p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects 3031215494, 9783031215490

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Bhargav Bhatt Martin Olsson Editors

p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects

Simons Symposia Series Editor Yuri Tschinkel, Courant Institute of Mathematical Sciences and Simons Foundation, New York University, New York, NY, USA

Working to foster communication and enable interactions between experts, volumes in the Simons Symposia series bring together leading researchers to demonstrate the powerful connection of ideas, methods, and goals shared by mathematicians, theoretical physicists, and theoretical computer scientists. Symposia volumes feature a blend of original research papers and comprehensive surveys from international teams of leading researchers in thriving fields of study. This blend of approaches helps to ensure each volume will serve not only as an introduction for graduate students and researchers interested in entering the field, but also as the premier reference for experts working on related problems. The Simons Foundation at its core exists to support basic, discovery-driven research in mathematics and the basic sciences, undertaken in pursuit of understanding the phenomena of our world without specific applications in mind. The foundation seeks to advance the frontiers of research in mathematics and the basic sciences by creating strong collaborations and fostering cross-pollination of ideas between investigators, leading to unexpected breakthroughs and a deeper understanding of the world around us.

Bhargav Bhatt • Martin Olsson Editors

p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects

Editors Bhargav Bhatt School of Mathematics Institute for Advanced Study Princeton, NJ

Martin Olsson Department of Mathematics University of California, Berkeley Berkeley, CA, USA

Department of Mathematics University of Michigan Ann Arbor, MI, USA Department of Mathematics Princeton Univeristy Princeton, NJ

ISSN 2365-9564 ISSN 2365-9572 (electronic) Simons Symposia ISBN 978-3-031-21549-0 ISBN 978-3-031-21550-6 (eBook) https://doi.org/10.1007/978-3-031-21550-6 Mathematics Subject Classification: 14F20, 14F30, 14F40, 14D10, 14G20, 14G22, 11G25 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains research articles related to the 2019 Simons Symposium on padic Hodge theory. To explain its context, recall that classical formulations of p-adic Hodge theory concern comparisons of cohomology with constant coefficients for smooth proper varieties. As in complex Hodge theory, a more refined understanding can be obtained by investigating the cohomology for varieties which are possibly non-compact or singular, considering cohomology with non-trivial coefficients, and contemplating non-abelian aspects. In recent years, there have been a number of structural developments that have helped make progress on these directions in p-adic Hodge theory. These include the development of new techniques, such as prismatic cohomology and the saturated de Rham–Witt complex, and the systematic use of adic and perfectoid spaces. The goal of this symposium was to understand some of these developments. The first article, by Abbes and Gros, gives an overview of their recent work on the Hodge–Tate spectral sequence. This spectral sequence, first introduced in the work of Faltings and later revisited by Scholze, is a key tool in p-adic Hodge theory akin to the Hodge–de Rham spectral sequence in complex geometry. Abbes and Gros develop a relative version of this spectral sequence using the Faltings topos, and a relative variant they develop. The article of Betts and Litt studies weight-monodromy conjecture for the prounipotent completion of the étale fundamental group. This is a certain pro-algebraic completion of the étale fundamental group. While still capturing nonabelian information, it has many properties similar to cohomology; in particular, one can study p-adic Hodge theory for these groups. In this article, the authors prove both weightmonodromy for the fundamental group and the semisimplicity of the Frobenius action. One feature of modern p-adic Hodge theory, especially after the advent of perfectoid spaces, is that Huber’s adic spaces play a key role, akin to the role played by complex analytic spaces in complex geometry. It is thus important to develop many tools in this context. The third article is concerned with a number of foundational problems in developing the theory of log geometry, in the sense of Fontaine, Illusie, and Kato, in the context of adic spaces. This is important for v

vi

Preface

several reasons including the study of open varieties and semistable degenerations. Among the contributions of this article is the development of the Kummer étale and pro-Kummer étale topologies of log adic spaces as well as the primitive comparison theorem in this context. In recent years, the introduction of prismatic cohomology has substantially improved our understanding of integral p-adic Hodge theory. The formalism of the prismatic site provides not only cohomology but also some natural coefficient systems called crystals. The fourth article, by Gros–Le Stum–Quirós, contributes to our understanding of prismatic crystals: they construct a functor from the category of q-crystals (a slight variant of the notion of a prismatic crystal) to a category of modules of a certain ring of twisted differential operators. One feature of this construction is that it gives a more explicit local description of q-crystals. The fifth article in the volume by Lurie revisits the moduli spaces of elliptic curves with level structure; a classical topic in arithmetic geometry. It was shown by Scholze that the tower of such moduli spaces naturally defines a perfectoid space over the perfectoid field obtained by adjoining the p-th power roots of unity to Qp . In the fifth article, Lurie proves an integral version of this statement. An interesting consequence of this is a moduli interpretation of the tilt of the perfectoid space constructed by Scholze. The book concludes with an article by Ogus on the saturated de Rham–Witt complex, originally introduced by Bhatt, Lurie, and Mathew. In the case of smooth varieties, this complex coincides with the classical de Rham–Witt complex studied by Illusie and others, but in the case of singular varieties it is a new object. Ogus studies the natural question of what information is captured by the saturated de Rham–Witt complex in the case of varieties with toric singularities—a natural first case of mildly singular varieties to consider. Ogus shows that in this case the saturated de Rham–Witt complex has good properties and gives a p-adic version of a complex already familiar in toric geometry mod p. Princeton, NJ, USA Berkeley, CA, USA

Bhargav Bhatt Martin Olsson

Contents

The Relative Hodge–Tate Spectral Sequence: An Overview . . . . . . . . . . . . . . . . Ahmed Abbes and Michel Gros

1

Semisimplicity of the Frobenius Action on π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Alexander Betts and Daniel Litt

17

Logarithmic Adic Spaces: Some Foundational Results . . . . . . . . . . . . . . . . . . . . . . Hansheng Diao, Kai-Wen Lan, Ruochuan Liu, and Xinwen Zhu

65

Twisted Differential Operators and q-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Michel Gros, Bernard Le Stum, and Adolfo Quirós Full Level Structures on Elliptic Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Jacob Lurie The Saturated de Rham–Witt Complex for Schemes with Toroidal Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Arthur Ogus

vii

List of Contributors

Ahmed Abbes Laboratoire Alexander Grothendieck, CNRS, IHES, Université Paris-Saclay, Bures-sur-Yvette, France L. Alexander Betts Harvard University, Cambridge, MA, USA Hansheng Diao Yau Mathematical Sciences Center, Tsinghua University, Beijing, China Michel Gros IRMAR, Université de Rennes, Rennes Cedex, France Kai-Wen Lan University of Minnesota, Minneapolis, MN, USA Bernard Le Stum IRMAR, Université de Rennes, Rennes Cedex, France Daniel Litt University of Toronto, Toronto, ON, Canada Ruochuan Liu Beijing International Center for Mathematical Research, Peking University, Beijing, China Jacob Lurie Institute for Advanced Study, Princeton, NJ, USA Arthur Ogus Department of Mathematics, University of California, Berkeley, CA, USA Adolfo Quirós Departamento de Matématicas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid, Spain Xinwen Zhu California Institute of Technology, Pasadena, CA, USA

ix

The Relative Hodge–Tate Spectral Sequence: An Overview Ahmed Abbes and Michel Gros

1 Introduction 1.1. Let K be a complete discrete valuation field of characteristic 0, with algebraically closed residue field of characteristic p > 0, OK the valuation ring of K, K an algebraic closure of K, OK the integral closure of OK in K. We denote by GK the Galois group of K over K, by OC the p-adic completion of OK , by mC the maximal ideal of OC and by C its field of fractions. We set S = Spec(OK ) and S = Spec(OK ) and we denote by s (resp. η, resp. η) the closed point of S (resp. generic point of S, resp. generic point of S). For any integer n ≥ 0, we set Sn = Spec(OK /pn OK ). For any S-scheme X, we set X = X ×S S and Xn = X ×S Sn .

(1)

The following statement, called the Hodge–Tate decomposition, was conjectured by Tate ([16] Remark page 180) and proved independently by Faltings [8, 9] and Tsuji [17, 18]. Theorem 1.2 For any proper and smooth η-scheme X and any integer n ≥ 0, there exists a canonical functorial GK -equivariant decomposition

A. Abbes Laboratoire Alexander Grothendieck, CNRS, IHES, Université Paris-Saclay, Bures-sur-Yvette, France e-mail: [email protected] M. Gros () IRMAR, Université de Rennes, Rennes Cedex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_1

1

2

A. Abbes and M. Gros

∼

Hne´ t (Xη , Qp ) ⊗Qp C →

n 

n−i Hi (X, ΩX/η ) ⊗K C(i − n).

(2)

i=0

The Hodge–Tate decomposition is equivalent to the existence of a canonical functorial GK -equivariant spectral sequence, the Hodge–Tate spectral sequence, i,j

j

i+j

E2 = Hi (X, ΩX/η ) ⊗K C(−j ) ⇒ He´ t (XK , Qp ) ⊗Qp C.

(3)

The two statements are equivalent by a theorem of Tate ([16] theo. 2). Indeed, the cohomology group H0 (GK , C(1)) vanishes, which implies that the spectral sequence degenerates at E2 . The cohomology group H1 (GK , C(1)) also vanishes, which implies that the abutment filtration splits. This Hodge–Tate spectral sequence, which one can guess implicitly in the work of Faltings [9], has been explicitly formulated only later by Scholze [14]. 1.3. We give in this note an overview of a recent work [2] leading to a generalization of the Hodge–Tate spectral sequence to morphisms. The latter takes place in Faltings topos. Its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. Using a different approach, Caraiani and Scholze ([7] 2.2.4) have constructed a relative Hodge–Tate filtration for proper smooth morphisms of adic spaces. Hyodo has also considered earlier a special case for abelian schemes [11]. Beyond the Hodge–Tate spectral sequences, we give in [2] complete proofs of Faltings’ main p-adic comparison theorems. The latter are essential in the construction of these spectral sequences. Although the absolute version of these theorems is rather well-understood, the relative version which was very roughly sketched by Faltings in the appendix of [9], has remained little studied. Scholze has proved similar results ([15] 1.3 and 5.12) in his setting of adic spaces and pro-étale topos. In a work in progress, we extend the relative Hodge–Tate spectral sequence to more general coefficients in relation with the p-adic Simpson correspondence [3]. This sheds new lights on the functoriality of the p-adic Simpson correspondence by proper (log)smooth pushforward (for a related result see the work of Liu and Zhu [13]).

2 The Local Version of the Relative Hodge–Tate Spectral Sequence 2.1. Let X = Spec(R) be an affine smooth1 S-scheme satisfying the following two conditions: 1 We treat in [2] schemes with toric singularities using logarithmic geometry, but for simplicity, we

consider in this overview only the smooth case.

The Relative Hodge–Tate Spectral Sequence: An Overview

3

(i) X is small in the sense of Faltings, that is to say it admits an étale S-morphism to a S-torus, X → Gdm,S = Spec(OK [T1±1 , . . . , Td±1 ]), for an integer d ≥ 0. (ii) Xs is non-empty. Let y a geometric point of Xη . We denote by Xη (resp. Xη∗ ) the connected component of Xη (resp. Xη ) containing the image of y and by (Vi )i∈I the universal cover of Xη∗ at y ([3] VI.9.7.3). We set Γ = π1 (Xη , y) and Δ = π1 (Xη∗ , y). For every i ∈ I , we denote by Xi = Spec(Ri ) the normalization of X = X ×S S in Vi . Vi

Xi (4)

Xη

X

The OK -algebras (Ri )i∈I form naturally an inductive system. We denote by R its inductive limit, R = lim Ri , −→

(5)

i∈I

and by  R its p-adic completion, that we equip with the natural actions of Γ . The R is an analog of the GK -representation OC . Γ -representation  Theorem 2.2 (Abbes and Gros [2] 6.9.6) Under the assumption of 2.1, for any projective and smooth morphism g : X → X, and every integer q ≥ 0, there exists q q (filr )0≤r≤q+1 , a canonical exhaustive decreasing filtration of He´ t (Xy , Zp )⊗Zp  R[ p1 ] q by  R[ 1 ]-representations of Γ , such that fil is zero and such that for every integer q+1

p

0 ≤ r ≤ q, we have a canonical Γ -equivariant exact sequence

1 q q q−r 0 → filr+1 → filr → Hr (X , ΩX /X ) ⊗R  R[ ](r − q) → 0. p

(6)

It amounts to saying that there exists a canonical Γ -equivariant spectral sequence 1 i,j j i+j E2 = Hi (X , ΩX /X ) ⊗R  R[ ](−j ) ⇒ He´ t (Xy , Qp ) ⊗Zp  R. p

(7)

Indeed, it follows from Faltings’ almost-purity theorem that, for every j = 0, the R[ p1 ](j )) is zero. Hence the spectral sequence (7) degenerates at E2 . group H0 (Γ,  R[ 1 ](1)) doesn’t vanish in general, the abutment However, as the group H1 (Γ,  p

filtration doesn’t split in general. 2.3. We conjectured the existence of the spectral sequence (7) in a first version of this work. Scholze immediately told us that he knew how to construct such a

4

A. Abbes and M. Gros

spectral sequence by using the relative Hodge–Tate filtration associated to a proper smooth morphism of adic spaces he developed with Caraiani ([7] 2.2.4). We also learned from Bhatt that he has a strategy to deduce the spectral sequence (7) from the general formalism of prismatic cohomology. He [10] constructed the relative Hodge–Tate filtration Theorem 2.2 in an even more general setting than that of 2.1. He deduced it from the global variant of our Hodge–Tate relative spectral sequence Theorem 3.5 and a cohomological descent result for Faltings’ topos he established. Our proof of Theorem 2.2, similar in spirit to his proof, has independently been suggested by the referee of [2]. To do this, we prove a cohomological descent result ([2], 4.6.30) that turns out to be a particular case of that of He [10].

