds for B. Let us view B as a filtered direct limit of sub-A-algebras of finite type (Bi )i∈I . Since for every i ∈ I, Bi is an integral domain and is finite over A, it is local and henselian. For all (i, j) ∈ I 2 such that i ≤ j, since the transition morphism Bi → Bj is finite, it is

III.4. ADEQUATE LOGARITHMIC SCHEMES

185

local. Consequently, B is local and henselian ([62] I § 3 Proposition 1). Suppose that A is strictly local. Since the homomorphism A → B is local, the residue field of B is an algebraic extension of that of A. It is therefore separably closed. Consequently, B is strictly local. Lemma III.3.6. Let X be a scheme, x a geometric point of X, X 0 the strict localization of X at x, Y an X-scheme, Y 0 = Y ×X X 0 , and f : Y 0 → Y the canonical projection. Then the canonical homomorphism f −1 (OY ) → OY 0 is an isomorphism of Y´e0t . We view X 0 as a cofiltered inverse limit of affine étale neighborhoods (Xi )∈I of x in X (cf. [2] VIII 4.5) and set Yi = Y ×X Xi for every i ∈ I. The scheme Y 0 is then the inverse limit of the schemes (Yi )∈I ([42] 8.2.5). Let y be a geometric point of Y 0 . For every i ∈ I, the canonical projection Yi → Y induces an isomorphism between the strict localizations of Yi and Y at the canonical images of y. On the other hand, it follows from ([39] 0.6.1.6) and ([62] I § 3 Proposition 1) that the strict localization of Y 0 at y is the inverse limit of the strict localizations of the Yi ’s at the canonical images of y. Consequently, the canonical projection Y 0 → Y induces an isomorphism between the strict localizations of Y 0 at y and Y at f (y); in other words, the canonical homomorphism OY,f (y) → OY 0 ,y is an isomorphism, giving the lemma. Lemma III.3.7. Let f : Y → X be an integral morphism of schemes, x a geometric point of X, and X 0 the strict localization of X at x. Suppose that Y is normal and that fx : Yx → x is a universal homeomorphism. Then X 0 ×X Y is normal and strictly local.

Viewing X 0 as a cofiltered inverse limit of affine étale neighborhoods (Xi )∈I of x in X ([2] VIII 4.5), X 0 ×X Y is canonically isomorphic to the inverse limit of the schemes (Xi ×X Y )i∈I ([42] 8.2.5). For every i ∈ I, Xi ×X Y is normal ([62] VII prop. 2). Since for every (i, j) ∈ I 2 such that i ≤ j, the morphism Xj → Xi is étale, every irreducible component of Xj ×X Y dominates an irreducible component of Xi ×X Y ([42] 2.3.5(ii)). It follows that X 0 ×X Y is normal ([39] 0.6.5.12(ii)). On the other hand, since X 0 ×X Y is integral over X 0 and fx is a universal homeomorphism, X 0 ×X Y is strictly local ([62] I § 3 Proposition 2). III.4. Adequate logarithmic schemes III.4.1. We will use in this chapter the conventions and notation of logarithmic geometry introduced in II.5. We denote by (S, MS ) the logarithmic trait fixed in (III.2.1). Proposition III.4.2. Let f : (X, MX ) → (S, MS ) be a smooth and saturated morphism (II.5.18) of fine logarithmic schemes, X ◦ the maximal open subscheme of X where the logarithmic structure MX is trivial, and j : X ◦ → X the canonical injection. Then: (i) The scheme X is S-flat and the scheme Xs is reduced. (ii) The logarithmic scheme (X, MX ) is regular ([51] 2.1 and [57] 2.2). (iii) The schemes X and X ×S S are normal and locally irreducible (III.3.1). (iv) The immersion j : X ◦ → X is schematically dominant, and we have canonical isomorphisms (III.4.2.1)

MXgp

(III.4.2.2)

MX

∼

× → j∗ (OX ◦ ), ∼

× → j∗ (OX ◦ ) ∩ OX .

In particular, the canonical homomorphism MX → OX is a monomorphism. (i) This follows from ([50] 4.5) and ([74] II 4.2). (ii) Since (X, MX ) is saturated by ([74] II 2.12), it is regular by virtue of ([51] 8.2); cf. also ([57] 2.3) and the proof of ([73] 1.5.1).

186

III. GLOBAL ASPECTS

(iii) The scheme X is normal by virtue of (ii) and ([51] 4.1); cf. also ([73] 1.5.1). By taking a direct limit, we deduce that the scheme X ×S S is normal (cf. the proof of II.6.3(iii)). Since X is locally noetherian, it is locally irreducible by III.3.2(ii). On the other hand, since X ×S S is S-flat, its generic points are the generic points of the scheme X ×S η, which is locally noetherian. Consequently, the set of generic points of X ×S S is locally finite, and X ×S S is locally irreducible by virtue of III.3.2(ii). (iv) This follows from (ii) and ([51] 11.6); cf. also ([57] 2.6). Lemma III.4.3. Let f : (X, MX ) → (S, MS ) be a smooth and saturated morphism of fine logarithmic schemes, (N, ι) a chart for (S, MS ), and x a geometric point of X over s. Then there exist an étale neighborhood U of x in X, a chart (P, γ) for (U, MX |U ), and a homomorphism of monoids ϑ : N → P such that the following conditions are satisfied: (i) ((P, γ), (N, ι), ϑ) is a chart for the restriction fU : (U, M |U ) → (S, MS ) of f to U (II.5.14); in other words, the diagram of homomorphisms of monoids (III.4.3.1)

γ

PO

/ Γ(U, MX ) O [ fU

ϑ ι

N

/ Γ(S, MS )

is commutative or, equivalently, the associated diagram of morphisms of logarithmic schemes (U, MX |U )

(III.4.3.2)

γa

fU

 (S, MS )

/ AP Aϑ

a

ι

 / AN

is commutative. (ii) P is toric; that is, P is fine and saturated and P gp is free over Z (II.5.1). (iii) The homomorphism ϑ is saturated (II.5.2). (iv) The homomorphism ϑgp : Z → P gp is injective, the torsion subgroup of the cokernel of ϑgp is of order prime to p, and the morphism of usual schemes U → S ×AN AP

(III.4.3.3)

deduced from (III.4.3.2) is étale. (v) There exists a subgroup A of P such that γ induces an isomorphism ∼

× . P/A → MX,x /OX,x

(III.4.3.4)

We adapt the proof of ([49] 4.1; cf. § 6). Set 1 e1 (III.4.3.5) Ω e S e = Ω(X,MX )/(S,MS ) , X/ and denote by λ ∈ Γ(X, MX ) the canonical image of ι(1) = π ∈ Γ(S, MS ) = OK − {0}. Let t1 , . . . , tr ∈ MX,x be such that d log(t1 ), . . . , d log(tr ) form a basis of the OX,x -module r+1 e1 and consider the homomorphism Ω e S,x e . Set H = N X/ ϕ : H → MX,x ,

(III.4.3.6) gp MX,x

Denote by α : homomorphism (III.4.3.7)

→

gp × MX,x /OX,x

(n1 , . . . , nr+1 ) 7→

r Y i=1

tni i · λnr+1 .

the canonical projection and by L the image of the

gp × α ◦ ϕgp : H gp → MX,x /OX,x .

III.4. ADEQUATE LOGARITHMIC SCHEMES

187

gp × /OX,x is a free Z-module of finite Since MX is fine and saturated ([74] II 2.12), MX,x type. By step 2 of ([49] page 331), we see that the cokernel of α ◦ ϕgp is annihilated gp × /OX,x by an integer invertible in OX,x . Hence there exist a Z-basis e1 , . . . , ed of MX,x and integers f1 , . . . , fd such that the ef11 , . . . , efdd form a Z-basis of L, that fi divides fi+1 for every 1 ≤ i ≤ d − 1, and that fd is invertible in OX,x . Let F1 , . . . , Fd ∈ H gp and gp ee1 , . . . , eed ∈ MX,x ei ) = ei for every 1 ≤ i ≤ d. be such that α(ϕgp (Fi )) = efi i and α(e fi × × gp Then there exists ui ∈ OX,x such that ϕ (Fi ) = ui eei . Since OX,x is fi -divisible, fi × there exists vi ∈ OX,x such that vi = ui . Replacing eei by vi eei , we may assume that gp gp × ϕgp (Fi ) = eefi i . We denote by β : MX,x /OX,x → MX,x the splitting of α defined by gp β(ei ) = eei for every 1 ≤ i ≤ d, by ρ : H → L the surjective homomorphism induced by α ◦ ϕgp , by σ : L → H gp the splitting of ρ defined by σ(efi i ) = Fi for every 1 ≤ i ≤ d, by M the kernel of ρ, and by τ : H gp → M the homomorphism that to each h ∈ H gp gp × associates h − σ(ρ(h)). We set G = M ⊕ MX,x /OX,x and

(III.4.3.8)

gp gp × φ : G = M ⊕ MX,x /OX,x → MX,x ,

(m, t) 7→ φ(m, t) = ϕgp (m) · β(t).

gp . One immediately verifies this for the We have φ ◦ (τ ⊕ α ◦ ϕgp ) = ϕgp : H gp → MX,x elements of M and for the elements (Fi )1≤i≤d . Set P = φ−1 (MX,x ). By ([50] 2.10), there exist an étale neighborhood U of x in X and a homomorphism γ : P → Γ(U, MX ) that is a chart for (U, MX |U ) and whose stalk γx : P → MX,x at x is induced by φ. The homomorphism τ ⊕α◦ϕgp : H gp → G induces a homomorphism H → P and consequently a homomorphism ϑ : N → P that makes diagram (III.4.3.1) commutative. gp × Since the homomorphism G → MX,x /OX,x induced by φ is surjective, P gp = G and P is integral. It immediately follows from the definition that there exists a subgroup A of P such that γx induces an isomorphism

(III.4.3.9)

∼

× . P/A → MX,x /OX,x

× Consequently, A = P × . Since MX,x is saturated, MX,x /OX,x is saturated and hence P is saturated (II.5.1). It follows that P is toric. The homomorphism ϑ is saturated by virtue of (III.4.3.9) and ([74] I 3.16). The homomorphism τ ⊕ α ◦ ϕgp : H gp → G is injective and its cokernel is isomorphic to that of α ◦ ϕgp . It follows that the homomorphism ϑgp : Z → P gp is injective and that the torsion subgroup of coker(ϑgp ) is of order prime to p. Step 4 of ([49] § 6 page 332) shows that the morphism of usual schemes U → S ×AN AP deduced from (III.4.3.2) is étale.

Definition III.4.4. Let f : (X, MX ) → (S, MS ) be a morphism of logarithmic schemes, (P, γ) a chart for (X, MX ), (N, ι) a chart for (S, MS ), and ϑ : N → P a homomorphism of monoids such that ((P, γ), (N, ι), ϑ) is a chart for f (II.5.14); in other words, such that the diagram of homomorphisms of monoids (III.4.4.1)

PO

γ

f[

ϑ ι

N

/ Γ(X, MX ) O / Γ(S, MS )

188

III. GLOBAL ASPECTS

is commutative or, equivalently, that the associated diagram of morphisms of logarithmic schemes (III.4.4.2)

(X, MX )

γa

/ AP

f

 (S, MS )

Aϑ

a

ι

 / AN

is commutative. We say that the chart ((P, γ), (N, ι), ϑ) is adequate if the following conditions are satisfied: (i) The monoid P is toric; that is, P is fine and saturated and P gp is free over Z. (ii) The homomorphism ϑ is saturated. (iii) The homomorphism ϑgp : Z → P gp is injective, the torsion subgroup of the cokernel ϑgp is of order prime to p, and the morphism of usual schemes (III.4.4.3)

X → S ×AN AP

deduced from (III.4.4.2) is étale. (iv) Set λ = ϑ(1) ∈ P and (III.4.4.4) (III.4.4.5)

L = HomZ (P gp , Z), H(P ) = Hom(P, N).

Note that H(P ) is a fine, saturated, and sharp monoid, and that the canonical homomorphism H(P )gp → Hom((P ] )gp , Z) is an isomorphism ([58] I 2.2.3), where P ] denotes the quotient P/P × (II.5.1). We suppose that there exist h1 , . . . , hr ∈ H(P ) that are Z-linearly independent in L, such that (III.4.4.6)

r X ai hi | (a1 , . . . , ar ) ∈ Nr }, ker(λ) ∩ H(P ) = { i=1

where we view λ as a homomorphism L → Z. Note that every morphism of logarithmic schemes that admits an adequate chart is smooth and saturated (II.5.25 and [74] Chapter II 3.5). Lemma III.4.5. Let X be an affine scheme, U a schematically dense open subscheme of X, j : U → X the canonical injection, M the multiplicative submonoid j∗ (OU× ) ∩ OX of j∗ (OU ), and λ ∈ Γ(X, M ). Denote by Γ(X, M )λ the localization of the monoid Γ(X, M ) by λ ([58] I 1.4.4) and by Xλ the open subscheme of X where the canonical image of λ in Γ(X, OX ) is invertible. Then the canonical homomorphism (III.4.5.1)

Γ(X, M )λ → Γ(Xλ , M )

is an isomorphism. Indeed, set A = Γ(X, OX ) and let us identify λ with an element of A and Γ(X, M ) (resp. Γ(Xλ , M )) with the multiplicative submonoid of A (resp. Aλ ) formed by the elements f such that f |U is a unit. One immediately verifies that for every monoid P , every homomorphism u : Γ(X, M ) → P such that u(λ) is invertible factors uniquely through Γ(Xλ , M ), giving the lemma. Proposition III.4.6. Let f : (X, MX ) → (S, MS ) be a smooth and saturated morphism of fine logarithmic schemes, x a geometric point of X over s, X 0 the strict localization of X at x, and MX 0 the logarithmic structure inverse image of MX on X 0 (II.5.10). Then the following conditions are equivalent:

III.4. ADEQUATE LOGARITHMIC SCHEMES

189

(a) There exists an étale neighborhood U of x in X such that the restriction fU : (U, MX |U ) → (S, MS ) of f to U admits an adequate chart (III.4.4). (b) There exists an étale neighborhood U of x in X such that the scheme Uη is smooth over η and that the logarithmic structure MX |Uη on Uη is defined by a divisor with strict normal crossings on Uη . (c) The scheme Xη0 is regular and the logarithmic structure MX 0 |Xη0 is defined by a divisor with strict normal crossings on Xη0 . The implication (a)⇒(b) follows from II.6.3(v); note that conditions (C1 ) and (C2 ) of II.6.2 do not play any role in the proof of II.6.3(v). We denote by ν : X 0 → X the canonical morphism and by X ◦ the maximal open subscheme of X where the logarithmic structure MX is trivial. We set X 0◦ = X 0 ×X X ◦ and denote by j : X ◦ → X and j 0 : X 0◦ → X 0 the canonical injections. The canonical homomorphism ν −1 (OX ) → OX 0 is an isomorphism of X´e0 t (III.3.6). Consequently, the canonical homomorphism (III.4.6.1)

ν −1 (MX ) → MX 0

× is also an isomorphism. We have MX = j∗ (OX ◦ ) ∩ OX by virtue of III.4.2(iv). Since ν is universally locally acyclic, it follows that

(III.4.6.2)

× MX 0 = j∗0 (OX 0◦ ) ∩ OX 0 .

The implication (b)⇒(c) follows. Let us show the implication (c)⇒(a). By III.4.3, after replacing X by an étale neighborhood of x in X, if necessary, we may assume that f admits a chart ((P, γ), (N, ι), ϑ) satisfying conditions (i), (ii), and (iii) of III.4.4 and that there exists a subgroup A of P such that γ induces an isomorphism (III.4.6.3)

∼

× P/A → MX,x /OX,x .

Set λ = ϑ(1) ∈ P and (III.4.6.4) (III.4.6.5)

L = HomZ (P gp , Z), H(P ) = Hom(P, N).

Note that H(P ) is a fine, saturated, and sharp monoid, and that the canonical homomorphism H(P )gp → Hom((P ] )gp , Z) is an isomorphism ([58] I 2.2.3), where P ] denotes the quotient P/P × . Let F be the face of P generated by λ, that is, the set of elements α ∈ P such that there exist β ∈ P and n ∈ N such that α + β = nλ ([58] I 1.4.2). Denote by P/F the quotient of P by F (cf. [58] I 1.1.5). The canonical homomorphism (III.4.6.6)

Hom(P/F, N) → ker(λ) ∩ H(P ),

where we view λ as a homomorphism L → Z, is an isomorphism. It therefore suffices to show that the monoid P/F is free of finite type. Let G be the face of MX,x generated by γ(λ). In view of (III.4.6.3), it also suffices to show that the monoid MX,x /G is free of finite type. We have a canonical isomorphism (III.4.6.7)

∼

MX,x /G → (G−1 MX,x )] .

On the other hand, we have MX,x = Γ(X 0 , MX 0 ) (III.4.6.1), and the canonical homomorphism (III.4.6.8)

G−1 Γ(X 0 , MX 0 ) → Γ(Xη0 , MX 0 )

is an isomorphism by III.4.5 and (III.4.6.2). The desired implication then follows from the fact that the monoid Γ(Xη0 , MX] 0 ) is free of finite type in view of (c).

190

III. GLOBAL ASPECTS

Definition III.4.7. We say that a morphism f : (X, MX ) → (S, MS ) of fine logarithmic schemes is adequate if it is smooth and saturated, if the morphism of underlying schemes X → S is of finite type, and if for every geometric point x of X over s, the equivalent conditions of III.4.6 are satisfied. The notion of adequate morphism of logarithmic schemes corresponds to Faltings’ notion of morphism with toroidal singularities. In his terminology, the morphism f is said to be small if it admits an adequate chart, if X is affine and connected, and if Xs is nonempty, in other words, if the conditions of II.6.2 are satisfied. Proposition III.4.8. Let f : (X, MX ) → (S, MS ) be an adequate morphism of fine logarithmic schemes, K 0 a finite extension of K, OK 0 the integral closure of OK in K 0 , and S 0 = Spec(OK 0 ). We endow S 0 with the logarithmic structure MS 0 defined by its closed point. We have a canonical morphism (S 0 , MS 0 ) → (S, MS ). Set (X 0 , MX 0 ) = (X, MX ) ×(S,MS ) (S 0 , MS 0 ),

(III.4.8.1)

where the product is taken in the category of logarithmic schemes. Then the canonical projection f 0 : (X 0 , MX 0 ) → (S 0 , MS 0 ) is adequate.

Indeed, f 0 is smooth and saturated ([74] II 2.11). Since X 0 = X ×S S 0 , it is of finite type over S 0 . On the other hand, condition III.4.6(b) is clearly satisfied at every geometric point of X 0 over the closed point of S 0 . III.5. Variations on the Koszul complex III.5.1. In this section, (X, A) denotes a ringed topos. For every morphism of A-modules u : E → F , there exists on the bigraded algebra S(E) ⊗A Λ(F ) (III.2.7) (anticommutative in the second degree; [45] I 4.3.1.1) a unique A-derivation du : S(E) ⊗A Λ(F ) → S(E) ⊗A Λ(F )

(III.5.1.1)

of bidegree (−1, 1) such that for all local sections x1 , . . . , xn of E (n ≥ 1) and y of ∧(F ), we have (III.5.1.2)

du ([x1 ⊗ · · · ⊗ xn ]⊗y)

=

n X [x1 ⊗ · · · ⊗ xi−1 ⊗ xi+1 ⊗ · · · ⊗ xn ] ⊗ (u(xi ) ∧ y), i=1

du (1⊗y)

(III.5.1.3)

=

0.

It moreover satisfies du ◦ du = 0. For every integer n ≥ 0, the homogeneous part of degree n of S(E) ⊗A Λ(F ) gives a complex (III.5.1.4)

0 → Sn (E) → Sn−1 (E) ⊗A F → · · · → E ⊗A ∧n−1 (F ) → ∧n (F ) → 0.

The algebra S(E) ⊗A Λ(F ) endowed with the derivation du depends functorially on u. If A is a Q-algebra, identifying S(E) with the divided power algebra Γ(E) of E, du identifies with the A-derivation of Γ(E) ⊗A Λ(F ) defined in ([45] I 4.3.1.2(b)). It then follows from ([45] I 4.3.1.6) that if u is an isomorphism of flat A-modules and if n > 0, the sequence (III.5.1.4) is exact. III.5.2. (III.5.2.1)

Let 0→A→E→F →0

be an exact sequence of flat A-modules. By ([45] I 4.3.1.7), it induces, for every integer n ≥ 1, an exact sequence (III.2.7) (III.5.2.2)

0 → Sn−1 (E) → Sn (E) → Sn (F ) → 0.

III.5. VARIATIONS ON THE KOSZUL COMPLEX

191

We deduce from this an exact sequence (III.5.2.3)

0 → Sn−1 (F ) → Sn (E)/Sn−2 (E) → Sn (F ) → 0.

We thus define a functor Sn from the category Ext(F, A) of extensions of F by A to the category Ext(Sn (F ), Sn−1 (F )) of extensions of Sn (F ) by Sn−1 (F ). Taking the groups of isomorphism classes of the objects of these categories, we obtain a homomorphism, which we denote also by (III.5.2.4)

Sn : Ext1A (F, A) → Ext1A (Sn (F ), Sn−1 (F )).

III.5.3. Let F be a locally projective A-module of finite type (III.2.8) and n an integer ≥ 1. The local-to-global spectral sequence of Ext ([2] V 6.1) gives an isomorphism (III.5.3.1)

∼

Ext1A (F, A) → H1 (X, H omA (F, A)).

Likewise, since the A-module Sn (F ) is locally free of finite type, we have a canonical isomorphism (III.5.3.2)

∼

Ext1A (Sn (F ), Sn−1 (F )) → H1 (X, H omA (Sn (F ), Sn−1 (F ))).

On the other hand, we denote by (III.5.3.3)

Jn : H omA (F, A) → H omA (Sn (F ), Sn−1 (F ))

the morphism that for every U ∈ Ob(X), associates with each morphism u : F |U → A|U the restriction to Sn (F )|U of the derivation du of S(F |U ) defined in (III.5.1.1). This induces a pairing (III.5.3.4)

H omA (F, A) ⊗A Sn (F ) → Sn−1 (F ).

Proposition III.5.4. For every locally projective A-module of finite type F and every integer n ≥ 1, the diagram (III.5.4.1)

Ext1A (F, A)  H1 (X, H omA (F, A))

/ Ext1 (Sn (F ), Sn−1 (F )) A

Sn

Jn

 / H1 (X, H omA (Sn (F ), Sn−1 (F )))

where Sn is the morphism (III.5.2.4), Jn is the morphism (III.5.3.3), and the vertical arrows are the isomorphisms (III.5.3.1) and (III.5.3.2), is commutative. We first recall ([2] V 3.4) that the Cartan–Leray spectral sequence associated with the coverings of the final object of X induce an isomorphism ∼ ˇ 1 (X, H omA (F, A)) → (III.5.4.2) H H1 (X, H omA (F, A)), where the source is the Čech cohomology group ([2] V (2.4.5.4)). We can explicitly describe the isomorphism ∼ ˇ1 (III.5.4.3) Ext1A (F, A) → H (X, H omA (F, A))

composed of (III.5.3.1) and the inverse of (III.5.4.2), as follows. Let (III.5.4.4)

ν

0→A→E→F →0

be an exact sequence of A-modules. Since F is locally projective of finite type, there exists a family U = (Ui )i∈I of objects of X, epimorphic over the final object, such that for every i ∈ I, there exists an (A|Ui )-linear section ϕi : F |Ui → E|Ui of ν|Ui . For every (i, j) ∈ I 2 , setting Ui,j = Ui × Uj , the difference ϕi,j = ϕi |Ui,j − ϕj |Ui,j defines a morphism from F |Ui,j to A|Ui,j . The collection (ϕi,j ) is a 1-cocycle for the covering ˇ 1 (X, H omA (F, A)) is the canonical U with coefficients in H omA (F, A) whose class in H

192

III. GLOBAL ASPECTS

image of the extension (III.5.4.4) (that is, its image by the isomorphism (III.5.4.3)). For any i ∈ I, we denote by ψin the composition Sn (F )|Ui

(III.5.4.5)

Sn (ϕi )

/ Sn (E)|Ui

/ (Sn (E)/Sn−2 (E))|Ui ,

where the second arrow is the canonical projection. This is clearly a splitting over Ui of the exact sequence (III.5.2.3) 0 → Sn−1 (F ) → Sn (E)/Sn−2 (E) → Sn (F ) → 0

(III.5.4.6)

n deduced from (III.5.4.4). For every (i, j) ∈ I 2 , the difference ψi,j = ψin |Ui,j − ψjn |Ui,j n n−1 n defines a morphism from S (F )|Ui,j to S (F )|Ui,j . The collection (ψi,j ) is a 1-cocycle n n−1 for the covering U with coefficients in H omA (S (F ), S (F )) whose class in the group ˇ 1 (X, H omA (Sn (F ), Sn−1 (F ))) is the canonical image of the extension (III.5.4.6). H For every (i, j) ∈ I 2 and all local sections x1 , . . . , xn of F |Ui,j ,

(III.5.4.7)

n ψi,j ([x1 ⊗ · · · ⊗ xn ])

= ψin ([x1 ⊗ · · · ⊗ xn ]) − ψjn ([x1 ⊗ · · · ⊗ xn ]) =

[(ϕj (x1 ) + ϕi,j (x1 )) ⊗ · · · ⊗ (ϕj (xn ) + ϕi,j (xn ))]

−[ϕj (x1 ) ⊗ · · · ⊗ ϕj (xn )] mod (Sn−2 (E)) n X = ϕi,j (xα )[x1 ⊗ · · · ⊗ xα−1 ⊗ xα+1 ⊗ · · · ⊗ xn ] α=1

= Jn (ϕi,j )([x1 ⊗ · · · ⊗ xn ]).

The proposition follows. III.5.5. (III.5.5.1)

Let 0→A→E→F →0

be an exact sequence of locally projective A-modules of finite type and n, q integers ≥ 0. By III.5.2, the exact sequence (III.5.5.1) induces an exact sequence (III.5.5.2)

0 → Sn (F ) → Sn+1 (E)/Sn−1 (E) → Sn+1 (F ) → 0.

On the other hand, the pairing (III.5.3.4) induces a pairing (III.5.5.3)

H1 (X, H omA (F, A)) ⊗A(X) Hq (X, Sn+1 (F )) → Hq+1 (X, Sn (F )).

It immediately follows from III.5.4 that the boundary map (III.5.5.4)

Hq (X, Sn+1 (F )) → Hq+1 (X, Sn (F ))

of the long exact sequence of cohomology deduced from the short exact sequence (III.5.5.2) is induced by the cup product with the class of the extension (III.5.5.1) by the pairing (III.5.5.3). Denote by (III.5.5.5)

∂ : Γ(X, F ) → H1 (X, A)

the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (III.5.5.1). It again follows from III.5.4 (more precisely, from (III.5.4.7)) that we have a commutative diagram (III.5.5.6)

Sn+1 (Γ(X, F )) α

 Sn (Γ(X, F )) ⊗A(X) H1 (X, A)

/ Γ(X, Sn+1 (F ))  / H1 (X, Sn (F ))

III.5. VARIATIONS ON THE KOSZUL COMPLEX

193

where α is the restriction to Sn+1 (Γ(X, F )) of the A(X)-derivation d∂ of S(Γ(X, F )) ⊗A(X) ∧(H1 (X, A))

defined in (III.5.1.1) with respect to the morphism ∂, the top (resp. bottom) horizontal arrow is the canonical morphism (resp. the morphism induced by the cup product) and the right vertical arrow is the morphism (III.5.5.4) for q = 0. Using the associativity of the cup product, we deduce that the diagram (III.5.5.7)

/ Hq (X, Sn+1 (F ))

Sn+1 (Γ(X, F )) ⊗A(X) Hq (X, A) α⊗id

 Sn (Γ(X, F )) ⊗A(X) H1 (X, A) ⊗A(X) Hq (X, A) id⊗∪

 Sn (Γ(X, F )) ⊗A(X) Hq+1 (X, A)

 / Hq+1 (X, Sn (F ))

where ∪ is the cup product of the A(X)-algebra ⊕i≥0 Hi (X, A), the horizontal morphisms are induced by the cup product, and the right vertical arrow is the morphism (III.5.5.4), is commutative. III.5.6. (III.5.6.1)

Let f : (X, A) → (Y, B) be a morphism of ringed topos and 0→A→E→F →0

an exact sequence of locally projective A-modules of finite type. We denote by (III.5.6.2)

u : f∗ (F ) → R1 f∗ (A)

the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (III.5.6.1). By III.5.2, the exact sequence (III.5.6.1) induces, for every integer n ≥ 0, an exact sequence

(III.5.6.3)

0 → Sn (F ) → Sn+1 (E)/Sn−1 (E) → Sn+1 (F ) → 0.

Proposition III.5.7. Under the assumptions of III.5.6, for all integers n, q ≥ 0, we have a commutative diagram (III.5.7.1)

Sn+1 (f∗ (F )) ⊗B Rq f∗ (A)

/ Rq f∗ (Sn+1 (F ))

α⊗id

 Sn (f∗ (F )) ⊗B R1 f∗ (A) ⊗B Rq f∗ (A) id⊗∪

 Sn (f∗ (F )) ⊗B Rq+1 f∗ (A)

∂

 / Rq+1 f∗ (Sn (F ))

where ∂ is the boundary map of the long exact sequence of cohomology deduced from the short exact sequence (III.5.6.1), α is the restriction to Sn+1 (f∗ (F )) of the B-derivation du of S(f∗ (F )) ⊗B ∧(R1 f∗ (A)) defined in (III.5.1.1) with respect to the morphism u (III.5.6.2), ∪ is the cup product of the B-algebra ⊕i≥0 Ri f∗ (A), and the horizontal morphisms are induced by the cup product. Indeed, Rq f∗ (Sn+1 (F )) is the sheaf on X (for the canonical topology) associated with the presheaf that with each V ∈ Ob(Y ) associates Hq (f ∗ (V ), Sn+1 (F )), and d is induced by the boundary map (III.5.7.2)

Hq (f ∗ (V ), Sn+1 (F )) → Hq+1 (f ∗ (V ), Sn (F ))

194

III. GLOBAL ASPECTS

of the long exact sequence of cohomology deduced from the short exact sequence (III.5.6.3). The proposition then follows from (III.5.5.7). III.6. Additive categories up to isogeny Definition III.6.1. Let C be an additive category. (i) A morphism u : M → N of C is called an isogeny if there exist an integer n 6= 0 and a morphism v : N → M in C such that v ◦ u = n · idM and u ◦ v = n · idN . (ii) An object M of C is said to be of finite exponent if there exists an integer n 6= 0 such that n · idM = 0. Let us complete the terminology and make a few remarks. III.6.1.1. The family of isogenies of C allows a bilateral fractional computation ([45] I 1.4.2). We call category of objects of C up to isogeny, and denote by CQ , the localized category of C with respect to isogenies. We denote by (III.6.1.2)

F : C → CQ ,

M 7→ MQ

the localization functor. One easily verifies that for all M, N ∈ Ob(C), we have (III.6.1.3)

HomCQ (MQ , NQ ) = HomC (M, N ) ⊗Z Q.

In particular, the category CQ is additive and the localization functor is additive. An object M of C is of finite exponent if and only if MQ is zero. III.6.1.4. If C is an abelian category, then the category CQ is abelian and the localization functor F : C → CQ is exact. In fact, CQ identifies canonically with the quotient category of C by the thick subcategory E of objects of finite exponent. Indeed, denote by C/E the quotient category of C by E and by T : C → C/E the canonical functor ([34] III §1). For every M ∈ Ob(E), we have F (M ) = 0. Consequently, there exists a unique functor F 0 : C/E → CQ such that F = F 0 ◦ T . On the other hand, for every M ∈ Ob(C) and every integer n 6= 0, T (n · idM ) is an isomorphism. Hence there exists a unique functor T 0 : CQ → C/E such that T = T 0 ◦ F . We immediately see that T 0 and F 0 are equivalences of categories quasi-inverse to each other. III.6.1.5. Every additive (resp. exact) functor between additive (resp. abelian) categories C → C0 extends uniquely to an additive (resp. exact) functor CQ → C0Q compatible with the localization functors. III.6.1.6. If C is an abelian category, the localization functor C → CQ transforms injective objects into injective objects ([34] III Corollary 1 to Proposition 1). In particular, if C has enough injectives, the same holds for CQ . III.6.2. Let (X, A) be a ringed topos. We denote by Mod(A) the category of A-modules of X and by ModQ (A), instead of Mod(A)Q , the category of A-modules of X up to isogeny (III.6.1.1). The tensor product of A-modules induces a bifunctor (III.6.2.1)

ModQ (A) × ModQ (A) → ModQ (A),

(M, N ) 7→ M ⊗AQ N,

making ModQ (A) into a symmetric monoidal category with AQ as identity element. The objects of ModQ (A) will also be called AQ -modules. This terminology is justified by viewing AQ as a monoid of ModQ (A). If M and N are two AQ -modules, we denote by HomAQ (M, N ) the group of morphisms from M to N in ModQ (A). The bifunctor “sheaf of morphisms” on the category of A-modules of X induces a bifunctor (III.6.2.2)

ModQ (A) × ModQ (A) → ModQ (A),

(M, N ) 7→ H omAQ (M, N ).

The bifunctors (III.6.2.1) and (III.6.2.2) inherit the same exactness properties as the bifunctors on the category Mod(A) that gave birth to them.

III.6. ADDITIVE CATEGORIES UP TO ISOGENY

III.6.3. (III.6.3.1) (III.6.3.2)

195

For any morphism of ringed topos f : (Y, B) → (X, A), we denote also by f ∗ : ModQ (A) → ModQ (B), f∗ : ModQ (B) → ModQ (A),

the functors induced by the functors inverse image under f and direct image by f , so that the first is a left adjoint of the second. The first functor is exact and the second is left exact. We denote by (III.6.3.3) (III.6.3.4)

Rf∗ : D+ (ModQ (B)) → D+ (ModQ (A)), Rq f∗ : ModQ (B) → ModQ (A), (q ∈ N),

the right derived functors of f∗ (III.6.3.2). This notation does not lead to any confusion with that of the right derived functors of the functor f∗ : Mod(B) → Mod(A), because the localization functor Mod(B) → ModQ (B) is exact and transforms injective objects into injective objects. Definition III.6.4. Let (X, A) be a ringed topos. We say that an AQ -module M is flat (or AQ -flat) if the functor N 7→ M ⊗AQ N from the category ModQ (A) to itself is exact. We can make the following remarks. III.6.4.1. Let M be an A-module. Then MQ is AQ -flat if and only if for every injective morphism of A-modules u : N → N 0 , the kernel of u ⊗ idM is of finite exponent (III.6.1.4). III.6.4.2. If M is a flat A-module, then MQ is AQ -flat. III.6.4.3. Let B be an A-algebra and M a flat AQ -module. Then M ⊗AQ BQ is BQ -flat. III.6.4.4. Let B be an A-algebra such that the functor Mod(A) → Mod(B),

N 7→ N ⊗A B

is exact and faithful. Then an AQ -module M is flat if and only if the BQ -module M ⊗AQ BQ is flat. III.6.5. Let (X, A) be a ringed topos and U an object of X. We denote by jU : X/U → X the localization of X at U . For any F ∈ Ob(X), we will denote the sheaf jU∗ (F ) also by F |U . The topos X/U will be ringed by A|U . Since the extension by zero functor jU ! : Mod(A|U ) → Mod(A) is exact and faithful ([2] IV 11.3.1), it induces an exact and faithful functor that we denote also by (III.6.5.1)

jU ! : ModQ (A|U ) → ModQ (A),

P 7→ jU ! (P ),

and that we also call the extension by zero. It is a left adjoint of the functor (III.6.3.1) (III.6.5.2)

jU∗ : ModQ (A) → ModQ (A|U ).

For any AQ -module M , we will denote the (A|U )Q -module jU∗ (M ) also by M |U . For every A-module N , we have, by definition, (N |U )Q = NQ |U . Lemma III.6.6. Let (X, A) be a ringed topos and U an object of X. Then: (i) For every flat (AQ |U )-module P , jU ! (P ) is AQ -flat. (ii) For every flat AQ -module M , jU∗ (M ) is (AQ |U )-flat. Indeed, it immediately follows from ([2] IV 12.11) that for every AQ -module M and every (AQ |U )-module P , we have a functorial canonical isomorphism (III.6.6.1)

∼

jU ! (P ⊗(AQ |U ) jU∗ (M )) → jU ! (P ) ⊗AQ M.

(i) This follows from (III.6.6.1) and the fact that the functors jU∗ and jU ! are exact.

196

III. GLOBAL ASPECTS

(ii) It follows from (III.6.6.1) and the fact that the functor jU ! is exact that the functor (III.6.6.2)

ModQ (A|U ) → ModQ (A),

P 7→ jU ! (P ⊗(AQ |U ) jU∗ (M ))

is exact. Since the functor jU ! is moreover faithful (III.6.5), it follows that the functor P 7→ P ⊗(AQ |U ) jU∗ (M ) on the category ModQ (A|U ) is exact; the statement follows. Lemma III.6.7. Let (X, A) be a ringed topos, (Ui )1≤i≤n a finite covering of the final object of X, and M, N two AQ -modules. For all 1 ≤ i, j ≤ n, we set Uij = Ui × Uj . Then: (i) The diagram of maps of sets (III.6.7.1) Y Hom(AQ |Ui ) (M |Ui , N |Ui ) ⇒ HomAQ (M, N ) → 1≤i≤n

Y 1≤i,j≤n

Hom(AQ |Ui,j ) (M |Uij , N |Uij )

is exact. (ii) The module M is zero if and only if for every 1 ≤ i ≤ n, M |Ui is zero. (iii) The module M is AQ -flat if and only if for every 1 ≤ i ≤ n, M |Ui is (AQ |Ui )flat. (i) Let M ◦ , N ◦ be two A-modules such that M = MQ◦ and N = NQ◦ , and u, v : M ◦ → N two A-linear morphisms. Suppose that for every 1 ≤ i ≤ n, we have uQ |Ui = vQ |Ui . Then there exists an integer m 6= 0 such that m · u|Ui = m · v|Ui . It follows that m · u = m · v, giving the left exactness of (III.6.7.1). On the other hand, let, for every 1 ≤ i ≤ n, ui : M ◦ |Ui → N ◦ |Ui be an (A|Ui )-linear morphism such that (ui,Q )1≤i≤n is in the kernel of the double arrow of (III.6.7.1). Then there exists an integer m0 6= 0 such that for every 1 ≤ i, j ≤ n, we have m0 ·ui |Uij = m0 ·uj |Uij . Consequently, the morphisms (m0 · ui )1≤i≤n glue to an A-linear morphism w : M ◦ → N ◦ . It is clear that (ui,Q )1≤i≤n is the canonical image of m0−1 wQ , giving the exactness in the middle of (III.6.7.1). (ii) Indeed, M is zero if and only if idM = 0. The statement therefore follows from (i). (iii) Indeed, the condition is necessary by virtue of III.6.6(ii), and it is sufficient in view of (ii) and ([2] IV 12.11). ◦

III.6.8. Let (X, A) be a ringed topos and E an A-module. A Higgs A-isogeny with coefficients in E is a quadruple (III.6.8.1)

(M, N, u : M → N, θ : M → N ⊗A E)

consisting of two A-modules M and N and two A-linear morphisms u and θ satisfying the following property: there exist an integer n 6= 0 and an A-linear morphism v : N → M such that v ◦ u = n · idM , u ◦ v = n · idN and that (M, (v ⊗ idE ) ◦ θ) and (N, θ ◦ v) are Higgs A-modules with coefficients in E (II.2.8). Note that u induces an isogeny of Higgs modules from (M, (v ⊗ idE ) ◦ θ) to (N, θ ◦ v) (III.6.1), whence the terminology. Let (M, N, u, θ), (M 0 , N 0 , u0 , θ0 ) be two Higgs A-isogenies with coefficients in E. A morphism from (M, N, u, θ) to (M 0 , N 0 , u0 , θ0 ) consists of two A-linear morphisms α : M → M 0 and β : N → N 0 such that β ◦ u = u0 ◦ α and (β ⊗ idE ) ◦ θ = θ0 ◦ α. We denote by HI(A, E) the category of Higgs A-isogenies with coefficients in E. It is an additive category. We denote by HIQ (A, E) the category of objects of HI(A, E) up to isogeny. III.6.9. Let (X, A) be a ringed topos, E an A-module, and (M, N, u, θ) a Higgs A-isogeny with coefficients in E. For every i ≥ 1, we denote by (III.6.9.1)

θi : M ⊗A ∧i E → N ⊗A ∧i+1 E

III.6. ADDITIVE CATEGORIES UP TO ISOGENY

197

the A-linear morphism defined for all local sections m of M and ω of ∧i E by θi (m ⊗ ω) = θ(m) ∧ ω. We denote by (III.6.9.2)

θi : MQ ⊗AQ (∧i E)Q → MQ ⊗AQ (∧i+1 E)Q

the morphism of ModQ (A) composed of the image of θi and the inverse of the image of u ⊗ id∧i+1 E . Let v : N → M be an A-linear morphism and n a nonzero integer such that v ◦ u = n · idM and that (M, (v ⊗ idE ) ◦ θ) is a Higgs A-module with coefficients in E. We denote by (III.6.9.3)

ϑi : M ⊗A ∧i E → M ⊗A ∧i+1 E

the A-linear morphism induced by (v ⊗ idE ) ◦ θ (II.2.8.3). The canonical image of ϑi in ModQ (A) is then equal to n · θi . It follows that θi+1 ◦ θi = 0 (cf. II.2.8.2). The Dolbeault complex of (M, N, u, θ), denoted by K• (M, N, u, θ), is the cochain complex of ModQ (A) (III.6.9.4)

θ

θ

0 1 MQ ⊗AQ EQ −→ MQ ⊗AQ (∧2 E)Q → . . . , MQ −→

where MQ is placed in degree zero and the differentials are of degree one. We thus obtain a functor from the category HI(A, E) to the category of complexes of ModQ (A). Every isogeny of HI(A, E) induces an isomorphism of the associated Dolbeault complexes. The functor “Dolbeault complex” therefore induces a functor from HIQ (A, E) to the category of complexes of ModQ (A). III.6.10. Let (X, A) be a ringed topos, B an A-algebra, and λ ∈ Γ(X, A). A λisoconnection with respect to the extension B/A (or simply λ-isoconnection when there is no risk of confusion) is a quadruple (III.6.10.1)

(M, N, u : M → N, ∇ : M → Ω1B/A ⊗B N )

where M and N are B-modules, u is an isogeny of B-modules (III.6.1), and ∇ is an A-linear morphism such that for all local sections x of B and t of M , we have (III.6.10.2)

∇(xt) = λd(x) ⊗ u(t) + x∇(t).

For every B-linear morphism v : N → M for which there exists an integer n such that u◦v = n·idN and v◦u = n·idM , the pairs (M, (id⊗v)◦∇) and (N, ∇◦v) are modules with (nλ)-connections (II.2.10), and u is a morphism from (M, (id ⊗ v) ◦ ∇) to (N, ∇ ◦ v). We say that the λ-isoconnection (M, N, u, ∇) is integrable if there exist a B-linear morphism v : N → M and an integer n 6= 0 such that u ◦ v = n · idN , v ◦ u = n · idM , and that the (nλ)-connections (id ⊗ v) ◦ ∇ on M and ∇ ◦ v on N are integrable. Let (M, N, u, ∇), (M 0 , N 0 , u0 , ∇0 ) be two λ-isoconnections. A morphism from (M, N, u, ∇) to (M 0 , N 0 , u0 , ∇0 ) consists of two B–linear morphisms α : M → M 0 and β : N → N 0 such that β ◦ u = u0 ◦ α and (id ⊗ β) ◦ ∇ = ∇0 ◦ α. III.6.11. Let f : (X 0 , A0 ) → (X, A) be a morphism of ringed topos, B an A-algebra, B an A0 -algebra, α : f ∗ (B) → B 0 a homomorphism of A0 -algebras, λ ∈ Γ(X, A), and (M, N, u, ∇) a λ-isoconnection with respect to the extension B/A. Denote by λ0 the canonical image of λ in Γ(X 0 , A0 ), by d0 : B 0 → Ω1B 0 /A0 the universal A0 -derivation of B 0 , and by 0

(III.6.11.1)

γ : f ∗ (Ω1B/A ) → Ω1B 0 /A0

the canonical α-linear morphism. We immediately see that (f ∗ (M ), f ∗ (N ), f ∗ (u), f ∗ (∇)) is a λ0 -isoconnection with respect to the extension f ∗ (B)/A0 , which is integrable if (M, N, u, ∇) is. There exists a unique A0 -linear morphism (III.6.11.2)

∇0 : B 0 ⊗f ∗ (B) f ∗ (M ) → Ω1B 0 /A0 ⊗f ∗ (B) f ∗ (N )

198

III. GLOBAL ASPECTS

such that for all local sections x0 of B 0 and t of f ∗ (M ), we have (III.6.11.3)

∇0 (x0 ⊗ t) = λ0 d0 (x0 ) ⊗ f ∗ (u)(t) + x0 (γ ⊗ idf ∗ (N ) )(f ∗ (∇)(t)).

The quadruple (B 0 ⊗f ∗ (B) f ∗ (M ), B 0 ⊗f ∗ (B) f ∗ (N ), idB 0 ⊗f ∗ (B) f ∗ (u), ∇0 ) is a λ0 -isoconnection with respect to the extension B 0 /A0 , which is integrable if (M, N, u, ∇) is. III.6.12. Let (X, A) be a ringed topos, B an A-algebra, λ ∈ Γ(X, A), and (M, N, u, ∇) an integrable λ-isoconnection with respect to the extension B/A. Suppose ∼ that there exist an A-module E and a B-isomorphism γ : E ⊗A B → Ω1B/A such that for every local section ω of E, we have d(γ(ω ⊗ 1)) = 0. Denote by θ : M → E ⊗A N the morphism induced by ∇ and γ. Then (M, N, u, θ) is a Higgs A-isogeny with coefficients in E (cf. II.2.12). Let (M 0 , N 0 , u0 , θ0 ) be a Higgs A-isogeny with coefficients in E. There exists a unique A-linear morphism (III.6.12.1)

∇0 : M ⊗A M 0 → Ω1B/A ⊗B N ⊗A N 0

such that for all local sections t of M and t0 of M 0 , we have (III.6.12.2)

∇0 (t ⊗ t0 ) = ∇(t) ⊗A u0 (t0 ) + (γ ⊗B idN ⊗A N 0 )(u(t) ⊗A θ0 (t0 )).

The quadruple (M ⊗A M 0 , N ⊗A N 0 , u ⊗ u0 , ∇0 ) is an integrable λ-isoconnection. III.6.13. Let A be an adic ring, I an ideal of definition of A, λ ∈ A, and B an adic A-algebra, that is, B is an A-algebra that is complete and separated for the (IB)-adic topology. Recall that the canonical topology on the B-module Ω1B/A is deduced from b1 that on B ([42] 0.20.4.5). We denote by Ω its Hausdorff completion and denote also B/A

by (III.6.13.1)

b1 d: B → Ω B/A

the universal continuous A-derivation of B. An adic λ-isoconnection (or I-adic λisoconnection) with respect to the extension B/A is a quadruple (III.6.13.2)

b1 ⊗ b (M, N, u : M → N, ∇ : M → Ω B/A B N )

where M and N are B-modules that are complete and separated for the (IB)-adic topologies, u is an isogeny of B-modules (III.6.1), and ∇ is an A-linear morphism such that for all x ∈ B and t ∈ M , we have (III.6.13.3)

b ∇(xt) = λd(x)⊗u(t) + x∇(t).

For every B-linear morphism v : N → M for which there exists an integer n such that b u◦v = n·idN and v◦u = n·idM , the pairs (M, (id⊗v)◦∇) and (N, ∇◦v) are modules with b ◦ ∇) to (N, ∇ ◦ v). adic (nλ)-connections (II.2.14), and u is a morphism from (M, (id⊗v) We say that the adic λ-isoconnection (M, N, u, ∇) is integrable if there exist a B-linear morphism v : N → M and an integer n 6= 0 such that u ◦ v = n · idN , v ◦ u = n · idM , b ◦ ∇ on M and ∇ ◦ v on N are integrable (cf. and that the adic (nλ)-connections (id⊗v) II.2.14). Let (M, N, u, ∇), (M 0 , N 0 , u0 , ∇0 ) be two adic λ-isoconnections with respect to the extension B/A. A morphism from (M, N, u, ∇) to (M 0 , N 0 , u0 , ∇0 ) consists of two B-linear b ◦ ∇ = ∇0 ◦ α. morphisms α : M → M 0 and β : N → N 0 such that β ◦ u = u0 ◦ α and (id⊗β)

III.6. ADDITIVE CATEGORIES UP TO ISOGENY

199

III.6.14. Let A be an adic ring, λ ∈ A, B an adic A-algebra, and (M, N, u, ∇) an integrable adic λ-isoconnection with respect to the extension B/A. Suppose that the following conditions are satisfied: (i) A admits an ideal of definition of finite type I, and if we set A1 = A/I and B1 = B ⊗A A1 , the B1 -module Ω1B1 /A1 is of finite type. (ii) There exist a free A-module of finite type E and a B-linear isomorphism ∼ b1 γ : E ⊗A B → Ω B/A such that γ(E) ⊂ d(B). b1 ⊗ b1 b Note that Ω B/A B N = ΩB/A ⊗B N = E ⊗A N . We denote by θ : M → E ⊗A N the morphism induced by ∇ and γ. It then follows from (II.2.16) that (M, N, u, θ) is a Higgs A-isogeny with coefficients in E. III.6.15. Recall that S denotes the formal scheme Spf(OC ) (III.2.1). Let X be a formal S -scheme locally of finite presentation (cf. [1] 2.3.15), J a coherent ideal of definition of X, and F an OX -module. The formal scheme X is therefore idyllic ([1] 0 2.6.13). Following ([1] 2.10.1), we call rigid closure of F and denote by Hrig (F ), the OX -module (III.6.15.1)

0 Hrig (F ) = lim H omOX (J n , F ). −→

n≥0

This notion does not depend on the ideal J . On the other hand, we set (III.6.15.2)

1 F [ ] = FQp = F ⊗Zp Qp . p

0 Since pOX is an ideal of definition for X, the canonical morphism FQp → Hrig (F ) is an isomorphism by ([1] 2.10.5). We say that F is rig-null if the canonical morphism F → FQp is zero (cf. [1] 2.10.1.4). Consider the following conditions:

(i) FQp = 0. (ii) F is rig-null. (iii) There exists an integer n ≥ 1 such that pn F = 0.

By ([1] 2.10.10), we then have (iii)⇒(i)⇔(ii). Moreover, if X is quasi-compact and if F is of finite type, the three conditions are equivalent. It follows that if X is quasi-compact and if F is of finite type, for every OX -module G , the canonical homomorphism (III.6.15.3)

HomOX (F , G ) ⊗Zp Qp → HomOX [ p1 ] (FQp , GQp )

is injective. Lemma III.6.16. Let X be a formal S -scheme of finite presentation. We denote by Modcoh (OX ) (resp. Modcoh (OX [ p1 ])) the category of coherent OX -modules (resp. OX [ p1 ]modules) and by Modcoh Q (OX ) the category of coherent OX -modules up to isogeny. Then the canonical functor (III.6.16.1)

1 Modcoh (OX ) → Modcoh (OX [ ]), p

induces an equivalence of abelian categories (III.6.16.2)

1 ∼ coh Modcoh (OX [ ]). Q (OX ) → Mod p

F 7→ FQp ,

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III. GLOBAL ASPECTS

First note that the functor (III.6.16.1) is well-defined by virtue of ([1] 2.10.24(i)) and that it induces an exact functor 1 coh (III.6.16.3) Modcoh (OX [ ]). Q (OX ) → Mod p This is essentially surjective by virtue of ([1] 2.10.24(ii)). Let us show that it is fully faithful. Let F , G be two coherent OX -modules. The canonical homomorphism (III.6.16.4)

HomOX (F , G ) ⊗Zp Qp → HomOX [ p1 ] (FQp , GQp )

is injective by (III.6.15.3). On the other hand, for every OX -linear morphism v : F → GQp , there exists an integer n ≥ 0 such that v(pn F ) is contained in the image of the canonical morphism cG : G → GQp . Since Gtor = ker(cG ) is coherent ([1] 2.10.14), there exists an integer m ≥ 0 such that pm Gtor = 0. It follows that there exists an OX -linear morphism w : F → G such that cG ◦ w = pn+m v. The homomorphism (III.6.16.4) is therefore surjective; the lemma follows. Lemma III.6.17. Let X = Spf(R) be an affine formal S -scheme of finite presentation and F a coherent OX [ p1 ]-module. Then F is a locally projective OX [ p1 ]-module of finite type (III.2.8) if and only if Γ(X, F ) is a projective R[ p1 ]-module of finite type. Recall that the ring R[ p1 ] is noetherian ([1] 1.10.2(i)). By III.6.16 and ([1] 2.7.2), there exists a coherent R-module M such that F = (M ∆ )Qp . We have Γ(X, F ) = MQp by virtue of ([1] (2.10.5.1)). Let us first suppose that F is a locally projective OX [ p1 ]-module of finite type and let us show that Γ(X, F ) is a projective R[ p1 ]-module of finite type. By ([1] 2.7.4, (2.10.5.1) and 5.1.11), there exists a faithfully flat R-algebra R0 that is topologically of finite presentation such that the R0 -module Γ(X, F ) ⊗R R0 is projective of finite type. We deduce from this by faithfully flat descent that Γ(X, F ) is a projective R[ p1 ]-module of finite type. Next, let us suppose that Γ(X, F ) is a projective R[ p1 ]-module of finite type and let us show that F is a locally projective OX [ p1 ]-module of finite type. In view of ([1] 1.10.2(iii) and (2.10.5.1)), we may assume that the R[ p1 ]-module Γ(X, F ) is free of finite type. Hence there exist an integer n ≥ 0 and an R-linear morphism Rn → M whose kernel and cokernel are torsion. Consequently, F is a free OX [ p1 ]-module of finite type. Lemma III.6.18. Let A be a ring, t ∈ A, and M an A-module of finite type. We assume that A is complete and separated for the (tA)-adic topology, that t is not a zero divisor c the Hausdorff completion in A, and that Mt is a projective At -module. We denote by M c induces an of M for the (tA)-adic topology. Then the canonical morphism M → M ∼ c isomorphism Mt → Mt . Indeed, denote by M 0 the image of the canonical morphism M → Mt . There exist an integer n ≥ 1 and an injective At -linear morphism u : Mt → Ant . Since t is not a zero divisor in A, there exists an integer i ≥ 0 such that M 0 ⊂ u−1 (t−i A)n . Consequently, M 0 is contained in a free A-module of finite type, and is therefore separated for the (tA)-adic topology. On the other hand, M and M 0 are complete for the (tA)-adic topologies by ([11] Chapter III § 2.12 Corollary 1 to Proposition 16). It follows that the canonical c is surjective and that M 0 is complete and separated for the (tA)morphism M → M adic topology. Consequently, the canonical surjective morphism M → M 0 factors into c → M 0 . Since Mt → Mt0 is an isomorphism, Mt → M ct is also an isomorphism. M →M

III.6. ADDITIVE CATEGORIES UP TO ISOGENY

201

III.6.19. Let X be a formal S -scheme of finite presentation and E an OX -module. We denote by HM(OX [ p1 ], EQp ) the category of Higgs OX [ p1 ]-modules with coefficients in EQp (II.2.8), by HMcoh (OX [ p1 ], EQp ) the full subcategory made up of the Higgs modules whose underlying OX [ p1 ]-module is coherent, by HI(OX , E ) the category of Higgs OX isogenies with coefficients in E (III.6.8), and by HIcoh (OX , E ) the full subcategory made up of the quadruples (M , N , u, θ) such that the OX -modules M and N are coherent. These are additive categories. We denote by HIQ (OX , E ) (resp. HIcoh Q (OX , E )) the catecoh gory of objects of HI(OX , E ) (resp. HI (OX , E )) up to isogeny (III.6.1.1). We have a functor (III.6.19.1) 1 HI(OX , E ) → HM(OX [ ], EQp ), (M , N , u, θ) 7→ (MQp , (u−1 Qp ⊗ idEQp ) ◦ θQp ). p Lemma III.6.20. Under the assumptions of III.6.19, let moreover (M , N , u, θ) and (M 0 , N 0 , u0 , θ0 ) be two objects of HI(OX , E ) such that M and N are OX -modules of e and (M 0 , θe0 ) their respective images by the functor (III.6.19.1). finite type, and (MQp , θ) Qp Then the canonical homomorphism (III.6.20.1) HomHI(OX ,E ) ((M , N , u, θ), (M 0 , N 0 , u0 , θ0 )) ⊗Zp Qp → e (MQ , θe0 )) HomHM(OX [ p1 ],EQp ) ((MQp , θ), p is injective. Indeed, let α : M → M 0 and β : N → N 0 be two OX -linear morphisms defining a morphism from (M , N , u, θ) to (M 0 , N 0 , u0 , θ0 ) in HI(OX , E ), whose image αQp by the homomorphism (III.6.20.1) is zero. Since (α(M ))Qp = 0 and M is of finite type over OX , there exists an integer n ≥ 0 such that pn α = 0 (III.6.15). Likewise, since βQp = 0, there exists an integer m ≥ 0 such that pm β = 0. The lemma follows.

Lemma III.6.21. Under the assumptions of III.6.19, suppose moreover that E is coherent. Then the functor 1 coh (III.6.21.1) HIcoh (OX [ ], EQp ) Q (OX , E ) → HM p induced by (III.6.19.1) is an equivalence of categories.

Let N be a coherent OX [ p1 ]-module, θ a Higgs OX [ p1 ]-field on N with coefficients in EQp . By ([1] 2.10.24(ii)), there exist a coherent OX -module N and an isomorphism ∼ u : NQp → N . We may assume that N has no p-torsion ([1] 2.10.14). By the proof of III.6.16, there exist an integer n ≥ 0 and an OX -linear morphism ϑ : N → N ⊗OX E such that the diagram /N (III.6.21.2) N ϑ

 N ⊗OX E



pn θ

/ N ⊗OX [ 1 ] EQp p

where the horizontal arrows are induced by u, is commutative. After multiplying ϑ by a power of p, if necessary, we may assume that ϑ ∧ ϑ = 0 (III.6.15.3) (that is, that ϑ is a Higgs field). Since N has no p-torsion, the morphism ϑ factors into two OX -linear morphisms (III.6.21.3)

N

νn

/ pn N

ϑn

/ N ⊗OX E ,

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III. GLOBAL ASPECTS

where νn is the isomorphism induced by the multiplication by pn on N . The composition (III.6.21.4)

pn N

ϑn

/ N ⊗OX E

/ (pn N ) ⊗O E X

νn ⊗idE

is then also a Higgs field. Denote by ιn : pn N → N the canonical injection. We have ϑ ◦ ιn = pn ϑn , so that the diagram (III.6.21.5)

pn N

ιn

/N

/N

ϑn



 N ⊗OX E

θ

/ N ⊗OX [ 1 ] EQp p

is commutative. Consequently, (pn N , N , ιn , ϑn ) is an object of HIcoh (OX , E ) whose image by the functor (III.6.19.1) is isomorphic to (N, θ). The functor (III.6.21.1) is therefore essentially surjective. We know by III.6.20 that it is faithful. Let us show that e it is full. Let (M , N , u, θ) and (M 0 , N 0 , u0 , θ0 ) be two objects of HIcoh (OX , E ), (MQp , θ) 0 0 0 e and (MQp , θ ) their respective images by the functor (III.6.21.1), and λ : MQp → MQp an OX [ 1 ]-linear morphism such that (λ ⊗ idE ) ◦ θe = θe0 ◦ λ. By the proof of III.6.16, there p

Qp

exist an integer n ≥ 0 and an OX -linear morphism α : M → M 0 such that the diagram (III.6.21.6)

M α

 M0

/ MQp pn λ

 / MQ0 p

where the horizontal arrows are the canonical morphisms, is commutative. By ([1] 2.10.22(i) and 2.10.10), there exists an integer m ≥ 0 such that pm annihilates the kernel and cokernel of u. Hence there exists an OX -linear morphism β : N → N 0 such that β ◦ u = u0 ◦ (p2m α). Denote by c : N 0 ⊗OX E → NQ0p ⊗OX [ p1 ] EQp the canonical morphism. Since (N 0 ⊗OX E )tor = ker(c) is coherent ([1] 2.10.14), there exists an integer q ≥ 0 such that pq ker(c) = 0. The relation (λ ⊗ idEQp ) ◦ θe = θe0 ◦ λ then implies that ((pq β) ⊗ idE ) ◦ θ = θ0 ◦ (pq+2m α). The functor (III.6.21.1) is therefore full. Proposition III.6.22. Let X be a formal S -scheme of finite presentation, f : X0 → X a faithfully flat morphism of finite presentation ([1] 5.1.7), X00 = X0 ×X X0 , and p1 , p2 : X00 → X0 the canonical projections. (i) Let F and G be two coherent OX [ p1 ]-modules, F 0 and G 0 their inverse images on X0 , and F 00 and G 00 their inverse images on X00 . Then the diagram of maps of sets (III.6.22.1)

HomOX [ p1 ] (F , G ) → HomOX0 [ p1 ] (F 0 , G 0 ) ⇒ HomOX00 [ p1 ] (F 00 , G 00 )

defined by the base change functors by f , p1 , and p2 is exact. (ii) For every coherent OX [ p1 ]-module F 0 , every descent datum on F 0 with respect to f is effective. This follows from the proof of ([1] 5.11.12).

III.7. INVERSE SYSTEMS OF A TOPOS

203

Remark III.6.23. Let X be a formal S -scheme of finite presentation and F , G two coherent OX [ p1 ]-modules. Denote by Xrig the rigid space associated with X ([1] 4.1.6), by rig Xrig ([1] 4.4.1), and by ad the admissible topos of X

1 %X : (Xrig ad , OXrig ) → (Xzar , OX [ ]) p

(III.6.23.1)

the canonical morphism of ringed topos ([1] (4.7.5.1)). By ([1] 2.10.24(ii), 4.7.8, and 4.7.28), the adjunction morphism F → %X∗ (%∗X (F )) is an isomorphism. Consequently, the map (III.6.23.2)

HomOXrig (%∗X (F ), %∗X (G )) → HomOX [ p1 ] (F , G ),

u 7→ %X∗ (u)

is bijective; it is the adjunction isomorphism. Statement III.6.22(i) then immediately follows from ([1] 5.11.12(i)). Statement III.6.22(ii) does not formally result from the statement of ([1] 5.11.12(ii)), but does result from the same proof. III.7. Inverse systems of a topos III.7.1. In this section, X denotes a U-topos and I a U-small category (III.2.5). We always consider X as endowed with its canonical topology, which makes it into a U-site. We endow X × I with the total topology associated with the constant fibered site X × I → I with fiber X (cf. VI.7.1 and [2] VI 7.4.1), which makes it into a U-site. Recall ([2] VI 7.4.7) that the topos of sheaves of U-sets on X × I is canonically equivalent to ◦ the category Hom(I ◦ , X) of functors from I ◦ to X, which we also denote by X I . In ◦ particular, X I is a U-topos. This last fact can be seen directly ([2] IV 1.2). We refer to ◦ VI.7.4 for the description of the rings and modules of X I . For any i ∈ Ob(I), we denote by (III.7.1.1)

αi! : X → X × I

the functor that associates with each object F of X the pair (F, i). Since this is cocontinuous ([2] VI 7.4.2), it defines a morphism of topos ([2] IV 4.7) (III.7.1.2)

◦

αi : X → X I .

◦

By ([2] VI 7.4.7), for all F ∈ Ob(X I ) and i ∈ Ob(I), we have (III.7.1.3)

αi∗ (F ) = F (i).

We denote also by (III.7.1.4)

αi! : X → X I

◦

◦

the composition of αi! (III.7.1.1) and the canonical functor X × I → X I . For all j ∈ Ob(I) and F ∈ Ob(X), we have (III.7.1.5)

αi! (F )(j) = F × (HomI (j, i))X ,

where (HomI (j, i))X is the constant sheaf on X with value HomI (j, i). By ([2] VI 7.4.3(4)), the functor αi! (III.7.1.4) is a left adjoint of αi∗ . For every morphism f : i → j of I, we have a morphism (III.7.1.6)

ρf : αi → αj ,

◦

defined at the level of the inverse images, for each F ∈ Ob(X I ), by the morphism F (j) → F (i) induced by f ([2] VI (7.4.5.2)). If f and g are two composable morphisms of I, then we have ρgf = ρg ρf .

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III. GLOBAL ASPECTS

Remarks III.7.2. (i) Let U ∈ Ob(X), i ∈ Ob(I). A family ((Un , in ) → (U, i))n∈N is covering for the total topology on X × I if and only if it is refined by a family ((Vm , i) → (U, i))m∈M , where (Vm → U )m∈M is a covering family of X ([2] VI 7.4.2(1)). ◦ (ii) It follows from (i) that a sieve R of X I is universal strict epimorphic, that is, ◦ covers the final object of X I for the canonical topology, if and only if for every i ∈ Ob(I), there exists a refinement (Ui,n )n∈Ni of the final object of X such that for every n ∈ Ni , αi! (Ui,n ) is an object of R (III.7.1.4). (iii) Suppose that fibered products are representable in I. The total topology on X × I then coincides with the covanishing topology with respect to the constant fibered site X × I → I with fiber X, when we endow I with the chaotic topology (VI.5.4). In other words, the total topology on X × I is generated by the pretopology formed by the vertical coverings (VI.5.3 and VI.5.7). ◦

Remarks III.7.3. (i) The direct U-limits (resp. the finite inverse limits) in X I can ◦ be computed term-wise; in other words, for every functor ϕ : N → X I such that the ◦ category N is U-small (resp. finite), if F is an object of X I that represents the direct (resp. inverse) limit of ϕ, then for every i ∈ Ob(I), the direct (resp. inverse) limit of αi∗ ◦ ϕ is representable by αi∗ (F ). (ii) Let ι be a final object of I. By (III.7.1.5), for every j ∈ Ob(I), αj∗ αι! is the identity functor of X. Hence by virtue of (i), αι! is left exact, and the pair (αι! , αι∗ ) forms a morphism of topos ◦

βι : X I → X.

(III.7.3.1)

The morphism βι αι : X → X is isomorphic to the identity morphism (cf. [2] VI 7.4.12). (iii) Let i be an object of I that is not final. Then there exists j ∈ Ob(I) such that HomI (j, i) is not a singleton. In particular, αj∗ αi! does not transform the final object into the final object (III.7.1.5). Consequently, αi! is not left exact, and the pair (αi! , αi∗ ) does not form a morphism of topos, unlike what was stated in ([22] line 20 page 59). Nevertheless, the isomorphism (4.4) and the spectral sequence (4.5) of loc. cit. are correct by virtue of (VI.7.7 and VI.7.8). III.7.4.

The functor λ∗ : X → X I

(III.7.4.1)

◦

that associates with each object F of X the constant functor I ◦ → X with value F is left exact by virtue of III.7.3(i). It admits as a right adjoint the functor ◦

λ∗ : X I → X

(III.7.4.2)

that associates with a functor I ◦ → X its inverse limit ([2] II 4.1(3)). The pair (λ∗ , λ∗ ) therefore defines a morphism of topos ◦

λ : X I → X.

(III.7.4.3)

III.7.5. Every morphism of U-topos f : X → Y induces a Cartesian morphism of constant fibered topos over I f × idI : X × I → Y × I.

(III.7.5.1)

By ([2] VI 7.4.10), this induces a morphism of topos ◦

(III.7.5.2) ◦

◦

f I : XI → Y I

◦

such that for every F ∈ X I , we have ([2] VI (7.4.9.2)) (III.7.5.3)

◦

(f I )∗ (F ) = f∗ ◦ F.

III.7. INVERSE SYSTEMS OF A TOPOS

205

◦

We immediately see that for every G ∈ Y I , we have ◦

(f I )∗ (G) = f ∗ ◦ G.

(III.7.5.4) ◦

◦

Denote by Rq (f I )∗ (q ∈ N) the right derived functors of the functor (f I )∗ for ◦ abelian groups. By VI.7.7, for every abelian group F of X I and every i ∈ Ob(I), we have a functorial canonical isomorphism ∼

◦

Rq (f I )∗ (F )(i) → Rq f∗ (F (i)).

(III.7.5.5)

Proposition III.7.6. Assume that fibered products are representable in I; let moreover U be an object of X, jU ! : X/U → X the canonical functor, jU : X/U → X the localization ◦ ◦ ◦ morphism of X at U , and jλ∗ (U ) : (X I )/λ∗ (U ) → X I the localization morphism of X I at λ∗ (U ). Then: (i) The total topology on X/U × I is induced by the total topology on X × I by the functor jU ! × idI . (ii) There exists a canonical equivalence of topos ◦

(III.7.6.1) ◦

∼

◦

h : (X/U )I → (X I )/λ∗ (U ) ,

◦

such that (jU )I = jλ∗ (U ) ◦ h (III.7.5.2). In particular, for every F ∈ Ob(X I ), F × λ∗ (U ) identifies with the functor I ◦ → X/U , i 7→ F (i) × U . First note that the canonical topology on X/U is induced by the canonical topology on X by the functor jU ! , and that in this case the “extension by the empty set” functor identifies with the functor jU ! ([2] IV 1.2, III 3.5 and 5.4), whence the notation. (i) The total topology on X ×I is generated by the pretopology formed by the vertical coverings by III.7.2(iii), and likewise for X/U × I. The statement therefore follows from ([2] III 3.3 and II 1.4). (ii) For all V ∈ Ob(X) and i ∈ Ob(I), we have a canonical isomorphism (III.7.1.3) ∼

λ∗ (U )(V × i) = HomX I ◦ (αi! (V ), λ∗ (U )) → HomX (V, αi∗ (λ∗ (U ))) = U (V ).

(III.7.6.2)

Consequently, the functor jU ! × idI : X/U × I → X × I factors canonically into X/U × I

(III.7.6.3)

e ∼

/ (X × I)/λ∗ (U )

0 jλ ∗ (U )

/ X ×I ,

where e is an equivalence of categories and jλ0 ∗ (U ) is the canonical functor. By (i) and ([2] III 5.4), e induces an equivalence of topos ◦

(III.7.6.4) ◦

∼

◦

h : (X/U )I → (X I )/λ∗ (U ) .

For every F ∈ Ob(X I ), we have jλ∗∗ (U ) (F ) = F ◦jλ0 ∗ (U ) by ([2] III 2.3 and 5.2(2)). Hence ◦

h∗ (jλ∗∗ (U ) (F )) = F ◦ (jU ! × idI ). Since F ◦ (jU ! × idI ) = ((jU )I )∗ (F ) (III.7.5.4), it follows ◦

that (jU )I = jλ∗ (U ) ◦ h.

III.7.7. Let J be a U-small ordered set. We denote also by J the category defined by J, that is, the category made up of the elements of J, with at most one arrow with given source and target, and for all i, j ∈ J, the set HomJ (i, j) is nonempty if and only if ◦ i ≤ j. It is often convenient to use for the objects F : J ◦ → X of X J the index notation (Fj )j∈J or even (Fj ), where Fj = F (j) for every j ∈ J. We say that an object (Fj ) ◦ of X J is strict if for all elements i ≥ j of J, the transition morphism Fi → Fj is an epimorphism. In this work, we will limit ourselves to the case where J is either the ordered set of natural numbers N, or one of the ordered subsets [n] = {0, 1, . . . , n}. Note that in each of these cases, fibered products are representable in J.

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III. GLOBAL ASPECTS

Lemma III.7.8. Let n be an integer ≥ 0 and ιn : X ×[n] → X ×N the canonical injection functor. Then: (i) The total topology on X × [n] is induced by the total topology on X × N by the functor ιn . (ii) The functor ιn is continuous and cocontinuous for the total topologies. Denote by ◦

ϕn : X [n] → X N

(III.7.8.1)

◦

◦

the associated morphism of topos. For every F = (Fi )i∈N ∈ Ob(X N ), we have ϕ∗n (F ) = (Fi )i∈[n] .

(III.7.8.2) ◦

◦

(iii) The functor ϕ∗n : : X N → X [n] admits as left adjoint the functor ◦

◦

ϕn! : X [n] → X N ,

(III.7.8.3)

◦

defined for every F = (Fi )i∈[n] ∈ Ob(X [n] ) by ϕn! (F ) = (Fi )i∈N ,

(III.7.8.4)

where for every i ≥ n + 1, Fi = ∅ is the initial object of X. (i) The total topology on X ×N is generated by the pretopology formed by the vertical coverings, by III.7.2(iii), and likewise for X × [n]. The statement therefore follows from ([2] III 3.3 and II 1.4). b (resp. (X × N)∧ , resp. (X × [n])∧ ) the category of presheaves of (ii) Denote by X U-sets on X (resp. X × N, resp. X × [n]). We then have an equivalence of categories (III.7.1.1) (III.7.8.5)

∼ b N◦ b (X × N)∧ → X = Hom(N◦ , X), ∧

F 7→ (F ◦ αi! )i∈N ,

and likewise for (X × [n]) . By composition, the functor ιn induces the functor b N◦ → X b [n]◦ , b ι∗n : X

(Fi )i∈N 7→ (Fi )i∈[n] .

This admits as right adjoint the functor b [n]◦ → X b N◦ , (III.7.8.7) b ιn∗ : X

(Fi )i∈[n] 7→ (Fi )i∈N ,

b [n]◦ → X b N◦ , b ιn! : X

(Fi )i∈[n] 7→ (Fi )i∈N ,

(III.7.8.6)

where for every i ≥ n + 1, Fi = Fn and the morphism Fi → Fi−1 is the identity of Fn . The functor b ι∗n admits as left adjoint the functor (III.7.8.8)

b It is clear that b where for every i ≥ n + 1, Fi = ∅ is the initial object of X. ι∗n transforms sheaves on X × N into sheaves on X × [n] and that b ιn∗ transforms sheaves on X × [n] into sheaves on X × N. Consequently, in is continuous and cocontinuous. The continuity also follows from (i). The formula (III.7.8.2) is a consequence of (III.7.8.6) and ([2] III 2.3). (iii) This follows from (III.7.8.8) and ([2] III 1.3). Lemma III.7.9. Let U be an object of X, jU : X/U → X the localization morphism of ◦ ◦ ◦ X at U , n ∈ N, and j(U,n) : (X N )/αn! (U ) → X N the localization morphism of X N at αn! (U ) (III.7.1.4). We then have a canonical equivalence of topos (III.7.9.1)

◦

∼

◦

h : (X/U )[n] → (X N )/αn! (U )

such that j(U,n) ◦ h is the composition (III.7.9.2)

◦

ϕn

◦

(X/U )[n] −→ (X/U )N

(jU )N

◦

◦

−→ X N ,

III.7. INVERSE SYSTEMS OF A TOPOS

207

where the first arrow is the morphism (III.7.8.1) and the second arrow is the morphism (III.7.5.2). ∼

Denote by u : αn! (U ) → λ∗ (U ) the adjoint of the canonical isomorphism U → ◦ ∗ αn (λ∗ (U )) and by α en! (U ) the image of (U, n) by the canonical functor X/U ×N → (X/U )N (III.7.1.4). By III.7.6(ii), we have a canonical equivalence of topos ◦

∼

◦

g : (X/U )N → (X N )/λ∗ (U ) .

(III.7.9.3)

In view of (III.7.6.3) and ([2] III 5.4), we have g(e αn! (U )) = u. We may therefore restrict ourselves to the case where U is the final object eX of X by virtue of III.7.6(ii) and ([2] IV 5.6). The canonical injection functor ιn : X × [n] → X × N factors canonically into (III.7.9.4)

X × [n]

ν ∼

/ (X × N)/(eX ,n)

0 jn

/ X ×N ,

where ν is an equivalence of categories and jn0 is the canonical functor. By III.7.8(i) and ([2] III 5.4), ν induces an equivalence of topos ∼

◦

◦

h : X [n] → (X N )/αn! (eX ) .

(III.7.9.5)

◦

∗ For every F = (Fi )i∈N ∈ Ob(X N ), we have j(e (F ) = F ◦ jn0 ([2] III 2.3 and 5.2(2)). X ,n) ∗ ∗ Hence h (j(eX ,n) (F )) = F ◦ ιn . Since F ◦ ιn = (Fi )i∈[n] = ϕ∗n (F ) (III.7.8.2), it follows that j(eX ,n) ◦ h = ϕn .

Proposition III.7.10 (cf. VI.7.9). Let n be an integer ≥ 0, A = (Ai )i∈[n] a ring of ◦ ◦ X [n] , and M = (Mi )i∈[n] an A-module of X [n] . For every integer q ≥ 0, we then have a canonical isomorphism ∼

◦

Hq (X [n] , M ) → Hq (X, Mn ).

(III.7.10.1)

◦

Proposition III.7.11 (cf. VI.7.10). Let A = (An )n∈N be a ring of X N , U an object of ◦ X, and M = (Mn )n∈N an A-module of X N . For every integer q ≥ 0, we then have a canonical and functorial exact sequence (III.7.11.1)

0 → R1 lim Hq−1 (U, Mn ) → Hq (λ∗ (U ), M ) → lim Hq (U, Mn ) → 0, ←−

←−

n∈N◦

n∈N◦

where we have set H−1 (U, Mn ) = 0 for every n ∈ N. Indeed, we may restrict ourselves to the case where U is the final object of X (III.7.6), in which case the proposition is a special case of VI.7.10. ◦

III.7.12. Let A = (An )n∈N be a ring of X N , and M = (Mn )n∈N and N = (Nn )n∈N ◦ two A-modules of X N . We then have a bifunctorial canonical isomorphism (III.7.12.1)

∼

M ⊗A N → (Mn ⊗An Nn )n∈N .

Indeed, for every n ∈ N, we have a canonical isomorphism ([2] IV 13.4) (III.7.12.2)

∼

αn∗ (M ⊗A N ) → αn∗ (M ) ⊗α∗n (A) αn∗ (N ).

Likewise, for every integer q ≥ 0, we have functorial canonical isomorphisms (III.2.7) (III.7.12.3) (III.7.12.4)

∼

SqA (M ) → (SqAn (Mn ))n∈N , ∼

∧qA (M ) → (∧qAn (Mn ))n∈N . ◦

Lemma III.7.13. Let A = (An )n∈N be a ring of X N and M = (Mn )n∈N an A-module ◦ of X N . Then M is A-flat if and only if for every integer n ≥ 0, Mn is An -flat.

208

III. GLOBAL ASPECTS

Indeed, the condition is necessary by virtue of (III.7.1.3) and ([2] V 1.7.1), and it is sufficient by III.7.3(i) and (III.7.12.1). ◦

Proposition III.7.14. Let A = (An )n∈N be a ring of X N and M = (Mn )n∈N a strict ◦ A-module of X N (III.7.7). Then M is of finite type over A if and only if for every integer n ≥ 0, Mn is of finite type over An . We clearly only have to show that the condition is sufficient (III.7.1.3). Suppose that for every integer n ≥ 0, the An -module Mn is of finite type. Let n be an integer ≥ 0, and U ∈ Ob(X) such that Mn |U is generated over An |U by a finite number of sections s1 , . . . , s` ∈ Γ(U, Mn ). Since M is strict, for every integer 0 ≤ i ≤ n, Mi |U is generated over Ai |U by the canonical images of the sections s1 , . . . , s` in Γ(U, Mi ). By virtue of III.7.9, we have a canonical equivalence of categories (III.7.14.1)

◦

∼

◦

h : (X/U )[n] → (X N )/αn! (U ) ◦

such that the inverse image h∗ (A|αn! (U )) is isomorphic to the ring (Ai |U )i∈[n] of (X/U )[n] ◦ and the inverse image h∗ (M |αn! (U )) is isomorphic to the module (Mi |U )i∈[n] of (X/U )[n] . By III.7.10, we have a canonical isomorphism (III.7.14.2)

◦

∼

Γ((X/U )[n] , (Mi |U )i∈[n] ) → Γ(U, Mn ).

In view of III.7.3(i), the module (Mi |U )i∈[n] is then generated over (Ai |U )i∈[n] by the sections s1 , . . . , s` ∈ Γ(U, Mn ). For every integer m ≥ 0, let (Um,j )j∈Jm be a refinement of the final object of X such that for every j ∈ Jm , Mm |Um,j is generated over Am |Um,j by a finite number of sections of Γ(Um,j , Mm ). By the above, for every m ∈ N and every j ∈ Jm , the module M |αm! (Um,j ) is generated over A|αm! (Um,j ) by a finite number of sections of Γ(Um,j , Mm ). On the other hand, the family (αm! (Um,j ))m∈N,j∈Jm is a refinement of the ◦ final object of X N by virtue of III.7.2(ii). Consequently, M is of finite type over A. ◦

Definition III.7.15. Let n be an integer ≥ 0 and A = (Ai )i∈[n] a ring of X [n] . We ◦ say that an A-module (Mi )i∈[n] of X [n] is adic if for all integers i and j such that 0 ≤ i ≤ j ≤ n, the morphism Mj ⊗Aj Ai → Mi deduced from the transition morphism Mj → Mi is an isomorphism. ◦

Definition III.7.16. Let A = (Ai )i∈N be a ring of X N . We say that an A-module ◦ (Mi )i∈N of X N is adic if for all integers i and j such that 0 ≤ i ≤ j, the morphism Mj ⊗Aj Ai → Mi deduced from the transition morphism Mj → Mi is an isomorphism. The notions of adic modules defined here are special cases of the notion of coCartesian modules introduced in VI.7.11. Lemma III.7.17. Suppose that X has enough points. Let moreover R be a ring of X ˘ the and J an ideal of R. For any integer n ≥ 0, we set Rn = R/J n+1 . We denote by R N◦ N◦ ˘ ring (Rn )n∈N of X . Then an adic R-module (Mn )n∈N of X is of finite type if and only if the R0 -module M0 is of finite type. We clearly only have to show that the condition is sufficient. Suppose that the R0 module M0 is of finite type. Let ϕ : X → Ens be a fiber functor and n an integer ≥ 1. Since we have ϕ(Rn ) = ϕ(R)/(ϕ(J))n+1 and ϕ(M0 ) = ϕ(Mn )/(ϕ(J)ϕ(Mn )), ϕ(Mn ) is of finite type over ϕ(Rn ) by ([1] 1.8.5). Consequently, Mn is of finite type over Rn , giving the lemma by virtue of III.7.14.

III.7. INVERSE SYSTEMS OF A TOPOS

209 ◦

III.7.18. Let f : Y → X be a morphism of U-topos, A = (An )n∈N a ring of X N , ◦ ◦ B = (Bn )n∈N a ring of Y N , and u : A → (f N )∗ (B) a ring homomorphism. We view ◦ ◦ ◦ f N : Y N → X N as a morphism of ringed topos (by A and B, respectively). For modules, ◦ we use the notation (f N )−1 to denote the inverse image in the sense of abelian sheaves, ◦ and we keep the notation (f N )∗ for the inverse image in the sense of modules. Giving u is equivalent to giving, for every n ∈ N, a ring homomorphism un : An → f∗ (Bn ) such that these homomorphisms are compatible with the transition morphisms of A and B ◦ (III.7.5.3). The homomorphism (f N )−1 (A) → B adjoint to u corresponds to the system of homomorphisms f ∗ (An ) → Bn adjoint to un (n ∈ N) (III.7.5.4). For any n ∈ N, we denote by (III.7.18.1)

fn : (Y, Bn ) → (X, An )

the morphism of ringed topos defined by f and un . By (III.7.5.4) and (III.7.12.1), for ◦ every A-module M = (Mn )n∈N of X N , we have a functorial canonical isomorphism (III.7.18.2)

∼

◦

(f N )∗ (M ) → (fn∗ (Mn )). ◦

Consequently, if M is adic, the same holds for (f N )∗ (M ). ◦

Lemma III.7.19. Let n be an integer ≥ 0, A = (Ai )i∈[n] a ring of X [n] , and (Mi )i∈[n] ◦ an A-module of X [n] . The A-module M is locally projective of finite type (III.2.8) if and only if M is adic and the An -module Mn is locally projective of finite type. First, suppose that M is locally projective of finite type over A. The An -module Mn is then locally projective of finite type (III.7.1.3). Let us show that M is adic. By III.7.2(ii), there exists a refinement (Uj )j∈J of the final object of X such that for every j ∈ J, M |αn! (Uj ) is a direct summand of a free A|αn! (Uj )-module of finite type. We have λ∗ (Uj ) = αn! (Uj ) (III.7.1.5). In view of III.7.6(ii), we may then restrict ourselves to the case where M is a direct summand of a free A-module of finite type. Hence there exist an ∼ integer d ≥ 1, an A-module N = (Ni )i∈[n] , and an A-linear isomorphism Ad → M ⊕ N . It follows that for all integers i and j such that 0 ≤ i ≤ j ≤ n, the canonical morphisms Mj ⊗Aj Ai → Mi and Nj ⊗Aj Ai → Ni are isomorphisms; in other words, M and N are adic. Next, suppose that M is adic and that the An -module Mn is locally projective of finite type. Let us show that the A-module M is locally projective of finite type. In view of III.7.6(ii), we may restrict ourselves to the case where Mn is a direct summand of a free An -module of finite type. Hence there exist an integer d ≥ 1, an An -module Nn , ∼ and an An -linear isomorphism Adn → Mn ⊕ Nn . This induces an A-linear isomorphism (III.7.19.1)

∼

Ad → M ⊕ (Nn ⊗An Ai )i∈[n] .

◦

Proposition III.7.20. Let A = (An )n∈N be a ring of X N and M = (Mn )n∈N an A◦ module of X N . The A-module M is locally projective of finite type (III.2.8) if and only if M is adic and for every integer n ≥ 0, the An -module Mn is locally projective of finite type. First, suppose that M is locally projective of finite type over A. For every integer n ≥ 0, the (Ai )i∈[n] -module (Mi )i∈[n] is locally projective of finite type by III.7.8(ii). It follows that M is adic and that for every integer n ≥ 0, the An -module Mn is locally projective of finite type, by virtue of III.7.19. Next, suppose that M is adic and that for every integer n ≥ 0, the An -module Mn is locally projective of finite type. Let n be an integer ≥ 0 and U ∈ Ob(X). By III.7.9, we have a canonical equivalence of topos (III.7.20.1)

◦

∼

◦

h : (X/U )[n] → (X N )/αn! (U )

210

III. GLOBAL ASPECTS ◦

such that the inverse image h∗ (A|αn! (U )) is isomorphic to the ring (Ai |U )i∈[n] of (X/U )[n] ◦ and the inverse image h∗ (M |αn! (U )) is isomorphic to the module (Mi |U )i∈[n] of (X/U )[n] . Consequently, the A|αn! (U )-module M |αn! (U ) is locally projective of finite type by virtue of III.7.19. In view of III.7.2(ii), we deduce from this that the A-module M is locally projective of finite type. III.7.21. Let Y be a connected scheme, y a geometric point of Y , Yf´et the finite étale topos of Y (III.2.9), Bπ1 (Y,y) the classifying topos of the profinite group π1 (Y, y), and νy : Yf´et → Bπ1 (Y,y)

(III.7.21.1)

the fiber functor of Yf´et at y (III.2.10). Let R be a ring of Yf´et . Set Ry = νy (R), which is a ring endowed with the discrete topology and with a continuous action of π1 (Y, y) by by the p-adic Hausdorff completion of Ry that we ring homomorphisms. We denote by R endow with the p-adic topology and with the action of π1 (Y, y) induced by that on Ry . By ([11] III § 2.11 Proposition 14 and Corollary 1; cf. also [1] 1.8.7), for every integer n ≥ 1, we have by ' Ry /pn Ry . by /pn R R

(III.7.21.2)

by is therefore continuous. The action of π1 (Y, y) on R For any topological ring A endowed with a continuous action of π1 (Y, y) by ring homomorphisms, we denote by Repcont A (π1 (Y, y)) the category of continuous A-representations of π1 (Y, y) (II.3.1). Consider the following categories: cont (a) Repdisc Ry (π1 (Y, y)): the full subcategory of RepRy (π1 (Y, y)) made up of the continuous Ry -representations of π1 (Y, y) for which the topology is discrete; (b) p-ad(Repdisc Ry (π1 (Y, y))): the category of p-adic inverse systems of the category disc RepRy (π1 (Y, y))—an inverse system (Mn )n∈N of Repdisc Ry (π1 (Y, y)) is said to be p-adic if for every integer n ≥ 0, pn+1 Mn = 0 and for all integers m ≥ n ≥ 0, the morphism Mm /pn+1 Mm → Mn induced by the transition morphism Mm → Mn is an isomorphism ([38] V 3.1.1); disc (c) p-adft (Repdisc Ry (π1 (Y, y))): the full subcategory of p-ad(RepRy (π1 (Y, y))) made up of the p-adic inverse systems of finite type—we say that a p-adic inverse system (Mn )n∈N of Repdisc Ry (π1 (Y, y)) is of finite type if for every integer n ≥ 0, Mn is an Ry -module of finite type, or, equivalently, if M0 is an Ry -module of finite type ([1] 1.8.5); (d) Repp-aft (π1 (Y, y)): the full subcategory of Repcont b (π1 (Y, y)) made up of the b R R y

y

by -representations of finite type of π1 (Y, y)—an R by -representation of p-adic R π1 (Y, y) is said to be p-adic of finite type if it is continuous for the p-adic by -module is separated of finite type. Note that topology and if the underlying R b every Ry -module of finite type is complete for the p-adic topology ([11] Chapter III § 2.11 Corollary 1 to Proposition 16). The inverse limit defines a functor (III.7.21.3)

cont p-ad(Repdisc b (π1 (Y, y)) Ry (π1 (Y, y))) → RepR y

that induces an equivalence of categories (III.7.21.4)

∼

p-aft p-adft (Repdisc (π1 (Y, y)). Ry (π1 (Y, y))) → RepR b y

III.8. FALTINGS RINGED TOPOS

211

b Indeed, for every object (Mn ) of p-adft (Repdisc Ry (π1 (Y, y))), the Ry -module c = lim Mn M ←− n∈N

c/pn+1 M c ' Mn by ([11] III § 2.11 is of finite type and for every n ∈ N, we have M Proposition 14 and Corollary 1). By restricting the functor νy (III.7.21.1) to R-modules, we obtain an equivalence of categories that we denote also by ∼

νy : Mod(R) → Repdisc Ry (π1 (Y, y)).

(III.7.21.5)

˘ the ring (R/pn+1 R)n∈N of Y N◦ and by Modad (R) ˘ (resp. Modaft (R)) ˘ We denote by R f´ et N◦ ˘ ˘ ˘ the category of adic R-modules (resp. adic R-module of finite type) of Yf´et . An RN◦ module (Mn )n∈N of Yf´et is adic (resp. adic of finite type) if and only if the inverse system (νy (Mn )) of Repdisc Ry (π1 (Y, y)) is p-adic (resp. p-adic of finite type by virtue of III.7.14). The functors (III.7.21.5) and (III.7.21.3) therefore induce a functor ˘ → Repcont Modad (R) b (π1 (Y, y)) R

(III.7.21.6)

y

and an equivalence of categories ∼

˘ → Repp-aft (π1 (Y, y)). Modaft (R) b R

(III.7.21.7)

y

III.8. Faltings ringed topos III.8.1.

For this section, we fix a commutative diagram of morphisms of schemes

(III.8.1.1)

/X Y @ @@ @@ ~ h @@ @  X j

such that X is normal and locally irreducible (III.3.1) and that j is a quasi-compact open immersion. Note that X and therefore Y are étale-locally connected by III.3.2(iii). For any X-scheme U , we set (III.8.1.2)

U = U ×X X

and UY = U ×X Y.

We denote by (III.8.1.3)

´ /X π : E → Et

the Faltings fibered U-site associated with the morphism h (VI.10.1). Recall that the objects of E are the morphisms of schemes V → U over h such that the morphism U → X is étale and that the morphism V → UY is finite étale. Let (V 0 → U 0 ) and (V → U ) be two objects of E. A morphism from (V 0 → U 0 ) to (V → U ) consists of an X-morphism U 0 → U and a Y -morphism V 0 → V such that the diagram (III.8.1.4)

V0

/ U0

 V

 /U

is commutative. The functor π is then defined for every (V → U ) ∈ Ob(E) by (III.8.1.5)

π(V → U ) = U.

212

III. GLOBAL ASPECTS

´ /X ), the fiber of E over U can be canonically identified with the For every U ∈ Ob(Et finite étale site of UY (III.2.9). We denote by (III.8.1.6)

´ f/U → E, αU ! : Et Y

V 7→ (V → U )

the canonical functor (VI.5.1.2). We denote by ´ /X F → Et

(III.8.1.7)

´ /X ) is canonithe fibered U-topos associated with π. The fiber of F over every U ∈ Ob(Et cally equivalent to the finite étale topos (UY )f´et of UY (III.2.9) and the inverse image func´ /X identifies with the functor (fY )∗ : (UY )f´et → tor for every morphism f : U 0 → U of Et f´ et 0 (UY )f´et inverse image under the morphism of topos (fY )f´et : (UY0 )f´et → (UY )f´et (VI.9.3). We denote by ´ /X )◦ F∨ → (Et

(III.8.1.8)

´ /X ) the category the fibered category obtained by associating with each U ∈ Ob(Et 0 ´ /X the functor (fY )f´et∗ : (U 0 )f´et → (UY )f´et , and with each morphism f : U → U of Et Y (UY )f´et direct image by the morphism of topos (fY )f´et . We denote by ´ /X )◦ P ∨ → (Et

(III.8.1.9)

´ /X ) the category the fibered category obtained by associating with each U ∈ Ob(Et ∧ ´ f/U ) of presheaves of U-sets on Et ´ f/U , and with each morphism f : U 0 → U of (Et Y Y ´ /X the functor Et (III.8.1.10)

´ f/U 0 )∧ → (Et ´ f/U )∧ (fY )f´et∗ : (Et Y Y

´ f/U → Et ´ f/U 0 . obtained by composing with the inverse image functor fY+ : Et Y Y b the category of presheaves of U-sets on E. We then have III.8.2. We denote by E an equivalence of categories (VI.5.2) (III.8.2.1)

b E F

→

7→

◦ ∨ ´ Hom(Et ´ /X )◦ ((Et/X ) , P )

{U 7→ F ◦ αU ! }.

From now on, we will identify F with the section {U 7→ F ◦ αU ! } that is associated with it by this equivalence. e We endow E with the covanishing topology defined by π (VI.5.3) and denote by E the topos of sheaves of U-sets on E. The resulting site and topos are called the Faltings site and topos associated with h (VI.10.1). If F is a presheaf on E, we denote by F a the associated sheaf. By VI.5.11, the functor (III.8.2.1) induces a fully faithful functor (III.8.2.2)

◦ ∨ e → Hom ´ ´ E (Et/X )◦ ((Et/X ) , F )

whose essential images consists of the sections {U 7→ FU } satisfying a gluing condition. ´ f/Y → E is continuous and left exact (VI.5.32). It III.8.3. The functor αX! : Et therefore defines a morphism of topos (VI.10.6.3) (III.8.3.1)

e → Yf´et . β: E

The functor (III.8.3.2)

´ /X → E, σ + : Et

U 7→ (UY → U )

III.8. FALTINGS RINGED TOPOS

213

is continuous and left exact (VI.5.32). It therefore defines a morphism of topos (VI.10.6.4) e → X´et . σ: E

(III.8.3.3) On the other hand, the functor

´ /Y , Ψ+ : E → Et

(III.8.3.4)

(V → U ) 7→ V

is continuous and left exact (VI.10.7). It therefore defines a morphism of topos e Ψ : Y´et → E.

(III.8.3.5)

We have canonical morphisms ((VI.10.8.3) and (VI.10.8.4)) σ∗ β∗

(III.8.3.6) (III.8.3.7)

→ Ψ∗ h´e∗t , → Ψ∗ ρ∗Y ,

where ρY : Y´et → Yf´et is the canonical morphism (III.2.9.1). If X is quasi-separated and Y is coherent, (III.8.3.7) is an isomorphism by virtue of VI.10.9(iii). Remark III.8.4. It immediately follows from VI.5.10 that {U 7→ UY } is a sheaf on E. e and we have canonical isomorphisms It is therefore a final object of E ∼

III.8.5.

∼

σ ∗ (X) → {U 7→ UY } ← β ∗ (Y ).

(III.8.4.1)

Consider a commutative diagram Y0

(III.8.5.1)

 Y

h0

h

/ X0  /X

e 0 the and denote by E 0 the Faltings topos associated with the morphism h0 and by E 0 topos of sheaves of U-sets on E (VI.10.1). Then we have a continuous left exact functor (VI.10.12) (III.8.5.2)

Φ+ : E → E 0 ,

(V → U ) 7→ (V ×Y Y 0 → U ×X X 0 ).

It defines a morphism of topos e 0 → E. e Φ: E

(III.8.5.3)

III.8.6. We denote by D the covanishing site associated with the functor ´ /X → Et ´ /Y induced by h (VI.4.1). The topos of sheaves of U-sets on D is the h+ : Et ←

covanishing topos X´et ×X´et Y´et of the morphism h´et : Y´et → X´et induced by h (VI.3.12 and VI.4.10). Every object of E is naturally an object of D. We thus define a fully faithful functor ρ+ : E → D.

(III.8.6.1)

It is continuous and left exact (VI.10.15). It therefore defines a morphism of topos ←

(III.8.6.2) ←

e ρ : X´et ×X´et Y´et → E.

Giving a point X´et ×X´et Y´et is equivalent to giving a pair of geometric points x of X and y of Y and a specialization arrow u from h(y) to x, that is, an X-morphism u : y → X(x) , where X(x) denotes the strict localization of X at x (VI.10.18). Such a point will be denoted by (y x) or by (u : y x). We denote by ρ(y x) its image by e ρ, which is therefore a point of E.

214

III. GLOBAL ASPECTS

If X and Y are coherent, then as (y

x) goes through the family of points of e x) is Y´et , the family of fiber functors of E associated with the points ρ(y conservative by virtue of VI.10.21.

← X´et ×X´et

III.8.7. Suppose that X is strictly local, with closed point x. We denote by Escoh the full subcategory of E made up of the (V → U )’s such that the morphism U → X is separated and coherent, which we endow with the topology induced by that on E. The canonical injection functor Escoh → E then induces by restriction an equivalence of e and the topos of sheaves of U-sets on Escoh (VI.10.4). categories between E For any étale, separated, and coherent morphism U → X, we denote by U f the disjoint union of the strict localizations of U at the points of Ux ; it is an open and closed subscheme of U that is finite over X ([42] 18.5.11). By VI.10.23, the functor ´ f/Y , (V → U ) 7→ V ×U U f (III.8.7.1) θ+ : Escoh → Et is continuous and left exact. It therefore defines a morphism of topos e (III.8.7.2) θ : Yf´et → E. We have a canonical isomorphism (VI.10.24.3) ∼

βθ → idYf´et .

(III.8.7.3)

We deduce from this a base change morphism (VI.10.24.4) β∗ → θ∗ .

(III.8.7.4)

This is an isomorphism by virtue of VI.10.27; in particular, the functor β∗ is exact. III.8.8. Let x be a geometric point of X, X 0 the strict localization of X at x, Y = Y ×X X 0 , and h0 : Y 0 → X 0 the canonical projection. We denote by E 0 the Faltings e 0 the topos of sheaves of U-sets on E 0 , by site associated with the morphism h0 , by E e 0 → Yf´0et (III.8.8.1) β0 : E 0

the canonical morphism (III.8.3.1), by

e0 → E e Φ: E

(III.8.8.2)

the functoriality morphism (III.8.5.3), and by

e0 θ : Yf´0et → E

(III.8.8.3)

the morphism (III.8.7.2). We denote by

e → Yf´0et ϕx : E

(III.8.8.4)

the composed functor θ∗ ◦ Φ∗ . If X and Y are coherent, then as x goes through the set of geometric points of X, the family of functors ϕx is conservative by virtue of VI.10.32. We denote by Vx the category of x-pointed étale X-schemes ([2] VIII 3.9) or, equivalently, the category of neighborhoods of the point of X´et associated with x in the site ´ /X ([2] IV 6.8.2). For any object (U, p : x → U ) of Vx , we denote also by p : X 0 → U Et the morphism deduced from p ([2] VIII 7.3) and by pY : Y 0 → UY

(III.8.8.5)

e we have a its base change by h. By VI.10.37, for every sheaf F = {U 7→ FU } of E, functorial canonical isomorphism (III.8.8.6)

lim −→

(U,p)∈V◦ x

∼

(pY )∗f´et (FU ) → ϕx (F ).

III.8. FALTINGS RINGED TOPOS

III.8.9. by

215

We denote by B the presheaf on E defined for every (V → U ) ∈ Ob(E), B((V → U )) = Γ(U , OU )

(III.8.9.1)

and by B a the associated sheaf. By VI.5.34(ii) and with the notation conventions of III.2.9, we have a canonical isomorphism ∼

σ ∗ (~∗ (OX )) → B a .

(III.8.9.2)

V

III.8.10. For any (V → U ) ∈ Ob(E), we denote by U the integral closure of U = U ×X X in V . For every morphism (V 0 → U 0 ) → (V → U ) of E, we have a

canonical morphism U

0V 0

→U

(III.8.10.1)

V0

V

that fits into a commutative diagram /

 V

U /

/

0V 0

 U

V

0

/ U0

 /U

 /U

U

We denote by B the presheaf on E defined for every (V → U ) ∈ Ob(E), by (III.8.10.2)

V

B((V → U )) = Γ(U , OU V ).

´ /X ), we set (III.8.1.6) For any U ∈ Ob(Et B U = B ◦ αU ! .

(III.8.10.3)

Remarks III.8.11. Let (V → U ) be an object of E. Then: (i) Since V is integral over UY , the canonical morphism V → UY ×U U

V

is an

V

isomorphism. In particular, the canonical morphism V → U is a schematically dominant open immersion. V (ii) The scheme U is normal and locally irreducible (III.3.1). Indeed, U and V are normal and locally irreducible by III.3.3. Let U0 be an open subscheme of V U having only finitely many irreducible components. Then U0 ×U U is the finite sum of the integral closures of U0 in the generic points of V that lie over U0 , each of which is an integral normal scheme by virtue of ([40] 6.3.7). (iii) For every étale U -scheme U 0 , setting V 0 = V ×U U 0 , the canonical morphism (III.8.10.1) U

(III.8.11.1)

0V 0

→U

V

V

×U U 0

is an isomorphism. Indeed, U ×U U 0 is normal and locally irreducible by (ii) V and III.3.3, and the canonical morphism V 0 → U ×U U 0 is a schematically dominant open immersion by virtue of (i) and ([42] 11.10.5). The assertion follows because U

0V 0

identifies with the integral closure of U

V

×U U 0 in V 0 .

Lemma III.8.12. Let (V → U ) be an object of E, f : W → V a torsor for the étale W V topology on V under a finite constant group G. We denote by f : U → U the morphism W induced by f . Then the natural action of G on U is admissible ([37] V Definition 1.7) V W and (U , f ) is a quotient of U by G; in other words, the canonical morphism (III.8.12.1)

OU V → f ∗ (OU W )G

216

III. GLOBAL ASPECTS

is an isomorphism. In particular, the canonical morphism (III.8.12.2)

V

Γ(U , OU V ) → Γ(U

W

, OU W )G

is an isomorphism. W

Indeed, the action of G on U is admissible by virtue of ([37] V Corollary 1.8). W Denote by Z the quotient of U by G. By III.8.11(i) and ([37] Proposition 1.9), f ∼ induces an isomorphism V → UY ×U Z, and the morphism V → Z is a schematically dominant open immersion. Since Z is integral over U , we deduce from this that the V morphism Z → U induced by f is an isomorphism, giving the isomorphism (III.8.12.1). V The isomorphism (III.8.12.2) follows because the functor Γ(U , −) on the Zariski topos V of U is left exact. ´ /X and y a geometric point of UY . Since the III.8.13. Let U be an object of Et scheme U is locally irreducible (III.3.3), it is the sum of the schemes induced on its irre? ducible components. We denote by U the irreducible component of U (or, equivalently, its connected component) containing y. Likewise, UY is the sum of the induced schemes ? on its irreducible components. We set V = U ×X Y , which is the irreducible component of UY containing y. We denote by Bπ1 (V,y) the classifying topos of the profinite group π1 (V, y), by (Vi )i∈I the normalized universal cover of V at y (III.2.10), and by (III.8.13.1)

∼

νy : Vf´et → Bπ1 (V,y) ,

F 7→ lim F (Vi ) −→ i∈I

the fiber functor of Vf´et at y (III.2.10.3). For each i ∈ I, (Vi → U ) is naturally an object Vi of E. We can therefore consider the filtered inverse system of schemes (U )i∈I . We set (III.8.13.2)

y

Vi

RU = lim Γ(U , OU Vi ), −→ i∈I

which is a continuous discrete representation of π1 (V, y) by virtue of III.8.12; in other words, it is an object of Bπ1 (V,y) . y

Remark III.8.14. We keep the assumptions of III.8.13 and denote by U the inverse Vi limit in the category of U -schemes of the filtered inverse system (U )i∈I , which exists by virtue of ([42] 8.2.3). By ([2] VI 5.3), if U is coherent, we have a canonical isomorphism (III.8.14.1)

y

y

∼

Γ(U , OU y ) → RU .

Lemma III.8.15. Under the assumptions of III.8.13, B U (III.8.10.3) is a sheaf for the ´ f/U , and we have a canonical isomorphism étale topology of Et Y (III.8.15.1)

∼

y

νy (B U |V ) → RU .

´ f/V is a sheaf. It suffices to show that the restriction of the presheaf B U to the site Et The isomorphism (III.8.15.1) will then immediately follow from the definition. Denote by ´ f/V → Bπ (V,y) (III.8.15.2) µ+ : Et y

1

´ f/V ), we have a functorial the fiber functor at y (III.2.10.2). For every W ∈ Ob(Et isomorphism (III.8.15.3)

∼

µy+ (W ) → lim HomV (Vi , W ). −→ i∈I

III.8. FALTINGS RINGED TOPOS

217

We deduce from this a morphism that is functorial in W : Γ(U

(III.8.15.4)

W

y

, OU W ) → HomBπ1 (V,y) (µ+ y (W ), RU ).

This is an isomorphism by virtue of III.8.12, giving the desired statement. Proposition III.8.16. The presheaf B on E is a sheaf for the covanishing topology. ´ /X . For any (i, j) ∈ I 2 , Let (V → U ) ∈ Ob(E) and (Ui → U )i∈I be a covering of Et Vi

set Vi = V ×U Ui , Uij = Ui ×U Uj , and Vij = Uij ×U V . We have U i ' U

Vij U ij

V

V

×U Ui and V

' U ×U Uij by III.8.11(iii). Viewing OU V as a sheaf on the étale topos of U , the Vi

V

étale covering (U i → U )i∈I induces an exact sequence of maps of sets Y Y (III.8.16.1) B((V → U )) → B((Vi → Ui )) ⇒ B((Vij → Uij )). (i,j)∈I 2

i∈I

The proposition follows in view of III.8.15 and VI.5.10. e associated with X. By III.8.17. From now on, we will say that B is the ring of E III.8.9, the canonical homomorphism B → B induces a homomorphism σ ∗ (~∗ (OX )) → e → X´et (III.8.3.3) as a morB (cf. III.2.9). Unless mentioned otherwise, we view σ : E phism of ringed topos (by B and ~∗ (OX ), respectively). For modules, we use the notation σ −1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation σ ∗ for the inverse image in the sense of modules. Remark III.8.18. The presheaf B is not in general a sheaf for the topology on E originally defined by Faltings in ([26] page 214). Indeed, suppose that ~ is integral, that j is not closed, and that there exists an affine open immersion Z → X such that Y = Z ×X X, where j : Y → X is the canonical projection. Consider two affine schemes ´ /X and a surjective morphism U 0 → U such that U 0 → UY is finite and U and U 0 of Et Y 0 U → U is not integral. We could, for example, take for U an affine open subscheme of X such that the open immersion jU : UY → U is not closed and for U 0 the sum of U and UZ . Set V = UY0 . It is clear that (V → U ) is an object of E and that (V → U 0 ) → (V → U ) is a covering for the topology on E defined by Faltings in ([26] page 214). But the sequence (III.8.18.1)

B((V → U )) → B((V → U 0 )) ⇒ B((V → U 0 ×U U 0 ))

is not exact. Indeed, let us suppose that it is. The diagram (III.8.18.2)

∆

V → U 0 → U 0 ×U U 0 ⇒ U 0 ,

where ∆ is the diagonal morphism and the double arrow represents the two canonical projections, is commutative. Consequently, the double arrow in (III.8.18.1) is made up of two copies of the same morphism. Hence the canonical map (III.8.18.3)

B((V → U )) → B((V → U 0 )) 0

0

is an isomorphism. Since Γ(U , OU 0 ) ⊂ B((V → U 0 )), it follows that Γ(U , OU 0 ) is 0 0 integral over Γ(U , OU ), and consequently that U → U is integral because U and U are affine, which contradicts the assumptions. Proposition III.8.19. Suppose that X and X are strictly local and let y be a geometric ← point of Y . Denote by x the closed point of X and by (y x) the point of X´et ×X´et Y´et defined by the unique specialization arrow from h(y) to x (III.8.6). Then: e (III.8.6.2) is a normal (i) The stalk B ρ(y x) of B at the point ρ(y x) of E strictly local ring.

218

III. GLOBAL ASPECTS

(ii) We have a canonical isomorphism ∼

(~∗ (OX ))x → Γ(X, OX ).

(III.8.19.1) (iii) The homomorphism

(~∗ (OX ))x → B ρ(y

(III.8.19.2)

induced by the canonical homomorphism σ tive and local.

x) −1

(~∗ (OX )) → B (III.8.17) is injec-

(i) First note that Y is integral. Let (Vi )i∈I be the normalized universal cover of Y at the point y (III.2.10). By VI.10.36, we have a canonical isomorphism ∼

lim B((Vi → X)) → B ρ(y

(III.8.19.3)

x) .

−→ i∈I

Vi

For every i ∈ I, the scheme X is normal, integral, and integral over X (III.8.11). It is therefore strictly local by virtue of III.3.5. On the other hand, for all (i, j) ∈ I 2 with Vj Vi j ≥ i, the transition morphism X → X is integral and dominant. In particular, the transition homomorphism B((Vi → X)) → B((Vj → X)) is local. It follows that the ring B ρ(y x) is local, normal, and henselian ([39] 0.6.5.12(ii) and [62] I § 3 Proposition 1). Since the homomorphism Γ(X, OX ) → B ρ(y x) is integral and therefore local, the residue field of B ρ(y x) is an algebraic extension of that of Γ(X, OX ). It is therefore separably closed. (ii) This immediately follows from the fact that X is strictly local. (iii) Recall that we have a canonical isomorphism (VI.10.18.1) (III.8.19.4)

∼

(σ −1 (~∗ (OX )))ρ(y

x)

→ ~∗ (OX )x .

In view of (ii) and (III.8.19.3), the stalk of the canonical homomorphism σ −1 (~∗ (OX )) → B at ρ(y x) identifies with the canonical homomorphism (III.8.19.5)

Γ(X, OX ) → B ρ(y

x)

= lim B((Vi → X)), −→ i∈I

which is clearly injective and integral, and therefore local. III.8.20.

Consider a commutative diagram Y0

(III.8.20.1)

j0

g0

/

X

0

~0

/ X0 g

g

 Y

j

 /X

~

 /X

0

and set h0 = ~0 ◦ j 0 . Suppose that X is normal and locally irreducible and that j 0 is e 0 ) the Faltings site (resp. a quasi-compact open immersion. We denote by E 0 (resp. E 0 e 0 associated with topos) associated with the morphism h0 (III.8.2), by B the ring of E 0 X (III.8.17), and by e0 → E e Φ: E

(III.8.20.2)

0

the functoriality morphism (III.8.5.3). For any (V 0 → U 0 ) ∈ Ob(E 0 ), we set U = 0

U 0 ×X 0 X and denote by U (III.8.20.3)

0V

0

0

0

the integral closure of U in V 0 , so that

B ((V 0 → U 0 )) = Γ(U

0V 0

, OU 0V 0 ).

III.8. FALTINGS RINGED TOPOS

219

For any (V → U ) ∈ Ob(E), set V 0 = V ×Y Y 0 and U 0 = U ×X X 0 , so that (V 0 → U 0 ) is an object of E 0 and that we have a commutative diagram (III.8.20.4)

Y0 o

/

V0 

 Y o

U

/

0 

 /U

 V

X

0

 /X

We deduce from this a morphism U

(III.8.20.5)

0V 0

V

→U ,

e and consequently a ring homomorphism of E 0

B → Φ∗ (B ).

(III.8.20.6)

0

From now on, we will consider Φ as a morphism of ringed topos (by B and B, respectively). For modules, we use the notation Φ−1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation Φ∗ for the inverse image in the sense of modules. Lemma III.8.21. We keep the assumptions of III.8.20 and moreover suppose that g is étale and that the two squares in diagram (III.8.20.1) are Cartesian, so that (Y 0 → X 0 ) is an object of E. Then: (i) The morphism Φ−1 (B) → B

(III.8.21.1)

0

adjoint to the morphism (III.8.20.6) is an isomorphism. e 0 , B 0 ) → (E, e B) identifies with the local(ii) The morphism of ringed topos Φ : (E e B) at σ ∗ (X 0 ). ization morphism of (E, e0 → E e identifies with the localization morphism Indeed, the morphism of topos Φ : E ∗ 0 0 0 a e of E at σ (X ) = (Y → X ) by virtue of VI.10.14. It therefore suffices to show e →E e 0 identifies with the restriction functor by the first statement. The functor Φ∗ : E 0 the canonical functor E → E. For every (V → U ) ∈ Ob(E 0 ), we have a canonical isomorphism (III.8.21.2)

0 ∼

U ×X 0 X → U ×X X = U .

We deduce from this an isomorphism (that is functorial in (V → U )) (III.8.21.3)

0

∼

Φ−1 (B)((V → U )) → B ((V → U )).

It remains to show that this is adjoint to the morphism (III.8.20.6). Set U 0 = U ×X X 0 and V 0 = V ×Y Y 0 . The structural morphisms U → X 0 and V → Y 0 induce sections U → U 0 and V → V 0 of the canonical projections U 0 → U and V 0 → V . We deduce from this a commutative diagram (III.8.21.4)

/U

V

idV

 V0  V

/

 0 U   /U

idU

220

III. GLOBAL ASPECTS 0V 0

V

V

→ U whose composition is the identity of and consequently morphisms U → U V U . The composition (III.8.21.5) 0 0 0 Φ−1 (B)((V → U )) → Φ−1 (Φ∗ (B ))((V → U )) = B ((V 0 → U 0 )) → B ((V → U )),

where the first arrow is induced by (III.8.20.6) and the last arrow is the adjunction morphism, is therefore the isomorphism (III.8.21.3); the first statement follows.

Lemma III.8.22. We keep the assumptions of III.8.20 and moreover suppose that X 0 is the strict localization of X at a geometric point x and that the two squares in diagram (III.8.20.1) are Cartesian. Let (V → U ) be an object of E and U 0 = U ×X X 0 , V 0 = V ×Y Y 0 . Then the canonical morphism U

0V 0

V

→ U ×X X 0 (III.8.20.5) is an isomorphism.

Indeed, we have a commutative diagram with Cartesian squares (III.8.22.1)

V0  V

/

0

/ U0

/ X0

 /U

 /U

 /X

U

0V 0

V

We can therefore identify U with the integral closure of U ×X X 0 in V 0 . It then V suffices to show that the canonical morphism V 0 → U ×X X 0 is a quasi-compact and V schematically dominant open immersion and that U ×X X 0 is normal and locally irreducible. The first statement follows from III.8.11(i) and ([42] 11.10.5). Since the second statement is local, to prove it we may restrict to the case where X is affine. We consider X 0 as a cofiltered inverse limit of affine étale neighborhoods (Xi )∈I of x in X (cf. [2] V VIII 4.5). Then U ×X X 0 is canonically isomorphic to the inverse limit of the schemes V V (U ×X Xi )i∈I ([42] 8.2.5). Since each U ×X Xi is normal by III.8.11(ii) and ([62] V VII Proposition 2), U ×X X 0 is normal by ([39] 0.6.5.12(ii)). On the other hand, since 0 X is normal and locally irreducible by assumption, the same holds for V 0 by III.3.3. It V follows that U ×X X 0 is locally irreducible by virtue of III.3.4, giving the lemma. Proposition III.8.23. We keep the assumptions of III.8.20 and moreover suppose that ~ is coherent, that X 0 is the strict localization of X at a geometric point x, and that the two squares in diagram (III.8.20.1) are Cartesian. Then the morphism (III.8.23.1)

Φ−1 (B) → B

0

adjoint to the morphism (III.8.20.6) is an isomorphism. We choose an affine open neighborhood X0 of the image of x in X and denote by I the category of x-pointed étale X0 -schemes that are affine over X0 (cf. [2] VIII 3.9 and 4.5). We denote by (III.8.23.2)

E →I

the fibered U-site defined in (VI.11.2.2): the fiber of E over an object U of I is the Faltings site associated with the canonical projection hU : UY → U , and the inverse image functor 0 associated with a morphism f : U 0 → U in I is the functor Φ+ f : EU → EU defined in (III.8.5.2). We denote by (III.8.23.3)

F →I

the fibered U-topos associated with E /I. The fiber of F over an object U of I is the topos EeU of sheaves of U-sets on the covanishing site EU , and the inverse image functor

III.8. FALTINGS RINGED TOPOS

221

with respect to a morphism f : U 0 → U of I is the inverse image functor under the morphism of topos Φf : EeU 0 → EeU defined in (III.8.5.3). We denote by F ∨ → I◦

(III.8.23.4)

the fibered category obtained by associating with each object U of I the category FU = EeU and with each morphism f : U 0 → U of I the direct image functor by the morphism of topos Φf . Recall (VI.10.14) that for every U ∈ Ob(I), EeU is canonically equivalent to e/(U →U )a , where (UY → U )a denotes the sheaf associated with (UY → U ). the topos E Y e 0 is canonically equivalent to the inverse limit of By virtue of VI.11.3, the topos E the fibered topos F /I. We endow E with the total topology ([2] VI 7.4.1) and denote by Top(E ) the topos of sheaves of U-sets on E . Then we have a canonical morphism (VI.11.4.2) e 0 → Top(E ) $: E

(III.8.23.5)

and a canonical commutative diagram of functors (VI.11.4.3) (III.8.23.6)

/ Homcart/I ◦ (I ◦ , F ∨ ) _

∼

e0 E $∗

 Top(E )

∼

 / HomI ◦ (I ◦ , F ∨ )

where the horizontal arrows are equivalences of categories and the right vertical arrow is the canonical injection (cf. [2] VI 8.2.9). For any object U of I, we denote by B/U the ring of EeU associated with U (III.8.17). For every morphism f : U 0 → U of I, we have a canonical homomorphism B /U → Φf ∗ (B /U 0 ) (III.8.20.6). These homomorphisms satisfy a compatibility relation for the composition of morphisms of I of type ([1] (1.1.2.2)). They therefore define a ring of Top(E ) that we denote by {U 7→ B /U } (not to be confused with the ring B = {U 7→ e (III.8.10.3)). For every U ∈ Ob(I), we have a canonical morphism gU : X 0 → U B U } of E that induces a morphism of ringed topos (III.8.20.6) (III.8.23.7)

e 0 , B 0 ) → (EeU , B /U ). ΦU : (E 0

The collection of homomorphisms B /U → ΦU ∗ (B ) defines a ring homomorphism of Top(E ) (III.8.23.8)

0

{U 7→ B /U } → $∗ (B ).

Let us show that the adjoint homomorphism (III.8.23.9)

$∗ ({U 7→ B /U }) → B

0

is an isomorphism. Since the functor $∗ is fully faithful (III.8.23.6), it suffices to show that the induced homomorphism (III.8.23.10)

0

$∗ ($∗ ({U 7→ B /U })) → $∗ (B )

is an isomorphism. In view of ([2] VI 8.5.3), this is equivalent to showing that for every U ∈ Ob(I), the canonical homomorphism of EeU (III.8.23.11)

lim −→

f : U 0 →U

0

Φf ∗ (B /U 0 ) → ΦU ∗ (B ),

where the limit is taken over the morphisms f : U 0 → U of I, is an isomorphism.

222

III. GLOBAL ASPECTS

Let (V1 → U1 ) be an object of E/(UY →U ) such that the morphism U1 → U is coherent. The canonical morphism V1

U 1 ×U X 0 →

(III.8.23.12)

V1

lim ←−

U 0 ∈Ob(I/U )

U 1 ×U U 0 V1

is an isomorphism by ([42] 8.2.5). Since ~ is coherent, the scheme U 1 is coherent. By ([2] VI 5.2), III.8.11(iii), and III.8.22, it follows that the canonical homomorphism (III.8.23.13)

lim −→

U 0 ∈Ob(I/U )

0

B /U 0 ((V1 ×U U 0 → U1 ×U U 0 )) → B ((V1 ×U X 0 → U1 ×U X 0 ))

is an isomorphism. On the other hand, since h is coherent, the sheaf (V1 → U1 )a of EeU associated with (V1 → U1 ) is coherent by VI.10.5(i). Hence by virtue of VI.10.4, VI.10.5(ii), and ([2] VI 5.3), the isomorphism (III.8.23.13) shows that (III.8.23.11) is an isomorphism. We deduce from the isomorphism (III.8.23.9) and from (VI.11.4.4) that the canonical homomorphism (III.8.23.14)

lim Φ−1 U (B /U ) → B

0

−→

i∈I ◦

is an isomorphism. For every U ∈ Ob(I), the canonical morphism U → X induces a morphism of ringed topos e B). (III.8.23.15) ρU : (EeU , B /U ) → (E,

Since the homomorphism ρ−1 U (B) → B /U is an isomorphism by virtue of III.8.21(i), the proposition follows from the isomorphism (III.8.23.14). III.9. Faltings topos over a trait III.9.1. In this section, S denotes the trait fixed in III.2.1. We fix a coherent S-scheme X and an open subscheme X ◦ of Xη . For any X-scheme U , we set U ◦ = U ×X X ◦ .

(III.9.1.1)

Recall that for every S-scheme Y , we have set Y = Y ×S S and for every integer n ≥ 1, Yn = Y ×S Sn (III.2.1.1). We denote by j : X ◦ → X and ~ : X → X the canonical morphisms. We suppose that j is quasi-compact and that X is normal and locally ◦ irreducible (III.3.1). Note that X and X are coherent and étale-locally connected by III.3.2(iii). We propose to apply the constructions of III.8 to the morphisms in the top row of the following commutative diagram: (III.9.1.2)

X

◦

jX

/X  /S

 η

~



/X  /S

e the Faltings site (resp. topos) associated with the morphism We denote by E (resp. E) ◦ e associated with X (III.8.17), which is ~ ◦ jX : X → X (III.8.2), and by B the ring of E then an OK -algebra. We denote by e σ: E

(III.9.1.3) (III.9.1.4)

←

ρ : X´et ×X´et

◦ X ´et

→ X´et , e → E,

the canonical morphisms (III.8.3.3) and (III.8.6.2), respectively.

III.9. FALTINGS TOPOS OVER A TRAIT

For any integer n ≥ 0, we set (III.9.1.5)

223

B n = B/pn B.

´ /X ), we set B U = B ◦ αU ! (III.8.1.6) and For any U ∈ Ob(Et (III.9.1.6)

B U,n = B U /pn B U .

Note that the canonical homomorphism B U,n → B n ◦ αU ! is not in general an isomorphism; this is why we will not use the notation B n,U . Nevertheless, the correspondences {U 7→ pn B U } and {U 7→ B U,n } naturally form presheaves on E (III.8.2.1), and the canonical morphisms (III.9.1.7) (III.9.1.8)

{U 7→ pn B U }a {U 7→ B U,n }a

→ pn B, → Bn,

e are isomorwhere the terms on the left-hand side denote the associated sheaves in E, phisms by VI.8.2 and VI.8.9. Lemma III.9.2. The ring B is OK -flat. V

For every (V → U ) ∈ Ob(E), since V is an η-scheme, U is S-flat by virtue of III.8.11(i). Consequently, B does not have any OK -torsion and is therefore OK -flat ([11] Chapter VI § 3.6 Lemma 1). III.9.3. Since Xη is a subobject of the final object X of X´et ([2] IV 8.3), σ ∗ (Xη ) = e (III.8.3.3). Recall that the topos (X → Xη )a is subobject of the final object of E e/σ∗ (X ) is canonically equivalent to the Faltings topos associated with the morphism E η ◦ X → Xη (VI.10.14). Denote by ◦

(III.9.3.1)

e/σ∗ (X ) → E e γ: E η

e at σ ∗ (Xη ), which we identify with the functoriality morthe localization morphism of E phism induced by the canonical injection Xη → X (III.8.5). We then have a sequence of three adjoint functors (III.9.3.2)

e/σ∗ (X ) → E, e γ! : E η

e→E e/σ∗ (X ) , γ∗ : E η

e/σ∗ (X ) → E, e γ∗ : E η

in the sense that for any two consecutive functors in the sequence, the one on the right is the right adjoint of the other. The functors γ! and γ∗ are fully faithful ([2] IV 9.2.4). es the closed subtopos of E e complement of the open subtopos σ ∗ (Xη ), We denote by E e made up of the sheaves F such that γ ∗ (F ) is a final that is, the full subcategory of E e object of E/σ∗ (Xη ) ([2] IV 9.3.5), and by (III.9.3.3)

es → E e δ: E

es → E e is the the canonical embedding, that is, the morphism of topos such that δ∗ : E ∗ e canonical injection functor. For any F ∈ Ob(E), we set Fs = δ (F ). e Pt(E e/σ∗ (X ) ), and Pt(E es ) the categories of points of E, e We denote by Pt(E), η e e E/σ∗ (Xη ) , and Es , respectively, and by (III.9.3.4)

e/σ∗ (X ) ) → Pt(E) e u : Pt(E η

es ) → Pt(E) e and v : Pt(E

the functors induced by γ and δ, respectively. These functors are fully faithful and every e belongs to the essential image of exactly one of these functors ([2] IV 9.7.2). point of E

224

III. GLOBAL ASPECTS

es ), the canonical projection Remark III.9.4. By definition, for every F ∈ Ob(E (III.9.4.1)

σ ∗ (Xη ) × δ∗ (F ) → σ ∗ (Xη )

e in other is an isomorphism. Hence there exists a unique morphism σ ∗ (Xη ) → δ∗ (F ) of E; ∗ ∗ e words, δ (σ (Xη )) is an initial object of Es . ←

◦

x) be a point of X´et ×X´et X ´et (III.8.6). Then ρ(y x) Lemma III.9.5. (i) Let (y belongs to the essential image of u (resp. v) (III.9.3.4) if and only if x lies over η (resp. s). e/σ∗ (X ) (resp. E es ) defined by the family of points (ii) The family of points of E η e x) of E such that x lies over η (resp. s) is conservative. ρ(y (i) Indeed, ρ(y x) belongs to the essential image of u (resp. v) if and only if (σ ∗ (Xη ))ρ(y x) is a singleton (resp. empty). On the other hand, we have a canonical isomorphism (VI.10.18.1) (III.9.5.1)

(σ ∗ (Xη ))ρ(y

∼

x)

→ (Xη )x ,

giving the desired statement. (ii) This follows from (i), VI.10.21, and ([2] IV 9.7.3). e the following properties are Lemma III.9.6. For every sheaf F = {U 7→ FU } of E, equivalent: es . (i) F is an object of E ´ /X , FU is a final object of U ◦f´et , that is, is representable by (ii) For every U ∈ Et η ◦ U . ← ◦ (iii) For every point (y x) of X´et ×X´et X ´et (III.8.6) such that x lies over η, the stalk Fρ(y x) of F at ρ(y x) is a singleton. Indeed, by VI.5.38, we have a canonical isomorphism ∼ ´ /X )). (III.9.6.1) γ ∗ (F ) → {U 0 7→ FU 0 }, (U 0 ∈ Ob(Et η

0◦ e/σ∗ (X ) (III.8.4), conditions ´ /X ), is a final object of E Since {U 0 7→ U }, for U 0 ∈ Ob(Et η η (i) and (ii) are equivalent. On the other hand, conditions (i) and (iii) are equivalent by virtue of III.9.5(ii).

es . Lemma III.9.7. For every n ≥ 0, the ring B n (III.9.1.5) is an object of E ←

◦

Indeed, for every point (y x) of X´et ×X´et X ´et (III.8.6) such that x lies over η, the canonical homomorphism σ −1 (OX ) → B (III.8.17) induces a homomorphism OX,x → B ρ(y x) (VI.10.18.1). Consequently, p is invertible in B ρ(y x) , giving the lemma by virtue of III.9.6. III.9.8. We denote by a : Xs → X and b : Xη → X the canonical injections. The e/σ∗ (X ) being canonically equivalent to the Faltings topos associated with the topos E η ◦ morphism X → Xη , we denote by (III.9.8.1)

e/σ∗ (X ) → Xη,´et ση : E η

the canonical morphism (III.8.3.3). By (VI.10.12.6), the diagram (III.9.8.2)

e/σ∗ (X ) E η

ση

γ

 e E

/ Xη,´et b

σ

 / X´et

III.9. FALTINGS TOPOS OVER A TRAIT

225

is commutative up to canonical isomorphism. By virtue of ([2] IV 9.4.3), there exists a morphism es → Xs,´et , σs : E

(III.9.8.3)

unique up to isomorphism, such that the diagram (III.9.8.4)

es E

σs

/ Xs,´et

σ

 / X´et

a

δ

 e E

is commutative up to isomorphism. By definition, we have a canonical isomorphism (III.9.8.5)

∼

σ ∗ ◦ a∗ → δ∗ ◦ σs∗ .

es and every integer Since the functors a∗ and δ∗ are exact, for every abelian group F of E i ≥ 0, we have a canonical isomorphism (III.9.8.6)

∼

a∗ (Ri σs∗ (F )) → Ri σ∗ (δ∗ F ).

III.9.9. For any integer ≥ 1, denote by a : Xs → X, an : Xs → Xn , ιn : Xn → X, and ιn : X n → X the canonical injections. Since the residue field of OK is algebraically closed, there exists a unique S-morphism s → S. This induces closed immersions a : Xs → X and an : Xs → X n that lift a and an , respectively. a

(III.9.9.1)

Xs

an

/X n

ιn

~n

Xs

an

 / Xn

/' X ~

ιn

 /7 X

a

The canonical homomorphism σ −1 (~∗ (OX )) → B (III.8.17) induces a homomorphism (III.9.1.5) (III.9.9.2)

σ −1 (ιn∗ (~n∗ (OX n ))) → B n .

Since an is a universal homeomorphism, we may view OX n as a sheaf of Xs,´et (cf. III.2.9). We can then identify the rings ιn∗ (~n∗ (OX n )) and a∗ (OX n ) of X´et and consequently the es (III.9.8.5). Since B n is an object of rings σ −1 (ιn∗ (~n∗ (OX n ))) and δ∗ (σs∗ (OX n )) of E es (III.9.7), we may view (III.9.9.2) as a homomorphism of E es E (III.9.9.3)

σs∗ (OX n ) → B n .

The morphism σs (III.9.8.3) is therefore underlying a morphism of ringed topos, which we denote by (III.9.9.4)

es , B n ) → (Xs,´et , O ). σn : ( E Xn

226

III. GLOBAL ASPECTS

˘ the ring (B e N◦ III.9.10. We denote by B ˘ the ring n+1 )n∈N of Es (III.7.7) and by OX N◦ N◦ (OX n+1 )n∈N of Xs,zar or of Xs,´et , depending on the context. This abuse of notation does not lead to any confusion (cf. III.2.9). By III.7.5, the morphisms (σn+1 )n∈N (III.9.9.4) induce a morphism of ringed topos ˘ → (X N◦ , O ). e N◦ , B) (III.9.10.1) σ ˘ : (E s,´ et

s

˘ X

For every integer n ≥ 1, we denote by (III.9.10.2)

un : (Xs,´et , OX n ) → (Xs,zar , OX n )

the canonical morphism of ringed topos (cf. III.2.9 and III.9.9). Then the morphisms (un+1 )n∈N induce a morphism of ringed topos (III.9.10.3)

◦

◦

N N u ˘ : (Xs,´ ˘ ). ˘ ) → (Xs,zar , OX et , OX

We denote by X the formal scheme p-adic completion of X. The topological space underlying X is canonically isomorphic to Xs . It is ringed by the sheaf of topological rings OX inverse limit of the sheaves of pseudo-discrete rings (OX n+1 )n∈N ([39] 0.3.9.1). We denote by ◦

N λ : Xs,zar → Xs,zar

(III.9.10.4)

the morphism defined in (III.7.4.3). The sheaf of rings λ∗ (OX˘ ) is canonically isomorphic to the sheaf of rings (without topologies) underlying OX ([39] 0.3.9.1 and 0.3.2.6). We then view λ as a morphism of ringed topos with structure sheaves OX˘ and OX , respectively. We denote by ˘ → (X e N◦ , B) (III.9.10.5) > : (E ,O ) s,zar

s

X

the composed morphism of ringed topos λ ◦ u ˘◦σ ˘.

III.9.11. Let g : X 0 → X be a coherent morphism and X 0B an open subscheme of X 0◦ = X ◦ ×X X 0 . For any X 0 -scheme U 0 , we set U 0B = U 0 ×X 0 X 0B .

(III.9.11.1)

0

We denote by j 0 : X 0B → X 0 the canonical injection and by ~0 : X → X 0 the canonical 0 morphism (III.9.1.1). Suppose that X is normal and locally irreducible. We denote by E 0 e 0 ) the Faltings site (resp. topos) associated with the morphism ~0 ◦j 0 0 : X 0B → X 0 (resp. E X 0 e 0 associated with X 0 (III.8.17), and by (III.8.2), by B the ring of E (III.9.11.2)

e 0 , B 0 ) → (X´e0 t , ~0∗ (O 0 )) σ 0 : (E X

es0 the closed subtopos the canonical morphism of ringed topos (III.8.17). We denote by E e 0 complement of the open subtopos σ 0∗ (Xη0 ), by of E es0 → E e0 δ0 : E

(III.9.11.3) the canonical embedding, and by (III.9.11.4)

0 es0 → Xs,´ σs0 : E et

0

the canonical morphism of topos (III.9.8.3). For any integer n ≥ 1, we set B n = 0 0 B /pn B , and denote by (III.9.11.5)

0 es0 , B 0n ) → (Xs,´ σn0 : (E et , OX 0 ) n

0

the morphism of ringed topos induced by σ (III.9.9.4).

III.9. FALTINGS TOPOS OVER A TRAIT

227

We denote by 0

e B) e 0 , B ) → (E, Φ : (E

(III.9.11.6)

the morphism of ringed topos deduced from g by functoriality (III.8.20). By (VI.10.12.6) and the definitions III.8.17 and III.8.20, the diagram of morphisms of ringed topos

σ0

/ (E, e B)

Φ

0

e0 , B ) (E

(III.9.11.7)

σ



(X´e0 t , ~0∗ (OX 0 ))



/ (X´et , ~∗ (O )) X

g

where g is the morphism induced by g, is commutative up to canonical isomorphism, in the sense of ([1] 1.2.3). We have a canonical isomorphism Φ∗ (σ ∗ (Xη )) ' σ 0∗ (Xη0 ) (III.8.5.2). By virtue of ([2] IV 9.4.3), there consequently exists a morphism of topos es0 → E es , Φs : E

(III.9.11.8)

unique up to isomorphism, such that the diagram es0 E

(III.9.11.9) δ

/E e

Φs

s

0

 e E0

δ

 /E e

Φ

is commutative up to isomorphism, and even 2-Cartesian. It follows from (VI.10.12.6) and ([2] IV 9.4.3) that the diagram of morphisms of topos es0 E

(III.9.11.10)

σs0

/E es

Φs

σs



0 Xs,´ et

gs



/ Xs,´et

is commutative up to canonical isomorphism. 0 The canonical homomorphism Φ−1 (B) → B induces a homomorphism Φ∗s (B n ) → 0 B n . The morphism Φs is therefore underlying a morphism of ringed topos, which we denote by es0 , B 0n ) → (E es , B n ). Φn : (E

(III.9.11.11)

It follows from (III.9.11.7) and (III.9.11.10) that the diagram of morphisms of ringed topos 0

es0 , B n ) (E

(III.9.11.12)

0 σn

 0 (Xs,´ et , OX 0 ) n

/ (E es , B n )

Φn

σn gn

 / (Xs,´et , OX ) n

where g n is the morphism induced by g, is commutative up to canonical isomorphism, in the sense of ([1] 1.2.3). ˘ 0 = (B 0 ) e 0N◦ . By III.7.5, the morphisms (Φ Set B , which is a ring of E ) n+1 n∈N

s

define a morphism of ringed topos (III.9.11.13)

˘ ˘ 0 ) → (E ˘ : (E es0N◦ , B esN◦ , B). Φ

n+1 n∈N

228

III. GLOBAL ASPECTS 0

We denote by X0 the formal scheme p-adic completion of X and by ˘ 0 ) → (X0 , O 0 ) es0N◦ , B >0 : (E X zar

(III.9.11.14)

the morphism of ringed topos defined in (III.9.10.5) with respect to (X 0 , X 0B ). It immediately follows from (III.9.11.12) and from the functorial character of the morphisms u ˘ (III.9.10.3) and λ (III.9.10.4) that the diagram of morphisms of ringed topos ˘ Φ

˘ 0) es0N◦ , B (E

(III.9.11.15)

>0

˘ / (E esN◦ , B) >



0 (Xs,zar , OX0 )

g



/ (Xs,zar , OX )

where g is the morphism induced by g, is commutative up to canonical isomorphism, in ˘ F and every integer the sense of ([1] 1.2.3). We deduce from this, for every B-module q ≥ 0, a base change morphism ˘ ∗ (F )). (III.9.11.16) g∗ (Rq >∗ (F )) → Rq >0∗ (Φ

Lemma III.9.12. We keep the assumptions of III.9.11 and moreover suppose that g is étale and X 0B = X 0◦ . Then Φs (III.9.11.8) is canonically isomorphic to the localization es at σs∗ (Xs0 ). morphism of E 0B

First note that (X → X 0 ) is an object of E and that the associated sheaf is none ∼ other than σ ∗ (X 0 ). On the other hand, we have a canonical isomorphism δ ∗ (σ ∗ (X 0 )) → e/σ∗ (X 0 ) → E e (resp. js : (E es )/σ∗ (X 0 ) → E es ) the localσs∗ (Xs0 ) (III.9.8.4). Denote by j : E s s ∗ 0 ∗ 0 e at σ (X ) (resp. of E es at σ (X )). By ([2] IV 5.10), the morphism ization morphism of E s s δ induces a morphism es )/σ∗ (X 0 ) → E e/σ∗ (X 0 ) (III.9.12.1) δ/σ∗ (X 0 ) : (E s

s

that fits into a commutative diagram up to canonical isomorphism es )/σ∗ (X 0 ) (E s s

(III.9.12.2)

δ/σ∗ (X 0 )

js

/E es δ



 /E e

j

e/σ∗ (X 0 ) E

e0 This diagram is in fact 2-Cartesian by ([2] IV 5.11). On the other hand, the topos E e/σ∗ (X 0 ) are canonically equivalent and the morphism Φ identifies with j by virtue and E of VI.10.14. Since the diagram (III.9.11.9) is also 2-Cartesian, the lemma follows. Lemma III.9.13. We keep the assumptions of III.9.11 and moreover suppose that X 0B = X 0◦ and that one of the following two conditions is satisfied: (i) g is étale; (ii) X 0 is the strict localization of X at a geometric point x. 0

Then for every integer n ≥ 1, the homomorphism Φ∗s (B n ) → B n is an isomorphism. This follows from III.8.21(i) and III.8.23. Proposition III.9.14. We keep the assumptions of III.9.11 and moreover suppose that esN◦ → E es the morphism of topos g is étale and X 0B = X 0◦ , and denote also by λ : E ˘ defined in (III.7.4.3). Then Φ (III.9.11.13) is canonically isomorphic to the localization ˘ at λ∗ (σ ∗ (X 0 )). e N◦ , B) morphism of the ringed topos (E s

s

s

III.10. HIGGS–TATE ALGEBRAS

229

◦ e 0N◦ → E e N◦ identifies with the localization Indeed, the morphism of topos ΦN s : Es s esN◦ at λ∗ (σs∗ (Xs0 )), by III.9.12 and III.7.6(ii). On the other hand, the morphism of E ◦ ˘ is an isomorphism by virtue of III.9.13 and ˘ →B canonical homomorphism (ΦN )∗ (B)

s

(III.7.5.4); the proposition follows.

Corollary III.9.15. We keep the assumptions of III.9.11 and moreover suppose that g ˘ is an open immersion and X 0B = X 0◦ . Then for every B-module F and every integer q ≥ 0, the base change (III.9.11.16) ˘ ∗ (F )) g∗ (Rq >∗ (F )) → Rq >0∗ (Φ

(III.9.15.1) is an isomorphism.

Indeed, the squares of the diagram of morphisms of topos (III.9.15.2)

λ

esN◦ E

/E es

◦

σs

σsN



◦

λ

N Xs,´ et



/ Xs,´et

◦

usN



us ◦

N Xs,zar

λ



/ Xs,zar

where us is the canonical morphism (III.2.9.2) and λ (abusively) denotes the morphisms defined in (III.7.4.3), are commutative up to canonical isomorphisms (III.7.5.4). Writing ◦ N◦ T = λ ◦ uN s ◦ σs , which is the morphism of topos underlying > (III.9.10.5), we deduce from this an isomorphism (III.9.15.3)

∼

T∗ (Xs0 ) → λ∗ (σs∗ (Xs0 )).

It then follows from (III.9.11.15) and III.9.14 that >0 identifies with the morphism >/Xs0 (cf. [2] IV 5.10); the corollary follows. III.10. Higgs–Tate algebras III.10.1. (III.10.1.1)

In this section, (S, MS ) denotes the logarithmic trait fixed in III.2.1 and f : (X, MX ) → (S, MS )

an adequate morphism of logarithmic schemes (III.4.7). We denote by X ◦ the maximal open subscheme of X where the logarithmic structure MX is trivial; it is an open subscheme of Xη . For any usual X-scheme U , we set (III.10.1.2)

U ◦ = U ×X X ◦ .

ˇ and for every Recall that for every S-scheme Y , we have set Y = Y ×S S, Yˇ = Y ×S S ◦ integer n ≥ 1, Yn = Y ×S Sn (III.2.1.1). We denote by j : X → X and ~ : X → X the canonical morphisms. To alleviate the notation, we set (III.10.1.3)

1 e1 Ω X/S = Ω(X,MX )/(S,MS ) ,

which we view as a sheaf of Xzar or X´et , depending on the context (cf. III.2.9). For any integer n ≥ 1, we set (III.10.1.4)

e 1 ⊗O O , e1 =Ω Ω X X/S Xn X n /S n

230

III. GLOBAL ASPECTS

which we view as a sheaf of Xs,zar or Xs,´et , depending on the context (cf. III.2.9 and ˇ with the logarithmic structures M and M inverse images III.9.9). We endow X and X ˇ X X of MX . We then have canonical isomorphisms (III.2.1) (III.10.1.5) (III.10.1.6)

∼

(X, MX ) → (X, MX ) ×(S,MS ) (S, MS ), ∼ ˇ M )→ ˇ M ), (X, (X, M ) × (S, ˇ X

X

ˇ S

(S,MS )

where the product is taken indifferently in the category of logarithmic schemes or in that of fine logarithmic schemes. ˇ M ) (III.2.3) consists of a smooth A smooth (A2 (S), MA2 (S) )-deformation of (X, ˇ X morphism of fine logarithmic schemes (III.10.1.7)

e M e ) → (A2 (S), M fe: (X, A2 (S) ) X

ˇ M )-isomorphism and an (S, ˇ S (III.10.1.8)

∼ ˇ M )→ e M e) × (X, (X, ˇ (A2 (S),M X X

A2 (S) )

ˇ M ), (S, ˇ S

where the product is taken indifferently in the category of logarithmic schemes or in that of fine logarithmic schemes (cf. [50] 3.14). For the remainder of this section, we assume e M e ), which we fix. that there exists such a deformation (X, X III.10.2. By III.4.2(iii), the schemes X and X are normal and locally irreducible. On the other hand, since X is noetherian, j is quasi-compact. We can therefore apply the constructions of III.9 to the morphisms in the top row of the following commutative diagram: (III.10.2.1)

X

◦

jX

/X  /S

 η

~



/X  /S

We denote by ´ /X π : E → Et

(III.10.2.2)

◦

the Faltings fibered U-site associated with the morphism h = ~ ◦ jX : X → X (III.8.1). e the topos of sheaves of We endow E with the covanishing topology and denote by E e U-sets on E (III.8.2), by B the ring of E associated with X (III.8.17), and by (III.10.2.3) (III.10.2.4)

←

e → X´et , σ: E

◦ e ρ : X´et ×X´et X ´et → E,

es the closed subtopos the canonical morphisms (III.8.3.3) and (III.8.6.2). We denote by E ∗ e of E complement of the open subtopos σ (Xη ) (III.9.3), by (III.10.2.5)

es → E e δ: E

the canonical embedding, and by (III.10.2.6)

es → Xs,´et σs : E

the morphism of topos induced by σ (III.9.8.3). For any integer n ≥ 0, we set (III.10.2.7)

B n = B/pn B.

III.10. HIGGS–TATE ALGEBRAS

231

´ /X ), we set B U = B ◦ αU ! (III.8.1.6) and For any U ∈ Ob(Et (III.10.2.8)

B U,n = B U /pn B U .

Note that the canonical homomorphism B U,n → B n ◦ αU ! is not in general an isomores . If n ≥ 1, we denote phism (cf. (III.9.1.8)). Recall (III.9.7) that B n is a ring of E by (III.10.2.9)

es , B n ) → (Xs,´et , O ) σn : ( E Xn

the canonical morphism of ringed topos (III.9.9.4). ´ /X made up of the affine III.10.3. We denote by P the full subcategory of Et schemes U such that the morphism (U, MX |U ) → (S, MS ) induced by f admits an ´ /X . adequate chart (III.4.4). We endow P with the topology induced by that of Et Since X is noetherian and therefore quasi-separated, any object of P is coherent over X. ´ /X and is Consequently, P is a U-small family that topologically generates the site Et stable under fibered products. We denote by πP : E P → P

(III.10.3.1)

the fibered site deduced from π (III.10.2.2) by base change by the canonical injection ´ /X . We endow EP with the covanishing topology defined by πP and we functor P → Et eP the topos of sheaves of U-sets on EP . By VI.5.21 and VI.5.22, the topology denote by E on EP is induced by that on E through the canonical projection functor EP → E, and the latter induces by restriction an equivalence of categories ∼ e e→ E EP .

(III.10.3.2)

◦

y

Remark III.10.4. Let U be an object of P, y a generic geometric point of U , and RU the ring defined in (III.8.13.2). Since the schemes U and U are locally irreducible (III.3.3), they are the sums of the schemes induced on their irreducible components. Denote by ? ◦ U c (resp. U ) the irreducible component of U (resp. U ) containing y. Likewise, U is ?◦ ? the sum of the schemes induced on its irreducible components and U = U ×X X ◦ ◦ is the irreducible component of U containing y. Then U c is naturally an object of y y ?◦ P over U , and the canonical homomorphism RU → RU c is a π1 (U , y)-equivariant y isomorphism. If Usc = ∅, then RU is an K-algebra. Suppose Usc 6= ∅, so that the morphism (U c , MX |U c ) → (S, MS ) induced by f satisfies the conditions of II.6.2. The y ?◦ algebra RU endowed with the action of π1 (U , y) then corresponds to the algebra R endowed with the action of ∆ introduced in II.6.7 and II.6.10; whence the notation. The ? coordinate ring of the affine scheme U corresponds to the algebra R1 in loc. cit. by II.6.8(i). III.10.5. We denote by Q the full subcategory of P (III.10.3) made up of the connected affine schemes U such that there exists a fine and saturated chart M → Γ(U, MX ) for (U, MX |U ) that induces an isomorphism (III.10.5.1)

∼

× M → Γ(U, MX )/Γ(U, OX ).

This chart is a priori independent of the adequate chart required in the definition of the ´ /X . It follows from objects of P. We endow Q with the topology induced by that of Et ´ /X . We denote by II.5.17 that Q is a topologically generating subcategory of Et (III.10.5.2)

πQ : EQ → Q

232

III. GLOBAL ASPECTS

the fibered site deduced from π (III.10.2.2) by base change by the canonical injection ´ /X . The canonical projection functor EQ → E is fully faithful and the functor Q → Et category EQ is U-small and topologically generates the site E. We endow EQ with the e is then equivalent to the topology induced by that on E. By restriction, the topos E category of sheaves of U-sets on EQ ([2] III 4.1). Notice that in general, since Q is not stable under fibered products, we cannot speak of the covanishing topology on EQ associated with πQ , and even less apply VI.5.21 and VI.5.22. III.10.6.

bQ the category of presheaves of U-sets on EQ and by We denote by E ∨ PQ → Q◦

(III.10.6.1)

∧ ´ ◦ the fibered category obtained by associating with each U ∈ Ob(Q) the category (Et f/U ) 0 ´ ◦ , and with each morphism f : U → U of Q the functor of presheaves of U-sets on Et f/U ◦ f f´et∗ :

(III.10.6.2)

∧ ∧ ´ ´ ◦ 0◦ (Et f/U ) → (Etf/U )

´ ´ ◦ → Et 0◦ under the obtained by composing with the inverse image functor Et f/U f/U ◦

0◦

◦

∨ is the fibered category morphism f : U → U deduced from f ; in other words, PQ ◦ on Q deduced from the fibered category (III.8.1.9) by base change by the canonical ´ /X . For any U ∈ Ob(Q), we denote by αU ! : Et ´ ◦ injection functor Q → Et f/U → EQ the canonical functor (III.8.1.6). By ([37] VI 12; cf. also [1] 1.1.2), we have an equivalence of categories ∼ ∨ bQ → (III.10.6.3) E HomQ◦ (Q◦ , PQ )

F

7→

{U 7→ F ◦ αU ! }.

We will, from now on, identify F with the section {U 7→ F ◦ αU ! } that is associated with it by this equivalence. Since EQ is a topologically generating subcategory of E, the “associated sheaf” functor on EQ induces a functor that we denote also by (III.10.6.4)

bQ → E, e E

F 7→ F a .

b (III.8.2.1) and FQ = {U 7→ ´ /X )) be an object of E Let F = {W 7→ GW } (W ∈ Ob(Et b GU } (U ∈ Ob(Q)) the object of EQ obtained by restricting F to EQ . It immediately follows from ([2] II 3.0.4) and from the definition of the “associated sheaf” functor ([2] II e 3.4) that we have a canonical isomorphism of E (III.10.6.5)

∼

(FQ )a → F a .

Remark III.10.7. Let F = {U 7→ FU } be a presheaf on EQ . For each U ∈ Ob(Q), ◦ denote by FUa the sheaf of U f´et associated with FU . Then {U 7→ FUa } is a presheaf on bQ , inducing an EQ and we have a canonical morphism {U 7→ FU } → {U 7→ FUa } of E isomorphism between the associated sheaves. This assertion does not follow directly from VI.5.17 because Q is not stable under fibered products. However, the proof is similar. The only point that needs to be verified is the isomorphism (VI.5.17.3). Let G = {W 7→ GW } e and GQ = {U 7→ GU } (U ∈ Ob(Q)) the object of E bQ ´ /X )) be an object of E (W ∈ Ob(Et ◦ obtained by restricting G to EQ . For every U ∈ Ob(Q), GU is a sheaf of U f´et (III.8.2.2). Consequently, the map (III.10.7.1)

HomEbQ ({U 7→ FUa }, {U 7→ GU }) → HomEbQ ({U 7→ FU }, {U 7→ GU })

induced by the canonical morphism {U 7→ FU } → {U 7→ FUa } is an isomorphism. The assertion follows.

III.10. HIGGS–TATE ALGEBRAS ←

233

◦

x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over III.10.8. Let (y s and X 0 the strict localization of X at x. Recall that giving a neighborhood of the ´ /X (resp. P (III.10.3), resp. Q (III.10.5)) is point of X´et associated with x in the site Et equivalent to giving an x-pointed étale X-scheme (resp. of P, resp. of Q) ([2] IV 6.8.2). These objects naturally form a cofiltered category, which we denote by Vx (resp. Vx (P), resp. Vx (Q)). The categories Vx (P) and Vx (Q) are U-small, and the canonical injection ´ /X induce fully faithful cofinal functors Vx (Q) → Vx (P) → Vx . functors Q → P → Et For any object (U, p : x → U ) of Vx , we denote also by p : X 0 → U the X-morphism deduced by p ([2] VIII 7.3) and we set ◦

(III.10.8.1) 0

p◦ = p ×X X : X

0◦

◦

→U .

By virtue of III.3.7, X is normal and strictly local (and in particular integral). ◦ 0◦ x) lifts to a X -morphism v : y → X and The X-morphism u : y → X 0 defining (y 0◦ therefore induces a geometric point of X that we (abusively) denote also by y. For any (U, p) ∈ Ob(Vx ), we (abusively) denote also by y the geometric point p◦ (v(y)) of ◦ U . Since U is locally irreducible (III.3.3), it is the sum of the schemes induced on ? its irreducible components. Denote by U the irreducible component of U containing ◦ y. Likewise, U is the sum of the schemes induced on its irreducible component and ?◦ ? ◦ U = U ×X X ◦ is the irreducible component on U containing y. The morphism 0◦ ◦ ?◦ p◦ : X → U therefore factors through U . We denote by e → X 0◦ ϕx : E f´ et

(III.10.8.2)

the canonical functor (III.8.8.4) and by 0◦

∼

νy : X f´et → Bπ1 (X 0◦ ,y)

(III.10.8.3) 0◦

the fiber functor of X f´et at y (III.2.10.3). By (III.8.8.6), we have a canonical isomorphism ∼

ϕx (B) →

(III.10.8.4)

(p◦ )∗f´et (B U ),

lim −→

(U,p)∈V◦ x

◦

where B U is the sheaf of U f´et defined in (III.8.10.3). In view of (III.8.15.1), we deduce from this an isomorphism of OK -algebras of Bπ1 (X 0◦ ,y) ∼

νy (ϕx (B)) →

(III.10.8.5)

lim −→

y

RU ,

(U,p)∈V◦ x

y

where RU is the OK -algebra of Bπ1 (U ?◦ ,y) defined in (III.8.13.2). By VI.10.31 and VI.9.9, the ring underlying νy (ϕx (B)) is canonically isomorphic to the stalk B ρ(y x) . Remark III.10.9. For every geometric point x of X over s, there exists a point (y ←

◦

x) of X´et ×X´et X ´et (III.8.6). Indeed, denote by X 0 the strict localization of X at x. 0 By III.3.7, X is normal and strictly local (and in particular integral). Since X ◦ is 0◦ schematically dense in X by III.4.2(iv), X is integral and nonempty ([42] 11.10.5). Let 0◦ 0◦ ◦ v : y → X be a geometric point of X . We denote also by y the geometric point of X and by u : y → X 0 the X-morphism induced by v. We thus obtain a point (y x) of ←

◦

X´et ×X´et X ´et .

←

◦

Proposition III.10.10. Let (y x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over s and X 0 the strict localization of X at x. Then:

234

III. GLOBAL ASPECTS

(i) The stalk B ρ(y x) of B at ρ(y x) is a normal and strictly local ring. (ii) The stalk ~∗ (OX )x of ~∗ (OX ) at x is a normal and strictly local ring. (iii) The homomorphism ~∗ (OX )x → B ρ(y

(III.10.10.1)

x)

induced by the canonical homomorphism σ −1 (~∗ (OX )) → B (III.8.17) is injective and local. Denote by j 0 : X 0◦ → X 0 the canonical injection and by g, g, and ~0 the canonical arrows of the following Cartesian diagram: (III.10.10.2)

X

/X

g

0

~0



 X0

~

 /X

g

0

By III.3.7, X is normal and strictly local (and in particular integral). We denote by e 0 ) the Faltings site (resp. topos) associated with the morphism h0 = ~0 ◦ E 0 (resp. E 0 0◦ 0 e 0 associated with X 0 (III.8.17). We jX → X 0 (III.8.2) and by B the ring of E 0 : X denote by e0 σ0 : E

(III.10.10.3) (III.10.10.4)

ρ0 :

← X´e0 t ×X´e0 t

→ X´e0 t ,

0◦

e0 , → E

X ´et

the canonical morphisms (III.8.3.3) and (III.8.6.2), respectively, and by (III.10.10.5)

e 0 , B 0 ) → (E, e B) Φ : (E

the morphism of ringed topos deduced from g by functoriality (III.8.20). The X0◦ morphism u : y → X 0 defining (y x) induces an X 0 -morphism v : y → X . We 0◦ (abusively) denote also by x the closed point of X 0 , by y the geometric point of X ←

0◦

defined by v, and by (y x) the point of X´e0 t ×X´e0 t X f´et defined by u. The points e are then canonically isomorphic (VI.10.17). ρ(y x) and Φ(ρ0 (y x)) of E 0 (i) Since the canonical homomorphism Φ−1 (B) → B is an isomorphism by III.8.23, it induces an isomorphism (III.10.10.6)

∼

B ρ(y

x)

0

→ B ρ0 (y

x) .

The statement then follows from III.8.19(i). 0 (ii) By III.3.6, the canonical morphism g −1 (OX ) → OX 0 is an isomorphism of X ´et . By ([2] VIII 5.2), we deduce from this a canonical isomorphism (III.10.10.7)

0

∼

~∗ (OX )x → Γ(X , OX 0 ).

The statement follows by virtue of III.3.7. (iii) The diagram of morphisms of topos (III.10.10.8)

e0 E

Φ

σ0

 X´e0 t

/E e σ

g

 / X´et

III.10. HIGGS–TATE ALGEBRAS

235

is commutative up to canonical isomorphism (VI.10.12.6). Moreover, the diagram (III.10.10.9)

σ 0−1 (g −1 (~∗ (OX )))

Φ−1 (σ −1 (~∗ OX ))

/ Φ−1 (B)

/ σ 0−1 (~0∗ (O 0 )) X

 / 0 B

c

b

 σ 0−1 (~0∗ (g −1 (OX )))

a

where c is the base change morphism with respect to the diagram (III.10.10.2) and the other arrows are the canonical morphisms, is commutative. Since ~ is integral, c is an isomorphism ([2] VIII 5.6). On the other hand, a is an isomorphism (III.3.6) and b is an isomorphism (III.8.23). The statement then follows from III.8.19(iii). es . es is locally ringed by B s = B|E Corollary III.10.11. (i) The topos E e (ii) For every integer n ≥ 1, σn : (Es , B n ) → (Xs,´et , OX n ) (III.10.2.9) is a morphism of locally ringed topos. This follows from III.9.5, III.10.10, and ([2] IV 13.9). Proposition III.10.12. The absolute Frobenius endomorphism of B 1 is surjective. ←

◦

Let (y x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over s and that y ◦ is a generic geometric point of X , and X 0 the strict localization of X at x. Using the notation of III.10.8, we have a canonical isomorphism (III.10.8.5) (III.10.12.1)

∼

νy (ϕx (B 1 )) →

y

y

RU /pRU .

lim −→

(U,p)∈Vx (P)◦

By the functoriality of the isomorphism (III.8.8.6), the latter is compatible with the y y absolute Frobenius endomorphisms of B 1 and RU /pRU . For every (U, p) ∈ Ob(Vx (P)), y y the absolute Frobenius endomorphism of RU /pRU is surjective by virtue of III.10.4 and II.9.10. The proposition follows by virtue of III.9.5 and III.9.7. III.10.13. Let Y be an object of Q (III.10.5) such that Ys 6= ∅ and y a geometric ◦ point of Y . Since Y is locally irreducible (III.3.3), it is the sum of the schemes induced on ? its irreducible components. We denote by Y the irreducible component of Y containing ◦ y. Likewise, Y is the sum of the schemes induced on its irreducible components, and y ?◦ ? ◦ Y = Y ×X X ◦ is the irreducible component of Y containing y. We denote by RY the b y its p-adic Hausdorff completion. We set (III.2.2.1) ring defined in (III.8.13.2) and by R Y

y

y

RRy = lim RY /pRY ,

(III.10.13.1)

←−p

Y

x7→x

y

b Fontaine’s homomorphism defined in (II.9.3.4). and we denote by θY : W(RRy ) → R Y Y We set (III.10.13.2)

y

A2 (RY ) = W(RRy )/ ker(θY )2 , Y

and denote also by θY : Finally, we set

y A2 (RY

y

b the homomorphism induced by θ (II.9.3.5). ) → R Y Y y

(III.10.13.3)

Y

(III.10.13.4)

y Yb

(III.10.13.5)

A2 (Y )

y

y

=

Spec(RY ),

=

b ), Spec(R Y

=

Spec(A2 (RY )).

y y

236

III. GLOBAL ASPECTS y

Note that since Y is affine, Y is none other than the scheme defined in III.8.14. We y y endow Y (resp. Yb ) with the logarithmic structure MY y (resp. M b y ) inverse image of Y

y

MX (II.5.10) and A2 (Y ) with the logarithmic structure MA2 (Y y ) defined as follows. Let QY be the monoid and qY : QY → W(RRy ) the homomorphism defined in II.9.6 (denoted Y by Q and q in loc. cit.) by taking for u the canonical homomorphism Γ(Y, MX ) → y y Γ(Y , MY y ). We denote by MA2 (Y y ) the logarithmic structure on A2 (Y ) associated y

with the prelogarithmic structure defined by the homomorphism QY → A2 (RY ) induced by qY . The homomorphism θY then induces a morphism (II.9.6.4) y y iY : (Yb , M b y ) → (A2 (Y ), MA2 (Y y ) ).

(III.10.13.6)

Y

y

The logarithmic scheme (A2 (Y ), MA2 (Y y ) ) is fine and saturated and iY is an exact ?◦

closed immersion. Indeed, all fiber functors of Y f´et being isomorphic, it suffices to show this assertion in the case where y is localized at a generic point of Y . In view of III.10.4, the notation above then corresponds to that introduced in II.9.11, with the exception of MA2 (Y y ) , which rather corresponds to the logarithmic structure M 0 in loc. cit. y A2 (Y ) But since Y is an object of Q, the latter is canonically isomorphic to the logarithmic structure MA2 (Y y ) introduced in II.9.12 by virtue of II.9.13; whence the assertion (and notation). We set b y , ξR by ) e 1 (Y ) ⊗O (Y ) R TyY = Hom b y (Ω Y Y X/S X

(III.10.13.7)

RY

y

b y (cf. III.2.3). We denote b -module with ξ −1 Ω e 1 (Y ) ⊗O (Y ) R and identify the dual R Y Y X X/S y y e y the O y -module associated with Ty , and by Ty by Yb zar the Zariski topos of Yb , by T b Y Y Y Y

y the Yb -bundle associated with its dual, in other words, (III.2.7)

(III.10.13.8)

b y )). e 1 (Y ) ⊗O (Y ) R TyY = Spec(S b y (ξ −1 Ω Y X/S X RY

y e the open subscheme of A2 (Y y ) Let U be a Zariski open subscheme of Yb and U defined by U . We denote by LYy (U ) the set of morphisms represented by dotted arrows that complete the diagram

(III.10.13.9)

(U, M b y |U )

iY |U

 ˇ (X, MXˇ )  ˇ M ) (S, ˇ S

e, M / (U

y

A2 (Y )

Y

e) |U

 / (X, e M e) X

iS

  / (A2 (S), M A2 (S) )

in such a way that it remains commutative. By II.5.23, the functor U 7→ LYy (U ) is a y b y -module of affine functions on L y (cf. e y -torsor of Yb zar . We denote by F y the R T Y Y Y Y

III.10. HIGGS–TATE ALGEBRAS

237

II.4.9). It fits into a canonical exact sequence (II.4.9.1) b y → 0. b y → F y → ξ −1 Ω e 1 (Y ) ⊗O (Y ) R 0→R Y Y X/S X Y

(III.10.13.10)

This sequence induces, for every integer m ≥ 1, an exact sequence (III.5.2.2) (III.10.13.11)

y b y ) → 0. m −1 e 1 ΩX/S (Y ) ⊗OX (Y ) R 0 → Sm−1 (FYy ) → Sm Y b y (FY ) → S b y (ξ by RY

RY

RY

b y -modules (Sm (F y )) The R Y Y m∈N therefore form a filtered direct system, whose direct by RY

limit y CYy = lim Sm b y (FY )

(III.10.13.12)

−→

m≥0

RY

b y -algebra. By II.4.10, the Yb y -scheme is naturally endowed with a structure of R Y LyY = Spec(CYy )

(III.10.13.13)

y is naturally a principal homogeneous TyY -bundle on Yb that canonically represents LYy . e M e ) fixed Note that LYy , FYy , CYy , and LyY depend on the choice of the deformation (X, X in III.10.1. y ?◦ The group π1 (Y , y) has a natural left action on the logarithmic schemes (Yb , M b y ) y

Y

?◦

and (A2 (Y ), MA2 (Y y ) ), and the morphism iY is π1 (Y , y)-equivariant (cf. II.9.11). Proe y with a canonical structure of π1 (Y ?◦ , y)-equivariant ceeding as in II.10.4, we endow T Y ?◦ y e y -torsor (cf. O y -module and L with a canonical structure of π1 (Y , y)-equivariant T Y

b Y

Y

?◦

II.4.18). By II.4.21, these two structures induce a π1 (Y , y)-equivariant structure on the b y -semi-linear action of π (Y ?◦ , y) O y -module associated with F y , or, equivalently, an R b Y

Y

1

Y

?◦

FYy

on such that the morphisms in the sequence (III.10.13.10) are π1 (Y , y)-equivariant. ?◦ We deduce from this an action of π1 (Y , y) on CYy by ring automorphisms that is comby . patible with its action on R Y

Lemma III.10.14. Under the assumptions of III.10.13, the actions of π1 (Y FYy and on CYy are continuous for the p-adic topologies.

?◦

, y) on

?◦

Indeed, in view of III.8.15 and the fact that all fiber functors of Y f´et are isomorphic (III.2.10.3), we may assume that y is localized at a generic point of Y . We denote by ˇ ([42] 18.1.2) and by M (resp. e the unique étale morphism that lifts Yˇ → X Ye → X Yˇ ˇ e M ) the logarithmic structure on Y (resp. Y ) inverse image of M (resp. M ). Let U e Y

ˇ X

e X

y e the open subscheme of A2 (Y y ) defined by U . be a Zariski open subscheme of Yb and U The set LYy (U ) is then canonically isomorphic to the set of morphisms represented by

238

III. GLOBAL ASPECTS

dotted arrows that complete the diagram (U, M b y |U )

(III.10.14.1)

iY |U

e, M / (U

 (Yˇ , MYˇ )  ˇ (S, MSˇ )

y

A2 (Y )

Y

e) |U

 / (Ye , M e ) Y

iS

  / (A2 (S), M A2 (S) )

b y -algebra C y endowed with the in such a way that it remains commutative. The R Y Y ?◦ action of π1 (Y , y) therefore identifies with the Higgs–Tate algebra associated with (Y, MY , Ye , MYe ) defined in II.10.5 (cf. III.10.4). The lemma follows by virtue of II.10.4. III.10.15. Let Y be an object of Q (III.10.3) such that Ys 6= ∅ and n an integer ≥ 0. If A is a ring and M an A-module, we denote also by A (resp. M ) the constant ◦ ◦ sheaf with value A (resp. M ) of Y f´et . Since the scheme Y is locally irreducible, it is the sum of the schemes induced on its irreducible components. Let W be an irreducible ◦ component of Y and Π(W ) its fundamental groupoid (VI.9.10). In view of III.8.15 and VI.9.11, the sheaf B Y |W of Wf´et defines a functor y

y 7→ RY .

Π(W ) → Ens,

(III.10.15.1) We deduce from this a functor

Π(W ) → Ens,

(III.10.15.2)

y 7→ FYy /pn FYy .

By III.10.14, for every geometric point y of W , FYy /pn FYy is a discrete continuous representation of π1 (W, y). Consequently, by virtue of VI.9.11, the functor (III.10.15.2) defines a (B Y,n |W )-module FW,n of Wf´et , unique up to canonical isomorphism, where B Y,n = B Y /pn B Y (III.10.2.8). By descent ([35] II 3.4.4), there exists a B Y,n -module ◦ FY,n of Y f´et , unique up to canonical isomorphism, such that for every irreducible com◦ ponent W of Y , we have FY,n |W = FW,n . The exact sequence (III.10.13.10) induces an exact sequence of B Y,n -modules e 1 (Y ) ⊗O (Y ) B Y,n → 0. 0 → B Y,n → FY,n → ξ −1 Ω X/S X

(III.10.15.3)

This induces, for every integer m ≥ 1, an exact sequence (III.5.2.2) m−1 0 → SB (FY,n ) → Sm B Y,n

The B Y,n -modules (Sm B (III.10.15.4)

Y,n

Y,n

(FY,n ) → Sm B

Y,n

e 1 (Y ) ⊗O (Y ) B Y,n ) → 0. (ξ −1 Ω X/S X

(FY,n ))m∈N therefore form a direct system whose direct limit m CY,n = lim SB −→

Y,n

(FY,n )

m≥0

◦

is naturally endowed with a structure of B Y,n -algebra of Y f´et . Note that FY,n and CY,n e M e ) fixed in III.10.1. depend on the choice of the deformation (X, X

III.10. HIGGS–TATE ALGEBRAS

239

III.10.16. Let g : Y → Z be a morphism of Q such that Ys 6= ∅, y a geometric ◦ point of Y , and z = g(y). Recall that Y and Z are sums of the schemes induced on ? their irreducible components (III.3.3). We denote by Y the irreducible component of Y ? ? ? containing y and by Z the irreducible component of Z containing z, so that g(Y ) ⊂ Z . ◦ We take again the notation of III.10.13 for Y and for Z. The morphism g ◦ : Y → ◦ ?◦ ?◦ Z induces a group homomorphism π1 (Y , y) → π1 (Z , z). The canonical morphism ?◦ (g ◦ )∗f´et (B Z ) → B Y induces a π1 (Y , y)-equivariant ring homomorphism (III.8.15.1) z

y

RZ → RY

(III.10.16.1)

y ?◦ b z . Since g is and consequently a π1 (Y , y)-equivariant morphism of schemes h : Yb → Z ?◦ ey ) e z → h∗ (T étale, we have a canonical π1 (Y , y)-equivariant O b z -linear morphism u : T Z Y Z y ] ∗ ez e such that the adjoint morphism u : h (TZ ) → TY is an isomorphism. It immediately follows from the definitions (III.10.13.9) that we have a canonical u-equivariant and ?◦ π1 (Y , y)-equivariant morphism

v : LZz → h∗ (LYy ).

(III.10.16.2)

?◦

By II.4.22, the pair (u, v) induces a π1 (Y

b z -linear isomorphism , y)-equivariant and R Y

∼ by FYy → FZz ⊗ b z R Y

(III.10.16.3)

RZ

and, consequently, a π1 (Y

?◦

b z -linear morphism , y)-equivariant and R Z FZz → FYy

(III.10.16.4) that fits into a commutative diagram (III.10.16.5)

0

/ bz RZ

/ Fz Z

/ ξ −1 Ω e1

bz ⊗OX (Z) R Z

/0

0

 / by RY

 / Fy Y

 by / ξ −1 Ω 1 e X/S (Y ) ⊗OX (Y ) RY

/0

We deduce from this a π1 (Y

?◦

b z -algebras , y)-equivariant homomorphism of R Z

(III.10.16.6) We denote by Π(Y spectively, and by (III.10.16.7)

?◦

X/S (Z)

?◦

CZz → CYy .

) and Π(Z ) the fundamental groupoids of Y γ : Π(Y

?◦

?◦

?◦

and Z , re-

?◦

) → Π(Z )

´ ´ ?◦ → Et ?◦ the functor induced by the inverse image functor Et f/Z f/Y . For any integer ?◦

?◦

n ≥ 0, we denote by FY,n : Π(Y ) → Ens and FZ,n : Π(Z ) → Ens the functors asso?◦ ?◦ ?◦ ?◦ ciated by VI.9.11 with the objects FY,n |Y of Y f´et and FZ,n |Z of Z f´et , respectively. The morphism (III.10.16.4) clearly induces a morphism of functors (III.10.16.8)

FZ,n ◦ γ → FY,n .

By VI.9.11, we deduce from this a (g ◦ )∗f´et (B Z,n )-linear morphism (III.10.16.9)

(g ◦ )∗f´et (FZ,n ) → FY,n

240

III. GLOBAL ASPECTS

and therefore by adjunction a B Z,n -linear morphism FZ,n → g ◦f´et∗ (FY,n ).

(III.10.16.10)

It follows from (III.10.16.5) that the diagram (III.10.16.11) e 1 (Z) ×O (Z) (g ◦ )∗ (B Z,n ) / (g ◦ )∗ (FZ,n ) / ξ −1 Ω / (g ◦ )∗ (B Z,n ) 0 f´ et X f´ et X/S f´ et

0

 / B Y,n

 (Y ) × OX (Y ) B Y,n X/S

 / FY,n

e1 / ξ −1 Ω

/0

/0

is commutative. We deduce from this a homomorphism of (g ◦ )∗f´et (B Z,n )-algebras (g ◦ )∗f´et (CZ,n ) → CY,n ,

(III.10.16.12)

and therefore by adjunction a homomorphism of B Z,n -algebras CZ,n → g ◦f´et∗ (CY,n ).

(III.10.16.13)

III.10.17. For any integer n ≥ 0 and any object Y of Q such that Ys = ∅, we set CY,n = FY,n = 0. The exact sequence (III.10.15.3) still holds in this case, because B Y is a K-algebra. The morphisms (III.10.16.10) and (III.10.16.13) are then defined for every morphism of Q, and they verify cocycle relations of the type ([1] (1.1.2.2)). III.10.18. Let r be a rational number ≥ 0, n an integer ≥ 0, and Y an ob◦ (r) ject of Q. We denote by FY,n the extension of B Y,n -modules of Y f´et deduced from FY,n (III.10.15.3) by inverse image under the morphism of multiplication by pr on e 1 (Y ) ⊗O (Y ) B Y,n , so that we have a canonical exact sequence of B Y,n -modules ξ −1 Ω X X/S (r)

e 1 (Y ) ⊗O (Y ) B Y,n → 0. 0 → B Y,n → FY,n → ξ −1 Ω X/S X

(III.10.18.1)

For every integer m ≥ 1, this induces an exact sequence of B Y,n -modules (III.5.2.2) (r)

0 → Sm−1 (FY,n ) → Sm B B Y,n

The B Y,n -modules (Sm B (III.10.18.2)

(r)

Y,n

(FY,n ) → Sm B

Y,n

e 1 (Y ) ⊗O (Y ) B Y,n ) → 0. (ξ −1 Ω X/S X

(r)

Y,n

(FY,n ))m∈N therefore form a direct system whose direct limit (r)

(r)

CY,n = lim Sm B −→

Y,n

(FY,n )

m≥0

◦

is naturally endowed with a structure of B Y,n -algebra of Y f´et . For all rational numbers r ≥ r0 ≥ 0, we have a canonical B Y,n -linear morphism (III.10.18.3)

0

(r)

(r 0 )

r,r aY,n : FY,n → FY,n 0

e 1 (Y ) ⊗O (Y ) B Y,n and that extends the that lifts the multiplication by pr−r on ξ −1 Ω X X/S identity of B Y,n (III.10.18.1). It induces a homomorphism of B Y,n -algebras (III.10.18.4)

0

(r)

(r 0 )

r,r αY,n : CY,n → CY,n .

III.10. HIGGS–TATE ALGEBRAS

241

III.10.19. Let r be a rational number ≥ 0, n an integer ≥ 0, and g : Y → Z a morphism of Q. The diagram (III.10.16.11) induces a (g ◦ )∗f´et (B Z,n )-linear morphism (r)

that fits into a commutative diagram (III.10.19.2) / (g ◦ )∗ (B Z,n ) / (g ◦ )∗ (F (r) ) 0 f´ et f´ et Z,n

0

 / B Y,n

(r)

(g ◦ )∗f´et (FZ,n ) → FY,n

(III.10.19.1)

/0

e 1 (Z) ×O (Z) (g ◦ )∗ (B Z,n ) / ξ −1 Ω X f´ et X/S

 / F (r) Y,n

 (Y ) × OX (Y ) B Y,n X/S

/0

e1 / ξ −1 Ω

We deduce from this by adjunction a B Z,n -linear morphism (r)

(r)

FZ,n → (g ◦ )f´et∗ (FY,n ).

(III.10.19.3)

We also deduce from this a morphism of (g ◦ )∗f´et (B Z,n )-algebras (r)

(r)

(g ◦ )∗f´et (CZ,n ) → CY,n ,

(III.10.19.4)

and therefore by adjunction a morphism of B Z,n -algebras (r)

(r)

CZ,n → (g ◦ )f´et∗ (CY,n ).

(III.10.19.5)

The morphisms (III.10.19.3) and (III.10.19.5) satisfy cocycles relations of the type ([1] (1.1.2.2)). Lemma III.10.20. For every rational number r ≥ 0, every integer n ≥ 0, and every morphism g : Y → Z of Q, the morphisms (III.10.19.1) and (III.10.19.4) induce isomorphisms (r)

(III.10.20.1)

(g ◦ )∗f´et (FZ,n ) ⊗(g◦ )∗

(III.10.20.2)

(g ◦ )∗f´et (CZ,n ) ⊗(g◦ )∗

f´ et (BZ,n )

(r)

f´ et (BZ,n )

∼

(r)

∼

(r)

B Y,n

→ FY,n ,

B Y,n

→ CY,n .