3 The Global Version of the Relative Hodge–Tate Spectral Sequence 3.1. Let X be a smooth S-scheme (see Footnote 1). We denote by E the category of morphisms V → U above the canonical morphism Xη → X, that is, commutative diagrams V

U (8)

Xη

X

such that U is étale over X and the canonical morphism V → Uη is finite étale. It is useful to consider the category E as fibered by the functor ´ /X , (V → U ) → U, π : E → Et

(9)

´ /X is canonically over the étale site of X. The fiber of π above an object U of Et ´ f/U of finite étale morphisms over Uη . We equip it with equivalent to the category Et η the étale topology and denote by Uη,f´et the associated topos. If Uη is connected and if y is a geometric point of Uη , then the topos Uη,f´et is equivalent to the classifying topos of the profinite group π1 (Uη , y), i.e., the category of discrete sets equipped with a continuous left action of π1 (Uη , y). We equip E with the covanishing topology ([3] VI.1.10), that is the topology generated by coverings {(Vi → Ui ) → (V → U )}i∈I of the following two types : (v) Ui = U for all i ∈ I and (Vi → V )i∈I is a covering; (c) (Ui → U )i∈I is a covering and Vi = V ×U Ui for all i ∈ I .

The Relative Hodge–Tate Spectral Sequence: An Overview

5

 and call Faltings The resulting site is called Faltings site of X. We denote by E topos of X the topos of sheaves of sets on E. It is an analogue of the covanishing ←

topos Xe´ t ×Xe´ t Xη,´et ([3] VI.4). To give a sheaf F on E amounts to give: ´ /X , a sheaf FU of Uη,f´et , namely the restriction of F to (i) for any object U of Et the fiber of π above U ; ´ /X , a morphism γf : FU → fη∗ (FU ). (ii) for any morphism f : U → U of Et These data should satisfy a cocycle condition for the composition of morphisms and ´ /X ([3] VI.5.10). Such a sheaf will be denoted a gluing condition for coverings of Et by {U → FU }. There are three canonical morphisms of topos Xη,´et ψ

X´et

σ

E

(10) β

Xη,f´et

such that ´ /X ), σ ∗ (U ) = (Uη → U )a , ∀ U ∈ Ob(Et

(11)

´ f/X ), β ∗ (V ) = (V → X)a , ∀ V ∈ Ob(Et η

(12)

∗

ψ (V → U ) = V , ∀ (V → U ) ∈ Ob(E),

(13)

where the exponent a means the associated sheaf. The morphisms σ and β in the diagram (10) are the analogues of the first and second projections of the covanishing ←

topos Xe´ t ×Xe´ t Xη,´et . The morphism ψ is an analogue of the co-nearby cycles morphism ([3] VI.4.13) Any specialization map y  x from a geometric point y of Xη to a geometric  denoted by ρ(y  x) ([3] VI.10.18). The point x of X, determines a point of E collection of these points is conservative ([3] VI.10.21). Proposition 3.2 (Abbes and Gros [2] 4.4.2) For any locally constant constructible torsion abelian sheaf F of Xη,´et , we have Ri ψ∗ (F ) = 0 for any i ≥ 1. This statement is a consequence of the fact that for any geometric point x of X over s, denoting by X the strict localization of X at x, Xη is a K(π, 1) scheme ([3] VI.9.21), i.e., if y is a geometric point of Xη , for any locally constant constructible torsion abelian sheaf F on X η and any i ≥ 0, we have an isomorphism ∼

Hi (Xη , F ) → Hi (π1 (Xη , y), Fy ).

(14)

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A. Abbes and M. Gros

This property was proved by Faltings ([8] Lemma 2.3 page 281), generalizing results of Artin ([5] XI). It was further generalized by Achinger to the log-smooth case ([4] 9.5). V 3.3. For any object (V → U ) of E, we denote by U the integral closure of U in V and we set V

B(V → U ) = Γ (U , OU V ).

(15)

The presheaf on E defined above is in fact a sheaf ([3] III.8.16). We write B = {U → BU } (cf. 3.1). For any étale X-scheme U which is affine, the stalk of the sheaf BU of Uη,f´et at a geometric point y of Uη , is the representation R U of π1 (Uη , y) defined in (5) for U . For any specialization map y  x, we have Bρ(yx) = lim R U , −→ U ∈Vx

(16)

where Vx is the category of x-pointed étale X-schemes U which are affine. 3.4. For any topos T , projective systems of objets of T indexed by the ordered ◦ set of natural numbers N, form a topos that we denote by T N ([3] III.7). For any integer n ≥ 0, we set Bn = B/pn B. To take into account p-adic ˘ = (B ) N◦ . We work in topology, we consider the OC -algebra B n n≥1 of the topos E ˘ of B-modules ˘ up to isogeny ([3] III.6.1), which is a global the category Mod (B) Q

analogue of the category of  R[ p1 ]-representations of Δ considered in 2.1.

Theorem 3.5 (Abbes and Gros [2] 6.7.5) Let g : X → X be a smooth projective morphism. We denote by (17) the morphisms induced by gη and ψ (10), and by Z˘ p the Zp -algebra (Z/pn Z)n≥1 ◦ ˘ -modules of X N . Then, we have a canonical spectral sequence of B η,´et

Q

i,j j ˘ (−j ) ⇒ ψ˘ (Ri+j g˘ (Z˘ )) ⊗ B ˘ . E2 = σ ∗ (Ri g∗ (ΩX /X )) ⊗σ ∗ (OX ) B ∗ η∗ p Q Zp Q (18)

The projectivity condition on g is used in Theorem 4.4 below. It should be possible to replace it by the properness of g. The spectral sequence (18) is called the relative Hodge–Tate spectral sequence. We can easily prove that it is GK -equivariant for the natural GK -equivariant structures on the topos and objects that appear. We deduce the following.

The Relative Hodge–Tate Spectral Sequence: An Overview

7

Proposition 3.6 (Abbes and Gros [2] 6.7.13) Under the assumptions of Theorem 3.5, the relative Hodge–Tate spectral sequence (18) degenerates at E2 . Remark 3.7 Using a different approach, Caraiani and Scholze have constructed a relative Hodge–Tate filtration for proper smooth morphisms of adic spaces ([7] 2.2.4).

4 Faltings’ Main p-adic Comparison Theorems 4.1. The assumptions and notation of Sect. 3 are in effect in this section. We denote by OK the limit of the projective system (OK /pOK )N whose transition morphisms are the iterates of the absolute Frobenius endomorphism of OK /pOK , OK = lim OK /pOK . ←−

(19)

N

It is a perfect complete non-discrete valuation ring of height 1 and characteristic p. p We fix a sequence (pn )n≥0 of elements of OK such that p0 = p and pn+1 = pn for any n ≥ 0. We denote by  the associated element of OK and we set ξ = [ ] − p in the ring W(OK ) of p-typical Witt vectors of OK . We have a canonical isomorphism ∼

1

OC (1) → p p−1 ξ OC .

(20)

Theorem 4.2 (Faltings [9] and Abbes and Gros [2] 4.8.13) Assume that X is proper over S. Let i, n be integers ≥ 0, F a locally constant constructible sheaf of (Z/pn Z)-modules of Xη,´et . Then, the kernel and cokernel of the canonical morphism  ψ∗ (F ) ⊗Zp B) Hi (Xη,´et , F ) ⊗Zp OC → Hi (E,

(21)

are annihilated by mC . We say that morphism (21) is an almost isomorphism. This is Faltings’ main p-adic comparaison theorem from which he derived all comparaison theorems between p-adic étale cohomology and other p-adic cohomologies. It is also the main ingredient in the construction of the absolute Hodge–Tate spectral sequence (3). We revisit in [2] Faltings’ proof of this important result providing more details. It is based on Artin-Schreier exact sequence for the “perfection” of the ring B1 = B/pB. One of the main ingredients is a structural statement for almost étale ϕ-modules on OK verifying certain conditions, including an almost finiteness condition in the sense of Faltings. In our application to the cohomology of Faltings’

8

A. Abbes and M. Gros

topos ringed by the “perfection” of B1 , the proof of this last condition results from the combination of three ingredients: (i) local calculations of Galois cohomology using Faltings’ almost-purity theorem ([9], [3] II.8.17); (ii) a fine study of almost finiteness conditions for quasi-coherent sheaves of modules on schemes; (iii) Kiehl’s result on the finiteness of cohomology of a proper morphism ([12] 2.9’a) (cf. [1] 1.4.7). 4.3. Next, we explain Faltings’ construction of the absolute Hodge–Tate spectral sequence (3). Assume that X is proper over S. By Proposition 3.2, for any i, n ≥ 0, we have a canonical isomorphism ∼  ψ∗ (Z/pn Z)). Hi (Xη,´et , Z/pn Z) → Hi (E,

(22)

It is not difficult to see that the canonical morphism Z/pn Z → ψ∗ (Z/pn Z) is an isomorphism. Then, by Faltings’ main p-adic comparison Theorem 4.2, we have a canonical morphism  Bn ), Hi (Xη,´et , Z/pn Z) ⊗Zp OC → Hi (E,

(23)

 Bn ), we use the Cartan-Leray which is an almost isomorphism. To compute Hi (E,  → Xe´ t (10), spectral sequence for the morphism σ : E i,j  Bn ). E2 = Hi (Xe´ t , Rj σ∗ (Bn )) ⇒ Hi+j (E,

(24)

We deduce the absolute Hodge–Tate spectral sequence (3) using the following global analogue of Faltings’ computation of Galois cohomology. Theorem 4.4 (Abbes and Gros [2] 6.3.8) There exists a canonical homomorphism of graded OXn -algebras of Xs,´et ∧ (ξ −1 ΩX1

n /S n

) → ⊕i≥0 Ri σ∗ (Bn ),

(25)

where ξ is the element of W(OK ) defined in 4.1, whose kernel (resp. cokernel) is 2d

2d+1

annihilated by p p−1 mK (resp. p p−1 mK ), where d = dim(X/S). We prove this result using Kummer theory over the special fiber of Faltings  B). ringed topos (E, 4.5. Let g : X → X be a smooth (see Footnote 1) morphism. We associate to X objects similar to those associated to X in Sect. 3 and we equip them with a prime . We have a commutative diagram

The Relative Hodge–Tate Spectral Sequence: An Overview

Xη,´et

ψ

gη

Xη,´et

σ

E

9

X´et g

Θ ψ

E

σ

(26)

X´et

where Θ is defined, for any object (V → U ) of E, by Θ ∗ (V → U ) = (V ×X X → U ×X X )a , where the exponent homomorphism

a

(27)

means the associated sheaf. We have also a canonical ring

B → Θ∗ (B ).

(28)

Theorem 4.6 (Faltings [9] § 6 and Abbes and Gros [2] 5.7.4) Assume that g : X → X is projective. Let i, n be integers ≥ 0, F a locally constant constructible . Then, the canonical morphism sheaf of (Z/pn Z)-modules of Xη,´ et

ψ∗ (Ri gη∗ (F )) ⊗Zp B → Ri Θ∗ (ψ∗ (F ) ⊗Zp B )

(29)

is an almost isomorphism. Observe that the sheaves Ri gη∗ (F ) (i ≥ 0) are locally constant constructible on Xη by smooth and proper base change theorems. Faltings formulated this relative version of his main p-adic comparison theorem in [9] and he very roughly sketched a proof in the appendix. Some arguments have to be modified and the actual proof in [2] requires much more work. It is based on a fine study of the local structure of certain almost-étale ϕ-modules which is interesting in itself ([2] 5.5.20). The projectivity condition on g is used to prove an almost finiteness result for almost coherent modules. We rely on the finiteness results of [6] instead of those of [12]. It should be possible to replace the projectivity condition on g just by the properness of g. Remark 4.7 Scholze has generalized Theorem 4.2 to rigid varieties following the same strategy ([15] 1.3). He has also proved an analogue of Theorem 4.6 in his setting of adic spaces and pro-étale topos ([15] 5.12). He deduces it from the absolute case using a base change theorem of Huber. 4.8. We keep the assumptions of Theorem 4.6. Let n be an integer ≥ 0. Since the canonical morphism Z/pn Z → ψ∗ (Z/pn Z) is an isomorphism, in order to construct the relative Hodge–Tate spectral sequence (18), we are led by Theorem 4.6 to compute the cohomology sheaves Rq Θ∗ (Bn ) (q ≥ 0). Inspired by the absolute case (4.3), the problem is then to find a natural factorization of Θ, to which we can apply the Cartan-Leray spectral sequence. Consider the commutative diagram

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A. Abbes and M. Gros

of morphisms of topos E σ

τ

π

E ×X´et X´et

X´et

(30)

g

Ξ σ

E

X´et

 X X is in fact a relative Faltings topos, We prove that the fiber product of topos E× e´ t e´ t whose definition was inspired by oriented products of topos, beyond the covanishing topos which inspired our definition of the usual Faltings topos.