Indeed, the first isomorphism results from the diagram (III.10.19.2) and the fact that the canonical morphism e 1 (Z) ⊗O (Z) OX (Y ) → Ω e 1 (Y ) (III.10.20.3) Ω X/S

X/S

X

is an isomorphism. The second isomorphism follows from the first. III.10.21. Let r be a rational number ≥ 0 and n an integer ≥ 0. By III.10.19, the correspondences (III.10.21.1)

(r)

(r)

{Y 7→ FY,n } and {Y 7→ CY,n },

(Y ∈ Ob(Q)),

define presheaves on EQ (III.10.5.2) of modules and algebras, respectively, with respect to the ring {Y 7→ B Y,n }. Let (III.10.21.2)

Fn(r)

(III.10.21.3)

Cn(r)

(r)

= {Y 7→ FY,n }a , (r)

= {Y 7→ CY,n }a ,

(r)

e (III.10.6.4). By (III.9.1.8) and (III.10.6.5), Fn is be the associated sheaves in E a B n -module; we call it the Higgs–Tate B n -extension of thickness r associated with e M e ). Likewise, Cn(r) is a B n -algebra; we call it the Higgs–Tate B n -algebra of (f, X, X e M e ). We set Fn = Fn(0) and Cn = Cn(0) , and call thickness r associated with (f, X, X

242

III. GLOBAL ASPECTS

these the Higgs–Tate B n -extension and the Higgs–Tate B n -algebra, respectively, associe M e ). ated with (f, X, X For all rational numbers r ≥ r0 ≥ 0, the morphisms (III.10.18.3) induce a B n -linear morphism 0

0

(r) (r ) ar,r n : Fn → Fn .

(III.10.21.4)

The homomorphisms (III.10.18.4) induce a homomorphism of B n -algebras 0

0

αnr,r : Cn(r) → Cn(r ) .

(III.10.21.5)

For all rational numbers r ≥ r0 ≥ r00 ≥ 0, we have 00

0

00

ar,r = anr ,r ◦ ar,r n n

(III.10.21.6)

0

00

0

00

0

and αnr,r = αnr ,r ◦ αnr,r .

Proposition III.10.22. Let r be a rational number ≥ 0 and n an integer ≥ 1. Then: (r) (r) es . (i) The sheaves Fn and Cn are objects of E

(ii) We have a canonical locally split exact sequence of B n -modules ( (III.10.1.4) and (III.10.2.9)) e1 ) → 0. 0 → B n → Fn(r) → σn∗ (ξ −1 Ω X n /S n

(III.10.22.1)

It induces, for every integer m ≥ 1, an exact sequence of B n -modules (III.2.7) (III.10.22.2)

m−1 m −1 e 1 0 → SB (Fn(r) ) → σn∗ (Sm (Fn(r) ) → SB ΩX n /S n )) → 0. OX (ξ n

n

In particular, the B n -modules

n

(r) (Sm (Fn ))m∈N Bn

form a filtered direct system.

(iii) We have a canonical isomorphism of B n -algebras ∼

Cn(r) → lim Sm (Fn(r) ). B

(III.10.22.3)

−→

n

m≥0

(iv) For all rational numbers r ≥ r0 ≥ 0, the diagram (III.10.22.4)

/B n

0

ar,r n

/B n

0

e1 / σn∗ (ξ −1 Ω X

/ F (r) n  / F (r0 ) n

0

n /S n

·pr−r

 e1 / σn∗ (ξ −1 Ω X

)

/0

)

/0

0

n /S n

where the horizontal arrows are the exact sequences (III.10.22.1) and the right 0 vertical arrow denotes the multiplication by pr−r , is commutative. Moreover, 0 0 the morphisms ar,r and αnr,r are compatible with the isomorphisms (III.10.22.3) n 0 for r and r . (r)

(r)

(i) Indeed, since FY,n = CY,n = 0 for every Y ∈ Ob(Q) such that Ys = ∅, we have

(r)

(r)

Fn |σ ∗ (Xη ) = Cn |σ ∗ (Xη ) = 0 by virtue of ([2] III 5.5). (ii) We have a canonical isomorphism (III.9.8.4) (III.10.22.5)

∼

e1 e 1 ) ⊗σ−1 (O ) B n . ) → σ −1 (ξ −1 Ω σn∗ (ξ −1 Ω X/S X X n /S n

e1 Hence by virtue of VI.5.34(ii), VI.8.9, VI.5.17, and (III.10.6.5), σn∗ (ξ −1 Ω ) is the X n /S n e sheaf of E associated with the presheaf on EQ defined by the correspondence (III.10.22.6)

e 1 (Y ) ⊗O (Y ) B Y,n }, {Y 7→ ξ −1 Ω X/S X

(Y ∈ Ob(Q)).

III.10. HIGGS–TATE ALGEBRAS

243

The exact sequence (III.10.22.1) then follows from (III.10.18.1) and (III.10.19.2) because the “associated sheaf” functor is exact ([2] II 4.1). It is locally split because the B n e1 module σn∗ (ξ −1 Ω ) is locally free of finite type. The exact sequence (III.10.22.2) X n /S n follows by (III.5.2.2). (iii) We easily deduce from III.10.7 and ([2] IV 12.10) that for every integer m ≥ 0, (r) m SB (Fn ) is the sheaf associated with the presheaf n

{Y 7→ Sm B

(III.10.22.7)

(r)

Y,n

(Y ∈ Ob(Q)).

(FY,n )},

The proposition follows in view of III.10.7 and the fact that the “associated sheaf” functor commutes with direct limits ([2] II 4.1). (iv) This immediately follows from the proofs of (ii) and (iii). III.10.23. Let r be a rational number ≥ 0 and n an integer ≥ 1. By III.10.22, we (r) have a canonical Cn -linear isomorphism ∼

e1 Ω1C (r) /B → σn∗ (ξ −1 Ω ) ⊗Bn Cn(r) . X n /S n

(III.10.23.1)

n

n

(r)

The universal B n -derivation of Cn B n -derivation

corresponds via this isomorphism to the unique

(r) ∗ −1 e 1 d(r) ΩX n /S n ) ⊗Bn Cn(r) n : Cn → σn (ξ

(III.10.23.2)

(r)

e1 ) (III.10.22.1). It follows that extends the canonical morphism Fn → σn∗ (ξ −1 Ω X n /S n 0 from III.10.22(iv) that for all rational numbers r ≥ r ≥ 0, we have 0

0

0

0

(r ) pr−r (id ⊗ αnr,r ) ◦ d(r) ◦ αnr,r . n = dn

(III.10.23.3)

III.10.24. Let Y be an object of Q (III.10.5) such that Ys 6= ∅ and y a geometric ◦ point of Y . We take again the notation of (III.10.13). For any rational number r ≥ 0, y,(r) b y -modules deduced from F y (III.10.13.10) by we denote by F the extension of R Y

Y

Y

r

inverse image under the morphism of multiplication by p on b y -modules so that we have an exact sequence of R Y

e 1 (Y ξ −1 Ω X/S

by , ) ⊗OX (Y ) R Y

b y → F y,(r) → ξ −1 Ω b y → 0. e 1 (Y ) ⊗O (Y ) R 0→R Y Y X/S X Y

(III.10.24.1)

b y -modules (III.5.2.2) This induces for every integer m ≥ 1 an exact sequence of R Y y,(r)

(III.10.24.2) 0 → Sm−1 (FY by RY

y,(r)

) → Sm b y (FY RY

b y ) → 0. −1 e 1 ) → Sm ΩX/S (Y ) ⊗OX (Y ) R Y b y (ξ RY

y

b -modules (Sm (F y,(r) )) In particular, the R m∈N form a filtered direct system whose Y Y by RY

direct limit (III.10.24.3)

y,(r)

CY

y,(r)

= lim Sm b y (FY −→

m≥0

),

RY

b y -algebra. is naturally endowed with a structure of R Y y,(r) y,(r) It follows from III.10.16 that the formation of FY and CY is functorial in the pair (Y, y). More precisely, let g : Y → Z be a morphism of Q and z the image of y by

244

III. GLOBAL ASPECTS

the morphism g ◦ : Y diagram (III.10.24.4)

◦

◦

→ Z . It immediately follows from III.10.16 that the canonical

0

/ bz RZ

/ F z,(r) Z

bz / ξ −1 Ω e 1 (Z) ⊗O (Z) R Z X X/S

/0

0

 / by RY

 / F y,(r) Y

 by / ξ −1 Ω e 1 (Y ) ⊗O (Y ) R Y X X/S

/0

is commutative. Since the canonical morphism (III.10.24.5)

e 1 (Z) ⊗O (Z) OX (Y ) → Ω e 1 (Y ) Ω X/S X/S X

is an isomorphism, we deduce from this that the canonical morphisms z,(r)

(III.10.24.6)

FZ

(III.10.24.7)

CZ

by ⊗bz R Y

→ FY

by ⊗bz R Y

→ CY

y,(r)

RZ

z,(r)

y,(r)

RZ

,

,

are isomorphisms. Since Y is locally irreducible (III.3.3), it is the sum of the schemes induced on its ? irreducible components. We denote by Y the irreducible component of Y containing ◦ y. Likewise, Y is the sum of the schemes induced on its irreducible components, and ?◦ ? ◦ Y = Y ×X X ◦ is the irreducible component of Y containing y. We denote by ?◦ Bπ1 (Y ?◦ ,y) the classifying topos of the profinite group π1 (Y , y) and by ?◦ ∼

νy : Y f´et → Bπ1 (Y ?◦ ,y)

(III.10.24.8)

the fiber functor at y (III.2.10.3). We then have a canonical ring isomorphism (III.8.15.1) νy (B Y |Y

(III.10.24.9)

?◦

y

∼

) → RY .

Since νy is exact and commutes with direct limits, for every integer n ≥ 0, we have y y canonical isomorphisms of RY -modules and of RY -algebras, respectively, (III.10.24.10) (III.10.24.11)

(r)

νy (FY,n |Y

?◦

∼

y,(r)

) → FY

?◦ (r) νy (CY,n |Y )

∼

→

y,(r)

/pn FY

,

y,(r) y,(r) CY /pn CY . ←

◦

III.10.25. Let (y x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over s and X 0 the strict localization of X at x. We take again the notation of III.10.8. Set (III.10.25.1)

y

RX 0 =

lim −→

y

RU ,

(U,p)∈V◦ x

y b y the p-adic Hausdorff where RU is the ring defined in (III.8.13.2). We denote by R X0 y completion of RX 0 . We have a canonical isomorphism (III.10.8.5)

(III.10.25.2)

∼

y

νy (ϕx (B)) → RX 0 .

III.10. HIGGS–TATE ALGEBRAS

For any rational number r ≥ 0, we set (III.10.25.3)

y,(r)

FX 0

=

y,(r)

−→

FU

lim

CU

lim

y,(r)

CX 0

=

y,(r)

−→

by , ⊗by R X0 RU

(U,p)∈Vx (Q)◦

(III.10.25.4)

245

by 0, ⊗by R X RU

(U,p)∈Vx (Q)◦

y,(r) b y -module defined in (III.10.24.1) and C y,(r) is the R b y -algebra deis the R where FU U U U y,(r) y,(r) fined in (III.10.24.3). We denote by CbX 0 the p-adic Hausdorff completion of CX 0 . By (III.10.24.6) and (III.10.24.7), for every object (U, p) of Vx (Q), the canonical homomorphisms y,(r)

(III.10.25.5)

FU

(III.10.25.6)

CU

by ⊗by R X0

→ FX 0 ,

by 0 ⊗by R X

→ CX 0 ,

RU

y,(r)

RU

y,(r)

y,(r)

are isomorphisms. Remark III.10.26. Under the assumptions of III.10.25, for every integer n ≥ 1, the canonical morphisms (III.10.26.1)

lim −→

y,(r)

FU

y,(r)

→ FX 0 /pn FX 0 ,

y,(r)

→ CX 0 /pn CX 0 ,

/pn FU

(U,p)∈Vx (Q)◦

(III.10.26.2)

y,(r)

CU

lim −→

/pn CU

(U,p)∈Vx (Q)◦

y,(r)

y,(r)

y,(r)

y,(r)

are isomorphisms. ←

◦

Lemma III.10.27. Let (y x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over s, X 0 the strict localization of X at x, and r a rational number ≥ 0. We take again 0 the notation of III.10.8 and III.10.25; moreover, we set R10 = Γ(X , OX 0 ) and denote by b0 its p-adic Hausdorff completion. Then: R 1 y

(i) The rings R10 and RX 0 are OK -flat, normal, and integral domains, and the y

canonical homomorphism R10 → RX 0 is injective and integral. b y 0 , C y,(r) , and Cby,(r) are O -flat. b0 , R (ii) The rings R C X 1 X0 X0

y

y

(iii) For every integer n ≥ 1, the canonical homomorphism R10 /pn R10 → RX 0 /pn RX 0 is injective. b y 0 is injective, and the p-adic topology b0 → R (iv) The canonical homomorphism R 1

X

b y [ 1 ] (II.2.2). b0 [ 1 ] is induced by the p-adic topology on R on R X0 p 1 p

(i) The ring R10 is normal and an integral domain by virtue of III.3.7, and it is clearly y OK -flat. For every object (U, p) of Vx (P), the ring RU (III.8.13.2) is OK -flat, normal, ? and an integral domain by III.8.11 and ([39] 0.6.5.12(ii)). Moreover, denoting by U the ? irreducible component of U containing y, the canonical homomorphism Γ(U , OU ? ) → y

RU is injective and integral. One immediately verifies that for every morphism (U 0 , p0 ) → y y y (U, p) of Vx (P), the canonical homomorphism RU → RU 0 is injective. Consequently, RX 0 0 is OK -flat, normal, and an integral domain. Since X is integral by III.3.7, we have

(III.10.27.1)

R10 '

lim −→

(U,p)∈Vx (P)◦

?

Γ(U , OU ? ).

246

III. GLOBAL ASPECTS y

The canonical homomorphism R10 → RX 0 is therefore injective and integral. (ii) By ([11] Chapter III §2.11 Proposition 14 and Corollary 1), for every n ≥ 1, we have b10 /pn R b10 ' R10 /pn R10 . R

(III.10.27.2)

b0 be such that pα = 0 and α its class in R0 /pn R0 (n ≥ 1). Since R0 is Let α ∈ R 1 1 1 1 b0 /pn R b0 . We deduce from this that OK -flat, it follows from (III.10.27.2) that α ∈ pn−1 R 1 1 b0 = {0}. Consequently, p is not a zero divisor in R b0 , and hence R b0 is OC -flat α ∈ ∩n≥0 pn R 1 1 1 y b 0 is O -flat. Consequently, ([11] Chapter VI §3.6 Lemma 1). We prove similarly that R C

X

y,(r)

y,(r)

CX 0

is OC -flat (III.10.25.6). As above, we deduce from this that CbX 0 is OC -flat. (iii) This immediately follows from (i). (iv) The first assertion immediately follows from (iii). By (iii) and (III.10.27.2), b y = pn R b0 . Since R b0 is OC -flat, it follows that b0 ∩ pn R for every n ≥ 1, we have R X0 1 1 1 y 0 1 nb n b0 b R [ ] ∩ p RX 0 = p R , giving the second assertion. 1 p

1

Proposition III.10.28. Let x be a point of X over s, X 0 the strict localization of X at x, r a rational number ≥ 0, and n an integer ≥ 0. Then, with the notation of (III.10.8), 0◦ we have canonical isomorphisms of X f´et lim

(III.10.28.1)

−→

(U,p)∈Vx (Q)◦

lim

(III.10.28.2)

−→

(U,p)∈Vx (Q)◦

(r)

∼

(r)

∼

(p◦ )∗f´et (CU,n ) → ϕx (Cn(r) ), (p◦ )∗f´et (FU,n ) → ϕx (Fn(r) ).

Note that the proposition does not follow directly from VI.10.37 because Q is not stable under fibered products. Set (III.8.2.1) (III.10.28.3)

Cn(r) = {U 7→ CU },

´ /X ). U ∈ Ob(Et

By virtue of VI.10.37, we have a functorial canonical isomorphism ∼

ϕx (Cn(r) ) →

(III.10.28.4)

(p◦ )∗f´et (CU ).

lim −→

(U,p)∈V◦ x

We may obviously replace Vx by the category Vx (Q) (III.10.8). To prove (III.10.28.1), it therefore suffices to prove that the canonical morphism (III.10.28.5)

γ:

lim −→

(U,p)∈Vx (Q)◦

(r)

(p◦ )∗f´et (CU,n ) →

lim −→

(p◦ )∗f´et (CU )

(U,p)∈Vx (Q)◦

0◦

0◦

is an isomorphism. Let y be a geometric point of X and φy : X f´et → Ens the fiber 0◦ functor associated with the point ρX 0◦ (y) of X f´et (III.2.9.1). It also suffices to show that φy (γ) is an isomorphism (VI.9.6). ◦ We denote also by y the geometric point of X induced by y, by u : y → X 0 the ← ◦ canonical X-morphism, and by (y x) the point of X´et ×X´et X ´et defined by u. Let Qρ(y x) be the category of ρ(y x)-pointed objects of EQ (III.10.5); in other words, the category of triples ((V → U ), p, q) consisting of an object (V → U ) of EQ , an X◦ morphism p : x → U , and an X -morphism q : y → V such that if we denote also by

III.10. HIGGS–TATE ALGEBRAS

247

p : X 0 → U the X-morphism induced by p, the diagram u

y

(III.10.28.6)

/ X0

q

p

 V

 /U

is commutative. By VI.10.20, Qρ(y x) is canonically equivalent to the category of neighborhoods of ρ(y x) in EQ ([2] IV 6.8.2). It is therefore cofiltered and for every presheaf F = {U 7→ FU } on EQ , we have a functorial canonical isomorphism ([2] IV (6.8.4)) (F a )ρ(y

(III.10.28.7)

∼

x)

→

FU (V ).

lim −→

((V →U ),p,q)∈Q◦ ρ(y

x)

We have a functor α : Qρ(y

(III.10.28.8)

x)

→ Vx (Q),

((V → U ), p, q) 7→ (U, p).

For every (U, p) ∈ Ob(Vx (Q)), the fiber of α over (U, p) is canonically equivalent to the ◦ y category D(U,p) of p◦ (y)-pointed, finite étale U -schemes (III.10.8.1). In view of (VI.9.3.4) and ([2] IV (6.8.4)), φy (γ) identifies with the canonical map (III.10.28.9) (r) φy (γ) : lim lim CU,n (V ) → lim lim CU (V ). −→

−→

−→

(U,p)∈Vx (Q)◦ (V,q)∈(D y )◦ (U,p)

−→

(U,p)∈Vx (Q)◦ (V,q)∈(D y )◦ (U,p)

We may clearly replace each of the double direct limits above with a direct limit on the ◦ category Qρ(y x) . It then follows from (III.10.28.7) that φy (γ) is bijective; this gives the isomorphism (III.10.28.1). The proof of (III.10.28.2) is similar. ←

◦

Corollary III.10.29. Let (y x) be a point of X´et ×X´et X ´et such that x lies over s, X 0 the strict localization of X at x, n an integer ≥ 0, and r a rational number ≥ 0. Then, using the notation of III.10.8 and III.10.24, we have canonical isomorphisms of Bπ1 (X 0◦ ,y) (III.10.29.1) (III.10.29.2)

∼

νy (ϕx (Cn(r) )) →

CU

lim

FU

y,(r)

/pn CU

,

(U,p)∈Vx (Q)◦

∼

νy (ϕx (Fn(r) )) →

y,(r)

−→

lim

−→

y,(r)

y,(r)

/pn FU

.

(U,p)∈Vx (Q)◦

This follows from III.10.28, (III.10.24.10), and (III.10.24.11). Remarks III.10.30. We keep the assumptions of III.10.29 and moreover suppose that n ≥ 1. (i) The isomorphisms (III.10.25.2), (III.10.29.1), and (III.10.29.2) are compatible with each other. (ii) We have a canonical isomorphism of Bπ1 (X 0◦ ,y) (III.10.30.1)

∼

e1 νy (ϕx (σn∗ (ξ −1 Ω ))) → X n /S n

lim −→

(U,p)∈V◦ x

y

y

e 1 (U ) ⊗O (U ) (R /pn R ). ξ −1 Ω U U X/S X

e1 Indeed, by (III.10.22.5), VI.5.34(ii), VI.8.9, and VI.5.17, σn∗ (ξ −1 Ω ) is the X n /S n e sheaf of E associated with the presheaf on EQ defined by the correspondence (III.10.30.2)

e 1 (U ) ⊗O (U ) B U,n }, {U 7→ ξ −1 Ω X/S X

´ /X )). (U ∈ Ob(Et

The assertion therefore follows from VI.10.37 and (III.8.15.1).

248

III. GLOBAL ASPECTS

(iii) The image of the exact sequence (III.10.22.1) by the composed functor νy ◦ ϕx identifies with the inverse limit on the category Vx (Q)◦ of the exact sequences (III.10.30.3) y y y,(r) y,(r) e 1 (U ) ⊗O (U ) (Ry /pn Ry ) → 0 → ξ −1 Ω 0 → RU /pn RU → FU /pn FU U U X/S X deduced from (III.10.24.1). (iv) The image of the isomorphism (III.10.22.3) by the composed functor νy ◦ ϕx identifies with the isomorphism y,(r)

CU

lim

(III.10.30.4)

−→

y,(r) ∼

/pn CU

(U,p)∈Vx (Q)◦

→

y,(r)

lim Sm y (FU R

lim −→

−→

(U,p)∈Vx (Q)◦ m≥0

U

y,(r)

/pn FU

)

deduced from (III.10.24.3). The proof is similar to that of III.10.28 and is left to the reader. ˘ the ring (B e N◦ We denote by B ˘ the ring n+1 )n∈N of Es (III.7.7), by OX N◦ −1 e 1 of Xs,´ (not to be confused with O (III.10.1.2)), and by ξ Ω ˘ ˘ the ˇ et X

III.10.31. (OX n+1 )n∈N

−1 e 1 OX ΩX ˘ -module (ξ

X/S

◦

n+1 /S n+1

N )n∈N of Xs,´ et (III.10.1.4). We denote by

˘ → (X N◦ , O ) esN◦ , B) σ ˘ : (E ˘ s,´ et X

(III.10.31.1)

the morphism of ringed topos induced by the (σn+1 )n∈N (III.10.2.9). Let r be a rational number ≥ 0. For all integers m ≥ n ≥ 1, we have a canonical (r) (r) B m -linear morphism Fm → Fn and a canonical homomorphism of B m -algebras (r) (r) Cm → Cn , compatible with the exact sequence (III.10.22.1) and the isomorphism (III.10.22.3) and such that the induced morphisms (III.10.31.2)

(r) Fm ⊗Bm B n → Fn(r)

(r) and Cm ⊗Bm B n → Cn(r)

are isomorphisms. These morphisms form compatible systems when m and n vary, so (r) (r) ˘ of that (Fn+1 )n∈N and (Cn+1 )n∈N are inverse systems. We call Higgs–Tate B-extension (r) ˘ (r) ˘ e thickness r associated with (f, X, MXe ), and denote by F , the B-module (Fn+1 )n∈N ˘ e M ), and e N◦ . We call Higgs–Tate B-algebra of thickness r associated with (f, X, of E s

e X

(r) ˘ ˘ esN◦ . These are adic B-modules denote by C˘(r) , the B-algebra (Cn+1 )n∈N of E (III.7.16). By III.7.3(i), (III.7.5.4), and (III.7.12.1), the exact sequence (III.10.22.1) induces an ˘ exact sequence of B-modules

(III.10.31.3)

˘ → F˘ (r) → σ e 1˘ 0→B ˘ ∗ (ξ −1 Ω

˘ X/S

) → 0.

˘ e1 e1 Since the OX -module Ω ˘ ∗ (ξ −1 Ω ˘ X/S is locally free of finite type, the B-module σ

˘ X/S

) is

locally free of finite type and the sequence (III.10.31.3) is locally split. By (III.5.2.2), it ˘ induces, for every integer m ≥ 1, an exact sequence of B-modules (III.10.31.4)

−1 e 1 ˘ (r) ) → σ 0 → Sm−1 (F˘ (r) ) → Sm ˘ ∗ (Sm Ω˘ ˘ (F O ˘ (ξ ˘ B

B

X

˘ X/S

)) → 0.

˘ ˘ (r) ))m∈N form a filtered direct system. By III.7.3(i), In particular, the B-modules (Sm ˘ (F B ˘ (III.7.12.3), and (III.10.22.3), we have a canonical isomorphism of B-algebras (III.10.31.5)

∼ ˘ (r) ). C˘(r) → lim Sm ˘ (F −→

m≥0

B

III.10. HIGGS–TATE ALGEBRAS

249

˘ and the Set F˘ = F˘ (0) and C˘ = C˘(0) . We call these the Higgs–Tate B-extension ˘ e Higgs–Tate B-algebra, respectively, associated with (f, X, MXe ). For all rational numbers 0 ˘ r ≥ r0 ≥ 0, the morphisms (ar,r ) (III.10.21.4) induce a B-linear morphism n∈N

n

0 0 a ˘r,r : F˘ (r) → F˘ (r ) .

(III.10.31.6)

0 ˘ The homomorphisms (αnr,r )n∈N (III.10.21.5) induce a homomorphism of B-algebras 0 0 α ˘ r,r : C˘(r) → C˘(r ) .

(III.10.31.7)

For all rational numbers r ≥ r0 ≥ r00 ≥ 0, we have 00

0

00

a ˘r,r = a ˘r ,r ◦ a ˘r,r

(III.10.31.8)

0

00

0

00

0

and α ˘ r,r = α ˘ r ,r ◦ α ˘ r,r .

(r)

The derivations (dn+1 )n∈N (III.10.23.2) define a morphism (III.10.31.9)

e 1˘ d˘(r) : C˘(r) → σ ˘ ∗ (ξ −1 Ω

˘ X/S

) ⊗ ˘ C˘(r) , B

˘ that is none other than the universal B-derivation of C˘(r) . It extends the canonical (r) ∗ −1 e 1 ˘ morphism F → σ ˘ (ξ Ω ˘ ˘ ). For all rational numbers r ≥ r0 ≥ 0, we have X/S

0 0 0 0 pr−r (id ⊗ α ˘ r,r ) ◦ d˘(r) = d˘(r ) ◦ α ˘ r,r .

(III.10.31.10)

Remarks III.10.32. Let r be a rational number ≥ 0 and n an integer ≥ 1.

(r)

(r)

m (Fn ) → Cn and (i) For every integer m ≥ 0, the canonical morphisms SB n m ˘ (r) (r) S ˘ (F ) → C˘ are injective. Indeed, for every integer m0 ≥ m, the canonical B

0

(r)

(r)

m morphism SB (Fn ) → Sm (Fn ) is injective (III.10.22.2). Since filtered Bn n e N◦ , Sm (Fn(r) ) → Cn(r) is direct limits commute with finite inverse limits in E s

Bn

injective. The second assertion follows from the first by III.7.3(i). (r) (r) (r) (r) e1 (ii) We have σn∗ (ξ −1 Ω ) = dn (Fn ) ⊂ dn (Cn ) (III.10.23.2). Consequently, X /S n

n

(r)

e1 the derivation dn is a Higgs B n -field with coefficients in σn∗ (ξ −1 Ω ) by X n /S n II.2.12. e 1 ) = d˘(r) (F˘ (r) ) ⊂ d˘(r) (C˘(r) ) (III.10.31.9). Consequently, (iii) We have σ ˘ ∗ (ξ −1 Ω ˘ ˘ X/S

˘ e1 with coefficients in σ ˘ ∗ (ξ −1 Ω the derivation d˘(r) is a Higgs B-field ˘

˘ X/S

).

Proposition III.10.33. For every rational number r ≥ 0, the functor (III.10.33.1)

˘ → Mod(C˘(r) ), Mod(B)

M 7→ M ⊗ ˘ C˘(r) B

˘ is exact and faithful; in particular, C˘(r) is B-flat. e1 Since the OX -module Ω X/S is locally free of finite type, the exact sequence (III.10.31.3) is locally split. A local splitting of this sequence induces, for every integer m ≥ 0, a local splitting of the exact sequence (III.10.31.4). We deduce from this ˘ ˘ → C˘(r) admits locally that Sm (F˘ (r) ) is B-flat and that the canonical homomorphism B ˘ B

sections. The proposition follows in view of (III.10.31.5).

250

III. GLOBAL ASPECTS

III.11. Cohomological computations III.11.1. We keep the assumptions and general notation of III.10 in this section. ˘ the We moreover denote by d = dim(X/S) the relative dimension of X over S, by B ◦ ◦ N N◦ esN (III.7.7), and by O ˘ the ring (O ring (B n+1 )n∈N of E et , X n+1 )n∈N of Xs,zar or of Xs,´ X depending on the context (cf. III.2.9 and III.9.9). For all integers i, n ≥ 1, we set i e1 ei Ω X/S = ∧ (ΩX/S )

(III.11.1.1) ei We denote by ξ −i Ω ˘

˘ X/S

e1 ei = ∧i (Ω ). and Ω X n /S n X n /S n

−i e i the OX ˘ -module (ξ ΩX

n+1 /S n+1

)n∈N . We have a canonical iso-

morphism (III.7.12.4) ∼

e i˘ ξ −i Ω

(III.11.1.2)

e 1˘ → ∧i (ξ −1 Ω

˘ X/S

˘ X/S

).

For any integer n ≥ 1, we denote by

es , B n ) → (Xs,´et , O ) σn : ( E Xn

(III.11.1.3)

the canonical morphism of ringed topos (III.10.2.9), by un : (Xs,´et , OX n ) → (Xs,zar , OX n )

(III.11.1.4)

the canonical morphism of ringed topos (III.2.9), and by es , B n ) → (Xs,zar , O ) τn : (E Xn

(III.11.1.5)

the composition un ◦ σn . We denote by ˘ N◦ esN◦ , B) σ ˘ : (E → (Xs,´ (III.11.1.6) ˘ ), et , OX ◦

◦

N N u ˘ : (Xs,´ ˘ ) → (Xs,zar , OX ˘ ), et , OX

(III.11.1.7)

˘ N◦ esN◦ , B) τ˘ : (E → (Xs,zar , OX ˘ ),

(III.11.1.8)

the morphisms of ringed topos induced by the (σn+1 )n∈N , (un+1 )n∈N , and (τn+1 )n∈N , respectively, so that τ˘ = u ˘◦σ ˘. Recall that we have set S = Spf(OC ) (III.2.1). We denote by X the formal scheme p-adic completion of X. It is a formal S -scheme of finite presentation ([1] 2.3.15). It ei is therefore idyllic ([1] 2.6.13). We denote by ξ −i Ω X/S the p-adic completion of the −i e i −i e i O -module ξ Ω ⊗O O ([1] 2.5.1). We have a canonical isomorphism =ξ Ω X

X/S

X/S

X

X

([1] 2.5.5(ii)) (III.11.1.9)

∼

i −1 e 1 ei ξ −i Ω ΩX/S ). X/S → ∧ (ξ

We denote by (III.11.1.10)

◦

N λ : (Xs,zar , OX˘ ) → (Xs,zar , OX )

the morphism defined in (III.9.10.5) and by (III.11.1.11)

˘ → (X esN◦ , B) > : (E s,zar , OX )

the composition λ ◦ τ˘. For modules, we use the notation >−1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation >∗ for the inverse image in the sense of modules. We do likewise for σ ˘ and τ˘. For every OX -module N of Xs,zar , we have a canonical isomorphism (III.11.1.12)

∼

>∗ (N ) → τ˘∗ ((N /pn+1 N )n∈N ).

In particular, >∗ (N ) is adic (III.7.18).

III.11. COHOMOLOGICAL COMPUTATIONS

III.11.2. (III.11.2.1)

251

For every integer n ≥ 1, the adjunction morphism of Xs,´et e1 e1 → σn∗ (σn∗ (ξ −1 Ω )) ξ −1 Ω X n /S n X n /S n

induces an OX n -linear morphism (III.11.2.2)

e1 e1 ξ −1 Ω → τn∗ (σn∗ (ξ −1 Ω )). X n /S n X n /S n

Likewise, in view of (III.7.5.3), the adjunction morphism (III.11.2.3)

e 1˘ ξ −1 Ω

˘ X/S

e 1˘ →σ ˘∗ (˘ σ ∗ (ξ −1 Ω

))

e 1˘ → τ˘∗ (˘ σ ∗ (ξ −1 Ω

)).

˘ X/S

induces an OX ˘ -linear morphism (III.11.2.4)

e 1˘ ξ −1 Ω

˘ X/S

˘ X/S

The latter induces an OX -linear morphism (III.11.2.5)

e1 e 1˘ ξ −1 Ω σ ∗ (ξ −1 Ω X/S → >∗ (˘

˘ X/S

)).

Note that the adjoint morphism (III.11.2.6)

e1 ) → σ e 1˘ >∗ (ξ −1 Ω ˘ ∗ (ξ −1 Ω X/S

˘ X/S

)

is an isomorphism by (III.11.1.12) and the remark following (III.2.9.5). We denote by (III.11.2.7)

e1 ∂n : ξ −1 Ω → R1 σn∗ (B n ) X n /S n

the OX n -linear morphism of Xs,´et composed of the morphism (III.11.2.1) and the boundary map of the long exact sequence of cohomology deduced from the canonical exact sequence (III.10.22.1) (III.11.2.8)

e1 0 → B n → Fn → σn∗ (ξ −1 Ω ) → 0. X n /S n

We denote by e 1˘ ∂˘ : ξ −1 Ω

(III.11.2.9)

˘ X/S

˘ → R1 σ ˘∗ (B)

◦

N the OX ˘ -linear morphism of Xs,´ et composed of the morphism (III.11.2.3) and the boundary map of the long exact sequence of cohomology deduced from the canonical exact sequence (III.10.31.3)

(III.11.2.10)

˘ → F˘ → σ e 1˘ 0→B ˘ ∗ (ξ −1 Ω

˘ X/S

) → 0.

In view of (III.7.5.4), (III.7.5.5), and (III.7.12.1), we can identify ∂˘ with the morphism (∂n )n≥1 . We denote by (III.11.2.11)

˘ 1 e1 δ : ξ −1 Ω X/S → R >∗ (B)

the OX -linear morphism of Xs,zar composed of the morphism (III.11.2.5) and the boundary map of the long exact sequence of cohomology deduced from the canonical exact sequence (III.10.31.3) (III.11.2.12)

˘ → F˘ → σ e 1˘ 0→B ˘ ∗ (ξ −1 Ω

Proposition III.11.3. Let n be an integer ≥ 1.

˘ X/S

) → 0.

252

III. GLOBAL ASPECTS

(i) There exists a unique homomorphism of graded OX n -algebras of X n,´et e1 ) → ⊕i≥0 Ri σn∗ (B n ) ∧ (ξ −1 Ω X n /S n

(III.11.3.1)

whose degree one component is the morphism ∂n (III.11.2.7). Moreover, its 2d+1 2d kernel is annihilated by p p−1 mK and its cokernel is annihilated by p p−1 mK (III.11.1). (ii) For every integer i ≥ d + 1, Ri σn∗ (B n ) is almost zero.

Let x be a geometric point of X over s, X 0 the strict localization of X of x, and e → X 0◦ ϕx : E f´ et

(III.11.3.2)

the functor (III.8.8.4). By VI.10.30 and (III.9.8.6), for every integer i ≥ 0, we have a canonical isomorphism 0◦

∼

(III.11.3.3) 0

(Ri σn∗ (B n ))x → Hi (X f´et , ϕx (B n )).

By virtue of III.3.7, X is normal and strictly local (and in particular integral). Since 0 0◦ 0◦ X is S-flat, X is integral and nonempty. Let v : y → X be a generic geometric point 0◦ of X and 0◦

∼

νy : X f´et → Bπ1 (X 0◦ ,y)

(III.11.3.4)

◦

the associated fiber functor (III.2.10.3). We denote also by y the geometric point of X and by u : y → X 0 the X-morphism induced by v. We thus obtain a point (y x) of ←

◦

X´et ×X´et X ´et . We denote by Vx (resp. Vx (Q)) the category of neighborhoods of the point of X´et ´ /X (resp. Q (III.10.5)). For any (U, p : x → U ) ∈ Ob(Vx ), associated with x in the site Et 0 we denote also by p : X → U the morphism deduced from p and we set (III.11.3.5)

◦

p◦ = p ×X X : X

0◦

◦

→U .

◦

We (abusively) denote also by y the geometric point p◦ (v(y)) of U . Note that y is ◦ localized at a generic point of U because p◦ is plat. Since U is locally irreducible (III.3.3), it is the sum of the schemes induced on its irreducible components. Denote ? ◦ by U the irreducible component of U containing y. Likewise, U is the sum of the ?◦ ? schemes induced on its irreducible components, and U = U ×X X ◦ is the irreducible ◦ ?◦ component of U containing y. The morphism p◦ factors through U . We have a canonical isomorphism (III.10.8.4) (III.11.3.6)

∼

ϕx (B) →

(p◦ )∗f´et (B U ).

lim −→

(U,p)∈V◦ x

It induces, for every integer i ≥ 0, an isomorphism (III.11.3.7)

0◦

∼

Hi (X f´et , ϕx (B n )) →

lim −→

◦

Hi ((U f´et )/U ?◦ , B U,n ).

(U,p)∈V◦ x

Indeed, if (Z, q) is an object of Vx such that Z is affine, then it suffices to apply VI.11.10 to the full subcategory of (Vx )/(Z,q) made up of the (U, p)’s such that U is affine. In view of (III.8.15.1) and (VI.9.8.6), we deduce from (III.11.3.3) and (III.11.3.7) an isomorphism (III.11.3.8)

∼

(Ri σn∗ (B n ))x →

lim −→

?◦

y

y

Hi (π1 (U , y), RU /pn RU ).

(U,p)∈V◦ x

The isomorphisms (III.11.3.3) and (III.11.3.7) are clearly compatible with cup products. The same therefore holds for (III.11.3.8).

III.11. COHOMOLOGICAL COMPUTATIONS

253

0

On the other hand, since X is strictly local (III.3.7), it identifies with the strict localization of X at x. We therefore have a canonical isomorphism ? ∼ e1 e1 (III.11.3.9) Ω (U ), → lim Ω X n /S n ,x

e1 where we view Ω X

n /S n

X n /S n

−→

(U,p)∈V◦ x

as a sheaf of X ´et . ?

(i) Let (U, p) be an object of Vx (Q). Note that the schemes U , U , and U are affine. y ?◦ The exact sequence of RU -representations of π1 (U , y) (III.11.3.10)

y

y

y

y

e 1 (U ) ⊗O (U ) (RU /pn RU ) → 0 0 → RU /pn RU → FUy /pn FUy → ξ −1 Ω X/S X ?

deduced from (III.10.13.10), induces an OX n (U )-linear morphism ?

?◦

y

y

e1 α(U,p) : ξ −1 Ω (U ) → H1 (π1 (U , y), RU /pn RU ). X n /S n

(III.11.3.11)

We can describe this morphism explicitly in view of III.10.4. Indeed, by II.10.16, we have a commutative diagram (III.11.3.12) α(U,p) ? e1 / H1 (π (U ?◦ , y), Ry /pn Ry ) ξ −1 Ω (U ) 1 X n /S n U U O a

c

 ?◦ y 1 −1 y H (π1 (U , y), ξ RU (1)/pn ξ −1 RU (1))

1 1 y y / H1 (π (U ?◦ , y), p p−1 RU /pn+ p−1 RU ) 1

−b ∼

where a is induced by the morphism defined in (II.8.13.2), b is induced by the isomorphism (III.11.3.13)

1 ∼ b y (1) → by R p p−1 ξ R U U

y

1

y

defined in II.9.18, and c is induced by the canonical injection p p−1 RU → RU . ? By virtue of II.8.17(i), there exists a unique homomorphism of graded OX n (U )algebras (III.11.3.14)

?

?◦

y

y

e1 ∧ (ξ −1 Ω (U )) → ⊕i≥0 Hi (π1 (U , y), ξ −i RU (i)/pn ξ −i RU (i)) X n /S n

whose component in degree one is the morphism a. This is almost injective and its 1 cokernel is annihilated by p p−1 mK . We deduce from this that there exists a unique ? homomorphism of graded OX n (U )-algebras (III.11.3.15)

?

?◦

y

y

e1 ∧ (ξ −1 Ω (U )) → ⊕i≥0 Hi (π1 (U , y), RU /pn RU ) X n /S n

whose component in degree one is α(U,p) . A chase on the diagram (III.11.3.12) shows that 2d

?◦

y

y

the kernel of (III.11.3.15) is annihilated by p p−1 mK . Since Hi (π1 (U , y), RU /pn RU ) is almost zero for every i ≥ d + 1 by virtue of II.8.17(ii), the cokernel of (III.11.3.15) is 2d+1 annihilated by p p−1 mK . On the other hand, by III.10.30(iii), the image of the exact sequence (III.11.2.8) by the functor νy ◦ ϕx identifies with the direct limit on the category Vx (Q)◦ of the exact sequences (III.11.3.10). Consequently, by VI.10.30(iii), the stalk of the morphism ∂n (III.11.2.7) at x identifies with the direct limit on the category Vx (Q)◦ of the morphisms α(U,p) . We deduce from this the existence (and uniqueness) of the homomorphism (III.11.3.1) in view of III.9.5(ii). The stalk of the latter at x identifies with the direct limit on the category Vx (Q)◦ of the homomorphisms (III.11.3.15). The statement follows because filtered direct limits are exact. (ii) This follows from (III.11.3.8), III.9.5(ii), and II.8.17(ii).

254

III. GLOBAL ASPECTS

Corollary III.11.4. (i) There exists a unique homomorphism of graded OX˘ -algebras of N◦ Xs,´ et e 1˘ ∧ (ξ −1 Ω

(III.11.4.1)

˘ X/S

˘ ) → ⊕i≥0 Ri σ ˘∗ (B)

whose component in degree one is induced by the morphism ∂˘ (III.11.2.9). Moreover, its 2d+1 2d kernel is annihilated by p p−1 mK and its cokernel is annihilated by p p−1 mK . ˘ is almost zero. (ii) For every integer i ≥ d + 1, Ri σ ˘ (B) ∗

This follows from III.11.3, III.7.3(i), and (III.7.5.5). Proposition III.11.5. (i) For all integers i ≥ 1 and j ≥ 1, the OX ˘ -module ˘ Ri u ˘∗ (Rj σ ˘∗ (B)) 4d+1

is annihilated by p p−1 mK . (ii) For every integer q ≥ 0, the kernel of the morphism ˘ →u ˘ Rq τ˘∗ (B) ˘∗ (Rq σ ˘∗ (B))

(III.11.5.1)

induced by the Cartan–Leray spectral sequence is annihilated by p cokernel is annihilated by p

q(4d+1) p−1

(d+1)(4d+1) p−1

mK and its

mK .

(i) Indeed, it follows from III.11.3(i) and ([2] VII 4.3) that for every integer n ≥ 1, the 4d+1 OX n -modules Ri un∗ (Rj σn∗ (B n )) are annihilated by p p−1 mK . The statement follows in view of (III.7.5.5). (ii) We may clearly restrict ourselves to the case where q ≥ 1. Consider the Cartan– Leray spectral sequence ([2] V 5.4) ˘ ⇒ Ri+j τ˘ (B), ˘ i Ei,j ˘∗ (Rj σ ˘∗ (B)) ∗ 2 =R u

(III.11.5.2)

˘ Note that Eq is the kernel and denote by (Eqi )0≤i≤q the abutment filtration on Rq τ˘∗ (B). 1 4d+1 p−1 m of the morphism (III.11.5.1). We know that Ei,j is annihilated by p 2 K for all i ≥ 1 and j ≥ 0 by (i) and that it is almost zero for all i ≥ 0 and j ≥ d + 1 by virtue of III.11.4(ii). We deduce from this that Eqi /Eqi+1 = Ei,q−i ∞

(III.11.5.3) 4d+1

is annihilated by p p−1 mK for every i ≥ 1 and that it is almost zero for every i ≤ q −d−1. (4d+1)(d+1)

Consequently, Eq1 is annihilated by p p−1 mK ; the first assertion follows. On the other 0,q hand, we have E0,q ∞ = Eq+2 , and the cokernel of the morphism (III.11.5.1) identifies with the cokernel of the composition of the canonical injections (III.11.5.4)

0,q 0,q E0,q q+2 → Eq+1 → · · · → E2 . 4d+1

The cokernel of each of these injections is annihilated by p p−1 mK by (i); the second assertion follows. N◦ Lemma III.11.6. Let M˘ = (Mn+1 )n∈N be an OX ˘ -module of Xs,zar such that for all integers m ≥ n ≥ 1, Mn is a quasi-coherent OX n -module on X n and the canonical morphism Mm → Mn is surjective. Then Ri λ∗ (M˘) (III.11.1.10) is zero for every integer i ≥ 1.

III.11. COHOMOLOGICAL COMPUTATIONS

255

Indeed, by ([2] V 5.1), Ri λ∗ (M˘) is the sheaf associated with the presheaf on Xs,zar i ∗ ˘ that to a Zariski open subscheme U of Xs associates the OX ˘ (U )-module H (λ (U ), M ). By virtue of III.7.11, we have a canonical exact sequence (III.11.6.1)

0 → R1 lim Hi−1 (U, Mn ) → Hi (λ∗ (U ), M˘) → lim Hi (U, Mn ) → 0. ←−

←−

n≥1

n≥1

From now on, we identify the Zariski sites of Xs and X 1 (III.9.9). Let U be an affine open subscheme of X 1 . For every integer n ≥ 1, U determines an affine open subscheme of X n ([39] 2.3.5). Consequently, Hi (U, Mn ) = 0 and for all integers m ≥ n ≥ 1, the canonical morphism (III.11.6.2)

H0 (U, Mm ) → H0 (U, Mn )

is surjective, so that the inverse system of abelian groups (H0 (U, Mn+1 ))n∈N satisfies the Mittag–Leffler condition. We deduce from this that Hi (λ∗ (U ), M˘) = 0 ([47] 1.15 and [63] 3.1). The lemma follows because the affine open subschemes of X 1 form a topologically generating family for the Zariski site of X 1 . Proposition III.11.7. For every integer j ≥ 0, set αj =

(d+1+j)(4d+1)+6d+1 . p−1

Then:

˘ is annihilated (i) For all integers i ≥ 1 and j ≥ 0, the OX˘ -module Ri λ∗ (Rj τ˘∗ (B)) αj by p mK . (ii) For every integer q ≥ 0, the kernel and cokernel of the morphism

(III.11.7.1)

˘ → λ (Rq τ˘ (B)) ˘ Rq >∗ (B) ∗ ∗

induced by the Cartan–Leray spectral sequence (III.11.1.11), are annihilated by Pq−1 p j=0 αj mK . ◦

N (i) Indeed, by virtue of III.11.4(i), we have an OX ˘ -linear morphism of Xs,zar

(III.11.7.2)

ej ξ −j Ω ˘

˘ X/S

˘ →u ˘∗ (Rj σ ˘∗ (B)), 4d+1

2d

whose kernel is annihilated by p p−1 mK and whose cokernel is annihilated by p p−1 mK . In 6d+1 ˘ view of III.11.6, we deduce from this that Ri λ (˘ u (Rj σ ˘ (B))) is annihilated by p p−1 m . ∗

∗

∗

On the other hand, by III.11.5(ii), we have an OX ˘ -linear morphism

(III.11.7.3)

˘ →u ˘ ˘∗ (Rj σ ˘∗ (B)) Rj τ˘(B)

whose kernel is annihilated by p p

j(4d+1) p−1

K

(d+1)(4d+1) p−1

mK and whose cokernel is annihilated by

mK . The statement follows. (ii) The proof is similar to that of III.11.5(ii). We may clearly assume q ≥ 1. Consider the Cartan–Leray spectral sequence (III.11.7.4)

˘ ⇒ Ri+j > (B), ˘ i j Ei,j ˘∗ (B)) ∗ 2 = R λ∗ (R τ

˘ Note that Eq is the kernel and denote by (Eqi )0≤i≤q the abutment filtration on Rq >∗ (B). 1 αj of the morphism (III.11.7.1). By (i), Ei,j is annihilated by p m for all i ≥ 1 and j ≥ 0. 2 K The same then holds for (III.11.7.5)

Eqi /Eqi+1 = Ei,q−i ∞

256

III. GLOBAL ASPECTS Pq−1

for every i ≥ 1. Hence Eq1 is annihilated by p j=0 αj mK . On the other hand, we have 0,q E0,q ∞ = Eq+2 , and the cokernel of the morphism (III.11.7.1) identifies with the cokernel of the composition of the canonical injections (III.11.7.6)

0,q 0,q E0,q q+2 → Eq+1 → · · · → E2 .

For every integer m such that 2 ≤ m ≤ q + 1, the cokernel of the canonical injection 0,q αq−m+1 E0,q mK by (i); the desired statement follows. m+1 → Em is annihilated by p Corollary III.11.8. There exists a unique isomorphism of graded OX [ p1 ]-algebras (III.11.8.1)

∼ ˘ 1] e 1 [ 1 ]) → ∧ (ξ −1 Ω ⊕i≥0 Ri >∗ (B)[ X/S p p

whose component in degree one is the morphism δ ⊗Zp Qp (III.11.2.11).

Indeed, the homomorphism (III.11.4.1) induces a homomorphism of graded OX algebras (III.11.8.2)

˘ e 1 ) → ⊕i≥0 λ∗ (˘ ∧ (ξ −1 Ω u∗ (Ri σ ˘∗ (B))) X/S

whose kernel and cokernel are rig-null by III.11.4(i) (cf. III.6.15). On the other hand, the morphisms (III.11.8.3)

˘ → ⊕ λ (Ri τ˘ (B)) ˘ → ⊕ λ (˘ ˘ i ⊕i≥0 Ri >∗ (B) ˘∗ (B))) i≥0 ∗ ∗ i≥0 ∗ u∗ (R σ

induced by the Cartan–Leray spectral sequences are homomorphisms of graded OX algebras (cf. [41] 0.12.2.6) whose kernels and cokernels are rig-null by virtue of III.11.5(ii) and III.11.7(ii). We obtain the desired isomorphism (III.11.8.1) by applying the functor − ⊗Zp Qp .

Lemma III.11.9. Let M be a locally free OX -module of finite type and q an integer ≥ 0. For every integer n ≥ 1, set Mn = M ⊗OX OX n , and let M˘ = (Mn+1 )n∈N , which we N◦ view as a sheaf of Xs,´ et (III.2.9). We then have a canonical isomorphism (III.11.9.1)

∼ ˘ → M ⊗OX Rq >∗ (B) Rq >∗ (˘ σ ∗ (M˘)).

∼ Indeed, we have a canonical isomorphism u ˘∗ (M˘) → M˘. On the other hand, by ∼ virtue of ([1] 2.8.5), we have a canonical isomorphism M → λ∗ (M˘). The adjunction morphism M˘ → σ ˘∗ (˘ σ ∗ (M˘)) then induces an OX -linear morphism

M → >∗ (˘ σ ∗ (M˘)).

(III.11.9.2)

We deduce from this, by cup product, an OX -linear morphism (III.11.9.3)

˘ → Rq > (˘ ∗ ˘ M ⊗OX Rq >∗ (B) ∗ σ (M )).

To see that this is an isomorphism, we can reduce to the case where M = OX (since the question is local for the Zariski topology on X), in which case the statement is immediate. III.11.10. (III.11.10.1)

By ([45] I 4.3.1.7), the canonical exact sequence (III.10.31.3) ˘ → F˘ → σ e 1˘ ˘ ∗ (ξ −1 Ω 0→B

˘ X/S

)→0

induces, for every integer m ≥ 1, an exact sequence (III.2.7) (III.11.10.2)

−1 e 1 ˘ 0 → Sm−1 (F˘ ) → Sm ˘ ∗ (Sm Ω˘ ˘ (F ) → σ O ˘ (ξ ˘ B

B

X

˘ X/S

)) → 0.

III.11. COHOMOLOGICAL COMPUTATIONS

257

m−i ˘ ˘ We endow Sm (F ))i∈N . We then have ˘ (F ) with the exhaustive decreasing filtration (S ˘ B B a canonical exact sequence (III.11.10.3) −1 e 1 −1 e 1 ˘ m−2 (F˘ ) → σ ˘ ∗ (Sm 0→σ ˘ ∗ (Sm−1 Ω ˘ ˘ )) → Sm Ω ˘ ˘ )) → 0. ˘ (F )/S ˘ O ˘ (ξ O ˘ (ξ B

X/S

X

B

X/S

X

For all integers i and j, set i+j −1 e 1 >∗ (˘ σ ∗ (S−i Ω˘ Ei,j 1 =R O ˘ (ξ

(III.11.10.4)

˘ X/S

X

))),

and denote by i+1,j i,j di,j 1 : E1 → E1

(III.11.10.5)

the morphism induced by the exact sequence (III.11.10.3). By III.11.9, we have a canonical OX -linear isomorphism ∼ ˘ → e 1 ) ⊗ Ri+j > (B) (III.11.10.6) S−i (ξ −1 Ω Ei,j . OX

X/S

∗

1

In view of III.11.8, we deduce from this an isomorphism 1 ∼ e 1 [ 1 ]) → e 1 [ 1 ]) ⊗O [ 1 ] ∧i+j (ξ −1 Ω Ei,j (III.11.10.7) S−i (ξ −1 Ω 1 [ ]. X/S X/S X p p p p Proposition III.11.11. We have a commutative diagram (III.11.11.1)

e 1 [ 1 ]) e 1 [ 1 ]) ⊗O [ 1 ] ∧i+j (ξ −1 Ω S−i (ξ −1 Ω X/S p X/S p X p φi,j

where φ

is the restriction of the

1 / Ei,j 1 [p] di,j 1 ⊗Zp Qp



e 1 [ 1 ]) ⊗O [ 1 ] ∧i+j+1 (ξ −1 Ω e 1 [ 1 ]) S−i−1 (ξ −1 Ω X/S p X/S p X p i,j

∼

OX [ p1 ]-derivation

∼

 / Ei+1,j [1] 1 p

of

e 1 [ 1 ]) ⊗O [ 1 ] ∧(ξ −1 Ω e 1 [ 1 ]) S(ξ −1 Ω X/S X/S X p p p

(III.11.11.2)

e 1 [ 1 ], and the horizontal defined in (III.5.1.1) relative to the identity morphism of ξ −1 Ω X/S p arrows are the isomorphisms (III.11.10.7). Indeed, by virtue of III.5.7 and in view of the morphism (III.11.2.5), we have a commutative diagram ˘ e 1 ) ⊗O Ri+j >∗ (B) S−i (ξ −1 Ω X X/S

(III.11.11.3)

α⊗id −i−1

S

e1 ) (ξ −1 Ω X/S

˘ ⊗ Ri+j > (B) ˘ ⊗OX R >∗ (B) OX ∗

S

e1 ) (ξ −1 Ω X/S

/ Ei,j 1



1

id⊗∪ −i−1

∼

 ˘ ⊗OX Ri+j+1 >∗ (B)

∼

di,j 1

 / Ei+1,j 1

e 1 ) of the OX -derivation dδ of the where the morphism α is the restriction to S−i (ξ −1 Ω X/S ˘ defined in (III.5.1.1) relative to the morphism δ e 1 ) ⊗ ∧(R1 > (B)) algebra S(ξ −1 Ω X/S

OX

∗

˘ and the horizontal (III.11.2.11), ∪ is the cup product of the OX -algebra ⊕i≥0 Ri >∗ (B), arrows are the isomorphisms (III.11.10.6). The proposition follows.

258

III. GLOBAL ASPECTS

Proposition III.11.12. Let m be an integer ≥ 1. Then: (i) The morphism 1 ˘ 1 (III.11.12.1) >∗ (Sm−1 (F˘ ))[ ] → >∗ (Sm ˘ (F ))[ ] ˘ B B p p induced by (III.11.10.2) is an isomorphism. (ii) For every integer q ≥ 1, the morphism 1 ˘ 1 (III.11.12.2) Rq >∗ (Sm−1 (F˘ ))[ ] → Rq >∗ (Sm ˘ (F ))[ ] ˘ B B p p induced by (III.11.10.2) is zero. For all integers i and j, we set (III.11.10.4)  i−m,j+m E1 i,j (III.11.12.3) E = m 1 0

if i ≥ 0, if i < 0.

We denote by i,j m E1

(III.11.12.4)

˘ ⇒ Ri+j >∗ (Sm ˘ (F )) B

˘ ˘ the spectral sequence of hypercohomology of the filtered B-module Sm ˘ (F ) (III.11.10.2) whose differentials (III.11.12.5)

i,j m d1

B

are given by (III.11.10.5)  i−m,j+m d1 if i ≥ 0, i,j m d1 = 0 of i < 0.

˘ For any integer q ≥ 0, denote by (m Eqi )i∈Z the abutment filtration on Rq >∗ (Sm ˘ (F )), so B that we have (III.11.12.6)

i,q−i m E∞

= m Eqi /m Eqi+1 .

We then have ( (III.11.12.7) and

0,q m E∞

q m Ei

=

˘ Rq >∗ (Sm ˘ (F )) B 0

if i ≤ 0, if i ≥ m + 1,

⊂ m E0,q 1 . We deduce from this that the image of the canonical morphism

(III.11.12.8)

˘ Rq >∗ (Sm−1 (F˘ )) → Rq >∗ (Sm ˘ (F )) ˘ B

B

q 0,q m E1 and that its cokernel is m E∞ . On the other hand, it follows from III.11.11 and III.5.1 that for all integers i and q satisfying one of the following conditions: (i) q = 0 and i < m, (ii) q ≥ 1 and i ≥ 1, we have 1 i,q−i 1 [ ] = 0. (III.11.12.9) [ ] = m Ei,q−i m E∞ 2 p p The proposition follows.

is

Proposition III.11.13. Let r, r0 be two rational numbers such that r > r0 > 0. Then: (i) For every integer n ≥ 1, the canonical homomorphism (III.10.21.3) (III.11.13.1)

OX n → σn∗ (Cn(r) )

(r)

is almost injective. We denote its cokernel by Hn .

III.11. COHOMOLOGICAL COMPUTATIONS

259

(ii) There exists a rational number α > 0 such that for every integer n ≥ 1, the morphism 0

Hn(r) → Hn(r )

(III.11.13.2)

0

(r 0 )

(r)

induced by the homomorphism αnr,r : Cn → Cn (III.10.21.5) is annihilated by pα . (iii) There exists a rational number β > 0 such that for all integers n, q ≥ 1, the canonical morphism 0

Rq σn∗ (Cn(r) ) → Rq σn∗ (Cn(r ) )

(III.11.13.3)

is annihilated by pβ .

Let n and q be two integers such that n ≥ 1 and q ≥ 0, x a geometric point of X over s, X 0 the strict localization of X at x, and e → X 0◦ ϕx : E f´ et

(III.11.13.4)

the functor (III.8.8.4). By VI.10.30 and (III.9.8.6), we have a canonical isomorphism 0◦

∼

(III.11.13.5) 0

(Rq σn∗ (Cn(r) ))x → Hq (X f´et , ϕx (Cn(r) )).

By virtue of III.3.7, X is normal and strictly local (and in particular integral). Since 0 0◦ 0◦ X is S-flat, X is integral and nonempty. Let v : y → X be a generic geometric point 0◦ of X and 0◦

∼

νy : X f´et → Bπ1 (X 0◦ ,y)

(III.11.13.6)

◦

the associated fiber functor (III.2.10.3). We denote also by y the geometric point of X and by u : y → X 0 the morphism induced by v. We thus obtain a point (y x) of ←

◦

X´et ×X´et X ´et . We denote by Vx (resp. Vx (Q)) the category of neighborhoods of the point of X´et ´ /X (resp. Q (III.10.5)). For any (U, p : x → U ) ∈ Ob(Vx ), associated with x in the site Et 0 we denote also by p : X → U the morphism deduced from p, and we set ◦

p◦ = p ×X X : X

(III.11.13.7)

0◦

◦

→U .

◦

We (abusively) denote also by y the geometric point p◦ (v(y)) of U . Note that y is ◦ localized at a generic point of U because p◦ is flat. Since U is locally irreducible (III.3.3), ? it is the sum of the schemes induced on its irreducible components. Denote by U the ◦ irreducible component of U containing y. Likewise, U is the sum of the schemes induced ?◦ ? ◦ on its irreducible components, and U = U ×X X ◦ is the irreducible component of U ?◦ containing y. The morphism p◦ factors through U . By (III.10.29.1), we have a canonical isomorphism of Bπ1 (X 0◦ ,y) ∼

νy (ϕx (Cn(r) )) →

(III.11.13.8)

lim −→

y,(r)

CU

y,(r)

/pn CU

.

(U,p)∈Vx (Q)◦

By virtue of VI.11.10 and (VI.9.8.6), this induces an isomorphism (III.11.13.9)

0◦

∼

Hq (X f´et , ϕx (Cn(r) )) →

?◦

−→

y,(r)

Hq (π1 (U , y), CU

lim

y,(r)

/pn CU

).

(U,p)∈Vx (Q)◦

We deduce from (III.11.13.5) and (III.11.13.9) an isomorphism (III.11.13.10)

∼

(Rq σn∗ (Cn(r) ))x →

lim −→

(U,p)∈Vx (Q)◦

?◦

y,(r)

Hq (π1 (U , y), CU

y,(r)

/pn CU

).

260

III. GLOBAL ASPECTS

On the other hand, we prove, as in (III.11.3.9), that we have a canonical isomorphism (III.11.13.11)

∼

(OX n )x →

?

OX n (U ),

lim −→

(U,p)∈V◦ x

where we view OX n on the left as a sheaf of Xs,´et and on the right as a sheaf of X ´et . The stalk of the morphism (III.11.13.1) at x identifies with the direct limit on the category Vx (Q)◦ of the canonical morphism ?

y,(r)

OX n (U ) → (CU

(III.11.13.12)

y,(r) π1 (U ?◦ ,y)

/pn CU

)

.

Since filtered direct limits are exact, the proposition then follows from II.12.7, in view of III.10.4 and the proof of III.10.14. Corollary III.11.14. Let r, r0 be two rational numbers such that r > r0 > 0. Then: N◦ (i) The canonical homomorphism of Xs,´ et (III.10.31) OX ˘∗ (C˘(r) ) ˘ →σ

(III.11.14.1)

is almost injective. We denote its cokernel by H˘ (r) . (ii) There exists a rational number α > 0 such that the morphism 0 H˘ (r) → H˘ (r )

(III.11.14.2)

0 0 induced by the canonical homomorphism α ˘ r,r : C˘(r) → C˘(r ) (III.10.31.7) is annihilated by pα . (iii) There exists a rational number β > 0 such that for every integer q ≥ 1, the N◦ canonical morphism of Xs,´ et

(III.11.14.3) is annihilated by pβ .

0 Rq σ ˘∗ (C˘(r) ) → Rq σ ˘∗ (C˘(r ) )

This follows from III.11.13, III.7.3(i), and (III.7.5.5). Lemma III.11.15. Let r, r0 be two rational numbers such that r > r0 > 0. Then: N◦ (III.10.31.1) (i) The canonical homomorphism of Xs,zar ˘∗ (C˘(r) ) OX ˘ →τ

(III.11.15.1)

is almost injective. We denote its cokernel by K˘(r) . (ii) There exists a rational number α > 0 such that the morphism 0 K˘(r) → K˘(r )

(III.11.15.2)

0 0 induced by the canonical homomorphism α ˘ r,r : C˘(r) → C˘(r ) (III.10.31.7) is annihilated by pα . (iii) For every integer q ≥ 1, there exists a rational number β > 0 such that the N◦ canonical morphism of Xs,zar

(III.11.15.3) is annihilated by pβ .

0 Rq τ˘∗ (C˘(r) ) → Rq τ˘∗ (C˘(r ) )

Let us use the notation of III.11.14 and moreover denote by N˘(r) and M˘(r) the kernels of the morphisms (III.11.14.1) and (III.11.15.1), respectively. (i) Since M˘(r) = u ˘∗ (N˘(r) ), the statement follows from III.11.14(i). 1 (ii) Since R u ˘∗ (OX ˘ ) = 0 by (III.7.5.5) and ([2] VII 4.3), we have an exact sequence (III.11.15.4)

0 → R1 u ˘∗ (N˘(r) ) → K˘(r) → u ˘∗ (H˘ (r) ).

III.11. COHOMOLOGICAL COMPUTATIONS

261

The statement then follows from III.11.14(i)-(ii). (iii) Consider the Cartan–Leray spectral sequence (III.11.15.5)

r

i Ei,j ˘∗ (Rj σ ˘∗ (C˘(r) )) ⇒ Ri+j τ˘∗ (C˘(r) ) 2 =R u

and denote by (r Eqi )0≤i≤q the abutment filtration on Rq τ˘∗ (C˘r) ), so that we have r

(III.11.15.6)

Eqi /r Eqi+1 = r Ei,q−i . ∞

For every integer 0 ≤ i ≤ q + 1, set ri = r0 + (q + 1 − i)(r − r0 )/(q + 1). By III.11.14(iii), for every integer 0 ≤ i ≤ q − 1, there exists a rational number βi > 0 such that the canonical morphism ri

(III.11.15.7)

→ ri+1 Ei,q−i Ei,q−i 2 2

is annihilated by pβi . The same then holds for the morphism ri Ei,q−i → ri+1 Ei,q−i . On ∞ ∞ q the other hand, R u ˘∗ (OX ˘ ) = 0 by (III.7.5.5) and ([2] VII 4.3). We deduce from this that the canonical morphism (III.11.15.8)

0

0

Rq u ˘∗ (˘ σ∗ (C˘(r ) )) → Rq u ˘∗ (H˘ (r ) )

is almost injective by III.11.14(i). Consequently, by virtue of III.11.14(ii), there exists a rational number βq > 0 such that the canonical morphism rq

(III.11.15.9)

rq+1 q,0 Eq,0 E2 2 →

is annihilated by pβq . The same then holds Pq for the morphism desired statement follows by taking β = i=0 βi .

rq

Eq,0 ∞ →

rq+1

Eq,0 ∞ . The

Proposition III.11.16. Let r, r0 be two rational numbers such that r > r0 > 0. Then: (i) The canonical homomorphism (III.10.31.1) OX → >∗ (C˘(r) )

(III.11.16.1)

is injective. We denote its cokernel by L (r) . (ii) There exists a rational number α > 0 such that the morphism 0

L (r) → L (r )

(III.11.16.2)

0 0 induced by the canonical homomorphism α ˘ r,r : C˘(r) → C˘(r ) (III.10.31.7) is annihilated by pα . (iii) For every integer q ≥ 1, there exists a rational number β > 0 such that the canonical morphism

(III.11.16.3) is annihilated by pβ .

0 Rq >∗ (C˘(r) ) → Rq >∗ (C˘(r ) )

Let us use the notation of III.11.15 and moreover denote by M˘(r) the kernel of the morphism (III.11.15.1). (i) The kernel of the morphism (III.11.16.1) is canonically isomorphic to λ∗ (M˘(r) ) (III.11.1.10). It is therefore almost zero by virtue of III.11.15(i). Since OX is OC -flat by III.4.2(i) (rig-pure in the terminology of [1] 2.10.1.4), the morphism (III.11.16.1) is injective. (ii) Since R1 λ∗ (OX ˘ ) = 0 by III.11.6, we have an exact sequence (III.11.16.4)

0 → R1 λ∗ (M˘(r) ) → L (r) → λ∗ (K˘(r) ).

The statement then follows from III.11.15(i)-(ii).

262

III. GLOBAL ASPECTS

(iii) The proof is similar to that of III.11.15(iii). Consider the Cartan–Leray spectral sequence r i,j (III.11.16.5) E = Ri λ∗ (Rj τ˘∗ (C˘(r) )) ⇒ Ri+j >∗ (C˘(r) ), 2

r

and denote by (

Eqi )0≤i≤q

the abutment filtration on Rq >∗ (C˘r) ), so that we have r

(III.11.16.6)

Eqi /r Eqi+1 = r Ei,q−i . ∞

For every integer 0 ≤ i ≤ q + 1, set ri = r0 + (q + 1 − i)(r − r0 )/(q + 1). By III.11.15(iii), for every integer 0 ≤ i ≤ q − 1, there exists a rational number βi > 0 such that the canonical morphism ri

(III.11.16.7)

→ ri+1 Ei,q−i Ei,q−i 2 2

is annihilated by pβi . The same then holds for the morphism ri Ei,q−i → ri+1 Ei,q−i . On ∞ ∞ q the other hand, R u ˘∗ (OX ˘ ) = 0 by III.11.6. We deduce from this that the canonical morphism 0 0 (III.11.16.8) Rq λ∗ (˘ τ∗ (C˘(r ) )) → Rq λ∗ (K˘(r ) ) is almost injective by III.11.15(i). Consequently, by virtue of III.11.15(ii), there exists a rational number βq > 0 such that the canonical morphism rq

(III.11.16.9)

rq+1 q,0 Eq,0 E2 2 →

rq

is annihilated by pβq . The same then holds Pq for the morphism desired statement follows by taking β = i=0 βi .

Eq,0 ∞ →

rq+1

Eq,0 ∞ . The

Corollary III.11.17. Let r, r0 be two rational numbers such that r > r0 > 0. (i) The canonical homomorphism 1 1 (III.11.17.1) ur : OX [ ] → >∗ (C˘(r) )[ ] p p admits (as an OX [ p1 ]-linear morphism) a canonical left inverse 1 1 v r : >∗ (C˘(r) )[ ] → OX [ ]. p p

(III.11.17.2) (ii) The composition

0 1 ur0 1 1 vr >∗ (C˘(r) )[ ] −→ OX [ ] −→ >∗ (C˘(r ) )[ ] p p p is the canonical homomorphism. (iii) For every integer q ≥ 1, the canonical morphism 0 1 1 (III.11.17.4) Rq >∗ (C˘(r) )[ ] → Rq >∗ (C˘(r ) )[ ] p p is zero.

(III.11.17.3)

Indeed, by III.11.16(i)-(ii), ur is injective and there exists a unique OX [ p1 ]-linear morphism 0 1 1 (III.11.17.5) v r,r : >∗ (C˘(r) )[ ] → OX [ ] p p 0 r0 r,r 0 such that u ◦ v is the canonical homomorphism >∗ (C˘(r) )[ p1 ] → >∗ (C˘(r ) )[ p1 ]. Since 0

0

0

0

we have ur ◦ v r,r ◦ ur = ur , we deduce from this that v r,r is a left inverse of ur . One immediately verifies that it does not depend on r0 , giving statements (i) and (ii). Statement (iii) immediately follows from III.11.16(iii).

III.11. COHOMOLOGICAL COMPUTATIONS

263

Corollary III.11.18. The canonical homomorphism 1 1 (III.11.18.1) OX [ ] → lim >∗ (C˘(r) )[ ] −→ p p r∈Q >0

is an isomorphism and for every integer q ≥ 1,

1 lim Rq >∗ (C˘(r) )[ ] = 0. −→ p r∈Q

(III.11.18.2)

>0

III.11.19. Recall that for every rational number r ≥ 0 and every integer n ≥ 1, (r) the universal B n -derivation of Cn (III.10.23.2) (r) ∗ −1 e 1 d(r) ΩX n /S n ) ⊗Bn Cn(r) n : Cn → σn (ξ

(III.11.19.1)

e1 is a Higgs B n -field with coefficients in σn∗ (ξ −1 Ω X

) (III.10.32). We denote the Dol-

beault complex of the Higgs B n -module the augmented Dolbeault complex

(II.2.8.2) by K• (Cn , pr dn ) and

n /S n (r) (r) (Cn , pr dn )

(r)

1 (r) r (r) 2 (r) r (r) B n → K0 (Cn(r) , pr d(r) n ) → K (Cn , p dn ) → K (Cn , p dn ) → . . . ,

(III.11.19.2)

(r)

where B n is placed in degree −1 and the differential B n → Cn e • (Cn(r) , pr d(r) morphism, by K n ). For all rational numbers r ≥ r0 ≥ 0, we have (III.10.23.3) 0

(III.11.19.3) 0

(r)

(r)

where αnr,r : Cn a morphism

(r 0 )

→ Cn

(III.11.19.4)

0

0

is the canonical homo-

0

r (r ) pr (id ⊗ αnr,r ) ◦ d(r) ◦ αnr,r , n = p dn

0

is the homomorphism (III.10.21.5). Consequently, αnr,r induces 0 e • (C (r) , pr d(r) ) → K e • (C (r0 ) , pr0 d(r0 ) ). νnr,r : K n n n n

Proposition III.11.20. For all rational numbers r > r0 > 0, there exists a rational number α ≥ 0 such that for all integers n and q with n ≥ 1, the morphism (III.11.20.1) is annihilated by pα .

0 e • (C (r) , pr d(r) )) → Hq (K e • (C (r0 ) , pr0 d(r0 ) )) Hq (νnr,r ) : Hq (K n n n n

Let n be an integer ≥ 1, x a geometric point of X over s, X 0 the strict localization of X at x, and (III.11.20.2)

e → X 0◦ ϕx : E f´ et 0

the functor (III.8.8.4). By virtue of III.3.7, X is normal and strictly local (and in 0 0◦ 0◦ particular integral). Since X is S-flat, X is integral and nonempty. Let v : y → X be a generic geometric point, and (III.11.20.3)

0◦

∼

νy : X f´et → Bπ1 (X 0◦ ,y)

◦

the associated fiber functor (III.2.10.3). We denote also by y the geometric point of X and by u : y → X 0 the morphism induced by v. We thus obtain a point (y x) of ←

◦

X´et ×X´et X ´et . We denote by Vx (resp. Vx (Q)) the category of neighborhoods of the point of X´et ´ /X (resp. Q (III.10.5)). For any (U, p : x → U ) ∈ Ob(Vx ), associated with x in the site Et we denote also by p : X 0 → U the morphism deduced from p, and we set (III.11.20.4)

◦

p◦ = p ×X X : X

0◦

◦

→U .

264

III. GLOBAL ASPECTS ◦

We (abusively) denote also by y the geometric point p◦ (v(y)) of U . Note that y is ◦ localized at a generic point of U because p◦ is flat. Since U is locally irreducible (III.3.3), ? it is the sum of the schemes induced on its irreducible components. Denote by U the ◦ irreducible component of U containing y. Likewise, U is the sum of the schemes induced ?◦ ? ◦ on its irreducible components, and U = U ×X X ◦ is the irreducible component of U ?◦ containing y. The morphism p◦ factors through U . By (III.10.8.5) and (III.10.29.1), we have canonical isomorphisms of Bπ1 (X 0◦ ,y) (III.11.20.5) (III.11.20.6)

∼

νy (ϕx (B n )) →

y

y

RU /pn RU ,

lim −→

(U,p)∈V◦ x

∼

νy (ϕx (Cn(r) )) →

y,(r)

CU

lim −→

y,(r)

/pn CU

.

(U,p)∈Vx (Q)◦

The rings underlying these representations are canonically isomorphic to the stalks of (r) B n and Cn at ρ(y x) (VI.10.31 and VI.9.9). On the other hand, we have canonical isomorphisms (III.11.13.11) and (III.11.3.9) ∼

(OX n )x →

(III.11.20.7)

∼

e1 ) → (Ω X n /S n x

(III.11.20.8)

e1 where we view OX n and Ω X

n /S n

lim −→

?

OX n (U ),

(U,p)∈V◦ x

lim −→

(U,p)∈V◦ x

? e1 (U ), Ω X n /S n

on the left as sheaves of Xs,´et and on the right as

sheaves of X ´et . These modules are canonically isomorphic to the stalks of σs∗ (OX n ) and e1 σs∗ (Ω ) at ρ(y x) (VI.10.18.1). It follows from III.10.30 that the stalk of the X /S n

n

(r)

derivation dn (III.10.23.2) at ρ(y x) identifies with the direct limit on the category y n y ◦ Vx (Q) of the universal (RU /p RU )-derivations (III.11.20.9)

y,(r)

CU

y,(r)

/pn CU

y,(r)

e 1 (U ) ⊗O (U ) (C → ξ −1 Ω X/S X U

y,(r)

/pn CU

).

es is conservative. Since filtered direct x) of E By III.9.5, the family of points ρ(y limits are exact, the proposition then follows from II.12.3(i), in view of III.10.4 and the proof of III.10.14. III.11.21. (III.11.21.1)

For any rational number r ≥ 0, we denote also by e 1 ) ⊗ ˘ C˘(r) d˘(r) : C˘(r) → >∗ (ξ −1 Ω X/S B

˘ the B-derivation induced by d˘(r) (III.10.31.9) and the isomorphism (III.11.2.6), which we ˘ ˘ identify with the universal B-derivation of C˘(r) . It is a Higgs B-field with coefficients in ∗ −1 e 1 • ˘(r) r ˘(r) > (ξ Ω ) (III.10.32). We denote by K (C , p d ) the Dolbeault complex of the X/S

˘ e • (C˘(r) , pr d˘(r) ) the augmented Dolbeault complex Higgs B-module (C˘(r) , pr d˘(r) ) and by K (III.11.21.2) ˘ → K0 (C˘(r) , pr d˘(r) ) → K1 (C˘(r) , pr d˘(r) ) → · · · → Kn (C˘(r) , pr d˘(r) ) → . . . , B ˘ is placed in degree −1 and the differential B ˘ → C˘(r) is the canonical homowhere B morphism. For all rational numbers r ≥ r0 ≥ 0, we have (III.10.31.10)

(III.11.21.3)

0 0 0 0 pr (id ⊗ α ˘ r,r ) ◦ d˘(r) = pr d˘(r ) ◦ α ˘ r,r ,

III.11. COHOMOLOGICAL COMPUTATIONS

265

0 0 0 where α ˘ r,r : C˘(r) → C˘(r ) is the homomorphism (III.10.31.7). Consequently, α ˘ r,r induces a morphism of complexes 0 e • (C˘(r) , pr d˘(r) ) → K e • (C˘(r0 ) , pr0 d˘(r0 ) ). (III.11.21.4) ν˘r,r : K

˘ ˘ the category of B-modules esN◦ up to isogeny (III.6.2), of E We denote by ModQ (B) e • (C˘(r) , pr d˘(r) ) the images of the complexes K• (C˘(r) , pr d˘(r) ) and by K•Q (C˘(r) , pr d˘(r) ) and K Q ˘ • ˘(r) r ˘(r) e and K (C , p d ) in ModQ (B). Proposition III.11.22. For all rational numbers r > r0 > 0 and every integer q, the canonical morphism (III.11.21.4) (III.11.22.1) is zero.

0

e • (C˘(r) , pr d˘(r) )) → Hq (K e • (C˘(r0 ) , pr0 d˘(r0 ) )) Hq (˘ νQr,r ) : Hq (K Q Q

This follows from III.11.20 and III.7.3(i). Corollary III.11.23. Let r, r0 be two rational numbers such that r > r0 > 0. (i) The canonical morphism ˘ → H0 (K• (C˘(r) , pr d˘(r) )) (III.11.23.1) ur : B Q Q admits a canonical left inverse

˘ . v r : H0 (K•Q (C˘(r) , pr d˘(r) )) → B Q

(III.11.23.2) (ii) The composition

0

(III.11.23.3)

vr ˘ ur 0 • ˘(r 0 ) r ˘(r 0 ) H0 (K•Q (C˘(r) , pr d˘(r) )) −→ B , p d )) Q −→ H (KQ (C

is the canonical morphism. (iii) For every integer q ≥ 1, the canonical morphism 0 0 0 (III.11.23.4) Hq (K• (C˘(r) , pr d˘(r) )) → Hq (K• (C˘(r ) , pr d˘(r ) )) Q

Q

is zero.

Indeed, consider the canonical commutative diagram (without the dotted arrow) (III.11.23.5)

/ H0 (K• (C˘(r) , pr d˘(r) ))

ur

˘ B Q

v r,r

˘ B Q

/ / H0 (K e • (C˘(r) , pr d˘(r) )) Q

Q

w ur

0

0

$ r,r

0

0

 / H0 (K• (C˘(r0 ) , pr0 d˘(r0 ) )) Q

H0 (˘ νQr,r )

 / / H0 (K e • (C˘(r0 ) , pr0 d˘(r0 ) )) Q

0

It follows from III.11.22 that ur and consequently ur are injective, and that there exists a 0 0 0 0 0 0 unique morphism v r,r as above such that $r,r = ur ◦v r,r . Since we have ur ◦v r,r ◦ur = 0 0 ur , it follows that v r,r is a left inverse of ur . One immediately verifies that it does not depend on r0 , giving statements (i) and (ii). Statement (iii) immediately follows from III.11.22. Corollary III.11.24. The canonical morphism ˘ → lim H0 (K• (C˘(r) , pr d˘(r) )) (III.11.24.1) B Q

Q

−→

r∈Q>0

is an isomorphism, and for every integer q ≥ 1, (III.11.24.2) lim Hq (K• (C˘(r) , pr d˘(r) )) = 0. −→

r∈Q>0

Q

266

III. GLOBAL ASPECTS

This follows from III.11.23. Remark III.11.25. Filtered direct limits are not a priori representable in the category ˘ ModQ (B). III.12. Dolbeault modules III.12.1. We keep the assumptions and general notation of III.10 in this section. ˘ the category of B-modules ˘ esN◦ (III.10.31), by We moreover denote by Mod(B) of E ˘ (resp. Modaft (B)) ˘ the full subcategory made up of adic B-modules ˘ Modad (B) (resp. ˘ ˘ B-modules of finite type) (III.7.16), and by ModQ (B) (resp. ˘ ad ˘ aft ˘ ad ˘ ModQ (B), resp. ModQ (B)) the category of objects of Mod(B) (resp. Mod (B), ˘ up to isogeny (III.6.1.1). The category Mod (B) ˘ is then abelian and resp. Modaft (B)) adic

Q

the canonical functors (III.12.1.1)

˘ ˘ ad ˘ Modaft Q (B) → ModQ (B) → ModQ (B)

are fully faithful. e1 We denote by X the formal S -scheme p-adic completion of X and by ξ −1 Ω X/S the −1 e 1 −1 e 1 p-adic completion of the OX -module ξ ΩX/S = ξ ΩX/S ⊗OX OX (cf. III.2.1). We

denote by Modcoh (OX ) (resp. Modcoh (OX [ p1 ])) the category of coherent OX -modules (resp. OX [ p1 ]-modules) of Xs,zar (III.6.15). Let ˘ → (X e N◦ , B) (III.12.1.2) > : (E s,zar , OX ) s

be the morphism of ringed topos defined in (III.11.1.11). The functor >∗ induces an additive left exact functor that we denote also by ˘ → Mod(O [ 1 ]). (III.12.1.3) >∗ : ModQ (B) X p ∗ By III.6.16, the functor > induces an additive functor that we denote also by 1 ˘ (III.12.1.4) >∗ : Modcoh (OX [ ]) → Modaft Q (B). p ˘ -module G , we have a bifunctorial For every coherent O [ 1 ]-module F and every B X p

Q

canonical homomorphism (III.12.1.5)

Hom ˘ (>∗ (F ), G ) → HomOX [ p1 ] (F , >∗ (G )), BQ

˘ -linear that is injective by (III.6.15.3) and III.6.16. We abusively call adjoint of a B Q ∗ morphism > (F ) → G its image by the homomorphism (III.12.1.5). We denote by 1 ˘ (III.12.1.6) R>∗ : D+ (ModQ (B)) → D+ (Mod(OX [ ])), p 1 ˘ (III.12.1.7) Rq >∗ : ModQ (B) → Mod(OX [ ]), (q ∈ N), p the right derived functors of the functor >∗ (III.12.1.3). This notation does not lead ˘ → to any confusion with that of the right derived functors of the functor > : Mod(B) ∗

˘ → Mod (B) ˘ is exact and transMod(OX ), because the localization functor Mod(B) Q forms injective objects into injective objects.

III.12. DOLBEAULT MODULES

267

˘ and q an integer ≥ 0. The III.12.2. Let M be an OX -module, N a B-module, adjunction morphism M → >∗ (>∗ (M )) and the cup product induce a bifunctorial morphism M ⊗OX Rq >∗ (N ) → Rq >∗ (>∗ (M ) ⊗ ˘ N ).

(III.12.2.1)

B

We can make the following remarks: (i) For every OX -module M 0 , the composition (III.12.2.2)

/ M ⊗OX Rq >∗ (>∗ (M 0 ) ⊗ ˘ N ) M ⊗OX M 0 ⊗OX Rq >∗ (N ) B VVVV VVVV VVVV VVVV VVVV  + q R >∗ (>∗ (M ⊗OX M 0 ) ⊗ ˘ N ) B

of the morphisms induced by the morphisms (III.12.2.1) with respect to M and M 0 is none other than the morphism (III.12.2.1) with respect to M ⊗OX M 0 . (ii) When q = 0, the morphism (III.12.2.1) is the composition (III.12.2.3)

M ⊗OX >∗ (N ) → >∗ (>∗ (M ⊗OX >∗ (N ))) → >∗ (>∗ (M ) ⊗ ˘ N ), B

where the first arrow is the adjunction morphism and the second arrow is induced by the canonical morphism >∗ (>∗ (N )) → N . Its adjoint >∗ (M ⊗OX >∗ (N )) → >∗ (M ) ⊗ ˘ N

(III.12.2.4)

B

is therefore induced by the canonical morphism >∗ (>∗ (N )) → N . ˘ -module, and q an inteIII.12.3. Let F be a coherent OX [ p1 ]-module, G a B Q ger ≥ 0. In view of III.6.16, the morphism (III.12.2.1) induces a bifunctorial morphism F ⊗OX [ p1 ] Rq >∗ (G ) → Rq >∗ (>∗ (F ) ⊗ ˘ G ).

(III.12.3.1)

BQ

We can make the following remarks: (i) Let F 0 be a coherent OX [ p1 ]-module. It follows from III.12.2(i) that the composition (III.12.3.2)

/ F ⊗OX [ 1 ] Rq >∗ (>∗ (F 0 ) ⊗ ˘ G ) F ⊗OX [ p1 ] F 0 ⊗OX [ p1 ] Rq >∗ (G ) p BQ WWWWW WWWWW WWWWW WWWWW +  q R >∗ (>∗ (F ⊗OX [ p1 ] F 0 ) ⊗ ˘ G ) BQ

of the morphisms induced by the morphisms (III.12.3.1) with respect to F and F 0 is none other than the morphism (III.12.3.1) with respect to F ⊗OX [ p1 ] F 0 . ˘ -linear morphism, (ii) Let L be a coherent OX [ p1 ]-module, u : >∗ (L ) → G a B Q and v : L → >∗ (G ) the adjoint morphism (III.12.1.5). It then follows from III.12.2(ii) and III.6.16 that the morphism (III.12.3.3)

F ⊗OX [ p1 ] L → >∗ (>∗ (F ) ⊗ ˘ G ) BQ

induced by (III.12.3.1) and v is the adjoint of the morphism (III.12.3.4) induced by u.

>∗ (F ⊗OX [ p1 ] L ) → >∗ (F ) ⊗ ˘ G BQ

268

III. GLOBAL ASPECTS

˘ Lemma III.12.4. (i) Let M be a locally free OX -module of finite type, N a B-module, and q an integer ≥ 0. Then the canonical morphism (III.12.2.1) M ⊗OX Rq >∗ (N ) → Rq >∗ (>∗ (M ) ⊗ ˘ N )

(III.12.4.1)

B

is an isomorphism. ˘ (ii) Let F be a locally projective OX [ p1 ]-module of finite type (III.2.8), G a B Q module, and q an integer ≥ 0. Then the canonical morphism (III.12.3.1) F ⊗OX [ p1 ] Rq >∗ (G ) → Rq >∗ (>∗ (F ) ⊗ ˘ G )

(III.12.4.2)

BQ

is an isomorphism. We only prove (ii); the proof of (i) is similar and simpler. There exists a Zariski open covering (Ui )i∈I of X such that for every i ∈ I, the restriction of F to (Ui )s is a direct summand of a free (OX |Ui )[ p1 ]-module of finite type. In view of III.9.15, we may then restrict to the case where F is a direct summand of a free OX [ p1 ]-module of finite type, and even to the case where F is a free OX [ p1 ]-module of finite type, in which case the assertion is obvious. III.12.5.

e 1 ) the category of Higgs OX -isogenies with We denote by HI(OX , ξ −1 Ω X/S e 1 ) the full subcategory made (III.6.8) and by HIcoh (OX , ξ −1 Ω

e1 ξ −1 Ω X/S

coefficients in X/S up of the quadruples (M , N , u, θ) such that M and N are coherent OX -modules. These e 1 ) (resp. HIcoh (OX , ξ −1 Ω e 1 )) are additive categories. We denote by HIQ (OX , ξ −1 Ω Q X/S X/S e 1 )) up to isogeny e 1 ) (resp. HIcoh (OX , ξ −1 Ω the category of objects of HI(OX , ξ −1 Ω X/S

X/S

(III.6.1.1). e 1 , we mean a Higgs OX [ 1 ]By a Higgs OX [ p1 ]-module with coefficients in ξ −1 Ω X/S p 1 −1 e 1 module with coefficients in ξ ΩX/S [ p ] (II.2.8). From now on, we will leave the Higgs field out of the notation of a Higgs module when we do not need it explicitly. We e 1 ) the category of Higgs OX [ 1 ]-modules with coefficients denote by HM(OX [ p1 ], ξ −1 Ω X/S p e1 e 1 ) the full subcategory made up of Higgs and by HMcoh (OX [ 1 ], ξ −1 Ω in ξ −1 Ω X/S

p

X/S

modules whose underlying OX [ p1 ]-module is coherent. The functor (III.6.19.1) (III.12.5.1)

e 1 ) → HM(OX [ 1 ], ξ −1 Ω e1 ) HI(OX , ξ −1 Ω X/S X/S p −1 (M , N , u, θ) 7→ (MQp , (id ⊗ uQp ) ◦ θQp )

induces a functor (III.12.5.2)

e 1 ) → HM(OX [ 1 ], ξ −1 Ω e 1 ). HIQ (OX , ξ −1 Ω X/S X/S p

By III.6.21, this induces an equivalence of categories (III.12.5.3)

1 ∼ −1 e 1 e 1 ). HIcoh ΩX/S ) → HMcoh (OX [ ], ξ −1 Ω Q (OX , ξ X/S p

e1 Definition III.12.6. A Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S is a Higgs 1 1 −1 e 1 OX [ p ]-module with coefficients in ξ ΩX/S whose underlying OX [ p ]-module is locally projective of finite type (III.2.8).

III.12. DOLBEAULT MODULES

III.12.7. (III.12.7.1)

269

Let r be a rational number ≥ 0. We denote also by e 1 ) ⊗ ˘ C˘(r) d˘(r) : C˘(r) → >∗ (ξ −1 Ω X/S

B

˘ the B-derivation induced by d˘(r) (III.10.31.9) and the isomorphism (III.11.2.6), which we ˘ ˘ of C˘(r) . It is a Higgs B-field with coefficients in identify with the universal B-derivation ∗ −1 e 1 r > (ξ Ω ) (III.10.32). We denote by Ξ the category of integrable pr -isoconnections X/S

˘ (III.6.10). It is an additive category. We denote by with respect to the extension C˘(r) /B r r ΞQ the category of objects of Ξ up to isogeny (III.6.1.1). By III.6.12 and III.10.32(iii), ˘ e 1 ) (III.6.8). In every object of Ξr is a Higgs B-isogeny with coefficients in >∗ (ξ −1 Ω X/S

particular, we can associate, functorially, with every object of ΞrQ a Dolbeault complex ˘ (cf. III.6.9). in Mod (B) Q

Consider the functor (III.12.7.2)

˘ → Ξr , Sr : Mod(B)

M 7→ (C˘(r) ⊗ ˘ M , C˘(r) ⊗ ˘ M , id, pr d˘(r) ⊗ id), B

B

and denote also by ˘ → Ξr Sr : ModQ (B) Q

(III.12.7.3)

the induced functor. Consider, on the other hand, the functor ˘ (F , G , u, ∇) 7→ ker(∇), (III.12.7.4) K r : Ξr → Mod(B), and denote also by ˘ K r : ΞrQ → ModQ (B)

(III.12.7.5)

the induced functor. It is clear that the functor (III.12.7.2) is a left adjoint of the functor (III.12.7.4). Consequently, the functor (III.12.7.3) is a left adjoint of the functor (III.12.7.5). e1 , By III.6.12, if (N , N 0 , v, θ) is a Higgs OX -isogeny with coefficients in ξ −1 Ω X/S (III.12.7.6) (C˘(r) ⊗ ˘ >∗ (N ), C˘(r) ⊗ ˘ >∗ (N 0 ), id⊗ ˘ >∗ (v), pr d˘(r) ⊗>∗ (v)+id⊗>∗ (θ)) B

B

B

is an object of Ξr . We thus obtain a functor

(III.12.7.7)

e 1 ) → Ξr . >r+ : HI(OX , ξ −1 Ω X/S

By (III.12.5.3), this induces a functor that we denote also by 1 e 1 ) → ΞrQ . (III.12.7.8) >r+ : HMcoh (OX [ ], ξ −1 Ω X/S p Let (F , G , u, ∇) be an object of Ξr . In view of III.12.4(i), ∇ induces an OX -linear morphism (III.12.7.9)

e1 >∗ (∇) : >∗ (F ) → ξ −1 Ω X/S ⊗OX >∗ (G ).

We easily deduce from III.12.3(i) that (>∗ (F ), >∗ (G ), >∗ (u), >∗ (∇)) is a Higgs OX e 1 . We thus obtain a functor isogeny with coefficients in ξ −1 Ω X/S (III.12.7.10)

e 1 ). >r+ : Ξr → HI(OX , ξ −1 Ω X/S

The composition of the functors (III.12.7.10) and (III.12.5.1) induces a functor that we denote also by 1 e 1 ). (III.12.7.11) >r+ : ΞrQ → HM(OX [ ], ξ −1 Ω X/S p

270

III. GLOBAL ASPECTS

It is clear that the functor (III.12.7.7) is a left adjoint of the functor (III.12.7.10). e 1 )) and A ∈ Ob(Ξr ), We deduce from this that for all N ∈ Ob(HMcoh (OX [ p1 ], ξ −1 Ω Q X/S we have a bifunctorial canonical homomorphism HomΞrQ (>r+ (N ), A ) → HomHM(OX [ 1 ],ξ−1 Ω e1

(III.12.7.12)

p

X/S

, >r+ (A )),

) (N

which is injective by III.6.20 and III.6.21. We abusively call adjoint of a morphism >r+ (N ) → A of ΞrQ its image by the homomorphism (III.12.7.12). Let r, r0 be two rational numbers such that r ≥ r0 ≥ 0 and (F , G , u, ∇) ˘ By (III.11.21.3), an integrable pr -isoconnection with respect to the extension C˘(r) /B. ˘ morphism there exists a unique B-linear III.12.8.

0 e 1 ) ⊗ ˘ C˘(r0 ) ⊗ ˘(r) G ∇0 : C˘(r ) ⊗C˘(r) F → >∗ (ξ −1 Ω X/S C

(III.12.8.1)

B

˘(r0 )

such that for all local sections x of C 0

and s of F , we have

0 ∇0 (x0 ⊗C˘(r) s) = pr d˘(r ) (x0 ) ⊗C˘(r) u(s) + x0 ⊗C˘(r) ∇(s). 0

(III.12.8.2)

0 0 0 The quadruple (C˘(r ) ⊗C˘(r) F , C˘(r ) ⊗C˘(r) G , id ⊗C˘(r) u, ∇0 ) is an integrable pr -isocon0 ˘ We thus obtain a functor nection with respect to the extension C˘(r ) /B. 0

0

r,r : Ξr → Ξr .

(III.12.8.3)

This induces a functor that we denote also by 0

0

r,r : ΞrQ → ΞrQ .

(III.12.8.4)

˘ to Ξr (resp. from We have a canonical isomorphism of functors from Mod(B) 0 ˘ to Ξr ) Mod (B) 0

Q

Q

∼

0

0

r,r ◦ Sr −→ Sr .

(III.12.8.5)

e1 ) On the other hand, we have a canonical isomorphism of functors from HI(OX , ξ −1 Ω X/S 0 0 e 1 ) to Ξr ) to Ξr (resp. from HMcoh (OX [ 1 ], ξ −1 Ω X/S

p

r,r 0



(III.12.8.6)

◦>

r+

Q

∼

0

−→ >r + .

The diagram

r,r 0

α ˘

e1 ) ⊗ ˘ G / >∗ (ξ −1 Ω X/S

∇

F

(III.12.8.7)

B

0

id⊗ ˘ α ˘ r,r ⊗C˘(r) id

⊗C˘(r) id

 (r 0 ) ˘ C ⊗C˘(r) F

B

∇0

 e 1 ) ⊗ ˘ C˘(r0 ) ⊗ ˘(r) G / >∗ (ξ −1 Ω X/S C B

is clearly commutative. We deduce from this a canonical morphism of functors from Ξr ˘ (resp. from Ξr to Mod (B)) ˘ to Mod(B) Q Q K

(III.12.8.8)

r

→K

r0

0

◦ r,r .

e 1 ) (resp. We also deduce a canonical morphism of functors from Ξr to HI(OX , ξ −1 Ω X/S 1 r −1 e 1 from Ξ to HM(OX [ ], ξ Ω )) Q

(III.12.8.9)

p

X/S

0

0

>r+ −→ >r+ ◦ r,r .

III.12. DOLBEAULT MODULES

271

For every rational number r00 such that r0 ≥ r00 ≥ 0, we have a canonical isomorphism 00 00 of functors from Ξr to Ξr (resp. from ΞrQ to ΞrQ ) 0

00

0

∼

00

r ,r ◦ r,r → r,r .

(III.12.8.10)

Remark III.12.9. Under the assumptions of III.12.8, 0 0 0 (C˘(r ) ⊗C˘(r) F , C˘(r ) ⊗C˘(r) G , id ⊗C˘(r) u, pr−r ∇0 ) 0 ˘ deduced from is the integrable pr -isoconnection with respect to the extension C˘(r ) /B r,r 0 (F , G , u, ∇) by extension of the scalars by α ˘ , defined in III.6.11. This shift can be explained by the fact that the canonical homomorphism Ω1˘(r) ˘ → Ω1˘(r0 ) ˘ identifies

C

/B

C

/B

with

0 0 e 1 ) ⊗ ˘ C˘(r) → >∗ (ξ −1 Ω e 1 ) ⊗ ˘ C˘(r0 ) . pr−r id ⊗ α ˘ r,r : >∗ (ξ −1 Ω X/S X/S

(III.12.9.1)

B

B

˘ Definition III.12.10. Let M be an object of Modaft Q (B) (III.12.1) and N a Higgs e1 OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S (III.12.6). (i) Let r be a rational number > 0. We say that M and N are r-associated if there exists an isomorphism of ΞrQ ∼

α : >r+ (N ) → Sr (M ).

(III.12.10.1)

We then also say that the triple (M , N , α) is r-admissible. (ii) We say that M and N are associated if there exists a rational number r > 0 such that M and N are r-associated. Note that for all rational numbers r ≥ r0 > 0, if M and N are r-associated, they are r0 -associated, in view of (III.12.8.5) and (III.12.8.6). ˘ -module is an object of Modaft (B) ˘ for which Definition III.12.11. (i) A Dolbeault B Q Q 1 −1 e 1 there exists an associated Higgs OX [ p ]-bundle with coefficients in ξ ΩX/S . e1 is solvable if it (ii) We say that a Higgs OX [ 1 ]-bundle with coefficients in ξ −1 Ω X/S

p

admits an associated Dolbeault module. ˘ the full subcategory of Modaft (B) ˘ made up of We denote by ModDolb (B) Q Q ˘ -modules, and by HMsol (O [ 1 ], ξ −1 Ω e 1 ) the full subcategory of Dolbeault B Q X p X/S e 1 ) made up of solvable Higgs OX [ 1 ]-bundles with coefficients in HM(OX [ p1 ], ξ −1 Ω X/S p e1 . ξ −1 Ω X/S

˘ -module is B ˘ -flat (III.6.4). Proposition III.12.12. Every Dolbeault B Q Q ˘ -module, N a Higgs O [ 1 ]-bundle with coefficients in Let M be a Dolbeault B Q X p ∼ e 1 , r a rational number > 0, and α : >r+ (N ) → ξ −1 Ω Sr (M ) an isomorphism of ΞrQ . X/S (r) (r) Since the OX [ 1 ]-module N is locally free of finite type, the C˘ -module >∗ (N )⊗ ˘ C˘ p

is flat by III.6.7(iii). It follows that M ⊗ ˘ BQ by virtue of III.6.4.4 and III.10.33.

Q

BQ

Q

(r) (r) ˘ -flat C˘Q is C˘Q -flat. Consequently, M is B Q

272

III. GLOBAL ASPECTS

˘ -module M and all rational numbers r ≥ r0 ≥ 0, the III.12.13. For every B Q morphism (III.12.8.9) and the isomorphism (III.12.8.5) induce a morphism 0

0

>r+ (Sr (M )) → >r+ (Sr (M ))

(III.12.13.1)

e 1 ). We thus obtain a filtered direct system (>r (Sr (M )))r∈Q . of HM(OX [ p1 ], ξ −1 Ω + ≥0 X/S We denote by H the functor (III.12.13.2)

˘ → HM(O [ 1 ], ξ −1 Ω e 1 ), H : ModQ (B) X X/S p

M 7→ lim >r+ (Sr (M )). −→

r∈Q>0

e 1 ) and all rational numbers r ≥ r0 ≥ 0, For every object N of HM(OX [ p1 ], ξ −1 Ω X/S the morphism (III.12.8.8) and the isomorphism (III.12.8.6) induce a morphism K r (>r+ (N )) → K

(III.12.13.3)

r0

0

(>r + (N ))

˘ We thus obtain a filtered direct system (K r (>r+ (N )))r≥0 . Recall of ModQ (B). (III.11.25) that filtered direct limits are not a priori representable in the category ˘ Mod (B). Q

e1 ) Lemma III.12.14. We have a canonical isomorphism of HM(OX [ p1 ], ξ −1 Ω X/S 1 ∼ ˘ ). (OX [ ], 0) → H (B Q p

(III.12.14.1) This follows from III.11.18.

Lemma III.12.15. Let r be a rational number ≥ 0 and N a Higgs OX [ p1 ]-bundle with e1 (III.12.6). We have a canonical isomorphism coefficients in ξ −1 Ω X/S

(III.12.15.1)

∼ ˘ )) → γ r : N ⊗OX [ p1 ] >r+ (Sr (B >r+ (>r+ (N )) Q

e 1 ), where the left-hand side is the tensor product of Higgs modules of HM(OX [ p1 ], ξ −1 Ω X/S (II.2.8.8). Moreover, we have the following properties: (i) The morphism (III.12.15.2)

N → >r+ (>r+ (N ))

˘ )) is the adjoint induced by γ r and the canonical morphism OX [ p1 ] → >r+ (Sr (B Q of the identity of >r+ (N ) (III.12.7.12). (ii) For every rational number r0 such that r ≥ r0 ≥ 0, the diagram (III.12.15.3)

˘ )) N ⊗OX [ p1 ] >r+ (Sr (B Q  0 0 ˘ N ⊗OX [ p1 ] >r+ (Sr (B Q ))

γr

γr

0

/ >r+ (>r+ (N ))  / >r0 (>r0 + (N )) +

where the vertical arrows are induced by the morphism (III.12.8.9) and the isomorphisms (III.12.8.5) and (III.12.8.6), is commutative.

III.12. DOLBEAULT MODULES

273

Indeed, by III.12.4(ii), we have canonical isomorphisms of OX [ p1 ]-modules ∼

(r)

(III.12.15.4)

N ⊗OX [ p1 ] >∗ (C˘Q ) →

(III.12.15.5)

e1 ξ −1 Ω X/S

BQ

∗

⊗OX >∗ (> (N ) ⊗ ˘

BQ

∼

→ (III.12.15.6)

(r) >∗ (>∗ (N ) ⊗ ˘ C˘Q ),

⊗OX N

e1 ξ −1 Ω X/S

(r) C˘Q )

˘(r) e1 >∗ (>∗ (ξ −1 Ω X/S ⊗OX N ) ⊗ ˘ CQ ), BQ

(r) ⊗OX [ p1 ] >∗ (C˘Q ) ∼

→

˘(r) e1 >∗ (>∗ (ξ −1 Ω X/S ⊗OX N ) ⊗ ˘ CQ ). BQ

The third isomorphism is induced by the first two by III.12.3(i). Moreover, in view of the bifunctoriality of the isomorphism (III.12.4.2), the diagram (III.12.15.7) (r) / >∗ (>∗ (N ) ⊗ ˘ C˘(r) ) 1 >∗ (C˘ N ⊗ ) OX [ p ]

(r) θ⊗id+pr id⊗>∗ (d˘Q )

BQ

Q

Q

>∗ (>∗ (θ)⊗id+pr id⊗d˘(r) )



e1 / >∗ (>∗ (ξ −1 Ω X/S

(r) e1 1 >∗ (C˘ ξ −1 Ω ] Q ) X/S ⊗OX N ⊗OX [ p

 (r) ⊗OX N ) ⊗ ˘ C˘Q ) BQ

where θ is the Higgs field of N , is commutative. We then take for γ (III.12.15.1) the isomorphism (III.12.15.4). Statement (i) follows from III.12.2(ii) and III.6.21. Statement (ii) is a consequence of the bifunctoriality of the isomorphism (III.12.4.2). r

III.12.16. Let r be a rational number > 0 and (M , N , α) an r-admissible triple. For any rational number r0 such that 0 < r0 ≤ r, we denote by 0

∼

0

0

αr : >r + (N ) → Sr (M )

(III.12.16.1) 0

0

the isomorphism of ΞrQ induced by r,r (α) and the isomorphisms (III.12.8.5) and (III.12.8.6), and by 0

0

0

β r : N → >r+ (Sr (M ))

(III.12.16.2) its adjoint (III.12.7.12).

Proposition III.12.17. Under the assumptions of III.12.16, let moreover r0 , r00 be two rational numbers such that 0 < r00 < r0 ≤ r. Then: (i) The composition βr

0

0

0

N −→ >r+ (Sr (M )) −→ H (M ),

(III.12.17.1)

where the second arrow is the canonical morphism (III.12.13.2), is an isomorphism that does not depend on r0 . (ii) The composition (III.12.17.2)

0

0

∼

βr

00

00

00

>r+ (Sr (M )) −→ H (M ) −→ N −→ >r+ (Sr (M )),

where the first arrow is the canonical morphism (III.12.13.2) and the second arrow is the inverse of the isomorphism (III.12.17.1), is the canonical morphism (III.12.13.1). (i) For any rational number 0 < t ≤ r, we denote by (III.12.17.3)

∼ ˘ )) → γ t : N ⊗OX [ p1 ] >t+ (St (B >t+ (>t+ (N )) Q

274

III. GLOBAL ASPECTS

e 1 ), and by the isomorphism (III.12.15.1) of HM(OX [ p1 ], ξ −1 Ω X/S ∼ ˘ )) → >t+ (St (M )) δ t : N ⊗OX [ p1 ] >t+ (St (B Q

(III.12.17.4)

the composition >t+ (αt ) ◦ γ t . The diagram δr

0 0 ˘ N ⊗OX [ p1 ] >r+ (Sr (B Q ))

(III.12.17.5)

 00 00 ˘ N ⊗OX [ p1 ] >r+ (Sr (B Q ))

δr

0

00

/ >r0 (Sr0 (M )) +  / >r00 (Sr00 (M )) +

where the vertical arrows are the canonical morphisms (III.12.13.1), is commutative by virtue of III.12.15(ii). The isomorphisms (δ t )0r+ (Sr (B Q )) NNN NNN NNN NNN '  ˘ ) N ⊗OX [ p1 ] H (B Q

δr

0

δ

/ >r0 (Sr0 (M )) +  / H (M )

0 0 ˘ 0 where ιr is induced by the canonical morphism OX [ p1 ] → >r+ (Sr (B Q )) and the vertical arrows are the canonical morphisms. By III.12.15(i), we have 0

0

0

0

0

0

0

δ r ◦ ιr = >r+ (αr ) ◦ γ r ◦ ιr = β r .

(III.12.17.8)

The statement follows by virtue of III.12.14. (ii) This follows from (III.12.17.5), (III.12.17.7), and III.11.17(ii). ˘ -module M , H (M ) (III.12.13.2) is a Corollary III.12.18. For every Dolbeault B Q solvable Higgs OX [ p1 ]-bundle associated with M . In particular, H induces a functor that we denote also by ˘ → HMsol (O [ 1 ], ξ −1 Ω e 1 ), M 7→ H (M ). (III.12.18.1) H : ModDolb (B) X Q X/S p ˘ -module M , there exist a rational number Corollary III.12.19. For every Dolbeault B Q r r > 0 and an isomorphism of ΞQ ∼

α : >r+ (H (M )) → Sr (M )

(III.12.19.1)

satisfying the following properties. For any rational number r0 such that 0 < r0 ≤ r, denote by 0

(III.12.19.2) 0

∼

0

0

αr : >r + (H (M )) → Sr (M ) 0

the isomorphism of ΞrQ induced by r,r (α) and the isomorphisms (III.12.8.5) and (III.12.8.6), and by (III.12.19.3)

0

0

0

β r : H (M ) → >r+ (Sr (M ))

its adjoint (III.12.7.12). Then:

III.12. DOLBEAULT MODULES

275 0

(i) For every rational number r0 such that 0 < r0 ≤ r, the morphism β r is a left 0 0 0 inverse of the canonical morphism $r : >r+ (Sr (M )) → H (M ). 0 00 (ii) For all rational numbers r and r such that 0 < r00 < r0 ≤ r, the composition 0

$r

0

0

βr

00

00

00

>r+ (Sr (M )) −→ H (M ) −→ >r+ (Sr (M ))

(III.12.19.4)

is the canonical morphism. Remark III.12.20. Under the assumptions of III.12.19, the isomorphism α is not a priori uniquely determined by (M , r), but for every rational number 0 < r0 < r, the 0 morphism αr (III.12.19.2) depends only on M , on which it depends functorially (cf. the proof of III.12.26). III.12.21. Let r be a rational number > 0 and (M , N , α) an r-admissible triple. To avoid any ambiguity with (III.12.16.1), we denote by α ˇ : Sr (M ) → >r+ (N )

(III.12.21.1)

the inverse of α in ΞrQ . For any rational number r0 such that 0 < r0 ≤ r, we denote by 0

∼

0

0

α ˇ r : Sr (M ) → >r + (N )

(III.12.21.2)

0

0

α) and the isomorphisms (III.12.8.5) and the isomorphism of ΞrQ induced by r,r (ˇ (III.12.8.6), and by 0 βˇr : M → K

(III.12.21.3)

r0

0

(>r + (N ))

the adjoint morphism. Proposition III.12.22. Under the assumptions of III.12.21, let moreover r0 , r00 be two rational numbers such that 0 < r00 < r0 ≤ r. Then:

(i) The direct limit V (N ) of the direct system (K t (>t+ (N )))t∈Q>0 (III.12.13.3) ˘ is representable in Mod (B). Q

(ii) The composition 0 βˇr

M −→ K

(III.12.22.1)

r0

0

(>r + (N )) −→ V (N ),

where the second arrow is the canonical morphism, is an isomorphism, which does not depend on r0 . (iii) The composition (III.12.22.2)

K

r0

∼

0

00 βˇr

(>r + (N )) −→ V (N ) −→ M −→ K

r 00

(>r

00

+

(N )),

where the first arrow is the canonical morphism and the second arrow is the inverse of the isomorphism (III.12.22.1), is the canonical morphism (III.12.13.3). ˘ -flat by III.12.12, for every rational number t ≥ 0, we have a (i) Since M is B Q ˘ canonical isomorphism of Mod (B) Q

(III.12.22.3)

∼ ˘ )) → γ t : M ⊗ ˘ K t (St (B K t (St (M )). Q BQ

We denote by (III.12.22.4)

∼ ˘ )) → δ t : M ⊗ ˘ K t (St (B K t (>t+ (N )) Q BQ

276

III. GLOBAL ASPECTS

the composition K t (ˇ αt ) ◦ γ t . The diagram r0

M ⊗˘ K

(III.12.22.5)

BQ

 r 00

M ⊗˘ K BQ

δr

0 ˘ (Sr (B Q ))

δr

˘ )) (Sr (B Q 00

0

00

/K

/K

r0

r 00

0

(>r + (N ))  00 (>r + (N ))

where the vertical arrows are induced by the morphism (III.12.8.8) and the isomorphisms (III.12.8.5) and (III.12.8.6), is clearly commutative. The statement then follows from III.11.23. (ii) By III.11.23, the canonical morphism ˘ )) (III.12.22.6) M → lim M ⊗ K t (St (B ˘ BQ

−→

t∈Q>0

Q

is an isomorphism. The isomorphisms (δ t )0r + (N ))

 / V (N )

δ

M

r0

0 ˘ → K r0 (Sr0 (B ˘ )) and the unlabeled where ιr is induced by the canonical morphism B Q Q arrow is the canonical morphism, is commutative. One immediately verifies that 0 0 0 0 0 0 0 (III.12.22.9) δ r ◦ ιr = K r (ˇ αr ) ◦ γ r ◦ ιr = βˇr .

The statement follows. (iii) This follows from (III.12.22.5) and III.11.23(ii). Corollary III.12.23. We have a functor (III.12.23.1) 1 ˘ e 1 ) → ModDolb V : HMsol (OX [ ], ξ −1 Ω (B), Q X/S p

N 7→ lim K r (>r+ (N )). −→

r∈Q>0

e 1 ), V (N ) is associated with N . Moreover, for every object N of HMsol (OX [ p1 ], ξ −1 Ω X/S Corollary III.12.24. For every solvable Higgs OX [ p1 ]-bundle N with coefficients in e 1 , there exist a rational number r > 0 and an isomorphism of Ξr ξ −1 Ω Q X/S ∼

α ˇ : Sr (V (N )) → >r+ (N )

(III.12.24.1)

satisfying the following properties. For any rational number r0 such that 0 < r0 ≤ r, denote by 0

(III.12.24.2) 0

∼

0

0

α ˇ r : Sr (V (N )) → >r + (N ) 0

the isomorphism of ΞrQ induced by r,r (ˇ α) and the isomorphisms (III.12.8.5) and (III.12.8.6), and by 0 0 0 (III.12.24.3) βˇr : V (N ) → K r (>r + (N ))

III.12. DOLBEAULT MODULES

277

its adjoint. Then: 0 (i) For every rational number r0 such that 0 < r0 ≤ r, the morphism βˇr is a right 0 0 0 inverse of the canonical morphism $r : K r (>r + (N )) → V (N ). 0 00 (ii) For all rational numbers r and r such that 0 < r00 < r0 ≤ r, the composition

(III.12.24.4)

K

r0

0

$r

00 βˇr

0

(>r + (N )) −→ V (N ) −→ K

r 00

(>r

00

+

(N ))

is the canonical morphism. Remark III.12.25. Under the assumptions of III.12.24, the isomorphism α ˇ is not a priori uniquely determined by (N , r), but for every rational number 0 < r0 < r, the 0 morphism α ˇ r (III.12.24.2) depends only on N , on which it depends functorially (cf. the proof of III.12.26). Theorem III.12.26. The functors (III.12.18.1) and (III.12.23.1) (III.12.26.1)

˘ o ModDolb (B) Q

H V

/

e1 ) HMsol (OX [ p1 ], ξ −1 Ω X/S

are equivalences of categories quasi-inverse to each other. ˘ H (M ) is a solvable Higgs O [ 1 ]-bundle asFor every object M of ModDolb (B), X p Q sociated with M , by virtue of III.12.18. We choose a rational number rM > 0 and an isomorphism of ΞrQM (III.12.26.2)

∼

αM : >rM + (H (M )) → SrM (M )

satisfying the properties of III.12.19. For any rational number r such that 0 < r ≤ rM , we denote by (III.12.26.3)

∼

r αM : >r+ (H (M )) → Sr (M )

the isomorphism of ΞrQ induced by rM ,r (αM ) and the isomorphisms (III.12.8.5) and (III.12.8.6), by (III.12.26.4) (III.12.26.5)

∼

α ˇ M : SrM (M ) → ∼

r α ˇM : Sr (M ) →

>rM + (H (M )),

>r+ (H (M )),

r the inverses of αM and αM , respectively, and by

(III.12.26.6) (III.12.26.7)

r βM : H (M ) → >r+ (Sr (M )), r βˇM : M → K r (>r+ (H (M ))),

r r r the adjoint morphisms of αM and α ˇM , respectively. Note that α ˇM is induced by rM ,r  (ˇ αM ) and the isomorphisms (III.12.8.5) and (III.12.8.6). By III.12.22(ii), the composition

(III.12.26.8)

βˇr

M M −→ K r (>r+ (H (M ))) −→ V (H (M )),

where the second arrow is the canonical morphism, is an isomorphism, which a priori depends on αM but not on r. Let us show that this isomorphism depends only on M (but not on the choice of αM ) and that it depends on it functorially. It suffices to ˘ and every rational number show that for every morphism u : M → M 0 of ModDolb (B) Q

278

III. GLOBAL ASPECTS

0 < r < inf(rM , rM 0 ), the diagram of ΞrQ αrM

>r+ (H (M ))

(III.12.26.9)

Sr (u)

>r+ (H (u))

 >r+ (H (M 0 ))

/ Sr (M )

αrM 0

 / Sr (M 0 )

is commutative. Let r, r0 be two rational numbers such that 0 < r < r0 < inf(rM , rM 0 ). Consider the diagram 0

0

0

>r+ (Sr (M ))

(III.12.26.10) 0

0



r βM

/ H (M )

0

>r+ (Sr (u))

0

$M 0

 / H (M 0 )

/ >r+ (Sr (M )) >r+ (Sr (u))

H (u)

r0

>r+ (Sr (M 0 )) 0

r $M

r βM 0

 / >r+ (Sr (M 0 ))

0

r r where $M and $M It follows from III.12.19(ii) that 0 are the canonical morphisms. r0 the large rectangle is commutative. Since the left square is commutative and $M 0 is surjective by III.12.19(i), the right square is also commutative. The desired assertion follows in view of the injectivity of (III.12.7.12). ˘ e 1 ), V (N ) is a Dolbeault B Likewise, for every object N of HMsol (OX [ 1 ], ξ −1 Ω Q p

X/S

module associated with N , by virtue of III.12.23. We choose a rational number rN > 0 and an isomorphism of ΞrQN (III.12.26.11)

∼

α ˇ N : SrN (V (N )) → >rN + (N )

satisfying the properties of III.12.24. For any rational number r such that 0 < r ≤ rN , we denote by (III.12.26.12)

∼

r α ˇN : Sr (V (N )) → >r+ (N )

the isomorphism of ΞrQ induced by rN ,r (ˇ αN ) and the isomorphisms (III.12.8.5) and (III.12.8.6), by (III.12.26.13) (III.12.26.14)

∼

αN : >rN + (N ) → SrN (V (N )), ∼

r αN : >r+ (N ) → Sr (V (N )),

r the inverses of α ˇ M and α ˇN , respectively, and by

(III.12.26.15) (III.12.26.16)

r βˇN : V (N ) → K r (>r+ (N )), r βN : N → >r+ (Sr (V (N ))),

r r the adjoint morphisms of α ˇN and αN , respectively. By III.12.17(i), the composition

(III.12.26.17)

βr

N −→ >r+ (Sr (V (N ))) −→ H (V (N )),

where the second arrow is the canonical morphism, is an isomorphism, which a priori depends on α ˇ N but not on r. Let us show that this isomorphism depends only on N (but not on the choice of α ˇ N ) and that it depends on it functorially. It suffices to show e 1 ) and every rational that for every morphism v : N → N 0 of HMsol (OX [ p1 ], ξ −1 Ω X/S

III.12. DOLBEAULT MODULES

279

number 0 < r < inf(rN , rN 0 ), the diagram of ΞrQ α ˇ rN

Sr (V (N ))

(III.12.26.18)

/ >r+ (N )

Sr (V (v))

>r+ (v)

 Sr (V (N 0 ))

α ˇ rN 0

 / >r+ (N 0 )

is commutative. Let r, r0 be two rational numbers such that 0 < r < r0 < inf(rN , rN 0 ). ˘ Consider the diagram of Mod (B) Q

0

(III.12.26.19) K

r0

K

r0

(>r

0+

K 0

r0

(>

r0 +

(v))

(N ))

r $N

r βˇM

/ V (N )

/ K r (>r+ (N ))

V (v)



0

(>r + (N 0 ))

r0 $N 0

 / V (N 0 )

r βˇN 0



K r (>r+ (v))

/ K r (>r+ (N 0 ))

0

r r where $N and $N 0 are the canonical morphisms. It follows from III.12.24(ii) that the r0 large rectangle is commutative. Since the left square is commutative and $N is invertible on the right by III.12.24(i), the right square is also commutative; the desired assertion follows. ←

◦

x) be a point of X´et ×X´et X ´et (III.8.6) such that x lies over III.12.27. Let (y 0 0 b0 its p-adic Hausdorff s, X the strict localization of X at x, R10 = Γ(X , OX 0 ), and R 1 completion. For any coherent OX -module F , we (abusively) set (III.12.27.1)

0

Fx = lim Γ(X n , F ⊗OX OX 0 ). ←−

n

n∈N◦

b0 ) induces a functor that we By III.6.16, the resulting functor Modcoh (OX ) → Mod(R 1 denote also by 1 b10 [ 1 ]), F 7→ Fx . (III.12.27.2) Modcoh (OX [ ]) → Mod(R p p es . For any object F = (Fn )n∈N of E esN◦ , we By III.9.5(i), ρ(y x) is a point of E (abusively) set (III.12.27.3)

Fρ(y

x)

= lim (Fn )ρ(y ←−

x) .

n∈N◦

e N◦ → Ens is not a priori a fiber functor. Let us Take care that the resulting functor E s take again the notation of III.10.25. By (III.10.25.2), VI.10.31, and VI.9.9, we have a canonical isomorphism ˘ B ρ(y

(III.12.27.4)

x)

y ∼ b →R X0 .

The functor (III.12.27.3) induces functors that we denote also by y

˘ → Mod(B)

b 0 ), M 7→ M Mod(R ρ(y x) , X y 1 ˘ b 0 [ ]), M 7→ M (III.12.27.6) ModQ (B) → Mod(R ρ(y x) . X p By III.10.26, III.10.29, VI.10.31, and VI.9.9, for every rational number r ≥ 0, we have a canonical isomorphism (III.12.27.5)

(III.12.27.7)

(r) C˘ρ(y

∼

x)

y,(r)

→ CbX 0 .

280

III. GLOBAL ASPECTS

Lemma III.12.28. Under the assumptions of III.12.27, let moreover F be a coherent OX -module, (U, p : x → U ) an object of Vx (III.10.8) such that U is affine, and U the formal scheme p-adic completion of U . Then: b0 -module Fx is of finite type, and is complete and separated for the p-adic (i) The R 1 topology. (ii) We have a canonical and functorial isomorphism (III.12.28.1)

∼

b10 . b OU (U ) R Fx → Γ(U , F ⊗OX OU )⊗

(iii) We have a canonical and functorial isomorphism (III.12.28.2)

(>∗ (F ))ρ(y

x)

∼ by 0. b Rb0 R → Fx ⊗ X 1

(i) This follows from ([11] Chapter III § 2.11 Proposition 14). (ii) This immediately follows from the definition. (iii) By ([2] VIII 5.2 and VII 5.8), we have a canonical isomorphism (III.12.28.3)

0

∼

(F ⊗OX OX n )x → Γ(X n , F ⊗OX OX 0 ), n

where we view F ⊗OX OX n on the left-hand side as a sheaf of Xs,´et (III.9.9). In view of (III.2.9.5), (III.11.1.12), and (VI.10.18.1), we deduce from this a canonical and functorial isomorphism (III.12.28.4)

(>∗ (F ))ρ(y

0

∼

x)

y

→ lim (Γ(X n , F ⊗OX OX 0 ) ⊗R10 RX 0 ). ←−

n

n∈N◦

The statement follows in view of ([11] Chapter III § 2.11 Corollary 1 to Proposition 14). III.12.29. We keep the assumptions and notation of III.12.27 and moreover set R0 = Γ(X 0 , OX 0 ) and (III.12.29.1)

e 1 (X 0 ) = Γ(X 0 , Ω e 1 ⊗O OX 0 ). Ω X X/S X/S

e 1 (X 0 ) we mean a For any R0 -algebra A, by a Higgs A-module with coefficients in ξ −1 Ω X/S e 1 (X 0 ) ⊗R0 A (II.2.8). In view of (III.12.5.3) Higgs A-module with coefficients in ξ −1 Ω X/S e 1 (X 0 ) is free of finite type, the functor (III.12.27.2) and the fact that the R0 -module Ω X/S induces a functor (III.12.29.2) 1 e 1 ) → HM(R b10 [ 1 ], ξ −1 Ω e 1 (X 0 )), (N , θ) 7→ (Nx , θx ). HMcoh (OX [ ], ξ −1 Ω X/S X/S p p Let r be a rational number ≥ 0. By (III.10.25.6), we have a canonical isomorphism (III.12.29.3)

Ω1 y,(r) CX 0

∼

y

b /R X0

y,(r)

e 1 (X 0 ) ⊗R0 C 0 . → ξ −1 Ω X/S X

We denote by (III.12.29.4)

y,(r)

dC y,(r) : CX 0 X0

y,(r)

e 1 (X 0 ) ⊗R0 C 0 → ξ −1 Ω X/S X

b y 0 -derivation of C y,(r) (cf. III.10.25) and by the universal R X X0 (III.12.29.5)

y,(r) e 1 (X 0 ) ⊗R0 Cby,(r) dCby,(r) : CbX 0 → ξ −1 Ω X/S X0 X0

e 1 (X 0 ) being free of finite type). its extension to the p-adic completions (the R0 -module Ω X/S b y 0 -fields with coefficients in ξ −1 Ω e 1 (X 0 ) by II.2.12 and II.2.16. These are Higgs R X X/S

III.12. DOLBEAULT MODULES

281

We denote by Ξaft,r the full subcategory of Ξr (III.12.7) made up of integrable ˘ such that the C˘(r) pr -isoconnections (F , G , u, ∇) with respect to the extension C˘(r) /B 0r modules F and G are adic of finite type (III.7.16), and by Ξ the category of integrable y y,(r) b /R 0 (III.6.13). These are p-adic pr -isoconnections with respect to the extension Cb X0

Ξaft,r Q

X

Ξ0r Q

additive categories. We denote by and the categories of objects of Ξaft,r and 0r Ξ up to isogeny (III.6.1.1). In view of ([11] Chapter III § 2.11 Proposition 14 and Corollary 1), the functor (III.12.27.3) induces an additive functor Ξaft,r → Ξ0r .

(III.12.29.6)

b y 0 -isogeny with coefficients in ξ −1 Ω e 1 (X 0 ). By III.6.14, every object of Ξ0r is a Higgs R X X/S We therefore have a functor (III.12.29.7) b y 1 −1 Ω e 1 (X 0 )), (M, N, u, ∇) 7→ (MQ , (id ⊗ u−1 ) ◦ ∇Q ). Ξ0r Q → HM(RX 0 [ ], ξ p p X/S Qp p We deduce from this a functor φρ(y

(III.12.29.8)

x) :

b y 0 [ 1 ], ξ −1 Ω e 1 (X 0 )). Ξaft,r → HM(R X X/S Q p

˘ of finite type. By ([11] Chapter III § 2.11 Proposition 14 Let M be an adic B-module y b and Corollary 1), the R 0 -module M is of finite type, and is complete and separated ρ(y

X

x)

for the p-adic topology. We have a canonical and functorial isomorphism (III.12.7.3) (III.12.29.9)

φρ(y

∼ r x) (S (M)) →

(r),y b b y Mρ(y ((CbX 0 ⊗ RX 0

r b Qp ). x) )Qp , (p dCby,(r) ⊗id) X0

Lemma III.12.30. Under the assumptions of III.12.27 and III.12.29, let moreover e 1 . Then: (N , θ) be a Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S b0 [ 1 ]-module Nx is projective of finite type. (i) The R 1 p (ii) For every rational number r ≥ 0, we have a canonical and functorial isomorphism (III.12.30.1)

φρ(y

r+ (N x) (>

∼ (r),y , θ)) → (CbX 0 ⊗Rb0 Nx , pr dCby,(r) ⊗ id + id ⊗ θx ). X0

1

Indeed, let N be a coherent OX -module such that NQp = N (III.6.16). (i) Let (U, p : x → U ) be an object of Vx (P) and U the formal scheme p-adic completion of U . By III.12.28(ii), we have a canonical isomorphism (III.12.30.2)

∼ b10 → b OU (U ) R Γ(U , N ⊗OX OU )⊗ Nx .

We have Γ(U , N ⊗OX OU ) ⊗Zp Qp = Γ(U , N ⊗OX OU ) ([1] (2.10.5.1)). By III.6.17, the OU (U )[ p1 ]-module Γ(U , N ⊗OX OU ) is projective of finite type. We deduce from this, by virtue of III.6.18 and III.10.27(ii), that the morphism (III.12.30.3)

b0 [ 1 ] → Nx Γ(U , N ⊗OX OU ) ⊗OU (U ) R 1 p

induced by (III.12.30.2) is an isomorphism; the desired statement follows. (ii) By III.12.28(iii), we have a canonical and functorial isomorphism (III.12.30.4)

(>∗ (N) ⊗ ˘ C˘(r) )ρ(y B

x)

∼ b Rb0 CbX(r),y → Nx ⊗ . 0

The statement then follows from (i), III.6.18, and III.10.27(ii).

1

282

III. GLOBAL ASPECTS

Lemma III.12.31. Under the assumptions of III.12.27 and III.12.29, let moreover M ˘ 1 −1 e 1 ΩX/S , be an object of Modaft Q (B), (N , θ) a Higgs OX [ p ]-bundle with coefficients in ξ r a rational number > 0, and ∼

α : >r+ (N , θ) → Sr (M )

(III.12.31.1)

an isomorphism of ΞrQ (III.12.7). Then: b y 0 [ 1 ]-module M (i) The R ρ(y X p (ii) The isomorphism φρ(y αρ(y

(III.12.31.2)

is projective of finite type. by,(r) x) (α) (III.12.29.8) induces a CX 0 -linear isomorphism

x) :

x)

∼ y,(r) y,(r) CbX 0 ⊗Rb0 Nx → CbX 0 ⊗ b y Mρ(y RX 0

1

x)

b y 0 -modules with coefficients in ξ −1 Ω e 1 (X 0 ), where Nx is endowed of Higgs R X X/S y,(r) with the Higgs field θx (III.12.29.2), CbX 0 is endowed with the Higgs field pr dCby,(r) (III.12.29.5), and Mρ(y x) is endowed with the zero Higgs field. X0

˘ such that M = M . By (III.12.29.9) Indeed, let M be an object of Modaft (B) Qp y,(r) b and III.12.30, φρ(y x) (α) is a CX 0 -linear isomorphism ∼ y,(r) y,(r) b b y Mρ(y CbX 0 ⊗Rb0 Nx → (CbX 0 ⊗

(III.12.31.3)

RX 0

1

x) )

⊗Zp Qp

b y 0 -modules with coefficients in ξ −1 Ω e 1 (X 0 ), where Nx is endowed with the of Higgs R X X/S y,(r) Higgs field θx , Cb 0 is endowed with the Higgs field pr d by,(r) , and Mρ(y x) is endowed CX 0

X

b y 0 , which exists b y 0 -augmentation u : Cby,(r) → R with the zero Higgs field. Consider an R X X X0 by (III.10.25.6). By virtue of III.6.18, III.10.27(ii), III.12.30(i), and (III.12.31.3), the canonical morphism y

b 0 ⊗ y,(r) (Cby,(r) ⊗ b b y Mρ(y (R X X0 Cb X0

RX 0

x) )) ⊗Zp

by 0⊗ b by,(r) (CbXy,(r) b b y Mρ(y Qp → (R ⊗ 0 X C RX 0

X0

x) )) ⊗Zp

Qp

is an isomorphism. The right-hand side identifies canonically with Mρ(y x) ⊗Zp Qp . The isomorphism (III.12.31.3) therefore induces, by base change by u, an isomorphism (III.12.31.4)

∼ by ⊗ N → R Mρ(y x b0 X0 R 1

x)

⊗Zp Qp = Mρ(y

x) .

b y 0 [ 1 ]-module M Consequently, the R ρ(y x) is projective of finite type by virtue of X p III.12.30(i). By III.6.18 and III.10.27(ii), the canonical morphism (III.12.31.5)

y,(r) (CbX 0 ⊗ b y Mρ(y RX 0

x) )

y,(r) b b y Mρ(y ⊗Zp Qp → (CbX 0 ⊗ RX 0

x) )

⊗Zp Qp

is therefore an isomorphism, giving the lemma. ˘ -module M and every integer q ≥ 0, we have a Lemma III.12.32. For every flat B Q functorial canonical isomorphism (III.11.21) (III.12.32.1)

∼ lim Rq >∗ (M ⊗ ˘ K•Q (C˘(r) , pr d˘(r) )) → Rq >∗ (M ), −→

r∈Q>0

BQ

where Rq >∗ (M ⊗ ˘ K•Q (C˘(r) , pr d˘(r) )) denotes the hypercohomology of the functor >∗ BQ (III.12.1.3) with respect to the complex M ⊗ ˘ K• (C˘(r) , pr d˘(r) ). BQ

Q

III.12. DOLBEAULT MODULES

283

Indeed, the spectral sequence of hypercohomology of the functor >∗ induces, for every rational number r ≥ 0, a functorial spectral sequence (III.12.32.2) r i,j E2 = Ri >∗ (M ⊗ ˘ Hj (K•Q (C˘(r) , pr d˘(r) ))) ⇒ Ri+j >∗ (M ⊗ ˘ K•Q (C˘(r) , pr d˘(r) )). BQ

BQ

By III.11.23(iii), for all integers i ≥ 0 and j ≥ 1 and all rational numbers r > r0 > 0, the canonical morphism r

(III.12.32.3)

0

i,j

r Ei,j 2 → E2

is zero. We therefore have r

lim

(III.12.32.4)

−→

Ei,j 2 = 0.

r∈Q>0

On the other hand, it follows from III.11.23(ii) that the canonical morphisms ˘ → H0 (K• (C˘(r) , pr d˘(r) )), B Q Q

(III.12.32.5)

for r ∈ Q>0 , induce an isomorphism

∼

Ri >∗ (M ) → lim

(III.12.32.6)

−→

r

Ei,0 2 .

r∈Q>0

Since filtered direct limits are representable in Mod(OX [ p1 ]) and commute with finite inverse limits ([2] II 4.3), the lemma follows. e X/S and Lemma III.12.33. Let N be a Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω • q an integer ≥ 0. Denote by K (N ) the Dolbeault complex of N (II.2.8.2) and for any rational number r ≥ 0, by K• (>r+ (N )) the Dolbeault complex of >r+ (N ) (III.12.7). We then have a functorial canonical isomorphism ∼

lim Rq >∗ (K• (>r+ (N ))) → Hq (K• (N )),

(III.12.33.1)

−→

r∈Q>0

q

•

r+

where R >∗ (K (> (N ))) denotes the hypercohomology of the functor >∗ (III.12.1.3) with respect to the complex K• (>r+ (N )). Let r be a rational number > 0, and i, j two integers ≥ 0. In view of III.12.4(i), d˘(r) (III.11.21.1) induces an OX -linear morphism j ˘(r) ), e1 δ j,(r) : Rj >∗ (C˘(r) ) → ξ −1 Ω X/S ⊗OX R >∗ (C

(III.12.33.2)

e 1 . We denote which is clearly a Higgs OX -field on Rj >∗ (C˘(r) ) with coefficients in ξ −1 Ω X/S j,(r)

by θ the Higgs OX [ p1 ]-field on N and by ϑtot = θ⊗id+pr id⊗δ j,(r) the total Higgs OX [ p1 ]field on N ⊗O Rj >∗ (Cb(r) ) (II.2.8.8). By III.12.4(ii), we have a canonical OX [ 1 ]-linear p

X

isomorphism (III.12.33.3)

∼ j,(r) Rj >∗ (Ki (>r+ (N ))) → Ki (N ⊗OX Rj >∗ (C˘(r) ), ϑtot ),

which is compatible with the differentials of the two Dolbeault complexes. On the other hand, we have a functorial canonical spectral sequence (III.12.33.4)

r

j i r+ Ei,j (N ))) ⇒ Ri+j >∗ (K• (>r+ (N ))). 1 = R >∗ (K (>

By III.11.18 and (III.12.33.3), for every i ≥ 0, we have a canonical isomorphism (III.12.33.5)

lim −→

r∈Q>0

r

∼

i Ei,0 1 → K (N , θ),

284

III. GLOBAL ASPECTS

and for every j ≥ 1, we have lim

(III.12.33.6)

−→

r

Ei,j 1 = 0.

r∈Q>0

Moreover, the isomorphisms (III.12.33.5) (for i ∈ N) form an isomorphism of complexes. The lemma follows ([2] II 4.3). ˘ -module and q an integer ≥ 0. Denote Theorem III.12.34. Let M be a Dolbeault B Q • by K (H (M )) the Dolbeault complex of the Higgs OX [ p1 ]-bundle H (M ) (II.2.8.2). We then have a functorial canonical isomorphism of OX [ p1 ]-modules ∼

Rq >∗ (M ) → Hq (K• (H (M ))).

(III.12.34.1)

Indeed, H (M ) is a solvable Higgs OX [ p1 ]-bundle associated with M by virtue of III.12.18. We choose a rational number rM > 0 and an isomorphism of ΞrQM ∼

αM : >rM + (H (M )) → SrM (M )

(III.12.34.2)

satisfying the properties of III.12.19. For any rational number r such that 0 < r < rM , we denote by ∼

r αM : >r+ (H (M )) → Sr (M )

(III.12.34.3)

the isomorphism of ΞrQ induced by rM ,r (αM ) and the isomorphisms (III.12.8.5) and r (III.12.8.6). By the proof of III.12.26, αM depends only on M (but not on αM ), and depends on it functorially. We denote by K• (>r+ (H (M ))) the Dolbeault complex of ˘ (cf. III.12.7). Since M is B ˘ -flat by III.12.12, αr induces >r+ (H (M )) in Mod (B) Q

M

Q

an isomorphism (III.11.21)

∼ K• (>r+ (H (M ))) → M ⊗ ˘ K•Q (C˘(r) , pr d˘(r) ).

(III.12.34.4)

BQ

We deduce from this a functorial canonical isomorphism of OX [ p1 ]-modules (III.12.34.5)

∼ lim Rq >∗ (K• (>r+ (H (M )))) → lim Rq >∗ (M ⊗ ˘ K•Q (C˘(r) , pr d˘(r) )), −→

−→

r∈Q>0

r∈Q>0

BQ

where Rq >∗ (−) denotes the hypercohomology of the functor >∗ . The theorem follows in view of III.12.32 and III.12.33. III.13. Dolbeault modules on a small affine scheme III.13.1. We keep the assumptions and general notation of III.10 and III.12 in this section. We moreover suppose that X is an object of Q (III.10.5), in other words, that the following conditions are satisfied: (i) X is affine and connected; (ii) f : (X, MX ) → (S, MS ) admits an adequate chart (III.4.4); (iii) there exists a fine and saturated chart M → Γ(X, MX ) for (X, MX ) inducing an isomorphism (III.13.1.1)

∼

× M → Γ(X, MX )/(X, OX ).

We set R = Γ(X, OX ), R1 = R ⊗OK OK , and (III.13.1.2)

e1 e1 Ω R/OK = Γ(X, ΩX/S ).

c1 the p-adic Hausdorff completion of R1 , by δ : E es → E e the canonical We denote by R embedding (III.10.2.5), and by e → X ◦f´et (III.13.1.3) β: E

III.13. DOLBEAULT MODULES ON A SMALL AFFINE SCHEME

285

the canonical morphism (III.8.3.1). For any integer n ≥ 1, we denote by (III.13.1.4)

es , B n ) → (X ◦f´et , B X,n ) βn : (E

the morphism of ringed topos defined by the morphism of topos β ◦δ and by the canonical homomorphism B X,n → β∗ (B n ) (cf. III.10.2). Recall that the latter is not in general an ◦ ◦ ˘ the ring (B ) of (X )N and by isomorphism (III.9.1.8). We denote by B X

(III.13.1.5)

X,n+1 n∈N

f´ et

˘ → ((X ◦ )N , B ˘ ) e N , B) β˘ : (E X s f´ et ◦

◦

the morphism of ringed topos induced by the morphisms (βn+1 )n∈N . If A is a ring and M an A-module, we denote also by A (resp. M ) the constant sheaf ◦ ◦ ◦ with value A (resp. M ) of X f´et or (X f´et )N , depending on the context. Proposition III.13.2. For every coherent OX -module N , we have a canonical and ˘ isomorphism functorial B-linear (III.13.2.1)

∼ ˘ ∗ β˘∗ (N (X) ⊗R c1 B X ) → > (N ).

For any integer n ≥ 1, we set Nn = N /pn N , which we view as an OX n -module of Xs,´et or X´et , depending on the context (cf. III.2.9 and III.9.9). By (III.11.1.12) and the remark following (III.2.9.5), we have a canonical isomorphism (III.13.2.2)

∼

∗ >∗ (N ) → (σn+1 (Nn+1 ))n∈N .

For every integer n ≥ 1, we have a canonical isomorphism (III.9.8.4) (III.13.2.3)

∼

σn∗ (Nn ) → σ −1 (Nn ) ⊗σ−1 (OX

n

)

Bn.

Hence by virtue of VI.5.34(ii), VI.8.9, and VI.5.17, the B n -module σn∗ (Nn ) is the sheaf e associated with the presheaf of E (III.13.2.4)

{U 7→ Nn (Us ) ⊗OX

n

(Us )

B U,n },

´ /X )). (U ∈ Ob(Et

´ /X , we have canonical isomorphisms For every affine object U of Et (III.13.2.5) (III.13.2.6) (III.13.2.7)

∼

N (Us ) → N (X) ⊗R c1 OX (Us ), ∼

Nn (Us ) → N (Us )/pn N (Us ), ∼

OX n (Us ) → OX (Us )/pn OX (Us ).

On the other hand, by virtue of VI.5.34(i), VI.8.9, and VI.5.17, the B n -module ˘ e βn∗ (N (X) ⊗R c1 B X,n ) is the sheaf of E associated with the presheaf (III.13.2.8)

{U 7→ N (X) ⊗R c1 B U,n },

´ /X )). (U ∈ Ob(Et

By (III.10.6.5), we deduce from this a canonical and functorial B n -linear isomorphism (III.13.2.9)

∼

∗ βn∗ (N (X) ⊗R c1 B X,n ) → σn (Nn ).

The proposition follows in view of (III.7.5.4) and (III.13.2.2).

286

III. GLOBAL ASPECTS

(r) ˘ -module Let r be a rational number ≥ 0. We denote by F˘X the B X (r) (r) ˘ -algebra (C (r) ) (cf. III.10.18). By III.7.3(i) and (FX,n+1 )n∈N and by C˘X the B X X,n+1 n∈N ˘ (III.7.12.1), we have an exact sequence of B -modules

III.13.3.

X

(III.13.3.1)

˘ ˘ → F˘ (r) → ξ −1 Ω e1 0→B X R/OK ⊗R B X → 0. X

˘ -algebras In view of III.7.3(i) and (III.7.12.3), we have a canonical isomorphism of B X (r) ∼ ˘ (r) C˘X → lim Sm ˘ (FX ).

(III.13.3.2)

BX

−→

m≥0

0

For all rational numbers r ≥ r0 ≥ 0, the morphisms (ar,r X,n+1 )n∈N (III.10.18.3) induce

˘ -linear morphism aB X

0

0

˘ (r) ˘ (r ) a ˘r,r X : FX → FX .

(III.13.3.3)

r,r 0 ˘ -algebras The homomorphisms (αX,n+1 )n∈N (III.10.18.4) induce a homomorphism of B X 0

0

(r) (r ) r,r α ˘X : C˘X → C˘X .

(III.13.3.4)

For all rational numbers r ≥ r0 ≥ r00 ≥ 0, we have (III.13.3.5)

00

0

00

r ,r a ˘r,r =a ˘X ◦a ˘r,r X X

0

00

0

00

0

r,r r ,r r,r and α ˘X =α ˘X ◦α ˘X .

(r) We have a canonical C˘X -linear isomorphism

(III.13.3.6)

Ω1˘(r)

∼

˘ C X /B X

(r)

˘ e1 → ξ −1 Ω R/OK ⊗R CX .

˘ -derivation of C˘(r) corresponds via this isomorphism to the unique The universal B X X ˘ B -derivation X

(III.13.3.7)

(r) (r) ˘(r) e1 d˘X : C˘X → ξ −1 Ω R/OK ⊗R CX

(r) ˘ e1 that extends the canonical morphism F˘X → ξ −1 Ω R/OK ⊗R B X (III.13.3.1). Since

(III.13.3.8)

˘ ˘(r) ˘ (r) ˘(r) ˘(r) e1 ξ −1 Ω R/OK ⊗R B X = dX (FX ) ⊂ dX (CX ),

(r) ˘ -field with coefficients in ξ −1 Ω e1 the derivation d˘X is a Higgs B X R/OK by II.2.12. For all rational numbers r ≥ r0 ≥ 0, we have

(III.13.3.9)

0

0

(r)

(r 0 )

0

r,r r,r pr−r (id ⊗ α ˘X ) ◦ d˘X = d˘X ◦ α ˘X .

Proposition III.13.4. For every rational number r ≥ 0, the canonical morphisms (III.13.4.1) (III.13.4.2)

∼ (r) β˘∗ (F˘X ) → F˘ (r) , ∼ (r) β˘∗ (C˘X ) → C˘(r) ,

0

are isomorphisms. Moreover, for all rational numbers r ≥ r0 ≥ 0, the morphisms β˘∗ (˘ ar,r X ) 0 0 0 r,r ˘ r,r (III.10.31.7), reand β˘∗ (˘ αX ) identify with the morphisms a ˘r,r (III.10.31.6) and α spectively. ´ /X ), we denote by gU : U → X the canonical morphism. By virtue For any U ∈ Ob(Et (r) of VI.5.34(i), VI.8.9, and VI.5.17, for every integer n ≥ 1, the B n -module βn∗ (FX,n ) is

III.13. DOLBEAULT MODULES ON A SMALL AFFINE SCHEME

287

canonically isomorphic to the sheaf associated with the presheaf on E defined by the correspondence (III.13.4.3)

(r)

{U 7→ (g ◦U )∗f´et (FX,n ) ⊗(g◦ )∗

U f´ et (BX,n )

B U,n }.

For every Y ∈ Ob(Q), the canonical homomorphism (III.13.4.4)

(r)

(r)

(g ◦Y )∗f´et (FX,n ) ⊗(g◦ )∗ Y

f´ et (BX,n )

B Y,n → FY,n

is an isomorphism by virtue of III.10.20. Consequently, the morphism (r)

βn∗ (FX,n ) → Fn(r)

(III.13.4.5)

(r)

(r)

adjoint to the canonical morphism FX,n → βn∗ (Fn ), is an isomorphism by (III.10.6.5). We deduce from this, in view of (III.7.5.4) and (III.7.12.1), that the canonical morphism (III.13.4.1) is an isomorphism. We prove likewise that the canonical homomorphism (III.13.4.2) is an isomorphism; we can also deduce this from (III.13.4.1).The last assertion is obvious by adjunction. (r) Remark III.13.5. It follows from III.13.4 that for every rational number r ≥ 0, β˘∗ (d˘X ) identifies with the derivation d˘(r) (III.10.31.9). We can construct the identification explicitly as follows. By the proof of III.10.22(ii), for every integer n ≥ 1, the diagram

(III.13.5.1)

(r)

FX,n  (r) β∗ (Fn )

e1 / ξ −1 Ω R/OK ⊗R B X,n  e1 / β∗ (σn∗ (ξ −1 Ω X

n /S n

))

where the vertical arrows are the canonical morphisms and the horizontal arrows come from the exact sequences (III.10.18.1) and (III.10.22.1), is commutative. We deduce from this by adjunction a commutative diagram (III.13.5.2)

(r)

βn∗ (FX,n )

e1 / βn∗ (ξ −1 Ω R/OK ⊗R B X,n )  e1 / σn∗ (ξ −1 Ω X



(r) Fn

n /S n

)

whose vertical arrows are isomorphisms, by the proofs of III.13.2 and III.13.4. Consequently, the diagram (III.13.5.3)

(r) β˘∗ (C˘X )



C˘(r)

(r) β˘∗ (d˘X )

d˘(r)

/ β˘∗ (ξ −1 Ω ˘(r) e1 R/OK ⊗R CX ) e1 /σ ˘ ∗ (ξ −1 Ω ˘



˘ X/S

) ⊗ ˘ C˘(r) B

where the vertical arrows are the isomorphisms induced by (III.13.2.1) and (III.13.4.2) is commutative.

288

III. GLOBAL ASPECTS

˘ ) the category III.13.6. Let r be a rational number ≥ 0. We denote by Mod(B X ˘ -modules, by Θr the category of integrable pr -isoconnections with respect to the of B X (r) ˘ (III.6.10), and by Sr the functor extension C˘ /B X

(III.13.6.1)

SrX

X

X

˘ ) → Θr , M 7→ (C˘(r) ⊗ r ˘(r) ˘(r) : Mod(B ˘ M , CX ⊗ ˘ M , id, p dX ⊗ id). X X BX

BX

e1 By III.6.12, if (N , N , v, θ) is a Higgs OX -isogeny with coefficients in ξ −1 Ω X/S (III.12.5), 0

(III.13.6.2)

(r) ˘(r) c N 0 (X), id ⊗ c v, pr d˘(r) ⊗ v + id ⊗ θ) (C˘X ⊗R c1 N (X), CX ⊗R X R1 1

is an object of Θr . We thus obtain a functor (III.13.6.3)

r+ e 1 ) → Θr . >X : HI(OX , ξ −1 Ω X/S

Proposition III.13.7. For every rational number r ≥ 0, the diagrams of functors (III.13.7.1)

˘ ) Mod(B X

SrX

/ Θr

β˘∗

β˘∗

 ˘ Mod(B) (III.13.7.2)

Sr

 / Ξr >r+

e 1 ) X / Θr HIcoh (OX , ξ −1 Ω OX/S OOO OOO β˘∗ OOO OO'  >r+ Ξr

where the inverse image functor under β˘ for pr -isoconnections is defined in (III.6.11), are commutative up to canonical isomorphisms. This follows from III.13.2, III.13.4, and III.13.5. 1 and r > 0, Proposition III.13.8. Let , r be two rational numbers such that  > r+ p−1 N an S -flat coherent OX -module, and θ a Higgs OX -field on N with coefficients in e1 ξ −1 Ω X/S such that

(III.13.8.1)

e1 θ(N ) ⊂ p ξ −1 Ω X/S ⊗OX N .

e1 . We denote also by N the Higgs OX -isogeny (N , N , id, θ) with coefficients in ξ −1 Ω X/S ◦ N◦ ˘ Then there exist an adic B X -module of finite type M of (X f´et ) and an isomorphism of Θr (III.13.8.2)

∼

r >r+ X (N ) → SX (M ).

We may clearly assume that Xs is nonempty, so that (X, MX ) satisfies the assump◦ tions of II.6.2. Let y be a generic geometric point of X . Since X is locally irreducible (III.3.1), it is the sum of the schemes induced on its irreducible components. We denote ◦ by X hyi the irreducible component of X containing y. Likewise, X is the sum of the ◦ schemes induced on its irreducible components, and X hyi = X hyi ×X X ◦ is the irreducible ◦ ◦ component of X containing y. We set R1y = Γ(X hyi , OX ) and ∆y = π1 (X hyi , y). We by the p-adic Hausdorff completion of Ry , by B∆ the classifying topos of ∆y , denote by R y 1 1 by (III.13.8.3)

◦

∼

νy : X hyi,f´et → B∆y

III.13. DOLBEAULT MODULES ON A SMALL AFFINE SCHEME

289

y b y its pthe fiber functor at y (III.2.10.3), by RX the ring defined in (III.8.13.2), and by R X y,(r) b y -algebra defined in (III.10.24.3), the R adic Hausdorff completion. We denote by C X

X

y,(r) by CbX its p-adic Hausdorff completion, by

(III.13.8.4)

y,(r)

dC y,(r) : CX X

y,(r)

e1 → ξ −1 Ω R/OK ⊗R CX

b y -derivation of C y,(r) , and by the universal R X X (III.13.8.5)

y,(r) by,(r) e1 dCby,(r) : CbX → ξ −1 Ω R/OK ⊗R CX X

e1 its extension to the completions (note that the R-module Ω R/OK is free of finite type). It follows from III.8.15 and the definitions that we have canonical isomorphisms

(III.13.8.7)

◦

∼

◦

∼

νy (B X |X hyi ) →

(III.13.8.6)

(r)

y

RX , y,(r)

νy (CX,n |X hyi ) → CX y

y,(r)

/pn CX

.

y,(r)

The objects ∆y , R1y , RX , and CX correspond to the objects ∆, R1 , R, and C (r) defined in II, by taking κ e = y in II.6.7. From now on, we will use the constructions of II.13. Set Ny = Γ(X hyi,s , N ) and denote by e1 θy : Ny → ξ −1 Ω R/OK ⊗R Ny

(III.13.8.8)

by -field with coefficients in ξ −1 Ω e1 the Higgs R 1 R/OK induced by θ, which is -quasi-small in by the sense of II.13.4. We associate with it, by the functor (II.13.10.10), a quasi-small R 1 y,(r) representation ϕy of ∆y on Ny . By II.13.17, we have a ∆y -equivariant Cb -isomorphism X

y y,(r) b of modules with p-adic p -connections with respect to the extension CbX /R , r

(III.13.8.9)

∼ b Rby CbXy,(r) → b Rby CbXy,(r) , uy : Ny ⊗ Ny ⊗ 1

1

y,(r) CbX

where is endowed with the canonical action of ∆y and with the p-adic pr -connection r p dCby,(r) , the module Ny in the source is endowed with the trivial action of ∆y and with X by -field θy , and the module Ny in the target is endowed with the action ϕy of the Higgs R 1 by -field. ∆y and with the zero Higgs R 1

◦

There exists an inverse system (Ln+1 )n∈N of R1 -modules of X f´et such that for every ◦ generic geometric point y of X , we have an isomorphism of inverse systems of R1y representations of ∆y , (III.13.8.10)

◦

∼

(νy (Ln+1 |X hyi ))n∈N → (Ny /pn+1 Ny , ϕy )n∈N .

This immediately follows from the definition of the functor (II.13.10.10) (cf. VI.9.8). By III.2.11, Ln is of finite type on R1 . For all integers m ≥ n ≥ 1, the morphism Lm /pn Lm → Ln induced by the transition morphism Lm → Ln is an isomorphism. We set (III.13.8.11)

M = (Ln+1 ⊗R1 B X,n+1 )n∈N .

˘ -module of finite type of (X ◦ )N◦ by virtue of III.7.14. The isomorphisms It is an adic B X f´ et (r) (III.13.8.9) induce a C˘X -linear isomorphism (III.13.8.12)

˘(r) ∼ u : N (X) ⊗R c1 CX → M ⊗ ˘

BX

(r) C˘X

290

III. GLOBAL ASPECTS

such that the diagram / M ⊗˘

u

˘(r) N (X) ⊗R c1 CX

(III.13.8.13)

BX

(r) θ⊗id+pr id⊗d˘X

e1 ξ −1 Ω R/OK

 ˘(r) ⊗R N (X) ⊗R c1 CX

id⊗u



(r) C˘X

(r) pr id⊗d˘X

e1 / ξ −1 Ω R/OK ⊗R M ⊗ ˘

BX

(r) C˘X

is commutative, giving the proposition. Corollary III.13.9. Under the assumptions of III.13.8, if NQp is a locally projective OX [ p1 ]-module of finite type, (NQp , θQp ) is a solvable Higgs OX [ p1 ]-bundle. This follows from III.13.7 and III.13.8. III.14. Inverse image of a Dolbeault module under an étale morphism III.14.1. We keep the assumptions and general notation of III.10 and III.12 in this section. Let, moreover, g : X 0 → X be an étale morphism of finite type. We endow X 0 with the logarithmic structure MX 0 inverse image of MX and we denote by f 0 : (X 0 , MX 0 ) → (S, MS ) the morphism induced by f and g. Note that f 0 is adequate (III.4.7) and that X 0◦ = X ◦ ×X X 0 is the maximal open subscheme of X 0 where the log0 ˇ 0 with the logarithmic structures arithmic structure M 0 is trivial. We endow X and X X

MX 0 and MXˇ 0 inverse images of MX 0 . There exists essentially a unique étale morphism e0 → X e that fits into a Cartesian diagram (III.10.1) ge : X /X e0

ˇ0 X

(III.14.1.1)

ˇ g

g e

 ˇ X

 /X e

e 0 with the logarithmic structure M e 0 inverse image of M e , so that (X e 0, M e0 ) We endow X X X X 0 ˇ , M 0 ). is a smooth (A (S), M )-deformation of (X 2

A2 (S) 0 e0

ˇ X

We associate with (f , X , MXe 0 ) objects analogous to those defined in III.10 and e M e ), which we denote by the same symbols equipped with an exponent 0 . III.12 for (f, X, X We denote by (III.14.1.2) (III.14.1.3)

e0 Φ: E es0 Φs : E

e → E, es , → E

the morphisms of topos (III.8.5.3) and (III.9.11.8) induced by functoriality by g. By e at σ ∗ (X 0 ). Moreover, the canonical VI.10.14, Φ identifies with the localization of E 0 −1 homomorphism Φ (B) → B is an isomorphism by virtue of III.8.21(i). For every integer n ≥ 1, Φs is underlying a canonical morphism of ringed topos (III.9.11.11) (III.14.1.4)

0

es0 , B n ) → (E es , B n ). Φn : (E 0

Since the homomorphism Φ∗s (B n ) → B n is an isomorphism (III.9.13), there is no difference for B n -modules between the inverse image under Φs in the sense of abelian sheaves and the inverse image under Φn in the sense of modules. The diagram of morphisms of

III.14. INVERSE IMAGE OF A DOLBEAULT MODULE UNDER AN ÉTALE MORPHISM

291

ringed topos (III.9.11.12) 0

es0 , B n ) (E

(III.14.1.5)

Φn

0 σn

 0 (Xs,´ , et OX 0 )

/ (E es , B n ) σn

gn

n

 / (Xs,´et , OX ) n

where g n is the morphism induced by g, is commutative up to canonical isomorphism. III.14.2. Since every object of E 0 is naturally an object of E, we denote by  : E 0 → E the canonical functor. This factors through an equivalence of categories ∼

E 0 → E/(X 0◦ →X 0 ) ,

(III.14.2.1)

´ /X 0 , where we view E 0◦ that is even an equivalence of categories over Et /(X →X 0 ) as an ´ ´ (Et/X 0 )-category by base change of the canonical fibration π : E → Et/X (III.10.2.2). By VI.5.38, the covanishing topology on E 0 is induced by that on E through . Consequently,  is continuous and cocontinuous ([2] III 5.2). Moreover, Φ identifies with the localization e at σ ∗ (X 0 ) = (X 0◦ → X 0 )a (VI.10.14). In particular, Φ∗ is none other morphism of E ´ /X 0 made than the restriction functor by . We denote by Q0 the full subcategory of Et up of the objects that are in Q (III.10.5) and by 0 0 0 πQ 0 : EQ0 → Q

(III.14.2.2)

´ /X 0 . the fibered category deduced by base change from the canonical fibration π 0 : E 0 → Et 0 → E that fits into a commutative The functor  therefore induces a functor Q : EQ 0 Q diagram up to canonical isomorphism 0 EQ 0

(III.14.2.3)

Q

u0

 E0

/ EQ u



 /E

where u and u0 are the canonical projection functors. The functors u and u0 are fully 0 faithful, and the category EQ (resp. EQ 0 ) is U-small and topologically generates the site 0 E (resp. E ) by III.10.5. It immediately follows from (III.14.2.1) that Q factors through an equivalence of categories ∼

(III.14.2.4) 0◦

0 0◦ EQ 0 → (EQ ) /b u∗ (X →X 0 ) ,

0◦

where u b∗ (X → X 0 ) is the presheaf on EQ deduced from (X → X 0 ) by restriction 0 0 by u. We endow EQ (resp. EQ 0 ) with the topology induced by that on E (resp. E ). e→E e 0 is essentially surjective, the topology on E 0 0 is induced Since the functor Φ∗ : E Q by that on EQ by Q ([2] II 2.2). Consequently, Q is continuous and cocontinuous ([2] III 5.2). III.14.3. Let Y be an object of Q0 (III.14.2) such that Ys 6= ∅, y a geometric point ◦ ? of Y , and Y the irreducible component of Y containing y (III.3.3). Consider the objects e M e ). Note that the ring associated with Y in III.10.13 and III.10.15 relative to (f, X, X y y by y on A (Y ) (III.10.13.5), and the R R (III.8.13.2), the logarithmic structure M A2 (Y )

Y

TyY

2

Y

e M e) (III.10.13.7) do not change whether we use f or f 0 . Replacing (f, X, X y 0y y b 0 e0 e by (f , X , MXe 0 ), we denote by LY the TY -torsor of Y zar defined in (III.10.13.9), by module

292

III. GLOBAL ASPECTS

b y -module defined in (III.10.13.10), and by C 0y the R b y -algebra defined in FY0y the R Y Y Y y e the open subscheme of (III.10.13.12). Let U be a Zariski open subscheme of Yb and U y A2 (Y ) defined by U . Consider the commutative diagram (without the dotted arrow) ˇ g

ˇ 0, M 0 ) / (X ˇ X

(U, M b y |U )

(III.14.3.1)

Y

ˇ M ) / (X, ˇ X

iY |U

 e, M e) y |U (U A2 (Y )

 / (X e 0, M e0 ) X

ψ

 / (X, e M e) X

g e

Since ge is étale, the map LY0y (U ) → LYy (U ),

(III.14.3.2)

is bijective. We deduce from this a π1 (Y

?◦

ψ 7→ ge ◦ ψ,

b y -linear isomorphism , y)-equivariant and R Y ∼

FYy → FY0y ,

(III.14.3.3)

which fits into a commutative diagram (III.14.3.4)

0

/ by RY

/ Fy

0

/ by RY

 / F 0y Y

/ ξ −1 Ω e1

X/S (Y

Y

by ) ⊗OX (Y ) R Y

 by / ξ −1 Ω e 1 0 (Y ) ⊗O 0 (Y ) R Y X /S X

/0

/0

where the horizontal lines are the exact sequences (III.10.13.10). The isomorphism ?◦ b y -algebras (III.14.3.3) induces a π (Y , y)-equivariant isomorphism of R 1

(III.14.3.5)

Y

∼ CYy →

CY0y .

e M e ) by (f 0 , X e 0 , M e 0 ), we denote by Let n be an integer ≥ 1. Replacing (f, X, X X 0 0 ◦ ◦ 0 the B Y -module of Y f´et defined in (III.10.15.3) and by CY,n the B Y -algebra of Y f´et ◦ defined in (III.10.15.4). By III.8.21(ii), we have a canonical ring isomorphism of Y f´et

0 FY,n

(III.14.3.6)

∼

0

BY → BY .

In view of the above, we have a canonical B Y -linear isomorphism (III.14.3.7)

∼

0 FY,n → FY,n .

We deduce from this an isomorphism of B Y -algebras (III.14.3.8)

∼

0 CY,n → CY,n .

III.14.4. Let n be an integer ≥ 1 and r a rational number ≥ 0. By ([2] III 2.3(2)), 0 since the canonical functor Q : EQ 0 → EQ is cocontinuous (III.14.2), the isomorphisms 0

(III.14.3.7) induce an isomorphism of B n -modules

(III.14.4.1)

∼

ρn : Φ∗n (Fn ) → Fn0

III.14. INVERSE IMAGE OF A DOLBEAULT MODULE UNDER AN ÉTALE MORPHISM

293

that fits into a commutative diagram (III.14.4.2)

0

/ Φ∗ (B n ) n

/ Φ∗n (Fn )

/ B0 n

 / Fn0

e1 / Φ∗n (σn∗ (ξ −1 Ω

X n /S n

ρn

0

))

 e1 0 / σn0∗ (ξ −1 Ω ) X /S n

/0

/0

n

where the horizontal lines are the exact sequences deduced from (III.10.22.1) (cf. the proof of III.10.22(ii)). Likewise, the isomorphisms (III.14.3.8) induce an isomorphism of 0 B n -algebras ∼

γn : Φ∗n (Cn ) → Cn0 ,

(III.14.4.3)

which is compatible with ρn via the isomorphisms (III.10.22.3). 0 By (III.14.4.2), ρn induces a B n -linear isomorphism ∗ (r) 0(r) ρ(r) n : Φn (Fn ) → Fn

(III.14.4.4)

that fits into a commutative diagram (III.14.4.5)

/ Φ∗ (B n ) n

0

/ Φ∗ (F (r) ) n n

e1 / Φ∗n (σn∗ (ξ −1 Ω X

ρ(r) n

/ B0 n

0

n /S n

 e1 0 / σn0∗ (ξ −1 Ω ) X /S

 / F 0(r) n

n

n

))

/0

/0

where the horizontal lines are the exact sequences deduced from (III.10.22.1). We deduce 0 from this an isomorphism of B n -algebras ∼

γn(r) : Φ∗n (Cn(r) ) → Cn0(r) .

(III.14.4.6)

III.14.5. We denote by X (resp. X0 ) the formal scheme p-adic completion of X 0 (resp. X ), by g : X0 → X

(III.14.5.1) 0

the extension of g : X → X to the completions, and by (III.14.5.2)

˘ 0 ) → (E ˘ ˘ : (E es0N◦ , B esN◦ , B) Φ

the morphism of ringed topos induced by the morphisms (Φn )n≥1 (III.14.1.4) (cf. III.7.5). We denote also by (III.14.5.3)

˘ → Mod (B ˘ 0) ˘ ∗ : ModQ (B) Φ Q

˘ By III.9.14, Φ ˘ is canonically isomorthe functor induced by the inverse image under Φ. ◦ ˘ at λ∗ (σ ∗ (X 0 )), where e N , B) phic to the localization morphism of the ringed topos (E s s s N◦ e e λ : Es → Es is the morphism of topos defined in (III.7.4.3). Consequently, there is ˘ ˘ in the sense of abelian no difference for B-modules between the inverse image under Φ sheaves and the inverse image in the sense of modules. The diagram of morphisms of

294

III. GLOBAL ASPECTS

ringed topos ˘ 0) es0N◦ , B (E

(III.14.5.4)

>0

˘ Φ

˘ / (E esN◦ , B) >



0 (Xs,zar , OX0 )



/ (Xs,zar , OX )

g

where > and >0 are the morphisms of ringed topos defined in (III.11.1.11), is commutative up to canonical isomorphism (III.9.11.15). Since the canonical morphism e 1 ) → ξ −1 Ω e1 0 g∗ (ξ −1 Ω X/S X /S

(III.14.5.5)

is an isomorphism, it induces by (III.14.5.4) an isomorphism ∼ ˘ ∗ (>∗ (ξ −1 Ω e 1 )) → e 1 0 ). >0∗ (ξ −1 Ω δ: Φ X/S X /S

(III.14.5.6)

Let r be a rational number ≥ 0. In view of (III.7.5.4) and (III.7.12.1), the isomor(r) ˘ 0 -linear isomorphism (III.14.4.4) induce a B phisms (ρ ) n

n≥1

∼ ˘ 0(r) ˘ ∗ (F˘ (r) ) → ρ˘(r) : Φ F .

(III.14.5.7)

(r) ˘ 0 -algebras Likewise, the isomorphisms (γn )n≥1 (III.14.4.6) induce an isomorphism of B ∼ ˘0(r) ˘ ∗ (C˘(r) ) → γ˘ (r) : Φ C .

(III.14.5.8)

It immediately follows from (III.14.4.5) that the diagram γ ˘ (r)

˘ ∗ (C˘(r) ) Φ

(III.14.5.9)

˘ ∗ (d˘(r) ) Φ



/ C˘0(r) 

(r)

d˘0(r)

e 1 0 ) ⊗ ˘ 0 C˘0(r) ˘ ∗ (>∗ (ξ −1 Ω e 1 ) ⊗ ˘ C˘(r) ) δ⊗˘γ / >0∗ (ξ −1 Ω Φ X /S X/S B

B

where d˘(r) and d˘0(r) are the derivations (III.12.7.1), is commutative. For all rational numbers r ≥ r0 ≥ 0, the diagram ˘ ∗ (C˘(r) ) Φ

(III.14.5.10)

γ ˘ (r)

0 ˘ ∗ (α Φ ˘ r,r )

 ∗ ˘(r 0 ) ˘ Φ (C )

0

/ C˘0(r) α ˘ 0r,r

γ ˘

(r 0 )

0

 / C˘0(r0 )

0

where α ˘ r,r and α ˘ 0r,r are the canonical homomorphisms (III.10.31.7), is commutative. III.14.6. (III.14.6.1)

We (abusively) denote by 1 e 1 ) → HM(OX0 [ 1 ], ξ −1 Ω e1 0 ) g∗ : HM(OX [ ], ξ −1 Ω X/S X /S p p

the inverse image functor for the Higgs modules (II.2.9) induced by g and the canonical morphism (III.14.5.5). We define likewise an inverse image functor (III.12.5) (III.14.6.2)

e 1 ) → HI(OX0 , ξ −1 Ω e 1 0 ). g∗ : HI(OX , ξ −1 Ω X/S X /S

This induces a functor that we denote also by (III.14.6.3)

e 1 ) → HIQ (OX0 , ξ −1 Ω e 1 0 ). g∗ : HIQ (OX , ξ −1 Ω X/S X /S

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295

The diagram of functors (III.14.6.4)

e1 / HM(OX [ 1 ], ξ −1 Ω

e1 ) HIQ (OX , ξ −1 Ω X/S

X/S )

p

g∗

g∗

 e1 0 ) / HM(OX0 [ 1 ], ξ −1 Ω X /S p

 e1 0 ) HIQ (OX0 , ξ −1 Ω X /S

where the horizontal arrows are the functors (III.12.5.2) is commutative up to canonical isomorphism. III.14.7. Let r be a rational number ≥ 0. By III.6.11 and (III.14.5.9), for every ˘ ∗ (F ), Φ ˘ ∗ (G ), Φ ˘ ∗ (u), Φ ˘ ∗ (∇)) identifies with an obobject (F , G , u, ∇) of Ξr (III.12.7), (Φ 0r (r) ject of Ξ through the isomorphisms γ˘ (III.14.5.8) and δ (III.14.5.6). We deduce from this a functor that we denote also by ˘ ∗ : Ξr → Ξ0r . (III.14.7.1) Φ This induces a functor that we denote also by ˘ ∗ : ΞrQ → Ξ0r (III.14.7.2) Φ Q. The diagrams of functors

(III.14.7.3)

˘∗ Φ

/ Ξr

Sr

˘ Mod(B)

˘∗ Φ



S0r

0

˘ ) Mod(B



/ Ξ0r

where the horizontal arrows are the functors (III.12.7.2), and (III.14.7.4)

e1 ) HI(OX , ξ −1 Ω X/S g∗

/ Ξr

>r+

˘∗ Φ



e1 0 ) HI(OX0 , ξ −1 Ω X /S

>0r+



/ Ξ0r

where the horizontal arrows are the functors (III.12.7.7), are clearly commutative up to canonical isomorphisms. By III.9.14, the diagram of functors (III.14.7.5)

Ξr

K

˘ / Mod(B)

r

Φ∗

 Ξ0r

K

0r

/



˘∗ Φ 0

˘ ) Mod(B

where K r and K 0r are the functors (III.12.7.4), is commutative up to canonical isomorphism. The base change morphism relative to the diagram (III.14.5.4) induces a morphism e1 0 ) of functors from Ξr to HI(OX0 , ξ −1 Ω X /S (III.14.7.6)

˘∗ g∗ ◦ >r+ → >0r + ◦Φ ,

where >r+ and >0r + are the functors (III.12.7.10). By ([2] XVII 2.1.3), this is the adjoint of the morphism ∼ ˘∗ ˘ ∗, (III.14.7.7) >0r+ ◦ g∗ ◦ >r+ → Φ ◦ >r+ ◦ >r+ → Φ

296

III. GLOBAL ASPECTS

where the first arrow is the isomorphism underlying the diagram (III.14.7.4) and the sece1 ) ond arrow is the adjunction map. Consequently, for every object N of HI(OX , ξ −1 Ω X/S and every object F of Ξr , the diagram of maps of sets (III.14.7.8) r / HomHI(OX ,ξ−1 Ω e1 HomΞr (>r+ (N ), F ) ) (N , >+ (F )) X/S

a

b



/ HomHI(O

˘ ∗ (F )) HomΞ0r (>0r+ (g∗ (N )), Φ

X0 ,ξ

−1 Ω e1 0 X /S

 ∗ ˘∗ (g (N ), >0r + (Φ (F ))) )

where the horizontal arrows are the adjunction isomorphisms, a is induced by the functor ˘ ∗ and the isomorphism underlying the diagram (III.14.7.4), and b is induced by the Φ functor g∗ and the morphism (III.14.7.6), is commutative. For all rational numbers r ≥ r0 ≥ 0, the diagram of functors Ξr

(III.14.7.9)

˘∗ Φ



Ξ0r

r,r

0

0r,r 0



/ Ξr0 

˘∗ Φ

/ Ξ0r0

where the horizontal arrows are the functors (III.12.8.3), is commutative up to canonical isomorphism. It immediately follows from (III.14.5.10) that the diagram of morphisms of functors / g∗ ◦ >r0 ◦ r,r0 g∗ ◦ >r+ (III.14.7.10) +

>0r +

 ˘∗ ◦Φ

/ >0r0 ◦ r,r0 ◦ Φ ˘∗ +

0 >0r +

 0 ˘ ◦ Φ∗ ◦ r,r

where the horizontal arrows are induced by the morphism (III.12.8.9), the vertical arrows are induced by the morphism (III.14.7.6), and the identification indicated by the symbol = comes from the diagram (III.14.7.9), is commutative. Consequently, the composition ∼ ˘∗ ◦ S → ˘ ∗, (III.14.7.11) g∗ ◦ >r ◦ Sr → >0r ◦ Φ >0r ◦ S0r ◦ Φ +

+

+

where the first arrow is induced by (III.14.7.6) and the second arrow is the isomorphism underlying the diagram (III.14.7.3), induces by direct limit, for r ∈ Q>0 , a morphism of ˘ to HM(O 0 [ 1 ], ξ −1 Ω e1 functors from Mod (B) ) X

Q

p

X0 /S

˘ ∗, g ◦H →H 0◦Φ ∗

(III.14.7.12)

where H and H 0 are the functors (III.12.13.2).

Proposition III.14.8. Suppose that g is an open immersion. Then: (i) For every rational number r ≥ 0, the morphism (III.14.7.6) is an isomorphism. It makes commutative the diagram of functors (III.14.8.1)

Ξr

>r+

g∗

˘∗ Φ

 Ξ0r

e1 ) / HI(OX , ξ −1 Ω X/S

>0r +

 e1 0 ) / HI(OX0 , ξ −1 Ω X /S

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297

(ii) The morphism (III.14.7.12) is an isomorphism. It makes commutative the diagram of functors ˘ ModQ (B)

(III.14.8.2)

˘∗ Φ

H

e1 ) / HM(OX [ 1 ], ξ −1 Ω X/S p g∗



˘ 0) ModQ (B

 e1 0 ) / HM(OX0 [ 1 ], ξ −1 Ω X /S p

H0

(i) This follows from III.9.15. (ii) This follows from (i) and the definitions. ˘ -module and N a solvable Higgs O [ 1 ]Proposition III.14.9. Let M be a Dolbeault B Q X p 0 ˘ −1 e 1 ∗ ∗ ˘ (M ) is a Dolbeault B -module and g (N ) bundle with coefficients in ξ Ω . Then Φ X/S

Q

e 1 0 . If, moreover, M and is a solvable Higgs OX0 [ p1 ]-bundle with coefficients in ξ −1 Ω X /S ˘ ∗ (M ) and g∗ (N ) are associated N are associated, then Φ e1 0 ˘∗ Indeed, g∗ (N ) is a Higgs OX0 [ p1 ]-bundle with coefficients in ξ −1 Ω X /S and Φ (M ) 0 ˘ is an object of Modaft Q (B ). Suppose that there exist a rational number r > 0 and an isomorphism of ΞrQ ∼

α : >r+ (N ) → Sr (M ).

(III.14.9.1)

˘ ∗ (α) induces an isomorphism of Ξ0r In view of (III.14.7.3) and (III.14.7.4), Φ Q ∼

˘ ∗ (M )), α0 : >0r+ (g∗ (N )) → S0r (Φ

(III.14.9.2) giving the proposition. III.14.10.