5 Relative Faltings Topos 5.1. We keep the assumption and notation of 4.5. We denote by G the category of morphisms (W → U ← V ) above the canonical morphisms X → X ← Xη , that is, commutative diagrams W

U

V (31)

X

X

Xη

such that W is étale over X , U is étale over X and the canonical morphism V → Uη is finite étale. We equip it with the topology generated by coverings {(Wi → Ui ← Vi ) → (W → U ← V )}i∈I

(32)

of the following three types : (a) Ui = U , Vi = V for all i ∈ I and (Wi → W )i∈I is a covering; (b) Wi = W , Ui = U for all i ∈ I and (Vi → V )i∈I is a covering; (c) diagrams W

U

V (33)

W

U

V

where U → U is any morphism and the right square is Cartesian.

The Relative Hodge–Tate Spectral Sequence: An Overview

11

The resulting site is called relative Faltings site of the morphism g : X → X. We  and call relative Faltings topos of g the topos of sheaves of sets on G. denote by G ←

It is an analogue of the oriented product of topos Xe ´ t ×Xe´ t Xη,´et ([3] VI.3). There are two canonical morphisms (34) defined by ´ /X ), π ∗ (W ) = (W → X ← Xη )a , ∀ W ∈ Ob(Et

(35)

´ f/X ), λ∗ (V ) = (X → X ← V )a , ∀ V ∈ Ob(Et η

(36)

where the exponent a means the associated sheaf. They are the analogues of the first ←

and second projections of the oriented product Xe ´ t ×Xe´ t Xη,´et .  is canonically equivalent to Faltings topos E  (3.1). Hence, by If X = X, G  → functoriality of relative Faltings topos, we have a natural factorization of Θ : E  into E g

τ  −→ E.   −→ G E

(37)

These morphisms fit into the following commutative diagram of morphisms of topos β

E σ

X´et

Xη,f´et gη

τ π

G

λ

Xη,f´et

(38)

g

g

β

X´et

σ

E

We prove that the lower left square is Cartesian (4.8). We have a canonical morphism ←

  : Xe ´ t ×Xe´ t Xη,´et → G. ←

(39)

To give a point of Xe ´ t ×Xe´ t Xη,´et amounts to give a geometric point x of X , a geometric point y of Xη and a specialization map y  g(x ). We denote (abusively)  is such a point by (y  x ). We prove that the collection of points (y  x ) of G conservative. 5.2. Let x be a geometric point of X , X be the strict localization of X at x and  the relative Faltings topos of X the strict localization of X at g(x ). We denote by G

12

A. Abbes and M. Gros

 → Xη,f´et the canonical morphism the morphism X → X induced by g, by λ : G   (34) and by Φ : G → G the functoriality morphism. There is a canonical section θ of λ, X η,f´et

θ

G λ

id

(40)

X η,f´et We prove that the base change morphism induced by this diagram λ∗ → θ ∗

(41)

 → X η,f´et . φx = θ ∗ ◦ Φ ∗ : G

(42)

is an isomorphism. We set

←

If y is a geometric point of Xη , we obtain naturally a point (y  x ) of Xe ´ t ×Xe´ t  we have a canonical functorial isomorphism Xη,´et . Then, for any sheaf F of G, ∼

F(yx ) → φx (F )y .

(43)

Proposition 5.3 (Abbes and Gros [2] 3.4.34) Under the assumptions of 5.2, for  and any q ≥ 0, we have a canonical isomorphism any abelian sheaf F of G ∼

Rq π∗ (F )x → Hq (Xη,f´et , φx (F )).

(44) ←

Corollary 5.4 (Abbes and Gros [2] 6.5.17) Let (y  x ) be a point of Xe ´ t ×Xe´ t Xη,´et , X the strict localization of X at x , X the strict localization of X at g(x ), g : X → X the morphism induced by g,  → X ϕx : E η,f´et

(45)

 and the canonical morphism analogue of (42). Then, for any abelian group F of E any q ≥ 0, we have a canonical functorial isomorphism ∼

(Rq τ∗ (F ))(yx ) → Rq g η,f´et∗ (ϕx (F ))y .

(46)

 5.5. We consider the following ring of G, !



B = τ∗ (B ).

(47)

The Relative Hodge–Tate Spectral Sequence: An Overview

13

!

!

We have canonical homomorphisms B → g∗ (B ) and h¯ ∗ (OX ) → π∗ (B ), where

h¯ : X → X is the canonical projection. Hence, we may consider g and π as morphisms of ringed topos. ←

!

For any point (y  x ) of Xe ´ t ×Xe´ t Xη,´et , we prove that the ring B(yx ) is normal and strictly henselian. Moreover, the canonical homomorphism OX ,x → !

B(yx ) is local and injective. 5.6. Assume that X = Spec(R) and X = Spec(R ) are affine. Let y be a geometric point of Xη , Δ = π1 (Xη , y ), (Wj )j ∈J the universal cover of Xη at y , y = gη (y ), Δ = π1 (Xη , y) and (Vi )i∈I the universal cover of Xη at y. For every i ∈ I , (Vi → X) is naturally an object of E and for every j ∈ J , (Wj → X ) is naturally an object of E . We set R = lim B(Vi → X), −→

(48)

i∈I





R = lim B (Wj → X ). −→

(49)

j ∈J

We recover the OK -algebras defined in (5). We equip them with the natural actions of Δ and Δ . For every i ∈ I , there exists a canonical X -morphism y → X ×X Vi . We denote by Vi the irreducible component of X ×X Vi containing y and by Πi the corresponding subgroup of Δ . Then, (Vj → X ) is naturally an object of E . We set

!

R = lim B (Vi → X ), −→

(50)

i∈I

Π=



(51)

Πi .

i∈I !



We have canonical homomorphisms R → R → R . For any geometric point x and any specialization map y  g(x ), we prove that we have a canonical isomorphism (determined by the choice of y ) !

∼

B(yx ) →

lim −→

x →U →U

!

R U →U ,

(52)

where the inductive limit is taken over the category of morphisms x → U → U over x → X → X, with U affine étale over X and U affine étale over X, and ! R U →U is the corresponding ring (50). Proposition 5.7 (Abbes and Gros [2] 5.2.29) We keep the assumptions of 5.6 and we assume moreover that g fits into a commutative diagram (see Footnote 1)

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A. Abbes and M. Gros

X

ι

Gdm,S

g

γ

X

ι

(53)

Gdm,S

where the morphisms ι and ι are étale, d and d are integers ≥ 0 and γ is a smooth homomorphism of tori over S. Let n be an integer ≥ 0. Then, !

(i) There exists a canonical homomorphism of graded R -algebras !

!





∧ (ΩR1 /R ⊗R (R /pn R )(−1)) → ⊕i≥0 Hi (Π, R /pn R ),

(54)

1

which is almost injective and its cokernel is killed by p p−1 mK . !





(ii) The R -module Hi (Π, R /pn R ) is almost finitely presented for all i ≥ 0, and it almost vanishes for all i ≥ r + 1, where r = dim(X /X). This is a relative version of Faltings’ computation of Galois cohomology of R, that relies on his almost purity theorem ([9] Theorem 4 page 192, [3] II.6.16). The statement can be globalized using Kummer theory on the special fiber of the ringed  , B ) into the following. topos (E Theorem 5.8 (Abbes and Gros [2] 6.6.4) For any integer n ≥ 1, there exists a !  canonical homomorphism of graded B -algebras of G ∧ (π ∗ (ξ −1 Ω 1

X n /X n



)) → ⊕i≥0 Ri τ∗ (Bn ),

(55) !

 B) → where π ∗ denotes the pullback by the morphism of ringed topos π : (G, (Xe ´ t , h¯ ∗ (OX )), whose kernel (resp. cokernel) is annihilated by p p

2r+1 p−1

mK ), where r =

2r p−1

mK (resp.

dim(X /X).

5.9. Next we consider the Cartan-Leray spectral sequence

i,j



E2 = Ri g∗ (Rj τ∗ (Bn )) ⇒ Ri+j Θ∗ (Bn ).

(56)

Taking into account Theorem 5.8, to obtain the relative Hodge–Tate spectral sequence (18), we need to prove a base change theorem relatively to the Cartesian diagram G

π

g

E

X´et g

σ

X´et

(57)

The Relative Hodge–Tate Spectral Sequence: An Overview

15

Theorem 5.10 (Abbes and Gros [2] 6.5.5) Assume g proper. Then, for any torsion abelian sheaf F of Xe ´ t and any q ≥ 0, the base change morphism σ ∗ (Rq g∗ (F )) → Rq g∗ (π ∗ (F ))

(58)

is an isomorphism. The proof is inspired by a base change theorem for oriented products due to Gabber. It reduces to proper base change theorem for étale topos. Proposition 5.11 (Abbes and Gros [2] 6.5.29) For any integer n ≥ 0, the canonical homomorphism !

Bn OX OX → Bn ,

(59)

where the exterior tensor product of rings is relative to the Cartesian diagram (57), is an almost isomorphism. Theorem 5.12 (Abbes and Gros [2] 6.5.31) Assume that the morphism g is proper. Then, there exists an integer N ≥ 0 such that for any integers n ≥ 1 and q ≥ 0, and any quasi-coherent OXn -module, the kernel and cokernel of the base change morphism σ ∗ (Rq g∗ (F)) → Rq g∗ (π ∗ (F)),

(60)

where σ ∗ and π ∗ denote the pullbacks in the sense of ringed topos, are annihilated by pN . Proposition 5.13 (Abbes and Gros [2] 6.5.32) Let n, q be integers ≥ 0, F a quasicoherent OXn -module which is Xn -flat (1). Assume that the morphism g is proper and that for any integer i ≥ 0, the OXn -module Ri g∗ (F) is locally free (of finite type). Then, the base change morphism σ ∗ (Rq g∗ (F)) → Rq g∗ (π ∗ (F)),

(61)

where σ ∗ and π ∗ denote the pullbacks in the sense of ringed topos, is an almost isomorphism. 5.14. Let n, q be integers ≥ 0. Since the canonical morphism Z/pn Z → is an isomorphism, we deduce from Theorem 4.6, for any q ≥ 0, a canonical morphism

ψ∗ (Z/pn Z)



ψ∗ (Rq gη∗ (Z/pn Z)) ⊗Zp B → Rq Θ∗ (Bn ),

(62)

which is an almost isomorphism. The relative Hodge–Tate spectral sequence (18) is then deduced from the Cartan-Leray spectral sequence (56) using Theorems 5.8 and 5.12.

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A. Abbes and M. Gros

Acknowledgments We would like first to convey our deep gratitude to G. Faltings for the continuing inspiration coming from his work on p-adic Hodge theory. We also thank very warmly O. Gabber and T. Tsuji for the exchanges we had on various aspects discussed in this work. Their invaluable expertise has enabled us to avoid long and unnecessary detours. The first author (A.A) thanks the University of Tokyo and Tsinghua University for their hospitality during several visits where parts of this work have been developed and presented. He expresses his gratitude to T. Saito and L. Fu for their invitations. The authors sincerely thank also B. Bhatt and M. Olsson for their invitation to the second Simons symposium (April 28–May 4, 2019) on p-adic Hodge theory.

References 1. A. ABBES, Éléments de géométrie rigide. Volume I. Construction et étude géométrique des espaces rigides, Progress in Mathematics Vol. 286, Birkhäuser (2010). 2. A. ABBES, M. GROS, Les suites spectrales de Hodge–Tate, preprint (2020) arxiv:2003.04714. To appear in Astérisque. 3. A. ABBES, M. GROS, T. TSUJI, The p-adic Simpson correspondence, Ann. of Math. Stud., 193, Princeton Univ. Press (2016). 4. P. ACHINGER, K(π, 1)-neighborhoods and comparison theorems, Compositio Math. 151 (2015), 1945–1964. 5. M. ARTIN, A. GROTHENDIECK, J. L. VERDIER, Théorie des topos et cohomologie étale des schémas, SGA 4, Lecture Notes in Math. Tome 1, 269 (1972); Tome 2, 270 (1972); Tome 3, 305 (1973), Springer-Verlag. 6. P. BERTHELOT, A. GROTHENDIECK, L. ILLUSIE, Théorie des intersections et théorème de Riemann-Roch, SGA 6, Lecture Notes in Math. 225 (1971), Springer-Verlag. 7. A. CARAIANI, P. SCHOLZE, On the generic part of the cohomology of compact unitary Shimura varieties, Annals of Mathematics 186 (2017), 649–766. 8. G. FALTINGS, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255–299. 9. G. FALTINGS, Almost étale extensions, Cohomologies p-adiques et applications arithmétiques. II, Astérisque 279 (2002), 185–270. 10. T. HE, Cohomological descent for Faltings’ p-adic Hodge theory and applications, preprint (2021), arXiv:2104.12645. 11. O. HYODO, On the Hodge–Tate decomposition in the imperfect residue field case, J. Reine Angew. Math. 365 (1986), 97–113. 12. R. KIEHL, Ein “Descente”-Lemma und Grothendiecks Projektionssatz für nichtnoethersche Schemata, Math. Ann. 198 (1972), 287–316. 13. R. LIU, X. ZHU, Rigidity and a Riemann-Hilbert correspondence for p-adic local systems, Invent. math. 207 (2017), 291–343. 14. P. SCHOLZE, Perfectoid spaces: A survey, Current Developments in Mathematics (2012), 193– 227. 15. P. SCHOLZE, p-adic Hodge theory for rigid-analytic varieties, Forum of Mathematics, Pi, 1 (2013). 16. J. TATE, p-divisible groups, Proceedings of a Conference on Local Fields (Driebergen, 1966), Springer (1967), Berlin, 158–183. 17. T. TSUJI, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. math. 137 (1999), 233–411. 18. T. TSUJI, Semi-stable conjecture of Fontaine-Jannsen: a survey, in Cohomologies p-adiques et applications arithmétiques (II), Astérisque 279 (2002), 323–370.