˘ ∗ induces a functor By III.14.9, Φ

(III.14.10.1) and g∗ induces a functor (III.14.10.2)

˘ → ModDolb (B ˘ 0 ), ˘ ∗ : ModDolb (B) Φ Q Q

1 e 1 ) → HMsol (OX0 [ 1 ], ξ −1 Ω e 1 0 ). g∗ : HMsol (OX [ ], ξ −1 Ω X/S X /S p p

Proposition III.14.11. (i) The diagram of functors (III.14.11.1)

˘ ModDolb (B) Q ˘∗ Φ



˘ 0) ModDolb (B Q

H

e1 / HMsol (OX [ 1 ], ξ −1 Ω

X/S )

p

H0



g∗

e1 0 ) / HMsol (OX0 [ 1 ], ξ −1 Ω X /S p

where H and H 0 are the functors (III.12.18.1) is commutative up to canonical isomorphism. (ii) The diagram of functors (III.14.11.2)

e1 ) HMsol (OX [ p1 ], ξ −1 Ω X/S g∗



e1 0 ) HMsol (OX0 [ p1 ], ξ −1 Ω X /S

V

V0

˘ / ModDolb (B) Q 

˘∗ Φ

/ ModDolb (B ˘ 0) Q

298

III. GLOBAL ASPECTS

where V and V 0 are the functors (III.12.23.1) is commutative up to canonical isomorphism. ˘ M and H (M ) are associated by virtue of (i) For every object M of ModDolb (B), Q III.12.18. We choose a rational number rM > 0 and an isomorphism of ΞrQM (III.14.11.3)

∼

αM : >rM + (H (M )) → SrM (M )

satisfying the properties of III.12.19. For any rational number r such that 0 < r ≤ rM , we denote by (III.14.11.4)

∼

r αM : >r+ (H (M )) → Sr (M )

the isomorphism of ΞrQ induced by rM ,r (αM ) (III.12.8.4) and the isomorphisms ˘ ∗ (αM ) induces (III.12.8.5) and (III.12.8.6). In view of (III.14.7.3) and (III.14.7.4), Φ 0rM an isomorphism of ΞQ (III.14.11.5)

∼ 0 ˘ ∗ (M )). αM : >0rM + (g∗ (H (M ))) → S0rM (Φ

˘ ∗ (αr ) induces an isomorphism of Ξ0r Likewise, Φ Q M (III.14.11.6)

∼ 0r ˘ ∗ (M )), αM : >0r+ (g∗ (H (M ))) → S0r (Φ

0 that we can also deduce from 0rM ,r (αM ) by (III.14.7.9). We denote by

(III.14.11.7)

0r 0r ˘ ∗ βM : g∗ (H (M )) → >0r + (S (Φ (M )))

˘ 0 -module and g∗ (H (M )) ˘ ∗ (M ) is a Dolbeault B its adjoint (III.12.7.12). By III.14.9, Φ Q e 1 0 , associated with Φ ˘ ∗ (M ). is a solvable Higgs OX0 [ p1 ]-bundle with coefficients in ξ −1 Ω X /S Consequently, by virtue of III.12.17(i), the composition 0r

(III.14.11.8)

βM 0r ˘ ∗ 0 ˘∗ g∗ (H (M )) −→ >0r + (S (Φ (M ))) −→ H (Φ (M )),

where the second arrow is the canonical morphism (III.12.13.2), is an isomorphism that depends a priori on αM but not on r. By the proof of III.12.26, for every mor˘ and every rational number r such that 0 < r < (B) phism u : M → M 0 of ModDolb Q r inf(rM , rM 0 ), the diagram of ΞQ (III.14.11.9)

>r+ (H (M ))

αrM

Sr (u)

>r+ (H (u))

 >r+ (H (M 0 ))

/ Sr (M )

αrM 0

 / Sr (M 0 )

is commutative. We deduce from this that the composed isomorphism (III.14.11.8) (III.14.11.10)

∼

˘ ∗ (M )) g∗ (H (M )) → H 0 (Φ

depends only on M (but not on the choice of αM ) and that it depends on it functorially; whence the statement. (ii) The proof is similar to that of (i) and is left to the reader. ˘ -module. Remarks III.14.12. Let M be a Dolbeault B Q (i) The canonical morphism (III.14.7.12) (III.14.12.1)

˘ ∗ (M )) g∗ (H (M )) → H 0 (Φ

III.15. FIBERED CATEGORY OF DOLBEAULT MODULES

299

is an isomorphism; it is the isomorphism underlying the commutative diagram (III.14.11.1). Indeed, let us use the notation of the proof of III.14.11(i). We moreover denote by r βM : H (M ) → >r+ (Sr (M ))

(III.14.12.2)

r the adjoint morphism of αM . It follows from (III.14.7.8) that the morphism 0r βM (III.14.11.7) is equal to the composition (III.14.12.3)

g∗ (H (M ))

r g∗ (βM ) ∗ /

g (>r+ (Sr (M )))

/ >0r (Φ ˘ ∗ (Sr (M ))) +

∼

/ >0r (S0r (Φ ˘ ∗ (M ))) , +

where the second arrow is the morphism (III.14.7.6) and the last arrow is the isomorphism underlying the diagram (III.14.7.3). On the other hand, the direct r limit of the morphisms βM , for r ∈ Q>0 , is the identity, and the direct limit 0r of the morphisms βM , for r ∈ Q>0 , is equal to the composed isomorphism (III.14.11.8) underlying the commutative diagram (III.14.11.1). (ii) Let r be a rational number > 0 and ∼

α : >r+ (H (M )) → Sr (M )

(III.14.12.4)

an isomorphism of ΞrQ satisfying the properties of III.12.19. By (i), (III.14.7.3), ˘ ∗ (α) with an isomorphism and (III.14.7.4), we can identify Φ ∼ ˘ ∗ (M ))) → ˘ ∗ (M )). α0 : >0r+ (H 0 (Φ S0r (Φ

(III.14.12.5)

0

˘ -module by III.14.9. It immedi˘ ∗ (M ) is a Dolbeault B On the other hand, Φ Q 0 ately follows from III.12.17 that α satisfies the properties of III.12.19. III.15. Fibered category of Dolbeault modules III.15.1. We keep the assumptions and general notation of III.10 and III.12 in this section. We denote by ψ the composition ι´et σs λ e esN◦ −→ ψ: E Es −→ Xs,´et −→ X´et ,

(III.15.1.1)

where λ is the morphism of topos defined in (III.7.4.3), σs is the canonical morphism of topos (III.10.2.6), and ι : Xs → X is the canonical injection. For any object U ´ /X , we denote by fU : (U, MX |U ) → (S, MS ) the morphism induced by f , and of Et ˇ so that (U ˇ → X, e , M e |U e ) is a e → X e the unique étale morphism that lifts U by U X ˇ ˇ smooth (A (S), M )-deformation of (U , M |U ). The localization of the ringed topos ˇ X

A2 (S)

2

˘ at ψ ∗ (U ) is canonically equivalent to the analogous ringed topos associated with e N , B) (E s ˘ fU by virtue of III.9.14. For every rational number r ≥ 0, the restriction of the B-algebra ˘ ∗ (U ))-algebra associC˘(r) over ψ ∗ (U ) is canonically isomorphic to the analogous (B|ψ ◦

˘ ∗ (U )) e , M e |U e ), by (III.14.5.8). We denote by Mod(B|ψ ated with the deformation (U X ˘ ∗ (U )) the category ˘ ∗ (U ))-modules of (E esN◦ )/ψ∗ (U ) , by ModQ (B|ψ the category of (B|ψ ˘ ∗ (U ))-modules up to isogeny, by Ξr the category of integrable pr -isoconnections of (B|ψ U

˘ ∗ (U )) (cf. III.12.7), by Ξr the category with respect to the extension (C˘(r) |ψ ∗ (U ))/(B|ψ U,Q Dolb ˘ r ∗ of objects of Ξ up to isogeny, and by Mod (B|ψ (U )) the category of Dolbeault U

Q

˘ ∗ (U )) -modules with respect to the deformation (U e , M e |U e ) (cf. III.12.11). By (B|ψ Q X

300

III. GLOBAL ASPECTS

´ /X , the restriction functor III.14.9, for every morphism g : U 0 → U of Et (III.15.1.2)

˘ ∗ (U 0 )), ˘ ∗ (U )) → Mod (B|ψ ModQ (B|ψ Q

M 7→ M |ψ ∗ (U 0 ),

induces a functor (III.15.1.3)

˘ ∗ (U )) → ModDolb (B|ψ ˘ ∗ (U 0 )). ModDolb (B|ψ Q Q

We denote by (III.15.1.4) (III.15.1.5)

ΞrU → ΞrU 0 , ΞrU,Q → ΞrU 0 ,Q ,

A 7→ A|ψ ∗ (U 0 ), B 7→ B|ψ ∗ (U 0 ),

the restriction functors defined in (III.14.7.1) and (III.14.7.2), respectively. Lemma III.15.2. Let r be a rational number ≥ 0, A, B two objects of ΞrX,Q , and (Ui )i∈I an étale covering of X. For all (i, j) ∈ I 2 , we set Uij = Ui ×X Uj . Then the diagram of maps of sets Y HomΞrX,Q (A, B) → (III.15.2.1) HomΞrU ,Q (A|ψ ∗ (Ui ), B|ψ ∗ (Ui )) i

i∈I

⇒

Y

HomΞrU

ij ,Q

(A|ψ ∗ (Uij ), B|ψ ∗ (Uij ))

(i,j)∈I 2

is exact. Indeed, since X is quasi-compact, we may assume that I is finite, in which case the assertion easily follows from III.6.7. ˘ the fibered (and even split [37] VI § 9) (E esN◦ )III.15.3. We denote by MOD(B) ˘ e N◦ ([35] II 3.4.1). It is a stack over E e N◦ by ([35] II 3.4.4). over E category of B-modules s

s

´ coh/X the full subcategory of Et ´ /X made up of étale schemes of finite We denote by Et presentation over X and by (III.15.3.1)

˘ → Et ´ coh/X MOD0 (B)

˘ ([37] VI § 3) by ψ ∗ ◦ ε, where ψ is the morphism (III.15.1.1) the base change of MOD(B) ´ coh/X → X´et is the canonical functor. This is also a stack by ([35] II 3.1.1). and ε : Et We deduce from this a fibered category (III.15.3.2)

˘ → Et ´ coh/X , MOD0Q (B)

˘ ∗ (U )) and the inverse ´ coh/X is the category ModQ (B|ψ whose fiber over an object U of Et 0 ´ coh/X is the restriction functor (III.15.1.2). image functor under a morphism U → U of Et Note that this is not a priori a stack. It induces a fibered category (III.15.3.3)

˘ → Et ´ coh/X MODDolb (B) Q

˘ ∗ (U )) and the ´ coh/X is the category ModDolb whose fiber over an object U of Et (B|ψ Q 0 ´ coh/X is the restriction functor inverse image functor under a morphism U → U of Et (III.15.1.3). ˘ Proposition III.15.4. Let M be an object of Modaft Q (B) and (Ui )i∈I a covering of ˘ ∗ (U )) -module ´ coh/X . Then M is Dolbeault if and only if for every i ∈ I, the (B|ψ Et i Q ∗ M |ψ (Ui ) is Dolbeault.

III.15. FIBERED CATEGORY OF DOLBEAULT MODULES

301

Indeed, the condition is necessary by virtue of III.14.9. Suppose that for every i ∈ I, M |ψ ∗ (Ui ) is Dolbeault and let us show that M is Dolbeault. Since X is quasi-compact, we may assume that I is finite. For any i ∈ I, denote by Xi the formal scheme padic completion of U i . For any (i, j) ∈ I 2 , set Uij = Ui ×X Uj and denote by Xij the formal scheme p-adic completion of U ij . By virtue of III.14.11(i), we have a canonical e1 isomorphism of Higgs OXij [ p1 ]-modules with coefficients in ξ −1 Ω Xij /S ∼

Hi (M |ψ ∗ (Ui )) ⊗OXi OXij → Hij (M |ψ ∗ (Uij )),

(III.15.4.1)

ei , M e |U ei ) and where Hi and Hij are the functors (III.12.18.1) associated with (fUi , U X e e (fUij , Uij , MXe |Uij ), respectively. We deduce from this a descent datum δ on the Higgs modules (Hi (M |ψ ∗ (Ui )))i∈I relative to the étale covering (Xi → X)i∈I . Since the latter is effective by III.6.22, there exist a Higgs OX [ p1 ]-bundle N and for every i ∈ I, an e1 isomorphism of Higgs OX [ 1 ]-modules with coefficients in ξ −1 Ω i

Xi /S

p

∼

N ⊗OX OXi → Hi (M |ψ (Ui )),

(III.15.4.2)

∗

that induce the descent datum δ. For any (i, j) ∈ I 2 and any rational number r > 0, we denote by >r+ and Sri (resp. i r+ r ei , M e |U ei ) >ij and Sij ) the functors (III.12.7.10) and (III.12.7.2) associated with (fUi , U X e e (resp. (fUij , Uij , MXe |Uij )). For every i ∈ I, we choose a rational number ri > 0 and an isomorphism of ΞrUii ,Q ∼

αi : >ri i + (Hi (M |ψ ∗ (Ui ))) → Sri i (M |ψ ∗ (Ui ))

(III.15.4.3)

satisfying the properties of III.12.19. For every (i, j) ∈ I 2 , M |ψ ∗ (Uij ) is Dolbeault by virtue of III.14.9. By (III.14.7.3), (III.14.7.4), and III.14.11(i), αi |ψ ∗ (Uij ) identifies with an isomorphism ∼

αi |ψ ∗ (Uij ) : >riji + (Hij (M |ψ ∗ (Uij ))) → Sriji (M |ψ ∗ (Uij )).

(III.15.4.4)

This satisfies the properties of III.12.19, in view of III.14.12. For any rational number r such that 0 < r ≤ ri , we denote by ri i ,r : ΞrUii ,Q → ΞrUi ,Q the functor (III.12.8.4) associated ei , M e |U ei ) and by with (fU , U i

(III.15.4.5)

X

∼

∗ r ∗ αir : >r+ i (Hi (M |ψ (Ui ))) → Si (M |ψ (Ui ))

the isomorphism of ΞrUi ,Q induced by ri i ,r (αi ) and the isomorphisms (III.12.8.5) and (III.12.8.6). By (III.14.7.3) and (III.14.7.4), we can identify αir with an isomorphism (III.15.4.6)

∼

αir : >r+ (N )|ψ ∗ (Ui ) → Sr (M )|ψ ∗ (Ui ).

It follows from the proof of III.12.26 that for every rational number 0 < r < inf(ri , rj ), we have in ΞrUij ,Q (III.15.4.7)

αir |ψ ∗ (Uij ) = αjr |ψ ∗ (Uij ).

By virtue of III.15.2, for every rational number 0 < r < inf(ri , i ∈ I), the isomorphisms (αir )i∈I glue to an isomorphism of ΞrQ (III.15.4.8) Consequently, M is Dolbeault.

∼

αr : >r+ (N ) → Sr (M ).

Proposition III.15.5. The following conditions are equivalent:

302

III. GLOBAL ASPECTS

(i) The fibered category (III.15.3.3) ˘ → Et ´ coh/X MODDolb (B) Q

(III.15.5.1)

is a stack ([35] II 1.2.1). ´ coh/X , denoting by U (resp. for every (ii) For every covering (Ui → U )i∈I of Et i ∈ I, by Ui ) the formal scheme p-adic completion of U (resp. U i ), a Higgs e1 OU [ p1 ]-bundle N with coefficients in ξ −1 Ω U /S is solvable if and only if for 1 e1 every i ∈ I, the Higgs OU [ ]-bundle N ⊗O OU with coefficients in ξ −1 Ω i

U

p

Ui /S

i

is solvable.

´ coh/X . For every (i, j) ∈ I 2 , set Uij = Ui ×X Uj . Let (Ui → U )i∈I be a covering of Et Denote by U the formal scheme p-adic completion of U and by H ? and V ? the functors e , M e |U e ). For any i ∈ I, denote by (III.12.18.1) and (III.12.23.1) associated with (fU , U X Ui the formal scheme p-adic completion of U i and by Hi and Vi the functors (III.12.18.1) ei , M e |U ei ). and (III.12.23.1) associated with (fUi , U X Let us first show (i)⇒(ii). Let N be a Higgs OU [ p1 ]-bundle with coefficients in e1 . If N is solvable, then for every i ∈ I, N ⊗O OU is solvable by III.14.9. ξ −1 Ω U /S

U

i

Conversely, suppose that for every i ∈ I, N ⊗OU OUi is solvable, and let us show that N ˘ |ψ ∗ (U )-module. is solvable. For every i ∈ I, M = V (N ⊗ O ) is a Dolbeault B i

i

OU

Ui

Q

i

By III.14.11(ii), the canonical descent datum on the Higgs bundles (N ⊗OU OUi )i∈I relative to the étale covering (Ui → U )i∈I induces a descent datum δ on the Dolbeault modules (Mi )i∈I relative to the covering (Ui → U )i∈I . Since the latter is effective by (i), ˘ |ψ ∗ (U ))-module M and for every i ∈ I, an isomorphism of there exists a Dolbeault (B Q

˘ |ψ ∗ (U )-modules B Q i

∼

M |ψ ∗ (Ui ) → Mi

(III.15.5.2)

that induce the descent datum δ. By virtue of III.12.26 and III.14.11(i), we have a ∼ canonical isomorphism of Higgs OU [ p1 ]-bundles H ? (M ) → N . Consequently, N is solvable. ˘ |ψ ∗ (U ))-modules M and M 0 , the diagram of Next, let us show (ii)⇒(i). For all (B Q maps of sets Y (III.15.5.3) Hom ˘ ∗ (M , M 0 ) → Hom ˘ ∗ (M |ψ ∗ (Ui ), M 0 |ψ ∗ (Ui )) BQ |ψ (U )

⇒

Y (i,j)∈I 2

BQ |ψ (Ui )

i∈I

Hom ˘

BQ |ψ ∗ (Uij )

(M |ψ ∗ (Uij ), M 0 |ψ ∗ (Uij ))

is exact. Indeed, since U is quasi-compact, we may assume that I is finite, in which case the assertion follows from III.6.7. ˘ |ψ ∗ (U ))-module and let δ be a descent For every i ∈ I, let Mi be a Dolbeault (B Q i datum on (Mi )i∈I relative to the covering (Ui → U )i∈I . Let us show that δ is effective. By assumption, for every i ∈ I, Ni = Hi (Mi ) is a solvable Higgs OUi [ p1 ]-bundle with e1 coefficients in ξ −1 Ω . In view of III.14.11(i), δ induces a descent datum γ on the Ui /S

Higgs bundles (Ni )i∈I relative to the étale covering (Ui → U )∈I . Since the latter is effective by III.6.22, there exist a Higgs OU [ p1 ]-bundle N and for every i ∈ I, an isomorphism of Higgs OUi [ p1 ]-modules (III.15.5.4)

∼

N ⊗OU OUi → Ni ,

III.15. FIBERED CATEGORY OF DOLBEAULT MODULES

303

that induce the descent datum γ. By (ii), N is solvable. Consequently, M = V ? (N ) ˘ |ψ ∗ (U ))-module. By III.12.26 and III.14.11(ii), for every i ∈ I, we is a Dolbeault (B Q ∼ ˘ |ψ ∗ (U ))-modules M |ψ ∗ (U ) → M , that induce have a canonical isomorphism of (B i

Q

i

i

the descent datum δ, proving the assertion.

Definition III.15.6. Let (N , θ) be a Higgs OX [ p1 ]-bundle with coefficients in e1 . ξ −1 Ω X/S

(i) We say that (N , θ) is small if there exist a coherent sub-OX -module N of N 1 such that that generates it over OX [ p1 ] and a rational number ε > p−1 e1 θ(N) ⊂ pε ξ −1 Ω X/S ⊗OX N.

(III.15.6.1)

(ii) We say that (N , θ) is locally small if there exists an open covering (Ui )i∈I of Xs such that for every i ∈ I, (N |Ui , θ|Ui ) is small. e1 Remark III.15.7. Suppose that X is affine and that the OX -module Ω X/S is free of b and R b1 their finite type. Set R = Γ(X, OX ) and R1 = R ⊗O O and denote by R K

K

p-adic Hausdorff completions. Let (N , θ) be a Higgs OX [ p1 ]-bundle with coefficients e 1 . Set N = Γ(X, N ), which is a projective R b1 [ 1 ]-module of finite type by in ξ −1 Ω X/S p III.6.17, and denote also by e 1 (X) ⊗R N (III.15.7.1) θ : N → ξ −1 Ω X/S

b1 [ 1 ]-field induced by θ. Then (N , θ) is small if and only if there exist a the Higgs R p b1 -module of finite type N ◦ of N that generates it over R b1 [ 1 ] and a rational number sub-R p 1 ε > p−1 such that e 1 (X) ⊗R N ◦ . θ(N ◦ ) ⊂ pε ξ −1 Ω X/S

(III.15.7.2)

Indeed, the condition is necessary by virtue of ([1] (2.10.5.1)) and it is sufficient in view of (III.12.5.3) and ([1] 1.10.2). Proposition III.15.8. Every solvable Higgs OX [ p1 ]-bundle (N , θ) with coefficients in e1 ξ −1 Ω is locally small. X/S

e1 Indeed, we may restrict to the case where X is affine and the OX -module Ω X/S is free of rank d (III.14.9). Let us then show that (N , θ) is small. Set R = Γ(X, OX ) b and R b1 their p-adic Hausdorff completions. Let and R1 = R ⊗OK OK and denote by R −1 e 1 e 1 . Set N = Γ(X, N ), which is a ω1 , . . . , ωd ∈ Γ(X, ξ ΩX/S ) be an OX -basis of ξ −1 Ω X/S b1 [ 1 ]-module of finite type by III.6.17, and denote also by projective R p

(III.15.8.1)

e 1 (X) ⊗R N θ : N → ξ −1 Ω X/S

b1 [ 1 ]-field induced by θ. We write the Higgs R p (III.15.8.2)

θ=

d X i=1

θ i ⊗ ωi ,

b1 [ 1 ]-endomorphisms of N that commute pairwise. where the θi ’s are R p ˘ and Let r be a rational number > 0, M an object of Modaft (B), Q

(III.15.8.3)

r+

α: >

∼

r

(N ) → S (M )

304

III. GLOBAL ASPECTS ←

◦

x) be a point of X´et ×X´et X ´et (III.8.6) an isomorphism of ΞrQ (III.12.10). Let (y 0 such that x lies over s and X the strict localization of X at x. We take again the notation of III.12.27 and III.12.29; however, to emphasize the dependence on x, we 0 bx,1 the p-adic Hausdorff set Rx = Γ(X 0 , OX 0 ) and Rx,1 = Γ(X , OX 0 ) and denote by R 0 0 0 b , respectively). By III.12.31, α induces a completion of Rx,1 (instead of R , R1 , and R 1 y,(r) b CX 0 -linear isomorphism ∼ y,(r) y,(r) CbX 0 ⊗Rbx,1 Nx → CbX 0 ⊗ b y Mρ(y

(III.15.8.4)

RX 0

x)

b y 0 -modules with coefficients in ξ −1 Ω e 1 (X 0 ) (III.12.29.1), where Nx is enof Higgs R X X/S y,(r) b dowed with the Higgs field θx (III.12.29.2), C 0 is endowed with the Higgs field pr d by,(r) CX 0

X

(III.12.29.5), and Mρ(y x) is endowed with the zero Higgs field (III.12.27.6). Let F be a coherent OX -module such that N = FQp (III.6.16), so that N = Γ(X, F ) ⊗Zp Qp ([1] (2.10.5.1)). By virtue of III.6.18, III.10.27(ii), and III.12.28(ii), we have a canonical isomorphism ∼ bx,1 . (III.15.8.5) Nx → N ⊗ b R R1

b1 [ 1 ]-module N (resp. the We will from now on identify these two modules. Endow the R p b y 0 [ 1 ]-module N ⊗ by bx,1 [ 1 ]-module Nx , resp. the R R x bx,1 RX 0 ) with the p-adic topology X p R p y

b 0 . Indeed, (II.2.2). Note that the p-adic topology on Nx is induced by that on Nx ⊗Rbx,1 R X 1 b since Nx is projective of finite type over Rx,1 [ ] (III.12.30), we may restrict ourselves p

bx,1 [ 1 ], for to the case where Nx is free of finite type, or even to the case where Nx = R p which the assertion has been established in III.10.27(iv). Q For every z ∈ N and every n = (n1 , . . . , nd ) ∈ Nd , p−r|n| ( 1≤i≤d n1i ! θini )(z) tends bx,1 when |n| tends to infinity. This follows from the isomorphism to 0 in Nx = N ⊗ b R R1

(III.15.8.4) by the same proof as II.13.24, in view of (III.10.25.6). Let π be a uniformizer of OK , (xi )i∈I the generic points of Xs , and for every i ∈ I, let xi be a generic point of X localized at xi . Since X is S-flat and Xs is reduced (III.4.2), for every integer n ≥ 1, the canonical homomorphism Y (III.15.8.6) R/π n R → Rxi /π n Rxi i∈I

is injective. We deduce from this that the canonical homomorphism Y (III.15.8.7) R1 /pn R1 → Rxi ,1 /pn Rxi ,1 i∈I

Q

b1 → b is injective. The homomorphism R i∈I Rxi ,1 is therefore injective. On the other hand, by ([11] Chapter III §2.11 Proposition 14 and Corollary 1), we have (III.15.8.8)

b1 /pn R b1 ' R1 /pn R1 , R

b1 ∩ pn (⊕i∈I R bx ,1 ) = pn R b1 and and likewise for the Rxi ,1 . We deduce from this that R i consequently that b1 [ 1 ] ∩ pn (⊕i∈I R bx ,1 ) = pn R b1 . (III.15.8.9) R i p b1 [ 1 ]-module N is projective of finite type, it follows that the p-adic topology Since the R p Q bx ,1 . on N is induced by the product of the p-adic topologies on i∈I N ⊗Rb1 R i

III.15. FIBERED CATEGORY OF DOLBEAULT MODULES

305

It follows from the above that for every z ∈ N and every n = (n1 , . . . , nd ) ∈ Nd , Y 1 θni )(z) (III.15.8.10) p−r|n| ( ni ! i 1≤i≤d

b1 -module of finite type of tends to 0 in N when |n| tends to infinity. Let N0 be a sub-R 1 1 1 b N that generates it over R1 [ p ] and ε a rational number such that p−1 < ε < r + p−1 . (r−ε)n Since the sequence p n! tends to 0 in O when n tends to infinity, for every z ∈ N , C Q p−ε|n| ( 1≤i≤d θini )(z) tends to 0 in N when |n| tends to infinity. We can therefore b1 -module consider the sub-R X Y (III.15.8.11) N◦ = p−ε|n| ( θini )(N0 ) n∈Nd

1≤i≤d

b1 and it generates N over R b1 [ 1 ]. Since we have of N . It is of finite type over R p (III.15.8.12)

e 1 (X) ⊗R N ◦ , θ(N ◦ ) ⊂ pε ξ −1 Ω X/S

e1 (N , θ) is a small Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S (III.15.7). Proposition III.15.9. Suppose that X is an object of Q (III.10.5), in other words, that the following conditions are satisfied: (i) X is affine and connected; (ii) f : (X, MX ) → (S, MS ) admits an adequate chart (III.4.4); (iii) there exists a fine and saturated chart M → Γ(X, MX ) for (X, MX ) inducing an isomorphism (III.15.9.1)

∼

× M → Γ(X, MX )/Γ(X, OX ).

e1 Then every small Higgs OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S is solvable. This statement is mentioned here as a reminder (III.13.9). Corollary III.15.10. If the conditions of III.15.5 are satisfied, every locally small Higgs e1 OX [ p1 ]-bundle with coefficients in ξ −1 Ω X/S is solvable.

CHAPTER IV

Cohomology of Higgs isocrystals Takeshi Tsuji IV.1. Introduction In [27], G. Faltings established a p-adic analogue of the Simpson correspondence between small Higgs bundles and small generalized representations or vector bundles on Faltings topos. In his theory and also in another approach by A. Abbes and M. Gros (cf. Chapters II and III), we need to assume the existence of a certain first infinitesimal smooth deformation of the relevant smooth variety and to choose and fix one such deformation to develop the theories. The purpose of this chapter is to remove this restriction by giving an interpretation of small Higgs bundles in a way similar to the interpretation of modules with integrable connections in terms of stratifications and crystals on crystalline sites (cf. [3]). Let us recall the basic settings in the p-adic Simpson correspondence. Let V be a complete discrete valuation ring of mixed characteristic (0, p) whose residue field k is algebraically closed, let K be its field of fractions, and let V be the integral closure of V in a fixed algebraic closure K of K. Following [29] and [31], we define RV to be the perfection limFrob V /pV and consider the ring of Witt vectors A(V ) := W (RV ), which is endowed ←− with a natural surjective homomorphism θ : W (RV ) → Vb (:= limm V /pm V ) characterized ←− m by θ([a]) = limm→∞ e apm . Here a = (an )n∈N ∈ RV , e am denotes a lifting of am ∈ V /pV in V and [a] denotes the Teichmüller representative of a. It is known that the kernel of θ is generated by a nonzero divisor ξ. We define AN (V ) to be W (RV )/ξ N W (RV ). Let X be a smooth scheme separated of finite type over V , and we consider the p-adic formal scheme X1 := X ×Spec(V ) Spf(Vb ). A (rational) Higgs bundle on X1 which we consider is a locally finitely generated projective OX1 ,Qp (= OX1 ⊗Zp Qp )-module M endowed with an OX1 ,Qp -linear homomorphism (called a Higgs field) θ : M → M ⊗OX1 ξ −1 Ω1 b such that θ∧θ = 0, where ξ −1 denotes (ξA2 (V ))−1 ⊗ b . In [27] and in Chapters V

X1 /V

II and III of this volume, a smooth lifting of X1 over A2 (V ) is fixed. In fact, we will consider, more generally, a regular scheme X flat separated of finite type over V and a horizontal divisor D on X such that the union of D and the special fiber Xk of X is a reduced divisor with normal crossings, and always work in the category of log schemes and log formal schemes giving X the log structure defined by D + Xk . A Higgs field will take its values in the tensor product with the differential module with log poles. To simplify the exposition, we assume that X is smooth over V and ignore log structures in the introduction.1 1Note that the definition of the site (X /A(V ))r 1 HIGGS without log structures is different from that with

log structures because we work with big sites. We need to restrict to an object with a strict morphism z : T1 → X1 to have the same one. It causes no problem except that the sites are functorial only for strict morphisms. 307

308

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

We start by explaining the idea of interpreting Higgs fields in terms of stratifications. Let A denote the category of A(V )-algebras A endowed with an ideal I containing ξA, and let A 0 denote the full subcategory of A consisting of (A, I) such that I = ξA and A is ξ-torsion free. Then the inclusion functor A 0 → A has a left adjoint functor D : A → A 0 , which is simply given by associating to (A, I) the A-subalgebra A[ Iξ ] of A[ 1ξ ] generated by xξ −1 (x ∈ I) (a partial blow up along I). This construction, which we call Higgs envelope, plays the role of PD-envelopes for nilpotent Higgs fields as follows. Let B be a smooth ring over V with coordinates t1 , . . . , td ∈ B × , and assume that we are given e × which are liftings of e of B over A(V ) and coordinates e t1 , . . . , e td ∈ B a smooth lifting B e e over A(V ), let t1 , . . . , td . Let B(ν) (ν ∈ N) be the tensor product of ν + 1 copies of B e e e IB(ν) be the kernel of B(ν) → B, let C(ν) be the image of (B(ν), IB(ν) ) under the functor e e e e D above, and put C(ν) = C(ν)/ξ C(ν). Then, for either of the two B-algebra structures, ∼ C(1) defined by xi 7→ ((e we have an isomorphism B[x1 , . . . , xd ] = ti ⊗ 1 − 1 ⊗ e ti )ξ −1 mod ξ). We have a similar description of C(2). This allows us to show that, defining a Higgs stratification on a B-module M similarly as an HPD stratification using / / o / C(2), / C(1) o Bo / the data of a Higgs stratification is equivalent to the data of a nilpotent Higgs field on M , i.e., a B-linear homomorphism θ : M → M ⊗B ξ −1 Ω1B/V such that θ ∧ θ = 0 and P the endomorphisms θi (1 ≤ i ≤ d) of M defined by θ = 1≤i≤d θi ⊗ ξ −1 d log(ti ) are nilpotent. Working with p-adic formal schemes over Spf(AN (V )), we can give a similar interpretation of a Higgs field on a locally finitely generated projective OX1 ,Qp -module M satisfying the following convergence condition: For local coordinates t1 , . . . , td and the endomorphism θi defined in the same way as above, we have Y 1 θni (x) → 0 ni ! i 1≤i≤d

P

as 1≤i≤d ni → ∞ for a local section x of M (cf. the proof of Theorem IV.3.4.16). This condition is weaker than the condition “small” by Faltings, which will be necessary when we consider the cohomology later. Having the interpretation above, we are naturally lead to introducing an analogue (X1 /A(V ))∞ HIGGS of (big) crystalline sites, whose object is simply a pair (T• , z) of a direct system consisting of p-adic formal schemes TN over Spf(AN (V )) and closed immersions TN → TN +1 over Spf(AN +1 (V )), and a morphism z : T → X over Spf(Vb ) such that T are flat over Spf(Z ), the kernel of O → O 1

1

N

p

TN

Tn

(N > n) is generated by ξ n and the multiplication by ξ on OTN +1 induces an injective morphism OTN → OTN +1 (cf. Definition IV.3.1.1). A Higgs bundle satisfying the convergence condition above is interpreted as an “isocrystal” on the site (X1 /A(V ))∞ HIGGS , which we call a Higgs isocrystal (cf. Definition IV.3.3.1), when we are given a compatible system of smooth liftings XN → Spf(AN (V )) of X1 , which exists if X is affine (cf. Theorem IV.3.4.16). Next we explain the idea behind the interpretation of the p-adic Simpson correspondence by Faltings in terms of Higgs isocrystals. Suppose first that X is connected and affine, has nonempty special fiber, and admits coordinates consisting of invertible functions. Let A = Γ(X, OX ), let K be an algebraic closure containing K of the field of fractions K := Frac(A) of A, let Kur denote the union of finite extensions L of K contained in K such that the integral closure of A ⊗V K in L is étale, and let A denote the integral closure of A in Kur . Put s = Frac(K). Then A and its p-adic completion

IV.1. INTRODUCTION

309

b are naturally endowed with continuous actions of ∆ := π (X , s) ∼ Gal(Kur /KK), A = X 1 K b -representation of ∆ , we mean a finitely where X = X × Spec(K). By an A K

Spec(V )

X

Qp

b -module endowed with a continuous semi-linear action of ∆ . generated projective A Qp X It is known, by the theory of almost étale extensions, that the absolute Frobenius of A/pA is surjective (cf. Theorem IV.5.3.6 (2)), and one can apply the same construction as AN (V ) to A, obtaining p-adically complete AN (V )-algebras AN (A) such that b (cf. IV.5.1). The direct system Spf(A (A)) with the natural morphism ∼ A A1 (A) = N b → X becomes an object of the site (X /A(V ))∞ Spf(A (A)) = Spf(A) endowed 1

1

1

HIGGS

with an action of ∆X . Now, simply by evaluating Higgs isocrystals on this object, we obtain a functor from the category of Higgs isocrystals on (X1 /A(V ))∞ HIGGS (finite on b X ) to that of A -representations of ∆ . We prove that this functor is fully faithful 1

X

Qp

(cf. Theorem IV.5.3.3). The continuity of the action of ∆X on the above evaluation and the fully faithfulness of the functor follow from the description of the functor in terms of a “period ring” explained below (cf. Corollary IV.5.2.13). Choose a compatible system of smooth liftings XN of X1 over AN (V ), and let D = (DN ) be the Higgs envelope of X1 in (XN ×Spf(AN (V )) Spf(AN (A)))N ∈N>0 , which is an object of (X1 /A(V ))∞ HIGGS . Then the ring A1 (A) = Γ(D1 , OD1 ) has both a continuous action of ∆X (Proposition IV.5.2.6) and a Higgs field, and plays the role of “period ring” for the local p-adic Higgs correspondence as follows. For a Higgs isocrystal F (finite on X1 ) and the Higgs bundle (M, θ) corresponding to b -representation of ∆ associated to F is given by F via the lifting (X ) , the A N N

X

Qp

(A1 (A) ⊗Γ(X1 ,OX1 ) Γ(X1 , M))θ=0 (cf. Proposition IV.5.2.12). We also see that a Higgs bundle (M, θ) is “admissible” with respect to the period ring A1 (A) if and only if θ satisfies the convergence condition above (cf. Proposition IV.5.3.10). If we start with a general X and a Higgs isocrystal on (X1 /A(V ))∞ HIGGS , we obtain a compatible system of representations on étale affine connected X-schemes U satisfying the conditions on X in the previous paragraph. One can interpret the compatible system, as Faltings did in [27], in terms of “vector bundles” on the Faltings site of X when the Higgs isocrystal admits a suitable lattice globally (cf. IV.6.4). We prove that the functor thus obtained is fully faithful (Theorem IV.6.4.9). For a small Higgs vector bundle (M, θ) on X1 , Faltings also proved, in [27], that the hypercohomology of the complex (M ⊗OX1 ξ −• Ω• b , θ• ) is canonically isomorphic to X1 / V

the Galois cohomology of the corresponding representation on X in the local case, and to the cohomology of the corresponding sheaf on the Faltings site of X in the global case. It is natural to ask whether one can also interpret the cohomology of the above complex in terms of the site (X1 /A(V ))∞ HIGGS and prove a comparison theorem of cohomologies for our Higgs isocrystals. We give positive answers to them as follows. Following an analogy with the comparison between crystalline cohomology and de Rham cohomology, one can construct a “linearization” of the complex above (when a compatible system of smooth liftings XN of X1 over AN (V ) is given). However the complex does not give a resolution of the corresponding Higgs isocrystal, i.e., an analogue of Poincaré lemma does not hold; a complex of the form

−• • • Ω ,θ ) (B1 [x1 , . . . , xd ]∧ Qp ⊗ b ξ V

Ω = ⊕1≤i≤d Vb d log(ti ),

q

q

Ω = ∧ Ω,

θ(xi ) = ξ −1 d log(ti )

310

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

appears for an affine object (Spf(BN ))N of (X1 /A(V ))∞ HIGGS and it does not give a resolution of B1 unless d = 0. Here ∧ denotes the p-adic completion. The B1 -algebra B1 [x1 , . . . , xd ]∧ in the above complex is obtained as the ring of the reduction mod ξ of the Higgs envelope of Spf(B1 ) in (Spf(BN ) ×Spf(AN (V )) XN )N . As it is well-known (cf. [55]), we obtain a resolution of B1 if we replace B1 [x1 , . . . , xd ]∧ by the weak completion B[x1 , . . . , xd ]† . With this fact in mind, we introduce the notion of Higgs envelope of level r (cf. Definition IV.2.2.10) for r ∈ N>0 , which gives a subring ( ) P Y X n P ni −[ r i ] an xi an ∈ B1,Qp , p an → 0 ( ni → ∞) d n∈N

1≤i≤d

of B1 [x1 , . . . , xd ]∧ Qp in the above setting (cf. Proposition IV.2.3.17 and Lemma IV.2.3.14), whose union over r is the weak completion tensored with Qp . Based on this notion, we further introduce an inverse system of sites (X1 /A(V ))rHIGGS (r ∈ N) (cf. Definition IV.3.1.1) and the notion of a Higgs isocrystal on (X1 /A(V ))rHIGGS (cf. Definition IV.3.3.1), and give an interpretation of the above cohomology in terms of the cohomology of the inverse limit (X1 /A(V ))HIGGS of the sites (X1 /A(V ))rHIGGS (cf. IV.4.5). If we are given a compatible system of smooth liftings XN of X1 over AN (V ), then the category of Higgs isocrystals on (X1 /A(V ))rHIGGS is equivalent to the category of Higgs bundles on X1 satisfying the convergence condition P Y 1 i ni θni (x) → 0 p−[ r ] ni ! i 1≤i≤d

(cf. Theorem IV.3.4.16). A Higgs bundle (M, θ) on X1 is small in the sense of Faltings (cf. Definition IV.3.6.5) if and only if M admits a “lattice” globally (as an OX1 ,Qp -module) and satisfies the convergence condition above for some r ∈ N>0 ; the existence of a “lattice” always holds if X is affine and M is finitely generated and projective (cf. Proposition IV.3.6.6 and Corollary IV.3.6.4). For a Higgs isocrystal F on (X1 /A(V ))rHIGGS , the corresponding Higgs bundle (M, θ) on X1 and the inverse image F † of F on (X1 /A(V ))HIGGS , we prove that the analogue of linearization gives a resolution of F † and that the derived direct image of F † under the canonical morphism of topos to (X1 )´e∼t is canonically isomorphic to the complex M ⊗OX1 ξ −• Ω• b (cf. Corollary IV.4.5.8). X1 / V

We also prove that, for a Higgs isocrystal F on (X1 /A(V ))rHIGGS admitting a global lattice of its pull-back F ∞ on (X1 /A(V ))∞ HIGGS (cf. Definition IV.3.5.4), the cohomology of the pull-back F † of F on (X1 /A(V ))HIGGS is canonically isomorphic to the cohomology of the sheaf on the Faltings site of X corresponding to F. The key local ingredients in the proof are as follows. One can define a period ring A1s (A) of level s similarly as b -representation V of ∆ A (A) using the Higgs envelope of level s. Then, for the A 1

Qp

X

corresponding to F, M = Γ(X1 , M) and A1 = Γ(X1 , OX1 ), we obtain a direct system of complexes of ∆X -modules (V → ξ −• M ⊗A1 A1s (A) ⊗A Ω•A )s≥r such that the transition morphisms are homotopic to zero as morphism of complexes of topological modules (cf. the proof of Proposition IV.5.2.15). On the other hand, one can show, by using the theory of almost étale extensions by Faltings (cf. Theorem IV.5.3.4), that ( (A1 )Qp (if q = 0), q lim Hcont (∆X , A1s (A)Qp ) = −→ 0 (if q > 0), s

IV.1. INTRODUCTION

311

which is an analogue of the computation of Galois cohomology of the period ring for Hodge-Tate representations by O. Hyodo in [43] (1.2).

In IV.2, we develop a general theory of Higgs envelopes. We define Higgs envelopes of level r (r ∈ N>0 ∪ {∞}) and prove their existence for p-adically complete algebras in IV.2.1 and then for p-adic fine log formal schemes in IV.2.2. In IV.2.3, we give an explicit description of Higgs envelopes for power series rings and sections of smooth morphisms. In IV.2.4, we study the relation between the differential module and the Higgs envelope of the diagonal immersion for a smooth morphism of p-adic fine log formal schemes. In IV.2.5, we study the relation between Higgs envelopes and torsors of deformations (which play a key role in the approach by Abbes and Gros in Chapters II and III) under a certain general setting. In IV.3, we introduce the notion of Higgs isocrystals of level r (r ∈ N>0 ∪ {∞}) and the sites on which they live. We define the site (X/B)rHIGGS in IV.3.1 for r ∈ N>0 ∪ {∞}. After preliminaries on finitely generated and projective OX,Qp -modules and their lattices on a p-adic formal scheme X in IV.3.2, we define in IV.3.3 Higgs isocrystals of level r and also Higgs crystals, which provide us the notion of lattices on Higgs isocrystals in IV.3.5. In IV.3.4, we give a description of a Higgs isocrystal and also of a Higgs crystal in terms of a (generalized) Higgs vector bundle on the Higgs envelope of an embedding of X into a smooth p-adic fine log formal scheme (similarly as the case of crystalline sites). In IV.3.6, we compare our convergence conditions on Higgs fields with the condition “small” (divisibility by a certain rational power of p) by Faltings. In IV.4, we study the relation between the cohomology of Higgs isocrystals of finite level and that of the corresponding (generalized) Higgs vector bundles. We first define the projection morphism of topos to the étale topos in IV.4.1 in the same way as crystalline sites. In IV.4.2, we introduce the notion of linearizations for our site, and prove that the linearization of a finitely generated projective module is acyclic under the projection to the étale site. We prove a Poincaré lemma for Higgs isocrystals of finite level in IV.4.3, and after preliminaries on the coherence of relevant topos and morphisms of topos in IV.4.4, which are necessary for the computation of the cohomology of the inverse limit of the sites (X/B)rHIGGS (r ∈ N), we prove a comparison theorem of cohomologies in IV.4.5. We discuss the local p-adic Simpson correspondence in IV.5. After preliminaries on the log structure and the continuity of the action of Galois group for the ring AN (A) = W (RA )/ξ N W (RA ) associated to a certain kind of affine fs (=fine and saturated) log scheme in IV.5.1, we construct a functor from the category of Higgs isocrystals to that b -representations for a sufficiently small affine fs log scheme U whose underlying of A Qp scheme is normal and of finite type over V and has reduced special fiber. We also construct a canonical homomorphism from the cohomology of a Higgs isocrystal of finite b -representation. In level to the continuous Galois cohomology of the corresponding A Qp IV.5.3, we prove that the functor is fully faithful and the comparison homomorphism of cohomologies is an isomorphism in the case where U has semi-stable reduction. We also study the admissibility of Higgs bundles with respect to the period ring associated to a compatible system of smooth liftings (UN ) of U1 . In IV.5.4, we compare our “period rings” with the Higgs-Tate algebras introduced in II.10.3 and II.12.1. In the final section IV.6, we study the global p-adic Simpson correspondence. A sheaf on our Faltings site of an fs log scheme X of finite type over V is equivalent to the data assigning to each strict étale affine log scheme U over X a sheaf on the finite étale site of the log trivial locus of the geometric generic fiber UK = U ×Spec(K) Spec(K) of U satisfying a certain gluing condition for strict étale coverings (Uα → U )α . We restrict

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ourselves to affine U because we also work with two variants simultaneously where UK is replaced by the geometric generic fiber of the Spec of the p-adic henselization or p-adic completion of Γ(U, OU ). We then assume that the underlying scheme of X is normal, separated of finite type over V and has reduced special fiber. After preliminaries on Faltings sites in IV.6.1, IV.6.2, and IV.6.3, we construct a functor from the category of Higgs crystals to that of sheaves on the Faltings site of X and prove its fully faithfulness (up to isogeny) in the semi-stable reduction case in IV.6.4. After some preliminaries on the cohomology of the projection to the étale site in IV.6.5 and inverse systems and direct systems of sheaves in IV.6.6, we construct a comparison homomorphism from the cohomology of a Higgs isocrystal of finite level (admitting a lattice) to that of the corresponding sheaf on the Faltings site in IV.6.7 and prove that it is an isomorphism in the semi-stable reduction case in IV.6.8. Acknowledgments. The author learned the work on a p-adic Simpson correspondence by G. Faltings [27] and obtained the idea of Higgs (iso)crystals and the proof of the Galois acyclicity of the “overconvergent period ring” [75] through his partial participation (in January 2009) in the workshop on the paper [27] held at the University of Rennes I in 2008-2009. He was away from the topic for a few years after the workshop, and then started to work on it again in the occasion of the preparation of his talk at the summer school: Higgs bundles on p-adic curves and representation theory, held at Mainz University in 2012; he introduced the notion of Higgs envelopes and Higgs isocrystals of finite level and established a theory interpreting the cohomology of Higgs bundles in terms of a certain site, which is the main theme of this chapter. The author would like to thank the organizers of the workshop and the summer school and, in particular, Ahmed Abbes, who kindly invited me to both of them. Without these two occasions, the works in this chapter would never have been accomplished. The project of this book was initiated by Ahmed Abbes and Michel Gros, and later they kindly invited the author to join them by writing the aforementioned results. Some of the results in this chapter were obtained through his discussions with them during his stay at IHES (Institut Hautes Études Scientifiques) in September 2013 and March 2014. The author would like to express his sincere gratitude to Ahmed Abbes and Michel Gros for their kindness and constant support for writing this article, and to IHES for its hospitality. The work of this chapter was financially supported by JSPS Grants-in-Aid for Scientific Research, Grant Number 24540009. Notation. Throughout this chapter, we fix a universe U and a universe V such that U ∈ V , and we only consider groups, rings, schemes, formal schemes, and log schemes which belong to U . By a topos (resp. a fibered topos), we mean a V -topos (resp. a fibered V -topos). For a sheaf of rings O (resp. an inverse system of sheaves of rings (Om )m∈N on a site C), we write Mod(C, O) (resp. Mod(C, O• )) for the category of sheaves of O-modules (resp. inverse systems of sheaves of Om -modules). For abelian groups, rings, sheaves of abelian groups, and sheaves of rings, the subscript Q means ⊗Z Q. Similarly the subscript Qp means ⊗Zp Qp for Zp -modules, Zp algebras, etc. We say that a log scheme is affine if its underlying scheme is affine. We say that a morphism of log schemes is affine (resp. of finite type) if its underlying morphism of schemes is affine (resp. of finite type). In IV.2–IV.4, we fix a Zp -algebra R with a principal ideal I generated by a regular element such that R/I is p-torsion free and p-adically complete and separated, which implies that, for every N ∈ N>0 , R/I N is p-torsion free and p-adically complete and separated. For N ∈ N>0 , let RN denote R/I N and let SN denote the affine formal scheme associated to RN with the p-adic topology. Let ξ be a generator of I.

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313

IV.2. Higgs envelopes IV.2.1. p-adically complete rings. In this subsection, we prove the existence of the Higgs envelope of level r (r ∈ N>0 ∪ {∞}) for an inverse system of p-adically complete and separated RN -algebras with surjective transition maps (Proposition IV.2.1.8, Definition IV.2.1.9). Definition IV.2.1.1. (1) We define A to be the category of R-algebras with a decreasing filtration F n A ⊂ A (n ∈ N) by ideals such that F 0 A = A and ξ n A ⊂ F n A for every n ∈ N. A morphism in A is a homomorphism of R-algebras compatible with the filtrations. r (2) For r ∈ N>0 (resp. r = ∞), we define Aalg to be the full subcategory of A n consisting of (A, F A) ∈ Ob A satisfying the following conditions: (i) The rings A and A/F n A (n ∈ N>0 ) are p-torsion free. (ii) The ring A is ξ-torsion free. (iii) For every s ∈ N ∩ [0, r] (resp. s ∈ N), we have pF s A ⊂ ξ s A (resp. F s A = ξ s A). (iv) The inverse image of F n A under ξ : A → A is F n−1 A for every n ∈ N>0 . r (3) For r ∈ N>0 ∪ {∞}, we define Apr to be the full subcategory of Aalg consisting n r of (A, F A) ∈ Ob Aalg satisfying the following condition: (i) The rings A and A/F n A (n ∈ N>0 ) are p-adically complete and separated. (4) For r ∈ N>0 ∪ {∞}, we define A r to be the full subcategory of Apr consisting of (A, F n A) ∈ Ob Apr satisfying the following condition: (i) The natural homomorphism A → limn A/F n A is an isomorphism. ←− r For an object (A, F n A) of Aalg and n ∈ N>0 , the multiplication by ξ induces an injective homomorphism A/F n−1 A → A/F n A, which is denoted by [ξ] in the following. Proposition IV.2.1.2. For every r ∈ N>0 ∪ {∞}, the inclusion functors A r → Apr → r Aalg → A have left adjoint functors.

c be lim M/pm M . Lemma IV.2.1.3. Let M be a module and let M ←−m c is p-torsion free, the projection M c→ (1) Assume that M is p-torsion free. Then M ∼ m m mc = c → M/p M for every m ∈ N, and the M/p M induces an isomorphism M /p M − c is an isomorphism. c/pm M c → lim M natural homomorphism M ←−m (2) Let N ⊂ M be a submodule such that the quotient L := M/N is p-torsion free, b (resp. L) b be lim N/pm N (resp. lim L/pm L). Then the sequence 0 → N b→ and let N ←−m ←−m c→L b → 0 is exact. M

Proof. Since M is p-torsion free, the homomorphism pm : M → M induces injective homomorphisms M/pN M → M/pm+N M whose cokernel is isomorphic to M/pm M . By pm c− c → M/pm M → 0, which implies taking limN , we obtain an exact sequence 0 → M −→ M ←− (1). The claim (2) is obtained by taking limm of the exact sequences 0 → N/pm N → ←− M/pm M → L/pm L → 0. 

r Lemma IV.2.1.4. Let r ∈ N>0 . Any object (A, F n A) of Aalg has the following properties: (1) Let n ∈ N and let m be the smallest integer satisfying nr−1 ≤ m. Then we have m n p F A ⊂ ξ n A ⊂ F n A. In particular, we have F n AQ = ξ n AQ . 0 0 (2) For any n, n0 ∈ N, we have F n A · F n A ⊂ F n+n A.

Proof. (1) It is enough to prove pF n+s A ⊂ ξ s F n A for n ∈ N and s ∈ N ∩ [0, r]. Let a ∈ pF n+s A. Then a ∈ pF s A ⊂ ξ s A. Since the inverse image of F n+s A by ξ s : A → A is F n A, we obtain a ∈ ξ s F n A.

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(2) Since A and A/F n A are p-torsion free, the last equality of (1) implies A∩ξ n AQ = F A. This immediately implies the claim.  n

r Proof of Proposition IV.2.1.2. Left adjoint of Aalg → A : Let (A, F n A) be an

1 object of A . Let B be the A-subalgebra of A[ pξ ] generated by p−1 and aξ −n (n ∈ n N>0 , a ∈ F A), and define the decreasing filtration F n B (n ∈ N) to be ξ n B. The natural homomorphism A → B is compatible with the filtrations. For an A-subalgebra A0 of B, we define the decreasing filtration F n A0 by ideals to be A0 ∩ F n B. Then (A0 , F n A0 ) is r an object of A and satisfies the conditions (i), (ii), (iv) in the definition of Aalg because × 0 n 0 n p ∈ B , ξ is regular in B, and the homomorphism A /F A → B/F B is injective. Let S r be the set of all A-subalgebras such that pF s A0 ⊂ ξ s A0 for every s ∈ N ∩ [0, r] if r ∈ N>0 , and F s A0 = ξ s A0 for every s ∈ N if r = ∞. The set S r contains B, and r any A0 ∈ S r with the filtration F n A0 is an object of Aalg . Let A0 be the intersection 0 s ∩A0 ∈S r A . Since p and ξ are regular in B, we have ξ A0 = ∩ξ s A0 , F s A0 = ∩F s A0 , r and pF s A0 = ∩pF s A0 . Hence A0 ∈ S r . We assert that (A0 , F n A0 ) ∈ Ob Aalg with the n n morphism (A, F A) → (A0 , F A0 ) in A satisfies the desired universal property, i.e., any r morphism f : (A, F n A) → (C, F n C) in A with (C, F n C) ∈ Ob Aalg uniquely factors as g

(A, F n A) → (A0 , F n A0 ) − → (C, F n C). Let us prove the existence of g. We note that 1 1 1 ] are injective. Let fe: A[ pξ ] → C[ pξ ] the natural homomorphisms C → C[ p1 ] → C[ pξ 1 1 n n be the homomorphism induced by f . Then F C[ p ] = ξ C[ p ] (cf. Lemma IV.2.1.4 (1)) implies fe(B) ⊂ C[ 1 ]. Let fp be the homomorphism B → C[ 1 ] induced by fe. Let A0 p

p

be the A-subalgebra fp−1 (C) of B. For n ∈ N, we have fp (F n A0 ) ⊂ fp (A0 ) ∩ fp (ξ n B) ⊂ C ∩ ξ n C[ p1 ] = C ∩ F n C[ p1 ] = F n C. We claim that A0 is contained in S r , which implies that fp (A0 ) ⊂ C and fp induces the desired morphism g. If r ∈ N>0 (resp. r = ∞), we have fp (pF s A0 ) ⊂ pF s C ⊂ ξ s C for all s ∈ N ∩ [0, r] (resp. fp (F s A0 ) ⊂ F s C = ξ s C for all s ∈ N). For any x ∈ F s B = ξ s B, fp (x) ∈ ξ s C implies x ∈ ξ s A0 because ξ is regular ∼ = 1 1 in C[ p1 ]. Hence (A0 , F n A0 ) ∈ S r . The uniqueness of g follows from A[ pξ ] → A0 [ pξ ] and 1 the injectivity of C → C[ pξ ]. c) denote the kernel Left adjoint of Apr → A r : For a module M , let M [pm ] (resp. M alg

r of pm : M → M (resp. limm M/pm M ). Let (A, F n A) be an object of Aalg . We give ←− n b b the R-algebra A the decreasing filtration F A (n ∈ N) by ideals defined by the image n A → A. b We assert that (A, b F n A) b is an object of A r [ of the natural homomorphism F p and has the desired universal property. By Lemma IV.2.1.3 (2), the homomorphism nA → F nA n A. \ b is an isomorphism and the quotient A/F b nA b is isomorphic to A/F [ F n b n b n The latter implies ξ A ⊂ F A. Applying Lemma IV.2.1.3 (1) to A and A/F A, we b and A/F b nA b are p-torsion free and p-adically complete and separated. Since see that A A/F 1 A is p-torsion free and p(F 1 A/ξA) = 0, we have (A/ξA)[pm ] = (A/ξA)[p] and (A/(F n A + ξA))[pm ] = (A/(F n A + ξA))[p] for m ∈ N>0 . This implies that the image of the kernel of ξ : A/pm+1 A → A/pm+1 A (resp. [ξ] mod pm+1 : (A/F n−1 A)/pm+1 → (A/F n A)/pm+1 ) in A/pm A (resp. (A/F n−1 A)/pm ) is 0. By taking the inverse limit b→A b is injective and induces an injective over m, we see that the homomorphism ξ : A n−1 b n b b b homomorphism A/F A → A/F A. If r ∈ N>0 (resp. r = ∞), taking the p-adic p

ξs

ξs

=

=

=

completion of F s A − → pF s A ⊂ ξ s A ← − A (s ∈ N ∩ [0, r]) (resp. A −∼ → F s A (s ∈ N)), we ∼ ∼

b ⊂ ξsA b (resp. F s A b = ξ s A). b For any morphism f : (A, F n A) → (B, F n B) in obtain pF s A r n r b F n A) b is obtained by taking the Aalg with (B, F B) ∈ Ob Ap , a factorization through (A,

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315

p-adic completion of f and using B ∼ B/pm B and F n B ∼ F n B/pm F n B (n ∈ = lim = lim ←−m ←−m N) (cf. Lemma IV.2.1.3 (2) for the last isomorphism). The uniqueness of the factorization ∼ ∼ = = b m b A (cf. Lemma IV.2.1.3 (1)) and B → limm B/pm B. follows from A/pm A → A/p ←− r r n r Left adjoint of A → Ap . Let (A, F A) be an object of Ap . Let B be the A-algebra limN A/F N A endowed with the decreasing filtration by ideals F n B = limN F n A/F N A. ←− ←− By taking the inverse limit of the exact sequence 0 → F n A/F N A → A/F N A → ∼ = A/F n A → 0, we obtain an isomorphism B/F n B → A/F n A. This implies (B, F n B) ∈ n r Ob A . We assert that (B, F B) ∈ Ob A and the morphism (A, F n A) → (B, F n B) in Apr satisfies the desired universal property. Since A/F N A (N ∈ N) are p-torsion free, B and B/F n B are p-torsion free. By taking the inverse limit of the injective homomorphisms [ξ] : A/F N A → A/F N +1 A (N ∈ N>0 ), we see that ξ is regular in B. p If r ∈ N>0 (resp. r = ∞), taking the inverse limit of F s A/F N A − → p(F s A/F N A) ⊂ ∼ =

s

N

[ξ]s

ξ (A/F A) ←− − A/F ∼ =

N −s

s

N

[ξ]s

A for s ∈ N ∩ [0, r] (resp. F A/F A = ξ s (A/F N A) ←− − ∼ =

A/F N −s A for s ∈ N), we obtain pF s B ⊂ ξ s B (resp. F s B = ξ s B). The exact sequences pm

∼ =

0 → A/F N A → A/F N A → (A/F N A)/pm → 0 induce an isomorphism B/pm B → limN (A/F N A)/pm . Taking limm , we obtain limm B/pm B ∼ lim (A/F N A)/pm ∼ = lim = ←− ←− ←− ←−m ←−N N m ∼ N limN limm (A/F A)/p = limN A/F A = B. The remaining conditions are verified ←− ←− n ←− using B/F B ∼ = A/F n A. For any morphism f : (A, F n A) → (C, F n C) in Apr with (C, F n C) ∈ Ob A r , a factorization through (B, F n B) is obtained by taking the inverse limit of the homomorphisms A/F N A → C/F N C induced by f and using C ∼ = ∼ = limN C/F N C. The uniqueness of the factorization follows from C → limN C/F N C and ←− ←− A/F N A ∼  = B/F N B. Definition IV.2.1.5. (1) We define Ap,• to be the category of inverse systems of Ralgebras (AN )N ∈N>0 such that AN (N ∈ N>0 ) are p-adically complete and separated, ξ N AN = 0 for N ∈ N>0 , and the transition maps π : AN +1 → AN (N ∈ N>0 ) are surjective. (2) For r ∈ N>0 (resp. r = ∞), we define A•r to be the full subcategory of Ap,• consisting of (AN ) ∈ Ob Ap,• satisfying the following conditions. (i) The rings AN (N ∈ N>0 ) are p-torsion free. (ii) For every s ∈ N ∩ [0, r] (resp. s ∈ N) and N ∈ N>0 , we have pF s AN ⊂ ξ s AN (resp. F s AN = ξ s AN ), where F n AN (n ∈ N) is the decreasing filtration of AN by ideals defined by F 0 AN = AN , F n AN = Ker(π : AN → An ) (1 ≤ n ≤ N ) and F n AN = 0 (n > N ). (iii) The kernel of the homomorphism AN → AN ; x 7→ ξx is F N −1 AN for every N ∈ N>0 .

The inverse system R1 ← R2 ← R3 ← · · · is an object of A•r for every r ∈ N>0 ∪{∞}. For an object (AN ) of A•r , the multiplication by ξ on AN induces an injective homomorphism AN −1 → AN , which is denoted by [ξ] in the following.

Lemma IV.2.1.6. Let r ∈ N>0 ∪ {∞}. Any object (AN ) of A•r have the following properties. (1) The kernel of AN +M → AN +M ; x 7→ ξ N x is F M AN +M for N, M ∈ N>0 . (2) Assume r ∈ N>0 , let n ∈ N and let m be the smallest integer such that nr−1 ≤ m. Then, for N ∈ N>0 , we have pm F n AN ⊂ ξ n AN ⊂ F n AN . In particular, (F n AN )Q = ξ n AN,Q . 0 0 (3) For n, n0 ∈ N and N ∈ N>0 , we have F n AN · F n AN ⊂ F n+n AN .

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Proof. (1) For positive integers n ≤ N , the inverse image of F n−1 An by AN → An is F n−1 AN . Hence, by the condition (iii) in the definition of A•r , the inverse image of F n AN under ξ : AN → AN is F n−1 AN . This implies the claim. (2) It suffices to prove pF n+s AN ⊂ ξ s F n AN for s ∈ N ∩ [0, r] and n, N ∈ N such that n + s < N . For a ∈ pF n+s AN , we have a ∈ pF s AN ⊂ ξ s AN . Since the inverse image of F n+s AN under ξ s : AN → AN is F n AN by the proof of (1), we have a ∈ ξ s F n AN . (3) The claim is obvious if r = ∞. Assume r ∈ N>0 . For n, N ∈ N>0 , AN and AN /F n AN ∼ = An are p-torsion free. Hence F n AN = F n AN,Q ∩ AN , and the claim follows from the last equality of (2).  Lemma IV.2.1.7. (1) For (AN ) ∈ Ob A•r , the R-algebra A = limN AN with the de←− creasing filtration F n A = limN F n AN by ideals is an object of A r . This construc←− tion gives an equivalence of categories A•r → A r , whose quasi-inverse is given by (A, F n A) 7→ (A/F N A). (2) The functor Ap,• → A defined by (AN ) 7→ (limN AN , limN F n AN ) is fully faith←− ←− ful. Proof. (1) Let (AN ) ∈ Ob A•r and let (A, F n A) be as in the claim. We prove (A, F n A) ∈ Ob A r . By taking limN of the exact sequences 0 → F n AN → AN → An → ←− 0, we obtain an isomorphism A/F n A ∼ = An . This implies (A, F n A) ∈ Ob A . Since AN (N ∈ N>0 ) are p-torsion free, A and A/F n A are p-torsion free. By taking limN of the ←− injective homomorphisms [ξ] : AN → AN +1 , we see that ξ is regular in A. If r ∈ N>0 ∼ ∼ = = (resp. r = ∞), by taking limN of F s AN − → pF s AN ⊂ ξ s AN ←−− AN −s for s ∈ N ∩ [0, r] ←− p [ξ]s ∼ =

(resp. F s AN = ξ s AN ←−− AN −s for s ∈ N), we obtain pF s A ⊂ ξ s A (resp. F s A = s [ξ]

pm

s

ξ A). By taking limN of the exact sequences 0 → AN −−→ AN → AN /pm AN → 0, ←∼ − = A /pm AN , we obtain A/pm A → limN AN /pm AN . Taking limm and using AN ∼ = lim ←−m N ←− ←− m ∼ we obtain limm A/p A = limN AN = A. The remaining conditions immediately follow ←− ←− from A/F n A ∼ = An . For an object (A, F n A) ∈ Ob A r , it is immediate to see that (A/F N A)N ∈N>0 ∈ Ob A•r just noting that F n (A/F N A) is the image of F n A in A/F N A. It is straightforward to verify that the two functors are quasi-inverses of each other. (2) Let (AN ) be an object of Ap,• , and put (A, F n A) = (limN AN , limN F n AN ). ←− ←− Then, by the same argument as in the beginning of the proof of (1), we have An ∼ = A/F n A n compatible with n, and (A, F A) ∈ Ob A . For another object (BN ) of Ap,• and the object (B, F n B) = (limN BN , limN F n BN ) of A , we see that the map ←− ←− HomA ((A, F n A), (B, F n B)) −→ HomAp,• ((AN ), (BN )) N

f mod F sending f to (AN ∼ = A/F N A −−−−−−−→ B/F N B ∼ = BN )N is the inverse of the map HomAp,• ((AN ), (BN )) → HomA ((A, F n A), (B, F n B)) defined by the functor in question. 

Combining Lemma IV.2.1.7 and Proposition IV.2.1.2, we obtain the following. Proposition IV.2.1.8. For every r ∈ N>0 ∪ {∞}, the inclusion functor A•r → Ap,• has r a left adjoint functor DHiggs : Ap,• → A•r . r Definition IV.2.1.9. For r ∈ N>0 ∪ {∞} and (AN ) ∈ Ob Ap,• , we call DHiggs ((AN )) the Higgs envelope of (AN ) of level r.

Lemma IV.2.1.10. Let (fN ) : (AN ) → (A0N ) be a morphism in Ap,• . Let AN,m , A0N,m and fN,m be the reduction mod pm of AN , A0N and fN , respectively, and assume that

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317

the natural homomorphism A0N +1,m ⊗AN +1,m AN,m → A0N,m is an isomorphism for every m, N ∈ N>0 . Let r ∈ N>0 ∪ {∞}. (1) If fN,m is flat for every m, N ∈ N>0 and (AN ) ∈ Ob A•r , then (A0N ) ∈ Ob A•r . (2) If fN,m is faithfully flat for every m, N ∈ N>0 and (A0N ) ∈ Ob A•r , then (AN ) ∈ Ob A•r . Proof. Since AN (resp. A0N ) is p-adically complete and separated, AN (resp. A0N ) is flat over Zp if and only if AN,m (resp. A0N,m ) is flat over Z/pm Z for every m ∈ N>0 . If fN,m is flat (resp. faithfully flat) and AN,m (resp. A0N,m ) is flat over Z/pm Z, then A0N,m (resp. AN,m ) is flat over Z/pm Z. Hence we may assume that AN and A0N are flat over Zp . Assume that fN,m is flat for every m, N ∈ N>0 . Let n, N, m be positive integers ∼ = such that n < N . Then since A0N,m ⊗AN,m An,m − → A0n,m , we have exact sequences 0 → F n AN /pm F n AN ⊗AN,m A0N,m → A0N,m → A0n,m → 0, 0 → F n A0N /pm F n A0N → A0N,m → A0n,m → 0. ∼ =

Hence we have an isomorphism F n AN /pm F n AN ⊗AN,m A0N,m → F n A0N /pm F n A0N . Now the claim follows from Sublemma IV.2.1.11 below.  Sublemma IV.2.1.11. Let (AN ) be an object of Ap,• and assume that AN is p-torsion free for every N ∈ N>0 . Let r ∈ N>0 (resp. r = ∞). Then (AN ) is an object of A•r if and only if the following two conditions hold. (i) For N, m, s ∈ N>0 such that s < N , the composition of F N −s AN /pm F N −s AN → ξs

AN /pm AN → AN /pm AN is 0, and hence ξ s : AN /pm AN → AN /pm AN induces a homomorphism gN,m,s : AN −s /pm AN −s → F s AN /pm F s AN . (ii) For N, m ∈ N>0 and s ∈ N ∩ [0, r] (resp. s ∈ N) such that s < N , p · Cok(gN,m,s ) = 0 and the homomorphism Ker(gN,m+1,s ) → Ker(gN,m,s ) induced by the projection AN −s /pm+1 AN −s → AN −s /pm AN −s is 0 (resp. the homomorphism gN,m,s is an isomorphism).

Proof. The necessity follows from Lemma IV.2.1.6 (1) and p · (F s AN /ξ s AN ) = 0 (resp. F s AN = ξ s AN ) for s ∈ N∩[0, r] (resp. s ∈ N). Let us prove the sufficiency. By the ξs

condition (i), the composition of F N −s AN → AN → F s AN is 0. Hence ξ s : AN → F s AN induces a homomorphism gN,s : AN −s → F s AN . Since gN,s = limm gN,m,s , the condition ←− (ii) implies that Cok(gN,s ) = F s AN /ξ s AN is annihilated by p and gN,s is injective (resp. gN,s is an isomorphism) for N ∈ N>0 and s ∈ N ∩ [0, r] (resp. s ∈ N) such that s < N. 

IV.2.2. p-adic fine log formal schemes. In this subsection, we prove the existence of the Higgs envelope of level r (r ∈ N>0 ∪ {∞}) and its compatibility with flat morphisms for a direct system of p-adic fine log formal schemes over SN satisfying certain conditions (Proposition IV.2.2.9, Definition IV.2.2.10). By a p-adic formal scheme, we mean an adic formal scheme X such that pOX is an ideal of definition. Let X be a p-adic formal scheme. We have a canonical morphism X → Spf(Zp ). Let Xm denote X ×Spf(Zp ) Spec(Z/pm Z). We define a fine log structure M on X to be a family of fine log structures Mm on Xm (m ∈ N>0 ) and exact closed immersions (Xm , Mm ) → (Xm+1 , Mm+1 ) (m ∈ N>0 ) extending the natural closed immersions Xm → Xm+1 . We define a p-adic fine log formal scheme to be a p-adic formal scheme endowed with a fine log structure. We say that a p-adic fine log formal scheme is affine if its underlying formal scheme is affine. A morphism of p-adic fine log formal schemes f : (X, M ) → (Y, N ) is a family of morphisms of fine log schemes

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fm : (Xm , Mm ) → (Ym , Nm ) (m ∈ N>0 ) compatible with the exact closed immersions (Xm , Mm ) → (Xm+1 , Mm+1 ) and (Ym , Nm ) → (Ym+1 , Nm+1 ). Note that giving a morphism of p-adic formal schemes X → Y is equivalent to giving a family of morphisms of schemes Xm → Ym (m ∈ N>0 ) compatible with the closed immersions Xm ,→ Xm+1 and Ym ,→ Ym+1 (cf. [1] Proposition 2.2.2). Let f : (X, M ) → (Y, N ) be a morphism of p-adic fine log formal schemes. We say that f is smooth (resp. étale, resp. an exact closed immersion, resp. a closed immersion, resp. strict, resp. integral) if fm is smooth (resp. étale, resp. ...) for every m ∈ N>0 . We say that f is an open immersion if fm is strict and the morphism of schemes underlying fm is an open immersion for every m ∈ N>0 . We say that f is an immersion if it is the composition g ◦ h of an open immersion g and a closed immersion h. We say that f is affine if the morphism of p-adic formal schemes underlying f is affine. Finite inverse limits are representable in the category of p-adic fine log formal schemes. In the following, a p-adic fine log formal scheme is denoted by a single letter such as X, Y , T . For a p-adic fine log formal scheme X, we write (MXm ) for the fine log structure of X. For a closed immersion i : X → Y of p-adic fine log formal schemes and n ∈ N>0 , the nth infinitesimal neighborhood Dm of the reduction mod pm of i for m ∈ N>0 gives a p-adic fine log formal scheme D with a factorization X → D → Y of i. We call D the nth infinitesimal neighborhood of i. For a p-adic fine log formal scheme X, we define the site X´et (resp. XÉT ) to be the category of p-adic fine log formal schemes strict étale over X (resp. p-adic fine log formal schemes over X) endowed with the topology associated to the pretopology defined by Cov(U ) = {(uα : Uα → U )α∈A |uα is strict étale for every α ∈ A and U = ∪α∈A uα (Uα )} for U ∈ Ob X´et (resp. Ob XÉT ). For any morphism of p-adic fine log formal schemes 0 f : X 0 → X, the functor X´et → X´e0 t (resp. XÉT → XÉT ); U 7→ U ×X X 0 defines a ∼ 0∼ morphism of sites, and hence a morphism of topos f´et : X´e0∼ t → X´ et (resp. fÉT : XÉT → ∼ 0 XÉT ) (cf. [2] I Proposition 5.4 4), III Proposition 1.3 5)). The functor XÉT → XÉT ; (u : U 0 → X 0 ) 7→ (f ◦ u : U 0 → X) is cocontinuous (cf. [2] III Définition 2.1) and is a 0 left adjoint of the functor XÉT → XÉT considered above. Hence we have a canonical ∗ 0 functorial isomorphism (fÉT F)(u : U → X 0 ) ∼ = F(f ◦ u : U 0 → X) for F ∈ Ob XÉT and 0 0 0 u : U → X ∈ Ob XÉT (cf. [2] III Proposition 2.5). For a p-adic fine log formal scheme X, we define the sheaf of rings OX (resp. the sheaf of monoids MX ) on X´et by Γ(U, OX ) = limm Γ(Um , OUm ) (resp. Γ(U, MX ) = ←− limm Γ(Um , MXm )). Let f : X → Y be a morphism of p-adic fine log formal schemes. ←− The morphism of topos f´et naturally extends to a morphism of ringed topos (X´et , OX ) → (Y´et , OY ), which is also denoted by f´et . For a sheaf of OY -modules F on Y´et , we write f −1 (F) (resp. f ∗ (F)) for the inverse image of F as a sheaf of abelian groups (resp. f −1 (F) ⊗f −1 (OY ) OX ). For a sheaf of OX -modules F on X´et and a sheaf of OY -modules G on Y´et , we abbreviate F ⊗f −1 (OY ) f −1 (G) to F ⊗OY G. We define the sheaf of OX -modules ΩqX/Y on X´et by Γ(U, ΩqX/Y ) = limm Γ(Um , ΩqXm /Ym ). For ←− t = (tm ) ∈ Γ(U, MX ) = limm Γ(Um , MXm ), we define d log t ∈ Γ(U, Ω1X/Y ) to be ←− (d log tm )m∈N>0 . If f is smooth, ΩqX/Y is a locally free OX -module of finite type, and strict étale locally, there exist t1 , . . . , td ∈ Γ(X, MX ) such that d log ti (1 ≤ i ≤ d) form a basis of Ω1X/Y . We call such t1 , . . . , td log coordinates of X over Y . We endow SN = Spf(RN ) with the trivial log structure. Definition IV.2.2.1. (1) We define C to be the category of sequences of morphisms Y1 → Y2 → Y3 → · · · → YN → YN +1 → · · · of p-adic fine log formal schemes YN

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319

over SN compatible with the closed immersions SN → SN +1 and satisfying the following conditions. (i) The morphism Y1 → Y2 is an immersion, and for each N ≥ 2, the morphism YN → YN +1 is an exact closed immersion. (ii) For each N ≥ 2, the morphism of schemes underlying the reduction mod p of YN → YN +1 is a nilpotent immersion. A morphism (YN ) → (YN0 ) in C is a family of morphisms fN : YN → YN0 over SN for N ∈ N>0 compatible with the immersions YN → YN +1 and YN0 → YN0 +1 . (2) For r ∈ N>0 (resp. r = ∞), we define C r to be the full subcategory of C consisting of (YN ) ∈ Ob C satisfying the following conditions. (i) The morphism Y1 → Y2 is an exact closed immersion. (ii) The morphism p : OYN → OYN is injective for every N ∈ N>0 . (iii) For every s ∈ N ∩ [0, r] (resp. s ∈ N) and N ∈ N>0 , we have pF s OYN ⊂ ξ s OYN (resp. F s OYN = ξ s OYN ), where F n OYN is the decreasing filtration of OYN by ideals defined by F 0 OYN = OYN , F n OYN = Ker(OYN → OYn ) (1 ≤ n ≤ N ), and F n OYN = 0 (n > N ). (iv) For every N ∈ N>0 , the kernel of ξ : OYN → OYN is F N −1 OYN . (3) Let Y = (YN ) be an object of C , let X be a p-adic fine log formal scheme over S1 , and let i : X → Y1 be an immersion over S1 . Then we write (i : X ,→ Y ) for the i object of C defined by the composition of X → − Y1 → Y2 and the exact closed immersions YN → YN +1 (N ≥ 2). For an object Y = (YN ) of C r , the multiplication by ξ on OYN induces an injective morphism OYN −1 → OYN , which is denoted by [ξ] in the following.

Definition IV.2.2.2. Let f = (fN ) : Y = (YN ) → Y 0 = (YN0 ) be a morphism in C . (1) We say that f is Cartesian if the morphism YN0 → YN0 +1 ×YN +1 YN induced by f is an isomorphism for every N ∈ N>0 . (2) We say that f is smooth (resp. étale, resp. strict, resp. integral, resp. affine, resp. an exact closed immersion, resp. an open immersion) if fN is smooth (resp. étale, resp. strict, resp. integral, resp. affine, resp. an exact closed immersion, resp. an open immersion) for every N ∈ N>0 . (3) We say that a family of morphisms (uα : Yα → Y ) in C is a strict étale covering (resp. a Zariski covering) if uα is strict étale (resp. an open immersion) and Cartesian for every α ∈ A and ∪α∈A uα,N (Yα,N ) = YN for every N ∈ N>0 (or equivalently, for N = 2). Remark IV.2.2.3. For Y = (YN ) ∈ C and a strict étale morphism V2 → Y2 , there exists a strict étale Cartesian morphism U = (UN ) → Y = (YN ) in C with U2 = V2 , and it is unique up to a unique isomorphism. By [39] Proposition (5.1.9), we see that, if U2 is affine and Y1 → Y2 is a closed immersion, then UN is affine for every N ∈ N>0 .

Let r denote a positive integer or ∞ in the following. Let Y = (YN ) be an object of C such that Y2 is affine and Y1 → Y2 is a closed immersion. Then, by Remark IV.2.2.3, YN is affine, and AN = Γ(YN , OYN ) defines an object A = (AN ) of Ap,• . Lemma IV.2.2.4. Let the notation and assumption be as above. If Y is an object of C r , then A is an object of Ob A•r . [ξ]s

Proof. By taking Γ(YN , −) of the factorization OYN  OYN −s −− → ξ s OYN ,→ OYN ∼ =

of the multiplication by ξ s on OYN , we obtain Γ(YN , ξ s OYN ) = ξ s AN . Similarly, since ∼ = the multiplication by p on F s OYN induces an isomorphism F s OYN → pF s OYN , we have Γ(YN , pF s OYN ) = pΓ(YN , F s OYN ) = pF s AN . Hence A satisfies the condition (ii) in the definition of A•r . The other two conditions immediately follow from the definition of C r .

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 Corollary IV.2.2.5. Let Y be an object of C r , then the morphism of schemes underlying the reduction mod p of the exact closed immersion Y1 → Y2 is nilpotent. Proof. If a morphism U → Y in C is an open immersion and Cartesian, then U is an object of C r . Hence, if U2 is affine, (Γ(UN , OUN ))N is an object of A•r by Lemma IV.2.2.4. By Lemma IV.2.1.6 (3), we have Γ(U2 , F 1 OU2 ) · Γ(U2 , F 1 OU2 ) = 0. 

By Corollary IV.2.2.5, for Y = (YN ) ∈ C r and a strict étale morphism V1 → Y1 , there exists a strict Cartesian morphism U = (UN ) → Y = (YN ) in C r with U1 = V1 , which is unique up to a unique isomorphism. If U1 is affine, then UN is affine for every N ∈ N>0 . Lemma IV.2.2.6. Let Y and A be as before Lemma IV.2.2.4. Then Y is an object of C r if and only if the morphism Y1 → Y2 is an exact closed immersion and A ∈ Ob A•r . Proof. The necessity follows from Lemma IV.2.2.4. Let us prove the sufficiency. Let V2 → Y2 be a strict étale morphism such that V2 is affine. As in Remark IV.2.2.3, there exists a strict étale Cartesian morphism U = (UN ) → Y = (YN ) in C such that U2 = V2 . By applying Lemma IV.2.1.10 (1) to the morphism (AN ) → (Γ(UN , OYN )) in Ap,• , we obtain (Γ(UN , OYN )) ∈ Ob A•r . Varying V2 , we see that (YN ) satisfies the conditions (ii), (iii), and (iv) in the definition of the category C r .  By Lemma IV.2.2.6, the sequence S = (S1 ,→ S2 ,→ S3 ,→ · · · ) is an object of C r for every r ∈ N>0 ∪ {∞}.

Lemma IV.2.2.7. (1) Let f : Y 0 → Y be a Cartesian morphism in C such that the morphism of schemes underlying the reduction mod pm of fN : YN0 → YN is flat for every N, m ∈ N>0 . If Y is an object of C r , then Y 0 is also an object of C r . (2) Let (uα : Yα → Y )α∈A be a family of strict Cartesian morphisms in C such that A is finite, the morphism of schemes underlying the reduction mod pm of uα,N : Yα,N → YN is flat and quasi-compact for every N, m ∈ N>0 and α ∈ A, and ∪α∈A uα,N (Yα,N ) = YN for every N ∈ N>0 . If Yα is an object of C r for every α ∈ A, then Y is an object of C r .

Proof. (1) If Y ∈ Ob C r , Y10 ,→ Y20 is an exact closed immersion since f is Cartesian. Since the question is Zariski local on Y2 and Y20 , we may assume that Y2 and Y20 are affine. By Lemma IV.2.2.4, the inverse system (Γ(YN , OYN ))N is an object of A•r . By the assumption on f and Y , we see that the morphism (Γ(YN , OYN ))N → (Γ(YN0 , OYN0 ))N in Ap,• satisfies the assumption in Lemma IV.2.1.10 (1). Hence (Γ(YN0 , OYN0 ))N is an object of A•r . By Lemma IV.2.2.6, Y 0 is an object of C r . (2) By fpqc descent of closed immersions of schemes (cf. [42] Proposition (2.7.1) (xii)), the morphism of schemes underlying the reduction mod pm of Y1 → Y2 is a closed immersion. Then we see that Y1 → Y2 is an exact closed immersion since uα is strict, Yα,1 → Yα,2 is an exact closed immersion, and ∪α∈A uα,1 (Yα,1 ) = Y1 . Since the question is Zariski local on Y2 and Yα,2Q , we may assume that Y2 and Yα,2 are affine. By r Lemma IV.2.2.4, the inverse system ( α∈A Γ(Yα,N , OYα,N ))N Q is an object of A• . By the assumption on (uα )α , the morphism (Γ(YN , OYN ))N → ( α∈A Γ(Yα,N , OYα,N ))N in Ap,• satisfies the assumption in Lemma IV.2.1.10 (2). Hence (Γ(YN , OYN ))N ∈ Ob A•r , and Lemma IV.2.2.6 implies Y ∈ Ob C r .  Lemma IV.2.2.8. Let f : Y 0 → Y and g : Y 00 → Y be two morphisms in C . Then YN000 = YN00 ×YN YN0 (N ∈ N>0 ) with the induced morphisms YN000 → YN000+1 define an object Y 000 of C and Y 000 with the canonical morphisms Y 000 → Y 0 and Y 000 → Y represents the fiber product of Y 0 → Y ← Y 00 in C .

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321

Proof. In the category of fine log schemes, closed immersions, open immersions, exact closed immersions, and nilpotent immersions in underlying schemes are stable under base changes. This implies the first claim. The second claim is obvious.  By Lemma IV.2.2.8, we see that Cartesian morphisms in C are stable under base changes. Proposition IV.2.2.9. (1) The inclusion functor C r → C has a right adjoint functor r DHiggs : C → C r. r (2) For Y ∈ C , the morphism DHiggs (Y )1 → Y1 defined by the adjunction morphism r r DHiggs (Y ) → Y is affine and strict. If Y1 → Y2 is a closed immersion, then DHiggs (Y ) → Y is affine. (3) Let f : Y 0 → Y be a strict Cartesian morphism in C such that the morphism of schemes underlying the reduction mod pm of fN : YN0 → YN is flat for every N, m ∈ N>0 . r r Then the natural morphism DHiggs (Y 0 ) → DHiggs (Y ) ×Y Y 0 in C is an isomorphism.

r Definition IV.2.2.10. For an object Y of C , we call DHiggs (Y ) the Higgs envelope of Y of level r.

Corollary IV.2.2.11. Finite fiber products and finite products are representable in C r . Proof. Since S is a final object of C r , it suffices to prove the representability of finite fiber products. Let T 0 → T ← T 00 be morphisms in C r , and let T 000 be the fiber r product of this diagram in C (cf. Lemma IV.2.2.8). Then DHiggs (T 000 ) gives the fiber r product in C .  In the rest of this subsection, we prove Proposition IV.2.2.9. Definition IV.2.2.12. Let Y be an object of C . We say that a pair (T, u) of an object T of C r and a morphism u : T → Y in C is an affine Higgs envelope of Y of level r if it satisfies the following two conditions. (a) The morphism u is affine. (b) For every T 0 ∈ C r , the map HomC r (T 0 , T ) → HomC (T, Y ); f 7→ u ◦ f is bijective.

Lemma IV.2.2.13. Let Y be an object of C and let u : T → Y be an affine Higgs envelope of Y of level r. Let Ye → Y be a strict Cartesian morphism in C such that the morphism of schemes underlying the reduction mod pm of YeN → YN is flat for every m, N ∈ N>0 . Then the base change u e : Te → Ye of u by Ye → Y is an affine Higgs envelope e of Y of level r. Proof. Since u is affine, u e is affine. By the assumption on Ye → Y , the morphism e T → T is strict Cartesian and the underlying morphism of schemes of the reduction mod pm of TeN → TN is flat for every m, N ∈ N>0 . By Lemma IV.2.2.7 (1), we see that Te is an object of C r . For any T 0 ∈ Ob C r , we have bijections ∼ ∼ HomC (T 0 , Te) − → HomC (T 0 , T ) ×Hom (T 0 ,Y ) HomC (T 0 , Ye ) − → HomC (T 0 , Ye ) C

because HomC (T 0 , T ) → HomC (T 0 , Y ) is bijective by the assumption on u.



Proposition IV.2.2.14. Let Y be an object of C such that Y2 is affine and Y1 → Y2 is an exact closed immersion. Then Y has an affine Higgs envelope of level r. Proof. Let A = (AN ) be the object of Ap,• defined by AN = Γ(YN , OYN ), and let r B = (BN ) be the object DHiggs (A) of A•r (cf. Proposition IV.2.1.8). We have the adjunction morphism A → B. By Lemma IV.2.1.6 (3), the square of F N BN +1 = Ker(BN +1 → BN ) is 0. Hence Spf(BN ) endowed with the inverse image of MYN defines an object

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T = (TN ) of C . Since Y1 → Y2 is an exact closed immersion, T is an object of C r by Lemma IV.2.2.6. We prove that the affine morphism u : T → Y is an affine Higgs envelope of level r. Let T 0 be an object of C r and let f : T 0 → Y be a morphism in C . It suffices to prove that there exists a unique morphism g : T 0 → T such that u ◦ g = f . Since u is a strict morphism, we see that giving a morphism g as above is equivalent to giving a compatible system of AN -algebra homomorphisms BN → Γ(TN0 , OTN0 ). If TN0 are affine, the inverse system (Γ(TN0 , OTN0 ))N is an object of A•r by Lemma IV.2.2.6, and r there exists a unique such compatible system by the definition of DHiggs . In the general 0 0 0 0 0 case, choose Zariski coverings (Tα → T )α∈A and (Tαβ;γ → Tα ×T 0 Tβ )γ∈Γαβ ((α, β) ∈ A2 ) 0 0 in C (cf. Definition IV.2.2.2 (3)) such that Tα,N and Tαβ;γ,N are affine. Then Tα0 and 0 r Tαβ;γ are objects of C . Hence we are reduced to the affine case by using the inverse system of exact sequences of AN -algebras: ! Y YY 0 0 0 0 0 Γ(TN , OTN0 ) → Γ(Tα,N , OTα,N )⇒ Γ(Tαβ;γ,N , OTαβ;γ,N ) .  α

αβ

γ

N

Lemma IV.2.2.15. Let f : Y → Y be an étale morphism in C such that f1 : Y10 → Y1 is an isomorphism. Then for any T ∈ C r , the morphism HomC (T, Y 0 ) → HomC (T, Y ); g 7→ f ◦ g is bijective. 0

Proof. Let h : T → Y be a morphism in C . For N ∈ N>0 , the morphism TN → TN +1 is an exact closed immersion and the underlying morphism of schemes of its reduction mod pm is a nilpotent immersion for every m ∈ N>0 (cf. Corollary IV.2.2.5). Since YN0 +1 → YN +1 is étale, this implies that any morphism TN → YN0 over YN has a unique lifting TN +1 → YN0 +1 over YN +1 . Since f1 is an isomorphism, we see that there exists a unique morphism g : T → Y 0 such that f ◦ g = h.  Lemma IV.2.2.16. Let Y be an object of C such that Y1 → Y2 is a closed immersion, g f and let y be a point on Y1 . Then there exist morphisms Y 00 − → Y0 − → Y in C such that 00 00 0 y is contained in the image of Y1 , YN and YN are affine for every N ∈ N>0 , f is strict étale and Cartesian, g is étale, g1 : Y100 → Y10 is an isomorphism, and Y100 → Y200 is an exact closed immersion. Proof. For N = 1, 2, let YN,1 denote the reduction mod p of YN . Then there exists 0 a strict étale morphism of fine log schemes f2,1 : Y2,1 → Y2,1 such that y is contained 0 0 0 in the image of Y1,1 := Y1,1 ×Y2,1 Y2,1 → Y1,1 , Y2,1 is affine, and the closed immersion i00 1,1

g2,1

0 00 0 00 0 0 is affine, g2,1 is étale, has a factorization Y1,1 −−→ Y2,1 −−→ Y2,1 , where Y2,1 Y1,1 → Y2,1 0 00 0 0 is also and i1,1 is an exact closed immersion. Since Y1,1 → Y2,1 is a closed immersion, Y1,1 0 affine. By [50] Proposition (3.14), there exist a unique strict étale lifting f2 : Y2 → Y2 of f2,1 and a unique étale lifting g2 : Y200 → Y20 of g2,1 . Put Y10 := Y1 ×Y2 Y20 . Then, since g2 is i00

g2

1 Y200 −→ Y20 . The morphism i001 étale, the factorization g2,1 ◦ i001,1 has a unique lifting Y10 −→ 0 0 is an exact closed immersion because Y1 → Y2 is a closed immersion and i001,1 is an exact closed immersion. Put Y100 = Y10 . By [50] Proposition (3.14), the morphism f2 : Y20 → Y2 extends uniquely to a strict étale Cartesian morphism Y 0 → Y in C , and the morphisms id : Y100 → Y10 and g2 : Y200 → Y20 extend uniquely to an étale morphism g : Y 00 → Y 0 in C such that the morphism YN00 → YN00+1 ×YN0 +1 YN0 is an isomorphism for every integer N ≥ 2. 

By combining Proposition IV.2.2.14 with Lemmas IV.2.2.15 and IV.2.2.16, we obtain the following proposition.

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Proposition IV.2.2.17. Let Y be an object of C such that Y1 → Y2 is a closed immersion. Then, for every point y of Y1 , there exists a strict étale Cartesian morphism Y 0 → Y in C such that y is contained in the image of Y10 and there exists an affine Higgs envelope of Y 0 of level r. g

f

Proof. Let Y 00 − →Y0 − → Y be as in Lemma IV.2.2.16. Then Y 00 has an affine Higgs envelope of level r by Proposition IV.2.2.14, and it is also an affine Higgs envelope of Y 0 of level r by Lemma IV.2.2.15.  Lemma IV.2.2.18. Let (uα : Yα → Y )α∈A be a family of morphisms in C satisfying one of the following two conditions. (a) The family (uα ) is a Zariski covering. (b) The family (uα ) is a strict étale covering, the set A is finite, and the morphism of schemes underlying the reduction mod pm of uα,N : Yα,N → YN is quasi-compact for every α ∈ A and m, N ∈ N>0 . Put Yαβ = Yα ×Y Yβ for α, β ∈ A and Yαβγ = Yα ×Y Yβ ×Y Yγ for α, β, γ ∈ A. (1) Let Z → Y and W → Y be morphisms in C , and let Zα and Wα (resp. Zαβ and Wαβ ) be the base changes of Z and W by Yα → Y (resp. Yαβ → Y ). Then the following sequence is exact. Y Y HomY (Z, W ) → HomYα (Zα , Wα ) ⇒ HomYαβ (Zαβ , Wαβ ) α∈A

(α,β)∈A2

(2) Assume that we are given an affine morphism Zα → Yα in C for each α ∈ A and ∼ = an isomorphism ιαβ : Zβ ×Yβ Yαβ − → Zα ×Yα Yαβ over Yαβ for each α, β ∈ A such that (ιαβ ×Yαβ Yαβγ ) ◦ (ιβγ ×Yβγ Yαβγ ) = ιαγ ×Yαγ Yαβγ for every (α, β, γ) ∈ A3 . Then there ∼ =

exist an affine morphism Z → Y in C and an isomorphism ια : Z ×Y Yα − → Zα over Yα for each α ∈ A such that ιαβ ◦ (ιβ ×Yβ Yαβ ) = ια ×Yα Yαβ . Furthermore the above (Z → Y, ια ) is unique up to a unique isomorphism.

Lemma IV.2.2.19. Let (uα : Xα → X)α∈A be a family of strict morphisms of log schemes satisfying one of the following two conditions. (a) The family of morphisms of schemes underlying (uα )α∈A is an open covering. (b) The set A is finite. The family of morphisms of schemes underlying (uα )α∈A is a covering by quasi-compact étale morphisms. Put Xαβ = Xα ×X Xβ for α, β ∈ A, and Xαβγ = Xα ×X Xβ ×X Xγ for α, β, γ ∈ A. (1) Let U → X and V → X be morphisms of log schemes. Let Uα and Vα (resp. Uαβ and Vαβ ) be the base changes of U and V by Xα → X (resp. Xαβ → X). Then the following sequence is exact. Y Y HomX (U, V ) → HomXα (Uα , Vα ) ⇒ HomXαβ (Uαβ , Vαβ ). α∈A

(α,β)∈A2

(2) Assume that we are give a morphism of log schemes Uα → Xα whose underlying ∼ = morphism of schemes is affine for each α ∈ A, and an isomorphism ιαβ : Uβ ×Xβ Xαβ → 2 Uα ×Xα Xαβ over Xαβ for each (α, β) ∈ A such that (ιαβ ×Xαβ Xαβγ )◦(ιβγ ×Xβγ Xαβγ ) = ιαγ ×Xαγ Xαβγ . Then there exists a morphism of log schemes U → X and an isomorphism ια : U ×X Xα ∼ = Uα over Xα for each α such that the underlying morphism of schemes of U → X is affine, and ιαβ ◦ (ιβ ×Xβ Xαβ ) = ια ×Xα Xαβ . Furthermore the above (U → X, ια ) is unique up to unique isomorphisms. ˚ Proof. For a log scheme Z (resp. a morphism of log schemes h : Z 0 → Z), let Z ˚ (resp. h) denote the underlying scheme of Z (resp. the morphism of schemes underlying h).

324

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS f

g

(1) The morphism V → X has a factorization V − → V − → X such that the underlying morphism of f is the identity map and the morphism g is strict. Let Vα → V α → Xα and Vαβ → V αβ → Xαβ denote the base changes of the above factorization by Xα → X and Xαβ → X, respectively. Let ? denote ∅, α or αβ. Then we ˚ ). By [37] VIII Théorème 5.2, this implies that ˚? , V have HomX (U? , V ? ) = Hom ˚ (U ? ?

X?

the claim holds for HomX? (U? , V ? ). Hence it suffices to prove that, for any morphism h : U → V over X and its base changes hα and hαβ by Xα → X and Xαβ → X, the claim also holds for HomV ? (U? , V? ) = HomMV ? (MV? , h?∗ (MU? )). This is an immediate ˚´et with respect consequence of the étale descent of morphisms of sheaves of monoids on V ˚ ˚ to the covering (Vα → V )α∈A . (2) The uniqueness follows from (1). Let us prove the existence. By fpqc descent of affine schemes (cf. [37] VIII Théorème 2.1), there exists an affine morphism of schemes ∼ = ˚ ˚ → X ˚ and an isomorphism ˚ ˚ ×˚ X ˚α − ˚α for each α ∈ A such f˚: U ια : U → Uα over X X ∗ ˚ ˚ ια ×X that ˚ ιαβ ◦ (˚ ιβ ×X ια (MUα ). Then the descent data ˚α Xαβ . Put Mα = ˚ ˚β Xαβ ) = ˚ ∼ =

(ιαβ )αβ is equivalent to isomorphisms of log structures ταβ : Mβ |U − → Mα |U ˚× ˚ X ˚ ˚× ˚ X ˚ X αβ X αβ (α, β ∈ A) satisfying the cocycle condition and the compatibility with the morphisms f˚α−1 (MXα ) → Mα (α ∈ A) induced by Uα → Xα . Here f˚α denotes the morphism ˚α → X ˚α . By gluing of sheaves on the étale site U ˚´et with respect to an étale ˚ ×˚ X U X ∼ = ˚, an isomorphism τα : M | ˚ ˚ − → Mα covering, we obtain a log structure M on U U ×X ˚ Xα

of log structures for each α ∈ A, and a morphism of log structures f˚−1 (MX ) → M compatible with ταβ and f˚α−1 (MXα ) → Mα (α ∈ A). These data give a morphism of log ∼ = ˚, M ) → X and an isomorphism ια : U ×X Xα − schemes f : U = (U → Uα over Xα for each α ∈ A satisfying the desired properties. 

Proof of Lemma IV.2.2.18. We obtain the claim (1) just by applying Lemma IV.2.2.19 (1) to the reduction mod pm of XN → YN and WN → YN for each m, N ∈ N>0 . Similarly we can apply Lemma IV.2.2.19 (2) to the reduction mod pm of ιαβ,N for m, N ∈ N>0 , obtaining the claim (2) by fpqc descent of immersions of schemes ([42] Proposition (2.7.1)).  Lemma IV.2.2.20. Let (uα : Yα → Y )α∈A be as in Lemma IV.2.2.18. If Yα has an affine Higgs envelope of level r for every α ∈ A, then Y also has an affine Higgs envelope of level r. Proof. Let Yα and Yαβγ be as in Lemma IV.2.2.18. Let Tα → Yα be an affine Higgs envelope of Yα of level r. By Lemma IV.2.2.13, Tδ ×Yδ Yαβ for δ ∈ {α, β} (resp. Tδ ×Yδ Yαβγ for δ ∈ {α, β, γ}) is an affine Higgs envelope of Yαβ (resp. Yαβγ ) of level r. Hence there ∼ = exists a unique Yαβ -isomorphism ιαβ : Tβ ×Yβ Yαβ → Tα ×Yα Yαβ for each (α, β) ∈ A2 satisfying the cocycle condition in Lemma IV.2.2.18 (2). Therefore there exists an affine ∼ = morphism T → Y in C and a Yα -isomorphism ια : Yα ×Y T → Tα for each α ∈ A compatible with ιαβ . By Lemma IV.2.2.7 (2), T is an object of C r . Let f : T 0 → Y be a morphism in C such that T 0 ∈ Ob C r . Then, for each α ∈ A (resp. (α, β) ∈ A2 ), there exists a unique Yα -morphism (resp. Yαβ -morphism) T 0 ×Y Yα → T ×Y Yα (∼ = Tα ) (resp. T 0 ×Y Yαβ → T ×Y Yαβ (∼ = Tα ×Yα Yαβ )). By Lemma IV.2.2.18 (1), we see that there exists a unique Y -morphism T 0 → T . Hence T → Y is an affine Higgs envelope of level r.  Proof of Proposition IV.2.2.9. Let Y ∈ Ob C . There exists a Cartesian open immersion f : Y 0 → Y in C such that f1 = id and Y10 → Y20 is a closed immersion. For any

IV.2. HIGGS ENVELOPES

325

T 0 ∈ C r , the morphism of topological spaces underlying T10 → TN0 is a homeomorphism for every N ∈ N>0 (cf. Corollary IV.2.2.5). This implies that the map HomC (T 0 , Y 0 ) → HomC (T 0 , Y ) induced by f is bijective. Hence we may assume that Y1 → Y2 is a closed immersion. Let U → Y be an open immersion such that U2 is the complement of the image of Y1 in Y2 . Then U1 = ∅, and ∅ → U is an affine Higgs envelope of level r. Combining this with Proposition IV.2.2.17, we obtain a strict étale covering (Yα → Y )α∈A such that Yα has an affine Higgs envelope of level r. If YN are affine, then by Lemma IV.2.2.13, we may assume that Yα,N are affine and A is finite. By Lemma IV.2.2.20, we see that Y has an affine Higgs envelope of r. The general case is reduced to the affine case by choosing a Zariski covering (Uβ → Y )β∈B such that Uβ,N are affine and applying Lemma IV.2.2.20. This completes the proof of (1) and (2). The claim (3) follows from Lemma IV.2.2.13.  IV.2.3. Rings of power series. For r ∈ N>0 ∪{∞}, we give an explicit description of the Higgs envelope of level r of a ring of power series with coefficients in an object A = (AN ) of A•r (Proposition IV.2.3.15). We also prove an analogous result for p-adic fine log formal schemes as follows. Let f : Y 0 → Y be a smooth Cartesian morphism in C (Definitions IV.2.2.1, IV.2.2.2) such that Y1 is affine, and let i : Y1 → Y10 be a section of f1 over S1 . Under the assumption on the existence of certain log coordinates of f , we give an explicit description of the Higgs envelope of level r of (i : Y1 ,→ Y 0 ) (Definition IV.2.2.1 (3)) in terms of the Higgs envelope of level r of Y and the log coordinates (Proposition IV.2.3.17). r Definition IV.2.3.1. For r ∈ N>0 (resp. r = ∞), let Malg,• denote the category of inverse systems (MN )N ∈N>0 of R-modules satisfying the following conditions: (0) For each N ∈ N>0 , ξ N MN = 0 and the transition map MN +1 → MN is surjective. (i) The module MN is p-torsion free for every N ∈ N>0 . (ii) For every s ∈ N ∩ [0, r] (resp. s ∈ N) and N ∈ N>0 , we have pF s MN ⊂ ξ s MN (resp. F s MN = ξ s MN ), where F n MN (n ∈ N) is the decreasing filtration of MN by RN -submodules defined by F 0 MN = MN , F n MN = Ker(MN → Mn ) (1 ≤ n ≤ N ), and F n MN = 0 (n > N ). (iii) For every N ∈ N>0 , the kernel of ξ : MN → MN is F N −1 MN . r Let M = (MN ) be an object of Malg,• . Since MN is p-torsion free, the natural homomorphism MN → MN,Q is injective. We regard MN as a submodule of MN,Q . Since MN /F n MN ∼ = Mn is p-torsion free for n ∈ N ∩ [1, N ], we have F n MN = MN ∩ F n MN,Q in MN,Q for n ∈ N.

r Lemma IV.2.3.2. Let r ∈ N>0 ∪ {∞} and let M = (MN ) be an object of Malg,• . n (1) For s, n, N ∈ N>0 such that s ≤ n ≤ N , the inverse image of F MN under the morphism ξ s : MN → MN is F n−s MN . (2) Assume r ∈ N>0 . For s ∈ N ∩ [0, r], n ∈ N and N ∈ N>0 , we have pF n+s MN ⊂ s n ξ F MN .

Proof. The same as the proof of Lemma IV.2.1.6.



r Lemma IV.2.3.3. Let r ∈ N>0 ∪ {∞} and let M = (MN ) be an object of Malg,• . (1) For N, M ∈ N>0 , the homomorphism ξ M : MN +M → MN +M decomposes as [ξ M ]

MN +M → MN −−−→ MN +M with [ξ M ] an injective homomorphism. (2) Assume r ∈ N>0 . For n ∈ N, N ∈ N>0 , and the smallest integer m such that nr−1 ≤ m, we have pm F n MN ⊂ ξ n MN ⊂ F n MN . In particular, we have (F n MN )Q = ξ n (MN )Q .

326

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

(3) For n, m ∈ N and N ∈ N>0 such that m ≤ n, we have ξ m MN ∩ F n MN = ξ F n−m MN . m

Proof. (1) and (3) (resp. (2)) follow(s) from Lemma IV.2.3.2 (1) (resp. (2)).



r Let r ∈ N>0 ∪ {∞} and let M = (MN ) be an object of Malg,• . For n ∈ N>0 and n n a submodule P (resp. an element x) of ξ (MN +n )Q = (F MN +n )Q , we define ξ −n P ∼ = (resp. ξ −n x) to be the inverse image of P (resp. x) by the isomorphism [ξ n ]Q : (MN )Q − → ξ n (MN +n )Q . We assume r ∈ N>0 until Lemma IV.2.3.13. For m, n ∈ N and N ∈ N>0 , we define (m) ,n the RN -submodule MN r of MN,Q by (m)r ,n

MN

:= p−n1 ξ n−n2 F n2 MN +max{0,n2 −n} ,

n−m where n1 = min{[ n−m r ], 0} and n2 = n − m − n1 r. If n ≤ m + r − 1, then n1 = [ r ] ≤ 0 and 0 ≤ n2 ≤ r − 1. If n ≥ m, we have n1 = 0 and n2 = n − m, which imply (m) ,n MN r = ξ m F n−m MN . The homomorphism (MN +1 )Qp → (MN )Qp induces a surjective (m) ,n (m) ,n homomorphism MN +1r → MN r .

Lemma IV.2.3.4. For m, n ∈ N and N ∈ N>0 such that m ≥ n, we have p[

m−n ]+1 r

(m)r ,n

ξ n MN ⊂ MN

⊂ p[

m−n ] r

ξ n MN .

Proof. Put n − m = n1 r + n2 (n1 ∈ Z, n2 ∈ Z ∩ [0, r − 1]). We have [ m−n r ] = −n1 (m) ,n (if n2 = 0), −n1 − 1 (if n2 6= 0). If n2 = 0, then we have MN r = p−n1 ξ n MN . If n2 6= 0, then we have ξ n2 MN 0 ⊂ F n2 MN 0 ⊂ p−1 ξ n2 MN 0 (N 0 = N + max{0, n2 − n}), (m) ,n  which implies p−n1 ξ n MN ⊂ MN r ⊂ p−n1 −1 ξ n MN . Lemma IV.2.3.5. Let m, n ∈ N and define m1 , m2 , n2 ∈ N, and n1 ∈ Z by m = m1 r + m2 , 0 ≤ m2 ≤ r − 1, n1 = min{[ n−m r ], 0} and n2 = n − m − n1 r. Then n − n2 < 0 if and only if 0 ≤ n ≤ m2 − 1. In this case, we have n2 − n = r − m2 and n1 = −1 − m1 . In particular, 1 ≤ n2 − n ≤ r − 1 and −n1 ≥ 1. Proof. If n ≥ m, then n1 = 0 and n − n2 = m ≥ 0. Assume n < m. Put n = m2 + l1 r + l2 (l1 , l2 ∈ Z, 0 ≤ l2 ≤ r − 1). Then n − m = (l1 − m1 )r + l2 , which implies n1 = l1 − m1 , n2 = l2 and n − n2 = m2 + l1 r. On the other hand, 0 ≤ n = m2 + l1 r + l2 ≤ r − 1 + l1 r + r − 1 implies l1 ≥ −1. Hence n − n2 < 0 if and only if l1 = −1, which is equivalent to n ≤ m2 − r + r − 1 = m2 − 1. Now the remaining assertions are obvious.  r Lemma IV.2.3.6. For M = (MN ) ∈ Ob Malg,• and t, s, N ∈ Z such that N > 0, t ≥ 0, t s n and t + s ≥ 0, we have [ξ ](ξ F MN +max{0,−s} ) = ξ t+s F n MN +t .

Proof. If s ≥ 0, the claim follows from the surjectivity of ξ s F n MN +t → ξ s F n MN . If s < 0, using [ξ t ] = [ξ t+s ][ξ −s ], we see that the left-hand side coincides with [ξ t+s ](F n MN −s ) = ξ t+s F n MN +t .  r Lemma IV.2.3.7. Let M = (MN ) be an object of Malg,• . Let N be a positive integer. (m) ,n

(1) We have MN r ⊂ F n MN for n, m ∈ N. (2) Let n, m ∈ N, and put n1 = min{[ n−m r ], 0} and n2 = n − m − n1 r. Then we have (m)r ,n (m) ,n+1 MN ∩ F n+1 MN ⊂ MN r . The equality holds if n ≥ m or 0 ≤ n2 ≤ r − 2. (m+1)r ,n (m) ,n (3) For n, m ∈ N, we have MN ⊂ MN r .

IV.2. HIGGS ENVELOPES

327

Proof. (1) Put n1 = min{[ n−m r ], 0}(≤ 0) and n2 = n − m − n1 r(≥ 0). If n − n2 ≥ 0, (m)r ,n n−n2 n2 then MN ⊂ξ F MN ⊂ F n MN by Lemma IV.2.3.3 (3). If n − n2 < 0, then 1 ≤ n2 − n ≤ r − 1 and −n1 ≥ 1 by Lemma IV.2.3.5. Hence, by Lemma IV.2.3.2 (2), we have (m)r ,n

[ξ n2 −n ](MN

) ⊂ pF n2 MN +n2 −n ⊂ ξ n2 −n F n MN +n2 −n = [ξ n2 −n ](F n MN ).

], 0} and n02 = (n + 1) − m − n01 r. We have n01 = n1 (2) Put n01 = min{[ (n+1)−m r or n1 + 1, and n01 = n1 + 1 holds if and only if [ (n+1)−m ] = [ n−m r r ] + 1 ≤ 0, which is 0 equivalent to n < m and n2 = r − 1. Choose N ∈ N such that N 0 ≥ n2 − n. By (1) and 0 0 (m) ,n F n+1 MN = F n+1 MN,Q ∩MN , it suffices to prove that [ξ N ](MN r )∩[ξ N ](F n+1 MN,Q ) 0 (m) ,n+1 is equal to (resp. contained in) [ξ N ](MN r ) if n01 = n1 (resp. n01 = n1 +1). By Lemma IV.2.3.6 and Lemma IV.2.3.3 (2)(3), the former is equal to p−n1 ξ N

0

+n−n2

F n2 MN +N 0 ∩ F N

0

+n+1

MN +N 0 ,Q = p−n1 ξ N

0

+n−n2

F n2 +1 MN +N 0 ,

which coincides with the latter if n01 = n1 . If n01 = n1 + 1, we have 0

(m)r ,n+1

[ξ N ](MN

) = p−n1 −1 ξ N

0

+n−n2 +r

F n2 +1−r MN +N 0

by Lemma IV.2.3.6. Hence the claim follows from Lemma IV.2.3.2 (2). n−(m+1) 0 ], 0}, n02 = (3) Put n1 = min{[ n−m r ], 0}, n2 = n − m − n1 r, n1 = min{[ r n − (m + 1) − n01 r. Then (n01 , n02 ) = (n1 , n2 − 1) or (n1 − 1, n2 + (r − 1)). Choose N 0 ∈ N such that N 0 + n − n2 ≥ 0 and N 0 + n − n02 ≥ 0. Then, by Lemma IV.2.3.6, we have 0

0

(m)r ,n

) = p−n1 ξ N +n−n2 F n2 MN +N 0 , ( 0 0 p−n1 ξ N +n−n2 +1 F n2 −1 MN +N 0 (m+1) ,n r N [ξ ](MN )= 0 p−n1 +1 ξ N +n−n2 −(r−1) F n2 +(r−1) MN +N 0 [ξ N ](MN

(if n01 = n1 ), (if n01 = n1 − 1).

By Lemma IV.2.3.3 (3) and Lemma IV.2.3.2 (2), we obtain 0

(m+1)r ,n

[ξ N ](MN

0

(m)r ,n

) ⊂ [ξ N ](MN



).

r Let M = (MN ) ∈ Malg,• . For m ∈ N and N ∈ N>0 , we define the RN -submodule P (m) ,n (m)r of MN to be n≥0 MN r . The homomorphism MN +1,Q → MN,Q induces a MN (m)

(m)r

surjective homomorphism MN +1r → MN

(m) (MN r ). F n MN,Q .

. We write M (m)r for the inverse system (m)r

We define the decreasing filtration F n MN

(m)r

(n ∈ N) on MN

(m)r

to be MN

∩

r Corollary IV.2.3.8. Let M = (MN ) be an object of Malg,• . P (m) (m) ,n0 r (1) For n, m ∈ N and N ∈ N>0 , we have F n MN = n0 ≥n MN r . P (m) (m) ,n (2) For m ∈ N and N ∈ N>0 , we have MN r = m≥n≥0 MN r . r Proposition IV.2.3.9. Let M = (MN ) be an object of Malg,• . (m)r r . (1) For m ∈ N, the inverse system M is an object of Malg,• (m)r (1)r (m+1)r (2) For m ∈ N, we have (M ) =M . r Lemma IV.2.3.10. For an object M = (MN ) of Malg,• , M (1)r is again an object of r Malg,• .

Proof. By Corollary IV.2.3.8 (2), we have (1)r

MN

(1)r ,0

= MN

(1)r ,1

+ MN

= pξ −(r−1) F r−1 MN +r−1 + ξMN .

328

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS (1)

(1)

(1)

Since MN r ⊂ MN , it remains to prove pF s MN r ⊂ ξ s MN r for s ∈ N ∩ [1, r]. By (1) Corollary IV.2.3.8 (1) and Lemma IV.2.3.7 (2), we have F s MN r = ξF s−1 MN . By Lemma IV.2.3.3 (3), we obtain (1)

(1)r ,0

).



(m)r ,n

=

[ξ r−1 ](pF s MN r ) = pξ r F s−1 MN +r−1 ⊂ pξ s F r−1 MN +r−1 = [ξ r−1 ](ξ s MN

r Lemma IV.2.3.11. For M = (MN ) ∈ Ob Malg,• and m, n ∈ N, we have ξMN (m+1)r ,n+1 MN .

Proof. Put n1 = min{[ n−m r ], 0} and n2 = n − m − n1 r. Since (n + 1) − (m + 1) = n − m, we have (m)r ,n

MN

(m+1)r ,n+1

MN

= p−n1 ξ n−n2 F n2 MN +max{0,n2 −n} ,

= p−n1 ξ n+1−n2 F n2 MN +max{0,n2 −(n+1)} .

Choose N 0 ∈ N such that N 0 + n − n2 ≥ 0. Then, by Lemma IV.2.3.6, we have 0

(m)r ,n

[ξ N ](ξMN

) = p−n1 ξ N

0

+n+1−n2

0

(m+1)r ,n+1

F n2 MN +N 0 = [ξ N ](MN

), 

which implies the claim.

r Lemma IV.2.3.12. For M = (MN ) ∈ Ob Malg,• and n ∈ N, n ≥ r − 1, we have (m) ,n

(m+1)r ,n−(r−1)

r ⊂ MN pξ −(r−1) MN +(r−1)

. Furthermore the equality holds if n < m + r.

n−(r−1)−(m+1) 0 ], 0} = Proof. Put n1 = min{[ n−m r ], 0}, n2 = n−m−n1 r, n1 = min{[ r n−m 0 0 min{[ r ]−1, 0} and n2 = n−(r−1)−(m+1)−n1 r. Then we have (n01 , n02 ) = (n1 −1, n2 ) 0 0 0 if [ n−m r ] ≤ 0 (⇔ n < m + r), and (n1 , n2 ) = (n1 , n2 − r) otherwise. Choose N ∈ N 0 0 such that N + n − n2 − (r − 1) ≥ 0 and N − (r − 1) ≥ 0. It suffices to prove that 0 0 (m)r ,n (m+1)r ,n−(r−1) [ξ N ](pξ −(r−1) MN +(r−1) ) is equal to (resp. contained in) [ξ N ](MN ) if n01 = 0 n1 − 1 (resp. n1 = n1 ). By Lemma IV.2.3.6, the former is

p[ξ N

0

−(r−1)

(m)r ,n

](MN

) = p−n1 +1 ξ N

0

−(r−1)+n−n2

F n2 MN +N 0 ,

which coincides with the latter if n01 = n1 − 1. If n01 = n1 , we have 0

(m+1)r ,n−(r−1)

[ξ N ](MN

) = p−n1 ξ N

0

+n−n2 +1

F n2 −r MN +N 0

by Lemma IV.2.3.6, which implies the claim by Lemma IV.2.3.2 (2).



Proof of Proposition IV.2.3.9. Let (1)m (resp. (2)m ) denote the claim (1) (resp. (2)) for an m ∈ N. Then (1)0 is trivial by M (0)r = M . (1)m and (2)m imply (1)m+1 by Lemma IV.2.3.10. Hence it suffices to prove that (1)m implies (2)m . Assume that (1)m holds. By Corollary IV.2.3.8, we have (m)r (1)r

(MN

)

(m)

(m)

r =pξ −(r−1) F r−1 MN +(r−1) + ξMN r X X (m)r ,n (m) ,n = pξ −(r−1) MN +r−1 + ξMN r .

n≥r−1

n≥0

Hence the claim (2)m follows from Lemmas IV.2.3.11 and IV.2.3.12. Note r − 1 < m + r.  Let A = (AN ) be an object of A•r (Definition IV.2.1.5 (2)). Then the underr lying inverse system of R-modules is an object of Malg,• . Hence, by applying the (m)r

above construction, we obtain objects A(m)r = (AN (m)

(m) ,0

r ) (m ∈ N) of Malg,• . Since

AN ⊃ AN r ⊃ AN r ⊃ p−n1 AN , where −m = n1 r + n2 (n1 ∈ Z, n2 ∈ Z ∩ [0, r − 1]), (m) we see that AN r is p-adically complete and separated.

IV.2. HIGGS ENVELOPES

329

Lemma IV.2.3.13. Let the notation and assumption be as above. (m) (m0 ) (m+m0 )r (1) We have AN r · AN r ⊂ AN for m, m0 ∈ N and N ∈ N>0 . (1)r (m)r (m+1)r (2) We have AN · AN ⊃ pAN for m ∈ N and N ∈ N>0 . (m0 ) ,n0

(m) ,n

(m+m0 ) ,n+n0

r Proof. (1) It suffices to prove AN r · AN r ⊂ AN for n, n0 , m, m0 ∈ n−m N and N ∈ N>0 . Put n1 = min{[ r ], 0} and n2 = n − m − n1 r. We define (n01 , n02 ) and (n001 , n002 ) similarly using (n0 , m0 ) and (n+n0 , m+m0 ), respectively. Put l = n001 −(n1 +n01 ). Then we have l ≥ 0 and n2 + n02 = n002 + lr. Choose N 0 ∈ N such that N 0 + n − n2 ≥ 0 (m)r ,n (m0 )r ,n0 and N 0 + n0 − n02 ≥ 0. Since the images of AN +N under the projection 0 and AN +N 0

(m)r ,n

AN +N 0 → AN are AN 0

(m)r ,n

[ξ 2N ](AN

(m0 )r ,n0

· AN

(m0 )r ,n0

and AN

0

, respectively, we have (m0 ) ,n

0

(m) ,n

r N ) = [ξ N ](AN +N ](AN +Nr 0 ) 0 ) · [ξ

= p−n1 ξ N

0

+n−n2

00

⊂ p−n1 +l ξ 2N 00

⊂ p−n1 ξ 2N 0

0

0

0

F n2 AN +2N 0 · p−n1 ξ N

+(n+n0 )−n00 2 −lr

+(n+n0 )−n00 2

+n0 −n02

0

F n2 AN +2N 0

00

F n2 +lr AN +2N 0

00

F n2 AN +2N 0

(m+m0 )r ,n+n0

= [ξ 2N ](AN

0

)

by Lemma IV.2.1.6 (3), Lemma IV.2.3.6, and Lemma IV.2.3.2 (2). (1) (1) ,0 (m) (1) (2) Since AN r ⊃ AN r = pξ −(r−1) F r−1 AN +r−1 ⊃ pAN , we obtain AN r · AN r ⊃ (m) (m+1)r pAN r ⊃ pAN from Lemma IV.2.3.7 (3).  (m)

Let r ∈ N>0 ∪ {∞} and let A = (AN ) be an object of A•r . If r = ∞, we define AN r to be AN for m ∈ N and N ∈ N>0 . Let d ∈ N>0 , and define the object B = (BN ) of Ap,• over A = (AN ) by B1 = A1 , BN = AN {T1 , . . . , Td }(= limm AN /pm AN [T1 , . . . , Td ]) ←− (N ≥ 2), B2 → B1 ; Ti 7→ 0, and BN +1 → BN ; Ti 7→ Ti (N ≥ 2). We define the RN -submodule AN [W1 , . . . , Wd ]r of AN [W1 , . . . , Wd ] to be M (|m|) AN r W m , m∈Nd

Q which is an AN -subalgebra by Lemma IV.2.3.13 (1). Here W m = 1≤i≤d Wimi and |m| = m1 + · · · + md for m = (m1 , . . . , md ) ∈ Nd . We define AN {W1 , . . . , Wd }r to be the p-adic completion of AN [W1 , . . . , Wd ]r . The natural homomorphism AN +1 {W1 , . . . , Wd }r → AN {W1 , . . . , Wd }r is surjective, and hence we obtain an object (AN {W1 , . . . , Wd }r ) of Ap,• , which is denoted by A{W1 , . . . , Wd }r in the following. We define a morphism u : B → A{W1 , . . . , Wd }r in Ap,• over A by BN = AN {T1 , . . . , Td } → AN {W1 , . . . , Wd }r ; Ti 7→ ξWi , (1)r ,1

Note that we have ξ ∈ ξAN = AN

N ≥ 2.

(1)

⊂ AN r .

Lemma IV.2.3.14. Assume r ∈ N>0 . Then, for N ∈ N>0 , we have   X  |m| (AN {W1 , . . . , Wd }r )Q = am W m ∈ AN,Q [[W1 , . . . , Wd ]] lim p−[ r ] am = 0 .   |m|→∞ d m∈N

m

(m)r

Proof. By Lemma IV.2.3.4 and Lemma IV.2.3.7 (1), we have p[ r ]+1 AN ⊂ AN +1 [ m−N p r ] AN for m ≥ N − 1.

⊂ 

330

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proposition IV.2.3.15. Under the notation and assumption as above, A{W1 , . . . , Wd }r r r is an object of A•r and DHiggs (B) with the adjunction morphism B → DHiggs (B) (Proposition IV.2.1.8) is isomorphic to u : B → A{W1 , . . . , Wd }r . Proof. To simplify the notation, we abbreviate W1 , . . . , Wd to W in the notation (|m|) (|m|) AN {W1 , . . . , Wd }r , A{W1 , . . . , Wd }, etc. Since AN r (resp. F n AN r ) is p-adically n complete and separated, an element of AN {W }r (resp. F (AN {W }r )) is written uniquely P (|m|) (|m|) in the form m∈Nd am W m where am ∈ pl(|m|) AN r (resp. pl(|m|) F n AN r ) for some sequence l(m) ∈ N (m ∈ N) tending to ∞ as m → ∞. Hence, by applying Proposition IV.2.3.9 (1) to A in the case r ∈ N>0 , we see that A{W }r is an object of A•r . Let C = (CN ) be an object of A•r and let f = (fN ) : B → C be a morphism in A•r . It suffices to prove that there exists a unique morphism g = (gN ) : A{W }r → C such that f = g ◦u. Since fN +1 (Ti ) ∈ F 1 CN +1 ⊂ ξ ·CN +1,Q by Lemma IV.2.3.3 (2) if r ∈ N>0 , there exists a unique wi,N ∈ CN,Q such that [ξ](wi,N ) = fN +1 (Ti ). These elements form an inverse system (wi,N ) ∈ limN CN,Q . We define the compatible system of AN -algebra ←− homomorphisms hN : AN [W ]Q → CN,Q by hN (Wi ) = wi,N . We assert hN (AN [W ]r ) ⊂ CN . If r = ∞, F 1 CN +1 = ξCN +1 implies wi,N ∈ CN . Hence the claim is trivial. Assume Q Q m mi r ∈ N>0 . For m = (mi ) ∈ Nd , let wN (resp. T m ) denote 1≤i≤d wi,N (resp. 1≤i≤d Timi ). m For m ∈ Nd such that 0 ≤ |m| ≤ r, we have p[ξ |m| ]wN = pfN +|m| (T m ) ∈ pF |m| CN +|m| ⊂ m ξ |m| CN +|m| by Lemma IV.2.1.6 (3). Hence pwN ∈ CN . We also have ξwi,N = fN (Ti ) ∈ (m) ,n (m) ,n F 1 CN . Since fN (AN r ) ⊂ CN r by definition, the proof of the above claim is (m) ,n m reduced to showing that CN r wN ⊂ CN for m, n ∈ N, N ∈ N>0 , and m ∈ Nd such (m)r ,n that |m| = m. Put n1 = min{[ n−m = r ], 0} and n2 = n − m − n1 r. Recall CN m −n1 n−n2 n2 p ξ F CN +max{0,n2 −n} . If n − n2 ≥ |m| = m, the claim follows from ξ m wN ∈ F m CN and −n1 ≥ 0. If 0 ≤ n − n2 ≤ |m| = m, choosing a decomposition m = m0 + m00 m00 m0 (m0 , m00 ∈ Nd ) such that |m00 | = n−n2 , we obtain ξ n−n2 wN ∈ F n−n2 CN and p−n1 wN ∈ CN because |m0 | = m − (n − n2 ) = −n1 r. This implies the claim. Suppose n − n2 < 0. Then we have 1 ≤ n2 − n ≤ r − 1 and −n1 ≥ 1 by Lemma IV.2.3.5. By Lemma IV.2.3.6, we have (m) ,n m m [ξ r ](CN r wN ) = p−n1 ξ r+n−n2 F n2 CN +r · wN +r . Put l = r + n − n2 (≥ 0). Then m − l = (−n1 − 1)r ≥ 0. Choosing a decomposition m0 m = m0 + m00 (m0 , m00 ∈ Nd ) such that |m00 | = l, we obtain p−n1 −1 wN ∈ CN and m00

pξ l F n2 CN +r · wN +r ⊂ pF l CN +r F n2 CN +r ⊂ pF l+n2 CN +r ⊂ ξ r F n CN +r (m) ,n

m

by Lemma IV.2.1.6 (3) and Lemma IV.2.3.2 (2). Hence [ξ r ](CN r wN ) ⊂ [ξ r ](F n CN ), which implies the claim. Now, by taking the p-adic completion of hN |AN [W ]r : AN [W ]r → CN , we obtain a morphism g : A{W }r → C such that f = g ◦ u. It remains to prove the uniqueness. 0 Let g 0 = (gN ) : A{W }r → C be a morphism in A•r such that f = g 0 ◦ u. Then, letting 0 0 gN,Q denote the morphism AN {W }r,Q → CN,Q induced by gN , we have fN +1 (Ti ) = 0 0 0 0 ξgN +1,Q (Wi ) = [ξ]gN,Q (Wi ), which implies gN,Q (Wi ) = wi,N . Hence gN |AN [W ]r = 0 gN |AN [W ]r . By taking the p-adic completion, we obtain gN = gN .  Lemma IV.2.3.16. For N, l ∈ N>0 , we have the following equality, where p-adic completion.  ∧ M (|m|+l) r W m . (AN {W1 , . . . , Wd }r )(l)r =  AN m∈Nd

∧

denotes the

IV.2. HIGGS ENVELOPES

331

r Proof. The claim is trivial if r = ∞. Assume r ∈ N>0 . For M ∈ Ob Malg,• (m)

and m, l ∈ N, the submodule MN r of MN is the sum of finite number of submod(m) ,n ules MN r (0 ≤ n ≤ m) by Corollary IV.2.3.8 (2), and we have (M (m)r )(l)r = M (m+l)r by Proposition IV.2.3.9. Hence the claim follows fromPthe fact that an element of F n (AN {W1 , . . . , Wd }r ) is written uniquely in the form m∈Nd am W m where (|m|)r

am ∈ pl(|m|) F n AN

for some sequence l(m) ∈ N (m ∈ N) tending to ∞ as m → ∞. 

Proposition IV.2.3.17. Let f : Y 0 → Y be a morphism in the category C (Definition IV.2.2.1 (1)) such that f1 : Y10 → Y1 is the identity, Y1 is affine, fN : YN0 → YN (N ≥ 2) are smooth, and the morphisms YN0 → YN0 +1 ×YN +1 YN (N ≥ 2) induced by f are isomorphisms. We further assume that there exist ti = (ti,N ) ∈ limN Γ(YN0 , MYN0 ) and ←− si = (si,N ) ∈ limN Γ(YN , MYN ) (1 ≤ i ≤ d) such that {d log(ti,N )}1≤i≤d is a basis of ←− r Ω1Y 0 /YN for N ≥ 2 and si,1 = ti,1 . Let r ∈ N>0 ∪ {∞}, and let D and D0 be DHiggs (Y ) N r 0 0 and DHiggs (Y ), respectively, for which DN and DN (N ∈ N>0 ) are affine by Proposition IV.2.2.9 (2) and the remark after Corollary IV.2.2.5. Put AN = Γ(DN , ODN ) and A0N = r 0 0 0 0 ), which define objects A = (AN ) and A = (A , ODN Γ(DN N ) of A• (cf. Lemma IV.2.2.4). For i ∈ {1, 2, . . . , d} and N ∈ N, N ≥ 2, let ui,N denote the unique element of 1 + 0 ∗ 0 ), where pY 0 denotes , MDN (si,N )ui,N in Γ(DN F 1 A0N such that p∗Y 0 ,N (ti,N ) = p∗Y 0 ,N fN 0 0 the canonical morphism D → Y . Then we have a canonical isomorphism ∼ =

A{W1 , . . . , Wd }r − → A0

in A•r over A such that the image of Wi in (A0N )Q is ξ −1 (ui,N +1 − 1).

Proof. For any T ∈ Ob C r , the natural map HomC (T, D ×Y Y 0 ) → HomC (T, Y 0 ) is bijective because HomC (T, D) → HomC (T, Y ) is bijective. Hence we may replace Y 0 → Y with D ×Y Y 0 → D and assume D = Y . For an integral monoid P , let Zp {P } denote the p-adic completion of Zp [P ], and let Spf(Zp ){P } denote the p-adic formal scheme Spf(Zp {P }) endowed with the log structure associated to the inclusion map P ,→ Zp {P }. Let 1i (1 ≤ i ≤ d) denote the element of Nd whose i-th component is 1 and other components are 0. Then, for N ∈ N such that N ≥ 2, we have a commutative diagram of p-adic fine log formal schemes YN0 fN

 YN

/ Spf(Zp ){Nd ⊕ Nd }  / Spf(Zp ){Nd },

where the lower (resp. upper) horizontal morphism is defined by Nd → Γ(YN , MYN ); 1i 7→ ∗ (si,N ), ti,N ) and the right vertical si,N (resp. Nd ⊕ Nd → Γ(YN0 , MYN0 ); (1i , 0), (0, 1i ) 7→ fN morphism is defined by Nd → Nd ⊕ Nd ; a 7→ (a, 0). Let YN00 be the fiber product of Spf(Zp ){Nd ⊕ Nd } → Spf(Zp ){Nd } ← YN for N ∈ N, N ≥ 2. Then the above diagram induces an étale morphism YN0 → YN00 . Hence, by Lemma IV.2.2.15, we may replace YN0 with YN00 for N ≥ 2 and assume that the above diagram is Cartesian. Let Q be the inverse image of Nd by Zd ⊕ Zd → Zd ; (a, b) 7→ a + b. For N ∈ N, N ≥ 2, let YeN0 be the fiber product of YN0 → Spf(Zp ){Nd ⊕ Nd } ← Spf(Zp ){Q}. Then the natural morphism YeN0 → YN0 is étale, and the morphism Q → Γ(Y10 , MY10 ); (−1i , 1i ), (0, 1i ) 7→ 1, ti,1 defines a factorization Y10 → Ye200 → Y20 because ti,1 = si,1 on Y10 = Y1 . Putting Ye10 = Y10 , we obtain an object Ye 0 = (YeN0 ) of C and the étale morphism Ye 0 → Y 0 induces an isomorphism r DHiggs (Ye 0 ) ∼ = D0 by Lemma IV.2.2.15. Since Ye 0 → Y is strict, the closed immersion

332

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Ye10 → Ye20 is exact. Let vi,N denote the image of (−1i , 1i ) ∈ Q× in Γ(YeN0 , OY× e 0 ). Then the N

r 0 pull-back of vi,N on DHiggs (Ye 0 )N ∼ is ui,N . On the other hand, for N ≥ 2, YeN0 is = DN affine and we have an isomorphism

AN [V1 , V1−1 , . . . , Vd , Vd−1 ]∧ ∼ = Γ(YeN0 , OYe 0 ) N

sending Vi to vi,N , where ∧ denotes the p-adic completion. Replacing YeN0 (N ≥ 2) with the open p-adic fine log formal scheme of YeN0 defined by the open formal subscheme Spf(AN {V1 − 1, . . . , Vd − 1}) of the underlying p-adic formal scheme of YeN0 , we obtain an ∼ = r r object Ye 00 of C with DHiggs (Ye 00 ) − → DHiggs (Ye 0 ). Now the claim follows from the proof of Proposition IV.2.2.14 and Proposition IV.2.3.15.  IV.2.4. Differential modules. For r ∈ N>0 ∪{∞}, B ∈ Ob C r (Definition IV.2.2.1 (2)), a smooth Cartesian morphism Y → B in C (Definition IV.2.2.2 (1), (2)), and an immersion i : X → Y1 of p-adic fine log formal schemes, we construct a canonical derivation of the structure sheaf of the Higgs envelope of level r of (i : X → Y ) (Definitions IV.2.2.1 (3), IV.2.2.10) with values in the differential module of Y /B (see (IV.2.4.8)), and study its properties (Lemma IV.2.4.10, Proposition IV.2.4.13). Lemma IV.2.4.1. Let r ∈ N>0 ∪ {∞} and let T = (TN ) be an object of C r . (1) For N, M ∈ N>0 , the morphism ξ N : OTN +M → OTN +M factors as OTN +M → [ξ N ]

OTM −−−→ OTN +M with [ξ N ] injective. (2) Assume r ∈ N>0 . For n, l ∈ N and the smallest integer m such that nr−1 ≤ m, we have pm F l+n OTN ⊂ ξ n F l OTN . In particular, we have F n OTN ,Q = ξ n OTN ,Q . Proof. (1) By the same argument as the proof of Lemma IV.2.1.6 (1), we see that the inverse image of F n OTN under ξ : OTN → OTN is F n−1 OTN for positive integers n ≤ N. (2) As in the proof of Lemma IV.2.1.6 (2), we have pF n+s OTN ⊂ ξ s OTN ∩F n+s OTN = s n ξ F OTN for s ∈ N ∩ [0, r] and n, N ∈ N such that n + s < N .  Let r ∈ N>0 ∪ {∞} and let T be an object of C r . For n ∈ N>0 and a submodule P of ξ n OTN +n ,Q = F n OTN +n ,Q , we define ξ −n P to be the inverse image of P under the ∼ =

isomorphism [ξ n ]Q : OTN ,Q − → ξ n OTN +n ,Q . Let m ∈ N and N ∈ N>0 . If r ∈ N>0 , we (m)r define the subsheaf OTN of OTN ,Q to be X p−n1 ξ n−n2 F n2 OTN +max{0,n2 −n} , n∈N

where n1 =

min{[ n−m r ], 0}

(m)r

and n2 = n − m − n1 r. We have OTN

⊃ p−m1 OTN , where (m)r

−m = m1 r + m2 (m1 ∈ Z, m2 ∈ Z ∩ [0, r − 1]). If r = ∞, we define OTN

= OTN .

Lemma IV.2.4.2. Let m ∈ N and N ∈ N>0 . (m) (1) The sheaf OTN r is an ideal of OTN . (2) Assume that T1 is affine, and let A = (AN ) be the object of A•r defined by (m) AN = Γ(TN , OTN ) (cf. Lemma IV.2.2.4 (1)). Then the ideal AN r of AN coincides with (m) Γ(TN , OTN r ). (m+1)r

(3) We have pOTN

(1)

(m)r

⊂ OTN r · OTN

(m+1)r

⊂ OTN

.

Sublemma IV.2.4.3. Assume that r ∈ N>0 and T1 is affine, and let A = (AN ) be the object of A•r defined by AN = Γ(TN , OTN ).

IV.2. HIGGS ENVELOPES

333

(1) For n ∈ N and m, N ∈ N>0 , the OTN /pm OTN -module F n OTN /pm F n OTN is quasi-coherent and the natural morphism F n AN /pm F n AN → Γ(TN , F n OTN /pm F n OTN ) is an isomorphism. (2) For m, N ∈ N>0 and s ∈ N ∩ [0, r], the OTN /pm OTN -module pξ −s F n+s OTN +s /pm F n+s OTN

is quasi-coherent and the natural morphism pξ −s F n+s AN +s /pm F n+s AN → Γ(TN , pξ −s F n+s OTN +s /pm F n+s OTN ) is an isomorphism. Proof. (1) Since OTN and OTn are p-torsion free, we have an exact sequence 0 → F n OTN /pm F n OTN → OTN /pm OTN → OTn /pm OTn → 0.

We also have an exact sequence with OTN replaced by AN . Hence the claim follows from the fact that OTN /pm OTN and OTn /pm OTn are quasi-coherent OTN /pm OTN -modules. (2) Note that we have pF n+s OTN ⊂ pξ −s F n+s OTN +s ⊂ F n OTN by Lemma IV.2.4.1 (2), and similar inclusions for A. We have a commutative diagram F n OTN pm F n OTN

O

pξ −s F n+s OTN +s pm F n OTN

∼

?

/

ξ s F n OTN +s pm ξ s F n OTN +s

/

pF n+s OTN +s pm ξ s F n OTN +s

∼ [ξ s ]

s

[ξ ]

O

?

o

p

F n+s OTN +s pm F n+s OTN +s

and a similar diagram for A. We obtain the claim by comparing the Γ(TN , −) of the diagram for OT• with the diagram for A and noting that F n OTN /pm F N OTN and F n+s OTN +s /pm F n+s OTN +s are quasi-coherent by (1).  Proof of Lemma IV.2.4.2. The claim is obvious if r = ∞. We assume r ∈ N>0 . (1) The same as the proof of Lemma IV.2.3.7 (1) using Lemma IV.2.4.1 (2) and ξ m F n OTN ⊂ F n+m OTN which follows from Lemma IV.2.4.1 (1). (m)r (m)r (2) Let l be a positive integer such that −l ≤ [ −m r ], and let OTN ,l (resp. AN,l ) be the (m)r

image of OTN

(m)r

pl ATN ⊂ ATN

(m)r

(resp. AN

(m)

) in OTN /pl OTN (resp. AN /pl AN ). We have pl OTN ⊂ OTN r ,

and the morphism pl AN → Γ(TN , pl OTN ) is an isomorphism. Hence (m)

(m)

it suffices to prove that AN,l r → Γ(TN , OTN ,lr ) is an isomorphism. For n ∈ N, put (m)

r n1 = min{[ n−m r ], 0} and n2 = n − m − n1 r. Then OTN ,l is the sum of the images of l morphisms of OTN /p OTN -modules

p−n1 −1

pξ n−n2 F n2 OTN +n−n2 /pl F n OTN −−−−−→ OTN /pl OTN p

−n1 n−n2

ξ

F n2 OTN /pl F n2 OTN −−−−−−−→ OTN /pl OTN

(if n − n2 < 0), (if n − n2 ≥ 0).

Hence the claim follows from Sublemma IV.2.4.3. (3) By applying (2) to a strict étale Cartesian morphism U → T in C such that U1 is affine, we are reduced to Lemma IV.2.3.13.  Let r ∈ N>0 ∪ {∞} and let B ∈ Ob C r . Let X be a p-adic fine log formal scheme over B1 , let Y → B be a smooth Cartesian morphism in C (Definition IV.2.2.2), and let X → Y1 be an immersion over B1 . For ν ∈ N, let Y (ν) denote the fiber product over B of ν + 1 copies of Y (cf. Lemma IV.2.2.8). The immersion X → Y (ν)2 and the exact closed immersion Y (ν)N → Y (ν)N +1 (N ≥ 2) define an object of C . Let D(ν) denote

334

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

r its image under the functor DHiggs in Proposition IV.2.2.9 (1). We write D for D(0). Let ∆D : D → D(1) be the exact closed immersion induced by the diagonal morphisms YN → Y (1)N (N ≥ 2), and let pi (i = 1, 2) denote the morphism D(1) → D induced by the i-th projection Y (1)N → YN (N ≥ 2). Let pij ((i, j) = (1, 2), (2, 3), (1, 3)) denote the morphism D(2) → D(1) induced by the morphism defined by i-th and j-th projections Y (2)N → Y (1)N (N ≥ 2), and let qi (i = 1, 2, 3) denote the morphism D(2) → D induced by the i-th projection Y (2)N → YN (N ≥ 2).

Remark IV.2.4.4. Let Y 0 → Y be a strict étale Cartesian morphism, let X 0 be X×Y1 Y10 , and define Y 0 (ν) and D0 (ν) in the same way as Y (ν) and D(ν) using Y 0 and X 0 → Y10 . Then, by applying Lemma IV.2.2.15 to Y 0 (ν) → Y 0 ×B Y (ν − 1) and Proposition IV.2.2.9 (3) to Y 0 ×B Y (ν − 1) → Y (ν) and X 0 → X, we see that the natural morphism D0 (ν)1 → D(ν)1 ×X X 0 is an isomorphism and the morphism D0 (ν) → D(ν) is strict étale and Cartesian. This will allow us to discuss properties about D(ν) étale locally on Y1 . Let D(1)1N denote the first infinitesimal neighborhood of ∆D,N : DN → D(1)N (cf. the beginning of IV.2.2). Since the reduction mod p of DN → D(1)1N is a nilpotent immersion, we may regard a sheaf on D(1)1N,´et as a sheaf on D1,´et . We define the sheaf Ω1DN /BN on D1,´et by Ω1DN /BN := Ker(OD(1)1N → ODN )/(p-tor).

The two ODN -module structures on Ω1DN /BN defined by p∗1 and p∗2 coincide. Let Y (1)1N be the first infinitesimal neighborhood of the diagonal immersion YN → Y (1)N . We have a canonical isomorphism ∼ Ker(OY (1)1 → OY ). Ω1 = YN /BN

D(1)1N

N

N

(1)1N

defined by the natural morphisms DN → YN and →Y Hence the morphism D(1)N → Y (1)N induces an ODN -linear morphism ODN ⊗OYN Ω1YN /BN −→ Ω1DN /BN .

(IV.2.4.5)

Proposition IV.2.4.6. Let the notation and assumption be as above. Let N be an integer ≥ 2. (1) The morphism (IV.2.4.5) induces an injective morphism ODN −1 ⊗OYN −1 Ω1YN −1 /BN −1 −→ Ω1DN /BN . Furthermore, there exists a unique isomorphism ∼ =

ξOSN ⊗OSN −1 Ω1DN −1 /BN −1 ⊗ Q −→ ODN −1 ⊗OYN −1 Ω1YN −1 /BN −1 ⊗ Q such that the following diagram is commutative. / Ω1D

ξOSN ⊗OSN Ω1DN /BN ⊗ Q

ξOSN ⊗OSN −1

 Ω1DN −1 /BN −1 ⊗ Q

N /BN

∼ =

O

⊗Q

? / ODN −1 ⊗OY Ω1YN −1 /BN −1 ⊗ Q. N −1

(2) The isomorphism in (1) induces an isomorphism (1) ξOSN ⊗OSN −1 Ω1DN −1 /BN −1 ∼ = ODNr−1 ⊗OYN −1 Ω1YN −1 /BN −1 .

Proof. By Remark IV.2.4.4, the question is strict étale local on Y1 and we may assume that Y1 is affine, X → Y1 is a closed immersion, and there exist (ti,N )N ∈ limN Γ(YN , MYN ) (1 ≤ i ≤ d) such that d log ti,N (1 ≤ i ≤ d) is a basis of Ω1YN /BN for ←−

IV.2. HIGGS ENVELOPES

335

every N . By Proposition IV.2.2.9 (2), DN and DN (1) are affine. Let A and A(1) be the objects of A•r defined by their coordinate rings (cf. Lemma IV.2.2.4). The p-adic fine log formal scheme D(1)1N is also affine. Let A(1)1N denote Γ(D(1)1N , OD(1)1N ). Let pY (1) = (pY (1),N )N denote the natural morphism D(1) → Y (1). Since p∗Y (1),1 p∗2,1 (ti,1 ) = p∗Y (1),1 p∗1,1 (ti,1 ) on D(1)1 and D(1)1 → D(1)N is an exact closed immersion, there exists a unique ui,N ∈ 1 + F 1 A(1)N such that p∗Y (1),N p∗2,N (ti,N ) = ui,N · p∗Y (1),N p∗1,N (ti,N ). Put wi,N := ξ −1 (ui,N +1 − 1) ∈ A(1)N,Q . By Proposition IV.2.3.17, we have an isomorphism ∼ = A{W1 , . . . , Wd }r → A(1) over A with respect to p∗1 : A → A(1) such that the image of Wi in A(1)N,Q is wi,N . By Lemma IV.2.3.13, we have (|m|)

(|m|)

m

m

p(⊕m∈Nd ,|m|=2 AN r wN )∧ ⊂ Ker(AN (1) → A(1)1N ) ⊂ (⊕m∈Nd ,|m|=2 AN r wN )∧ , Q m mi where wN = 1≤i≤d wi,N and ∧ denotes the p-adic completion. Hence we have an isomorphism ∼ =

(1)

Ker(A(1)1N → AN )/(p-tor) −→ ⊕1≤i≤d AN r wi,N ,

where wi,N denotes the image of wi,N in A(1)1N . Applying this to all strict étale Cartesian morphisms Y 0 → Y and X ×Y1 Y10 such that Y10 is affine, we obtain an isomorphism ∼ =

(1)

Ω1DN /BN −→ ⊕1≤i≤d OYN r wi,N (cf. Lemma IV.2.4.2 (2)). Now the claim follows from the fact that the image of d log ti,N ∈ Ω1YN /BN in Ω1DN /BN is ξ · wi,N .  ∼ =

For N ∈ N>0 , we have an isomorphism [ξ] : OSN − → ξOSN +1 . Hence ξOSN +1 is a free OSN -module of rank 1. For n ∈ N>0 and an OYN -module M, we define ξ −n M to be (ξOSN +1 )⊗(−n) ⊗OSN M. For x ∈ M, we define ξ −n x to be ξ ⊗(−n) ⊗ x. By Proposition IV.2.4.6 (2), we have a canonical isomorphism (IV.2.4.7)

(1) Ω1DN /BN ∼ = ξ −1 ODNr ⊗OYN Ω1YN /BN .

Hence the OBN -linear derivation ODN → Ω1DN /BN ; x 7→ p∗2,N (x) − p∗1,N (x) induces an OBN -linear derivation (IV.2.4.8)

(1)

θ : ODN −→ ξ −1 ODNr ⊗OYN Ω1YN /BN .

Since the derivation d : OYN → Ω1YN /BN is defined by d(x) = p∗2 (x) − p∗1 (x), we see that the following diagram is commutative. (IV.2.4.9)

ODN O

p−1 Y,N (OYN )

θ

/ ξ −1 O(1)r ⊗O Ω1 Y DN O N YN /BN d

1 / p−1 Y,N (ΩYN /BN ),

where pY = (pY,N )N denotes the natural morphism D → Y and the right vertical morphism is defined by ω 7→ ξ −1 (ξ) ⊗ ω. This implies that one can define a morphism (1)

θq : ξ −q ODN ⊗OYN ΩqYN /BN −→ ξ −q−1 ODNr ⊗OYN Ωq+1 YN /BN by θq (ξ −q a ⊗ ω) = ξ −q θ(a) ∧ ω + ξ −q a ⊗ dq ω.

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Lemma IV.2.4.10. (1) For m, q ∈ N, we have (m)

(m+1)r

θq (ξ −q ODN r ⊗OYN ΩqYN /BN ) ⊂ ξ −q−1 ODN

⊗OYN Ωq+1 YN /BN .

(2) For q ∈ N, we have θq+1 ◦ θq = 0. We will give a proof of Lemma IV.2.4.10 after the proof of Proposition IV.2.4.13. Now assume that Y1 is affine, X → Y1 is a closed immersion, and there exist ti = (ti,N ) ∈ limN Γ(YN , MYN ) (1 ≤ i ≤ d) such that {d log(ti,N )}1≤i≤d is a basis of ←− Ω1YN /BN for every N ∈ N>0 . Let AD (ν) = (AD (ν)N ) be the object of A•r defined by AD (ν)N = Γ(D(ν)N , OD(ν)N ) (cf. Lemma IV.2.2.4). We write AD for AD (0). Since the images of p∗1,N p∗Y,N (ti,N ), p∗2,N p∗Y,N (ti,N ) ∈ Γ(D(1)N , MD(1)N ) in Γ(D(1)1 , MD(1)1 ) coincide and D(1)1 → D(1)N is an exact closed immersion, there exists a unique ui,N ∈ 1+F 1 AD (1)N such that p∗1,N p∗Y,N (ti,N )ui,N = p∗2,N p∗Y,N (ti,N ) in Γ(D(1)N , MD(1)N ). Put wi,N = ξ −1 (ui,N +1 − 1) ∈ AD (1)N,Q . By Proposition IV.2.3.17, either of the two homomorphisms p∗1,N , p∗2,N : AD,N → AD (1)N induces an isomorphism ∼ =

AD,N {W1 , . . . , Wd }r −→ AD (1)N

(IV.2.4.11)

sending Wi to wi,N (after ⊗Q), and any of the three homomorphisms qi∗ : AD,N → AD (2)N (i = 1, 2, 3) induces an isomorphism ∼ =

AD,N {W1 , . . . , Wd , W10 , . . . , Wd0 }r −→ AD (2)N

(IV.2.4.12)

sending Wi and Wi0 to p∗12 (wi,N ) and p∗23 (wi,N ). the endomorphisms θi : AD,N → AD,N (1 ≤ i ≤ d) by the formula P We define P θ(x) = −1 d ξ θ (x) ⊗ d log(t ). For m = (m , . . . , m ) ∈ N , we set |m| = i,N 1 d 1≤i≤d Q i 1≤i≤d mi and m! = 1≤i≤d (mi )!. Proposition IV.2.4.13. The endomorphisms θi of AD,N (1 ≤ i ≤ d) have the following properties. (1) θi ◦ θj Q = θj ◦ θiQ(i 6= j). Put θm = 1≤i≤d 0≤j≤mi −1 (θi − jξ) for m = (m1 , . . . , md ) ∈ Nd . (m)

(|m|+m)

r 1 (2) For x ∈ AD,Nr and m ∈ Nd , m! θm (x) is contained in AD,N , and there exists a sequence of non-negative integers l(n) (n ∈ N) such that limn→∞ l(n) = ∞ and 1 l(|m|) (|m|+m)r AD,N . m! θm (x) ∈ p

(m)

(m)r

(3) For x ∈ AD,Nr , we have the following equality in AD (1)N p∗2,N (x) =

.

X 1 Y mi p∗1,N (θm (x)) wi,N . m! d

m∈N

1≤i≤d

Proof. We abbreviate pi,N , pi,j,N and qi,N to pi , pij and qi , respectively. We define (|m|) the homomorphisms Θm : AD,N → AD,N r (m ∈ Nd ) by the equality X m p∗2 (x) = p∗1 (Θm (x))wN (x ∈ AD,N ) m∈Nd

m

mi 1≤i≤d wi,N for m = (l)r (|m|+l) Θm (AD,N ) ⊂ AD,N r by

in AD (1)N , where wN = (l)

Q

(l)

r (m1 , . . . , md ) ∈ Nd . Since p∗2 (AD,N )⊂

AD (1)N r , we have Lemma IV.2.3.16. We have θi = Θ1i (1 ≤ i ≤ d) by definition. For x ∈ AD,N , we have the following two expansions of q3∗ (x)

IV.2. HIGGS ENVELOPES

337

in AD (2)N . q3∗ (x) =

l

X

q2∗ (Θl (x))p∗23 (wN ) =

l∈Nd

q3∗ (x) =

l

X l,n∈Nd

X

n

q1∗ (Θn ◦ Θl )p∗23 (wN )p∗12 (wN ),

m

q1∗ (Θm (x))p∗13 (wN ).

m∈Nd

Since p∗13 (ui,N +1 ) = p∗23 (ui,N +1 )p∗12 (ui,N +1 ), we have p∗13 (wi,N ) = ξp∗23 (wi,N )p∗12 (wi,N ) + l n p∗23 (wi,N ) + p∗12 (wi,N ). Comparing the coefficients of p23 (wN )p12 (wN ) for n, l ∈ Nd , we obtain

Θn ◦ Θl =

X n=m1 +m3 , l=m2 +m3

(m1 + m2 + m3 )! |m3 | ξ Θm1 +m2 +m3 . m1 !m2 !m3 !

Hence we have θi ◦ θj = Θ1i +1j = θj ◦ θi (i 6= j) and θi ◦ Θm = (mi + 1)Θm+1i + mi ξΘm ⇐⇒ Θm+1i = for m ∈ Nd . By induction on |m|, we see Θm = (3).

1 m! θm

1 (θi − mi ξ) ◦ Θm mi + 1

for m ∈ Nd , which implies (2) and 

Proof of Lemma IV.2.4.10. By Remark IV.2.4.4, the question is étale local on (m) (m+1) Y1 . Hence we see θ(ODN r ) ⊂ ξ −1 ODN r ⊗OYN Ω1YN /BN by Lemma IV.2.4.2 (2) and (m)

(1)

(m)

Proposition IV.2.4.13 (2). Therefore the claim (1) follows from ξODN r ⊂ ODNr · ODN r ⊂ (m+1) ODN r

(Lemma IV.2.4.2 (3)). The claim (2) follows from Proposition IV.2.4.13 (1). 

IV.2.5. Torsors of deformations and Higgs envelopes. Let B be an object of C ∞ , let Y be an object of C ∞ over B, let X → B be a smooth Cartesian morphism in C , and let z : Y1 → X1 be a morphism of p-adic fine log formal schemes over B1 . Assume that Y1 is affine. Let TZar be the sheaf of OY1 -modules HomOY1 (z ∗ (ξ −1 Ω1X1 /B1 ), OY1 ) on the Zariski site (Y1 )Zar . Then, following II.10.3, one can construct a TZar -torsor LZar from Y → B ← X and z. In this subsection, we prove that the principal homogeneous space L associated to LZar (cf. Proposition II.4.10) is canonically isomorphic to the first component D1 of the Higgs envelope D of level ∞ of Y1 ,→ X ×B Y (Theorem IV.2.5.2). We also give a description (Proposition IV.2.5.16) of the first components of the Higgs envelopes of finite level in terms of the symmetric tensor product of the module of affine functions on LZar (cf. II.4.7), which will allow us, in IV.5.4, to compare the period ring of finite level r defined in IV.5.2 with the p-adic completion of the Higgs-Tate algebra of depth 1/r defined in II.12.1. We prove the compatibility with the canonical derivations (Proposition IV.2.5.19) and functoriality (Proposition IV.2.5.25) for the isomorphism L∼ = D1 . For a p-adic fine log formal scheme Z (resp. a morphism f of p-adic fine log formal schemes), let n Z (resp. n f ) denote the reduction mod pn of Z (resp. f ) in this subsection. As in II.10.3, we consider the following torsor. Let U be an open p-adic fine log formal subscheme of Y1 , which means an open formal subscheme of the formal scheme underlying e be the open p-adic Y1 endowed with the restriction of the log structure of Y1 , and let U fine log formal subscheme of Y2 defined by U . Let LZar (U ) denote the set of morphisms e → X2 over B2 whose composition with the natural closed immersion U ,→ U e coincides U z with the composition U ⊂ Y1 − → X1 ,→ X2 . This construction is obviously functorial on U , and the presheaf of sets LZar : (Y1 )Zar → (Sets) on the Zariski site (Y1 )Zar of Y1 admits a natural structure of a torsor under TZar := HomOY1 (z ∗ (ξ −1 Ω1X1 /B1 ), OY1 ), where z ∗ (−) = OY1 ⊗z−1 (OX1 ) z −1 (−) by [50] Proposition (3.9) (cf. II.5.23). Note that

338

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS ∼ =

→ Ker(OY2 → OY1 ) by the multiplication by ξ on OY1 induces an isomorphism OY1 − Definition IV.2.2.1 (2) (iii), (iv), OY1 and OY2 are p-torsion free by Definition IV.2.2.1 (2) (ii), and the natural homomorphism z ∗ (ξ −1 Ω1X1 /B1 ) → limn (n z ∗ (ξ −1 Ω1n X1 /n B1 )) is ←− an isomorphism. For each n ∈ N>0 , we put n TZar := HomOn Y1 (n z ∗ (ξ −1 Ω1n X1 /n B1 ), On Y1 ) and define the n TZar -torsor n LZar on (n Y1 )Zar = (Y1 )Zar as follows. For an open fine log subscheme e of n Y2 defined by U , we define n LZar (U ) U of n Y1 and the open fine log subscheme U e → n X2 over n B2 whose composition with the natural to be the set of morphisms U nz e closed immersion U ,→ U coincides with the composition of U ⊂ n Y1 → n X1 → n X2 . For n ∈ N>0 , we have a natural morphism n+1 LZar → n LZar equivariant with respect to the natural homomorphism n+1 TZar → n TZar . Let n T be the vector bundle Spec(Sym•On Y1 (n z ∗ (ξ −1 Ω1n X1 /n B1 ))) associated to n TZar endowed with the inverse image of the log structure of n Y1 . Then, by applying the construction in II.4.9, we obtain an n T -principal homogeneous space n L in the category of strict fine log schemes over n Y1 locally trivial with respect to the Zariski topology on n Y1 , and a canonical isomorphism n LZar

∼ =

→ ϕn Y1 (n L)

of n TZar -torsors, where ϕn Y1 is the functor defined as (II.4.3.1). By II.4.13, we have a natural isomorphism n+1 T ×n+1 Y1 n Y1 ∼ = n T of group fine log schemes and an isomorphism of n T -principal homogeneous spaces n+1 L ×n+1 Y1 n Y1 ∼ = n L. By taking the inverse limit with respect to n, we obtain a group p-adic fine log formal scheme T over Y1 and a T -principal homogeneous space L in the category of strict p-adic fine log formal schemes over Y1 locally trivial with respect to the Zariski topology on Y1 . Let PHS(T /Y1 ) denote the category of T -principal homogeneous spaces in the category of strict p-adic fine log formal schemes over Y1 locally trivial with respect to the Zariski topology on Y1 . Let Tors(TZar , (Y1 )Zar ) denote the category of TZar -torsors on (Y1 )Zar . Similarly as (II.4.3.1), we have a canonical functor ϕY1 : PHS(T /Y1 ) −→ Tors(TZar , (Y1 )Zar )

defined by Z 7→ HomY1 (−, Z). By the above construction, we have a canonical isomorphism ∼ = LZar −→ ϕY1 (L). Note that for p-adic fine log formal schemes Z and Z 0 , the natural map Hom(Z, Z 0 ) → limn Hom(n Z, n Z 0 ) is bijective by the definition of fine log structures on p-adic formal ←− schemes in the beginning of IV.2.2 and [1] Proposition 2.2.2. By applying Proposition II.4.4 to n T and n TZar on n Y1 , and then II.4.13 to the morphisms n Y1 → n+1 Y1 , we obtain the following proposition. Proposition IV.2.5.1. The functor ϕY1 above is fully faithful. The morphism z : Y1 → X1 and the identity morphism of Y1 induce an immersion ∞ ze : Y1 → Y1 ×B1 X1 . We define the object D ∈ Ob C ∞ to be DHiggs (e z : Y1 ,→ X ×B Y ) (cf. Definition IV.2.2.1 (3) and Proposition IV.2.2.9 (1)). The purpose of this subsection is to prove the following isomorphism and study its properties (cf. Propositions IV.2.5.16, IV.2.5.19, and IV.2.5.25). Theorem IV.2.5.2. There exists a canonical isomorphism of p-adic fine log formal ∼ = schemes D1 − → L over Y1 . We will regard D1 as a T -principal homogeneous space by the action of T on D1 induced by that on L via the canonical isomorphism in Theorem IV.2.5.2.

IV.2. HIGGS ENVELOPES

339

Lemma IV.2.5.3. The canonical morphism D → Y in C ∞ is strict smooth Cartesian and D1 is affine. Proof. We may replace X with X ×B Y and z with ze : Y1 → X1 ×B1 Y1 , and assume that Y = B. By Proposition IV.2.2.9 (3), the claim is strict étale local on Y1 . Hence the claim follows from Proposition IV.2.2.9 (2) and Proposition IV.2.3.17.  Let (AffSmStr/Y1 )Zar denote the category of strict smooth affine p-adic fine log formal schemes over Y1 endowed with the Zariski topology, and let (Y1 )ZarAff denote the category of open affine p-adic fine log formal schemes of Y1 endowed with the Zariski topology. ∼ = The restriction functor induces an equivalence of categories of topos (Y1 )∼ → (Y1 )∼ Zar − ZarAff . By Lemma IV.2.5.3, D1 is an object of (AffSmStr/Y1 )Zar . Let T denote the sheaf on u (AffSmStr/Y1 )Zar defined by T (U → Y1 ) = Γ(U, HomOU (u∗ z ∗ (ξ −1 Ω1X1 /B1 ), OU )). We construct a natural extension of the TZar -torsor LZar to a T -torsor L on (AffSmStr/Y1 )Zar and prove that L is canonically isomorphic to the sheaf represented by D1 . First let us define the torsor L. For (u : U → Y1 ) ∈ Ob (AffSmStr/Y1 ), let Lift(U/Y2 ) denote the category defined as follows. An object is a smooth strict morphism of p-adic ∼ = e e → Y2 with an isomorphism U − fine log formal schemes u e: U → U ×Y2 Y1 over Y1 (i.e., a smooth strict lifting of U → Y1 ). A morphism is a morphism of p-adic fine log formal schemes over Y2 compatible with the isomorphisms from U in the obvious sense. e of Lift(U/Y2 ), we write F 1 O e for the kernel of O e → O U . For an object U n nU nU By Definition IV.2.2.1 (2), the multiplication by ξ on On Y2 induces an isomorphism ∼ = e be an object of Lift(U/Y2 ). Then, since the → F 1 On Y2 for each n ∈ N>0 . Let U On Y1 − morphism of schemes underling n U → n Y1 is flat, we see that the multiplication ξ on On U2 induces an isomorphism (IV.2.5.4)

∼ =

On U −→ F 1 On Ue .

Lemma IV.2.5.5. The category Lift(U/Y2 ) is nonempty. Any two objects of Lift(U/Y2 ) are isomorphic to each other. Furthermore every morphism in Lift(U/Y2 ) is an isomorphism. Proof. The first two claims follow from [50] Proposition (3.14). By (IV.2.5.4), a ∼ = e0 → U e in Lift(U/Y2 ) induces an isomorphism v ∗ : F 1 O e → F 1 On Ue 0 for morphism v : U nU each n ∈ N>0 . This implies that n v (n ∈ N>0 ) and hence v are isomorphisms.  e → Y2 of Lift(U/Y2 ), we define LU (U e ) to be the set of morphisms For each object u e: U e e U → X2 over B2 such that the composition with U ,→ U coincides with the composition u z of U → Y1 → X1 ,→ X2 . By Sublemma IV.3.4.2 (2) and Corollary IV.2.2.5, the set e ) is nonempty. A morphism v : U e0 → U e in Lift(U/Y2 ), which is an isomorphism by LU (U Lemma IV.2.5.5, induces a bijection e ) → LU (U e 0 ); g 7→ g ◦ v. v ∗ : LU (U We have id∗ = id and (w ◦ v)∗ = v ∗ ◦ w∗ for two composable morphisms v and w in Lift(U/Y2 ). Thus we obtain a functor LU : Lift(U/Y2 ) → (Sets). e0 → U e be two morphisms in Lift(U/Y2 ) with the same Lemma IV.2.5.6. Let v, v 0 : U ∗ e ) → LU (U e 0 ). source and target. Then we have v = v 0∗ : LU (U Proof. By (IV.2.5.4), the two homomorphisms F 1 On Ue → F 1 On Ue 0 induced by v and v coincide. On the other hand, for an element ze of LY (Y2 ), which exists by Sublemma 0

340

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

e ) under the two maps in the lemma coincide. IV.3.4.2 (2), the image of ze ◦ u e ∈ LU (U Hence the claim follows from [50] Proposition (3.9) (cf. II.5.23).  We define the set L(U ) by L(U ) := Γ(Lift(U/Y2 ), LU ). e ) is bijective for By Lemmas IV.2.5.5 and IV.2.5.6, the projection map L(U ) → LU (U e of Lift(U/Y2 ). By [50] Proposition (3.9) (cf. II.5.23) and (IV.2.5.4), the set any object U e e of Lift(U/Y2 ) is naturally regarded as a Γ(U, T )-torsor. For a morLU (U ) for an object U ∼ = 0 e →U e in Lift(U/Y2 ), the bijection v ∗ : LU (U e) − e 0 ) is an isomorphism phism v : U → LU (U of Γ(U, T )-torsors. Hence L(U ) has a natural structure of a Γ(U, T )-torsor. The Γ(U, T )-torsor L(U ) is functorial on U as follows. Let v : U 0 → U be a morphism e (resp. U e 0 ) be an object of Lift(U/Y2 ) (resp. Lift(U 0 /Y2 )). in (AffSmStr/Y1 )Zar , and let U e → Y2 , there exists a Y2 -morphism ve : U e0 → U e Then, by the smoothness of the morphism U 0 ∗ 0 e e which lifts v : U → U . By composing with ve, we obtain a map ve : LU (U ) → LU (U ); g 7→ g ◦ ve. e ) → L U (U e 0) Lemma IV.2.5.7. For two liftings ve and ve0 of v, the maps ve∗ , ve0∗ : LU (U coincide. Proof. By (IV.2.5.4), we have the following commutative diagrams. On U

∼ =

/ F 1O e nU v e∗

v∗



On U 0

v e0∗

  / F 1O e0 . nU

∼ =

Hence the right two vertical homomorphisms coincide and the same argument as the proof of Lemma IV.2.5.6 shows the claim.  By Lemmas IV.2.5.5, IV.2.5.6, and IV.2.5.7, we see that the composition of ∼ ∼ v e = = e ) −→ e 0 ) ←− L(U ) −→ LU (U LU (U L(U 0 ) ∗

e, U e 0 , and ve. We denote the composition by L(v). By does not depend on the choice of U the construction of the bijection in [50] Proposition (3.9), we see that L(v) is compatible with the torsor structures via v ∗ : Γ(U, T ) → Γ(U 0 , T ). Thus we obtain a T -torsor L on (AffSmStr/Y1 )Zar . Let D be an object of C ∞ defined before Theorem IV.2.5.2, and let D denote the sheaf on (AffSmStr/Y1 )Zar represented by the object D1 of (AffSmStr/Y1 )Zar . Theorem IV.2.5.8. There exists a canonical isomorphism of sheaves L ∼ = D on the site (AffSmStr/Y1 )Zar . Theorem IV.2.5.8 and Proposition IV.2.5.1 immediately imply Theorem IV.2.5.2 as follows. Proof of Theorem IV.2.5.2. By Theorem IV.2.5.8, D1 has naturally a structure of an object of PHS(T /Y1 ), and there exist canonical isomorphisms ϕY1 (L)|(Y1 )ZarAff ∼ = LZar |(Y1 )ZarAff ∼ = ϕY1 (D1 )|(Y1 )ZarAff of TZar |(Y1 )ZarAff -torsors. Since the restriction functor ∼ (Y1 )∼ Zar → (Y1 )ZarAff is an equivalence of categories, Proposition IV.2.5.1 implies that there exists a unique isomorphism L ∼ = D1 in PHS(T /Y1 ) inducing the composition of the above isomorphisms. 

IV.2. HIGGS ENVELOPES

341

By the transport of structures via the canonical isomorphism in Theorem IV.2.5.8, the sheaf D acquires a T -torsor structure. By the proof of Theorem IV.2.5.2, it induces the same T -action on D1 as the one defined after Theorem IV.2.5.2. Let U1 be an object of (AffSmStr/Y1 )Zar and let f : U → Y be a smooth strict Cartesian morphism in C with the given U1 → Y1 over S1 . We have U ∈ Ob C ∞ by Lemma IV.2.2.7 (1). Then we have natural maps (IV.2.5.9) (IV.2.5.10)

HomY (U, (e z : Y1 ,→ X ×B Y )) −→ LU1 (U2 ), HomY (U, D) −→ HomY1 (U1 , D1 ).

Proposition IV.2.5.11. (1) The two maps (IV.2.5.9) and (IV.2.5.10) are surjective. (2) There exists a unique bijective map ∼ =

ιU : LU1 (U2 ) −→ HomY1 (U1 , D1 ) such that the following diagram is commutative, where the bottom horizontal bijection is induced by the adjunction morphism D → (e z : Y1 ,→ X ×B Y ) in C . LU1 (U2 ) O

∼ =

HomC/Y (U, (e z : Y1 ,→ X ×B Y )) o

/ HomY1 (U1 , D1 ) O ∼ =

∞ (U, D). HomC/Y

Proof. (1) The morphism (IV.2.5.10) is surjective by Lemma IV.2.5.3 and Sublemma IV.3.4.2. By definition, giving an element of LU1 (U2 ) is equivalent to giving a morphism U2 → X2 ×B2 Y2 over Y2 such that the composition with U1 → U2 coincides z e

with the composition of U1 → Y1 → X2 ×B2 Y2 . Hence the morphism (IV.2.5.9) is also surjective by Sublemma IV.3.4.2. (2) By the description of the element of LU1 (U2 ) in the proof of (1), we may replace X by X ×B Y and z : Y1 → X1 by the section Y1 → X1 ×B1 Y1 induced by z, and assume Y = B. Let (Yα → Y )α∈A be a strict étale covering such that Yα,1 is affine and A is a finite set, and let Yαβ denote Yα ×Y Yβ for (α, β) ∈ A2 . Let Uα and Xα (resp. Uαβ and Xαβ ) denote the base change of U and X under Yα → Y (resp. Yαβ → Y ), and define the set LUα,1 (Uα,2 ) (resp. LUαβ,1 (Uαβ,2 )) in the same way as LU1 (U2 ) by using Uα → Yα ← Xα and Yα,1 → Xα,1 (resp. Uαβ → Yαβ ← Xαβ andQYαβ,1 → Xαβ,1 ). Q Then, by Lemma IV.2.2.19 (1), we see that the sequence LU1 (U2 ) → α LUα,1 (Uα,2 ) ⇒ αβ LUαβ,1 (Uαβ,2 ) is exact. Proposition IV.2.2.9 (3), Lemma IV.2.2.18, and Lemma IV.2.2.19 imply similar exact sequences for the other three terms of the diagram in question. Hence the claim holds for (U, Y, X) if it holds for (Uα , Yα , Xα ) and (Uαβ , Yαβ , Xαβ ). By Corollary IV.2.2.5 and Lemma IV.2.2.15, we also see that every term in the diagram does not change if we replace X with an open p-adic fine log formal scheme containing the image of Y1 . Hence it suffices to prove the claim in the case where XN (N ≥ 1) are affine, there exist (ti,N )N ∈ limN Γ(XN , MXN ) (1 ≤ i ≤ d) such that d log(ti,N ) (1 ≤ i ≤ d) is a basis of ←− Ω1XN /YN for every N . Let Spf(Zp ){Nd } be the p-adic fine log formal scheme defined as in the proof of Proposition IV.2.3.17, and let YN {Nd } denote YN ×Spf(Zp ) Spf(Zp ){Nd }. Then the YN morphism XN → YN {Nd } induced by Nd → Γ(XN , MXN ); 1i 7→ ti,N is étale and defines a morphism X → Y {Nd } in C . Hence we have the following bijections, where iU,1 denotes

342

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

the morphism U1 → U2 and Γ(U, MU ) denotes the inverse limit limN Γ(UN , MUN ). ←−

LU1 (U2 ) → {(ai )i ∈ Γ(U2 , MU2 )d | i∗U,1 (ai ) = f1∗ z ∗ (ti,1 )}; g 7→ (g ∗ (ti,2 ))i ,

HomY (U, (z : Y1 ,→ X)) → {((ai,N )N )i ∈ Γ(U, MU )d | ai,1 = f1∗ z ∗ (ti,1 )} h 7→ ((h∗N (ti,N ))N ≥2 )i .

Choose a lifting (si,N )N ∈ limN Γ(YN , MYN ) of si,1 := z ∗ (ti,1 ). Then, by Proposition ←− IV.2.3.17, we obtain isomorphisms Γ(YN , OYN ){W1,N , . . . , Wd,N } ∼ = Γ(DN , ODN ) compatible with N . By Proposition IV.2.2.9 (2), we also see that the morphism D → Y is strict. Hence, we have the following bijections, where Γ(U, OU ) denotes limN Γ(UN , OUN ). ←− HomY1 (U1 , D1 ) → Γ(U1 , OU1 )d ; ϕ 7→ (ϕ∗ (Wi,1 )),

∗ HomY (U, D) → Γ(U, OU )d ; ψ 7→ (ψN (Wi,N ))N .

Let h ∈ HomY (U, (z : Y1 ,→ X)), let ψ be the corresponding Y -morphism U → D, and let ((ai,N )N )i ∈ Γ(U, MU )d (resp. ((bi,N )N )i ∈ Γ(U, OU )d ) be the image of h (resp. ψ) under the bijections above. Then, by the construction of the isomorphism in Proposition IV.2.3.17, we have an equality ai,2 = f2∗ (si,2 ) · (1 + [ξ]bi,1 ) ∼ =

in Γ(U2 , MU2 ), where [ξ] denotes the isomorphism Γ(U1 , OU1 ) → Ker(Γ(U2 , OU2 ) → Γ(U1 , OU1 )) induced by the multiplication by ξ on OU2 (cf. Definition IV.2.2.1 (2) (iii) (iv)). Hence ai,2 and bi,1 determine each other. This completes the proof by (1) and the descriptions of LU1 (U2 ) and HomY1 (U1 , D1 ) above.  ∼ =

Lemma IV.2.5.12. The composition of the morphism ιU with the projection L(U1 ) − → LU1 (U2 ) is independent of the choice of U . Proof. Let U 0 → Y be another strict smooth Cartesian morphism in C ∞ with = U1 . Then, by the smoothness of U → Y and Sublemma IV.3.4.2 (2), there exists a morphism v : U 0 → U over Y such that v1 is the identity of U10 = U1 . Put CN = 0 0 , OUN0 ). Then, by Lemma IV.2.2.6, we have an exact Γ(UN , OUN ) and CN = Γ(UN U10

(0)

[ξ]

(0)

(0)

sequence 0 → CN −1 −→ CN → C1 → 0. Hence we see that vN is an isomorphism by induction on N . Now the claim follows from the commutative diagram below. LU1 (U2 ) o ∗ ∼ = v2

 LU1 (U20 ) o

HomY (U, (Y1 ,→ X ×B Y )) o

∼ =

−◦v ∼ =

HomY (U, D)

/ HomY (U1 , D1 ) 1

−◦v ∼ =



HomY (U 0 , (Y1 ,→ X ×B Y )) o

∼ =

 HomY (U 0 , D)

/ HomY (U10 , D1 ). 1 

Let ιU1 denote the canonical bijection ∼ =

L(U1 ) −→ HomY1 (U1 , D1 ) obtained by the composition in Lemma IV.2.5.12. Now Theorem IV.2.5.8 follows from the following lemma.