Semisimplicity of the Frobenius Action on π1 L. Alexander Betts and Daniel Litt

1 Introduction Let K be a finite extension of Qp with residue field of size q, and fix a prime  = p. Fix a choice of geometric Frobenius ϕK ∈ GK and an element σ ∈ IK of inertia which generates tame inertia. If X/K is a smooth proper variety with good reduction, then the Galois action on the étale cohomology Hiet´ (XK , Q ) is unramified, and the eigenvalues of ϕK are all q-Weil numbers of weight i. The strong Tate Conjecture also predicts that the action of ϕK should be semisimple. In general—X not necessarily smooth, proper, or of good reduction—the behavior of the Galois representations Hiet´ (XK , Q ) is less well-understood, and is the subject of several major conjectures. Conjecture 1 (Weight–Monodromy) Let N be the nilpotent endomorphism of Hiet´ (XK , Q ) given by 1e log(σ e ) where σ e is any power of σ acting unipotently on Hiet´ (XK , Q ). Let 0 ≤ W0 Hiet´ (XK , Q ) ≤ W1 Hiet´ (XK , Q ) ≤ · · · ≤ W2i Hiet´ (XK , Q ) = Hiet´ (XK , Q )

L. A. Betts Harvard University, Cambridge, MA, USA e-mail: [email protected] https://lalexanderbetts.net/ D. Litt () University of Toronto, Toronto, ON, Canada e-mail: [email protected] https://www.daniellitt.com/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_2

17

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L. A. Betts and D. Litt

be the weight filtration on Hiet´ (XK , Q ) constructed by Deligne [13, §6]. Then for i (X , Q ) is pure of weight j (see Definition 3 for a all j , the representation grW  K j Het ´ 1 precise definition). Conjecture 2 (Frobenius-Semisimplicity) ϕK acts semisimply on Hiet´ (XK , Q ). When X is smooth and proper, the first of these conjectures is due to Deligne [12, 31]; it is a folklore theorem2 that this implies the general case. The second of these conjectures follows from the strong Tate Conjecture and the Rapoport–Zink spectral sequence when X is smooth and proper with semistable reduction, and de Jong’s theory of alterations allows one to extend this to the case of arbitrary reduction. We will shortly explain (Corollary 1) why one should believe Conjecture 2 in general. There are also analogous conjectures in the case  = p regarding the action of the crystalline Frobenius on Dpst (Hiet´ (XK , Qp )) (see e.g. [29, Conjecture 3.27]). The main aim of this paper is to prove analogues of the above conjectures when the cohomology Hiet´ (XK , Q ) is replaced by the Q -pro-unipotent étale fundamental groupoid π1Q (XK ) of XK . This object, whose formal definition we will recall in Sect. 3, consists of affine Q -schemes π1Q (XK ; x, y) for every x, y ∈ X(K), endowed with an action of GK . These come endowed with various structure maps; in particular each π1Q (XK ; x, x) is a pro-unipotent group over Q . The Lie algebras Lie(π1Q (XK ; x, x)) and the affine rings O(π1Q (XK ; x, y)) admit natural weight filtrations compatible with their induced Galois actions. We will prove the following two results, along with their analogues in the case  = p.

Theorem 1 (Weight–Monodromy for the Fundamental Groupoid) Suppose that X/K is smooth and geometrically connected. Then: Q 1. grW −n Lie(π1 (XK ; x, x)) is pure of weight −n, for all n and all x ∈ X(K); and Q  2. grW n O(π1 (XK ; x, y)) is pure of weight n, for all n and all x, y ∈ X(K).

Theorem 2 (Frobenius-Semisimplicity for the Fundamental Groupoid) Suppose that X/K is smooth and geometrically connected. Then ϕK acts semisimply on Lie(π1Q (XK ; x, x)) and O(π1Q (XK ; x, y)) for all x, y ∈ X(K). The crystalline version of Theorem 1 already appears in work of Vologodsky [34, Theorem 26], to whom this work owes a great debt; we present two proofs, both different to Vologodsky’s and more direct.

1 There are two competing definitions of the word “pure” in the literature: one referring to representations, all of whose Frobenius eigenvalues are Weil numbers of a single weight, and one only requiring this condition on each graded piece of the monodromy filtration. In this paper, we will work exclusively with the latter. 2 A proof of this was outlined to the first author by Tony Scholl; to the best of our knowledge there is no published proof of this fact.

Semisimplicity for π1

19

Remark 1 In [11, Conjecture 3.10], the authors conjecture a number of independence of  results for Weil–Deligne representations associated to unipotent fundamental groups. Our Theorem 2 above implies that the “weak” and “strong” versions of these conjectures are equivalent. In particular, Chiarellotto–Lazda prove the weak forms of their conjectures for smooth projective curves over mixed characteristic local fields, which, by Theorem 2, implies the strong form of their conjectures.

1.1 Canonical Splittings The main technical result in this paper is a pure linear algebra lemma (Definition 4), which shows that weight–monodromy conditions can be used to overcome “extension problems” when studying Frobenius actions. If (V , W• ) is a filtered representation of GK , then it is not in general true that semisimplicity of the Frobenius action on the graded pieces grW n V implies semisimplicity of the action on V itself. However, if each grW n V is pure of weight n, we will show that in fact there is a canonical ϕK -equivariant way to split the W• -filtration on V . In particular, in this case Frobenius-semisimplicity of each grW n V does imply Frobenius-semisimplicity of V . As well as proving our Frobenius-semisimplicity Theorem 2, this also proves the following relation between the weight–monodromy conjecture and the Frobeniussemisimplicity conjecture. Corollary 1 Suppose that the weight–monodromy Conjecture 1 holds for all varieties over K, and that the Frobenius-semisimplicity Conjecture 2 holds for all smooth proper varieties over K. Then the Frobenius-semisimplicity Conjecture 2 holds for all varieties over K. i (X , Q ) is a subquotient of the degree Proof If X/K is a variety, then each grW  K j Het ´ j étale cohomology of a smooth proper variety, and hence Frobenius-semisimple. The canonical ϕK -equivariant splitting provided by the weight–monodromy conjecture for X implies that Hiet´ (XK , Q ) is Frobenius-semisimple.

The canonical splittings we construct also have further consequences for the fundamental groupoids of smooth geometrically connected varieties, in that they provide, for each x, y ∈ X(K), a canonical ϕK -invariant path px,y ∈ π1Q (XK ; x, y)(Q )ϕK . The analogues of these paths in the case  = p already appear in the work of Vologodsky, and in the good reduction case they play a central role in the theory of iterated Coleman integration [6]; in the case  = p these paths were used in [4] to explicitly calculate the non-abelian Kummer map associated to a smooth hyperbolic curve X. Our construction of these canonical paths provides a unified explanation for these phenomena. We remark that in the archimedean setting, i.e. in the category of mixed Hodge structures, the existence of similar canonical splittings is well-known. The weight

20

L. A. Betts and D. Litt

filtration on a mixed Hodge structure is canonically split over C by the Deligne splitting, and there is even a variant of this splitting which is defined over R [10, Proposition 2.20]. Thus there are also canonical choices of R-pro-unipotent Betti paths between any two points in a smooth connected variety X/C. The study of these paths will be taken up by the second author in forthcoming work.

Applications As a further application of our main theorems, we apply these results in Sect. 4 to study the geometry of local Bloch–Kato Selmer varieties arising in the ChabautyKim program. These are three sub-presheaves H1e (GK , U ) ⊆ H1f (GK , U ) ⊆ H1g (GK , U ) of the continuous Galois cohomology presheaf H1 (GK , U ) associated Q

¯ and are all to the Qp -pro-unipotent étale fundamental group U = π1 p (XK , x), representable by affine schemes over Qp [23, Proposition 3][24, Lemma 5]. The relevance of the local Selmer varieties to Diophantine geometry and the Chabauty– Kim method comes via a certain non-abelian Kummer or higher Albanese map X(K) → H1 (GK , U )(Qp ) . When X is a smooth projective curve with good reduction, then the image of this map is contained in H1e (GK , U ) = H1f (GK , U ), which is known to be an affine space over Qp [25, Proposition 1.4]. In the absence of properness or good reduction assumptions, the image is contained in H1g (GK , U ), but not in general in H1f . Our contribution in this paper is to describe the geometry of H1g (GK , U ) (for X not necessarily proper, of arbitrary dimension, and with no restrictions on the reduction type). In fact, we find that its geometry is as good as could be hoped: it is canonically isomorphic to the product of H1e (GK , U ) and an explicit (pro-finitedimensional) vector space V(Dpst (U ))ϕ=1 , and therefore also an affine space. As an illustration, we are able to write down explicit dimension formulae in the case that X is a curve.

2 Weil–Deligne Representations In this section, we fix a finite extension K of Qp with residue field k, along with an algebraic closure K. We write WK for the Weil group of K, i.e. the subgroup of the absolute Galois group GK consisting of elements acting on the residue field k via an integer power of the absolute Frobenius σ : x → x p , and we write v : WK → Z for the unique homomorphism such that w ∈ WK acts on k via σ v(w) . We fix a geometric Frobenius ϕK ∈ WK , i.e. an element such that v(ϕK ) = −f (K/Qp ). The following definition is standard.

Semisimplicity for π1

21

Definition 1 Let E be a field of characteristic 0. A Weil representation with coefficients in E is a representation ρ : WK → Aut(V ) of WK on a finite dimensional E-vector space V on which the inertia group IK acts through a finite quotient. A Weil–Deligne representation with coefficients in E consists of a Weil representation V endowed with an E-linear endomorphism N ∈ End(V ), called the monodromy operator, such that ρ(w) ◦ N ◦ ρ(w)−1 = pv(w) · N for all w ∈ WK . It follows from this condition that N is necessarily nilpotent. We denote the category of Weil–Deligne representations by RepE ( WK ). The category RepE ( WK ) has a canonical tensor product making it into a neutral Tannakian category, where the tensor V1 ⊗ V2 is endowed with the tensor product WK -action, and with the endomorphism NV1 ⊗ 1 + 1 ⊗ NV2 . Example 1 The Weil–Deligne representation E(1) has underlying vector space E, trivial monodromy operator N , and the Weil group acts via w : x → pv(w) x. As explained in [17], Weil–Deligne representations arise naturally from Galois representations, both -adic and p-adic. Example 2 Let  be a prime distinct from p, and choose a generator  t ∈ Q (1). Let t : IK → Q (1) denote the -adic tame character w → is a fully faithful exact ⊗-functor

n

w(p1/ ) . n p1/ n∈N

Then there

RepQ ,cts (GK ) → RepQ ( WK ) from the category of continuous Q -linear3 representations of GK to the category of Weil–Deligne representations. This functor is defined as follows. If (V , ρ0 ) is a continuous Q -linear representation of GK , there is an open subgroup IL ≤ IK acting unipotently on V by Grothendieck’s -adic Monodromy Theorem. We let N denote the endomorphism of V such that   ρ0 (g) = exp t −1 t (g)N for all g ∈ IL . We define an action ρ of WK on V by    n n ρ ϕK g = ρ0 (ϕK g) exp −t −1 t (g)N

3 One

can equally well work with continuous E-linear representations for any algebraic extension E/Q . However, in our context only Q -linear representations will appear, so we restrict to this case.