IV.2. HIGGS ENVELOPES

343

Lemma IV.2.5.13. For a morphism v1 : U10 → U1 in (AffSmStr/Y1 )Zar , the following diagram is commutative. L(U1 )

∼ = ιU1

−◦v1

L(v1 )

 L(U10 )

/ HomY1 (U1 , D1 )

∼ = ιU 0

 / HomY (U10 , D1 ) 1

1

Proof. Let U → Y and U 0 → Y be strict smooth Cartesian liftings of U1 → Y1 and → Y1 , respectively. By the smoothness of U → Y and Sublemma IV.3.4.2 (2), there exists a lifting v : U 0 → U of v1 . Then the claim follows from the same kind of diagram as in the proof of Lemma IV.2.5.12 induced by v.  U10

z e

r Next let us consider the object Dr (r ∈ N>0 ) of C r defined by Dr := DHiggs (Y1 ,→ X ×B Y ) (cf. Proposition IV.2.2.9 (1)). By Proposition IV.2.2.9 (2), D1r is affine. Let AY1 , AL , AD1 , and AD1r denote the rings of coordinates of the underlying affine p-adic formal schemes of Y1 , L, D1 , and D1r , respectively. Let n ΩZar and ΩZar denote the sheaves HomOn Y1 (n TZar , On Y1 ) and HomOY1 (TZar , OY1 ). Let n F be the sheaf of affine functions on the n TZar -torsor n LZar (II.4.9). Then we have an exact sequence of On Y1 -modules 0 → On Y1 → n F → n ΩZar → 0 compatible with n. By taking the global sections and the inverse limit over n, we obtain an exact sequence of AY1 -modules c

0 −→ AY1 −→ M −→ Γ(Y1 , ΩZar ) −→ 0. Lemma IV.2.5.14. Under the notation and the assumption above, the AY1 -module Γ(Y1 , ΩZar ) is finitely generated and projective. Proof. Put n P := Γ(Y1 , n ΩZar ), n AY1 := Γ(n Y1 , On Y1 ) and P := Γ(Y1 , ΩZar ). We ∼ = have P = limn (n P ) and n+1 P/pn − → n P . Since n ΩZar is a locally free On Y1 -module ←− of finite type, n P is a finitely generated projective n AY1 -module. Choose a surjective ⊕r 1 AY1 -linear homomorphism 1 ϕ : 1 AY1 → 1 P and then a compatible system of n AY1 ⊕r linear liftings n ϕ : n AY1 → n P . By Nakayama’s lemma, n ϕ is surjective. Since the natural homomorphisms Ker(n+1 ϕ) → Ker(n ϕ) are surjective, we see that there exists a compatible system of n AY1 -linear homomorphisms n s : n P → n A⊕r Y1 such that n ϕ ◦ n s = idn P . Letting s and ϕ denote the inverse limits of n s and n ϕ, we obtain an AY1 -linear ∼ = isomorphism P ⊕ Ker(ϕ) − → A⊕r  Y1 ; (a, b) 7→ s(a) + b. m For m ∈ N, let SA (M ) denote the m-th symmetric tensor product of the AY1 Y1 module M . Then the homomorphism c induces an injective AY1 -linear homomorphism m+1 m m cm : SA (M ) → SA (M ). Let CAY1 (M ) denote the direct limit limm (SA (M )), and Y1 Y1 Y1 −→ bA (M ) denote the p-adic completion of CA (M ). Then, by the construction of the let C Y1 Y1 T -torsor L, we have a canonical isomorphism ∼ = bA (M ) −→ C AL . Y1

(IV.2.5.15)

m Identifying SA (M ) with its image in CAY1 (M ) under the canonical injection, we define Y 1

(r)

the AY1 -subalgebra CAY (M ) by 1

(r)

CAY (M ) :=

X

1

m

m (M ). pd r e SA Y1

m∈N

Let

b (r) (M ) C A Y1

(r)

denote the p-adic completion of CAY (M ). 1

344

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proposition IV.2.5.16. For r ∈ N>0 , the natural homomorphisms AD1r → AD1 and b (r) (M ) → C bA (M ) are injective, and the composition of the isomorphism (IV.2.5.15) C Y1 AY 1

∼ =

and the isomorphism AL − → AD1 (Theorem IV.2.5.2) induces an isomorphism ∼ = b (r) (M ) −→ AD1r . C AY 1

Proof. Replacing X with X ×B Y and z with the morphism ze : Y1 → X1 ×B1 Y1 , we may assume that Y = B. Let n F 0 be the sheaf of affine functions on the n TZar torsor n DZar := ϕn Y1 (n D1 ), and let M 0 denote the AY1 -module limn Γ(Y1 , n F 0 ), which ←− is an extension of Γ(Y1 , ΩZar ) by AY1 . Let n L0 be the n T -principal homogeneous space associated to the n TZar -torsor n DZar (II.4.9) and put L0 = (n L0 )n≥1 , which is a T -principal homogeneous space. The isomorphism in Theorem IV.2.5.2 induces an isomorphism of ∼ = extensions M − → M 0 , and it suffices to prove the claim for M 0 , L0 , and the canonical isomorphism AL0 ∼ = AD1 instead of M , L, and AL ∼ = AD1 . By Lemma IV.2.5.14, we have an exact sequence M m+1 (∗) M d m e m (r) m 0 −→ pd r e SA (M 0 ) −→ p r SAY1 (M 0 ) −→ CAY (M 0 ) −→ 0, Y1 1

m∈N

m∈N

m+1 m m where (∗) is induced by (id, −cm ) : SA (M 0 ) → SA (M 0 ) ⊕ SA (M 0 ). It remains exact Y1 Y1 Y1 N after ⊗Z Z/p Z. For any strict étale covering Y = (uα : Yα → Y )α∈A such that ]A < ∞ and Yα,1 are affine, and any quasi-coherent On Y1 -module G on Y1,Zar , the Čech cohomolˇ q (Y, G) vanishes if q > 0 and H ˇ 0 (Y, G) = Γ(Y, G). Hence, by Propositions IV.2.5.22 ogy H and IV.2.5.24 below and Lemma IV.2.2.15, we see that the question is strict étale local on Y1 and we may assume that there exist (ti,N ) ∈ limN Γ(XN , MXN ) (1 ≤ i ≤ d) such ←− that d log(ti,N ) (1 ≤ i ≤ d) is a basis of Ω1XN /YN for every N ∈ N>0 . Choosing a lifting (si,N )N ∈ limN Γ(YN , MYN ) of z ∗ (ti,1 ) and applying Proposition IV.2.3.17, we obtain an ←− isomorphism AD1 ∼ = AY1 {W1 , . . . , Wd }r , and an isomorphism ∧  M |m| pd r e AY1 W m  . ArD1 ∼ = m∈Nd (m)r

m

= pd r e AY1 by F n AYN = ξ n AYN . Hence it suffices to prove ∼ = bA (M 0 ) − that the image of M 0 under the natural isomorphism C → AD1 is AY1 ⊕ Y1 (⊕1≤i≤d AY1 Wi ). We identify D with OY⊕d by the coordinates Wi (1 ≤ i ≤ d). Then, by the proof of 1 Proposition IV.2.5.11, we see that the action of f ∈ T (U ) on D(U ) = OU (U )⊕d is given by (ai )1≤i≤d 7→ (ai + f (ξ −1 d log(ti,1 )))1≤i≤d . This implies that the action of T on D1 is given by b AY AD1 ; Wi 7→ ξ −1 d log(ti,1 ) ⊗ 1 + 1 ⊗ Wi , AD1 −→ SAY1 (Ω)⊗ 1 Note that we have AY1

where Ω = Γ(Y1 , ΩZar ). We identify n DZar with On⊕d Y1 by the coordinates Wi (1 ≤ i ≤ d). ∼ =

Then it induces isomorphisms n F 0 − → On Y1 ⊕ (On Y1 )⊕d ; f 7→ (f (0), (f (1i ) − f (0))1≤i≤d ) 0 ∼ and n L = n Y1 [V1 , . . . , Vd ]. We identify ϕn Y1 (n L0 ) with On⊕d Y1 by the coordinates Vi . Then we see that the canonical isomorphism of n TZar -torsors n DZar ∼ = ϕn Y1 (n L0 ) (cf. (II.4.9.7)) ⊕r is simply given by the identity map of OY1 . On the other hand, the action of n T on n L0 is given by On Y1 [V1 , . . . , Vd ] → SOn Y1 (n ΩZar ) ⊗On Y1 On Y1 [V1 , . . . , Vd ]; Vi 7→ ξ −1 d log ti,1 ⊗ 1 + 1 ⊗ Vi

IV.2. HIGGS ENVELOPES

345 ∼ =

(cf. (II.4.9.4)). Hence the unique n T -equivariant isomorphism n D1 − → n L0 inducing the 0 above isomorphism n DZar ∼ = ϕY1 (n L ) (cf. II.4.4) is given by Vi 7→ Wi . This completes the proof.  ∼ =

For n ∈ N, we have a canonical On Y1 -linear isomorphism On T ⊗On Y1 n ΩZar → Ω1n T /n Y1 on (T1 )Zar equivariant with respect to the translation by any Zariski local section n Y1 ⊃ U → n T of n T → n Y1 . This induces a canonical On Y1 -linear isomorphism On L ⊗On Y1 ∼ =

→ Ω1n L/n Y1 on (L1 )Zar equivariant with respect to the translation by any Zariski local section of n T → n Y1 (cf. II.10.9). By Lemma IV.2.5.14, we see that ΩZar is a direct factor of a free OY1 -module of finite type. Hence the above isomorphism induces ∼ = an AL -linear isomorphism AL ⊗AY1 Γ(Y1 , ΩZar ) → Γ(L, Ω1L/Y1 ) and then an AY1 -linear derivation n ΩZar

(IV.2.5.17)

θL : AL → AL ⊗AY1 Γ(Y1 , ΩZar ).

We obtain the following AY1 -linear derivation from (IV.2.4.8). (IV.2.5.18)

θD1 : AD1 → AD1 ⊗AY1 Γ(Y1 , ΩZar ).

Proposition IV.2.5.19. The isomorphism AL ∼ = AD1 induced by that in Theorem IV.2.5.2 is compatible with the derivations (IV.2.5.17) and (IV.2.5.18). Proof. Replacing X and z with X ×B Y and ze : Y1 → X1 ×B1 Y1 , we may assume that Y = B. Let n F 0 , n DZar , M 0 , and L0 be as in the beginning of the proof of Proposition IV.2.5.16. It suffices to prove the claim for L0 instead of L and for the canonical isomorphism AL0 ∼ = AD1 . By Propositions IV.2.5.22 and IV.2.5.24 below, Lemma IV.2.2.15, and a similar argument as the proof of Proposition IV.2.5.16, we see that the question is strict étale local on Y1 and we may assume that there exist (ti,N ) ∈ limN Γ(XN , MXN ) (1 ≤ i ≤ d) as in the proof of Proposition IV.2.5.16. ←− We use the notation in loc. cit. Since θL0 |M 0 is induced by the canonical projection M 0 → Γ(Y1 , ΩZar ), we see θL0 (Vi ) = 1 ⊗ ξ −1 d log(ti,1 ) (1 ≤ i ≤ d) by the explicit description of the action of T on D1 given in loc. cit. Now the claim follows from θD1 (Wi ) = 1 ⊗ ξ −1 d log(ti,1 ) (1 ≤ i ≤ d).  We discuss the functoriality of the isomorphisms in Theorems IV.2.5.2 and IV.2.5.8. We define the category CTors as follows: An object X = (Y → B ← X, z) consists of morphisms Y → B ← X in C and a B1 -morphism z : Y1 → X1 of p-adic fine log formal schemes satisfying the conditions in the beginning of this subsection. A morphism α : (Y 0 → B 0 ← X 0 , z 0 ) → (Y → B ← X, z) is a triple (αY , αB , αX ) consisting of morphisms αY : Y 0 → Y , αB : B 0 → B, and αX : X 0 → X in C making the following diagrams commutative. (IV.2.5.20)

Y0 αY

 Y

/ B0 o αB

 /Bo

X0 αX

 X,

Y10

z0

αY,1

 Y1

/ X10 αX,1

z

 / X1 .

We often abbreviate αY to α in the following. For an object X of CTors , we write LX,Zar , TX,Zar , TX , LX,U , LX , TX , DX , and DX for LZar , TZar , T , LU , L, etc. constructed from X as above. We begin by studying the functoriality for a morphism α = (αY , αB , αX ) : X 0 = 0 (Y → B 0 ← X 0 ) → X = (Y → B ← X) satisfying one of the following conditions. (Cond 1) The morphism X 0 → X ×B B 0 induced by αX and αB is an isomorphism.

346

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

(Cond 2) B = B 0 , Y = Y 0 , and αB and αY are the identity morphisms. We first consider the case where α satisfies (Cond 1). Fiber products are representable in (AffSmStr/Y1 )Zar , and the functor (AffSmStr/Y1 )Zar → (AffSmStr/Y10 )Zar defined by the base change under the morphism α1 : Y10 → Y1 preserves finite fiber products, final objects, and Zariski open coverings. Hence the functor above is a morphism of sites and defines a morphism of topos, whose direct image functor will be simply denoted by α1∗ . Let u : U → Y1 be an object of (AffSmStr/Y1 )Zar , let u0 : U 0 → Y10 be the base change of u under the morphism α1 : Y10 → Y1 , and let αU : U 0 → U denote the natural mor∗ phism. By (Cond 1), the OX10 -linear homomorphism αX,1 Ω1X1 /B1 → Ω1X 0 /B 0 on (X10 )Zar 1 1 ∗ is an isomorphism, which induces an isomorphism αU (HomOU (u∗ z ∗ (ξ −1 Ω1X1 /B1 ), OU )) ∼ = ∗ ∗ ∗ −1 1 HomOU 0 (αU u z (ξ ΩX1 /B1 ), OU 0 ) ∼ = HomOU 0 (u0∗ z 0∗ (ξ −1 Ω1X10 /B10 ), OU 0 ). By taking Γ(U 0 , −), we obtain a homomorphism TX (U ) → TX 0 (U 0 ) = α1∗ TX 0 (U ). Varying U , we obtain a morphism αT∗ : TX → α1∗ (TX 0 ) compatible with the actions of OY1 and OY10 via the morphism OY1 → α1∗ OY10 . e → Y2 be an object of Let u : U → Y1 be an object of (AffSmStr/Y1 )Zar , let u e: U e 0 → Y 0 ) be the base change of u (resp. u Lift(U/Y2 ), and let u0 : U 0 → Y10 (resp. u e0 : U e) by 2 0 e α1 (resp. α2 ). By (Cond 1), for any g ∈ LX,U (U ), there exists a unique B2 -morphism e 0 → X 0 such that αX,2 ◦ g 0 : U e 0 → X2 coincides with the composition of U e0 → g0 : U 2 g e − e 0 ). Thus we obtain U → X2 . It is straightforward to verify that g 0 belongs to LX 0 ,U 0 (U ∗ 0 e e a map αL,Ue ,Ue 0 : LX,U (U ) → LX,U 0 (U ). By the definition of the torsor structures, we see that this morphism is Γ(U, αT∗ )-equivariant. We also see that this construction is functorial on the pair (u, u e) in the obvious sense. Hence the composition of LX (U ) ∼ = α∗

e ,U e0 U ∼ LX 0 (U 0 ) = α1∗ LX (U ) does not depend on the choice of e ) −−L, e 0) = LX,U (U −−−→ LX 0 ,U 0 (U e and induces an α∗ -equivariant morphism U T

∗ αL : LX −→ α1∗ LX 0 .

The morphisms αY,1 : Y10 → Y1 and αX ×αB αY : X 0 ×B 0 Y 0 → X ×B Y induce a morphism DX 0 → DX compatible with the morphism αY : Y 0 → Y . By (Cond 1), Proposition IV.2.2.9 (3), Lemma IV.2.2.18 (1), and Proposition IV.2.3.17, we see that the morphism αD : DX 0 → DX ×Y Y 0 induced by DX 0 → DX is an isomorphism, and we obtain a morphism ∗ αD : DX −→ α1∗ DX 0 . Proposition IV.2.5.21. For a morphism α : X 0 → X in CTors satisfying (Cond 1), the ∗ following diagram is commutative. In particular, αD is αT∗ -equivariant. LX

α∗ L

Thm. IV.2.5.8 ∼ =



DX

/ α1∗ LX 0 ∼ = Thm. IV.2.5.8

α∗ D

 / α1∗ DX 0 .

Proof. Let Z (resp. Z 0 ) denote X ×B Y (resp. X 0 ×B 0 Y 0 ). By (Cond 1), we have ∼ ∼ = = isomorphisms (Y10 ,→ Z 0 ) − → (Y1 ,→ Z) ×Y Y 0 and DX 0 − → DX ×Y Y 0 in the category C . Hence, for a strict smooth affine Cartesian morphism U → Y in C and its base change U 0 → Y 0 under the morphism αY : Y 0 → Y , we have the following commutative diagram,

IV.2. HIGGS ENVELOPES

347

∗ where the vertical morphisms except αL,U 0 are defined by the base change under αY . 2 ,U 2

LX,U1 (U2 ) o α∗ L,U

∼ =

0 2 ,U2

HomY (U, (Y1 ,→ Z)) o

 LX 0 ,U10 (U20 ) o

 HomY 0 (U 0 , (Y10 ,→ Z 0 )) o

∼ =

HomY (U, DX )

/ HomY1 (U1 , DX,1 )

 HomY 0 (U 0 , DX 0 )

 / HomY 0 (U10 , DX 0 ,1 ). 1

This implies that the diagram in the proposition commutes for the sections on U1 .



∗ The constructions of αT∗ and αL still work after taking the reduction mod pn , and we ∗ obtain a morphism n αT : n TX,Zar → α1∗ (n TX 0 ,Zar ) on (Y1 )Zar compatible with the actions ∗ of On Y1 and On Y10 , and an n αT∗ -equivariant morphism n αL : n LX,Zar → α1∗ (n LX 0 ,Zar ). ∼ =

The morphism n αT∗ induces an isomorphism On Y10 ⊗α−1 (O

n Y1 )

1

∼ =

α1−1 (n TX,Zar ) − → n TX 0 ,Zar .

By II.4.13, we obtain an isomorphism n αL : n LX 0 − → n LX ×n Y1 n Y10 of n TX 0 -principal homogeneous spaces, and then an isomorphism of TX 0 -principal homogeneous spaces ∼ =

αL : LX 0 −→ LX ×Y1 Y10 .

Proposition IV.2.5.22. For a morphism α : X 0 → X in CTors satisfying (Cond 1), ∼ = the following diagram is commutative. In particular, the isomorphism αD,1 : DX 0 ,1 − → DX,1 ×Y1 Y10 is TX 0 -equivariant. LX 0

∼ = αL

/ LX ×Y Y10 1

Thm. IV.2.5.2 ∼ =



DX 0 ,1

∼ = Thm. IV.2.5.2

∼ = αD,1

 / DX,1 ×Y1 Y10 .

Proof. By (II.4.13.8) and Proposition IV.2.5.21, we have the following commutative diagram on (Y1 )Zar . ∼ =

ϕY1 (LX ) o 

∼ =

α1∗ (ϕY10 (LX 0 )) o

∼ =

LX,Zar o 

Thm. IV.2.5.8

α∗ L

α1∗ (LX 0 ,Zar ) o

∼ = Thm. IV.2.5.8

ϕY1 (DX,1 )

 α1∗ (ϕY10 (DX 0 ,1 )).

Here the left (resp. right) vertical morphism is naturally induced by αL (resp. αD,1 ). By taking the adjoints of the vertical morphisms, we obtain a diagram, whose outer square is commutative. α1−1 (ϕY1 (DX,1 )) Thm. IV.2.5.2 ∼ =



α1−1 (ϕY1 (LX ))

/ ϕY1 (DX,1 ×Y1 Y10 ) Thm. IV.2.5.2 ∼ =

∼ =

/ ϕY1 (DX 0 ,1 )

Thm. IV.2.5.2 ∼ =



/ ϕY (LX ×Y Y10 ) 1 1

∼ =

 / ϕY1 (LX 0 ).

By using the action of TX 0 ,Zar , we see that the right square is also commutative. Hence the claim follows from Proposition IV.2.5.1.  Next we consider a morphism α : X 0 = (Y → X 0 ← B, z 0 ) → X = (Y → X ← B, z) satisfying (Cond 2). ∗ The OX10 -linear morphism αX,1 (Ω1X1 /B1 ) → Ω1X 0 /B1 induces an OY1 -linear morphism 1

αT ∗ : TX 0 → TX .

348

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

e of Lift(U/Y2 ), the composition For an object U of (AffSmStr/Y1 )Zar and an object U 0 e ) → LX,U (U e ), which with αX,2 : X2 → X2 induces an αT ∗ (U )-equivariant map LX 0 ,U (U is functorial on the pair (u, u e). Hence it defines an αT ∗ -equivariant morphism αL∗ : LX 0 → LX . The morphisms αY1 and αX × αY induce a morphism αD : DX 0 → DX . The composition with αD,1 gives a morphism αD∗ : DX 0 −→ DX . Proposition IV.2.5.23. For a morphism α : X 0 → X in CTors satisfying (Cond 2), the following diagram is commutative. In particular, αD∗ is αT ∗ -equivariant. LX 0

αL∗

∼ = Thm. IV.2.5.8

Thm. IV.2.5.8 ∼ =



DX 0

/ LX

αD∗

 / DX .

Proof. Let Z (resp. Z 0 ) denote Y ×B X (resp. Y ×B X 0 ). Let U → Y be a strict smooth affine Cartesian morphism in C . Then the compositions with X20 → X2 , (Y1 ,→ Z 0 ) → (Y1 ,→ Z), DX 0 → DX , and DX 0 ,1 → DX,1 give the following commutative diagram. LX 0 ,U1 (U2 ) o

HomY (U, (Y1 ,→ Z 0 )) o

 LX,U1 (U2 ) o

 HomY (U, (Y1 ,→ Z)) o

∼ =

∼ =

HomY (U, DX 0 )

/ HomY1 (U1 , DX 0 ,1 )

 HomY (U, DX )

 / HomY1 (U1 , DX,1 ).

This implies that the diagram in the proposition is commutative for sections on U1 .  We continue to assume that α satisfies (Cond 2). The constructions of αT ∗ and αL∗ above still work after taking the reduction mod pn , and we obtain an On Y1 -linear morphism n αT ∗ : n TX 0 ,Zar → n TX,Zar and an n αT ∗ -equivariant morphism n αL∗ : n LX 0 ,Zar → n LX,Zar . By II.4.12, we obtain a morphism n αT : n TX 0 → n TX and an n αT -equivariant morphism n αL : n LX 0 → n LX . These form a compatible system with respect to n and define a morphism αT : TX 0 → TX and an αT -equivariant morphism αL : LX 0 → LX . Proposition IV.2.5.24. For a morphism α : X 0 → X in CTors satisfying (Cond 2), the morphism αD,1 : DX 0 ,1 → DX,1 is αT -equivariant, and the following diagram is commutative. LX 0

αL

Thm. IV.2.5.2 ∼ =



DX 0 ,1

/ LX ∼ = Thm. IV.2.5.2

αD,1

 / DX,1 .

IV.2. HIGGS ENVELOPES

349

Proof. The first claim follows from Proposition IV.2.5.23. By (II.4.12.4) and Proposition IV.2.5.23, the following diagram is commutative. ϕY1 (LX 0 ) o

∼ =

LX 0 ,Zar o αL∗

ϕY1 (αL )

 ϕY1 (LX ) o

∼ =



∼ = Thm. IV.2.5.8

ϕY1 (DX 0 ,1 ) ϕY1 (αD,1 )

LX,Zar o

∼ = Thm. IV.2.5.8

 ϕY1 (DX,1 ).

Hence it suffices to prove that, for two αT -equivariant morphisms σ, σ 0 : LX 0 → LX , ϕY1 (σ) = ϕY1 (σ 0 ) implies σ = σ 0 . By considering the morphisms of sheaves on (AffSmStr/Y1 )Zar associated to σ and σ 0 , we see that there exists a unique TX -equivariant automorphism τ : LX → LX such that σ 0 = τ ◦ σ. The equality ϕY1 (σ) = ϕY1 (σ 0 ) implies ϕY1 (τ ) = id. By Proposition IV.2.5.1, we have τ = id, which implies σ = σ 0 .  Let α : X 0 = (Y 0 → B 0 ← X 0 , z 0 ) → (Y → B ← X, z) be a morphism in CTors . Let X 00 → B 0 denote the base change of X → B by αB : B 0 → B, let z 00 : Y10 → X100 αY,1 denote the B10 -morphism induced by Y10 −−−→ Y1 → X1 , and let X 00 denote the object αII

(Y 0 → B 0 ← X 00 , z 00 ) of CTors . Then the morphism α decomposes naturally as X 0 −−→ αI

X 00 −−→ X, and αI (resp. αII ) satisfies (Cond 1) (resp. (Cond 2)). We define the group morphism αT : TX 0 → TX ×Y1 Y10

∼ TX ×Y to be the composition of αTII : TX 0 → TX 00 with the natural isomorphism TX 00 = 1 ∼ = 0 II I 0 Y1 . Composing αL : LX 0 → LX 00 with αL : LX 00 → LX ×Y1 Y1 , we obtain an αT equivariant morphism αL : LX 0 → LX ×Y1 Y10 . On the other hand, the morphisms αY,1 : Y10 → Y1 and αX ×αB αY : X 0 ×B 0 Y 0 → X ×B Y induce a morphism DX 0 → DX and then a Y 0 -morphism αD : DX 0 → DX ×Y Y 0 . Proposition IV.2.5.25. For a morphism α : X 0 → X in CTors , the following diagram is commutative. In particular, the morphism αD,1 is αT -equivariant. LX 0

αL

Thm. IV.2.5.2 ∼ =



DX 0 ,1

/ LX ×Y Y10 1 ∼ = Thm. IV.2.5.2

αD,1

 / DX,1 ×Y1 Y10 .

II Proof. The morphism αD coincides with the composition of αD : DX 0 → DX 00 I 0 and αD : DX 00 → DX ×Y Y . Hence the claim immediately follows from Propositions IV.2.5.22 and IV.2.5.24. 

Corollary IV.2.5.26. For morphisms α : X 0 → X, α0 : X 00 → X 0 in CTors , and their 00 0 00 0 composition α00 := α ◦ α0 , we have αL = (αL × 1Y100 ) ◦ αL and αD = (αD × 1Y 00 ) ◦ αD . Proof. The second equality is obvious by the construction of morphisms, and it implies the first equality by Proposition IV.2.5.25. 

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

IV.3. Higgs isocrystals and Higgs crystals IV.3.1. Sites. For r ∈ N>0 ∪ {∞}, B ∈ Ob C r and a p-adic fine log formal scheme X over B1 , we define a site (X/B)rHIGGS (Definitions IV.3.1.1, IV.3.1.4), an analogue of the crystalline site for Higgs fields, on which Higgs isocrystals and Higgs crystals will be defined in IV.3.3. We have a description of sheaves on the site (X/B)rHIGGS in terms of certain compatible systems of étale sheaves on objects of (X/B)rHIGGS in a way similar to the crystalline site (see after Definition IV.3.1.4). We then study functoriality and two kinds of localizations: one with respect to a morphism of p-adic fine log formal schemes U → X and the other with respect to an object of the site. Definition IV.3.1.1. Let r ∈ N>0 ∪ {∞}, let B ∈ Ob C r , and let X be a p-adic fine log formal scheme over B1 . We define the category (X/B)rHIGGS as follows. An object r is a pair (T, z) of an object T of C/B and a B1 -morphism z : T1 → X. A morphism 0 0 0 r (T , z ) → (T, z) is a morphism u : T → T in C/B such that z ◦ u1 = z 0 . We say that a morphism u : (T 0 , z 0 ) → (T, z) in (X/B)rHIGGS is étale (resp. strict, resp. Cartesian) if the underlying morphism u : T 0 → T in C r is étale (resp. strict, resp. Cartesian). Proposition IV.3.1.2. Let r, B, and X be as in Definition IV.3.1.1. (1) Finite fiber products and nonempty finite products are representable in the category (X/B)rHIGGS . (2) Finite fiber products are compatible with the forgetful functor (X/B)rHIGGS → r C /B. Proof. The fiber product of (T 0 , z 0 ) → (T, z) ← (T 00 , z 00 ) in (X/B)rHIGGS is reprer sented by the fiber product T 000 of T 0 → T ← T 00 in C/B (cf. Corollary IV.2.2.11) endowed z0

z 00

with the composition of T1000 → T10 → X, which coincides with that of T1000 → T100 → X. The product of two objects (T, z) and (T 0 , z 0 ) of (X/B)rHIGGS is represented by r T 00 := DHiggs (T1 ×X T10 ,→ T ×B T 0 ) z

endowed with the composition of T100 → T1 → X, which coincides with that of T100 → z

0

T10 → X.



For an object (T, z) of (X/B)rHIGGS , we define the set Cov((T, z)) to be   (i) uα is strict étale and Cartesian for all α ∈ A S (uα : (Tα , zα ) → (T, z))α∈A . (ii) α∈A uα,1 (Tα,1 ) = T1

Lemma IV.3.1.3. Let r, B, and X be as in Definition IV.3.1.1. The sets Cov((T, z)) for (T, z) ∈ Ob (X/B)rHIGGS satisfy the axiom of pretopology ([2] II Définition 1.3).

Proof. By Lemma IV.2.2.7 (1) and the construction of fiber products in C r given in the proof of Corollary IV.2.2.11, we see that Cov((T, z)) is stable under base change. The stability under composition and id(T,z) ∈ Cov((T, z)) are obvious.  Definition IV.3.1.4. Let r, B, and X be as in Definition IV.3.1.1. We endow the category (X/B)rHIGGS with the topology associated to the pretopology Cov((T, z)), (T, z) ∈ Ob (X/B)rHIGGS . We define the sheaf of rings OX/B,1 on (X/B)rHIGGS by OX/B,1 (T, z) = Γ(T1 , OT1 ). Let r, B, and X be as in Definition IV.3.1.1. Let F be a sheaf of sets (resp. OX/B,1 modules) on the site (X/B)rHIGGS . Then, for each object (T, z) of (X/B)rHIGGS , one can define a sheaf of sets (resp. OT1 -modules) F(T,z) on T1,´et by F(T,z) (v1 : U1 → T1 ) =

IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

351

F((U, z ◦ v1 )), where v : U → T is the unique strict étale Cartesian morphism in C r such that U1 → T1 is the given morphism v1 . For a morphism u : (T 0 , z 0 ) → (T, z) in (X/B)rHIGGS , we can define a morphism of sheaves of sets (resp. OT10 -modules) τu : u−1 1 (F(T,z) ) → F(T 0 ,z 0 )

(resp. τu : u∗1 (F(T,z) ) → F(T 0 ,z0 ) )

satisfying the following conditions: (i) For any morphism u : (T 0 , z 0 ) → (T, z) such that the underlying morphism u : T 0 → T in C is strict étale and Cartesian, the morphism τu is an isomorphism. v u (ii) For any composable morphisms (T 00 , z 00 ) − → (T 0 , z 0 ) − → (T, z), we have τv ◦ −1 ∗ v1 (τu ) = τuv (resp. τv ◦ v1 (τu ) = τuv ). The category of sheaves of sets (resp. OX/B,1 -modules) on (X/B)rHIGGS is equivalent to the category of (F(T,z) , τu ) satisfying the conditions (i) and (ii) above. Proposition IV.3.1.5. Let r, r0 ∈ N>0 ∪ {∞} such that r0 ≥ r. Let B (resp. B 0 ) be 0 an object of C r (resp. C r ) and let X (resp. X 0 ) be a p-adic fine log formal scheme over B1 (resp. B10 ). Let g : B 0 → B be a morphism in C , and let f : X 0 → X be a morphism of p-adic fine log formal schemes compatible with the morphism g1 : B10 → B1 . Then the functor r0 (X 0 /B 0 )HIGGS −→ (X/B)rHIGGS g◦h

h

defined by (T − → B 0 , z) 7→ (T −−→ B, f ◦ z) is cocontinuous (cf. [2] III Définition 2.1), and induces a morphism of topos (cf. [2] III Proposition 2.3) 0

∼ fHIGGS : (X 0 /B 0 )rHIGGS −→ (X/B)r∼ HIGGS .

∗ We have fHIGGS (OX/B,1 ) = OX 0 /B 0 ,1 .

Proof. By [2] III Définition 2.1 and [2] II Proposition 1.4, the cocontinuity follows h

0

from the following fact: Let (T − → B 0 , z) ∈ Ob (X 0 /B 0 )rHIGGS and let (uα : Tα → T )α∈A 0 be a strict étale covering in C . Note that Tα ∈ Ob C r by Lemma IV.2.2.7 (1). Let hα denote h ◦ uα : Tα → B 0 and let zα denote z ◦ uα,1 : Tα,1 → X 0 . Then a more 0 h h phism ψ : (Te − → B 0 , ze) → (T − → B 0 , z) in (X 0 /B 0 )r factors thorough a morphism HIGGS

h

h

α uα : (Tα −−→ B 0 , zα ) → (T − → B 0 , z) for some α ∈ A if and only if the morphism

g◦e h

g◦h

ψ : (Te −−→ B, f ◦ ze) → (T −−→ B, f ◦ z) in (X/B)rHIGGS factors through a morphism g◦hα

g◦h

uα : (Tα −−−→ B, f ◦ zα ) → (T −−→ B, f ◦ z) for some α ∈ A.



Proposition IV.3.1.6. Let r, r0 , B, B 0 , X, X 0 , f , and g be the same as in Proposition IV.3.1.5 and assume that the morphism X 0 → X ×B1 B10 induced by f and g1 is an isomorphism. Then the functor 0

0

f ∗ : (X/B)rHIGGS −→ (X 0 /B 0 )rHIGGS

r (T ×B B 0 ), z 0 ) is continuous (cf. [2] III Définition 1.1) and defined by (T, z) 7→ (DHiggs is a right adjoint of the functor in Proposition IV.3.1.5, where z 0 is the composition of z×id r0 DHiggs (T ×B B 0 )1 → T1 ×B1 B10 −−−→ X ×B1 B10 ∼ = X 0 . Hence the functor f ∗ defines a morphism of sites, and it induces the morphism of topos fHIGGS in Proposition IV.3.1.5 (cf. [2] III Proposition 2.5).

Proof. For T ∈ Ob (X/B)rHIGGS and (Tα → T )α∈A ∈ Cov(T ), we have (f ∗ (Tα ) → f (T ))α∈A ∈ Cov(f ∗ (T )) and f ∗ (Tα ×T Tβ ) → f ∗ (Tα )×f ∗ (T ) f ∗ (Tβ ) is an isomorphism by 0 Proposition IV.2.2.9 (3). Hence f ∗ is continuous. Let j be the functor (X 0 /B 0 )rHIGGS → (X/B)rHIGGS in Proposition IV.3.1.5. Let (T, z) ∈ Ob (X/B)rHIGGS , let (T 0 , z 0 ) be f ∗ (T, z) ∗

352

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS 0

r and let u be the composition of T 0 = DHIGGS (T ×B B 0 ) → T ×B B 0 → T , which induces 0 0 a morphism j(T , z ) → (T, z) functorial in (T, z). It suffices to prove that, for any 0 (Te0 , ze0 ) ∈ Ob (X 0 /B 0 )rHIGGS , the map

Hom(X 0 /B 0 )r0

HIGGS

((Te0 , ze0 ), (T 0 , z 0 )) −→ Hom(X/B)rHIGGS (j(Te0 , ze0 ), (T, z)) 0

r induced by u is bijective. By the adjoint property of DHIGGS , we see that the map

e0 , T ) r (T HomC r0 0 (Te0 , T 0 ) −→ HomC/B /B

induced by u is bijective. For a morphism ψ : Te0 → T 0 over B 0 , we see that ψ1 : Te10 → T10 is compatible with ze0 and z 0 if and only if u1 ◦ ψ1 : Te10 → T1 is compatible with ze0 and z noting that the morphism X 0 → X ×B1 B10 is an isomorphism. 

Proposition IV.3.1.7. Let r ∈ N>0 ∪{∞}, let B be an object of C r , and let X 0 → X be a strict étale morphism of p-adic fine log formal schemes over B1 . Let f ∗ : (X/B)rHIGGS → (X 0 /B)rHIGGS be the functor associating to (T, z) the strict étale lifting T 0 → T of T1 ×X X 0 → T1 endowed with the morphism T10 = T1 ×X X 0 → X 0 . Then f ∗ is continuous and a right adjoint of the functor in Proposition IV.3.1.5. Hence the functor f ∗ defines a morphism of sites and it induces the morphism of topos fHIGGS in Proposition IV.3.1.5. Proof. For T ∈ Ob (X/B)rHIGGS and (Tα → T )α∈A ∈ Cov(T ), we have (f ∗ (Tα ) → f (T ))α∈A ∈ Cov(f ∗ (T )) and f ∗ (Tα ×T Tβ ) → f ∗ (Tα ) ×f ∗ (T ) f ∗ (Tβ ) is an isomorphism. Hence f ∗ is continuous. For (T, z) ∈ Ob (X/B)rHIGGS and (T 0 , z 0 ) := f ∗ (T, z), (T 00 , z 00 ) ∈ Ob (X 0 /B)rHIGGS , the composition with f : X 0 → X induces a bijection HomX 0 (T100 , T10 ) → HomX (T100 , T1 ). Let fT be the canonical morphism T 0 → T . Then, for an X 0 -morphism v1 : T100 → T10 and a lifting u : T 00 → T of fT,1 ◦ v1 : T100 → T1 , there exists a unique lifting v : T 00 → T 0 of v1 such that u = fT ◦v. Hence the composition with fT induces a bijection ∼ = Hom(X 0 /B)rHIGGS ((T 00 , z 00 ), (T 0 , z 0 )) − → Hom(X/B)rHIGGS ((T 00 , f ◦ z 00 ), (T, z)).  ∗

Corollary IV.3.1.8. Let r ∈ N>0 ∪ {∞}, let B be an object of C r , let X be a p-adic fine log formal scheme over B1 , and let (uα : Xα → X)α∈A be a strict étale covering. Let Xαβ denote Xα ×X Xβ and let uαβ : Xαβ → X be the canonical morphism. Let Fi (i = 1, 2) be sheaves of OX/B,1 -modules on (X/B)rHIGGS , and put Fi,α := u∗α,HIGGS (Fi ) and Fi,αβ := u∗αβ,HIGGS (Fi ). Then the following sequence is exact. Y Hom(Xα /B)rHIGGS (F1,α , F2,α ) Hom(X/B)rHIGGS (F1 , F2 ) → α∈A

⇒

Y (α,β)∈A2

Hom(Xαβ /B)rHIGGS (F1,αβ , F2,αβ ).

Proof. For a strict étale morphism f : X 0 → X, (T, z) ∈ Ob (X/B)rHIGGS and (T , z 0 ) := f ∗ (T, z) ∈ Ob (X 0 /B)rHIGGS , the morphism 0

∗ Fi (T, z) → fHIGGS∗ fHIGGS Fi (T, z) = Fi (T 0 , f ◦ z 0 )

is induced by the natural morphism T 0 → T . On the other hand, for an object (T, z) of (X/B)rHIGGS , letting (Tα , zα ) := u∗α (T, z) and (Tαβ , zαβ ) := u∗αβ (T, z), we see that the family of canonical morphisms (Tα,1 → T1 )α∈A is a strict étale covering and the natural morphism Tαβ,1 → Tα,1 ×T1 Tβ,1 is an isomorphism. Hence the sequence Y Y Fi → uα,HIGGS∗ u∗α,HIGGS Fi ⇒ uαβ,HIGGS∗ u∗αβ,HIGGS Fi α∈A

is exact. This implies the claim.

(α,β)∈A2



IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

353

Proposition IV.3.1.9. Let r ∈ N>0 ∪ {∞}, let B be an object of C r , and let X be a p-adic fine log formal scheme over B1 . (1) Let u : U → X be a morphism of p-adic fine log formal schemes, and define the presheaf U ∧ on (X/B)rHIGGS by   f is a morphism of p-adic fine log formal schemes ∧ . U ((T, z)) = f : T1 → U such that u ◦ f = z Then the functor ((X/B)rHIGGS )/U ∧ → (U/B)rHIGGS defined by ((T, z), f ) 7→ (T, f ) for (T, z) ∈ Ob (X/B)rHIGGS and f ∈ U ∧ ((T, z)), is an isomorphism of sites. (2) Let (D, zD ) ∈ Ob (X/B)rHIGGS . Then the topology on ((X/B)rHIGGS )/(D,zD ) induced by that of (X/B)rHIGGS coincides with the topology associated to the pretopology defined by strict étale coverings of T for ((T, z) → (D, zD )) ∈ Ob ((X/B)rHIGGS )/(D,zD ) . This follows from the following lemma. Lemma IV.3.1.10. Let C be a U -site defined by a pretopology Cov(X) (X ∈ Ob C) and let U be a presheaf on C with values in the category of U -sets. For an object X → U of C/U , we define the set Cov(X → U ) of families of morphisms in C/U with target X → U by Cov(X → U ) = {(fα : (Xα → U ) → (X → U ))α∈A |(fα : Xα → X)α∈A ∈ Cov(X)}.

(1) The data Cov(X → U ) ((X → U ) ∈ Ob C/U ) is a pretopology on C/U . (2) The topology T on C/U induced by that of C coincides with the topology T 0 on C/U defined by the pretopology Cov(X → U ) ((X → U ) ∈ Ob C/U ).

Proof. The proof of (1) is straightforward. Let us prove (2). Let R be a sieve of an object u : X → U of C/U . Then, by [2] III Proposition 5.2 (1), R is a covering for the topology T if and only if the sieve j! R of X(= j! (X → U )) is a covering, where j denotes the functor C/U → C forgetting morphisms to U . For an object X 0 of C, we have the following commutative diagram. a HomC/U ((u0 : X 0 → U ), (u : X → U )) HomC (X 0 , X) O 0 0 u : X →U O ? a ? 0 R(u : X 0 → U ). j R(X) !

u0 : X 0 →U

By [2] II Proposition 1.4, j! R is a covering if and only if there exists (fα : Xα → X)α∈A ∈ Cov(X) such that fα ∈ (j! R)(Xα ), which is equivalent to (fα : (u ◦ fα : Xα → U ) → (u : X → U ))α∈A ∈ R(u ◦ fα : X → U ) by the commutative diagram above. By using [2] II Proposition 1.4 again, we see that j! R is a covering if and only if R is a covering for the topology T 0 .  Let r, B, and X be the same as in Proposition IV.3.1.9 and let (D, zD ) be an object of (X/B)rHIGGS . Then, by Proposition IV.3.1.9, we see that the category of sheaves of sets on (((X/B)rHIGGS )/(D,zD ) )∼ is naturally equivalent to the category of the following data: a sheaf of sets F((T,z),v) on T1,´et for each object v : (T, z) → (D, zD ) of ((X/B)rHIGGS )/(D,zD ) and a morphism τf : f1−1 (F(T,z),v ) → F(T 0 ,z0 ),v0 for each morphism f : ((T 0 , z 0 ), v 0 ) → ((T, z), v) in ((X/B)rHIGGS )/(D,zD ) which satisfy the following two conditions: (i) The morphism τf is an isomorphism if the underlying morphism f : T 0 → T is strict étale and Cartesian. g (ii) We have τg ◦ g1−1 (τf ) = τf ◦g for any two composable morphisms ((T 00 , z 00 ), v 00 ) − → f

((T 0 , z 0 ), v 0 )− → ((T, z), v).

354

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

With the description above, one can define a morphism of topos ∼ πD : (((X/B)rHIGGS )/(D,zD ) )∼ → D1,´ et

as follows. The direct image functor πD∗ is defined by πD∗ F = F(D,zD ),id . The inverse −1 ∗ ∼ image πD G of G ∈ Ob D1,´ et is defined by the data v1 (G) for each v : (T, z) → (D, zD ) and ∼ =

the canonical isomorphism f1−1 v1−1 (G) − → (v10 )−1 (G) for each morphism f : ((T 0 , z 0 ), v 0 ) → ∗ ((T, z), v). It is obvious that the functor πD preserves finite inverse limits, and the pair ∗ (πD , πD∗ ) defines the desired morphism of topos πD by the following lemma. ∗ Lemma IV.3.1.11. The functor πD is a left adjoint of the functor πD∗ .

Proof. It suffices to construct a functorial isomorphism ∗ Hom(G, πD∗ F) ∼ G, F) = Hom(πD

∼ r ∼ for G ∈ Ob D1,´ et and F ∈ Ob (((X/B)HIGGS )/(D,zD ) ) . One can construct a map F from the left to the right by associating to ϕ : G → πD∗ F = F(D,zD ),id , the morphism ∗ G → F defined by τv ◦ v −1 (ϕ) : v −1 (G) → v −1 (F(D,zD ),id ) → F(T,z),v for v : (T, z) → πD ∗ (D, zD ). One can define the map G in the opposite direction by sending ψ : πD G → F to ψ(D,zD ),id : G → F(D,zD ),id = πD∗ F. It is straightforward to check that F ◦ G = id and G ◦ F = id. 

Let jD be the canonical morphism of topos (((X/B)rHIGGS )/(D,zD ) )∼ → (X/B)r∼ HIGGS . ∗ preserves injective sheaves of abelian groups. By [2] V Proposition 4.11 1), the functor jD On the other hand, the direct image functor πD∗ is obviously exact by its construction. Hence, for any sheaf of abelian groups F on (((X/B)rHIGGS )/(D,zD ) )∼ , we have an isomorphism (IV.3.1.12)

∗ RΓ((D, zD ), F) ∼ F) ∼ = RΓ(((D, zD ), id), jD = RΓ(D1,´et , F(D,zD ) ),

∗ F = F(D,zD ) for the second isomorphism. where we use πD∗ jD

IV.3.2. Lattices of projective modules. Let X be a p-adic formal scheme flat over Zp . In this subsection, we study integral structures of finitely generated projective OX,Qp -modules. Recall that we always work on X´et . Let Xétaff be the full subcategory of X´et consisting of affine p-adic formal schemes étale over X endowed with the topology associated to the pretopology defined by étale coverings. Then the topology of Xétaff coincides with the one induced by the topology of X´et , and the inclusion functor Xétaff → X´et induces an ∼ ∼ equivalence of categories X´e∼t − → Xétaff (cf. [2] III Corollaire 3.3, Théorème 4.1). For any U ∈ Ob Xétaff , the functor Γ(U, −) from the category of sheaves of abelian groups on Xétaff (or on X´et ) to that of abelian groups commutes with filtered direct limits and direct sums. Let A be Γ(X, OX ), which is p-adically complete and separated and p-torsion free. Let PM(OX,Qp ) (resp. PM(AQp )) denote the category of finitely generated projective OX,Qp (resp. AQp )-modules. Lemma IV.3.2.1. (1) For M ∈ Ob PM(OX,Qp ), we have Γ(X, M) ∈ Ob PM(AQp ). (2) For M ∈ Ob PM(AQp ), the presheaf M of OX,Qp -modules on Xétaff defined by Γ(U, M)= M ⊗A Γ(U, OU ) (U ∈ Ob Xétaff ) is a finitely generated and projective sheaf of OX,Qp -modules on Xétaff . (3) The constructions of (1) and (2) define equivalences of categories between PM(OX,Qp ) and PM(AQp ) inverse to each other.

IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

355

Proof. (1) There exist an object M0 of PM(OX,Qp ) and an isomorphism of OX,Qp ⊕r modules M ⊕ M0 ∼ . By taking Γ(X, −) of the isomorphism, we obtain the claim. = OX,Q p (2) There exist an A-module M 0 and an isomorphism of A-modules M ⊕ M 0 ∼ = A⊕r Qp . ⊕r ∼ 0 ∼ For U ∈ Ob Xétaff , we have an isomorphism Γ(U, M)⊕(M ⊗A Γ(U, OU )) = Γ(U, OU )Qp = ⊕r ⊕r Γ(U, OX,Q ). Hence M is a sheaf and isomorphic to a direct summand of OX,Q . p p (3) For M and M as in (2), we have Γ(X, M) = M by definition. Hence it remains to prove that, for M ∈ Ob PM(OX,Qp ) and U ∈ Ob Xétaff , the natural homomorphism Γ(X, M) ⊗A Γ(U, OU ) → Γ(U, M) is an isomorphism. This is reduced to the case M = OX,Qp by taking M0 ∈ Ob PM(OX,Qp ) as in (1).  It is natural to consider finitely generated A-modules M with MQp projective over AQp as integral structures of objects of PM(AQp ). This is justified by Lemma IV.3.2.3 below. Let LPM(A) denote the category of such A-modules. For M ∈ Ob LPM(A), let Mtor denote the kernel of the natural homomorphism M → MQp . Lemma IV.3.2.2. Let M ∈ Ob LPM(A). (1) There exists an N ∈ N such that pN Mtor = 0. (2) The A-module M is p-adically complete and separated. Proof. (1) Choose a surjective homomorphism f : A⊕r → M of A-modules. Since MQp is projective, there exists a decomposition A⊕r Qp = M1 ⊕ M2 as AQp -modules such that M2 = KerfQp . Since M1 and M2 are finitely generated AQp -modules, there exist finitely generated A-submodules Mi◦ ⊂ Mi,Qp (i = 1, 2) such that M1◦ ⊕ M2◦ ⊂ A⊕r ◦ and Mi,Q = Mi (i = 1, 2). There exist N1 , N2 ∈ N such that pN1 A⊕r ⊂ M1◦ ⊕ M2◦ p N2 and p f (M2◦ ) = 0. For any x ∈ A⊕r such that fQp (x) = 0, we have pN1 x ∈ M2◦ and pN1 +N2 f (x) ∈ pN2 f (M2◦ ) = 0. (2) Put M = Im(M → MQp ), which is an object of LPM(A). By (1), there exists an exact sequence 0 → Mtor → limm M/pm M → limm M /pm M → 0. Hence we are ←− ←− reduced to the case where M is p-torsion free. Then there exists a p-torsion free M 0 ∈ Ob LPM(A) and an isomorphism MQp ⊕ MQ0 p ∼ = A⊕r Qp . Choosing N ∈ N such that N ⊕r 0 −N ⊕r p A ⊂ M ⊕ M ⊂ p A , we see that M is p-adically complete and separated.  For an additive category A , we define the additive category AQ by Ob AQ = Ob A and HomAQ (x, y) = Hom(x, y) ⊗Z Q for x, y ∈ Ob A .

Lemma IV.3.2.3. The natural functor LPM(A) → PM(AQp ) defined by M 7→ MQp ∼ induces an equivalence of categories LPM(A)Q − → PM(AQp ). Proof. Since the functor LPM(A) → PM(AQp ) is obviously essentially surjective, it remains to prove that the natural homomorphism HomA (M, M 0 ) ⊗Z Q −→ HomAQp (MQp , MQ0 p )

is an isomorphism for M, M 0 ∈ Ob LPM(A). This follows from Lemma IV.3.2.2 (1) for M 0 . For the surjectivity, note that pN : M 0 → M 0 factors through Im(M 0 → MQ0 p ) for a sufficiently large N ∈ N by the lemma.  For a general p-adic affine formal scheme X, we don’t have a general theory relating coherent (or finitely generated) OX -modules and certain A-modules. Therefore we will interpret an object M of LPM(A) in terms of OX -modules by taking the inverse limit of the quasi-coherent OXm -modules associated to M/pm M . First we interpret objects of LPM(A) in terms of inverse systems of Am -modules. We consider an inverse system of Am -modules (Mm )m∈N> 0 such that M1 is a finitely

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generated A1 -module, the transition maps induce isomorphisms Mm+1 ⊗Am+1 Am − → Mm (m ∈ N>0 ), and (limm Mm )Qp is a projective AQp -module. Let LPM(A• ) be the category ←− of inverse systems of Am -modules as above. A morphism is a compatible system of Am linear homomorphisms. Proposition IV.3.2.4 (cf. [39] Chap. 0 Proposition 7.2.9). The functor LPM(A) → LPM(A• ) defined by M 7→ (M ⊗A Am )m∈N>0 is an equivalence of categories. A quasiinverse is given by (Mm ) 7→ limm Mm . ←− ∼ =

→ limm (M ⊗A Am ) by Lemma Proof. For an object M of LPM(A), we have M − ←− IV.3.2.2 (2). Hence (M ⊗A Am )m is an object of LPM(A• ). Conversely, for an object (Mm ) of LPM(A• ), the inverse limit M := limm Mm is a finitely generated A-module ←− and the natural homomorphism M/pm M → Mm (m ∈ N>0 ) is an isomorphism by [39] Chap. 0 Proposition (7.2.9). Hence M is an object of LPM(A) and the functor (Mm ) 7→ M is a quasi-inverse.  We define the category LPM(OX• ) as follows. An object is an inverse system (Mm )m∈N>0 of quasi-coherent OXm -modules such that Γ(X1 , M1 ) is a finitely generated ∼ =

A1 -module, the transition maps induce isomorphisms Mm+1 ⊗OXm+1 OXm − → Mm , and (limm Γ(X, Mm ))Qp is a projective AQp -module. A morphism is a compatible system of ←− OXm -linear homomorphisms. Then we have a canonical equivalence of categories (IV.3.2.5)

∼

LPM(A• ) −→ LPM(OX• )

fm ) and (Mm ) 7→ (Γ(X, Mm )). defined by (Mm ) 7→ (M Let LPM(OX ) be the full subcategory of the category of OX -modules consisting of OX -modules M such that Mm := M ⊗OX OXm is a quasi-coherent OXm -module for every m ∈ N>0 , Γ(X, M1 ) is a finitely generated A1 -module, the natural morphism M → limm Mm is an isomorphism, and Γ(X, M)Qp is a finitely generated and projective ←− AQp -module. Lemma IV.3.2.6. (1) The functor LPM(OX ) → LPM(OX• ); M 7→ (M ⊗OX OXm )m is an equivalence of categories. Its quasi-inverse is given by (Mm ) 7→ (limm Mm ). ←− (2) For M ∈ Ob LPM(OX ) and U ∈ Ob Xétaff , the natural Γ(U, OU )-linear ho∼ = momorphism Γ(X, M) ⊗A Γ(U, OU ) − → Γ(U, M) is an isomorphism and Γ(U, M) is p-adically complete and separated. Proof. (1) It is obvious that (M ⊗OX OXm )m ∈ LPM(OX• ). To prove that the functor in the opposite direction is well-defined and gives a quasi-inverse, it suffices to prove that, for (Mm ) ∈ LPM(OX• ), the natural morphism (limn Mn )⊗OX OXm → Mm ←− is an isomorphism. Let U ∈ Ob Xétaff and put A0 = Γ(U, OX ). Put M := limm Mm , ←− Mm = Γ(X, Mm ) and M := limm Mm . Then by (IV.3.2.5), Proposition IV.3.2.4, and ←− Lemma IV.3.2.2 (2), we see that M ⊗A A0 is p-adically complete and separated. Hence we ∼ = ∼ M ⊗A A0 , which implies Γ(U, M)/pm Γ(U, M) → have Γ(U, M) = limm Mm ⊗Am A0m = ←− ∼ = Mm ⊗Am A0m ∼ → Mm . = Γ(U, Mm ). Varying U , we obtain M/pm M − (2) We have (M ⊗OX OXm ) ∈ Ob LPM(OX• ) and M∼ lim = ←−m (M ⊗OX OXm ) by (1). Hence we obtain (2) from the above proof of (1).  Proposition IV.3.2.7. (1) For M ∈ LPM(OX ), we have Γ(X, M) ∈ LPM(A). The functor LPM(OX ) → LPM(A) defined by M 7→ Γ(X, M) is an equivalence of categories.

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(2) For M ∈ LPM(OX ), we have MQp ∈ PM(OX,Qp ). The functor LPM(OX ) → ∼ PM(OX,Qp ) defined by M 7→ MQp induces an equivalence of categories LPM(OX )Q − → PM(OX,Qp ). ∼

Proof. (1) The equivalence of categories LPM(OX ) − → LPM(A) obtained by the composition of the equivalences of categories in Proposition IV.3.2.4, (IV.3.2.5), and Lemma IV.3.2.6 is given by M 7→ limm Γ(X, M/pm M), and the latter A-module is ←− isomorphic to Γ(X, limm M/pm M) ∼ = Γ(X, M). ←− (2) By combining Lemma IV.3.2.1, Lemma IV.3.2.3, and the equivalence of categories in (1), we obtain a functor LPM(OX ) → PM(OX,Qp ) inducing an equivalence ∼ of categories LPM(OX )Q → PM(OX,Qp ). By Lemma IV.3.2.1, we see that the image of M ∈ Ob LPM(OX ) under this functor is given by U 7→ Γ(X, M) ⊗A Γ(U, OU )Qp (U ∈ Ob Xétaff ). Hence the claim follows from Lemma IV.3.2.6 (2).  Lemma IV.3.2.8. Let f : X 0 → X be a morphism of affine p-adic formal schemes flat over Zp , and put A = Γ(X, OX ) and A0 = Γ(X 0 , OX 0 ). Then the inverse image functors PM(OX,Qp ) → PM(OX 0 ,Qp ), PM(AQp ) → PM(A0Qp ), LPM(A) → LPM(A0 ), LPM(A• ) → LPM(A0• ), LPM(OX• ) → LPM(OX•0 ), and LPM(OX ) → LPM(OX 0 ) 0 , and F 7→ are defined by ⊗f −1 (OX,Qp ) OX 0 ,Qp , ⊗AQp A0Qp , ⊗A A0 , ⊗Am A0m , ⊗f −1 (OXm ) OXm 0 limm (F ⊗f −1 (OX ) OXm ). Furthermore these functors are compatible with the equivalences ←− of categories in Lemma IV.3.2.1, Lemma IV.3.2.3, Proposition IV.3.2.4, (IV.3.2.5), Lemma IV.3.2.6, and Proposition IV.3.2.7 Proof. For M ∈ Ob PM(OX,Qp ) (resp. M ∈ Ob PM(AQp )), there exist M0 ∈ ⊕r Ob PM(OX,Qp ) (resp. M 0 ∈ Ob PM(A0Qp )) and an isomorphism M ⊕ M0 ∼ = OX,Q p ⊕r 0 0 ∼ (resp. M ⊕ M = AQp ). By taking ⊗f −1 (OX,Qp ) OX 0 ,Qp (resp. ⊗AQp AQp ), we see that M ⊗OX,Qp OX 0 ,Qp (resp. M ⊗AQp A0Qp ) is an object of PM(OX 0 ,Qp ) (resp. PM(A0Qp )). Hence the first three functors are well-defined and compatible with the equivalences of categories in Lemma IV.3.2.1 and Lemma IV.3.2.3. For (Mm ) ∈ Ob LPM(A• ) and M = limm Mm ∈ Ob LPM(A), M ⊗A A0 is p-adically complete and separated by Lemma ←− IV.3.2.2 (2). Hence we have an isomorphism M ⊗A A0 ∼ (M ⊗ A0 ). This = lim ←−m m Am m implies the well-definedness of the fourth and fifth functors and the compatibility with the equivalences of categories in Proposition IV.3.2.4 and (IV.3.2.5). Now the claims for the last functor and the equivalence of categories in Lemma IV.3.2.6 are obvious.  Definition IV.3.2.9. (1) For a p-adic formal scheme X flat over Zp , we define the category LPMloc (OX ) to be the category of OX -modules which belong to LPM(OX ) strict étale locally on X. (2) For a morphism f : X 0 → X of p-adic log formal schemes flat over Zp , we define ∗ the functor fb∗ : LPMloc (OX ) → LPMloc (OX 0 ) by fb∗ (M) = limm fm (M ⊗OX OXm ). ←− (This is well-defined by Lemma IV.3.2.8.) For an object M of LPMloc (OX ), the sheaf of OXm -modules Mm := M ⊗OX OXm is quasi-coherent and the natural morphism M → limm Mm is an isomorphism. For two ←− composable morphisms g : X 00 → X 0 and f : X 0 → X of p-adic fine log formal schemes flat over Zp , we have a canonical isomorphism of functors gb∗ ◦ fb∗ ∼ ◦ g)∗ . = (f[ Lemma IV.3.2.10. Let M be an object of LPM(OX ) and let M be the image of M → MQp . (1) The OX -module M is an object of LPM(OX ). (2) There exists N ∈ N such that pN · Ker(M → M) = 0.

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(3) Let M and M be the images Γ(X, M) and Γ(X, M) of M and M under the equivalence of categories in Proposition IV.3.2.7 (1). Then the natural homomorphism M/Mtor → M is an isomorphism. Proof. Put M = Γ(X, M) and M = M/Mtor , which is an object of LPM(A) by 0 Proposition IV.3.2.7 (1), and let M be the object of LPM(OX ) corresponding to M by loc. cit. By Lemma IV.3.2.2 (1), there exists N ∈ N such that pN Mtor = 0. Under this notation, it suffices to prove that, for every U ∈ Ob Xétaff , the homomorphism 0 0 Γ(U, M) → Γ(U, M ) is surjective, its kernel is annihilated by pN , and Γ(U, M ) is ptorsion free. Put AU = Γ(U, OU ). By Lemma IV.3.2.6 (2), we have Γ(U, M) ∼ = M ⊗A AU 0 and Γ(U, M ) ∼ = M ⊗A AU , and these modules are p-adically complete and separated. Hence the first and second claims hold. Since M is p-torsion free, the multiplication by p on M induces an injective homomorphism M /pm−1 M → M /pm M . By taking the extension of scalars by the flat homomorphism A/pm A → AU /pm AU and then limm , we ←− see that M ⊗A AU is p-torsion free.  Lemma IV.3.2.11. Let f : X 0 → X be a morphism of affine p-adic formal schemes flat over Zp , let M be an object of LPM(OX ) and let M be the image of M → MQp , which is an object of LPM(OX ) by Lemma IV.3.2.10. If the reduction mod pm of f is flat for every m ∈ N>0 , then, via the identification (fb∗ (M))Q = f ∗ (MQ ) = f ∗ (MQ ) = fb∗ (M)Q (cf. Lemma IV.3.2.8), we have fb∗ (M) = Im(fb∗ (M) → fb(M)Q ). In particular, if M is p-torsion free, then fb∗ (M) is p-torsion free. Proof. Put A = Γ(X, OX ) and A0 = Γ(X 0 , OX 0 ). By Lemma IV.3.2.10 and Lemma IV.3.2.8, it suffices to prove the corresponding claim for the functor LPM(A) → LPM(A0 ) : M 7→ M ⊗A A0 . For M ∈ Ob LPM(A) and M = Im(M → MQ ), the homomorphism M ⊗A A0 → M ⊗A A0 is surjective and the kernel is annihilated by a power of p. Since M is p-torsion free, the multiplication by p on M /pm M factors through an injective homomorphism M /pm−1 M → M /pm M . Since A/pm A → A0 /pm A0 is flat, it induces an injective homomorphism M ⊗A (A0 /pm−1 A0 ) → M ⊗A (A0 /pm A0 ). By taking the inverse limit over m, we see that M ⊗A A0 is p-torsion free. This completes the proof.  IV.3.3. Higgs isocrystals and Higgs crystals. For r ∈ N>0 ∪{∞} (resp. r = ∞), B ∈ Ob C r , and a p-adic fine log formal scheme X over B1 , we define Higgs isocrystals of level r (resp. Higgs crystals) on X/B. Definition IV.3.3.1. Let r ∈ N>0 ∪ {∞}, let B ∈ C r , and let X be a p-adic fine log formal scheme over B1 . A Higgs isocrystal on X/B of lever r (or a Higgs isocrystal on (X/B)rHIGGS ) is a sheaf F of OX/B,1,Q -modules on (X/B)rHIGGS such that the corresponding system of sheaves of OT1 ,Q -modules F(T,z) on T1,´et for (T, z) ∈ Ob (X/B)rHIGGS satisfies the following conditions. (i) For any morphism u : (T 0 , z 0 ) → (T, z) in (X/B)rHIGGS , τu : u∗1 (F(T,z) ) → F(T 0 ,z0 ) is an isomorphism. (ii) There exists a strict étale covering (Xα → X) satisfying the following condition: For any object (T, z) of (X/B)rHIGGS such that T1 is affine and z factors through Xα → X for some α, the OT1 ,Q -module F(T,z) is finitely generated and projective. We define a morphism of Higgs isocrystals on X/B of level r to be a morphism of sheaves of OX/B,1,Q -modules. Let HCrQp (X/B) denote the category of Higgs isocrystals on X/B of level r. For a p-adic fine log formal scheme U over X, we say that a Higgs isocrystal F on X/B of level r is finite on U if there exists a strict étale covering (Xα →

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359

X) for F as in condition (ii) such that the morphism U → X factors through Xα → X for some α. We write HCrQp ,U -fin (X/B) for the full subcategory of HCrQp (X/B) consisting of objects finite on U . Definition IV.3.3.2. Let B be an object of C ∞ and let X be a p-adic fine log formal scheme over B1 . A Higgs crystal on X/B is a sheaf F of OX/B,1 -modules on (X/B)∞ HIGGS such that the corresponding system of sheaves of OT1 -modules F(T,z) on T1,´et for (T, z) ∈ Ob (X/B)∞ HIGGS satisfies the following conditions. (i) For every (T, z) ∈ Ob (X/B)∞ HIGGS , the OT1 -module F(T,z) is an object of loc LPM (OT1 ) (cf. Definition IV.3.2.9). (ii) For every morphism u : (T 0 , z 0 ) → (T, z) in (X/B)∞ HIGGS , the natural morphism ∗ u b (F(T,z) ) → F(T 0 ,z0 ) is an isomorphism. (iii) There exists a strict étale covering (Xα → X) satisfying the following condition: For any (T, z) ∈ Ob (X/B)∞ HIGGS such that T1 is affine and z factors through Xα → X for some α, the OT1 -module F(T,z) is an object of LPM(OT1 ). We define a morphism of Higgs crystals on X/B to be a morphism as sheaves of OX/B,1 -modules. Let HCZp (X/B) denote the category of Higgs crystals on X/B. For a p-adic fine log formal scheme U over X, we say that a Higgs crystal F on X/B is finite on U if there exists a strict étale covering (Xα → X) for F as in the condition (iii) such that the morphism U → X factors through Xα → X for some α. We write HCZp ,U -fin (X/B) for the full subcategory of HCZp (X/B) consisting of objects finite on U . Remark IV.3.3.3. We can define a Higgs crystal on (X/B)rHIGGS for r ∈ N>0 in the same way as Definition IV.3.3.2. However it is not useful as an integral structure of a Higgs crystal of level r as we will see in Remark IV.3.4.13. IV.3.4. Local Description. Let r ∈ N>0 ∪ {∞}, let B be an object of C r , let X be a p-adic fine log formal scheme over B1 , let Y → B be a smooth Cartesian morphism in C , and let X → Y1 be an immersion over B1 . Let D denote the Higgs envelope of level r of X ,→ Y . In this subsection, we give a description of a Higgs isocrystal of level r on X/B in terms of an OD1 ,Qp -module on D1 with a derivation taking its values in ξ −1 ΩY1 /B1 compatible with the canonical derivation on OD1 (IV.2.4.8). We also give a similar description of a Higgs crystal when r = ∞. See Theorem IV.3.4.16. Similarly to the crystalline site, the descriptions are given via a kind of stratifications on modules on D1 defined in terms of the Higgs envelopes of the diagonal immersions of X into the fiber products of copies of Y . Lemma IV.3.4.1. For any object (T, z) of (X/B)rHIGGS such that T1 is affine, a morphism z : T1 → X extends to a morphism in C/B from T to the object (X ,→ Y ) of C (cf. Definition IV.2.2.1 (3)). Proof. This immediately follows from Corollary IV.2.2.5 and Sublemma IV.3.4.2 (2) below.  Sublemma IV.3.4.2. (1) Let j : U → U be an exact closed immersion of fine log schemes whose underlying morphism of schemes is a nilpotent immersion. Let W0

s0

/V f

i

 W

W

s

 /U

0

s0

i

 W

s

/V  /U

f

be commutative diagrams of fine log schemes such that the right diagram is the base change of the left one by the morphism j, f is smooth, i is an exact closed immersion

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whose underlying morphism of schemes is a nilpotent immersion, and the underlying scheme of W is affine. Then, for any morphism g : W → V such that f ◦ g = s and g ◦ i = s0 , there exists a lifting g : W → V of g such that f ◦ g = s and g ◦ i = s0 . (2) Consider a commutative diagram of p-adic fine log formal schemes W0

s0

i

 W

s

/V  /U

f

such that f is smooth, i is an exact closed immersion, the morphism of schemes underlying the reduction mod p of i is a nilpotent immersion, and the underlying formal scheme of W is affine. Then there exists a morphism g : W → V such that f ◦ g = s and g ◦ i = s0 . 0

Proof. (1) Let I (resp. I) be an ideal of OW (resp. OW ) defining W 0 in W (resp. W in W ). By factorizing the exact closed immersion i : W 0 → W into a sequence of exact closed immersions defined by I n for n ∈ N>0 , we are reduced to the case I 2 = 0. By the assumptions on i and f , there exists a morphism g : W → V such that f ◦ g = s and g ◦ i = s0 . Since Ω1V /U is locally free of finite type and W is affine, the homomorphism HomOW 0 (OW 0 ⊗OV Ω1V /U , I) → HomOW (OW 0 ⊗OV Ω1V /U , I) is surjective. By [50] Proposition (3.9), we can modify g in such a way that the base change of g by j coincides with g. (2) We obtain the claim by applying (1) repeatedly to the reduction mod pn of the diagram in question.  We define Y (ν), D(ν) (ν ∈ N), D, ∆D , pi , pij , and qi in the same way as after the proof of Lemma IV.2.4.2. Definition IV.3.4.3. (1) We define the category HSrQp (X, Y /B) as follows. An object ∼ =

is a pair (M, ε) of an OD1 ,Q -module and an OD(1)1 ,Q -linear isomorphism ε : p∗2 (M) − → p∗1 (M) satisfying the following conditions. (i) There exists a strict étale covering (Xα → X)α∈A such that M|D1,α (D1,α = D1 ×X Xα ) is a finitely generated projective OD1,α ,Q -module for every α ∈ A. ∼ =

(ii) (a) ∆∗D (ε) = idM . (b) p∗12 (ε) ◦ p∗23 (ε) = p∗13 (ε) : q3∗ (M) −→ q1∗ (M). A morphism f : (M, ε) → (M0 , ε0 ) in HSrQp (X, Y /B) is an OD1 ,Q -linear morphism f : M → M0 such that ε0 ◦ p∗2 (f ) = p∗1 (f ) ◦ ε. (2) Suppose that r = ∞. We define the category HSZp (X, Y /B) as follows. An object ∼ =

is a pair (M, ε) of an object M of LPMloc (OD1 ) and an isomorphism ε : pb∗2 (M) − → pb∗1 (M) in LPMloc (OD(1)1 ) satisfying the following conditions. (i) There exists a strict étale covering (Xα → X)α∈A such that Xα is affine and M|D1,α (D1,α = D1 ×X Xα ) is an object of LPM(OD1,α ) for every α ∈ A. ∼ = b ∗ (ε) = idM . (b) pb∗ (ε) ◦ pb∗ (ε) = pb∗ (ε) : qb∗ (M) −→ qb1∗ (M). (ii) (a) ∆ 3 12 23 13 D 0 0 A morphism f : (M, ε) → (M , ε ) in HSZp (X, Y /B) is a morphism f : M → M0 in LPMloc (OD1 ) such that ε0 ◦ pb∗2 (f ) = pb∗1 (f ) ◦ ε. Proposition IV.3.4.4. (1) There exists a canonical equivalence of categories between HCrQp (X/B) and HSrQp (X, Y /B). (2) Suppose r = ∞. Then there exists a canonical equivalence of categories between HCZp (X/B) and HSZp (X, Y /B).

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361

Proof. Standard argument using Lemma IV.3.4.1 and the adjoint property of the r functor DHiggs .  Let M be an OD1 -module endowed with an additive morphism θ : M → ξ −1 M ⊗OY1 such that θ(ax) = aθ(x) + x ⊗ θ(a) for local sections a ∈ OD1 and x ∈ M. (See (IV.2.4.8) for θ on OD1 .) Since the right vertical morphism in (IV.2.4.9) is 0 if N = 1, the morphism θ is p−1 Y,1 (OY1 )-linear, where pY = (pY,N ) denotes the natural morphism D → Y . Hence we can define an additive morphism Ω1Y1 /B1

θq : ξ −q M ⊗OY1 ΩqY1 /B1 → ξ −q−1 M ⊗OY1 Ωq+1 Y1 /B1

by θq (ξ −q x ⊗ ω) = ξ −q θ(x) ∧ ω. Lemma IV.3.4.5. If θ1 ◦ θ = 0, then θq+1 ◦ θq = 0 for every q ∈ N. 0

0

0

Proof. For q, q 0 ∈ N, y ∈ ξ −q M⊗OY1 ΩqY1 /B1 , and η ∈ ΩqY1 /B1 , we have θq+q (ξ −q y∧ 0

η) = ξ −q (θq (y)) ∧ η. Hence, for x ∈ M and ω ∈ ΩqY1 /B1 , we have θq+1 ◦ θq (ξ −q x ⊗ ω) = θq+1 (ξ −q θ(x) ∧ ω) = ξ −q θ1 (θ(x)) ∧ ω = 0.  Lemma IV.3.4.6. Let M be as above, and suppose that Y1 is affine and we are given (ti,N ) ∈ limN Γ(YN , MYN ) (1 ≤ i ≤ d) such that {d log(ti,N )}1≤i≤d is a basis of Ω1YN /BN ←− for every N . Let θi (1 ≤ i ≤ d) be the endomorphisms of M defined by θ(x) = P −1 θi (x) ⊗ d log(ti,1 ) for x ∈ M. 1≤i≤d ξ (1) We have θ1 ◦ θ = 0 if and only if θi ◦ θj = θj ◦ θi for every i, j ∈ {1, 2, . . . , d}. (2) If θ1 ◦ θ = 0, then we have X m! θm1 (a)θm2 (x) θm (ax) = m !m ! 1 2 m +m =m 1

2

for a ∈ OD1 and Q x ∈ M. Here we define the endomorphisms θi of OD1 in the same way as M and θm = 1≤i≤d θimi for m = (mi ) ∈ Nd . Proof. (1) follows from the formula X θ1 ◦ θ(x) = ξ −2 (θi ◦ θj − θj ◦ θi )(x) ⊗ (d log ti,1 ∧ d log tj,1 ) 1≤i0 . In the case I N N (resp. II), we also assume that M0 is a finitely generated and projective OD10 ,Q -module (resp. an object of LPM(OD10 )). Let AD0 (ν) = (AD0 (ν)N ) be the object of A•r defined by AD0 (ν)N = Γ(D0 (ν)N , OD0 (ν)N ). We write AD0 for AD0 (0). Put M 0 = Γ(D10 , M). 0 0∗ 0 For i ∈ {1, 2} (resp. i ∈ {1, 2, 3}), let p0∗ i M (resp. qi M ) denote the scalar extension of 0 0∗ 0∗ 0∗ M by pi,1 : AD0 ,1 → AD0 (1)1 (resp. qi,1 : AD0 ,1 → AD0 (2)1 ), and let p0∗ i (resp. qi ) also 0 0∗ 0 0 0∗ 0 denote the natural homomorphism M → pi M (resp. M → qi M ). Note that, in the 0 0 0 0∗ 0 0 0∗ case II, we have p0∗ b0∗ bi,1 M0 ) and these are i M = Γ(D (1)1 , p i,1 M ) and qi M = Γ(D (2)1 , q p-adically complete and separated by Lemma IV.3.2.8 and Lemma IV.3.2.2 (2). In the case I, choose a finitely generated AD0 ,1 -submodule M 0◦ of M 0 such that MQ00 = 0 M . Since M 0 is a finite projective AD0 ,1,Q -module and hence a direct summand of a finite free AD0 ,1,Q -module, the isomorphisms (IV.2.4.11) and (IV.2.4.12) induce isomorphisms ∧

 (IV.3.4.8)

0 ∼ p0∗ i M =

M

(|m|) AD0 ,1 r M 0◦ W m 

m∈Nd

qj0∗ M 0 ∼ =

(i = 1, 2),

∧

 (IV.3.4.9)

⊗Q

M

l,n∈Nd

(|l+n|)r

AD0 ,1

M 0◦ W n W 0l  ⊗ Q

(j = 1, 2, 3),

IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

363

where ∧ denotes the p-adic completion. Similarly, in the case II, we obtain isomorphisms ∧  M 0 ∼ (IV.3.4.10) M 0W m (i = 1, 2), p0∗ i M = m∈Nd

∧

 (IV.3.4.11)

qj0∗ M 0 ∼ =

M

M 0 W n W 0l 

(j = 1, 2, 3).

l,n∈Nd

We define endomorphisms Θ0m (m ∈ Nd ) of M 0 by X Y mi 0 ε0 (p0∗ p0∗ wi,1 2 (x)) = 1 (Θm (x)) ⊗ m∈Nd

1≤i≤d

(x ∈ M 0 ).

We define endomorphisms θi0 (1 ≤ i ≤ d) of M 0 by the formula θ0 (x) = d log(ti,1 ) for x ∈ M 0 .

P

1≤i≤d

ξ −1 θi0 (x)⊗

Lemma IV.3.4.12. Let notation and assumption be as above. In the case I (resp. II), we endow M 0 with the p-adic topology defined by M 0◦ (resp. the p-adic topology). Assume that M0 is p-torsion free in the case II. (1) The isomorphism ε0 satisfies the condition (ii) (b) in Definition IV.3.4.3 if and Q 0mi 1 only if θi0 ◦ θj0 = θj0 ◦ θi0 (i 6= j) and Θ0m = m! (m ∈ Nd ). 1≤i≤d θi 0 0 (2) If the equivalent conditions in (1) hold, then (M , θ ) satisfies the following convergence: Q −1 0mi 1 (Conv) For any x ∈ M 0 , p−[|m|r ] m! (x) converges to 0 as |m| → ∞, 1≤i≤d θi −1 where |m|r = 0 if r = ∞. (3) In the case II, if the equivalent conditions in (1) hold, then (M 0 , θ0 ) satisfies the following integrality: Q 0mi 1 (Int) For any x ∈ M 0 and m ∈ Nd , we have m! (x) ∈ M 0 . 1≤i≤d θi Proof. The claim (3) immediately follows from (1). We prove (1) and (2). (1) To simplify the notation, we omit 0 from the notation M 0 , Θ0m , θi0 , ε0 , p0i , p0ij , 0 to pi , pij , and qi , respectively. We have and qi0 . We also abbreviate p0i,N , p0ij,N , and qi,N θi = Θ1i by definition. For x ∈ M , we have the following equalities in q3∗ (M ): ! X ∗ ∗ l ∗ ∗ ∗ ∗ p12 (ε) ◦ p23 (ε)(q3 (x)) = p12 (ε) q2 (Θl (x)) ⊗ p23 (w ) l∈Nd

=

X

q1∗ (Θn

l,n∈Nd

p∗13 (ε)(q3∗ (x)) =

X m∈Nd

◦ Θl (x)) ⊗ p∗23 (wl )p∗12 (wn )

∗ q1∗ (Θm (x)) ⊗ q13 (wm ),

Q mi where wm = 1≤i≤d wi,1 for m = (mi ) ∈ Nd . Since p∗13 (ui,2 ) = p∗23 (ui,2 )p∗12 (ui,2 ) and ∗ ξAD0 (2)1 = 0, we have p13 (wi,1 ) = p∗23 (wi,1 ) + p∗12 (wi,1 ). Hence, by (IV.3.4.9) and (IV.3.4.11), the condition (ii) (b) in Definition IV.3.4.3 is equivalent to Θn ◦ Θl =

(n + l)! Θn+l n!l!

(n, l ∈ Nd ).

These equalities imply θi ◦ θj = Θ1i +1j = θj ◦ θi (i 6= j) and θi ◦ Θm = (mi + 1)Θm+1i ⇐⇒ Θm+1i =

1 θi ◦ Θm mi + 1

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

for m ∈ Nd and i ∈ {1, 2, . . . , d}. This implies the necessity. The sufficiency is obvious. m m (m) (m) ,0 (2) If r ∈ N>0 , we have p[ r ]+1 AD0 ,1 ⊂ AD0 ,1r = AD0 ,1r ⊂ p[ r ] AD0 ,1 by Lemma (m)

IV.2.3.4. If r = ∞, we have AD0 ,1r = AD0 ,1 by definition. Hence the claim follows from (IV.3.4.8) and (IV.3.4.10).  Remark IV.3.4.13. If we define a Higgs crystal on (X/B)rHIGGS similarly Q as Definition 0mi 1 (x) ∈ IV.3.3.2, then the integrality in Lemma IV.3.4.12 (3) becomes m! 1≤i≤d θi (|m|)

AD0 ,1 r M 0 by (IV.2.4.11). In the case X → B1 is smooth and integral, X = Y1 and B ∈ (1)

r C ∞ , which is typical, we have AD0 ,1 = pξ −(r−1) F r−1 AYr0 = pAX 0 and hence θi (M 0 ) ⊂ 0 0 pM , where AYN0 = Γ(YN , OYN0 ) and AX 0 = Γ(X 0 , OX 0 ). This implies that, if p ≥ 5 0 (resp. p = 3), the pair (MQ0 , θQ ) automatically satisfies the convergence condition in Lemma IV.3.4.12 (2) for every level ≥ 2 (resp. ≥ 3).

Definition IV.3.4.14. We define HBrQp ,conv (X, Y /B) (resp. HBZp ,conv (X, Y /B) when r = ∞) to be the full subcategory of HMrQp (X, Y /B) (resp. HMZp (X, Y /B)) (cf. Definition IV.3.4.7) consisting of (M, θ) satisfying the following condition: There exists a strict étale covering (Yα → Y )α∈A in C and (ti,N,α )N ∈ limN Γ(Yα,N , MYα,N ) (1 ≤ i ≤ dα ) for each ←− α ∈ A such that (i) d log ti,N,α (1 ≤ i ≤ dα ) is a basis of Ω1Yα,N /BN for every N ∈ N>0 , (ii) Yα,1 is affine, (iii) the restriction of M to Dα,1 = Yα,1 ×Y1 D1 is a finitely generated projective ODα,1 ,Q -module (resp. a p-torsion free object of LPM(ODα,1 )), (iv) the pair (Γ(Dα,1 , M), Γ(Dα,1 , θ)) satisfies (Conv) (resp. and (Int)) in Lemma IV.3.4.12 with respect to (ti,N,α )N (1 ≤ i ≤ dα ).

Remark IV.3.4.15. Let (M, θ) be an object of the category HBrQp ,conv (X, Y /B) (resp. HBZp ,conv (X, Y /B)) and let (Yα → Y ) be as in Definition IV.3.4.14. Then, by Lemma IV.3.4.6 (2) and Proposition IV.2.4.13 (2), we see that, for any strict étale Cartesian morphism Y 0 → Yα such that Y10 is affine and the pull-back of (ti,N,α ) on Y 0 , the pair (Γ(D10 , M), Γ(D10 , θ)) (D10 = Y10 ×Y1 D1 ) also satisfies (Conv) (resp. and (Int)) in Lemma IV.3.4.12. Theorem IV.3.4.16. In the case I (resp. II), Proposition IV.3.4.4 and the construction of θ after Definition IV.3.4.7 give an equivalence of categories from the category HCrQp (X/B) (resp. the full subcategory of HCZp (X/B) consisting of F such that FD is p-torsion free) to the category HBrQp ,conv (X, Y /B) (resp. HBZp ,conv (X, Y /B)).

Remark IV.3.4.17. Suppose r = ∞, let (M, ε) be an object of HSZp (X, Y /B), and let M be the image of M → MQp , which is an object of LPMloc (OD1 ) and satisfies the condition (2) (i) in Definition IV.3.4.3 by Lemma IV.3.2.10. By (IV.2.4.11), Remark IV.2.4.4, and r = ∞, we see that pi,1 : D(1)1 → D1 and pij,1 : D(2)1 → D(1)1 are flat after taking the reduction mod pm for every m ∈ N>0 . Hence, by Lemma IV.3.2.11, the ∼ = isomorphism ε induces an isomorphism ε : pb∗2 M − → pb∗1 M and the pair (M, ε) becomes a p-torsion free object of HSZp (X, Y /B). Proof of Theorem IV.3.4.16. By Proposition IV.3.4.4, it suffices to prove that the category HSrQp (X, Y /B) (resp. the full subcategory of HSZp (X, Y /B) consisting of (M, ε) such that M is p-torsion free) is equivalent to the category HBrQp ,conv (X, Y /B) (resp. HBZp ,conv (X, Y /B)) via the construction of θ after Definition IV.3.4.7. For an object (M, ε) of HSrQp (X, Y /B) (resp. HSZp (X, Y /B) such that M is p-torsion free), we see that the associated pair (M, θ) satisfies the conditions in Definitions IV.3.4.7 and

IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

365

IV.3.4.14 by Lemmas IV.3.4.12 and IV.3.4.6 (1). Let us construct a quasi-inverse functor. Let (M, θ) be an object of HBrQp ,conv (X, Y /B) (resp. HBrZp ,conv (X, Y /B)). First let us consider the case where Y1 is affine, M is a finitely generated projective OD1 ,Q -module (resp. an object of LPM(OD1 )), and there exist (ti,N ) ∈ limN Γ(YN , MYN ) ←− (1 ≤ i ≤ d) such that {d log ti,N } is a basis of Ω1YN /BN and (M, θ) := (Γ(D1 , M), Γ(D1 , θ)) satisfies (Conv) (resp. and (Int)) in Lemma IV.3.4.12. Put AN = Γ(DN , ODN ) and A(ν)N = Γ(D(ν)N , OD(ν)N ) (ν ∈ N). We define wi,N ∈ A(1)N as before (IV.2.4.11). Let p∗i , qj∗ , p∗ij , and ∆∗ denote the ring homomorphisms among A1 , A(1)1 , and A(2)1 induced by the morphisms pi,1 , qj,1 , pij,1 , and ∆D,1 among D1 , D(1)1 , and D(2)1 . Let p∗i M (resp. qj∗ M ) denote the scalar extension of M by pi (resp. qj ) and let p∗i (resp. qj∗ ) also denote the natural homomorphism M → p∗i M (resp. M → p∗j M ). Then we have isomorphisms similar to (IV.3.4.8) and (IV.3.4.9) for a finitely generated A1 -submodule M ◦ of M such that MQ◦ = M (resp. (IV.3.4.10) and (IV.3.4.11)). We define the endoP morphisms θi (1 ≤ i ≤ d) of M by θ(x) = 1≤i≤d ξ −1 θi (x) ⊗ d log(ti,1 ) as usual. Since (m)

m

m

p[ r ]+1 A1 ⊂ A1 r ⊂ p[ r ] A1 if r ∈ N>0 by Lemma IV.2.3.4, one can define an additive map α : M → p∗1 M by   X 1 m m ∗ w · p1 θ (x) , α(x) = m! d m∈N

Q mi where w = 1≤i≤d wi,1 and θm = 1≤i≤d θimi for m = (mi ) ∈ Nd . Since p∗2 (a) = P m ∗ 1 m m∈Nd w · p1 ( m! θ (a)) for a ∈ A1 by Proposition IV.2.4.13 (3), we obtain the equal∗ ity p2 (a)α(x) = α(ax) for a ∈ A1 and x ∈ M from Lemma IV.3.4.6 (2). Hence the morphism α induces an A(1)1 -linear homomorphism ε : p∗2 M → p∗1 M . It is clear that the scalar extension of ε by ∆∗ : A(1)1 → A is the identity. By the same argum

Q

ment as the proof of Lemma IV.3.4.12, we see that ε satisfies the cocycle condition p∗12 (ε) ◦ p∗23 (ε) = p∗13 (ε) : q3∗ M → q1∗ M . Let τ (resp. ι) be the morphism D(1) → D(2) ∼ = (resp. the isomorphism D(1) → D(1)) induced by the morphism τY : Y (1) → Y (2) ∼ = (resp. the isomorphism ιY : Y (1) → Y (1)) defined by q1 ◦ τY = q3 ◦ τY = p1 and q2 ◦ τY = p2 (resp. p1 ◦ ιY = p2 and p2 ◦ ιY = p1 ). Let τ ∗ (resp. ι∗ ) denote the ho∼ = momorphism A(2)1 → A(1)1 (resp. the isomorphism A(1)1 → A(1)1 )) induced by τ (resp. ι). Then, by taking the extension of scalars of the above cocycle condition, we see ι∗ (ε)

ε

that the composition of p∗1 M −−−→ p∗2 M − → p∗1 M is the identity. This implies that ε is an isomorphism. By Lemma IV.3.4.12 (1), we see that ε constructed above is independent of the choice of (ti,N ). In the general case, by applying the above argument to Yα → Y in the condition of Definition IV.3.4.14 and using Lemma IV.3.2.1 (3), Proposition IV.3.2.7 (1), and Lemma ∼ = IV.3.2.8, we obtain isomorphisms εα : p∗2 M|D(1)1 ×Y Yα − → p∗1 M|D(1)1 ×Y Yα , which glue ∼ =

and give an isomorphism ε : p∗2 M − → p∗1 M satisfying the conditions (ii) (a) and (b) in Definition IV.3.4.3 (cf. Remark IV.3.4.15). Thus we obtain a functor in the opposite direction and Lemma IV.3.4.12 (1) implies that this gives a quasi-inverse.  IV.3.5. Lattices of Higgs isocrystals. For B ∈ Ob C ∞ and a morphism of p-adic fine log formal schemes X → B1 , we define a kind of an integral structure on a Higgs isocrystal of level r (r ∈ N>0 ) on X/B (Definition IV.3.5.4), which is justified by the two preliminary propositions: IV.3.5.1 and IV.3.5.3 when X → B1 is smooth and integral. Proposition IV.3.5.1. Let B ∈ Ob C ∞ and let X be a p-adic fine log formal scheme over B1 such that the morphism of schemes underlying the reduction mod p of X → B1

366

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

is of finite type. Then the functor HCZp (X/B)Q −→ HC∞ Qp (X/B); F 7→ FQp is fully faithful. Proposition IV.3.5.2. Let B ∈ Ob C ∞ and let X be an affine p-adic fine log formal scheme over B1 . Assume that there exists a smooth Cartesian morphism Y → B and an immersion X → Y1 over B1 satisfying the following condition: There exist (ti,N ) ∈ limN Γ(YN , MYN ) (1 ≤ i ≤ d) such that {d log(ti,N )}1≤i≤d is a basis of Ω1YN /BN for every ←− N ∈ N>0 . Then the functor HCZp ,X-fin (X/B)Q −→ HC∞ Qp ,X-fin (X/B); F 7→ FQp

is an equivalence of categories. Proof. We apply Theorem IV.3.4.16 to Y and X ,→ Y1 in the assumption. Then the essential image of HC∞ Qp ,X-fin (X/B) (resp. the full subcategory of HCZp ,X-fin (X/B) consisting of F such that FD is p-torsion free) under the equivalence of categories in Theorem IV.3.4.16 is the full subcategory consisting of (M, θ) such that M ∈ PM(OD1 ,Q ) (resp. LPM(OD1 )). By the proof of the theorem, we also see that such (M, θ) satisfies (Conv) (resp. (Conv) and (Int)) in Lemma IV.3.4.12 for r = ∞ with respect to (ti,N )N (1 ≤ i ≤ d). By Proposition IV.3.4.4, Lemma IV.3.2.8, and Proposition IV.3.2.7 (2), we see that the functor in the proposition is fully faithful. Let (M, θ) be an object of HBQp ,conv (X, Y /B) such that M ∈ PM(OD1 ,Qp ), put M = Γ(D1 , M), let θ also P denote Γ(D1 , θ), and define the endomorphisms θi (1 ≤ i ≤ d) of M by θ(x) = 1≤i≤d ξ −1 θi (x)d log(ti,1 ). Choose a finitely generated AD,1 -submodule M ◦0 such that MQ◦0p = M . Since θi satisfy (Conv) in Lemma IV.3.4.12 for r = ∞, Lemma IV.3.4.6 (2) Q and Proposition IV.2.4.13 (2) imply that there exists a finite S ⊂ Nd such that P subset Q mi mi 1 1 d ◦ ◦0 ◦0 ◦0 ∈ N \S. Hence M = θ (M ) ⊂ M for all m 1≤i≤d i m∈Nd m! 1≤i≤d θi (M ) m! Q mi 1 ◦ ◦ is a finitely generated AD,1 -module and satisfies m! 1≤i≤d θi (M ) ⊂ M for every m ∈ Nd . Let M◦ be the object of LPM(OD1 ) corresponding to M ◦ (cf. Proposition IV.3.2.7 (1)). Then the natural morphism M◦ → M is injective (cf. Lemma IV.3.2.10), M◦Qp ∼ = M, the image of M◦ in M is stable under θ (cf. Lemma IV.3.2.6 (2), Lemma IV.3.4.6 (2), and Proposition IV.2.4.13 (2)), and M◦ with the induced θ satisfies (Conv) in Lemma IV.3.4.12. This completes the proof of the essential surjectivity.  Proof of Proposition IV.3.5.1. By the assumption on X, there exists a strict étale covering (Xα → X)α∈A such that F is Xα -finite and Xα satisfies the assumption of Proposition IV.3.5.2 for each α ∈ A. Furthermore, for each α, β ∈ A, there exists an affine open covering Xαβ = ∪γ∈Γαβ Xαβ;γ . For Xα ,→ Yα satisfying the condition in Proposition IV.3.5.2, Xαβ;γ ,→ Yα ×B Yβ also satisfies the condition. Hence we obtain the proposition by combining Corollary IV.3.1.8 and Proposition IV.3.5.2.  Proposition IV.3.5.3. Let B ∈ C ∞ and let X → B1 be a smooth integral morphism of p-adic fine log formal schemes. Then, for every r ∈ N>0 , the functor HCrQp (X/B) −→ HC∞ Qp (X/B)

r induced by the inclusion functor (X/B)∞ HIGGS ,→ (X/B)HIGGS is fully faithful.

Proof. By Corollary IV.3.1.8, we are reduced to the case where X is affine. Choose a smooth integral Cartesian morphism Y → B in C such that Y1 = X. Then we can apply Theorem IV.3.4.16 to X and id : X → Y1 . Since the reduction mod pm of the underlying morphism of formal schemes of YN → BN is flat, we have Y ∈ C ∞ by Lemma IV.2.2.7

IV.3. HIGGS ISOCRYSTALS AND HIGGS CRYSTALS

367

r (1) and hence DHiggs (X1 ,→ Y ) = Y for every r ∈ N>0 ∪ {∞}. Now the proposition follows from Theorem IV.3.4.16. 

Definition IV.3.5.4. Let B ∈ Ob C ∞ and let X → B1 be a morphism of p-adic fine log formal schemes. For r ∈ N>0 , we define the category HCrZp (X/B) as follows. An object is a triple consisting of an object F of HCrQp (X/B), an object F ◦ of HCZp (X/B), and ∼ =

. A morphism in (F, F ◦ , ιF ) → (G, G ◦ , ιG ) is a an isomorphism ιF : FQ◦ p − → F|(X/B)∞ HIGGS ◦ ◦ ◦ ιF = ιG ◦ fQ◦p . pair of morphisms f : F → G and f : F → G ◦ such that f |(X/B)∞ HIGGS Under the notation in Definition IV.3.5.4, we have a natural functor HCrZp (X/B) −→ HCZp (X/B)

defined by (F, F ◦ , ιF ) 7→ F ◦ and (f, f ◦ ) 7→ f ◦ . This is an equivalence of categories if r = ∞. If X → B1 is smooth and integral, this functor is fully faithful by Proposition IV.3.5.3. The following diagram is commutative up to canonical isomorphism. (IV.3.5.5)

HCrZp (X/B)

/ HCZp (X/B)

 HCrQp (X/B)

 / HC∞ (X/B). Qp

Here the left vertical functor is defined by (F, F ◦ , ιF ) 7→ F and (f, f ◦ ) 7→ f . If the morphism of schemes underlying the reduction mod p of X → B1 is of finite type, then we see that the functor HCrZp (X/B)Q → HCrQp (X/B) defined by (F, F ◦ , ιF ) 7→ F is fully faithful by Proposition IV.3.5.1. IV.3.6. Convergence and divisibility of Higgs fields. We discuss the relation between the convergence condition on a Higgs field in Lemma IV.3.4.12 (2) and the divisibility of a Higgs field by a positive rational power of p considered in [27]. Let V be a complete discrete valuation ring of mixed characteristic (0, p), let K be an algebraic closure of the field of fractions K of V , let V be the integral closure of V in K, and let Vb be the p-adic completion of V . Let v be a valuation of V , and for α ∈ Q>0 , let pα denote an element of V such that v(pα ) = αv(p). Let U → B1 be a smooth integral morphism of p-adic fine log formal schemes over Spf(Vb ) such that the morphism p : OB1 → OB1 is injective, the underling formal scheme of U is affine, and there exist t1 , . . . , td ∈ Γ(U, MU ) such that {d log(ti )|1 ≤ i ≤ d} is a basis of Ω1U/B1 . Choose ti as above, and put A = Γ(U, OU ) and Ω1 = Γ(U, Ω1U/B1 ). The assumption implies that A is p-torsion free. Let M be a finitely generated projective AQp -module and let θ : M → M ⊗A ξ −1 Ω1 be a Higgs field of M , i.e., an A-linear homomorphism such that θ1 ◦ θ = 0, where θ1 is defined as before Lemma IV.3.4.5. We define the endomorphisms θi of M by P θ(x) = 1≤i≤d θi (x) ⊗ ξ −1 d log(ti ) (x ∈ M ). For n = (n1 , . . . , nd ) ∈ Nd , let |n|, n! and P Q Q θn denote 1≤i≤d ni , 1≤i≤d ni ! and 1≤i≤d θini , respectively. As in Lemma IV.3.4.12 (2), we consider the following convergence of θ for r ∈ N>0 ∪ {∞}. (Conv)r

For any x ∈ M , p−r

−1

|n| 1 n n! θ (x)

(n ∈ Nd ) converges to 0 as |n| → ∞.

Here we understand r−1 |n| to be 0 if r = ∞. If we are given an object B ∈ C ∞ such that B1 is the p-adic fine log formal scheme above and a smooth Cartesian morphism Y → B such that Y1 → B1 is the morphism U → B1 above, then the category of Higgs isocrystals on (U/B)rHIGGS finite on U is equivalent to the category of pairs (M, θ) satisfying (Conv)r

368

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

(cf. Theorem IV.3.4.16 together with its proof, and Lemma IV.3.2.1). Note that we have r DHIGGS (Y ) = Y because Y is an object of C ∞ by Lemma IV.2.2.7 (1). 1 , we first consider a similar convergence condition as follows. For α ∈ Q> p−1 (Conv)0α αn

For any x ∈ M , p−α|n| θn (x) (n ∈ Nd ) converges to 0 as |n| → ∞.

Then since pn! (resp. pn! αn ) converges to 0 as n → ∞ if α > obtain the following implications between the two conditions.

1 p−1

(resp. α
0 and α ∈ Q> p−1

1 + 1r . (1) (Conv)r implies (Conv)0α if α < p−1 1 (2) (Conv)0α implies (Conv)r if p−1 + 1r < α.

1 , we consider the following condition as in [27] Definition 2. For α ∈ Q> p−1

p

−α

(Div)α There exists a finitely generated A-submodule M ◦ such that MQ◦p = M and θ(M ◦ ) ⊂ M ◦ ⊗A ξ −1 Ω1 .

1 . Proposition IV.3.6.2. Let α, β ∈ Q> p−1 0 (1) (Div)α implies (Conv)β if β < α. (2) (Conv)0α implies (Div)α .

Proof. (1) follows from (p−β θi )n (M ◦ ) ⊂ p(α−β)n M ◦ . Let us prove (2). Suppose that (M, θ) satisfies (Conv)0α . Choose a finitely generated A-submodule M ◦0 of M such that M ◦0 ⊗Zp Qp = M and let xλ , λ ∈ Λ, ]Λ < ∞ be generators of M ◦0 over A. Then (Conv)0α implies that there exists N ∈ N such that for every n ∈ Nd satisfying since θi are A-linear, that |n| ≥ N , we have p−α|n| θn (xλ ) ∈ M ◦0 , λ ∈ Λ, which implies, P p−α|n| θn (M ◦0 ) ⊂ M ◦0 . Hence the A-submodule M ◦ := n∈Nd p−α|n| θn (M ◦0 ) is finitely generated, M ◦ generates M over AQp , and for any 1 ≤ i ≤ d, we have X p−α θi (M ◦ ) = p−α θi p−α|n| θn (M ◦0 ) ⊂ M ◦ . n∈Nd

Thus we see that (M, θ) satisfies (Div)α .



Following [27] p.852, we define a small Higgs field as follows. Definition IV.3.6.3. Let U → B1 , A and Ω1 be as above, and let M be a finitely generated projective AQp -module. We say that a Higgs field θ : M → M ⊗A ξ −1 Ω1 is 1 . small if it satisfies (Div)α for some α ∈ Q> p−1 Corollary IV.3.6.4. Let M be a finitely generated projective AQp -module and let θ be a Higgs field M → M ⊗A ξ −1 Ω1 on M . Then θ is small if and only if it satisfies (Conv)r for some r ∈ N>0 .

Proof. This immediately follows from Lemma IV.3.6.1 and Proposition IV.3.6.2. 

In the global case, we have a similar equivalence assuming the existence of a “lattice” of the underlying module as follows. Let X → B1 be a smooth integral morphism of p-adic fine log formal schemes over Spf(Vb ) such that the morphism p : OB1 → OB1 is injective. Let M be an OX,Qp -module locally finitely generated and projective, and let θ : M → M ⊗OX ξ −1 Ω1X/B1 be a Higgs field, i.e., an OX,Qp -linear morphism such that θ1 ◦ θ = 0, where θ1 is defined as before Lemma IV.3.4.5. Following [27] p.852 and p.855, we define a small Higgs field as follows. (See also II.13.1.)

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Definition IV.3.6.5. Let M and θ be as above. We say that θ is small if there exist α ∈ 1 and an OX -submodule M◦ ⊂ M such that M◦ ∈ Ob LPMloc (OX ) (cf. Definition Q> p−1 IV.3.2.9 (1)), M◦Qp = M, and p−α θ(M◦ ) ⊂ M◦ ⊗OX ξ −1 Ω1X/B1 . For r ∈ N>0 ∪ {∞}, let HBrQp ,conv (X/B1 ) denote the category whose object is an OX,Qp -module locally finitely generated and projective M with a Higgs field θ : M → M ⊗OX ξ −1 Ω1X/B1 satisfying (Conv)r strict étale locally on X, i.e., there exists a strict étale covering (Uα → X)α∈A and tα,i ∈ Γ(Uα , MX ) (1 ≤ i ≤ dα ) such that {d log(tα,i )|1 ≤ i ≤ dα } is a basis of Ω1Uα /B1 , Uα is affine, M|Uα is a finitely generated projective OUα ,Qp module, and the pair (Γ(Uα , M), Γ(Uα , θ)) satisfies (Conv)r with respect to tα,i . If we are given an object B ∈ C ∞ such that B1 is the p-adic fine log formal scheme above and a smooth Cartesian morphism Y → B such that Y1 → B1 is the morphism X → B1 above, then the category HBrQp ,conv (X/B1 ) is equivalent to the category of Higgs isocrystals HCrQp (X/B) on (X/B)rHIGGS by Theorem IV.3.4.16. Proposition IV.3.6.6. Let M be an OX,Qp -module locally finitely generated and projective, and let θ : M → M ⊗OX ξ −1 Ω1X/B1 be a Higgs field on M. We further assume that there exists an OX -submodule M◦0 ⊂ M such that M◦0 ∈ Ob LPMloc (OX ) and r M◦0 Qp = M. Then θ is small if and only if (M, θ) ∈ Ob (HBQp ,conv (X/B1 )) for some r ∈ N>0 . Proof. The necessity follows from Corollary IV.3.6.4. Let us prove the sufficiency. 1 1 Let (M, θ) be an object of HBrQp ,conv (X/B1 ). Let α ∈ Q such that p−1 < α < p−1 + 1r and let U → X be a strict étale morphism satisfying the conditions on Uα for (M, θ) in the definition of HBrQp ,conv (X/B1 ). We further assume that M◦0 |U ∈ LPM(OU ) (cf. Remark IV.3.4.15). Then choosing ti ∈ Γ(U, MU ) (1 ≤ i ≤ d) such that {d log(ti )} is a basis of Ω1U/B1 and applying Lemma IV.3.6.1 (1) and the proof of Proposition IV.3.6.2 (2) to the pair (Γ(U, M), Γ(U, θ)) and Γ(U, M0◦ ), we obtain an OU -submodule M◦U of M|U such that M◦U ∈ LPM(OU ), M◦U,Qp = M|U , and p−α θ|U (M◦U ) ⊂ M◦U ⊗OU ξ −1 Ω1U/B1 (cf. Lemma IV.3.2.1 (3) and Proposition IV.3.2.7 (1)). We see that the sheaf M◦U does not 0 0 depend on the choice of ti as follows. If you make another choice t0i and define Q θi using0 tnii, 0 then θi is a linear combination of θj with coefficients in Γ(U, OU ) and hence 1≤i≤d (θi ) Q (n = (ni ) ∈ Nd ) is a linear combination of 1≤i≤d θimi for m = (mi ) ∈ Nd such that |m| = |n|. This implies that the OU -submodule of M|U defined by using t0i is contained in M◦U . By exchanging ti and t0i in this argument, we see that the two submodules coincide. Thus the above M◦U ’s glue together and give an OX -submodule M◦ of M such that M◦Qp = M, M◦ ∈ LPMloc (OX ), and p−α θ(M◦ ) ⊂ M◦ ⊗OX ξ −1 Ω1X/B1 .  IV.4. Cohomology of Higgs isocrystals IV.4.1. Projections to étale sites. Let r ∈ N>0 ∪ {∞}, let B ∈ Ob C r , and let X be a p-adic fine log formal scheme over B1 . We construct a morphism of topos ∼ UX/B : (X/B)r∼ HIGGS → XÉT (Proposition IV.4.1.2) in a way similar to the crystalline site (cf. [3] III 4.4) and prove its functoriality (Proposition IV.4.1.3). For a sheaf of sets F on (X/B)rHIGGS , we define the presheaf UX/B∗ (F) on XÉT by UX/B∗ (F)(u : U → X) = Γ((U/B)rHIGGS , u∗HIGGS (F)). Lemma IV.4.1.1. Under the notation and assumption above, UX/B∗ (F) is a sheaf on XÉT .

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Proof. Put G = UX/B∗ (F). For an object u : U → X of XÉT , giving a section x of G(U ) is equivalent to giving a section x(T,z) of Γ((T, u ◦ z), F) for each (T, z) ∈ Ob (U/B)rHIGGS such that for every morphism f : (Te, ze) → (T, z) in (U/B)rHIGGS , we have f ∗ (z(T,z) ) = z(Te,ez) . With this description, the pull-back v ∗ : G(U ) → G(U 0 ) by a morphism v : U 0 → U in XÉT is given by v ∗ (x)(T 0 ,z0 ) = x(T 0 ,v◦z0 ) . Let (uα : Uα → U )α∈A be a strict étale covering of (u : U → X) ∈ Ob XÉT . Put Uαβ = Uα ×U Uβ and let pα , pβ Q denote the projections Uαβ → Uα , Uβ . It suffices to Q prove that the sequence G(U ) → α∈A G(Uα ) ⇒ (α,β)∈A2 G(Uαβ ) is exact. Let (T, z) be an object of (U/B)rHIGGS . Let Tα,1 be T1 ×U Uα , let vα,1 , zα denote the projections Tα,1 → T1 , Uα , and let vα : Tα → T be the unique strict étale Cartesian lifting of vα,1 . Put Uαβ = Uα ×U Uβ (resp. Tαβ = Tα ×T Tβ ), and let qα , qβ denote the projections Tαβ → Tα , Tβ . Let zαβ : Tαβ,1 → Uαβ be the morphism induced by zα and zβ . Then (Tα , zα ) (resp. (Tαβ , zαβ )) is an object of (Uα /B)rHIGGS (resp. (Uαβ /B)rHIGGS ), and the morphism vα (resp. qα , resp. qβ ) defines a morphism (Tα , uα ◦ zα ) → (T, z) (resp. (Tαβ , pα ◦ zαβ ) → (Tα , zα ), resp. (Tαβ , pβ ◦ zαβ ) → (Tβ , zβ )). Since (vα : Tα → T )α∈A is a strict étale covering, we have an exact sequence Y Y (∗) Γ((T, u ◦ z), F) → Γ((Tα , u ◦ uα ◦ zα ), F) ⇒ Γ((Tαβ , u ◦ uαβ ◦ zαβ ), F), (α,β)∈A2

α∈A

Q where uαβ = uα ◦ pα = uβ ◦ pβ . This implies that G(U ) → α∈A G(Uα ) is injective. Q Suppose that (xα ) ∈ α∈A Γ(Uα , G) satisfies p∗α (xα ) = p∗β (xβ ) for every (α, β) ∈ A2 . Put xαβ = p∗α (xα ) = p∗β (xβ ). Then the image of the element (xα,(Tα ,zα ) )α in the middle term of (∗) under the two maps are both (xαβ,(Tαβ ,zαβ ) )αβ . Hence there exists a unique x(T,z) ∈ Γ((T, u ◦ z), F) such that vα∗ (x(T,z) ) = xα,(Tα ,zα ) . Since this construction is functorial in (T, z), we see that x(T,z) defines a section x of Γ(U, G). Now it remains to show that the image of x in Γ(Uγ , G) (γ ∈ A) is xγ , i.e., for any object (T, z 0 ) of (Uγ /B)rHIGGS , we have x(T,uγ ◦z0 ) = xγ,(T,z0 ) in Γ((T, u◦uγ ◦z 0 ), F). Put z = uγ ◦z 0 : T1 → U . By the construction of x, it suffices to prove that the image of xγ,(T,z0 ) under the morphism vα∗ : Γ((T, u ◦ z), F) → Γ((Tα , u ◦ uα ◦ zα ), F) is xα,(Tα ,zα ) . The factorization z = uγ ◦ z 0 : T1 → Uγ → U induces a factorization zα = pα ◦ zα0 : Tα,1 → Uαγ → Uα , and vα defines a morphism (Tα , pγ ◦ zα0 ) → (T, z 0 ). Hence we have xα,(Tα ,zα ) = xαγ,(Tα ,zα0 ) = vα∗ (xγ,(T,z0 ) ).  ∗ For a sheaf of sets G on XÉT , we define the sheaf of sets UX/B (G) on (X/B)rHIGGS ∗ by UX/B (G)((T, z)) = G(z : T1 → X). ∗ Proposition IV.4.1.2. The functor UX/B is a left adjoint of UX/B∗ and left exact, so ∗ that the pair (UX/B , UX/B∗ ) defines a morphism of topos ∼ UX/B : (X/B)r∼ HIGGS −→ XÉT . ∼ Proof. Let F ∈ Ob (X/B)r∼ HIGGS and G ∈ Ob XÉT . We can define a map ∗ F : Hom(UX/B (G), F) → Hom(G, UX/B∗ F) ∗ as follows. Let ϕ be a morphism UX/B (G) → F. For (u : U → X) ∈ Ob (XÉT ) and r (T, z) ∈ Ob (U/B)HIGGS , the morphism ϕ defines a map z∗

∗ ψU,(T,z) : Γ(u : U → X, G) −→ Γ(u ◦ z : T1 → X, G) = Γ((T, u ◦ z), UX/B G) Γ((T,u◦z),ϕ)

−−−−−−−−→ Γ((T, u ◦ z), F).

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For a morphism f : (T 0 , z 0 ) → (T, z) in (U/B)rHIGGS , we have f ∗ ◦ ψU,(T,z) = ψU,(T 0 ,z0 ) . Hence, by the description of sections of UX/B∗ F in the proof of Lemma IV.4.1.1, we see that (ψU,(T,z) )(T,z) defines a morphism ψU : Γ(U, G) → Γ(U, UX/T ∗ F). For a morphism v : U 0 → U in XÉT and (T 0 , z 0 ) ∈ Ob (U 0 /B)rHIGGS , we also have ψU 0 ,(T 0 ,z0 ) ◦ v ∗ = ψU,(T 0 ,v◦z0 ) . This implies that v ∗ ◦ ψU = ψU 0 ◦ v ∗ and (ψU )U defines a morphism ψ : G → UX/B∗ F. We define F (ϕ) to be ψ. This construction is obviously functorial in F and ∗ G. Conversely we can define a map G : Hom(G, UX/B∗ F) → Hom(UX/B G, F) as follows. r Let ψ be a morphism G → UX/B∗ F. Then, for (T, z) ∈ Ob (X/B)HIGGS , ψ defines a map Γ(z : T1 →X,ψ)

∗ ∗ ϕ(T,z) : UX/B G((T, z)) = G(z : T1 → X) −−−−−−−−−→ Γ((T1 /B)rHIGGS , zHIGGS F)

∗ −→ Γ((T, idT1 ), zHIGGS F) = Γ((T, z), F).

For a morphism f : (T 0 , z 0 ) → (T, z), we see that f ∗ ◦ ϕ(T,z) = ϕ(T 0 ,z0 ) ◦ f ∗ . Hence ∗ (ϕ(T,z) )(T,z) defines a morphism ϕ : UX/B G → F. We define G(ψ) to be ϕ. Now it is straightforward to verify F ◦ G = id and G ◦ F = id.  We define the morphism of topos ∼ uX/B : (X/B)r∼ HIGGS −→ X´ et

∼ to be the composition of UX/B with the canonical morphism of topos XÉT → X´e∼t (defined by the inclusion functor X´et → XÉT , which is a morphism of sites).

Proposition IV.4.1.3. Let r, r0 , B, B 0 , X, X 0 , g, and f be the same as in Proposition IV.3.1.5. Then the following diagram of topos is commutative up to canonical isomorphism. (IV.4.1.4)

0

∼ (X 0 /B 0 )rHIGGS UX 0 /B 0



0∼ XÉT

fHIGGS

fÉT

/ (X/B)r∼ HIGGS 

UX/B

/ X∼ . ÉT

∗ Proof. It suffices to prove that there exists a canonical isomorphism fHIGGS ◦ ∗ ∗ ∗ ∼ 0 0 ∼ UX/B U ◦ f , which is verified as follows: For F ∈ Ob X and (T , z ) ∈ = X 0 /B 0 ÉT ÉT 0 Ob (X 0 /B 0 )rHIGGS , we have ∗ ∗ ∗ fHIGGS (UX/B F)((T 0 , z 0 )) = UX/B F((T 0 , f ◦ z 0 )) = F(f ◦ z 0 : T10 → X), ∗ ∗ ∗ UX F)((T 0 , z 0 )) = fÉT F(z 0 : T10 → X 0 ) ∼ = F(f ◦ z 0 : T10 → X). 0 /B 0 (f ÉT



IV.4.2. Linearizations. Let r ∈ N>0 , let B ∈ Ob C r , let X be a p-adic fine log formal scheme over B1 , let Y → B be a smooth Cartesian morphism in C (Definition r IV.2.2.2), and let i : X → Y1 be an immersion over B1 . Let D denote DHiggs (i : X ,→ Y ) (cf. Definition IV.2.2.1 (3) and Proposition IV.2.2.9 (1)). The object D of C r with the canonical morphism zD : D1 → X is an object of (X/B)rHIGGS . For a sheaf of OD1 ,Q -modules F on D1,´et , we define a sheaf of OX/B,1,Q -modules LY (F) on (X/B)rHIGGS , which we call the linearization of F. We follow the construction of the linearization on the crystalline site given in [5] 6.10.1 Remark. We then prove that the derived direct image of LY (F) under the projection to the small étale site X´et gives the direct image of F by zD : D1 → X when F is finitely generated and projective strict étale locally on X (Proposition IV.4.2.1). The necessity of the last condition on F stems from the fact that the morphism of topological spaces underlying

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r DHIGGS (Z)1 → Z1 for Z ∈ C is not a homeomorphism in general, which is different from the case of divided power envelopes. Except this point, one can adapt the argument via Čech-Alexander complex in [3] V. Théorème 1.2.5 and Proposition 2.2.2. i) to our context without difficulty. Let F be a sheaf of OD1 ,Q -modules on D1,´et . Let (T, z) be an object of (X/B)rHIGGS . z

i

The identity morphism of T1 and the morphism T1 − → X → − Y1 induce an immersion r iT : T1 → T1 ×B1 Y1 . Let DT denote DHiggs (iT : T1 ,→ T ×B Y ), and let pD : DT → D be the morphism in C r induced by z : T1 → X and the projection T ×B Y → Y . Let pT : DT → T be the morphism induced by id : T1 → T1 and the projection T ×B Y → T . We define LY (F)((T, z)) to be Γ(DT,1 , p∗D1 (F)) regarded as an Γ(T1 , OT1 ,Q )-module via the ring homomorphism p∗T1 : Γ(T1 , OT1 ,Q ) → Γ(DT,1 , ODT ,1 ,Q ). This construction is functorial in (T, z), and, by Proposition IV.2.2.9 (3), defines a sheaf of OX/B,1.Q -modules LY (F) on (X/B)rHIGGS ; for an object (T, z) of (X/B)rHIGGS , the sheaf of OT1 ,Q -modules LY (F)T on (T1 )´et is given by pT,1∗ p∗D,1 (F).

Proposition IV.4.2.1. Let r, B, X, Y , X → Y1 , D, and zD be as above. Let F be a sheaf of OD1 ,Q -modules on D1,´et . Assume that there exists a strict étale covering (Xα → X)α∈A such that F|D1,α (D1,α = Xα ×X D1 ) is a finitely generated and projective OD1,α ,Q -module for every α ∈ A. Then we have a canonical isomorphism ∼ zD∗ (F). RuX/B∗ (LY (F)) = Lemma IV.4.2.2. Let X = Spf(A) be a p-adic fine log formal scheme and let F be a finitely generated projective OX,Q -module on X´et . Then we have H q (X´et , F) = 0 for q > 0. ◦

For an abelian category A , let A N denote the category of inverse system of objects ◦ of A indexed by N (cf. III.7, IV.6.6). If A has enough injectives, then A N has enough N◦ injectives (cf. [47] (1.1) Proposition a)). An object (An ) of A is injective if and only if An (n ∈ N) are injective and the transition maps An+1 → An (n ∈ N) are split surjections (cf. [47] (1.1) Proposition b)). In particular, if an additive functor f : A → B between abelian categories preserves injectives, then the induced functor ◦ ◦ ◦ f N : A N → B N preserves injectives. Lemma IV.4.2.3. Let C be a site and let S(C, Z) be the category of sheaves of abelian ◦ groups on C. Let (Fn ) be an object of S(C, Z)N and assume that there exists a subset S ⊂ Ob C satisfying the following conditions. (i) For every X ∈ S , H q (X, Fn ) = 0 (q > 0) and R1 limn H 0 (X, Fn ) = 0. ←− (ii) For every X ∈ Ob C, there exists a covering (Xα → X)α∈A such that Xα ∈ S for all α ∈ A. Then we have Rq limn (Fn ) = 0 (q > 0). ←− ◦

Proof. Let (Fn ) → (In• ) be an injective resolution in S(C, Z)N . Let X ∈ S . Since (q, n ∈ N) are injective, the assumption (i) implies that Γ(X, Fn ) → Γ(X, In• ) is a q q resolution. Since In+1 → Inq (q, n ∈ N) are split surjections, Γ(X, In+1 ) → Γ(X, Inq ) are surjective. Hence we have H q (limn Γ(X, In• )) = 0 (q ≥ 2) and H 1 (limn Γ(X, In• )) = ←− ←− R1 limn Γ(X, Fn ) = 0. Now the assumption (ii) implies the claim because Rq limn Fn = ← − ←− Hq (limn (In• )).  ←− Proof of Lemma IV.4.2.2. Since F is a direct summand of a finitely generated free OX,Q -module, it suffices to prove H q (X´et , OX,Q ) = 0 for q > 0. By Lemma IV.4.4.6 (4) and [2] VI Corollaire 5.2, we have H q (X´et , OX,Q ) = H q (X´et , OX ) ⊗ Q. The presheaf Gn (n ∈ N>0 ) on X´et defined by Γ(Y, G) = Γ(Yn , OYn ), Yn = Y ×Spf(Zp ) Spec(Z/pn Z)

Inq

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373

is a sheaf and we have OX = lim Gn . If Y is affine, then we have H q (Y´et , Gn ) ∼ = ←− H q (Yn,´et , OYn ) ∼ = H q (Yn,Zar , OYn ) = 0 for q > 0 (cf. [2] VII Proposition 4.3 for the second isomorphism) and H 0 (Y´et , Gn+1 ) → H 0 (Y´et , Gn ) is surjective. Hence, by Lemma IV.4.2.3, ◦ we have RΓ(X´et , OX ) ∼ ((G ) ) ∼ R lim ◦R(Γ(X´et , −)N )((Gn )n ) ∼ = RΓ(X´et , −) ◦ R lim = ←−n n n = ←−n 0 0 ∼ R limn (H (X´et , Gn ))n = limn H (X´et , Gn ).  ←− ←− Proof of Proposition IV.4.2.1. For a strict étale morphism u : U → X, we can define a natural morphism ΦU : Γ(U, zD∗ F) = Γ(D1 ×X U, F) −→ Γ((U/B)rHIGGS , u∗HIGGS (LY (F)))

as follows. For (T, z) ∈ Ob (U/B)rHIGGS , we have

Γ((T, z), u∗HIGGS (LY (F))) = Γ(DT,1 , p∗D,1 F),

where pD : DT → D is the morphism in C r associated to (T, u ◦ z) as in the definition of LY (F). The morphism z : T1 → U gives rise to a factorization DT,1 → D1 ×X U → D1 of pD,1 , which induces a natural morphism Γ(D1 ×X U, F) → Γ((T, z), u∗HIGGS LY (F)). Observing that this morphism is functorial in (T, z), we obtain the desired morphism. Varying U , we also obtain a morphism zD∗ F → uX/B∗ LY (F). For a strict étale morphism u : U → X, the functor r,∼ u∗HIGGS : (X/B)r,∼ HIGGS → (U/B)HIGGS

preserves injective sheaves of abelian groups by Proposition IV.3.1.9 (1) and [2] V Proposition 4.11 1). Hence it suffices to prove that H q ((U/B)rHIGGS , u∗HIGGS LY (F)) = 0 (q > 0) and that the morphism ΦU constructed above is an isomorphism for u satisfying the following conditions. There exists an X-morphism U → Xα for some α, there exists a strict étale Cartesian lifting YU → Y of u : U → X, and U is affine. For an object r r (T, z) of (U/B)rHIGGS , the natural morphism DHiggs (T1 ,→ T ×B YU ) → DHiggs (T1 ,→ T ×B Y ) is an isomorphism by Lemma IV.2.2.15. Hence we have a natural isomorr phism u∗HIGGS LY (F) ∼ (U ,→ YU )1 ∼ = LYU (v1∗ F), where v1 denotes the morphism DHiggs = D1 ×X1 U1 → D1 . By replacing X with U , we may assume that X = U and F is a finitely generated projective OD1 ,Q -module. Let ε denote the natural functor (X/B)rHIGGS → (X/B)r∼ HIGGS . By Lemma IV.3.4.1 (and [2] II Proposition 4.3 2)), we see that ε(D, zD ) is a covering of the final object e of (X/B)r∼ HIGGS . We define D(ν) (ν ∈ Z) in the same way as before Remark IV.2.4.4 and let zD(ν) denote the natural morphism D(ν)1 → X. Then ε(D(ν), zD(ν) ) represents the product of ν + 1 copies of ε(D, zD ). By Proposition IV.2.2.9 (2), D(ν)1 is affine. Now, by applying [2] V Corollaire 3.3 to the covering ε(D, zD ) → e and using (IV.3.1.12), we obtain a spectral sequence E2a,b = H a (ν 7→ H b (D(ν)1,´et , LY (F)(D(ν),zD(ν) ) )) =⇒ H a+b ((X/B)rHIGGS , LY (F)).

r (T1 ,→ T ×B For any object (T, z) of (X/B)rHIGGS , the morphism pT,1 : DT,1 = DHiggs Y )1 → T1 is affine by Proposition IV.2.2.9 (2). Hence, if T1 is affine, we obtain H q (T1,´et , LY (F)(T,z) ) = 0 for q > 0 from Lemma IV.4.2.2. Hence we have E2a,b = 0 for b > 0 and obtain H a ((X/B)r , LX (F)) ∼ = H a (ν 7→ Γ(D(ν)1,´et , LY (F)(D(ν),z ) )). HIGGS

D(ν)

We see that the morphism D(ν) ×D Y → Y (ν) ×B Y = Y (ν + 1) induces an ∼ = r isomorphism DHiggs (D(ν)1 ,→ D(ν) ×B Y ) − → D(ν + 1) by verifying that the source satisfies the universal property defining the target. Hence we have an isomorphism Γ(D(ν)1,´et , LY (F)(D(ν),zD(ν) ) ) ∼ = Γ(D(ν + 1)1,´et , p∗ν+2,1 F), where pν+2 denotes the morphism D(ν +1) → D defined by the projection to the (ν +2)-th component Y (ν +1) → Y .

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With this description, the differential morphism from the degree ν term to the degree (ν+1) term is given by the alternating sum of the pull-backs by the morphisms D(ν+2) → D(ν + 1) defined by forgetting one of the components except the last one. We also see that the composition Γ(D1,´et , F) → Γ((X/B)rHIGGS , LY (F)) → Γ(D1,´et , LY (F)(D,zD ) ) ∼ = Γ(D(1)1,´et , p∗2 F) is given by the pull-back by the projection to the second component. Hence the claim follows from [3] V Lemme 2.2.1.  IV.4.3. Poincaré lemma. Let r, B, X, Y , i : X → Y1 , and (D, zD ) be the same as the beginning of IV.4.2. For a Higgs isocrystal F of level r on X/B, we construct a complex on (X/B)rHIGGS (IV.4.3.1) F → LY (FD ) → LY (ξ −1 FD ⊗OY1 Ω1Y1 /B1 ) → · · · → LY (ξ −q FD ⊗OY1 ΩqY1 /B1 ) → · · · and study its properties: a compatibility with the complex (ξ −• FD ⊗OY1 Ω•Y1 /B1 , θ• ) on (D1 )´et (Lemma IV.4.3.2) and the vanishing of H• of the complex (IV.4.3.1) after raising the level from r to r + 1 (Proposition IV.4.3.4), which may be regarded as a Poincaré lemma. Let (T, z) be an object of (X/B)rHIGGS , and define DT , pD , and pT as in IV.4.2. Then by applying Lemma IV.2.4.10 to a smooth Cartesian morphism Y ×B T → T and the immersion T1 → T1 ×B1 Y1 , we obtain a complex (1)

(q)

−1 p−1 ODTr,1 ⊗OY1 Ω1Y1 /B1 → · · · → ξ −q ODTr ⊗OY1 ΩqY1 /B1 → · · · T,1 (OT1 ) → ODT ,1 → ξ 1

−1 whose differential maps are p−1 T,1 (OT1 )-linear. Hence, by taking pT,1 (FT )⊗p−1 (OT ) , and

using the isomorphisms

∼ = p∗T,1 (FT ) →

∼ =

FDT ←

T ,1

p∗D,1 (FD ),

θ0

1

we obtain a complex θ1

∗ ∗ −1 p−1 FD ⊗OY1 Ω1Y1 /B1 ) −→ · · · T,1 (FT ) → pD,1 (FD ) −→ pD,1 (ξ θ q−1

θq

· · · −−−→ p∗D,1 (ξ −q FD ⊗OY1 ΩqY1 /B1 ) −→ · · · . By taking Γ(DT , −) and varying (T, z), we obtain the desired complex (IV.4.3.1). We have a complex (ξ −q FD ⊗OY1 ΩqY1 /B1 , θq )q∈N by Theorem IV.3.4.16 and Lemma IV.3.4.5. Lemma IV.4.3.2. Under the notation and assumption as above, the following diagram is commutative for every q ∈ N. uX/B∗ (LY (ξ −q FD ⊗OY1 ΩqY1 /B1 )) O

uX/S∗ (θ q )

/ uX/B∗ (LY (ξ −q−1 FD ⊗OY Ωq+1 )) Y1 /B1 1 O

Prop. IV.4.2.1 ∼ =

zD∗ (ξ −q FD ⊗OY1 ΩqY1 /B1 )

Prop. IV.4.2.1 ∼ = q

zD∗ (θ )

/ zD∗ (ξ −q−1 FD ⊗OY Ωq+1 ) Y1 /B1 1

Proof. It suffices to prove that Γ(U, −) of the diagram is commutative for any strict étale morphism u : U → X which has a strict étale Cartesian lifting YU → Y . Let (T, z) be an object of (U/B)rHIGGS . It is enough to prove that the following diagrams are commutative, where the vertical morphisms are defined as in the first paragraph of the

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375

proof of Proposition IV.4.2.1. Γ(U1 ×X1 D1 , ξ −q FD ⊗ ΩqY1 /B1 )  Γ((T, u ◦ z), LY (ξ −q FD ⊗ ΩqY1 /B1 ))

Γ(U1 ×X1 D1 ,θ q )

Γ((T,u◦z),θ q )

/ Γ(U1 ×X1 D1 , ξ −q−1 FD ⊗ Ωq+1 ) Y1 /B1

 / Γ((T, u ◦ z), LY (ξ −q−1 FD ⊗ Ωq+1 )), Y1 /B1

where ⊗ denotes ⊗OY1 . By the definition of θq (q ≥ 2) before Lemma IV.2.4.10 and Lemma IV.3.4.5, the claim is reduced to the case q = 0. We can construct D(ν) (resp. DU (ν), resp. DT (ν), resp. DU,T (ν)) from Y /B and X ,→ Y1 (resp. YU /B and U ,→ YU,1 , resp. (T ×B Y )/T and T1 ,→ T1 ×B1 Y1 , resp. (T ×B YU )/T and T1 ,→ T1 ×B1 YU,1 ) as before Remark IV.2.4.4. We omit (ν) if ν = 0. Let DU (1) (resp. DU,T (1)) be the first infinitesimal neighborhood of DU,1 ,→ DU (1)1 (resp. DU,T,1 ,→ DU,T (1)1 ). The natural morphism DU (ν) → D(ν) is strict étale and ∼ = Cartesian, and it induces an isomorphism DU (ν)1 − → D(ν)1 ×X U (cf. Remark IV.2.4.4). Hence the upper horizontal morphism in the diagram above for q = 0 is induced by the morphism p∗2 − p∗1 : FDU → FDU (1) ⊗ODU (1)1 ODU (1) /(p-tor) on D1 ×X U ∼ = DU,1 . The natural morphisms DU,T (ν) → DT (ν) are isomorphisms by Lemma IV.2.2.15. Hence the lower horizontal morphism for q = 0 is induced by p∗2 − p∗1 : FDU,T → FDU,T (1) ⊗ODU,T (1) ODU,T (1) /(p-tor) and the isomorphisms FDU,T ∼ = FDT ∼ = LY (FD )T . Hence the commutativity of the diagram for q = 0 follows from the compatibility of the natural morphisms DU,T (1) → DU (1) and DU,T → DU with the projections pi : DU,T (1) → DU,T and DU (1) → DU for i = 1, 2.  r+1 Next we prove a kind of Poincaré lemma. Put B 0 = DHiggs (B), X 0 = X ×B1 B10 , 0 0 0 0 0 and Y = Y ×B B . We have a B1 -immersion X → Y1 induced by the B1 -immersion r X → Y1 . Let µ : (X 0 /B 0 )r+1 HIGGS → (X/B)HIGGS denote the morphism of topos defined r+1 0 by the natural morphisms X → X and B 0 → B. Let D0 be DHiggs (X 0 ,→ Y 0 ) and let µD : D0 → D be the morphism induced by X 0 → X and Y 0 → Y . r+1 0 0 For any object (T, z) of (X 0 /B 0 )r+1 HIGGS , define DT to be DHiggs (T1 ,→ T ×B 0 Y ) r as in the definition of linearization. Regarding (T, z) as an object of (X/B)HIGGS , we may also define DT , and we have a natural morphism µT : DT0 → DT compatible with µD : D0 → D. Hence for a sheaf of OD1 ,Q -modules G, the morphism µT induces a 0 morphism Γ(DT,1 , p∗D,1 (G)) → Γ(DT,1 , p∗D0 ,1 (µ∗D (G))). This construction is functorial in (T, z). Hence varying (T, z), we obtain a canonical morphism

(IV.4.3.3)

µ∗ (LY (G)) −→ LY 0 (µ∗D (G)).

Proposition IV.4.3.4. Let F be a Higgs isocrystal on (X/B)rHIGGS , and let F 0 be the −q pull-back µ∗ (F) on (X 0 /B 0 )r+1 FD ⊗OY1 HIGGS . Then the morphisms (IV.4.3.3) for ξ q ΩY1 /B1 (q ∈ N) define a morphism of complexes µ∗ F F0

/ µ∗ (LY (FD ))

/ µ∗ (LY (ξ −1 FD ⊗ Ω1Y /B )) 1 1

 / LY 0 (F 0 0 ) D

 0 1 / LY 0 (ξ −1 FD 0 ⊗ ΩY 0 /B 0 ) 1

1

/ µ∗ (LY (ξ −2 FD ⊗ Ω2Y

1 /B1

))

 0 2 / LY 0 (ξ −2 FD 0 ⊗ ΩY 0 /B 0 ) 1

1

/ ··· / ··· ,

where ⊗ in the upper (resp. lower) complex denotes ⊗OY1 (resp. ⊗OY 0 ). Furthermore the 1 morphism induces a zero morphism on each Hq (q ∈ Z).

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proof. Let zD , zD0 , and µX denote the natural morphisms D1 → X, D10 → X 0 , and X 0 → X, respectively. Then we have isomorphisms µ∗D (F(D,zD ) ) ∼ = F(D0 ,µX ◦zD0 ) = ∼ =

q 0 0 F(D → ΩqY 0 /B 0 , which induce an isomorphism F(D 0 ,z 0 ) and OY 0 ⊗OY ΩY /B − 0 ,z 0 ) ⊗OY 0 1 1 1 1 D D 1 1 1 q q ∗ ∼ ΩY 0 /B 0 = µD (F(D,zD ) ⊗OY1 ΩY1 /B1 ). Hence (IV.4.3.3) defines the vertical morphisms. 1

1

Let (T, z) be an object of (X 0 /B 0 )r+1 HIGGS , let YT (ν) (ν ∈ N) be the fiber product of s (ν + 1) copies of Y ×B T = Y 0 ×B 0 T over T , and let DT (ν) (resp. DT0 (ν)) be DHiggs (T1 ,→ 0 YT (ν)) for s = r (resp. s = r+1). Then we have a morphism DT (•) → DT (•) of simplicial objects of C r compatible with the morphisms to YT (•). Hence, by the definition of θ and θq , the following diagrams are commutative. −q µ−1 ODT ,1 ,Q ⊗OY1 ΩqY1 /B1 ) T,1 (ξ

 ξ −q ODT0 ,1 ,Q ⊗OY 0 ΩqY 0 /B 0 1

1

θq

θq

1

/ µ−1 (ξ −q−1 ODT ,1 ,Q ⊗OY Ωq+1 ) T,1 Y1 /B1 1  / ξ −q−1 OD0 ,Q ⊗O 0 Ωq+1 Y 0 /B 0 Y T ,1 1

1

1

, where p0T denotes the morphism DT0 → T , we see that By taking p0−1 T,1 (FT )⊗p0−1 T ,1 (OT ) Γ((T, z), −) of the diagram in the proposition is commutative. Let us prove the second claim. Choose a strict étale covering (uα : Xα → X)α∈A for F as in Definition IV.3.3.1 (ii). Let u : U → X be a strict étale morphism satisfying the following conditions: The morphism u factors through uα for some α ∈ A and there exist a strict étale Cartesian lifting YU → Y of u and ti = (ti,N ) ∈ limN Γ(YU,N , MYU,N ) ←− (1 ≤ i ≤ d) such that d log ti,N (1 ≤ i ≤ d) form a basis of Ω1YU,N /BN for every N ∈ N>0 . 0 0 s Then, for (T, z) ∈ Ob (U 0 /B 0 )r+1 HIGGS , U = U ×B B , the natural morphisms DHiggs (T1 ,→ s YU (ν) ×B T ) → DHiggs (T1 ,→ Y (ν) ×B T ) for s = r, r + 1 are isomorphisms by Lemma IV.2.2.15. Hence we may replace X, Y , and F by U , YU , and u∗HIGGS F. We will prove that Γ(T, −) of the morphism in the proposition is homotopic to 0 by a Γ(T1 , OT1 )-linear homotopy when T1 is affine. Suppose that T1 is affine, which implies that DT,1 and 0 DT,1 are affine by Proposition IV.2.2.9 (2). Put A1 = Γ(T1 , OT1 ), C = Γ(DT,1 , ODT ,1 ), 0 0 C = Γ(DT,1 , ODT0 ,1 ), and M = Γ(T, FT ). Then, we have Γ(DT,1 , p∗T,1 (FT )) = C ⊗A1 M 0 0 and Γ(DT,1 , p0∗ T,1 (FT )) = C ⊗A1 M since FT is a finitely generated projective OT1 ,Q module. Hence the claim is reduced to the case F = OX/B,1,Q . Choose a lifting (si,N ) ∈ limN Γ(TN , MTN ) of the image of ti,1 under Γ(Y1 , MY1 ) → Γ(X, MX ) → Γ(T1 , MT1 ). ←− Then, by Proposition IV.2.3.17, we obtain isomorphisms C ∼ = A1 {W1 , . . . , Wd }r and C0 ∼ = A1 {W1 , . . . , Wd }r+1 compatible with the natural homomorphism C → C 0 . With this description, θ is characterized by θ(Wi ) = ξ −1 d log ti,1 (1 ≤ i ≤ d). Put ωi = ξ −1 d log ti,1 . Then, by Lemma IV.2.3.14, we can define the desired A1,Q -linear homotopy by ! X am W m = a0 k0 m∈Nd

for am ∈ A1,Q such that p kq

|m| −[ r ]

am → 0 as |m| → 0, and ! X X 1 m am W ωi1 ∧ · · · ∧ ωiq = am W m+1i1 ωi2 ∧ · · · ∧ ωiq m + 1 i1 d d

m∈N

m∈N mi =0 for i 0, 1 ≤ i1 < . . . < iq ≤ d, and am ∈ A1,Q such that p−[

p

|m|+1 −[ r+1 ]

1

mi1 +1 am

→ 0 as |m| → 0.

|m| r ]

am → 0. Note 

IV.4. COHOMOLOGY OF HIGGS ISOCRYSTALS

377

IV.4.4. Coherence. In this subsection, we study coherence of topos (X/B)r∼ HIGGS , ∼ XÉT and morphisms between them, which will be used to describe the cohomology of the inverse limit of the sites (X/B)rHIGGS (r ∈ N>0 ) as the direct limit of the cohomology of each (X/B)rHIGGS in IV.4.5. We say that a morphism of p-adic fine log formal schemes f : X → Y is quasi-compact (resp. quasi-separated) if the morphism of schemes underlying the reduction mod pm of f is quasi-compact (resp. quasi-separated) for every m ∈ N>0 . We say that f is coherent if it is quasi-compact and quasi-separated. Quasi-compact (resp. quasi-separated) morphisms of p-adic fine log formal schemes are stable under compositions and base changes (cf. [42] Proposition (1.1.2) (ii), (iii), Proposition (1.2.2) (ii), (iii)). Note that for a scheme with a coherent log structure X, the morphism of schemes underlying X int → X is a closed immersion, and hence quasi-compact and quasi-separated. Let f : X → Y and g : Y → Z be two composable morphisms of p-adic fine log formal schemes. If g ◦ f is quasi-compact and g is quasi-separated, then f is quasi-compact (cf. [42] Proposition (1.2.4)). If g ◦ f is quasi-separated, then f is quasi-separated (cf. [42] Proposition (1.2.2) (v)). In particular, if X is an affine p-adic fine log formal scheme, then any morphism X → Y of p-adic fine log formal schemes is quasi-separated. We say that a morphism f : Y 0 → Y in the category C is quasi-compact (resp. quasiseparated, resp. coherent) if the morphism of p-adic fine log formal schemes fN : YN0 → YN is quasi-compact (resp. quasi-separated, resp. coherent) for every N ∈ N>0 . Since the morphisms of schemes underlying the reduction mod p of Y2 → YN and Y20 → YN0 are nilpotent immersions, the condition for N = 2 implies that for every N ≥ 2. Similarly, if f is a morphism in the subcategory C r , then the condition for N = 1 implies the condition for every N ∈ N>0 (cf. Corollary IV.2.2.5). Let r ∈ N>0 ∪ {∞}, let B ∈ Ob C r , and let X be a p-adic fine log formal scheme over B1 . We define (X/B)rHIGGS,coh to be the full subcategory of (X/B)rHIGGS consisting of (T, z) such that the structure morphism T → B is coherent. We endow (X/B)rHIGGS,coh with the topology induced by that of (X/B)rHIGGS . X´e∼t ,

Lemma IV.4.4.1. The full subcategory (X/B)rHIGGS,coh of (X/B)rHIGGS is stable under finite fiber products. If the morphism of p-adic fine log formal schemes X → B1 is quasi-separated, then it is also stable under finite products. Proof. By the proofs of Proposition IV.3.1.2 (1) and Corollary IV.2.2.11, the fiber product of (T 0 , z 0 ) → (T, z) ← (T 00 , z 00 ) in (X/B)rHIGGS is represented by T 000 = r DHiggs (T 0 ×T T 00 ). Suppose that the morphisms T, T 0 , T 00 → B are coherent. Then the morphism T 0 → T and its base change T 0 ×T T 00 → T 00 are coherent. Since T 000 → T 0 ×T T 00 is affine by Proposition IV.2.2.9 (2), the composition T 000 → T 0 ×T T 00 → T 00 → B is coherent. By the proof of Proposition IV.3.1.2 (1), the product of (T, z) and (T 0 , z 0 ) ∈ r (T1 ×X T10 ,→ T ×B T 0 ). Ob (X/B)rHIGGS in (X/B)rHIGGS is represented by T 00 = DHiggs 0 Suppose that T, T → B are coherent and X → B1 is quasi-separated. Then z : T1 → X and its base change T1 ×X T10 → T10 are coherent. Since T100 → T1 ×X T10 is affine by Proposition IV.2.2.9 (2), the composition of T100 → T1 ×X T10 → T10 → B1 is coherent.  For an object (T, z) of (X/B)rHIGGS,coh , we define Covcoh ((T, z)) to be {(uα : (Tα , zα ) → (T, z))α∈A ∈ Cov((T, z))|(Tα , zα ) ∈ Ob (X/B)rHIGGS,coh for all α ∈ A}. Lemma IV.4.4.2. Assume that the morphism B → S is quasi-separated. (1) An object (T, z) of (X/B)rHIGGS such that T1 is affine belongs to (X/B)rHIGGS,coh . (2) The sets Covcoh ((T, z)) for (T, z) ∈ Ob (X/B)rHIGGS,coh satisfy the axiom of pretopology, and the topology of (X/B)rHIGGS,coh is induced by this pretopology.

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

(3) The continuous functor (X/B)rHIGGS,coh → (X/B)rHIGGS induces an equivalence ∼ r∼ of categories (X/B)r∼ HIGGS → (X/B)HIGGS,coh . Proof. (1) If T1 is affine, then TN are affine and the morphism T → S is coherent. Hence, if B → S is quasi-separated, the morphism T → B is coherent. (2) The first claim follows from the fact that the subcategory (X/B)rHIGGS,coh is stable under fiber products (Lemma IV.4.4.1). By [2] III Corollaire 3.3, it also implies that a family of morphisms U = (uα : (Tα , zα ) → (T, z))α∈A in (X/B)rHIGGS,coh is a covering if and only if there exists a refinement U 0 ∈ Cov((T, z)) of U. By (1), every V ∈ Cov((T, z)) admits a refinement by a V 0 ∈ Covcoh ((T, z)). Hence, we may replace Cov((T, z)) with Covcoh ((T, z)) in the above equivalence. (3) This follows from (1) and [2] III Théorème 4.1.  Proposition IV.4.4.3. Assume that the morphism B → S is coherent, and the morphism X → B1 is quasi-separated. (1) The topos (X/B)r∼ HIGGS is algebraic ([2] VI Définition 2.3). (2) If the morphism X → B1 is quasi-compact, then the topos (X/B)r∼ HIGGS is coherent. Proof. Put C = (X/B)rHIGGS,coh to simplify the notation. By Lemma IV.4.4.2 (3) and the assumption on B → S, it suffices to prove the claims for C ∼ . Let ε denote the canonical functor C → C ∼ . (1) For any (T, z) ∈ Ob C, the composition of T1 → B1 → S1 is quasi-compact by the assumption on B → S. Since S1 is quasi-compact, T1 is quasi-compact, which implies that (T, z) is a quasi-compact object of the site C ([2] VI Définition 1.1) by Lemma IV.4.4.2 (2). By Lemma IV.4.4.1 and Proposition IV.3.1.2 (1), fiber products are representable in C. Hence, by [2] VI Corollaire 2.1.1, ε(T ) is a coherent object of C ∼ for any T ∈ Ob C. By Lemma IV.4.4.1, Proposition IV.3.1.2 (1) and the assumption on X → B1 , we see that T × T 0 are representable in C for any T, T 0 ∈ Ob C, which implies that ε(T ) × ε(T 0 ) = ε(T × T 0 ) is coherent. Now the claim follows from one of the defining properties [2] VI Proposition 2.2 (ii bis) of an algebraic topos. (2) By (1), it suffices to prove that the final object e of C ∼ is coherent (cf. [2] VI Définition 2.3). By the proof of (1), ε(T ) and ε(T ) × ε(T 0 ) are coherent for any T , T 0 ∈ Ob C. Hence, by [2] VI Corollaire 1.17, it suffices to prove that there exists a covering of e of the form (ε(Tα ) → e)α∈A such that Tα ∈ Ob C and A is a finite set. Since X → B1 , B1 → S1 , and S1 are quasi-compact, X is quasi-compact. Hence there exists a strict étale covering (Xα → X)α∈A such that A is a finite set, Xα is affine, and there exists a chart Pα → Γ(X α , MX α ) of MX α , where X α = Xα ×Spf(Zp ) Spec(Z/pZ). Choose a surjective morphism of monoids hα : Ndα → Pα for some dα ∈ N, and a surjective homomorphism of R1 /pR1 -algebras fα : R1 /pR1 [Tλ ; λ ∈ Λα ] → Γ(X α , OX α ). Let Yα,N (resp. Zα,N ) be Spf(RN [Ndα ]∧ ) (resp. Spf(RN [Ndα ][Tλ ; λ ∈ Λα ]∧ )) endowed with the log structure naturally defined by Ndα . Here ∧ denotes the p-adic completion. Then Yα,N and Zα,N naturally define objects Y and Z of C with morphisms Z → Y → S in C . The morphism Y → S is smooth and Cartesian. The morphisms hα and fα induce an S1 -closed immersion X α → Zα,1 over S1 , which is lifted to an S1 -closed immersion iα : Xα → Zα because Yα,1 → S1 is smooth. Let (T, z) be an object of C such that T1 is affine (and hence TN are affine), and suppose that there exist α ∈ A and an X-morphism v : T1 → Xα . Then, by Sublemma IV.3.4.2 (2), we see that the morphism T1 → Xα → Y1,α over S1 has a lifting T → Yα over S, which obviously i ◦v

α admits a factorization T → Zα → Yα lifting T1 −− −→ Zα,1 → Yα,1 . Thus we obtain a morphism T → (Xα ,→ Zα ×S B) over B compatible with v : T1 → Xα . Hence

IV.4. COHOMOLOGY OF HIGGS ISOCRYSTALS

379

r Tα = DHiggs (Xα ,→ Zα ×S B) with the natural morphism zα : Tα,1 → X gives the desired covering of e. Note that Tα,1 is affine since Tα,1 → Xα is affine by Proposition IV.2.2.9 (2), which implies that (Tα , zα ) is an object of C by Lemma IV.4.4.2 (1). 

Proposition IV.4.4.4. Let the notation and assumption be the same as in Proposition IV.3.1.6. If the morphisms B, B 0 → S are coherent, then the morphism of topos r0 ∼ fHIGGS : (X 0 /B 0 )HIGGS → (X/B)r∼ HIGGS is coherent. 0

r Proof. For an object (T, z) of (X/B)rHIGGS,coh , the morphism DHiggs (T ×B B 0 ) → 0 T ×B B is affine, in particular, coherent by Proposition IV.2.2.9 (2). Hence the functor 0 ∗ f ∗ in Proposition IV.3.1.6 induces a functor fcoh : (X/B)rHIGGS,coh → (X 0 /B 0 )rHIGGS,coh . For a morphism u : (T 0 , z 0 ) → (T, z) in (X/B)rHIGGS,coh such that u is strict étale, the r0 r0 natural morphism DHIGGS (T 0 ×B B 0 ) → DHIGGS (T ×B B 0 ) ×T T 0 is an isomorphism by ∗ Proposition IV.2.2.9 (3). Hence fcoh is continuous by Lemma IV.4.4.2 (2). By Lemma ∗ IV.4.4.2 (3) and Proposition IV.3.1.6, the functor fcoh is a morphism of site, and it ∗ suffices to prove that the morphism of topos associated to fcoh is coherent. By Lemma IV.4.4.1 and Proposition IV.3.1.2 (1), fiber products are representable in (X/B)rHIGGS,coh 0 and (X 0 /B 0 )rHIGGS,coh . On the other hand, for every object (T, z) of (X/B)rHIGGS,coh 0 (resp. (X 0 /B 0 )rHIGGS,coh ), T1 is quasi-compact because B1 → S1 (resp. B10 → S1 ) and S1 are quasi-compact. Hence T is a quasi-compact object. Now the claim follows from [2] VI Corollaire 3.3. 

Next we discuss coherence of projections to étale sites. Let X be a p-adic fine log formal scheme. We define X´et,coh (resp. XÉT,coh ) to be the full subcategory of X´et (resp. XÉT ) consisting of u : U → X such that U is quasi-compact and u is coherent. We endow X´et,coh (resp. XÉT,coh ) with the topology induced by that of X´et (resp. XÉT ). Lemma IV.4.4.5. The full subcategory X´et,coh of X´et is stable under finite fiber products and nonempty finite products, and the full subcategory XÉT,coh of XÉT is stable under finite fiber products. Proof. For morphisms U 0 → U ← U 00 in X´et,coh (resp. XÉT,coh ), the morphism U → U and its base change U 0 ×U U 00 → U 00 are coherent because U → X is quasiseparated and U 0 → X is coherent. Since U 00 is quasi-compact and U 00 → X is coherent, this implies that U 0 ×U U 00 is quasi-compact and U 0 ×U U 00 → X is coherent. For two objects U and U 0 of X´et,coh , the morphism U ×X U 0 → U is coherent since U 0 → X is coherent. Hence U ×X U 0 is quasi-compact and U ×X U 0 → X is coherent because U is quasi-compact and U → X is coherent.  0

Lemma IV.4.4.6. Assume that X → Spf(Zp ) is quasi-separated. Let ? denote ´et or ÉT. (1) The sets   (i) (Uα → U )α∈A is a strict étale covering. Covcoh (U ) = (Uα → U )α∈A (ii) Uα ∈ Ob X?,coh . for U ∈ Ob X?,coh satisfy the axiom of pretopology, and it induces the topology of X?,coh . ∼ (2) The natural morphism of topos X?∼ → X?,coh is an equivalence of categories. ∼ (3) Any object of X?,coh represented by an object of X?,coh is coherent. (4) The topos X?∼ is algebraic. If X is quasi-compact, then the topos X?∼ is coherent. ∼ (5) The canonical morphism of topos XÉT → X´e∼t is coherent.

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proof. (1) (2) An object U → X of X? with U affine is an object of X?,coh by the assumption on X → Spf(Zp ). Hence one can prove the claims in the same way as the proof of Lemma IV.4.4.2 (2) and (3) by using Lemma IV.4.4.5 and [2] III Corollaire 3.3, Théorème 4.1. (3) By (1) and the definition of X?,coh , every object of X?,coh is quasi-compact. Hence the claim follows from Lemma IV.4.4.5 and [2] VI Corollaire 2.1.1. ∼ (4) By (2), it suffices to prove the claims for X?,coh . Let ε denote the canonical ∼ functor X?,coh → X?,coh . Then, by (3) and Lemma IV.4.4.5, ε(U ) and ε(U ) × ε(U 0 ) = ∼ ε(U ×X U 0 ) are coherent for any U, U 0 ∈ Ob X?,coh . Hence X?,coh is algebraic by the defining property [2] VI Proposition 2.2 (ii bis) of algebraic topos. If X is quasi-compact, ∼ then X ∈ Ob X?,coh , which implies that the final object ε(X) of X?,coh is coherent. (5) The inclusion functor ι : X´et → XÉT induces a functor ιcoh : X´et,coh → XÉT,coh . By (1), the functor ιcoh is continuous. By Lemma IV.4.4.5, finite inverse limits are representable in X´et,coh and the functor ιcoh is left exact. Hence ιcoh is a morphism of site (cf. [2] I Proposition 5.4 4), III Proposition 1.3 5)). By (1), every object of X´et,coh and XÉT,coh is quasi-compact. Hence the claim follows from (2), Lemma IV.4.4.5, and [2] VI Corollaire 3.3.  Lemma IV.4.4.7. Let f : X 0 → X be a morphism of p-adic fine log formal schemes. If X, X 0 → Spf(Zp ) are quasi-separated and f is quasi-compact, then the morphisms of ∼ 0∼ ∼ topos f´et : X´e0∼ t → X´ et and fÉT : XÉT → XÉT are coherent. Proof. Let ? denote ´et or ÉT. Since f is quasi-compact, the functor f ∗ : X? → ∗ 0 7→ U ×X X 0 induces a functor fcoh : X?,coh → X?,coh . By Lemma IV.4.4.6 (1), the ∗ functor fcoh is continuous. Hence it is a morphism of sites by Lemma IV.4.4.6 (2). Since 0 every object of X?,coh and X?,coh is quasi-compact by Lemma IV.4.4.6 (1), the claim follows from Lemma IV.4.4.6 (2), Lemma IV.4.4.5, and [2] VI Corollaire 3.3. 

X?0 ; U

Proposition IV.4.4.8. Let r ∈ N>0 ∪ {∞}, let B ∈ Ob C r , and let X be a p-adic fine log formal scheme over B1 . Assume that X → B1 is quasi-separated and B → S is ∼ coherent. Then the morphism of topos UX/B : (X/B)r∼ HIGGS → XÉT is coherent. Proof. Let εX/B and εX denote the canonical functors (X/B)rHIGGS → (X/B)r∼ HIGGS ∼ and XÉT → XÉT , respectively. By Lemma IV.4.4.2 (3) and the proof of Proposition IV.4.4.3 (1), the objects εX/B (T, z), (T, z) ∈ Ob (X/B)rHIGGS,coh are coherent and generate (X/B)r∼ HIGGS . Similarly, by Lemma IV.4.4.6 (2) and (3), the objects εX (U ), ∼ U ∈ Ob XÉT,coh are coherent and generate XÉT . Hence, by [2] VI Proposition 3.2, it ∗ suffices to prove that UX/B (εX (U )) is coherent for U ∈ Ob XÉT,coh . ∗ We have Γ((T, z), UX/B (εX (U ))) = Γ(T1 , εX (U )) = HomX (T1 , U ) for an object (T, z) r of (X/B)HIGGS (cf. [17] IV Proposition 6.3.1 (iii)). Since U is quasi-compact, there exists a strict étale covering (uα : Uα → U )α∈A such that Uα is affine, A is a finite set, and there exists a chart Pα → Γ(U α , MU α ), where U α = Uα ×Spf(Zp ) Spec(Z/pZ) for each α ∈ A. As in the proof of Proposition IV.4.4.3 (2), we can construct (Tα , zα ) ∈ Ob ((X/B)rHIGGS,coh ) with a morphism wα : Tα,1 → Uα over X satisfying the following property: For any (T, z) ∈ Ob (X/B)rHIGGS such that T1 is affine and any morphism w : T1 → Uα over X, there exists a morphism f : (T, z) → (Tα , zα ) such that zα ◦ f1 = w. This property ∗ (εX (U )))α∈A defined by the implies that the family of morphisms (εX/B (Tα , zα ) → UX/B composition vα := uα ◦ zα : Tα,1 → Uα → U is a covering (cf. [2] II Proposition 4.3 2)). Let (Tαβ , vαβ ) be the product of (Tα , vα ) and (Tβ , vβ ) in the category (U/B)rHIGGS,coh (cf. Lemma IV.4.4.1, Proposition IV.3.1.2 (1)). Note that the composition of U → X → B is quasi-separated because U → X is coherent. Let zαβ denote the composition of

IV.4. COHOMOLOGY OF HIGGS ISOCRYSTALS

381

vαβ

Tαβ,1 −−→ U → X. Then we see that the object (Tαβ , zαβ ) represents the fiber product ∗ of (Tα , zα ) → UX/B (εX (U )) ← (Tβ , zβ ) in the category of presheaves on (X/B)rHIGGS ∗ (cf. [2] I Proposition 5.11 and Proposition IV.3.1.9 (1)). Hence εX/B (Tα , zα )×UX/B (εX (U )) εX/B (Tβ , zβ ) = ε(Tαβ , zαβ ) is coherent. Now the claim follows from [2] VI Corollaire 1.17.  IV.4.5. Cohomology. Let r0 ∈ N>0 and let B ∈ Ob C r0 . For each integer r ≥ r0 , r let B r denote DHiggs (B r0 ). We have a sequence of morphisms · · · → B r+1 → B r → · · · → B r0 +1 → B r0 = B. We assume that B → S is coherent. By Proposition IV.2.2.9 (2), the morphism B r → B r0 for r ≥ r0 is affine, in particular, coherent. Let X → B1 be a coherent morphism of p-adic fine log formal schemes. For an integer r ≥ r0 , let X r be the base change X ×B1 B1r . Suppose that we are given a smooth Cartesian morphism Y → B in C and an immersion X → Y1 over B1 , and let Dr (r ≥ r0 ) denote the Higgs envelope of level r of X r → Y r := Y ×B B r . Then, for r ∈ N, r ≥ r0 and a Higgs isocrystal F of level r on s s s s s X r /B r , one obtains an object (FD s , θ) of HBQ ,conv (X , Y /B ) associated to the pullp s s s s back F of F on (X /B )HIGGS by the equivalence of categories in Theorem IV.3.4.16 for each s ∈ N, s ≥ r. In this subsection, we prove a comparison theorem (Theorem IV.4.5.6) between the cohomology of the pull-back of F on the inverse limit of the sites s • (X s /B s )sHIGGS (s ≥ r0 ) and the complexes ξ −• FD s ⊗OY s ΩY s /B s (s ≥ r) associated to 1 1 1 s (FD s , θ) by Lemma IV.3.4.5 We define the category (X • /B • )HIGGS as follows. An object is a pair (r, (T, z)) of an integer r ≥ r0 and an object (T, z) of (X r /B r )rHIGGS . For two objects (r, (T, z)) and (r0 , (T 0 , z 0 )), there is no morphism (r0 , (T 0 , z 0 )) → (r, (T, z)) if r0 < r. If r0 ≥ r, a morphism (r0 , (T 0 , z 0 )) → (r, (T, z)) is a morphism f : T 0 → T in C r compatible with the 0 0 morphisms X r → X r and B r → B r . Let I r0 denote the opposite category of the category associated to the ordered set {r ∈ N|r ≥ r0 } (cf. IV.6.6). Then we have a natural functor π : (X • /B • )HIGGS → I r0 defined by ((T, z), r) 7→ r. The fiber of π over r ∈ I r0 can be identified with the category (X r /B r )rHIGGS . Lemma IV.4.5.1. The pair ((X • /B • )HIGGS , π) is a fibered site over I r0 . Proof. Let r, r0 ∈ I r0 such that r0 ≥ r and let m : r0 → r be the unique morphism. Then, by Proposition IV.3.1.6, the inverse image functor m∗ : (X r /B r )rHIGGS → 0 0 0 (X r /B r )rHIGGS is given by the morphism of sites induced by the natural morphisms 0 0 X r → X r and B r → B r as in loc. cit.  Let (X ∞ /B ∞ )HIGGS be the inverse limit of the fibered site π : (X • /B • )HIGGS → I r0 ([2] VI Définition 8.2.5). The associated topos (X ∞ /B ∞ )∼ HIGGS is an inverse limit of • • ∼/I r0 r0 the fibered topos (X /B ) → I ([2] VI Théorème 8.2.3 2)). The composition of (X r /B r )rHIGGS → (X • /B • )HIGGS → (X ∞ /B ∞ )HIGGS is a morphism of site ([2] VI Théorème 8.2.3 1)). Let µr denote the associated morphism of topos (X ∞ /B ∞ )∼ HIGGS → 0 r0 r0 r0 ∼ r r r∼ (X r /B r )r∼ HIGGS . For integers r ≥ r ≥ r0 , let µrr 0 : (X /B )HIGGS → (X /B )HIGGS be the morphism of topos defined by the inverse image functor between the fibers of (X • /B • )HIGGS over r and r0 , which coincides with the morphism of topos induced by the 0 0 morphisms X r → X r and B r → B r (cf. the proof of Lemma IV.4.5.1). Then we have a ∼ = canonical isomorphism µrr0 ◦µr0 ∼ = µr . We have a natural isomorphism µ∗rr0 (OX r /B r ,1 ) → OX r0 /B r0 ,1 of sheaves of rings. We define the sheaf of rings OX ∞ /B ∞ ,1 on (X ∞ /B ∞ )HIGGS

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

to be limr≥r µ∗r (OX r /B r ,1 ), which is canonically isomorphic to µ∗r (OX r /B r ,1 ) for any −→ 0 r ≥ r0 . By Proposition IV.4.4.3, (X r /B r )r∼ HIGGS is coherent for every r ≥ r0 . By Proposition IV.4.4.4, the morphism µrr0 is coherent. Hence, by [2] VI Corollaire 8.7.7, we have a canonical isomorphism ∼ lim H q ((X s /B s )HIGGS , µ∗ (F)) (IV.4.5.2) H q ((X ∞ /B ∞ )HIGGS , µ∗r F) = rs −→ s≥r

for an integer r ≥ r0 and a sheaf of OX r /B r ,1,Q -modules F on (X r /B r )rHIGGS . Let X´e•t → I r0 be the fibered site defined by the sequence of morphisms of p-adic fine log formal schemes · · · → Xr+1 → Xr → · · · → Xr0 +1 → Xr0 = X.

• r0 Let X´e∞ t denote the inverse limit of the fibered site X´ et r→ I . The associated topos ∞ ∼ • ∼/I 0 (X )´et is an inverse limit of the fibered topos (X )´et → I r0 . Let µr : (X ∞ )´e∼t → r ∼ s ∼ r ∼ (X )´et (r ≥ r0 ) and µrs : (X )´et → (X )´et (s ≥ r ≥ r0 ) be the natural morphisms of

topos. By Proposition IV.4.1.3, the morphisms of topos uX r /B r (r ≥ r0 ) define a Cartesian morphism of fibered topos over I r0 ∼/I r0

r0

uX • /B • : (X • /B • )HIGGS −→ (X´e•t )∼/I .

∞ ∼ Let uX ∞ /B ∞ : (X ∞ /B ∞ )∼ denote the morphism of topos induced by HIGGS → (X´ et ) uX • /B • . By Propositions IV.4.4.3, IV.4.4.4, IV.4.4.8, and Lemmas IV.4.4.6, IV.4.4.7, the r ∼ topos (X r /B r )r∼ are coherent for r ≥ r0 , and the morphisms of topos HIGGS and (X´ et ) s s s∼ r r r∼ (X /B )HIGGS → (X /B )HIGGS , (X´est )∼ → (X´ert )∼ , and uX r /B r : (X r /B r )r∼ HIGGS → (X´ert )∼ are coherent for s ≥ r ≥ r0 . Hence, by [2] VI Corollaire 8.7.5, we have a canonical isomorphism (IV.4.5.3) Rq uX ∞ /B ∞ ∗ (µ∗r (F)) ∼ µ∗ (Rq uX r /B r ∗ (µ∗rs (F)) = lim −→ s s≥r

for an integer r ≥ r0 and a sheaf of OX r /B r ,1,Q - modules F on (X r /B r )rHIGGS . Let Y → B be a smooth Cartesian morphism in the category C and let i : X → Y1 be an immersion over B1 . For an integer r ≥ r0 , let Y r → B r denote the base change of Y → B by the morphism B r → B in the category C , and let ir : X r → Y1r be the base r change of i by B1r → B1 . Let Dr be DHiggs (ir : X r ,→ Y r ) and let zDr be the natural r r r morphism D1 → X over B1 . Let r be an integer ≥ r0 and let F be a Higgs isocrystal on (X r /B r )rHIGGS . We will give a description of RuX ∞ /B ∞ ∗ (µ∗r (F)) in terms of the complexes (ξ −• µ∗rs (F)Ds ⊗OY s 1 Ω•Y s /B1 , θ• ) (s ≥ r). 1

r • • r0 Let (X • /B • )≥r by the inclusion HIGGS → I be the base change of (X /B )HIGGS → I r r0 ∞ ∞ functor I → I . The site (X /B )HIGGS is also the direct limit of the fibered site r • • ≥r∼ (X • /B • )≥r HIGGS → I . Let (X /B )HIGGS be the topos associated to the total site of • • ≥r∼ • • ≥r the fibered site (X /B )HIGGS → I r . Let Q : (X ∞ /B ∞ )∼ HIGGS → (X /B )HIGGS be ≥r the morphism of topos induced by the functor (X • /B • )HIGGS → (X ∞ /B ∞ )HIGGS . For s ≥ r, let F s denote the Higgs isocrystal µ∗rs F on (X s /B s )sHIGGS , and let F • denote s s s0 the sheaf in (X • /B • )≥r∼ HIGGS defined by F and the natural morphisms F → µss0 ∗ F (s0 ≥ s ≥ r) (cf. [2] VI Proposition 7.4.7). Let F † be Q∗ (F • ), which is canonical isomorphic to lims≥r µ∗s (F s ) and also to µ∗r (F r ) (cf. [2] VI Proposition 8.5.2). −→ By Proposition IV.4.3.4, we obtain a complex

(IV.4.5.4)

s • (F s −→ LY s (ξ −• (FD s ) ⊗OY s ΩY s /B s ))s≥r 1 1 1

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

383

in (X • /B • )≥r∼ HIGGS . By the second claim of Proposition IV.4.3.4 and [2] VI Proposition 8.5.2, we obtain the following. Proposition IV.4.5.5. The pull-back of (IV.4.5.4) by the morphism of topos Q gives a resolution of F † (∼ = µ∗r F) s • F † −→ Q∗ ((LY s (ξ −• (FD s ) ⊗OY s ΩY s /B s ))s≥r ). 1 1 1

By [2] VI Théorème 8.7.3, we have a canonical isomorphism Rq uX ∞ /B ∞ ∗ Q∗ (G) ∼ µ∗ Rq uX s /B s ∗ G s = lim −→ s s≥r

for a sheaf of abelian groups G = (G s )s≥r on (X • /B • )≥r∼ HIGGS . Hence, by Proposition IV.4.2.1 and Lemma IV.4.3.2, we obtain the following. Theorem IV.4.5.6. Let r be an integer ≥ r0 , let F be a Higgs isocrystal on the site (X r /B r )rHIGGS , and let F s (s ≥ r) denote the pull-back of F on (X s /B s )sHIGGS , which is a Higgs isocrystal. Then there exists a canonical isomorphism s • • RuX ∞ /B ∞ ∗ µ∗r (F) ∼ (µ∗ (z s (ξ −• FD s ⊗OY s ΩY s /B s , θ ))). = lim 1 1 −→ s D ∗ 1 s≥r

If B is an object of C , then B is contained in C r for every r ∈ N>0 . Hence, we may assume r0 = 1 and we have B r = B and X r = X for every r ∈ N>0 . The functor µr∗ : (X ∞ )´e∼t → X´ert is an equivalence of categories for r ∈ N>0 . Hence we may regard ∼ uX∞ /B∞ as a morphism of topos (X ∞ /B ∞ )∼ HIGGS → X´ et . ∞

Corollary IV.4.5.7. Assume that r0 = 1 and B is an object of C ∞ . Let r be a positive integer, let F be a Higgs isocrystal on (X/B)rHIGGS , and let F s (s ≥ r) denote the pull-back of F on (X/B)sHIGGS . Then there exists a canonical isomorphism s • • RuX ∞ /B ∞ ∗ µ∗r (F) ∼ (z s (ξ −• FD s ⊗OY ΩY /B , θ )). = lim 1 1 1 −→ D ∗ s≥r

If X → B1 is smooth and integral and i : X → Y1 is an isomorphism, then YN → BN is smooth and integral. In particular, the morphism of schemes underlying the reduction mod pm of YN → BN is flat for every N, m ∈ N>0 . Hence, if B is an object of C ∞ , then for every r ∈ N>0 , Y is an object of C r by Lemma IV.2.2.7 (1), and therefore we have Y = Dr and X ∼ = Y1 = D1r . Corollary IV.4.5.8. Assume that r0 = 1, B is an object of C ∞ , X → B1 is smooth and integral, and i : X → Y1 is an isomorphism. Let r be a positive integer, let F be a Higgs isocrystal on (X/B)rHIGGS , and let (M, θ) be the Higgs vector bundle on X associated to F via Y and i (cf. Theorem IV.3.4.16). Then there exists a canonical isomorphism ∼ (ξ −• M ⊗O Ω• RuX ∞ /B ∞ ∗ µ∗ (F) = , θ• ). r

X

X/B1

IV.5. Representations of the fundamental group IV.5.1. The p-adic fine log formal schemes DN (U ) and U . Let U be an affine fs log scheme over Zp satisfying the following conditions: The ring A = Γ(U, OU ) is a noetherian normal ring flat over Zp , and U satisfies the conditions in [73] Lemma 1.3.2, i.e., there exists a chart α : PU → MU such that P is a finitely generated and saturated × monoid and the morphism P → Γ(U, MU )/Γ(U, OU ) induced by α is an isomorphism. Let A be one of the following three A-algebras: (Case I) The algebra A itself. (Case II) The henselization of A with respect to the ideal pA. (Case III) The p-adic completion of A.

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Then A is noetherian and, in the cases I and II, it is normal. In the case III, we also assume that A is normal, which holds if A is of finite type over a complete noetherian local ring (cf. [42] Scholie (7.8.3) (ii), (iii), (v)). Let U be Spec(A) endowed with the inverse image of MU . Let Utriv be the open subscheme of Spec(A) defined by {x ∈ Spec(A)|MU ,x = OU×,x }. If P is generated by a1 , . . . , ar ∈ P , we have Utriv = Spec(A[α(a1 a2 · · · ar )−1 ]). Put Atriv = Γ(Utriv , OUtriv ). Let s → Utriv be a geometric point of Utriv such that the image s of s in Utriv is of codimension 0 and K := Γ(s, Os ) is an algebraic closure of the residue field K := κ(s) of Utriv at s. We further assume that p is not invertible on the connected component of U containing s, which always holds in the cases II and III. We define Kur to be the union of all finite extensions L of K contained in K such that the integral closures of Atriv,Qp b in L are étale over A . Let A denote the integral closure of A in Kur and let A triv,Qp

b= denote the p-adic completion of A. We have A/pA = 6 0 and hence A 6 0 by assumption. ur Let G(U,s) denote the Galois group Gal(K /K), which is canonically isomorphic to the fundamental group of the connected component of Utriv,Qp containing s with respect to the base point s → Utriv,Qp . Let RA be the inverse limit of F

F

F

F

A/pA ←− A/pA ←− A/pA ←− A/pA ←− · · · , where F denotes the absolute Frobenius of A/pA. We have a canonical ring homomorb characterized by θ([x]) = lim xf pm for x = (x , x , . . .) ∈ R , phism θ : W (RA ) → A m 0 1 A m→∞

xn ∈ A/pA, where [ ] : A → W (RA ) denotes the Teichmüller representative and e denotes a lifting of an element of A/pA to A. The kernel of θ is generated by a nonzero divisor ξ (cf. [29] 2.4. Proposition, [25] II b), [73] Corollary A2.2). One can construct a generator as follows. Let νn ∈ A be a compatible system of pn -th roots of −p, i.e., ν0 = −p and (νn+1 )p = νn , and let −p be (νn mod p)n≥0 ∈ RA . Then the element e = θ(W (R )), which is ξ := [−p] + p of W (R ) generates the ideal Ker(θ). Put A p

A

A

p-adically complete and separated (cf. [73] Sublemma A2.12). For N ∈ N>0 , let AN (A) denote W (RA )/ξ N W (RA ), which is also p-adically complete and separated. b = A. e Lemma IV.5.1.1. In the cases II and III, we have A

Proof. It suffices to prove that the absolute Frobenius of A/pA is surjective (cf. [31] 2 1.2.2, [25] II b), [73] Lemma A 1.1). We prove that, for a ∈ A, all solutions of xp −px = a in K are contained in A. Choose a finite extension L of K contained in Kur such that a ∈ 2 L, and let B be the integral closure of A in L. Then the ring B 0 := B[X]/(X p − pX − a) is integral over A, which implies that the pair (B 0 , pB 0 ) is henselian. Hence the image of 2 2 0 0 (X p − pX − a)0 = p(−1 + pX p −1 ) in BQ is invertible, BQ /BQp is étale, and hence all p p 2

solutions of xp − px − a = 0 in K are contained in A.



Lemma IV.5.1.2. The action of G(U,s) on AN (A) is continuous with respect to the p-adic topology of AN (A). Proof (cf. the proof of [73] Lemma 1.4.4). It suffices to prove that the action of G(U,s) on AN (A)/pm AN (A) = Wm (RA )/ξ N Wm (RA ) is continuous with respect to the discrete topology, which is reduced to proving that the stabilizer of the image of [x] in AN (A)/pm is open for any x ∈ RA . For the polynomials Sn (X0 , . . . , Xn , Y0 , . . . , Yn ) ∈ Z[X0 , . . . , Xn , Y0 , . . . , Yn ] (n ∈ N) defining the addition of Witt vectors, let ϕn (X0 , Y0 )

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

385

denote Sn (X0 , 0, . . . , 0, Y0 , 0, . . . , 0). Then we have n

n

n

(X0 )p + (Y0 )p = ϕ0 (X0 , Y0 )p + pϕ1 (X0 , Y0 )p

n−1

+ · · · + pn ϕn (X0 , Y0 ),

which implies ϕ0 (X0 , Y0 ) = X0 + Y0 and ϕn (X0 , 0) = ϕn (0, Y0 ) = 0 for n ≥ 1. Hence, for y ∈ RA and M ∈ N>0 , the sum [x] + [(−p)M y] in Wm (RA ) is written in the form (x + (−p)M y, (−p)M y1 , . . . , (−p)M ym−1 )

=[x + (−p)M y] + p[(−p)p

−1

M

−1

(y1 )p ] + · · · + pm−1 [(−p)p

−(m−1)

M

(ym−1 )p

N +m

−(m−1)

]

m

= 0 in AN (A)/p . Hence, (y1 , . . . , ym−1 ∈ RA ). Since [−p] = ξp − p, we have [−p] m−1 m if M ≥ p (N + m), the morphism [ ] : RA → AN (A)/p factors through the quotient RA /(−p)M RA . For any l ∈ N, the l-th power of the absolute Frobenius induces an ∼ =

l

→ RA /(−p)RA , and the projection to the first component isomorphism RA /(−p)p RA − induces an injection RA /(−p)RA ,→ A/pA (cf. [73] Lemma A2.1). Hence the action of G(U,s) on RA /(−p)M RA is continuous with respect to the discrete topology for any M ∈ N>0 . This completes the proof.  Lemma IV.5.1.3 (cf. [73] Lemma 1.4.1). (1) Let Atriv be the integral closure × in Kur . Then, for any a ∈ Atriv , all solutions of xp = a in K are contained in b factors through (2) The composition of the morphisms Γ(U, MU ) → A → A

of Atriv × Atriv . e A.

Proof. (1) Choose a finite extension L of K contained in Kur such that a ∈ L, let B be the integral closure of A in L, and put Btriv = Atriv ⊗A B. Since a and a−1 are × integral over Atriv , we have a ∈ Btriv , which implies that Btriv,Qp [X]/(X p − a) is étale ×

over Btriv,Q . Hence every p-th root b ∈ K of a is contained in Atriv . Note that b and b−1 are integral over Atriv . × × (2) The image of Γ(U, MU ) in K is contained in Atriv ∩ A. For any a ∈ Atriv ∩ A, choose a compatible system of pn -th roots an ∈ K of a. Then an ∈ A by (1), and we e have a = θ((a mod p) ) ∈ A.  n

n≥0

e endowed with the log structure We define U to be the p-adic formal scheme Spf(A) e e induces its action on U . Let Q associated to Γ(U, MU ) → A. The action of G(U,s) on A be the fiber product of the diagram of integral monoids f

f

f

lim(A\{0} ← − A\{0} ← − A\{0} ← − · · · ) −→ A\{0} ←− Γ(U, MU ), ←− where f is the morphism defined by f (x) = xp and the left map is the projection to the first component. For N ∈ N>0 , we define DN (U ) to be the p-adic formal scheme Spf(AN (A)) endowed with the log structure associated to the composition of []

Q −→ lim(A\{0}) −→ RA −→ AN (A). ←− f

The natural actions of G(U,s) on Q and AN (A) define its action on DN (U ). For N ∈ N>0 , the surjective homomorphism AN +1 (A) → AN (A) and the identity map of Q induces a morphism iU ,N : DN (U ) → DN +1 (U ) compatible with the actions of G(U,s) . The e induced by θ and the projection Q → Γ(U, M ) induce a isomorphism A (A) → A 1

U

morphism iU : U → D1 (U ) compatible with the actions of G(U,s) .

Lemma IV.5.1.4 (cf. [73] Lemma 1.4.2). The log structures of U and DN (U ) (N ∈ N>0 ) are fine and saturated, the morphism iU is an isomorphism, and the morphisms iU ,N (N ∈ N>0 ) are exact closed immersions.

386

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proof. Let U n and DN (U )n denote the reduction mod pn of U and DN (U ). Since = × ∼ the chart α in the condition on U induces an isomorphism P × Γ(U, OU )− → Γ(U, MU ), the log structure MU is fine and saturated and the composition P → Γ(U, MU ) → Γ(U n , MU n ) gives the chart of MU n . Since P × = {1} and P is fine and saturated, P gp ×

is torsion free. Hence by Lemma IV.5.1.3 (1), there exists a lifting P gp → limn Atriv ←− × A , which induces a lifting of the composition of P gp → Γ(U, MU )gp → A× → triv triv g : P → limf (A\{0}) and then a morphism h := (g, Γ(U, α)) : P → Q. Let G be the fiber ←− × × × product of the diagram of groups limf A → A ← Γ(U, OU ). Then h and the natural ←− ∼ inclusion G ,→ Q induce an isomorphism P × G = Q. Indeed, for any ((xn ), x) ∈ Q, × there exists a unique decomposition x = au (a ∈ P, u ∈ Γ(U, OU )), which implies the n × −1 p injectivity. Furthermore (xn g(a)n ) =(the image of u) implies (xn g(a)−1 n ) ∈ A , and we obtain the desired decomposition ((xn ), x) = h(a) · ((xn g(a)−1 n ), u). Now we see that the log structure of DN (U ) is fine and saturated and P → Q → Γ(DN (U )n , MDN (U )n ) gives the chart of MDN (U )n . The claims on iU and iU ,N are obvious because the projection Q → Γ(U, MU ) is compatible with h and Γ(U, α).  We identify U with D1 (U ) by the isomorphism iU in the following. b be the p-adic formal completion of U . Here and hereafter, the p-adic formal Let U completion of a fine log scheme X means the p-adic formal completion of the underlying scheme endowed with the inverse image of MX on the reduction mod pn of the completion b factors through A, e which is always for each n ∈ N>0 . If the homomorphism A → A true in the cases II and III by Lemma IV.5.1.1, the identity map of Γ(U, MU ) and the e induce a morphism of p-adic fine log formal schemes homomorphism limn A/pn A → A ←− b. U →U The above constructions are functorial in (U, s) as follows. Let U 0 be another fs 0 log scheme satisfying the same conditions as U . We define A0 , A0 , U 0 , and Utriv using 0 0 0 U in the same way as A, A, etc. Let s → Utriv be a geometric point satisfying the 0 0 0 same conditions as s, and define A , G(U 0 ,s0 ) , U , DN (U ), iU 0 ,N using U 0 in the same way as A, G(U,s) , etc. Let f : U 0 → U be a morphism over Zp and let h be a path f

0 − → Utriv to s → Utriv , where f denotes the morphism induced by f . from s0 → Utriv Then the morphism f induces a homomorphism u : A → A0 and u : A → A0 , and the 0 path h induces a homomorphism u : A → A compatible with u. For any σ ∈ G(U 0 ,s0 ) , there exists ρh (σ) ∈ G(U,s) uniquely such that σ ◦ u = u ◦ ρh (σ) (cf. [37] I Corollaire 5.4). This correspondence defines a continuous homomorphism ρh : G(U 0 ,s0 ) → G(U,s) . The morphisms Γ(U, MU ) → Γ(U 0 , MU 0 ) and RA → RA0 induced by f and u naturally 0 0 give a morphism f : U → U and f N : DN (U ) → DN (U ) compatible with the actions of G(U,s) and G(U 0 ,s0 ) via ρh , and with the exact closed immersions iU ,N and iU 0 ,N . We have f = f 1 .

IV.5.2. Representations associated to Higgs crystals and Higgs isocrystals. Let V be a complete discrete valuation ring of mixed characteristic (0, p) such that the residue field k of V is algebraically closed, and let K be the field of fractions of V . Choose and fix an algebraic closure K of K, and let V denote the integral closure of V in K. We take the ring W (RV ) and the ideal Ker(θ : W (RV ) → Vb ) as the base (R, I) of our theory of Higgs isocrystals and Higgs crystals. See IV.5.1 for RV and θ. Let Σ be

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

387

Spec(V ) endowed with the trivial log structure or the canonical log structure (i.e., the log structure defined by the closed point). The p-adic fine log formal schemes DN (Σ) over SN = Spf(AN (V )) and the exact closed immersions iΣ,N : DN (Σ) → DN +1 (Σ) compatible with the closed immersions SN → SN +1 define an object of C ∞ (cf. Lemma b be the p-adic completion of IV.2.2.6), which is denoted by D(Σ) in the following. Let Σ b Σ. We have a natural morphism Σ → Σ of p-adic fine log formal schemes. Lemma IV.5.2.1. Let A be a flat finitely generated V -algebra such that A⊗V k is reduced and A is normal. Let A be one of A (Case I), the henselization of A with respect to the ideal pA (Case II), and the p-adic completion of A (Case III). (1) TheQring A is noetherian and normal. (2) Let i∈I Ai be the decomposition of A into the product of finite number of normal domains. For i ∈ I such that Ai /pAi 6= 0 (which always holds in the cases II and III), Ai ⊗V V is a normal domain. Proof. (1) This is trivial in the case I and follows from [62] XI §2 (resp. [42] Scholie (7.8.3) (ii), (iii), (v)) in the case II (resp. III). (2) It suffices to prove that Ai ⊗V V 0 is a normal domain for the integer ring V 0 of any finite extension K 0 of K. We write B, BV 0 , and Bk for Ai , Ai ⊗V V 0 , and Ai ⊗V k. Note that BV 0 ⊗V 0 k ∼ = Bk is reduced. We first prove that BV 0 is normal. Since BV 0 is noetherian, it suffices to prove that BV 0 satisfies (R1 ) and (S2 ). Since BV 0 ⊗V 0 K 0 = B ⊗V K 0 is étale over B ⊗V K, BV 0 ⊗V 0 K 0 is normal. Hence it remains to check the conditions for a prime ideal p of BV 0 containing p. Since Bk is reduced, Bk satisfies (R0 ) and (S1 ). Put p = pBk . Since V 0 → BV 0 is flat, we have ht p = ht p − 1. If ht p = 1, then ht p = 0, which implies that (Bk )p is regular, i.e., a field. Hence the maximal ideal of (BV 0 )p is generated by a uniformizer of V 0 , and (BV 0 )p is regular. If ht p ≥ 2, then ht p ≥ 1, which implies depth(Bk )p ≥ 1. Since a uniformizer of V 0 is a nonzero divisor in (BV 0 )p , this implies depth(BV 0 )p ≥ 2. Thus we see that BV 0 is normal. In the cases II and III, the pairs (B, pB) and (BV 0 , pBV 0 ) are henselian because B and ∼ = BV 0 are finite over A ([62] XI §2 Proposition 2). Hence we have π0 (Spec(B ⊗V V 0 )) → ∼ = π0 (Spec(Bk )) ← π0 (Spec(B)). In the case I, since BV 0 → B⊗V K 0 is injective, it suffices to prove that K is algebraically closed in Frac(B). Let L be a finite extension of K contained in Frac(B). Then the ring of integers VL of L is contained in B because B is normal. Since VL → B is faithfully flat by the assumption Ai /pAi 6= 0, the homomorphism VL ⊗V k → Bk is injective. Hence VL ⊗V k is reduced, i.e., a field, and therefore L = K because k is algebraically closed.  In this subsection, we consider an fs log scheme U over Σ satisfying the conditions in the first paragraph in IV.5.1. We further assume that the underlying scheme of U is of finite type over Spec(V ) and its special fiber is reduced. Let A, A, U, and Utriv be as in the beginning of IV.5.1, let UK,triv be Utriv ×Spec(K) Spec(K), and let s → UK,triv be a geometric point such that the composition with UK,triv → Utriv satisfies the condition e , A, in the beginning of IV.5.1. We keep the notation in IV.5.1 such as K, K, Kur , G (U,s)

b U , and DN (U ). In the case I, we further assume that the composition of A → A → A e Recall that we have A b =A e in the cases II and III (Lemma IV.5.1.1). factors through A. b. Hence in all cases, we have a canonical morphism U → U The morphism s → UK,triv induces an extension K → K of the inclusion map K → K. Hence, by the last paragraph of IV.5.1, we have a continuous homomorphism G(U,s) = Gal(Kur /K) → GΣ = Gal(K/K), morphisms DN (U ) → DN (Σ) compatible

388

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

with the actions of G(U,s) and GΣ , and a commutative diagram (IV.5.2.2)

U = D1 (U )

/U b

 Σ = D1 (Σ)

 / Σ. b

The p-adic fine log formal schemes DN (U ) over SN and the exact closed immersions iU ,N : DN (U ) → DN +1 (U ) define an object of C ∞ (cf. Lemma IV.2.2.6), which is denoted by D(U ). The morphisms DN (U ) → DN (Σ) define a morphism D(U ) → D(Σ) in b × b Σ. Then the commutative diagram (IV.5.2.2) induces a morphism C ∞ . Let U1 be U Σ zU : U → U1 over Σ, and the pair (D(U ), zU ) becomes an object of (U1 /D(Σ))∞ HIGGS . Let ∆(U,s) be the kernel of the continuous homomorphism G(U,s) → GΣ , which coincides with Gal(Kur /KK) and is canonically isomorphic to the fundamental group of the connected component of UK,triv containing the image of s with respect to the base point s → UK,triv . Then ∆(U,s) acts on the object (D(U ), zU ). Let X → Σ be a morphism of fine log schemes whose underlying morphism of schemes is of finite type, and suppose that we are given a morphism h : U → X over Σ. Let b b →X b be the p-adic completion of h, and let h1 : U1 → X1 be the base change of h: U b b Let r ∈ N>0 ∪ {∞} and let F be a Higgs isocrystal on h by the morphism Σ → Σ. r (X1 /D(Σ))rHIGGS finite on U1 (cf. Definition IV.3.3.1). We define V(U,s),HIGGS (F) by r V(U,s),HIGGS (F) = Γ((D(U ), h1 ◦ zU ), F),

e -module. The action of ∆ which is a finitely generated projective A Qp (U,s) on (D(U ), zU ) r induces a semi-linear action of ∆(U,s) on V(U,s),HIGGS (F). We have an obvious identity r r (F) = V(U,s),HIGGS (h∗1,HIGGS (F)). For r0 ∈ N>0 ∪ {∞} such that r0 > r and V(U,s),HIGGS 0

the morphism of topos µr,r0 : (X1 /D(Σ))rHIGGS → (X1 /D(Σ))rHIGGS , we have 0

r r V(U,s),HIGGS (µ∗r,r0 (F)) = V(U,s),HIGGS (F).

Similarly for a Higgs crystal F on X/D(Σ) finite on U1 (cf. Definition IV.3.3.2), we can e e (cf. IV.3.2) endowed with a define an A-module T(U,s),HIGGS (F) belonging to LPM(A) semi-linear action of ∆(U,s) by T(U,s),HIGGS (F) := Γ((D(U ), h1 ◦ zU ), F). r We will construct a “period ring” for the functors V(U,s),HIGGS and T(U,s),HIGGS . Let Y be a smooth fine log scheme over Σ and let i be an immersion X → Y over Σ. Let Yb be the p-adic completion of Y and let Y1 be Yb × b Σ. Then the morphism Y1 → Σ is smooth Σ

and the immersion i induces an immersion i1 : X1 → Y1 over Σ. We further assume that we are given a compatible system of smooth liftings YN → DN (Σ) (N ∈ N>0 ) of Y1 → Σ. The smooth p-adic fine log formal schemes YN over DN (Σ) and the closed immersions YN → YN +1 define an object Y• of C with a smooth Cartesian morphism zU h1 i1 Y → D(Σ). The morphism U −−→ U1 −→ X1 −→ Y1 and the identity morphism of U define an immersion iU ,Y : U → Y1 ×Σ U . For r ∈ N>0 ∪ {∞}, we define the object r r DX,Y (U ) = (DX,Y,N (U )) of C r by r r DX,Y (U ) = DHiggs (iU ,Y : U ,→ Y• ×D(Σ) D(U )).

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

389

r (U ). By Proposition The action of ∆(U,s) on (D(U ), zU ) induces its action on DX,Y r r r IV.2.2.9 (2), DX,Y,N (U ) is affine. We define the object AX,Y (A) = (AX,Y,N (A)) of A•r by r r AX,Y,N (A) = Γ(DX,Y,N (U ), OD r (U ) ) X,Y,N

(cf. Lemma IV.2.2.6), which is naturally endowed with the action of ∆(U,s) . We abbrer r viate AX,Y (−) and DX,Y (−) to A r (−) and D r (−) if there is no risk of confusion. In order to study some properties of ANr (A), we assume that there exist a strict étale j

fine log scheme Y 0 over Y and a factorization U → Y 0 → Y of i ◦ h : U → Y such that Y 0 satisfies the following conditions. Condition IV.5.2.3. (i) The underlying scheme of Y 0 is affine. (ii) There exist ti ∈ Γ(Y 0 , MY 0 ) (1 ≤ i ≤ d) such that d log ti (1 ≤ i ≤ d) form a basis of Ω1Y 0 /Σ . Then there exists a compatible system of strict étale liftings YN0 → YN of Y10 := 0 Yb 0 ×Σ b Σ → Y1 uniquely up to a unique isomorphism. We see that YN is affine, there exist (ti,N )N ∈ limN Γ(YN0 , MYN0 ) (1 ≤ i ≤ d) such that ti,1 coincides with the image of ←− ti , and d log(ti,N ) (1 ≤ i ≤ d) also form a basis of Ω1Y 0 /D (Σ) for each N ∈ N>0 . N

N

Remark IV.5.2.4. If a smooth p-adic fine log formal scheme Y 0 over Σ satisfies the condition (i), then there always exists a compatible system of smooth liftings YN0 → DN (Σ) ([50] Proposition (3.14) (1)). Let iU ,Y 0 be the immersion U → Y10 ×Σ U induced by j. Then the morphism Y•0 → Y• r r induces an isomorphism DX,Y (U ) ∼ (iU ,Y 0 : U → Y•0 ×D(Σ) D(U )) by Lemma = DHiggs IV.2.2.15. We will first give an explicit description of ANr (A). Let si be the image of ti under j ∗ : Γ(Y 0 , MY 0 ) → Γ(U, MU ) and let Q be the monoid limf A\{0} ×A\{0} Γ(U, MU ) ←− used in the definition of the log structure of DN (U ) in IV.5.1. By Lemma IV.5.1.3 (1), there exists (si,n )n∈N ∈ limf A\{0} such that the pair ((si,n ), si ) becomes an element of ←− Q. Choose such a (si,n ) and let si,N be the image of (si , (si,n )) in Γ(DN (U ), MDN (U ) ). Since si,1 coincides with the image of si , we can apply Proposition IV.2.3.17 to idU , iU ,Y 0 , the inverse image of ti,N to YN0 ×DN (Σ) DN (U ) and si,N , and obtain an isomorphism (IV.5.2.5)

∼ =

→ A r (A) A(A){W1 , . . . , Wd }r −

for r ∈ N>0 ∪ {∞}, where the image of Wi (1 ≤ i ≤ d) in ANr (A)Qp is given as follows: Let πY 0 and πD(U ) denote the natural morphisms D r (U ) → Y•0 , D(U ). Then πY∗ 0 ,N (ti,N ) ∗ and πD(U (s ) have the same inverse image in Γ(D1r (U ), MD r (U ) ), and there exists a ),N i,N 1

∗ unique ui,N ∈ 1 + F 1 ANr (A) such that πY∗ 0 ,N (ti,N ) = πD(U (s )ui,N . Let wi,N be the ),N i,N

unique element of ANr (A)Qp such that [ξ](wi,N ) = ui,N +1 − 1. Then the image of Wi in ANr (A)Qp is wi,N . Proposition IV.5.2.6. Suppose that h : U → X satisfies the assumption before Condition IV.5.2.3. Then, for r ∈ N>0 ∪ {∞} and N ∈ N>0 , the action of ∆(U,s) on ANr (A) is continuous with respect to the p-adic topology. Proof. It suffices to prove that the action of ∆(U,s) on ANr (A)/pm is continuous with respect to the discrete topology. By (IV.5.2.5), this is further reduced to proving m that, for m ∈ Nd and a ∈ AN (A)(|m|)r , the stabilizer of the image of awN in ANr (A)/pm Q m mi is open. Here wN = 1≤i≤d wi,N . Choose c ∈ N such that pc AN (A) ⊂ AN (A)(ν)r for

390

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

every ν ∈ N, 0 ≤ ν ≤ |m|, and a generator ε = (εn ) of Zp (1) = limn µpn (A). Let ε ←− be (εn mod p) ∈ RA . For g ∈ ∆(U,s) , define ηi (g) = (ηi,n (g)) ∈ Zp = limn Z/pn by ←− −η (g) g(si,n ) = si,n εn i,n . Then we have g(wi,N ) = [εηi (g) ]wi,N + ξ −1 ([εηi (g) ] − 1).

(IV.5.2.7)

l

Choose l ∈ N such that pl ≥ N + 1. Then, since [ε] − 1 ∈ F 1 AN +1 (A), we have [ε]p = 1 l+m+c−1 = 1 in AN +1 (A)/pm+c , which implies ξ −1 ([εη ] − 1) ∈ in AN +1 (A)/p. Hence [ε]p l+m+c−1 m+c Zp . By Lemma IV.5.1.2, there exists an open subgroup p AN (A) for η ∈ p ∆0 ⊂ ∆(U,s) such that g(a) − a ∈ pc+m AN (A)(⊂ pm AN (A)(|m|)r ) for every g ∈ ∆0 and ηi (∆0 ) ∈ pl+m+c−1 Zp for 1 ≤ i ≤ d. For g ∈ ∆0 , the above description of g(wi,N ) implies m

m

m

m

g(a)(g(wN ) − wN ) ∈ pc+m ⊕|n|≤|m| AN (A)wN ⊂ pm ANr (A). m

m

m

m

Hence g(awN ) − awN = g(a)(g(wN ) − wN ) + (g(a) − a)wN ∈ pm ANr (A).



For q ∈ N, put Ωq = Γ(U, (i ◦ h)∗ ΩqY /Σ ) which is isomorphic to Γ(U, j ∗ ΩqY 0 /Σ ) and hence finite free over A by Condition IV.5.2.3. By applying (IV.2.4.8) and Lemma IV.2.4.10 to the smooth Cartesian morphism Y• ×D(Σ) D(U ) → D(U ) and the immersion iU ,Y : U → Y1 ×Σ U , we obtain a derivation θ : ANr (A) −→ ξ −1 ANr (A) ⊗A Ω1

(IV.5.2.8) and a complex

AN (A) → (ξ −q ANr (A) ⊗A Ωq , θq )q∈N .

(IV.5.2.9)

Proposition IV.5.2.10. Suppose that h : U → X satisfies the assumption before Condition IV.5.2.3. (1) For r ∈ N>0 ∪ {∞} and N ∈ N>0 , we have ANr (A)θ=0 = AN (A). (2) For r ∈ N>0 , the natural homomorphism ANr (A) → ANr+1 (A) induces a morphism of complexes from the complex (IV.5.2.9) for ANr (A) to the complex (IV.5.2.9) for ANr+1 (A). Furthermore, it becomes homotopic to zero after taking ⊗Zp Qp , and a homotopy is given by AN (A)Qp -linear continuous homomorphisms. Proof. (1) Under the description (IV.5.2.5) of AN (A), θ of AN (A) is given by θ(wi,N ) = ξ −1 ui,N d log ti,N because θ(ui,N +1 ) = ui,N +1 d log(ti,N +1 ). This immediately implies the claim. (2) Put Y = Y• ×D(Σ) D(U ), let Y (ν) (ν ∈ N) be the fiber product over D(U ) of (ν+1) s

s copies of Y , and let D (ν) (s ∈ {r, r + 1}, ν ∈ N) be DHIGGS (U ,→ Y (ν)). Then we have r+1 r a natural morphism D (•) → D (•) of simplicial objects of C r /D(Σ) compatible with the natural morphisms to Y (•). Hence the construction of θ and θq implies the first claim. Put ωi = θ(Wi ) = ξ −1 ui,N d log ti,N . Then, since θ(ωi ) = ξ −1 ui,N d log ti,N ∧d log ti,N = 0, we can construct the desired homotopy in the same way as the proof of Proposition IV.4.3.4 using Lemma IV.2.3.14.  r (X1 ,→ Y• ). By applying (IV.2.4.8) to Y• → D(Σ) and X1 ,→ Y1 , Let Dr be DHiggs we obtain a derivation θ : OD1r → ξ −1 OD1r ⊗OY Ω1Y /Σ . From the commutative diagram

(IV.5.2.11)

(U ,→ Y• ×D(Σ) D(U ))

/ D(U )

 (X1 ,→ Y• )

 / D(Σ)

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

391

−1 in C , we obtain a morphism πD : D r (U ) → Dr in C r and a homomorphism πD,1 (OD1r ) → OD r (U ) compatible with θ. 1 Assume that h : U → X is strict étale and put ArD,1 = Γ(D1r ×X1 U1 , OD1r ). Then since z

πD,1

U D1r (U ) −−−→ D1r → X1 canonically factors through U1 as D1r (U ) → U −−→ U1 → X1 , we r r obtain a canonical homomorphism AD1 → A1 (A) compatible with θ.

Proposition IV.5.2.12. Assume that h : U → X is strict étale and satisfies the condition before Condition IV.5.2.3. Let r ∈ N>0 ∪{∞} (resp. r = ∞), let F be a Higgs isocrystal on (X1 /D(Σ))rHIGGS finite on U1 (resp. a Higgs crystal on X1 /D(Σ) finite on U1 such that FD∞ is p-torsion free), and let (M, θ) be the object of HBrQp ,conv (X1 , Y• /D(Σ)) (resp. HBZp ,conv (X1 , Y• /D(Σ))) associated to F by Theorem IV.3.4.16. Let M be the ArD,1 -module Γ(D1r ×X1 U1 , M), let θ also denote the morphism Γ(D1r ×X1 U1 , θ) : M → r M ⊗A Ω1 , and let V (F) denote V(U,s),HIGGS (F) (resp. T(U,s),HIGGS (F)). We define the r endomorphism θ on A1 (A) ⊗ e V (F) by θ ⊗ idV (F ) . Then one can define a ∆(U,s) A equivariant homomorphism θ : A1r (A) ⊗ArD,1 M −→ ξ −1 A1r (A) ⊗ArD,1 M ⊗A Ω1

by θ(a ⊗ m) = a ⊗ θ(m) + θ(a) ⊗ m (a ∈ A1r (A), m ∈ M ) and there exists a canonical ∆(U,s) -equivariant A1r (A)-linear isomorphism ∼ =

A1r (A) ⊗ e V (F) −→ A1r (A) ⊗ArD,1 M A

e -linear compatible with θ and functorial in F. In the case of isocrystal, it induces an A Qp isomorphism V (F) ∼ = (M ⊗ArD,1 A1r (A))θ=0 . Proof. The first claim follows from the compatibility of the homomorphism ArD,1 → r DHiggs (U ,→ Y• ×D(Σ) D(U )), let zDr be the composition of

r A1r (A) with θ. Let D be r D1 → U → U1 → X1 , and

morphisms

let zDr be the canonical morphism D1r → X1 . Then we have r

(D(U ), h1 ◦ zU ) ←− (D , zDr ) −→ (Dr , zDr ) in (X1 /D(Σ))rHIGGS compatible with the actions of ∆(U,s) . Hence we obtain the following ∆(U,s) -equivariant A1r (A)-linear isomorphisms (cf. Lemma IV.3.2.8), where M = r Γ((D , zDr ), F). ∼ =

∼ =

A1r (A) ⊗ e V (F) −→ M ←− A1r (A) ⊗ArD,1 M. A

Note that FDr |D1r ×X1 U1 , FDr and FD(U ) belong to PM(−) (resp. LPM(−)) by the finiteness of F on U1 . Let F be the inverse image of F on (U /D(U ))rHIGGS and let (M, θ) be the object of HMrQp (U , Y• ×D(Σ) D(U )/D(U )) (resp. HMZp (U , Y• ×D(Σ) D(U )/D(U ))) r

associated to F. (See after Definition IV.3.4.7.) Then we have M = Γ(D1 , M) and r obtain a homomorphism Γ(D1 , θ) : M → M ⊗A Ω1 , which is also denoted by θ. By the commutative diagram (IV.5.2.11), we see that the above isomorphisms are compatible with θ ⊗ idV (F ) , θ and θ. The last claim follows from Proposition IV.5.2.10 (1).  For a p-adically complete and separated algebra C and a finitely generated CQp module M , we define the p-adic topology of M to be the topology defined by pm M ◦ (m ∈ N), where M ◦ is a finitely generated C-submodule of M such that MQ◦p = M . The subset W of M is open if and only if, for any w ∈ W , there exists an m ∈ N such that w + pm M ◦ ⊂ W . The topology does not depend on the choice of M ◦ and M becomes a topological CQp -module.

392

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Corollary IV.5.2.13. Let F be a Higgs isocrystal on (X1 /D(Σ))rHIGGS (resp. a Higgs r (F) (resp. crystal on X1 /D(Σ)) finite on U1 . Then the action of ∆(U,s) on V(U,s),HIGGS T(U,s),HIGGS (F)) is continuous with respect to the p-adic topology. Proof. Put r = ∞ in the second case. By replacing F with h∗1,HIGGS F, we may assume X = U . Since U is affine and we have a chart PU → MU globally, there exists a Σ-closed immersion X = U → Y into a smooth p-adic fine log scheme Y over Σ satisfying Condition IV.5.2.3. Choose a compatible system of smooth liftings YN → DN (Σ) r (cf. Remark IV.5.2.4) and define the period ring A1r (A). Put W := V(U,s),HIGGS (F) (resp. T(U,s),HIGGS (F)). By Proposition IV.5.2.12, we have a ∆(U,s) -equivariant A1r (A)∼ =

linear isomorphism A1r (A) ⊗ e W − → A1r (A) ⊗ArD,1 M . Since W is a finitely generated A e -module (i.e., a direct summand of a finite free module) (resp. W is an projective A Qp

e mA e → A ∞ (A)/pm A ∞ (A) is faithfully flat), we see that e and A/p object of LPM(A) 1 1 the natural homomorphism W → A1r (A) ⊗ e W is injective and the p-adic topology of A W is induced by the p-adic topology of A1r (A) ⊗ e W . On the other hand, in the case A of isocrystals, the p-adic topology of A1r (A) ⊗ArD,1 M is defined by pm (A1r (A) · M ◦ ) (m ∈ N), where M ◦ is a finitely generated ArD,1 -submodule of M such that MQ◦p = M . Hence the claim follows from Proposition IV.5.2.6.  e e LPM Let RepPM cont (∆(U,s) , AQp ) (resp. Repcont (∆(U,s) , A)) denote the category of finitely e -modules (resp. objects of LPM(A)) e W endowed with a semigenerated projective A Qp

linear action of ∆(U,s) continuous with respect to the p-adic topology of W . By Corollary IV.5.2.13, we obtain a functor e r V(U,s),HIGGS : HCrQp ,U1 -fin (X1 /D(Σ)) −→ RepPM cont (∆(U,s) , AQp ), e T(U,s),HIGGS : HCZp ,U1 -fin (X1 /D(Σ)) −→ RepLPM cont (∆(U,s) , A).

e We will need the following lemma in IV.6.4. We define the category RepLPM cont (A• ) as e m A-modules e e := A/p endowed with semifollows. An object is an inverse system of A m e -modules linear actions of ∆ for m ∈ N such that the underlying inverse system of A m

(U,s)

e ). A morphism is a morphism in LPM(A e ) compatible with is contained in LPM(A • • the actions of ∆(U,s) . e e LPM e Lemma IV.5.2.14. (1) The functor RepLPM e Am )m cont (A) → Repcont (A• ); T 7→ (T ⊗A is an equivalence of categories. A quasi-inverse is given by (Mm ) 7→ limm Mm . ←− e → RepLPM (A e ) is an equivalence of categories. (2) The functor RepLPM (A) cont

Q

cont

Qp

Proof. The claim (1) follows from Proposition IV.3.2.4. By Lemma IV.3.2.2 (1), we see that the functor in (2) is fully faithful. It remains to prove that it is essentially surjece ) and choose a generator x , . . . , x ∈ W tive. Let W be an object of RepPM (∆ ,A cont

(U,s)

Qp

1

r

e -module and let T be the A-module e as an A generated by xi (1 ≤ i ≤ r). Then, for Qp each i ∈ {1, . . . , r}, the map ∆(U,s) → W/T ; g 7→ (g(xi ) mod T ) is continuous, ∆(U,s) is e compact, and W/T is discrete. Hence the image of the map is finite and the A-submodule

generated by g(xi ) (g ∈ ∆(U,s) , i ∈ {1, . . . , r}) is finitely generated and stable under the action of ∆(U,s) . This completes the proof of (2). 

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

393

Proposition IV.5.2.15. Let r ∈ N>0 and let F be a Higgs isocrystal on (U1 /D(Σ))rHIGGS finite on U1 . Let (U1 /D(Σ))HIGGS denote the inverse limit of the fibered site (s 7→ (U1 /D(Σ))sHIGGS ) defined as in IV.4.5, and let F † be the pull-back of F on (U1 /D(Σ))HIGGS . Then there exists a canonical morphism • r RΓ((U1 /D(Σ))HIGGS , F † ) −→ Ccont (∆(U,s) , VU,HIGGS (F))

(IV.5.2.16)

• in D+ (C-Vect) functorial in F. Here Ccont (∆(U,s) , −) denotes the inhomogeneous continuous cochain complex and C denotes the completion of K with respect to the valuation.

Proof. Put X = U and choose X ,→ Y and YN (N ∈ N>0 ) as in the proof of Corollary IV.5.2.13. For an integer s ≥ r, let F s denote the inverse image of F r r s on (U1 /D(Σ))sHIGGS . Put W = V(U,s),HIGGS (F) = V(U,s),HIGGS (F s ) (s ≥ r), AD1s = s s s s s s Γ(D1 , OD1s ), and M = Γ(D1 , FDs ) (s ≥ r), where D = DHIGGS (X ,→ Y• ) as before Proposition IV.5.2.12. From Proposition IV.5.2.12 and (IV.5.2.9), we obtain morphisms of complexes ∼ =

W −→ ξ −• A1s (A) ⊗ e W ⊗A Ω• −→ ξ −• A1s (A) ⊗ADs M s ⊗A Ω• A

1

compatible with s in the obvious sense. By Proposition IV.5.2.10 (2), we obtain a quasiisomorphism • • Ccont (∆(U,s) , W ) −→ lim Ccont (∆(U,s) , ξ −• A1s (A) ⊗ADs M s ⊗A Ω• ). −→ 1

(IV.5.2.17)

s≥r

Let zDs be the canonical morphism D1s → U1 . Since U1 and zDs are affine, Lemma IV.4.2.2 implies that the natural morphism s • ξ −• M s ⊗A Ω• −→ RΓ(U1,´et , zDs ∗ (ξ −• FD )) s ⊗OY Ω Y1 /Σ 1

is an isomorphism in D+ (C-Vect). By Lemma IV.4.4.6 (4), [2] VI Corollaire 5.2 and Corollary IV.4.5.7, we obtain an isomorphism (IV.5.2.18) RΓ((U1 /D(Σ))HIGGS , F † ) ∼ (ξ −• M s ⊗A Ω• ). = RΓ(U1,´et , RuU1 /D(Σ)∗ F † ) ∼ = lim −→ s≥r

We have a natural morphism of complexes from the last term of (IV.5.2.18) to the target of (IV.5.2.17), which induces the desired morphism. We need to verify the independence of the choice of (i : X → Y, Y• ). Choose another (i0 : X → Y 0 , Y•0 ). By considering (X → Y ×Σ Y 0 , Y• ×D(Σ) Y•0 ), we are reduced to the case where we are given morphisms f : Y 0 → Y and f• : Y•0 → Y• compatible with i, i0 , 0 b 0 b Σ. Let Ds be Ds Y1 = Yb ×Σ b Σ, and Y1 = Y ×Σ HIGGS (U ,→ Y• ×D(Σ) D(U )) and let s

0s

s

M be Γ(D , F s ). We define A10s (A), D , M 0s , etc. in the same way as those without using (i0 , Y•0 ). Then the independence follows from the commutative diagram A1s (A) ⊗ e W

∼ =

/ Ms o

A

 A10s (A) ⊗ e W A

∼ =

/

 0s o M

∼ =

A1s (A) ⊗ADs M s 1

∼ =

 A 0s (A) ⊗AD0s M 0s 1

0

394

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

compatible with θ’s and the commutative diagram s • zDs ∗ (ξ −• FD ) lim s ⊗OY Ω 1 Y1 /Σ 2 −→s≥r

RuU1 /D(Σ)∗ (F † ) +

 s • lims≥r zD0s ∗ (ξ −• FD ). 0s ⊗O 0 Ω 0 Y1 Y1 /Σ −→

Here the vertical morphisms are the ones naturally induced from f and f• , and the second commutative diagram is derived from the following commutative diagram. 2

Prop.IV.4.5.5

F† Prop.IV.4.5.5

,

s • Q∗ ((LY? (ξ −• FD s ⊗OY Ω 1 Y

1 /Σ

))s≥r )

 s • Q∗ ((LY?0 (ξ −• FD )) ). 0s ⊗O 0 Ω 0 Y Y /Σ s≥r 1

1

 IV.5.3. Semi-stable reduction case. We keep the notation and the assumption in IV.5.2 and assume that U ×Spec(V ) Spec(k) is nonempty. In the case I (resp. the cases II and III), we further assume that U (resp. U ×Spec(V ) Spec(k)) is connected. Then A and A ⊗V V are normal domains and the homomorphism A ⊗V V → A induced by s → UK,triv is injective (cf. Lemma IV.5.2.1). We further assume that the log structure of Σ is defined by the closed point and U → Σ satisfies the following condition.

Condition IV.5.3.1. The morphism U → Σ admits a chart (NΣ → MΣ , Nd+1 → U d+1 d+1 MU , h : N → N ) such that the morphism U → Spec(Z)[N ] ×Spec(Z)[N] Σ induced by P the chart is strict étale and the morphism h is given by N → Nd+1 ; 1 7→ 1≤i≤e 1i for some integer e ≥ 1. Here, for a monoid P , Spec(Z)[P ] denotes Spec(Z[P ]) endowed with the log structure associated to P ,→ Z[P ], and 1i denotes the element of Nd+1 whose i-th component is 1 and other components are 0.

We choose and fix such a chart as in Condition IV.5.3.1 in the following. Under this assumption, we will prove the following theorems. Theorem IV.5.3.2 (cf. [27] §3). Let r ∈ N>0 and let F be a Higgs isocrystal on (U1 /D(Σ))rHIGGS finite on U1 . Let (U1 /D(Σ))∼ HIGGS denote the inverse limit of the fibered site (s 7→ (U1 /D(Σ))sHIGGS ) defined as in IV.4.5, and let F † be the pull-back of F on (U1 /D(Σ))HIGGS . Then the morphism (IV.5.2.16) in Proposition IV.5.2.15 is an isomorphism in D+ (C-Vect). Theorem IV.5.3.3 (cf. [27] Theorem 3). For r ∈ N>0 ∪ {∞}, the functor e ) Vr : HCr (U /D(Σ)) → RepPM (∆ ,A (U,s),HIGGS

Qp ,U1 -fin

1

cont

(U,s)

Qp

is fully faithful. Put X = U . Since U is smooth over Σ, we can use Y = U and a compatible system of smooth liftings YN (N ∈ N>0 ) of Y1 to define the rings ANr (A) (r ∈ N>0 ∪{∞}, N ∈ N>0 ). We will also discuss the admissibility of Higgs vector bundles with respect to the period ring A1r (A) in Proposition IV.5.3.10. Let ti ∈ Γ(Y, MY ) = Γ(U, MU ) (0 ≤ i ≤ d) be the image of 1i+1 ∈ Nd+1 under the chart Nd+1 → MU . Then ti (1 ≤ i ≤ d) satisfy Condition IV.5.2.3 (ii). Choosing a lifting U

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

395

(ti,N ) ∈ limN Γ(YN , MYN ) of the image of ti in Γ(Y1 , MY1 ), we obtain the isomorphism ←− (IV.5.2.5). We will derive Theorems IV.5.3.2 and IV.5.3.3 from the following theorem. Theorem IV.5.3.4. With the above notation, we have: (1) A1∞ (A)∆(U,s) = (A ⊗V V )∧ , ( 0 i r (2) limr∈N Hcont (∆(U,s) , A1 (A)Qp ) = −→ >0 (A ⊗V V )∧ Qp ∧ where denotes the p-adic completion.

if i > 0, if i = 0,

Let π denote the image of 1 ∈ N in Γ(Σ, MΣ ) ⊂ Γ(Σ, OΣ ) = V by the chart of MΣ , which is a uniformizer of V . We have an étale homomorphism V [T0 , . . . , Td ]/(T0 · · · Te−1 − π) → A; Ti 7→ ti . Choose a compatible system (πn ) of pn -th roots of π in V , let Vn be V [πn ], and define An to be the cofiber product of Vn [T0,n , . . . , Td,n ]/(T0,n · · · Te−1,n − πn ) ←− V [T0 , . . . , Td ]/(T0 · · · Te−1 − π) −→ A, n

where the left homomorphism is defined by Ti 7→ (Ti,n )p . The ring An is normal and flat over Vn , Spec(An /πn An ) is reduced, and the morphism Spec(An /πn An ) → Spec(A/πA) is a homeomorphism. Observing that the extension of residue fields for A ⊗V Vn → An at height one primes containing p are purely inseparable of degree pnd , we see that An is a normal domain. Note that A ⊗V Vn is a normal domain by Lemma IV.5.2.1. We put An = A ⊗A An , which is the henselization of (An , pAn ) (resp. the p-adic completion of An ) in the case II (resp. III). We have a natural injective homomorphism An ⊗Vn V → An+1 ⊗Vn+1 V . By Lemma IV.5.2.1, we see that An ⊗Vn V is a normal domain. Since AQp → An,Qp is finite étale, there exists an injective homomorphism An ⊗Vn V ,→ A compatible with n. We choose and fix such an embedding, regard An ⊗Vn V as a subring of A, and define A∞ to be the union of An ⊗Vn V . We write ti,n for the image of Ti,n in A∞ . For n ∈ N, let Kn denote the field of fractions of An ⊗Vn V , and put K∞ := ∪n∈N Kn . Then, for n ∈ N ∪ {∞}, Kn is a Galois extension of KK. Let ∆n denote its Galois group. For n ∈ N, we have an isomorphism ∆n ∼ = µ⊕d pn defined by g 7→ (εi,n ), g(ti,n ) = ti,n εi,n (1 ≤ i ≤ d). This isomorphism is compatible with n and induces an isomorphism ∼ =

∆∞ −→ Zp (1)⊕d .

(IV.5.3.5)

Let H be the kernel of the homomorphism ∆(U,s) → ∆∞ . We use the following theorem of Faltings which is a consequence of his almost purity theorem for almost étale extensions. Let m denote the maximal ideal of V . Theorem IV.5.3.6 (G. Faltings [26]). (1) For n ∈ N>0 , H q (H, A/pn A) (q > 0) is annihilated by m and the cokernel of the injective homomorphism A∞ /pn A∞ → H 0 (H, A/pn A) is annihilated by m. (2) The absolute Frobenius of A/pA is surjective in the case I (i.e., A = A). In e =A b (cf. Lemma IV.5.1.1). particular, we have A We define wi,1 ∈ A1r (A) appearing in the isomorphism (IV.5.2.5) by using ti and ((ti,n ), ti ) ∈ Q. We abbreviate wi,1 to wi in the following. Since F n AN (A) = ξ n AN (A), m b we have A (A)(m)r = pd r e A (cf. IV.2.3) for r ∈ N and m ∈ N, where dxe (x ∈ R) 1

>0

m to be 0 for m ∈ N in denotes the smallest integer l satisfying x ≤ l. We understand ∞ Q d m the following. For m = (m1 , m2 , . . . , md ) ∈ N , we write w for 1≤i≤d wimi . Then, for

396

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

r ∈ N>0 ∪ {∞}, we have A1r (A) =

(IV.5.3.7)

M

p

|m| d r e

!∧ b m Aw

.

m∈Nd

For r ∈ N>0 ∪ {∞}, we define the subalgebras A1r,◦ (A) and A1r,◦ (A∞ ) of A1r (A)Qp as follows. X  |m| r,◦ b m r d r am w ∈ A1 (A)Qp am ∈ p A for all m ∈ N , A1 (A) = m∈Nd

A1r,◦ (A∞ ) =

X m∈Nd

 |m| am wm ∈ A1r (A)Qp am ∈ p r Ab∞ for all m ∈ Nd .

By (IV.5.2.7), we see that A1r,◦ (A) and A1r,◦ (A∞ ) are stable under the action of ∆(U,s) , and the action on the latter factors through ∆∞ . We have A1∞ (A) = A1∞,◦ (A) and A1r (A) ⊂ A1r,◦ (A) ⊂ p−1 A1r (A) if r ∈ N>0 . We endow A1r,◦ (A∞ )Qp with the p-adic topology induced by A1r,◦ (A∞ ). Since the intersection of A1r,◦ (A∞ )Qp and A1r,◦ (A) is A1r,◦ (A∞ ), it is the same as the topology induced by that of A1r (A)Qp . Theorem IV.5.3.6 (1) above implies the following. Corollary IV.5.3.8. For r ∈ N>0 ∪ {∞} and i ∈ N, the kernel and the cokernel of the homomorphism i i Hcont (∆∞ , A1r,◦ (A∞ )) −→ Hcont (∆(U,s) , A1r (A))

are annihilated by m.

Proof. The action of H on wi is trivial. Hence, by Theorem IV.5.3.6 and the Hochschild-Serre spectral sequence for H ⊂ ∆(U,s) , we see that the kernel and the cokernel of the homomorphism H i (∆∞ , A1r,◦ (A∞ )/pn ) −→ H i (∆(U,s) , A1r,◦ (A)/pn )

are annihilated by m. Since A1r,◦ (A) is p-adically complete and separated, we have the following exact sequence (cf. [47] §2): i 0 −→ R1 lim H i−1 (∆(U,s) , A1r,◦ (A)/pn ) → Hcont (∆(U,s) , A1r,◦ (A)) ← − n

−→ lim H i (∆(U,s) , A1r,◦ (A)/pn ) → 0. ←− n

We have the same exact sequence for A∞ and ∆∞ . Hence the kernel and the cokernel of the homomorphism i i (∆(U,s) , A1r,◦ (A)) Hcont (∆∞ , A1r,◦ (A∞ )) −→ Hcont

are annihilated by m.



The isomorphism (IV.5.3.5) induces an isomorphism Homcont (∆∞ , µp∞ (V )) ∼ = (Qp /Zp )⊕d . For α ∈ (Qp /Zp )⊕d , we denote by χα the corresponding character of ∆∞ . Q m The ring V [T0,n , . . . , Td,n ]/( 0≤i≤e−1 Ti,n −πn ) is a free V -module with basis T•,n := Q mi d+1 − ((N>0 )e × Nd+1−e ). For 0≤i≤d Ti,n (m = (m0 , m1 , . . . , md ) ∈ S ), where S = N n d l = (l1 , l2 , . . . , ld ) ∈ (N ∩ [0, p [) , put   m −m0 ≡ li mod pn (1 ≤ i ≤ e−1), Sln := (m0 , m1 , . . . , md ) ∈ S i . mi ≡ li mod pn (e ≤ i ≤ d).

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

397

m

Then S is a disjoint union of Sln (l ∈ (N ∩ [0, pn [)d ) and ⊕m∈Sln V · T•,n is stable under the action of V [T0 , . . . , Td ]/(T0 · · · Te−1 − π). This implies that A∞ has a decomposition as an A ⊗V V -module M A∞ = A∞,α , α∈(Qp /Zp )⊕d

where A∞,α is ∆∞ -stable and g(x) = χα (g)x for x ∈ A∞,α and g ∈ ∆∞ . We have A∞,0 = A ⊗V V . For a ∈ N, 0 ≤ a ≤ d, let Ia (resp. Ja ) denote the submodule of (Qp /Zp )⊕d (resp. the subset of Nd ) consisting of elements whose first a components are 0. For r ∈ N>0 ∪ {∞} and a ∈ N ∩ [0, d], we define a A r to be the ∆∞ -stable submodule !∧ M |m| m p r A∞,α w (m,α)∈Ja ×Ia

of

A1r,◦ (A∞ ),

where

∧

denotes the p-adic completion. We have 0A

r

= A1r,◦ (A∞ ),

dA

r

= (A ⊗V V )∧ .

Choose a generator ε = (εn ) of Zp (1)(V ) and let γi ∈ ∆∞ be the element corresponding to the element of Zp (1)⊕d whose i-th component is ε and other components are 0. Proposition IV.5.3.9. Let a be an integer such that 1 ≤ a ≤ d. (1) For r ∈ N>0 ∪ {∞}, the γa -invariant part of a−1 A r is a A r . (2) For r ∈ N>1 , there exists an integer l such that pl · a−1 A r−1 is contained in the image of γa − 1 : a−1 A r → a−1 A r . Proof. The action of γa on wi (i 6= a) is trivial. Hence it suffices to prove the r following two claims. Let Aa,α be the submodule !∧ M ν p r A∞,α waν ν∈N

A1r,◦ (A∞ ),

which is stable under the action of γa . r is 0 if χα (γa ) 6= 1 and Ab∞,α if (i) For r ∈ N>0 ∪ {∞}, the γa -invariant part of Aa,α χα (γa ) = 1. r−1 (ii) For r ∈ N>1 , there exists an integer l independent on α such that pl Aa,α is r r contained in the image of γa − 1 : Aa,α → Aa,α . For aν ∈ Ab∞,α (ν ∈ N), aν → 0 (ν → ∞), we have X ν  X ν (γa − 1) p r aν waν = p r bν waν , of

ν∈N

ν∈N

bν = (χα (γa ) − 1)aν +

X µ>ν

−1

1

p r (µ−ν) χα (γa )aµ

  µ µ−ν η ν

by (IV.5.2.7), where η is the element (ξ ([ε−1 ] − 1) mod ξ) ∈ V . We have vp (η) = where the valuation vp of V is normalized by vp (p) = 1. Proof of (i). Suppose bν = 0 for all ν ∈ N. If χα (γa ) 6= 1, then vp (χα (γa ) − 1) ≤ and we have   X 1 µ µ−ν aν = −(χα (γa ) − 1)−1 p r (µ−ν) χα (γa )aµ η . ν µ>ν

1 p−1 , 1 p−1 ,

398

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

For any l ∈ N, there exists νl ∈ N such that aν ∈ pl Ab∞,α for all ν ≥ νl . Then, by the above equality for 0 ≤ ν ≤ νl − 1, we see aν ∈ pl Ab∞,α for all ν ∈ N. Hence aν ∈ ∩l∈N pl Ab∞,α = 0 for all ν ∈ N. If χα (γa ) = 1, then we have X 1 η µ−ν−1 (ν + 1)!aν+1 = − p r (µ−ν−1) µ!aµ . (µ − ν)! µ≥ν+2

1 p−1 ,

m−1

we have η /m! ∈ V for m ∈ N>0 . For any l ∈ N, there exists νl ∈ N l b such that (ν + 1)!aν+1 ∈ p A∞,α for all ν ≥ νl . By the above equality for 0 ≤ ν ≤ νl − 1, we see (ν + 1)!aν+1 ∈ pl Ab∞,α for all ν ∈ N. Hence (ν + 1)!aν+1 ∈ ∩l∈N pl Ab∞,α = 0 for all ν ∈ N. r is divisible by χα (γa ) − 1 and Proof of (ii). For the case χα (γa ) 6= 1, γa − 1 on Aa,α Since vp (η) =

1

it is congruent to the identity modulo p r . Hence (χα (γa ) − 1)−1 (γa − 1) is bijective and P ν 1 we may take l = 1 since vp (χα (γa ) − 1) ≤ p−1 . Suppose χα (γa ) = 1. Let ν p r−1 cν waν , r−1 . It suffices to show that there exists an integer l ≥ 1 cν ∈ Ab∞,α be an element of Aa,α depending only on r such that the equality X µ 1 1 ν η µ−ν pl p r ν!p( r−1 − r )ν cν = p r µ!aµ (µ − ν)! µ>ν ν 1 1 has a solution aν ∈ Ab∞,α such that aν → 0 as ν → ∞. Set c0ν = pl ν!p( r−1 − r )ν p− p−1 cν ν and a0ν = ν!p− p−1 aν . Then the above equation is written as X 1 1 1 1 η µ−ν−1 p( r + p−1 )(µ−ν−1) a0µ c0ν = p r + p−1 η . (µ − ν)!

µ≥ν+1

The endomorphism of (⊕ν∈N>0 A∞,α )∧ defined by X 1 1 (xν ) 7→ (yν ), yν = p( r + p−1 )(µ−ν) xµ µ≥ν

1

η µ−ν (µ − ν + 1)!

1

is congruent to the identity map modulo p r + p−1 . Hence the above equation has a solution 1 1 a0ν ∈ Ab∞,α , ν ∈ N>0 such that a0ν → 0 as ν → ∞ if c0ν ∈ p r + p−1 η Ab∞,α and c0ν → 0 as ν p−1 ν → ∞. Since p ∈ V , we have aν ∈ Ab∞,α and aν → 0 as ν → ∞ for such a solution ν!

a0ν . Since vp (η) =

1 p−1

1

1

ν

and ordp (ν!p( r−1 − r )ν p− p−1 ) → ∞ as ν → ∞, it is enough to take 1

1

ν

an integer l such that inf{ordp (ν!p( r−1 − r )ν p− p−1 )|ν ∈ N} −

1 r

−

2 p−1

≥ −l.



Proof of Theorem IV.5.3.4. The equality (1) follows from Proposition IV.5.3.9 (1) and Corollary IV.5.3.8. Let us prove (2). By Corollary IV.5.3.8, it suffices to prove ( 0 if i > 0, r,◦ i lim Hcont (∆∞ , A1 (A∞ )Qp ) = ∧ −→ (A ⊗ V ) if i = 0. V Qp r

For r ∈ N>0 , define the complex Car (0 ≤ a ≤ d) inductively by C0r = A1r,◦ (A∞ ) and γa −1

r • r −−−→ Ca−1 ). Then the complex Ccont (∆∞ , A1r,◦ (A∞ )) is isomorphic to Car = fiber(Ca−1 r + Cd in the derived category D (Zp -Mod). By using Proposition IV.5.3.9, one can show that the natural homomorphism r lim a AQrp −→ lim Ca,Q p −→ −→ r r

is a quasi-isomorphism by induction on a. The quasi-isomorphism for a = d implies the desired claim. 

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

399

Proof of Theorem IV.5.3.2. We apply the construction of (IV.5.2.16) in the proof of Proposition IV.5.2.15 to Y = X = U and YN (N ∈ N>0 ). We follow the notation in loc. cit. Then we have AD1s = (A ⊗V V )∧ and M s = M r for all s ≥ r, and M r is a finitely generated projective (A ⊗V V )∧ Qp -module. Hence Theorem IV.5.3.4 implies that the following natural morphism of complexes is a quasi-isomorphism. • ξ −• M r ⊗A Ω• = lim ξ −• M s ⊗A Ω• −→ lim Ccont (∆(U,s) , ξ −• A1s (A)⊗(A⊗V V )∧ M s ⊗A Ω• ). −→ −→ s≥r

s≥r

Combining with the quasi-isomorphisms (IV.5.2.17) and (IV.5.2.18), we obtain the theorem.  Proof of Theorem IV.5.3.3. Let F be a Higgs isocrystal on (U1 /A(Σ))rHIGGS finite on U1 . By Proposition IV.5.2.12, we have a functorial ∆(U,s) -equivariant A1r (A)linear isomorphism compatible with θ A r (A)Q ⊗ b V r (F) ∼ = A r (A)Q ⊗ ∧ M. 1

p

AQp

1

(U,s),HIGGS

(A⊗V V )Qp

p

By Theorem IV.5.3.4, we have (A1r (A)Qp )∆(U,s) = (A ⊗V V )∧ Qp . Since M is a finitely ∧ -module, we obtain an (A ⊗ generated projective (A ⊗V V )∧ V V )Qp -linear isomorphism Qp M∼ = (A1r (A)Qp ⊗A b

Qp

r V(U,s),HIGGS (F))∆(U,s)



compatible with θ.

We discuss the admissibility of Higgs vector bundles with respect to the period ring A1r (A). Let A1 denote the p-adic completion of A ⊗V V and let Ω1 denote Γ(U, Ω1U/Σ ). Let M be a finitely generated projective A1,Qp -module and let θ : M → M ⊗A Ω1 be a Higgs field, i.e., an A1,Qp -linear homomorphism such that θ1 ◦ θ = 0, where θ1 is defined as before Lemma IV.3.4.5. Let r ∈ N>0 ∪ {∞}. If θ satisfies the condition (Conv)r in IV.3.6, then it comes from a Higgs isocrystal on (U1 /D(Σ))rHIGGS finite on U1 by Theorem IV.3.4.16. Hence b ) and the natural ,A Vr (M ) := (A r (A) ⊗ M )θ=0 belongs to RepPM (∆ HIGGS

A1

1

cont

(U,s)

A1r (A)-linear homomorphism

Qp

r A1r (A) ⊗ b VHIGGS (M ) −→ A1r (A) ⊗A1 M A

is an isomorphism by Proposition IV.5.2.12 and Corollary IV.5.2.13. One can prove the converse as follows. Proposition IV.5.3.10. Let r ∈ N>0 ∪ {∞}, let M be a finitely generated projective A1,Qp -module, and let θ : M → M ⊗A Ω1 be a Higgs field. Suppose that there exists an b -module W and an A r (A)-linear isomorphism A Qp

1

∼ =

A1r (A) ⊗ b W −→ A1r (A) ⊗A1 M A

b -module, compatible with the Higgs fields. Then W is a finitely generated projective A Qp ∼ =

the above homomorphism induces an isomorphism W − → (A1r (A) ⊗A1 M )θ=0 , and the Higgs field θ of M satisfies the condition (Conv)r in IV.3.6.

Proof. We use the description (IV.5.3.7) of A1r (A). By taking the reduction mod b -linear isomorphism (w1 , . . . , wd ) of the isomorphism in the assumption, we obtain an A Qp (IV.5.3.11)

b W ∼ = A ⊗A1 M,

400

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

b -module. From Proposition which implies that W is a finitely generated projective A Qp IV.5.2.10 (1), we obtain the second claim. It remains to prove the last claim. Since M is a finitely generated projective A1,Qp -module, we have  X b ⊗ M, p−dr−1 |m|e x → 0 (|m| → ∞) . A1r (A) ⊗A1 M = wm ⊗ xm xm ∈ A A1 m m∈Nd

Let x ∈ M and let v ∈ W be the image of 1 ⊗ x under the isomorphism (IV.5.3.11). Then the image of 1 ⊗ v under the isomorphism in the assumption of the proposition is written as X b ⊗ M, p−dr−1 |m|e x → 0 (|m| → ∞). wm ⊗ xm , x0 = x, xm ∈ A A1 m m∈Nd

Since this section is annihilated by θi (1 ≤ i ≤ d), θi (wi ) = ξ −1 d log(ti ), and θi (wj ) = 0 (j 6= i), we have θi (xm ) = −(mi + 1)xm+1i for 1 ≤ i ≤ d and m ∈ Nd , where 1i denotes the element of Nd whose i-th component is 1 and other components are 0. This implies Q Q mi mi 1 −dr −1 |m|e 1 that xm = (−1)|m| m! 1≤i≤d θi (x) and hence p 1≤i≤d θi (x) converges to m! b ⊗ M as |m| → ∞. Since M is a finitely generated projective module over A 0 in A A1

1,Qp

b A b is injective, this implies that the sequence also and the homomorphism A1 /pA1 → A/p converges in M . This completes the proof. 

IV.5.4. Comparison with the approach via Higgs-Tate torsors. In this subr (A) defined before Condition IV.5.2.3 with section, we compare the period ring AX,Y,1 (r) the algebras C and C (r ∈ Q>0 ) introduced in II.10.3 and II.12.1. b and D(Σ) be the same as the beginning of IV.5.2. We consider the Let Σ, Σ, Σ, case where the log structure of Σ is defined by its closed point. Let X → Σ be a smooth morphism of fine and saturated log schemes satisfying the conditions in II.6.2 strict étale locally on X, and let h : U → X be a strict étale morphism such that the morphism U → Σ satisfies the conditions mentioned above. We further assume that U satisfies the conditions in [73] Lemma 1.3.2 (cf. the beginning of IV.5.1). Then U satisfies the conditions in the beginning of IV.5.1. We consider (Case I), i.e., A = A and U = U (cf. IV.5.1). As in IV.5.2, choose a geometric point s → UK,triv whose image is e by Proposition b = A of codimension 0. Then, with the notation in IV.5.1, we have A

II.9.10, which is a consequence of the almost purity theorem by G. Faltings (cf. the proof of Lemma IV.5.1.1). Hence, as in IV.5.2, we have a morphism D(U ) → D(Σ) in C ∞ compatible with the actions of G(U,s) and GΣ via the natural morphism G(U,s) → GΣ b × b Σ. and a Σ-morphism zU : U = D1 (U ) → U1 = U Σ Suppose that we are given a smooth Cartesian lifting X• = (XN )N ∈N>0 → D(Σ) of X1 → Σ = D1 (Σ). Then we may apply the construction of the period ring before r r (U )) of C r for r ∈ N>0 ∪ {∞} is Condition IV.5.2.3; the object DX,X (U ) = (DX,X • ,N • r defined to be DHiggs (U ,→ X• ×D(Σ) D(U )), and the period ring AX,X• ,1 (U ) is defined r to be the coordinate ring of DX,X (U ). • ,1 We may also apply IV.2.5 to D(U ) → D(Σ) ← X• and the composition of zU : U → U U1 and h1 : U1 → X1 . We obtain a group p-adic fine log formal scheme TX and a 2 U U TX2 -principal homogenous space LX2 over U . This principal homogeneous space is a natural formal scheme analogue of that associated to the Higgs-Tate torsor defined in U II.10.3. The action of ∆(U,s) (= Ker(G(U,s) → GΣ )) on U induces an action on TX 2 U U U and an equivariant action on the TX -principal homogeneous space L . Put C = X X 2 2 2

IV.5. REPRESENTATIONS OF THE FUNDAMENTAL GROUP

401

b U U Γ(LU X2 , OLU ) and define the A-submodule of “affine functions” MX2 of CX2 as before X2

b Lemma IV.2.5.14, which is a finitely generated projective A-module by Lemma IV.2.5.14. U U As before Proposition IV.2.5.16, we define C b (MX2 ) to be the direct limit limm S m b (MX2 ) A −→ A b ,→ M U , C (r) (M U ) (r ∈ N ) to be its subring whose transition maps are induced by A X2

P

m∈N

p

dm r e

U Sm b (MX2 ), A

X2

b A

>0

b b (M U ) and C b (r) (M U ) to be their p-adic completions. and C X2 X2 b A

A

b b (M U ) = C U These are naturally endowed with the actions of ∆(U,s) and we have C X2 X2 A b b U U by the construction of LX2 . The A-algebra CX2 is naturally endowed with the A-linear U U → CX ⊗A Ω1 (cf. (IV.2.5.17)), where Ω1 = Γ(U, h∗ (Ω1X/Σ )). derivation θ : CX 2 2 By Theorem IV.2.5.2 and Proposition IV.2.5.22, we have a natural ∆(U,s) -equivariant isomorphism (U ). TU ∼ = D∞ X,X•,1

X2

It induces a ∆(U,s) -equivariant isomorphism ∼ =

U ∞ CX −→ AX,X (U ), 2 • ,1

(IV.5.4.1)

which is compatible with the derivations θ’s by Proposition IV.2.5.19. By Proposition IV.2.5.16, it further induces a ∆(U,s) -equivariant isomorphism ∼ = U r b (r) (MX C ) −→ AX,X (U ) b 2 • ,1

(IV.5.4.2)

A

for r ∈ N>0 . b b (M U ) and C b (r) (M U ) with the algebras C and Now let us compare the algebras C X2 X2 b A

A

C (r) in II.10.3 and II.12.1. Under the notation and the assumption in IV.5.1, we can define fine and saturated ˇ and D (U ˇ ) by replacing Spf(A) e and Spf(A (A)) with Spec(A) e and log schemes U N N Spec(AN (A)) in the definition of U and DN (U ) in IV.5.1. Note that the proof of Lemma IV.5.1.4 still works after this modification. We have natural actions of G(U,s) on these ˇ → D (U ˇ ), and G log schemes, a G -equivariant isomorphism i : U -equivariant ˇ U

(U,s)

1

(U,s)

ˇ) → D ˇ ˇ ˇ exact closed immersions iUˇ ,N : DN (U ˇ. N +1 (U ). We identify U with D1 (U ) via iU b = A, e then we have a natural strict morphism U ˇ → U. If A Let us return to the settings in this subsection. By applying the above construction ˇ D (Σ) ˇ and i ˇ , D (U ˇ ) and i to (Σ, Spec(K)) (resp. (U, s)), we obtain Σ, (resp. U ˇ ˇ ,N ). N N Σ,N U ˇ ˇ , we have natural morphisms D (U ) → D (Σ) Since s is a geometric point of U N

K,triv

N

ˇ h ˇ1 → compatible with iΣ,N and iUˇ ,N and also with the actions of G(U,s) and GΣ . Let U ˇ h ˇ → Σ. Since A e =A b as ˇ be the base change of U → ˇ1 → Σ X X → Σ by the morphism Σ ˇ ˇ ˇ mentioned above, we have a natural strict morphism U → U1 over Σ. ˇ endowed with an isomorphism X ˇ → D (Σ) ˇ × Choose a smooth morphism X 2

2

2

ˇ ∼ Xˇ over Σ. ˇ Then we can apply II.10.3 to the commutative diagram Σ = 1 /X ˇ2

ˇ U  ˇ Σ

iΣ,1 ˇ

 / D2 (Σ),

ˇ D2 (Σ)

402

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS ˇ

ˇ

ˇ

U U U obtaining a vector bundle TX ˇ 2 and a TX ˇ 2 -principal homogeneous space LX ˇ 2 over the ˇ underlying scheme of U . Precisely speaking, we have to work with the diagram in which ˇ 2 is replaced by the unique strict étale lifting U ˇ2 → X ˇ 2 of U ˇ1 → X ˇ 1 , but the resulting X principal vector bundle and principal homogeneous space are canonically identified with ˇ ˇ ˇ ˇ U U U U ˇ ), and TX ˇ and LX ˇ above (cf. the proof of Lemma III.10.14). Put CX ˇ := Γ(LX ˇ , OLU 2

2

2

ˇ CXUˇ 2

ˇ U FX ˇ2

2

ˇ X 2

ˇ b consisting of affine functions. The A-module CXUˇ is

be the submodule of 2 ˇ ˇ b naturally endowed with an A-linear derivation θ : CXUˇ → CXUˇ ⊗A Ω1 (cf. II.10.9). Let 2 2 ˇ ˇ U U b C denote the p-adic completion of C . let

ˇ2 X

ˇ2 X

The morphisms U → U1 → X1 → Σ and D(U ) → D(Σ) are naturally identified with ˇ and the compatible ˇ →U ˇ1 → X ˇ1 → Σ the p-adic formal completion of the morphisms U ˇ ˇ system of morphisms DN (U ) → DN (Σ) (N ∈ N>0 ). Under this identification, the p-adic ˇ 2 is regarded as a smooth lifting of X1 over D2 (Σ). Suppose that formal completion of X we are given an isomorphism between the above completion and the lifting X2 of X1 U U , LU which we have chosen. Then by the construction of TX X2 , and MX2 , we see that 2 ˇ b ⊗ Ω1 by A b endowed with M U is canonically isomorphic to F U as extensions of ξ −1 A A

ˇ2 X

X2

actions of ∆(U,s) . Combined with (IV.5.4.1), this isomorphism induces ∆(U,s) -equivariant isomorphisms ˇ ∼ ∼ = b = U U ∞ CbXUˇ 2 −→ C b (MX2 ) = CX2 −→ AX,X,1 (U )

(IV.5.4.3)

A

compatible with the derivations θ.

ˇ ,(r) U

As in II.12.1, we define FXˇ (r ∈ Q>0 ) to be the pull-back of the extension 2 ˇ b ⊗ Ω1 → 0 by pr (ξ −1 A b → F U → ξ −1 A b ⊗ Ω) ⊂ ξ −1 A b ⊗ Ω1 , and C Uˇ ,(r) to be 0→A A A A ˇ2 ˇ2 X X ˇ ,(r) ˇ ,(r) ˇ U U b the A-subalgebra limm S m ) of CXUˇ . Let CbXˇ denote the p-adic completion of ˇ2 b (FX 2 −→ 2 A ˇ ,(r) ˇ ˇ U C . If we identify C U with C b (M U ) by the isomorphism induced by F U ∼ = MU , ˇ2 X

then we have

ˇ ,(1/r) U pCXˇ 2

ˇ2 X

⊂

A (r) U C b (MX2 ) A

ˇ2 X

X2

⊂

ˇ ,(1/r) U CXˇ 2

X2

for r ∈ N>0 . By (IV.5.4.2), we see that

the isomorphism (IV.5.4.3) induces an injective homomorphism (IV.5.4.4)

ˇ ,(1/r) U

r AX,X,1 (U ) ,→ CbXˇ

2

b is annihilated by p. whose cokernel as a homomorphism of A-modules By Proposition IV.2.5.22, we see that the homomorphisms (IV.5.4.3) and (IV.5.4.4) are functorial with respect to X-morphisms between U ’s. IV.6. Comparison with Faltings cohomology IV.6.1. Faltings site. Let V , k, K, K, V , and Σ be the same as in the beginning of IV.5.2. Let U be an fs log scheme over Σ whose underlying scheme is affine and of finite type over Spec(V ). Let AU denote Γ(U, OU ). Similarly as in IV.5.1, let AU denote one of the following three AU -algebras: (Case I) The algebra AU itself. (Case II) The henselization of AU with respect to the ideal pAU . (Case III) The p-adic completion of AU .

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

403

For an extension L of K contained in K, put AU,L := AU ⊗V L, let UL denote Spec(AU,L ) endowed with the inverse image of MU , and let UL,triv denote the open subscheme of Spec(AU,L ) defined by {x ∈ Spec(AU,L )|MUL ,x = OU×L ,x }. By using charts, we see that the morphism UL,triv → UL is affine strict étale locally on UL . Hence UL,triv → UL is affine, which implies that UL,triv is affine (cf. [42] Proposition (2.7.1)). Let AU,L,triv denote Γ(UL,triv , OUL,triv ). We consider the category P defined as follows. An object of P is a triple U = (U, V, v) consisting of an fs log scheme U over Σ whose underlying scheme is affine and of finite type over Spec(V ) and a finite étale morphism v : V → UK,triv . A morphism f = (f, g) : U0 = (U 0 , V 0 , v 0 ) → U = (U, V, v) is a pair of a morphism f : U 0 → U over Σ and a morphism g : V 0 → V such that v ◦ g = fK,triv ◦ v 0 , where fK,triv denotes the 0 morphism UK,triv → UK,triv induced by f . The composition of two morphisms is defined in the obvious way. We often abbreviate (U, V, v) to (U, V) in the following. The fiber product of (U 0 , V 0 ) → (U, V) ← (U 00 , V 00 ) in P is representable as follows: Let U 000 be the fiber product of U 0 → U ← U 00 , and let V 000 be the fiber product 000 000 000 of V 0 ×U 0 UK,triv → V ×UK,triv UK,triv ← V 00 ×U 00 UK,triv , which is finite étale K,triv K,triv 000 000 000 over UK,triv . Then the object (U , V ) of P with the natural morphisms (U 000 , V 000 ) → (U 0 , V 0 ), (U 00 , V 00 ) represents the fiber product. We say that a morphism f : (U 0 , V 0 ) → (U, V) in P is horizontal (resp. vertical) if the 0 morphism V 0 → V ×Utriv,K Utriv,K (resp. U 0 → U ) induced by f is an isomorphism. By the above explicit construction of fiber products, we see that f is horizontal if and only if 0 the morphism (U 0 , V 0 ) → (U, V) ×(U,Utriv,K ) (U 0 , Utriv,K ) induced by f is an isomorphism.

For an object U = (U, V) of Ob P, we define Covh (U) (resp. Covv (U)) to be the set of families of horizontal (resp. vertical) morphisms ((fα , gα ) : (Uα , Vα ) → (U, V))α∈A such that (fα : Uα → U )α∈A (resp. (gα : Vα → V)α∈A ) is a strict étale covering (resp. a finite étale covering). We call a family of morphisms belonging to Covh (U) (resp. Covv (U)) a horizontal strict étale covering (resp. a vertical finite étale covering) of U. Horizontal strict étale coverings and vertical finite étale coverings are stable under base changes and compositions. The latter means that, for (Uα → U)α∈A ∈ Covh (U) (resp. Covv (U)) and (Uαβ → Uα )β∈Bα ∈ Covh (Uα ) (resp. Covv (Uα )), we have (Uαβ → U)α∈A,β∈Bα ∈ Covh (U) (resp. Covv (U)). Put Cov(U) = Covh (U) ∪ Covv (U). Let X be an fs log scheme over Σ whose underlying scheme is separated of finite type over Spec(V ). We define the sites (P/X)´et-f´et and (P/X)f´et associated to X as follows. We first define the category P/X as follows. An object is a pair (U, u) of an object U of P and a strict étale morphism u : U → X over Σ. A morphism f = (f, g) : (U0 , u0 ) → (U, u) is a morphism U0 → U in P such that u ◦ f = u0 . The fiber product of (U0 , u0 ) → (U, u) ← (U00 , u00 ) in P/X is represented by U000 = (U 000 , V 000 ) := U0 ×U U00 with the natural morphism u000 : U 000 → X. Similarly the product of (U, u) and (U0 , u0 ) in P/X is representable as follows. Put U = (U, V) and U0 = (U 0 , V 0 ). Let U 00 = U ×X U 0 , which is affine because the underlying scheme of X is separated by assumption, and let V 00 be the fiber product 00 00 00 of V ×UK,triv UK,triv UK,triv and V 0 ×U 0 over UK,triv . Then the pair U00 = (U 00 , V 00 ) is K,triv an object of P and U00 with the natural morphism U 00 → X represents the product of (U, u) and (U0 , u0 ). If X is affine, then P/X is naturally regarded as a full subcategory of the category P/(X,XK,triv ) of objects of P over (X, XK,triv ). We often omit u in the notation (U, u) and write simply U. For an object (U, u) of P/X, a family of morphisms (Uα → U)α∈A ∈ Cov(U) is naturally regarded as a family of morphisms in P/X. We define (P/X)´et-f´et (resp. (P/X)f´et ) to be the category P/X endowed with the topology generated by Cov(U) (resp. Covv (U)).

404

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Let Xétaff be the category of affine fs log schemes strict étale over X endowed with the topology associated to the pretopology defined by strict étale covering. The category P/X with the functor (P/X) → Xétaff ; (U, V) 7→ U is a fibered category such that the fiber over the object U of Xétaff is canonically identified with (UK,triv )f´et . For a morphism f : U 0 → U in Xétaff , the inverse image functor between the fibers is given by the base 0 change under the morphism fK,triv : UK,triv → UK,triv , which is continuous and defines a morphism of sites. Thus P/X with the finite étale topology on each fiber becomes a fibered site and the site (P/X)f´et defined above is the category P/X with the total topology (cf. [2] VI Définition 7.4.1, Proposition 7.4.2). The fibered site P/X and the strict étale site Xétaff satisfies the conditions (i), (ii), and (iii) in VI.5.1 and (P/X)´et-f´et is the category P/X with the covanishing topology associated to the fibered site and the strict étale topology of Xétaff defined in VI.5.3. The sets Cov(U) (resp. Covv (U)) for (U, u) ∈ P/X are stable under base changes in P/X. Hence, by [2] II Corollaire 2.3, sheaves on (P/X)´et-f´et (resp. (P/X)f´et ) are characterized as follows. Lemma IV.6.1.1 (cf. Proposition VI.5.10). A presheaf F on P/X is a sheaf on the site (P/X)´et-f´et (resp. (P/X)f´et ) if and only if for every U ∈ Ob (P/X) and (Uα → U)α∈A ∈ Cov(U) (resp. Covv (U)), the following sequence is exact Y Y F(U) → F(Uα ) ⇒ F(Uα ×U Uβ ). α∈A

(α,β)∈A2

The identity functor (P/X)f´et → (P/X)´et-f´et is continuous. Hence the identity functor (P/X)´et-f´et → (P/X)f´et is cocontinuous and these functors induce a morphism of topos (cf. [2] III Proposition 2.5) (IV.6.1.2)

vP/X : (P/X)´e∼t-f´et → (P/X)∼ f´ et .

Under the interpretation in terms of total topology and covanishing topology above, this coincides with the morphism δ in (VI.5.16.1). For F ∈ Ob ((P/X)∼ f´ et ), the inverse image ∗ vP/X F is the sheaf on (P/X)´e∼t-f´et associated to F regarded as a presheaf on (P/X)´et-f´et .

For U ∈ Ob P, let Covhf (U) (resp. Covvf (U)) be the subset of Covh (U) (resp. Covv (U)) consisting of (Uα → U)α∈A such that A is a finite set. Put Covf (U) = Covhf (U)∪Covvf (U). Since U and UK,triv is affine, for any (Uα → U)α∈A ∈ Cov(U), there exists a finite subset A0 of A such that (Uα → U)α∈A0 belongs to Covf (U). Hence the topology of (P/X)´et-f´et (resp. (P/X)f´et ) is generated by Covf (U) (resp. Covvf (U)), and Lemma IV.6.1.1 with Cov(U) (resp. Covv (U)) replaced by Covf (U) (resp. Covvf (U)) still holds.

Proposition IV.6.1.3. Let U be an object of P/X and let R be a sieve of U. Then R is a covering sieve for the topology of (P/X)´et-f´et if and only if there exist (fα : Uα → U)α∈A ∈ Covhf (U) and (gαβ : Uαβ → Uα )β∈Bα ∈ Covvf (Uα ) for each α ∈ A such that fα ◦ gαβ ∈ R(Uαβ ) for every α ∈ A and β ∈ Bα . Proof. Let J(U) be the set of sieves R of U such that there exist fα and gαβ as in the proposition satisfying fα ◦ gαβ ∈ R(Uαβ ). It is sufficient to prove that J(U) satisfies the axiom of topology in [2] II Définition 1.1. The condition T3) follows from (idU : U → U) ∈ Covhf (U). The condition T1) follows from the stability of horizontal strict étale coverings (resp. vertical finite étale coverings) by base change. It remains to prove T2), i.e., for R ∈ J(U) and a sieve R0 of U, if for any U0 ∈ Ob (P/X) and any morphism U0 → R the sieve R0 ×U U0 belongs to J(U0 ), then R0 belongs to J(U). The assumption R ∈ J(U) means that there exist (fα : Uα → U)α∈A ∈ Covhf (U) and (gαβ : Uαβ → Uα )β∈Bα ∈ Covvf (Uα ) for each α ∈ A such that the composition fα ◦

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

405

gαβ : Uαβ → U factors through R. By assumption, we have R0 ×U Uαβ ∈ J(Uαβ ), which implies that there exist (fαβ;γ : Uαβ;γ → Uαβ )γ∈Γαβ ∈ Covhf (Uαβ ) and (gαβ;γδ : Uαβ;γδ → Uαβ;γ )δ∈∆αβ;γ ∈ Covvf (Uαβ;γ ) for each γ ∈ Γαβ such that Uαβ;γδ → Uαβ factors through R0 ×U Uαβ . Then we see that the composition Uαβ;γδ → Uαβ;γ → Uαβ → Uα → U factors through R0 . We will exchange the order of the vertical finite étale coverings and the horizontal strict étale coverings in the middle of the composition above. Let Uα , Uαβ , and Uαβ;γ denote the first component of Uα , Uαβ , and Uαβ;γ , respectively. Note that the morphism Uαβ → Uα induced by gαβ is an isomorphism. Since Bα is a finite 0 set, there exists a strict étale covering (Uακ → Uα )κ∈Kα , ]Kα < ∞ such that for every 0 κ ∈ Kα and β ∈ Bα , there exist γ = γ(κ, β) ∈ Γαβ and a morphism Uακ → Uαβ;γ v 0 0 0 over Uα . For κ ∈ Kα , let (Uακ;β → Uακ )β∈Bα ∈ Covf (Uακ ) be the base change of 0 0 ) → (Uα , Uα,triv,K ). (Uαβ → Uα )β∈Bα ∈ Covvf (Uα ) by the morphism (Uακ , Uακ,triv,K

Then (U0ακ → Uα )κ∈Kα belongs to Covhf (Uα ) and we have the following commutative diagram whose three squares are Cartesian: U0ακ;β

/ Uαβ o

 U0ακ

 / Uα

 0 (Uακ , Uακ,triv,K )

 / (Uα , Uα,triv,K ) o

Uαβ;γ

 (Uαβ;γ , Uαβ;γ,triv,K )

0 → Uαβ;γ over Uα inHence for κ ∈ Kα , β ∈ Bα , and γ = γ(κ, β) ∈ Γαβ , a morphism Uακ 0 0 0 duces a morphism Uακ;β → Uαβ;γ over Uαβ . Let (Uακ;βδ → Uακ;β )δ∈∆αβ;γ ∈ Covvf (U0ακ;β ) be the base change of (Uαβ;γδ → Uαβ;γ )δ∈∆αβ;γ ∈ Covvf (Uαβ;γ ) by a Uαβ -morphism U0ακ;β → Uαβ;γ as above. Then we have (U0ακ;βδ → U0ακ )β,δ ∈ Covvf (U0ακ ) and (U0ακ → U)α,κ ∈ Covhf (U), and their composition (U0ακ;βδ → U) refines (Uαβ;γδ → U). Hence R0 ∈ J(U). 

Corollary IV.6.1.4 (cf. Proposition VI.5.38). Let f : X 0 → X be a strict étale morphism of fs log schemes over Σ such that X 0 satisfies the same condition as X. Let X 0∧ denote the presheaf on (P/X) defined by X 0∧ ((U, V), u) = HomX (U, X 0 ). Then the ∼ ∼ = = canonical isomorphism of categories (P/X 0 )´et-f´et → (P/X)´et-f´et /X 0∧ (resp. (P/X 0 )f´et → 0∧ (P/X)f´et /X ) defines an isomorphism of sites, where the target is endowed with the topology induced by that of (P/X)´et-f´et (resp. (P/X)f´et ). Proof. It suffices to prove that the topology of (P/X 0 )´et-f´et (resp. (P/X 0 )f´et ) is induced by the functor j : (P/X 0 ) → (P/X); (U0 , u0 ) 7→ (U0 , f ◦ u0 ) and the topology of (P/X)´et-f´et (resp. (P/X)f´et ). Let (U0 , u0 ) ∈ Ob (P/X 0 ) and let R be a sieve of (U0 , u0 ). By [2] III Proposition 5.2 1), it suffices to prove that R is a covering sieve for the topology of (P/X 0 )´et-f´et (resp. (P/X 0 )f´et ) if and only if j! R is a covering sieve of (U0 , f ◦ u0 ) for the topology of (P/X)´et-f´et (resp. (P/X)f´et ). For (U, u) ∈ Ob (P/X), we have G (j! (U0 , u0 ))(U, u) = Hom((U, u), (U0 , f ◦ u0 )) = Hom((U, v), (U0 , u0 )), v∈HomX (U,X 0 )

(j! R)(U, u) =

G

R(U, v).

v∈HomX (U,X 0 )

By Proposition IV.6.1.3 (resp. By definition), the sieve R is a covering if and only if there exist (fα : U0α → U0 ) ∈ Covhf (U0 ) and (gαβ : U0αβ → U0α ) ∈ Covvf (U0α ) (resp. there

406

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

exists (fα : U0α → U0 ) ∈ Covv (U0 )) such that fα ◦ gαβ (resp. fα ) factors through R in (P/X 0 )∧ . By the above description of j! R, the last condition is equivalent to saying that fα ◦ gαβ (resp. fα ) factors through j! R in (P/X)∧ . By applying Proposition IV.6.1.3 to (P/X)´et-f´et (resp. By the definition of (P/X)f´et ), we obtain the claim.  By Corollary IV.6.1.4, the functor (P/X 0 )´et-f´et → (P/X)´et-f´et ; (U0 , u0 ) → (U0 , f ◦ u0 ) is continuous and cocontinuous (cf. [2] III Proposition 5.2 2) and induces a morphism of topos (IV.6.1.5)

fP,´et-f´et : (P/X 0 )´e∼t-f´et −→ (P/X)´e∼t-f´et .

Similarly, we obtain a morphism of topos (IV.6.1.6)

∼ fP,f´et : (P/X 0 )∼ f´ et −→ (P/X)f´ et .

Proposition IV.6.1.7. Let f : X 0 → X be as in Corollary IV.6.1.4 and assume that X 0 is affine. We define the functor f ∗ : (P/X) → (P/X 0 ) by ((U, V), u) 7→ ((U 0 , V 0 ), u0 ), 0 , and u0 is the natural morphism U 0 → X 0 . where U 0 = X 0 ×X U , V 0 = V ×UK,triv UK,triv Then the functor f ∗ is continuous with respect to the topologies of (P/X)´et-f´et and (P/X 0 )´et-f´et (resp. (P/X)f´et and (P/X 0 )f´et ) and it is a right adjoint of the functor j : (P/X 0 ) → (P/X); (U0 , u0 ) 7→ (U0 , f ◦ u0 ). Hence f ∗ defines a morphism of sites (P/X 0 )´et-f´et → (P/X)´et-f´et (resp. (P/X 0 )f´et → (P/X)f´et ) and induces the morphism of topos fP,´et-f´et (resp. fP,f´et ) above (cf. [2] III Proposition 2.5). Proof. The functor f ∗ preserves finite fiber products and induces Cov(U, u) → Cov(f ∗ (U, u)) (resp. Covv (U, u) → Covv (f ∗ (U, u))) for any (U, u) ∈ Ob (P/X). Hence f ∗ is continuous. Let (U, u) ∈ Ob (P/X), (U0 , u0 ) := f ∗ (U, u), and (U00 , u00 ) ∈ Ob (P/X 0 ). Put U = (U, V), U0 = (U 0 , V 0 ), and U00 = (U 00 , V 00 ). Then the composition with the natural morphism fU : U 0 → U induces a bijection HomX 0 (U 00 , U 0 ) → HomX (U 00 , U ). Given a morphism U 00 → U 0 over X 0 , the composition with the natural morphism fV : V 0 → V induces a bijection HomU 0 (V 00 , V 0 ) → HomUtriv,K (V 00 , V). Hence the composition with triv,K

∼ =

fU and fV induces a bijection Hom((U00 , u00 ), (U0 , u0 )) → Hom((U00 , f ◦ u00 ), (U, u)).



We will later interpret the representations constructed in IV.5.2 in terms of sheaves on (P/X)´et-f´et . Since the representations are constructed only for sufficiently small affine fs log schemes, we need to work with a full subcategory of P/X of the following type. Let C be a full subcategory of Xétaff such that for any U ∈ Ob Xétaff , there exists a strict étale covering (Uα → U )α∈A such that Uα ∈ Ob C for every α ∈ A. We define P/C to be the full subcategory of P/X consisting of (U, V) such that U ∈ Ob C and define the site (P/C)´et-f´et (resp. (P/C)f´et ) to be the category P/C endowed with the topology induced by that of (P/X)´et-f´et (cf. [2] III §3) (resp. the pretopology Covv (U) (U ∈ Ob (P/C)f´et )). For any (U, V) ∈ (P/X)´et-f´et , a strict étale covering (Uα → U )α∈A by objects of C induces a horizontal strict étale covering ((U, V)×(U,UK,triv ) (Uα , Uα,K,triv ) → (U, V))α∈A by objects of P/C. Hence the inclusion functor w : (P/C) → (P/X) induces an equivalence of categories (cf. [2] III Théorème 4.1) (IV.6.1.8)

∼

ws : (P/X)´e∼t-f´et → (P/C)´e∼t-f´et .

The inclusion functor w : (P/C)f´et → (P/X)f´et is continuous. Using (IV.6.1.8), we also see that the identity functor (P/C)f´et → (P/C)´et-f´et is continuous. Hence the identity functor (P/C)´et-f´et → (P/C)f´et is cocontinuous and these two functors induce a morphism of topos (cf. [2] III Proposition 2.5) (IV.6.1.9)

vP/C : (P/C)´e∼t-f´et → (P/C)∼ f´ et .

The inverse image functor is given by the sheafification.

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

407

Let f : X 0 → X be a strict étale morphism of fs log schemes over Σ such that X 0 0 satisfies the same condition as X. Let C 0 be the full subcategory of Xétaff such that 0 0 every object of Xétaff admits a strict étale covering by objects of C and, for every object u0 : U 0 → X 0 of C 0 , the composite f ◦ u0 is an object of C. Lemma IV.6.1.10. Let the notation and the assumption be as above. Then the functor (P/C 0 )´et-f´et → (P/C)´et-f´et (resp. (P/C 0 )f´et → (P/C)´et-f´et ) defined by (u0 : U 0 → X 0 , V) 7→ (f ◦ u0 , V) is continuous and cocontinuous. Proof. We first prove the claim for (P/C)´et-f´et and (P/C 0 )´et-f´et . We have a commutative diagram of categories (P/C 0 )´et-f´et

w0

j0

 (P/C)´et-f´et

/ (P/X 0 )´et-f´et j

w

 / (P/X)´et-f´et ,

where w and w0 are inclusion functors and j and j 0 are defined by the composition with f . By Corollary IV.6.1.4 and [2] III Proposition 5.2 2), the functor j is continuous and cocontinuous. Hence, by (IV.6.1.8) applied to w and w0 , we see that j 0 is continuous. By the proof of [2] III Théorème 4.1, the functor w0 is cocontinuous. Hence the composition w ◦ j 0 = j ◦ w0 is cocontinuous. Let b j 0∗ (resp. w b∗ ) denote the functor (P/C)´e∧t-f´et → 0 ∧ ∧ ∧ (P/C )´et-f´et (resp. (P/X)´et-f´et → (P/C)´et-f´et ) defined by the composition with j 0 (resp. w), b∗ ) denote its right adjoint. Then, for a sheaf F on (P/C 0 )´et-f´et , w b∗b and let b j∗0 (resp. w j∗0 F is a sheaf by [2] III Proposition 2.2. Since w is fully faithful, we see that the adjunction j∗0 F is a morphism w b∗ w b∗b j∗0 F is an isomorphism (cf. [2] I Proposition 5.6). Hence b j∗0 F → b 0 sheaf on (P/C)´et-f´et because w is continuous. By [2] III Proposition 2.2, j is cocontinuous. Next we prove the second case. The topologies are induced by the pretopologies defined by vertical finite étale coverings and the functor in question is compatible with f the fiber products of U0 → U ← U00 with f vertical. This implies the continuity. For an object (U, u) of P/C 0 , the functor (P/C 0 )/(U, u) → (P/C)/(U, f ◦ u) is fully faithful. Hence, by [2] II Proposition 1.4, we see that the functor (P/C 0 )f´et → (P/C)f´et satisfies the property defining cocontinuity (cf. [2] III Définition 2.1).  We obtain from Lemma IV.6.1.10 a sequence of three adjoint functors between ∼ (P/C 0 )´e∼t-f´et and (P/C)´e∼t-f´et (resp. (P/C 0 )∼ f´ et and (P/C)f´ et ) (IV.6.1.11)

0

0

0

C,C C,C ∗ C,C fP,´ et-f´ et! , fP,´ et-f´ et , fP,´ et-f´ et∗

0

0

0

C,C C,C ∗ C,C (resp. fP,f´ et! , fP,f´ et , fP,f´ et∗ ).

These are equivalences of categories if X 0 = X and f = idX in the first case. The pairs C,C 0 ∗ C,C 0 C,C 0 ∗ C,C 0 (fP,´ et-f´ et , fP,´ et-f´ et∗ ) and (fP,f´ et , fP,f´ et∗ ) define morphisms of topos (IV.6.1.12) (IV.6.1.13)

0

C,C 0 ∼ ∼ fP,´ et-f´ et −→ (P/C)´ et-f´ et , et-f´ et : (P/C )´ 0

C,C 0 ∼ ∼ fP,f´ et −→ (P/C)f´ et , et : (P/C )f´

0 which coincide with fP,´et-f´et and fP,f´et if C = Xétaff and C 0 = Xétaff . We see that the following diagram of topos is commutative up to canonical isomorphism by looking at

408

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

the corresponding diagram of sites and cocontinuous morphisms. 0

(P/C 0 )´e∼t-f´et

(IV.6.1.14)

C,C fP,´ et-f´ et

vP/C0

 (P/C 0 )∼ f´ et

0

0

/ (P/C)∼ ´ et-f´ et vP/C

 / (P/C)∼ . f´ et

C,C 0 fP,f´ et

0

0

C,C C,C C,C 0 We write ιC,C P,f´ et and ιP,´ et-f´ et for fP,f´ et and fP,´ et-f´ et if X = X and f = idX . We abbreviate 0 the superscript C, C if there is no risk of confusion in the following. For an object U = (U, V) of P/X, let AV denote the integral closure of AU in Γ(V, OV ).

Lemma IV.6.1.15. The presheaf V 7→ AV on (UK,triv )f´et is a sheaf. Proof. Since V 7→ Γ(V, OV ) is a sheaf on (UK,triv )f´et , it suffices to prove the following claim. For a finite étale covering (Vα → V)α∈A , a section x ∈ Γ(V, OV ) is integral over AU if its image xα in Γ(Vα , OVα ) is integral over AU for every α ∈ A. Since V is affine, there exists a finite subset A0 of A such that (Vα → V)α∈A0 is still a covering. For each α ∈ A0 , choose a Q monic polynomial fα ∈ AU [X]Qsuch that fα (xα ) = 0. Then, we have f (x) = 0 for f = α∈A0 fα because Γ(V, OV ) → α∈A0 Γ(Vα , OVα ) is injective.  We define the sheaf OUintK,triv on (UK,triv )f´et by OUintK,triv (V) = AV , and define the sheaf

v OP/C

v on (P/C)f´et by OP/C (U, V) = OUintK,triv (V) = AV . Let OP/C denote the sheaf on v ∗ v (P/C)´et-f´et associated to OP/C . We have OP/C = vP/C (OP/C ). v For m ∈ N, we define OP/C,m (resp. OP/C,m ) to be the quotient OP/C /pm OP/C v v /pm OP/C ) as a sheaf on (P/C)´et-f´et (resp. (P/C)f´et ). We have a canonical (resp. OP/C isomorphism ∗ v vP/C (OP/C,m )∼ = OP/C,m . ? ? ? ? We write OP/X and OP/X,m (? = v, ∅) for OP/C and OP/C,m if C = Xétaff .

Lemma IV.6.1.16. Assume that AU is flat over V , AU is normal, and AU ⊗V k is v v reduced. Then, in the cases I and III, we have OP/C = OP/C , i.e., OP/C is a sheaf on (P/C)´et-f´et . Proof. By Lemma IV.6.1.15 and the remark before Proposition IV.6.1.3, it suffices h to prove the following. For U = (U, V) ∈ Ob (P/C), (Uα = Q (Uα , Vα ) → U)α∈A Q ∈ Covf (U), and Uαβ = (Uαβ , Vαβ ) := Uα ×U Uβ , the sequence AV → α∈A AVα ⇒ (α,β)∈A2 AVαβ e =U e e =U U to simplify the notation. is exact. Put U K,triv , Uα = Uα,K,triv , and αβ,K,triv Q αβ Q In the cases I and III, the sequence AU → α∈A AUα ⇒ (α,β)∈A2 AUαβ is exact, and it remains exact after ⊗V V . By Lemma IV.5.2.1, we see that AU ⊗V V , AUα ⊗V V , and AUαβ ⊗V V are products of finite numbers of normal domains. Hence, AUe , AUeα , and AUeαβ are also products of finite numbers of normal domains, and the following sequence is exact. Y Y (∗) AUe → AUeα ⇒ AUeαβ . (α,β)∈A2

α∈A

e, Oe) → We also see that the sequence Γ(U U

Q

eα , O e ) ⇒ Γ(U Uα

Q

eαβ , O e ) Γ(U Uαβ eα × e V is exact. Since the morphism Uα → U is horizontal, the natural morphisms Vα → U U eαβ × e V are isomorphisms. Since Vα → U eα is flat, this implies that the and Vαβ → U U α∈A

(α,β)∈A2

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

409

Q Q sequence Γ(V, OV ) → α∈A Γ(Vα , OVα ) ⇒ (α,β)∈A2 Γ(Vαβ , OVαβ ) is also exact. Hence it remains to prove that an element x ∈ Γ(V, OV ) is integral over AUe if its pull-back in Γ(Vα , OVα ) is integral over AUeα for every α ∈ A. Let xα be the image of x in Γ(Vα , OVα ), eα , O e )[T ] be the characteristic polynomial of xα over U eα . Since and let Φα (T ) ∈ Γ(U Uα

AUeα is a product of finite number of normal domains, the coefficients of Φα (T ) lie in AUeα . The inverse images of Φα (T ) and Φβ (T ) in AUeαβ [T ] coincide because they are both eαβ . By the the characteristic polynomial of the inverse image of x in Γ(Vαβ , OVαβ ) over U exact sequence (∗), there exists Φ(T ) ∈ AUe [T ] whose pull-back in AUeα [T ] is Φα (T ). The vanishing Φα (xα ) = 0 for every α implies Φ(x) = 0 and hence x is integral over AUe .  For f : X 0 → X, C, and C 0 as before Lemma IV.6.1.10, we have 0

C,C ∗ ∼ fP,´ et-f´ et (OP/C,m ) = OP/C 0 ,m

by (IV.6.1.14). Hence we have a diagram of ringed topos commutative up to canonical isomorphism 0

(IV.6.1.17)

((P/C 0 )´e∼t-f´et , OP/C 0 ,m )

C,C fP,´ et-f´ et

vP/C0

 v ((P/C 0 )∼ f´ et , OP/C 0 ,m )

C,C 0 fP,f´ et

/ ((P/C)´e∼t-f´et , OP/C,m ) 

vP/C

v / ((P/C)∼ f´ et , OP/C 0 ,m ).

If X 0 = X and f = id, we have the following equivalences of categories which are quasiinverse of each other, where Mod((P/C (0) )´et-f´et , OP/C (0) ,• ) denotes the category of inverse systems of OP/C (0) ,m -modules for m ∈ N>0 . 0

(IV.6.1.18)

Mod((P/C 0 )´et-f´et , OP/C 0 ,• ) o

ιC,C P,´ et-f´ et∗ 0∗ ιC,C P,´ et-f´ et

/ Mod((P/C) ´ et-f´ et , OP/C,• ).

Proposition IV.6.1.19. Let (fα : Xα → X)α∈A be a strict étale covering such that Xα is affine. Let Xαβ denote Xα ×X Xβ and let fαβ : Xαβ → X denote the natural morphism. Then, for a sheaf of OP/X,m -modules Fi (i = 1, 2), the following sequence is exact. Y ∗ ∗ HomOP/X,m (F1 , F2 ) → HomOP/Xα ,m (fα,´ et-f´ et F1 , fα,´ et-f´ et F2 ) α∈A

⇒

Y (α,β)∈A2

∗ ∗ HomOP/Xαβ ,m (fαβ,´ et-f´ et F1 , fαβ,´ et-f´ et F2 ).

Proof. For a strict étale morphism f : X 0 → X with X 0 affine, an object (U, u) of (X/P)´et-f´et , and (U0 , u0 ) := f ∗ (U, u) (cf. Proposition IV.6.1.7), the morphism Fi (U, u) → f´et-f´et∗ f´e∗t-f´et Fi (U, u) = Fi (U0 , f ◦ u0 ) is induced by the natural morphism (U0 , f ◦ u0 ) → (U, u). On the other hand, for (U, u) ∈ Ob (P/X)´et-f´et , letting Uα and Uαβ denote the ∗ ∗ objects of P underlying fα,´ et-f´ et (U, u) and fαβ,´ et-f´ et (U, u), the family of natural morphisms (Uα → U)α∈A is a horizontal strict étale covering and the natural morphism Uαβ → Uα ×U Uβ is an isomorphism. Hence the sequence Y Y ∗ ∗ Fi → fα,´et-f´et∗ fα,´ fαβ,´et-f´et∗ fαβ,´ et-f´ et Fi ⇒ et-f´ et Fi α∈A

is exact. This implies the claim.

(α,β)∈A2



410

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Definition IV.6.1.20. Let C be a V -linear abelian category. (1) We say that an object x of C is almost zero if a · idx = 0 for all a ∈ m. (2) We say that a morphism f : x → y in C is an almost isomorphism if Ker(f ) and Cok(f ) is almost zero. Lemma IV.6.1.21. Let C be a V -linear abelian category. (1) Let x → y → z be an exact sequence in C. If x and z are almost zero, then y is almost zero. (2) If morphisms f : x → y and g : y → z in C are almost isomorphisms, then g ◦ f is also an almost isomorphism. (3) If f : x → y is an almost isomorphism in C, then for every a ∈ m, there exists a morphism g : y → x such that g ◦ f = a · idx and f ◦ g = a · idy . Proof. It is straightforward to prove (1) and (2). Let us prove (3). Let f : x → y be an almost isomorphism in C and let a ∈ m. Choose b, c ∈ m such that a = bc. Since b · idCokf = 0, b · idy factors through ϕb : y → Im(f ). Since c · idKerf = 0, c · idx factors through ψc : Imf → x. The composition ψc ◦ϕb : y → x satisfies the desired property.  Corollary IV.6.1.22. Let C be a site and let f : F → G be a morphism of sheaves of V -modules on C. Then f is an almost isomorphism of sheaves of V -modules if and only if it is an almost isomorphism of presheaves of V -modules. Proposition IV.6.1.23. Let X and C be as above. Let F be a presheaf of V -modules on (P/X)´et-f´et satisfying the following two conditions: (a) For any U ∈ Ob (P/C) and (Uα → U) ∈ Covvf (U) such that Uα ∈ Ob (P/C), the morphism Y Y F(U) → Ker( F(Uα ) ⇒ F(Uα ×U Uβ )) α∈A

(α,β)∈A2

is an almost isomorphism. (b) For any U ∈ Ob (P/C), (Uα → U)α∈A ∈ Covhf (U), and (Uαβ;γ → Uα ×U Uβ )γ∈Γαβ ∈ Covhf (Uα ×U Uβ ) for (α, β) ∈ A2 such that Uα , Uαβ;γ ∈ Ob (P/C), the morphism Y Y Y F(U) → Ker( F(Uα ) ⇒ F(Uαβ;γ )) α∈A

(α,β)∈A2 γ∈Γαβ

is an almost isomorphism. Then for the sheaf F a on the site (P/C)´et-f´et associated to F, the canonical morphism F(U) → F a (U) is an almost isomorphism for every U ∈ Ob (P/C). Proof. By Proposition IV.6.1.3 and the construction of the sheafification in [2] II §3, it suffices to prove that for (fα ) and (gαβ ) as in Proposition IV.6.1.3, the morphism Y Y F(U) −→ Ker( F(Uαβ ) ⇒ F(Uαβ ×U Uα0 β 0 )) αβ

αβ,α0 β 0

is an almost isomorphism. (If L denotes the functor defined in [2] II 3.0.5, then the above claim implies that F → LF is an almost isomorphism. Hence LF also satisfies the conditions (a) and (b). Applying the above claim again, we see that LF → LLF = F a is an almost isomorphism.) The conditions (a) and (b) on F immediately imply that the above Q homomorphism is almost injective. Assume that (xαβ ) ∈ αβ F(Uαβ ) is contained in the kernel. Let ε ∈ Q>0 . Since the restrictions of xαβ and xαβ 0 on Uαβ ×Uα Uαβ 0 coincide, we see that there exists xα ∈ F(Uα ) such that xα |Uαβ = pε xαβ for every β ∈ Bα by the condition (a). For each (α, α0 ) ∈ A2 , choose (Uαα0 ;γ → Uα ×U Uα0 )γ∈Bαα0 ∈ Covhf (Uα ×U Uα0 ) such that Uαα0 ;γ ∈ Ob C. For (β, β 0 ) ∈ Bα ×Bα0 , let (Uαβ,α0 β 0 ;γ → Uαβ ×U Uα0 β 0 )γ∈Bαα0 be

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

411

the base change of the above covering. Since (Uαβ ×U Uα0 β 0 → Uα ×U Uα0 )(β,β 0 )∈Bα ×Bα0 belongs to Covvf (Uα ×U Uα0 ), we see that Uαβ,α0 β 0 ;γ ∈ Ob C and the family of natural morphisms (Uαβ,α0 β 0 ;γ → Uαα0 ;γ )(β,β 0 )∈Bα ×Bα0 belongs to Covvf (Uαα0 ;γ ). Since the restrictions of xαβ and xα0 β 0 on Uαβ,α0 β 0 ;γ coincide by assumption, we see that the restrictions of pε xα and pε xα0 on Uαα0 ;γ coincide by the condition (a). Hence there exists x ∈ F(U) whose restriction on Uα is p2ε xα by the condition (b).  Corollary IV.6.1.24. Let X and C be as above, let C 0 be a full subcategory of Xétaff containing C, let F be a presheaf of V -modules on (P/C 0 )´et-f´et satisfying the conditions (a) and (b) in Proposition IV.6.1.23, and let F a be the sheaf on (P/C 0 )´et-f´et associated to F. Then the homomorphism F(U) → F a (U) is an almost isomorphism for every U ∈ Ob (P/C).

Proof. Give P/C 0 and P/X the topologies defined by the identity coverings. Then the inclusion functor (P/C 0 ) → (P/X) induces a diagram of categories and adjoint pairs of functors, commutative up to canonical isomorphisms (cf. Lemma IV.6.1.10 and [2] I Proposition 5.1). Here a denotes the sheafification functors. Mod((P/C 0 ), V ) O i

o

u∗ u−1

a

 o Mod((P/C 0 )´et-f´et , V )

/ Mod((P/X), V ) O i

u´et-f´et∗ u´e−1 t-f´ et

a

 / Mod((P/X)´et-f´et , V ).

For U ∈ Ob (P/C 0 ), we have (u−1 F)(U) = limU→U0 ,U0 ∈Ob (P/C 0 ) F(U0 ) = F(U). Hence −→ u−1 F satisfies the conditions in Proposition IV.6.1.23 and the adjunction map F → u∗ u−1 F is an isomorphism. From the above diagram, we obtain the following commutative diagram. / iaF / iu´et-f´et∗ u−1 aF FF ´ et-f´ et FF FF ∼ FF = ∼ = FF "  / u∗ iau−1 F. u∗ u−1 F The upper right morphism is an isomorphism because u´et-f´et∗ is an equivalence of categories. The restriction of the bottom arrow on P/C is an almost isomorphism by Proposition IV.6.1.23. Hence it also holds for F → iaF.  IV.6.2. A description of sheaves on (P/C)f´et in terms of representations. In order to interpret the representations constructed in IV.5.2 in terms of sheaves on (P/C)´et-f´et , we give a description of sheaves on (P/C)f´et in terms of compatible system of modules on “universal coverings” of Utriv,K under the assumption that the underlying scheme of X is normal and has reduced special fiber. We first start by reviewing a description of sheaves on (Utriv,K )f´et . To avoid the dependence on the choice of geometric points, we work with the category of all geometric points whose images are of codimension 0 instead of the fundamental group at one geometric point (cf. Lemma VI.9.11, [37] V Proposition 5.8). Let U be an fs log scheme over Σ satisfying the following conditions: The underlying scheme of U is affine and normal. The morphism of schemes underlying U → Σ is flat of finite type and its special fiber is reduced. We define AU , AU , and UL,triv as in the beginning of IV.6.1. Let s → UK,triv be a geometric point of UK,triv such that the image sg of s is of codimension 0 and Γ(s, Os ) is an algebraic closure of the residue field κ(sg ) at sg . Let

412

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

s be the image of s in UK,triv and let κ(s) be the residue field at s. Then we have an ∼ = isomorphism κ(s) ⊗K K → κ(sg ) in the cases II and III by Lemma IV.5.2.1, and an algebraic closure of K in κ(s) is finite over K in the case I. Let SK,s be the set of all finite Galois extensions L of κ(s) contained in κ(s) such that the normalization of UK,triv in L is étale over UK,triv . We define the set Ss to be {Lκ(sg )|L ∈ SK,s }. For L ∈ Ss , let UL denote the normalization of UK,triv in L (cf. [40] Proposition (6.3.4)). Choose L0 ∈ SK,s such that L = L0 κ(sg ) and let L be an algebraic closure of K in L, which is finite over K because an algebraic closure of K in κ(s) is finite over K. Let UL,L0 be the normalization of UL,triv in L0 . Then UL,L0 is finite étale over UL,triv and the natural morphism UL,L0 ×UL,triv UK,triv → UL is an isomorphism. This implies that UL → UK,triv is finite étale for L ∈ Ss and we have an isomorphism G G ∼ = (IV.6.2.1) σ × 1: UL0 → UL0 ×UL UL0 σ

σ∈Gal(L0 /L)

for L, L0 ∈ Ss such that L ⊂ L0 . Let V → UK,triv be a finite étale morphism. Then it comes from a finite étale morphism VL → UL,triv for some finite extension L ⊂ K of K. Hence we see that there exists L ∈ Ss such that V ×UK,triv UL is isomorphic to the disjoint union of a finite number of copies of UL as a UL -scheme. This property will allow us to use the inverse system of finite étale coverings (UL )L∈Ss as a “universal covering” of the connected component of Utriv,K containing sg . We define κ(s)ur to be the union of L ∈ Ss (which coincides with the union of L ∈ SK,s ) and sur to be Spec(κ(s)ur ). Let ∆s denote the Galois group Gal(κ(s)ur /κ(sg )). For another geometric point s0 → UK,triv satisfying the same conditions, we define a morphism from s0 to s to be a morphism h : s0ur → sur compatible with the morphisms s0ur , sur → Utriv,K . Let (UK,triv )gpt denote the category of s → Utriv,K and morphisms defined above. By definition, we have a canonical isomorphism ∆s ∼ = Aut(UK,triv )gpt (s)◦ for s ∈ (UK,triv )gpt . Now we are ready to give a description of sheaves on (UK,triv )f´et . Associated to a sheaf of sets F on (UK,triv )f´et , one can define a presheaf of sets sU (F) on (UK,triv )gpt by sU (F)(s) = lim F(UL ); −→ L∈Ss

∼ =

a morphism h : s0 → s in (UK,triv )gpt induces an isomorphism h∗ : κ(s)ur → κ(s0 )ur over ∼ =

κ(s) = κ(s0 ), and then an isomorphism Uh∗ (L) → UL for each L ∈ Ss , which induces an ∼ = isomorphism sU (F)(s) → sU (F)(s0 ). By construction, the natural left action of ∆s (∼ =

Aut(s)◦ ) on sU (F)(s) is continuous. Having this observation, we define (UK,triv )∧,cont gpt to be the full subcategory of (UK,triv )∧ gpt consisting of presheaves G such that the action of ∆s on G(s) is continuous for every s ∈ Ob (UK,triv )gpt . Then the above construction gives a functor ∧,cont sU : (Utriv,K )∼ . f´ et −→ (Utriv,K )gpt

One can also construct a functor in the opposite direction as follows. Let G ∈ Ob (UK,triv )∧,cont . For V ∈ Ob (UK,triv )f´et , we define the presheaf GV on (UK,triv )gpt by gpt GV (s) = Map(Hom UK,triv (sur , V), G(s)). For a morphism h : s0 → s, we define the morphism GV (h) : GV (s) → GV (s0 ) by composing with the bijection Hom(s0ur , V) → Hom(sur , V) induced by h and G(h) : G(s) → G(s0 ). For a morphism g : V 0 → V in (UK,triv )f´et , the map g ◦ − : Hom(sur , V 0 ) → Hom(sur , V)

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

413

induces a morphism of presheaves GV → GV 0 . We define the presheaf rU (G) on (UK,triv )f´et associated to G by rU (G)(V) = Γ((UK,triv )gpt , GV ) =

lim ←−

(UK,triv )gpt

GV .

This construction is obviously functorial in G. Let P be a finite subset of Ob (UK,triv )gpt such that the natural map P → π0 (UK,triv ) is bijective. Then the evaluation at s ∈ P induces an isomorphism Y ∼ = (IV.6.2.2) rU (G)(V) −→ Map∆s (Hom UK,triv (sur , V), G(s)). s∈P

Lemma IV.6.2.3. The presheaf rU (G) is a sheaf on (UK,triv )f´et . Proof. For any finite number of finite étale schemes Vλ (λ ∈ Λ) over UK,triv , Q we have rU (G)(tλ∈Λ Vλ ) = λ∈Λ rU (G)(Vλ ). Hence it suffices to prove the following claim. For any finite étale surjective morphism V 0 → V of finite étale schemes over UK,triv and s ∈ Ob (UK,triv )gpt , the sequence GV (s) → GV 0 (s) ⇒ GV 0 ×V V 0 (s) is exact. This follows from Hom(sur , V 0 ×V V 0 ) = Hom(sur , V 0 ) ×Hom(sur ,V) Hom(sur , V 0 ) and the surjectivity of the map Hom(sur , V 0 ) → Hom(sur , V).  By Lemma IV.6.2.3, we obtain a functor → (UK,triv )∼ rU : (UK,triv )∧,cont f´ et . gpt Proposition IV.6.2.4. The functors rU and sU defined above are equivalences of categories which are canonically quasi-inverse of each other. e for U Proof. We write U K,triv to simplify the notation. We prove that there exist canonical and functorial isomorphisms F ∼ = G for F ∈ = rU ◦ sU (F) and sU ◦ rU (G) ∼ ∧,cont ∼ e e e e Ob (U )f´et and G ∈ Ob (U )gpt . For V ∈ Ob Uf´et , s ∈ Ob Ugpt , and a morphism u : sur → e , there exists an L ∈ Ss such that u factors through a U e -morphism uL : UL → V V over U uniquely, and the morphism F(V) → sU (F)(s) = limL∈S F(UL ) defined by F(uL ) is −→ s independent of the choice of L. Let F(u) denote the morphism. Then we obtain a canonical morphism F(V) → sU (F)V (s) = Map(HomUe (sur , V), sU (F)(s)) by x 7→ (u 7→ F(u)(x)). It is straightforward to check that this map is compatible with the morphism egpt and defines a morphism sU (F)V (s) → sU (F)V (s0 ) induced by a morphism s0 → s in U e Φ(V) : F(V) → rU ◦ sU (F)(V) = Γ(Ugpt , sU (F)V ). One can also verify that this defines ef´et is a morphism of sheaves Φ : F → rU ◦ sU (F). We show that Φ(V) for V ∈ Ob U an isomorphism. Choose P as before (IV.6.2.2). Then for each s and sufficiently large L ∈ Ss , UL ×Ue V is isomorphic to the disjoint union of a finite number of copies of UL and hence we have a canonical isomorphism ∼ =

F(UL ×Ue V)∆L −→ Map∆L (HomUe (UL , V), F(UL )); x 7→ (f 7→ F((idUL , f ))(x)), Q where ∆L = Gal(L/κ(sg )). By taking s∈P limL∈S and using (IV.6.2.1), we obtain −→ s Y Y F(V) ∼ lim F(UL ×Ue V)∆L ∼ Map∆s (HomUe (sur , V), sU (F)(s)). = = −→ s∈P L∈Ss

s∈P

We see that this coincides with Φ(V) via the description (IV.6.2.2) of rU ◦ sU (F)(V).

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IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

e )∧,cont and s ∈ U egpt , we have canonical isomorphisms For G ∈ (U gpt ∼ =

sU ◦ rU (G)(s) = lim rU (G)(UL ) −→ lim Map∆s (HomUe (sur , UL ), G(s)) −→ −→ L∈Ss

L∈Ss

∼ =

∼ =

−→ lim Map∆L (HomUe (UL , UL ), G(s)Gal(κ(s)/L) ) −→ lim G(s)Gal(κ(s)/L) = G(s), −→ −→ L∈Ss

L∈Ss

where the second (resp. fourth) isomorphism is defined by evaluating at s (resp. idUL ) and the last equality is a consequence of the continuity of the action of ∆s on G(s). These isomorphisms are obviously functorial in G. One can verify the compatibility with the morphisms sU ◦ rU (G)(s) → sU ◦ rU (G)(s0 ) and G(s) → G(s0 ) induced by a morphism s0 → s simply going back to the definition of the functors sU and rU .  Let X be an fs log scheme over Σ satisfying the following conditions: The underlying scheme is normal, the morphisms of schemes underlying X → Σ is flat, separated, and of finite type, and its special fiber is reduced. Let C be a full subcategory of Xétaff such that any object of Xétaff admits a strict étale covering by objects of C. We describe sheaves on (P/C)f´et in terms of presheaves on the category Cgpt defined as follows. An object of Cgpt is a pair (U, s) of U ∈ Ob C and s ∈ Ob (UK,triv )gpt . A morphism (f, h) : (U 0 , s0 ) → (U, s) is a pair of a morphism f : U 0 → U in C and a morphism h : s0ur → sur compatible with the 0 morphism UK,triv → UK,triv induced by f . A morphism (f, h) : (U 0 , s0 ) → (U, s) induces a homomorphism h∗ : κ(s)ur → κ(s0 )ur and then a compatible system of morphisms hL : Uh0 ∗ (L)·κ(s0g ) → UL for L ∈ Ss .

∧,cont ∧ Let Cgpt be the full subcategory of Cgpt consisting of G such that for each (U, s) ∈ Ob Cgpt , the action of ∆(U,s) := Gal(κ(s)/κ(sg ))(∼ = AutCgpt ((U, s))◦ ) on G(U, s) is continuous. Then one can define a functor ∧,cont sC : (P/C)∼ f´ et −→ (Cgpt )

by sC (F)(U, s) = lim F(U, UL ) (= sU (F|(Utriv,K )f´et )(s)); −→ L∈Ss

0

0

for a morphism (f, h) : (U , s ) → (U, s), the compatible system of morphisms hL (L ∈ Ss ) induces a morphism sC (F)(U, s) → sC (F)(U 0 , s0 ). ∧,cont For G ∈ Ob (Cgpt ) and (U, V) ∈ Ob (P/C), we define the presheaf G(U,V) on (C/U )gpt by G(U,V) (U 0 , s) = Map(Hom UK,triv (sur , V), G(U 0 , s));

for a morphism (f, h) : (U 00 , s0 ) → (U 0 , s) in (C/U )gpt , we define the map G(U,V) (U 0 , s) → G(U,V) (U 00 , s0 ) by the composition with G(f, h) : G(U 0 , s) → G(U 00 , s0 ) and the inverse of the bijection − ◦ h : HomUK,triv (sur , V) → HomUK,triv (s0ur , V). For a morphism f = (f, g) : (U 0 , V 0 ) → (U, V) in P/C, let f ∗ (G(U,V) ) denote the composition of G(U,V) with the functor (C/U 0 )gpt → (C/U )gpt induced by f . Then we obtain a morphism Gf : f ∗ (G(U,V) ) → G(U 0 ,V 0 ) by the composition with g ◦ − : Hom U 0 (sur , V 0 ) → Hom UK,triv (sur , V). Now K,triv one can define a presheaf rC (G) on (P/C)f´et by rC (G)(U, V) := Γ((C/U )gpt , G(U,V) ),

rC (G)(f) := Γ((C/U 0 )gpt , Gf ) ◦ f ∗ : Γ((C/U )gpt , G(U,V) ) −→ Γ((C/U 0 )gpt , f ∗ (G(U,V) ))

−→ Γ((C/U 0 )gpt , G(U 0 ,V 0 ) ).

∧,cont Lemma IV.6.2.5. Let G ∈ Ob Cgpt .

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

415

(1) For (U, V) ∈ Ob (P/C), the following restriction map is bijective.

rC (G)(U, V) = G(U,V) ((C/U )gpt ) → G(U,V) ((Utriv,K )gpt ) = rU (G|(UK,triv )gpt ).

(2) The presheaf rC (G) is a sheaf on (P/C)f´et . Proof. (2) follows from (1) and Lemma IV.6.2.3. Let us prove (1). The injectivity follows from the fact that for any (U 0 , s0 ) ∈ Ob (C/U )gpt , there exists s ∈ (UK,triv )gpt and a morphism (U 0 , s0 ) → (U, s) in (C/U )gpt . For any two morphisms h, h1 : (U 0 , s0 ) → ∼ =

ur (U, s), (U, s1 ), there exists a unique isomorphism σ : sur such that h = σ ◦ h1 . This 1 →s implies the surjectivity. 

By Lemma IV.6.2.5, we obtain a functor ∧,cont rC : Cgpt → (P/C)∼ f´ et .

Proposition IV.6.2.6. The functors rC and sC defined above are equivalences of categories which are canonically quasi-inverse of each other. Proof. We prove that there exist canonical and functorial isomorphisms rC sC (F) ∼ = ∼ G for F ∈ Ob (P/C)∼ and G ∈ Ob (C ∧,cont ). For U ∈ Ob C and F and sC rC (G) = gpt f´ et e =U U , we have canonical isomorphisms K,triv

(rC sC (F))|Uef´et = rU (sC (F)|Uegp ) ∼ = rU sU (F|Uef´et ) ∼ = F|Uef´et

by the definition of sC , Lemma IV.6.2.5 (1), and Proposition IV.6.2.4. For (f : U 0 → ef´et , any morphism u : sur → V over U e factors through U, s) ∈ Ob (C/U )gpt and V ∈ Ob U 0 a morphism uL : UL → V for a sufficiently large L ∈ Ss , and the morphism F(U, V) → limL∈S F(U 0 , UL0 ) =: F(U 0 , s) induced by uL depends only on u, which we denote by −→ s F(f, u). Then, for (U, V) ∈ Ob (P/C), the above isomorphism ∼ =

F(U, V) −→ rC sC (F)(U, V) =

(f :

lim ←−

U 0 →U,s)∈Ob (C/U )gpt

Map(HomUe (sur , V), F(U 0 , s))

is given by x 7→ (u 7→ F(f, u)(x)). For a morphism (f, g) : (U 0 , V 0 ) → (U, V) in (P/C)f´et , a 00 morphism f 0 : U 00 → U 0 in C, s ∈ Ob (UK,triv )gpt , and a morphism h : s → V 0 compatible with f 0 , we have F(f 0 , h) ◦ F(f, g) = F(f ◦ f 0 , g ◦ h). This implies that the above isomorphism is functorial in (U, V) and defines the desired isomorphism. ∧,cont Let G ∈ Ob (Cgpt ). By the proof of Proposition IV.6.2.4 and Lemma IV.6.2.5, we have a canonical isomorphism ∼ = Map∆s (HomUe (sur , UL ), G(U, s)) −→ G(U, s) sC rC (G)(U, s) ∼ = lim −→ L∈Ss

defined by the evaluation at the canonical morphism sur → UL . Via the first isomorphism, the restriction map by (f, h) : (U 0 , s0 ) → (U, s) of the source is given by the composition with the morphisms HomUe 0 (s0ur , Uh0 ∗ (L)κ(s0g ) ) → HomUe (sur , UL ) and G(U, s) → G(U 0 , s0 ) induced by (f, h), and the first morphism preserves the canonical morphisms. Hence the above isomorphism is functorial in (U, s) and gives the desired isomorphism.  Definition IV.6.2.7. Let X be an fs log scheme over Σ satisfying the following conditions: The underlying scheme is normal, the morphism of schemes underlying X → Σ is flat, separated, and of finite type, and its special fiber is reduced. Let C be a full subcategory of Xétaff such that any object of Xétaff admits a strict étale covering by objects of C. v (1) We define OCgpt and OCgpt ,m (m ∈ N) to be the ring objects sC (OP/C ) and v ∧,cont sC (OP/C,m ) of (Cgpt ) .

416

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

(2) For m ∈ N, we define Mod(Cgpt , OCgpt ,m ) to be the category of OCgpt ,m -modules in (Cgpt )∧,cont . We define Mod(Cgpt , OCgpt ,• ) to be the category of inverse systems of OCgpt ,m -modules for m ∈ N. (3) For m ∈ N, we define Modcocart (Cgpt , OCgpt ,m ) to be the full subcategory of Mod(Cgpt , OCgpt ,m ) consisting of G such that for every morphism (f, h) : (U 0 , s0 ) → (U, s) in Cgpt , the natural homomorphism OCgpt ,m (U 0 , s0 ) ⊗OCgpt ,m (U,s) G(U, s) −→ G(U 0 , s0 )

is an isomorphism. We define Modcocart (Cgpt , OCgpt ,• ) to be the full subcategory of Mod(Cgpt , OCgpt,• ) consisting of (Gm )m∈N with Gm an object of Modcart (Cgpt , OCgpt ,m ) for every m ∈ N. v (4) For m ∈ N, we define Modcocart ((P/C)f´et , OP/C,m ) to be the full subcategory v of Mod((P/C)f´et , OP/C,m ) consisting of F such that rC (F) belongs to the category v Modcocart (Cgpt , OCgpt ,m ). We define Modcocart ((P/C)f´et , OP/C,• ) to be the full subcatv egory of Mod((P/C)f´et , OP/C,• ) consisting of (Fm )m∈N such that Fm is an object of v Modcocart ((P/C)f´et , OP/C,m ) for every m ∈ N. We have OCgpt ,m = OCgpt /pm OCgpt and OCgpt (U, s) is the integral closure of AU in κ(s)ur . Proposition IV.6.2.8. Let X and C be as in Definition IV.6.2.7 and assume that X ∈ Ob C, X is connected, X ×Spec(Zp ) Spec(Z/pZ) is connected (resp. nonempty) in the cases II and III (resp. the case I), and XK,triv is nonempty. Let t ∈ Ob (XK,triv )gpt , and let Repcont (∆t , OCgpt ,m (X, t)) denote the category of OCgpt ,m (X, t)-modules with the discrete topology endowed with a continuous semi-linear action of Gt . Then the functor evt : Modcocart (Cgpt , OCgpt ,m ) −→ Repcont (∆t , OCgpt ,m (X, t)); G 7→ G(X, t) is an equivalence of categories. Proof. We write O for OCgpt ,m to simplify the notation. For any (U, s) ∈ Ob Cgpt , there exists a morphism (U, s) → (X, t) because XK,triv is connected by Lemma IV.5.2.1. Hence the functor evt is faithful. Let G1 and G2 be two objects of Modcocart (Cgpt , O) and let ϕ : G1 (X, t) → G2 (X, t) be an O(X, t)-linear ∆t -equivariant homomorphism. Then, for each (U, s) ∈ Ob Cgpt , choosing a morphism (f, h) : (U, s) → (X, t), one obtains an O(U, s)-linear homomorphism ψ(U, s) : G1 (U, s) → G2 (U, s) by the composition of ∼ =

id⊗ϕ

∼ =

G1 (U, s) ←−−−−− O(U, s) ⊗O(X,t) G1 (X, t) −−−→ O(U, s) ⊗O(X,t) G2 (X, t) −−−−−→ G2 (U, s). G1 (f,h)

G2 (f,h)

0

Let (f, h ) : (U, s) → (X, t) be another morphism in Cgpt . (Note that f is unchanged because X is the final object of C.) Then there exists a unique σ ∈ ∆t such that h0 = σ ◦ h and we have Gi (f, h0 ) = Gi (f, h) ◦ σ : Gi (X, t) → Gi (U, s) (i = 1, 2). Hence, the ∆t -equivariance of ϕ implies that ψ(U, s) is independent of the choice of (f, h). Now it is straightforward to verify that ψ(U, s) defines a morphism ψ : G1 → G2 and ψ(X, t) = ϕ. Thus we see that evt is full. It remains to prove that evt is essentially surjective. Let G ∈ Ob Repcont (∆t , O(X, t)). For (U, s) ∈ Ob Cgpt , let I(U,s) denote the set of morphisms h : sur → tur compatible with the structure morphism f : U → X. The set I(U,s) is a ∆t -torsor under the right action h 7→ σ ◦ h. We define the O(U, s)-module G(U, s) to be   Y ∗ (xh ) ∈ h (G) (id ⊗ σ)(xσ◦h ) = xh for all σ ∈ ∆t , h∈I(U,s)

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

417

where h∗ (G) = O(U, s) O(f,h) ⊗O(X,t) G. For a morphism (u, w) : (U 0 , s0 ) → (U, s) in Cgpt , one can define a morphism G(u, w) : G(U, s) → G(U 0 , s0 ) by sending (xh ) to (yh0 ) defined by yh◦w = (w∗ ⊗ id)(xh ) (h ∈ I(U,s) ). (Note that I(U,s) → I(U 0 ,s0 ) ; h 7→ h ◦ w is an isomorphism of ∆t -torsors.) It is straightforward to verify that G(id) = id and G((u, w) ◦ (u0 , w0 )) = G(u0 , w0 ) ◦ G(u, w). For (U, s) ∈ Ob Cgpt and a morphism (f, h) : (U, s) → (X, t), we have a continuous homomorphism rh : ∆s → ∆t defined by rh (τ ) ◦ h = h ◦ τ ∼ = and the projection to the h-component induces an O(U, s)-linear isomorphism G(U, s) → ∗ h (G), through which the action of ∆s on G(U, s) is transferred to the action of ∆s on h∗ (G) defined by τ ⊗ rh (τ ) (τ ∈ ∆s ). The latter action is continuous. Thus we obtain an ∼ = object G of Modcocart (Cgpt , O) and a ∆t -equivariant isomorphism G(X, t) → id∗ (G) = G.  Let X and C be as in Definition IV.6.2.7, let X 0 → X be a strict étale morphism of fs 0 log schemes over Σ, and let C 0 be a full subcategory of Xétaff such that X 0 and C 0 satisfy 0 the same conditions as X and C and, for every (u : U → X ) ∈ Ob C 0 , f ◦ u : U → X is an 0 → Cgpt ; (u : U → X 0 , s) 7→ object of C. Then the composition with the functor fgpt : Cgpt (f ◦ u : U → X, s) induces the following functor. 0

C,C ∗ ∧,cont 0 fgpt : Cgpt −→ (Cgpt )∧,cont .

We see that the diagram ∧,cont

(Cgpt )

C,C fgpt

0∗

∼ rC0

∼ rC

 (P/C)∼ f´ et

0 / (Cgpt )∧,cont

C,C 0 ∗ fP,f´ et

 / (P/C 0 )∼

f´ et

is commutative up to canonical isomorphism by going back to the constructions of rC C,C 0 ∗ and rC 0 and using Lemma IV.6.2.5 (1). See (IV.6.1.13) for the definition of fP,f´ et . We 0

C,C ∗ ∗ abbreviate fgpt to fgpt if there is no risk of confusion in the following. ∗ v ∗ ∼ v 0 We have fgpt (OCgpt ,m ) ∼ = OCgpt ,m and fP,f´ et (OP/C,m ) = OP/C 0 ,m and the above diagram induces the following diagrams commutative up to canonical isomorphism.

(IV.6.2.9)

Modcocart (Cgpt , OCgpt ,m )

∗ fgpt

∼ rC0

∼ rC

 v Modcocart ((P/C)f´et , OP/C,m ) (IV.6.2.10)

Modcocart (Cgpt , OCgpt ,• )

0 / Modcocart (Cgpt 0 , OCgpt ,m )

∗ fP,f´ et

 v / Modcocart ((P/C 0 )f´et , OP/C 0 ,m ),

∗ fgpt

∼ rC0

∼ rC

 v ) Modcocart ((P/C)f´et , OP/C,•

0 / Modcocart (Cgpt 0 , OCgpt ,• )

∗ fP,f´ et

 v / Modcocart ((P/C 0 )f´et , OP/C 0 ,• ).

IV.6.3. Sheaves of OP/C -modules in the semi-stable reduction case. In this subsection, we study sheaves of OP/C -modules when X has semi-stable reduction. Assume that the log structure of Σ is defined by the closed point, and let U be an affine fs log scheme over Σ satisfying Condition IV.5.3.1. We keep the notation in IV.6.1 and IV.6.2. Note that U satisfies the conditions in IV.6.2. We abbreviate AU and AU

418

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

e e e ). Choose and fix a chart e for U to A and A and write U K,triv . Let A denote Γ(U , OU as in Condition IV.5.3.1, let π denote the image of 1 ∈ N in Γ(Σ, MΣ ) ⊂ Γ(Σ, OΣ ) = V by the chart of MΣ , and let ti denote the image of 1i+1 ∈ Nd+1 by the chart of MU . Then we have an étale homomorphism V [T0 , . . . , Td ]/(T0 · · · Te−1 − π) → A; Ti 7→ ti and e = A ⊗V K[(T0 · · · Td )−1 ]. Choose a compatible system (πn ) of n!-th roots of π in V , A let Vn be V [πn ], and define An to be the cofiber product of Vn [T0,n , . . . , Td,n ]/(T0,n · · · Te−1,n − πn ) ←− V [T0 , . . . , Td ]/(T0 · · · Te−1 − π) −→ A,

where the left homomorphism is defined by Ti 7→ (Ti,n )n! . Then the ring An is normal and flat over Vn and An /πn An is reduced. We put An = A ⊗A An , which is the henselization of (An , pAn ) (resp. the p-adic completion of An ) in the case II (resp. III). In particular, An is noetherian and normal. The natural homomorphism An ⊗Vn V → An+1 ⊗Vn+1 V en = (An ⊗V K)[(T0 · · · Td )−1 ] and U en = Spec(A en ). We have U e =U e0 is injective. Put A n and ±1 en = A e⊗ A K[T ±1 ;1≤i≤d] K[Ti,n ; 1 ≤ i ≤ d]. i

en → U e is finite étale. If we put ∆n = Gal(K(Ti,n ; 1 ≤ i ≤ d)/K(Ti ; 1 ≤ i ≤ d)), Hence U en /U e and it induces an isomorphism tσ × 1 : tσ∈∆ we have a natural action of ∆n on U n ∼ = en . Hence, for any sheaf F on U ef´et and any V ∈ Ob (U ef´et ), we have an en → U en × e U U U isomorphism ∼ = en )∆n . F(V) −→ F(V ×Ue U

(IV.6.3.1)

By applying Lemma IV.5.2.1 to An /Vn , we see that An ⊗Vn V is a product of finite number of normal domains, which implies int e OU e (Un ) = An ⊗Vn V .

(IV.6.3.2)

en is bounded when n varies. Hence it is The number of the connected components of U en ) → π0 (U en ) is bijective for stable when n → ∞, i.e., there exists n0 ∈ N such that π0 (U 0 ±1 e is étale. every n ≥ n0 . In the case I, this follows from the fact that K[Ti ; 1 ≤ i ≤ d] → A e In the cases II and III, the number of the connected components of Un coincides with that of Spec(An /πn An ) = Spec(An /πn An ) by Lemma IV.5.2.1. Hence the claim follows from the facts that k[Ti ; 0 ≤ i ≤ d]/(T0 · · · Te−1 ) → A/πA is étale and that the number of irreducible components of the special fiber of k[Ti,n ; 0 ≤ i ≤ d]/(T0,n · · · Te−1,n ) is e for every n. egpt , there exists an extension A en → κ(s)ur of A e → κ(sg ) and its image For s ∈ U is independent on the choice of extension. Let κ(sn ) denote the field generated by the e of s → U e such that the natural image. Let P be a finite set of liftings v : s → limn U ←− n en ), sending v to the connected component containing its image in U en , map P → π0 (U is bijective for n ≥ n0 . For n ≥ n0 , each v ∈ P induces an open and closed immersion en . By taking the union of these immersions, we obtain an isomorphism Uκ(sn ) → U G ∼ = e (IV.6.3.3) Uκ(sn ) −→ U n ×U e Uκ(sg ) P

e in L (cf. IV.6.2). for n ≥ n0 . Recall that, for L ∈ Ss , UL denotes the integral closure of U We use the following theorem of Faltings. Theorem IV.6.3.4 (cf. [26] 2b). Let notation and assumption be as above. Then for any en )f´et , the homomorphism lim Oint (U en ) → lim Oint (U en × e V) n1 ∈ N and V ∈ Ob (U 1 Un1 −→n Ue −→n Ue is an almost étale covering (cf. Definition V.7.1).

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

419

egpt and Corollary IV.6.3.5. Let notation and assumption be as above, and let s ∈ Ob U ef´et . V ∈ Ob U (1) Let L ∈ Ss , and put κ(s∞ ) = ∪n κ(sn ) and Ln = κ(sn )L (n ∈ N ∪ {∞}). Then int int the homomorphism limn OU e Uκ(sn ) ) → limn OU e ULn ) is an almost étale e (V ×U e (V ×U −→ −→ Gal(L∞ /κ(s∞ ))-covering (cf. Definition V.12.2, Lemma V.12.3). int e et . Then the following homomorphism is (2) Let F be a sheaf of OU e -modules on Uf´ an almost isomorphism int (V ×Ue UL ) −→ lim F(V ×Ue UL ). lim F(V ×Ue Uκ(sn ) ) ⊗lim Oint lim OU (V×U e Uκ(sn ) ) − −→ −→ → e − →n Ue n

L∈Ss

L∈Ss

Proof. (1) Choose n1 ∈ N such that n1 ≥ n0 and L∞ ∼ = Ln1 ⊗κ(sn1 ) κ(s∞ ). Then for any n ∈ N, n ≥ n1 , we have Uκ(sn ) ×Uκ(n1 ) ULn1 ∼ = ULn . Choose a lifting v : s → e of s → U e . Then it induces a compatible system of open and closed immersions limn U ←− n en × e Uκ(s ) by en and hence a compatible system of isomorphisms Uκ(s ) ∼ Uκ(s ) → U =U n

n

Un1

n1

int Oint (ULn ) is an (IV.6.3.3). By Theorem IV.6.3.4, we see that limn OU e (Uκ(sn ) ) → lim −→n Ue −→ almost étale Gal(L∞ /κ(s∞ ))-covering (cf. Proposition V.12.9). Hence its base change by int limn OU Oint (V ×Ue Uκ(sn ) ) is an almost étale Gal(L∞ /κ(s∞ ))-covering e (Uκ(sn ) ) → lim −→ −→n Ue and is almost isomorphic to the homomorphism in the claim. (2) Let L ∈ Ss and let Ln be as in (1). Then by (IV.6.2.1), we have an isomorphism ∼ =

lim F(V ×Ue Uκ(sn ) ) −→ (lim F(V ×Ue ULn ))Gal(L∞ /κ(s∞ )) . −→ −→ n n

By (1) and almost Galois descent (cf. Proposition V.12.5), we obtain an almost isomorphism ≈

int lim F(V ×Ue Uκ(sn ) ) ⊗lim Oint lim OU e ULn ) −→ lim F(V ×U e ULn ). (V×U e (V ×U e Uκ(sn ) ) − e −→ → −→ U n − → n n n

Varying L, we obtain the claim.



∧,cont v v Recall that OCgpt denotes the ring object sC (OP/C ) of Cgpt corresponding to OP/C (cf. Definition IV.6.2.7).

Proposition IV.6.3.6. Let X be an fs log scheme satisfying Condition IV.5.3.1 strict étale locally and the underlying scheme of X is separated. Let C be a full subcategory of Xétaff satisfying the following two conditions: (a) Every object of Xétaff admits a strict étale covering by objects of C. (b) Every object U of C satisfies Condition IV.5.3.1. v Let F be an object of Modcocart ((P/C)f´et , OP/C,m ) (cf. Definition IV.6.2.7). Then F satisfies the condition (b) of Proposition IV.6.1.23. Lemma IV.6.3.7. Let X and C be as in Proposition IV.6.3.6 and let F be an object of v Modcocart ((P/C)f´et , OP/C,m ). Let (f, g) : (U 0 , V 0 ) → (U, V) be a horizontal morphism in e and U e 0 (resp. A and A0 ) denote the schemes U and U 0 (resp. the P/C, and let U K,triv

K,triv

V -algebras AU and AU 0 ) associated to U and U 0 (cf. the beginning of IV.6.1). Choose a chart of U → Σ as in Condition IV.5.3.1 and define a compatible system of finite étale en → U e (n ∈ N) as in the beginning of this subsection. Put U en0 = U en × e U e 0. morphisms U U Then the following natural homomorphism is an almost isomorphism. en ) ⊗A A0 −→ lim F(U 0 , V 0 × e 0 U en0 ). lim F(U, V ×Ue U U −→ −→ n

n

420

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

egpt (resp. s0 ∈ Ob U e 0 ), we define κ(sn ) (resp. κ(s0 )) as Proof. For s ∈ Ob U gpt n before Theorem IV.6.3.4. Let F(U, V, s) (resp. F(U, V, s∞ )) denote the direct limit v limL∈S F(U, V ×Ue UL ) (resp. limn F(U, V ×Ue Uκ(sn ) )) and define similarly for OP/C and −→ −→ s ∼ = for (U 0 , V 0 ). Since we have an isomorphism t(f, idUL ) : tf ∈HomUe (UL ,V) UL → V ×Ue UL for a sufficiently large L ∈ Ss , we have a canonical isomorphism M F(U, V, s) ∼ sC F(U, s). = ur HomU e (s ,V)

v v We have the same type of descriptions of OP/C (U, V, s), F(U 0 , V 0 , s0 ) and OP/C (U 0 , V 0 , s0 ). Choose a morphism (f, h) : (U 0 , s0 ) → (U, s) over f . Then it induces homomorphisms v v (U, V, s) → OP/C (U 0 , V 0 , s0 ) and the above descripF(U, V, s) → F(U 0 , V 0 , s0 ) and OP/C 0 0 tions are compatible via the map HomUe 0 (s , V ) → HomUe (s, V) induced by f , g and h (cf. the definition of the functor sC in IV.6.2). Hence the condition on F implies that the homomorphism ∼ =

v 0 0 0 0 0 0 v F(U, V, s) ⊗OP/C (U,V,s) OP/C (U , V , s ) −→ F(U , V , s )

is an isomorphism. By Corollary IV.6.3.5 (2), we obtain an almost isomorphism v 0 0 0 v F(U, V, s∞ ) ⊗OP/C (U,V,s∞ ) OP/C (U , V , s ) ≈

v 0 0 0 v −→ F(U 0 , V 0 , s0∞ ) ⊗OP/C (U 0 ,V 0 ,s0∞ ) OP/C (U , V , s ). v v Since OP/C (U 0 , V 0 , s0∞ ) → OP/C (U 0 , V 0 , s0 ) is almost faithfully flat by Corollary IV.6.3.5 v (1) (cf. Proposition V.8.11 and Definition V.12.2), we may replace OP/C (U 0 , V 0 , s0 ) by 0 v 0 0 0 OP/C (U , V , s∞ ) in the above almost isomorphism. Varying s and s and using (IV.6.3.3), we obtain an almost isomorphism: ≈

v 0 0 0 0 v F(U, V∞ ) ⊗OP/C (U,V∞ ) OP/C (U , V∞ ) −→ F(U , V∞ ),

en ) and similarly for the other three terms. where F(U, V∞ ) denotes limn F(U, V ×Ue U −→ On the other hand, since (f, g) is horizontal, we see that the natural morphism (V ×Ue = 0 en ) × e U e0 ∼ e0 U e 0 Un is an isomorphism, which implies, by Theorem IV.6.3.4, that Un n ← V ×U the homomorphism v v e 0 ) −→ Ov (U 0 , V 0 ) OP/C (U, V∞ ) ⊗lim Ov (U,Uen ) lim OP/C (U 0 , U n ∞ P/C −→ − →n P/C n

is an almost isomorphism (cf. Proposition V.7.11). Furthermore, from (IV.6.3.2) for U and U 0 , we obtain an isomorphism ∼ = v v e 0 ). en ) ⊗A A0 −→ (U 0 , U lim OP/C (U, U lim OP/C n −→ −→ n

n

Combining with the above two almost isomorphisms, we obtain the lemma.



Proof of Proposition IV.6.3.6. Let U = (U, V), Uα = (Uα , Vα ) (α ∈ A), and e denote U Uαβ;γ = (Uαβ;γ , Vαβ;γ )γ∈Γαβ be as in Proposition IV.6.1.23 (b). Let U K,triv eα and U eαβ;γ similarly. Choose a chart as in Condition associated to U and define U en → U e . Put U eα,n = U en × e U eα and U eαβ;γ,n := U en × e U eαβ;γ . By IV.5.3.1 and define U U U (IV.6.3.1), it suffices to prove that the following homomorphism is an almost isomorphism   Y Y Y F(U, V∞ ) −→ Ker  F(Uα , Vα,∞ ) ⇒ F(Uαβ;γ , Vαβ;γ,∞ ) , α∈A

(α,β)∈A2 γ∈Γαβ

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

421

e?,n ) for ? = ∅, α, αβ; γ. Let A, Aα , Aαβ , and where F(U? , V?,∞ ) = limn F(U? , V? ×Ue? U −→ Aαβ;γ denote AU , AUQ , AUα ×U Uβ , and AUαβ;γ . ThenQnoting A/pm A = AU /pm AU , etc., α m we see that A/p → α∈A Aα /pm and Aαβ /pm → γ∈Γαβ Aαβ;γ /pm are faithfully flat and Aαβ /pm ∼ = Aα /pm ⊗Aα /pm Aβ /pm . Hence we obtain the claim from Lemma IV.6.3.7 applied to Uα → U and Uαβ;γ → U.  Corollary IV.6.3.8. Let X and C be the same as in Proposition IV.6.3.6. Then the v natural homomorphism OP/C,m → OP/C,m is an almost isomorphism. Proposition IV.6.3.9. Let X and C be as in Proposition IV.6.3.6. Then the functor ∗ v (vP/C )Q : Modcocart ((P/C)f´et , OP/C,• )Q → Mod((P/C)´et-f´et , OP/C,• )Q

is fully faithful. v Proof. Let F• and G• be objects of Modcocart ((P/C)f´et , OP/C,• ). It suffices to prove that the homomorphism ∗ ∗ v HomOP/C,• (F• , G• ) −→ HomOP/C,• (vP/C (F• ), vP/C (G• ))

is an almost isomorphism. By Proposition IV.6.3.6 and Corollary IV.6.1.24, the adjunc∗ tion morphism G• → vP/C∗ vP/C G• is an almost isomorphism. This implies the claim.  IV.6.4. Higgs crystals and modules on Faltings sites. Let V , k, K, K, V , Σ, b Σ, Σ, and D(Σ) be the same as in the beginning of IV.5.2. We also take the base ring R of our theory of Higgs crystals as in loc. cit. Throughout this subsection, let X denote an fs log scheme over Σ satisfying the following conditions: the underlying scheme is normal, the morphism of schemes underlying X → Σ is flat, separated, and of finite type, its special fiber is reduced, and the log structure of X is trivial at every codimension zero point. In the case I (cf. IV.5.1 and IV.6.1), we further assume that the log structure of Σ is defined by the closed point and b denote the p-adic completion of X satisfies Condition IV.5.3.1 strict étale locally. Let X b X and let X1 denote X ×Σ b Σ as in IV.5.2. Definition IV.6.4.1. Let X be as above. (1) We define CX to be the full subcategory of Xétaff consisting of U satisfying the following conditions: (a) The underlying scheme of U is affine. (b) There exists a chart α : PU → MU such that P is a finitely generated and × saturated monoid, and the morphism P → Γ(U, MU )/Γ(U, OU ) induced by α is an isomorphism (cf. [73] Lemma 1.3.2). (c) In the case I, U satisfies Condition IV.5.3.1. (2) For a covering sieve R of X with respect to the site Xétaff , we define CR to be the full subcategory of CX consisting of u : U → X such that u ∈ R(U ). By [73] Lemma 1.3.3, any object of Xétaff admits a strict étale covering by objects of CR for any R. Let (h : U → X, s) be an object of (CX )gpt and let AU be as in the beginning of IV.6.1. If p is not invertible on the connected component of Spec(AU ⊗V V ) (cf. Lemma IV.5.2.1) containing the image sg of s, which always holds in the cases II and III, then (U, s) satisfies the conditions after Lemma IV.5.2.1 (cf. Theorem IV.5.3.6 (2)). In this case, we write (D(U, s), z(U,s) ) for the object (D(U ), zU ) of (U1 /D(Σ))∞ HIGGS constructed in IV.5.2. If p is invertible on the connected component of Spec(AU ⊗V V ) containing sg , then we define (D(U, s), z(U,s) ) to be ∅. By the functoriality mentioned in the last

422

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

paragraph of IV.5.1, we see that the construction of (D(U, s), z(U,s) ) is functorial in (U, s) and obtain a functor DX : (CX )gpt −→ (X1 /D(Σ))∞ HIGGS by associating (D(U, s), h1 ◦ z(U,s) ) to (h : U → X, s). Definition IV.6.4.2. Let R be a covering sieve of X with respect to the site Xétaff . We say that F ∈ Ob (HCZp (X1 /D(Σ))) (cf. Definition IV.3.3.2) is R-finite if F(T,z) ∈ Ob LPM(OT1 ) for every (T, z) ∈ Ob (X1 /D(Σ))∞ HIGGS (cf. IV.3.2) such that T1 is affine and z factors through u1 : U1 → X1 for some u : U → X ∈ Ob Xétaff satisfying u ∈ R(U ). We write HCZp ,R-fin (X1 /D(Σ)) for the full subcategory of HCZp (X1 /D(Σ)) consisting of R-finite objects. Remark IV.6.4.3. Since X1 ×Σ Spec(k) = X ×Spec(V ) Spec(k), every object of (X1 )´et admits a strict étale covering by objects coming from Xétaff by base change. Hence, for every F ∈ Ob (HCZp (X1 /D(Σ))), there exists a covering sieve R of X such that F is R-finite. Let R be a covering sieve of X with respect to the site Xétaff , and let F be an R-finite Higgs crystal on X1 /D(Σ). Then, for each (U, s) ∈ Ob (CR )gpt , the evaluation of F on R DX (U, s), denoted TX,gpt (F)(U, s), is an object of LPM(OCR,gpt (U, s)) (cf. Proposition R R IV.3.2.7 (1)). Put TX,gpt,m (F)(U, s) := TX,gpt (F)(U, s) ⊗Z Z/pm Z for m ∈ N. Then R the left action of ∆s on TX,gpt,m (F)(U, s) induced by the right action of ∆s on (U, s) is continuous by Corollary IV.5.2.13. The morphism (U 0 , s0 ) → (U, s) in CR induces an isomorphism ∼ =

R R OCR,gpt ,m (U 0 , s0 ) ⊗OCR,gpt ,m (U,s) TX,gpt,m (F)(U, s) −→ TX,gpt,m (F)(U 0 , s0 )

compatible with m (cf. Lemma IV.3.2.8 and Definition IV.3.3.2). Thus we obtain an obR R ject TX,gpt,• (F) = (TX,gpt,m (F))m of Modcocart (CR,gpt , OR,gpt,• ) (cf. Definition IV.6.2.7 (3)). This construction is obviously functorial in F and we obtain a functor (IV.6.4.4)

R TX,gpt,• : HCZp ,R-fin (X1 /D(Σ)) −→ Modcocart (CR,gpt , OCR,gpt ,• ).

Composing with the equivalence of categories rCR (cf. Proposition IV.6.2.6), the functor ∗ v vP/C : Modcocart ((P/CR )f´et , OP/C ) −→ Mod((P/CR )´et-f´et , OP/CR ,• ) R R ,•

and the equivalence of categories ∼

R ιX,C et-f´ et , OP/CR ,• ) −→ Mod((P/X)´ et-f´ et , OP/X,• ) P,´ et-f´ et∗ : Mod((P/CR )´

(cf. (IV.6.1.18)), we obtain a functor (IV.6.4.5)

R TX,• : HCZp ,R-fin (X1 /D(Σ)) −→ Mod((P/X)´et-f´et , OP/X,• ).

Lemma IV.6.4.6. Let f : X 0 → X be a strict étale morphism of fs log schemes over Σ such that X 0 satisfies the same conditions as X. Let R (resp. R0 ) be a covering 0 ) such that for every sieve of X (resp. X 0 ) with respect to the site Xétaff (resp. Xétaff 0 0 0 0 0 0 u : U → X ∈ R (U ), the composition f ◦ u is contained in R(U 0 ). Then the following diagram is commutative up to canonical isomorphism. HCZp ,R-fin (X1 /D(Σ))

R TX,•

∗ f1,HIGGS

 HCZp ,R0 -fin (X10 /D(Σ))

/ Mod((P/X)´et-f´et , OP/X,• ) ∗ fP,´ et-f´ et

R0 TX 0 ,•

 / Mod((P/X 0 )´et-f´et , OP/X 0 ,• ).

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

423

∗ Proof. By assumption, u0 ∈ CR0 implies f ◦ u0 ∈ CR . Define fgpt and fgpt as before (IV.6.2.9). Then the diagram

CR,gpt O

DX

/ (X1 /D(Σ))∞ O HIGGS

DX 0

/ (X 0 /D(Σ))∞ 1 HIGGS

fgpt

CR0 ,gpt

is commutative, which implies that the diagram HCZp (X1 /D(Σ)) ∗ f1,HIGGS



HCZp (X10 /D(Σ))

R TX,gpt,•

R0 TX 0 ,gpt,•

/ Modcocart (CR,gpt , OCR,gpt ,• ) 

∗ fgpt

/ Modcocart (CR0 ,gpt , OR0 ,gpt,• )

is commutative. Hence the claim follows from (IV.6.2.10) and (IV.6.1.17).



Corollary IV.6.4.7. For two covering sieves R0 ⊂ R of X with respect to the site Xétaff , we have a canonical isomorphism of functors: 0

R TX,• |HCZ

p ,R-fin (X1 /D(Σ))

R ∼ : HCZp ,R-fin (X1 /D(Σ)) −→ Mod((P/X)´et-f´et , OP/X,• ). = TX,•

By Corollary IV.6.4.7 and Remark IV.6.4.3, we obtain a functor TX,• : HCZp (X1 /D(Σ)) −→ Mod((P/X)´et-f´et , OP/X,• ).

(IV.6.4.8)

Theorem IV.6.4.9. Assume that the log structure of Σ is defined by the closed point and X satisfies Condition IV.5.3.1 strict étale locally. Then the functor (TX,• )Q : HCZp (X1 /D(Σ))Q → Mod((P/X)´et-f´et , OP/X,• )Q is fully faithful. Proof. First let us prove that the functor restricted to HCZp ,X-fin (X1 /D(Σ)) is fully faithful when X is an object of CX and satisfies the conditions on U in the beginning of IV.5.3. By Propositions IV.6.3.9 and IV.6.2.8, it suffices to prove that the functor X (TX,gpt,• )(−)(X, t)Q : HCZp ,X-fin (X1 /D(Σ))Q → Repcont (∆t , OCX ,gpt,• (X, t))Q

is fully faithful for t ∈ Ob (XK,triv )gpt . This follows from Lemma IV.5.2.14, Proposition IV.3.5.1, and Theorem IV.5.3.3. Let R be a covering sieve of X with respect to the site Xétaff . Choose a strict étale covering (Xα → X)α∈A and a strict étale covering (Xαβ;γ → Xαβ )γ∈Γαβ of Xαβ := Xα ×X Xβ such that A and Γαβ are finite sets, and Xα and Xαβ;γ are objects of CR and satisfy the conditions on X in the previous paragraph. Let Fi (i = 1, 2) be objects of HCZp ,R-fin (X/D(Σ)) and let Fi,α (resp. Fi,αβ;γ ) be their pull-backs on Xα (resp. Xαβ;γ ), which are Xα (resp. Xαβ;γ )-finite. We may apply the above special case of the theorem to Fi,α and Fi,αβ;γ . Hence, by Corollary IV.3.1.8, Proposition IV.6.1.19, and Lemma IV.6.4.6, we see that the homomorphism Hom(F1 , F2 )Q → Hom((TX,• )Q (F1 ), (TX,• )Q (F2 )) is an isomorphism. 

424

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

IV.6.5. Projections to étale sites. Let X be an fs log scheme over Σ whose underlying scheme is separated and of finite type over V . In this subsection, we give a way to describe the higher direct images by a natural projection from the Faltings site (P/X)´et-f´et to the étale site Xétaff in terms of the cohomology on (UK,triv )f´et for each U ∈ Xétaff (cf. Proposition IV.6.5.22). Let Xtraff denote the category Xétaff endowed with the topology defined by the identity covering. Every presheaf is a sheaf on Xtraff , i.e., (Xtraff )∼ = (Xétaff )∧ . The identity functor Xtraff → Xétaff is continuous and the identity functor Xétaff → Xtraff is cocontinuous. By [2] III Proposition 2.5, these two functors induce a morphism of topos (IV.6.5.1)

∼ ∼ ∧ vX : Xétaff → Xtraff (= Xétaff ).

∗ The functor vX∗ is the inclusion and the functor vX is the sheafification. The functor Xétaff → (P/X)´et-f´et (resp. Xtraff → (P/X)f´et ); U 7→ (U, UK,triv ) is continuous because it preserves finite fiber products and the image of a strict étale covering is a horizontal strict étale covering. The functor (P/X)´et-f´et → Xétaff (resp. (P/X)f´et → Xtraff ); (U, V) → U is a left adjoint of the above functor. Hence it is cocontinuous and these two functors induce a morphism of topos (cf. [2] III Proposition 2.5)

(IV.6.5.2)

h ∼ ∼ πX,´ et : (P/X)´ et-f´ et −→ Xétaff

h ∧ (resp. πX : (P/X)∼ f´ et → Xétaff ).

Under the interpretation of the topology of (P/X)´et-f´et in terms of covanishing topology, the above functor coincides with the morphism σ in (VI.5.32.7) if X is affine. We see that the diagram of topos (IV.6.5.3)

(P/X)´e∼t-f´et

vP/X

h πX,´ et

 ∼ Xétaff

vX

/ (P/X)∼ f´ et 

h πX

/ X∧ étaff

is commutative up to canonical isomorphism by looking at the corresponding diagram of sites and continuous (or cocontinuous) morphisms. Similarly, if X is affine, then the functor XK,triv → (P/X)´et-f´et (resp. (P/X)f´et ) defined by V 7→ (XK,triv , V) is continuous and defines a morphism of sites because it preserves finite inverse limits and the image of a finite étale covering is a vertical finite étale covering. Thus we obtain a diagram of topos commutative up to canonical isomorphism (IV.6.5.4)

vP/X

/ (P/X)∼ (P/X)´e∼t-f´et f´ et OOO OOO v OOO πX v OO' πX,´ et  (XK,triv )∼ f´ et .

v Under the interpretation in terms of covanishing topology, the morphism πX,´ et coincides with the morphism β in (VI.5.32.2).

Lemma IV.6.5.5. Let X be as above. (1) The topos (P/X)´et-f´et is coherent and the topos (P/X)f´et is algebraic. If X is affine, then the topos (P/X)f´et is coherent. ∼ ∧ (2) The topos Xétaff is coherent and the topos Xétaff is algebraic. If X is affine, then ∧ the topos Xétaff and (XK,triv )f´et are coherent. h h (3) The morphisms of topos πX,´ et and πX are coherent. v v (4) If X is affine, then the morphisms of topos πX,´ et and πX are coherent.

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

425

Proof. For a site C, let ε denote the canonical functor C ∧ → C ∼ . (1) By Proposition IV.6.1.3 (resp. Since the topology of (P/X)f´et is defined by Covvf (−)), every object of the site (P/X)´et-f´et (resp. (P/X)f´et ) is quasi-compact (cf. [2] VI Définition 1.1). Since finite fiber products and finite products are representable in P/X, we see that ε(U) and ε(U) × ε(U0 )(= ε(U × U0 )) are coherent for any objects U and U0 of P/X by [2] VI Corollaire 2.1.1. Hence the topos (P/X)´et-f´et (resp. (P/X)f´et ) satisfies the condition [2] VI Proposition 2.2 (ii) bis defining algebraic topos. If X is affine, then the final object ε((X, XK,triv )) is coherent, and hence the topos in question is coherent. In the general case, there exists a strict étale covering (Uα → X)α∈A such that Uα is affine and A is a finite set. Letting Uα = (Uα , Uα,K,triv ) ∈ Ob (P/X), we see that the finite set (ε(Uα ))α∈A of objects of (P/X)´e∼t-f´et is a covering of the final object and the product ε(Uα ) × ε(Uβ ) is coherent as we have seen above. Hence the final object of (P/X)´e∼t-f´et is coherent by [2] VI Corollaire 1.17. (2) Every object of the site Xétaff (resp. (XK,triv )f´et , resp. Xtraff ) is quasi-compact. Since finite fiber products and finite products are representable, the same argument as (1) shows that the associated topos is algebraic and the coherence follows from the existence of finite family of strict étale covering of X by objects of the site (resp. XK,triv ∈ Ob (XK,triv )f´et , resp. X ∈ Ob Xétaff ). (3) (4) Since finite fiber products are representable and every object is quasi-compact in every relevant category, the claim follows from [2] VI Corollaire 3.3.  Let C be a full subcategory of Xétaff such that every object of Xétaff admits a strict étale covering by objects of C. Let C´et (resp. Ctr ) denote the category C endowed with the topology induced by that of Xétaff via the inclusion functor (resp. the trivial topology). By [2] III Théorème 4.1, the restriction functor induces an equivalence of categories (IV.6.5.6)

∼

∼ Xétaff −→ C´e∼t .

The identity functor Ctr → C´et is continuous and the identity functor C´et → Ctr is cocontinuous. By [2] III Proposition 2.5, these two functors induce a morphism of topos (IV.6.5.7)

vC : C´e∼t −→ C ∧ .

The functor C´et → (P/C)´et-f´et (resp. Ctr → (P/C)f´et ); U 7→ (U, UK,triv ) is continuous by (IV.6.1.8) (resp. obviously) and the functor (P/C)´et-f´et → C´et (resp. (P/C)f´et → Ctr ); (U, V) 7→ U is its left adjoint. Hence, by [2] III Proposition 2.5, the latter functor is cocontinuous and this adjoint pair of functors induces a morphism of topos (IV.6.5.8)

h ∼ ∼ πC,´ et : (P/C)´ et-f´ et −→ C´ et

∧ (resp. πCh : (P/C)∼ f´ et −→ C ).

We also have a canonical isomorphism of morphisms of topos h (IV.6.5.9) πCh ◦ vP/C ∼ = vC ◦ πC,´ et , which follows from the corresponding isomorphism for continuous (or cocontinuous) morphisms of sites. Since X1 ×Spf(V ) Spec(k) ∼ = X ×Spec(V ) Spec(k), we may regard sheaves on (X1 )´et as sheaves on X´et with supports in X ×Spec(V ) Spec(k). Put OX1 ,m = OX1 /pm OX1 . We define OC1 ,m to be the restriction of OX1 ,m on C´et . Let OC1 ,m also denote vC∗ OC1 ,m . Then, for U ∈ Ob C, we have a canonical homomorphism of V -algebras v Γ(U, OX ) ⊗V V = AU ⊗V V −→ Γ((U, UK,triv ), OP/C ) = AU ⊗V V

functorial in U (cf. Lemma IV.5.2.1 for the last equality). Since Γ(U, OC1 ,m ) = (AU ⊗V V )/pm , this induces h v OC1 ,m −→ πC∗ (OP/C,m )

426

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

and then v h ∗ h v ∼ h (OP/C,m ). −→ πC,´ OC1 ,m = vC∗ OC1 ,m −→ vC∗ πC∗ OP/C,m et∗ et∗ vP/C (OP/C,m ) = πC,´

Thus we obtain a commutative diagram of ringed topos ((P/C)´e∼t-f´et , OP/C,m )

(IV.6.5.10)

vP/C

v / ((P/C)∼ f´ et , OP/C,m )

h πC,´ et

 (C´e∼t , OC1 ,m )

h πC

 / (C ∧ , OC1 ,m ).

vC

∧,cont h via rC and sC (cf. PropoThe functors πC∗ and πCh∗ are described in terms of Cgpt ∧,cont h sition IV.6.2.6) as follows. We define a functor πC,gpt∗ : Cgpt −→ C ∧ by h (πC,gpt∗ G)(U ) = Γ((C/U )gpt , G|U ),

where G|U denotes the composition of G with the forgetful functor (C/U )gpt → Cgpt . For a morphism f : U 0 → U , G|U 0 coincides with the composition of G|U with the functor fgpt : (C/U 0 )gpt → (C/U )gpt induced by f . Therefore we obtain a map Γ((C/U )gpt , G|U ) → Γ((C/U 0 )gpt , G|U 0 ) by composition with fgpt . By the construction of rC , we have a canonical isomorphism h h πC∗ ◦ rC ∼ . = πC,gpt∗

(IV.6.5.11)

∧,cont h∗ We define a functor πC,gpt : C ∧ −→ Cgpt by h∗ (πC,gpt F)(U, s) = F(U ),

h∗ (πC,gpt F)(f, h) = F(f ).

Then we have a canonical isomorphism h∗ sC ◦ πCh∗ ∼ . = πC,gpt

(IV.6.5.12)

Let f : X 0 → X be a strict étale morphism of fs log schemes over Σ such that X 0 0 0 satisfies the same condition as X. Then the functor Xétaff → Xétaff (resp. Xtraff → 0 0 0 0 Xtraff ); (u : U → X) 7→ (f ◦ u : U → X) is continuous and cocontinuous and induces a 0∼ ∼ 0∧ ∧ morphism of topos fétaff : Xétaff → Xétaff (resp. ftraff : Xétaff → Xétaff ). We see that the following diagrams of topos are commutative up to canonical isomorphisms by looking at the corresponding diagrams of sites and cocontinuous morphisms. (IV.6.5.13) (P/X 0 )´e∼t-f´et h πX 0 ,´ et



0 (Xétaff )∼

fP,´et-f´et

fétaff

/ (P/X)∼ ´ et-f´ et 

h πX,´ et

∼ / Xétaff ,

(P/X 0 )∼ f´ et h πX 0



0 (Xétaff )∧

fP,f´et

ftraff

/ (P/X)∼ f´ et 

h πX

/ X∧ . étaff

We also have a canonical isomorphism of morphisms of topos (IV.6.5.14)

vX ◦ fétaff ∼ = ftraff ◦ vX 0 .

0 0 If X 0 is affine, then the functor Xétaff → Xétaff (resp. Xtraff → Xtraff ); U 7→ U ×X X 0 is 0 continuous and a right adjoint of the cocontinuous functor u 7→ f ◦ u0 above. Hence it defines a morphism of sites and induces the morphism of topos fétaff (resp. ftraff ) (cf. [2] III Proposition 2.5). We can also verify the commutativity of the diagram (IV.6.5.13) by looking at the corresponding diagrams of sites and continuous morphisms (cf. Proposition IV.6.1.7).

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

427

Similarly, if X and X 0 are affine, then we have the following diagrams of topos 0 → commutative up to canonical isomorphisms, where f denotes the morphism XK,triv XK,triv induced by f . (IV.6.5.15) (P/X 0 )´e∼t-f´et v πX 0 ,´ et



0 )∼ (XK,triv f´ et

fP,´et-f´et

ff´et

/ (P/X)∼ ´ et-f´ et 

fP,f´et

(P/X 0 )∼ f´ et

v πX,´ et

v πX 0

/ (XK,triv )∼ f´ et ,



/ (P/X)∼ f´ et 

/ (XK,triv )∼ f´ et .

ff´et

0 (XK,triv )∼ f´ et

v πX

Lemma IV.6.5.16. Let f : X 0 → X be as above and assume that X 0 is affine. Then the morphisms of topos fP,´et-f´et : (P/X 0 )´e∼t-f´et → (P/X)´e∼t-f´et and fP,f´et : (P/X 0 )∼ f´ et → (P/X)∼ are coherent. f´ et Proof. The same as the proof of Proposition IV.6.5.5 (3) using Proposition IV.6.1.7.  0 Let C and C 0 be full subcategories of Xétaff and Xétaff such that every object of Xétaff 0 (resp. Xétaff ) admits a strict étale covering by objects of C (resp. C 0 ) and u0 : U 0 → X 0 ∈ Ob C 0 implies f ◦ u0 ∈ Ob C. Then we see that the functor C´e0 t → C´et ; u0 7→ f ◦ u0 is continuous and cocontinuous by the same argument as the proof of Lemma IV.6.1.10 0 → Ctr ; u0 7→ f ◦ u0 is obviously continuous and cocontinuous. Hence and the functor Ctr similarly as fétaff , ftraff , (IV.6.5.13), and (IV.6.5.14), we obtain morphisms of topos 0 C,C 0 0∼ ∼ f´eC,C : C 0∧ → C ∧ and canonical isomorphisms of morphisms of et → C´ et and ftr t : C´ topos 0

0

C,C h ∼ C,C ◦ π h0 , πC,´ et ◦ fP,´ C ,´ et et-f´ et = f´ et

(IV.6.5.17)

0

0

0

0

C,C ∼ C,C πCh ◦ fP,f´ ◦ πCh0 et = ftr

C,C ∼ vC ◦ f´eC,C = ftr ◦ vC 0 . t

(IV.6.5.18) 0

We have f•C,C ∗ OC1 ,m = OC10 ,m (• = ´et, tr) and the above isomorphisms (IV.6.5.17) and (IV.6.5.18) are naturally extended to isomorphisms of morphisms of ringed topos (cf. (IV.6.1.17) and (IV.6.5.10)). There are also canonical isomorphism of functors (IV.6.5.19)

∼ =

0

0

∗ C,C ∗ h h f´eC,C ◦ πC,´ et∗ −→ πC 0 ,´ et∗ ◦ fP,´ t et-f´ et , 0

∼ =

∼ =

0

0

C,C ∗ C,C ∗ h ftr ◦ πC∗ −→ πCh0 ∗ ◦ fP,f´ et 0

C,C ∗ C,C ∗ h ftr ◦ πC,gpt∗ −→ πCh0 ,gpt∗ ◦ fgpt .

The first and the second isomorphisms are derived from the corresponding isomorphisms of continuous morphisms of sites. The third isomorphism follows from Lemma IV.6.2.5 (1) for V = UK,triv . Lemma IV.6.5.20. Assume that X is affine. The functor (XK,triv )f´et → (P/X)f´et ; V 7→ v ∼ (X, V) is cocontinuous. In particular, the functor πX∗ : (P/X)∼ f´ et → (XK,triv )f´ et is exact (cf. [2] III Proposition 2.3 1)). Proof. Since the functor V 7→ (X, V) is fully faithful, the claim immediately follows from the definition of the topologies of the source and target.  Corollary IV.6.5.21. Let F be a sheaf of abelian groups on (P/X)f´et . Then the natural v homomorphism H q ((XK,triv )f´et , πX∗ F)) → H q ((P/X)f´et , F) is an isomorphism for every q ∈ N. In the rest of this subsection, we prove the following proposition.

428

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proposition IV.6.5.22 (cf. Théorème VI.10.40). Let X be as above. For an object F of D+ ((P/X)f´et , Z), the following base change morphism with respect to the diagram (IV.6.5.3) is an isomorphism. ∗ h h ∗ vX RπX∗ F −→ RπX,´ et∗ vP/X F.

We prove Proposition IV.6.5.22 by looking at the stalks at every geometric point of X. Let ν : x → X be a geometric point and choose a filtered inverse system (Xλ , νλ )λ∈Λ of affine strict étale fs log schemes over X with liftings νλ : x → Xλ of ν such that limλ Γ(Xλ , Oλ ) = OX,x . Let I be the category defined by Ob I = Λ and ]HomI (λ, µ) = 1 −→ if λ ≥ µ and 0 otherwise. We define the category Ie by adding a final object e to I, i.e., Ie = I t {e} and e is the final object of Ie . We set Xe = X. Then the inverse system defines a functor X e : Ie → Xétaff . We define the category P/X e as follows. An object is a pair (i, (U, u)) ∈ Ob Ie × Ob (P/Xi ) and a morphism (i0 , (U0 , u0 )) → (i, (U, u)) is a pair of a morphism m : i0 → i (which is unique if it exists) and a morphism U0 → U in P compatible with the morphism X(m) : Xi0 → Xi . Then we have a natural functor (P/X e ) → Ie , whose fiber over i ∈ Ob Ie is naturally identified with P/Xi , and P/X e becomes a fibered category over Ie . For a morphism m : i0 → i in Ie , the inverse image functor is given by the functor X(m)∗ : (P/Xi ) → (P/Xi0 ) (cf. Proposition IV.6.1.7). Hence giving the topology defined by Cov(−) (resp. Covv (−)), we obtain a fibered site (P/X e )´et-f´et (resp. (P/X e )f´et ) over Ie . We define the fibered site (P/X)´et-f´et (resp. (P/X)f´et ) over I using X := X e |I : I → Xétaff , which coincides with the base change of (P/X e )´et-f´et (resp. (P/X e )f´et ) by the inclusion functor I → Ie . We can define e similarly the fibered sites (X e )étaff and (X e )traff over Ie and the fibered site X f´ et over I e e whose fibers are (Xi )étaff , (Xi )traff , and (Xi )f´et , where Xi denotes the scheme Xi,K,triv associated to Xi . From (IV.6.5.3), (IV.6.5.4), (IV.6.5.13), (IV.6.1.14), and (IV.6.5.15), we obtain the following diagrams of fibered sites over Ie and I which are commutative up to canonical isomorphisms. Every functor is Cartesian. (P/X e )´et-f´et

(IV.6.5.23)

h πX

vP/X e

h πX

et e ,´

 (X e )étaff

/ (P/X e )f´et

vX e

e

 / (X )traff , e

vP/X

/ (P/X)f´et (P/X)´et-f´et MMM MMM v πX MMM v MM&  πX,´ et e . X f´ et

e∞ be Let X∞ be Spec(OX,x ) endowed with the inverse image of MX and let X the inverse limit of Xλ,K,triv . Then, by [2] VI Propositions 6.4, 6.5, 8.3.3 and [42] Théorème (8.8.2), Théorème (8.10.5), Proposition (17.7.8), we see that the inverse lime its limI (X e )étaff , limI (X e )traff , and limI X f´ et are canonically equivalent to the sites ←− e ←− ←− e e X∞,étaff , X∞,traff , and X∞,f´et . Hence by taking the inverse limit of (IV.6.5.23) (cf. [2] VI Définition 8.2.5), we obtain the diagram of sites commutative up to canonical isomorphisms (cf. [2] VI Lemme 8.2.2, 8.2.7). (IV.6.5.24) v P/X e

− limI (P/X e )´et-f´et← ←− e πh ← −X e ,´et



X∞,étaff

vX ∞

/ lim (P/X e )f´et ←−Ie 

πh ← −X e

/ X∞,traff ,

v P/X

− / lim (P/X)f´et limI (P/X)´et-f´et ← ←− ←−I PPP PPP PPP πv ← −X PPP πv P(  ← −X,´et e∞,f´et . X

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

429

For the fibered site π : C → J and an object x of C, we write [x] for the image of x in limJ C. For a morphism m : j → π(x), the canonical morphism m∗ (x) → x induces ←− ∼ = an isomorphism [m∗ (x)] → [x]. Lemma IV.6.5.25. (1) The functor Γ(X∞ , −) : (X∞ )∼ étaff → V -Sets is exact. ∗ (2) The canonical morphism of functors Γ(X∞ , −) ◦ vX → Γ(X∞ , −) is an isomor∞ phism. ∗ is the Proof. (1) follows from Γ(X∞ , F) ∼ = Fx . (2) follows from the fact that vX ∞ sheafification with respect to the strict étale topology and that every strict étale covering of X∞ is refined by the trivial covering. 

e∞,f´et → lim(P/X)´et-f´et and X e∞,f´et → lim(P/X)f´et Lemma IV.6.5.26. The functors X ←− ←− I I v v and π in (IV.6.5.24) are cocontinuous. In defining the morphisms of sites ← π X e t X,´ ← − − ∼ ∼ ∼ v v e π (P/X)f´et → particular, the functors ← π (P/X)´et-f´et → X ∞,f´ et and ← −X,∗ : lim −X,´et∗ : lim ←− ←− I I e∼ X are exact (cf. [2] III Proposition 2.3 1)). ∞,f´ et

Proof. We prove the cocontinuity of the first functor and then explain how one can simplify the argument to prove the claim for the second functor. Since the functor ei )f´et → (P/Xi )´et-f´et ; V 7→ (Xi , V) is fully faithful, we see that the functor limX e (X → ←− f´et I lim(P/X)´et-f´et is fully faithful by the explicit construction of the direct limit of a fibered ←− I ei,f´et ). For j ∈ Ob I such site in [2] VI Propositions 6.4, 6.5. Let i ∈ Ob I and Vi ∈ Ob (X e that j ≥ i, put Vj = Vi × e Xj and Uj = (Xj , Vj ) ∈ Ob (P/Xj ). Let R be a covering Xi

sieve of [i, Ui ] on (limI P/X)´et-f´et . Since finite fiber products are representable in P/Xi ←− and the inverse image functors between the fibers of P/X preserve finite fiber products, we may apply [2] VI Proposition 8.3.3 to (P/X)´et-f´et . Hence, by Proposition IV.6.1.3, we see that there exist j ≥ i, (Uα → Uj )α∈A ∈ Covhf (Uj ), and (Uαβ → Uα )β∈Bα ∈ ∼ Covvf (Uα ) such that the composition [j, Uαβ ] → [j, Uj ] → [i, Ui ] is a section of R. Put Uα = (Uα , Vα ). Since (Uα ×Xi X∞ → X∞ )α is a strict étale covering and Γ(X∞ , OX∞ ) is strictly henselian local, there exist α and a section X∞ → Uα ×Xj X∞ , which comes from a section Xj 0 → Uα ×Xj Xj 0 for some j 0 ≥ j, j 0 ∈ Ob I. This implies that the inverse image of (Uαβ → Uj )αβ by the unique morphism j 0 → j is refined by a vertical étale covering ((Xj 0 , Vγ ) → Uj 0 )γ . Thus we see that R is refined by a sieve generated by ∼ = 0 0 e [j 0 , (Xj 0 , Vγ )] → [j 0 , Uj 0 ] → [i, Ui ]. Let V∞ be an object of limI X f´ et and let U∞ be its ←− 0 0 image in limI (P/X)f´et . Then, for a morphism V∞ → [i, Vi ] and its image U∞ → [i, Ui ], ←− the latter factors through [j 0 , (Xj 0 , Vγ )] if and only if the former factors through [j 0 , Vγ ] ∼ = by the fully faithfulness mentioned above. Furthermore ([j 0 , Vγ ] → [j 0 , Vj 0 ] → [i, Vi ])γ e is a covering by [2] VI Proposition 8.3.3. Thus we see that the functor limI X f´ et → ←− limI (P/X)´et-f´et satisfies the defining property of cocontinuous functor. ←− In the case of the second functor, we can take a vertical finite étale covering (Uα → Uj ) for some j ≥ i from the beginning, and the refinement is not necessary.  Lemma IV.6.5.27. For a sheaf of sets F on limI (P/X)f´et , the morphism ←− v v v v π π (adj) : π F −→ π v v ∗P/X F ∼ v ∗P/X F =← ← −X,´et∗ ← −X,∗ ← −X,∗ ← −P/X∗ ← − −X,´et∗ ← − is an isomorphism.

430

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

v ∗P/X F is a sheafification of F for the topology Proof. The morphism F → ← v P/X∗ ← − − e (P/X)´et-f´et is continuous and of limI (P/X)´et-f´et . Since the functor limI X f´ et → lim ←− ←−I ←− v v cocontinuous by Lemma IV.6.5.26, the morphism ← π F →← π v P/X∗ ← v ∗P/X F is the X,∗ X,∗ ← − − − − v sheafification of ← π −X,∗ F (cf. [2] III Proposition 2.3 2)), which is an isomorphism because v π  ← −X,∗ F is a sheaf. Corollary IV.6.5.28. For F ∈ D+ (limI (P/X)f´et , Z), the following natural morphism ←− is an isomorphism. RΓ(lim(P/X)f´et , F) −→ RΓ(lim(P/X)´et-f´et , ← v ∗P/X F). − ←− ←− I

I

Proof. By Lemma IV.6.5.26 and Lemma IV.6.5.27, we obtain isomorphisms ∼ =

e∞ )f´et , π vX,´et∗ v ∗ F) v ∗P/X F) ←− RΓ((X RΓ(lim(P/X)´et-f´et , ← − ← − ← −P/X ←− I

∼ =

∼ = e∞ )f´et , π vX,∗ F) −→ ←− RΓ((X RΓ(lim(P/X)f´et , F) ← − ←− I

and it is straightforward to verify that the composition coincides with the natural pullback morphism.  For a fibered site π : C → J where J belongs to the universe U and associated to a filtered ordered set, we use the following notation. For i, j ∈ Ob J such that j ≥ i, let µij denote the morphism of topos Cj∼ → Ci∼ induced by the inverse image functor by the unique morphism j → i. For i ∈ Ob J, let µi denote the morphism of topos (limJ C)∼ → Ci∼ induced by the natural functor Ci → limJ C which defines a morphism ←− ←− of sites. We have a canonical isomorphism µij ◦ µj ∼ = µi for j ≥ i. (See [2] VI Théorème 8.2.3.) By (IV.6.5.23), we see that the functors µi and µij are compatible with the morphisms of topos induced by those of sites in the first (resp. second) diagram in (IV.6.5.24) and the morphisms of topos in (IV.6.5.3) (resp. (IV.6.5.4)) for Xi , i ∈ Ie (resp. i ∈ I). In particular, we have the following diagram commutative up to canonical isomorphisms. (IV.6.5.29) iii4 iiii (limI (P/X e )´et-f´et )∼ ←− e µe

πh ← −X e ,´et

 (X∞ )∼ étaff

(P/X)´e∼t-f´et v ← −P/X e h πX,´ et

 ∼ Xétaff 4 h µe hhhh hh hhhh vX ∞

vP/X

/ (P/X)∼ f´ et 4 j j j j j j j h / (lim (P/X e )f´et )∼ πX I ←− e  vX / X∧ 4 étaff h πX µe iiii i ← − e  iiii / (X∞ )∧ étaff µe

Lemma IV.6.5.30. For F ∈ D+ ((P/X)´et-f´et , Z) and G ∈ D+ ((P/X)f´et , Z), the following base change morphisms are isomorphisms. ∗ h h µ∗e RπX,´ π et∗ F −→ R← −X e ,´et∗ µe F,

h h ∗ µ∗e RπX∗ G −→ R← π −X e ∗ µe G.

Proof. By Corollary IV.6.1.4, the pull-back morphisms h h ∗ µ∗ei RπX,´ et∗ (F) −→ RπXi ,´ et∗ µei (F),

h h µ∗ei RπX∗ (G) −→ RπX µ∗ (G) i ∗ ei

are isomorphisms for every i ∈ Ob I. By Lemma IV.6.5.5 (1), (2), (3) and Lemma h IV.6.5.16, we may apply [2] VI Corollaire 8.7.5 to the morphisms of fibered sites πX,´ et h and πX , and obtain isomorphisms in the lemma. 

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

431

Proof of Proposition IV.6.5.22. It suffices to prove that the stalk at every geometric point x → X is an isomorphism. Since Gx = Γ(X∞,étaff , G) = RΓ(X∞,étaff , G) for a sheaf of abelian groups G on X∞,étaff (cf. Lemma IV.6.5.25), it suffices to prove that the morphism ∗ h h ∗ RΓ(X∞,étaff , µ∗e vX RπX∗ F) −→ RΓ(X∞,étaff , µ∗e RπX,´ et∗ vP/X F) h h is an isomorphism. We write µ, v, π, ← v , and ← π v • , and ← π − − for µe , v• , π• , ← − −• with • = X, etc. to simplify the notation. Then, by considering the compositions of the base change morphisms along the vertical faces in the diagram (IV.6.5.29), we obtain the following commutative diagram.

µ∗ v ∗ Rπ∗ F

/ µ∗ Rπ∗ v ∗ F

ϕ

v ∗ µ∗ Rπ∗ F ← −

/ v ∗ R π ∗ µ∗ F ← − ← −

/ R π ∗ µ∗ v ∗ F ← − ψ

/ R π ∗ v ∗ µ∗ F ← − ← −

The upper right and the lower left morphisms are isomorphisms by Lemma IV.6.5.30. After taking RΓ((X∞ )étaff , −), the lower right morphisms are put in the following commutative diagram. RΓ(X∞,étaff ,ψ)

/ RΓ(X∞,étaff , R π ∗ v ∗ µ∗ F) ← − ← − f3 fffff f f f f fff fffff fffff ∗ RΓ(X∞,traff , R← π −∗ µ F)

∗ RΓ(X∞,étaff , ← v ∗R π −∗ µ F) O− ←

The vertical morphism is an isomorphism by Lemma IV.6.5.25 and the right slanted morphism is an isomorphism by Corollary IV.6.5.28. Hence RΓ(X∞,étaff , ψ) is an isomorphism, which implies that RΓ(X∞,étaff , ϕ) is an isomorphism.  IV.6.6. Topos of inverse systems of sheaves and direct systems of sheaves. In this subsection, we review basic facts on the topos of inverse systems of sheaves and the topos of direct systems of sheaves (cf. III.7). Let C be a site, and let Λ be a U -small ordered set. We regard Λ as a category as follows. The set of objects is Λ and HomΛ (λ, µ) is empty if λ > µ and consists of one element if λ ≤ µ. Let Λ◦ denote the opposite category of Λ. The product category C × Λ with the projection C × Λ → Λ is a fibered site (cf. [2] VI 7.2.1); every fiber is identified with the site C and the inverse image functor is given by the identity. We write C Λ for the fibered site C × Λ endowed with the total topology (cf. [2] VI 7.4.1). Then the presheaf F on C Λ is a sheaf if and only if its restriction Fλ on the fiber over λ ∈ Λ is a sheaf on C for every λ ∈ Λ (cf. [2] VI Proposition 7.4.4). Hence we have isomorphisms of categories. ∼ =

(IV.6.6.1)

(C Λ )∼ −→ HomΛ◦ (Λ◦ , C ∼ × Λ◦ ),

(IV.6.6.2)

(C Λ )∧ −→ HomΛ◦ (Λ◦ , C ∧ × Λ◦ )

∼ =

defined by F 7→ (Fλ ) (cf. [2] VI Proposition 7.4.7). The target of the first (resp. the second) functor is the category of inverse systems of sheaves (resp. presheaves) of sets on ◦ ◦ C indexed by Λ, which is denoted by C ∼Λ (resp. C ∧Λ ) following III.7.1.