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for n ∈ Z and g ∈ IK . The tuple (V , ρ, N) is the Weil–Deligne representation associated to V . There is an alternative construction of the Weil–Deligne representation due to Fontaine which avoids the choice of Frobenius ϕK [17, §2.2]. If we write B for the Q -linear Tate module of the Tate elliptic curve Gm /pZ , then B(−1) is an extension of Q (−1) by Q and hence we may form the direct limit Bst, := lim Symn (B(−1)), where the transition maps Symn (B(−1)) → Symn+1 (B(−1)) − → are induced by the inclusion Q → B(−1). There is a canonical Q -algebra structure on Bst, , namely the polynomial algebra Q [x] where x ∈ B(−1) is any d lift of t −1 ∈ Q (−1), as well as a nilpotent derivation N = − dx . For a continuous -adic representation V of GK , one sets Dpst (V ) := lim (B ⊗ V )IL , − →L/K st, which is a Weil–Deligne representation with respect to the natural action of WK and the induced monodromy operator N. This construction yields an isomorphic Weil– Deligne representation to that constructed earlier, depending only on the choice of t. Specifically, if we choose x to be an eigenvector for the chosen Frobenius ϕK (of eigenvalue q), then the map Bst,  Q sending x to 0 induces an isomorphism ∼ Dpst (V ) → V of Weil–Deligne representations. Example 3 Let L/K be a Galois extension, not necessarily finite, and let L0 denote the maximal subfield of L which is unramified over Qp . A discrete (ϕ, N, GL|K )module [18, §4.2.1] consists of a finite-dimensional L0 -vector space V endowed with a σ -linear Frobenius automorphism ϕ : V → V , an L0 -linear endomorphism N : V → V , and a semilinear action ρ0 : GL|K → AutQp (V ) such that: – the action of GL|K has open point-stabilisers and commutes with ϕ and N ; and – we have N ◦ ϕ = p · ϕ ◦ N. This definition comes from the study of potentially semistable representations in abstract p-adic Hodge theory. Given a discrete (ϕ, N, GL|K )-module (V , ϕ, N, ρ0 ), we obtain an L0 -linear Weil–Deligne representation whose underlying vector space and monodromy operator are V and N, and whose representation of WK is given by ρ(w) = ρ0 (w)ϕ −v(w) . We thus obtain a faithful, exact and conservative ⊗-functor Mod(ϕ, N, GL|K ) → RepL0 ( WK ) from the category of discrete (ϕ, N, GL|K )modules to the category of L0 -linear Weil–Deligne representations. Precomposing with the Dieudonné functor Dst,L : RepQp ,dR (GK ) → Mod(ϕ, N, GL|K ), we also obtain an exact ⊗-functor from the category of de Rham (=potentially semistable [5, Théorème 0.7]) representations to the category of L0 -linear Weil–Deligne representations. In the particular case that L = K or L = K, we denote the

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functor Dst,L simply by Dst or Dpst respectively; the latter functor is faithful and conservative. If P is a property of (filtered) Weil–Deligne representations and  is a prime, we shall say that a Q -linear (filtered) Galois representation V has property P just when its associated Weil–Deligne representation has property P ; when  = p this means we assume that V is de Rham. For example, we say that a Weil– Deligne representation is semistable [17, §1.3.7] just when the action of IK is trivial. This corresponds to the usual notions of semistability on Q -linear representations, namely unipotence of the IK -action when  = p and Bst -admissibility when  = p. The following two properties—being Frobenius-semisimple and mixed— play a central role in this paper. Definition 2 An E-linear Weil–Deligne representation V is said to be Frobeniussemisimple just when the action of the geometric Frobenius ϕK on V is semisimple, or equivalently just when every element of WK acts semisimply. For a de Rham Qp linear representation V of GK , this is the same as the action of crystalline Frobenius ϕ on Dpst (V ) being semisimple (as a Qp -linear automorphism). Definition 3 A q-Weil number of weight i in an algebraically closed field E of characteristic 0 is an element α ∈ E which is algebraic over Q ⊆ E and satisfies |ι(α)| = q i/2 for every complex embedding ι : Q → C. Given a Weil–Deligne representation V over a characteristic 0 field E, we write VEi for the largest ϕK -stable subspace of VE such that all the generalized eigenvalues of ϕK |V i are q-Weil numbers of weight i, where q is the size of the residue field E

of K. By Galois descent, VEi is the base change of a subspace V i defined over E. It follows from the definition that N(V i ) ⊆ V i−2 . We

say that a Weil–Deligne representation V is pure of weight i just when j V = j V (i.e. all the eigenvalues of ϕK are q-Weil numbers) and the map ∼

N j : V i+j → V i−j is an isomorphism for all j ≥ 0. We say that a Weil–Deligne representation V endowed with an increasing filtration · · · ⊆ Wi V ⊆ Wi+1 V ⊆ . . . by Weil–Deligne subrepresentations is mixed just when W• is exhaustive and separated and grW i V is pure of weight i for all i. The filtration W• V is called the weight filtration of a mixed Weil–Deligne representation V ; its set of weights is the set wt(V ) := {i ∈ Z : grW i V = 0} .

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The collection of mixed Weil–Deligne representations naturally forms a sym metric monoidal category Repmix E ( WK ), whose morphisms are filtered maps of Weil–Deligne representations, and whose tensor product ⊗ and tensor unit 1 are defined in the usual way. A pure Weil–Deligne representation V of weight i can be viewed as a mixed Weil–Deligne representation by endowing it with the filtration where Wj V = V for j ≥ i and Wj V = 0 for j < i. Remark 2 The subspaces V i ⊆ V defined in Definition 3 are stable under the action of WK and do not depend on the choice of geometric Frobenius ϕK . Indeed, for any there is an n ∈ N such that ρ(ϕ )n = ρ(ϕ )n , and other geometric Frobenius ϕK K K we can equivalently describe VEi as the largest subspace of VE on which all the generalized eigenvalues of ρ(ϕK )n are q-Weil numbers of weight ni. Remark 3 Let V be a filtered de Rham representation of GK on a finite dimensional Qp -vector space, and suppose that V is mixed, which for us means that the Qnr p -linear Weil–Deligne representation associated to Dpst (V ) is mixed. Then the K0 -linear Weil–Deligne representation associated to Dst (V ) is also mixed. Indeed, Qnr p ⊗K0 Dst (V ) is the inertia-invariant subspace of the Weil–Deligne representation associated to Dpst (V ), and hence is mixed since taking invariants under actions of finite groups is exact in characteristic 0 vector spaces. Remark 4 In what follows, our Weil–Deligne representations will not necessarily be finite-dimensional, and will sometimes be either ind-finite-dimensional or pro-finite-dimensional (i.e. a direct limit or inverse limit of finite-dimensional Weil–Deligne representations, respectively). With suitable adaptations all of the definitions, constructions and results of this section also apply to ind-finite-dimensional and pro-finite-dimensional representations (by functoriality). In fact, all the results of this section except Theorem 3 are stated so as to be true verbatim for ind-finitedimensional Weil–Deligne representations, and the corresponding statements in the pro-finite-dimensional case are just the duals. The following result is well-known. Theorem 3 (cf. [34, Proposition 20]) The category Repmix E ( WK ) is a neutral mix Tannakian category over E, and the forgetful functor RepE ( WK ) → RepE ( WK ) is exact, conservative and compatible with the tensor structure. Morphisms in Repmix E ( WK ) are strict for the weight filtration.

Proof (Sketch) Compatibility with the tensor structure is easy to check, and conservativity will be a consequence of exactness, since the forgetful functor reflects zero objects. For the remainder, is suffices to prove that any morphism f : V1 → V0 of mixed representations is strict and that its kernel and cokernel are again mixed when endowed with the subspace and quotient filtrations, respectively.

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We begin by proving this in the case that V0 and V1 are pure of weights i0 and i1 , respectively, viewed as mixed Weil–Deligne representations as described in Definition 3. In this case, the claim amounts to showing that f = 0 if i0 = i1 , and that ker(f ) and coker(f ) are pure of weight i0 = i1 otherwise. If i1 < i0 , then f must carry Wi1 V1 = V1 into Wi1 V0 = 0, so f = 0 in this case. We suppose henceforth that i1 ≥ i0 . Now the functor V → V j picking out the weight j generalized eigenspace is exact for all j , and hence for all j ≥ 0 we have a commuting diagram

with exact rows. The middle and right-hand vertical maps are an isomorphism ∼ and injective, respectively, by purity of V0 and V1 , and hence N j : ker(f )i1 +j → ker(f )i1 −j is an isomorphism. Thus ker(f ) is pure of weight i1 ; the dual argument establishes that coker(f ) is pure of weight i0 and we are done in the case i0 = i1 . Finally, in the case i1 > i0 , we see from the equal-weight case that the image of f is pure of weight i0 , while its coimage is pure of weight i1 . But these have the same underlying representation, which is only possible if this is zero (e.g. since the weight of a non-zero pure representation is the average weight of its generalized −1 ϕK -eigenvalues). Hence f = 0 in this case too. f

Now we deal with the general case. We view V1 → V0 as a filtered chain complex in the category of Weil–Deligne representations, with V0 in degree 0. The associated (homological) spectral sequence [28, Theorem XI.3.1] has first page given by

1 Ei,j

⎧  W ⎪ gri f if i + j = 0, ⎪ ⎨coker  = ker grW if i + j = 1, i f ⎪ ⎪ ⎩0 else,

and degenerates to

∞ Ei,j

⎧ W ⎪ ⎪ ⎨gri (coker(f )) if i + j = 0, = grW if i + j = 1, i (ker(f )) ⎪ ⎪ ⎩0 else.

The differentials on the first page all vanish, since they are morphisms of pure Weil–Deligne representations whose domain has strictly higher weight than the codomain. The same argument establishes that all differentials on higher pages 1 ∞ . In particular, grW (ker(f )) and also vanish, and hence we have Ei,j = Ei,j i

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grW i (coker(f )) are both pure of weight i for all i, so that ker(f ) and coker(f ) are mixed. Strictness of f also follows from degeneration page, since  W at the first W coker(f ) → ker(f ) → ker gr f and gr this ensures that the natural maps grW • • •  coker grW • f are isomorphisms. Proposition 1 (cf. [34, Lemma 21]) In a W -strict short exact sequence 0 → V1 → V → V2 → 0 of filtered Weil–Deligne representations, if V1 and V2 are mixed, so too is V . Proof Taking W -graded pieces, it suffices to prove that any extension of Weil– Deligne representations V1 , V2 which are both pure of weight i is again of weight i. The induced sequence j

j

0 → V1 → V j → V2 → 0 between the weight j generalized Frobenius eigenspaces is exact for each j , and hence we are done by the five-lemma applied to N j −i .

2.1 The Canonical Splitting In what follows, we will need several basic facts about the structure theory of mixed Weil–Deligne representations, most notably that their weight filtrations have canonical splittings compatible with their Weil group actions (which is all we will need in Sect. 3). This will be an immediate consequence of the following lemma describing to what extent one can lift maps between associated gradeds of mixed Weil–Deligne representations. Lemma 1 Let V1 and V2 be mixed Weil–Deligne representations, and let W W grW • f : gr• V1 → gr• V2 be a morphism of graded Weil–Deligne representations. Then there exists a unique linear map f : V1 → V2 satisfying the following properties: 1. f is WK -equivariant and preserves the W -filtration; 2. the associated W -graded of f is the map grW • f ; and 3. for every r > 0, the map r    r (−1)s NVr−s ◦ f ◦ NVs 1 2 s s=0

is W -filtered of degree −r − 1, i.e. takes Wi V1 into Wi−r−1 V2 for every i.

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Moreover, the assignment grW • f → f is linear, and compatible with composition and tensor products. Proof Let us say that a linear map f : V1 → V2 is a weak morphism just when it satisfies conditions (1) and (3) above. In other words, a weak morphism is an element f ∈ W0 Hom(V1 , V2 )WK such that N r (f ) ∈ W−r−1 Hom(V1 , V2 ) for all r > 0, where N denotes the monodromy operator on Hom(V1 , V2 ) = V1∗ ⊗ V2 . It follows from this description that composites and tensor products of weak morphisms are weak morphisms, so it suffices to prove that every morphism W W grW • f : gr• V1 → gr• V2 is induced by a unique weak morphism f : V1 → V2 . To prove this, it suffices to prove that for every mixed Weil–Deligne representaWK ,N =0 , there is a unique f ∈ W V WK lifting tion V and every element f ∈ grW 0 0 V r f such that N (f ) ∈ W−r−1 V for all r > 0; applying this to V = Hom(V1 , V2 ) yields the desired result. Let −i denote the lowest weight of V —if i ≤ 0 then all the weights of V are non-negative and the result is trivial. In general, we proceed by induction on i, and write V as an extension  0 → grW −i V → V → V → 0  are all > −i. It follows from the inductive hypothesis that where the weights of V W (V ) has a unique lift to an element f ∈ W0 V WK such that f ∈ grW (V ) = gr 0 0  for all r > 0. Since f is WK -fixed, it lies in V 0 , so we may N r (f) ∈ W−r−1 V 0 0  further lift f to some f ∈ V . Since f 0 is a lift of f, we have that f 0 ∈ W0 V and that N r (f 0 ) ∈ W−r−1 V for ˜ all r < i. We also have N i (f 0 ) ∈ grW −i V : if i = 1 this follows since f = f lies in the kernel of N on V˜ by assumption, while if i > 1 this follows since N i−1 (f 0 ) ∈ W −2i , and so by purity there 0 0 i 0 grW −i V already. Since f ∈ V , we have N (f ) ∈ gr−i V 0 i 1 i 0 0 1 is a unique f 1 ∈ grW −i V such that N (f ) = N (f ). It follows that f := f − f 0 r is the unique element of W0 V which maps to f and satisfies N (f ) ∈ W−r−1 V for all r > 0. Unicity implies that f is also WK -fixed, so the lemma is proved. Definition 4 Let V be a mixed Weil–Deligne representation. The canonical splitting of the weight filtration is the WK -equivariant linear isomorphism ∼

V → grW • V ∼

W W obtained by applying Lemma 1 to the evident isomorphism grW • V → gr• gr• V . In other words, it is the WK -equivariant map f uniquely characterised by the fact that it takes grW i V into Wi V , and that for any vi ∈ Wi V we have r      s  W r r−s (−1)s (grW f N (vi ) ∈ gri−j −1 V • N) s s=0

j ≥r

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W for all r > 0, where grW i N denotes the induced monodromy operator on gri V . It follows from Lemma 1 that this splitting is functorial and compatible with tensor products.