432

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Let u : C Λ → C be the projection functor. Then the right adjoint u b∗ : (C Λ )∧ → C ∧ ∗ ∧ Λ ∧ of the functor u b : C → (C ) ; G 7→ G ◦ u is given by u b∗ F = lim Fλ . ←− λ∈Λ

Indeed, for F ∈ Ob (C ) and G ∈ Ob C , we have Hom(G ◦u, F) ∼ = Hom((G)λ , (Fλ )λ ) ∼ = Hom(G, limΛ Fλ ). Since the inverse limit preserves sheaves, we see that the functor u is ←− cocontinuous (cf. [2] III Proposition 2.2) and defines a morphism of topos Λ ∧

(IV.6.6.3)

∧

l : (C Λ )∼ → C ∼ , ← −

l ∗ (F) = lim Fλ , ← − ←− Λ

l ∗ (G) = G ◦ u. ← −

Let f : C ∼ → D∼ be a morphism of topos. Then the compositions with f∗ ×idΛ◦ and ◦ ◦ ◦ ∗ f ×idΛ◦ define a morphism of topos f Λ : C ∼Λ → D∼Λ ; the left exactness of the inverse image functor follows from the fact that the natural morphism (limI F)(λ) → limI (F(λ)) ←− ←− ◦ ◦ (λ ∈ Λ) is an isomorphism for an inverse system F : I ◦ → C ∼Λ (or D∼Λ ) over a U small category I. Combining with the isomorphisms (IV.6.6.1) for C and D, we obtain ◦ a morphism of topos (C Λ )∼ → (DΛ )∼ , which is also denoted by f Λ in the following. By looking at the compositions of inverse image functors, we see that the following diagram of topos is commutative up to canonical isomorphism. (C Λ )∼

(IV.6.6.4)

fΛ

l ← −

/ C∼

◦



(DΛ )∼

f l ← −



/ D∼ .

If we replace Λ with Λ◦ , we obtain a topos of direct systems of sheaves on C indexed by Λ as follows. The product C × Λ◦ with the projection C × Λ◦ → Λ◦ is a fibered site, ◦ and we write C Λ for the fibered site C × Λ◦ with the total topology. Then the presheaf Λ◦ F of sets on C is a sheaf if and only if its restriction on the fiber over λ is a sheaf on C for every λ ∈ Λ, and we have an isomorphism f categories defined by F 7→ (Fλ ) ◦

∼ =

◦

∼ =

(IV.6.6.5)

(C Λ )∼ −→ HomΛ (Λ, C ∼ × Λ),

(IV.6.6.6)

(C Λ )∧ −→ HomΛ (Λ, C ∧ × Λ).

The target of the first (resp. the second) functor is the category of direct systems of sheaves (resp. presheaves) on C indexed by Λ, which is denoted by C ∼Λ (resp. C ∧Λ ). ◦ Let v : C Λ → C be the projection functor, which is continuous by the above ◦ ◦ characterization of sheaves on C Λ . The left adjoint vb! : (C Λ )∧ → C ∧ of the functor ◦ vb∗ : C ∧ → (C Λ )∧ ; G 7→ G ◦ v is given by vb! F = lim Fλ . −→ λ∈Λ

Indeed, for F ∈ Ob (C ) and G ∈ Ob C ∧ , we have bijections Hom(F, G ◦ v) ∼ = Hom((Fλ )λ , (G)λ ) ∼ F , G). = Hom(lim −→Λ λ Suppose, from now on, that Λ is a filtered ordered set. Then the functor vb! is exact. Therefore the functor v defines a morphism of sites and hence a morphism of topos Λ◦ ∧

(IV.6.6.7)

◦

∼ Λ ∼ → −l : C → (C ) ,

→ −l ∗ G = G ◦ v,

∗ lim Fλ . → −l F = − → λ∈Λ

∗ In the description of → −l ∗F, we consider the direct limit as sheaves of sets. Note that the inverse image functor → −l gives the direct limit of sheaves, while the direct image functor

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

433

l ∗ gives the inverse limit of sheaves. We have ← − (IV.6.6.8)

∗ ∼ → −l ◦ → −l ∗ = idC .

A morphism of topos f : C ∼ → D∼ induces a morphism of topos f Λ : C ∼Λ → D∼Λ whose direct (resp. inverse) image functor is defined by the composition with f∗ × idΛ (resp. f ∗ × idΛ ). Combining with the isomorphisms (IV.6.6.5) for C and D, we obtain a ◦ ◦ morphism of topos (C Λ )∼ → (DΛ )∼ , which is also denoted by f Λ . By looking at the compositions of the direct image functors, we see that the following diagram of topos is commutative up to canonical isomorphism. (IV.6.6.9)

l → −

C∼ f

 D∼

l → −

/ (C Λ◦ )∼ 

fΛ

/ (DΛ◦ )∼ .

b Σ, and D(Σ) be the IV.6.7. Comparison morphism. Let V , k, K, K, V , Σ, Σ, same as in the beginning of IV.5.2 and take the ring W (RV ) as the base ring R of our theory of Higgs crystals as in loc. cit. Let X be an fs log scheme over Σ satisfying the b and X1 as in loc. cit. conditions in the beginning of IV.6.4, and define X Let (X1 /D(Σ))HIGGS be the inverse limit of the fibered site (r 7→ (X1 /D(Σ))rHIGGS ) r∼ defined as in IV.4.5. Let µr : (X1 /D(Σ))∼ HIGGS → (X1 /D(Σ))HIGGS (r ∈ N>0 ) be the canonical morphism of topos and let OX1 /D(Σ),1 be the sheaf of rings limr µ∗r (OX1 /D(Σ),1 ) −→ (∼ = µ∗s (OX1 /D(Σ),1 ), s ∈ N>0 ). The compatible system of morphisms of ringed topos ∼ uX1 /D(Σ) : ((X1 /D(Σ))r∼ HIGGS , OX1 /D(Σ),1 ) → ((X1 )´ et , OX1 ) (r ∈ N>0 ) induces a morphism of ringed topos ∼ uX1 /D(Σ) : ((X1 /D(Σ))∼ HIGGS , OX1 /D(Σ),1 ) −→ ((X1 )´ et , OX1 ).

Under the notation in IV.6.6, we regard the inverse system of sheaves of rings OP/X,• = (OP/X,m )m∈N as a sheaf of rings on (P/X)´eNt-f´et by the isomorphism (IV.6.6.1). Similarly, we regard the inverse system of sheaves of rings OX1 ,• = (OX1 ,m )m∈N as a sheaf of rings N on (X1 )´eNt and also on Xétaff . Then we have morphisms of ringed topos (cf. (IV.6.5.10)) ◦

h,N πX,´ et

l

← − N∼ ∼ ((P/X)´eN∼ t-f´ et , OP/X,• ) −−−→ (Xétaff , OX1 ,• ) −→ (Xétaff , OX1 ).

Let (F, F ◦ ) be an object of HCrZp (X1 /D(Σ)) (cf. Definition IV.3.5.4). In this subsection, we will construct a canonical morphism (IV.6.7.1)

◦

h,N ◦ RuX1 /D(Σ)∗ (µ∗r F) −→ Q ⊗ R← l ∗ RπX,´ et∗ TX,• (F ) −

in D+ (Xétaff , OX1 ,Q ). See IV.6.4 for the definition of the functor TX,• . By taking RΓ(Xétaff , −), we obtain a canonical morphism in D+ (C-Mod) (IV.6.7.2)

RΓ((X1 /D(Σ))HIGGS , µ∗r F) −→ Q ⊗ RΓ((P/X)´eNt-f´et , TX,• (F ◦ )).

Here C denotes the completion of K with respect to the valuation. We begin with some preliminaries. Let C be a full subcategory of Xétaff such that every object of Xétaff admits a strict étale covering by objects of C. We define the functors

434

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

ηP/C and ηC by the following compositions: (IV.6.7.3) ∼

v ) ηP/C : Mod(Cgpt , OCgpt ,• ) −→ Mod((P/C)f´et , OP/C,• rC

∗ vP/C

∼

−−−→ Mod((P/C)´et-f´et , OP/C,• ) −−−−−−→ Mod((P/X)´et-f´et , OP/X,• ), idX,C P,´ et-f´ et∗

v∗

∼

C (IV.6.7.4) ηC : Mod(Ctr , OC1 ,• ) −→ Mod(C´et , OC1 ,• ) −−−→ Mod(Xétaff , OX1 ,• ).

id´eX,C t∗

Then we have a natural morphism of functors h h ηC ◦ πC,gpt∗ −→ πX,´ et∗ ◦ ηP/C

(IV.6.7.5) defined by the composition ∼ =

∗ h ∗ h h ∗ id´eX,C −−−−−−→ id´eX,C −−−−−−→ id´eX,C et∗ ◦ vP/C ◦ rC t∗ ◦ vC ◦ πC,gpt∗ − t∗ ◦ vC ◦ πC∗ ◦ rC − t∗ ◦ πC,´ (IV.6.5.11)

(IV.6.5.10)

∼ =

X,C h ∗ −−−−−−−→ πX,´ et∗ ◦ idP,´ et-f´ et∗ ◦ vP/C ◦ rC . (IV.6.5.17)

We define the functor ηC,Q by v∗

∼

C ηC,Q : Mod(Ctr , OC1 ,Q ) −→ Mod(C´et , OC1 ,Q ) −−−→ Mod(Xétaff , OX1 ,Q ).

(IV.6.7.6)

id´eX,C t∗

Then we have a morphism of functors ηC,Q ◦ (Q ⊗ lim) −→ (Q ⊗ lim) ◦ ηC ←− ←− defined by the composition (IV.6.7.7)

∼

∗ ∗ id´eX,C ) −→ id´eX,C ) ◦ vC∗ −→ (Q ⊗ lim) ◦ id´eC,X t∗ ◦ vC ◦ (Q ⊗ lim t∗ ◦ (Q ⊗ lim t∗ ◦ vC ←− ←− ←− (cf. (IV.6.6.4) for the first morphism). By combining (IV.6.7.5) and (IV.6.7.7), we obtain a morphism of functors h h ηC,Q ◦ (Q ⊗ lim) ◦ πC,gpt∗ −→ (Q ⊗ lim) ◦ πX,´ et∗ ◦ ηP/C . ←− ←− The morphisms (IV.6.7.5) and (IV.6.7.7) are compatible with the pull-backs by a strict étale morphism as follows. Let X and C be as above, let f : X 0 → X be a strict étale 0 0 morphism, and let C 0 be a full subcategory of Xétaff such that every object of Xétaff admits a strict étale covering by objects of C 0 and that u : U 0 → X 0 ∈ Ob (C 0 ) implies f ◦ u ∈ Ob (C). We obtain the following isomorphism from (IV.6.2.10), (IV.6.1.17), and (IV.6.1.18).

(IV.6.7.8)

∼ =

0

C,C ∗ ∗ fP,´ et-f´ et ◦ ηP/C −→ ηP/C 0 ◦ fgpt .

(IV.6.7.9)

Lemma IV.6.7.10. Let f , C, and C 0 be as above. Then the following diagrams of functors are commutative, where the isomorphisms (A) are induced by (IV.6.5.18) and (IV.6.5.6). h f´e∗t ◦ ηC ◦ πC,gpt∗

∼ = (A)

/ ηC 0 ◦ f C,C 0 ∗ ◦ π h tr C,gpt∗

∼ = (IV.6.5.19)

(IV.6.7.5)



h f´e∗t ◦ πX,´ et∗ ◦ ηP/C

∼ = (IV.6.5.19)

h ∗ / πX 0 ,´ et∗ ◦ fP,´ et-f´ et ◦ ηP/C

∼ = (IV.6.7.9)

C,C 0 ∗ / ηC 0 ◦ π h0 C ,gpt∗ ◦ fgpt



(IV.6.7.5)

C,C 0 ∗ / πh 0 X ,´ et∗ ◦ ηP/C 0 ◦ fgpt

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

f´e∗t ◦ ηC,Q ◦ (Q ⊗ lim) ←−

∼ = (A)

/ ηC 0 ,Q ◦ f C,C 0 ∗ ◦ (Q ⊗ lim) tr ←−

∼ =

/ ηC 0 ,Q ◦ (Q ⊗ lim) ◦ f C,C 0 ∗ tr ←−

(IV.6.7.7)

 f´e∗t ◦ (Q ⊗ lim) ◦ ηC ←−

∼ =

/ (Q ⊗ lim) ◦ f´e∗t ◦ ηC ←−

∼ = (A)

435



(IV.6.7.7)

/ (Q ⊗ lim) ◦ ηC 0 ◦ f C,C 0 ∗ tr ←−

Proof. For the first diagram, it suffices to prove three similar diagrams obtained X (0) ,C (0) ∗ X (0) ,C (0) ∗ ∗ ∗ by replacing (ηP/C (0) , ηC (0) ) with (rC (0) , id), (vP/C ) are (0) , vC (0) ), and (idP,´ et-f´ et , id´ et commutative. The diagram for (rC (0) , id) commutes by the constructions of (IV.6.2.10), (IV.6.5.19), and (IV.6.5.11). The commutativity of the other two diagrams follows from C,C 0 the compatibility of base change morphisms with compositions applied to v··· ◦ f··· = 0 0 0 0 C,C C,C X ,C f··· ◦ v··· and idX,C ◦ f = f ◦ id . For the second diagram, it suffices to prove ··· ··· ··· ··· (0)

(0)

that the two similar diagrams obtained by replacing ηC (0) with vC∗(0) and id´eXt ,C ∗ are commutative, which also follows from the compatibility of base change morphisms with compositions. 

Now let us construct the morphism (IV.6.7.1). We first consider the case where we are given a smooth fine log scheme Y over Σ and an immersion X → Y over Σ. Let Yb be the p-adic completion of Y , let Y1 be Yb ×Σ b Σ and let i1 denote the immersion X1 → Y1 over Σ induced by i. We further assume that we are given a smooth Cartesian morphism Y• → D(Σ) in the category C (cf. Definitions IV.2.2.1 and IV.2.2.2) which lifts Y1 → Σ. r Let Dr (r ∈ N>0 ) be DHiggs (i : X1 ,→ Y• ) and let zDr be the natural morphism D1r → s X1 . let (M , θ) (s ≥ r) denote the object of HBsQp ,conv (X1 , Y• /D(Σ)) corresponding to µ∗rs (F) ∈ Ob (HCsQp (X1 /D(Σ))) by the equivalence of categories in Theorem IV.3.4.16. Then, by Corollary IV.4.5.7, we have a canonical isomorphism (IV.6.7.11) RuX1 /D(Σ)∗ µ∗r (F) ∼ z s (ξ −• Ms ⊗OX i∗ (Ω•Y /Σ )) = lim −→ D ∗ s≥r

in D+ ((X1 )´et , OX1 ,Q ). Using (IV.6.7.11), we construct the morphism (IV.6.7.1) along the N N◦ following lines. Let (OP/X,• )N denote the sheaf of rings → −l ∗ (OP/X,• ) on ((P/X)´et,f´et ) (cf. (IV.6.6.7)), which corresponds to the constant direct system defined by OP/X,• via N the equivalence of categories (IV.6.6.5). Similarly we define the sheaves of rings OX and 1 ,• ◦ ◦ N N N N OX1 on (Xétaff ) and Xétaff , respectively. We first construct a morphism of complexes ◦ N of OP/X,• -modules on ((P/X)´eNt-f´et )N ?+1 ◦ ◦ → −l ∗ TX,• (F ) −→ TX,• ΩY (F ) by a method similar to the construction of V → ξ −• A1s (A) ⊗ e V ⊗A Ω• (s ∈ N) in the A proof of Proposition IV.5.2.15. Then we prove that (IV.6.7.12) induces an isomorphism

(IV.6.7.12)

(IV.6.7.13)

◦

∼ =

◦

h,N h,N N ?+1 ◦ ∗ ◦ Q ⊗ R← l ∗ RπX,´ lN ∗ R(πX,´ et∗ TX,• (F ) −→ → et )∗ (TX,• ΩY (F )) − −l Q ⊗ R← −

in D+ (Xétaff , OX1 ,Q ). Finally we construct a canonical morphism of direct systems of complexes of OX1 ,Q -modules (IV.6.7.14)

◦

h,N s ◦ (zDs ∗ (ξ −• Ms ⊗OX i∗ Ω•Y /Σ ))s≥r −→ (Q ⊗ ← l ∗ πX,´ et∗ TX,• ΩY (F ))s≥r . −

By taking the direct limit of (IV.6.7.14) and combining with (IV.6.7.11) and (IV.6.7.13), we obtain (IV.6.7.1). Let C Y be the full subcategory of CX (cf. Definition IV.6.4.1 (1)) consisting of U ∈ Ob CX satisfying the following condition: There exists a strict étale morphism Y 0 → Y

436

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

such that Y 0 satisfies Condition IV.5.2.3 and the morphism U → X factors through Y and r ∈ N>0 , by using the object DX (U, s) = X ×Y Y 0 → X. For (h : U → X, s) ∈ Cgpt r (D(U, s), h1 ◦ z(U,s) ) of (X1 /D(Σ))∞ (cf. IV.6.4), we define the object DX,Y (U, s) = HIGGS r r (DX,Y,N (U, s)) of C by r r DX,Y (U, s) := DHiggs (D(U, s)1 ,→ Y• ×D(Σ) D(U, s)) r (cf. the construction of DX,Y (U ) before Condition IV.5.2.3). This construction is funcr Y torial in (U, s) and we can define a ring object AX,Y,m of (Cgpt )∧,cont by m r r r AX,Y,m (U, s) := Γ(DX,Y,1 (U, s), ODX,Y,1 (U,s) )/p , Y ,m -algebra (cf. Proposition IV.5.2.6). For a sheaf H on X´ which is an OCgpt et , we also Y r write H for the object of Cgpt defined by (U, s) 7→ H(U ). Then AX,Y,m is also an OX1 algebra. The construction of the complex (IV.5.2.9) is functorial in (U, s). By Proposition Y Y ,• ) , OCgpt IV.5.2.10 (2), we obtain a direct system of complexes in Mod(Cgpt

(IV.6.7.15)

r+1 −q Y ,• −→ (ξ (OCgpt AX,Y,• ⊗OX i∗ (ΩqY /Σ ), θq )q∈N )r∈N .

Choose a covering sieve R of X in Xétaff such that F ◦ is R-finite (cf. Definition IV.6.4.2) and F(T,z) ∈ Ob PM(OT1 ,Qp ) for every (T, z) ∈ Ob (X1 /D(Σ))rHIGGS such that T1 is affine and z factors through u1 : U1 → X1 for some u : U → X ∈ Ob Xétaff satisfying u ∈ R(U ). See Remark IV.6.4.3 for the existence of such an R. Let C be a full subcategory of Xétaff contained in both C Y and CR (cf. Definition IV.6.4.1 (2)) such that every object of Xétaff admits a strict étale covering by objects of C and that, for a morphism U 0 → U in Xétaff , U ∈ Ob C and U 0 ∈ Ob CR imply U 0 ∈ Ob C. By taking the tensor product of R Y ,• |C TX,gpt,• (F ◦ )|Cgpt (cf. (IV.6.4.4)) and (IV.6.7.15)|Cgpt over OCgpt gpt , we obtain a direct system of complexes in Mod(Cgpt , OCgpt ,• ) (IV.6.7.16)

s+1 R (TX,gpt,• (F ◦ )|Cgpt −→ TC,gpt,• ΩY (F ◦ ))s∈N .

Lemma IV.6.7.17. For r ∈ N>0 , there exists Nr ∈ N>0 such that the homomorphism from Hq of the r-th term of (IV.6.7.16) to Hq of the (r + 1)-th term is annihilated by pNr for every q ∈ Z. Proof. It suffices to prove that there exists an integer Nr determined only by p and r such that the natural morphism from the complex (IV.5.2.9) for A1r (A) to the complex (IV.5.2.9) for A1r+1 (A) is homotopic to 0 by an A1 (A)-linear homotopy after multiplied m by pNr . Since F n AN (A) = ξ n AN (A), we have A1 (A)(m)r = pd r e A1 (A). Hence, by the construction of the homotopy in the proof of Proposition IV.5.2.10 (2), it suffices to show m+1 m that the p-adic valuation of m01+1 pd r e p−d r+1 e is bounded for 0 ≤ m0 ≤ m. This follows m+1 from lim (− logp (m + 1) + d m  r e − d r+1 e) = ∞. m→∞

By applying ηP/C (IV.6.7.3) to (IV.6.7.16), we obtain (IV.6.7.12). Note that the R image of TX,gpt,• (F ◦ )|Cgpt under ηP/C is canonically isomorphic to TX,• (F ◦ ) by (IV.6.7.9) applied to f = idX and C ⊂ CR . We also see that the above complex does not depend on the choice of R and C up to canonical isomorphisms. Note that if we choose another C 0 , then C ∩ C 0 also satisfies the conditions on C.

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

437

N Lemma IV.6.7.18. Regard (IV.6.7.12) as a complex of OP/X,• -modules. Then its image under the composition of the following functors are 0. ◦

h,N N R(πX,´ et )∗

◦

Rl

◦

N ∗

◦

− N N N N N ) −−−−−−→ D+ ((Xétaff D+ (((P/X)´eNt-f´et )N , OP/X,• )N , OX ) −−← − → D+ (Xétaff , OX ) 1 1 ,• l

◦

∗

→ − N N −−→ D+ (Xétaff , OX )− −→ D+ (Xétaff , OX1 ,Q ). 1 ,Q Q⊗

N Proof. Let K = (Ks )s∈N denote (IV.6.7.12) regarded as a complex of OP/X,• modules. Since the functor ηP/C is exact, Lemma IV.6.7.17 implies that the transition map Hq (Ks ) → Hq (Ks+1 ) is annihilated by pNs for every s ∈ N. We have a spectral sequence h,N◦ N b h,N◦ N a+b E2a,b = Ra (← l ◦ πX,´ (← l ◦ πX,´ et )∗ H (K) =⇒ R et )∗ (K) − − ◦

◦

a,b h,N N a b in Mod(Xétaff ,→ l ◦ πX,´ et )∗ H (Ks ) −l ∗ OX1 ), and the s-th term of E2 is given by R (← − ∗ (cf. [2] VI the first paragraph of the proof of Lemme 8.7.2). Hence we have → −l Q ⊗ ◦



h,N N Ra+b (← l ◦ πX,´ et )∗ (K) = 0. −

The upper (resp. lower) square of the diagram below is commutative up to canonical isomorphisms by (IV.6.6.9) (resp. (IV.6.6.4)). D+ ((P/X)´eNt-f´et , OP/X,• )

l → −∗

/ D+ (((P/X)´eNt-f´et )N◦ , ON P/X,• )

◦

◦

h,N RπX,´ et∗

 N D+ (Xétaff , OX1 ,• ) Rl∗ ← −



D+ (Xétaff , OX1 )

h,N N R(πX,´ et )∗

l → −∗

 / D+ ((X N )N◦ , ON ) X1 ,• étaff

l → −∗



RlN ← −∗

/ D+ (X N◦ , ON ). X1 étaff

◦ Hence by (IV.6.6.8), we see that the image of → of the −l ∗ TX,• (F ) under the composition h,N◦ ◦ functors in Lemma IV.6.7.18 is canonically isomorphic to Q ⊗ R← l ∗ RπX,´et∗ TX,• (F ). By − Lemma IV.6.7.18, we obtain the isomorphism (IV.6.7.13). It remains to construct (IV.6.7.14). Let (U, s) ∈ Ob Cgpt . By Proposition IV.5.2.12 s and the choice of C, we have a canonical ∆(U,s) -equivariant limm AX,Y,m (U, s)-linear ←− isomorphism

(IV.6.7.19)

s Ms (D1s ×X1 U1 ) ⊗ODs (D1s ×X1 U1 ) lim AX,Y,m (U, s) ←− 1 m

∼ =

−→ Q ⊗

R lim(TX,gpt,m (F ◦ )(U, s) ← − m

s ⊗OCgpt,m (U,s) AX,Y,m (U, s))

compatible with θ. By construction, this isomorphism is compatible with s and functorial with respect to (U, s). Hence, the above isomorphism (IV.6.7.19) induces a morphism of direct systems of complexes of OC1 ,Q (:= OX1 ,Q |C )-modules on Ctr h s (IV.6.7.20) (zDs ∗ (ξ −• Ms ⊗OX i∗ Ω•Y /Σ )|C )s≥r −→ (Q ⊗ lim πC,gpt∗ TC,gpt,m ΩY (F ◦ ))s≥r . ← − m

By taking ηC,Q of (IV.6.7.20) and using (IV.6.7.8), we obtain (IV.6.7.14). By Lemma IV.6.7.10 for f = idX , we see that the morphism (IV.6.7.14) obtained above is independent of the choice of C. Thus we obtain the morphism (IV.6.7.1). We can verify the independence of the choice of (i : X → Y, Y• ) as follows. Choose another (i0 : X → Y 0 , Y•0 ). By considering (X → Y ×Σ Y 0 , Y• ×D(Σ) Y•0 ), we are reduced to

438

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

the case where there exist morphisms g : Y 0 → Y and g• : Y•0 → Y• compatible with i, i0 , 0 b 0 b Σ. Then the independence follows from the functoriality of Y1 ∼ = Yb ×Σ b Σ, and Y1 = Y ×Σ 0 (IV.6.7.12) and (IV.6.7.14) below applied to f = idX , g, g• and C = C 0 ⊂ C Y ∩ C Y ∩ CR . Let i0 : X 0 → Y 0 , Y•0 be another data satisfying the same assumptions as i : X → Y, Y• , and let f : X 0 → X and g : Y 0 → Y be morphisms over Σ such that g ◦ i0 = i ◦ f and f is strict étale. Let g• : Y•0 → Y• be a morphism over D(Σ) such that g1 = gb ×Σ b Σ. 0 Let R0 be a covering sieve of X 0 in Xétaff defined by R0 (u : U → X 0 ) = R(f ◦ u : U → X). ∗ Let (F 0 , F 0◦ ) be f1,HIGGS (F, F ◦ ). Assume that there exist full subcategories C ⊂ CR ∩C Y 0 0 Y and C ⊂ CR0 ∩ C satisfying the condition before Lemma IV.6.7.10. By the proof of Lemma IV.6.4.6, we have a canonical isomorphism ∼ =

0

0

C,C ∗ R 0 fgp (TX,gp,• (F ◦ )|Cgpt ) −→ TXR0 ,gp,• (F 0◦ )|Cgpt .

(IV.6.7.21)

Hence by the construction of (IV.6.7.16), we see that the morphisms g and g• naturally 0 0 , OCgpt induce a direct system of morphisms in Mod(Cgpt ,• ). 0

C,C ∗ s+1 0◦ (fgpt (TC,gpt,• ΩY (F ◦ )) −→ TCs+1 ))s∈N 0 ,gpt,• ΩY 0 (F

(IV.6.7.22)

such that (IV.6.7.21) and (IV.6.7.22) are compatible with (IV.6.7.16). By taking ηP/C of ◦ (IV.6.7.22) and using (IV.6.7.9), we obtain a morphism on ((P/X 0 )´eNt-f´et )N ?+1 ?+1 ∗ ◦ 0◦ fP,´ et-f´ et (TX,• ΩY (F )) −→ TX 0 ,• ΩY 0 (F )

(IV.6.7.23)

such that the following diagram is commutative. (IV.6.7.24) ∼ ∼ = = ∗ ◦ ∗ ◦ / l ∗ (TX 0 ,• (F 0◦ )) / l ∗ fP,´ fP,´ et-f´ et (TX,• (F )) et-f´ et → → − −l ∗ (TX,• (F )) − Lemma IV.6.4.6 → (IV.6.7.12)



?+1 ∗ ◦ fP,´ et-f´ et (TX,• ΩY (F ))

(IV.6.7.12)

 0◦ / T ?+1 X 0 ,• ΩY 0 (F ).

(IV.6.7.23)

We define D0s and M0s similarly as Ds and Ms using i0 : X 0 → Y 0 , Y•0 and (F 0 , F 0◦ ). Then, by the construction of (IV.6.7.20), we see that the composition of ∼ =

0

0

C,C ∗ C,C ∗ s h s ftr (Q ⊗ lim πC,gpt∗ TC,gpt,m ΩY (F ◦ )) −−−−−−−→ Q ⊗ lim πCh0 ,gpt∗ fgpt TC,gpt,m ΩY (F ◦ ) ← − ← − (IV.6.5.19) m m (IV.6.7.22)

−−−−−−−→ Q ⊗ lim πCh0 ,gpt∗ TCs0 ,gpt,m ΩY 0 (F 0◦ ) ← − m and the natural morphism 0

C,C ∗ ftr (zDs ∗ (ξ −• Ms ⊗OX i∗ Ω•Y /Σ )|C ) −→ (zD0s ∗ (ξ −• M0s ⊗OX 0 i0∗ Ω•Y 0 /Σ ))|C 0

are compatible with (IV.6.7.20). By taking ηC 0 ,Q and using Lemma IV.6.7.10, we see that h the following diagram is commutative, where ← π πh . −X (0) ,´et∗,Q denotes Q ⊗ lim ←−m X (0) ,´et∗ (IV.6.7.25) f´e∗t (zDs ∗ (ξ −• Ms ⊗OX i∗ Ω•Y /Σ ))

(IV.6.7.14)

s / f´e∗t π hX,´et∗,Q (TX,m ΩY (F ◦ )) ← − ∼ = (IV.6.5.19)



 zD0s ∗ (ξ −• M0s ⊗OX 0 i0∗ Ω•Y 0 /Σ )

h ∗ s π ← −X 0 ,´et∗,Q fP,´et-f´et (TX,m ΩY (IV.6.7.14)



(F ◦ ))

(IV.6.7.23)

/ π hX 0 ,´et∗,Q (TXs 0 ,m ΩY 0 (F 0◦ )). ← −

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

439

Now let us construct the morphism (IV.6.7.1) for a general X. We reduce it to the special case above by using étale cohomological descent. Let ∆ be the category whose objects are ordered sets [ν] = {0, 1, . . . , ν} (ν ∈ N) and whose morphisms are non-decreasing maps. Recall that a simplicial object of a category C means a functor ∆◦ → C. Choose a strict étale hypercovering X = (X [ν] )ν∈N of X such that X [ν] are quasi-compact, an immersion of simplicial p-adic fine log formal schemes i = (i[ν] ) : X ,→ Y = (Y [ν] ) over Σ such that each Y [ν] is smooth over Σ, and a [ν] [ν] simplicial object Y • = (Y• ) of C over D(Σ) such that each Y• is smooth and Cartesian over D(Σ) and Y 1 coincides with (Yb [ν] ×Σ b Σ)ν∈N . Let X 1 denote the simplicial p-adic [ν] [ν] b fine log formal scheme (X ×Σ b Σ)ν∈N over Σ and let i1 = (i1 ) denote the immersion X 1 → Y 1 induced by i. One can define a fibered ringed topos over Xétaff whose fiber over U ∈ Ob Xétaff is 0 ((P/U )´eN∼ t-f´ et , OP/U,• ) and whose inverse image functor by a morphism U → U in Xétaff is the inverse image functor of ringed topos (cf. (IV.6.1.12), (IV.6.1.17)). By taking the base change of the fibered ringed topos by the functor X : ∆◦ → Xétaff and considering the category of sections of the associated ringed ∆-topos, we obtain a ringed topos bis ((P/X)´eN∼ Définition (1.2.1), (1.2.5), Proposition (1.2.12), Définit-f´ et , OP/X,• ) (cf. [2] V tion (1.3.1), (1.3.4)). Note that a fibered topos over a category D is the same as “une catégorie bifibrée en duaux de topos au-dessus de D” in [2] Vbis Définition (1.2.2). An object [ν] [ν] on (P/X [ν] )´eNt-f´et of (P/X)´eN∼ t-f´ et is a data (G , ρs )[ν]∈Ob ∆,s∈Mor∆ consisting of a sheaf G N −1 [ν] [µ] for each ν ∈ N and a morphism ρs : (sP,´et-f´et ) (G ) → G for each (s : [ν] → [µ]) ∈ Mor(∆), where s denotes the morphism X [µ] → X [ν] corresponding to s, such that t s −1 ρid = id and ρst = ρs ◦ (sN (ρt ) for two composable morphisms [κ] → [ν] → P,´ et-f´ et ) ◦ ∼ N ), (X étaff , OX 1 ), [µ]. Similarly, one can define ringed topos ((P/X)´eNt-f´et )N ∼ , OP/X,• N N (X N∼ étaff , OX 1 ,• ), ((X étaff )

◦

∼

r,∼ N , OX ), (X ∼ 1,´ et , OX 1 ), ((X 1 /D(Σ))HIGGS , OX 1 ,•

) and 1 /D(Σ),1 ∼ ((X 1 /D(Σ))HIGGS , OX /D(Σ),1 ) by using the fibered ringed topos over Xétaff defined by 1 ◦ N ∼ ((P/U )´eNt-f´et )N ∼ , OP/U,• ), (Uétaff , OU1 ), etc. for U ∈ Ob Xétaff . We may naturally regard ∼ an object of X 1,´et as an object of X ∼ étaff . ∆ be the constant simplicial object of Xétaff defined by X. Then the morphism Let CX ∆ of functors X → CX induces morphisms of ringed topos θ : ((P/X)´eN∼ t-f´ et , OP/X,• ) → N∼ ∼ bis ((P/X)´et-f´et , OP/X,• ), θ : (X ∼ , O ) → (X , O ), etc. (cf. [2] V (2.1.3), (2.2.1)). X X étaff 1 étaff 1 ∼ For (Xétaff , OX1 ), we have an isomorphism of functors ∼ =

∗

idD+ (Xétaff ,OX1 ) −→ Rθ∗ θ .

(IV.6.7.26)

By (IV.6.5.17) and the remark following it, we see that the morphisms of ringed h,N◦ N∼ N∼ topos πU,´ et-f´ et , OP/U,• ) → (Uétaff , OU1 ,• ) for U ∈ Ob Xétaff induce a Cartesian et : ((P/U )´ morphism between the corresponding fibered ringed topos and then a morphism of ringed topos ◦

h,N N∼ N∼ πX,´ et-f´ et , OP/X,• ) −→ (X étaff , OX 1 ,• ) et : ((P/X)´

compatible with θ’s (cf. [2] Vbis Proposition (1.2.15), Lemme (1.2.16)). Similarly one can construct morphisms of ringed topos ◦

h,N N N N (πX,´ et ) : (((P/X)HIGGS )

◦

∼

◦

N N , OP/X,• ) −→ ((X N étaff )

∼

N , OX ), 1 ,•

∼ uX/D(Σ) : ((X/D(Σ))?∼ HIGGS , OX/D(Σ),1 ) −→ (X 1,´ et , OX 1 ),

r∼ µr? : ((X/D(Σ))?∼ HIGGS , OX/D(Σ),1 ) −→ ((X/D(Σ))HIGGS , OX/D(Σ),1 )

440

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

compatible with θ’s. Here ? = s ∈ N or ∅ and assume s ≥ r for the last morphism. By (IV.6.6.4) the morphisms of topos ← l (resp. ← l N ) induce a morphism of ringed topos − − ◦

◦

∼ N ∼ N ∼ N N , OX ) → (X N (X N∼ étaff , OX 1 ,• ) → (X étaff , OX 1 ) (resp. ((X étaff ) étaff , OX 1 )), which is 1 ,• also denoted by ← l (resp. ← l N ). Similarly, by (IV.6.6.9), the morphisms of topos → − − −l ◦

N N induce morphisms of ringed topos ((P/X)´eN∼ t-f´ et , OP/X,• ) → (((P/X)´ et-f´ et )

∼

◦

N , OP/X,• ),

N ∼ N (X ∼ étaff , OX 1 ) → (X étaff , OX 1 ), etc., which are also denoted by → −l . They are compatible with θ’s. Let r ∈ N>0 and let (F, F ◦ ) be an object of HCrZp (X1 /D(Σ)). Let (F [ν] , F [ν],◦ ) be [ν]

the pull-back of (F, F ◦ ) by the morphism X1 → X1 . Then the sheaves of OP/X [ν] ,• modules TX [ν] ,• (F [ν],◦ ) and the isomorphisms s∗HIGGS (TX [ν] ,• (F [ν],◦ )) → TX [µ] ,• (F [µ],◦ ) (cf. Lemma IV.6.4.6) for morphisms s : [ν] → [µ] in ∆ define a sheaf of OP/X,• -modules ◦ on (P/X)´eN,∼ t-f´ et , which is denoted by TX,• (F ). By Lemma IV.6.4.6, we have a canonical isomorphism ∗ θ (TX,• (F ◦ )) ∼ (IV.6.7.27) = TX,• (F ◦ ). ?+1 N [ν],◦ -modules TX,• Similarly we obtain a complex of OP/X,• ΩY (F ◦ ) from TX?+1 ) [ν] ,• ΩY [ν] (F

and (IV.6.7.23) applied to the morphism X [µ] → X [ν] associated to a morphism s : [ν] → [µ] in ∆. By (IV.6.7.24), we see that the morphisms (IV.6.7.12) for X [ν] ,→ Y [ν] and ◦ N -modules on ((P/X)´eNt-f´et )N ∼ F [ν],◦ define a morphism of complexes of OP/X,• (IV.6.7.28) h Let RπX∗

◦ ◦ ?+1 → −l ∗ TX,• (F ) −→ TX,• ΩY (F ). denote the composition (cf. Lemma IV.6.7.18). ◦

◦

h,N N ∗ N N ∼ N + lN , OP/X,• ) → D+ (X ∼ ∗ ◦ R(πX,´ et-f´ et ) étaff , OX 1 ,Q ). et )∗ : D (((P/X)´ → −l ◦ Q ⊗ ◦R← − Then, since the four functors appearing above can be computed fiber by fiber (by [2] Vbis Corollaire (1.3.12) for the right derived functors), Lemma IV.6.7.18 implies that (IV.6.7.28) induces an isomorphism ∼ =

◦ ◦ ?+1 h h (→ RπX∗ −l ∗ TX,• (F )) −→ RπX∗ (TX,• ΩY (F ))

h in D+ (X ∼ 1,étaff , OX 1 ,Q ). By the same argument as after Lemma IV.6.7.18, we see RπX∗ ◦ ◦ h,N ∼ l ∗ ◦ RπX,´et∗ and obtain an isomorphism → −l ∗ = Q ⊗ ◦R← −

(IV.6.7.29)

∼ =

◦

h,N ◦ ◦ ?+1 h Q ⊗ R← l ∗ RπX,´ et∗ TX,• (F ) −→ RπX∗ (TX,• ΩY (F )). − ∗

◦

◦

∗

h,N h,N Lemma IV.6.7.30. The base change morphism θ R← l ∗ RπX,´ l ∗ RπX,´ et∗ → R← et∗ θ be− − ∼ + N + tween the functors from D ((P/X)´et-f´et , OP/X,• ) to D (X étaff , OX 1 ) is an isomorphism. ◦

h,N Proof. Since the right derived functors R← l ∗ and RπX,´ et∗ appearing in the target − bis are computed fiber by fiber (cf. [2] V Corollaire (1.3.12)), it suffices to prove the h,N◦ base change theorems for R← l−∗ and Rπ−,´ et∗ with respect to a strict étale morphism f : X 0 → X. By Corollary IV.6.1.4 (resp. Lemma IV.3.1.10 applied to Xétaff and X 0 ), [47] (1.1) Proposition b), and [2] V Proposition 4.11 1), the inverse image of an injective sheaf of OX/P,• (resp. OX1 ,• )-modules by f is again injective. Hence it suffices to verify h,N◦ the base change theorems for the functors ← l ∗ and π−,´  et∗ , which are obvious. −

From (IV.6.7.27) and (IV.6.7.29) and Lemma IV.6.7.30, we obtain an isomorphism

(IV.6.7.31)

∗

◦

∼ =

h,N ◦ ?+1 ◦ h θ (Q ⊗ R← l ∗ RπX,´ et∗ TX,• (F )) −→ RπX∗ (TX,• ΩY (F )). −

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

441

s Let F denote the sheaf of OX/D(Σ),1,Q -modules θ∗ (F) on (X/D(Σ))r∼ HIGGS . Let F ,

F [ν],s , and F s for s ∈ N≥r (resp. F † , F [ν],† , and F † ) be the inverse image of F, F [ν] , and F by the inverse image functors µ∗rs (resp. µ∗r ). The sheaf F [ν],s (resp. F [ν],† ) is [ν] canonically isomorphic to the inverse image of F s (resp. F † ) by the morphism X1 → X1 , ∗ ∗ and we have canonical isomorphisms F s ∼ = θ (F s ) and F † ∼ = θ (F † ). Hence the sheaf s † F (resp. F ) is canonically isomorphic to the sheaf defined by the system (F [ν],s )ν (resp. (F [ν],† )ν ). [ν] [ν] s Let D[ν],s (s ∈ N>0 ) be DHiggs (X1 ,→ Y• ), let z [ν],s be the natural morphism [ν],s

[ν]

[ν]

[ν]

D1 → X1 , and let (M[ν],s , θ) (s ≥ r) denote the object of HBsQp ,conv (X1 , Y• /D(Σ)) corresponding to F [ν],s by Theorem IV.3.4.16. We define the complex CY [ν] (M[ν],s ) of [ν],s OX [ν] ,Q -modules to be z∗ (M[ν],s ⊗O [ν] Ω• [ν] ). For a morphism τ : [ν] → [µ], we Y1

1

Y1 /Σ

have a natural morphism τ ∗1,´et (CY [ν] (M[ν],s )) → CY [µ] (M[µ],s ) compatible with s and [µ]

[ν]

with compositions. Here τ 1 denotes the morphism X1 → X1 corresponding to τ . These data define a direct system of complexes of OX 1 ,• -modules (CY (Ms ))s≥r on X ∼ 1,´ et , which we also regard as that on X ∼ étaff . Now, by (IV.6.7.25), the morphisms (IV.6.7.14) for [ν] X [ν] ,→ Y [ν] , Y• and (F [ν],s , F [ν]◦ ) induce a morphism of complexes of OX 1 ,Q -modules ?+1 h lim CY (Ms ) −→ πX∗ (TX,• ΩY (F ◦ )) −→

(IV.6.7.32)

s

on

X∼ étaff ,

where

h πX∗

denotes the composition ◦

h,N ∗ N N N lN ∗ ◦ (πX,´ et-f´ et ) et∗ ) : (((P/X)´ → −l ◦ Q ⊗ ◦← −

◦

∼

N ) → (X ∼ , OP/X,• étaff , OX 1 ,Q ).

Proposition IV.6.7.33. There exists a canonical isomorphism in D+ (X ∼ étaff , OX 1 ,Q ) RuX

1 /D(Σ)∗

(F † ) ∼ C (Ms ). = lim −→ Y s

Proof. By Proposition IV.4.5.5, we have a resolution F [ν]† −→ lim µ∗s (LY [ν] (ξ −• M[ν],s ⊗O [ν] Ω•Y [ν] /Σ )) −→ • Y1 1 s

for each ν ∈ N. By construction, we have a canonical morphism τ ∗1,HIGGS (LY [ν] (ξ −• M[ν],s ⊗O

[ν] Y1

•

[µ] X1

Ω•Y [ν] /Σ )) −→ LY [µ] (ξ −• M[µ],s ⊗O

[µ] Y1

•

1

Ω•Y [µ] /Σ ) 1

[ν] X1

→ corresponding to a morphism τ : [ν] → [µ]. This for the morphism τ : morphism is compatible with s, the compositions of τ ’s, and the resolution of F [ν]† above. Thus we obtain a resolution F † → C Y• (M) of F † as an OX /D(Σ),1,Q -module on 1

(X 1 /D(Σ))∼ HIGGS . Since the right derived functor RuX 1 /D(Σ)∗ can be computed fiber by fiber ([2] Vbis Corollaire (1.3.12)), applying the argument before Theorem IV.4.5.6 to each X[ν] , we obtain RuX

1 /D(Σ)∗

(C Y• (M)) ∼ = uX

1 /D(Σ)∗

(C Y• (M)) ∼ C (Ms ). = lim −→ Y



s

Proposition IV.6.7.34. The following base change morphism in D+ (X ∼ étaff , OX 1 ,Q ) is an isomorphism ∗

θ RuX1 /D(Σ)∗ (F † ) −→ RuX

∗

1 /D(Σ)∗

θ (F † ) ∼ = RuX

1 /D(Σ)∗

(F † ).

442

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

Proof. Since RuX /D(Σ)∗ can be computed fiber by fiber (cf. [2] Vbis Corollaire 1 (1.3.12)), it suffices to prove that the base change morphism ∗ q † q ∗ † f1,´ et (R uX1 /D(Σ)∗ F ) → R uX 0 /D(Σ)∗ (f1,HIGGS (F )) 1

is an isomorphism for a strict étale morphism f : X 0 → X such that X 0 is affine. As before Theorem IV.4.5.6, we have an isomorphism ∼ =

lim Rq uX1 /D(Σ)∗ F s −→ Rq uX1 /D(Σ)∗ F † −→

s≥r

∗ and a similar isomorphism for X 0 and f1,HIGGS (F s ). Thus we are reduced to proving the base change theorem for RuX1 /D(Σ)∗ : D+ ((X1 /D(Σ))rHIGGS , OX1 /D(Σ),1,Q ) → D+ (X1,´et , OX1 ,Q ) with respect to f1 : X10 → X1 . By Proposition IV.3.1.9 (1) and [2] V Proposition 4.11 1), the inverse image of an injective sheaf of OX1 /D(Σ),1,Q -modules by f1 is injective. Hence it suffices to prove the base change theorem for uX1 /D(Σ)∗ , which is obvious. 

From (IV.6.7.32), Proposition IV.6.7.33, and Proposition IV.6.7.34, we obtain a morphism ∗

◦ ?+1 h θ RuX1 /D(Σ)∗ (F † ) −→ RπX,´ et∗ (TX,• ΩY (F ))

(IV.6.7.35)

in D+ (X ∼ étaff , OX 1 ,Q ). Composing (IV.6.7.35) with the inverse of (IV.6.7.31), we obtain a morphism (IV.6.7.36)

∗

∗

◦

h,N ◦ θ RuX1 /D(Σ)∗ (F † ) −→ θ (Q ⊗ R← l ∗ RπX,´ et∗ TX,• (F )). −

By taking Rθ∗ and using (IV.6.7.26), we obtain the morphism (IV.6.7.1).

IV.6.8. Comparison theorem. We keep the notation in the previous subsection. In this subsection, we will prove the following theorem. Theorem IV.6.8.1. Assume that the log structure of Σ is defined by the closed point and X satisfies Condition IV.5.3.1 strict étale locally. Then, for any r ∈ N>0 and an object (F, F ◦ ) of HCrZp (X1 /D(Σ)), the morphisms (IV.6.7.1) and (IV.6.7.2) are isomorphisms. The claim for (IV.6.7.2) obviously follows from that of (IV.6.7.1). By the construction of (IV.6.7.1), we are reduced to the case where X is affine and satisfies Condition IV.5.3.1, F and F ◦ are finite on X1 (cf. Definitions IV.3.3.1, IV.3.3.2), and X ∈ Ob CX b × b Σ has a (cf. Definition IV.6.4.1 (1)), which we assume in the following. Then X1 = X Σ smooth Cartesian lifting X• → D(Σ) and we can construct the morphism (IV.6.7.1) by using id : X → X and X• . Let C denote C X = CX in the following (cf. the construction r r of AX,Y,m in IV.6.7). By the construction of AX,X,• before (IV.6.7.15), the morphisms N z(U,s) : D(U, s)1 → U1 for (U, s) ∈ Cgpt induce a morphism of OC1 ,• -modules on Ctr (IV.6.8.2)

h r OC1 ,• −→ πC,gpt∗ AX,X,•

compatible with r ∈ N>0 . By applying ηC (IV.6.7.4) and using (IV.6.7.5), we obtain a N morphism of OX1 ,• -modules on Xétaff (IV.6.8.3)

◦

h,N r OX1 ,• −→ πX,´ et∗ ηP/C (AX,X,• )

compatible with r ∈ N>0 . We will derive Theorem IV.6.8.1 from the isomorphism (IV.6.7.19) and the following theorem, which is an analogue of Theorem IV.5.3.4.

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

443

Theorem IV.6.8.4. The following morphism in D+ (Xétaff , OX1 ,Q ) induced by (IV.6.8.3) is an isomorphism. ◦

h,N N ?+1 ∗ ∗ OX1 ,Q ∼ lN =→ ∗ R(πX,´ et )∗ ηP/C (AX,X,• ). −l Q ⊗ → −l ∗ OX1 −→ → −l Q ⊗ R← −

Proposition IV.6.8.5. Let U , d, s, A, A1r (A) be as in IV.5.3. Let AV be A ⊗V V . Then there exist a positive integer N depending only on d and positive integers Mr (r ∈ N>0 ) depending only on d and r such that, for any r, m, i ∈ N>0 , the following natural homomorphisms are annihilated by pMr . H i (∆(U,s) , A1r (A)/pm ) −→ H i (∆(U,s) , A1r+N (A)/pm ),

Ker(AV /pm → H 0 (∆(U,s) , A1r (A)/pm )) → Ker(AV /pm → H 0 (∆(U,s) , A1r+N (A)/pm )),

Cok(AV /pm → H 0 (∆(U,s) , A1r (A)/pm )) → Cok(AV /pm → H 0 (∆(U,s) , A1r+N (A)/pm )).

Proof. Let A1r,◦ (A) and A1r◦ (A∞ ) be as before Corollary IV.5.3.8. Then as in the proof of Corollary IV.5.3.8, the kernel and cokernel of H q (∆∞ , A1r,◦ (A∞ )/pm ) → H q (∆(U,s) , A1r,◦ (A)/pm ) are annihilated by m for every q ∈ N. We define a A r (a ∈ N ∩ [0, d], r ∈ N>0 ) and γi ∈ ∆∞ (i ∈ N ∩ [1, d]) as before Proposition IV.5.3.9. We define the complexes Car (a ∈ N ∩ [0, d], r ∈ N>0 ) as in the proof of Theorem IV.5.3.4. Then RΓ(∆∞ , A1r,◦ (A∞ )/pm ) is canonically isomorphic to Cdr /pm Cdr in D+ (Z/pm -Mod). Hence it suffices to prove the following claim; the claim for a = d implies the proposition. r For a ∈ N ∩ [0, d] and r ∈ N>0 , let C a be the mapping fiber of a A r → Car . Then there exist Na and Mr,a for a ∈ N ∩ [0, d] and r ∈ N>0 such that the following holds: The r

r+Na

natural homomorphism H q (C a ) → H q (C a ) is annihilated by pMr,a for a ∈ N ∩ [0, d], r ∈ N>0 , and q ∈ Z. We prove the claim by induction on a. The claim is trivial if a = 0 because 0 A r = C0r . r Let a ∈ N ∩ [1, d] and assume that the claim holds for C a−1 . By the construction of Car , r r r C a is canonically isomorphic to the mapping fiber of (a A r → Ca−1 ) −→ (0 → Ca−1 ). r r r r Hence, letting Da (resp. E a ) denote the mapping fiber of γa −1 : C a−1 → C a−1 (resp. the complex

a−1 A

r

γa −1

/a A r −−−→ a A r → 0 → · · · ), we obtain a short exact sequence r

r

r

0 −→ C a −→ Da −→ E a −→ 0.

r

r+2Na−1

) By the induction hypothesis, we see that the homomorphism H q (Da ) → H q (Da r is annihilated by p2Mr,a−1 for every q ∈ Z. By Proposition IV.5.3.9, we have H q (E a ) = r r+1 0 (q 6= 1) and the homomorphism H 1 (E a ) → H 1 (E a ) is annihilated by pla,r for r r+2Na−1 +1 some la,r ∈ N. Hence the homomorphism H q (C a ) → H q (C a ) is annihilated by p2Mr,a−1 +la,r .  Let us consider the following commutative diagram of ringed topos. (IV.6.8.6) ((P/C)´eN∼ t-f´ et , OP/C,• ) h πC,´ et iiii tiii id´eX,C t (C´eN∼ , O ) C1 ,• t vP/C

vC

 N∼ (Ctr , OC1 ,• )

 v ((P/C)N∼ f´ et , OP/C,• ) h i πC ii tiiii X,C idtr

idX,C P,´ et-f´ et

/ ((P/X)N∼ , OP/X,• ) ´ et-f´ et hhh shhhhh / (X N∼ , OX ,• ) vP/X 1 étaff h πX,´ et

idX,C P,f´ et vX

 / (X N∼ , OX ,• ) tr 1

 v / ((P/X)N∼ f´ et , OP/X,• ) h πX h hh thhhhh

444

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

X,C ∗ ∗ Lemma IV.6.8.7. (1) The base change morphisms vP/X idX,C P,f´ et∗ −→ idP,´ et-f´ et∗ vP/C and

X,C ∗ ∗ vX idX,C tr∗ −→ id´ et∗ vC = ηC are isomorphisms. (2) The morphism (IV.6.7.5) coincides with the composition of the following morphisms, where (∗) is the base change morphism for the right vertical face in (IV.6.8.6). ∼ ∼ = = h ∗ h ∗ h ∼ ∗ h X,C rC −−−−−−→ vX idX,C ←−− vX idX,C ηC πC,gpt∗ tr∗ πC∗ rC = vX πX∗ idP,f´ tr∗ πC,gpt∗ − et∗ (1)

(IV.6.5.11)

∼ =

(∗)

X,C h h ∗ −→ πX,´ −→ πX,´ et∗ vP/X idP,f´ et∗ ηP/C . et∗ rC − (1)

Proof. (1) We prove that the first morphism is an isomorphism. Since idX,C∗ P,´ et-f´ et is an equivalence of categories, it suffices to prove that it is an isomorphism after tak∼ = ∗ X,C∗ X,C ∗ ing idX,C∗ P,´ et-f´ et ◦ and composing with idP,´ et-f´ et idP,´ et-f´ et∗ vP/C → vP/C . The morphism thus

adj X,C X,C∗ X,C ∗ ∗ ∼ ∗ obtained coincides with idX,C∗ P,´ et-f´ et vP/X idP,f´ et∗ = vP/C idP,f´ et idP,f´ et∗ −→ vP/C . Hence it

X,C suffices to prove that adj : idX,C∗ P,f´ et idP,f´ et∗ → id is an isomorphism, which is reduced to ∗ ∼ = \ \ X,C X,C id et∗ → id, i.e., the fully faithfulness of the functor (P/C) → (P/X) (cf. [2] I P,f´ et idP,f´ Proposition 5.6). One can prove the second isomorphism by the same argument. et, tr, (2) We write v, π, and ι for v• , π•h , and idX,C ? , where • = X, C, etc. and ? = ´ etc. to simplify the notation. Then by considering the compositions of the base change morphisms for vertical faces in the diagram (IV.6.8.6), we obtain the following commuv tative diagram of the functors from Mod((P/C)f´et , OP/C,• ) to Mod(Xétaff , OX1 ,• ).

/ ι ∗ π∗ v ∗

ι∗ v ∗O π∗ ∼ = (1)

v ∗ ι∗ π∗

∼ =

/ π∗ ι∗ v ∗ O ∼ = (1)

∼ =

/ v ∗ π∗ ι∗

/ π∗ v ∗ ι∗ . 

By composing with rC from the left, we obtain the claim.

Proposition IV.6.8.8. Let N and Mr (r ∈ N>0 ) be as in Proposition IV.6.8.5. Then, for any r, m, i ∈ N>0 , the following natural morphisms are annihilated by pMr . ◦

◦

h,N r+N r i h,N Ri πX,´ et∗ (ηP/C (AX,X,• )) −→ R πX,´ et∗ (ηP/C (AX,X,• )), ◦

◦

h,N h,N r+N r Ker(OX1 ,• → πX,´ et∗ (ηP/C (AX,X,• ))) −→ Ker(OX1 ,• → πX,´ et∗ (ηP/C (AX,X,• ))), ◦

◦

h,N h,N r+N r Cok(OX1 ,• → πX,´ et∗ (ηP/C (AX,X,• ))) −→ Cok(OX1 ,• → πX,´ et∗ (ηP/C (AX,X,• ))).

X,C r r v Proof. Let A•,f´ et , OP/X,• ). By et be the object idP,f´ et∗ rC AX,X,• of Mod((P/X)f´ X,C ∗ r ∼ Lemma IV.6.8.7 (1), we have isomorphisms OX1 ,• ∼ idtr∗ OC1 ,• and ηP/C AX,X,• = vX = ∗ r vP/X A•,f´et . On the other hand, the morphism (IV.6.8.2) induces a morphism ∼ = X,C h X,C h r r r ∼ h idX,C et . tr∗ OC1 ,• −→ idtr∗ πC,gpt∗ AX,X,• −→ idtr∗ πC∗ rC AX,X,• = πX∗ A•,f´ ∗ Lemma IV.6.8.7 (2) implies that the morphism (IV.6.8.3) is obtained by taking vX of ∗ h r the above morphism and composing with the base change morphism vX πX∗ A•,f´ → et h ∗ r πX,´ v A . et∗ P/X •,f´ et h We have a canonical isomorphism Γ(U, Ri πX G) ∼ = Ri Γ((UK,triv )f´et , G|UK,triv ) for ∗ U ∈ Ob Xétaff and a sheaf of abelian groups G on (P/X)f´et (cf. [2] VI the first paragraph of the proof of Lemme 8.7.2). Hence Proposition IV.6.8.5 implies that the following

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

445

0 morphisms are annihilated by pMr after restricting to Ctr , where C 0 is the full subcategory of C consisting of U satisfying the condition in the beginning of IV.5.3. ◦

◦

h,N r+N r i h,N Ri πX∗ (A•,f´ et ) −→ R πX∗ (A•,f´ et ),

◦

◦

h,N h,N X,C r+N r Ker(idX,C et )) −→ Ker(idtr∗ OC1 ,• → πX∗ (A•,f´ tr∗ OC1 ,• → πX∗ (A•,f´ et )), ◦

◦

h,N h,N X,C r+N r Cok(idX,C et )) −→ Cok(idtr∗ OC1 ,• → πX∗ (A•,f´ tr∗ OC1 ,• → πX∗ (A•,f´ et )). X,C∗ X,C idX,C Note that the adjunction morphisms idX,C∗ tr∗ → id and idP,f´ tr et idP,f´ et∗ → id are iso∗ morphisms (cf. the proof of Lemma IV.6.8.7). Now, by taking vX and using Proposition IV.6.5.22, we obtain the claim. Note that every object of Xétaff admits a strict étale covering by objects of C 0 . 

Lemma IV.6.8.9. Let C be a site, let Z/p• be the sheaf of rings on C N defined by the inverse system of sheaves of rings (Z/pm )m∈N on C, and let (Z/p• )N denote the sheaf • N N◦ • N of rings → −l ∗ (Z/p ) on (C ) . Let K be the complex of (Z/p ) -modules bounded below ◦

on (C N )N , which corresponds to a direct system of complexes of Z/p• -modules (Kr )r∈N on C N . Assume that there exist N ∈ N and Mr ∈ N (r ∈ N) such that the morphism Hi (Kr ) → Hi (Kr+N ) is annihilated by pMr for every r ∈ N and i ∈ Z. Then we have ∗ lN ∗ (K) = 0. → −l Q ⊗ R← − Proof. We have a spectral sequence

b a+b N E2a,b = Ra ← lN l ∗ (K) ∗ H (K) =⇒ R − ← −

◦

a,b a in Mod(C N , → l ∗ Hb (Kr ). By the − −l ∗ Zp ), and the r-th component of E2 is given by R ← l ∗ Hb (Kr+N ) vanishes. assumption on Kr , the morphism Q ⊗ Ra ← l ∗ Hb (Kr ) → Q ⊗ Ra ← − − ∗ a+b N Hence → l (K) = 0.  −l Q ⊗ R ← −

Proof of Theorem IV.6.8.4. Consider a distinguished triangle ◦

h,N N ?+1 N OX −→ R(πX,´ et )∗ (ηP/C (AX,X,• )) −→ K −→ 1 ,• ◦

N N ). Then Proposition IV.6.8.8 implies that K satisfies the condition in D+ ((Xétaff )N , OX 1 ,• ∗  in Lemma IV.6.8.9. Hence → lN ∗ (K) = 0. −l Q ⊗ R← −

Proof of Theorem IV.6.8.1. Let (M, ε) be the object of HSZp (X1 , X• /D(Σ)) corresponding to F ◦ by the equivalence of categories in Proposition IV.3.4.4 (2), and let M be the image of M → MQp , which is naturally endowed with the structure ε of an object of HSZp (X1 , X• /D(Σ)) induced by ε (cf. Remark IV.3.4.17). Since we assume that F ◦ is finite on X1 , M and M are objects of LPM(OX1 ). Let (M, θ) denote the object of HBZp (X1 , X• /D(Σ)) corresponding to (M, ε) by Theorem IV.3.4.16. For an integer s ≥ r, the object of HBsQp (X1 , X• /D(Σ)) corresponding to the object µ∗rs (F) of HCsQp (X1 /D(Σ)) is canonically isomorphic to (MQp , θQp ). ∧ h ∧ Let Cgpt be the category of presheaves on Cgpt , and define the functors πgpt∗ : Cgpt → ∧ h∗ ∧ ∧ C and πgpt : C → Cgpt in the same way as before (IV.6.5.11) and (IV.6.5.12), reh∗ h spectively. The functor πgpt naturally becomes a left adjoint of πgpt∗ . We define the s s s s presheaves of rings AX,X on Cgpt by AX,X (U, s) = Γ(DX,X,1 (U, s), ODX,X,1 (U,s) ) (cf. the r s s X definition of AX,Y,m before (IV.6.7.15)). Put A T = AX,X ⊗ObC TX,gpt (F ◦ ) and gpt h∗ bC denotes the presheaf of A Ms = A s ⊗πh∗ O πgpt (M|C ) for s ∈ N, s ≥ r. Here O X,X

gpt

C1

gpt

446

IV. COHOMOLOGY OF HIGGS ISOCRYSTALS

rings limm OCgpt,m on Cgpt . Then, from Proposition IV.5.2.12, we obtain an isomorphism ←− s of complexes of AX,X,Q -modules on Cgpt p ∼ =

h∗ • s h∗ • c : A TQsp ⊗πgpt h∗ (O | ) πgpt (Ω h∗ (O | ) πgpt (Ω X/Σ |C ) −→ A MQp ⊗πgpt X/Σ |C ) X C X C

compatible with s. We assert that there exists N ∈ N such that pN c((A T s )/p-tor) ⊂ (A Ms )/p-tor and pN c−1 ((A Ms )/p-tor) ⊂ (A T s )/p-tor for every s. This is reduced to the claim for s = r and the section on (X, t) ∈ Cgpt because we have M(U ) = X X bC (U ) for M(X) ⊗OC1 (X) OC1 (U ) and TX,gpt (F ◦ )(U, s) = TX,gpt (F ◦ )(X, t) ⊗ObC (X) O gpt gpt

an object (U, s) of Cgpt over (X, t) (cf. Lemma IV.3.2.8). This claim is obvious because X bC (X))). The M(X) (resp. TX,gpt (X, t)) is an object of LPM(OC1 (X)) (resp. LPM(O gpt 0 last fact also implies that there exists N ∈ N such that the p-torsion parts of A Ms and 0 A T s are annihilated by pN for every s ≥ r by Lemma IV.6.8.10. Now, by using Lemma IV.6.8.11, we obtain morphisms h∗ • A T s ⊗πgpt h∗ (O | ) πgpt (Ω X/Σ |C ) o X C 0

ϕ ψ

/ A Ms ⊗ h∗ h∗ • πgpt (OX |C ) πgpt (ΩX/Σ |C )

0

0

0

such that ϕQ = pN +N c, ψQ = pN +N c−1 , ϕ ◦ ψ = p2(N +N ) id, and ψ ◦ ϕ = p2(N +N ) id. ∧,cont , OCgpt ,• ) obtained by Let A T•s Ω (resp. A Ms• Ω) denote the complex in Mod(Cgpt taking the reduction mod pm of the source (resp. target) of the morphism ϕ above. Let M• , ϕ• , and ψ• denote the inverse systems obtained from M, ϕ, and ψ by taking the reduction mod pm (m ∈ Z). Since (M/pm M)|C coincides with the reduction mod pm of M|C as a presheaf, we obtain morphisms of complexes in Mod(Ctr , OC1 ,• ) h πC,gpt∗ (ϕ• )

h h (M• ⊗OX Ω•X/Σ )|C −→ πC,gpt∗ (A Ms• Ω) −−−−−−−→ πC,gpt∗ (A T•s Ω) 0

compatible with s. The morphism (IV.6.7.20) is obtained by applying p−N −N · Q ⊗ ← l∗ − to the composition of the above morphisms. By taking ηC and composing with (IV.6.7.5), we obtain morphisms of complexes in Mod(Xétaff , OX1 ,• ) h πX,´ et∗ ηP/C (ϕ• )

h s h s M• ⊗OX Ω•X/Σ −→ πX,´ −−−−−−−−−→ πX,´ et∗ ηP/C (A M• Ω) − et∗ ηP/C (A T• Ω). 0

The morphism (IV.6.7.14) is obtained by taking p−N −N · Q ⊗ ← l ∗ of the composition − of the above morphisms. Hence it suffices to prove that the following morphism is an isomorphism for each q ∈ N. ◦

h,N ≥r ∗ ∗ l ∗ ≥r (M• ⊗OX ΩqX/Σ )s≥r −→ → l ∗ ≥r R(πX,´ (ηP/C (A Ms• Ωq ))s≥r . et )∗ → −l Q ⊗ ← − −l Q ⊗ R← − The last claim follows from Theorem IV.6.8.4 since M• ⊗OX ΩqX/Σ is a direct factor ⊕n of OX for some n ∈ N in the category LPM(OX1,• )Q by Lemma IV.3.2.6 (1) and 1,• Proposition IV.3.2.7 (2).  N

N

N

Lemma IV.6.8.10. Let A be a p-adically complete and separated algebra flat over Zp . Then for any object M of LPM(A), there exists an integer N satisfying the following condition. For any A-algebra A0 flat over Zp , the kernel of M ⊗A A0 → (M ⊗A A0 )Qp is annihilated by pN . Proof. By Lemma IV.3.2.2 (1), there exists N1 ∈ N such that pN1 (Ker(M → MQp )) = 0. Choose a surjective A-linear homomorphism f : A⊕r → M . Since MQp is a projective AQp -module, there exists an AQp -linear homomorphism g : MQp → A⊕r Qp g

such that fQp ◦ g = idMQp . Let h denote the composition M → MQp − → A⊕r Qp . Then

IV.6. COMPARISON WITH FALTINGS COHOMOLOGY

447

there exists N2 ∈ N such that pN2 h(M ) ⊂ A⊕r . The composition pN1 f ◦ pN2 h coincides with pN1 +N2 idM . By taking the scalar extension by A → A0 , we obtain a factorization M ⊗A A0 → (A0 )⊕r → M ⊗A A0 of the multiplication by pN1 +N2 on M ⊗A A0 .  Lemma IV.6.8.11. Let M be a Zp -module, let N ∈ N, let M be the image of M → MQp , and assume that Ker(M → MQp ) is annihilated by pN . Then the morphism pN : M → M uniquely factors through M → M . Proof. Obvious.



CHAPTER V

Almost étale coverings Takeshi Tsuji

V.1. Introduction In this chapter, we explain the theory of almost étale coverings introduced by G. Faltings in [24] I 2 and [26] 1, 2a. We can find a more systematic and complete argument in the book [32], where they work with the category of modules “modulo almost isomorphisms.” In this chapter, we try to follow faithfully the original approach by G. Faltings, and to give detailed proofs of the results in [26] 1, 2a and the beginning of 2c, adding some preliminaries if necessary. We do not discuss the so-called “almost purity theorem” for a variety with a certain type of log smooth reduction proved in [26] 2b; we refer the reader to II.6 for a detailed explanation of the theorem. In the introduction, we give a history of almost étale extensions briefly. The idea of “almost étaleness” dates back to the work [71] of Tate. Let K be a complete discrete valuation field of mixed characteristic (0, p) with a perfect residue field, let K be an algebraic closure of K, let C be the completion of K, let Kn be the subfield of K generated by pn -th roots of unity over K, and let K∞ denote the union ∪n∈N Kn . Let OK and OK∞ denote the rings of integers of K and OK∞ , respectively. Let v denote a valuation of K. Then, for any finite extension L of K contained in K, letting Ln := LKn , he proved that the valuation of a generator of the relative different DLn /Kn (n ∈ N) converges to 0 as n → ∞, i.e., the extension L∞ /K∞ is “almost étale.” (Precisely speaking, he considered an arbitrary ramified Zp -extension, and also determined the behavior of the valuations more precisely.) Using this almost étaleness, he computed the continuous Galois cohomology H q (Gal(K/K), C(r)) (q = 0, 1, r ∈ Z) and applied it to a p-divisible group over OK ; in particular, he proved the Hodge-Tate decomposition for a p-divisible group over OK . The above “almost étaleness” implies the vanishing of the higher continuous Galois cohomology H q (Gal(K/K∞ ), C(r)) (q > 0) (cf. Proposition V.12.8) and thus the computation of H q (Gal(K/K), C(r)) is reduced to that of H q (Gal(K∞ /K), C(r)) which is easier. The above results on the almost étaleness and the computation of the Galois cohomology were generalized to the imperfect residue field case by O. Hyodo [44], to an affine smooth scheme over OK having invertible coordinate functions and then to a certain affine log smooth scheme over OK by G. Faltings in [24] and [26], respectively. See also [23] for the case of curves. For a finite homomorphism A → B of flat algebras over OK∞ such that A[ p1 ] → B[ p1 ] is étale, an idempotent eB/A of (B ⊗A B)[ p1 ] is defined by the diagonal immersion, which is open and closed. In [24], Faltings defined the almost étaleness by the condition that mK∞ eB/A is contained in the image of B ⊗A B and the trace map B[ p1 ] → A[ p1 ] maps B into A. Here mK∞ denotes the maximal ideal of OK∞ . With this definition, he proved that, for a smooth algebra A over OK with invertible co−∞ −∞ ordinates t1 , . . . , td , A∞ := AOK ⊗OK OK∞ [t±p , . . . , t±p ] and a finite étale algebra 1 d 1 C over A∞ [ p ], the normalization of A∞ in C is almost étale over A∞ . This theorem is 449

450

V. ALMOST ÉTALE COVERINGS

called the “almost purity theorem” because the proof is based on the almost étaleness over the completion of the localization at a codimension one point lying on the special fiber. Using this generalization, he proved the Hodge-Tate decomposition of the étale cohomology of a proper smooth variety, and also gave, in [25], a proof of the crystalline conjecture: Ccrys by Fontaine. Actually the proof of the above almost purity theorem for a smooth ring in [24] has a gap in the Lefschetz argument. An alternative proof was given by Faltings in [26] being generalized to a log smooth setting. In the proof, he needed the notion of almost étaleness for algebras over W ( lim OK /pOK ) on which p is nilpotent and the theory of ←− xp ←x almost étale extension in [24] was generalized to a more general setting applicable to such algebras. Finally we should mention that the almost purity theorem also plays a key role in the theory of p-adic Simpson correspondence by G. Faltings [27]: the main theme of this book and that the almost purity theorem by Faltings was generalized to more general rings by P. Scholze in [64] Notation. As in [26] 1, let V denote a commutative ring endowed with a sequence of principal ideals mα ( V indexed by α ∈ Λ+ , where Λ is a subgroup of Q dense in R and Λ+ = {α ∈ Λ|α > 0}. Choose a generator π α of mα for each α ∈ Λ+ . We further assume that π α (α ∈ Λ+ ) are nonzero divisors in V and, for α, β ∈ Λ+ , we have ∗ π α · π β = uα,β · π α+β , uα,β ∈ V . Set m := ∪α∈Λ+ mα . V.2. Almost isomorphisms Definition V.2.1. (1) We say that a homomorphism of V -modules M → N is an almost isomorphism if the kernel and the cokernel are annihilated by m. We denote it ≈ by M → N . (2) We say that a V -module M is almost zero if it is annihilated by m. We denote it by M ≈ 0. f

g

For homomorphisms of V -modules L → M → N , if two of f , g, and g ◦ f are almost isomorphisms, then so is the rest. This follows from the fact that, for any α ∈ Λ+ , there exist α1 , α2 ∈ Λ+ such that α = α1 + α2 . Lemma V.2.2. Let R be a V -algebra and let f : M → N be a homomorphism of Rmodules such that the kernel and the cokernel are annihilated by mα for an α ∈ Λ+ . Then, there exists an R-homomorphism g : N → M such that f ◦ g = π 3α · idN and g ◦ f = π 3α · idM .

Proof. Since π α ·Ext1R (N, Ker(f )) = 0 and π α ·N ⊂ f (M ), there exists g 0 : N → M such that f ◦ g 0 = (π α )2 · idN . Then f ◦ g 0 ◦ f = (π α )2 · f and, since π α · Ker(f ) = 0, (π α g 0 ) ◦ f = (π α )3 · idM . Hence, for g = π α · g 0 , we have g ◦ f = (π α )3 · idM and f ◦ g = (π α )3 · idN . 

Proposition V.2.3. Let R be a V -algebra and let f : M → M 0 , g : N → N 0 be two almost isomorphisms of R-modules. Then, the homomorphisms f ⊕g, f ⊗R g, ∧rR f (r ∈ Z, r ≥ 0), and HomR (M 0 , N ) → HomR (M, N 0 ); ϕ 7→ g ◦ ϕ ◦ f are almost isomorphisms. Similarly i for TorR i and ExtR . By Proposition V.2.3, for a V -module M , the canonical homomorphisms (V.2.4)

M = HomV (V , M ) → HomV (m, M )

m ⊗V M → V ⊗V M = M

V.2. ALMOST ISOMORPHISMS

451

are almost isomorphisms. Proposition V.2.5. Let f : M → N be a homomorphism of V -modules. (1) f is an almost isomorphism if and only if the homomorphism HomV (m, M ) → HomV (m, N ) induced by f is an isomorphism. (2) f is an almost isomorphism if and only if the homomorphism m⊗V M → m⊗V N induced by f is an isomorphism. Proof. The sufficiency is trivial by the remark before the proposition. The necessity follows from Lemma V.2.6 below.  Lemma V.2.6. Let M be a V -module almost zero. Then we have TorVi (m, M ) = 0 and ExtiV (m, M ) = 0 for all integers i ≥ 0. α1 Proof. For any α ∈ Λ+ and m ∈ M , we have π α ⊗ m = u−1 ⊗ π α2 · m = 0 α1 ,α2 π + in m ⊗V M , where α1 , α2 ∈ Λ such that α = α1 + α2 . Hence m ⊗V M = 0. Similarly, α2 for f ∈ HomV (m, M ) and α ∈ Λ+ , we have f (π α ) = π α1 · f (u−1 α1 ,α2 π ) = 0. Hence HomV (m, M ) = 0. Since mα is a free V -module of rank 1 and m = limα∈Λ+ mα , m is −→ flat over V . Hence TorVi (m, M ) = 0 for i > 0. Choose a strictly decreasing sequence αn ∈ Λ+ (n = 0, 1, . . .) converging to 0. Then, we can construct a free resolution of m: Φ

Ψ

0 ←− m ←− ⊕n≥0 V ←− ⊕n≥0 V ←− 0, P where Φ((xn )) = n≥0 π αn xn , Ψ((xn )) = (yn ), y0 = x0 , yn = xn − an−1 xn−1 (n ≥ 1), and an ∈ m (n ≥ 0) is defined by π αn = an · π αn+1 . Since the endomorphism Ψ∗ on HomV (⊕n≥0 V , M ) is the identity if M ≈ 0, this implies ExtVi (m, M ) = 0 for i ≥ 0.  Proposition V.2.5 implies that, for any V -module M , the canonical homomorphisms (see (V.2.4)) HomV (m, M ) −→ HomV (m, HomV (m, M )) m ⊗V (m ⊗V M ) −→ m ⊗V M

are isomorphisms. For a V -algebra R, we can define a product on HomV (m, R) by HomV (m, R) × HomV (m, R) → HomV (m ⊗V m, R ⊗V R) → HomV (m, R), ∼

where the first map is (f, g) 7→ f ⊗ g and the second one is induced by m ⊗V m → m and the multiplication R ⊗V R → R. We have the formula (f · g)(xy) = f (x)g(y) for f, g ∈ HomV (m, R) and x, y ∈ m. This product makes HomV (m, R) an R-algebra; the structure homomorphism is the canonical homomorphism R → HomV (m, R); a 7→ (x 7→ x · a). For a V -algebra homomorphism R → S, the induced homomorphism HomV (m, R) → HomV (m, S) is a ring homomorphism. Remark V.2.7. Let R be a V -algebra such that the canonical homomorphism R → HomV (m, R) is an isomorphism. Then, for two free R-modules L and M of finite rank, if ≈ we are given an R-module N , an almost isomorphism f : N → L, and a homomorphism g : N → M , then they induce a homomorphism from L to M ∼

∼

∼

L → HomV (m, L) ← HomV (m, N ) → HomV (m, M ) ← M by Proposition V.2.5.

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V.3. Almost finitely generated projective modules Proposition V.3.1. Let R be a V -algebra and let M be an R-module. Then the following conditions are equivalent: (i) ExtiR (M, N ) ≈ 0 for all R-modules N and all integers i > 0. (ii) Ext1R (M, N ) ≈ 0 for all R-modules N . (iii) For any surjective homomorphism f : L → N and homomorphism g : M → N of R-modules, and any α ∈ Λ+ , there exists a homomorphism of R-modules h : M → L such that f ◦ h = π α · g. (iv) For any α ∈ Λ+ , there exist a free R-module L and homomorphisms of R-modules f : M → L, g : L → M such that g ◦ f = π α · idM . Proof. (i)⇒(ii)⇒(iii) are trivial. (iii)⇒(iv): Choose a free R-module L and a surjection g : L → M . Then (iii) implies that, for any α ∈ Λ+ , there exists f : M → L such that g ◦ f = π α · idM . (iv)⇒(i): The composite of the homomorphisms ExtiR (M, N ) → ExtiR (L, N ) → ExtiR (M, N ) induced by f and g in the condition (iv) is the multiplication by π α and the middle term vanishes for i > 0.  Definition V.3.2. Let R be a V -algebra and let M be an R-module. We say that M is almost projective if M satisfies the equivalent conditions of Proposition V.3.1. Proposition V.3.3. (1) Let R be a V -algebra and let M → M 0 be an almost isomorphism of R-modules. Then, M is almost projective over R if and only if M 0 is almost projective. (2) Let R → R0 be a homomorphism of V -algebras and let M be an almost projective R-module. Then M ⊗R R0 is an almost projective R0 -module. ≈

≈

Proof. If M → M 0 , then ExtiR (M, N ) → ExtiR (M 0 , N ) (i > 0) for any R-module N (Proposition V.2.3). This implies (1). The claim (2) is trivial by the condition (iv) of Proposition V.3.1.  Definition V.3.4. Let R be a V -algebra and let M be an R-module. We say that M is almost finitely generated if, for any α ∈ Λ+ , there exist a finitely generated Rmodule N and homomorphisms of R-modules ϕα : M → N , ψα : N → M such that ψα ◦ ϕα = π α · idM and ϕα ◦ ψα = π α · idN . We say that M is almost finitely generated projective if it is almost finitely generated and almost projective. Lemma V.3.5. Let R be a V -algebra and let M be an R-module. Then M is almost finitely generated if and only if, for any α ∈ Λ+ , there exist a free R-module L of finite rank and a homomorphism of R-modules f : L → M such that π α · Cok(f ) = 0. Proof. The necessity is trivial. For f : L → M in the condition in Lemma V.3.5, N = f (L) is finitely generated and the multiplication by π α on M factors through N . Hence the inclusion N ,→ M and π α : M → N (⊂ M ) satisfies the condition of Definition V.3.4.  Proposition V.3.6. (1) Let R be a V -algebra and let f : M → M 0 be an almost isomorphism of R-modules. Then M is almost finitely generated if and only if M 0 is almost finitely generated. (2) Let R → R0 be a homomorphism of V -algebras and let M be an R-module. If M is almost finitely generated over R, then M ⊗R R0 is almost finitely generated over R0 . Proof. (1) follows from Lemma V.2.2 and (2) is trivial by definition.



V.4. TRACE

453

Proposition V.3.7. Let R be a V -algebra and let M be an R-module. Then, M is almost finitely generated projective if and only if, for any α ∈ Λ+ , there exist a free Rmodule L of finite rank and homomorphisms of R-modules f : M → L, g : L → M such that g ◦ f = π α · idM . Proof. The sufficiency follows from the condition (iv) of Proposition V.3.1 and Lemma V.3.5. Let us prove the necessity. By Lemma V.3.5, for any α ∈ Λ+ , there exist a free R-module L of finite rank and a homomorphism of R-modules g : L → M such that π α · Cok(g) = 0. Since the image of π α idM : M → M is contained in Im(g) and M is almost finitely generated and projective, there exists f : M → L such that g ◦ f = (π α )2 · idM .  Proposition V.3.8. Let R be a V -algebra and let M , N be almost finitely generated projective R-modules. Then HomR (M, N ), M ⊗R N , and ∧iR M (i ≥ 1) are also almost finitely generated projective. Proof. For α ∈ Λ+ , by Proposition V.3.7, there exist free R-modules of finite rank f1 g1 f2 g2 L1 , L2 and homomorphisms of R-modules M → L1 → M , N → L2 → N such that g1 ◦ f1 = π α · idM , g2 ◦ f2 = π α · idN . Then the composites of the homomorphisms HomR (M, N ) → HomR (L1 , L2 ) → HomR (M, N ) M ⊗R N → L1 ⊗R L2 → M ⊗R N ∧iR M → ∧iR L1 → ∧iR M

induced by f1 , g1 , f2 , and g2 are the multiplication by (π α )2 , (π α )2 , and (π α )i , respectively, and the middle modules are free R-modules of finite rank. By Proposition V.3.7, this completes the proof.  Proposition V.3.9. Let R → R0 be an almost isomorphism of V -algebras and let M be an R-module. Then R is almost projective (resp. almost finitely generated) over R if and only if M 0 := M ⊗R R0 is almost projective (resp. almost finitely generated) over R0 . Proof. The necessity is already proved (Proposition V.3.3 and Proposition V.3.6). Let us prove the sufficiency. Since the canonical homomorphism M → M 0 is an almost isomorphism (Proposition V.2.3), it suffices to prove that M 0 is almost projective (resp. almost finitely generated) over R by Proposition V.3.3 (resp. Proposition V.3.6). First note that L → L ⊗R R0 is an almost isomorphism for any free R-module L. By Lemma V.3.5, this implies the claim in the second case. In the first case, for α ∈ Λ+ , choose a free R-module L and R0 -linear homomorphisms f : M 0 → L ⊗R R0 and g 0 : L⊗R R0 → M 0 such that g 0 ◦f 0 = π α ·idM 0 . Then, by the above remark and Proposition ≈ V.2.3, ExtiR (L ⊗R R0 , N ) → ExtiR (L, N ) = 0 (i > 0) and hence (π α )2 · ExtiR (M 0 , N ) = 0 (i > 0) for any R-module N . Varying α, we see that M 0 is almost projective over R. 

V.4. Trace Lemma V.4.1. Let R be a V -algebra and let M and N be almost finitely generated projective R-modules. Then, the natural homomorphism M ⊗R HomR (N, R) → HomR (N, M ) is an almost isomorphism.

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V. ALMOST ÉTALE COVERINGS

Proof. For α ∈ Λ+ , choose Li , fi , gi (i = 1, 2) as in the proof of Proposition V.3.8. Then, we have a commutative diagram M ⊗R HomR (N, R) −−−−→ HomR (N, M )     y y L1 ⊗R HomR (L2 , R) −−−−→ HomR (L2 , L1 )     y y M ⊗R HomR (N, R) −−−−→ HomR (N, M ), where the vertical homomorphisms are induced by f1 , g1 , f2 , g2 . The composites of the vertical homomorphisms are the multiplication by (π α )2 and the middle homomorphism is an isomorphism. Hence the kernel and the cokernel of the homomorphism in question are annihilated by (π α )2 .  Let R be a V -algebra and let M be an almost finitely generated projective R-module. We define the trace map trR : EndR (M ) −→ HomV (m, R)

functorial on M as the composite:

EndR (M ) → HomV (m, EndR (M )) ∼

← HomV (m, M ⊗R HomR (M, R)) → HomV (m, R).

Here, the second homomorphism is an isomorphism by Lemma V.4.1 and Proposition V.2.5 (1), and the last homomorphism is induced by M ⊗R HomR (M, R) → R; m ⊗ f 7→ f (m). Proposition V.4.2. (1) Let R be a V -algebra and let M be a finitely generated projective R-module. Then, the trace map defined above coincides with the usual trace map EndR (M ) → R followed by the canonical homomorphism R → HomV (m, R). (2) Let R → R0 be a homomorphism of V -algebra and let M be an almost finitely generated projective R-module. Let f ∈ EndR (M ) and let f 0 ∈ EndR0 (M ⊗R R0 ) be the base change of f . Then trR0 (f 0 ) is the image of trR (f ) under the homomorphism HomV (m, R) → HomV (m, R0 ). 

Proof. Trivial by definition.

Proposition V.4.3. Let R be a V -algebra, let f : M → M 0 be an almost isomorphism of almost finitely generated projective R-modules, and let g and g 0 be R-linear endomorphisms of M and M 0 , respectively, such that f ◦ g = g 0 ◦ f . Then trR (g) = trR (g 0 ). Especially, if M ≈ 0, then the trace map trR : EndR (M ) → HomV (m, R) is 0. Proof. This follows from the commutative diagrams ∼

HomV (m, M ⊗R M ∗ ) ←−−−− HomV (m, M ⊗R M 0∗ )    o y y HomV (m, R) and

←−−−− HomV (m, M 0 ⊗R M 0∗ ) ∼

HomV (m, M ⊗R M ∗ ) ←−−−− HomV (m, M ⊗R M 0∗ )    o oy y ∼

HomV (m, M 0 ⊗R M ∗ ) ←−−−− HomV (m, M 0 ⊗R M 0∗ ),

V.5. RANK AND DETERMINANT

455



where M ∗ = HomV (M, R) and M 0∗ = HomV (M 0 , R).

Proposition V.4.4. Let R be a V -algebra and let M1 , M2 be two almost finitely generated projective R-modules. Then, for any R-linear endomorphisms f1 and f2 of M1 and M2 , respectively, we have trR (f ⊗ g) = trR (f ) · trR (g)

(resp. trR (f ⊕ g) = trR (f ) + trR (g)).

Proof. The first case follows from the commutative diagram: EndR (M1 ) ⊗R EndR (M2 ) x  

−−−−→

(M1 ⊗R M1∗ ) ⊗R (M2 ⊗R M2∗ )

∼ = (M1 ⊗R M2 ) ⊗R (M1∗ ⊗R M2∗ )   y

EndR (M1 ⊗R M2 ) x  

−−−−→ (M1 ⊗R M2 ) ⊗R (M1 ⊗R M2 )∗   y ∼

R ⊗R R The second case is easy.

−−−−→

R. 

V.5. Rank and determinant To avoid the confusion, we will denote by trcl (resp. detcl ) the usual trace (resp. determinant) map for a projective (resp. free) module of finite rank. The following relation between usual determinant and trace will play a key role. Proposition V.5.1. Let R be a commutative ring and let M be a free R-module of finite rank. Then, for any R-linear endomorphism f of M , we have P∞ cl s detcl R (1 + f ) = s=0 trR (∧ f ). Note ∧s f = 0 if s ≥ rankR M + 1 and hence the right-hand sum is finite. Proof. Explicit computation. Choose a basis {ei |1 ≤ i ≤ r} of M , and describe (∧ (1 + f ))(e1 ∧ · · · ∧ er ) (= (detcl R (1 + f )) · e1 ∧ · · · ∧ er ) in terms of the matrix associated to ∧s f with respect to the base {ei1 ∧ · · · ∧ eis |1 ≤ i1 < · · · < is ≤ r} of ∧s M .  r

Let R be a V -algebra. The canonical injection HomV (m, R[T ]) ,→ HomV (m, R)[[T ]] is a ring homomorphism and we will regard the former as a subring of the latter in the following. For an almost finitely generated projective R[T ]-module M and an R[T ]-linear endomorphism F of M, we set (V.5.2)

det(1 + T · F ) :=

∞ X s=0

 =

T s · trR[T ] (∧s F ) ∈ HomV (m, R)[[T ]]

∞ X s=0

 trR[T ] (∧ (T · F )) . s

If M = M ⊗R R[T ] for an almost finitely generated projective R-module M and F arises from an R-linear endomorphism f of M by the base change R → R[T ], then, by Proposition V.4.2 (2), we have (V.5.3)

det(1 + T · F ) =

∞ X i=0

T i · trR (∧i f ).

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V. ALMOST ÉTALE COVERINGS

Proposition V.5.4. Let R be a V -algebra, let M be an almost finitely generated projective R-module, let f and g be R-linear endomorphisms of M , and let F and G denote the R[T ]-linear endomorphisms f ⊗ idR[T ] and g ⊗ idR[T ] of M ⊗R R[T ]. Then we have det(1 + T · F ) · det(1 + T · G) = det(1 + T · (F + G + T · (F ◦ G))).

Lemma V.5.5. Let R be a V -algebra and let ϕ : M → N and ψ : N → M be homomorphisms of almost finitely generated projective R-modules such that ψ ◦ ϕ = a · idM for an a ∈ R. Then, for an R-linear endomorphism f of M , we have for any integer s ≥ 0.

trR (∧s (ϕ ◦ f ◦ ψ)) = as · trR (∧s f )

Proof. Since ∧s ψ ◦ ∧s ϕ = as · id∧s M and ∧s (ϕ ◦ f ◦ ψ) = ∧s ϕ ◦ ∧s f ◦ ∧s ψ, it suffices to prove the lemma for s = 1, in which case, it follows from the commutative diagram: EndR (M ) ←−−−− M ⊗R HomR (M, R) −−−−→   ϕ◦−◦ψ ϕ⊗(−◦ψ) y y

R  a· y

EndR (N ) ←−−−− N ⊗R HomR (N, R) −−−−→ R.



+

Proof of Proposition V.5.4. Let α ∈ Λ and choose a free R-module L of finite rank and R-linear homomorphisms ϕ : M → L, ψ : L → M such that ψ ◦ ϕ = π α · idM (Proposition V.3.7). Set f 0 := ϕ◦f ◦ψ, g 0 := ϕ◦g◦ψ ∈ EndR (L) and let F 0 , G0 be the base changes of f 0 , g 0 by R → R[T ]. Let HomV (m, R)[[T ]] → P κα be the ring P homomorphism α n HomV (m, R)[[T ]] defined by κα ( n an T n ) = a (π T ) . Then, applying Lemma n n V.5.5 to ϕ, ψ, f (resp. g) and using (V.5.3), we obtain κα (det(1 + T · F )) = det(1 + T · F 0 ) (resp. κα (det(1 + T · G)) = det(1 + T · G0 )). On the other hand, by (V.5.2), Proposition V.5.1, and Proposition V.4.2 (1), we have det(1 + T · F 0 ) · det(1 + T · G0 ) = det(1 + T · (F 0 + G0 + T · F 0 ◦ G0 )).

We assert that the right-hand side coincides with κα (det(1 + T · (F + G + T · F ◦ G))). The base change of F + G + T · F ◦ G under R[T ] → R[T ]; T 7→ π α · T is F + G + π α T · F ◦ G. Hence, by Proposition V.4.2 (2), we obtain κα (det(1 + T · (F + G + T · F ◦ G))) =

∞ X (π α T )s trR[T ] (∧s (F + G + π α T · F ◦ G)). s=0

Let Φ and Ψ be the base changes of ϕ and ψ by R → R[T ]. Then Ψ ◦ Φ = π α · idM ⊗R R[T ] and Φ ◦ (F + G + π α T · F ◦ G) ◦ Ψ = F 0 + G0 + T · F ◦ G0 . Now the assertion follows from Lemma V.5.5. Thus we obtain the required equality modulo Ker(κα ). Since HomV (m, R) is mtorsion free, varying α, we obtain the exact equality.  Definition V.5.6. Let R be a V -algebra and let M be an almost finitely generated projective R-module. For an integer r ≥ 0, we say that M has rank ≤ r over R if ∧r+1 R M ≈ 0, and denote it by rankR M ≤ r. ≈

By Proposition V.2.3, if M → M 0 , then rankR M ≤ r if and only if rankR M 0 ≤ r. For a ring homomorphism R → R0 and M 0 := M ⊗R R0 , if rankR M ≤ r, then rankR M 0 ≤ r. ≈ ≈ r+1 The converse is also true if R → R0 . For the last claim, we use ∧r+1 R M → (∧R M ) ⊗R 0 ∼ r+1 0 R = ∧R0 M . We will only use the following consequence of Proposition V.5.4.

V.5. RANK AND DETERMINANT

457

Proposition V.5.7. Let R be a V -algebra, let M be an almost finitely generated projective R-module of rank ≤ r, and let f and g be two R-linear endomorphisms of M . Then, we have trR (∧r f ◦ g) = trR (∧r f ) · trR (∧r g). By setting f = g = idM , we obtain the following corollary. Corollary V.5.8. Let R, M be as in Proposition V.5.7. Then, trR (id∧r M ) is an idempotent of HomV (m, R). Proof of Proposition V.5.7. Let F , G ∈ EndR[T ] (M ⊗R R[T ]) denote the base changes of f and g. Since ∧s M ≈ 0 (s ≥ r+1), by Proposition V.4.3, trR (∧s f ), trR (∧s g), and trR[T ] (∧s (F + G + T · F ◦ G)) vanish for s ≥ r + 1. Hence, by Proposition V.5.4 and Pr (V.5.3), we see that trR (∧s f )·trR (∧s g) is the coefficient of T 2r in s=0 T s ·trR[T ] (∧s (F + G + T · F ◦ G)). For any f0 , fP 1 ∈ EndR (M ) and their base changes F0 , F1 by R → R[T ], s we have ∧s (F0 + T · F1 ) = t=0 T t · Gt,s , where Gt,s denotes the base change of the R-linear endomorphism gt,s : ∧s M → ∧s M defined by X gt,s (x1 ∧ · · · ∧ xs ) = fε1 (x1 ) ∧ · · · ∧ fεs (xs ). ε∈{0,1}s ]{i|εi =1}=t

Ps t Hence trR[T ] (∧s (F0 + T · F1 )) = t=0 T · trR (gt,s ). Now it is easy to see that the 2r r coefficient of T in question is trR (∧ f ◦ g).  Proposition V.5.9. Let R be a V -algebra and let M be an almost finitely generated projective R-module of rank ≤ r. Set L := ∧rR M and L∗ := HomR (L, R). Let t denote the homomorphism HomV (m, EndR (L)) → HomV (m, R) appearing in the definition of trR (V.4), and let h denote the homomorphism HomV (m, R) → HomV (m, EndR (L)) induced by R → EndR (L); a 7→ a · idL . Then t ◦ h = trR (idL ) · id and h ◦ t = id. Lemma V.5.10. Let R be a commutative ring, let M be an R-module, and let M ∗ denote HomR (M, R). Then, for any integer s ≥ 0, there exists a canonical R-linear homomorphism functorial on M : Φ : ∧s M ∗ −→ HomR (∧s M, R) characterized by Φ(f1 ∧ · · · ∧ fs )(x1 ∧ · · · ∧ xs ) =

X σ∈Ss

sgn(σ)fσ(1) (x1 ) ∧ · · · ∧ fσ(s) (xs ).

Furthermore, if M is free of finite rank, Φ is an isomorphism. Proof. Straightforward.



Lemma V.5.11. Let R be a V -algebra and let M be an almost finitely generated projective R-module. Then, for any integer s ≥ 0, the homomorphism Φ : ∧s M ∗ → HomR (∧s M, R) in Lemma V.5.10 is an almost isomorphism. Proof. The same argument as the proof of Lemma V.4.1.



Lemma V.5.12. Let R be a V -algebra and let M be an almost finitely generated projective R-module. Then the canonical homomorphism M → HomR (HomR (M, R), R) is an almost isomorphism. Proof. The same argument as the proof of Lemma V.4.1.



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V. ALMOST ÉTALE COVERINGS

Lemma V.5.13. Let M and N be two V -modules, let f , g : M → N be two V -linear homomorphisms, and let h : L → M be an almost isomorphism. Suppose that N is m-torsion free. Then, f = g if and only if f ◦ h = g ◦ h. 

Proof. Trivial.

Proof of Proposition V.5.9. For the first equality, by Lemma V.5.13, it suffices tr −·id to prove that the composite R →L EndR (L) →R HomV (m, R) is the canonical homomorphism ιR : R → HomV (m, R) (V.2.4) multiplied by trR (idL ), which is trivial. Let us prove the second equality. By Lemma V.5.13 and Lemma V.5.11, it suffices to prove that h◦trR and the canonical homomorphism ιEndR (L) : EndR (L) → HomV (m, EndR (L)) (V.2.4) coincides after composing with ≈

≈

idL ⊗Φ

Lemma V.4.1

κ : ∧r M ⊗R ∧r M ∗ −−−−→ L ⊗R L∗ −−−−−−−−→ EndR (L), where Φ is as in LemmaPV.5.10. Take µi ∈ M , λi ∈ M ∗ (1 ≤ i ≤ r) and define f ∈ EndR (M ) by f (x) = 1≤i≤r λi (x) · µi . Then we have κ((µ1 ∧ · · · ∧ µr ) ⊗ (λ1 ∧ · · · ∧ λr )) = ∧r f,

trR ◦ κ((µ1 ∧ · · · ∧ µr ) ⊗ (λ1 ∧ · · · ∧ λr )) = ιR (Φ(λ1 ∧ · · · ∧ λr )(µ1 ∧ · · · ∧ µr )). Hence it suffices to prove (∧r f )(φ1 ∧ · · · ∧ φr ) = Φ(λ1 ∧ · · · ∧ λr )(µ1 ∧ · · · ∧ µr ) · φ1 ∧ · · · ∧ φr modulo m-torsion for any φi ∈ M (1 ≤ i ≤ r). P Take νi ∈ M ∗ (1 ≤ i ≤ r) and define g ∈ EndR (M ) by g(x) = 1≤i≤r νi (x)φi . Then, trR (∧r f ) = ιR (Φ(λ1 ∧ · · · ∧ λr )(µ1 ∧ · · · ∧ µr )), trR (∧r g) = ι(Φ(ν1 ∧ · · · ∧ νr )(φ1 ∧ · · · ∧ φr )) and trR (∧r (f ◦ g)) = ιR (Φ(ν1 ∧ · · · ∧ νr )(∧r f (φ1 ∧ · · · ∧ φr ))). By Proposition V.5.7, we obtain ιR (Φ(ν1 ∧ · · · ∧ νr )(∧r f (φ1 ∧ · · · ∧ φr )))

=ιR (Φ(λ1 ∧ · · · ∧ λr )(µ1 ∧ · · · ∧ µr ) · Φ(ν1 ∧ · · · ∧ νr )(φ1 ∧ · · · ∧ φr )). Since we have almost isomorphisms ≈

L −→HomR (L∗ , R) (Lemma V.5.12) ≈

HomR (∧r M ∗ , R) (Lemma V.5.11, Proposition V.2.3) −→ ∗ Φ ≈

−→ HomR (∧r M ∗ , HomV (m, R)) (Proposition V.2.3),

ιR ◦−

varying νi (1 ≤ i ≤ r), we obtain the required congruence.



Definition V.5.14. Let R be a V -algebra and let M be an almost finitely generated projective R-module. For an integer r ≥ 0, we say that M has rank r over R if ∧r+1 R M ≈0 and trR (id∧r M ) = 1, and denote it by rankR M = r. ≈

By Proposition V.4.3, if M → M 0 , rankR M = r if and only if rankR M 0 = r. By Proposition V.4.2 (2), for a ring homomorphism R → R0 and M 0 := M ⊗R R0 , if rankR M = r, then rankR0 M 0 = r. By Proposition V.2.5, the converse is also true if ≈ R → R0 . Corollary V.5.15. Let R be a V -algebra, let M be an almost finitely generated projective R-module of rank ≤ r, and set L := ∧r M , L∗ := HomR (L, R). If trR (id∧r M ) = 1 (resp. 0), then the canonical homomorphism L ⊗R L∗ → R is an almost isomorphism (resp. rankR M ≤ r − 1, that is, L ≈ 0).

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Let R be a V -algebra and let M be an almost finitely generated projective Rmodule of rank ≤ r. Set e = trR (id∧r M ) and define Rr := HomV (m, R)/(e − 1), ∼ R0 := HomV (m, R)/(e). Then, by Corollary V.5.8, HomV (m, R) → Rr × R0 . Set 0 0 Mr := M ⊗R Rr and M := M ⊗R R . Then, by Proposition V.4.2 (2), trRr (id∧r Mr ) = 1 and trR0 (id∧rM 0 ) = 0. By Corollary V.5.15, rankRr (Mr ) = r and rankR0 M 0 ≤ r − 1. ∼

Repeating this procedure and using HomV (m, R) → HomV (m, HomV (m, R)), we obtain the following proposition.

Proposition V.5.16. Let R be a V -algebra and let M be an almost finitely generated projective R-module of rank ≤ r. Then, there exists a unique decomposition HomV (m, R) = Rr × · · · × R0 such that rankRi (M ⊗R Ri ) = i (0 ≤ i ≤ r). Proof. We have already proven the existence. Suppose that we have another decomposition Rr0 × · · · × R00 . Then the base change Mi,j (0 ≤ i, j ≤ r) of M to Ri,j := Ri ⊗HomV (m,R) Rj0 has rank i and j at the same time. Hence, by Corollary V.5.15, Ri,j ≈ 0 if i 6= j. On the other hand Ri,j is a direct factor of HomV (m, R). Hence Ri,j is m-torsion free and Ri,j = 0 if i 6= j, which implies Ri = Ri0 (0 ≤ i ≤ r).  By the uniqueness, the decomposition is compatible with base changes. For R, M and the decomposition in Proposition V.5.16, we define the determinant map detR : EndR (M ) −→ HomV (m, R) by detR (f ) = (trRi (∧i fi ))0≤i≤r , where fi denotes the base change of f to Ri . Note ∼ Ri → HomV (m, Ri ). We have detR (f ◦ g) = detR (f ) · detR (g) (Proposition V.5.7) and detR (idM ) = 1. The map detR is compatible with base changes. V.6. Almost flat modules and almost faithfully flat modules Definition V.6.1. Let R be a V -algebra and let M be an R-module. We say that M is almost flat if it satisfies the following equivalent conditions: (i) TorR i (M, N ) ≈ 0 for all R-modules N and all integers i ≥ 1. (ii) TorR 1 (M, N ) ≈ 0 for all R-modules N . (iii) For any injective homomorphism f : N1 → N2 of R-modules, the kernel of idM ⊗ f : M ⊗R N1 → M ⊗R N2 is almost zero. We say that an R-algebra is almost flat if it is almost flat as an R-module. ≈

Proposition V.6.2. (1) For a V -algebra R and an almost isomorphism M → M 0 of R-modules, M is almost flat if and only if M 0 is almost flat. (2) For a homomorphism R → R0 of V -algebra and an R-module M , if M is almost ≈ flat over R, then M 0 := M ⊗R R0 is almost flat over R0 . If R → R0 , then the converse is also true. Proof. (1) follows from Proposition V.2.3. For (2), use the condition (iii) in Def≈ inition V.6.1 and M ⊗R N → M 0 ⊗R N ∼ = M 0 ⊗R0 (N ⊗R R0 ) for an R-module N if ≈ R → R0 .  Proposition V.6.3. Let R be a V -algebra and let M be an R-module. If M is almost projective, then M is almost flat. Proof. This follows from the condition (iv) in Proposition V.3.1 similarly as the proof of (iv)⇒(i) of Proposition V.3.1. 

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Definition V.6.4. Let R be a V -algebra and let M be an R-module. We say that M is almost faithfully flat if it is almost flat and, for any R-module N , N ⊗R M ≈ 0 implies N ≈ 0. We say that an R-algebra is almost faithfully flat if it is almost faithfully flat as an R-module. ≈

Proposition V.6.5. (1) For a V -algebra R and an almost isomorphism M → M 0 of R-modules, M is almost faithfully flat if and only if M 0 is almost faithfully flat. (2) For a homomorphism of V -algebras R → R0 and an R-module M , if M is almost ≈ faithfully flat over R, then M 0 := M ⊗R R0 is almost faithfully flat over R0 . If R → R0 , the converse is also true. Proof. (1) follows from Proposition V.6.2 (1) and Proposition V.2.3. The first assertion of (2) is trivial. For the second assertion, we use almost isomorphisms M ⊗R ≈ ≈ N → R0 ⊗R (M ⊗R N ) ∼  = M 0 ⊗R0 (N ⊗R R0 ) and N → N ⊗R R0 . f

g

Proposition V.6.6. Let R → R0 → R00 be homomorphisms of V -algebras. If f and g are almost flat (resp. almost faithfully flat), then g ◦ f is also almost flat (resp. almost faithfully flat). Proof. The first case follows from the condition (iii) of Definition V.6.1 and Proposition V.2.3, and then the second case is trivial by definition.  Proposition V.6.7. Let R be a V -algebra and let M be an almost finitely generated projective R-module of rank r ≥ 1. Then M is almost faithfully flat over R. Lemma V.6.8. Let R be a V -algebra and let M be an almost flat R-module. Then M is almost faithfully flat if and only if, for any ideal I of R, M/IM ≈ 0 implies R/IR ≈ 0. Proof. The necessity is trivial. Let us prove the sufficiency. For an R-module N , suppose M ⊗R N ≈ 0. Then, for any x ∈ N , we have an injection R/I ,→ N ; a 7→ a · x, where I = {a ∈ R|a · x = 0}. The kernel of the induced homomorphisms M/IM → N ⊗R M is almost zero. Hence M/IM ≈ 0, which implies R/I ≈ 0, i.e., m · x = 0 by assumption.  Proof of Proposition V.6.7. Let I be an ideal of R and suppose M/IM ≈ 0. Then, since rankR/I (M/IM ) = r, we have trR (id∧r M/IM ) = 1. On the other hand, since M/IM ≈ 0, we have trR (id∧r M/IM ) = 0 by Proposition V.2.3 and Proposition V.4.3. Hence HomV (m, R/I) = 0 and R/I ≈ 0. By Lemma V.6.8 and Proposition V.6.3, M is almost faithfully flat.  The following lemma will be useful in V.8 and V.9. Lemma V.6.9. Let R be a V -algebra, let M be an almost flat R-module, and let C • be a complex of R-modules. Then, the canonical homomorphism H q (C • ) ⊗R M → H q (C • ⊗R M ) is an almost isomorphism for q ∈ Z. Especially, if H q (C • ) ≈ 0, then H q (C • ⊗R M ) ≈ 0, and the converse is also true if M is almost faithfully flat over R. Proof. Put Z q := Ker(dq : C q → C q+1 ) and B q := Im(dq−1 : C q−1 → C q ). Let C denote the complex C • ⊗R M and define Z 0q and B 0q similarly. Since the canonical homomorphism B q ⊗R M → B 0q is surjective and the kernel of B q ⊗R M → C 0q (= ≈ C q ⊗R M ) is almost zero, we have B q ⊗R M → B 0q . Comparing the exact sequence 0q 0q 0q+1 0→Z →C →B → 0 with the exact sequence Z q ⊗R M → C 0q → B q+1 ⊗R M → 0 ≈ in which the kernel of the left homomorphism is almost zero, we see Z q ⊗R M → Z 0q . This completes the proof.  0•

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V.7. Almost étale coverings Definition V.7.1. Let f : R → S be a homomorphism of V -algebras. We say that f is an almost étale covering if it satisfies the following conditions: (i) S is almost finitely generated projective of finite rank as an R-module. (ii) S is almost projective as an S ⊗R S-module. Lemma V.7.2. Let f : R → S be a surjective homomorphism of V -algebras and let S 0 denote the image of the homomorphism g : HomV (m, R) → HomV (m, S) induced by f . Then S is almost projective as an R-module if and only if the closed immersion Spec(S 0 ) ,→ Spec(HomV (m, R)) induced by g is an open immersion. Proof. First let us prove the sufficiency. The condition implies that S 0 is projective ≈ ≈ as a HomV (m, R)-module. Since S ⊗R HomV (m, R) → S 0 and R → HomV (m, R), S is almost projective as an R-module by Proposition V.3.3 (1) and Proposition V.3.9. Conversely suppose that S is almost projective as an R-module. Then, e = trR (idS ) is an idempotent (because rankR S ≤ 1) and we have HomV (m, R) = R1 × R0 , where R1 = HomV (m, R)/(e−1), R0 = HomV (m, R)/(e). We will prove HomV (m, I) = {0}×R0 for I = Ker(f ). By Proposition V.4.2 (2), the image of e in HomV (m, S) is trS (idS ) = 1. Hence HomV (m, I) ⊃ {0} × R0 . For ϕ ∈ HomV (m, I), ϕ(π α ) · e = trR (ϕ(π α ) · idS ) = 0 for any α ∈ Λ+ , which implies ϕ · e = 0. Hence HomV (m, I) ⊂ {0} × R0 .  By Lemma V.7.2, the condition (ii) in Definition V.7.1 can be replaced by (ii)0 The closed immersion Spec(HomV (m, S)) ,→ Spec(HomV (m, S ⊗R S)) is an open immersion. This implies that the above definition is compatible with the usual étale coverings as follows. Proposition V.7.3. Let f : R → S be a homomorphism of V -algebras and suppose that π α is invertible on R for some (or, equivalently, all) α ∈ Λ+ . Then f is an almost étale covering if and only if f is finite étale. Proof. The conditions (i) and (ii)0 are equivalent to saying that S is flat of finite presentation as an R-module ([11] II §5 n◦ 2 Corollaire 2) and the closed immersion Spec(S) ,→ Spec(S ⊗R S) is an open immersion. Hence the proposition follows from [42] Proposition (1.4.7), Corollaire (17.4.2) and Corollaire (17.6.2).  Proposition V.7.4. Let R be a V -algebra. ≈ (1) For an almost isomorphism of R-algebras S → S 0 , S is an almost étale covering over R if and only if S 0 is an almost étale covering over R. (2) For R-algebras S1 and S2 , S1 × S2 is an almost étale covering over R if and only if S1 and S2 are almost étale coverings over R. (3) For R-algebras S and R0 , if S is an almost étale covering over R, then the base change S 0 := S ⊗R R0 is an almost étale covering over R0 . The converse is also true if ≈ R → R0 . Proof. For (1), the equivalence in the condition (i) follows from Proposition V.3.6 (1), Proposition V.3.3 (1), and the remark after Definition V.5.6. By Proposition V.2.3, ≈ ≈ we have S ⊗R S → S 0 ⊗R S 0 and S ⊗S⊗R S (S 0 ⊗R S 0 ) → S 0 . Hence we obtain the equivalence in the condition (ii) from Proposition V.3.9 and Proposition V.3.3 (1). The proof of (2) is straightforward using ∧sR (S1 ⊕ S2 ) ∼ = ⊕s=s1 +s2 (∧sR1 S1 ) ⊗R (∧sR2 S2 ) and ExtiR (S1 ⊕ S2 , N ) ∼ = ExtiR (S1 , N ) ⊕ ExtiR (S2 , N ). The claim (3) immediately follows from Proposition V.3.3 (2), Proposition V.3.6 (2), Proposition V.3.9, and the remark after Definition V.5.6. 

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Proposition V.7.5. Let f : R → S be a homomorphism of V -algebra and suppose that f is an almost étale covering and rankR S = r. Then, there exist an almost faithfully flat ≈ R-algebra R0 and an almost isomorphism of R0 -algebras S ⊗R R0 → (R0 )r .

Proof. We prove by induction on r. The case r = 0 is trivial. Let r ≥ 1 and assume that the proposition is true for r − 1. The homomorphism R → HomV (m, S) is almost faithfully flat (Proposition V.6.7) and we have an almost isomorphism of HomV (m, S)≈ algebras HomV (m, S) ⊗R S → HomV (m, S ⊗R S). The last algebra is of the form HomV (m, S)×R0 by the remark after Lemma V.7.2. By Proposition V.7.4 (2) and Lemma V.7.6 below, HomV (m, S) → R0 is an almost étale covering and rankHomV (m,S) (R0 ) = r − 1. By Proposition V.6.6, we are reduced to the case r − 1.  Lemma V.7.6. Let R be a V -algebra and let M1 , M2 be two almost finitely generated projective R-modules. For integers r ≥ r1 ≥ 0, if rankR (M1 ⊕ M2 ) = r and rankR (M1 ) = r1 , then rankR (M2 ) = r − r1 .

Proof. Set M = M1 ⊕ M2 and r2 = r − r1 . Since ∧r+1 M ⊃ ∧r1 M1 ⊗R ∧r2 +1 M2 , we ≈ have ∧r1 M1 ⊗R ∧r2 +1 M2 ≈ 0. Taking (∧r1 M1 )∗ ⊗ − and using (∧r1 M1 )∗ ⊗R ∧r1 M1 → R (Corollary V.5.15), we obtain ∧r2 +1 M2 ≈ 0. Then, we have ∧r M ∼ = ⊕r=s1 +s2 ∧s1 ≈ r1 r2 s2 M1 ⊗R ∧ M2 ← ∧ M1 ⊗R ∧ M2 and 1 = trR (id∧r M ) = trR (id∧r1 M1 ⊗ id∧r2 M2 ) = trR (id∧r1 M1 ) · trR (id∧r2 M2 ) = trR (id∧r2 M2 ) by Proposition V.4.3 and Proposition V.4.4. 

Proposition V.7.7. Let f : R → S and g : S → T be homomorphisms of V -algebras. If f and g are almost étale coverings, then g ◦ f is also an almost étale covering. ϕ

ψ

Proof. For α ∈ Λ+ , choose integers r, s ≥ 0 and homomorphisms S → R⊕r → S η θ and T → S ⊕s → T as R-modules and as S-modules, respectively, such that ψ ◦ ϕ = π α · idS and θ ◦ η = π α · idT (Proposition V.3.7). Then the R-linear homomorphisms ϕ⊕s ◦ η : T → R⊕rs and θ ◦ ψ ⊕s : R⊕rs → T satisfy (θ ◦ ψ ⊕s ) ◦ (ϕ⊕s ◦ η) = (π α )2 · idT . Hence, T is almost finitely generated projective as an R-module. The homomorphism T ⊗R T → T factors as T ⊗R T → T ⊗S T → T and the first homomorphism is isomorphic to the base change of S ⊗R S → S by S ⊗R S → T ⊗R T . Hence, by Proposition V.3.3 (2) and the same argument as above, we see that T is almost projective as a T ⊗R T module. It remains to prove that T is of finite rank over R. By Proposition V.5.16 and Proposition V.7.5, there exist an almost faithfully flat R-algebra Rr × · · · × R0 and an ≈ almost isomorphism S ⊗R Ri → (Ri )i of Ri -algebras for each 0 ≤ i ≤ r. Then, for an integer t ≥ 0, rankR (T ) ≤ t if and only if rankRi ((T ⊗R Ri ) ⊗S⊗R Ri (Ri )i ) ≤ t for every 0 ≤ i ≤ r. Thus, we are reduced to the case S = Rr , in which case the assertion is trivial.  Proposition V.7.8. Let f and g be the same as in Proposition V.7.7. If f and g ◦ f are almost étale coverings, then g is also an almost étale covering. Proof. We consider the following commutative diagram, in which every square is co-Cartesian: g T ←−−−− S x x   hT  h g⊗1

i

2 T ⊗R S ←−−−− S ⊗R S ←−− −− S x x x    f iT  i1 

T

g

←−−−−

S

f

←−−−− R

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Since g ◦ f is an almost étale covering, so is g ⊗ 1 ◦ i2 by Proposition V.7.4 (3). Since f is an almost étale covering, S is almost finitely generated projective as an S ⊗R S-module. Since h is surjective, this implies that h and hence hT are almost étale coverings. By Proposition V.7.7, g = hT ◦ (g ⊗ 1) ◦ i2 is an almost étale covering.  Proposition V.7.9. Let p be a prime and let f : R → S be a homomorphism of V /pV algebra. Let S (p) be the base change of S by the absolute Frobenius FR : R → R of R and let FS/R : S (p) → S be the unique homomorphism of R-algebras such that the composite with S → S (p) is the absolute Frobenius FS of S. If f is an almost étale covering, then FS/R is an almost isomorphism. Proof. By Proposition V.5.16 and Proposition V.7.5, there exists an almost faith≈ fully flat R-algebra Rr × · · · × R0 and an almost isomorphism S ⊗R Ri → (Ri )i of Ri -algebras for each 0 ≤ i ≤ r. Since the relative Frobenius FS/R is compatible with base changes and functorial on S, we are reduced to the case S = Rr , in which case the proposition is trivial.  For π α -torsion free algebras, we have the following criterion for almost étale coverings, which was adopted as a definition in [24] 2.1 Definition. Proposition V.7.10. Let f : R → S be a homomorphism of V -algebras and assume that R and S are π α -torsion free for some (or equivalently all) α ∈ Λ+ . Let Rπ and Sπ denote the rings R[ πα1 0 ] = R[ π1α (α ∈ Λ+ )] and S[ πα1 0 ] = S[ π1α (α ∈ Λ+ )] (α0 ∈ Λ+ ). Then f is an almost étale covering if and only if it satisfies the following conditions: (i) Rπ → Sπ is finite étale. (ii) π α · trSπ /Rπ (S) ⊂ R for all α ∈ Λ+ . (iii) Let eSπ /Rπ ∈ Sπ ⊗Rπ Sπ be the idempotent corresponding to the image of the open and closed diagonal immersion Spec(Sπ ) ,→ Spec(Sπ ⊗Rπ Sπ ). Then π α · eSπ /Rπ is contained in the image of S ⊗R S for all α ∈ Λ+ . Note that the condition (i) implies that Sπ is projective of finite rank as an Rπ module ([42] Proposition (1.4.7) and [11] II §5 n◦ 2 Corollaire 2) and hence the trace map trSπ /Rπ is well-defined. Proof. For the necessity, the condition (i) (resp. (ii), resp. (iii)) follows from Proposition V.7.3 and Proposition V.7.4 (3) (resp. Proposition V.4.2, resp. the condition (ii)0 after Lemma V.7.2). Let us prove the sufficiency. Set e := eSπ /Rπ for simplicity. Since e corresponds to (1, 0) under the decomposition Sπ ⊗Rπ Sπ ∼ = Sπ × (Sπ ⊗Rπ Sπ )/(e) defined by e, for b ∈ S, the trace of (b⊗1)·e withP respect to Sπ → Sπ ⊗Rπ Sπ ; y 7→ 1⊗y is b. Hence, for α ∈ Λ+ , if we write π α ·e in the form 1≤i≤n xi ⊗yi , xi , yi ∈ S (the condition (i)), then P π α · b = 1≤i≤n trSπ /Rπ (bxi ) · yi . By the condition (ii), we can construct R-linear homoP morphisms ϕ : S → Rn ; b 7→ (π α · trSπ /Rπ (bxi ))i and ψ : Rn → S; (ai ) 7→ 1≤i≤n ai · yi , and ψ ◦ ϕ = (π α )2 · idS . By Proposition V.3.7, S is almost finitely generated projective as an R-module. By the condition (i), ∧r+1 Rπ Sπ = 0 for some integer r ≥ 0. Since ∧r+1 S is almost flat over R by Proposition V.3.8 and Proposition V.6.3, the kernel of R r+1 r+1 ∼ ∧r+1 S → ∧ S ⊗ R ∧ S = 0 is almost zero. Hence rankR S ≤ r. For the = R π π R R Rπ Sπ ⊗Rπ Sπ -linear section i : Sπ → Sπ ⊗Rπ Sπ ; a 7→ a ˜ · e of the surjective homomorphism m : Sπ ⊗Rπ Sπ → Sπ , π α ·i(S) is contained in the image S ⊗R S of S ⊗R S in Sπ ⊗Rπ Sπ for π α ·i

m

any α ∈ Λ+ by the condition (iii). Hence we have homomorphisms S −→ S ⊗R S → S of S ⊗R S-modules whose composite is π α · idS . On the other hand, by Proposition V.3.8 ≈ and Proposition V.6.3, S ⊗R S is almost flat over R and hence we have S ⊗R S → S ⊗R S ≈ and ExtiS⊗R S (S ⊗R S, N ) ← ExtiS⊗R S (S ⊗R S, N ) = 0 (i > 0) for any S ⊗R S-modules

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N . Therefore (π α )2 · ExtiS⊗R S (S, N ) = 0 (i > 0) for any S ⊗R S-module N and any α ∈ Λ+ .  Proposition V.7.11. Let f : R → S be a homomorphism of V -algebras and assume that f is an almost étale covering and that R is a normal domain and π α -torsion free for some (or equivalently all) α ∈ Λ+ . Let Rπ and Sπ denote the rings R[ πα1 0 ] = R[ π1α (α ∈ Λ)] and S[ πα1 0 ] = S[ π1α (α ∈ Λ)] (α0 ∈ Λ+ ), and let S 0 denote the normalization of R in ≈ Sπ . Then, the natural homomorphism S → Sπ induces an almost isomorphism S → 0 0 HomV (m, S ). Here we regard HomV (m, S ) as a subring of Sπ by the natural injection HomV (m, S 0 ) ,→ HomV (m, Sπ ) ∼ = Sπ . Especially R → S 0 is an almost étale covering (Proposition V.7.4 (1)). Note that Rπ → Sπ is finite étale by Proposition V.7.3 and Proposition V.7.4 (3). By [42] Proposition (17.5.7), Sπ and hence S 0 are normal. Proof. Since R → S is almost flat by Proposition V.6.3, the kernel of the homomorphism S → S ⊗R Rπ ∼ = Sπ is almost zero. Hence we may assume that S is π α -torsion + free for all α ∈ Λ . It suffices to prove m · S ⊂ S 0 and m · S 0 ⊂ S. For any α ∈ Λ+ , choose a homomorphism f : R⊕r → S (r ≥ 1) of R-modules whose cokernel is annihilated by π α (Lemma V.3.5). For any y ∈ S, there exists g ∈ EndR (R⊕r ) such that f ◦ g = (π α y · idS ) ◦ f . For the monic polynomial F (T ) := det(T − g) ∈ R[T ], we have F (g) = 0 and hence F (π α y) = 0, which implies π α y ∈ S 0 . Varying α and y, we obtain m · S ⊂ S 0 . Next let us prove m · S 0 ⊂ S. Let e ∈ Sπ ⊗Rπ Sπ be the idempotent corresponding to the open and closed immersion Spec(Sπ ) P → Spec(Sπ ⊗Rπ Sπ ). By Proposition V.7.10, for any α ∈ Λ+ , π α e is written in the form 1≤i≤n xi ⊗ yi , xi , yi ∈ m · S ⊂ S 0 . P As in the proof of Proposition V.7.10, we have π α · y = 1≤i≤n trSπ /Rπ (yxi ) · yi for any y ∈ S 0 . Since R is a normal domain, trSπ /Rπ (S 0 ) ⊂ R. Hence, we have π α y ∈ S. Varying α and y, we obtain m · S 0 ⊂ S  We need the following Proposition in V.12. Proposition V.7.12. Let f : R → S be a homomorphism of V -algebras. If f is an almost étale covering and almost faithfully flat (e.g., rkR S = r ≥ 1 (Proposition V.6.7)), then trS/R (S) ⊃ m · HomV (m, R). Proof. By Proposition V.4.2 (2), trS/R ⊗ 1 : S ⊗R S → HomV (m, R) ⊗R S followed ≈ by the almost isomorphism HomV (m, R) ⊗R S → HomV (m, S) becomes trS⊗R S/S , where we regard S ⊗R S as an S-algebra by y 7→ 1 ⊗ y. Since f is almost faithfully flat, it suffices to prove trS⊗R S/S (S ⊗R S) ⊃ m · HomV (m, S). Let e be the idempotent of HomV (m, S ⊗R S) corresponding to the open and closed immersion Spec(HomV (m, S)) ,→ Spec(HomV (m, S⊗R S)) (cf. the remark after Lemma V.7.2). Then, for any α ∈ Λ+ , there exists x ∈ S⊗R S whose image in HomV (m, S⊗R S) is π α ·e, and by Propositions V.4.2 (1), V.4.3, and V.4.4, we have trS⊗R S/S (x) = trHomV (m,S⊗R S)/S (π α e) = trHomV (m,S)/S (π α ) = trS/S (π α ) = π α . Varying α, we obtain the claim.  V.8. Almost faithfully flat descent I We discuss almost faithfully flat descent of some properties of modules and algebras. Proposition V.8.1. Let R → R0 be an almost faithfully flat homomorphism of V algebras, and let M be an R-module. If M 0 := M ⊗R R0 is almost finitely generated over R0 , then M is almost finitely generated over R.

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Proof (cf. [37]VIII Corollaire 1.11). We have M = limi∈I Mi , where Mi are finitely −→ generated R-submodules of M and I is a filtered ordered set. Then M 0 = limi∈I Mi0 , −→ where Mi0 = Mi ⊗R R0 . By assumption, for any α ∈ Λ+ , there exists a finitely generated R0 -submodule N 0 of M 0 such that π α · M 0 /N 0 = 0. Choose i ∈ I such that the image of Mi0 in M 0 contains N . Then M 0 /( the image of Mi0 ) ∼ = (M/Mi ) ⊗R S is annihilated by π α . By Lemma V.8.2 below, π 2α · M/Mi = 0.  Lemma V.8.2. Let R → R0 and M be as in Proposition V.8.1. Then the kernel of M → M ⊗R R0 is almost zero.

Proof. Let N denote the kernel. Then the homomorphism N ⊗R R0 → M ⊗R R0 is zero and its kernel is almost zero by the condition (iii) in Definition V.6.1. Hence N ⊗R R0 ≈ 0, which implies N ≈ 0.  Definition V.8.3. Let R be a V -algebra and let M be an R-module. We say that M is almost finitely presented if, for any α ∈ Λ+ , there exist a finitely presented Rmodule N and homomorphisms of R-modules ϕα : M → N , ψα : N → M such that ψα ◦ ϕα = π α · idM and ϕα ◦ ψα = π α · idN . By Lemma V.2.2, we have the following criterion. Lemma V.8.4. Let R be a V -algebra and let M be an R-module. Then M is almost finitely presented if and only if, for any α ∈ Λ+ , there exist a free R-module L of finite rank, a homomorphism of R-modules f : L → M , and a finitely generated R-submodule N of Ker(f ) such that Cok(f ) and Ker(f )/N are annihilated by π α .

Proposition V.8.5. Let R → R0 be an almost faithfully flat homomorphism of V algebras and let M be an R-module. If M ⊗R R0 is almost finitely presented over R0 , then M is almost finitely presented over R. Lemma V.8.6. Let R be a V -algebra, let M be an R-module, let L1 and L2 be two free Rmodules of finite rank, and let fi : Li → M (i = 1, 2) and g : L1 → L2 be homomorphisms of R-modules such that f2 ◦ g = f1 . Let α ∈ Λ+ and assume π α · Cok(g) = 0. (1) If there exists a finitely generated R-submodule N1 of Ker(f1 ) and β ∈ Λ+ such that π β ·Ker(f1 )/N1 = 0, then there exists a finitely generated R-submodule N2 of Ker(f2 ) such that π α+β · Ker(f2 )/N2 = 0. (2) If there exists a finitely generated R-submodule N2 of Ker(f2 ) and β ∈ Λ+ such that π β ·Ker(f2 )/N2 = 0, then there exists a finitely generated R-submodule N1 of Ker(f1 ) such that π 2α+β · Ker(f1 )/N1 = 0. Proof. By assumption, there exists s : L2 → L1 such that g ◦ s = π α · idL2 . We define the homomorphism h : L1 → Ker(g) by h(x) = π α ·x−s◦g(x). Since the restriction of h to Ker(g) is the multiplication by π α , the cokernel of h is annihilated by π α . On g the other hand, we have an exact sequence 0 → Ker(g) → Ker(f1 ) → Ker(f2 ) and the cokernel of the last homomorphism is killed by π α . The lemma follows easily from these facts.  Proof of Proposition V.8.5. Set M 0 := M ⊗R R0 . By Proposition V.8.1, M is almost finitely generated over R. For α ∈ Λ+ , choose a free R-module L of finite rank and a homomorphism of R-modules f : L → M such that π α · Cok(f ) = 0. Let L0 and f 0 denote the base change of L and f by R → R0 . We have π α · Cok(f 0 ) = 0 and ≈ Ker(f ) ⊗R R0 → Ker(f 0 ) (Lemma V.6.9). Choose a free R0 -module L01 of finite rank and a homomorphism of R0 -modules f10 : L01 → M 0 such that π α · Cok(f10 ) = 0 and there exists a finitely generated R0 -submodule N10 of Ker(f10 ) such that π α · Ker(f10 )/N10 = 0

466

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(Lemma V.8.4). Then there are homomorphisms g : L0 → L01 and h : L01 → L0 such that f10 ◦ g = π α · f 0 and f 0 ◦ h = π α · f10 . Choose such g and h. Then the composite of g + π α · idL01 : L0 ⊕ L01 → L01 (resp. π α · idL0 + h : L0 ⊕ L01 → L0 ) with f10 (resp. f 0 ) becomes π α · (f 0 + f10 ). Using Lemma V.8.6 twice, we see that there exists a finitely generated R0 -submodule N 0 of Ker(f 0 ) such that π 4α ·Ker(f 0 )/N 0 = 0. Hence, there exists a finitely generated R0 -submodule N 00 of (Ker(f )) ⊗R R0 such that π 5α · (Ker(f ) ⊗R R0 )/N 00 = 0. By the same argument as the proof of Proposition V.8.1, there exists a finitely generated R-submodule N of Ker(f ) such that π 6α · Ker(f )/N = 0. By Lemma V.8.4, M is almost finitely presented.  Proposition V.8.7. Let R → R0 be an almost faithfully flat homomorphism of V algebras and let M be an R-module. If M ⊗R R0 is an almost finitely generated projective R-module, then M is an almost finitely generated projective R-module. To prove this proposition, we need the following lemmas. Lemma V.8.8. Let R be a V -algebra and let M be an R-module. If M is almost finitely generated projective over R, then M is almost finitely presented over R. Proof. For α ∈ Λ+ , choose L, f , and g as in Proposition V.3.7. Then we have an R-linear homomorphism L → Ker(g) defined by x 7→ π α · x − f ◦ g(x), and its cokernel is annihilated by π α because its restriction to Ker(g) is the multiplication by π α . Hence, by Lemma V.8.4, M is almost finitely presented.  Lemma V.8.9. Let R → R0 be an almost flat homomorphism of V -algebras and let M and N be two R-modules. If M is finitely presented, then the canonical homomorphism HomR (M, N ) ⊗R R0 → HomR0 (M ⊗R R0 , N ⊗R R0 ) is an almost isomorphism. Proof. Choose a presentation L2 → L1 → M → 0 with L1 and L2 free R-modules of finite rank. Then we have a commutative diagram HomR (M, N ) ⊗R R0 −−−−→ Ker{HomR (L1 , N ) ⊗R R0 → HomR (L2 , N ) ⊗R R0 }    o y y HomR0 (M 0 , N 0 )

∼

−−−−→

Ker{HomR0 (L01 , N 0 ) → HomR0 (L02 , N 0 )},

where M 0 denotes M ⊗R R0 and similarly for N 0 , L01 , and L02 . By Lemma V.6.9, the upper horizontal homomorphism is an almost isomorphism and it implies the lemma.  Proof of Proposition V.8.7. By Lemma V.8.8 and Proposition V.8.5, M is almost finitely presented. For α ∈ Λ+ , choose a finitely presented R-module M and homomorphisms f : M1 → M and g : M → M1 such that f ◦ g = π α · idM and g ◦ f = π α · idM1 . Set M 0 := M ⊗R R0 and M10 := M1 ⊗R R0 . By assumption, for any R0 -module N 0 , we have ExtiR0 (M 0 , N 0 ) ≈ 0 (i > 0) and hence π 2α · ExtiR0 (M10 , N 0 ) = 0 (i > 0). Choose a free R-module L of finite rank and a surjective homomorphism h : L → M1 . Let C denote the cokernel of the homomorphism HomR (M1 , L) → HomR (M1 , M1 ) induced by h. Then, we have a commutative diagram whose two lines are exact: HomR (M1 , L) ⊗R R0

/ HomR (M1 , M1 ) ⊗R R0

/ C ⊗R R0

 HomR0 (M10 , L0 )

 / HomR0 (M10 , M10 )

 / Ext1 0 (M 0 , Ker(h0 )),

0

0

0

R

/0

1

where L = L ⊗R R and h = h ⊗ idR0 . By Lemma V.8.9, the left and middle vertical homomorphisms are almost isomorphisms. Hence, the right one is almost injective. By

V.9. ALMOST FAITHFULLY FLAT DESCENT II

467

Lemma V.8.2, π 3α · C = 0 and there exists s : M1 → L such that h ◦ s = π 3α · idM1 . This implies that, for any R-module N , π 3α · ExtiR (M1 , N ) = 0 (i > 0) and hence π 4α · ExtiR (M, N ) = 0 (i > 0). Varying α, we obtain the proposition.  Corollary V.8.10. Let R → R0 be an almost faithfully flat homomorphism of V algebras, and let S be an R-algebra. If S ⊗R R0 is an almost étale covering of R0 , then S is an almost étale covering of R. Proposition V.8.11. Let R → R0 be an almost faithfully flat homomorphism of V algebras and let M be an R-module. If M ⊗R R0 is almost flat (resp. almost faithfully flat) over R0 , then M is almost flat (resp. almost faithfully flat) over R. 

Proof. Straightforward. V.9. Almost faithfully flat descent II We will discuss almost faithfully flat descent of modules and algebras.

Proposition V.9.1. Let f : R → S be an almost faithfully flat homomorphism of V algebras and let M be an R-module. Then the sequence ε

1⊗d0

1⊗d1

0 −→ M −→ M ⊗R S −→ M ⊗R (S ⊗R S) −→

1⊗dq−1

· · · −→ M ⊗R (S ⊗q ) −→ M ⊗R (S ⊗(q+1) ) −→ · · · is almost exact, that is, the cohomology groups are almost zero, where dq−1 (y0 ⊗ · · · ⊗ yq−1 ) =

q X (−1)i y0 ⊗ · · · ⊗ yi−1 ⊗ 1 ⊗ yi ⊗ · · · ⊗ yq−1 . i=0

Proof. By taking the base change by f and using Lemma V.6.9, we are reduced to the case that there exists a homomorphism s : S → R such that s◦f = idR . Then, the Rlinear homomorphism k q : S ⊗(q+1) → S ⊗q defined by k q (y0 ⊗· · ·⊗yq ) = s(y0 )·y1 ⊗· · ·⊗yq satisfy k q+1 ◦ dq + dq−1 ◦ k q = 1 (q ≥ 0) and k 0 ◦ f = 1. Hence {1 ⊗ k q }q≥0 is a contracting homotopy.  f for Hom (m, M ) to simplify In the following until the end of V.9, we write M V the notation. Let us recall its fundamental properties briefly. We have a canonical ≈ f almost isomorphism of V -modules M → M functorial on M . If M → N is an almost f→N e is an isomorphism (Proposition V.2.5 isomorphism, the induced homomorphism M f f→M f in two ways: one by taking (1)). We can construct a canonical homomorphism M f f e . The coincidence is trivial e of M → M and the other by putting N = M in N → N e and, by the first construction, this becomes an isomorphism. If A is a V -algebra, A e has a canonical ring structure and A → A is a ring homomorphism. For a V -algebra e→B e is a ring homomorphism. For a V -algebra R and an homomorphism A → B, A f has a natural R-module structure and M → M f is R-linear. For an R-module M , M f e R-linear homomorphism M → N , M → N is also R-linear. For a V -algebra R, let R-Mod (resp. R-Alg) denote the category of R-modules g (resp. R-Alg) g denote the full subcategory of R-Mod (resp. R-algebras) and let R-Mod f (resp. R-Alg) consisting of R-modules M (resp. R-algebras A) such that M → M e (resp. A → A) are isomorphisms.

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V. ALMOST ÉTALE COVERINGS

g (resp. Proposition V.9.2. With the above notation, the functor R-Mod → R-Mod g associating M f (resp. A) e to an R-module M (resp. an R-algebra A) is R-Alg → R-Alg) a left adjoint of the forgetful functor. g the homomorphism HomR (M f, N ) → Proof. For M ∈ R-Mod and N ∈ R-Mod, f HomR (M, N ) induced by the canonical homomorphism M → M is bijective. Indeed, it e ) is m-torsion free, and any R-linear homomorphism ϕ : M → N is injective since N (∼ =N ϕ e ∼ f→ e← is the image of the composite M N N . We can prove the second case similarly.  For a homomorphism of V -algebras f : R → S, the functor S-Mod → R-Mod g → R-Mod g (resp. S-Alg → R-Alg) restricting the scalars by f induces a functor S-Mod g → R-Alg). g Let fe∗ denote the functor R-Mod g → S-Mod g (resp. R-Alg g → (resp. S-Alg g ^ ^ S-Alg) associating M ⊗R S to M (resp. A ⊗R S to A). Proposition V.9.3. With the above notation, the functor fe∗ is a left adjoint of the functor restricting the scalars by f . g and N ∈ S-Mod, g we have Proof. For any M ∈ R-Mod

∼ ∼ HomS (M^ ⊗R S, N ) −→ HomS (M ⊗R S, N ) −→ HomR (M, N ).

Similarly for the second case.



Corollary V.9.4. For any two composable homomorphisms of V -algebras f : R → S, g : S → T , there exists a canonical isomorphism ge∗ ◦ fe∗ ∼ ◦ f ∗ satisfying the obvious = g] cocycle condition. If we denote by f∗ the functor R-Mod → S-Mod; M 7→ M ⊗R S, we have fe∗ M = f] ∗M , f adj adj f ] the adjunction M → fe∗ M is given by M → f∗ M → f] ∗ M or M → M → f∗ M , and the following diagram is commutative g] ◦ f ∗M  o y

(g ^ ◦ f )∗ M  o y

^ ∼ ge∗ fe∗ M = g∗ f] ∗ M ←−−−−

g^ ∗ f∗ M .

g and fe∗ . g R-Alg, Now we will prove almost faithfully flat descent for R-Mod, Proposition V.9.5. Let f : R → S be an almost faithfully flat homomorphism of V algebras, let f0 and f1 be the homomorphisms S → S ⊗R S sending a to a ⊗ 1 and 1 ⊗ a, g the sequence respectively, and let g be f0 ◦ f = f1 ◦ f . Then, for any M ∈ R-Mod, M

/ fe M ∗

i0 i1

// ge M ∗

is exact, where iν (ν = 0, 1) denotes the adjunction map with respect to fν : fe∗ M −→ feν∗ fe∗ M ∼ = ge∗ M. Proof. By Proposition V.9.1, we have ≈

i −i

0 1 M −→ Ker(f∗ M −− −→ g∗ M ),

where iν is defined similarly. Since the functor ˜ is left exact, by taking ˜ of this almost isomorphism and using Proposition V.2.5 (1), we obtain the proposition. 

V.9. ALMOST FAITHFULLY FLAT DESCENT II

469

Corollary V.9.6. Let the notation and the assumption be the same as Proposition V.9.5. g (resp. A, B ∈ R-Alg), g the following sequence is exact: For any M, N ∈ R-Mod fe0∗

/ Hom (fe M, fe N ) ∗ S ∗

HomR (M, N )

// Hom

fe1∗

/ HomS-alg (fe∗ A, fe∗ B)

(resp. HomR-alg (A, B)

g∗ M, ge∗ N ) S⊗R S (e

// Hom g∗ A, ge∗ B)) S⊗R S-alg (e

fe0∗ fe1∗

Proof. The second case is reduced to the first one, which follows from Proposition V.9.5 since HomS (fe∗ M, fe∗ N ) ∼ g∗ M, ge∗ N ) ∼ = HomR (M, fe∗ N ) and HomS⊗R S (e = HomR (M, ge∗ N ) by Proposition V.9.2.  Proposition V.9.7. Let f , f0 , f1 , and g be as in Proposition V.9.5 and let f01 , f12 , and f02 denote the homomorphism S ⊗R S → S ⊗R S ⊗R S sending a0 ⊗ a1 to a0 ⊗ a1 ⊗ 1, g and suppose that we are given 1 ⊗ a0 ⊗ a1 , and a0 ⊗ 1 ⊗ a1 , respectively. Let M ∈ S-Mod ∼ an S ⊗R S-linear isomorphism φ : fe0∗ M → fe1∗ M such that fe12∗ (φ) ◦ fe01∗ (φ) = fe02∗ (φ). g endowed with an Then, up to unique isomorphisms, there exists a unique N ∈ R-Mod ∼ e isomorphism ι : f∗ N → M which makes the following diagram commute: fe0∗ fe∗ N o

fe0∗ (ι)

∼

∼

ge∗ N





fe1∗ (ι)

/ fe M 1∗

∼

fe0∗ M

/ fe fe N 1∗ ∗

φ

Lemma V.9.8. Let f : R → S be an almost flat homomorphism of V -algebras. Then, g (i = 1, 2, 3), the sequence for any exact sequence 0 → M1 → M2 → M3 with Mi ∈ R-Mod e e e 0 → f∗ M1 → f∗ M2 → f∗ M3 is exact. Proof. By Lemma V.6.9, we have ≈

f∗ M1 −→ Ker(f∗ M2 → f∗ M3 ). By Proposition V.2.5 (1) and the left exactness of ˜, we obtain the lemma.

 adj

Proof of Proposition V.9.7. Let α0 (resp. α1 ) be the homomorphism M −→ adj ∼ e f0∗ M → fe1∗ M (resp. M −→ fe1∗ M ), which is compatible with f0 (resp. f1 ), and define φ

the R-module N to be the kernel of α0 − α1 . Since the functor e is left exact, we have ∼ e g We can verify easily that the S-linear homomorphism N → N , that is, N ∈ R-Mod. ι : fe∗ N → M induced by the inclusion makes the diagram in the proposition commute. adj (Show the commutativity after composing with N −→ ge∗ N .) We will prove that ι is an isomorphism. Let g0 , g1 , and g2 be the homomorphisms S → S ⊗R S ⊗R S sending a ∈ S to a ⊗ 1 ⊗ 1, 1 ⊗ a ⊗ 1, and 1 ⊗ 1 ⊗ a, respectively. First, note that we have a commutative diagram of V -algebras whose squares are co-Cartesian: (∗)

R

f

f

 S

/S

f0 f1

f0

f1

 / S ⊗R S

// S ⊗ S R f01

f02 f12

 // S ⊗R S ⊗R S.

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V. ALMOST ÉTALE COVERINGS

Hence, by taking the base change by f of the exact sequence // fe1∗ M

α0

/M

N

α1

compatible with the first line, we obtain an exact sequence / fe M 0∗

fe∗ N

β0 β1

// e e f01∗ f1∗ M ∼ = ge1∗ M

compatible with the second line by Lemma V.9.8. By the following commutative diagram, adj β1 is fe0∗ M −→ fe12∗ fe0∗ M ∼ = ge1∗ M .

adj

/ fe M 1∗

adj

M

adj



adj

fe0∗ M

 ∼ e f M g e = 1∗ M ∼ = fe01∗ fe1∗ M. 12∗ 0∗

/ fe

adj ∼ Similarly, we see that β0 is the composite fe0∗ M −→ fe02∗ fe0∗ M ∼ = ge0∗ M −→ ge1∗ M . On fe01∗ φ

the other hand, since f1 is almost faithfully flat by Proposition V.6.5 (2), by applying Proposition V.9.5 to f1 , we obtain another exact sequence / fe M 1∗

M

// ge M 2∗

γ0 γ1

compatible with the second line of (∗), where γi (i = 1, 2) is the adjunction fe1∗ M → fei2∗ fe1∗ M ∼ = ge2∗ M . Now we can check that the following diagram is commutative and hence ι is an isomorphism. / fe M 0∗

fe∗ N

β0 β1

o φ

ι



 M

/ fe M 1∗

// ge1∗ M ∼ = fe12∗ fe0∗ M o fe12∗ φ

γ0 γ1

 // ge2∗ M ∼ = fe12∗ fe1∗ M

(For the commutativity of the left square, we consider the composite with adj : N → fe∗ N . For fe12∗ φ ◦ β0 = γ0 ◦ φ, we use the cocycle condition on φ.) It remains to prove the uniqueness. Suppose that we have (N1 , ι1 ), (N2 , ι2 ) satisfying the condition. Then, the following diagram is commutative for ν = 1, 2, where αi is as adj above and δi is fe∗ Nν → fei∗ fe∗ Nν ∼ = ge∗ Nν . δ0

fe∗ Nν

δ1

o ιν

 M

// ge N ∗ ν o fe1∗ ιν

α0 α1

 // e f1∗ M.

∼

By Proposition V.9.5, Nν → Ker(δ0 − δ1 ). Hence there exists a unique R-linear isomor∼ phism N1 → N2 compatible with ι1 and ι2 .  f be the full subcategory of R-Alg g consisting of A ∈ R-Alg g which is an almost Let R-Et étale covering over R (Definition V.7.1). For a homomorphism of V -algebras f : R → S, g is an almost isomorphism. the canonical homomorphism A ⊗R S → fe∗ A (A ∈ S-Alg)

V.10. LIFTINGS

471

f then fe∗ A ∈ S-Et f Hence, by Proposition V.7.4 (1), (3) and Corollary V.8.10, if A ∈ R-Et, and the converse is also true if f is almost faithfully flat. If we denote by R-Et, the full subcategory of R-Alg consisting of almost étale coverings over R, we have a functor f A → A, e which is a left adjoint of the forgetful functor. e : R-Et → R-Et; Corollary V.9.9. Let f , f0 , f1 , g, f01 , f12 , and f02 be the same as Proposition V.9.7. f and suppose that we are given an S-algebra isomorphism g (resp. S-Et) Let A ∈ S-Alg ∼ e e φ : f0∗ A → f1∗ A such that fe12∗ (φ)◦ fe01∗ (φ) = fe02∗ (φ). Then, up to unique isomorphisms, g (resp. R-Et) f endowed with an S-algebra isomorphism there exists a unique B ∈ R-Alg ∼ ι : fe∗ B → A, which makes the following diagram commute. fe0∗ fe∗ B o 

∼

ge∗ B

∼

/ fe fe B 1∗ ∗

fe0∗ (ι) ∼

fe0∗ A

φ



fe1∗ (ι)

/ fe A 1∗

Proof. Immediate from the construction of N and the proof of the uniqueness in the proof of Proposition V.9.7.  V.10. Liftings We study liftings of almost étale coverings with respect to nilpotent immersions. To do it, we first review the definition and some basic properties of Hochschild cohomology. Let A be a commutative ring and let B be any commutative A-algebra. Recall that giving a B ⊗A B-module M is equivalent to giving a module M endowed with left and right B-module structures which induce the same A-module structure and commute with each other (i.e., (x · m) · y = x · (m · y) for x, y ∈ B, m ∈ M ); the correspondence is given by the formula: (x ⊗ y) · m = (x · m) · y = x · (m · y). We define the chain complex C• (B/A) of B ⊗A B-modules as follows: The degree n part (n ≥ 0) is the (n + 2)-fold tensor product ⊗n B/A of B over A endowed with the B⊗A B-module structure: (x⊗y)·(b0 ⊗b1 ⊗· · ·⊗bn+1 ) = x · b0 ⊗ b1 ⊗ · · · ⊗ y · bn+1 . The differential dn : Cn (B/A) → Cn−1 (B/A) (n ≥ 1) is defined by n X dn (b0 ⊗ b1 ⊗ · · · ⊗ bn+1 ) = (−1)i b0 ⊗ · · · ⊗ bi bi+1 ⊗ · · · ⊗ bn+1 . i=0

If we regard B as a B ⊗A B-module by (x ⊗ y) · b = x · b · y, we have a B ⊗A B-linear augmentation ε : C0 (B/A) → B; b0 ⊗ b1 7→ b0 · b1 . ε

Lemma V.10.1. The complex C• (B/A) → B is homotopically equivalent to zero as a complex of left (resp. right) B-modules. Proof. A homotopy is explicitly given by B → C0 (B/A); b 7→ b ⊗ 1 (resp. 1 ⊗ b) and Cn (B/A) → Cn+1 (B/A); b0 ⊗ b1 ⊗ · · · ⊗ bn+1 7→ (−1)n+1 b0 ⊗ b1 ⊗ · · · ⊗ bn+1 ⊗ 1 (resp. 1 ⊗ b0 ⊗ b1 ⊗ · · · ⊗ bn+1 ).  For a B ⊗A B-module M , we define the Hochschild cohomology H ∗ (B/A, M ) to be the cohomology of the (cochain) complex HomB⊗A B (C• (B/A), M ). The degree n-part of this complex is naturally identified with the module HomA-multilin (B n , M )

472

V. ALMOST ÉTALE COVERINGS

of A-multilinear maps B n → M and the differential is given by

(dn g)(b1 , . . . , bn+1 ) = b1 · g(b2 , . . . , bn+1 )

(V.10.2)

+

n X (−1)i g(b1 , . . . , bi bi+1 , . . . , bn+1 ) i=1

+

(−1)n+1 g(b1 , . . . , bn ) · bn+1 .

Hence, we have (V.10.3)

H 0 (B/A, M ) = {m ∈ M | x · m = m · x for all x ∈ B},

(V.10.4) H 1 (B/A, M ) = {f ∈ HomA (B, M )| f (xy) = x·f (y)+f (x)·y}/{x 7→ x·m−m·x| m ∈ M }. Proposition V.10.5. For any B-module N , if we regard HomA (B, N ) as a B ⊗A Bmodule by ((x ⊗ y) · f )(b) = x · f (y · b) (f ∈ HomA (B, N ), x, y, b ∈ B), then we have ( 0 (i > 0), H i (B/A, HomA (B, N )) = N (i = 0). Proof. This follows from Lemma V.10.1 and Lemma V.10.6 below.



Lemma V.10.6. Let M be a B ⊗A B-module and let N be a B-module. Then there exists a canonical isomorphism functorial on M and N : HomB⊗A B (M, HomA (B, N )) ∼ = HomB-left (M, N ) ϕ,

ϕ 7→ ψ,

ψ(m) = (ϕ(m))(1)

(ϕ(m))(b) = ψ(m · b) ←[ ψ

Corollary V.10.7. If B is a projective B ⊗A B-module, then for any B ⊗A B-module M , we have H i (B/A, M ) = 0 (i > 0). Proof. For any two B ⊗A B-modules M and N , we define the B ⊗A B-module structure on HomB-right (M, N ) by ((x ⊗ y) · f )(m) = x · f (y · m), which is functorial on M and N . Then the natural isomorphisms M∼ = HomB-right (B, M ),

HomB-right (B ⊗A B, M ) ∼ = HomA (B, M )

are B ⊗A B-linear. Here we regard the right-hand side of the second isomorphism as a B ⊗A B-module in the same way as Proposition V.10.5 using the left B-module structure on M . By the assumption, B is a direct summand of B ⊗A B as a B ⊗A B-module. Hence the claim follows from Proposition V.10.5.  Corollary V.10.8. Let R → S be a homomorphism of V -algebras such that S is an almost projective S ⊗R S-module. Then, for any S ⊗R S-module M , the natural almost isomorphisms H i (S/R, M ) → H i (S/R, HomV (m, M )) (i > 0)

vanish. Especially, H i (S/R, M ) ≈ 0 (i > 0) and, if M → HomV (m, M ) is an isomorphism, H i (S/R, M ) = 0 (i > 0).

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473

Proof. By Lemma V.7.2, we have HomV (m, S ⊗R S) = HomV (m, S) × R1 . Let S 0 be the fiber product of HomV (m, S) → HomV (m, S ⊗R S) ← S ⊗R S in the category of S ⊗R S-modules. Then, we have homomorphisms S 0 → S ⊗R S → S of S ⊗R Smodules whose composite are almost isomorphisms. By a similar argument as the proof of Corollary V.10.7, we see that the homomorphisms H i (S/R, M ) → H i (S/R, HomS-right (S 0 , M )) ∼

vanish. Since HomV (m, M ) → HomV (m, HomS-right (S 0 , M )) by Propositions V.2.3 and V.2.5 (1), this implies the claim.  Corollary V.10.9. Let R → S be as in Corollary V.10.8. Then, for any S-module M , the natural almost isomorphism DerR (S, M ) → DerR (S, HomV (m, M ))

vanishes. Especially, DerR (S, M ) ≈ 0 and, if M → HomV (m, M ) is an isomorphism, DerR (S, M ) = 0. Proof. Apply Corollary V.10.8 to M with the S ⊗R S-module structure induced by S ⊗R S → S; x ⊗ y 7→ xy and use (V.10.4).  Proposition V.10.10. Let SO

t

g

f

R

/T O

s

/T

be a commutative diagram of V -algebras such that g is almost surjective (i.e., g(T ) ⊃ π α T ∼ ∼ for all α ∈ Λ+ ), (Ker(g))2 = 0, and T → HomV (m, T ), T → HomV (m, T ). If S is almost projective as an S ⊗R S-module (resp. f is an almost étale covering), then there exists at most one (resp. unique) homomorphism u : S → T such that g ◦ u = t and u ◦ f = s. Proof. Set I := Ker(g). First let us prove the uniqueness. Suppose that we have two homomorphisms u1 , u2 satisfying the conditions. Then their difference u1 − u2 : S → ∼ I is an R-derivation, which vanishes by Corollary V.10.9 because I → HomV (m, I) by assumption. Here we regard I as an S-module via t. Next let us prove the existence when f is an almost étale covering. For any α ∈ Λ+ , since S is an almost projective R-module and g is almost surjective, there exists an R-linear lifting uα : S → T of π α t. Define the R-linear map vα : S × S → I by vα (x, y) = (π α )2 uα (x · y) − π α · uα (x) · uα (y). Then, using the fact that the action of π α · x (x ∈ S) on I coincides with that of uα (x) ∈ T , we see that the coboundary (V.10.2) of vα : dvα (x, y, z) = x · vα (y, z) − vα (xy, z) + vα (x, yz) − z · vα (x, y)

vanishes. By Corollary V.10.8, there exists an R-linear map hα : S → I such that vα (x, y) = dhα (x, y) = x · hα (y) − hα (xy) + hα (x) · y,

which implies (π α )3 uα (xy) − (π α )2 uα (x) · uα (y) = uα (x) · hα (y) − π α · hα (x · y) + hα (x) · uα (y).

If we set u0α := (π α )2 · uα + hα , which is a lifting of (π α )3 t, we obtain (π α )3 u0α (xy) = u0α (x) · u0α (y). For each α ∈ 3Λ+ , by multiplying u0α/3 by π α /(π α/3 )3 , we obtain an R-linear lifting wα : S → T of π α t such that (V.10.11)

π α · wα (y) = wα (x) · wα (y).

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Suppose that we have another such lifting w eα . Then the R-linear map π α ·(wα −w eα ) : S → I is an R-derivation, which vanishes by Corollary V.10.9. By replacing wα with wα/2 multiplied by π α (π α/2 )−1 for α ∈ 6Λ+ , we obtain a system of R-linear liftings (wα : S → β T )α∈6Λ+ of π α t satisfying (V.10.11) and wβ = ππα · wα for any α, β ∈ 6Λ+ , β ≥ α. Define the homomorphism u : S → HomV (m, T ) = T by u(x)(a · π α ) = a · wα (x) (x ∈ S, α ∈ 6Λ+ , a ∈ V ). We see easily that this is well-defined, R-linear, and multiplicative and satisfies g ◦ u = t. Since u(1) is an idempotent lifting of 1 ∈ T and I 2 = 0, we have u(1) = 1. Hence u is the required homomorphism.  To prove the existence of liftings of almost étale coverings, we first prove that of almost projective modules. Proposition V.10.12. Let R be a V -algebra, let I be an ideal of R such that I 2 = 0, and set R := R/I. Then, for any almost projective (resp. almost finitely generated projective) R-module M , there exists an almost projective (resp. almost finitely generated projective) ≈ R-module M with an almost isomorphism of R-modules M ⊗R R → M . Furthermore, if M ≈ 0, then ∧r+1 ∧r+1 R M ≈ 0 for any R-module M as above. R Proof. For each α ∈ Λ+ , choose a free (resp. finite free) R-module Lα and R-linear homomorphisms fα : M → Lα , gα : Lα → M (Lα := Lα ⊗R R) such that gα ◦fα = π α ·idM . Set eα := fα ◦ gα , which satisfies eα 2 = π α · eα . Choose a lifting eα ∈ EndR (Lα ) of eα . Then we have (e2α − π α · eα )2 = 0 and e03α := 3π α · (eα )2 − 2(eα )3 is a lifting of (π α )2 eα satisfying (e03α )2 = (π α )3 · e03α . For each α ∈ Λ+ , by choosing β ∈ Λ+ such that 3β ≤ α and replacing fα by π α (π β )−1 · f β , we may assume that eα has a lifting eα ∈ EndR (Lα ) satisfying e2α = π α · eα . Set Mα := Lα /(eα − π α )Lα . Then the endomorphism eα induces an R-linear map Mα → Lα such that the composite Mα → Lα → Mα is π α · idMα . Hence π α · ExtiR (Mα , −) = 0 and π α · ToriR (Mα , −) = 0 (i > 0). If we set Mα := Mα ⊗R R(∼ = Lα /(eα −π α )Lα ), then fα and gα induce R-linear maps ϕα : M → Mα and ψα : Mα → M satisfying ϕα ◦ ψα = π α · idMα and ψα ◦ ϕα = π α · idM . ψβ

ϕα

For α, β ∈ Λ+ , the composite Mβ → Mβ → M → Mα has a lifting τα,β : Mβ → Mα after multiplication by π β because π β · Ext1R (Mβ , −) = 0, and we have ψα ◦ τα,β = π α · π β · ψ β , where τα,β denotes the reduction of τα,β mod I. When α ≤ β/3, if we β

0 0 set τα,β := (ππα )3 τα,β , then we have (π α )2 ψα ◦ τα,β = (π β )2 ψβ . Choose a sequence αn ∈ Λ+ (n ≥ 0) such that αn+1 ≤ αn /3 and define M0 to be the direct limit of {Mαn , τα0 n+1 ,αn }n≥0 . The homomorphisms (π αn )2 ψαn induce ψ : M 0 (:= M0 ⊗R R) → M , which is an almost isomorphism. Consider the morphism of exact sequences: h

I ⊗R Mαn −−−n−→ Mαn −−−−→ Mαn −−−−→ 0       y y y h

I ⊗R M0 −−−0−→ M0 −−−−→ M0 −−−−→ 0. αn Since π αn · TorR · Ker(hn ) = 0 and Ker(h0 ) ≈ 0. Using the almost 1 (R, Mαn ) = 0, π isomorphism ψ, we see that the kernels (resp. the cokernels) of the left and the right vertical maps are killed by π 3αn (resp. π 4αn ). Hence the kernel and the cokernel of the middle map are killed by π 8αn . Since π αn · ExtiR (Mαn , −) = 0 (i > 0), varying n, we see that M0 is almost projective. If M is almost finitely generated Mαn is finitely generated,

V.10. LIFTINGS

475

and hence M0 is almost finitely generated. The last claim follows from the exact sequence ≈ r+1 r+1 r+1 M0 → ∧r+1  I ⊗R ∧r+1 R M0 → ∧R M0 → 0 and ∧R M0 → ∧R M . R Theorem V.10.13. Let R be a V -algebra, let I be an ideal of R such that I 2 = 0, and set R := R/I. Then, for any almost étale coverings S of R, there exists an almost étale ≈ covering S of R with an almost isomorphism S ⊗R R → S of R-algebras. ∼

Proof. We may assume S → HomV (m, S). (The general case is reduced to this ≈ case by taking the fiber product of S → HomV (m, S) ← S 0 for an almost étale lifting S 0 of HomV (m, S).) By Proposition V.10.12, there exists an almost finitely generated ≈ projective R-module M of finite rank with an almost isomorphism M ⊗R R → S of R∼ modules. By replacing M with HomV (m, M ), we may assume M → HomV (m, M ). We denote by m the multiplication of S. ≈ Let N denote the kernel of M → S. Since I ·M → N and N does not have m-torsion, N is an R-module. By taking HomV (m, −) of the almost isomorphisms

≈ ≈ ≈ N ← I · M ← I ⊗R M ∼ = I ⊗R (M ⊗R R) → I ⊗R S, we obtain an isomorphism N ∼ = HomV (m, I ⊗R S) and we can naturally regard N as an S-module. For α ∈ Λ+ ∪ {0}, if we are given an R-bilinear lifting mα : M × M → M of π α m (π 0 = 1), then, for x ∈ M , y ∈ N , and the image x ∈ S of x, we have

(V.10.14)

(π α · x) · y = mα (x, y) = mα (y, x).

By Proposition V.3.8, M ⊗R M is an almost projective R-module. Hence, for each α ∈ Λ+ , there exists an R-bilinear lifting mα : M × M → M of π α · m. Consider the R-multilinear map gα : M × M × M → N defined by gα (x, y, z) = mα (mα (x, y), z) − ≈ mα (x, mα (y, z)). gα (x, y, z) = 0 if one of x, y, z is contained in I · M . Since I · M → N 0 and N does not have m-torsion, this also holds for N . Hence, if we denote by S the 0 0 0 image of M in S, gα induces an R-multilinear map gα : S × S × S → N . This extends to an R-multilinear map S × S × S → HomV (m, N ) = N which sends (x, y, z) to the n defined by n(a(π β )3 ) = a · gα (π β x, π β y, π β z) (β ∈ Λ+ , a ∈ m). We denote this extension also by gα . The coboundary (V.10.2) of π α gα vanishes. Indeed, for each β ∈ Λ+ and 0 x, y, z, w ∈ S, if we choose a lifting xβ , yβ , zβ , wβ ∈ M of π β x, π β y, π β z, π β w ∈ S , we have (d(π α gα )(x, y, z, w))((π β )4 ) =mα (xβ , gα (yβ , zβ , wβ )) − gα (mα (xβ , yβ ), zβ , wβ )

+ gα (xβ , mα (yβ , zβ ), wβ ) − gα (xβ , yβ , mα (zβ , wβ ))

+ mα (gα (xβ , yβ , zβ ), wβ ) =0

by (V.10.14). By Corollary V.10.8, there exists an R-bilinear map hα : S × S → N such that π α · gα = dhα , which implies: (π α )2 mα (mα (x, y), z) − (π α )2 mα (x, mα (y, z))

=mα (x, hα (y, z)) − hα (mα (x, y), z) + hα (x, mα (y, z)) − mα (hα (x, y), z) h

α N. for x, y, z ∈ M by (V.10.14), where hα denotes the R-bilinear map M ×M → S ×S → α 2 α 3 + Now the lifting (π ) mα + hα of (π ) m is associative. For each α ∈ Λ , choosing β ∈ Λ+ such that 3β ≤ α and applying the above argument to β, we see that π α · m has an associative lifting. We choose such a lifting mα .

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V. ALMOST ÉTALE COVERINGS

For a ∈ M , the R-linear map ga : M → N ; x 7→ mα (a, x) − mα (x, a) induces an Rlinear map ga : S → HomV (m, N ) = N similarly as above. Then π α ·ga is an R-derivation. Indeed, for x, y ∈ S, β ∈ Λ+ , and liftings xβ , yβ ∈ M of π β x, π β y, we have π α {(ga (xy) − x · ga (y) − y · ga (x))((π β )2 )}

=ga (mα (xβ , yβ )) − mα (xβ , ga (yβ )) − mα (ga (xβ ), yβ ) = 0 by (V.10.14) and the associativity of mα . By Corollary V.10.9, π α · ga = 0 and hence π α · mα is commutative. For each α ∈ Λ+ , choosing β ∈ Λ+ such that 2β ≤ α and applying the above argument to β, we see that π α mα has an R-bilinear associative commutative lifting M × M → M . Suppose that we have two R-bilinear associative commutative liftings mα , m0α : M × M → M of π α m. Then the R-bilinear map g = m0α − mα : M × M → N induces an R-bilinear map g : S × S → N similarly as above. For x, y, z ∈ S, β ∈ Λ+ , and liftings xβ , yβ , zβ ∈ M of π β x, π β y, π β z, by (V.10.14), we have (d(π α g)(x, y, z))((π β )3 ) =m0α (xβ , g(yβ , zβ )) − g(mα (xβ , yβ ), zβ ) + g(xβ , mα (yβ , zβ )) − m0α (g(xβ , yβ ), zβ ) = 0. By Corollary V.10.8, there exists an R-linear map h : S → N such that π α g = dh. By (V.10.14), this implies (π α )2 m0α (x, y) − (π α )2 mα (x, y) = mα (x, h(y)) − h(m0α (x, y)) + mα (h(x), y), h

where h denotes M → S → N , and hence the homomorphism (V.10.15)

(π α )2 · idM + h : (M, (π α )2 m0α ) → (M, mα )

is multiplicative. For each α ∈ Λ+ , choose a commutative associative R-bilinear lifting mα to M of π α m and define nα : π 3α V ⊗V M × π 3α V ⊗V M → π 3α V ⊗V M ∼

by transporting (π α )2 mα via (π α )3 : M → π 3α V ⊗V M , which is compatible with m. For β α, β ∈ Λ+ such that α ≤ β/3, by applying the above claim to the two liftings ππα mα and mβ of π β m, we obtain an R-linear map hα,β : M → N such that (π β )2 · idM + β hα,β : (M, (π β )2 mβ ) → (M, ππα mα ) is multiplicative. Composing with the multiplicative β β ∼ map (ππα )3 : (M, ππα mα ) → (M, (π α )2 mα ) and transporting via (π γ )3 : (M, (π γ )2 mγ ) →

(π 3γ V ⊗V M, nγ ) (γ = α, β), we obtain an R-linear map kα,β : π 3β V ⊗V M → π 3α V ⊗V N such that 1 + kα,β : (π 3β V ⊗V M, nβ ) → (π 3α V ⊗V M, nα ) is multiplicative. Choose a sequence αn ∈ Λ+ (n ≥ 0) satisfying αn+1 ≤ αn /3 and define the R-module M 0 with the associative commutative R-bilinear map m0 : M 0 × M 0 → M 0 to be the direct limit of {(π 3αn V ⊗V M, nαn ), 1 + kαn+1 ,αn }n . We have a natural R-linear multiplicative map M 0 → S inducing an almost isomorphism ≈ ≈ M 0 ⊗R R → S. Note (π 3αn V ⊗V M ) ⊗R R → π 3αn V ⊗V S and 1 + kαn+1 ,αn is a lifting of 1 : π 3αn V ⊗V S → π 3αn+1 V ⊗V S. We assert that M 0 is an almost finitely generated projective R-module of finite rank. Set Mn := π 3αn V ⊗V M . If we denote by ¯ the reduction mod I of an R-module, we

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477

have a commutative diagram with exact lines: I ⊗R M n −−−−→   y I ⊗R M

0

Mn −−−−→   y

M n −−−−→ 0   y

−−−−→ M 0 −−−−→ M

0

−−−−→ 0.

Since Mn is almost projective and hence almost flat (Proposition V.6.3) over R, the two left horizontal homomorphisms are almost injective. Since the composite π 3αn V ⊗V M ∼ = 0 ≈ ≈ M n → M → S is induced by M → S, the kernel and the cokernel of the left and the right vertical maps are annihilated by π 4αn . Hence the kernel and the cokernel of the middle map are annihilated by π 9αn . Since Mn is an almost finitely generated projective 0 ≈ R-module, varying n, we see that so is M 0 . M 0 is of finite rank by M → S and the last claim of Proposition V.10.12. We set S := HomV (m, M 0 ) and define the R-bilinear map m : S × S → S by the composite of HomV (m, M ) × HomV (m, M 0 ) → HomV (m ⊗V m, M 0 ⊗V M 0 ) → HomV (m, M 0 ),

where the left map is defined by (f, g) 7→ f ⊗g and the right one is induced by m0 : M 0 ⊗V ∼ M 0 → M 0 and m ⊗V m → m; a ⊗ b 7→ a · b. We have m(f, g)(a · b) = m0 (f (a), g(b)) (a, b ∈ m). We see easily that m is associative commutative and the homomorphism S → HomV (m, S) = S induced by M 0 → S is multiplicative. We prove that 1S ∈ S has an idempotent lifting e ∈ S. Since S → S is almost surjective, for each α ∈ Λ+ , ≈ there exists a lifting fα ∈ S of π α · 1S . Set J := Ker(S → S). Then I · S → J and J does not have m-torsion, which implies m(J, J) = 0. Hence (fα2 − π α · fα )2 = 0 and 0 0 2 0 f3α := 3π α · fα2 − 2fα3 is a lifting of (π α )3 · 1S satisfying (f3α ) = (π α )3 · f3α . Choosing + β ∈ Λ such that 3β ≤ α and applying the above argument to β, we see that π α · 1S has a lifting eα such that e2α = π α · eα . If we have two liftings eα and e0α = eα + x α 0 (x ∈ J) of π α · 1S satisfying e2α = π α · eα , e02 α = π · eα , by taking the difference of the two α equations, we obtain (2eα − π ) · x = 0. Multiplying 2eα − π α , we obtain (π α )2 · x = 0, i.e., (π α )2 eα = (π α )2 e0α . Choose a lifting eα ∈ S of π α · 1S for each α ∈ Λ+ and define e ∈ HomV (m, S) = S by e(a · (π α )3 ) = a · (π α )2 · eα (a ∈ V , α ∈ Λ+ ), which is welldefined by the above uniqueness. We can verify easily that e is an idempotent lifting of 1S . By (V.10.14) with α = 0, the multiplication by e on J is the identity. Hence the multiplication by e on S is bijective, and for each x ∈ S, by choosing y ∈ S such that x = e · y, we see e · x = e · (e · y) = (e · e) · y = e · y = x. Thus S becomes an R-algebra ≈ with an almost isomorphism S ⊗R R → S. To prove that S is the required lifting, it is enough to prove that S is an almost projective S ⊗R S-module. Let e be the idempotent of HomV (m, S ⊗R S) corresponding to the “diagonal immersion” Spec(S) → Spec(HomV (m, S ⊗R S)) (Lemma V.7.2). Then, by Lemma V.10.16 below, e has a unique idempotent lifting e ∈ HomV (m, S ⊗R S) and its image in S is the unique idempotent lifting 1S of 1S . Let J1 (resp. J 1 ) be the ≈ kernel of HomV (m, S ⊗R S) → S (resp. HomV (m, S ⊗R S) → S). Since S ⊗R R → S and S is almost projective and hence almost flat (Proposition V.6.3) over R, we have ≈ ≈ J1 ⊗R R → J 1 . Since I · J1 → Ker(J1 → J 1 ) and e · J 1 = 0, we have π α · e · J1 ⊂ I · J1 and hence (π α )2 e · J1 = (π α e)2 · J1 = 0 for each α ∈ Λ+ . Since J1 does not have m-torsion, e · J1 = 0.  Lemma V.10.16. . Let R, I, and R be the same as in Theorem V.10.13. Let S be an R∼ algebra such that S → HomV (m, S) and let S be an R-algebra with an almost isomorphism

478

V. ALMOST ÉTALE COVERINGS ≈

∼

of R-algebras S ⊗R R → S such that S → HomV (m, S). Then, any idempotent e of S has a unique idempotent lifting e ∈ S. ≈

Proof. Set J := Ker(S → S). Then I ·S → J and J does not have m-torsion. Hence J = 0. The proof of the existence is the same as the construction of an idempotent lifting 1S to S in the proof Theorem V.10.13. If we have two idempotent liftings e and e0 = e + x (x ∈ J) of e. Then, by taking the difference of e2 = e and e02 = e0 , we obtain (2e − 1) · x = 0. Multiplying 2e − 1, we obtain x = 0.  2

f the category of almost étale coverings S of R such As in V.9, we denote by R-Et, ∼ that S → HomV (m, S). Corollary V.10.17. Let R, I, and R be the same as Theorem V.10.13. Then the functor f → R-Et; f S 7→ Hom (m, S ⊗R R) is an equivalence of categories. R-Et V f and put S = Hom (m, S ⊗R R) and T = Hom (m, T ⊗R Proof. Let S, T ∈ R-Et, V V R). Then, for any homomorphism of R-algebras h : S → T , there exists a unique homomorphism h : S → T of R-algebras inducing h by Proposition V.10.10, i.e., the functor in question is fully faithful. Note that T → T is almost surjective with a square-zero kernel. It is essentially surjective by Theorem V.10.13.  V.11. Group cohomology of discrete A-G-modules Let G be a profinite group and let A be a commutative ring. A discrete A-G-module is an A-module with an A-linear action of G continuous with respect to the discrete topology on M . Discrete A-G-modules form an abelian category, which we denote by A-G-discMod. Let M be an A-module endowed with the discrete topology. The induced module IndA,G (M ) is defined to be the A-module of continuous maps of G to M . The group G acts on IndA,G (M ) by the formula (g · f )(x) = f (x · g)

(f ∈ IndA,G (M ), g ∈ G, x ∈ G).

The A-module IndA,G (M ) with the above action of G is a discrete A-G-module and functorial on M . Thus we obtain an exact functor IndA,G : A-Mod → A-G-discMod. We say that a discrete A-G-module is induced if it is isomorphic to the induced module of an A-module. Proposition V.11.1. The functor IndA,G is a right adjoint of the forgetful functor. Especially IndA,G preserves injectives. Proof. For a discrete A-G-module M , we have a canonical and functorial morphism adjM : M → IndA,G (M ) of discrete A-G-modules which sends m to the map x 7→ x · m. For an A-module N , we can verify that the homomorphism HomA (M, N ) → HomA,G (M, IndA,G (N )); f 7→ IndA,G (f ) ◦ adjM

is an isomorphism. Indeed, the inverse is given by ϕ 7→ adj0N ◦ ϕ, where adj0N denotes the A-linear homomorphism IndA,G (N ) → N defined by ψ 7→ ψ(1).  Corollary V.11.2. (1) The category A-G-discMod has enough injectives. (2) An object of A-G-discMod is injective if and only if it is isomorphic to a direct factor of IndA,G (I) for an injective A-module I.

V.11. GROUP COHOMOLOGY OF DISCRETE A-G-MODULES

479

Proof. Let M be a discrete A-G-module and choose an A-linear injection f of M into an injective A-module I. Then, the morphism in A-G-discMod IndA,G (f ) ◦ adjM : M → IndA,G (I) is injective, where adjM is as in the proof of Proposition V.11.1. Furthermore, if M is injective, this admits a section. Now the corollary follows easily from Proposition V.11.1.  Let q HA (G, −) : A-G-discMod → A-Mod (q ≥ 0),

RΓA (G, −) : D+ (A-G-discMod) → D+ (A-Mod) be the right derived functors of the left exact functor ΓA (G, −) : A-G-discMod → A-Mod; M 7→ M G ,

where M G denotes the G-invariant part.

Lemma V.11.3. For an A-module M , we have a canonical and functorial isomorphism RΓA (G, IndA,G (M )) ∼ = M. ∼

Proof. We have a canonical and functorial isomorphism M → ΓA (G, IndA,G (M )) which sends m to the constant map x 7→ m (x ∈ G). Hence the claim follows from Proposition V.11.1 and the exactness of IndA,G .  Corollary V.11.4. Let A → B be a homomorphism of commutative rings. Then we have canonical isomorphisms of functors q q F ◦ HB (G, −) ∼ = HA (G, −) ◦ F (q ≥ 0), F ◦ RΓB (G, −) ∼ = RΓA (G, −) ◦ F,

where F denote the exact functors B-G-discMod → A-G-discMod, B-Mod → A-Mod restricting the action of B to A and the functors D+ (B-G-discMod) → D+ (A-G-discMod), D+ (B-Mod) → D+ (A-Mod) induced by them.

Proof. For any B-module M , we have F (IndB,G (M )) ∼ = IndA,G (F (M )). Hence, by Corollary V.11.2 (2) and Lemma V.11.3, the functor F : B-G-discMod → A-G-discMod sends injectives to objects acyclic with respect to ΓA (G, −).  q In the following, we abbreviate HA (G, −), RΓA (G, −) to H q (G, −), RΓ(G, −) if there is no risk of confusion. Let H be a closed normal subgroup of G. Then the groups H and G/H endowed with the induced topology are again profinite groups. We also denote by Γ(H, −) the left exact functor

A-G-discMod → A-G/H-discMod; M 7→ M H ,

and by H q (H, −) (q ≥ 0), RΓ(H, −) its right derived functors. This notation is justified by the following proposition. Proposition V.11.5. The following diagrams commute up to canonical isomorphisms of functors: H q (H,−)

A-G-discMod −−−−−−→ A-G/H-discMod     y y H q (H,−)

A-H-discMod −−−−−−→

A-Mod,

480

V. ALMOST ÉTALE COVERINGS RΓ(H,−)

D+ (A-G-discMod) −−−−−−→ D+ (A-G/H-discMod)     y y RΓ(H,−)

D+ (A-H-discMod) −−−−−−→

D+ (A-Mod).

Proof. This follows from Corollary V.11.2 (2), Lemma V.11.3, and the following Lemma V.11.6.  Lemma V.11.6. Induced discrete A-G-modules are induced discrete A-H-modules. Proof. Choose a continuous section s : G/H → G ([65] I Proposition 1). Then, we have a homeomorphism G ∼ = H × G/H; g 7→ (s(gH)−1 g, gH), s(g)h ← (h, g) compatible with the right action of H, where H acts on H × G/H by (h, g) 7→ (hk, g) (k ∈ H). Hence, there are isomorphisms of discrete A-H-modules IndA,G (M ) ∼ (H × G/H, M ) = Map cont

∼ = Mapcont (H, Mapcont (G/H, M ))

= IndA,H (Mapcont (G/H, M ))  Proposition V.11.7. The functor Γ(H, −) : A-G-Mod → A-G/H-Mod preserves injectives (resp. induced modules). Especially, we have a spectral sequence E1a,b = H a (G/H, H b (H, M )) =⇒ H a+b (G, M ) for a discrete A-G-module M . Proof. This follows from Γ(H, −) ◦ IndA,G ∼ = IndA,G/H and Corollary V.11.2 (2).  Finally, we will recall the explicit description of RΓ(G, −) in terms of the inhomogeneous cochain complex. Let M be a discrete A-G-module. For each integer q ≥ 0, we denote by K q (G, M ) the A-module of continuous maps of the (q + 1)-fold products Gq+1 to M endowed with the action of G defined by (g · f )(g0 , . . . , gq ) = g · f (g −1 g0 , . . . , g −1 gq ).

Then K q (G, M ) are discrete A-G-modules. We define the homomorphisms of discrete A-G-modules dq : K q (G, M ) → K q+1 (G, M ) (q ≥ 0) and ε : M → K 0 (G, M ) by the formulae (dq f )(g0 , . . . , gq+1 ) =

q+1 X (−1)i f (g0 , . . . , gˆi , . . . , gq+1 ), i=0

ε(m)(g0 ) = m. We have dq+1 ◦ dq = 0 (q ≥ 0) and d0 ◦ ε = 0. ε

Lemma V.11.8. The complex M → K • (G, M ) is homotopy equivalent to zero as a complex of A-modules. Proof. The homotopy between the identity and the zero endomorphisms of the complex in question is given by k q : K q (G, M ) → K q−1 (G, M ); (k q f )(g0 , . . . , gq−1 ) = f (1, g0 , . . . , gq−1 ) (q ≥ 1) and k 0 : K 0 (G, M ) → M ; k 0 f = f (1).  Lemma V.11.9. The discrete A-G-modules K q (G, M ) (q ≥ 0) are induced.

V.12. GALOIS COHOMOLOGY

481

Proof. By the change of variables ∼

Gq+1 → G × Gq ; (g0 , . . . , gq ) 7→ (g0 , g0−1 g1 , g0−1 g2 , . . . , g0−1 gq ),

we may replace the action of G on K q (G, M ) by

(g · f )(g0 , g1 , . . . , gq ) = g · f (g −1 g0 , g1 , . . . , gq ).

Then we have isomorphisms of discrete A-G-modules K q (G, M ) ∼ = Mapcont (G × Gq , M ) ∼ = IndA,G (Mapcont (Gq , M )),

where the first one is defined by f 7→ F (g0 , g1 , . . . , gq ) = g0 · f (g0−1 , g1 , . . . gq ) and the action of G on the middle module is defined by (g · ϕ)(g0 , g1 , . . . , gq ) = ϕ(g0 g, g1 , . . . , gq ).  The inhomogeneous cochain complex C • (G, M ) is defined as follows: The A-module C (G, M ) (q ≥ 0) is an A-module of continuous maps of Gq to M and the differential dq : C q (G, M ) → C q+1 (G, M ) (q ≥ 0) is defined by q

(dq f )(g1 , . . . , gq+1 ) = g1 · f (g2 , . . . , gq+1 ) +

(V.11.10)

q X (−1)i f (g1 , . . . , gi · gi+1 , . . . , gq+1 ) i=1

+ (−1)q+1 f (g1 , . . . , gq ). By the restriction with respect to the continuous map Gq → Gq+1 ; (g1 , . . . , gq ) 7→ (1, g1 , g1 g2 , . . . , g1 g2 · · · gq ), we obtain an isomorphism (V.11.11)

∼

K • (G, M )G ←− C • (G, M ).

By Lemmas V.11.3, V.11.8, and V.11.9, we obtain a canonical and functorial isomorphism in D+ (A-Mod): (V.11.12) RΓ(G, M ) ∼ = C • (G, M ). V.12. Galois cohomology Throughout this section, let G denote a finite group. Lemma V.12.1. Let A be a commutative ring and let B be an A-algebra endowed with an action of G. Let M be an arbitrary B-module endowed with a semi-linear action of G (i.e., g(b · m) = g(b) · m for b ∈ B, m ∈ M , g ∈ G). Let PtrG denote the A-linear homomorphisms B → B G and M → M G defined by trG (x) = g∈G g(x) (x ∈ B or M ). (1) For any a ∈ A whose image in B G is contained in trG (B), a · H q (G, M ) = 0 (q > 0) and a(M G /trG (M )) = 0. Pr (2) Let a ∈ A and Prsuppose that there exist bi , ci ∈ B (1 ≤ i ≤ r) such that i=1 bi · ci = a · 1B and that i=1 bi · g(ci · b) = 0 for all b ∈ B, g ∈ G\{1}. Then, the kernel and the cokernel of ψ : B ⊗A M G → M ; b ⊗ x 7→ b · x are killed by a. ε

Proof. (1) As in V.11, there exists a canonical resolution M → K • (G, M ) as AG-modules (Lemma V.11.8) such that H i (G, K q (G, M )) = 0 (i > 0, q ≥ 0) (Lemma V.11.9 and Lemma V.11.3). By the B-module structure of M , the A-module structure on K q (G, M ) naturally extends to a B-module structure, the action of G on K q (G, M )

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V. ALMOST ÉTALE COVERINGS

becomes B-semi-linear, and the above resolution becomes B-linear. For any b ∈ B, the composite of the A-linear homomorphisms K • (G, M )G → K • (G, M ); x 7→ b · x and trG : K • (G, M ) → K • (G, M )G is the multiplication by a. Since H q (K • (G, M )G ) = H q (G, M ) and H q (K • (G, M )) = 0 (if q > 0), M (if q = 0), this implies (1). P r (2) Define the A-linear homomorphism ϕ : M → B ⊗A M G by ϕ(m) = i=1 bi ⊗ trG (ci m). Then we have ϕ ◦ ψ = a · 1B⊗A M G and ψ ◦ ϕ = a · 1M .  Definition V.12.2. Let R be a V -algebra. An R-algebra S endowed with an action of G is a G-covering (of R) if there exists an almost faithfully flat homomorphism R → R0 and ≈ Q 0 an almost isomorphism SQ = S ⊗R R0 → G R0 over R0 compatible with the actions of G. Here the action of G on G R0 is defined by g((xh )h∈G ) = (xhg )h∈G (g ∈ G, xh ∈ R0 ). Lemma V.12.3. Let R be a V -algebra, and let f : R → S be a G-covering of R. Then f is an almost étale covering and rkR S = |G|. Proof. The first claim follows from Proposition V.7.4 (1) and Corollary V.8.10. Let R → R0 be an almost faithfully flat homomorphism as in Definition V.12.2. Then, by Proposition V.4.2 (1) and the remark after Definition V.5.14, rkR0 S 0 = |G| where |G|+1 |G|+1 |G|+1 S 0 = S ⊗R R0 . Since (∧R S)⊗R R0 ∼ S ≈ 0. On the other = ∧R0 S 0 ≈ 0, we have ∧R 0 hand, the kernel of R → R is almost zero by Lemma V.8.2, and hence HomV (m, R) → HomV (m, R0 ) is injective. By Proposition V.4.2 (2), we obtain trS/R (id∧|G| S ) = 1.  R

Lemma V.12.4. Let R be a V -algebra and let S be an R-algebra endowed with an action Q of G. Then S is a G-covering of R if and only if the homomorphism S ⊗R S → G S defined by x ⊗ y 7→ (x · g(y))g∈G is an almost isomorphism and S is an almost faithfully flat R-algebra. If we regard S ⊗R S as anQS-algebra by S → S ⊗R S; y 7→ y ⊗ 1 and define the action of G on S ⊗R S (resp. G S) by x ⊗ y 7→ x ⊗ g(y) Q (x, y ∈ S, g ∈ G) (resp. as in Definition V.12.2), then the homomorphism S ⊗R S → G S in Lemma V.12.4 is an S-algebra homomorphism compatible with the actions of G. Proof. By the above remark, the sufficiency is trivial. Conversely suppose that S is a G-covering of R and choose a faithfully flatQ homomorphism R → R0 as in Definition 0 0 V.12.2. Then, one sees easily that S ⊗R0 S → G S 0 for the base change S 0 := S ⊗R R0 is an almost isomorphism and hence it also holds for S/R. By Proposition V.6.5 (1), R0 → S 0 is almost faithfully flat. By Proposition V.8.1, R → S is almost faithfully flat.  Proposition V.12.5. Let R be a V -algebra and let S be a G-covering of R. Then, for any S-module M endowed with a semi-linear action of G, the homomorphism S⊗R M G → M ; y ⊗ m 7→ y · m is an almost isomorphism. ≈ Q Proof. The almost isomorphism S ⊗R S → G S;Qx ⊗ y 7→ (x · g(y))g∈G (Lemma V.12.4) induces an isomorphism HomV (m, S ⊗R S) ∼ = G HomV (m, S) by Proposition V.2.5 (1). Let e be the idempotent of HomV (m, S ⊗R S) corresponding to (δ1,g )g∈G in the right-hand side, where δ1,g = 1 (if g = For any α ∈ Λ+ , choose P1), 0 (otherwise). 0 0 α 0 0 xi , yi ∈ S (1 ≤ i ≤ r) such that π e = 1≤i≤r xi ⊗ yi in HomV (m, S ⊗R S) and set P P xi := π α x0i , yi := π α yi0 . Then we have 1≤i≤r xi · yi = (π α )3 and 1≤i≤r xi · g(yi y) = 0 for g ∈ G\{1} and y ∈ S. By applying Lemma V.12.1 (2), we obtain the proposition.  Corollary V.12.6. Let R be a V -algebra and let S be a G-covering of R. Then the natural homomorphism R → S G is an almost isomorphism.

V.12. GALOIS COHOMOLOGY

483

Proof. By Proposition V.12.5, the homomorphism S G ⊗R S → S; x ⊗ y 7→ x · y is an almost isomorphism, and hence so is S → S G ⊗R S. Since R → S is almost faithfully ≈ flat by Lemma V.12.4, we have R → S G .  Lemma V.12.7. Let R be a V -algebra, letPS be a G-covering of R, and let trG denote the R-linear homomorphism S → S G ; y 7→ g∈G g(y). Then trG followed by the natural ≈

almost isomorphism S G → HomV (m, S G ) coincides with the composition of trS/R : S → ∼ HomV (m, R) and the isomorphism HomV (m, R) → HomV (m, S G ) (Corollary V.12.6, Proposition V.2.5 (1)).

Proof. If we regard S ⊗R S as an S-algebra by the homomorphism S → S ⊗R S; y 7→ ≈ Q y ⊗ 1, there exists an almost isomorphism of S-algebras S ⊗R S → G S; x ⊗ y 7→ (x · g(y))g∈G by Lemma V.12.4. By Propositions V.4.2 and V.4.3, for y ∈ S, the image of tr (y) in HomV (m, S) is trS⊗R S/S (1 ⊗ y) = trQG S/S ((g(y))g∈G ) = the image of P S/R  g∈G g(y). Proposition V.12.8. Let R be a V -algebra and let S be a G-covering of R. Then, for any S-module M with a semi-linear action of G, we have m · H q (G, M ) = 0 (q > 0) G G and P m · (M /trG (M )) = 0, where trG denotes the homomorphism M → M ; m 7→ g∈G g(m). Proof. This follows from Lemma V.12.1(1), Lemma V.12.7, and Proposition V.7.12. Note that R → S is almost faithfully flat by Lemma V.12.4. 

Proposition V.12.9. Let R be a normal domain over V which is π α -torsion free for all α ∈ Λ+ , let L be a finite Galois extension of K := Frac(R), and let S be the normalization of R in L. If R → S is an almost étale covering, then it is a Gal(L/K)-covering.

Proof. Set G := Gal(L/K) and let Rπ and Sπ be as in Proposition V.7.11. First let us prove rankR S = |G| and hence S is almost faithfully flat over R (Proposition V.6.7). By Proposition V.5.16, we have a decomposition HomV (m, R) = Rr × · · · × R0 , Rr 6= 0 such that rankRi (S ⊗R Ri ) = i (0 ≤ i ≤ r). By taking ⊗R Rπ , we obtain a decomposition Rπ ∼ = HomV (m, R) ⊗R Rπ = Rr,π × · · · × R0,π , where Ri,π = Ri [ π1α (α ∈ Λ+ )]. Since Ri are π α -torsion free and Rπ is a domain, we have Ri = 0 (0 ≤ i < r) and rankR S = r. Since Rπ → Sπ is finite étale (Propositions V.7.3, V.7.4 (3)), we have S ⊗R K = L and rankR S = rankK L = |G|. Q Now, by Lemma V.12.4, it suffices to prove that the homomorphism ϕ : S ⊗R S → G S; x ⊗ y 7→ x · g(y) of almost étale coverings of S is an almost isomorphism. After inverting π α (α ∈ Λ+ ), this becomes an isomorphism because both sides become normal rings finite étale over Sπ ([42] Proposition (17.5.7)) and the ∼ Q homomorphism becomes the isomorphism L ⊗K L → G L; x → x · g(y) after ⊗R K. By Proposition V.7.11, ϕ is an almost isomorphism. 

CHAPTER VI

Covanishing topos and generalizations Ahmed Abbes and Michel Gros VI.1. Introduction VI.1.1. The aim of this chapter is to consolidate the topogical foundation necessary for Faltings’ approach in p-adic Hodge theory [24, 26, 27]. Its genesis has been motivated by our work on the p-adic Simpson correspondence (cf. I–III). Schematically, Faltings’ approach consists of two steps. The first, of a local nature, is a generalization of the Galois techniques of Tate, Sen, and Fontaine to certain affine schemes over p-adic local fields. It uses, in an essential manner, his theory of almost étale extensions (cf. V). The second step, of a more global nature, links the p-adic étale cohomology of certain schemes over p-adic local fields with the Galois cohomology studied in the first step. To do this, Faltings uses, on the one hand, the notion of K(π, 1)-schemes [24] and, on the other hand, a new topos [26]. The latter has been explicitly defined rather late compared to the rest of the theory ([26] page 214), and in our opinion has not received the attention it deserves. VI.1.2. We have benefited from a letter from Deligne to Illusie [16] (prior to [26]). In this letter, Deligne suggests that the topos that Faltings needs should in reality be a covanishing topos, in other words, a special case of the oriented product of topos that he had introduced to develop the formalism of vanishing cycles in greater dimensions [46]. The first author of the present work noted in 2008 that the covanishing topology of Deligne differs in general from that considered by Faltings in ([26] page 214); however, the latter uses a characterization of sheaves that holds for covanishing topos (cf. VI.1.5).1 It turns out that this characterization is not in general satisfied by the topos initially considered by Faltings (cf. VI.10.2 for a counterexample). In fact, one of the main sheaves used by Faltings, namely the structural sheaf, is not in general a sheaf for the topology he defined in ([26] page 214); but it is a sheaf for the covanishing topology (cf. III.8.16 and III.8.18). In this chapter, we propose to correct the definition of the topos introduced by Faltings and to develop it following Deligne’s suggestion. It is this new topos that we will call Faltings topos; the one introduced in ([26] page 214) seems of little interest. VI.1.3. Since the results of this chapter are rather technical, we will now give a detailed summary of them. Let us begin by recalling the definition of oriented products e → Se and g : Ye → Se be two morphisms of U-topos, of topos due to Deligne. Let f : X ← e × e Ye is a U-topos endowed with where U is a fixed universe. The oriented product X S two morphisms (VI.1.3.1)

←

e × e Ye → X e p1 : X S

and

←

e × e Ye → Ye p2 : X S

1This problem does not seem to have been observed beforehand. 485

486

VI. COVANISHING TOPOS AND GENERALIZATIONS

and a 2-morphism (VI.1.3.2) ←

τ : gp2 → f p1 ,

e × e Ye , p1 , p2 , τ ) is universal in the 2-category of U-topos. We such that the quadruple (X S can explicitly construct for it an underlying site C from the data of U-sites X, Y , and e Ye , and S, e respectively, in which finite inverse limits are representable, S underlying X, and of two continuous left exact functors f + : S → X and g + : S → Y defining f and ← e × e Se the vanishing topos of f , g (cf. VI.3.1 and VI.3.7). Following Illusie, we call X S ← and Se ×Se Ye the covanishing topos of g. The first topos has been used by Deligne to develop the formalism of vanishing cycles of f , and has been studied by Gabber, Illusie [46], Laumon [53], and Orgogozo [60]. The second topos is the prototype of the Faltings topos. We can explicitly construct for it another underlying site D, simpler than C, that we call the covanishing site of g + (cf. VI.4.1 and VI.4.10). VI.1.4. Strictly speaking, the Faltings topos is not in fact a covanishing topos, but a special case of a more general notion that we develop in this chapter, and that we call generalized covanishing topos. Let I be a U-site, Ie the topos of sheaves of U-sets on I, π: E → I

(VI.1.4.1)

a cleaved and normalized fibered category over the category underlying I. We suppose that the following conditions are satisfied: (i) Fibered products are representable in I. (ii) For every i ∈ Ob(I), the fiber Ei of E over i is endowed with a topology making it into an U-site, and finite inverse limits are representable in Ei . We denote ei the topos of sheaves of U-sets on Ei . by E (iii) For every morphism f : i → j of I, the inverse image functor f + : Ej → Ei is continuous and left exact. It therefore defines a morphism of topos that we will ei → E ej . (abusively) denote also by f : E The functor π is in fact a fibered U-site (cf. VI.5.1). We call covanishing topology on E the topology generated by the families of coverings (Vn → V )n∈Σ of the following two types: (v) There exists i ∈ Ob(I) such that (Vn → V )n∈Σ is a covering family of Ei . (c) There exists a covering family of morphisms (fn : in → i)n∈Σ of I such that Vn is isomorphic to fn+ (V ) for every n ∈ Σ. The coverings of type (v) are called vertical, and those of type (c) are called Cartesian. The resulting site is called the covanishing site associated with the fibered site π; it is a e U-site. We call covanishing topos associated with the fibered site π, and denote by E, the topos of sheaves of U-sets on E (cf. VI.5.3). When I is endowed with the chaotic topology, that is, with the coarsest topology on I, we recover the total topology on E associated with the fibered site π (cf. VI.5.4). VI.1.5. We show (VI.5.10) that giving a sheaf F on E is equivalent to giving, for every object i of I, a sheaf Fi on Ei and, for every morphism f : i → j of I, a morphism Fj → f∗ (Fi ), these morphisms being subject to compatibility relations, such that for every covering family (fn : in → i)n∈Σ of I, if for every (m, n) ∈ Σ2 , we set imn = im ×i in and we denote by fmn : imn → i the canonical morphism, the sequence of morphisms of sheaves on Ei Y Y (VI.1.5.1) Fi → (fn )∗ (Fin ) ⇒ (fmn )∗ (Fimn ) n∈Σ

(m,n)∈Σ2

VI.1. INTRODUCTION

487

is exact. We will, from now on, identify F with the functor {i 7→ Fi } associated with it. We study the functoriality of the covanishing sites and topos with respect to the fibration π (VI.5.18 and VI.5.36) and by base change (VI.5.20). We also establish a number of coherence properties. We show (VI.5.28), among other things, that if there exists a full subcategory I 0 of I, U-small and topologically generating, made up of quasicompact objects stable under fibered products, such that for every i ∈ Ob(I 0 ), the topos ei is coherent and that for every morphism f : i → j of I 0 , the morphism of topos E ei → E ej is coherent, then the topos E e is locally coherent. If, moreover, the category f: E e is coherent. I admits a final object that belongs to I 0 , then the topos E VI.1.6. The link with Deligne’s covanishing topos is as follows. Let X and Y be two U-sites in which finite inverse limits are representable, and f + : X → Y a continuous e and Ye the topos of sheaves of U-sets on X and left exact functor. We denote by X e e Y , respectively, and by f : Y → X the morphism of topos defined by f + . Consider the category Fl(Y ) of morphisms of Y , and the “target functor” (VI.1.6.1)

Fl(Y ) → Y,

that makes Fl(Y ) into a cleaved and normalized fibered category over Y ; the fiber over any V ∈ Ob(Y ) is canonically equivalent to the category Y/V . Endowing each fiber Y/V with the topology induced by that of Y , Fl(Y )/Y becomes a fibered U-site, satisfying the conditions of VI.1.4. Let (VI.1.6.2)

π: E → X

be the fibered site deduced from Fl(Y )/Y by base change by the functor f + . The covanishing site E associated with π (VI.1.4) is canonically equivalent to the covanishing site D associated with the functor f + (VI.1.3); whence the terminology. The covanishing e associated with π is therefore canonically equivalent to the covanishing topos topos E ← e X ×Xe Ye associated with f (cf. VI.5.5). VI.1.7. We keep the assumptions of VI.1.4 and moreover suppose that finite inverse limits are representable in I; in view of VI.1.4(i), it is equivalent to requiring I to admit e analogues a final object ι, which we assume fixed from now on. We can then define for E of the canonical projections of the oriented product (VI.1.3.1). On the one hand, the canonical injection functor αι! : Eι → E is continuous and left exact (cf. VI.5.32). It therefore defines a morphism of topos (VI.1.7.1)

e→E eι , β: E

analogous to the second projection p2 (VI.1.3.1). On the other hand, we take a final object e of Eι , which exists by VI.1.4(ii), and which we assume fixed from now on. There then exists essentially a unique Cartesian section of π (VI.1.4.1) (VI.1.7.2)

σ+ : I → E

such that σ + (ι) = e. For every i ∈ Ob(I), σ + (i) is a final object of Ei . One easily verifies that σ + is continuous and left exact (cf. VI.5.32). It therefore defines a morphism of topos (VI.1.7.3)

e → I, e σ: E

analogous to the first projection p1 (VI.1.3.1).

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VI. COVANISHING TOPOS AND GENERALIZATIONS

VI.1.8. We keep the assumptions of VI.1.7 and moreover let V be an object of E, c = π(V ), and (VI.1.8.1)

$ : E/V → I/c

the functor induced by π. For every morphism f : i → c of I, the fiber of $ over the object (i, f ) of I/c is canonically equivalent to the category (Ei )/f + (V ) . Endowing I/c with the topology induced by that of I, and each fiber (Ei )/f + (V ) with the topology induced by that on Ei , $ becomes a fibered site satisfying the conditions of VI.1.4. We show (VI.5.38) that the covanishing topology on E/V is induced by that on E through the canonical functor E/V → E. In particular, the topos of sheaves of U-sets on the e/ε(V ) , where ε : E → E e is the canonical covanishing site E/V is canonically equivalent to E functor. VI.1.9. One of the main characteristics of Deligne’s covanishing topos is the existence of a morphism of co-nearby cycles (cf. VI.4.13). This also extends to the generalized covanishing topos introduced in VI.1.4. We keep the assumptions of VI.1.7 and moreover consider a U-site X and a continuous and left exact functor Ψ+ : E → X. We denote by e the topos of sheaves of U-sets on X and by X (VI.1.9.1)

e →E e Ψ: X

the morphism of topos associated with Ψ+ . We set u+ = Ψ+ ◦ σ + : I → X and (VI.1.9.2)

e → I. e u = σΨ : X

For every i ∈ Ob(I), the functor Ψ+ induces a functor Ψ+ i : Ei → X/u+ (i) . When we endow X/u+ (i) with the topology induced by that of X, Ψ+ i is exact and left continuous (cf. VI.6.1). It therefore defines a morphism of topos (VI.1.9.3)

e/u∗ (i) → E ei . Ψi : X

e →E eι . From the relation The morphism Ψι is none other than the composition βΨ : X ∗ ∗ ∗ Ψ β = Ψι , we deduce by adjunction a morphism (VI.1.9.4)

β ∗ → Ψ∗ Ψ∗ι .

Generalizing an important property of the covanishing topos of Deligne, we show (VI.6.3) that if for every i ∈ Ob(I), the adjunction morphism id → Ψi∗ Ψ∗i is an isomorphism, then the adjunction morphisms id → β∗ β ∗ and β ∗ → Ψ∗ Ψ∗ι are isomorphisms. VI.1.10. We keep the assumptions of VI.1.9 and moreover suppose that finite inverse limits are representable in X. We denote by $ : D → I the fibered sited associated with the functor u+ : I → X defined in (VI.1.6), and we endow D with the covanishing topology associated with $. We thus obtain the covanishing topos associated with the ← e By the universal the functor u+ (VI.1.3), whose topos of sheaves of U-sets is Ie ×Ie X. e e property of oriented products, the morphisms u : X → I and idXe and the 2-morphism idu define a morphism of topos, called morphism of co-nearby cycles (cf. VI.4.13) (VI.1.10.1)

←

e → Ie × e X. e ΨD : X I

On the other hand, the functors Ψ+ i for every i ∈ Ob(I) define a Cartesian I-functor ρ+ : E → D. The latter is continuous and left exact (cf. VI.6.4). It therefore defines a morphism of topos (VI.1.10.2)

←

e → E. e ρ : Ie ×Ie X

VI.1. INTRODUCTION

489

One immediately verifies that the squares of the diagram (VI.1.10.3)

Ie o

p1

p2

←

e Ie ×Ie X ρ

Ie o

σ

 e E

/X e Ψι

β

 /E eι

and the diagram (VI.1.10.4)

/ Ie × X e eD X Ie DD DD DD ρ Ψ DD D"  e E ΨD

←

are commutative up to canonical isomorphisms. e it amounts to VI.1.11. We keep the assumptions of VI.1.7 and let R be a ring of E; e giving, for every i ∈ Ob(I), a ring Ri of Ei , and for every morphism f : i → j of I, a ring homomorphism Rj → f∗ (Ri ), these homomorphisms being subject to compatibility and gluing relations (VI.1.5.1). We develop in VI.8 a number of results on the category of Re in particular, on the tensor product (VI.8.10) and the sheaf of morphisms modules of E, e R). (VI.8.12). We also study elementary cohomological invariants of the ringed topos (E, As an intermediary step, we revisit in VI.7 the formalism of ringed total topos ([2] VI 8.6). For the sake of simplification, suppose that the category I is U-small (cf. VI.8.4 for the general case). We denote by Top(E) the total topos associated with the fibered site π, that is, the topos of sheaves of U-sets on the total site E (cf. VI.7.1). We then have a canonical embedding of topos (VI.5.16.1) (VI.1.11.1)

e → Top(E) δ: E

e in Top(E). We consider it such that the functor δ∗ is the canonical inclusion functor of E as a morphism of ringed topos (by R and δ∗ (R), respectively), and we compute in VI.8.5 the right derived functors of δ∗ . The Cartan–Leray spectral sequence associated with δ then gives a spectral sequence that computes the higher direct images of a morphism of ringed generalized covanishing topos (VI.8.8). Note that the Cartan–Leray spectral e is the sequence associated with δ that computes the cohomology of an R-module of E same as that associated with β (VI.8.6). VI.1.12. The remainder of the chapter is devoted to studying a special case of the generalized covanishing topos, namely the Faltings topos. As a prelude, we develop in VI.9 a few results on the finite étale topos that we have not been able to find in the ´ /X its étale site, that is, the category of literature. For any scheme X, we denote by Et étale schemes over X (elements of U), endowed with the étale topology, and by X´et the ´ /X . We call finite étale site of X, and denote by Et ´ f/X , topos of sheaves of U-sets on Et ´ the full subcategory of Et/X made up of finite étale covers of X, endowed with the ´ /X . We call finite étale topos of X, and denote by Xf´et , topology induced by that of Et ´ f/X . The canonical injection functor Et ´ f/X → Et ´ /X the topos of sheaves of U-sets on Et induces a morphism of topos (VI.1.12.1)

ρX : X´et → Xf´et .

We show (VI.9.18) that if X is a coherent scheme having a finite number of connected components, then the adjunction morphism id → ρX∗ ρ∗X is an isomorphism; in particular,

490

VI. COVANISHING TOPOS AND GENERALIZATIONS

the functor ρ∗X : Xf´et → X´et is fully faithful. Its essential image consists of the filtered direct limits of locally constant constructible sheaves (VI.9.20); we also establish a variant for abelian torsion sheaves. We denote by Sch the category of schemes elements of U, by R the category of finite étale covers (that is, the full subcategory of the category of morphisms of Sch made up of finite étale covers), and by (VI.1.12.2)

R → Sch

the “target functor,” which makes R into a cleaved and normalized fibered category over ´ f/X . We Sch; the fiber over a scheme X is canonically equivalent to the category Et consider R/Sch as a fibered U-site by endowing each fiber with the étale topology. VI.1.13. Let f : Y → X be a morphism of schemes. We call Faltings fibered site associated with f the fibered U-site (VI.1.13.1)

´ /X π : E → Et

deduced from R/Sch (VI.1.12.2) by base change by the functor (VI.1.13.2)

´ /X → Sch, Et

U 7→ U ×X Y.

We can describe the category E explicitly as follows (cf. VI.10.1). The objects of E are the morphisms of schemes V → U over f : Y → X such that the morphism U → X is étale and that the morphism V → UY = U ×X Y is finite étale. Let (V 0 → U 0 ), (V → U ) be two objects of E. A morphism from (V 0 → U 0 ) to (V → U ) consists of an X-morphism U 0 → U and a Y -morphism V 0 → V such that the diagram (VI.1.13.3)

V0

/ U0

 V

 /U

is commutative. The functor π is then defined for every object (V → U ) of E by π(V → U ) = U . This clearly satisfies the conditions of VI.1.4 and VI.1.7. We can therefore apply to it the constructions developed above. We endow E with the covanishing topology associated with π. The resulting covanishing site is called the Faltings site associated e the with f ; it is a U-site. We call Faltings topos associated with f , and denote by E, topos of sheaves of U-sets on E. In fact, Faltings limits himself to the following case. Let K be a complete discrete valuation ring of characteristic 0, with perfect residue field of characteristic p > 0, OK the valuation ring of K, K an algebraic closure of K, X a separated OK -scheme of finite type, and X ◦ an open subscheme of X. Faltings only ◦ considers the case where f is the canonical morphism XK → X. Many results on the Faltings site and topos in VI.10 are direct applications of those established in VI.5 and VI.6 for generalized covanishing sites and topos. In particular, we find in this paragraph a study of the functoriality with respect to f (VI.10.12), and of the localization with respect to an object of E (VI.10.14). In the remainder of this introduction, we summarize other more specific properties. VI.1.14. We keep the assumptions of VI.1.13 and moreover suppose that X is ´ coh/X (resp. Et ´ scoh/X ) the full subcategory of Et ´ /X quasi-separated. We denote by Et made up of étale schemes of finite presentation over X (resp. étale separated schemes ´ /X ; of finite presentation over X), endowed with the topology induced by that on Et they are U-small sites. We denote by ? the symbol “coh” or “scoh.” Recall that the

VI.1. INTRODUCTION

491

´ ?/X is an equivalence restriction functor from X´et to the topos of sheaves of U-sets on Et of categories. We denote by ´ ?/X π? : E? → Et

(VI.1.14.1)

the fibered site deduced from π by base change by the canonical injection functor ´ ?/X → Et ´ /X , Et

(VI.1.14.2)

and by Φ : E? → E the canonical projection. We show (VI.5.21) that if we endow E? e? the topos of sheaves of with the covanishing topology defined by π? and denote by E ∼ e e→ U-sets on E? , the functor Φ induces by restriction an equivalence of categories E E? . Moreover, the covanishing topology on E? is induced by that on E through the functor Φ (VI.5.22). The subcategory Ecoh allows us to apply the coherence results established earlier to the Faltings topos. We show (VI.10.5), for example, that if X and Y are coherent, then e is coherent; in particular, it has enough points. We will see further on the the topos E interest of introducing the subcategory Escoh (VI.1.17). ´ /X with the final object VI.1.15. We keep the assumptions of VI.1.13 and endow Et X and E with the final object (Y → X). The functors αX! and σ + introduced in VI.1.7 are explicitly defined by (VI.1.15.1) (VI.1.15.2)

´ f/Y → E, αX! : Et ´ /X → E, σ + : Et

V 7→ (V → X),

U 7→ (UY → U ).

They are left exact and continuous. They therefore define two morphisms of topos e → Yf´et , β: E e → X´et . σ: E

(VI.1.15.3) (VI.1.15.4) On the other hand, the functor (VI.1.15.5)

´ /Y , Ψ+ : E → Et

(V → U ) 7→ V

is continuous and left exact (VI.10.7); it therefore defines a morphism of topos (VI.1.15.6)

e Ψ : Y´et → E.

´ /X , we can identify the morphism ΨU We have f´et = σΨ. For every object U of Et defined in (VI.1.9.3) with the canonical morphism ρUY : (UY )´et → (UY )f´et (VI.1.12.1); in particular, we have βΨ = ρY . From the isomorphism Ψ∗ β ∗ = ρ∗Y , we deduce by adjunction a morphism (VI.1.15.7)

β ∗ → Ψ∗ ρ∗Y .

Suppose, moreover, that X is quasi-separated and that Y is coherent and étale-locally connected (that is, for every étale morphism X 0 → X, every connected component of X 0 is an open set of X 0 ). We show (VI.10.9) that the adjunction morphisms id → β∗ β ∗ and β ∗ → Ψ∗ ρ∗Y are isomorphisms. ´ /X the VI.1.16. We keep the assumptions of VI.1.13. We denote by $ : D → Et + ´ ´ /Y defined in fibered site associated with the inverse image functor f : Et/X → Et VI.1.6, and we endow D with the covanishing topology associated with $. We thus obtain the covanishing site associated with the functor f + , whose topos of sheaves of ← U-sets is X´et ×X´et Y´et . Every object of E is naturally an object of D. We therefore have a fully faithful left exact functor ρ+ : E → D that is none other than the functor of the

492

VI. COVANISHING TOPOS AND GENERALIZATIONS

same name defined in the more general setting of VI.1.10. Since the latter is continuous and left exact, it defines a morphism of topos ←

e ρ : X´et ×X´et Y´et → E.

(VI.1.16.1)

We refer to VI.10.15 for more details. We show (VI.10.21) that if X and Y are coherent, ← e images by ρ of the points of X´et ×X Y´et is conservative. then the family of points of E ´ et ←

Note that giving a point of X´et ×X´et Y´et is equivalent to giving a pair of geometric points x of X and y of Y and a specialization arrow from f (y) to x, that is, an X-morphism y → X(x) , where X(x) denotes the strict localization of X at x (cf. VI.10.18).

VI.1.17. We keep the assumptions of VI.1.13 and moreover suppose that X is strictly local, with closed point x. For any separated étale X-scheme of finite presentation U , we denote by U f the disjoint sum of the strict localizations of U at the points of Ux ; it is an open and closed subscheme of U , which is finite over X (cf. VI.10.22). Consider the fibered site ´ scoh/X (VI.1.17.1) πscoh : Escoh → Et defined in (VI.1.14.1), and endow Escoh with the covanishing topology associated with πscoh . For every object (V → U ) of Escoh , V ×U U f = V ×UY UYf is a finite étale cover of Y . We thus obtain a functor ´ f/Y , (V → U ) 7→ V ×U U f . (VI.1.17.2) θ+ : Escoh → Et It is continuous and left exact (VI.10.23). It therefore defines a morphism of topos e θ : Yf´et → E,

(VI.1.17.3)

that we study following the approach introduced in VI.1.9 (cf. VI.10.24). We have a ∼ canonical isomorphism βθ → idYf´et , which induces a base change morphism β∗ → θ∗ .

(VI.1.17.4)

We show (VI.10.27) that the base change morphism β∗ → θ∗ is an isomorphism; in particular, the functor β∗ is exact. We deduce from this (VI.10.28) that for every sheaf e the canonical map F of E, e F ) → Γ(Yf´et , θ∗ F ) Γ(E,

(VI.1.17.5)

e the canonical map is bijective, and that for every abelian sheaf F of E, (VI.1.17.6) is bijective for every i ≥ 0.

e F ) → Hi (Yf´et , θ∗ F ) Hi (E,

VI.1.18. We keep the assumptions of VI.1.13 and moreover suppose that f is coherent. Let x be a geometric point of X, X the strict localization of X at x, Y = Y ×X X, and f : Y → X the canonical projection. We denote by E the Faltings site e the topos of sheaves of U-sets on E, and by associated with f , by E e θ : Y f´et → E

(VI.1.18.1)

the morphism of topos defined in (VI.1.17.3). Since the functor (VI.1.18.2)

Φ+ : E → E,

(V → U ) 7→ (V ×Y Y → U ×X X)

is continuous and left exact (cf. VI.10.12), it defines a morphism of topos (VI.1.18.3)

e → E. e Φ: E

VI.2. NOTATION AND CONVENTIONS

493

e we have a functorial canonical isomorWe show (VI.10.30) that for every sheaf F of E, phism (VI.1.18.4)

∼

σ∗ (F )x → Γ(Y f´et , θ∗ (Φ∗ F ));

e and every integer i ≥ 0, we have a functorial canonical and for every abelian sheaf F of E isomorphism (VI.1.18.5)

∼

Ri σ∗ (F )x → Hi (Y f´et , θ∗ (Φ∗ F )).

The proof of this result depends on the computation of an inverse limit of Faltings topos (VI.11.3) and its cohomological consequences (VI.11.6). In VI.10.40 we give a global variant of the isomorphism (VI.1.18.5). VI.1.19. Finally, let us point out that the generalized covanishing topos may be suitable for other applications, including rigid and henselian variants of the Faltings topos. We want to thank L. Illusie for having passed on Deligne’s letter [16] and his article [46], which have been the main source of inspiration for this work. After we sent him a first version of this work, O. Gabber sent us a copy of an email he had sent to L. Illusie in 2006 in which he defines the covanishing topology for a fibered site over a site and sketches statements that overlap some of our results. We are very grateful for this exchange and the thus offered prospects for further development. A large part of this work was written during the stay of the first author (A.A.) at the University of Tokyo during the autumn of 2010 and the winter of 2011. He wishes to thank this university for its hospitality. VI.2. Notation and conventions All rings in this chapter have an identity element; all ring homomorphisms map the identity element to the identity element. VI.2.1. For this entire chapter, we fix a universe U with an element of infinite cardinality. We call category of U-sets, and denote by Ens, the category of sets that are in U. It is a punctual U-topos that we also denote by Pt ([2] IV 2.2). Unless stated otherwise, the schemes in this chapter are assumed to be elements of the universe U. We denote by Sch the category of schemes elements of U. VI.2.2. For a category C , we denote by Ob(C ) the set of its objects, by C ◦ the opposite category, and for X, Y ∈ Ob(C ), by HomC (X, Y ) (or Hom(X, Y ) when there is no ambiguity) the set of morphisms from X to Y . If C and C 0 are two categories, we denote by Hom(C , C 0 ) the set of functors from C to C 0 , and by Hom(C , C 0 ) the category of functors from C to C 0 . Let I be a category and C , C 0 two categories over I ([37] VI 2). We denote by HomI (C , C 0 ) the set of I-functors from C to C 0 and by Homcart/I (C , C 0 ) the set of Cartesian functors ([37] VI 5.2). We denote by HomI (C , C 0 ) the category of I-functors from C to C 0 and by Homcart/I (C , C 0 ) the full subcategory made up of Cartesian functors. VI.2.3. Let C be a category. We denote by Cb the category of presheaves of U-sets on C , that is, the category of contravariant functors on C with values in Ens ([2] I 1.2). If C is endowed with a topology ([2] II 1.1), we denote by Ce the topos of sheaves of U-sets on C ([2] II 2.1). For an object F of Cb, we denote by C/F the following category ([2] I 3.4.0). The objects of C/F are the pairs consisting of an object X of C and a morphism u from X to

494

VI. COVANISHING TOPOS AND GENERALIZATIONS

F . If (X, u) and (Y, v) are two objects, a morphism from (X, u) to (Y, v) is a morphism g : X → Y such that u = v ◦ g.

VI.2.4. For a ringed topos (E , R), we denote by Mod(R) or Mod(R, E ) the category of (left) R-modules of E , by D(R) its derived category, and by D− (R), D+ (R), and Db (R) the full subcategories of D(R) made up of complexes with cohomology bounded from above, from below, and from both sides, respectively. VI.2.5. Following the conventions of ([2] VI), we use the adjective coherent as a synonym for quasi-compact and quasi-separated. VI.3. Oriented products of topos The notion of oriented products of topos, recalled below, is due to Deligne. It has been studied by Gabber, Illusie, Laumon, and Orgogozo [46, 53, 60]. VI.3.1. In this section, X, Y , and S denote U-sites ([2] II 3.0.2) in which finite inverse limits are representable, and f+ : S → X

and g + : S → Y e Ye , and Se the topos of sheaves of two continuous left exact functors. We denote by X, U-sets on X, Y , and S, respectively, by e → Se and g : Ye → Se (VI.3.1.2) f: X

(VI.3.1.1)

the morphisms of topos defined by f + and g + ([2] IV 4.9.2), respectively, and by εX : X → e εY : Y → Ye , and εS : S → Se the canonical functors. Let eX , eY , and eS be final X, objects of X, Y , and S, respectively, which exist by assumption. Since the canonical e Ye , and S, e functors are left exact, εX (eX ), εY (eY ), and εS (eS ) are final objects of X, respectively. We denote by C the category of triples (W, U → f + (W ), V → g + (W )),

where W is an object of S, U → f + (W ) is a morphism of X, and V → g + (W ) is a morphism of Y ; such an object will be denoted by (U → W ← V ). Let (U → W ← V ) and (U 0 → W 0 ← V 0 ) be two objects of C. A morphism from (U 0 → W 0 ← V 0 ) to (U → W ← V ) consists of three morphisms U → U 0 , V → V 0 , and W → W 0 of X, Y , and S, respectively, such that the diagrams (VI.3.1.3)

U0

/ f + (W 0 )

V0

/ g + (W 0 )

 U

 / f + (W )

 V

 / g + (W )

are commutative. It immediately follows from the definition and the fact that the functors f + and g + are left exact that finite inverse limits in C are representable. We endow C with the topology generated by the coverings {(Ui → Wi ← Vi ) → (U → W ← V )}i∈I

of the following three types: (a) Vi = V , Wi = W for every i ∈ I, and (Ui → U )i∈I is a covering family. (b) Ui = U , Wi = W for every i ∈ I, and (Vi → V )i∈I is a covering family. (c) I = {0 }, U 0 = U , and the morphism V 0 → V ×g+ (W ) g + (W 0 ) is an isomorphism (there is no condition on the morphism W 0 → W ).

VI.3. ORIENTED PRODUCTS OF TOPOS

495

e the topos Note that each of these families is stable under base change. We denote by C a of sheaves of U-sets on C. For a presheaf F on C, we denote by F the associated sheaf. Lemma VI.3.2. A presheaf F on C is a sheaf if and only if the following conditions are satisfied: (i) For every covering family (Zi → Z)i∈I of C of type (a) or (b), the sequence Y Y (VI.3.2.1) F (Z) → F (Zi ) ⇒ F (Zi ×Z Zj ) i∈I

(i,j)∈I×J

is exact. (ii) For every covering (U → W 0 ← V 0 ) → (U → W ← V ) of type (c), the map F (U → W ← V ) → F (U → W 0 ← V 0 )

(VI.3.2.2) is bijective.

Indeed, after extending the universe U, if necessary, we may assume that the categories X and Y are small ([2] II 2.7(2)). For every covering (U → W 0 ← V 0 ) → (U → W ← V ) of type (c), the diagonal morphism (U → W 0 ← V 0 ) → (U → W 0 ×W W 0 ← V 0 ×V V 0 )

is a covering of type (c) that equalizes the two canonical projections (U → W 0 ×W W 0 ← V 0 ×V V 0 ) ⇒ (U → W 0 ← V 0 ). The proposition therefore follows from ([2] II 2.3, I 3.5, and I 2.12). Remark VI.3.3. It follows from VI.3.2 that the morphism of sheaves associated with a covering of type (c) is an isomorphism; in particular, the topology on C is not always coarser than the canonical topology. VI.3.4.

The functors

(VI.3.4.1)

p+ 1 : X

(VI.3.4.2)

p+ 2 :

Y

→

→

C, C,

U 7→ (U → eZ ← eY ),

V 7→ (eX → eZ ← V ),

are left exact and continuous ([2] III 1.6). They therefore define two morphisms of topos ([2] IV 4.9.2) (VI.3.4.3) (VI.3.4.4)

e p1 : C e p2 : C

→

e X, → Ye .

On the other hand, we have a 2-morphism (VI.3.4.5)

τ : gp2 → f p1 ,

given by the following morphism of functors (gp2 )∗ → (f p1 )∗ : for every sheaf F on C and every W ∈ Ob(S), (VI.3.4.6)

g∗ (p2∗ (F ))(W ) → f∗ (p1∗ (F ))(W )

is the composition F (eX → eZ ← g + (W )) → F (f + (W ) → W ← g + (W )) → F (f + (W ) → eZ ← eY ), where the first arrow is the canonical morphism and the second arrow is the inverse of the isomorphism (VI.3.2.2).

496

VI. COVANISHING TOPOS AND GENERALIZATIONS

Remark VI.3.5. For every W ∈ Ob(S), the morphism τ : (f p1 )∗ (W ) → (gp2 )∗ (W ) (VI.3.4.5) is the morphism + + + a a (p+ 1 (f W )) → (p2 (g W )) ,

(VI.3.5.1) composed of

(f + W → eZ ← eY )a → (f + W → W ← g + W )a → (eX → eZ ← g + W )a , where the first arrow is the inverse of the canonical isomorphism (VI.3.3) and the second arrow is the canonical morphism. Lemma VI.3.6. For every object Z = (U → W ← V ) of C, we have a Cartesian diagram / p∗2 (V )

Za

(VI.3.6.1)

 p∗1 (U )

 / (gp2 )∗ (W )

i

where the unlabeled arrows are the canonical morphisms and i is the composition of the morphism p∗1 (U ) → (f p1 )∗ (W ) and the morphism τ : (f p1 )∗ (W ) → (gp2 )∗ (W ) (VI.3.4.5). We have a commutative canonical diagram of C with Cartesian squares Z

/ (f + W → W ← V )

 (U → W ← g + W )

 / (f + W → W ← g + W )

(VI.3.6.2)

/ p+ V 2

v

 / p+ (g + W ) 2

u



 / p+ (f + W )

p+ 1U

1

On the other hand, u is an isomorphism (VI.3.3) and τ : (f p1 )∗ W → (gp2 )∗ W is equal e is left exact, we deduce from to v a ◦ (ua )−1 (VI.3.5). Since the canonical functor C → C this that the diagram (VI.3.6.1) is Cartesian. a

e b : T → Ye two morphisms of topos, Theorem VI.3.7. Let T be a U-topos, a : T → X, and t : gb → f a a 2-morphism. Then there exists a triple ∼

∼

e α : p1 h → a, β : p2 h → b), (h : T → C,

unique up to unique isomorphism, consisting of a morphism of topos h and two isomorphisms of morphisms of topos α and β, such that the diagram (VI.3.7.1)

gp2 h

τ ∗h

g∗β

 gb

t

/ f p1 h 

f ∗α

/ fa

is commutative. The uniqueness of (h, α, β) is clear. Indeed, by VI.3.6, the “restriction” h+ : C → T of the functor h∗ to C is necessarily given, for any object Z = (U → W ← V ) of C, by (VI.3.7.2)

h+ (Z) = a∗ (U ) ×(gb)∗ (W ) b∗ (V ),

VI.3. ORIENTED PRODUCTS OF TOPOS

497

where the morphism a∗ (U ) → (gb)∗ (W ) is the composition t

a∗ (U ) → (f a)∗ (W ) → (gb)∗ (W ). Let us show that the resulting functor h+ is a morphism of sites and that the associated morphism of topos answers the question. The functor h+ is clearly left exact and transforms covering families of C of type (a) or (b) into covering families of T . On the other hand, if (U → W 0 ← V 0 ) → (U → W ← V ) is a covering of C of type (c), the square (VI.3.7.3)

b∗ V 0

/ b∗ V

 b∗ (g + W 0 )

 / b∗ (g + W )

is Cartesian, and consequently the morphism (VI.3.7.4)

h+ ((U → W 0 ← V 0 )) → h+ ((U → W ← V ))

is an isomorphism. Hence the functor h+ is continuous by virtue of VI.3.2. We deduce from this that h+ is a morphism of sites ([2] IV 4.9.4); it therefore defines a morphism e of topos h : T → C. ∼ ∼ We have canonical isomorphisms α : a∗ → h∗ p∗1 and β : b∗ → h∗ p∗2 whose “restrictions” to X and Y , respectively, are the tautological isomorphisms. To verify that the diagram (VI.3.7.1) is commutative, it suffices to show that its “restriction” to S is. For every W ∈ Ob(S), consider the diagram (VI.3.7.5)

h+ ((f + W → W ← g + W ))

u

/ (f a)∗ (W )

t

α(f ∗ W )

h+ ((f + W → W ← g + W ))

v

 / h∗ ((f p1 )∗ W )

/ (gb)∗ (W ) β(g ∗ W )

h∗ (τ )

 / h∗ ((gp2 )∗ W )

where u is the projection deduced from the formula (VI.3.7.2) and v is the canonical morphism. By definition, we have v = α(f ∗ W ) ◦ u. On the other hand, u0 = t ◦ u is the projection deduced from the formula (VI.3.7.2) and v 0 = h∗ (τ ) ◦ v is the canonical morphism. By definition, we have v 0 = β(g ∗ W ) ◦ u0 . Since u and v are isomorphisms, we deduce from this that the right square in (VI.3.7.5) is commutative. Hence the diagram (VI.3.7.1) is commutative. VI.3.8. Let X 0 , Y 0 , S 0 be three U-sites in which finite inverse limits are representable, and f 0+ : S 0 → X 0 , g 0+ : S 0 → Y 0 two continuous left exact functors. We denote e 0 , Ye 0 , and Se0 the topos of sheaves of U-sets on X 0 , Y 0 , and S 0 , respectively, and by by X (VI.3.8.1)

e 0 → Se0 f0 : X

and g 0 : Ye 0 → Se0

the morphisms of topos defined by f 0+ and g 0+ . We denote by C 0 the site associated e 0 the topos of sheaves of U-sets on with the functors (f 0+ , g 0+ ) defined in (VI.3.1), by C 0 0 0 0 0 0 0 e e e e C , by p1 : C → X and p2 : C → Y the canonical projections, and by τ 0 : g 0 p02 → f 0 p01

498

VI. COVANISHING TOPOS AND GENERALIZATIONS

the canonical 2-morphism (VI.3.4). Consider a diagram of morphisms of topos (VI.3.8.2)

e0 X

f0

/ Se0 o

u

g0

Ye 0

w

 e X

 / Se o

f

 Ye

g

v

and two 2-morphisms a : wf 0 → f u and b : gv → wg 0 .

(VI.3.8.3)

e0 → X e and vp0 : C e 0 → Ye and the composed By VI.3.7, the morphisms of topos up01 : C 2 2-morphism t gvp02

(VI.3.8.4)

b∗p02

h∗τ 0

/ wg 0 p02

a∗p01

/ wf 0 p01

/ f up01 ,

define a morphism of topos e0 → C e h: C

(VI.3.8.5) ∼

∼

and 2-isomorphisms α : p1 h → up01 and β : p2 h → vp02 making the following diagram commutative: gp2 h

(VI.3.8.6)

τ ∗h

/ f p1 h

t

 / f up01

f ∗α

g∗β

 gvp02

Corollary VI.3.9. Under the assumptions of VI.3.8, if u, v, and w are equivalences of topos, and a and b are 2-isomorphisms, then h is an equivalence of topos. This follows from VI.3.7. e depends only on the pair of morphisms of It follows from VI.3.9 that the topos C topos (f, g), up to canonical equivalence. This justifies the following terminology and notation. e is called the oriented product of X e and Ye over S, e and Definition VI.3.10. The topos C ← e × e Ye . denoted by X S Under the assumptions of VI.3.8, we denote the morphism h (VI.3.8.5) by ←

←

←

e 0 × e0 Ye 0 → X e × e Ye . u ×w v : X S S

(VI.3.10.1)

←

e × e Ye is equivalent to giving a pair of points Corollary VI.3.11. Giving a point of X S e e x : Pt → X and y : Pt → Y and a 2-morphism u : gy → f x. This follows from VI.3.7. ←

e × e Se is called the vanishing topos of f , Definition VI.3.12 ([46] 4.1). The topos X S ← and the topos Se × e Ye is called the covanishing topos of g. S

←

We will give in VI.4.10 a simpler description of Se ×Se Ye due to Deligne ([46] 4.6).

VI.3. ORIENTED PRODUCTS OF TOPOS

499

e Ye , and Se as U-sites endowed with the canonical topoloVI.3.13. We consider X, e Ye , and S, e respectively ([2] IV 1.2). On gies; the associated U-topos then identify with X, e and g ∗ : Se → Ye are clearly continuous and left the other hand, the functors f ∗ : Se → X exact. We can therefore consider the site C † associated with (f ∗ , g ∗ ) defined in VI.3.1. e † the topos of sheaves of U-sets on C † , by π + : X e → C† For the moment, denote by C 1 + e † and π2 : Y → C the functors defined in (VI.3.4.1) and (VI.3.4.2), by e† π1 : C e† π2 : C

(VI.3.13.1) (VI.3.13.2)

e → X, → Ye ,

the associated morphisms of topos, and by ν : gπ2 → f π1 the 2-morphism defined in (VI.3.4.5). The canonical functors εX , εY , and εS induce a left exact functor ϕ+ : C → C † .

(VI.3.13.3)

This transforms coverings of C of type (a) (resp. (b), resp. (c)) into coverings of C † of the same type. It then follows from VI.3.2 that for every sheaf F on C † , F ◦ ϕ+ is a sheaf on C. Consequently, ϕ+ is continuous. It therefore defines a morphism of topos e † → C. e ϕ: C

(VI.3.13.4) We have canonical isomorphisms ∼

+ ϕ+ ◦ p+ 1 → π1 ◦ εX

∼

+ and ϕ+ ◦ p+ 2 → π2 ◦ εY .

We deduce from this isomorphisms (VI.3.13.5)

∼

∼

π1 → p1 ϕ and π2 → p2 ϕ.

Moreover, it immediately follows from the definition (VI.3.4.5) that τ ∗ ϕ identifies with ν. Consequently, ϕ is an equivalence of topos by virtue of VI.3.7. Indeed, ϕ is the morphism of topos (VI.3.8.5) defined in VI.3.8 by taking for u, v, and w the identity e Ye , and S, e respectively. From now on, we identify C e † with the topos morphisms of X, ← e × e Ye through the equivalence ϕ, the morphism π1 (resp. π2 ) with p1 (resp. p2 ), and X S the 2-morphism ν with τ . VI.3.14. Let (F → H ← G) be an object of C † (VI.3.13). Recall ([2] IV 5.1) e/F is a U-topos, called the topos induced on F by X, e and that we that the category X e/F → X, e called the localization morphism of X e at F . have a canonical morphism jF : X e e e e Likewise, we have localization morphisms jG : Y/G → Y and jH : S/H → S. Denote by f 0 the composition (VI.3.14.1)

/X e/f ∗ (H)

e/F X

f/H

/ Se/H ,

e/f ∗ (H) at F → f ∗ (H) ([2] IV where the first arrow is the localization morphism of X 5.5), and the second arrow is the morphism deduced from f ([2] IV 5.10). We define the morphism g 0 : Ye/G → Se/H likewise. The squares of the diagram (VI.3.14.2)

e/F X

f0

jF

 e X

/ Se/H o

g0

jH

f

 / Se o

g

Ye/G  Ye

jG

500

VI. COVANISHING TOPOS AND GENERALIZATIONS ←

e/F × e Ye/G the oriare commutative up to canonical isomorphisms. We denote by X S/H e e e ented product of X/F and Y/G over S/H . The diagram (VI.3.14.2) then induces a canonical morphism (VI.3.10.1) ←

←

←

e/F × e Ye/G → X e × e Ye . jF ×jH jG : X S/H S

(VI.3.14.3) ←

←

e × e Ye associated with (F → H ← G) (cf. We denote by F ×H G the sheaf of X S VI.3.13). By VI.3.6, we have a canonical isomorphism ←

∼

F ×H G → p∗1 (F ) ×(gp2 )∗ (H) p∗2 (G),

(VI.3.14.4)

where the morphism p∗1 (F ) → (gp2 )∗ (H) is the composition of the morphism p∗1 (F ) → (f p1 )∗ (H) and the morphism τ : (f p1 )∗ (H) → (gp2 )∗ (H) (VI.3.4.5). We denote by j

(VI.3.14.5)

←

←

F ×H G

←

e × e Ye ) : (X S

←

/(F × H G)

←

e × e Ye →X S

←

e × e Ye at F ×H G. the localization morphism of X S By constructions analogous to (VI.3.14.1), the morphisms p1 and p2 induce morphisms ←

e × e Ye ) q1 : (X S

(VI.3.14.6)

←

/(F × H G)

←

e × e Ye ) q2 : (X S

(VI.3.14.7)

e/F , →X

←

/(F × H G)

→ Ye/G ,

that fit into a diagram with commutative squares up to canonical isomorphisms (VI.3.14.8)

e/F o X

q1

←

e × e Ye ) (X S j

jF

 eo X

/ Ye/G

q2 ←

/(F × H G)

← F ×HG

jG

 ← e X ×Se Ye

p1

 / Ye

p2

The 2-morphism τ : gp2 → f p1 (VI.3.4.5) and the commutative diagram / p∗2 G

←

(VI.3.14.9)

F ×H G  p∗1 F

/ p∗1 (f ∗ H)

 / p∗2 (g ∗ H)

τ

induce, for every L ∈ Ob(Se/H ), a functorial morphism (VI.3.14.10)

←

←

p∗1 (f ∗ L) ×p∗1 (f ∗ H) (F ×H G) → p∗2 (g ∗ L) ×p∗2 (g∗ H) (F ×H G).

Since p∗1 and p∗2 are left exact, we obtain a 2-morphism τ 0 : g 0 q2 → f 0 q1

(VI.3.14.11) such that jH ∗ τ 0 = τ ∗ j (VI.3.14.12)

←

F ×H G

. By VI.3.7, q1 , q2 , and τ 0 define a morphism ←

e × e Ye ) m : (X S

←

←

/(F × H G)

e/F × e Ye/G . →X S/H

Proposition VI.3.15. The morphism m is an equivalence of topos, and we have a canonical isomorphism (VI.3.15.1)

←

∼

(jF ×jH jG ) ◦ m → j

←

F ×H G

.

VI.3. ORIENTED PRODUCTS OF TOPOS

501

First note that the isomorphism (VI.3.15.1) follows from VI.3.7, in view of (VI.3.14.8) and the relation jH ∗ τ 0 = τ ∗ j ← . Denote by F ×H G

(VI.3.15.2)

† † j(F →H←G) : C/(F →H←G) → C

the canonical functor, by T the topology of C † (VI.3.13), and by T1 the topology of † C/(F →H←G) induced by T via j(F →H←G) . A family (Li → L)i∈I of morphisms of † † C/(F →H←G) is covering for T1 if and only if its image by j(F →H←G) is covering in C for ←

e × e Ye ) T ([2] III 5.2(1)). By virtue of ([2] III 5.4), (X S to the topos of

is canonically equivalent

←

/(F × H G) † sheaves of U-sets on the site (C/(F →H←G) , T1 ). ← e/F × e Ye/G , we consider site underlying the topos X S/H

e/F , Ye/G , and Se/H X e/F ) as U-sites endowed with the canonical topologies (cf. VI.3.13). For all F 0 ∈ Ob(X 0 0 0∗ 0 e/F corresponds to taking a and H ∈ Ob(Se/H ), taking a morphism F → f (H ) of X 0 ∗ 0 ∗ e morphism F → f (H ) of X over F → f (H). Consequently, the site associated with e/F and g 0∗ : Se/H → Ye/G defined in VI.3.1 identifies the pair of functors f 0∗ : Se/H → X † canonically with the category C/(F →H←G) , endowed with a topology T2 , which is a To define a

† priori coarser than the topology T1 . The identity functor of C/(F →H←G) then defines a morphism of sites

(VI.3.15.3)

† † id : (C/(F →H←G) , T1 ) → (C/(F →H←G) , T2 ).

Let us show that m is the morphism of topos associated with (VI.3.15.3). It follows from the proof of VI.3.7, in particular from (VI.3.7.2), that the restriction (VI.3.15.4)

←

† e e Ye ) m+ : C/(F →H←G) → (X ×Z

←

/(F × H G)

† of the functor m∗ is given, for every object (F 0 → H 0 ← G0 ) of C/(F →H←G) , by

(VI.3.15.5)

m+ ((F 0 → H 0 ← G0 )) = q1∗ (F 0 ) ×(g0 q2 )∗ (H 0 ) q2∗ (G0 ),

where the morphism q1∗ (F 0 ) → (g 0 q2 )∗ (H 0 ) is the composition of the morphism q1∗ (F 0 ) → (f 0 q1 )∗ (H 0 ) and the morphism τ 0 : (f 0 q10 )∗ (H 0 ) → (g 0 q2 )∗ (H 0 ) (VI.3.14.11). We have canonical isomorphisms ←

(VI.3.15.6)

q1∗ (F 0 ) ' p∗1 (F 0 ) ×p∗1 (F ) (F ×H G),

(VI.3.15.7)

q2∗ (G0 ) ' p∗2 (G0 ) ×p∗2 (G) (F ×H G),

(VI.3.15.8)

←

←

(g 0 q2 )∗ (H 0 ) ' (gp2 )∗ (H 0 ) ×(gp2 )∗ (H) (F ×H G).

Moreover, in view of (VI.3.14.9), the morphism q1∗ (F 0 ) → (g 0 q2 )∗ (H 0 ) comes from the composition τ

p∗1 (F 0 ) → (f p1 )∗ (H 0 ) → (gp2 )∗ (H 0 ). We deduce from this an isomorphism ([2] I 2.5.0) (VI.3.15.9)

m+ ((F 0 → H 0 ← G0 )) ' p∗1 (F 0 ) ×(gp2 )∗ (H 0 ) p∗2 (G0 ).

502

VI. COVANISHING TOPOS AND GENERALIZATIONS

Consequently, by virtue of VI.3.6 and ([2] III 5.5), the diagram (VI.3.15.10)

←

m∗

e/F × e Ye/G X S/H O

←

e × e Ye ) ← / (X S O /(F ×H G) † C/(F →H←G)

† C/(F →H←G)

where the vertical arrows are the canonical functors, is commutative. Hence m is the morphism of topos associated with (VI.3.15.3). To show that m is an equivalence of topos, it suffices to show that T1 = T2 , or that T1 is coarser than T2 , or that the canonical functor (VI.3.15.11)

† † j(F →H←G) : (C/(F →H←G) , T2 ) → (C , T )

is cocontinuous ([2] III 2.1). The functor j(F →H←G) is a left adjoint of the functor (VI.3.15.12)

† + † j(F (C/(F →H←G) : (C , T ) → →H←G) , T2 ), L 7→ L × (F → H ← G).

Recall that finite inverse limits are representable in C † . We show as above that the diagram ←

(VI.3.15.13)

←

e × e Ye X OS

C†

(jF × jH jG )∗

+ j(F →H←G)

←

/X e/F × e Ye/G S/H O / C† /(F →H←G)

where the vertical arrows are the canonical functors, is commutative. Consequently, + j(F →H←G) is continuous by virtue of ([2] III 1.6), and therefore j(F →H←G) is cocontinuous by ([2] III 2.5). VI.4. Covanishing topos VI.4.1. In this section, X and Y denote two U-sites in which finite inverse limits e and are representable, and f + : X → Y a continuous left exact functor. We denote by X e the morphism Ye the topos of sheaves of U-sets on X and Y , respectively, by f : Ye → X e and εY : Y → Ye the canonical functors. of topos associated with f + , and by εX : X → X Let eX and eY be final objects of X and Y , respectively, which exist by assumption. e and Since the canonical functors are left exact, εX (eX ) and εY (eY ) are final objects of X e Y , respectively. We denote by D the category of pairs (U, V → f + (U )), where U is an object of X and V → f + (U ) is a morphism of Y ; such an object will be denoted by (V → U ). Let (V → U ), (V 0 → U 0 ) be two objects of D. A morphism from (V 0 → U 0 ) to (V → U ) consists of two morphisms V 0 → V of Y and U 0 → U of X, such that the diagram V0

/ f + (U 0 )

 V

 / f + (U )

is commutative. It immediately follows from the definition and the fact that the functor f + is left exact that finite inverse limits in D are representable.

VI.4. COVANISHING TOPOS

503

We call covanishing topology on D the topology generated by the coverings {(Vi → Ui ) → (V → U )}i∈I of the following two types: (α) Ui = U for every i ∈ I, and (Vi → V )i∈I is a covering family. (β) (Ui → U )i∈I is a covering family, and for every i ∈ I, the canonical morphism Vi → V ×f + (U ) f + (Ui ) is an isomorphism. Note that each of these families is stable under base change. The resulting site is called b (resp. the covanishing site associated with the functor f + ; it is a U-site. We denote by D e the category of presheaves (resp. the topos of sheaves) of U-sets on D. We say that D) e is the covanishing topos associated with the functor f + . We will show in VI.4.10 that D the terminology does not lead to any confusion with that introduced in VI.3.12. For a presheaf F on D, we denote by F a the associated sheaf. Remark VI.4.2. The topology on D is generated by the coverings {(Vij → Ui ) → (V → U )}(i,j) satisfying the following conditions: (i) The family (Ui → U )i is covering. (ii) For every i, the family (Vij → V ×f + (U ) f + (Ui ))j is covering. Notice that in general, the family of these coverings is not stable under composition and therefore does not form a pretopology. VI.4.3. We denote by Yb the category of presheaves of U-sets on Y , and by Q the split category of presheaves of U-sets on Y ([35] I 2.6.1), that is, the fibered category on Y obtained by associating with each V ∈ Ob(Y ) the category (Y/V )∧ = Yb/V , and with each morphism h : V 0 → V of Y the functor h∗ : Yb/V → Yb/V 0 defined by composition with the functor Y/h : Y/V 0 → Y/V . Note that h∗ is also the base change in Yb by h. Since fibered products are representable in Y , h∗ admits a right adjoint, namely the “Weil restriction” functor h∗ : Yb/V 0 → Yb/V , defined, for all F ∈ Ob(Yb/V 0 ) and W ∈ Ob(Y/V ), by (VI.4.3.1)

h∗ (F )(W ) = F (W ×V V 0 ).

We denote by Q ∨ the cleaved and normalized fibered category over Y ◦ obtained by associating with each V ∈ Ob(Y ) the category Yb/V , and with each morphism h : V 0 → V of Y the functor h∗ ([1] 1.1.2), and by P P∨

(VI.4.3.2) (VI.4.3.3)

→ X → X◦

the fibered categories deduced from Q and Q ∨ , respectively, by base change by the functor f + : X → Y . By ([37] VI 12; cf. also [1] 1.1.2), we have an equivalence of categories (VI.4.3.4)

b D F

∼

→ HomX ◦ (X ◦ , P ∨ ) 7→ {U 7→ FU },

defined, for every (V → U ) ∈ Ob(D), by the relation (VI.4.3.5)

FU (V ) = F (V → U ).

From now on, we identify F with the section {U 7→ FU } that is associated with it by this equivalence.

504

VI. COVANISHING TOPOS AND GENERALIZATIONS

Proposition VI.4.4. A presheaf F = {U 7→ FU } on D is a sheaf if and only if the following conditions are satisfied: (i) For every U ∈ Ob(X), FU is a sheaf on Y/f + (U ) . (ii) For every covering family (Ui → U )i∈I of X, if for (i, j) ∈ I 2 , we set Uij = Ui ×U Uj and we denote by hi : f + (Ui ) → f + (U ) and hij : f + (Uij ) → f + (U ) the structural morphisms, then the sequence of morphisms of sheaves on Y/f + (U ) Y Y (VI.4.4.1) FU → hi∗ (FUi ) ⇒ hij∗ (FUij ) (i,j)∈I 2

i∈I

is exact. Indeed, after extending the universe U, if necessary, we may assume that the category X is small ([2] II 2.7(2)). The proposition then follows from ([2] II 2.3, I 3.5, I 2.12, and II 4.1(3)). Remark VI.4.5. Condition VI.4.4(ii) corresponds to saying that, for every (V → U ) ∈ Ob(D), if we set Vi = V ×f + (U ) f + (Ui ) and Vij = V ×f + (U ) f + (Uij ), the sequence of maps of sets Y Y (VI.4.5.1) FU (V ) → FUi (Vi ) ⇒ FUij (Vij ) i∈I

(i,j)∈I 2

is exact. Remarks VI.4.6. (i) For every object (V → U ) of D, the diagram (VI.4.6.1)

(V → U )a

/ (V → eX )a

 (f + (U ) → U )a

 / (f + (U ) → eX )a

e Indeed, the canonical functor D → D e is left exact. is Cartesian in D. (ii) Let W be an object of X and F = {U 7→ FU } the presheaf on D defined by (f + (W ) → W ). For every U ∈ Ob(X), FU is the constant presheaf on Y/f + (U ) with value HomX (U, W ). In particular, the topology on D is not in general coarser than the canonical topology. (iii) Let V be an object of Y , F = {U 7→ FU } the presheaf on D defined by (V → eX ). For every U ∈ Ob(X), FU is the presheaf V × f + (U ) on Y/f + (U ) . If the topologies on X and Y are coarser than the canonical topologies, F is a sheaf on D by virtue of VI.4.4. VI.4.7. Denote by C the site associated with the pair of functors (idX , f + ) defined in VI.3.1, and consider the functors (VI.4.7.1) (VI.4.7.2)

ι+ : D + : C

→ C, → D,

(V → U ) 7→ (U → U ← V ), (U → W ← V ) 7→ (V ×f + (W ) f + (U ) → U ).

It is clear that ι+ is a left adjoint of + , that the adjunction morphism id → + ◦ ι+ is an isomorphism (that is, ι+ is fully faithful), and that ι+ and + are left exact. Proposition VI.4.8. (i) The functors ι+ and + are continuous. (ii) The topology on D is induced by that on C through the functor ι+ .

VI.4. COVANISHING TOPOS

505

(i) The functor ι+ transforms covering families of D of type (α) into covering families of C of type (b), and covering families of D of type (β) into covering families of C: (VI.4.8.1)

/ Ui

Vi  V



7→

 /U

Ui

/ Ui o

 Ui

 /U o

 U

 /U o

Vi 

 V  V

Let G be a presheaf on C and F = {U 7→ FU } = G ◦ ι+ . For every (V → U ) ∈ Ob(D), we have (VI.4.8.2)

FU (V ) = G(U → U ← V ).

Consequently, if G is a sheaf on C, F is a sheaf on D by virtue of VI.3.2, VI.4.4, and (VI.4.8.1); hence ι+ is continuous. The functor + transforms coverings of C of type (a) (resp. (b)) into coverings of D of type (β) (resp. (α)), and coverings of C of type (c) into isomorphisms. Consequently, for every sheaf F on D, F ◦ + is a sheaf on C by virtue of VI.3.2; hence + is continuous. e ([2] (ii) We know that the topology on D is induced by the canonical topology on D e is a III 3.5); in other words, the topology on D is the finest such that every F ∈ Ob(D) sheaf. By (i), we can consider the functors (VI.4.8.3) (VI.4.8.4)

e ιs : C e s : D ∼

e → D, e → C,

G 7→ G ◦ ι+ ,

F 7→ F ◦ + .

∼

The adjunction isomorphism id → + ◦ ι+ induces an isomorphism ιs ◦ s → id. The functor ιs is therefore essentially surjective. Consequently, the topology on D is the e ιs (G) is a sheaf on D; whence the proposition. finest such that, for every G ∈ Ob(C), VI.4.9. Since the functors ι+ and + are continuous and left exact (VI.4.8), they define morphisms of topos ([2] IV 4.9.2) (VI.4.9.1) (VI.4.9.2)

e ι: C e : D

e → D, e → C.

The adjunction morphisms id → + ◦ ι+ and ι+ ◦ + → id induce morphisms ι∗ ◦ ∗ → id and id → ∗ ◦ ι∗ that make ι∗ into a right adjoint of ∗ . Proposition VI.4.10. The adjunction morphisms ι∗ ◦ ∗ → id and id → ∗ ◦ ι∗ are isomorphisms. In particular, ι (VI.4.9.1) and  (VI.4.9.2) are equivalences of topos quasiinverse to each other. Indeed, since the adjunction morphism id → + ◦ ι+ is an isomorphism, ι∗ ◦ ∗ → id is an isomorphism. On the other hand, the adjunction morphism ι+ ◦ + → id is defined, for every object (U → W ← V ) of C, by the canonical morphism (VI.4.10.1)

(U → U ← V ×f + (W ) f + (U )) → (U → W ← V ),

which is a covering of type (c). Since the morphism of sheaves associated with (VI.4.10.1) e (VI.3.3), id → ∗ ◦ ι∗ is an isomorphism. is an isomorphism in C

506

VI. COVANISHING TOPOS AND GENERALIZATIONS

VI.4.11.

The functors

(VI.4.11.1)

p+ 1 : X

(VI.4.11.2)

p+ 2 : Y

→

→

D, D,

U 7→ (f + (U ) → U ), V 7→ (V → eX ),

are left exact and continuous ([2] III 1.6). They therefore define two morphisms of topos ([2] IV 4.9.2) e p1 : D e p2 : D

(VI.4.11.3) (VI.4.11.4)

e → X, → Ye .

For every U ∈ Ob(X), the morphism (U → U ← f + (U ))a → (U → eX ← eY )a is e (VI.3.3). The morphisms p1 ◦ ι and p2 ◦ ι therefore identify with an isomorphism of C e→X e and p2 : C e → Ye defined in (VI.3.4); whence the terminology. The morphisms p1 : C 2-morphism (VI.3.4.5) τ : f p2 → p1

(VI.4.11.5)

is then defined by the following morphism of functors (f p2 )∗ → p1∗ : for every sheaf F on D and every U ∈ Ob(X), (VI.4.11.6)

f∗ (p2∗ (F ))(U ) → p1∗ (F )(U )

is the canonical map F (f + (U ) → eX ) → F (f + (U ) → U ). The 2-morphism τ induces a base change morphism f∗ → p1∗ p∗2

(VI.4.11.7) composed of f∗

/ f∗ p2∗ p∗2

τ ∗p∗ 2

/ p1∗ p∗2 ,

where the first morphism is deduced from the adjunction morphism. For every ring Λ, e Λ) the morphism (VI.4.11.7) induces a morphism of functors from D+ (Ye , Λ) to D+ (X, (VI.4.11.8)

Rf∗ → Rp1∗ p∗2 .

Proposition VI.4.12. (i) For every sheaf F on X, p∗1 (F ) is the sheaf associated with the presheaf {U 7→ F (U )} on D (VI.4.3.4), where for every U ∈ Ob(X), F (U ) is the constant presheaf on Y/f + (U ) with value F (U ). (ii) For every sheaf F on Y , p∗2 (F ) is the sheaf {U 7→ F × f ∗ (U )}. (iii) For every sheaf F on X, the morphism τ : p∗1 (F ) → (f p2 )∗ (F ) (VI.4.11.5) is induced by the morphism of presheaves on D defined, for every U ∈ Ob(X), by the morphism of presheaves on f + (U ) (VI.4.12.1)

F (U ) → f ∗ (F × U )

given, for every (V → U ) ∈ Ob(D), by the composition (VI.4.12.2)

F (U ) → (f ∗ F )(f + U ) → (f ∗ F )(V ).

(iv) The adjunction morphism id → p2∗ p∗2 is an isomorphism. Indeed, after extending the universe U, if necessary, we may assume that the categories X and Y are U-small ([2] II 3.6, and III 1.5). (i) By ([2] I 5.1 and III 1.3), the sheaf p∗1 (F ) is the sheaf on D associated with the presheaf G defined for (V → U ) ∈ Ob(D) by (VI.4.12.3)

G(V → U ) =

lim −→

(P,u)∈I ◦ (V →U )

F (P ),

VI.4. COVANISHING TOPOS

507

where I(V →U ) is the category of pairs (P, u) consisting of an object P of X and a morphism u : (V → U ) → (f + (P ) → P ) of D. This category admits as initial object the pair consisting of U and the canonical morphism (V → U ) → (f + (U ) → U ). We therefore have G(V → U ) = F (U ). (ii) The sheaf p∗2 (F ) is the sheaf on D associated with the presheaf H defined for (V → U ) ∈ Ob(D) by H(V → U ) =

(VI.4.12.4)

lim −→

F (Q),

(Q,v)∈J ◦ (V →U )

where J(V →U ) is the category of pairs (Q, v) consisting of an object Q of Y and a morphism v : (V → U ) → (Q → eX ) of D. This category admits as initial object the pair consisting of V and the canonical morphism (V → U ) → (V → eX ). We therefore have H(V → U ) = F (V ). Consequently, H = {U 7→ F × f ∗ (U )}, which is indeed a sheaf on D by virtue of VI.4.4. (iii) Denote by G the presheaf on D associated with F defined in (VI.4.12.3) and by H the sheaf on D associated with f ∗ (F ) defined in (VI.4.12.4). For every object (V → U ) of D, we have a functor I(V →U ) → J(V →U ) ,

(VI.4.12.5)

defined by (P, u) 7→ (f + (P ), v), where v is the composition u

(V → U ) → (f + (P ) → P ) → (f + (P ) → eX ). Then the composition lim −→

(P,u)∈I ◦ (V →U )

F (P ) →

lim −→

(P,u)∈I ◦ (V →U )

(f ∗ F )(f + P ) →

lim −→

(f ∗ F )(Q),

(Q,v)∈J ◦ (V →U )

where the first arrow is the canonical map and the second arrow is induced by the functor (VI.4.12.5), is equal to the map G(V → U ) → H(V → U )

(VI.4.12.6)

defined in (VI.4.12.2). Consequently, the morphism of sheaves p∗1 (F ) → (f p2 )∗ (F ) associated with (VI.4.12.6) is the adjoint of the morphism (VI.4.11.6), giving the proposition. (iv) Indeed, for every sheaf F on Y and every V ∈ Ob(Y ), the adjunction morphism F (V ) → (p∗2 F )(V → eX ) identifies with the identity morphism of F (V ) by virtue of (ii). VI.4.13. (VI.4.13.1)

The functor Ψ+ : D → Y,

(V → U ) 7→ V

is clearly left exact. For every sheaf F on Y , we have (VI.4.13.2)

F ◦ Ψ+ = {U 7→ F |f + (U )},

where for every morphism g : U 0 → U of X, if we set h = f + (g), the transition morphism (VI.4.13.3)

F |f + (U ) → h∗ (F |f + (U 0 ))

∼

is the adjoint of the canonical isomorphism h∗ (F |f + (U )) → F |f + (U 0 ). It follows from VI.4.4 that F ◦ Ψ+ is a sheaf on D. Consequently, Ψ+ is continuous. It therefore defines a morphism of topos (VI.4.13.4)

e Ψ : Ye → D

508

VI. COVANISHING TOPOS AND GENERALIZATIONS

such that p1 Ψ = f , p2 Ψ = idYe , and τ ∗ Ψ = idf , where τ is the 2-morphism (VI.4.11.5). e and id e and the Consequently, Ψ is the morphism defined by the morphisms f : Ye → X Y 2-morphism idf , in view of the universal property of oriented products (VI.3.7): (VI.4.13.5)

Ye FF xx FF id x f x FF Ye xx Ψ FF x x FF  x | x p1 " p2 ← o / Ye e e eF X y FF X ×Xe Y yy FF yy FF y FF yy f F" |yy e X

The morphism Ψ (or Ψ) is called the morphism of co-nearby cycles. From the relation p2∗ Ψ∗ = idYe , we obtain by adjunction a morphism p∗2 → Ψ∗ .

(VI.4.13.6)

Proposition VI.4.14. The morphism p∗2 → Ψ∗ (VI.4.13.6) is an isomorphism; in particular, the functor Ψ∗ is exact. Indeed, for every sheaf F on Y and every (V → U ) ∈ Ob(D), we have a commutative diagram (VI.4.14.1)

p∗2 (F )(V → eX )

/ Ψ∗ (F )(V → eX )

 p∗2 (F )(V → U )

 / Ψ∗ (F )(V → U )

where the horizontal arrows are the maps (VI.4.13.6) and the vertical arrows are the canonical maps. The latter are isomorphisms by virtue of VI.4.12(ii). On the other hand, the top horizontal arrow is induced by the morphism p2∗ p∗2 → p2∗ Ψ∗

(VI.4.14.2)

deduced from (VI.4.13.6). The composition of (VI.4.14.2) and the adjunction mor∼ phism id → p2∗ p∗2 is the canonical isomorphism id → p2∗ Ψ∗ . It then follows from VI.4.12(iv) that (VI.4.14.2) is an isomorphism. Consequently, the bottom horizontal arrow of (VI.4.14.1) is an isomorphism, giving the proposition. Proposition VI.4.15. (i) For every sheaf of sets F on Y , the morphism (VI.4.11.7) (VI.4.15.1)

f∗ (F ) → p1∗ p∗2 (F )

is an isomorphism. (ii) Let Λ be a ring. For every complex F of D+ (Ye , Λ), the morphism (VI.4.11.8) (VI.4.15.2)

Rf∗ (F ) → Rp1∗ p∗2 (F )

is an isomorphism. (i) Consider the commutative diagram (VI.4.15.3)

/ f∗ p2∗ p∗2 f∗ G GG GG GG GG b d #  f∗ p2∗ Ψ∗ a

τ ∗p∗ 2

τ ∗Ψ∗

/ p1∗ p∗2 

c

/ p1∗ Ψ∗

VI.4. COVANISHING TOPOS

509

where a is induced by the adjunction morphism and b and c are induced by (VI.4.13.6). Since d = b ◦ a is the isomorphism deduced from the relation p2 Ψ = idYe , (τ ∗ Ψ∗ ) ◦ d identifies with the isomorphism deduced from the relation p1 Ψ = f (VI.4.13). On the other hand, c is an isomorphism by virtue of VI.4.14, giving the statement. (ii) Let τe : Rf∗ Rp2∗ → Rp1∗ be the morphism induced by τ (VI.4.11.5). Since Ψ∗ is exact by virtue of VI.4.14, the diagram (VI.4.15.3) induces a commutative diagram (VI.4.15.4)

α / Rf∗ Rp2∗ p∗2 Rf∗ K KKK KKK β δ KKK %  Rf∗ Rp2∗ Ψ∗

τe∗p∗ 2

/ Rp1∗ p∗2 γ



/ Rp1∗ Ψ∗

τe∗Ψ∗

On the other hand, since Ψ∗ is exact, δ is the isomorphism deduced from the relation p2 Ψ = idYe , and consequently (e τ ∗ Ψ∗ ) ◦ δ identifies with the isomorphism deduced from the relation p1 Ψ = f (VI.4.13). Since γ is an isomorphism by virtue of VI.4.14, the statement follows. Remark VI.4.16. Proposition VI.4.15 and its proof are due to Gabber ([46] 4.9). It is a special case of a base change theorem for oriented topos ([46] 2.4), which, nevertheless, requires more restrictive coherence assumptions. VI.4.17. Let (B → A) be an object of D. We denote by jA : X/A → X and jB : Y/B → Y the canonical functors, and endow X/A and Y/B with the topologies induced by those on X and Y by the functors jA and jB , respectively. Denote by (X/A )∼ the topos of sheaves of U-sets on X/A . By ([2] III 5.2), the functor jA is continuous and cocontinuous. It therefore induces a sequence of three adjoint functors: ∗ e e → (X/A )∼ , jA∗ : (X/A )∼ → X e (VI.4.17.1) jA! : (X/A )∼ → X, jA :X

in the sense that for any two consecutive functors in the sequence, the one on the right is right adjoint to the other. The functor jA! factors through an equivalence of categories ∼ e a ∗ (X/A )∼ → X /Aa , where A = εX (A) ([2] III 5.4), and the pair (jA , jA∗ ) defines a e called localization morphism of X e at Aa , and likewise for e/Aa → X, morphism of topos X jB . Finite inverse limits are representable in X/A and Y/B . On the other hand, the functor (VI.4.17.2)

f 0+ : X/A → Y/B ,

U 7→ f + (jA (U )) ×f + (A) B

is left exact and continuous by virtue of ([2] III 1.6 and 3.3). The morphism of topos e/Aa f 0 : Ye/B a → X

(VI.4.17.3)

associated with f 0+ identifies with the composition Ye/B a

/ Ye/f ∗ (A)

f/Aa

/X e/Aa ,

where the first arrow is the localization morphism associated with B a → f ∗ (Aa ) ([2] IV 5.5) and the second arrow is the morphism deduced from f ([2] IV 5.10). e 0 ) the covanishing site (resp. topos) associated with the We denote by D0 (resp. D 0+ functor f . The functors jA and jB induce a functor (VI.4.17.4)

j(B→A) : D0 → D,

that factors through an equivalence of categories (VI.4.17.5)

∼

n : D0 → D/(B→A) .

510

VI. COVANISHING TOPOS AND GENERALIZATIONS

Proposition VI.4.18. Under the assumptions of VI.4.17, the covanishing topology on D0 is induced by the covanishing topology on D through the functor j(B→A) (VI.4.17.4); in particular, n (VI.4.17.5) induces an equivalence of topos ∼ e0 e /(B→A)a → m: D D.

(VI.4.18.1)

←

←

e and X e × e Ye and the topos D e 0 and X e/Aa × e e a We identify the topos D X X/Aa Y/B using the equivalences (VI.4.9.1). By VI.3.15, we have an equivalence of topos ←

←

∼ e e × e Ye )/(B→A)a → m : (X X/Aa ×Xe/Aa Ye/B a . X

(VI.4.18.2)

A priori, the covanishing topology on D0 is coarser than the topology induced by the covanishing topology on D by the functor j(B→A) . But it follows from the proof of VI.3.15, in particular from (VI.3.15.10), that the diagram ←

e a e/Aa × e X X/Aa Y/B O

(VI.4.18.3)

m∗

← / (X e × e Ye )/(B→A)a X O

n

D0

/ D/(B→A)

where the vertical arrows are the canonical functors, is commutative. We deduce from this, by ([2] III 3.5), that the covanishing topology on D0 is induced by the covanishing topology on D by the functor j(B→A) . Note that the equivalence (VI.4.18.1) induced by n identifies with the equivalence (VI.4.18.2) by virtue of (VI.4.18.3); whence the notation. Remark VI.4.19. We can give a direct proof of VI.4.18 that does not pass through VI.3.15, but that uses the same arguments. The proof becomes particularly simple when B = f + (A). We will treat this case directly in a more generalized setting (VI.5.38). ←

e × e Ye is equivalent to giving a VI.4.20. Recall (VI.3.11) that giving a point of X X e and y : Pt → Ye and a 2-morphism u : f (y) → x. Such a point pair of points x : Pt → X e and G ∈ Ob(Ye ), we will be denoted by (y → x), or by (u : y → x). For all F ∈ Ob(X) have functorial canonical isomorphisms (VI.4.20.1)

(p∗1 F )(y→x)

(VI.4.20.2)

(p∗2 G)(y→x)

∼

→ Fx , ∼

→ Gy .

By VI.3.7, the map (p∗1 F )(y→x) → (p∗2 (f ∗ F ))(y→x)

(VI.4.20.3)

induced by τ (VI.4.11.5), identifies canonically with the specialization morphism Fx → ← e and X e × e Ye by the equivalence (VI.4.9.2). Ff (y) defined by u. We identify the topos D X The isomorphisms (VI.4.13.6) and (VI.4.20.2) induce a functorial canonical isomorphism ∼

(Ψ∗ G)(y→x) → Gy .

(VI.4.20.4)

It follows from VI.3.7 and (VI.4.13.5) that we have a functorial canonical isomorphism ← e × e Ye of points of X X

(VI.4.20.5)

∼

Ψ(y) → (y → f (y)).

By VI.4.6(i), for every (V → U ) ∈ Ob(D), we have a functorial canonical isomorphism (VI.4.20.6)

∼

(V → U )a(y→x) → Uxa ×Ufa(y) Vya ,

VI.5. GENERALIZED COVANISHING TOPOS

511

where the exponent a denotes the associated sheaves, the map Vya → Ufa(y) is induced by the structural morphism V → f + (U ), and the map Uxa → Ufa(y) is the specialization morphism defined by u. VI.5. Generalized covanishing topos VI.5.1.

In this section, I denotes a U-site, Ie the topos of sheaves of U-sets on I,

and (VI.5.1.1)

π: E → I

a cleaved and normalized fibered category over the category underlying I ([37] VI 7.1). We suppose that the following conditions are satisfied: (i) Fibered products are representable in I. (ii) For every i ∈ Ob(I), the fiber Ei of E over i is endowed with a topology making it into a U-site, and finite inverse limits are representable in Ei . We denote by ei the topos of sheaves of U-sets on Ei . E (iii) For every morphism f : i → j of I, the inverse image functor f + : Ej → Ei is continuous and left exact. It therefore defines a morphism of topos that we ei → E ej ([2] IV 4.9.2). (abusively) denote also by f : E

Condition (i) will be strengthened from VI.5.32 onward. For every i ∈ Ob(I), we denote by (VI.5.1.2)

αi! : Ei → E

the canonical inclusion functor. The functor π is in fact a fibered U-site ([2] VI 7.2.1 and 7.2.4). We denote by (VI.5.1.3)

F →I

the fibered U-topos associated with π ([2] VI 7.2.6). The fiber of F over any i ∈ Ob(I) is ei , and the inverse image functor under any morphism canonically equivalent to the topos E f : i → j of I identifies with the inverse image functor under the morphism of topos ei → E ej . We denote by f: E (VI.5.1.4)

F ∨ → I◦

ei , and the fibered category obtained by associating with each i ∈ Ob(I) the category E ei → E ej direct image by the morphism with each morphism f : i → j of I the functor f∗ : E e e of topos f : Ei → Ej . We denote by (VI.5.1.5)

P∨ → I◦

bi of the fibered category obtained by associating with each i ∈ Ob(I) the category E b presheaves of U-sets on Ei , and with each morphism f : i → j of I the functor f∗ : Ei → bj obtained by composing with the inverse image functor f + : Ej → Ei . This notation E convention does not follow that of ([2] I 5.0); we make it so that the canonical I ◦ -functor F ∨ → P ∨ becomes compatible with inverse image functors. b the category of presheaves VI.5.2. Note that E is a U-category. We denote by E b of U-sets on E. Observe that since the category E is not naturally fibered over I, the bi for the fibers of P ∨ over I ◦ does not lead to any confusion. notation E

512

VI. COVANISHING TOPOS AND GENERALIZATIONS

By ([37] VI 12; cf. also [1] 1.1.2) and with the notation of (VI.2.2), we have an equivalence of categories (VI.5.2.1)

b E F

∼

→ HomI ◦ (I ◦ , P ∨ ) 7 → {i 7→ F ◦ αi! },

where αi! is the functor (VI.5.1.2). From now on, we identify F with the section {i 7→ F ◦ αi! } that is associated with it by this equivalence. VI.5.3. We call covanishing topology on E the topology generated by the families of coverings (Vn → V )n∈Σ of the following two types: (v) There exists i ∈ Ob(I) such that (Vn → V )n∈Σ is a covering family of Ei . (c) There exists a covering family of morphisms (fn : in → i)n∈Σ of I such that Vn is isomorphic to fn+ (V ) for every n ∈ Σ. The coverings of type (v) are called vertical, and those of type (c) are called Cartesian. The resulting site is called covanishing site associated with the fibered site π (VI.5.1.1); it is a U-site. We call covanishing topos associated with the fibered site π, and denote e the topos of sheaves of U-sets on E. We denote by by E, (VI.5.3.1)

e ε: E → E

the canonical functor. Example VI.5.4. Suppose that I is endowed with the trivial or chaotic topology, that is, with the coarsest of all topologies on I ([2] II 1.1.4). Note that under this assumption, requiring that I be a U-site amounts to requiring that I be equivalent to a U-small category. The total topology on E associated with the fibered site π ([2] VI 7.4.1) is generated by the vertical coverings, by virtue of ([2] VI 7.4.2(1)). It is therefore equal to the covanishing topology on E . Example VI.5.5. Let X and Y be two U-sites in which finite inverse limits are representable and f + : X → Y a continuous left exact functor. We associate with f + a fibered U-site (VI.5.5.1)

π: E → X

satisfying the conditions of VI.5.1, as follows. Consider the category Fl(Y ) of morphisms of Y , and the “target functor” (VI.5.5.2)

Fl(Y ) → Y,

which makes Fl(Y ) into a cleaved and normalized fibered category over Y : the fiber over any object V of Y is canonically equivalent to the category Y/V , and for every morphism h : V 0 → V of Y , the inverse image functor h∗ : Y/V → Y/V 0 is none other than the base change functor by h. Endowing each fiber Y/V with the topology induced by that of Y , Fl(Y )/Y becomes a fibered U-site satisfying the conditions of VI.5.1. We then take for π the fibered site deduced from Fl(Y )/Y by base change by the functor f + . The covanishing site E associated with the fibered site π (VI.5.3) is canonically equivalent to the covanishing site D associated with the functor f + (VI.4.1) by virtue of ([2] III 5.2(1)), whence the terminology. Lemma VI.5.6. (i) Fibered products are representable in E. (ii) The functors π and αi! (VI.5.1.2), for every i ∈ Ob(I), commute with fibered products. (iii) The family of vertical (resp. Cartesian) coverings of E is stable under base change.

VI.5. GENERALIZED COVANISHING TOPOS

513

(i) Consider a commutative diagram of E (VI.5.6.1)

X

/V

 U

 /W

x

/v

 u

 /w

over a commutative diagram of I (VI.5.6.2)

Then u ×w v is representable in I. Denote by U 0 , V 0 , and W 0 the inverse images of U , V , and W over u ×w v under the canonical morphisms from u ×w v to u, v, and w, respectively. Then U 0 ×W 0 V 0 is representable in Eu×w v . Since the inverse image functors of π are left exact (VI.5.1(iii)), the diagram (VI.5.6.1) uniquely determines a morphism X → U 0 ×W 0 V 0 over the canonical morphism x → u ×w v. Consequently, U 0 ×W 0 V 0 represents the fibered product U ×W V in E. (ii) & (iii) These immediately follow from the proof of (i). Remark VI.5.7. One easily verifies that the family of vertical (resp. horizontal) coverings forms a pretopology (VI.5.6). It is not the case for their union, which causes many difficulties. Lemma VI.5.8. Let (Vm → V )m∈M be a finite vertical covering of E and for every m ∈ M , let (Vm,n → Vm )n∈Nm be a Cartesian covering of E. Then, there exists a Cartesian covering (W` → V )`∈L such that for all m ∈ M and ` ∈ L, there exist n` ∈ Nm and a Vm -morphism Vm ×V W` → Vm,n` ; in particular, the covering (Vm ×V W` → V )m∈M,`∈L refines the covering (Vm,n → V )m∈M,n∈Nm . Indeed, for every m ∈ M , (π(Vm,n ) → π(V ))n∈Nm is a covering of I. Since M is finite and I is stable under fibered products, there exists a covering (f` : i` → π(V ))`∈L of I such that for all m ∈ M and ` ∈ L, there exist n` ∈ Nm and a π(V )-morphism gm,` : i` → π(Vm,n` ) of I. For any ` ∈ L, set W` = f`+ (V ). For every m ∈ M , the morphism gm,` then induces a Vm -morphism Vm ×V W` → Vm,n` , giving the lemma. Proposition VI.5.9 (IV.6.1.3). Suppose that for every i ∈ Ob(I), every object of Ei is quasi-compact. Then a sieve R of an object V of E is covering if and only if there exist a Cartesian covering (Vn → V )n∈N and for every n ∈ N , a finite vertical covering (Vn,m → Vn )n∈Mn such that for all n ∈ N and m ∈ Mn , we have a V -morphism Vn,m → R. For every object V of E, denote by J(V ) the set of sieves R of V in E satisfying the required property. By definition, every sieve of J(V ) is covering for the covanishing topology, and every sieve generated by a Cartesian (resp. vertical) covering of V belongs to J(V ). It therefore suffices to show that the J(V ), for V ∈ Ob(E), define a topology ([2] II 1.1). It is clear that V belongs to J(V ) (axiom (T3) of loc. cit.). The stability under base change (axiom (T1) of loc. cit.) follows from VI.5.6(iii). It remains to establish the local character (axiom (T2) of loc. cit.). Let V ∈ Ob(E), R and R0 two sieves of V such that R ∈ J(V ) and that for every W ∈ Ob(E) and every morphism W → R, the sieve R0 ×V W belongs to J(W ). Let us show that R0 belongs to J(V ). By assumption, there exist a Cartesian covering (Vn → V )n∈N and for every n ∈ N , a finite vertical covering (Vn,m → Vn )n∈Mn such that for all n ∈ N and m ∈ Mn , the

514

VI. COVANISHING TOPOS AND GENERALIZATIONS

composition Vn,m → V belongs to R. Moreover, for all n ∈ N and m ∈ Mn , there α exist a Cartesian covering (Vn,m → Vn,m )α∈An,m and for every α ∈ An,m , a finite vertical α,β α α α covering (Vn,m → Vn,m )β∈Bn,m such that for all α ∈ An,m and β ∈ Bn,m , the composition α,β 0 Vm,n → V belongs to R . By VI.5.8, for every n ∈ N , there exists a Cartesian covering (Wn,` → Vn )`∈Ln such that for all m ∈ Mn and ` ∈ Ln , there exist αn,m,` ∈ An,m and a αn,m,` Vn,m -morphism pn,m,` : Vn,m ×Vn Wn,` → Vn,m . For any n ∈ N , m ∈ Mn , and ` ∈ Ln , m,β αn,m,` the inverse image under p we denote by (Wn,` → Vn,m ×Vn Wn,` )β∈Bn,m n,m,` of the α

n,m,` α` ,β αn,m,` . It is clear that (W vertical covering (Vn,m → Vn,m )β∈Bn,m n,` → V )n∈N,`∈Ln is a

m,β αn,m,` Cartesian covering, and that for all n ∈ N and ` ∈ Ln , (Wn,` → Wn,` )m∈Mn ,β∈Bn,m α

n,m,` is a finite vertical covering. For all n ∈ N , ` ∈ Ln , m ∈ Mn , and β ∈ Bn,m , the β,m composition Wn,` → V belongs to R0 , giving the required property.

Proposition VI.5.10. A presheaf F = {i 7→ Fi } on E is a sheaf, if and only if the following conditions are satisfied: (i) For every i ∈ Ob(I), Fi is a sheaf on Ei . (ii) For every covering family (fn : in → i)n∈Σ of I, if for every (m, n) ∈ Σ2 , we set imn = im ×i in and we denote by fmn : imn → i the canonical morphism, then the sequence of morphisms of sheaves on Ei Y Y (VI.5.10.1) Fi → (fn )∗ (Fin ) ⇒ (fmn )∗ (Fimn ) (m,n)∈Σ2

n∈Σ

is exact. Indeed, after extending the universe U, if necessary, we may assume that the category I is small ([2] II 2.7(2)). The proposition then follows from VI.5.6 and ([2] II 2.3, I 3.5, I 2.12, and II 4.1(3)). Corollary VI.5.11. The functor (VI.5.2.1) induces an equivalence of categories between e and the full subcategory of HomI ◦ (I ◦ , F ∨ ) made up of sections {i 7→ Fi } such that for E every covering family (fn : in → i)n∈Σ of I, if for every (m, n) ∈ Σ2 , we set imn = im ×i in and we denote by fmn : imn → i the canonical morphism, the sequence of sheaves on Ei Y Y (VI.5.11.1) Fi → (fn )∗ (Fin ) ⇒ (fmn )∗ (Fimn ) (m,n)∈Σ2

n∈Σ

is exact. Corollary VI.5.12. For every i ∈ Ob(I), the functor αi! : Ei → E (VI.5.1.2) is continuous. Remark VI.5.13. Condition VI.5.10(ii) is equivalent to the following condition: (ii’) For every i ∈ Ob(I) and every covering sieve R of i in I, the canonical morphism (VI.5.13.1)

Fi →

lim ←−

u∗ (Fi0 ),

(i0 ,u)∈R◦

where u : i0 → i denotes the structural morphism, is an isomorphism. Indeed, after extending the universe U, if necessary, we may assume that the category ei , I is small. The assertion then follows from ([2] I 2.12) applied to the functor R◦ → E 0 (i , u) 7→ u∗ (Fi0 ). Remark VI.5.14. Suppose that every object of I is quasi-compact. Then Proposition VI.5.10 remains true if in (ii) we restrict to finite covering families (fn : in → i)n∈Σ

VI.5. GENERALIZED COVANISHING TOPOS

515

of I. Indeed, every Cartesian covering of E admits a finite covering family. Consequently, the topology on E is generated by the finite Cartesian coverings and the vertical coverings, giving the assertion. Remark VI.5.15. The functors αi! are cocontinuous for every i ∈ Ob(I) if and only if the covanishing topology of E is equal to its total topology associated with π (VI.5.4). Indeed, the latter is by definition the coarsest topology that makes the functors αi! continuous for every i ∈ Ob(I) ([2] VI 7.4.1), and also the finest topology that makes the functors αi! cocontinuous for every i ∈ Ob(I) ([2] VI 7.4.3(2)). The assertion therefore follows from VI.5.12. Note that the covanishing and total topologies on E are not in general equal, in view of VI.5.10. VI.5.16. Suppose that I is equivalent to a U-small category. It follows from VI.5.10 and ([2] VI 7.4.7) that the identity functor idE : E → E is continuous when we endow the source with the total topology associated with the fibered site π (VI.5.4) and the target with the covanishing topology. Denote by Top(E) the total topos associated with the fibered site π, that is, the topos of sheaves of U-sets on the total site E. We therefore have a canonical morphism of topos ([2] IV 4.9.2) (VI.5.16.1)

e → Top(E) δ: E

e in Top(E); it is an such that the functor δ∗ is the canonical inclusion functor of E embedding ([2] IV 9.1.1). Note that the diagram (VI.5.16.2)

e PP E PPP PPP PPP δ∗ PPP '  ∼ / HomI ◦ (I ◦ , F ∨ ) Top(E)

where the horizontal arrow is the canonical equivalence of categories ([2] VI 7.4.7) and the slanted arrow is induced by the functor (VI.5.2.1), is commutative up to canonical isomorphism. On the other hand, for every object F of Top(E), δ ∗ (F ) is canonically isomorphic to the sheaf associated with the presheaf F on the covanishing site E. Indeed, after extending U, if necessary, we may assume that the category E is U-small ([2] II 3.6 and III 1.5), in which case the assertion follows from ([2] I 5.1 and III 1.3). Lemma VI.5.17. Let F = {i 7→ Fi } be a presheaf on E. For each i ∈ Ob(I), denote by ei associated with Fi . Then {i 7→ F a } is a presheaf on E and we have Fia the sheaf of E i b inducing an isomorphism between the a canonical morphism {i 7→ Fi } → {i 7→ Fia } of E associated sheaves. For any morphism f : i0 → i of I, denote by γf : Fi → f∗ (Fi0 ) the transition morphism of F associated with f (VI.5.2.1). The canonical morphism Fi0 → Fia0 induces a morphism ei , γf induces a morphism f∗ (Fi0 ) → f∗ (Fia0 ). Since f∗ (Fia0 ) is a sheaf of E (VI.5.17.1)

γfa : Fia → f∗ (Fia0 ).

The morphisms γf satisfy cocycle relations of type ([1] (1.1.2.2)), deduced from the composition of the morphisms in I. These induce analogous cocycle relations for the morphisms γfa . Consequently, {i 7→ Fia } is a section of the fibered category P ∨ , and is therefore a presheaf on E (VI.5.2.1) (cf. [37] VI 12 or [1] 1.1.2). Moreover, we have a canonical morphism (VI.5.17.2)

{i 7→ Fi } → {i 7→ Fia }.

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VI. COVANISHING TOPOS AND GENERALIZATIONS

ei by virtue of Let G = {i 7→ Gi } be a sheaf on E. For every i ∈ Ob(I), Gi is a sheaf of E VI.5.10. We immediately deduce from this that the map (VI.5.17.3)

HomEb ({i 7→ Fia }, {i 7→ Gi }) → HomEb ({i 7→ Fi }, {i 7→ Gi })

induced by (VI.5.17.2) is an isomorphism. Consequently, the morphism (VI.5.17.2) induces an isomorphism between the associated sheaves. VI.5.18. Let π 0 : E 0 → I be a cleaved and normalized fibered U-site satisfying the conditions of VI.5.1, and Φ : E0 → E

(VI.5.18.1)

a Cartesian I-functor (VI.2.2). We endow E 0 with the covanishing topology defined by e 0 the topos of sheaves of U-sets on E 0 . We associate with π 0 π 0 , and we denote by E objects analogous to those associated with π, and we denote them by the same letters equipped with an exponent 0 . For any i ∈ Ob(I), we denote by Φi : Ei0 → Ei the functor b∗ : E bi → E b 0 the functor obtained by composing induced by Φ on the fibers at i, and by Φ i i with Φi . For every morphism f : j → i of I, we have an isomorphism of functors ∼

Φj ◦ fE+0 → fE+ ◦ Φi ,

(VI.5.18.2)

where fE+ and fE+0 are the inverse image functors of E and E 0 , respectively. It induces an isomorphism of functors ∼ b ∗i ◦ fE∗ → b ∗j , Φ fE 0 ∗ ◦ Φ

(VI.5.18.3)

where fE∗ and fE 0 ∗ are the inverse image functors of P ∨ and P 0∨ , respectively (VI.5.1.5). The isomorphisms (VI.5.18.2) satisfy a cocycle relation of type ([1] (1.1.2.2)), which induces an analogous relation for the isomorphisms (VI.5.18.3). By ([37] VI 12; cf. also [1] b ∗ therefore define a Cartesian I ◦ -functor 1.1.2), the functors Φ i P ∨ → P 0∨ .

(VI.5.18.4)

One immediately verifies that the diagram of functors (VI.5.18.5)

b E

∼

/ HomI ◦ (I ◦ , P ∨ )

∼

 / HomI ◦ (I ◦ , P 0∨ )

b∗ Φ

 b0 E

b ∗ is the functor defined by the composition with Φ, the right vertical arrow is dewhere Φ fined by the composition with (VI.5.18.4), and the horizontal arrows are the equivalences of categories (VI.5.2.1), is commutative up to canonical isomorphism. Consequently, for every presheaf F = {i 7→ Fi } on E, we have (VI.5.18.6)

b ∗ (F ) = {i 7→ Φ b ∗ (Fi )}. Φ i

Suppose that for every i ∈ Ob(I), the functor Φi is continuous. Then the functor b ∗ induces a functor Φi,s : E ei → E e 0 , that commutes with inverse limits ([2] III 1.2). It Φ i i b ∗ (F ) is a sheaf on follows from VI.5.10 and (VI.5.18.5) that for every sheaf F on E, Φ 0 E , and, consequently, that Φ is continuous for the covanishing topologies of E and E 0 . It therefore induces a functor ([2] III 1.1.1) (VI.5.18.7)

e→E e0 . Φs : E

ei → E e 0 is an equivalence of cateIf, moreover, for every i ∈ Ob(I), the functor Φi,s : E i gories, then Φs is an equivalence of categories by virtue of VI.5.11.

VI.5. GENERALIZED COVANISHING TOPOS

517

Remark VI.5.19. We consider the fibered topos F /I (VI.5.1.3) as a fibered U-site by endowing each fiber with the canonical topology. This clearly satisfies the conditions of VI.5.1. Denote by εI : E → F

(VI.5.19.1)

the canonical Cartesian I-functor, which induces on the fibers the canonical functors ei ([2] VI (7.2.6.7)). It follows from VI.5.18 that εI induces an equivalence εi : Ei → E between the covanishing topos associated with E/I and F /I. VI.5.20. Let I 0 be a U-site in which fibered products are representable and ϕ : I 0 → I a continuous functor that commutes with fibered products. We denote by π0 : E 0 → I 0

(VI.5.20.1)

the base change of π (VI.5.1.1) by ϕ, and by

Φ : E0 → E

(VI.5.20.2)

the canonical projection ([37] VI § 3). Then E 0 /I 0 is a fibered site satisfying the conditions of VI.5.1. The fibered U-topos F 0 → I 0 associated with π 0 is canonically I 0 equivalent to the fibered topos deduced from F /I by base change by ϕ. We denote by F 0∨ P 0∨

(VI.5.20.3) (VI.5.20.4)

→ I 0◦ , → I 0◦ ,

the fibered categories associated with π 0 , defined in (VI.5.1.4) and (VI.5.1.5), respectively. They are canonically I 0◦ -equivalent to the fibered categories deduced from F ∨ /I ◦ and P ∨ /I ◦ by base change by ϕ◦ . b 0 the category of presheaves of U-sets on E 0 . One immediately verifies We denote by E that the diagram of functors (VI.5.20.5)

b E

∼

/ HomI ◦ (I ◦ , P ∨ )

∼

 / HomI 0◦ (I 0◦ , P 0∨ )

b∗ Φ

 b0 E

b ∗ is the functor where the horizontal arrows are the equivalences of categories (VI.5.2.1), Φ defined by the composition with Φ, and the right vertical arrow is the canonical functor ([37] VI § 3), is commutative up to canonical isomorphism. Consequently, for every presheaf F = {i 7→ Fi } on E, we have b ∗ (F ) = {i0 7→ Fϕ(i0 ) }, (VI.5.20.6) Φ where for every i0 ∈ Ob(I 0 ), we have identified the fibers Ei00 and Eϕ(i0 ) . We endow E 0 with the covanishing topology associated with the fibered site π 0 , and e 0 the topos of sheaves of U-sets on E 0 . It immediately follows from VI.5.10 we denote by E b ∗ (F ) is a sheaf on E 0 , and consequently, and (VI.5.20.5) that for every sheaf F on E, Φ that the functor Φ is continuous. It therefore induces a functor ([2] III 1.1.1) e→E e0 . (VI.5.20.7) Φs : E Proposition VI.5.21. We keep the assumptions of VI.5.20 and moreover suppose that the following conditions are satisfied: (i) The category I 0 is U-small and the functor ϕ is fully faithful. (ii) The topology on I 0 is induced by that on I through the functor ϕ. (iii) Every object of I can be covered by objects coming from I 0 .

518

VI. COVANISHING TOPOS AND GENERALIZATIONS

e→E e 0 (VI.5.20.7) is an equivalence of categories. Then the functor Φs : E 0 For every object i of I, we denote by I/i the category of pairs (i0 , u), where i0 ∈ Ob(I 0 ) 0 and u : ϕ(i0 ) → i is a morphism of I. Let (i0 , u), (i00 , v) be two objects of I/i . A morphism 0 00 0 00 0 from (i , u) to (i , v) is a morphism w : i → i of I such that u = v ◦ ϕ(w). Note that 0 the functor I/i → I/i , (i0 , u) 7→ (ϕ(i0 ), u) is fully faithful. e and every object i of I, Let us first show that for every object F = {i 7→ Fi } of E the morphism of sheaves on Ei

Fi →

(VI.5.21.1)

u∗ (Fϕ(i0 ) )

lim ←−

(i0 ,u)∈I 0◦ /i

is an isomorphism. Indeed, it follows from VI.5.10 and the assumptions that the sequence Y Y Y Fi → u∗ (Fϕ(i0 ) ) ⇒ w∗ (Fϕ(j) ) 0 ) (i0 ,u)∈Ob(I/i

0 )2 (j,w)∈Ob(I 0 ((i0 ,u),(i00 ,v))∈Ob(I/i /ϕ(i0 )×

00 ) i ϕ(i )

is exact. On the other hand, the canonical morphism lim ←−

(i0 ,u)∈I 0◦ /i

u∗ (Fϕ(i0 ) ) → 

  ker 

Y 0 ) (i0 ,u)∈Ob(I/i

u∗ (Fϕ(i0 ) ) ⇒

Y

Y

0 )2 (j,w)∈Ob(I 0 ((i0 ,u),(i00 ,v))∈Ob(I/i /ϕ(i0 )×

 w∗ (Fϕ(j) ) 00 ) i ϕ(i )

is a monomorphism by ([2] II 4.1(3)), giving the assertion. The isomorphism (VI.5.21.1) shows that the functor Φs is fully faithful. Let us show e 0 . Since for every that Φs is essentially surjective. Let F 0 = {i0 7→ Fi00 } be a sheaf of E 0 e i ∈ Ob(I), the category I/i is U-small, we define the sheaf Fi of Ei by the formula ([2] II 4.1) (VI.5.21.2)

Fi =

lim ←−

u∗ (Fi00 ).

(i0 ,u)∈I 0◦ /i

ej → E ei commutes with inverse For every morphism f : j → i of I, the functor f∗ : E limits. We deduce from this that {i 7→ Fi } is a section of the fibered category P ∨ /I ◦ (VI.5.1.5) and is therefore a presheaf on E (VI.5.2.1). We clearly have a canonical ∼ isomorphism Φ∗ (F ) → F 0 (VI.5.20.6). It therefore suffices to show that F is a sheaf on E, or, equivalently, by VI.5.10 and VI.5.13, that for every i ∈ Ob(I) and every covering sieve R of i in I, the canonical morphism (VI.5.21.3)

Fi →

lim ←−

v∗ (Fj ),

(j,v)∈R◦

where v : j → i denotes the structural morphism, is an isomorphism. For any j ∈ Ob(I), 0 denote by J/j the full subcategory of I/j made up of pairs (j 0 , v) such that ϕ(j 0 ) is an 0 object of R. We have J/j = I/j if j ∈ Ob(R). Since the functors u∗ commute with inverse limits, it suffices to show that the canonical morphism (VI.5.21.4)

Fi →

lim ←−

(i0 ,u)∈J ◦ /i

u∗ (Fi00 )

VI.5. GENERALIZED COVANISHING TOPOS

519

0 is an isomorphism. For every (i0 , u) ∈ Ob(I/i ), J/ϕ(i0 ) is a covering sieve of i0 in I 0 ([2] III 1.6). Consequently, the canonical morphism

Fi00 →

(VI.5.21.5)

v∗ (Fj00 )

lim ←−

(j 0 ,v)∈J ◦ /ϕ(i0 )

is an isomorphism (VI.5.13.1). Applying the functor u∗ and taking the inverse limit over 0◦ the category I/i , we deduce from this that the canonical morphism lim

(VI.5.21.6)

←−

(i0 ,u)∈I 0◦ /i

u∗ (Fi00 ) →

lim ←−

v∗ (Fj00 )

(j 0 ,v)∈J ◦ /i

is an isomorphism, and consequently that (VI.5.21.4) is an isomorphism. Proposition VI.5.22. We keep the assumptions of VI.5.21 and moreover suppose that for every i0 ∈ Ob(I 0 ), the category Ei00 is U-small. Then the covanishing topology of E 0 is induced by that on E by the functor Φ (VI.5.20.2). The functor Φs (VI.5.20.7) admits a left adjoint Φs that extends Φ, that is, that fits into a diagram commutative up to isomorphism (VI.5.22.1)

EO

ε

/E e O Φs

Φ

E0

ε0

/E e0

where the horizontal arrows are the canonical functors ([2] III 1.2). Since Φs is an equivalence of categories by virtue of VI.5.21, Φs is a quasi-inverse of Φs . We deduce ∼ from this an isomorphism ε0 → Φs ◦ ε ◦ Φ. Consequently, the covanishing topology of E 0 e by the functor ε ◦ Φ ([2] III 3.5). is induced by the canonical topology of E On the other hand, it follows from the assumptions that the category E 0 is U-small and that the functor Φ is fully faithful. Hence by virtue of the comparison lemma ([2] III 4.1), if we endow E 0 with the topology induced by that on E, the restriction functor e to the category of sheaves of U-sets on E 0 is an equivalence of categories. The from E reasoning above then shows that the topology on E 0 induced by that on E is also induced e by the functor ε ◦ Φ, giving the proposition. by the canonical topology of E Remark VI.5.23. We keep the assumption of VI.5.21 and moreover suppose that for every i0 ∈ Ob(I 0 ), the category Ei00 is U-small. We can then deduce VI.5.21 from VI.5.22 and ([2] III 4.1), even though we proceeded in the other direction. Lemma VI.5.24. With every U-small category J and every functor e (VI.5.24.1) φ : J → E, j 7→ Fj = {i 7→ Fj,i } are canonically associated the following data: (i) a sheaf {i 7→ lim Fj,i } on E and a canonical isomorphism ←− j∈J

∼

{i 7→ lim Fj,i } → lim φ,

(VI.5.24.2)

←−

←−

j∈J

J

ei , and a canonical (ii) a presheaf {i 7→ lim Fj,i } on E, where the limits are taken in E −→ j∈J

isomorphism (VI.5.24.3)

∼

lim φ → {i 7→ lim Fj,i }a , −→ J

−→ j∈J

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VI. COVANISHING TOPOS AND GENERALIZATIONS

where the right-hand side is the sheaf on E associated with the presheaf {i 7→ lim Fj,i }. −→ j∈J

In particular, for every i ∈ I, the canonical functor (VI.5.24.4)

e→E ei , E

{i 7→ Fi } 7→ Fi

commutes with inverse U-limits. It immediately follows from (VI.5.2.1) and ([2] I 3.1) that the functor (VI.5.24.5)

b→E bi , E

{i 7→ Gi } 7→ Gi

commutes with inverse and direct U-limits. Moreover, with every U-small category J and every functor (VI.5.24.6)

b ψ : J → E,

j 7→ Gj = {i 7→ Gj,i }

are canonically associated two presheaves {i 7→ lim Gj,i } and {i 7→ lim Gj,i } on E, and −→ j∈J

←− j∈J

two canonical morphisms (VI.5.24.7)

{i 7→ lim Gj,i } → lim ψ, ←−

←−

J

j∈J

(VI.5.24.8)

lim ψ → {i 7→ lim Gj,i }. −→

−→

j∈J

J

The latter are therefore isomorphisms. Suppose that ψ is induced by a functor φ as in (VI.5.24.1). By ([2] II 4.1(3)), the isomorphism (VI.5.24.7) induces the isomorphism (VI.5.24.2); in particular, {i 7→ lim Gj,i } is a sheaf on E. On the other hand, by virtue of VI.5.17 and ([2] II 4.1), {i 7→ ←− j∈J

lim Fj,i } is a presheaf on E, and the isomorphism (VI.5.24.8) induces the isomorphism −→ j∈J

(VI.5.24.3). Proposition VI.5.25. Suppose that the following conditions are satisfied: (i) Every object of I is quasi-compact. ei is algebraic ([2] VI 2.3). (ii) For every object i of I, the topos E ei → E ej is (iii) For every morphism f : i → j of I, the morphism of topos f : E coherent ([2] VI 3.1). Then for every filtered U-small category J and every functor (VI.5.25.1)

e φ : J → E,

j 7→ Fj = {i 7→ Fj,i },

{i 7→ lim Fj,i } is a sheaf on E, and we have a canonical isomorphism −→ j∈J

(VI.5.25.2)

∼

lim φ → {i 7→ lim Fj,i }. −→ J

−→ j∈J

In particular, for every i ∈ I, the canonical functor (VI.5.25.3)

e→E ei , E

commutes with filtered direct U-limits.

{i 7→ Fi } 7→ Fi

VI.5. GENERALIZED COVANISHING TOPOS

521

In view of VI.5.24(ii), it suffices to show that {i 7→ lim Fj,i } is a sheaf on E. Let −→ j∈J

(fn : in → i)n∈Σ be a finite covering family of I. For any (m, n) ∈ Σ2 , we set imn = im ×i in and we denote by fmn : imn → i the canonical morphism. For every j ∈ J, the sequence Y Y (VI.5.25.4) Fj,i → (fn )∗ (Fj,in ) ⇒ (fmn )∗ (Fj,imn ) (m,n)∈Σ2

n∈Σ

is exact. Since Σ is finite, we deduce from this by taking the direct limit over J that the sequence Y Y lim (fmn )∗ (Fj,imn ) (VI.5.25.5) lim Fj,i → lim (fn )∗ (Fj,in ) ⇒ −→ j∈J

n∈Σ

−→

(m,n)∈Σ2

j∈J

−→ j∈J

is exact ([2] II 4.3(4)). Since the functors (fn )∗ and (fmn )∗ commute with filtered direct limits of sheaves of sets by virtue of ([2] VI 5.1 and VII 5.14), we deduce from this that the sequence Y Y (VI.5.25.6) lim Fj,i → (fn )∗ (lim Fj,in ) ⇒ (fmn )∗ (lim Fj,imn ) −→ j∈J

n∈Σ

−→ j∈J

−→

(m,n)∈Σ2

j∈J

is exact. The desired assertion follows in view of VI.5.14. Corollary VI.5.26. We keep the assumptions of VI.5.25 and moreover let V be an object of E and v its image in I. Then V is quasi-compact in E if and only if it is quasi-compact in Ev . It immediately follows from the definition of the covanishing topology (VI.5.3) that if V is quasi-compact in E, it is also quasi-compact in Ev . Let us show the converse ev the canonical functor. By VI.5.25, for every implication. Denote by εv : Ev → E filtered U-small category J and every functor e j 7→ Fj = {i 7→ Fj,i }, (VI.5.26.1) J → E, we can identify the canonical maps

lim HomEe (ε(V ), Fj ) → HomEe (ε(V ), lim Fj ),

(VI.5.26.2)

−→ j∈J

(VI.5.26.3)

−→ j∈J

lim HomEev (εv (V ), Fj,v ) → HomEev (εv (V ), lim Fj,v ). −→ j∈J

−→ j∈J

If V is quasi-compact in Ev , the map (VI.5.26.3) is injective by ([2] VI 1.2 and 1.23(i)). The same therefore holds for the map (VI.5.26.2). Consequently, V is quasi-compact in E by virtue of ([2] VI 1.23(i)). Proposition VI.5.27. Suppose that every object of I is quasi-compact and that for every i ∈ Ob(I), every object of Ei is quasi-compact. Then: ei is coherent. (i) For every object i of I, the topos E ei → E ej is (ii) For every morphism f : i → j of I, the morphism of topos f : E coherent. e in particular, the topos (iii) For every object V of E, ε(V ) is a coherent object of E; e E is locally coherent. e is coherent. (iv) If, moreover, the category E admits a final object e, then the topos E ei the canonical functor. For every U ∈ Ob(Ei ), εi (U ) is (i) Denote by εi : Ei → E e a coherent object of Ei by virtue of VI.5.1(ii) and ([2] VI 2.1). On the other hand, Ei admits a final object ei (VI.5.1(ii)). Since εi is left exact, εi (ei ) is the final object of

522

VI. COVANISHING TOPOS AND GENERALIZATIONS

ei . For every U ∈ Ob(Ei ), the diagonal morphism δ : εi (U ) → εi (U ) ×ε (e ) εi (U ) is the E i i image by εi of the diagonal morphism U → U ×ei U . Hence δ is quasi-compact because its source and target are coherent; in other words, εi (U ) is quasi-separated over εi (ei ). ei is coherent ([2] VI 2.3). Consequently, the topos E (ii) This follows from ([2] VI 3.3). (iii) Every object of E is quasi-compact by VI.5.9 (or by (i), (ii), and VI.5.26). Since fibered products are representable in E (VI.5.6), the proposition follows from ([2] VI 2.1). e is left exact, ε(e) is the final object of E. e (iv) Since the canonical functor ε : E → E For every V ∈ Ob(E), the diagonal morphism δ : ε(V ) → ε(V ) ×ε(e) ε(V ) is the image by ε of the diagonal morphism V → V ×e V (VI.5.6). Hence δ is quasi-compact because its source and target are coherent by (iii); in other words, ε(V ) is quasi-separated over ε(e). The proposition follows in view of (iii). Proposition VI.5.28. Suppose that the following conditions are satisfied: (i) There exists a U-small, topologically generating, full subcategory I 0 of I, made up of quasi-compact objects, and stable under fibered products. ei is coherent. (ii) For every object i of I 0 , the topos E ei → E ej is (iii) For every morphism f : i → j of I 0 , the morphism of topos f : E coherent. e is locally coherent. If, moreover, the category I admits a final object Then the topos E e is coherent. that belongs to I 0 , then the topos E By VI.5.21, we may restrict to the case where I = I 0 . By virtue of VI.5.19, it suffices to show the analogous proposition for the fibered site F /I. Denote by C the full subcategory of F made up of the objects that are coherent in their fibers. By condition (iii), the structural functor C → I makes C /I into a cleaved and normalized fibered category. The fiber Ci of C over an object i of I identifies with the full subcategory of ei made up of the coherent objects of E ei . It is generating in E ei , and is stable under E e fibered products by ([2] VI 2.2). Since the final object of Ei is an object of Ci by assumption, finite inverse limits are representable in Ci ([2] I 2.3.1). Endowing each fiber ei , C /I becomes a fibered site Ci with the topology induced by the canonical topology on E satisfying the conditions of VI.5.1. The covanishing topos associated with C /I and F /I are equivalent by VI.5.18. On the other hand, if I admits a final object ι, then the final object e of Cι , which exists by VI.5.1(ii), is a final object of C (this easily follows from VI.5.1(iii); cf. VI.5.32 below). We thus reduce to the statement of Proposition VI.5.27. e has enough points. Corollary VI.5.29. Under the assumptions of VI.5.28, the topos E This follows from ([2] VI § 9). e Ye be two coherent U-topos and f : Ye → X e a coherent Corollary VI.5.30. Let X, ← e × e Ye is coherent. In particular, it has enough morphism. Then the oriented product X X points. By VI.5.31 below, there exists a U-small and generating full subcategory X (resp. e (resp. Ye ), made up of coherent objects of X e (resp. Ye ) and stable under finite Y ) of X ∗ inverse limits, such that for every U ∈ Ob(X), f (U ) belongs to Y . The oriented product ← e × e Ye is equivalent to the covanishing topos associated with the functor f ∗ : X → Y X X (VI.4.10). On the other hand, for every object V of Y , the topos Ye/V is coherent ([2] VI 2.4.2), and for every morphism V 0 → V of Y , the localization morphism Ye/V 0 → Ye/V is coherent ([2] VI 3.3). The corollary therefore follows from VI.5.5 and VI.5.28.

VI.5. GENERALIZED COVANISHING TOPOS

523

e there exists a U-small Lemma VI.5.31. For every morphism of U-topos f : Ye → X, e (resp. Ye ), stable under finite inverse and generating full subcategory X (resp. Y ) of X e and Ye are limits such that for every U ∈ Ob(X), f ∗ (U ) belongs to Y . If, moreover, X coherent and f is coherent, then we may assume that X (resp. Y ) is made up of coherent objects. e Indeed, there exists a U-small and generating full subcategory X (resp. Y ) of X ∗ e (resp. Y ) such that for every U ∈ Ob(X), f (U ) belongs to Y . By the first argument of ([2] IV 1.2.3), after extending X, and then Y , if necessary, we may assume that they are stable under finite inverse limits. e and Ye are coherent and that f is coherent. By ([2] VI 2.1), Next, suppose that X e (resp. Ye ), made there exists a U-small and generating full subcategory X (resp. Y ) of X ∗ up of coherent objects, such that for every U ∈ Ob(X), f (U ) belongs to Y . On the e (resp. Ye ) made up of the coherent objects of X e other hand, the full subcategory of X (resp. Ye ) is stable under finite inverse limits ([2] VI 2.4.4). The argument of ([2] IV 1.2.3) then shows that after extending X, and then Y , if necessary, we may assume that they are stable under finite inverse limits. VI.5.32. For the remainder of this section, in addition to the assumptions made in VI.5.1, we suppose that the following condition is satisfied: (i’) Finite inverse limits are representable in I. In view of VI.5.1(i), this is equivalent to requiring that I admits a final object ([2] I 2.3.1). We fix from now on a final object ι of I and a final object e of Eι (which exists by VI.5.1(ii)). For every i ∈ Ob(I), we denote by (VI.5.32.1)

fi : i → ι

the canonical morphism. By VI.5.1(iii), fi+ (e) is a final object of Ei . Consequently, e is a final object of E, and finite inverse limits are representable in E by VI.5.6(i) and ([2] I 2.3.1). The canonical injection functor αι! : Eι → E commutes with fibered products (VI.5.6) and transforms a final object into a final object; it is therefore left exact. On the other hand, it is continuous (VI.5.12). It then defines a morphism of topos ([2] IV 4.9.2) (VI.5.32.2)

e→E eι . β: E

For every sheaf F = {i 7→ Fi } on E, we have a canonical isomorphism (VI.5.32.3)

∼

β∗ (F ) → Fι .

There exists essentially a unique Cartesian section of π (VI.5.1.1) (VI.5.32.4)

σ+ : I → E

such that σ + (ι) = e. It is defined, in view of ([1] 1.1.2), for every object i of I, by σ + (i) = fi+ (e) and for every morphism f : j → i of I by the canonical isomorphism (VI.5.32.5)

∼

σ + (j) → f + (σ + (i)).

For every i ∈ Ob(I), σ + (i) is a final object of Ei , and we have a canonical morphism of functors (VI.5.32.6)

idE → σ + ◦ π.

It immediately follows from the proof of VI.5.6 that σ + commutes with fibered products, and is therefore left exact. On the other hand, σ + is continuous by virtue of ([2] III 1.6).

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VI. COVANISHING TOPOS AND GENERALIZATIONS

It therefore defines a morphism of topos (VI.5.32.7)

e → I. e σ: E

Remark VI.5.33. The morphisms β (VI.5.32.2) and σ (VI.5.32.7) identify canonically with those defined from the fibered site F /I (VI.5.19). eι , β ∗ (F ) is the sheaf associated with the Lemma VI.5.34. (i) For every sheaf F of E ∗ presheaf {i 7→ fi F } on E, where for every morphism f : j → i of I, the transition morphism (VI.5.34.1)

fi∗ F → f∗ (fj∗ F ) ∼

is the adjoint of the canonical isomorphism f ∗ (fi∗ F ) → fj∗ F . Moreover, the adjunction morphism F → β∗ (β ∗ F ) factors through the morphism of presheaves on Eι (VI.5.34.2)

F → {i 7→ fi∗ F } ◦ αι!

defined by the identity of F . e σ ∗ (F ) is the sheaf associated with the presheaf {i 7→ F (i)} (ii) For every sheaf F of I, on E, where for any i ∈ Ob(I), we have denoted by F (i) the constant presheaf on Ei with value F (i). Indeed, after extending U, if necessary, we may assume that the categories I and Eι are U-small ([2] II 3.6 and III 1.5). (i) By ([2] I 5.1 and III 1.3), β ∗ (F ) is the sheaf associated with the presheaf G on E defined, for every V ∈ Ob(E), by (VI.5.34.3)

G(V ) =

lim −→

F (U ),

(U,u)∈A◦ V

where AV is the category of pairs (U, u) consisting of an object U of Eι and a morphism u : V → U of E. If i denotes the image of V in I, AV identifies with the category of pairs (U, u) consisting of an object U of Eι and a morphism u : V → fi+ (U ) of Ei . It then follows from VI.5.17 that β ∗ (F ) is the sheaf associated with the presheaf {i 7→ fi∗ F } on E defined by the transition morphisms (VI.5.34.1). The last assertion immediately follows from the above. (ii) Indeed, σ ∗ (F ) is the sheaf associated with the presheaf H on E defined, for every V ∈ Ob(E), by (VI.5.34.4)

H(V ) =

lim −→

F (i),

(i,u)∈B ◦ V

where BV is the category of pairs (i, u) consisting of an object i of I and a morphism u : V → σ + (i) of E. This category admits as initial object the pair consisting of π(V ) and the canonical morphism V → σ + (π(V )) (VI.5.32.6). We therefore have H(V ) = F (π(V )). Proposition VI.5.35. Suppose that the following conditions are satisfied: (i) There exists a U-small and generating full subcategory I 0 of I, made up of quasi-compact objects, and stable under finite inverse limits. ei is coherent. (ii) For every object i of I 0 , the topos E ei → E ej is (iii) For every morphism f : i → j of I 0 , the morphism of topos f : E coherent. Then the morphisms β (VI.5.32.2) and σ (VI.5.32.7) are coherent.

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525

In view of VI.5.33, proceeding as in the proof of VI.5.28, we can reduce to the case where every object of I is quasi-compact and for every i ∈ Ob(I), every object of Ei is quasi-compact. Consequently, every object of E is quasi-compact by virtue of VI.5.27. The proposition then follows from ([2] VI 3.3). VI.5.36. Let π 0 : E 0 → I be a cleaved and normalized fibered U-site satisfying the conditions of VI.5.1, and Φ+ : E 0 → E

(VI.5.36.1)

a Cartesian I-functor. We endow E 0 with the covanishing topology defined by π 0 , and e 0 the topos of sheaves of U-sets on E 0 . We associate with π 0 objects we denote by E analogous to those associated with π, and we denote them by the same letters equipped 0 with an exponent 0 . Suppose that for every i ∈ Ob(I), the functor Φ+ i : Ei → Ei , induced + by Φ on the fibers at i, is continuous and left exact. This then defines a morphism of topos ei → E ei0 , Φi : E

(VI.5.36.2)

characterized by Φi∗ (F ) = F ◦ Φi and Φ∗i extends Φ+ i ([2] IV 4.9.4). We know (VI.5.18) that the functor Φ+ is continuous for the covanishing topologies of E and E 0 . On the other hand, it commutes with fibered products (cf. the proof of VI.5.6(i)), and Φ+ (e) = Φ+ ι (e) is a final object of Eι0 and therefore of E 0 (VI.5.32). Consequently, Φ+ is left exact. It therefore defines a morphism of topos e→E e0 Φ: E

(VI.5.36.3)

characterized by Φ∗ (F ) = F ◦ Φ+ and Φ∗ extends Φ+ . By (VI.5.18.6), for every sheaf F = {i 7→ Fi } on E, we have (VI.5.36.4) VI.5.37.

Φ∗ (F ) = {i 7→ Φi∗ (Fi )}.

Let V be an object of E and c = π(V ). We denote by $ : E/V → I/c

(VI.5.37.1) the functor induced by π, and by

γV : E/V → E

(VI.5.37.2)

the canonical functor. For every morphism f : i → c of I, the fiber of $ over the object (i, f ) of I/c is canonically equivalent to the category (Ei )/f + (V ) . Endowing I/c with the topology induced by that of I, and each fiber (Ei )/f + (V ) with the topology induced by that of Ei , $ becomes a fibered site satisfying the conditions of VI.5.1. We then endow E/V with the covanishing topology associated with $. Proposition VI.5.38. Under the assumptions of VI.5.37, the covanishing topology of E/V is induced by that of E through the functor γV . In particular, the topos of sheaves e/ε(V ) . The restriction functor from E e to of U-sets on E/V is canonically equivalent to E e/ε(V ) is isomorphic to the functor E (VI.5.38.1)

e→E e/ε(V ) , E

{i 7→ Fi } 7→ {(i, f ) 7→ Fi × f ∗ (V )},

where (i, f ) denotes an object of I/c ; in other words, f : i → c is a morphism of I. Let F = {i 7→ Fi } be a sheaf on E. By ([2] III 5.4), we have a canonical isomorphism of presheaves on E/V (VI.5.38.2)

∼

F ◦ γV → {(i, f ) 7→ Fi × f ∗ (V )}.

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VI. COVANISHING TOPOS AND GENERALIZATIONS

For every morphism g : (i, f ) → (j, h) of I/c , the diagram of morphisms of topos ei )/f ∗ (V ) (E

(VI.5.38.3)

 e Ei

g$

/ (E ej )/h∗ (V )  /E ej

gπ

where g$ and gπ denote the morphisms of topos associated with g by $ and π, respectively, and the vertical arrows are the localization morphisms, is commutative up to canonical isomorphism ([2] IV 5.10). The transition morphism Fj × h∗ (V ) → g$∗ (Fi × f ∗ (V ))

(VI.5.38.4)

of the presheaf F ◦ γV is the composition

∼

Fj × h∗ (V ) → gπ∗ (Fi ) × h∗ (V ) → g$∗ (Fi × f ∗ (V )),

(VI.5.38.5)

where the first arrow is induced by the transition morphism of F , and the second arrow is the base change morphism associated with (VI.5.38.3) ([1] (1.2.2.2)), which is in fact an isomorphism. Let (i, f ) be an object of I/c and (gn : in → i)n∈Σ a covering family of I. For any (m, n) ∈ Σ2 , we set imn = im ×i in and we denote by gmn : imn → i the canonical ei → morphism. Set fn = f ◦ gn and fmn = f ◦ gmn . Since the restriction functor E e (Ei )/f ∗ (V ) admits a left adjoint, it commutes with inverse limits. Consequently, in view of (VI.5.38.5), the exact sequence of morphisms of sheaves on Ei Y Y (VI.5.38.6) Fi → (gn )π∗ (Fin ) ⇒ (gmn )π∗ (Fimn ) (m,n)∈Σ2

n∈Σ

ei )/f ∗ (V ) induces an exact sequence of morphisms of sheaves on (E Y Y ∗ (VI.5.38.7) Fi ×f ∗ (V ) → (gn )$∗ (Fin ×fn∗ (V )) ⇒ (gmn )$∗ (Fimn ×fmn (V )). (m,n)∈Σ2

n∈Σ

We deduce from this that F ◦ γV is a sheaf for the covanishing topology of E/V . Hence γV is continuous; in other words, the covanishing topology of E/V is coarser than the topology induced by that of E. To show the first assertion of the proposition, it suffices to show that γV is cocontinuous ([2] III 2.1). The functor γV is a left adjoint of the functor Φ+ : E → E/V ,

(VI.5.38.8)

W 7→ W × V.

Let G = {(i, f ) 7→ G(i,f ) } be a sheaf on E/V (where (i, f ) ∈ Ob(I/c )). For any i ∈ Ob(I), we denote by pi : i × c → c and qi : i × c → i the canonical projections, and by ei×c )/p∗ (V ) → E ei (VI.5.38.9) qi0 : (E i

the composition

qi ei×c )/p∗ (V ) −→ E ei×c −→ ei , (E E i where the first arrow is the localization morphism. We then have a canonical isomorphism 0 G ◦ Φ+ ' {i 7→ qi∗ (Gi×c )}.

(VI.5.38.10)

For every morphism f : j → i of I, the transition morphism

(VI.5.38.11)

is the composition

0 0 qi∗ (Gi×c ) → fπ∗ (qj∗ (Gj×c )) ∼

0 0 0 qi∗ (Gi×c ) → qi∗ ((f × idc )$∗ (Gj×c )) → fπ∗ (qj∗ (Gj×c )),

VI.6. MORPHISMS WITH VALUES IN A GENERALIZED COVANISHING TOPOS

527

where the first arrow comes from the transition morphism of F and the second arrow is the canonical isomorphism (VI.5.38.3). Let (in → i)n∈Σ be a covering family of I. 0 Since the functor qi∗ commutes with inverse limits, the gluing relation (VI.5.10.1) for G relative to the covering (in × c → i × c)n∈Σ of I/c implies the analogous relation for G ◦ Φ+ relative to the covering (in → i)n∈Σ . We deduce from this that G ◦ Φ+ is a sheaf on E by VI.5.10; hence Φ+ is continuous. Consequently, γV is cocontinuous by virtue of ([2] III 2.5), giving the first statement of the proposition. The second statement follows by virtue of ([2] III 5.4). The last statement follows from the above, in particular from (VI.5.38.2). VI.5.39.

Let c be an object of I, ec = σ + (c) = fc+ (e) (VI.5.32.1). The fibered site

(VI.5.39.1)

$ : E/ec → I/c

is deduced from the fibered site π by base change by the canonical functor I/c → I. If we endow I/c with the topology induced by that of I, the fibered site $ satisfies the conditions of VI.5.1. By virtue of VI.5.38, the covanishing topology of E/ec is induced by that of E through the canonical functor γc : E/ec → E. In particular, the topos of e/σ∗ (c) . The final object idc of I/c sheaves of U-sets on E/ec is canonically equivalent to E then defines a morphism of topos (VI.5.32.2) (VI.5.39.2)

e/σ∗ (c) → E ec . βc : E

We denote also by (VI.5.39.3)

e/σ∗ (c) → E e γc : E

e at σ ∗ (c). By VI.5.38 and (VI.5.32.3), for every sheaf the localization morphism of E F = {i 7→ Fi } on E, we have a canonical isomorphism (VI.5.39.4)

∼

βc∗ (γc∗ ({i 7→ Fi })) → Fc .

VI.6. Morphisms with values in a generalized covanishing topos VI.6.1. We keep the notation and conventions of VI.5, in particular, those introduced in VI.5.32. More precisely, I denotes a U-site and π : E → I a cleaved and normalized fibered U-site, such that the following conditions are satisfied: (i) Finite inverse limits are representable in I. (ii) For every i ∈ Ob(I), finite inverse limits are representable in Ei . (iii) For every morphism f : i → j of I, the inverse image functor f + : Ej → Ei is continuous and left exact. e the topos We endow E with the covanishing topology defined by π, and denote by E of sheaves of U-sets on E. We moreover fix a U-site X and a continuous and left exact functor (VI.6.1.1)

Ψ+ : E → X.

e the topos of sheaves of U-sets on X and by We denote by X e →E e (VI.6.1.2) Ψ: X

the morphism of topos associated with Ψ+ ([2] IV 4.9.2). We set (VI.6.1.3)

u+ = Ψ+ ◦ σ + : I → X,

where σ + is the functor defined in (VI.5.32.4). Since the functor u+ is continuous and left exact, we denote by e → Ie (VI.6.1.4) u = σΨ : X

528

VI. COVANISHING TOPOS AND GENERALIZATIONS

the associated morphism of topos. Let i be an object of I. Since σ + (i) is a final object of Ei , Ψ+ induces a functor Ψ+ i : Ei → X/u+ (i) .

(VI.6.1.5)

This commutes with fibered products (VI.5.6) and transforms a final object into a final object; it is therefore left exact. On the other hand, when we endow X/u+ (i) with the topology induced by that of X, Ψ+ i transforms a covering family into a covering family by ([2] III 1.6 and 5.2); it is therefore continuous. We denote by e/u∗ (i) → E ei Ψi : X

(VI.6.1.6)

the morphism of topos defined by Ψ+ i , and by

e/u∗ (i) → X e i : X

(VI.6.1.7)

e at u∗ (i) ([2] IV 5.1). For every morphism f : i0 → i of I, the localization morphism of X we denote by e/u∗ (i0 ) → X e/u∗ (i) (VI.6.1.8) f : X

the localization morphism associated with u+ (f ) : u+ (i0 ) → u+ (i) ([2] IV 5.5). Recall that we have fixed a final object ι of I (VI.5.32). Since u+ (ι) is a final object of X, Ψι is none other than the composition e →E eι . (VI.6.1.9) Ψι = βΨ : X

For any i ∈ Ob(I), we denote by fi : i → ι the canonical morphism (VI.5.32.1). We identify i and fi . From the relations Ψ∗ σ ∗ = u∗ and Ψ∗ β ∗ = Ψ∗ι , we obtain by adjunction morphisms σ∗ β∗

(VI.6.1.10) (VI.6.1.11) Lemma VI.6.2. of topos

→ Ψ∗ u∗ , → Ψ∗ Ψ∗ι .

(i) For every morphism f : i0 → i of I, the diagram of morphisms e/u∗ (i0 ) X

(VI.6.2.1)

f



e/u∗ (i) X

Ψi0

/E ei0 f

 /E ei

Ψi

is commutative up to a canonical isomorphism ∼

Ψi f → f Ψi0 .

(VI.6.2.2)

e we have a canonical isomorphism of E e (ii) For every object F of X, ∼

Ψ∗ (F ) → {i 7→ Ψi∗ (∗i F )},

(VI.6.2.3)

where for every morphism f : i0 → i of I, the transition morphism Ψi∗ (ji∗ F ) → f∗ (Ψi0 ∗ (∗i0 F ))

(VI.6.2.4) is the composition (VI.6.2.5)

∼

∼

Ψi∗ (∗i F ) → Ψi∗ (f ∗ (∗f (∗i F ))) → Ψi∗ (f ∗ (∗i0 F )) → f∗ (Ψi0 ∗ (∗i0 F )),

in which the first morphism comes from the adjunction morphism id → f ∗ ∗f , the second morphism is the canonical isomorphism, and the last morphism comes from (VI.6.2.2).

VI.6. MORPHISMS WITH VALUES IN A GENERALIZED COVANISHING TOPOS

529

eι , the adjunction morphism (iii) For every sheaf F of E β ∗ (F ) → Ψ∗ (Ψ∗ι F ) (VI.6.1.11) is induced by the morphism of presheaves on E (VI.6.2.6)

{i 7→ fi∗ F } → {i 7→ Ψi∗ (∗i (Ψ∗ι F ))}

ei defined, for every i ∈ Ob(I), by the morphism of E (VI.6.2.7)

fi∗ F → Ψi∗ (∗i (Ψ∗ι F ))

adjoint of the isomorphism (VI.6.2.2). (i) For every object V of Ei , we have a canonical isomorphism in E (VI.6.2.8)

∼

f + (V ) → V ×σ+ (i) σ + (i0 ).

Since Ψ+ is left exact, we deduce from this that the diagram of morphisms of functors (VI.6.2.9)

Ei

Ψ+ i

/ X/u+ (i)

f+

 Ei0

Ψ+ i0



+ f

/ X/u+ (i0 )

+ where + f is the base change functor by u (f ), is commutative up to canonical isomorphism. The assertion follows in view of the interpretation of the functor ∗f as a base change functor by the morphism u∗ (f ) ([2] III 5.4). (ii) This immediately follows from the definitions. (iii) Set β ∗ (F ) = {i 7→ Gi }. By (ii), the morphism β ∗ (F ) → Ψ∗ (Ψ∗ι F ) (VI.6.1.11) is ei defined, for every i ∈ Ob(I), by a morphism of E

(VI.6.2.10)

si : Gi → Ψi∗ (∗i (Ψ∗ι F )).

By virtue of VI.5.34(i), we have a canonical morphism of presheaves on E (VI.6.2.11)

{i 7→ fi∗ F } → {i 7→ Gi },

ei . The diagram defined, for every i ∈ Ob(I), by a morphism ti : fi∗ F → Gi of E (VI.6.2.12)

f ∗ (t )

ι i / f ∗ (Gι ) fi∗ F P i PPP PPP PPP ti PPP P'  Gi

fi∗ (sι )

/ f ∗ (Ψι∗ (Ψ∗ι F )) i

si

 / Ψi∗ (∗ (Ψ∗ι F )) i

where the vertical arrows are the adjoints of the transition morphisms, is commutative. On the one hand, sι identifies with the morphism (VI.6.2.13)

β∗ (β ∗ F ) → β∗ (Ψ∗ (Ψ∗ι F ))

induced by (VI.6.1.11). On the other hand, tι identifies with the adjunction morphism F → β∗ (β ∗ F ) by virtue of VI.5.34(i). Consequently, in view of the definition of (VI.6.1.11), sι ◦ tι is the adjunction morphism F → Ψι∗ (Ψ∗ι F ). It then follows from (ii) and (VI.6.2.12) that the morphism si ◦ ti is the composition (VI.6.2.14)

fi∗ F → fi∗ (Ψι∗ (Ψ∗ι F )) → Ψi∗ (∗i (Ψ∗ι F )),

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VI. COVANISHING TOPOS AND GENERALIZATIONS

where the first arrow comes from the adjunction morphism id → Ψι∗ Ψ∗ι and the second arrow is the base change morphism associated with the diagram (VI.6.2.1) (for f = fi ). By ([2] XVII 2.1.3), the adjoint of si ◦ ti is the composition (VI.6.2.15)

∼

Ψ∗i (fi∗ F ) → Ψ∗i (fi∗ (Ψι∗ (Ψ∗ι F ))) −→ ∗i (Ψ∗ι (Ψι∗ (Ψ∗ι F ))) → ∗i (Ψ∗ι F ),

where the first arrow comes from the adjunction morphism id → Ψι∗ Ψ∗ι , the second arrow is the isomorphism (VI.6.2.2), and the third arrow comes from the adjunction morphism Ψ∗ι Ψι∗ → id. This composed morphism is equal to the composition (VI.6.2.16)

∼

u

v

Ψ∗i (fi∗ F ) −→ ∗i (Ψ∗ι F ) −→ ∗i (Ψ∗ι (Ψι∗ (Ψ∗ι F ))) −→ ∗i (Ψ∗ι F ),

where the first arrow is the isomorphism (VI.6.2.2), u comes from the adjunction morphism id → Ψι∗ Ψ∗ι , and v comes from the adjunction morphism Ψ∗ι Ψι∗ → id. Since v ◦ u is the identity, the assertion follows. Proposition VI.6.3. Suppose that, for every i ∈ Ob(I), the adjunction morphism id → Ψi∗ Ψ∗i is an isomorphism. Then: eι , β ∗ (F ) is the sheaf on E defined by {i 7→ f ∗ F }. (i) For every sheaf F of E i (ii) The adjunction morphism id → β∗ β ∗ is an isomorphism. (iii) The adjunction morphism β ∗ → Ψ∗ Ψ∗ι (VI.6.1.11) is an isomorphism. (i) Indeed, for every i ∈ Ob(I), the morphism (VI.6.3.1)

fi∗ F → Ψi∗ (∗i (Ψ∗ι F ))

defined in (VI.6.2.7), is the composition ∼

fi∗ F → Ψi∗ (Ψ∗i (fi∗ F )) → Ψi∗ (∗i (Ψ∗ι F )),

where the first morphism comes from the adjunction morphism id → Ψi∗ Ψ∗i and the second from (VI.6.2.2). It is therefore an isomorphism. Consequently, the morphism b (VI.6.2.6) is an isomorphism of E (VI.6.3.2)

∼

{i 7→ fi∗ F } → {i 7→ Ψi∗ (∗i (Ψ∗ι F ))}.

Since the target of this morphism is a sheaf on E (VI.6.2.3), the same holds for its source. The proposition then follows from VI.5.34(i). (ii) This immediately follows from (i) and VI.5.34(i). (iii) This follows from VI.6.2(iii) and from the proof of (i). VI.6.4. For the remainder of this section, in addition to the general assumptions made in VI.6.1, we suppose that finite inverse limits are representable in X. We denote by (VI.6.4.1)

$: D → I

the fibered U-site associated with the functor u+ : I → X (VI.6.1.3) defined in VI.5.5: the fiber of D over any object i of I is the category X/u+ (i) , and for every morphism f : i0 → i of I, the inverse image functor + f : X/u+ (i) → X/u+ (i0 ) is the base change + functor by u (f ). We endow D with the covanishing topology associated with $. We thus obtain the covanishing site associated with the functor u+ defined in VI.4.1, whose ← e (VI.4.10). topos of sheaves of U-sets is Ie ×Ie X By ([37] VI 12; cf. also [1] 1.1.2), the functors Ψ+ i (VI.6.1.5) for every i ∈ Ob(I) and the isomorphisms (VI.6.2.2) define a Cartesian I-functor (VI.6.4.2)

ρ+ : E → D.

VI.6. MORPHISMS WITH VALUES IN A GENERALIZED COVANISHING TOPOS

531

For every V ∈ Ob(E), the canonical morphism V 7→ σ + (π(V )) (VI.5.32.6) induces a morphism Ψ+ (V ) → u+ (π(V )), and we have (VI.6.4.3)

ρ+ (V ) = (Ψ+ (V ) → π(V )).

The functor ρ+ transforms a final object into a final object and commutes with fibered products; it is therefore left exact. On the other hand, it is continuous by virtue of VI.5.18. It therefore defines a morphism of topos ([2] IV 4.9.2) ←

e → E. e ρ : Ie ×Ie X

(VI.6.4.4)

For every sheaf F = {i 7→ Fi } on D, we have (VI.6.4.5)

ρ∗ (F ) = {i 7→ Ψi∗ (Fi )}.

It immediately follows from the definitions that the squares of the diagram (VI.6.4.6)

Ie o

p1

p2

←

e Ie ×Ie X

/X e

ρ

Ie o

 e E

σ

Ψι

 /E eι

β

and the diagram D / Ie × X e eD X Ie DD DD DD ρ Ψ DD D"  e E

←

Ψ

(VI.6.4.7)

where ΨD is the morphism (VI.4.13.4), are commutative up to canonical isomorphisms. e ρ∗ (F ) is the sheaf associated Proposition VI.6.5. For every sheaf F = {i 7→ Fi } of E, with the presheaf on D defined by {i 7→ Ψ∗i (Fi )}, where for every morphism f : i0 → i of I, the transition morphism Ψ∗i (Fi ) → f ∗ (Ψ∗i0 Fi0 )

(VI.6.5.1) is the adjoint of the composition

∼

∗f (Ψ∗i Fi ) → Ψ∗i0 (f ∗ Fi ), → Ψ∗i0 Fi0 , where the first arrow is the isomorphism (VI.6.2.2) and the second arrow is induced by the transition morphisms of F . Indeed, after extending U, if necessary, we may assume that the category E is Usmall ([2] II 3.6 and III 1.5). By ([2] I 5.1 and III 1.3), the sheaf ρ∗ (F ) is the sheaf on D associated with the presheaf G defined, for every U ∈ Ob(D), by (VI.6.5.2)

G(U ) =

lim −→

F (V ),

(V,u)∈A◦ U

where AU is the category of pairs (V, u) consisting of an object V of E and a morphism u : U → ρ+ (V ) of D. We set i = $(U ). We consider U as an object of X/u+ (i) and we denote by BU the category of pairs (W, v) consisting of an object W of Ei and a u+ (i)-morphism v : U → Ψ+ i (W ) of X. The categories AU and BU are clearly cofiltered. Every object (W, v) of BU can be naturally considered as an object of AU . We thus define a fully faithful functor ϕ : BU → AU .

532

VI. COVANISHING TOPOS AND GENERALIZATIONS

For every object (V, u) of AU , u induces a morphism f : i → π(V ) of I and a u+ (i)+ + morphism v : U → Ψ+ i (f V ) of X, so that (f (V ), v) is an object of BU . Moreover, we + have a canonical morphism ϕ((f (V ), v)) → (V, u) of AU . It then follows from ([2] I 8.1.3(c)) that the functor ϕ◦ is cofinal. Consequently, ϕ induces an isomorphism (VI.6.5.3)

G(U ) '

lim −→

Fi (W ).

(W,v)∈B ◦ U

Hence by virtue of VI.5.17, ρ∗ (F ) is the sheaf on D associated with the presheaf {i 7→ Ψ∗i (Fi )} defined by the transition morphisms (VI.6.5.1). VI.7. Ringed total topos VI.7.1.

In this section, I denotes a category equivalent to a U-small category and π: E → I

(VI.7.1.1)

a cleaved and normalized fibered U-site ([2] VI 7.2.1). For any i ∈ Ob(I), we denote by Ei the fiber of E over i, which we always consider as endowed with the topology given ei the topos of sheaves of U-sets on Ei , and by by π, by E (VI.7.1.2)

αi! : Ei → E

the canonical inclusion functor. We denote by F →I

(VI.7.1.3)

the fibered U-topos associated with π ([2] VI 7.2.6) and by F ∨ → I◦

(VI.7.1.4)

ei , and the fibered category obtained by associating with each i ∈ Ob(I) the category E e e with each morphism f : i → j of I the functor f∗ : Ei → Ej direct image by the morphism ei → E ej . of topos f : E We endow E with the total topology associated with π ([2] VI 7.4.1); it is a U-site ([2] VI 7.4.3(3)). We denote by Top(E) the topos of sheaves of U-sets on E, called the total topos associated with π. For every i ∈ Ob(I), the functor αi! being cocontinuous ([2] 7.4.2), it defines a morphism of topos ([2] 4.7) (VI.7.1.5)

ei → Top(E). αi : E

Since, moreover, αi! is continuous, the functor αi∗ admits a left adjoint that we denote also by (VI.7.1.6)

ei → Top(E), αi! : E

which extends the functor αi! : Ei → E. Recall ([2] VI 7.4.5) that for every morphism f : i → j of I, the diagram (VI.7.1.7)

/ Top(E) x< xx x f xxα  xxx j ej E ei E

αi

is not commutative in general, but there exists a 2-morphism of topos (VI.7.1.8)

αi → αj f,

VI.7. RINGED TOTAL TOPOS

533

in other words, a morphism of functors f ∗ ◦ αj∗ → αi∗ , or by adjunction a morphism of functors αj∗ → f∗ ◦ αi∗ .

(VI.7.1.9)

The latter satisfy a cocycle relation of type ([1] (1.1.2.2)). They therefore induce a functor Top(E) → HomI ◦ (I ◦ , F ∨ ) F 7→ {i 7→ αi∗ (F )},

(VI.7.1.10)

which is in fact an equivalence of categories ([2] VI 7.4.7). From now on, we will identify F with the section {i 7→ Fi } that is associated with it. Definition VI.7.2. We say that an object F of Top(E) is Cartesian if the section {i 7→ Fi } of F ∨ → I ◦ that corresponds to it through the equivalence of categories (VI.7.1.10) is Cartesian, in other words, if for every morphism f : i → j of I, the transition morphism Fj → f∗ (Fi ) is an isomorphism. If the category I is cofiltered and U-small, then the inverse limit of the fibered topos F → I exists ([2] VI 8.2.3), and the underlying category is canonically equivalent to the subcategory of Cartesian objects of Top(E) ([2] VI 8.2.9). VI.7.3. Let F = {i 7→ Fi } and G = {i 7→ Gi } be two objects of Top(E) such that G is Cartesian. Then we have a canonical isomorphism (VI.7.3.1)

∼

HomTop(E) (F, G) → lim HomEi (Fi , Gi ), ←− i∈I

where for every morphism f : i → j of I, the transition morphism df : HomEi (Fi , Gi ) → HomEj (Fj , Gj )

of the inverse system that appears in (VI.7.3.1) associates with each morphism u : Mi → Ni the morphism df (u) composition of Mj

/ f∗ (Mi )

f∗ (u)

/ f∗ (Ni )

∼

/ Nj ,

where the first arrow is the transition morphism of M and the last arrow is the inverse ∼ of the transition isomorphism Nj → f∗ (Ni ) of N . VI.7.4. In addition to the data fixed above, we fix a sheaf of rings R of Top(E), in other words, a ringed structure on the fibered topos F in the terminology of ([2] VI ei , and for every 8.6.1). This corresponds to fixing, for every i ∈ Ob(I), a ring Ri of E morphism f : i → j of I, a ring homomorphism Rj → f∗ (Ri ), these homomorphisms being ei → E ej are subject to compatibility relations (VI.7.1.10). The morphisms of topos f : E therefore morphisms of ringed topos (by Ri and Rj , respectively). For modules, we use the notation f −1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation f ∗ for the inverse image in the sense of modules. ei → Top(E) On the other hand, for every i ∈ Ob(I), the morphism of topos αi : E (VI.7.1.5) is a morphism of ringed topos (by Ri and R, respectively). Note that since Ri = αi∗ (R), there is no difference for modules between the inverse image in the sense of abelian sheaves and the inverse image in the sense of modules. Giving a structure of left (resp. right) R-module on a sheaf M = {i 7→ Mi } of Top(E) is equivalent to giving, for every i ∈ Ob(I), a structure of left (resp. right) Ri -module on Mi such that for every morphism f : i → j of I, the transition morphism Mj → f∗ (Mi ) is Rj -linear.

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VI. COVANISHING TOPOS AND GENERALIZATIONS

Lemma VI.7.5. Let M = {i 7→ Mi } be a right R-module of Top(E) and N = {i 7→ Ni } a left R-module of Top(E). Then we have a bifunctorial canonical isomorphism of abelian sheaves on E ∼

M ⊗R N → {i 7→ Mi ⊗Ri Ni }.

(VI.7.5.1)

Indeed, for every i ∈ Ob(I), we have a canonical isomorphism ([2] IV 13.4) ∼

αi∗ (M ⊗R N ) → αi∗ (M ) ⊗α∗i (R) αi∗ (N ),

(VI.7.5.2) giving the lemma. VI.7.6. (VI.7.6.1)

Let π 0 : E 0 → I be a cleaved and normalized fibered U-site and Φ+ : E 0 → E

a Cartesian I-functor. We endow E 0 with the total topology associated with π 0 , and we denote by Top(E 0 ) the topos of sheaves of U-sets on E 0 . We associate with π 0 objects analogous to those associated with π (VI.7.1), which we denote by the same letters equipped with an exponent 0 . Suppose that for every i ∈ Ob(I), the functor 0 + Φ+ i : Ei → Ei , induced by Φ on the fibers at i, is a morphism from the site Ei to the 0 site Ei , and denote by (VI.7.6.2)

ei → E ei0 Φi : E

the associated morphism of topos. The morphisms Φi define a Cartesian morphism of fibered topos ([2] VI 7.1.5 and 7.1.7) (VI.7.6.3)

F → F 0.

By virtue of ([2] 7.4.10), Φ+ is a morphism from the total site E to the total site E 0 . It therefore defines a morphism of topos (VI.7.6.4)

Ψ : Top(E) → Top(E 0 )

such that for every sheaf F = {i 7→ Fi } on E, we have (VI.7.6.5)

Ψ∗ (F ) = {i 7→ Φi∗ (Fi )}.

Let R0 = {i 7→ Ri0 } be a ring of Top(E 0 ) and h : R0 → Ψ∗ (R) a ring homomorphism, so that Ψ : Top(E) → Top(E 0 ) is a morphism of ringed topos (by R and R0 , respectively). Giving h is equivalent to giving, for every i ∈ Ob(I), a ring homomorphism hi : Ri0 → ei → E e0 Φi∗ (Ri ) satisfying compatibility relations. In particular, for every i ∈ Ob(I), Φi : E i 0 n is a morphism of ringed topos (by Ri and Ri , respectively). We denote by R Ψ∗ and Rn Φi∗ (n ∈ N) the right derived functors of the functors (cf. VI.2.4) Ψ∗ : Mod(R, Top(E)) → Mod(R0 , Top(E 0 )), ei ) → Mod(R0 , E e 0 ). Φi∗ : Mod(Ri , E i i

Proposition VI.7.7. Under the assumptions of VI.7.6, let, moreover M = {i 7→ Mi } be an R-module of Top(E) and n an integer ≥ 0. We then have a functorial and canonical R0 -isomorphism (VI.7.7.1)

∼

Rn Ψ∗ (M ) → {i 7→ Rn Φi∗ (Mi )}.

Indeed, for every c ∈ Ob(I), the functor (VI.7.7.2)

e → Mod(Rc , E ec ), Mod(R, E)

M = {i 7→ Mi } 7→ Mc = αc∗ (M )

is additive and exact. On the other hand, for every injective R-module M = {i 7→ Mi }, Mc is flabby by virtue of ([2] VI 8.7.2). The proposition therefore follows from (VI.7.6.5).

VI.7. RINGED TOTAL TOPOS

535

Let us describe explicitly the transition morphisms of the sheaf {i 7→ Rn Φi∗ (Mi )}. The morphism of ringed fibered topos (F , R) → (F 0 , R0 ) induces for every morphism f : i → j of I a diagram of morphisms of ringed topos, commutative up to canonical isomorphism ([1] 1.2.3) (VI.7.7.3)

ei , Ri ) (E

Φi

fE 0

fE

 ej , Rj ) (E

/ (E e 0 , R0 ) i i

Φj

 / (E e 0 , R0 ) j j

where the vertical arrows, usually denoted by f , have been equipped with an index E or E 0 to distinguish them. We leave it to the reader to verify that the transition morphism associated with f is the composition (VI.7.7.4)

Rn Φj∗ (Mj ) → Rn Φj∗ (fE∗ Mi ) → Rn (Φj fE )∗ (Mi ) ' Rn (fE 0 Φi )∗ (Mi ) → fE 0 ∗ (Rn Φi∗ (Mi )),

where the first arrow comes from the transition morphism of M , the second arrow and the last arrow are induced by the Cartan–Leray spectral sequence ([2] V 5.4), and the third isomorphism comes from (VI.7.7.3). e we have Proposition VI.7.8 ([2] VI 7.4.15). For every R-module M = {i 7→ Mi } of E, a functorial and canonical spectral sequence (VI.7.8.1)

p q e p+q Ep,q (Top(E), M ). 2 = R lim H (Ei , Mi ) ⇒ H ←−

i∈I ◦

Consider the constant fibered topos $ : Ens × I → I with fiber the punctual topos Ens (VI.2.1). By (VI.7.1.10), the topos of sheaves of U-sets on the total site Ens × I associated with $ is equivalent to the category Ib of presheaves of U-sets on I. We have a Cartesian morphism of fibered topos (VI.7.8.2)

Φ : F → Ens × I

ei → Ens ([2] IV whose fiber over i ∈ Ob(I) is the canonical morphism of topos Φi : E 2.2). By ([2] VI 7.4.10), Φ defines a morphism of topos (VI.7.8.3)

Ψ : Top(E) → Ib

such that for every sheaf F = {i 7→ Fi } on E, we have (VI.7.8.4)

ei , Fi )}. Ψ∗ (F ) = {i 7→ Γ(E

We consider Ψ as a morphism of ringed topos (by R and Ψ∗ (R), respectively). By VI.7.7, for every R-module M = {i 7→ Mi } of Top(E) and every integer q ≥ 0, we have a functorial and canonical isomorphism (VI.7.8.5)

∼

ei , Mi )}. Rq Ψ∗ (M ) → {i 7→ Hq (E

On the other hand, for every abelian group N of Ib and every integer p ≥ 0, we have a functorial and canonical isomorphism ([2] V 2.3.1) (VI.7.8.6)

b N ) = Rp lim N (i). Hp (I, ←−

i∈I ◦

We then take for (VI.7.8.1) the Cartan–Leray spectral sequence associated with Ψ ([2] V 5.3).

536

VI. COVANISHING TOPOS AND GENERALIZATIONS

Corollary VI.7.9. Suppose that I admits a final object ι. Then for every R-module e and every integer n ≥ 0, we have a functorial and canonical M = {i 7→ Mi } of E isomorphism ∼ eι , Mι ). (VI.7.9.1) Hn (Top(E), M ) → Hn (E Indeed, with the notation of the proof of VI.7.8, the functor Ib → Ens, N 7→ b Γ(I, N ) = N (ι) is exact. Consequently, Rp lim = 0 for every p ≥ 1. The proposition then ←−

i∈I ◦

follows from VI.7.8. Corollary VI.7.10. Suppose that I is the filtered category defined by the ordered set of e and every integer natural numbers N. Then for every R-module M = {i 7→ Mi } of E n ≥ 0, we have a functorial and canonical exact sequence ei , Mi ) → Hn (Top(E), M ) → lim Hn (E ei , Mi ) → 0, (VI.7.10.1) 0 → R1 lim Hn−1 (E ←−

←−

i∈N◦

i∈N◦

en , Mn ) = 0 for every n ∈ N. where we set H−1 (E

This follows from VI.7.8 and from the fact that Rp lim = 0 for every p ≥ 2 ([47] 1.4 ←−

n∈N◦

and [63] 2.1). Definition VI.7.11. We say that an R-module M = {i 7→ Mi } of Top(E) is coCartesian if for every morphism f : i → j of I, the transition morphism f ∗ (Mj ) → Mi of M is an isomorphism, where f ∗ denotes the inverse image in the sense of modules (VI.7.4). e the topos of sheaves of U-sets on X, A a Example VI.7.12. Let X be a U-site, X e commutative ring of X, and J an ideal of A. We denote also by N the filtered category defined by the ordered set of natural numbers N. The total topos associated with the constant fibered U-site X × N → N with fiber X is canonically equivalent to the category e (VI.7.1.10), which we also denote by X e N◦ . We endow it with the ring Hom(N◦ , X) ˘ e N◦ is co-Cartesian if and A˘ = {n 7→ A/J n+1 }. An A-module M = {n 7→ Mn } of X only if for every integer n ≥ 0, the morphism Mn+1 ⊗A (A/J n+1 ) → Mn deduced from the transition morphism Mn+1 → Mn , is an isomorphism, in other words, if the inverse system (Mn )n∈N is J-adic in the sense of ([38] V 3.1.1). VI.7.13. Let M = {i 7→ Mi } and N = {i 7→ Ni } be two R-modules of Top(E) such that M is co-Cartesian. We then have a canonical isomorphism (VI.7.13.1)

∼

HomR (M, N ) → lim HomRi (Mi , Ni ), ←−

i∈I ◦

where for every morphism f : i → j of I, the transition morphism df : HomRj (Mj , Nj ) → HomRi (Mi , Ni )

of the inverse system that appears in (VI.7.13.1) associates with each Rj -morphism u : Mj → Nj the morphism df (u) composed of Mi

∼

/ f ∗ (Mj )

f ∗ (u)

/ f ∗ (Nj )

/ Ni , ∼

where the first arrow is the inverse of the transition isomorphism f ∗ (Mj ) → Mi of M and the last arrow is the transition morphism of N . In particular, if I admits a final object ι, we have a canonical isomorphism (VI.7.13.2)

∼

HomR (M, N ) → HomRι (Mι , Nι ).

VI.8. RINGED COVANISHING TOPOS

537

VI.8. Ringed covanishing topos VI.8.1. In this section, I denotes a U-site and π : E → I a cleaved and normalized fibered U-site over the category underlying I, such that the following conditions are satisfied: (i) Finite inverse limits are representable in I. (ii) For every i ∈ Ob(I), finite inverse limits are representable in Ei . (iii) For every morphism f : i → j of I, the inverse image functor f + : Ej → Ei is continuous and left exact. e the topos We endow E with the covanishing topology defined by π, and we denote by E of sheaves of U-sets on E. We fix a final object ι of I and a final object e of Eι , and we take again the notation introduced in VI.5, in particular, that introduced in VI.5.32. e By VI.5.10, it amounts to fixing, for every i ∈ Ob(I), a We also fix a ring R of E. e ring Ri of Ei , and for every morphism f : i → j of I, a ring homomorphism Rj → f∗ (Ri ), these homomorphisms being subject to compatibility (VI.5.2.1) and gluing relations ei → E ej are therefore morphisms of ringed (VI.5.10.1). The morphisms of topos f : E topos (by Ri and Rj , respectively). For modules, we use the notation f −1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation f ∗ for the inverse image in the sense of modules. e is equivalent to Giving a structure of R-module on a sheaf M = {i 7→ Mi } of E giving, for every i ∈ Ob(I), a structure of Ri -module on Mi such that for every morphism f : i → j of I, the transition morphism Mj → f∗ (Mi ) is Rj -linear. For every c ∈ Ob(I), the functor e → Mod(Rc , E ec ), {i 7→ Mi } 7→ Mc (VI.8.1.1) Mod(R, E) is clearly additive.

e F its Lemma VI.8.2. Let u : {i 7→ Mi } → {i 7→ Ni } be a morphism of R-modules of E, kernel, and G its cokernel. For every i ∈ Ob(I), denote by ui : Mi → Ni the Ri -morphism induced by u, and by Fi (resp. Gi ) its kernel (resp. cokernel). Then {i 7→ Fi } is an Re {i 7→ Gi } is an R-module of E, b and we have canonical R-isomorphisms module of E, (VI.8.2.1) (VI.8.2.2)

F

∼

→

{i 7→ Fi },

∼

{i 7→ Gi }a ,

G →

where {i 7→ Gi }a is the sheaf on E associated with the presheaf {i 7→ Gi }. In particular, the functor (VI.8.1.1) is left exact. b the category of R-modules of E, b by jR : Mod(R, E) e → Indeed, denote by Mod(R, E) b b e Mod(R, E) the canonical inclusion functor, and by aR : Mod(R, E) → Mod(R, E) the “associated sheaf” functor ([2] II 6.4). Then aR is left exact and commutes with direct limits, and jR commutes with inverse limits ([2] II 4.1). We deduce from this canonical R-isomorphisms (VI.8.2.3) (VI.8.2.4)

∼

jR (F ) → ker(jR (u)), ∼

G → aR (coker(jR (u))),

and likewise for the morphisms ui for every i ∈ Ob(I). In view of (VI.5.2.1) and ([2] I 3.1), we have ker(jR (u)) = {i 7→ Fi }. On the other hand, it follows from VI.5.17 that b and that we have a canonical R-isomorphism {i 7→ Gi } is an R-module of E (VI.8.2.5) giving the lemma.

∼

aR ({i 7→ Gi }) → aR (coker(jR (u))),

538

VI. COVANISHING TOPOS AND GENERALIZATIONS

VI.8.3. (VI.8.3.1)

For every c ∈ Ob(I), we denote by e/σ∗ (c) → E e γc : E

e at σ ∗ (c), and by the localization morphism of E (VI.8.3.2)

e/σ∗ (c) → E ec βc : E

the morphism defined in (VI.5.39.2). By (VI.5.39.4), the functor (VI.8.1.1) identifies with the composed functor βc∗ ◦ γc∗ . Consequently, βc is a morphism of ringed topos (by γc∗ (R) and Rc , respectively). For modules, we use the notation βc−1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation βc∗ for the inverse image in the sense of modules. We denote by Rn βc∗ (n ∈ N) the right derived functors of (VI.8.3.3)

e/σ∗ (c) ) → Mod(Rc , E ec ). βc∗ : Mod(γc∗ (R), E

The nth right derived functor of (VI.8.1.1) then identifies with the functor (Rn βc∗ ) ◦ γc∗ by virtue of ([2] V 4.11). Recall that β = βι , which we therefore consider as a morphism of ringed topos (by R and Rι , respectively). For modules, we use the notation β −1 to denote the inverse image in the sense of abelian sheaves, and we keep the notation β ∗ for the inverse image in the sense of modules. VI.8.4. Let V be a universe such that U ⊂ V and I ∈ V. The category E endowed with the total topology associated with the fibered site π is a V-site; but it is not in eV the topos of sheaves of V-sets on the covanishing site general a U-site. We denote by E E, by e→E eV : E

(VI.8.4.1)

the canonical inclusion functor, and by TopV (E) the topos of sheaves of V-sets on the total site E (VI.7.1). We consider the canonical morphism (VI.5.16.1) (VI.8.4.2)

eV → TopV (E) δ: E

as a morphism of ringed topos (by R and δ∗ (R), respectively). Since δ is an embedding, the adjunction morphism δ ∗ δ∗ → id is an isomorphism. Hence there is no difference for modules between the inverse image in the sense of abelian sheaves and the inverse image in the sense of modules. We denote by Rn δ∗ (n ∈ N) the right derived functors of (VI.8.4.3)

eV ) → Mod(δ∗ (R), TopV (E)). δ∗ : Mod(R, E

Recall that the functor  is exact and fully faithful on the categories of modules and transforms injective modules into injective modules ([2] V 1.9). Consequently, the nth right derived functor of δ∗ ◦  is canonically isomorphic to (Rn δ∗ ) ◦ . Proposition VI.8.5. Under the assumptions of VI.8.4, let moreover M be an R-module e of E. (i) For every integer n ≥ 0, we have a functorial canonical isomorphism of δ∗ (R)modules (VI.8.5.1)

∼

Rn δ∗ (M ) → {i 7→ Rn βi∗ (γi∗ (M ))}.

(ii) If M is an injective R-module, then δ∗ (M ) is an injective δ∗ (R)-module. (iii) For every integer n > 0, δ ∗ (Rn δ∗ (M )) = 0.

VI.8. RINGED COVANISHING TOPOS

539

ei,V the topos of sheaves of V-sets on Ei , by (i) For every i ∈ Ob(I), we denote by E ei → E ei,V the canonical inclusion functor, and by i : E ei,V → TopV (E) αi : E

(VI.8.5.2)

the canonical morphism (VI.7.1.5). The latter is a morphism of ringed topos (by Ri and δ∗ (R), respectively); recall (VI.7.4) that there is no difference for modules between the inverse image in the sense of abelian sheaves and the inverse image in the sense of modules. We therefore have a functorial canonical isomorphism ∼

αi∗ (Rn δ∗ (M )) → Rn (αi∗ ◦ δ∗ )(M ).

(VI.8.5.3)

On the other hand, by (VI.5.16.2) and (VI.5.39.4), we have a canonical isomorphism of functors ∼

αi∗ ◦ δ∗ ◦  → i ◦ βi∗ ◦ γi∗ .

(VI.8.5.4)

The statement follows in view of ([2] V 1.9 and 4.11(1)). We leave it to the reader to describe explicitly the transition morphisms of the sheaf {i 7→ Rn βi∗ (γi∗ (M ))} of TopV (E). (ii) Since the module (M ) is injective, we may restrict to the case where U = V. The statement then follows from ([2] V 0.2) because the functor δ∗ admits an exact left adjoint δ ∗ = δ −1 (VI.8.4). (iii) Since the functor δ ∗ = δ −1 is exact, we have a functorial canonical isomorphism ∼

δ ∗ (Rn δ∗ (M )) → Rn (δ ∗ ◦ δ∗ )(M ).

(VI.8.5.5)

On the other hand, since δ is an embedding, the adjunction morphism δ ∗ δ∗ → id is an isomorphism, giving the desired statement. e the Cartan–Leray spectral sequence asRemark VI.8.6. For every R-module M of E, sociated with δ that computes the cohomology of M ([2] V 5.3) reduces to that associated with β eι , Rq β∗ (M )) ⇒ Hp+q (E, e M ), (VI.8.6.1) Ep,q = Hp (E 2

by virtue of VI.7.9 and VI.8.5.

VI.8.7. Let π 0 : E 0 → I be a cleaved and normalized fibered U-site satisfying conditions (ii) and (iii) of VI.8.1 and (VI.8.7.1)

Φ+ : E 0 → E

a Cartesian I-functor. We endow E 0 with the covanishing topology defined by π 0 , and e 0 the topos of sheaves of U-sets on E 0 . We associate with π 0 objects we denote by E analogous to those associated with π, and we denote them by the same letters equipped 0 with an exponent 0 . Suppose that for every i ∈ Ob(I), the functor Φ+ i : Ei → Ei induced + by Φ on the fibers at i is continuous and left exact, and denote by ei → E ei0 (VI.8.7.2) Φi : E the associated morphism of topos. By virtue of VI.5.36, Φ+ is continuous and left exact. It therefore defines a morphism of topos e→E e0 (VI.8.7.3) Φ: E such that for every sheaf F = {i 7→ Fi } on E, we have

(VI.8.7.4)

Φ∗ (F ) = {i 7→ Φi∗ (Fi )}.

e 0 and h : R0 → Φ∗ (R) a ring homomorphism, so Let R0 = {i 7→ Ri0 } be a ring of E 0 e e that Φ : E → E is a morphism of ringed topos (by R and R0 , respectively). Giving h

540

VI. COVANISHING TOPOS AND GENERALIZATIONS

amounts to giving, for every i ∈ Ob(I), a ring homomorphism hi : Ri0 → Φi∗ (Ri ) satisfyei → E e0 ing compatibility relations (VI.5.18.3). In particular, for every i ∈ Ob(I), Φi : E i 0 n is a morphism of ringed topos (by Ri and Ri , respectively). We denote by R Φ∗ and Rn Φi∗ (n ∈ N) the right derived functors of e → Mod(R0 , E e 0 ), Φ∗ : Mod(R, E) ei ) → Mod(Ri0 , E ei0 ). Φi∗ : Mod(Ri , E

Proposition VI.8.8. Under the assumptions of VI.8.7, for every R-module M , we have a functorial and canonical spectral sequence (VI.8.8.1)

p q ∗ a p+q Ep,q Φ∗ (M ), 2 = {i 7→ R Φi∗ (R βi∗ (γi M ))} ⇒ R

where the source denotes the sheaf associated with the presheaf {i 7→ Rp Φi∗ (Rq βi∗ (γi∗ M ))} on E 0 . After extending U, if necessary, we may restrict to the case where I ∈ U ([2] II 3.6 and V 1.9). We denote by Top(E) and Top(E 0 ) the topos of sheaves of U-sets on the total sites E and E 0 , respectively (VI.7.1). The morphisms Φi define a Cartesian morphism of fibered topos F → F 0 . On the other hand, Φ+ is a morphism from the total site E to the total site E 0 ([2] 7.4.10). It therefore defines a morphism of topos (VI.8.8.2)

Ψ : Top(E) → Top(E 0 )

such that for every sheaf F = {i 7→ Fi } on E, we have (VI.8.8.3)

Ψ∗ (F ) = {i 7→ Φi∗ (Fi )}.