Example 4 Suppose that V is an extension of E by E(1) in the category of Weil– Deligne representations. We endow V with the filtration such that W0 V = V , W−1 V = W−2 V = E(1) and W−3 V = 0, so that V is a mixed Weil–Deligne representation. V admits a canonical choice of basis v0 , v2 , where v2 ∈ E(1) is the canonical generator and v0 is the unique ϕK -invariant lift of the canonical generator of to this basis, the actions of ϕK and N are given by the matrices  E. With  respect   1 0 00 and , respectively, for some λ ∈ E. 0 q −1 λ0 ∼

It follows from this description that the linear isomorphism E ⊕ E(1) → V defined by the basis v0 , v2 satisfies the conditions of Lemma 1, and hence is the canonical splitting of the weight filtration. Note that this splitting is not a splitting in the category of Weil–Deligne representations when λ = 0 above. The following corollary plays a crucial role in our proofs of Theorem 2. Corollary 2 (To Definition 4) Let V be a mixed Weil–Deligne representation. Then V is Frobenius-semisimple if and only if grW • V is Frobenius-semisimple.

2.2 Mixed Representations as Deformations of Pure Representations The canonical splitting of a mixed Weil–Deligne representation allows us to view every mixed Weil–Deligne representation as a deformation of a pure Weil–Deligne representation. Here, a pure Weil–Deligne representation means a graded Weil– Deligne representation whose ith graded piece is pure of weight i for all i. We will continue to write grW i V for the graded pieces of a pure Weil–Deligne representation,  for the monodromy which are direct summands of V . We will usually write N operator on a pure Weil–Deligne representation. The sense in which mixed Weil–Deligne representations are deformations of pure ones is made precise in the following definition. Definition 5 Let V be a pure Weil–Deligne representation. A collection of mixing W data δ for V consists of W -graded endomorphisms δr : grW • V → gr•−r V of degree −r for r > 0, satisfying • ρ(w) ◦ δr ◦ ρ(w)−1 = pv(w) · δr for all r > 0 and all w ∈ WK ; and r−1 • adr−1  (δr ) = 0 for all r > 0 (so in particular δ1 = 0). Here, adN  (−) denotes N  −]. the (r − 1)-fold iterate of the commutator map adN (−) = [N,

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Given mixing data δ 0 and δ 1 for pure Weil–Deligne representations V0 and V1 , there are natural mixing data on V0 ⊕ V1 and V0 ⊗ V1 , given by δr = δr0 ⊕ δr1 and δr = δr0 ⊗ 1 + 1 ⊗ δr1 , respectively. Now if δ is a collection of mixing data for a pure Weil–Deligne representation V , we define an associated mixed Weil–Deligne representation Vδ , whose underlying Weil representation is V , whose W -filtration is the filtration underlying the W  grading on V , and whose monodromy operator is Nδ = N + i>0 δi . It is easy to see that the associated graded of Vδ is V , and hence V is indeed a mixed Weil– Deligne representation. Lemma 2 The functor (V , δ) → Vδ defines a ⊗-equivalence from the category of pure Weil–Deligne representations with mixing data to the category of mixed Weil– Deligne representations. Proof Let us describe the inverse functor. Let V be a mixed Weil–Deligne representation, and identify V with its associated graded grW • V via the canonical  = grW splitting from Definition 4. For clarity, we will write N • N for the monodromy  operator on grW • V and N for the monodromy operator  on V . Now δ : = N − N is W -filtered of degree ≤ −1, so we write δ = r>0 δr with δr W -graded of degree −r. Showing that the δr satisfy the conditions in Definition 5 is equivalent to showing that ρ(w) ◦ δ ◦ ρ(w)−1 = pv(w) · δ for all w ∈ WK , and that adr−1  (δ) is N W -filtered of degree ≤ −r − 1 for all r > 0. The first of these follows immediately . from the commutation relations for N and N For the second, we proceed by strong induction on r > 0. Let εr ∈ End(V ) be given by r    r r−s ◦ N r . (−1)s N εr = s s=0

An easy calculation verifies that ε1 = δ and εr+1 = adN (εr ) + δ ◦ εr for all r > 0. sk −1 s1 −1 s2 −1 Thus εr is a linear combination of composites adN  (δ)  (δ) ◦ adN  (δ) ◦ · · · ◦ adN for positive integers si summing to r, and the coefficient of adr−1  (δ) in εr is 1. N

Assuming for the purposes of induction that ads−1  (δ) is W -filtered of degree ≤ N

−s − 1 for all s < r, we obtain that εr ≡ adr−1  (δ) modulo W−r−1 End(V ). But the N

construction of the canonical splitting ensures that εr ∈ W−r−1 , and so adr−1  (δ) ∈ N W−r−1 as claimed. It remains to check that this functor is inverse to the functor (V , δ) → Vδ . One direction—that V = Vδ for a mixed Weil–Deligne representation V where δ is the mixing data constructed above—is clear. For the other, we wish to show that if δ is mixing data for a pure Weil–Deligne representation V then the mixing data constructed on Vδ is δ again. For this, we reverse the above argument: the condition that adr−1  (δ) is W -filtered of degree ≤ −r − 1 for all r > 0 ensures that N r  r s r−s ◦ N r is W -filtered of degree ≤ −r − 1 for all r > 0, and hence s=0 s (−1) N δ

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the identity map Vδ → V is the canonical splitting of the weight filtration of Vδ . This implies that the mixing data constructed from Vδ is indeed δ.

2.3 The Tannaka Group of Mixed Weil–Deligne Representations To conclude this section, we will use the above structure theory of mixed Weil– Deligne representations to give an explicit description of the Tannaka group of mixed Weil–Deligne representations. For this, we write WKmix for the Tannaka group of Weil representations, all of whose generalized Frobenius eigenvalues are q-Weil numbers. Lemma 3 Let Gpure (resp. Gmix ) denote the Tannaka group of pure (resp. mixed) Weil–Deligne representations. 1. There is a canonical isomorphism Gpure ∼ = SL2  WKmix , where WK acts on SL2   1 0 ∈ GL2 . via conjugation by w → 0 pv(w) 2. Let B denote the standard 2-dimensional representation of SL2  WKmix , i.e. the   1 0 standard representation of SL2 with the Weil group acting via w → . 0 pv(w) ∼ U  Gpure , where U is the free Then there is a canonical isomorphism  Gmix = r 4 pro-unipotent group generated by r≥0 Sym (B)(1) and the action of Gpure = mix SL2  WK on U is the natural one. Remark 5 The above lemma says that any mixed Weil–Deligne representation V carries a canonical action of U  SL2  WKmix . The relationship between these two structures is as follows. 1. The restriction of the action to WKmix is the Weil group action on V . 2. There is a Frobenius-weight torus Gm → WKmix

. The grading on V corresponding to the action of this torus is the grading V = i V i , i.e. the action of λ ∈ Gm on V i is by multiplication by λi . (In fact, this defines the Frobenius-weight  −1  torus.)  λ 0 3. There is a weight torus Gm → SL2  WKmix given by λ → , ι(λ) 0 λ with ι the inclusion of the Frobenius-weight torus. The grading on V corresponding to the action of this torus is the canonical splitting of the weight filtration W• . In particular, the weight filtration on V is the one underlying this grading. One can check that the weight torus is central in SL2  WKmix .

this, we mean that U is the pro-unipotent group pro-representing the functor U →  r lim Hom( N r=0 Sym (B)(1), Lie(U )) from (finite-dimensional) unipotent groups to sets. − →N

4 By

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  10 4. The action of log(Y ), where Y = ∈ SL2 , is the W -graded monodromy 11  := grW operator N • N, viewed as an endomorphism of V via the canonical splitting of the weight filtration. 5. For r ≥ 2, let exp(δr ) ∈ Symr−2 (B)(1)−2 = E(1) denote the standard generator e1r−2 , where e1 is the first basis vector in B. We view exp(δr ) as an element of U. Then, as the notation suggests, the action of log(exp(δr )) is equal to δr where δ = (δr )r≥2 is the mixing data of V as in Definition5. In particular, the monodromy operator N is given by the action of log(Y ) + r≥2 δr . The following lemma explains the relevance of SL2 in the context of the weight– monodromy condition. Lemma

4 Let C denote the category of finite-dimensional graded vector spaces j endowed with an endomorphism N : V • → V •−2 of degree V = jV ∼

−2 such that N j : V j → V −j is an isomorphism for all j ≥ 0. Then C is neutral Tannakian, with Tannaka group (with respect to the evident fiber functor) canonically isomorphic to SL2 .

Proof If V is a representation of SL2 , we endow V with the grading V = j V j   λ 0 where Hλ := acts on V j by multiplication by λj . We let N denote the 0 λ−1   10 endomorphism given by log(Y ) with Y = . From the commutation relation 11 Hλ ◦ log(Y ) ◦ Hλ−1 = λ−2 log(Y ) we see that N is graded of degree −2, and we see, ∼ e.g. from the classification of irreducible representations of SL2 , that N j : V j → −j V is an isomorphism for all j ≥ 0. This construction provides a ⊗-functor F : Rep(SL2 ) → C. To show that F is an equivalence, it suffices to show that it induces a bijection between the sets of isomorphism classes of irreducible objects, and that every object of C decomposes as a direct sum of irreducible objects. For the first part, if Vr denotes the r + 1dimensional irreducible representation of SL2 (r ≥ 0), then F (Vr ) is generated under N by a single element v in degree r, subject to the relation N r+1 (v) = 0. This implies that F (Vr ) is irreducible, e.g. since any proper subobject would have to contain N r (v) but not v. Conversely, if V is an irreducible object of C, then let r be the greatest integer such that V r = 0 (so r ≥ 0). If v ∈ V r \ {0} then N r+1 (v) = 0 but N r (v) = 0, so that v spans a subobject of V isomorphic to F (Vr ). It follows that V = F (Vr ). Since the objects F (Vr ) are clearly non-isomorphic, we have established that F induces a bijection on isomorphism classes of irreducible objects. To establish that objects of C are completely decomposable, take any non-zero object V and let V = V /U be an irreducible quotient, isomorphic to F (Vr ) for some r r ≥ 0. Let v ∈ V \ {0} be a highest weight vector and let v ∈ V r be any lift of v. r+1 Since N (v) ∈ U −r−2 , there is some u ∈ U r+2 such that N r+1 (v) = N r+2 (u);

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replacing the lift v with v − N(u) if necessary, we may assume that N r+1 (v) = 0. But then we see that the quotient map V  V splits, via the map N j (v) → N j (v). Thus V is in fact a direct summand of V , which establishes complete reducibility. Proof of Lemma 3 (1) We describe a canonical ⊗-equivalence of categories between the category of pure Weil–Deligne representations and the category of representations of SL2  WKmix . In the one direction, if V is a pure Weil–Deligne

 W i+j  representation, then the grading V = and endomorphism N j i gri V endow V with the structure of an object of the category C from Lemma 4, and hence endows V with an action ρ : SL2 → GL(V ). If we let V (n) denote the object , then of C with the same vector space and grading but with endomorphism pn N the corresponding  action  of SL2 is given by precomposing the original action with 1 0 . Now the action of an element w ∈ WK can be viewed as conjugation by 0 pn ∼

an isomorphism V → V (v(w)) in C, and is hence equivariant for the SL2 -action. Unpacking this, this says that we have the commutation relation ρ(w) ◦ ρ(M) ◦ ρ(w)

−1

 =ρ

   1 0 1 0 ·M · 0 p−v(w) 0 pv(w)

for all w ∈ WK and M ∈ SL2 , so that the actions of WK and SL2 together induce an action of SL2  WKmix on V . In the other direction, given a representation V of SL2  WKmix , we obtain by  := log(Y ) restriction a representation of WK and a nilpotent endomorphism N −1 v(w)   satisfying ρ(w) ◦ N ◦ ρ(w) = p · N for all w ∈ WK (where Y is as in Remark 5). Now the weight torus Gm → SL2  WKmix is central, and so provides a

W decomposition V = i gri V in the category of Weil–Deligne representations. ∼ i−j is an j : grW V i+j → grW It then follows from Lemma 4 that the map N i i V isomorphism for all j ≥ 0 and all i, so that this provides V with the structure of a pure Weil–Deligne representation. These two constructions are evidently self-inverse. (2) It follows from Lemma 2 that the category of mixed Weil–Deligne representations is ⊗-equivalent to the category of pure Weil–Deligne representations together with morphisms Symr−2 (B)(1) → End(V ) of pure representations for every r ≥ 2 (where δr is the image of the generator in Symr−2 (B)(1)−2 = E(1)). Specifying these morphisms is equivalent to specifying a Gpure -equivariant morphism U → GL(V ), which provides the desired description of Gmix . Remark  6 In  Sect. 4, an important role will be played by the action of the element 11 X = ∈ SL2 , in particular its fixed vectors (the “highest weight vectors” 01 of the underlying SL2 -representation). One can verify that the X-fixed vectors of a mixed Weil–Deligne representation V are given by

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(V j )X = {v ∈ V j such that N r (v) ∈ Wj −r for all r ≥ 0} . Using this, one sees that the mixing data (δr )r>0 associated to a mixed Weil–Deligne representation V is X-fixed, i.e. we have X ◦ δr ◦ X−1 = δr for all r. Indeed, this is actually equivalent to the condition that adr−1  (δr ) = 0 (in the presence of the N −1 −v(w) δr for all w ∈ WK , which ensures that condition that ρ(w) ◦ δr ◦ ρ(w) = p −2 ). δr ∈ grW End(V ) −r Remark 7 If V is a W -filtered (ϕ, N, GL|K )-module which is mixed in the sense that its associated L0 -linear Weil–Deligne representation is mixed, then all of the constructions in this section are compatible with the crystalline Frobenius ϕ: for instance the canonical splitting (Definition 4) is ϕ-invariant, and the mixing operators δr (Definition 5) satisfy ϕ ◦ δr ◦ ϕ −1 = p−1 δr . Perhaps the easiest way to see this is to observe that ϕ provides an isomorphism of mixed Weil–Deligne representations from V to V with a rescaled monodromy operator N, and observe that all of our constructions are functorial and are unchanged (up to appropriate scaling factors) on rescaling N.