In particular, the diagram of morphisms of topos (VI.8.8.4)

e E

δ

/ Top(E)

δ0

 / Top(E 0 )

Ψ

Φ

 e0 E

is commutative up to canonical isomorphism. The homomorphism h : R0 → Φ∗ (R) induces a homomorphism h0 : δ∗0 (R0 ) → Ψ∗ (δ∗ (R)) that makes Ψ : Top(E) → Top(E 0 ) into a morphism of ringed topos (by δ∗ (R) and δ∗0 (R0 ), respectively). Since δ 0 is an embedding, the adjunction morphism δ 0∗ δ∗0 Φ∗ → Φ∗ is an isomorphism. ∼ In view of (VI.8.8.4), we deduce from this an isomorphism δ 0∗ Ψ∗ δ∗ → Φ∗ . Since the functor δ 0∗ = δ 0−1 is exact (VI.8.4), the Cartan–Leray spectral sequence ([2] V 5.4) then induces a functorial spectral sequence in M (VI.8.8.5)

0∗ p q p+q Ep,q Φ∗ (M ). 2 = δ (R Ψ∗ (R δ∗ (M ))) ⇒ R

By virtue of VI.7.7 and VI.8.5(i), we have a canonical isomorphism of δ∗0 (R0 )-modules (VI.8.8.6)

∼

Rp Ψ∗ (Rq δ∗ (M )) → {i 7→ Rp Φi∗ (Rq βi∗ (γi∗ M ))}.

On the other hand, for every object G of Top(E 0 ), δ 0∗ (G) is canonically isomorphic to the sheaf on the covanishing site E 0 associated with the presheaf G (cf. VI.5.16), giving the proposition. b (VI.5.2), M = {i 7→ Mi } a right Lemma VI.8.9. Let A = {i 7→ Ai } be a ring of E b N = {i 7→ Ni } a left A-module of E, b and Aa , M a , and N a the sheaves A-module of E, on E associated with A, M , and N , respectively. For every i ∈ Ob(I), we denote by bi . Then {i 7→ Mi ⊗A Ni } Mi ⊗Ai Ni the abelian group tensor product of Mi and Ni in E i

VI.8. RINGED COVANISHING TOPOS

541

is a presheaf on E, and denoting by {i 7→ Mi ⊗Ai Ni }a the associated sheaf on E, we have a bifunctorial canonical isomorphism of abelian sheaves on E (VI.8.9.1)

∼

M a ⊗Aa N a → {i 7→ Mi ⊗Ai Ni }a .

This follows from ([2] IV 12.10). e and N = {i 7→ Ni } Lemma VI.8.10. Let M = {i 7→ Mi } be a right R-module of E e Then we have a bifunctorial canonical isomorphism of abelian a left R-module of E. sheaves on E (VI.8.10.1)

∼

M ⊗R N → {i 7→ Mi ⊗Ri Ni }a ,

where for every i ∈ Ob(I), Mi ⊗Ri Ni is the abelian group tensor product of Mi and Ni in ei , and {i 7→ Mi ⊗R Ni }a is the sheaf on E associated with the presheaf {i 7→ Mi ⊗R Ni }. E i i This follows from VI.5.17 and VI.8.9. e is co-Cartesian Definition VI.8.11. We say that an R-module M = {i 7→ Mi } of E if the δ∗ (R)-module δ∗ (M ) is co-Cartesian in the sense of (VI.7.11), in other words, if for every morphism f : i → j of I, the transition morphism f ∗ (Mj ) → Mi of M is an isomorphism, where f ∗ denotes the inverse image in the sense of modules (VI.8.1). e Proposition VI.8.12. Let M = {i 7→ Mi } and N = {i 7→ Ni } be two R-modules of E such that M is co-Cartesian. Then we have a canonical R-isomorphism (VI.8.12.1)

∼

H omR (M, N ) → {i 7→ H omRi (Mi , Ni )}.

Indeed, by (VI.7.13.2), we have a canonical isomorphism (VI.8.12.2)

∼

HomR (M, N ) → HomRι (Mι , Nι ).

Let V be an object of E, c = π(V ), and (VI.8.12.3)

$ : E/V → I/c

the functor induced by π. For every morphism f : i → c of I, the fiber of $ over the object (i, f ) of I/c is canonically equivalent to the category (Ei )/f + (V ) . Endowing I/c with the topology induced by that of I, and each fiber (Ei )/f + (V ) with the topology induced by that of Ei , $ becomes a fibered site satisfying the conditions of VI.5.1. By virtue of VI.5.38, the covanishing topology of E/V associated with $ is induced by that of E through the canonical functor γV : E/V → E. In particular, the topos of sheaves e/ε(V ) . Moreover, we have a functorial of U-sets on E/V is canonically equivalent to E canonical isomorphic (VI.8.12.4)

∼

M |ε(V ) → {(i, f ) 7→ Mi |f ∗ (V )},

and likewise for N . Consequently, the R|ε(V )-module M |ε(V ) is co-Cartesian. Denoting ec the canonical functor, the isomorphisms (VI.8.12.2) and (VI.8.12.4) by εc : Ec → E induce an isomorphism (VI.8.12.5)

∼

HomR|ε(V ) (M |ε(V ), N |ε(V )) → HomRc |εc (V ) (Mc |εc (V ), Nc |εc (V )),

that is clearly functorial in V ∈ Ob(Ec ), giving the proposition ([2] IV 12.1).

Remark VI.8.13. Under the assumptions of VI.8.12, it immediately follows from VI.7.13 that for every morphism f : i → j of I, the transition morphism of the R-module H omR (M, N ) is the composition f ∗ (H omRj (Mj , Nj )) → H omRi (f ∗ (Mj ), f ∗ (Nj )) ∼

→ H omRi (Mi , f ∗ (Nj )) → H omRi (Mi , Ni ),

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VI. COVANISHING TOPOS AND GENERALIZATIONS

where f ∗ denotes the inverse image in the sense of modules, the first arrow is the canonical ∼ morphism, the second arrow is induced by the transition isomorphism f ∗ (Mj ) → Mi of ∗ M , and the last arrow is induced by the transition morphism f (Nj ) → Ni of N . eι Proposition VI.8.14. For every locally projective Rι -module of finite type Mι of E (that is, Mι is locally a direct summand of a free Rι -module of finite type), we have a functorial canonical isomorphism (VI.8.14.1)

∼

β ∗ (Mι ) → {i 7→ fi∗ (Mι )},

where β ∗ denotes the inverse image in the sense of modules (VI.8.3) and for every i ∈ Ob(I), fi : i → ι is the canonical morphism and fi∗ denotes the inverse image in the sense of modules (VI.8.1). In particular, β ∗ (Mι ) is a co-Cartesian R-module. By virtue of VI.5.34(i), VI.8.9, and VI.5.17, we have a canonical isomorphism (VI.8.14.2)

∼

β ∗ (Mι ) → {i 7→ fi∗ (Mι )}a ,

where the right-hand side is the sheaf on E associated with the presheaf {i 7→ fi∗ (Mι )}. eι It remains to show that M = {i 7→ fi∗ (Mι )} is a sheaf on E. Denote by ει : Eι → E the canonical functor. There exists a vertical covering (Vn → e)n∈Σ of E (that is, a covering family of Eι ) such that for every n ∈ Σ, Mι |ει (Vn ) is a direct summand of a free (Rι |ει (Vn ))-module of finite type. It suffices to show that for every n ∈ Σ, the restriction of M to the category E/Vn is a sheaf for the topology induced by that of E ([35] II 3.4.4). In view of VI.5.38, we can restrict to the case where Vn = e, in which case there exist an ∼ eι , and an Rι -isomorphism Rιd → Mι ⊕ Nι . We then integer d ≥ 0, an Rι -module Nι of E b have an isomorphism of E (VI.8.14.3)

∼

Rd → M ⊕ {i 7→ fi∗ (Nι )},

which implies that M and {i 7→ fi∗ (Nι )} are sheaves on E. VI.9. Finite étale site and topos of a scheme ´ /X the étale site of X, that is, the VI.9.1. For every scheme X, we denote by Et full subcategory of Sch/X (VI.2.1) made up of the étale schemes on X, endowed with the étale topology; it is a U-site. We denote by X´et the étale topos of X, that is, the topos of ´ /X . We denote by Et ´ coh/X (resp. Et ´ scoh/X ) the full subcategory sheaves of U-sets on Et ´ /X made up of the étale schemes of finite presentation over X (resp. separated étale of Et schemes of finite presentation over X), endowed with the topology induced by that of ´ /X ; these are U-small sites. If X is quasi-separated, the restriction functor from X´et to Et ´ coh/X (resp. Et ´ scoh/X ) is an equivalence of categories the topos of sheaves of U-sets on Et ([2] VII 3.1 and 3.2). ´ f/X VI.9.2. For every scheme X, we call finite étale site of X and denote by Et ´ the full subcategory of Et/X made up of the finite étale covers of X, endowed with the ´ /X ; it is a U-small site. We call finite étale topos of X and topology induced by that of Et ´ f/X . The étale topology on Et ´ f/X denote by Xf´et the topos of sheaves of U-sets on Et ´ f/X → Et ´ /X is clearly coarser than the canonical topology. The canonical injection Et induces a morphism of topos (VI.9.2.1)

ρX : X´et → Xf´et .

VI.9. FINITE ÉTALE SITE AND TOPOS OF A SCHEME

VI.9.3. (VI.9.3.1)

543

Let f : Y → X be a morphism of schemes. The inverse image functor f • : Sch/X → Sch/Y ,

X 0 7→ X 0 ×X Y

induces two morphisms of topos that we denote (to distinguish them from each other) by (VI.9.3.2) (VI.9.3.3)

→ X´et , → Xf´et .

f´et : Y´et ff´et : Yf´et

We will leave the indices out of the notation f´et and ff´et when there is no risk of ambiguity. One immediately verifies that the diagram of morphisms of topos (VI.9.3.4)

Y´et ρY



Yf´et

f´et

ff´et

/ X´et 

ρX

/ Xf´et

is commutative up to canonical isomorphism: (VI.9.3.5) VI.9.4. (VI.9.4.1)

∼

ρX f´et → ff´et ρY . Let f : Y → X be a finite étale cover and ´ f/Y → Et ´ f/X τf : Et

the functor defined by composition on the left with f . Then τf induces an equivalence ∼ ´ ´ f/Y is induced by that on ´ f/Y → (Etf/X )/(Y,f ) . The étale topology on Et of categories Et ´ Etf/X through the functor τf . By virtue of ([2] III 5.2), τf is continuous and cocontinuous. It therefore induces a sequence of three adjoint functors: (VI.9.4.2)

τf ! : Yf´et → Xf´et ,

τf∗ : Xf´et → Yf´et ,

τf ∗ : Yf´et → Xf´et ,

in the sense that for any two consecutive functors in the sequence, the one on the right is the right adjoint of the other. By ([2] III 5.4), the functor τf ! factors through an ∼ equivalence of categories (Yf´et ) → (Xf´et )/Y . The pair of functors (τf∗ , τf ∗ ) defines a morphism of topos Yf´et → Xf´et , called localization morphism of Xf´et at Y . On the other hand, τf∗ is left adjoint to the functor ff´et∗ direct image by the morphism of topos ff´et . The latter therefore identifies canonically with the localization morphism of Xf´et at Y . We refer to ([2] VII 1.6) for the analogous statements for the étale sites and topos. VI.9.5. We denote by R the category of finite étale covers (that is, the full subcategory of the category of morphisms of Sch made up of the finite étale covers) and by (VI.9.5.1)

R → Sch

the “target functor,” which makes R into a cleaved and normalized fibered category over ´ f/X , and for Sch: the fiber over any scheme X is canonically equivalent to the category Et + ´ ´ f/Y every morphism of schemes f : Y → X, the inverse image functor f : Etf/X → Et is none other than the base change functor by f . Endowing each fiber with the étale topology, R/Sch becomes a fibered U-site ([2] VI 7.2.1). We denote by (VI.9.5.2)

G → Sch

the fibered U-topos associated with R/Sch ([2] VI 7.2.6): the fiber of G over any scheme X is the topos Xf´et , and for every morphism of schemes f : Y → X, the inverse image

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VI. COVANISHING TOPOS AND GENERALIZATIONS

functor f ∗ : Xf´et → Yf´et is the inverse image functor under the morphism of topos ff´et (VI.9.3.3). We denote by (VI.9.5.3)

G ∨ → Sch◦

the fibered category obtained by associating with each scheme X the category GX = Xf´et , and with each morphism of schemes f : Y → X, the direct image functor ff´et∗ : Yf´et → Xf´et by the morphism of topos ff´et . Lemma VI.9.6. Let X be a scheme and ρX : X´et → Xf´et the canonical morphism (VI.9.2.1). Then, the family of fiber functors of Xf´et associated with the points ρX (x), when x goes through the family of geometric points of X, is conservative ([2] IV 6.4.0). Indeed, for any geometric point x of X, to give a neighborhood of the point ρX (x) of ´ f/X is equivalent to giving an x-pointed étale cover of X ([2] IV 6.8.2 Xf´et in the site Et and VIII 3.9). These objects form naturally a cofiltered category, that we denote by Px . For any object F of Xf´et , denoting the stalk of F at ρX (x) by Fx , we have a canonical functorial isomorphism (VI.9.6.1)

∼

Fx →

lim −→

F (U ).

(U,ξ)∈P ◦ x

Let u : F → G be a morphism of Xf´et such that for any geometric point x of X, the associated morphism ux : Fx → Gx is a monomorphism. Let’s prove that u is a ´ f/X ) and a, b ∈ F (X 0 ) such monomorphism. We need to prove that for any X 0 ∈ Ob(Et that u(a) = u(b), we have a = b. By VI.9.4 and (VI.9.3.4), we can assume X 0 = X. For any geometric point x of X, we have ax = bx because ux is a monomorphism. By (VI.9.6.1), there exists an object (Ux , ζx ) of Px such that a and b have the same image in F (Ux ). The family of morphisms (Ux → X)x being clearly a covering, we deduce that a = b. Therefore, u is a monomorphism. Assume furthermore that for any geometric point x of X, the morphism ux is an epimorphism. Let’s prove that u is an epimorphism. It is enough to show that for any ´ f/X ) and b ∈ G(X 0 ), there exits a ∈ F (X 0 ) such that b = u(a). We can X 0 ∈ Ob(Et assume again that X 0 = X. By (VI.9.6.1), for any point x of X, there exists an object (Ux , ξx ) of Px and a section ax ∈ F (Ux ) whose image by u in G(Ux ) is the restriction of b. Since u is a monomorphism, the sections ax coincide over Ux ×X Ux0 , for any geometric points x and x0 of X. They are therefore induced by a section a ∈ F (X), and we have u(a) = b as the restrictions to the Ux ’s coincide. Definition VI.9.7. Let X be a scheme. We say that X is locally connected if the underlying topological space is locally connected and that X is étale-locally connected if for every étale morphism X 0 → X, every connected component of X 0 is an open subset of X 0 . We can make the following remarks. VI.9.7.1. A scheme X is étale-locally connected if and only if every étale X-scheme is locally connected ([12] I § 11.6 Proposition 11). VI.9.7.2. The set of connected components of a topological space X is locally finite if and only if every connected component of X is open in X. This is the case if X is locally connected. VI.9.7.3. Let X be a scheme whose set of irreducible components is locally finite (for example a scheme whose underlying topological space is locally noetherian). Then: (i) For every étale morphism f : X 0 → X, the set of irreducible components of X 0 is locally finite. Indeed, since the question is local on X and X 0 , we can restrict

VI.9. FINITE ÉTALE SITE AND TOPOS OF A SCHEME

545

to the case where X and X 0 are affine, in which case f is quasi-compact and therefore quasi-finite. The assertion then follows from ([42] 2.3.6(iii)). (ii) It follows from (i) and ([39] 0.2.1.5(i)) that X is étale-locally connected. VI.9.8.

Let X be a connected scheme and x a geometric point of X. We denote

by (VI.9.8.1)

´ f/X → Ens ωx : Et

the fiber functor at x, which with each finite étale cover Y of X associates the set of geometric points of Y over x, by π1 (X, x) the fundamental group of X at x, and by Bπ1 (X,x) the classifying topos of the profinite group π1 (X, x), that is, the category of discrete U-sets endowed with a continuous left action of π1 (X, x) ([2] IV 2.7). Then ωx induces a fully faithful functor (VI.9.8.2)

´ µ+ x : Etf/X → Bπ1 (X,x)

with essential image the full subcategory C (π1 (X, x)) of Bπ1 (X,x) made up of the finite ´ f/X is covering sets ([37] V § 4 and § 7). On the other hand, a family (Yλ → Y )λ∈Λ of Et for the étale topology if and only if its image in Bπ1 (X,x) is surjective, or, equivalently, covering for the canonical topology of Bπ1 (X,x) . Consequently, the étale topology on ´ f/X is induced by the canonical topology of Bπ (X,x) ([2] III 3.3). Since the objects of Et 1 C (π1 (X, x)) form a generating family of Bπ1 (X,x) , the functor (VI.9.8.3)

µx : Bπ1 (X,x) → Xf´et

which with each object G of Bπ1 (X,x) (considered as a representable sheaf) associates its ´ f/X , is an equivalence of categories by virtue of ([2] IV 1.2.1). restriction to Et ´ f/X that prorepLet (Xi )i∈I be an inverse system on a filtered ordered set I in Et resents ωx , normalized by the fact that the transition morphisms Xi → Xj (i ≥ j) are ´ f/X is equivalent to a suitable epimorphisms and that every epimorphism Xi → X 0 of Et epimorphism Xi → Xj (j ≤ i). Such a pro-object is essentially unique. It is called the ´ f/X normalized universal cover of X at x or the normalized fundamental pro-object of Et at x. Note that the set I is U-small. Consider the functor (VI.9.8.4)

νx : Xf´et → Bπ1 (X,x) ,

F 7→ lim F (Xi ). −→ i∈I

´ f/X is canonically isomorphic to the functor µ+ , By definition, the restriction of νx to Et x and we have a canonical isomorphism of functors (VI.9.8.5)

∼

νx ◦ µx → id.

Since µx is an equivalence of categories, νx is an equivalence of categories, quasi-inverse to µx . We call it the fiber functor of Xf´et at x. The functor νx induces an equivalence between the category of sheaves of groups (resp. abelian sheaves) of Xf´et and the category of discrete π1 (X, x)-groups (resp. of discrete π1 (X, x)-Z-modules) (cf. II.3.1). In particular, for every abelian sheaf (resp. sheaf of groups) F of Xf´et and every integer n ≥ 0 (resp. n ∈ {0, 1}), we have a functorial canonical isomorphism (VI.9.8.6)

Hn (Xf´et , F ) ' Hn (π1 (X, x), νx (F )).

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VI. COVANISHING TOPOS AND GENERALIZATIONS

VI.9.9. Let X be a scheme whose set of connected components is locally finite, x a geometric point of X, ϕx the fiber functor of X´et associated with x, and Y the connected component of X containing x, which is then open in X (VI.9.7.2). We endow Y with the scheme structure induced by that of X and denote by j : Y → X the canonical injection, by νx : Yf´et → Bπ1 (Y,x) the fiber functor of Yf´et at x (VI.9.8.4), and by γ : Bπ1 (Y,x) → Ens the forgetful functor of the action of π1 (Y, x). Then we have a canonical isomorphism of functors (VI.9.9.1)

∼

∗ ϕx ◦ ρ∗X → γ ◦ νx ◦ jf´ et .

´ f/X ), we have ϕx (ρ∗ V ) = ϕx (V ), and this set identifies Indeed, for every V ∈ Ob(Et X with the geometric fiber Vx of V over x. Consequently, V is a neighborhood of the point ρX (x) of Xf´et ([2] IV 6.8.2) if and only if Vx is nonempty, or, equivalently, if and only if Y is contained in the image of V in X. Let (Yi )i∈I be a normalized universal cover of Y at x. For every i ∈ I, by definition ϕx (Yi ) contains a distinguished element. ´ f/X . We can therefore canonically consider Yi as a neighborhood of ρX (x) in the site Et Moreover, it follows from the equivalence of categories (VI.9.8.1) that the system (Yi )i∈I is cofinal in the (opposed category of the) category of neighborhoods of ρX (x) in the site ´ f/X . The isomorphism (VI.9.9.1) then follows from ([2] IV 6.8.3), VI.9.4, and from Et the definition (VI.9.8.4). VI.9.10. Let X be a connected scheme and Π(X) the fundamental groupoid of X (recall that a groupoid is a category whose morphisms are isomorphisms). The objects of Π(X) are the geometric points of X. For any geometric point x of X, we denote ´ f/X → Ens the corresponding fiber functor (VI.9.8.1). If x and x0 are two by ωx : Et geometric points of X, the set π1 (X, x, x0 ) of morphisms from x to x0 in Π(X) is the set of morphisms (or, equivalently, of isomorphisms) ωx → ωx0 of the associated fiber functors. By ([37] V 5.8), the functor (VI.9.10.1)

´ f/X → Hom(Π(X), Ens), µ+ : Et

Y 7→ (x 7→ ωx (Y )),

´ f/X and the category of functors ψ from induces an equivalence between the category Et Π(X) to Ens such that for every geometric point x of X, ψ(x) is a finite set endowed with a continuous action of π1 (X, x). For every geometric point x of X, the fiber functor ωx is left exact and transforms covering families into surjective families. It therefore extends to a fiber functor φx : Xf´et → Ens ([2] IV 6.3). The latter is deduced from the functor νx defined in (VI.9.8.4) by forgetting the action of π1 (X, x). In view of (VI.9.9.1), it therefore corresponds to the point ρX (x) of Xf´et . Interpreting Π(X) as the opposed category of the ´ f/X ([37] V 5.7), for all geometric category of normalized fundamental pro-objects of Et 0 points x and x of X, every morphism of π1 (X, x, x0 ) induces a morphism φx → φx0 of the associated fiber functors of Xf´et . We deduce from this a functor (VI.9.10.2)

Π(X) → Pt(Xf´et ),

x 7→ φx ,

where Pt(Xf´et ) is the category of points of Xf´et . It is an equivalence of categories by virtue of ([2] IV 4.9.4 and 7.2.5). Lemma VI.9.11. Under the assumptions of VI.9.10, the functor (VI.9.11.1)

ν : Xf´et → Hom(Π(X), Ens),

F 7→ (x 7→ φx (F )),

deduced from (VI.9.10.2) induces an equivalence between the category Xf´et and the category Φ of functors ϕ from Π(X) to Ens such that for every geometric point x of X, ϕ(x) is a discrete U-set endowed with a continuous left action of π1 (X, x).

VI.9. FINITE ÉTALE SITE AND TOPOS OF A SCHEME

547

Indeed, let x be a geometric point of X and Bπ1 (X,x) the classifying topos of the profinite group π1 (X, x). The functor (VI.9.11.2)

Φ → Bπ1 (X,x) ,

ϕ 7→ ϕ(x),

is clearly an equivalence of categories. On the other hand, the composition of the latter and the functor ν is the equivalence of categories (VI.9.11.3)

∼

νx : Xf´et → Bπ1 (X,x)

defined in (VI.9.8.4), giving the lemma.

Proposition VI.9.12. Let X be a coherent scheme (resp. a scheme with a finite number ´ f/X is coherent in Xf´et ; in particular, of connected components). Then every object of Et the topos Xf´et is coherent. ´ f/X , it suffices For the first assertion, since fibered products are representable in Et ´ to show that every object of Etf/X is quasi-compact ([2] VI 2.1). If X is coherent, every ´ f/X is a coherent scheme, and is therefore quasi-compact as an object of object of Et ´ the site Etf/X . Next, suppose that X has a finite number of connected components ´ f/X and every 1 ≤ i ≤ n, X1 , . . . , Xn . It suffices to show that for every object Y of Et Y ×X Xi is quasi-compact ([2] VI 1.3). We may therefore suppose that the image of Y in X is equal to X1 . After replacing X by X1 (VI.9.4), if necessary, we can restrict to the case where X is connected. It then follows from the equivalence of categories (VI.9.8.1) that Y is quasi-compact. ´ f/X , the second assertion follows Since finite inverse limits are representable in Et from the first and from ([2] VI 2.4.5). Corollary VI.9.13. For every scheme X whose set of connected components is locally finite, the topos Xf´et is algebraic. Indeed, the connected components (Xi )i∈I of X are open and closed in X (VI.9.7.2). ´ f/X . On the other hand, for every i ∈ I, Consequently, (Xi → X)i∈I is a covering of Et the topos (Xf´et )/Xi = (Xi )f´et is coherent by virtue of VI.9.12, and the morphism Xi → X is clearly quasi-separated in Xf´et . It then follows from the definition ([2] VI 2.3) that Xf´et is algebraic. Corollary VI.9.14. For every coherent scheme X, the morphism ρX : X´et → Xf´et is coherent. ´ f/X , ρ∗ (Y ) = Y is a coherent object of X´et . ConseIndeed, for every object Y of Et X quently, ρX is coherent by virtue of VI.9.12 and ([2] VI 3.2). Lemma VI.9.15. Let X be a scheme and F a locally constant and constructible torsion abelian sheaf of X´et , locally constant and constructible. Then: (i) There exist a finite étale cover Y → X and, letting ZY´et be the free Z-module of X´et generated by Y ([2] IV 11.3.3), a surjective homomorphism u : ZY´et → F . (ii) If ρX∗ (F ) = 0, then F = 0. ´ f/X ([2] IX 2.2). The identity (i) By descent, F is representable by an object Y of Et of F then defines a section e of F (Y ) and consequently a homomorphism u : ZY´et → F that is clearly surjective. (ii) Consider Y and u as in (i) and denote by ZYf´et the free Z-module of Xf´et generated by Y . We have ρ∗X (ZYf´et ) ' ZY´et by virtue of ([2] IV 13.4(b)). We deduce from this a surjective homomorphism u : ρ∗X (ZYf´et ) → F . Let v : ZYf´et → ρX∗ (F ) be the adjoint morphism of u. If ρX∗ (F ) = 0, then v = 0 and consequently u = 0, which implies F = 0.

548

VI. COVANISHING TOPOS AND GENERALIZATIONS

Lemma VI.9.16. Let X be a scheme whose set of connected components is locally finite, ´ f/X , F an abelian sheaf of Xf´et , and e ∈ F (Y ). Denote by ZYf´et the free Y an object of Et Z-module of Xf´et generated by Y ([2] IV 11.3.3) and by u : ZYf´et → F the homomorphism associated with e. Then: (i) Y is a locally constant and constructible sheaf of Z-modules of X´et . (ii) If u is an epimorphism, then ρ∗X (F ) is a locally constant and constructible sheaf of X´et . (i) We can restrict to the case where X is connected (VI.9.7.2). By (VI.9.8.1), there exists a surjective finite étale cover X 0 → X such that Y ×X X 0 is X 0 -isomorphic to a finite disjoint sum of copies of X 0 , giving the assertion. (ii) Proceeding as in (i), we can reduce to the case where X is connected and Y is X-isomorphic to a finite disjoint sum of copies of X, by virtue of (VI.9.3.4) and ([2] IV 13.4(b)). Hence e corresponds to sections e1 , . . . , en ∈ F (X) such that the induced homomorphism ZnXf´et → F is surjective. Consider a geometric point x of X, and take the notation of (VI.9.8). We deduce from this, in view of the equivalence of categories (VI.9.8.4), that νx (F ) is an abelian group of finite type, endowed with the trivial action of π1 (X, x). Consequently, F is a constant abelian sheaf of Xf´et with value νx (F ), and the same then holds for ρ∗X (F ). Lemma VI.9.17. Let X be a scheme whose set of connected components is locally finite ´ f/X )/F is filtered. and F a sheaf of Xf´et . Then the category (Et It suffices to show that the sums of two objects and the cokernels of double arrows ´ f/X )/F ([2] I 2.7.1). If U and V are two objects of (Et ´ f/X )/F , are representable in (Et the disjoint sum U t V is a finite étale cover of X that represents the sum of U and V in ´ f/X )/F . Let f, g : U ⇒ V be a double arrow of (Et ´ f/X )/F and G its cokernel in Xf´et . (Et ´ f/X . There exists an étale It suffices to show that G is representable by an object of Et ´ covering (Xi → X)i∈I of Etf/X such that, for every i ∈ I, Xi is connected (VI.9.7.2). By descent, it suffices to show that, for every i ∈ I, G|Xi is representable by an object of ´ f/X ([37] VIII 2.1 and 5.7, [35] II 3.4.4). We can therefore restrict to the case where Et i X is connected. We consider a geometric point x of X, take the notation of (VI.9.8) and ´ f/X with the category C (π1 (X, x)) by the equivalence of categories (VI.9.8.1). identify Et Since in every Galois category, finite direct limits are representable ([37] V 4.2), the cokernel W of the double arrow (f, g) is representable in C (π1 (X, x)). One immediately sees that W is also the cokernel of (f, g) in the topos Bπ1 (X,x) . Identifying Xf´et to Bπ1 (X,x) by the functor (VI.9.8.4), we then see that W represents G. Proposition VI.9.18. If X is a coherent scheme having a finite number of connected components, then the adjunction morphism id → ρX∗ ρ∗X is an isomorphism; in particular, the functor ρ∗X : Xf´et → X´et is fully faithful. Let G be a sheaf of Xf´et . By ([2] II 4.1.1), the canonical morphism of Xf´et

(VI.9.18.1)

lim −→

´ (Et f/X )/G

U →G

is an isomorphism. Since the functor ρ∗X commutes with direct limits and extends the ´ f/X → Et ´ /X , we deduce from this an isomorphism of X´et canonical injection functor Et (VI.9.18.2)

lim −→

´ (Et f/X )/G

∼

U → ρ∗X (G).

On the other hand, the topos X´et and Xf´et are coherent (VI.9.12) and the morphism ρX is coherent (VI.9.14). Hence the functor ρX∗ commutes with filtered direct limits.

VI.9. FINITE ÉTALE SITE AND TOPOS OF A SCHEME

549

Indeed, the proof of ([2] VI 5.1) also works for the direct image functor of sheaves of sets. ´ f/X )/G is filtered by VI.9.17, we deduce from this an isomorphism Since the category (Et (VI.9.18.3)

∼

lim −→

´ (Et f/X )/G

U → ρX∗ (ρ∗X (G)).

By functoriality, the adjunction morphism G → ρX∗ (ρ∗X G) is the direct limit of the ´ f/X )/G. It is therefore an isomorphism, identity morphisms idU , for the objects U of (Et giving the first part of the proposition. The second part is equivalent to the first by general properties of adjoint functors. Remark VI.9.19. Let X be a scheme and G a sheaf of Xf´et . By ([2] I 5.1 and III ´ /X defined, for every 1.3), ρ∗X (G) is the sheaf associated with the presheaf F on Et ´ V ∈ Ob(Et/X ), by (VI.9.19.1)

F (V ) =

lim −→

G(U ),

(U,u)∈I ◦ V

´ f/X and an Xwhere IV is the category of pairs (U, u) consisting of an object U of Et ∗ morphism u : V → U . Moreover, the adjunction morphism G → ρX∗ (ρX G) is induced by ´ f/X defined, for every U ∈ Ob(Et ´ f/X ), by the canonical the morphism of presheaves on Et isomorphism ∼

G(U ) → F (U );

(VI.9.19.2)

indeed, IU admits a final object, namely (U, idU ). Notice that, in general, this does not imply that the adjunction morphism id → ρX∗ ρ∗X is an isomorphism, because ρ∗X (G) is not in general equal to F . Corollary VI.9.20. Let X be a coherent scheme having a finite number of connected components and F a sheaf (resp. a torsion abelian sheaf ) of X´et . Then the following conditions are equivalent: (i) There exist a sheaf (resp. a torsion abelian sheaf ) G of Xf´et and an isomorphism F ' ρ∗X (G). (ii) F is the direct limit in X´et of a filtered direct system of locally constant and constructible sheaves (resp. torsion abelian sheaves). Let us show that (i) implies (ii). Let G be a sheaf of Xf´et . By (VI.9.18.2), we have an isomorphism of X´et ∼

lim

(VI.9.20.1)

−→

´ (Et f/X )/G

U → ρ∗X (G).

We deduce from this by VI.9.16(i) and VI.9.17 that ρ∗X (G) satisfies the non-resp. condition of (ii). ´ f/X )/G , denote Let G be a torsion abelian sheaf of Xf´et . For every object U of (Et by ZUf´et the free Z-module of Xf´et generated by U and by HU the image of the canonical morphism ZUf´et → G ([2] IV 11.3.3). The isomorphism (VI.9.18.1) induces a group isomorphism of Xf´et lim

(VI.9.20.2)

−→

´ U ∈(Et f/X )/G

∼

HU → G.

We deduce from this a group isomorphism of X´et (VI.9.20.3)

lim −→

´ (Et f/X )/G

ρ∗X (HU ) → ρ∗X (G).

550

VI. COVANISHING TOPOS AND GENERALIZATIONS

By VI.9.16(ii) and ([2] IX 1.2), ρ∗X (HU ) is a locally constant and constructible torsion ´ f/X )/G is filtered (VI.9.17), ρ∗ (G) satisfies abelian sheaf of X´et . Since the category (Et X the resp. condition of (ii). Let us show that (ii) implies (i). First, consider the case where F is a sheaf of sets. By descent, if F is locally constant and constructible, it is representable by an object Y ∼ ∗ ´ f/X ([2] IX 2.2); we therefore have an isomorphism F → of Et ρX (Y ), which shows the required property in the case under consideration. The general case follows because ρ∗X is fully faithful (VI.9.18) and commutes with direct limits. Next, consider the case where F is a locally constant and constructible torsion abelian sheaf. By VI.9.15(i), there exist a finite étale cover Y → X and a surjective homomorphism u : ZY´et → F , where ZY´et is the free Z-module of X´et generated by Y . Denote by ZYf´et the free Z-module of Xf´et generated by Y . We have ρ∗X (ZYf´et ) ' ZY´et by virtue of ([2] IV 13.4(b)). Let v : ZYf´et → ρX∗ (F ) be the adjoint morphism of u, G the image of v, w : ρ∗X (G) → F the adjoint of the canonical morphism G → ρX∗ (F ), and H the kernel of w. It is clear that w is surjective. On the other hand, ρX∗ (F ) is torsion by virtue of VI.9.12, VI.9.14, and ([2] IX 1.2(v)); the same then holds for G and ρ∗X (G). By VI.9.16(ii), ρ∗X (G) is a locally constant and constructible abelian sheaf of X´et . Consequently, H is a locally constant and constructible torsion abelian sheaf, by virtue of ([2] IX 2.1(ii) and 2.6). On the one hand, the sequence (VI.9.20.4)

0 → ρX∗ (H) → ρX∗ (ρ∗X (G)) → ρX∗ (F )

is exact. On the other hand, the adjunction morphism G → ρX∗ (ρ∗X (G)) is an isomorphism by virtue of VI.9.18. We deduce from this that ρX∗ (H) = 0 and consequently that H = 0 by virtue of VI.9.15(ii). Hence w is an isomorphism, which shows the required property in the case under consideration. Finally, the case where F is the direct limit in X´et of a filtered direct system of locally constant and constructible torsion abelian sheaves follows from the previous case because ρ∗X is fully faithful (VI.9.18) and commutes with direct limits. Definition VI.9.21. We say that a scheme X is K(π, 1) if for every invertible integer n in OX and every (Z/nZ)-module F of Xf´et , the adjunction homomorphism F → RρX∗ (ρ∗X F ) is an isomorphism. This notion seems reasonable only for schemes that satisfy the conclusion of VI.9.18, in particular for coherent schemes with a finite number of connected components. VI.10. Faltings site and topos VI.10.1. (VI.10.1.1)

In this section, f : Y → X denotes a morphism of schemes and ´ /X π : E → Et

the fibered U-site deduced from the fibered site of finite étale covers R/Sch (VI.9.5.1) by base change by the functor (VI.10.1.2)

´ /X → Sch, Et

U 7→ U ×X Y.

We say that π is the Faltings fibered site associated with f . We can describe the category E explicitly as follows. The objects of E are the morphisms of schemes V → U over f : Y → X such that the morphism U → X is étale and that the morphism V → UY = U ×X Y is finite étale. Let (V 0 → U 0 ), (V → U ) be two objects of E. A morphism from (V 0 → U 0 ) to (V → U ) consists of an X-morphism U 0 → U and a Y -morphism V 0 → V

VI.10. FALTINGS SITE AND TOPOS

551

such that the diagram (VI.10.1.3)

V0

/ U0

 V

 /U

is commutative. The functor π is then defined for every object (V → U ) of E, by (VI.10.1.4)

π(V → U ) = U.

Note that the fibered site π satisfies the conditions of VI.5.1 as well as condition VI.5.32(i’). We endow E with the covanishing topology associated with π (VI.5.3), in other words, the topology generated by the coverings {(Vi → Ui ) → (V → U )}i∈I of the following two types: (v) Ui = U for every i ∈ I, and (Vi → V )i∈I is an étale covering. (c) (Ui → U )i∈I is an étale covering and Vi = Ui ×U V for every i ∈ I.

The resulting covanishing site E is also called the Faltings site associated with f ; it is a b (resp. E) e the category of presheaves (resp. the topos of sheaves) U-site. We denote by E e of U-sets on E. We also call E the Faltings topos associated with f . If F is a presheaf on E, we denote by F a the associated sheaf. Remark VI.10.2. The category E was initially introduced by Faltings, but with a topology that is in general strictly finer than the covanishing topology, namely the topology generated by the families of morphisms {(Vi → Ui ) → (V → U )}i∈I such that (Vi → V )i∈I and (Ui → U )i∈I are étale coverings ([26] page 214). Indeed, if Y is empty, e is the empty topos; that is, it is equivalent to the punctual category ([2] IV 2.2 and E ´ /X ), UY is empty, and 4.4). This follows from VI.5.11 because for every U ∈ Ob(Et therefore (UY )f´et is the empty topos. However, if we endow E with the topology considered by Faltings, we obtain the topos X´et . This example also shows that the topology considered by Faltings does not in general satisfy Proposition VI.5.10, which nevertheless plays an essential role in his approach, whence the need to modify it as we have done in VI.10.1. We give in III.8.18 an example that illustrates another drawback of the topology considered by Faltings. VI.10.3. It follows from VI.5.6 and from the fact that E admits a final object that finite inverse limits are representable in E, that the functor π is left exact, and that the family of vertical (resp. Cartesian) coverings of E is stable under base change. In fact, the inverse limit of a diagram (V 00 → U 00 )

(VI.10.3.1)

(V 0 → U 0 )

 / (V → U )

of E is representable by the morphism (V 0 ×V V 00 → U 0 ×U U 00 ). Indeed, this morphism clearly represents the inverse limit of the diagram (VI.10.3.1) in the category of morphisms of schemes over f . It therefore suffices to show that it is an object of E, or, equivalently, that the morphism (VI.10.3.2)

V 0 ×V V 00 → UY0 ×UY UY00

552

VI. COVANISHING TOPOS AND GENERALIZATIONS

´ /Y , and is therefore étale. On the is a finite étale cover. It is clearly a morphism of Et other hand, the diagram (VI.10.3.3)

/ V 0 ×U V 00 Y

V 0 ×V V 00  V

∆V

 / V ×UY V

where ∆V is the diagonal embedding, is Cartesian. Since ∆V is a closed immersion, we immediately deduce from this that the morphism (VI.10.3.2) is finite, giving our assertion. VI.10.4. Let ? be one of the two symbols “coh” for coherent or “scoh” for separated and coherent, introduced in VI.9.1. We denote by (VI.10.4.1)

´ ?/X π? : E? → Et

the fibered site deduced from π by base change by the canonical injection functor (VI.9.1) (VI.10.4.2)

´ ?/X → Et ´ /X , ϕ : Et

and by (VI.10.4.3)

Φ : E? → E

the canonical projection. We endow E? with the covanishing topology defined by π? e? the topos of sheaves of U-sets on E? . By virtue of VI.5.21, if X is and denote by E quasi-separated, the functor Φ induces by restriction an equivalence of categories (VI.10.4.4)

∼ e e→ Φs : E E? .

Moreover, under the same assumption, the covanishing topology of E? is induced by that of E through the functor Φ, by VI.5.22. Proposition VI.10.5. Suppose that X and Y are coherent. Then: e (i) For every object (V → U ) of Ecoh , (V → U )a is a coherent object of E. e (ii) The topos E is coherent; in particular, it has enough points. ´ coh/X is quasi-compact. On the other hand, for every (i) Indeed, every object of Et ´ coh/Y ), since W is a coherent scheme, every object of Et ´ f/W is quasi-compact W ∈ Ob(Et by virtue of VI.9.12. The statement then follows from (VI.10.4.4) and VI.5.27(iii) (applied to the fibered site πcoh (VI.10.4.1)). (ii) This follows from VI.5.27(iv) and ([2] VI § 9). ´ /X with the final object X and E by the final object (Y → VI.10.6. We endow Et X). The functors αX! (VI.5.1.2) and σ + (VI.5.32.4) are then explicitly defined by (VI.10.6.1) (VI.10.6.2)

´ f/Y → E, αX! : Et ´ /X → E, σ + : Et

V 7→ (V → X),

U 7→ (UY → U ).

These are left exact and continuous (VI.5.32). Hence they define two morphisms of topos (VI.10.6.3) (VI.10.6.4)

e → Yf´et , β: E e → X´et . σ: E

For every sheaf F = {U 7→ FU } on E, we have β∗ (F ) = FX .

VI.10. FALTINGS SITE AND TOPOS

553

Lemma VI.10.7. (i) The functor ´ /Y , Ψ+ : E → Et

(VI.10.7.1)

(V → U ) 7→ V

is continuous and left exact; it therefore defines a morphism of topos e (VI.10.7.2) Ψ : Y´et → E. e (ii) For every sheaf F of Y´et , we have a canonical isomorphism of E ∼

Ψ∗ (F ) → {U 7→ ρUY ∗ (F |UY )}, ´ /X , ρUY : (UY )´et → (UY )f´et is the canonical morphism where for every object U of Et ´ /X , the transition morphism (VI.9.2.1), and for every morphism g : U 0 → U of Et

(VI.10.7.3)

(VI.10.7.4) is the composition (VI.10.7.5)

ρUY ∗ (F |UY ) → (gY )f´et∗ (ρUY0 ∗ (F |UY0 )) ∼

ρUY ∗ (F |UY ) → ρUY ∗ ((gY )´et∗ (F |UY0 )) → (gY )f´et∗ (ρUY0 ∗ (F |UY0 )),

in which the first arrow is induced by the adjunction morphism id → (gY )´et∗ (gY )´e∗t and the second arrow by (VI.9.3.4). Indeed, Ψ+ is clearly left exact (VI.10.3). On the other hand, for every sheaf F of b Y´et , we have a canonical isomorphism of E (VI.10.7.6)

∼

F ◦ Ψ+ → {U 7→ ρUY ∗ (F |UY )},

where the right-hand side is the presheaf on E defined by the transition morphisms ´ /X . For every (i, j) ∈ I 2 , set Vi = (VI.10.7.5). Let (Ui → U )i∈I be a covering of Et Ui ×X Y , Uij = Ui ×U Uj , and Vij = Uij ×X Y , and denote by hi : Vi → UY and hij : Vij → UY the structural morphisms. The sequence Y Y (VI.10.7.7) 0 → F |UY → (hi )´et∗ (F |Vi ) ⇒ (hij )´et∗ (F |Vij ) (i,j)∈I 2

i∈I

is exact. Since ρUY ∗ commutes with inverse limits, we deduce from this by (VI.9.3.4) that the sequence Y Y (VI.10.7.8) 0 → ρUV ∗ (F |UY ) → (hi )f´et∗ (ρVi ∗ (F |Vi )) ⇒ (hij )f´et∗ (ρVij ∗ (F |Vij )) (i,j)∈I 2

i∈I

is exact. Consequently, F ◦ Ψ+ is a sheaf on E by virtue of VI.5.10, giving the lemma. VI.10.8. Let us describe explicitly the constructions of (VI.6.1) for the functor Ψ+ defined in (VI.10.7.1). The composed functor ´ /X → Et ´ /Y (VI.10.8.1) Ψ+ ◦ σ + : Et

is none other than the inverse image functor by f : Y → X; we therefore have f´et = σΨ. ´ /X , the functor (VI.6.1.5) On the other hand, for every object U of Et (VI.10.8.2)

´ ´ Ψ+ U : Etf/UY → Et/UY

induced by Ψ+ , identifies with the canonical injection. We can therefore identify the morphism ΨU (VI.6.1.6) with the canonical morphism ρUY : (UY )´et → (UY )f´et (VI.9.2.1); ´ /X , the in particular, we have βΨ = ρY (VI.6.1.9). For every morphism g : U 0 → U of Et diagram (VI.6.2.1) then identifies with the diagram (VI.9.3.4). From the isomorphisms Ψ∗ σ ∗ = f´e∗t and Ψ∗ β ∗ = ρ∗Y , we deduce, by adjunction, morphisms (VI.10.8.3) (VI.10.8.4)

σ∗ β∗

→ Ψ∗ f´e∗t , → Ψ∗ ρ∗Y .

554

VI. COVANISHING TOPOS AND GENERALIZATIONS

Proposition VI.10.9. Suppose that X is quasi-separated, and that Y is coherent and ´ /X ), denote by hU : UY → Y the étale-locally connected (VI.9.7). For every U ∈ Ob(Et canonical projection. Then: (i) For every sheaf F of Yf´et , β ∗ (F ) is the sheaf on Ecoh defined by {U 7→ (hU )∗f´et F }. (ii) The adjunction morphism id → β∗ β ∗ is an isomorphism. (iii) The adjunction morphism β ∗ → Ψ∗ ρ∗Y (VI.10.8.4) is an isomorphism. First note that the three functors coh ´ (VI.10.9.1) αX! : Etf/Y → Ecoh , (VI.10.9.2) (VI.10.9.3)

+ σcoh : Ψ+ coh :

´ coh/X → Ecoh , Et ´ coh/Y , Ecoh → Et

V 7→ (V → X),

U 7→ (UY → U ),

(V → U ) 7→ V,

are well-defined, continuous, and left exact. The morphisms of topos they define identify with β, σ, and Ψ, respectively, in view of (VI.10.4.4) and VI.9.1. On the other hand, ´ coh/X , the scheme UY is coherent and locally connected. Consefor every object U of Et quently, the adjunction morphism id → ρUY ∗ ρ∗UY is an isomorphism by virtue of VI.9.18. The proposition therefore follows from VI.6.3. Proposition VI.10.10. Suppose that X and Y are coherent. Then the morphisms β, σ, and Ψ are coherent. ´ coh/X is coherent in X´et ; every object of Et ´ f/Y is coherent Indeed, every object of Et e in Yf´et (VI.9.12); for every object (V → U ) of Ecoh , (V → U )a is a coherent object of E by VI.10.5(i). The proposition therefore follows from ([2] VI 3.3), in view of the proof of VI.10.9. Remark VI.10.11. We have a 2-morphism (VI.10.11.1)

τ : ff´et β → ρX σ,

´ f/X ), such that for every sheaf F on E and every U ∈ Ob(Et

ff´et∗ (β∗ (F ))(U ) → ρX∗ (σ∗ (F ))(U )

is the canonical map F (UY → X) → F (UY → U ). VI.10.12.

Consider a commutative diagram of morphisms of schemes Y0

(VI.10.12.1)

f0

g0

/ X0 g

 Y

f

 /X

We denote by (VI.10.12.2)

´ /X 0 π 0 : E 0 → Et

the fibered Faltings site associated with f 0 (VI.10.1). We endow E 0 with the covanishing e 0 the topos of sheaves of U-sets on E 0 . For topology associated with π 0 and denote by E every object (V → U ) of E, the canonical morphism V ×Y Y 0 → U ×X X 0 is an object of E 0 . We thus define a functor (VI.10.12.3)

Φ+ : E → E 0 ,

(V → U ) 7→ (V ×Y Y 0 → U ×X X 0 ),

that is clearly left exact (VI.10.3). For every sheaf F = {U 0 7→ FU 0 } on E 0 , F ◦ Φ+ is the presheaf on E defined by (VI.10.12.4)

0 {U 7→ (gU )f´et∗ (FU ×X X 0 )},

VI.10. FALTINGS SITE AND TOPOS

555

´ /X ), g 0 : U ×X Y 0 → U ×X Y is the base change by g 0 . Let where for every U ∈ Ob(Et U ´ /X . Since the functor (g 0 )f´et∗ commutes with inverse (Ui → U )i∈I be a covering of Et U limits, the gluing relation of F with regard to the covering (Ui ×X X 0 → U ×X X 0 )i∈I (VI.5.10.1) implies the analogous relation for F ◦ Φ+ with regard to the covering (Ui → U )i∈I . Consequently, F ◦ Φ+ is a sheaf on E (VI.5.10), and Φ+ is continuous. The latter therefore defines a morphism of topos e 0 → E. e Φ: E

(VI.10.12.5)

It immediately follows from the definitions that the squares of the diagram (VI.10.12.6)

X´e0 t o

σ0

g´et

 X´et o

e0 E

β0

/ Y0 f´ et

Φ

 e E

σ



0 gf´ et

/ Yf´et

β

where β 0 and σ 0 are the canonical morphisms (VI.10.6.3) and (VI.10.6.4) relative to f 0 , are commutative up to canonical isomorphisms. On the other hand, the diagram Y´e0t

(VI.10.12.7)

g´e0 t

Ψ0



Y´et

/E e0 Φ

Ψ

 /E e

where Ψ0 is the morphism (VI.10.7.2) relative to f 0 , is commutative up to canonical isomorphism. Remark VI.10.13. We keep the assumptions of VI.10.12 and let F be an abelian sheaf e and i an integer ≥ 0. Then the diagram on E (VI.10.13.1)

/ Hi (E, e F)

Hi (Yf´et , β∗ F ) u

 0∗ Hi (Yf´0et , gf´ et (β∗ F )) v

 Hi (Yf´0et , β∗0 (Φ∗ F ))

w

 / Hi (E e 0 , Φ∗ F )

where the horizontal arrows come from Cartan–Leray spectral sequences ([2] V 5.3), u and w are the canonical morphisms, and v is induced by the base change morphism with respect to the right square of (VI.10.12.6) ([1] (1.2.2.2)) is commutative. VI.10.14. (VI.10.14.1)

Let (f 0 : Y 0 → X 0 ) be an object of E. We denote by ´ /X 0 π 0 : E 0 → Et

the fibered Faltings site associated with f 0 . Every object of E 0 is naturally an object of E. We thus define a functor (VI.10.14.2)

Φ : E 0 → E.

One immediately verifies that Φ factors through an equivalence of categories (VI.10.14.3)

∼

E 0 → E/(Y 0 →X 0 ) ,

556

VI. COVANISHING TOPOS AND GENERALIZATIONS

´ /X 0 ([37] VI 4.3), where we consider that is even an equivalence of categories over Et ´ /X 0 )-category by base change by the functor π. It then follows E/(Y 0 →X 0 ) as an (Et from VI.5.38 that the covanishing topology of E 0 is induced by that of E by the functor Φ. Consequently, Φ is continuous and cocontinuous ([2] III 5.2). It therefore defines a sequence of three adjoint functors: e 0 → E, e Φ! : E

(VI.10.14.4)

e→E e0 , Φ∗ : E

e 0 → E, e Φ∗ : E

in the sense that for any two consecutive functors in the sequence, the one on the right is right adjoint to the other. By ([2] III 5.4), the functor Φ! factors through an equivalence of categories ∼ e e0 → E E/(Y 0 →X 0 )a .

(VI.10.14.5)

e at (Y 0 → X 0 )a The pair of functors (Φ∗ , Φ∗ ) defines the localization morphism of E e e 0 → E. Φ: E

(VI.10.14.6)

Since Φ : E 0 → E is a left adjoint of the functor Φ+ : E → E 0 defined in (VI.10.12.3), the morphism (VI.10.14.6) identifies with the morphism defined in (VI.10.12.5), by virtue of ([2] III 2.5). VI.10.15.

We denote by ´ /X $ : D → Et

(VI.10.15.1)

´ /X → Et ´ /Y , defined in the fibered site associated with the inverse image functor f + : Et (VI.5.5). We endow D with the covanishing topology associated with $. We thus obtain the covanishing site associated with the functor f + (VI.4.1), whose topos of sheaves of ←

U-sets is X´et ×X´et Y´et (VI.4.10). Every object of E is naturally an object of D. We thus define a fully faithful and left exact functor ρ+ : E → D.

(VI.10.15.2)

´ /X ), the restriction of ρ+ to the fibers over U is none other than the For every U ∈ Ob(Et ´ f/U → Et ´ /U , in other words, the functor Ψ+ (VI.10.8.2). canonical injection functor Et Y Y U + The functor ρ therefore identifies with a functor with the same name, associated with the functor Ψ+ (VI.10.7.1) and defined in (VI.6.4.2). It is continuous and left exact. It therefore defines a morphism of topos (VI.6.4.4) ←

e ρ : X´et ×X´et Y´et → E.

(VI.10.15.3)

It immediately follows from the definitions that the squares of the diagram (VI.10.15.4)

X´et o

p1

p2

←

X´et ×X´et Y´et

/ Y´et

ρ

X´et o

σ

 e E

ρY β



/ Yf´et

where p1 and p2 are the canonical projections (VI.4.11), are commutative up to canonical isomorphisms. Moreover, we have a commutative diagram (VI.10.15.5)

ff´et βρ  ff´et ρY p2

τE ∗ρ

/ ρX σρ

 / ρX f´et p2 ρX ∗τD / ρX p1

VI.10. FALTINGS SITE AND TOPOS

557

where τD is the 2-morphism (VI.4.11.5), τE is the 2-morphism (VI.10.11.1), the vertical arrows are the isomorphisms underlying the diagram (VI.10.15.4), and the unlabeled horizontal arrow comes from (VI.9.3.4). On the other hand, the diagram (VI.10.15.6)

/ Y´et J JJ X´et ×X´et Y´et JJ JJ ρ J Ψ JJJ  J% e E ΨD

←

where ΨD is the morphism (VI.4.13.4), is commutative up to canonical isomorphism. Proposition VI.10.16. Suppose that the scheme X is quasi-separated and that the scheme Y is coherent. Then a family ((Vi → Ui ) → (V → U ))i∈I of morphisms of Ecoh (VI.10.4) is covering if and only if it is in D (VI.10.15). Fist note that since the covanishing topology of Ecoh is induced by that of E (VI.10.4), a family of morphisms with the same target in Ecoh is covering in Ecoh if and only if it is in E ([2] III 3.3). The morphism f induces a functor (VI.9.1) (VI.10.16.1)

+ ´ coh/X → Et ´ coh/Y . fcoh : Et

+ We denote by Dcoh the covanishing site associated with fcoh (VI.4.1). The canonical injection functor Dcoh → D is continuous and left exact. It induces an equivalence between the associated topos, by VI.3.9 and VI.4.10. Consequently, a family of morphisms with the same target in Dcoh is covering in Dcoh if and only if it is in D ([2] II 4.4). On the other hand, every object of Ecoh is naturally an object of Dcoh . We thus define a fully faithful and left exact functor

(VI.10.16.2)

ρ+ coh : Ecoh → Dcoh .

´ coh/X )-functor whose restriction to the fibers over any object U It is a Cartesian (Et ´ coh/X is none other than the canonical injection functor Et ´ f/U → Et ´ coh/U (cf. of Et Y Y VI.5.5). Hence ρ+ is continuous by virtue of VI.5.18. To show the proposition, it coh suffices to show that a family F = ((Vi → Ui ) → (V → U ))i∈I of morphisms of Ecoh is covering if and only if it is in Dcoh . The condition is necessary because ρ+ coh is continuous ([2] III 1.6). Conversely, suppose that the family F is covering in Dcoh and let us show ´ coh/X ), the scheme UY is coherent. Consequently, that it is in Ecoh . For every U ∈ Ob(Et ´ coh/U is quasi-compact. By VI.5.9, there consequently exists an étale every object of Et Y 0 covering (Uj0 → U )j∈J and for every j ∈ J, an étale covering (Vj,k → Uj0 ×U V )k∈Kj such that for all j ∈ J and k ∈ Kj , there exist ij,k ∈ I, a U -morphism Uj0 → Uij,k , and a 0 V -morphism Vj,k → Vij,k such that the diagram (VI.10.16.3)

0 Vj,k

/ Vij,k

 Uj0

 / Uij,k

is commutative. Set Wj,k = Uj0 ×Uij,k Vij,k . For every j ∈ J, (Wj,k → Uj0 ×U V )k∈Kj is a ´ f/U 0 × Y . Consequently, ((Vi → Ui ) → (V → U ))i∈I is a covering of Ecoh . covering of Et i X

558

VI. COVANISHING TOPOS AND GENERALIZATIONS

Remark VI.10.17. Under the assumptions of VI.10.12, the diagram of morphisms of topos (VI.10.17.1)

←

/X ← ´ et ×X´et Y´ et

Ξ

X´e0 t ×X´e0 t Y´e0t ρ0

ρ

 e E0

 /E e

Φ

where ρ and ρ0 are the canonical morphisms (VI.10.15.3), Φ is the morphism (VI.10.12.5), and Ξ is the morphism deduced from the functoriality of the covanishing topos (VI.3.8), is commutative up to canonical isomorphism. Indeed, by VI.4.6(i), for every (V → U ) ∈ Ob(D) (VI.10.15), we have a canonical isomorphism (VI.10.17.2)

∼

Ξ∗ ((V → U )a ) → (V ×Y Y 0 → U ×X X 0 )a . ←

VI.10.18. By VI.4.20 and ([2] VIII 7.9), giving a point of X´et ×X´et Y´et is equivalent to giving a pair of geometric points x of X and y of Y and a specialization arrow u from f (y) to x, that is, an X-morphism u : y → X(x) , where X(x) denotes the strict localization of X at x. Such a point will be denoted by (y x) or by (u : y x). We denote by e For all F ∈ Ob(X´et ) x) its image by ρ (VI.10.15.3), which is therefore a point of E. ρ(y and G ∈ Ob(Yf´et ), we have functorial canonical isomorphisms ∼

(VI.10.18.1)

(σ ∗ F )ρ(y

x)

(VI.10.18.2)

(β ∗ G)ρ(y

x)

→ Fx , ∼

→ (ρ∗Y G)y .

By (VI.10.15.5), for every H ∈ Ob(Xf´et ), the map (VI.10.18.3)

(σ ∗ (ρ∗X H))ρ(y

x)

∗ → (β ∗ (ff´ et H))ρ(y

x)

induced by τ (VI.10.11.1), identifies canonically with the specialization morphism defined by u (ρ∗X H)x → (ρ∗X H)f (y) . For every object (V → U ) of E, we have a functorial canonical isomorphism (VI.10.18.4)

(V → U )aρ(y

∼

x)

→ Ux ×Uf (y) Vy ,

the map Vy → Uf (y) is induced by V → U , and the map Ux → Uf (y) is the specialization morphism defined by u. This follows from (VI.4.20.6) and from the fact that ρ∗ extends ρ+ ([2] III 1.4). ´ /X ), x, z be two geometric points of X, u a speRemark VI.10.19. Let U ∈ Ob(Et cialization morphism from z to x, and u∗ : Ux → Uz the corresponding specialization morphism ([2] VIII 7.7). Then: (i) For every x0 ∈ Ux , z 0 = u∗ (x0 ) is the point of Uz corresponding to the composition (VI.10.19.1)

u

i

z −→ X(x) −→ U,

where i is the X-morphism defined by x0 ([2] VIII 7.3); more precisely, i is ∼ induced by the inverse of the canonical isomorphism U(x0 ) → X(x) , where U(x0 )

VI.10. FALTINGS SITE AND TOPOS

559

denotes the strict localization of U at x0 . Hence there exists a unique specialization morphism u0 from z 0 to x0 that fits into a commutative diagram z0

(VI.10.19.2)

 z

u0

/ U(x0 )

u

 / X(x)

where the vertical arrows are the canonical isomorphisms. (ii) If U is separated over X, the map u∗ is injective. Indeed, if x0 and x00 are two points of Ux such that u∗ (x0 ) = u∗ (x00 ) = z 0 , then we have two specializations from z 0 to x0 and x00 , respectively. In particular, the image of the canonical morphism z 0 → U ×X X(x) is contained in each of the open and closed subschemes U(x0 ) and U(x00 ) of U ×X X(x) ([42] 18.5.11), which is impossible. Q (iii) If U is finite over X, the map u∗ is bijective because U ×X X(x) = x0 ∈Ux U(x0 ) . VI.10.20. Let x be a geometric point of X, y a geometric point of Y , X(x) the strict localization of X at x, and u : y → X(x) an X-morphism, so that (y x) is a point ←

of X´et ×X´et Y´et (VI.10.18). We denote by Pρ(y x) the category of ρ(y x)-pointed objects of E, defined as follows. The objects of Pρ(y x) are the triples ((V → U ), ξ, ζ) consisting of an object (V → U ) of E, an X-morphism ξ : x → U , and a Y -morphism ζ : y → V such that, denoting again by ξ : X(x) → U the X-morphism induced by ξ ([2] VIII 7.3), the diagram y

(VI.10.20.1)

u

/ X(x)

ζ

ξ

 V

 /U

is commutative. Let ((V → U ), ξ, ζ), ((V 0 → U 0 ), ξ 0 , ζ 0 ) be two objects of Pρ(y x) . A morphism from ((V 0 → U 0 ), ξ 0 , ζ 0 ) to ((V → U ), ξ, ζ) is defined as a morphism (g : U 0 → U, h : V 0 → V ) of E such that g ◦ ξ 0 = ξ and h ◦ ζ 0 = ζ. It follows from (VI.10.18.4) and VI.10.19(i) that Pρ(y x) is canonically equivalent to the category of neighborhoods of ρ(y x) in E ([2] IV 6.8.2). It is therefore cofiltered and for every presheaf F = {U 7→ FU } on E, we have a functorial canonical isomorphism ([2] IV (6.8.4)) (VI.10.20.2)

(F a )ρ(y

∼

x)

→

FU (V ).

lim −→

((V →U ),ξ,ζ)∈P ◦ ρ(y

x)

If X is quasi-separated, we can replace in the limit above Pρ(y x) by the full subcategory coh Pρ(y x) made up of the objects ((V → U ), ξ, ζ) such that U is of finite presentation over ´ coh/X ), which is also cofiltered (cf. VI.10.4). X (that is, is an object of Et Proposition VI.10.21. Suppose that the schemes X and Y are coherent. Then when ← (y x) goes through the family of points of X´et ×X´et Y´et , the family of fiber functors of e associated with the points ρ(y E x) is conservative ([2] IV 6.4.0). ←

e such that for every point (y Let u : F → G be a morphism of E x) of X´et ×X´et Y´et , the corresponding morphism uρ(y x) : Fρ(y x) → Gρ(y x) is a monomorphism. Let us show that u is a monomorphism. We must show that for all (V → U ) ∈ Ob(Ecoh ) (VI.10.4) and a, b ∈ FU (V ) such that u(a) = u(b), we have a = b. In view of VI.10.14 and VI.10.17, we may assume (V → U ) = (Y → X). For every point (y x) of

560

VI. COVANISHING TOPOS AND GENERALIZATIONS ←

X´et ×X´et Y´et , we have aρ(y x) = bρ(y x) because uρ(y x) is a monomorphism. By coh (VI.10.20.2), there exists an object ((U(y x) → V(y x) ), ξ(y x) , ζ(y x) ) of Pρ(y x) such that a and b have the same images in FU(y x) (V(y x) ). On the other hand, since the topos X´et and Y´et are coherent and the morphism f : Y´et → X´et is coherent ([2] VI ←

3.3), the family of fiber functors of X´et ×X´et Y´et associated with the points (y x) is conservative, by VI.5.30 and VI.10.18. We deduce from this that the family of morphisms ((U(y x) → V(y x) ) → (Y → X))(y x) is covering in D (VI.10.15) ([2] IV 6.5). It is therefore covering in Ecoh by virtue of VI.10.16. Consequently, a = b and u is a monomorphism. ← x) of X´et ×X´et Y´et , the morphism Suppose, moreover, that for every point (y uρ(y x) is an epimorphism and let us show that the same holds for u. It suffices to show that for all (V → U ) ∈ Ob(Ecoh ) and b ∈ GU (V ), there exists a ∈ FU (V ) such that b = u(a). We may still assume that (V → U ) = (Y → X). By (VI.10.20.2), for every ←

x) of X´et ×X´et Y´et , there exist an object ((U(y x) → V(y x) ), ξ(y x) , ζ(y x) ) point (y coh of Pρ(y x) and a section a(y x) ∈ FU(y x) (V(y x) ) whose image by u in GU(y x) (V(y x) ) is the restriction of b. Since u is a monomorphism, the sections a(y x) coincide on (V(y0 x0 ) ×Y V(y x) → U(y0 x0 ) ×X U(y x) ), for all points (y x) and (y 0 x0 ) of ←

X´et ×X´et Y´et . They therefore come from a section a ∈ FX (Y ), and we have u(a) = b because the restrictions on the (U(y x) → V(y x) ) coincide.

VI.10.22. Suppose that X is strictly local with closed point x. For every U ∈ ´ scoh/X ) (VI.9.1), we denote by U f the disjoint sum of the strict localizations of U Ob(Et at the points of Ux ; it is an open and closed subscheme of U , and is finite over X ([42] 18.5.11). The correspondence U 7→ U f defines a functor (VI.10.22.1)

´ ´ ι+ x : Etscoh/X → Etf/X ,

that is clearly left exact and continuous. The associated morphism of topos ιx : Xf´et → X´et

(VI.10.22.2)

identifies with the morphism Ens → X´et defined by x. Indeed, the fiber functor at x induces an equivalence between the topos Xf´et and Ens because the group π1 (X, x) ´ scoh/X ), the canonical map Uxf → Ux is is trivial (VI.9.8.1), and for every U ∈ Ob(Et bijective. Let us consider the fibered site ´ scoh/X (VI.10.22.3) πscoh : Escoh → Et

defined in (VI.10.4), and endow Escoh with the covanishing topology associated with e πscoh . Recall that the topos of sheaves of U-sets on Escoh identifies canonically with E (VI.10.4.4). For every object (V → U ) of Escoh , V ×U U f = V ×UY UYf is a finite étale cover of Y . We thus obtain a functor ´ f/Y , (V → U ) 7→ V ×U U f . (VI.10.22.4) θ+ : Escoh → Et Lemma VI.10.23. We keep the assumptions of VI.10.22. (i) The functor θ+ is continuous and left exact; it therefore defines a morphism of topos e θ : Yf´et → E.

(VI.10.23.1)

(ii) For every sheaf F of Yf´et , we have a canonical isomorphism of sheaves on Escoh (VI.10.23.2)

∼

θ∗ (F ) → {U 7→ (jU )f´et∗ (F |UYf )},

VI.10. FALTINGS SITE AND TOPOS

561

´ scoh/X ), jU : U f → UY denotes the canonical injecwhere for every U ∈ Ob(Et Y ´ scoh/X , the transition morphism tion, and for every morphism g : U 0 → U of Et (VI.10.23.3) is the composition

(jU )f´et∗ (F |UYf ) → (gY )f´et∗ ((jU 0 )f´et∗ (F |UY0f )) ∼

(VI.10.23.4) (jU )f´et∗ (F |UYf ) → (jU )f´et∗ ((gYf )f´et∗ (F |UY0f )) → (gY )f´et∗ ((jU 0 )f´et∗ (F |UY0f )),

where g f : U 0f → U f is the image of g by the functor (VI.10.22.1), the first arrow is induced by the adjunction morphism id → (gYf )f´et∗ (gYf )∗f´et , and the second arrow is the canonical isomorphism.

One easily verifies that θ+ commutes with fibered products and transforms final objects into final objects. It is therefore left exact. On the other hand, for every sheaf F bscoh of Yf´et , we have a canonical isomorphism of E ∼

F ◦ θ+ → {U 7→ (jU )f´et∗ (F |UYf )},

(VI.10.23.5)

where the right-hand side is the presheaf on Escoh defined by the transition morphisms ´ scoh/X . For every (i, j) ∈ I 2 , set (VI.10.23.4). Let (Ui → U )i∈I be a covering of Et f f Vi = Ui ×X Y , Wi = Ui ×X Y , Uij = Ui ×U Uj , Vij = Uij ×X Y , and Wij = Uij ×X Y , f and denote by hi : Vi → UY , gi : Wi → UY , hij : Vij → UY , and gij : Wij → UYf the structural morphisms. Since the functor (VI.10.22.1) is left exact and continuous, the sequence Y Y (VI.10.23.6) 0 → F |UYf → (gi )f´et∗ (F |Wi ) ⇒ (gij )f´et∗ (F |Wij ) (i,j)∈I 2

i∈I

is exact. Since (jU )f´et∗ commutes with inverse limits, we deduce from this that the sequence (VI.10.23.7) Y Y 0 → (jU )f´et∗ (F |UYf ) → (hi )f´et∗ ((jUi )f´et∗ (F |Wi )) ⇒ (hij )f´et∗ ((jUij )f´et∗ (F |Wij )) (i,j)∈I 2

i∈I

is exact. Consequently, F ◦ θ giving the lemma.

+

is a sheaf by virtue of VI.5.10. Hence θ+ is continuous,

VI.10.24. We keep the assumptions of VI.10.22 and describe explicitly the constructions of (VI.6.1) for the functor θ+ defined in (VI.10.22.4). The composed functor ´ scoh/X → Et ´ f/Y (VI.10.24.1) θ+ ◦ σ + : Et is none other than the functor U 7→ UYf ; we therefore have σθ = ιx ff´et , where ιx is the ´ scoh/X , the functor morphism (VI.10.22.2). On the other hand, for every object U of Et (VI.6.1.5) ´ f/U → Et ´ f/U f (VI.10.24.2) θ+ : Et U

Y

Y

+

induced by θ is none other than the inverse image under the canonical morphism jU : UYf → UY . In particular, θ is a section of β; that is, we have a canonical isomorphism (VI.6.1.9) ∼

(VI.10.24.3) We obtain a base change morphism (VI.10.24.4) ∼

βθ → idYf´et . β∗ → θ∗ ,

composition of β∗ → β∗ θ∗ θ∗ → θ∗ , where the first arrow is induced by the adjunction morphism id → θ∗ θ∗ , and the second arrow by (VI.10.24.3).

562

VI. COVANISHING TOPOS AND GENERALIZATIONS

VI.10.25. We keep the assumptions of VI.10.22. The canonical morphism ρX : X´et → Xf´et (VI.9.2.1) identifies with the unique morphism of topos X´et → Ens ([2] IV 4.3). On the other hand, the composed morphism ιx ρX : X´et → X´et is defined by the morphism of sites ´ scoh/X → Et ´ scoh/X , U 7→ U f . (VI.10.25.1) Et ´ scoh/X ), then defines a 2-morphism The canonical injection U f → U , for U ∈ Ob(Et idX´et → ιx ρX .

(VI.10.25.2)

By VI.3.7, the morphisms of topos ιx ρX f´et : Y´et → X´et and idY´et , and the 2-morphism f´et → ιx ρX f´et induced by (VI.10.25.2), define a morphism of topos ←

γ : Y´et → X´et ×X´et Y´et

(VI.10.25.3)

such that p1 γ = ιx ρX f´et , p2 γ = idY´et , and τ ∗ γ is induced by (VI.10.25.2): (VI.10.25.4)

X´et o ιx ρX

 X´et

f´et

Y´et J JJ JJ id JJ γ JJ JJ  % p1 p2 ← o / Y´et X × Y ´ e t X ´ e t KK ´ et t KK tt KK tt t KK t KK tt f´et K% ytt X´et

We take again the notation of (VI.10.15) and denote by Dscoh the fibered site over ´ scoh/X deduced from D (VI.10.15.1) by base change by the canonical injection funcEt ´ scoh/X → Et ´ /X (VI.10.4). It follows from VI.4.6(i) that for every (V → U ) ∈ tor Et Ob(Dscoh ), we have a canonical isomorphism (VI.10.25.5)

∼

γ ∗ ((V → U )a ) → V ×U U f .

We deduce from this that the diagram (VI.10.25.6)

Y´et

/X ← ´ et ×X´et Y´ et

γ

ρY

ρ

 Yf´et

θ

 /E e

where ρ is the canonical morphism (VI.10.15.3), is commutative up to canonical isomorphism. e and Proposition VI.10.26. Under the assumptions of VI.10.22, for every sheaf F of E every geometric point y of Y , the map (VI.10.26.1)

(β∗ F )ρY (y) → (θ∗ F )ρY (y)

induced by the base change morphism (VI.10.24.4) is bijective. First, note that there exists a unique specialization from f (y) to x; we can therefore ←

consider the point (y x) of X´et ×X´et Y´et (VI.10.18). It is clear that the points γ(y) and (y x) are canonically isomorphic. It then follows from (VI.10.25.6) that the points e are canonically isomorphic. We denote by Cy the category of θ(ρY (y)) and ρ(y x) of E y-pointed finite étale Y -schemes, which we identify with the category of neighborhoods ´ f/Y ([2] IV 6.8.2), and by P scoh of ρY (y) in the site Et ρ(y x) the full subcategory of the

VI.10. FALTINGS SITE AND TOPOS

563

category Pρ(y x) (VI.10.20) made up of the ρ(y x)-pointed objects ((V → U ), ξ, ζ) of E such that U is separated and of finite presentation over X (that is, is an object ´ scoh/X ). It follows from (VI.10.18.4) and VI.10.19(i) that P scoh of Et ρ(y x) is canonically equivalent to the category of neighborhoods of the point ρ(y x) in the site Escoh . It is therefore cofiltered. Denote by ξ0 : x → X the canonical injection. We have a fully faithful functor scoh y : Cy → Pρ(y

(VI.10.26.2)

x) ,

(V, ζ : y → V ) 7→ ((V → X), ξ0 , ζ),

compatible with the canonical functor (VI.10.6.1) scoh ´ (VI.10.26.3) αX! : Etf/Y → Escoh , ∗

V 7→ (V → X).

The adjunction morphism F → θ∗ (θ F ) is defined, for every (V → U ) ∈ Ob(Escoh ), by the canonical map F (V → U ) → (θ∗ F )(V ×U U f ). Consequently, (VI.10.26.1) identifies with the map lim

(VI.10.26.4)

−→

(W,ζ)∈C ◦ y

F (W → X) → Fρ(y

x)

induced by the functor ◦y . Hence it suffices to show that ◦y is cofinal. Let ((V → U ), ξ, ζ) scoh be an object of Pρ(y x) . Denote also by ξ : X → U the X-morphism induced by ξ, so that the diagram /X y (VI.10.26.5) ζ

 V

 /U

ξ

is commutative. We deduce from this a Y -morphism ζ 0 : y → V ×U,ξ X that fits into a commutative diagram ζ0

/ V ×U,ξ X y HH HH HH HH HH ζ H$  V

(VI.10.26.6)

/X ξ

 /U

Consequently, (V ×U,ξ X, ζ 0 ) is an object of Cy , and the diagram (VI.10.26.6) induces a morphism scoh of Pρ(y sition.

y (V ×U,ξ X, ζ 0 ) = ((V ×U,ξ X → X), ξ0 , ζ 0 ) → ((V → U ), ξ, ζ) x) .

We deduce from this that y◦ is cofinal by ([2] I 8.1.3(c)), giving the propo-

e the Corollary VI.10.27. Suppose that X is strictly local, and denote by θ : Yf´et → E morphism of topos defined in (VI.10.23.1). Then, the base change morphism β∗ → θ∗ (VI.10.24.4) is an isomorphism; in particular, the functor β∗ is exact. It follows from VI.10.26 and VI.9.6. e the Corollary VI.10.28. Suppose that X is strictly local, and denote by θ : Yf´et → E morphism of topos defined in (VI.10.23.1). Then: e the canonical map (i) For every sheaf F of E, (VI.10.28.1) is bijective.

e F ) → Γ(Yf´et , θ∗ F ) Γ(E,

564

VI. COVANISHING TOPOS AND GENERALIZATIONS

e the canonical map (ii) For every abelian sheaf F of E, e F ) → Hi (Yf´et , θ∗ F ) Hi (E,

(VI.10.28.2)

is bijective for every i ≥ 0. (i) Indeed, the diagram

e F) o Γ(E,

(VI.10.28.3)

v

w ∼

Γ(Yf´et , β∗ F ) u



0

 Γ(Yf´et , β∗ (θ∗ (θ∗ F )))

e θ∗ (θ∗ F )) o ∼ Γ(E, RRR RRR RRR u0 RRR v0 RR)  Γ(Yf´et , θ∗ F ) w

where w, w0 , and v 0 are the canonical bijections, u and v are induced by the adjunction morphism id → θ∗ θ∗ , and u0 is induced by the isomorphism (VI.10.24.3), is commutative. Since u0 ◦u is bijective by virtue of VI.10.27, the same holds for v 0 ◦v, giving the assertion. (ii) Indeed, the diagram e F) o Hi (E,

(VI.10.28.4)

v

w

Hi (Yf´et , β∗ F ) u



0

 Hi (Yf´et , β∗ (θ∗ (θ∗ F )))

e θ∗ (θ∗ F )) o w Hi (E, SSS SSS SSS u0 SSS v0 SS)  Hi (Yf´et , θ∗ F )

where w, w0 , and v 0 are induced by the Cartan–Leray spectral sequence ([2] V 5.3), u and v are induced by the adjunction morphism id → θ∗ θ∗ , and u0 is induced by the isomorphism (VI.10.24.3), is commutative. On the other hand, u0 ◦ u is bijective, and the functor β∗ is exact by virtue of VI.10.27; hence w is bijective, giving the assertion. VI.10.29. Let x be a geometric point of X, X the strict localization of X at x, e the Faltings Y = Y ×X X, and f : Y → X the canonical projection. We denote by E topos associated with f (VI.10.1) and by (VI.10.29.1)

e θ : Y f´et → E

the morphism defined in (VI.10.23.1). The canonical morphism X → X induces by functoriality a morphism (VI.10.12.5) (VI.10.29.2)

e → E. e Φ: E

We denote by (VI.10.29.3) the composed functor θ∗ ◦ Φ∗ .

e→Y ϕx : E f´ et

Proposition VI.10.30. We keep the assumptions of VI.10.29 and moreover suppose that f is coherent. Then: e we have a functorial canonical isomorphism (i) For every sheaf F of E, (VI.10.30.1)

∼

σ∗ (F )x → Γ(Y f´et , ϕx (F )).

VI.10. FALTINGS SITE AND TOPOS

565

e and every integer i ≥ 0, we have a functorial (ii) For every abelian sheaf F of E canonical isomorphism ∼

Ri σ∗ (F )x → Hi (Y f´et , ϕx (F )).

(VI.10.30.2)

e and (iii) For every exact sequence of abelian sheaves 0 → F 0 → F → F 00 → 0 of E every integer i ≥ 0, the diagram Ri σ∗ (F 00 )x

/ Ri+1 σ∗ (F 0 )x

 Hi (Y f´et , ϕx (F 00 ))

 / Hi+1 (Y , ϕx (F 0 )) f´ et

(VI.10.30.3)

where the vertical arrows are the canonical isomorphisms (VI.10.30.2) and the horizontal arrows are the boundary maps of the long exact sequences of cohomology, is commutative. This proposition will be proved in VI.11.14. Remark VI.10.31. We keep the assumptions and notation of VI.10.29 and moreover let y be a geometric point of Y and u : y → X an X-morphism, so that (y x) is a ←

point of X´et ×X´et Y´et (VI.10.18). Denote by x e the closed point of X, by v : y → Y the morphism induced by u, by ye the geometric point of Y defined by v, by ρY : Y ´et → Y f´et the canonical morphism (VI.9.2.1), and by ψye : Y f´et → Ens the fiber functor associated with the point ρY (e y ) of Y f´et . The composed functor e → Ens ψye ◦ ϕx : E

(VI.10.31.1)

is then canonically isomorphic to the fiber functor associated with the point ρ(y e (VI.10.15.3). Indeed, the squares of the diagram of morphisms of topos E Y ´et

(VI.10.31.2)

/X ← ´ et ×X ´et Y ´ et

γ

ρ

ρY

 Y f´et

Ξ

θ

 /E e

x) of

/X ← ´ et ×X´et Y´ et ρ

Φ

 /E e

where ρ and ρ are the canonical morphisms (VI.10.15.3), γ is the morphism (VI.10.25.3), and Ξ is the morphism deduced from the functoriality of the covanishing topos (VI.3.8), are commutative up to canonical isomorphisms: the left square corresponds to the diagram (VI.10.25.6) and the right square to the diagram (VI.10.17.1). On the other hand, y ) to x e; we may therefore consider the there exists a unique specialization arrow from f (e ←

point (e y x e) of X ´et ×X ´et Y ´et . It is clear that γ(e y ) is canonically isomorphic to (e y x e) and that Ξ(e y x e) = (y x). Consequently, ρ(y x) is canonically isomorphic to y ))), giving the assertion. Φ(θ(ρY (e Proposition VI.10.32. Suppose that the schemes X and Y are coherent. For every geometric point x of X, let X(x) be the strict localization of X at x, Y(x) = Y ×X X(x) , and (VI.10.32.1)

e → (Y(x) )f´et ϕx : E

the functor defined in (VI.10.29.3). Then, the family of functors (ϕx ), when x goes through the set of geometric points of X, is conservative.

566

VI. COVANISHING TOPOS AND GENERALIZATIONS

e associated with the points of the form Indeed, the family of fiber functors of E ← ρ(y x), when (y x) goes through the points of X´et ×X´et Y´et , is conservative by virtue of VI.10.21. The proposition follows in view of VI.10.31. e is a Corollary VI.10.33. Under the assumptions of VI.10.32, a morphism u of E monomorphism (resp. epimorphism) if and only if for every geometric point x of X, ϕx (u) is one (VI.10.32.1). This follows from VI.10.32 and ([2] I 6.2(ii)). VI.10.34. We take the assumptions and notation of VI.10.29 and moreover denote by Cx the category of x-pointed étale X-schemes ([2] VIII 3.9), which we identify with ´ /X ([2] IV 6.8.2). It is a cofiltered the category of neighborhoods of x in the site Et category. For every object (U, ξ : x → U ) of Cx , we denote again by ξ : X → U the morphism deduced from ξ ([2] VIII 7.3) and by ξY : Y → U Y

(VI.10.34.1)

e/(U →U )a is canonically equivalent to the Faltings its base change by f . The topos E Y topos associated with the morphism fU : UY → U by (VI.10.14.5). Denote by e/(U →U )a → E e U : E Y

(VI.10.34.2)

e at (UY → U )a . The morphism ξ : X → U then induces the localization morphism of E by functoriality a morphism of topos (VI.10.12.5) e/(U →U )a . e →E Φξ : E Y

(VI.10.34.3) By (VI.10.12.6), the diagram (VI.10.34.4)

e E

Φξ

/E e/(U

Y

β



Y f´et

(ξY )f´et



→U )a βU

/ (UY )f´et

where β and βU are the canonical morphisms (VI.10.6.3), is commutative up to canonical isomorphism. We deduce from this a base change morphism (ξY )∗f´et βU ∗ → β ∗ Φ∗ξ .

(VI.10.34.5) By ([1] 1.2.4(i)), the composition (VI.10.34.6)

(ξY )∗f´et βU ∗ → β ∗ Φ∗ξ → θ∗ Φ∗ξ ,

where the second arrow is induced by (VI.10.24.4), is the base change morphism deduced from the canonical isomorphism (VI.10.24.3) (VI.10.34.7)

∼

βU ◦ Φξ ◦ θ → (ξY )f´et .

b F a = {U 7→ GU } the sheaf of E e associated Let F = {U 7→ FU } be an object of E, a ´ with F , and for every U ∈ Ob(Et/X ), FU the sheaf of (UY )f´et associated with FU . By VI.5.17, {U 7→ FUa } is a presheaf on E and we have a canonical morphism {U 7→ FU } → b inducing an isomorphism between the associated sheaves. We associate {U 7→ FUa } of E, with the presheaf {U 7→ FUa } the functor (VI.10.34.8)

Cx◦ → Y f´et ,

(U, ξ) 7→ (ξY )∗f´et (FUa ),

that with each morphism t : (U 0 , ξ 0 ) → (U, ξ) to Cx associates the composed morphism ∼

(ξY )∗f´et (FUa ) → (ξY0 )∗f´et ((tY )∗f´et FUa ) → (ξY0 )∗f´et (FUa 0 ),

VI.10. FALTINGS SITE AND TOPOS

567

where the first arrow is induced by the relation ξ = t ◦ ξ 0 and the second arrow comes from the transition morphism FUa → (tY )f´et∗ (FUa 0 ) of {U 7→ FUa }. Likewise, we associate with the sheaf F a = {U 7→ GU } the functor (VI.10.34.9)

Cx◦ → Y f´et ,

(U, ξ) 7→ (ξY )∗f´et (GU ).

The canonical morphism {U 7→ FUa } → {U 7→ GU } then induces a morphism of functors from Cx◦ to Y f´et : (VI.10.34.10)

(ξY )∗f´et (FUa ) → (ξY )∗f´et (GU ),

(U, ξ) ∈ Ob(Cx ).

By ([2] III 5.3), we have a canonical isomorphism ∼

βU ∗ (∗U (F a )) → GU .

(VI.10.34.11)

The morphisms (VI.10.34.5) and (VI.10.34.6) therefore induce two functorial morphisms (VI.10.34.12) (VI.10.34.13)

(ξY )∗f´et (GU ) → β ∗ (Φ∗ F a ),

(ξY )∗f´et (GU ) → ϕx (F a ).

By ([1] 1.2.4(i)), these are morphisms of inverse systems on the category Cx◦ (VI.10.34.9). In view of (VI.10.34.10), we deduce from this two functorial morphisms in F (VI.10.34.14)

lim −→

(U,ξ)∈C ◦ x

(VI.10.34.15)

lim −→

(U,ξ)∈C ◦ x

(ξY )∗f´et (FUa ) → β ∗ (Φ∗ F a ), (ξY )∗f´et (FUa ) → ϕx (F a ).

Remarks VI.10.35. We keep the assumptions of VI.10.34. (i) By ([2] XVII 2.1.3), the morphism (VI.10.34.6) is equal to the composition (VI.10.35.1)

∼

(ξY )∗f´et βU ∗ → θ∗ Φ∗ξ βU∗ βU ∗ → θ∗ Φ∗ξ ,

where the first arrow is induced by (VI.10.34.7) and the second arrow is induced by the adjunction morphism βU∗ βU ∗ → id. (ii) We choose an affine object (X0 , ξ0 ) of Cx and denote by I the category of ξ0 pointed étale X0 -schemes that are affine over X0 . The canonical functor I → Cx is then cofinal ([2] VIII 4.5). We may therefore replace in the direct limits in (VI.10.34.14) and (VI.10.34.15) the category Cx by I. Proposition VI.10.36. Under the assumptions of VI.10.34, let moreover y be a geomet← ric point of Y and u : y → X an X-morphism, so that (y x) is a point of X´et ×X´et Y´et (VI.10.18). We denote by v : y → Y the Y -morphism induced by u, by ye the geometric point of Y defined by v, by ρY : Y ´et → Y f´et the canonical morphism (VI.9.2.1), and by ψye : Y f´et → Ens the fiber functor associated with the point ρY (e y ). Let F = {U 7→ FU } a b e ´ /X ), be an object of E, F the sheaf of E associated with F , and for every U ∈ Ob(Et a FU the sheaf of (UY )f´et associated with FU . Then, we have a functorial and canonical isomorphism (VI.10.36.1)

(F a )ρ(y

∼

x)

→

lim −→

ψye((ξY )∗f´et (FUa )),

(U,ξ)∈C ◦ x

whose inverse identifies with the image of the canonical morphism (VI.10.34.15) by the functor ψye. b and that the canonical First, note that {U 7→ FUa } is naturally an object of E a morphism {U 7→ FU } → {U 7→ FU } induces an isomorphism between the associated

568

VI. COVANISHING TOPOS AND GENERALIZATIONS

sheaves, by VI.5.17. Let Pρ(y (VI.10.20). We have a functor (VI.10.36.2)

x)

be the category of ρ(y

φ : Pρ(y

x)

→ Cx ,

x)-pointed objects of E

((V → U ), ξ, ζ) 7→ (U, ξ).

For every (U, ξ) ∈ Ob(Cx ), the fiber of φ over (U, ξ) is canonically equivalent to the y e category D(U,ξ) of ξY (e y )-pointed finite étale UY -schemes (VI.10.34.1). The isomorphism (VI.10.20.2) therefore induces a functorial and canonical isomorphism (VI.10.36.3)

(F a )ρ(y

∼

x)

→

lim

lim

−→

−→

FU (V ).

e (U,ξ)∈C ◦ (V,ζ)∈(D y )◦ x (U,ξ)

In view of (VI.9.3.4) and ([2] IV (6.8.4)), for every (U, ξ) ∈ Ob(Cx ), we have a functorial canonical isomorphism (VI.10.36.4)

∼

ψye((ξY )∗f´et (FUa )) →

FU (V ),

lim −→

y e (V,ζ)∈(D )◦ (U,ξ)

giving the isomorphism (VI.10.36.1). On the other hand, ψye ◦ ϕx is the fiber functor of e associated with the point ρ(y E x), by virtue of VI.10.31. To establish the second assertion, it therefore suffices to show that for every object ((V → U ), ξ, ζ) of Pρ(y x) , the canonical map (VI.10.36.3) FU (V ) → (F a )ρ(y

(VI.10.36.5) is the composition (VI.10.36.6)

x)

FU (V ) → ψye((ξY )∗f´et (FUa )) → ψye(ϕx (F a )),

y e where the first arrow is defined by the object (V, ζ) of D(U,ξ) and the second arrow is the image by ψye of the composition of (VI.10.34.10) and (VI.10.34.13). We may restrict to e so that F a = FU . By localization (VI.10.14), we may the case where F is a sheaf of E, U assume U = X. Denote by prY : Y → Y the canonical projection. By VI.10.35(i), the image by ψye of the morphism (prY )∗f´et (FX ) → ϕx (F ) (VI.10.34.13) coincides with the map

(FX )ρY (y) → Fρ(y

(VI.10.36.7)

x)

induced by the adjunction morphism β ∗ (β∗ (F )) → F and the isomorphism (VI.10.18.2), giving the desired assertion. Corollary VI.10.37. We keep the assumptions of VI.10.34, and moreover let F = {U 7→ b F a the sheaf of E e associated with F , and for every U ∈ Ob(Et ´ /X ), FU } be an object of E, a FU the sheaf of (UY )f´et associated with FU . Then, the canonical morphism (VI.10.34.15) (VI.10.37.1)

lim −→

(U,ξ)∈C ◦ x

(ξY )∗f´et (FUa ) → ϕx (F a )

is an isomorphism. It follows from VI.10.36 and VI.9.6. Proposition VI.10.38. Under the assumptions of VI.10.34, for every sheaf F = {U 7→ e the canonical morphism (VI.10.34.14) FU } of E, (VI.10.38.1)

lim −→

(U,ξ)∈C ◦ x

(ξY )∗f´et (FU ) → β ∗ (Φ∗ F )

is an isomorphism. It follows from VI.10.27 and VI.10.37.

VI.10. FALTINGS SITE AND TOPOS

569

VI.10.39. Let F = {U 7→ FU } be a presheaf of abelian groups on E such that for ´ /X ), FU is a sheaf (of abelian groups) of (UY )f´et (for example, F is an every U ∈ Ob(Et e and let i be an integer ≥ 0. We denote by H i (F ) the sheaf of X´et abelian sheaf of E), ´ /X defined for every U ∈ Ob(Et ´ /X ) by the group associated with the presheaf on Et Hi ((UY )f´et , FU ),

(VI.10.39.1)

´ /X , by the composed map and for every morphism g : U 0 → U of Et (VI.10.39.2)

Hi ((UY )f´et , FU ) → Hi ((UY )f´et , (gY )f´et∗ (FU 0 )) → Hi ((UY0 )f´et , FU 0 ),

where the first arrow is induced by the transition morphism of F and the second arrow is induced by the Cartan–Leray spectral sequence ([2] V 5.3). e associated with F We denote by F a = {U 7→ GU } the sheaf (of abelian groups) of E e/(U →U )a is canonically equivalent to ´ /X ), the topos E ([2] III 6.4). For every U ∈ Ob(Et Y the Faltings topos associated with the morphism UY → U by (VI.10.14.5). We therefore have a canonical morphism of topos (VI.10.6.3) (VI.10.39.3)

e/(U →U )a → (UY )f´et . βU : E Y

By definition of the restriction functor ([2] III 5.3), we have a canonical isomorphism (VI.10.39.4)

∼

βU ∗ (F a |(UY → U )a ) → GU .

Consequently, the Cartan–Leray spectral sequence induces a functorial map in F (VI.10.39.5)

Hi ((UY )f´et , GU ) → Hi ((UY → U )a , F a ).

Composing with the map Hi ((UY )f´et , FU ) → Hi ((UY )f´et , GU ) induced by the canonical morphism F → F a , we obtain a functorial map in F (VI.10.39.6)

Hi ((UY )f´et , FU ) → Hi ((UY → U )a , F a ).

´ /X . The diagram of morphisms of topos Let g : U 0 → U be a morphism of Et (VI.10.39.7)

e/(U 0 →U 0 )a E Y j

βU 0

/ (U 0 )f´et Y (gY )f´et



e/(U →U )a E Y

βU



/ (UY )f´et

e/(U →U )a at (U 0 → U 0 )a , is commutative, up where j is the localization morphism of E Y Y to canonical isomorphism. One immediately verifies this on the corresponding diagram of morphisms of sites. On the other hand, the diagram (VI.10.39.8) u / v βU ∗ (j∗ (F a |(UY0 → U 0 )a )) ∼ / (gY )f´et∗ (βU 0 ∗ (F a |(UY0 → U 0 )a )) βU ∗ (F a |(UY → U )a ) GU

w

/ (gY )f´et∗ (GU 0 )

where u is induced by the adjunction morphism id → j∗ j ∗ , v is induced by the diagram (VI.10.39.7), w is the transition morphism of F a , and the vertical identifications come from the isomorphism (VI.10.39.4), is commutative. We deduce from this that the

570

VI. COVANISHING TOPOS AND GENERALIZATIONS

diagram (VI.10.39.9)

/ Hi ((UY → U )a , F a )

Hi ((UY )f´et , GU )  Hi ((UY )f´et , (gY )f´et∗ (GU 0 ))  Hi ((UY0 )f´et , GU 0 )

 / Hi ((U 0 → U 0 )a , F a ) Y

where the horizontal arrows are the maps (VI.10.39.5), the left vertical arrows are defined in (VI.10.39.2) (for F a ), and the right vertical arrow is the canonical map, is commutative. ´ /X . Taking Consequently, the map (VI.10.39.6) defines a morphism of presheaves on Et the associated sheaves, we obtain a morphism of abelian groups of X´et (VI.10.39.10)

H i (F ) → Ri σ∗ (F a ).

Theorem VI.10.40. Let F = {U 7→ FU } be a presheaf of abelian groups on E, x a geometric point of X, and X(x) the strict localization of X at x. Suppose that f is ´ /X ), FU is a sheaf of (UY )f´et . Then, for every coherent, and that for every U ∈ Ob(Et i i integer i ≥ 0, the stalk H (F )x → R σ∗ (F a )x of the morphism (VI.10.39.10) at x is an isomorphism. This theorem will be proved in VI.11.15. VI.11. Inverse limit of Faltings topos VI.11.1. We denote by M the category of morphisms of schemes and by D the category of morphisms of M. The objects of D are therefore commutative diagrams of morphisms of schemes (VI.11.1.1)

V

/U

 Y

 /X

where we consider the horizontal arrows as objects of M and the vertical arrows as morphisms of M; such an object will be denoted by (V, U, Y, X). We denote by E the full subcategory of D made up of the objects (V, U, Y, X) such that the morphism U → X is étale of finite presentation and that the morphism V → UY = U ×X Y is finite étale. The “target functor” (VI.11.1.2)

E → M,

(V, U, Y, X) 7→ (Y → X),

makes E into a cleaved and normalized fibered category. The fiber over an object f : Y → X of M is the category denoted by Ecoh in VI.10.4. For every commutative diagram of morphisms of schemes (VI.11.1.3)

Y0

f0

g0

 Y

/ X0 g

f

 /X

VI.11. INVERSE LIMIT OF FALTINGS TOPOS

571

the inverse image functor of (VI.11.1.2) associated with the morphism (g 0 , g) of M is the functor (VI.10.12.3) (VI.11.1.4)

(V → U ) 7→ (V ×Y Y 0 → U ×X X 0 ).

Φ+ : Ef → Ef 0 ,

Endowing each fiber of E/M with the covanishing topology (VI.10.4), E becomes a fibered U-site (cf. VI.10.12 and [2] VI 7.2.4). We denote by F→M

(VI.11.1.5)

the fibered U-topos associated with E/M ([2] VI 7.2.6): the fiber of F over an object e f of sheaves of U-sets on the covanishing site Ef , and f : Y → X of M is the topos E the inverse image functor relative to the morphism defined by the diagram (VI.11.1.3) ef → E e f 0 , inverse image under the morphism of topos Φ : E ef 0 → E ef is the functor Φ∗ : E + associated with the morphism of sites Φ (VI.11.1.4). We denote by F∨ → M◦

(VI.11.1.6)

the fibered category obtained by associating with each object f : Y → X of M the e f , and with each morphism defined by a diagram (VI.11.1.3) the functor category Ff = E e e ef 0 → E ef . Φ∗ : Ef 0 → Ef , direct image by the morphism of topos Φ : E VI.11.2. (VI.11.2.1)

Let I be an essentially small cofiltered category ([2] I 2.7 and 8.1.8) and ϕ : I → M,

i 7→ (fi : Yi → Xi )

a functor such that for every morphism j → i of I, the morphisms Yj → Yi and Xj → Xi are affine. We suppose that there exists i0 ∈ Ob(I) such that Xi0 and Yi0 are coherent. We denote by Eϕ Fϕ F∨ ϕ

(VI.11.2.2) (VI.11.2.3) (VI.11.2.4)

→ I → I → I◦

the site, topos, and fibered category deduced from E (VI.11.1.2), F (VI.11.1.5), and F∨ (VI.11.1.6), respectively, by base change by the functor ϕ. Note that Fϕ is the fibered topos associated with Eϕ ([2] VI 7.2.6.8). By ([42] 8.2.3), the inverse limits (VI.11.2.5)

X = lim Xi and Y = lim Yi ←−

←−

i∈Ob(I)

i∈Ob(I)

are representable in the category of schemes. The morphisms (fi )i∈I induce a morphism f : Y → X, that represents the inverse limit of the functor (VI.11.2.1). For every i ∈ Ob(I), we have a canonical commutative diagram (VI.11.2.6)

Y  Yi

f

fi

/X  / Xi

There corresponds to it an inverse image functor (VI.11.1.4) (VI.11.2.7)

Φ+ i : Efi → Ef ,

that is continuous and left exact, and consequently a morphism of topos ef → E ef . (VI.11.2.8) Φi : E i

We have a natural functor (VI.11.2.9)

Eϕ → Ef ,

572

VI. COVANISHING TOPOS AND GENERALIZATIONS

whose restriction to the fiber over any i ∈ Ob(I) is the functor Φ+ i (VI.11.2.7). This functor transforms Cartesian morphisms into isomorphisms. It therefore factors uniquely through a functor ([2] VI 6.3) lim Eϕ → Ef .

(VI.11.2.10)

−→ I◦

The I-functor Eϕ → Ef × I deduced from (VI.11.2.9) is a Cartesian morphism of fibered sites ([2] VI 7.2.2). It therefore induces a Cartesian morphism of fibered topos ([2] VI 7.2.7) e f × I → Fϕ . (VI.11.2.11) E e f equipped with the morphism (VI.11.2.11) is an Proposition VI.11.3. The topos E inverse limit of the fibered topos Fϕ /I ([2] VI 8.1.1). First, note that the functor (VI.11.2.10) is an equivalence of categories by virtue of ([42] 8.8.2, 8.10.5 and 17.7.8). Let T be a U-topos, h : T × I → Fϕ

(VI.11.3.1)

a Cartesian morphism of fibered topos over I. Denote by εI : Eϕ → Fϕ the canonical Cartesian functor ([2] VI (7.2.6.7)), and set h+ = h∗ ◦ εI : Eϕ → T × I.

(VI.11.3.2)

For every i ∈ Ob(I), we denote by (VI.11.3.3)

h+ i : Efi → T

the restriction of h+ to the fiber over i. By the equivalence of categories (VI.11.2.10) and ([2] VI 6.2), there exists essentially a unique functor g + : Ef → T

(VI.11.3.4)

such that h+ is isomorphic to the composition (VI.11.3.5)

g + ×idI

/ Ef × I

Eϕ

/ T × I,

where the first arrow is the functor deduced from (VI.11.2.9). Let us show that g + is a morphism of sites. For every object (V → U ) of Ef , there exist i ∈ Ob(I), an object (Vi → Ui ) of Efi , and an isomorphism of Ef (VI.11.3.6)

∼

(V → U ) → Φ+ i (Vi → Ui ).

+ + Since the functors h+ is left exact. i and Φi are left exact, we deduce from this that g On the other hand, every finite Cartesian (resp. vertical) covering of Ef (VI.5.3) is the inverse image of a Cartesian (resp. vertical) covering of Efi for an object i ∈ I, by virtue of ([42] 8.10.5(vi)). Since the schemes X and Y are coherent, we deduce from this that g + transforms Cartesian (resp. vertical) coverings of Ef into epimorphic families of T . Consequently, g + is continuous by virtue of VI.5.10. It therefore defines a morphism of topos ef (VI.11.3.7) g: T → E

such that h is isomorphic to the composition (VI.11.3.8)

T ×I

g×idI

/E ef × I

/ Fϕ ,

where the second arrow is the morphism (VI.11.2.11). Such a morphism g is essentially unique because the “restriction” g + : Ef → T of the functor g ∗ is essentially unique by the above, giving the proposition.

VI.11. INVERSE LIMIT OF FALTINGS TOPOS

573

VI.11.4. We endow Eϕ with the total topology ([2] VI 7.4.1) and denote by Top(Eϕ ) the topos of sheaves of U-sets on Eϕ . By ([2] VI 7.4.7), we have a canonical equivalence of categories (VI.2.2) ∼

Top(Eϕ ) → HomI ◦ (I ◦ , F∨ ϕ ).

(VI.11.4.1)

On the other hand, the natural functor Eϕ → Ef (VI.11.2.9) is a morphism of sites ([2] VI 7.4.4) and therefore defines a morphism of topos e f → Top(Eϕ ). (VI.11.4.2) $: E By virtue of VI.11.3 and ([2] VI 8.2.9), there exists an equivalence of categories Θ that fits into a commutative diagram (VI.11.4.3)

/ Homcart/I ◦ (I ◦ , F∨ ϕ) _

Θ ∼

ef E $∗

 Top(Eϕ )

 / HomI ◦ (I ◦ , F∨ ϕ)

∼

where the bottom horizontal arrow is the equivalence of categories (VI.11.4.1) and the right vertical arrow is the canonical injection. For every object F ∈ Ob (Top(Eϕ )), if {i 7→ Fi } is the corresponding section of HomI ◦ (I ◦ , F∨ ϕ ), we have a functorial canonical isomorphism ([2] VI 8.5.2) ∼

$∗ (F ) → lim Φ∗i (Fi ).

(VI.11.4.4)

−→

i∈I ◦

Corollary VI.11.5. Let F be a sheaf of Top(Eϕ ) and {i 7→ Fi } the associated section of HomI ◦ (I ◦ , F∨ ϕ ) under the equivalence of categories (VI.11.4.1). Then we have a functorial canonical isomorphism ∼ e f , Fi ) → e f , lim Φ∗ (Fi )). Γ(E (VI.11.5.1) lim Γ(E −→

i

−→

i

i∈I ◦

i∈I ◦

Corollary VI.11.6. Let F be an abelian sheaf of Top(Eϕ ) and {i 7→ Fi } the section of HomI ◦ (I ◦ , F∨ ϕ ) associated with it by the equivalence of categories (VI.11.4.1). Then for every integer q ≥ 0, we have a functorial canonical isomorphism ∼ e f , Fi ) → e f , lim Φ∗ (Fi )). (VI.11.6.1) lim Hq (E Hq (E −→

i

−→

i∈I ◦

i

i∈I ◦

Corollaries VI.11.5 and VI.11.6 follow from VI.11.3 and ([2] VI 8.7.7). Note that the conditions required in ([2] VI 8.7.1 and 8.7.7) are satisfied by virtue of VI.10.5 and ([2] VI 3.3, 5.1, and 5.2). VI.11.7. (VI.11.7.1) (VI.11.7.2) (VI.11.7.3)

We denote by Rϕ Gϕ Gϕ∨

→ I, → I, → I◦

the fibered site, topos, and category deduced, respectively, from the fibered site of finite étale covers R/Sch (VI.9.5.1), the fibered topos G /Sch (VI.9.5.2), and the fibered category G ∨ /Sch◦ (VI.9.5.3), by base change by the functor (VI.11.7.4)

I → Sch,

i 7→ Yi

induced by ϕ (VI.11.2.1). For every i ∈ Ob(I), we denote by (VI.11.7.5)

ti : Y → Yi

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VI. COVANISHING TOPOS AND GENERALIZATIONS

the canonical morphism (VI.11.2.5). We have a natural functor Rϕ → RY

(VI.11.7.6)

whose restriction to the fiber over i ∈ Ob(I) is given by the base change functor by the morphism ti RYi → RY ,

Yi0 7→ Yi0 ×Yi Y.

This functor transforms Cartesian morphisms into isomorphisms, and therefore factors uniquely through a functor lim Rϕ → RY .

(VI.11.7.7)

−→ I◦

The I-functor Rϕ → RY ×I deduced from (VI.11.7.6) is a Cartesian morphism of fibered sites ([2] VI 7.2.2). It therefore induces a morphism of fibered topos Yf´et × I → Gϕ .

(VI.11.7.8)

Lemma VI.11.8. The functor (VI.11.7.7) is an equivalence of sites when we endow the source with the topology of the direct limit of the fibered site Rϕ ([2] VI 8.2.5) and the target with the étale topology. Indeed, the functor (VI.11.7.7) is an equivalence of categories by virtue of ([42] 8.8.2, 8.10.5, and 17.7.8). Let i ∈ Ob(I), gi : Yi0 → Yi be a finite étale cover and g : Y 0 → Y the finite étale cover deduced from gi by base change by the morphism Y → Yi . By ([42] 8.10.5), g is surjective if and only if there exists a morphism j → i of I such that the finite étale cover gj : Yj0 → Yj deduced from gi by base change is surjective. The assertion concerning the topologies follows in view of ([2] VI 8.2.2 and III 1.6). Proposition VI.11.9. The topos Yf´et equipped with the morphism (VI.11.7.8) is an inverse limit of the fibered topos Gϕ /I. This follows from VI.11.8 and ([2] VI 8.2.3). Corollary VI.11.10. Let F be an abelian sheaf of the total topos of Rϕ and {i 7→ Fi } the section of HomI ◦ (I ◦ , Gϕ∨ ) associated with it. Then for every integer q ≥ 0, we have a functorial canonical isomorphism (VI.11.10.1)

∼

lim Hq ((Yi )f´et , Fi ) → Hq (Yf´et , lim (ti )∗f´et (Fi )). −→

−→

i∈I ◦

i∈I ◦

This follows from VI.11.9 and ([2] VI 8.7.7). Note that the conditions required in ([2] VI 8.7.1 and 8.7.7) are satisfied by virtue of VI.9.12 and ([2] VI 3.3, 5.1, and 5.2). VI.11.11.

We denote by

(VI.11.11.1)

e f → Yf´et β: E

and, for every i ∈ Ob(I), by (VI.11.11.2)

e f → (Yi )f´et βi : E i

the canonical morphisms (VI.10.6.3). It follows from (VI.10.12.6) and ([37] VI 12; cf. also [1] 1.1.2) that there exists essentially one Cartesian morphism of fibered topos (VI.11.11.3)

βϕ : Fϕ → Gϕ

VI.11. INVERSE LIMIT OF FALTINGS TOPOS

575

whose fiber over each i ∈ Ob(I) is the morphism βi . Moreover, the diagram of morphisms of fibered topos (VI.11.11.4)

ef × I E  Fϕ

β×id

/ Yf´et × I  / Gϕ

βϕ

where the vertical arrows are the morphisms (VI.11.2.11) and (VI.11.7.8), is commutative up to canonical isomorphism. We can therefore identify β with the morphism deduced from βϕ by taking the inverse limit in the sense of ([2] VI 8.1.4). Proposition VI.11.12. Let F be a sheaf of Top(Eϕ ) and {i 7→ Fi } the section of HomI ◦ (I ◦ , F∨ ϕ ) associated with it by the equivalence of categories (VI.11.4.1). Then we have a functorial canonical isomorphism ∼

lim (ti )∗f´et (βi∗ (Fi )) → β∗ ($∗ F ),

(VI.11.12.1)

−→

i∈I ◦

where $ is the morphism (VI.11.4.2) and ti is the morphism (VI.11.7.5). This follows from VI.11.11 and ([2] VI 8.5.9). Note that the conditions required in loc. cit. are satisfied by virtue of VI.10.5, VI.10.10, and ([2] VI 3.3 and 5.1). Remark VI.11.13. Let F be an abelian sheaf of Top(Eϕ ), {i → 7 Fi } the object of ) associated with it by the equivalence of categories (VI.11.4.1), and q an HomI ◦ (I ◦ , F∨ ϕ integer ≥ 0. It then follows from VI.10.13 that the diagram (VI.11.13.1)

lim Hq ((Yi )f´et , βi∗ (Fi )) −→

i∈I ◦

e f , Fi ) Hq (E / lim i −→ i∈I ◦

u

 lim Hq (Yf´et , (ti )∗f´et (βi∗ (Fi )))

w

−→

i∈I ◦

v

 Hq (Yf´et , β∗ ($∗ F ))

 / Hq (E e f , $∗ F )

where the horizontal arrows come from the Cartan–Leray spectral sequences, u is the canonical morphism, v is induced by (VI.11.12.1), and w is induced by (VI.11.4.4), is commutative. VI.11.14. We can now prove Proposition VI.10.30. We choose an affine étale neighborhood X0 of x in X. We denote by I the category of x-pointed étale X0 -schemes that are affine over X0 (cf. [2] VIII 3.9 and 4.5), and by ϕ : I → M the functor that with each object U of I, associates the canonical projection fU : UY → U . Then, f identifies canonically with the inverse limit of the functor ϕ. For every U ∈ Ob(I), the topos e/(U →U )a is canonically equivalent to the Faltings topos associated with the morphism E Y fU by (VI.10.14.5). Consequently, with the notation of this section, for every sheaf F of e {U 7→ F |(UY → U )a } is naturally a section of HomI ◦ (I ◦ , F∨ E, ϕ ). It therefore defines a sheaf of Top(Eϕ ) (VI.11.4.1). We have a functorial canonical isomorphism (VI.11.4.4) (VI.11.14.1)

∼

Φ∗ (F ) → $∗ ({U 7→ F |(UY → U )a }).

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VI. COVANISHING TOPOS AND GENERALIZATIONS

(i) By ([2] VIII 3.9 and 4.5), we have a canonical isomorphism (VI.11.14.2)

∼

σ∗ (F )x → lim Γ((UY → U )a , F ). −→

U ∈I ◦

This induces a functorial isomorphism ∼ e Φ∗ (F )), σ∗ (F )x → Γ(E,

(VI.11.14.3)

by virtue of VI.11.5. The statement follows in view of VI.10.28(i). (ii) This follows, like (i), from VI.11.6, VI.10.28(ii), and the canonical isomorphism ([2] V 5.1(1)) (VI.11.14.4)

∼

Ri σ∗ (F )x → lim Hi ((UY → U )a , F ). −→

U ∈I ◦

(iii) This immediately follows from the proof of (ii). VI.11.15. We can finally prove Theorem VI.10.40. We choose an affine étale neighborhood X0 of x in X. We denote by I the category of x-pointed étale X0 -schemes that are affine over X0 , and by ϕ : I → M the functor that with each object U of I, associates the canonical projection fU : UY → U . By ([2] IV (6.8.4)) and VI.11.10, we have canonical isomorphisms (VI.11.15.1)

∼

∼

H i (F )x → lim Hi ((UY )f´et , FU ) → Hi (Y f´et , lim (ξY )∗f´et FU ). −→

−→

U ∈I ◦

U ∈I ◦

By virtue of VI.10.30(ii), we have an isomorphism (VI.11.15.2)

∼

Ri σ∗ (F a )x → Hi (Y f´et , ϕx (F a )).

On the other hand, it follows from VI.11.13 and from the definitions that the diagram (VI.11.15.3)

H i (F )x

i (ξY )∗f´et FU ) / H (Y f´et , lim −→

∼

U ∈I ◦

u

 Ri σ∗ (F a )x

v ∼

 / Hi (Y , ϕx (F a )) f´ et

where the horizontal arrows are the isomorphisms (VI.11.15.1) and (VI.11.15.2), u is the stalk of the morphism (VI.10.39.10) at x, and v is induced by the isomorphism (VI.10.37.1), is commutative. Consequently, u is an isomorphism, giving the theorem.

Facsimile : A p-adic Simpson correspondence by Gerd Faltings Advances in Mathematics 198 (2005), 847–862

577

Advances in Mathematics 198 (2005) 847 – 862 www.elsevier.com/locate/aim

A p-adic Simpson correspondence Gerd Faltings∗ Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany Received 28 May 2004; accepted 25 May 2005 Communicated by Johan De Jong Dedicated to M. Artin on the occasion of his 70th birthday

Abstract For curves over a p-adic field we construct an equivalence between the category of Higgsbundles and that of “generalised representations” of the étale fundamental group. The definition of “generalised representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. © 2005 Elsevier Inc. All rights reserved. Keywords: Almost étale extensions; Higgs-bundles; p-adic Hodge theory

1. Introduction The purpose of this note is to construct Higgs-bundles associated to representations of the geometric fundamental group of a curve over a p-adic field K. It thus can be considered a p-adic analogue of the results of Simpson and Corlette (see [12]). The functor is fully faithful but it is difficult to characterise its image: namely the resulting Higgs-bundles are semistable of slope zero, but we do not know whether any such Higgs-bundle lies in the image (this is true for line-bundles on curves over p-adic local fields). Conversely we can construct for all Higgs-bundles so-called “generalised representations”, which form a category containing the usual representations as full subcategory. However, we do not know which of those come from genuine representations. ∗ Fax: +49 228 402 277.

E-mail address: [email protected]. 0001-8708/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2005.05.026

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G. Faltings / Advances in Mathematics 198 (2005) 847 – 862

Over a local field K and for line-bundles one can check that one gets all line-bundles of degree zero, so one can hope that over local fields all semistable Higgs-bundles of degree zero lie in the image. Recall that a Higgs-bundle on an algebraic manifold X is a pair (E,
), where E is a vectorbundle on X and
a global section of End(E) ⊗
X satisfying
∧
= 0 (that is in local coordinates the components of
commute). We also use variants where X is only logsmooth, or where
has coefficients in some Tate-twist. The latter corresponds (I assume) to a factor 2i in the classical complex setup. We should note that our functors depend on certain choices, the most important being that of an exponential function for the multiplicative group. This induces exponential functions on all commutative group schemes over K. Another choice is that we have to choose lifts to certain types of dual numbers, and different lifts amount to twists by “Higgs-line-bundles”. That is we do not obtain just one functor but a whole family of them, all related by such twists. The method used in the proofs is the theory of almost étale extensions (see [6], for a more systematic treatment [11]) which was developed by the author for applications in p-adic Hodge-theory. We develop a nonabelian Hodge–Tate-theory. What is still missing is an appropriate role for connections and Frobenius, which might result in a more powerful theory generalising Fontaine’s ideas. This work was inspired by the workshop on nonabelian Hodge-theory at MSRI Berkeley, at Easter 2002. We also mention the preprint [9] which uses similar techniques in a different setting.

2. Generalised representations We denote by V a complete discrete valuation-ring with perfect residue-field k of characteristic p > 0 and fraction-field K of characteristic 0. K¯ is the algebraic closure ¯ X is a proper V-scheme which has toroidal of K and V¯ the integral closure of V in K. singularities (as explained in [6, Chapter 2, Appendix 1]), for example X could be smooth or have semistable singularities. Further more D ⊂ X is a divisor which satisfies the conditions in [6]. Especially the generic fibre XK is smooth and DK a divisor with normal crossings. As in [6] X◦ = X − D. We have a topos X ◦ of sheaves on the situs whose objects consists of finite étale coverings of the generic fibres UK◦ of schemes U → X which are étale over X. The normalisation of OU in such covers defines a sheaf O¯ on X ◦ . Furthermore if L is the locally constant sheaf on XK◦ associated to a representation of the fundamental group ◦ ) on a finite V-module the associated maps 1 (XK
◦ 
◦  i ¯ ¯ H i XK ¯ , L ⊗ V → H XK¯ , L ⊗ O are almost isomorphisms, that is their kernels and cokernels are annihilated by the maximal ideal of V¯ (see [6, Chapter 4, Theorem 9] for curves also [4]). This remains ◦ because true if L is only a locally constant étale sheaf on the geometric fibre XK ¯

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such a sheaf is already defined over a finite extension of K. It follows that the functor ◦ ) on L ⊗ O¯ is fully faithful as a functor from continuous representations of 1 (XK ¯ ¯ finitely presented torsion V¯ -modules, to O-modules up to almost isomorphisms. (In the first category almost maps coincide with usual maps). Also the essential image is closed under extensions and deformations. Deformations arise if we have a family of representations over a complete local ring, and consider various base-changes to V¯ /(p s ). If one of them lies in the essential image so do all. In the following we restrict to representations on free modules over V¯ /(ps ) which correspond to vectorbundles over s ). We call the latter generalised representations. ¯ O/(p 3. The local structure of generalised representations Now assume given an affine U = Spec(R) ⊂ X which is small, that is R is étale over a toroidal model. By adjoining roots of characters of the torus we obtain a subextension R∞ of R¯ (the integral closure of R in the maximal étale cover of UK◦ ) ˆ d . We write it as the union which is Galois over R1 = R ⊗V V¯ with group ∞ = Z(1) of algebras Rn which have themselves toroidal singularities. Furthermore R¯ is almost ¯ étale over R∞ ([6, Section 2, Theorem 4]). This implies that each R-module with a ¯ continuous semilinear action of  = Gal(R/R) is almost induced from an R∞ -module s )) (with s ¯ with ∞ -action. Also we can compute the Galois-cohomology H i (, R/(p i any positive rational number): namely it is almost isomorphic to H (∞ , R∞ /(p s )) which in turn is the direct sum of H i (∞ , R ⊗V V¯ /(p s )) and of a direct summand annihilated by p 1/(p−1) (see [6, p. 206]). This results from the decomposition of R∞ into ∞ -eigenspaces where the contributions from nontrivial eigenspaces are annihilated by −1,  a nontrivial root of unity. Finally the (interesting) first summand is canonically ˜ i ⊗V V¯ /(p s ). Examples of such (locally identified with the logarithmic differentials
R/V defined) generalised representations are given by homomorphisms 
∞ → GL(r, R ⊗V V¯ /(p s )), we show that many others are close to these. ¯ R¯ r /(p s ) a generalised Lemma 1. Suppose  > 1/(p−1) is a rational number, and M

representation (it admits a semilinear -operation). (i) Suppose that M¯ is trivial modulo p2 . Then its reduction modulo ps− is given by a representation ∞ → GL(r.R ⊗V V¯ /(p s− )), and this representation is trivial modulo p . (ii) Suppose given two representations ∞ → GL(ri , R ⊗V V¯ /(p s )), trivial modulo p  , and an R¯ − -linear map between the associated generalised representations. Then its reduction modulo ps− is given by an R1 − ∞ -linear map of representations. Proof. For (ii) we consider the representation on the Hom-space, which we call M, and have to show that -invariants in M¯ = M ⊗ R¯ come from ∞ -invariants in M. This

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G. Faltings / Advances in Mathematics 198 (2005) 847 – 862

¯ is almost isomorphic to H 0 (∞ , M ⊗ R∞ ), and the latter follows because H 0 (, M) decomposes into a direct sum corresponding to the eigenspace decomposition of R∞ . On nontrivial eigenspaces some element of ∞ operates as the sum of a nontrivial root of unity  and of an endomorphism divisible by p  and the corresponding contribution is annihilated by p . For (i) we choose a positive rational  with  > 3 + 1/(p − 1), and show by induction over n that the assertion holds for the representation modulo p2+n . We may assume that s 3 + n. For n = 0 M¯ modulo p 2 is by assumption induced from a (trivial) M. Assume ¯ If we try to lift M to a we have found such an M modulo p 2+n , inducing M. 3  +n  representation modulo p we encounter an obstruction in H 2 (∞ , End(M)/(p )) 2 ¯ whose image in H (, End(M)/(p )) vanishes because M¯ lifts. As the induced map is almost injective (over R∞ it is a direct summand) the obstruction vanishes after multiplication by p , that is M modulo p 2+(n−1) lifts to a representation modulo p 3+(n−1) . Over R¯ the induced generalised representation differs from M¯ by a class  )). Again this cohomology is almost isomorphic to the direct ¯ in H 1 (, End(M)/(p sum of H 1 (∞ , End(M)/(p )) and terms annihilated by p 1/(p−1) . Hence our class becomes “constant” after multiplication by p −2 and vanishes after modifying the lift of M, which is now a lift from coefficients modulo p+(n+1) to coefficients modulo p 2+(n+1) . This finishes the proof.  Remarks. (i) The result extends to p-adic representations: this follows from the inductive method of liftings. (ii) For  > 1/(p − 1) the exponential and logarithmic series converge for arguments divisible by p  . Applying the logarithm to the images of generators of ∞ then defines ˜ R/V ⊗ V¯ (−1), divisible by p  , and endomorphisms of M or an element of End(M) ⊗
with commuting components. We shall see that this element is independent of the choices involved in the construction of R∞ , by defining an inverse functor which associates to such “Higgs-bundles” a generalised representation. Namely consider Fontaine’s rings Ainf (R) and Ainf (V ) associated to R and V (see [8]). Ainf (V ) surjects onto the p-adic completion Vˆ¯ and the kernel is a principal ideal with generator . Here we only need the quotient A2 (V ) = Ainf (V )/(2 ), which is an extension of Vˆ¯ by Vˆ¯ . The latter contains canonically Vˆ¯ (1) = p1/(p−1) Vˆ¯ . Similar ¯ hold for A2 (R). Also A2 (V ) and A2 (R) have natural assertions (with V¯ replaced by R) toroidal (or logarithmic) structures. Now we first lift R ⊗V Vˆ¯ to a log-smooth algebra R˜ over A2 (V ). Two such lifts are isomorphic but not canonically isomorphic. Namely the automorphism group of ˜ R/V , R ⊗V Vˆ¯ ). Also an étale a lift is the group of logarithmic derivations HomR (
map from Spec(R) to a toroidal model defines such an R˜ induced from the toroidal model. Next we lift R ⊂ Rˆ¯ to an A2 (V )-linear R˜ → A2 (R). Again two lifts differ by a ˆ¯ If M R ⊗ V¯ r /(p s ) denotes a free module together logarithmic derivation into R. V ˜ R/V such that
∧
= 0 (that is
defines with an endomorphism
∈ End(M) ⊗R


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a Higgs-bundle, or the components of
commute), and
is divisible by p for some  > 1/(p − 1), then the two pushforwards of M via different lifts R˜ → A2 (R) are canonically isomorphic: Use the fact that
behaves like a connection. For example if R is smooth over V, ti ∈ R form local coordinates with associated derivations *i = */*ti , and if the two ˆ¯ then an isomorphism is given by the Taylor-series lifts differ on ti by ui  ∈ R, 


(*)I (m)/I ! ⊗ uI .

I

Here m ∈ M, the sum is over all multi-indices I = (i1 , . . . , id ), and there is an obvious divided power structure to explain how to divide by the factorials. For more general logarithmic coordinates one has to use the logarithmic Taylorseries: for one variable t with derivative *˜ = t*/*t and two lifts differing by tu one obtains 

˜ n (m)/n! ⊗ un ,
(*)

and the same for several variables. Note that the pushforward is always the same module M ⊗ Rˆ¯ but that different lifts of R˜ induce nontrivial automorphisms of this module. ˆ¯ −1 . Finally everything Also
should be considered as an element of End(M) ⊗ R extends to p-adic Higgs-bundles, by passing to a p-adic limit. ˆ¯ Now the Galois-group  acts on lifts R˜ → A2 (R) and thus semilinearly on M ⊗ R. If we choose an étale map from Spec(R) to a toroidal model we obtain a lift R˜ mapping to A2 (R) by extracting p-power roots out of characters of the torus (which map to elements of R). Also we have an R∞ and  acts on the preferred lifting via ˆ d . That group acts in turn on the ith coordinate via its ith its quotient ∞ = Z(1) ˆ¯ ˆ¯ It follows that
induces the previous ˆ projection to Z(1) and the inclusion R(1) ⊂ R. ˆ¯ −1 ⊂ R(−1). ˆ¯ ˜ R/V (−1) via R element of End(M) ⊗R
As conclusion we get that for a s  Higgs-bundle modulo p with
divisible by p ( > 1/(p − 1)) we get an associated generalised representation modulo ps which will be trivial modulo p  for any  <  + 1/(p − 1). Conversely, if we start with a generalised representation modulo (ps ) which is trivial modulo p we obtain a bundle with endomorphism modulo (p s− ), and dividing this endomorphism by p1/(p−1) gives a Higgs-bundle modulo p s−−1/(p−1) . These procedures are inverse up to a loss of exponents s which can be bounded by any  > 2/(p − 1). Passing to the limit s → ∞ we obtain an equivalence of categories between generalised p-adic representations which are trivial modulo p for some  > 2/(p − 1), and p-adic Higgs-bundles with
divisible by p  for some  > 1/(p − 1). We formalise this as follows: Definition 2. A Higgs-module (M,
) (M a finitely generated free R-module,
∈ ˆ¯ −1 ) is called small if
is divisible by p for some  > End R (M) ⊗R
R/V ⊗R R 1/(p − 1). A generalised representation M¯ is called small if it is trivial modulo p 2 .

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Thus our local theory gives a bijection between small Higgs-modules and small generalised representations. There also exists a Qp -theory where we consider continuous -representations on ˆ¯ finitely generated projective R[1/p]-modules and projective Rˆ 1 [1/p]-modules with an ˜ X/V (−1),
∧
= 0. For simplicity we do this only endomorphism
with coefficients
for the case of curves, that is for relative dimension d = 1. ˆ V V¯ [1/p]-module Suppose first that we are given a finitely generated projective R ⊗ ˜ M with an
X/V (−1)-valued endomorphism
. Then the coefficients i of the chari i acteristic polynomial of
(that is the traces of M) are elements of
on ⊗i ˜ ˆ V V¯ [1/p]. If we assume that they are integral and divisible by p 2i  for
X/V ⊗ R ⊗ ˆ V V¯ -submodule M ◦ ⊂ M some  > 1/(p − 1) we can find a finitely generated R ⊗ 2  which is stable under
/p and which generates M. The previous constructions (using local lifts of R to A2 (V ) and to A2 (R)) then define a representation of  on the p-adic ˆ completion of R¯ · M. Less canonically the quotient ∞ = Z(1) acts by exponentiating
. Thus we get a functor from small Qp -Higgs-bundles (“small” is now defined in terms of divisibility of the coefficients i ) to generalised Qp -representations. ˆ¯ For the converse assume  operates semilinearly on a projective R[1/p]-module ˆ¯ M¯ and assume it is generated by a finitely generated R-submodule M¯ ◦ such that ◦ 2  M¯ is generated by elements which are -invariant modulo p M¯ ◦ , for some  > 1/(p − 1). Replacing  by a slightly smaller  we may replace M¯ ◦ by its ¯ ∞ ) and consider the problem with coefficients Rˆ ∞ . Then invariants under Gal(R/R ˆ on generators of M¯ ◦ we find a small lifting the action of a generator of ∞ = Z(1) n ˆ action of ∞ on some R∞ and a ∞ -linear surjection of this module onto M¯ ◦ . That is M¯ becomes the quotient of the Qp -object defined by an integral small generalised representation. Applying the same reasoning to the kernel we get a resolution by integral small generalised representations. The cokernel of the induced map on associated Higgs-bundles then defines the inverse functor. Both functors are fully faithful: using resolutions and duals we reduce to the previous results for integral objects. Thus: Theorem 3. The construction above defines, for small toroidal affines, an equivalence of categories between small generalised representations and small Higgs-bundles, or generalised Qp -representations and Higgs-bundles (both assumed small) on the generic fibre. We can use these methods to compare cohomologies, first in a locals setting but in such a way that it will globalise later. Suppose M is a finitely generated projective Rˆ 1 -module with a Higgs-field
such that all invariants i are divisible by p 2i  ,  > 1/(p − 1). Then we can find a sublattice M 0 ⊂ M with
(M 0 ) ⊆ ˜ R/V −1 , and a representation of  on M 0 which induces a generalised p 2 M 0 ⊗
¯ can be, up to almost isomor¯ The cohomology-groups H i (, M) representation M. ˆ d (1). Decomposphism, computed by reduction to R∞ and the action of ∞ = Z ing R∞ into eigenspaces we see that the contribution of nontrivial eigenspaces is

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annihilated by p 1/(p−1) eigenspace has the same cohomology as  , while the trivial ˜ R/V ⊗ Rˆ 0 −1 ) ⊗ M 0 , with exterior multiplication by
as the Koszul-complex ∗ (
differential. A map of complexes inducing this isomorphism can be constructed as follows:  n ˜ ˆ −1 Consider the symmetric algebra S = n  0 S (
R/V ⊗ R0  ). It has a Higgs ˜ R/V and the *i the dual field
= − *i ⊗ i , where the i are a local basis of
derivations (of degree −1) on the symmetric algebra. The Koszul-complex of −
S is a resolution of R0 . Also the Higgs-field
S has divided powers
nS /n!, and the associated exponential series is finite when applied to elements of S. ˆ Sˆ also has a Higgs-field and by the usual proceThe completed tensorproduct M 0 ⊗ dure we obtain -representations on the p-adic completion of the tensorproduct with ¯ This gives a resolution of M¯ in the category of continuous -representations, R. and as for any such resolution its -invariants map in the derived category to the complex representing cohomology (in fact the analogue also holds for all coefficients M 0 /p s M 0 , thus one can avoid continuous cohomology with Qp -coefficients). Finally the Koszul-complex for M 0 maps into the complex of invariants, by sending m to the  n sum n  0
(m)/n! in the p-adic completion of M 0 ⊗ S. For general Qp -representations we denote R0 = R ⊗V V¯ and by Rn the result of adjoining all roots of order n! of the toroidal coordinates, so that R0 ⊂ Rn ⊂ R∞ , and the Rn are still toroidal over V¯ . We now pass to an Rn where our representation ˜ ⊗i −i . becomes small, thus obtaining invariants i in the p-adic completion of
Rn /V ˆ˜ ⊗i These are invariant under Gal(Rn /R0 ) and thus define elements in
R/V ⊗R R0 [1/p]−i . We claim that they are independent of the choice of the Rn : Lemma 4. The invariants i do not depend on the choice of R∞ and commute with basechange. Proof. We use that we have found a lift of R to A2 (V ) and to A2 (R) such that for each ∈  and a in the lift we have (a) − a ∈ Rˆ 0 , and for in a subgroup n of finite index this is divisible by p n+ . Assume more generally that we have a finite ˆ if lies in the subgroup S subextension R0 ⊂ S ⊂ R¯ such that (a) − a ∈ p n+ S −1 −n r ˜ fixing S, and an element
∈ p
R/V ⊗ End(S ) . Then we can define an action r of S on Sˆ r and on Rˆ¯ by the Taylor-series above, which converges because of the assumptions on divisibility. We also assume that there are subgroups S,m ⊂ , of finite index, such that (a) − a is divisible by p m+n+ if ∈ S,m . Now assume that two such data define representations which become isomorphic after tensoring with Qp . We then claim that the two
’s have the same coefficients i in their characteristic polynomials. To show this we are allowed to enlarge the S’s and may assume that they coincide. The two lifts of R to A2 (R) then differ by ˆ¯ Its reduction modulo pm+n+ is invariant under a logarithmic derivation R → R. S,m . As the discriminant of the compositum of S and Rs over Rs (our original sequence) divides p for big s we can pass to this compositum and then have that after multiplication by p our S,m -invariants are S-linear combinations of invariants under

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¯ s ). However, from the explicit decomposition of R∞ into ∞ -eigenspaces Gal(R/R it follows that such invariants modulo pm+n+ are almost sums of elements of Rs  and elements annihilated by p1/(p−1) . Thus multiplying again by (say) p 2 we see that our invariant lifts. That means after replacing S by a smaller subgroup we may ˆ and assume that the two lifts differ by the sum of a derivation with values in S ˆ n+  ¯ Changing one of the lifts we get rid of the of a derivation with values in p R. first term. But then previous arguments imply that the two lifts give isomorphic representations, so we may assume that the two lifts coincide and only the
’s might differ. But then one concludes that an isomorphism between the two representations has coefficients in Sˆ ⊗ Qp and conjugates the two
’s. Namely to show that the coefficients are invariant under S localise and reduce to R a discrete valuation-ring, then replace R by S. This finally shows the claim.  For a generalised Qp -representation over R we can pass to some Rn where it is given by
acting on a projective Rˆ n [1/p]-module, then add a direct summand (with trivial
) to make it free of rank r. Thus our reasoning applies also to such modules and shows that the i are canonical. Finally the same type of reasoning shows that the definition of i ’s commutes with basechange: assume R → R is a logarithmic map, and we are given a generalised Qp -representation over R which induces such a representation over R . Then we can choose S and S as above, and assume that the map extends to S → S . Next we compare the two maps from an A2 (V )-lift of R to A2 (R ) obtained by either mapping to A2 (R) → A2 (R ), or first to a lift of R and then to A2 (R ). For any element a of the lift we obtain two 1-cocycles on  S with values in Sˆ  which are divisible by pn , and whose difference is a boundary. As before we then can assume that it is the boundary of an element divisible by pn . The restriction to  S of the pushforward is defined by the first cocycle, and a ˜ R/V −1 , and (after basechange S → S ) isomorphic to the Higgs-field
∈ End(S r ) ⊗
representation given by the second cocycle. However, this second cocyle is defined for all lifts of elements of R , not just of R, and we get the representation defined by the pushforward of
. But this is our claim. 4. Globalisation Again X is proper over Spec(V ), with toroidal singularities. A small generalised representation of its fundamental group is defined as a compatible system of small generalised representations on a covering of X by small affines. For example, it can ¯ on Vˆ¯ r which is trivial module p 2 , be defined by a representation of 1 (X ◦ ⊗V K)  > 1/(p − 1). However, a representation may induce locally small generalised representations without being trivial modulo some p-power. A small Higgs-bundle on X is ˜ X/V ⊗ Vˆ¯ −1 a vector-bundle E on X together with an endomorphism
∈ End(E) ⊗
which is divisible by p for some  > 1/(p − 1), such that
∧
= 0. Also Xˆ and

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Eˆ denote p-adic completions over Vˆ¯ , that is Xˆ is a formal scheme and Eˆ a formal vector-bundle on it. For any open Uˆ ⊂ Xˆ and any formal vectorfield ϑ ∈ (Uˆ , T˜X/V ) we obtain an automorphism exp(
(ϑ)) of Eˆ on Uˆ . Hence for any open covering Uˆ i of Xˆ and any 1-Cech-cocycle ϑij ∈ (Uˆ i ∩ Uˆ j , T˜X/V ) we get a functor exp(
(ϑij )) by formal Higgs-bundles by twisting Eˆ by the cocycle exp(
(ϑij )). These functors are multiplicative in {ϑij }, up to canonical isomorphism. Furthermore if ϑij = ϑi − ϑj is a boundary the exp(
(ϑi )) define an isomorphism between the identity and the functor given by {ϑij }. If we choose a covering Ui of X by small affines we have over each Ui equivalences of categories between small Higgs-bundles and small generalised representations. However, these depend on a choice of lifting Ui to A2 (V ) since automorphisms of such a lift act on small Higgs-bundles via our exp(
(ϑ))-construction. Only if we choose a lift of X to A2 (V ) we get a global equivalence of categories which depends on the lift: for two lifts we can choose local isomorphisms over each Ui . On the overlaps they differ by a vectorfield ϑij , and exp(
(ϑij )) describes the difference between the associated equivalences of categories. Furthermore change of local isomorphisms results in modifying {ϑij } by a coboundary. We also remark that after inverting p K lifts to A2 (V ), and thus if X is defined over K we can lift it if we invert p. We then could extend the theory to any such X if we only consider Higgs-bundles with
divisible by a sufficiently high p-power. Similarly for functoriality: if f : X → Y denotes a (logarithmic) map we have natural pullbacks f ∗ for Higgs-bundles as well as for generalised representations. If we lift X and Y to A2 (V ) and choose local lifts fi of f these will differ on the overlaps by vectorfields ϑij ∈ f ∗ (T˜Y /S ) which act on the pullbacks f ∗ (F,
) of small Higgs-bundles on Y. Then the two functors f ∗ differ by twisting by the corresponding cocycle. All in all we get: Theorem 5. This procedure defines (for liftable schemes) an equivalence of categories between small generalised representations and small Higgs-bundles. This also extends to the Qp -theory, and there the functor induces an isomorphism on cohomology (we have shown this locally, but the construction gives a global map). We cannot drop the adjective “small” as is demonstrated by explicit calculations for rank one bundles on curves: Namely suppose X is a smooth proper (geometrically connected) curve over V. Then ˆ 2g where g the abelianised fundamental group of X ⊗V K¯ is free and isomorphic to Z ˆ¯ denotes the genus of X. Its continuous K-representations are parametrised by the images of the generators which are elements of Vˆ¯ ∗ whose reduction modulo the maximal ideal is torsion, that is lie in the algebraic closure of Fp . This is always the case if k is finite, that is if K is a local field. The logarithm maps such elements surjectively onto ˆ¯ with kernel the roots of unity (K), ¯ and we obtain an exact sequence K, ˆ¯ ∗ ) → Hom( (X ), K) ˆ¯ → 0. ¯ → Hom(1 (X ¯ ), K 0 → Hom(1 (XK¯ ), (K)) 1 K K¯

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The first term coincides with the torsion-points in the Jacobian J of X, and the third ˆ¯ By Hodge–Tate theory with the homomorphisms from the Tate-module Tp (J ) into K. ˆ¯ ˆ¯ ⊕ (X,
[3] the latter coincides with the direct sum Lie(J ) ⊗V K X/V ) ⊗V K(−1). On the other hand from the logarithm-sequence for the Jacobian we obtain an exact sequence ˆ¯ → Lie(J ) ⊗ K ˆ ¯ tors → J (K) 0 → J (K) V ¯ → 0, ¯ This exact sequence where the middle term is the preimage of the torsion in J (k). ˆ¯ turns out to be the restriction of the first sequence to the direct summand Lie(J ) ⊗V K (proofs will follow from the considerations below. They amount to functoriality of the logarithm map for the homomorphism J [p ∞ ] → p∞ defined by an element of the Tate-module of J, or better of its dual which, however, coincides with J). Now assume that K is a local field, that is its residue-field is finite. If we choose ˆ¯ a splitting of this exact sequence over the second summand (X,
X/V ) ⊗V K(−1) we obtain an isomorphism between one-dimensional representations of 1 (XK¯ ) and ˆ¯ and (X,
ˆ¯ the product of J (K) that is a bijection (in rank one) X/V ) ⊗V K(−1), between representations and Higgs-bundles. Unfortunately there is no canonical splitting because the first exact sequence realises the universal extension of Tp (J )[1/p] by ˆ¯ − Gal(K/K)-modules, ¯ K so it cannot be trivial on a direct summand (see [10]). That ¯ is there exists no continuous Gal(K/K)-invariant bijection between representations and Higgs-bundles. Nevertheless we can construct such bijections if we choose an ˆ¯ exponential map for K: Namely the logarithm defines an exact sequence ˆ¯ → 0, ¯ → Gm (Vˆ¯ ) → K 0 → (K) where Gm (Vˆ¯ ) denotes the elements of Vˆ¯ ∗ which are torsion modulo the maximal ideal. An exponential map is a continuous right inverse of the logarithm, or a continuous splitting of this extension. It induces such a splitting for all commutative algebraic groups over K: Suppose G is such an algebraic group. Then the exponential and logarithm induce ˆ¯ and its Lie-algebra isomorphisms between sufficiently small open subgroups of G(K) ˆ¯ If G(K) ˆ¯ ⊆ G(K) ˆ¯ denotes the subgroup of elements g for which some multiple g⊗K K. ˆ¯ g n (n ∈ N nonzero) lies in this neighbourhood, we can extend the logarithm to G(K) and obtain an exact sequence ˆ¯ ˆ¯ → g ⊗ K. ˆ 0 → G(K) tors → G(K) K ¯ We show that the last map is surjective and construct a right inverse, as follows: First of all the logarithm is an isomorphism for additive or unipotent groups. We thus may divide the connected component of the identity of G by its maximal unipotent

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subgroup and may assume that G is semiabelian, using Rosenlicht’s theorem that G is an extension of an abelian variety by a linear group. Namely this operation does not change the torsion-points and thus allows us to lift splittings. Then passing to a finite extension of K we may assume that G has split semistable reduction and is ˜ by a group of periods : Y → G(K). ˜ over V the quotient of a semiabelian G We ˜ and may assume that G is semiabelian over V, that is G is an may replace G by G extension 0 → T → G → A → 0, with A an abelian variety and T a split torus. If Tp (G) denotes its Tate-module, then any element ∈ Hom(Tp (G), Zp (1)) in the Tate-module of its dual group defines over Vˆ¯ a homomorphism of p-divisible groups G[p ∞ ] → p∞ and thus also a map from ˆ¯ → G(k) ¯ to the 1-units in Vˆ¯ ∗ . We thus obtain a the kernel of the reduction map G(K) transformation from the kernel of the reduction-map to the kernel of the reduction map on Tp (G)(−1) ⊗ Vˆ¯ ∗ , and this induces an isomorphism on p-torsion. On the level of ˆ¯ ˆ¯ to T (G)⊗ K(−1). Lie-algebras we get a map from g⊗K K These maps are compatible p via the logarithm, by naturality. ˆ¯ ⊕ Lie(At ) ⊗ ˆ¯ Now by Hodge–Tate theory Tp (G) ⊗ K(−1) is isomorphic to g ⊗K K K ˆ¯ K(−1). Strictly speaking this has been shown as it stands only if G is defined over a finite extension of V, not if G is defined over Vˆ¯ . In general (G = A only defined ˆ¯ we obtain a subspace g ⊗ K ˆ ˆ¯ t over K) without a K ¯ with quotient Lie(A ) ⊗K K(−1), canonical lift. This follows from the proof in [3] by applying it to a universal family of abelian schemes A, and then specialising. By Galois-invariance (or for many other reasons) our map lands in the first factor and it is known to be an isomorphism onto the first direct summand. As the logarithm-map is surjective for G (it is p-divisible) and as we have an isomorphism on torsion-groups we get all in all an isomorphism ˆ¯ onto the preimage of g ⊗ K ˆ from the kernel of reduction on G(K) K ¯ in the kernel of ˆ ∗ reduction of Tp (G)(−1) ⊗ V¯ . Thus finally the exponential map on the multiplicative group induces such a map on G. These maps are functorial. We also remark that we obtain a certain uniformity in G, as follows: call an element of the Lie-algebra g small if there exists a rigid analytic homomorphism from the closed rigid unit disk (with addition) into G, with derivative at the origin the given ˆ¯ , and one checks that for the element. Such a map is unique, has values in G(K) multiplicative group Gm an element is small if and only if it has valuation > 1/(p − 1) (the exponential series has to converge). It then follows that there exists an integer n such that the exponential map is rigid analytic on any product of pn with a small element: reduce to G semiabelian over Vˆ¯ . Then an analytic homomorphism from the closed unit disk into G coincides with a map from the p-adic completion of the additive group into G, is trivial on the special fibre, the map from G(Vˆ¯ ) to Tp (G)(−1) ⊗ Vˆ¯ ∗ is analytic, and everything reduces to the case of the multiplicative group.

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As an application we can exponentiate cohomology-classes to line-bundles: consider a ˆ¯ ˜ ⊗i )⊗K(−i), semistable proper curve X over V and a sequence of elements i ∈ (X,
X/V for 1 i r. For example the i could be the coefficients of det(Tid −
) for a Higgsfield
, on a vectorbundle of rank r on the generic fibre of X. Associated to these invariants is an algebra A over X ˆ¯ , commutative and locally free of rank r. Namely it K ˆ¯ is the quotient of the symmetric algebra in the logarithmic tangent-bundle T˜ ⊗ K(1), X/V

under the monic polynomial with coefficients i . As a module A is the direct sum of ⊗i ˆ¯ powers T˜X/V ⊗ K(i), for 0 i r − 1. For example if the i come from a Higgs-field on E then E ˆ¯ becomes naturally an A-module. K ˆ¯ Now the Picard-group H 1 (X , A∗ ) consists of the K-points of a smooth algebraic ˆ¯ K

ˆ¯ with tangent-space H 1 (X , A), so we can exponentiate classes in the group over K, ˆ¯ K latter to get line-bundles. To avoid the complications due to automorphisms we first consider line-bundles trivialised at finitely many Vˆ¯ -points of X, up to isomorphism. If the number of points is positive and > (r − 1)(2 − 2g) then A has no global sections vanishing at all these points, so these objects have no automorphisms, and the moduliˆ¯ with tangent problem is representable by a smooth locally algebraic group over K space the first cohomology of X ˆ¯ with coefficients in the local sections of A which K vanish in the prescribed points. Let G denote the connected component of the identity of this moduli-space. The product of the fibres of A∗ maps to G (via change of trivialisations) and has as image a closed subgroup H (the kernel is given by the global sections of A∗ ). If g/h = H 1 (X ˆ¯ , A) denotes the Lie-algebra of the quotient G/H , choose a section K g/h → g of the projection, and the exponential map for G, to define a family of A-linebundles on X ˆ¯ parametrised by g/h. This family is p-adically continuous and additive K in the sense that the line-bundle corresponding to a sum is canonically isomorphic to the tensorproduct of the line-bundles corresponding to the factors, and these isomorphisms are compatible with associativity. Finally the difference between two sections of g → g/h lifts to a linear map from g into the product of the fibres of A in the prescribed points, and then exponentiates to a coherent family of isomorphisms between the associated line-bundles. Finally this coherent family of isomorphisms is canonical up to a multiplicative family of invertible global sections of A. Also for p-adically small elements of g/h the exponential coincides with the map defined by the exponential for some model for A on the p-adic formal scheme defined by X, by continuity. This construction allows us to construct a twisted pullback for Higgs-bundles on X ˆ¯ : namely for a map f : X → Y and a Higgs-bundle (E,
Y ) on Y ˆ¯ the usual K K pullback f ∗ (E), f ∗ (
Y ) admits an action of the algebra f ∗ (AY ) (AY is defined like the previous A, with X replaced by Y). Also the obstruction to lift f to A2 (V ) is an element of H 1 (X, f ∗ (T˜Y /V )) which exponentiates to an f ∗ (AY )-line-bundle. The twisted pullback f ◦ (E,
Y ) is defined as the tensorproduct of f ∗ (E,
Y ) with this linebundle. This functor depends on certain choices (trivialisations in suitable points) but different families give isomorphic functors. Also for compositions (fg)◦ is naturally

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isomorphic to g ◦ f ◦ , and we leave it to the reader to check the various compatibilities involved (later we consider actions of a finite group on X where compatibilities become easy to check because one can make all arbitrary choices invariant under this group). Finally there exists an integer n2, independent of X, such that if the i extend to a semistable model of X over Vˆ¯ (so A has an extension to this model) and if a class in H 1 (X, T˜X/V ) ⊗V Vˆ¯ (1) is divisible by p n , then our exponential coincides with the usual exponential of the class on the formal scheme defined by X. Now we come to our main result. Theorem 6. There exists an equivalence of categories between Higgs-bundles and genˆ¯ eralised representations, if we allow K-coefficients. Proof. For this we use the equivalence of categories for small representations and a descent argument. The latter uses the following construction of coverings of X. After passing to an extension of V choose a V-rational point in X and use it to embed the generic fibre XK into its Jacobian J. Then multiplication by p n on J induces a covering Xn,K of XK which has (after finite extension of K) a semistable model Xn mapping to X. If Jn,K denotes the Jacobian of Xn,K the induced map Jn,K → JK is divisible by pn , and so is by duality the pullback JK → Jn,K . Next consider the induced map on differentials on Néron-models which is identified with the pullback ˜ X/V ) → (Xn ,
˜ X /V ). Thus the pullback on differentials is divisible by p n (X,
n ˜ X/V on the generic fibre there exists on global sections. As global sections generate
˜ X/V generated by global sections contains an integer n0 such that the subsheaf of
n ˜ 0 p
X/V , and so the pullback-map on differentials is divisible by p n−n0 . Thus we can make generalised representations or Higgs-bundles small by pullback along such a covering. Now if we have a generalised representation on X we choose a finite Galois covering Y → X such that it becomes small on Y, that is it is given by a small Higgs-bundle there. The covering group acts via the f ◦ -action on this Higgs-bundle, and via the usual action if we twist by the obstruction to lift Y → X to A2 (V ). By descent we get a Higgs-bundle on X. Conversely given such a Higgs-bundle on X we choose Y such that it becomes small on Y, then twist the pullback by the inverse of the obstruction-class, and get a generalised representation on Y which descends to X.  Things are slightly more complicated if we allow open curves, that is the divisor at infinity D is nonempty: namely then the covering may ramify over D, and an equivariant action of the covering group does not always allow descent. We need that the (finite) stabiliser of a point acts trivially on the fibre in this point. In the A-linebundles used for the twisting, the action of inertia is given by the exponential of its derivative (the ˆ inertia-group is Z(1)), and to get descent we need the same for the action of inertia on the fibre of our generalised representation. The same reasoning gives the converse, so we get an equivalence between Higgs-bundles and representations satisfying the following condition on inertia.

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ˆ The action of the inertia group Z(1) on the fibre at a point in the boundary has a derivative Res(
) (it is the residue of the associated
). It then must be equal to the exponential (in the multiplicative group of the algebra generated by Res(
)) of its derivative. Especially the action factors over Zp (1). Our functors induce isomorphisms between the X 0 -cohomology of generalised representations and Higgs-cohomology of Higgs-bundles. This has been shown for small objects. The general result follows by descent once we show that Higgs-cohomology is invariant under our twists by A[1/p]-line-bundles. We only use that these line-bundles come from twists by vectorfields, and that they are in the image of the exponential map. The latter is used as follows. A is a quotient of the symmetric algebra S(T˜ ) under a monic polynomial (naturally ˜ ⊗r ) with coefficients i . Denote by B the algebra (of rank r+1) an element P ∈ S(T˜ )⊗
defined by multiplying this polynomial by T. Then our line-bundles lift to compatible systems of B[1/p]-line-bundles, trivialised on A/(
) = OX . We claim that the Koszul˜ X/V is up to quasi-isomorphism invariant under tensoring E with complex E → E ⊗
B-line-bundles M which are trivialised modulo
: Proposition 1. For two B-line-bundles M1 and M2 any isomorphism M1 /(
)

M2 /(
) induces a quasi-isomorphism on Koszul-complexes for E ⊗B Mi . Proof. Denote by N ⊂ M1 ⊕ M2 the elements which lie modulo
in the graph of the isomorphism. Then multiplication by
induces a map from M1 ⊕ M2 to ˜ X/V , and after tensoring (over B) with A the resulting complex becomes loN ⊗
cally free over A, and the two projections induce quasi-isomorphisms with the Koszulcomplexes of the Mi . Tensoring with E gives the desired zigzag of quasi-isomorphisms. Also the construction is transitive if we have three bundles Mi and compatible isomorphisms between their reductions modulo
(do the construction above with three summands).  5. Examples and open questions A natural question to ask is which Higgs-bundles come from actual representations of the fundamental group. If this representation is trivial modulo a sufficiently high p-power the associated Higgs-bundle has a model which is trivial modulo some positive p-power, so its restriction to the generic fibre is semistable (even without the Higgs-field) of slope zero. By descent it follows that in general the associated Higgs-bundle is semistable of slope zero (now we need the Higgs-field to construct the twisted pullback f ◦ ). Furthermore if a Higgs-bundle (on the generic fibre) has an integral model which is the trivial bundle modulo some p  for  > 0 we can (by pullback) assume that a Higgs-field is divisible by a high p-power. It then follows by deformation-theory that it is associated to an 1 -representation: for some small  (to take care of the “almost”) we first get this modulo p2− , by considering extensions of trivial representations or generalised representations. Then continue. This generalises a result in [1,2].

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Also if for a dominant (i.e. nonconstant) map YK → XK the pullback to Y of a generalised representation comes from a real representations, then the same holds already on X: we may assume that the cover is Galois, with group G. Then G operates on the locally constant system of Y, and the inertia acts trivially on the fibres at fixedpoints (by checking for the generalised representations). Thus the system descends. ˆ¯ come This suffices to show that all rank-one Higgs-bundles (L,
) with L ∈ J (K) from 1 -representations: if L is torsion it can be trivialised by pullback. Thus we reduce to L which is trivial on the special fibre. The resulting homomorphism ˆ ˆ¯ × (X,
) ⊗ K(−1) → Hom(1 (XK¯ ), Vˆ¯ ∗ ) J (K) X K ¯ coincides on the first factor with the previously defined map: namely this is true on ¯ the torsion, and this determines the map by Gal(K/K)-invariance. If X is only defined ˆ ¯ over K the same follows by an approximation argument. For the second factor one ˆ¯ can show that there the map is the exponential of a K-linear map into the Lie-algebra ˆ¯ ˆ 1 ¯ of the group of representations, that is into H (X, OX ) ⊗K K ⊕ (X,
X ) ⊗K K(−1). Furthermore the second component of our map is the identity, while the first one depends on the choice of lifting X to A2 (V ). Example. Sometimes Qp -representations L of the fundamental group are associated to filtered Frobenius-crystals E, that is there are functorial isomorphisms Bcrys (R) ⊗ L Bcrys (R) ⊗ E respecting Galois-action, filtration, and Frobenius (for small R’s). ˆ¯ The induced isomorphism on grF0 is Rˆ¯ ⊗ L ⊕i R[1/p](−i) ⊗ grFi (E) shows that L corresponds to the graded Higgs-bundle grF (E) “with Tate-twists”. Unipotent representations of the fundamental group correspond to unipotent Higgsbundles. Here “unipotent” means in both cases that the object is a successive extension of the trivial representation, respectively, Higgs-bundle. That follows easily from the comparison-theorem for cohomology as extensions are classified by H 1 . In fact using the method in [5] (proof of Theorem 5) one can even show that unipotent representations are (logarithmically) crystalline (see also [13]): for simplicity assume that our curve X has two V-rational points O and ∞. We consider representations of the fundamental-group of X − ∞ which has the advantage of being free. If Tp (J ) denotes the Tate-module of the Jacobian J of X we construct by induction unipotent smooth p-adic systems Ln on the generic fibre (X − {∞})K such that we have exact sequences 0 → Tp (J )⊗n → Ln → Ln−1 → 0. Furthermore in the fibre over 0 the quotient L0 = Zp lifts to a subsheaf of Ln , the projection Ln → L0 = Zp induces an isomorphism on homomorphisms into Zp (over (X − {∞})K¯ ), and the inclusion Tp (J )⊗n → Ln an isomorphism on extensions by Zp .

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All these hold for L0 = Zp . If we have constructed Ln−1 we know that its extensions (over XK¯ ) by the constant sheaf Tp (J )⊗n are classified by the corresponding H 1 which is isomorphic to H 1 ((X − {∞})K¯ , Hom(Tp (J )⊗n−1 , Tp (J )⊗n )) = Hom(Tp (J )⊗n , Tp (J )⊗n ),

and Ln is the class corresponding to the identity. From the long exact sequence of Ext-groups one derives the assertion about Ext i ((X − {∞})K¯ ; Ln , Zp ). Finally, the automorphisms of the extension are Tp (J )⊗n and they act simply transitively on the splittings in 0. Thus we get that Ln is unique up to unique isomorphism, and thus already defined over K. The same type of argument works on the crystalline side and constructs filtered (with degrees  0) crystals En , on the logarithmic crystalline site of X relative to Zp , with splittings at 0 as before. These are unique, either filtered or unfiltered, and thus Frobenius-crystals. Finally, the two objects correspond via the comparison theory. As any unipotent representation of the fundamental-group is a quotient of a direct sum of Ln ’s the assertion follows. References [1] [2] [3] [4] [5] [6] [8] [9] [10] [11] [12] [13]

C. Deninger, A. Werner, Vector bundles and p-adic representations I, M˘unster, 2003, preprint. C. Deninger, A. Werner, Bundles on p-adic curves and parallel transport, M˘unster, 2004, preprint. G. Faltings, Hodge–Tate structures and modular forms, Math. Ann. 278 (1987) 133–149. G. Faltings, Crystalline cohomology of semistable curves, and p-adic Galois-representations, J. Algebraic Geom. 1 (1992) 61–82. G. Faltings, Curves and their fundamental groups (following Grothendieck, Tamagawa and Mochizuki), Sem. Bourbaki 840, Astérisque 252 (1998) 131–150. G. Faltings, Almost étale extensions, Astérisque 279 (2002) 185–270. J.M. Fontaine, Le corps de périodes p-adiques, in: Périodes p-adiques, Astérisque, vol. 223, 1994. J.M. Fontaine, Arithmétique des représentations galoisiennes p-adiques, Orsay 2000-24, preprint. J.M. Fontaine, Presque Cp -représentations, Orsay 2002-12, preprint. O. Gabber, L. Ramero, Almost ring theory, Springer Lecture Notes, vol. 1800, Springer, Berlin, 2003. C.T. Simpson, Higgs bundles and local systems, Publ. Math. IHES 75 (1992) 5–95. V. Vologodsky, Hodge structure on the fundamental group and its application to p-adic integration, Moscow Math. J. 3 (2003) 205–247.

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Indexes

Index of Symbols

e1 Ω , 95 R/OK δ, 95 e 96 δ, E (Faltings extension), 98 Ξpn , Ξp∞ , 102

Chapter I, 1 Chapter II, 27 K, K, C, 28 b (p-adic Hausdorff completion; A an abelian A group), 28 A • KA • (u), K• (u, C), KA (u, C) (Koszul complexes of a linear form), 29 HM(A, E) (A a ring and E an A-module), 31 K• (M, θ) (Dolbeault complex of (M, θ)), 31 disc RepA (G), Repcont A (G), RepA (G), 35 e • (G, M ) (cochain complexes), 38 K• (G, M ), K A N (A an abelian category), 38 Sch (category of schemes), 50 PHB(G/X) (category of principal homogeneous G-bundles over X), 51 Tors(G, Xzar ) (category of G-torsors of Xzar ), 51 AM (M a monoid), 67 ex (x an element of a monoid), 67 Ω1(X,M )/(Y,M ) , 70 X Y (S, MS ), 71 (X, MX ), 72 R, 72 ϑ : N → P , λ = ϑ(1), 72 L, Lλ , 72 X ◦ , 72 α : P → R, 72 Kn , πn , 74 (Xn , MXn ), An , 74 Rn , R∞ , Rp∞ , R, 77 F , Fn , F∞ , Fp∞ , F , 77 Bn , B∞ , Hn , H∞ , 77 ∆, ∆∞ , ∆p∞ , 79 Γ, Γ∞ , Γp∞ , 79 Σ, Σ0 , 79 d log(πn ), 84 N(n) , N∞ , 85 an , a∞ , 85 P (n) , P∞ , t(n) ∈ P (n) , 87 αn , α∞ , 87 h , i : Γ∞ × P∞ → µ∞ (OK ), 93 M∞ , L , N∞ , N , 94 E∞ , 95

(ν)

Rp∞ , 103 RA , 111 b 111 θ : W(RA ) → A, A2 (A), 111 ξ ∈ W(RA ), 112 ˇ M ) → (A (S), M i : (S, ˇ S

S

2

A2 (S) ),

114

ξ −1 M (M an OC -module), 115 (Y, MY ), (Yb , MYb ), 116 A2 (Y ), 116 qY : QY → W(RR ), 116 iY : (Yb , MYb ) → (A2 (Y ), MA2 (Y ) ), 117 log([ ]) : Zp (1) → A2 (R), 119 ˇ M ), 120 (X, M ), (X, ˇ X

X

e M e ), 120, 143 (X, X e T, 121 T, T,

S = ⊕n≥0 S n , 121 L (Higgs–Tate torsor), 121 F (module of affine functions on L ), 121 C (Higgs–Tate algebra), 121 L, 121 g ψ, 122 ϕψ , 123 Du , exp(Du ), 124 σψ,ψ0 , 126 e0 , M e ), L0 , F0 , C0 , ψ0 ∈ L0 (Yb ), 127 (X X0 ϕ0 , 129, 135 S, 129, 135 S (r) , S † , 132 dS (r) , dSc(r) , dS † , 132 e • (Sb(r) , pr d c(r) ), 132 K S

F (r) , C (r) , C † , 143 0 0 αr,r , hr,r α , 143 dC (r) , dCb(r) , dC † , 143 e • (Cb(r) , pr d b(r) ), 143 K (r)

C

C0 , C0 , 144 H(M ), 146 V(N ), 146 599

600

INDEXES

α-small Repα-qsf (G), Repqsf (G), A A (G), RepA small RepA (G), 153 (G), 153 Repsmall A[ 1 ] p

c1 , ξ −1 Ω e1 HMβ-small (R ), 154 R/OK small c 1 −1 e1 HM (R1 [ p ], ξ Ω ), 154 R/OK P  d −1 ϕ = exp θi ⊗ χi , 157 i=1 ξ P θ = di=1 ξ −1 θi ⊗ d log(ti ), 157 S(r) , 157 expr (θ), 158 CHT , 175 Di (V ), 177 Chapter III, 179 K, K, C, 180 ˇ M ), 180 (S, MS ), (S, MS ), (S, ˇ S S , 180 ˇ X (X an S-scheme), 180 X, X, n RA , 181 A2 (A), 181 b 181 θ : A2 (A) → A, ξ ∈ W(RO ), 181 K ˇ M ) → (A (S), M iS : (S, ˇ 2 A2 (S) ), 181 S Mod(A, X) or Mod(A), 182 SA (M ) or S(M ), ∧A (M ) or ∧(M ), 182 ´ /X , X´et (étale site and topos), 182 Et ´ f/X , Xf´et (fine étale site and topos), 182 Et ρX : X´et → Xf´et , 182 uX : X´et → Xzar , 182 νx : Xf´et → Bπ1 (X,x) , 183 CQ , MQ , 194 M ⊗AQ N , H omAQ (M, N ), 194 HI(A, E), HIQ (A, E), 196 ◦ X I (X a topos, I a category), 203 ◦ αi : X → X I , 203 ◦ I λ : X → X, 204 ◦ f I (f a morphism of topos, I a category), 204 ◦ ◦ X N , X [n] , 205 X, X, Y , h, ~, j, 211 U , UY (U an X-scheme), 211 ´ /X , 211, 230 π : E → Et (V → U ), 211 F, F∨ , P ∨ , 211 b 212 E, e 212, 230 E, σ, 212, 230 β, 212 Ψ, 212 ← e 213, 230 ρ : X´et × X Y´et → E, ´ et

x), 213 ρ(y e 214 θ : Yf´et → E, B, 215 V U , 215 B, B U , 215 y RU , 216 X, X ◦ , 222 U ◦ = U ×X X ◦ , 222, 229

B n , B U,n , 222, 230 γ, 223 es , δ, 223, 230 E σs , ση , 224 a, an , an , ιn , ιn , 225 σn , 225, 230, 250 e N◦ , 226 E s ˘ B, 226 σ ˘ ,˘ u, 226, 250 X, 226, 250, 266 ˘ → (X e N◦ , B) > : (E s,zar , OX ), 226, 250, 266 s (X, MX ), X ◦ , 229 e1 , Ω e1 Ω , 229 X/S X n /S n ˇ (X, M ), (X, M ), 229 ˇ X

X

e M e ), 229 (X, X P, 231 Q, EQ , 231 Vx , Vx (P), Vx (Q), 233 (U, p), 233 e → X 0◦ , 233 ϕx : E f´ et y by Y , Y , 235 y

(A2 (Y ), MA

y

), 235

2 (Y ) LYy , FYy , CYy , 235 FY,n , CY,n (Y ∈ Ob(Q), n ∈ N), 238 (r) (r) FY,n , CY,n (Y ∈ Ob(Q), r ∈ Q≥0 , n ∈ 0 r,r 0 ar,r Y,n , αY,n , 240 (r) (r) Fn , Cn (r ∈ Q≥0 , n ∈ N), 241

Fn , Cn (n ∈ N), 241 0

0

r,r ar,r n , αn , 241 (r) dn , 243 y,(r) y,(r) , CY , 243 FY y

y,(r)

y,(r)

RX 0 , FX 0 , CX 0 , 244 ˘ → (X N◦ , O ), 248, 250 e N◦ , B) σ ˘ : (E ˘ s s,´ et X e 1 , 248 ξ −1 Ω ˘ S ˘ X/

F˘(r) , C˘(r) (r ∈ Q≥0 ), 248 F˘, C˘, 248 0

0

˘r,r , α a ˘ r,r , 248 (r) ˘ d , 248, 264 ei ξ −i Ω , 250 ˘ S ˘ X/

τ˘, 250 ˘ Modaft Q (B), 266 coh e1 HI (OX , ξ −1 Ω Q

X/S

), 268

e1 HMcoh (OX [ p1 ], ξ −1 Ω ), 268 X/S ΞrQ , 269 Sr , K r , >r+ , >r+ (r ∈ Q≥0 ), 269 0

r,r , 270 H , 272 V , 276

˘ 300 MODDolb (B), Q Chapter IV, 307 Mod(C, O), Mod(C, O• ), 312

N), 240

INDEXES r , A r , A r , 313 A , Aalg p Ap,• , A•r , 315 r DHiggs , 316, 321 C , C r , 318 Cartesian, 319 exact closed immersion, open immersion, 319 smooth, étale, strict, integral, affine, 319 strict étale covering, Zariski covering, 319 r Malg,• , 325 (m)r ,n

MN

, 326

(m)

MN r , 327 AN {W1 , . . . , Wd }r , A{W1 , . . . , Wd }r , 329 (m) OT r , 332 N Y (ν), D(ν), D, ∆D , pi , pij , qi , 333, 334 θ, θq of ODN , 335 θi , 336 LZar , n LZar , TZar , n TZar , 337, 338 n T , n L, T , L, 338 PHS(T /Y1 ), 338 Tors(TZar , (Y1 )Zar ), 338 ϕY1 : PHS(T /Y1 ) −→ Tors(TZar , (Y1 )Zar ), 338 (AffSmStr/Y1 )Zar , 339 (Y1 )ZarAff , 339 T , L, 339 Lift(U/Y2 ), 339 bA (M ), C (r) (M ), C b (r) (M ), CAY (M ), C Y1 AY1 AY1 1 343 CTors , 345 (X/B)rHIGGS , 350 Xétaff for an affine p-adic formal scheme X, 354 OX/B,1 , 350 0

∼ fHIGGS : (X 0 /B 0 )rHIGGS −→ (X/B)r∼ HIGGS , 351 ∼ , 354 πD : (((X/B)rHIGGS )/(D,zD ) )∼ → D1,´ et PM(OX,Qp ), PM(AQp ), 354 LPM(A), LPM(A• ), 355, 356 LPM(OX• ), LPM(OX ), 356 LPMloc (OX ), 357 HCrQp (X/B), HCrQp ,U -fin (X/B), 358, 359 HCZp (X/B), HCZp ,U -fin (X/B), 359 HSrQp (X, Y /B), HSZp (X, Y /B), 360 0 HMr (X, Y /B), 0 HM (X, Y /B), 361 Zp Qp HMrQp (X, Y /B), HMZp (X, Y /B), 361 HBrQp ,conv (X, Y /B), HBZp ,conv (X, Y /B), 364 HCrZp (X/B), 367 (Conv)r , (Conv)0α , (Div)α , 367, 368 ∼ UX/B : (X/B)r∼ HIGGS −→ XÉT , 370 r∼ ∼ uX/B : (X/B)HIGGS −→ X´et , 371 LY (−), 372 A, U , Utriv , Atriv , 383, 384 G(U,s) , 384 RA , 384 b 384 θ : W (R ) → A, A

AN (A), 384 U , DN (U ), iU : U → D1 (U ), 385

601

V , K, K, V , 386 b 386, 387 Σ, DN (Σ), D(Σ), Σ, UK,triv , 387

(D(U ), zU ), 388 ∆(U,s) , 388 r V(U,s),HIGGS (−), T(U,s),HIGGS (−), 388, 392 r r DX,Y (U ), AX,Y,N (A), D r (U ), ANr (A), 388, 389 e e LPM RepPM cont (∆(U,s) , AQp ), Repcont (∆(U,s) , A), 392 UL , UL,triv , AU,L , AU,L,triv , 403 P, 403 Covh (U), Covv (U), 403 horizontal morphism, vertical morphism, 403 horizontal strict étale covering, vertical finite étale covering, 403 P/X, (P/X)´et-f´et , (P/X)f´et , 403 Xétaff , 404 vP/X : (P/X)´e∼t-f´et → (P/X)∼ f´ et , 404 fP,´et-f´et , fP,f´et , 406 (P/C)´et-f´et , (P/C)f´et , 406 P/C, (P/C)´et-f´et , (P/C)f´et , 406 vP/C : (P/C)´e∼t-f´et → (P/C)∼ f´ et , 406 0

0

C,C C,C fP,´ et-f´ et , fP,f´ et , 407 v v OP/C , OP/C , OP/C,m , OP/C,m , 408 SK,s , Ss , 412 UL for L ∈ Ss , 412 , 412 (UK,triv )gpt , (UK,triv )∧,cont gpt Cgpt , 414 ∧,cont , 414 sC : (P/C)∼ f´ et −→ (Cgpt ) ∧,cont rC : Cgpt → (P/C)∼ , f´ et 415 OCgpt , OCgpt ,m , 415 Modcocart (Cgpt , OCgpt ,• ), 416 v Modcocart ((P/C)f´et , OP/C,• ), 416 0

C,C ∗ fgpt , 417 CX , CR , 421 DX : (CX )gpt −→ (X1 /D(Σ))∞ HIGGS , 422 HCZp ,R-fin (X1 /D(Σ)), 422 R R (−), T TX,gpt,• (−), TX,• X,• (−), 422, 423 Xtraff , 424 ∼ ∧ vX : Xétaff → Xétaff , 424 h ∼ ∼ , 424 πX,´et : (P/X)´et-f´et −→ Xétaff h ∼ ∧ πX : (P/X)f´et → Xétaff , 424 C´et , Ctr , 425 vC : C´e∼t −→ C ∧ , 425 h ∼ ∼ πC,´ et : (P/C)´ et-f´ et −→ C´ et , 425 ∧ , 425 πCh : (P/C)∼ −→ C f´ et OC1 ,m , 425 ∧,cont h πC,gpt∗ −→ C ∧ , 426 : Cgpt ∧,cont h∗ πC,gpt : C ∧ −→ Cgpt , 426 0

0

C,C f´eC,C , 427 t , ftr Λ l : (C )∼ → C ∼ , 432 ← − ◦ l : C ∼ → (C Λ )∼ , 432 − → ηP/C , ηC , 434 ?+1 s+1 TX,• ΩY (F ◦ ), TC,gpt,• ΩY (F ◦ ), 435, 436 Y C , 435

602

INDEXES

Chapter V, 449 R-Mod, 467 R-Alg, 467 g 467 R-Mod, g 467 R-Alg, R-Et, 471 f 470 R-Et, C• (B/A), 471 H ∗ (B/A, M ), 471 A-G-discMod, 478 Chapter VI, 485 C/F , 493 ←

e × e Ye (oriented product of topos), 498 X S π : E → I, 511 αi , 511 F , F ∨ , P ∨ , 511 b 511, 550 E, {i 7→ Fi }, 511 e 512, 550 E, ε, 512 Top(E), 515 δ, 515 σ, β, 523, 552 ´ /X , Et ´ coh/X , Et ´ scoh/X , 542 Et X´et (étale topos), 542 ´ f/X , Xf´et (finite étale site and topos), 542 Et ρX : X´et → Xf´et , 542 νx : Xf´et → Bπ1 (X,x) , 545 IndA,G , 478 ∗ (G, −), 479 HA RΓA (G, −), 479 K • (G, M ), 480 ´ /X , 550 π : E → Et (V → U ), 550 ecoh , E escoh (E a Faltings site), Ecoh , Escoh , E 552 e 553 Ψ : Y´et → E, ← e 556 ρ : X´et × X Y´et → E, ´ et

ρ(y x) point of a Faltings topos, 558 e 560 θ : Yf´et → E, e → Y , 564 ϕx : E f´ et

Alphabetical Index Additive category up to isogeny, 194 Adequate (logarithmic) chart, 187 Adequate morphism (of logarithmic schemes), 190 r-admissible triple, 271 Affine function on a torsor, 53 Almost étale covering, 461 G-covering, 482 Almost faithfully flat, 460 Almost finitely generated, 452 Almost finitely generated projective, 452 rank, 458 rank ≤r, 456 trace map, 454

Almost finitely presented, 465 Almost flat, 459 Almost isomorphism, 410, 450 Almost projective, 452 Almost zero, 410, 450 Associated, r-associated (modules), 271 Canonical projections of an oriented product of topos, 495 Characteristic invariants of a Higgs field, 31 Co-Cartesian module of a ringed covanishing topos, 541 of a ringed total topos, 536 Coherent (quasi-compact and quasi-separated), 494 Covanishing topology, 404, 424, 512 Coverings Cartesian —, 512 Vertical —, 512 (A2 (S), MA2 (S) )-deformation, 120, 229 Dolbeault complex of a Higgs isogeny, 196 of a Higgs module, 31 ˘ -module, 271 Dolbeault B Q

b 147 Dolbeault R-representation, b 1 ]-representation, 148 Dolbeault R[ p

Étale-locally connected scheme, 184, 544 Faltings extension, 98 Faltings fibered site, 211, 230, 550 Faltings site, 212, 402, 550 Faltings topos, 212, 550 Faltings’ almost purity theorem, 81 Fiber functor of a finite étale topos, 183, 545 Higgs OX [ p1 ]-bundle, 268 (Locally) small —, 303 Solvable —, 271 Higgs crystal, 359 finite on U , 359 R-finite, 422 Higgs envelope of level r, 316, 321 Higgs field, 31 Higgs isocrystal, 358 finite on U , 358 Higgs isogeny, 196 Higgs module, 31 c1 -module Higgs R β-quasi-small —, β-small —, quasi-small —, small —, 154 Solvable —, 147 c1 [ 1 ]-module Higgs R p Small —, 154 Solvable —, 149 Higgs–Tate algebra (C ), 123 algebra of thickness r (C (r) ), 143 extension (F ), 123

INDEXES

torsor (L ), 121 (r) Higgs–Tate B n -algebra of thickness r (Cn ), 241 ˘ Higgs–Tate B-algebra of thickness r (C˘(r) ), 248 Higgs–Tate B n -extension of thickness r (r) (Fn ), 241 ˘ Higgs–Tate B-extension of thickness r (F˘(r) ), 248 Hochschild cohomology, 471 Hodge–Tate Qp -representation, 177 Hyodo ring, 175 Induced A-G-module, 478 Inhomogeneous cochain complex, 480 (Integrable) λ-connection, 32 λ-isoconnection Adic integrable —, 198 Integrable —, 197 Isogeny, 194 K(π, 1) scheme, 550 Linearization, 371 Locally irreducible scheme, 184 Locally projective module of finite type, 182 Morphism of p-adic fine log formal schemes, 317 ΩqX/Y , d log t, 318 f´et , fÉT , 318 exact closed immersion, closed immersion, open immersion, immersion, 318 infinitesimal neighborhood, 318 log coordinates, 318 smooth, étale, strict, integral, affine, 318 Normalized universal cover, 183, 545 Oriented product of topos, 498 p-adic fine log formal scheme, 317 X´et , XÉT , 318 OX , MX , 318 affine, 317 p-adic formal scheme, 317 fine log structure on, 317 Poincaré lemma, 374 Principal homogeneous G-bundle, 51 ∆-equivariant —, 62 α-quasi-small, α-small, quasi-small, small A-representation (A a p-adic algebra), 153 Small Higgs field, 368, 369 Small A[ p1 ]-representation (A a p-adic algebra), 153 Tensor product of a λ-connection and a Higgs module, 33 of Higgs modules, 31 Topos

603

Covanishing —, 498 Faltings —, 550 Generalized covanishing —, 512 Total —, 515 Vanishing —, 498