3 Results on Semisimplicity and Weight–Monodromy As before, we fix K a finite extension of Qp , with residue field k and ring of integers OK . Let X be a geometrically connected K-variety, and let x ∈ X(K) be a rational point. Fixing an algebraic closure K¯ of K, we let x¯ be the geometric point of X associated to x.

3.1 The étale Fundamental Group Let  be a prime. We let π1 (XK¯ , x) ¯ be the pro- completion of the geometric étale ´ et fundamental group π1 (XK¯ , x). ¯ As x was a rational point of X, there is a natural ¯ action of Gal(K/K) on π1 (XK¯ , x). We let Z [[π1 (XK¯ , x)]] ¯ :=

lim ← −

Z [H ]

π1 (XK¯ ,x)H ¯

be the group ring of π1 (Xk¯ , x), ¯ where the inverse limit is taken over all finite groups arising as continuous quotients of π1 (Xk¯ , x). ¯ There is a natural augmentation map  : Z [[π1 (XK¯ , x)]] ¯ → Z

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(induced by the map g → 1, for g ∈ π1 (XK¯ , x)), ¯ and we let I be the kernel of this map—the augmentation ideal. As with any group algebra, there is a natural comultiplication map  Z [[π1 (XK¯ , x)]] Δ : Z [[π1 (XK¯ , x)]] ¯ → Z [[π1 (XK¯ , x)]] ¯ ⊗ ¯  denotes the completed tensor product.) sending a group element g to g ⊗ g. (Here ⊗ Finally, we set n Q [[π1 (XK¯ , x)]] ¯ = lim(Z [[π1 (XK¯ , x)]]/I ¯ ⊗ Q ). ← − n

The comultiplication map Δ induces a comultiplication on Q [[π1 (XK¯ , x)]]. ¯ We abuse notation to denote the augmentation ideal of Q [[π1 (XK¯ , x)]] ¯ by I. The category of topological Q [[π1 (XK¯ , x)]]-modules ¯ which are finite-dimensional as Q -vector spaces is equivalent to the category of continuous unipotent π1 (XK¯ , x)¯ representations on Q -vector spaces. The ring Q [[π1 (XK¯ , x)]] ¯ is thus the (topological) opposite Hopf algebra to the ring of functions on the Q -pro-unipotent fundamental group of X. If x¯1 , x¯2 are two geometric points of X, we let π1et´ (XK¯ ; x¯1 , x¯2 ) be the pro-finite set of “étale paths” from x¯1 to x¯2 (that is, the set of isomorphisms between the fiber functors associated to x¯1 , x¯2 ). This is a (right) torsor for the group π1et´ (XK¯ ; x¯1 ); let π1 (XK¯ ; x¯1 , x¯2 ) be the associated (right) torsor for π1 (XK¯ , x¯1 ). It is easy to check that the natural left action of π1et´ (XK¯ , x¯2 ) on π1et´ (XK¯ ; x¯1 , x¯2 ) descends to a left action of π1 (XK¯ , x¯2 ) on π1 (XK¯ ; x¯1 , x¯2 ). Let Z [[π1 (XK¯ ; x¯1 , x¯2 )]] :=

lim ← −

Z [H ],

π1 (XK¯ ;x¯1 ,x¯2 )H

where the inverse limit is taken over all finite sets with a continuous surjection from π1 (XK¯ ; x¯1 , x¯2 ). This is a free right module of rank one over Z [[π1 (XK¯ , x¯1 )]], and thus inherits an I-adic filtration; we let Q [[π1 (XK¯ ; x¯1 , x¯2 )]] := lim(Z [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Q ). ← − n

This vector space also has a natural filtration, which we call the I-adic filtration by an abuse of notation, defined by In = ker(Q [[π1 (XK¯ ; x¯1 , x¯2 )]] → Z [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Q ). We define a rational tangential basepoint of X to be a K((t))-point of X; the inclusion K → K((t)) allows one to view any K-point of X as a rational tangential

Semisimplicity for π1

35

basepoint. We let K((t)) be the usual algebraic closure of K((t)), namely the field of Puiseux series K((t Q )), which we fix for the rest of this paper. Now if x1 , x2 are rational tangential basepoints of X, the group Gal(K((t))/K((t))) acts on the triple (XK , x¯1 , x¯2 ), and hence by functoriality of π1 (XK¯ ; x¯1 , x¯2 ), on π1 (XK¯ ; x¯1 , x¯2 ). In particular, if x¯1 = x¯2 arise from a rational point x ∈ X(K), we ¯ obtain an action of Gal(K/K) on π1 (XK¯ , x). Suppose  = p. Then we set Π  (XK¯ ; x¯1 , x¯2 ) := Q [[π1 (XK¯ ; x¯1 , x¯2 )]]. We give Π  (XK¯ ; x¯1 , x¯2 ) the structure of a Weil–Deligne representation as in Example 2. Explicitly, for each n, Π  (XK¯ ; x¯1 , x¯2 )/In is a finite-dimensional vector space, and thus naturally admits the structure of a Weil–Deligne representation as in Example 2. The construction is functorial, giving Π  (XK¯ ; x¯1 , x¯2 ) the structure of a pro-finite-dimensional Weil–Deligne representation.

3.2 The Crystalline Setting Suppose  = p (the residue characteristic of K), and let x1 , x2 be rational tangential basepoints of X. Then we set p

Π p (XK¯ ; x¯1 , x¯2 ) := lim Dpst (Zp [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Qp ). ← − n

Π p (XK¯ ; x¯1 , x¯2 )

Remark 8 The object may be interpreted in terms of the logcrystalline fundamental group of the special fiber of a semi-stable model of (X, D), but we will not need this interpretation here. We abuse notation and set p

In := ker(Π p (XK¯ ; x¯1 , x¯2 ) → Dpst (Zp [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Qp ). As in Example 3, Π p (XK¯ ; x¯1 , x¯2 ) has the structure of a (pro-finite-dimensional) Weil–Deligne representation. p

Remark 9 We briefly explain why the Galois representation Zp [[π1 (XK¯ ; x¯1 , x¯2 )]]/ In ⊗ Qp is de Rham—this is proven in [7, Lemma 7.1] if x¯1 , x¯2 arise from rational points of X, but does not appear in the literature if these geometric points arise from rational tangential basepoints. We will require this for the proof of Theorem 4. Deligne and Goncharov [15, Proposition 3.4] construct a local system on X × X p whose fiber at a point (x¯1 , x¯2 ) is (Zp [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Qp )∨ , as a higher direct image of a sheaf on a diagram of schemes over X × X (strictly speaking, Deligne and Goncharov work in the Betti setting, but an identical construction works in the étale setting). The fiber of this local system at a K-point of X × X is de Rham

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by e.g. [7, Lemma 7.1] (or by Andreatta et al. [2, Theorem 1.4] in the case this point lies on the diagonal of X × X). Hence this local system is de Rham in the sense of [27] by Liu and Zhu [27, Theorem 1.3]. Now the result at tangential basepoints follows from the results of [16, Section 4.3], for example.

3.3 The Main Theorems Let X be a smooth geometrically connected variety over K, and x1 , x2 rational tangential basepoints of X. Let  be a prime, which may be equal to p. The main theorems of this section are: Theorem 4 (Weight–Monodromy) The Weil–Deligne representation Π  (XK¯ ; x¯1 , x¯2 ), with the canonical weight filtration (Definition 6 below), is mixed. Theorem 5 (Semisimplicity) Each element of WK acts semisimply on the Weil– Deligne representation Π  (XK¯ ; x¯1 , x¯2 ). Theorem 5 above admits the following down-to-earth reformulation. If  = p, the theorem says that every Frobenius element of Gal(K/K) acts semisimply on Z [[π1 (XK¯ ; x¯1 , x¯2 )]], or equivalently that every element of WK acts semisimply on Π  (XK¯ ; x¯1 , x¯2 )/In for all n (with the structure of a Weil–Deligne representation given by Example 2). If  = p, the theorem is the analogous statement for a K-linear power of the crystalline Frobenius. In both cases, the statement is equivalent to the semi-simplicity of the geometric Frobenius ϕK fixed at the beginning of Sect. 2. As an immediate corollary, we have Corollary 3 Let x be a rational tangential basepoint of X. Let gX be the Lie ¯ Then ( = p) gX is algebra of the Q -pro-unipotent completion of π1et´ (XK¯ , x). a mixed Weil–Deligne representation (with respect to the weight filtration defined below) and each element of WK acts semisimply on it, and ( = p) Dpst (gX ) is mixed and each element of WK acts semisimply on it. Proof The Lie algebra gK may be identified as the set of primitive elements in Π  (XK¯ ; x, ¯ x), ¯ i.e. the kernel of the map Δ − id ⊗1 − 1 ⊗ id, with the weight filtration inherited from Π  (XK¯ ; x, ¯ x) ¯ (Definition 6). The result is immediate.

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37

Preliminaries Before giving the proof, we will need to recall some lemmas, most of which are likely well-known to experts. Proposition 2 There is a canonical (Galois-equivariant) isomorphism ∼

π1 (XK¯ , x¯1 )ab −→ I/I2 . ∨ Moreover π1 (XK¯ , x¯1 )ab /π1 (XK¯ , x¯1 )ab [∞ ]  H1 (XK,´ ¯ et , Z ) canonically (in particular, as Galois modules).

Proof See [26, Proposition 2.4]. We will also need the following part of the Weight–Monodromy Conjecture. Proposition 3 Let Y be any smooth K-variety with dim(Y ) ≤ 2. Let i ∈ Z and let W• be the weight filtration on Hi (YK,´ ¯ et , Q ) [12]. Then the Weil–Deligne i representation H (YK,´ ¯ et , Q ) is mixed with positive weights. Proof Let Y be a simple normal crossings compactification of Y , which exists by resolution of singularities. For  = p, the case of smooth projective Y with semistable reduction is proven for  = p in [31, Satz 2.13]; the case  = p is proven by Mokrane [29, Corollaire 6.2.3] (Mokrane proves the log-crystalline statement; the statement here follows by applying the p-adic comparison theorem [33, Theorem 0.2]). The case of arbitrary smooth proper Y follows from Chow’s lemma and de Jong’s theory of alterations. Finally, the general case follows immediately from the Deligne spectral sequence, i.e. the Leray spectral sequence associated to the embedding Y → Y (see e.g. [22, pg. 2]), using Theorem 3 and Proposition 1. Proposition 4 Let Y be any smooth K-variety. 1. ( = p): ϕK acts semi-simply on H1 (YK,´ ¯ et , Q ). 2. ( = p): ϕK acts semi-simply on Dpst (H1 (YK,´ ¯ et , Qp )). Proof We may reduce to the case Y is quasi-projective by replacing Y with an affine open. By the Lefschetz hyperplane theorem, it suffices to prove this for Y a curve. Let Y by the smooth compactification of Y ; after extending K, we may assume Y has semistable reduction, and that Y \ Y is a disjoint union of rational points of Y . Then the result for Y smooth proper is immediate from the Rapoport–Zink spectral sequence (see [31, Satz 2.10] for the case  = p and [29, 3.23] for the case  = p, again using the p-adic comparison theorem [33, Theorem 0.2] to apply the statement) and the analogous fact for abelian varieties. To deduce the result for Y , note that H1 (Y K¯ , Q )  W1 H1 (YK¯ , Q ) → H1 (YK¯ , Q )

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splits ϕK -equivariantly by Definition 4; but H1 (YK¯ , Q )/W1 H1 (YK¯ , Q ) is isomorphic to a direct sum of copies of Q (−1), so the result follows.

3.4 Weight-Monodromy for π1 We are now ready to prove Theorem 4. Let Y be any smooth geometrically connected K-variety with simple normal crossings compactification Y ; let x1 , x2 be rational tangential basepoints of Y . Definition 6 (Weight Filtration on Π  ) We define the weight filtration on Π  (YK¯ ; x¯1 , x¯2 ). Let K = ker(Π  (YK¯ ; x¯1 , x¯2 ) → Π  (Y K¯ ; x¯1 , x¯2 )) W−1 = I W−2 = I2 + K and in general, W−i Π  (YK¯ ; x¯1 , x¯2 ) =



W−p Π  (YK¯ ; x¯1 , x¯2 ) · W−q Π  (YK¯ ; x¯1 , x¯1 ) for i > 2.

p+q=i,p,q>0

Remark 10 As with any weight filtration arising in algebraic geometry, we claim the filtration defined above is uniquely characterized as follows. Let R ⊂ K((t)) be a finitely-generated Z-algebra, Y an R-model of Y (that is, a flat R-scheme equipped with an isomorphism YK((t))  YK((t)) ). After possibly enlarging R, we may let y1 , y2 be R-points of Y so that the fiber functors associated to yi,K((t)) are Galois-equivariantly isomorphic to those associated to those associated to x¯i . Then there exists an open subset U of Spec(R[1/]) such that for any closed point p  of U , the associated Frobenius element acts on grW ¯ , y¯2,k(p) ¯ ) with ¯ ; y1,k(p) i Π (Yk(p) ¯ eigenvalues #k(p)-Weil numbers of weight i. Note that we spread out a model over K((t)), rather than over K, to deal with the case where the xi are rational tangential basepoints. In particular, the induced filtration on I/I2 agrees with the usual weight filtration coming from the identification with H1 (YK¯ , Q )∨ in Proposition 2 by construction; then the claim above follows by the multiplicativity of the weight filtration. We will use below the resulting compatibility with another description of the weight filtration arising from work of Deligne and Goncharov [15]. We now prove Theorem 4. Before beginning the proof, we will recall the following crucial result of Deligne and Goncharov. Strictly speaking, Deligne and Goncharov

Semisimplicity for π1

39

only prove this in the topological setting but the proof works identically in the algebraic setting; we state the result in the form we need. Theorem 6 (Deligne and Goncharov [15, Proposition 3.4]) Let X be a smooth, geometrically connected variety over K, and let  be a prime. Let a, b be geometric points of X, and for i ∈ {0, 1, · · · , n} let Yi ⊂ Xn be the subvariety given by Y0 := {(x1 , · · · , xn ) ⊂ Xn | x0 = a}, Yi := {(x1 , · · · , xn ) ⊂ Xn | xi = xi+1 if 1 ≤ i ≤ n − 1, and Yn := {(x1 , · · · , xn ) ⊂ Xn | xn = b}. For I ⊂ {0, 1 · · · , n} let YI :=



Yi .

i∈I

Let jI : YI → Xn be the natural inclusion and let VI := (jI )∗ Q be the pushforward of the constant lisse sheaf from YI to Xn . Then there is a natural complex Ka,b : Q →

 i∈{0,··· ,n}

Vi → · · · →



VI → · · · → V{0,··· ,n} → 0

I ⊂{0,··· ,n},|I |=p

such that • Hi (Xn , Ka,b ) = 0 for i < n • Hn (Xn , Ka,b ) = (Z [[π1 (XK¯ ; a, b)]]/In+1 ⊗ Q )∨ where Hi denotes the hypercohomology of the complexes above. Proof of Theorem 4 We explain how to deduce the theorem from Proposition 3. By Chow’s lemma, we may assume X is quasi-projective. First, note that by the Lefschetz hyperplane theorem [19, pg. 195] for fundamental groups, we may reduce to the case where dim(X) ≤ 2. In the case  = p, recall from [15, Proposition 3.4] (or Theorem 6) that (Π  (XK¯ ; x¯1 , x¯2 )/In )∨ may be computed as the hypercohomology of a complex of sheaves on Xn−1 ; each of these sheaves is a direct sum of sheaves of the form j∗ Q , where j : Xm → Xn−1 is a closed embedding. Thus, by Proposition 3, there is a spectral sequence whose E 1 term consists of mixed Weil–Deligne representations of the form Hi (Xm , Q ), where the weight filtration comes from the usual weight filtration on cohomology [12] (using the Künneth formula), and whose E ∞ page has on it grF• (Π  (XK¯ ; x¯1 , x¯2 )/In )∨ for some filtration F• . Now we may conclude the result by Theorem 3 and Proposition 1.

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In the case  = p, we may conclude once we know that the spectral sequence p indeed converges after applying Dpst , which follows as Zp [[π1 (XK¯ ; x¯1 , x¯2 )]]/In ⊗ Qp is de Rham (hence potentially semistable) by Remark 9.

3.5 Semisimplicity We now begin preparations for the proof of Theorem 5. The canonical splitting of the weight filtration from Definition 4 induces a WK -equivariant splitting of the natural quotient map Π  (YK¯ ; x¯1 , x¯2 ) → Π  (YK¯ ; x¯1 , x¯2 )/I. Definition 7 (Canonical Paths) We denote the image of 1 under this splitting by p(x1 , x2 ). This is WK -invariant by construction—in the case  = p it is moreover invariant under the crystalline Frobenius. Proposition 5 Let x1 , x2 , x3 be rational points or rational tangential basepoints of X. Then 1. p(x1 , x1 ) = 1, and 2. p(x2 , x3 ) ◦ p(x1 , x2 ) = p(x1 , x3 ). Proof (1) is immediate from the definition; (2) follows from compatibility with tensor products. Remark 11 In the case  = p, the paths p(x1 , x2 ) are Vologodsky’s canonical padic paths [34, Proposition 29]. In the case  = p, the paths p(x1 , x2 ) are the from [4, Remark 2.2.5], if Y is a curve. canonical -adic paths γxcan 1 ,x2 Proof of Theorem 5 This is a more involved variant of [26, Theorem 2.12]. By the Lefschetz hyperplane theorem for fundamental groups, we may without loss of generality assume dim(X) ≤ 1. Indeed, there exists a smooth curve C → X such that the induced map on fundamental groups is a surjection; it suffices to prove the theorem for C. So we assume dim(X) = 1 and let X be the connected smooth proper curve compactifying X. Let D = X \ X. Without loss of generality (by replacing K with a finite extension) we may assume D = {x1 , . . . , xn }, with the xi ∈ X(K) rational points of X. ¯ Recall that if  = p, we have fixed a Frobenius element ϕK ∈ Gal(K/K); if f (K/Q ) p  = p, we let ϕK = ϕ be the smallest power of the crystalline Frobenius which is K-linear (so the geometric Frobenius of the underlying Weil–Deligne representation). We first claim that it suffices to prove the theorem when x1 = x2 . Indeed, suppose we know the theorem for x1 . Then composition with p(x1 , x2 ) is a ϕK -equivariant isomorphism

Semisimplicity for π1

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Π  (XK¯ , x¯1 ) −→ Π  (XK¯ ; x¯1 , x¯2 ). So ϕ acts semisimply on Π  (XK¯ ; x¯1 , x¯2 ). Hence we may and do assume x1 = x2 = x for the rest of the proof. We now claim it suffices to show that the quotient map I → I/I2 splits ϕK -equivariantly. Indeed, let s : I/I2 → I be such a splitting; then the map

⊗n  s 2 ⊗n (I/I ) −→ Π  (XK¯ ; x¯1 , x¯2 ) n

has dense image. But ϕK acts semi-simply on (I/I2 ) by Propositions 2 and 4, so we may conclude the theorem. We now construct such a ϕK -equivariant splitting s. Step 1.

We first construct a splitting of the quotient map I → I/W−2 .

But this map splits ϕK -equivariantly by the formula in Definition 4; choose any ϕK -equivariant splitting s1 . Step 2. We now construct a splitting of the map W−2 Π  (XK¯ , x) ¯ → W−2 /I2 . For each i = 1, . . . , n, we choose a rational tangential basepoint yi ∈ X(K((t))) so that the associated K[[t]]-point of X (obtained via the valuative criterion for properness) specializes to xi . Let pi = p(x, yi ) be the canonical path arising from Definition 7. ( = p): Let γ be a topological generator of Gal(K((t))/K((t))) = π1 (Spec(K((t)), Spec(K((t)))) = Z (1). Then the maps ιi : yi → X induce maps ιi∗ : π1 (Spec(K((t)), Spec(K((t)))) → π1 (XK¯ , y¯i ). Let γi be the image of ιi∗ (γ ) − 1 in W−2 /I2 . Then the map

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ι∗ : Q (1){x1 ,··· ,xn } → W−2 /I2 (a1 , · · · , an )γ →



ai γi

is surjective and ϕK -equivariant; as ϕK acts semisimply on Q (1){x1 ,··· ,xn } , ι∗ splits ϕK -equivariantly, so it suffices to construct a ϕK -equivariant map s˜2 : Q (1){x1 ,...,xn } → I such that the diagram

W−2

Q (1){x1 ,··· ,xn } ss s˜2 sss ι∗ s s s  ysss / / W−2 /I2

commutes. Set s˜2 to be the map s˜2 : (a1 , · · · , an )γ →

n 

ai pi · log(ι∗ γ ) · pi−1

i=1

A direct computation shows that this gives the desired section; see the proof of [26, Theorem 2.12, p. 621-622] for an identical computation. Finally, let p be any section to ι∗ ; then we set s2 = s˜2 ◦ p. ( = p): Let β be a topological generator of Zp (1) and let Π p (yi ) := lim Dst (Zp [[Zp (1)]]/(β − 1)n ⊗ Qp ). ← − n

Now the Weil–Deligne representation K(1) := Dst (Qp (1)) has ϕK -action given by multiplication by q −1 = (#k)−1 and N ≡ 0. As (β − 1)/(β − 1)2  Zp (1), there is a ϕ-equivariant isomorphism Π p (yi ) 



K(i),

i≥0

where K(i) is the ϕK -module K(1)⊗i . Let γ be an element of Π p (yi ) such that ϕK (γ ) = qγ . Note that this element γ plays the role of log(γ ) in the ( = p) case of the group above; in particular, it is a primitive element of the Hopf algebra Π p (yi ) rather than a group-like element.

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The map ιi induces a map ιi∗ : Π p (yi ) → Π p (X, y¯i ); let γi = ιi∗ (γ ). Now the map ι∗ : K(1){x1 ,··· ,xn } → W−2 /I2 (a1 , · · · , an ) →



ai γi

is surjective and ϕK -equivariant; as ϕK acts semisimply on K(1){x1 ,··· ,xn } , ι∗ splits ϕK -equivariantly, so it suffices to construct a ϕK -equivariant map s˜2 : K(1){x1 ,··· ,xn } → I such that the diagram

W−2

K(1){x1 ,··· ,xn } s˜2 ttt t ι∗ tt tt  t yt / / W−2 /I2

commutes. Set s˜2 to be the map s˜2 : (a1 , · · · , an )γ →

n 

ai pi · γi · pi−1

i=1

Again, this gives the desired section by an argument identical to the proof of [26, Theorem 2.12, p. 621-622]. Finally, let p be any section to ι∗ ; then we set s2 = s˜2 ◦ p. Step 3. We now construct the desired ϕK -equivariant section s : I/I2 → I. namely, s : v → s1 (v mod W−2 ) + s2 (v − (s1 (v mod W−2 ) mod I2 )). This is ϕK -equivariant because the same is true for s1 , s2 , and is a section by direct computation. This completes the proof.

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4 Structure of Local Bloch–Kato Selmer Schemes We now turn to our application of these results to the study of local Bloch–Kato Selmer schemes. As before, let K be a finite extension of Qp , and let U be a Qp pro-unipotent group endowed with a continuous action of GK (see Remark 12). We assume moreover that U is de Rham. We have in mind that U is the Qp -prounipotent étale fundamental group of a smooth geometrically connected variety, but will not assume this in what follows. One can associate to U a continuous Galois cohomology presheaf H1 (GK , U ) of pointed sets on the category AffQp of affine Qp -schemes, namely the presheaf whose sections over some Spec(Λ) is the non-abelian Galois cohomology set H1 (GK , U (Λ)). The local Bloch–Kato Selmer presheaves (cf. [23, §2] & [7, Definition 1.2.1]) are three sub-presheaves H1e (GK , U ) ⊆ H1f (GK , U ) ⊆ H1g (GK , U ) ⊆ H1 (GK , U )

(1)

whose sections over some Spec(Λ) are given by   ϕ=1 H1e (GK , U )(Λ) = ker H1 (GK , U (Λ)) → H1 (GK , U (Bcris ⊗ Λ)) ,   H1f (GK , U )(Λ) = ker H1 (GK , U (Λ)) → H1 (GK , U (Bcris ⊗ Λ)) ,   H1g (GK , U )(Λ) = ker H1 (GK , U (Λ)) → H1 (GK , U (BdR ⊗ Λ)) . Remark 12 (On Topologies) Throughout this section, we will adhere to the standard conventions regarding topologies on Qp -linear objects. A finite-dimensional Qp vector space will be endowed with its natural p-adic topology. A general Qp -vector space, for instance O(U ) or a Qp -algebra Λ will be endowed with the inductive (co)limit topology over its finite-dimensional subspaces. A pro-finite-dimensional Qp -vector space—i.e. a pro-object in the category of finite-dimensional Qp vector spaces, such as Lie(U )—will be endowed with the inverse limit topology over its finite-dimensional quotients. More generally, if V = lim (Vi ) is a pro← −i finite-dimensional Qp -vector space, with each Vi finite-dimensional, then for each Qp -algebra Λ we endow the base change VΛ := lim (Λ ⊗Qp Vi ) with the inverse ← −i limit of the inductive limit topologies on each Λ ⊗Qp Vi . If U is a Qp -pro-unipotent group and Λ is a Qp -algebra, then the logarithm ∼ map provides a bijection U (Λ) → Lie(U )Λ , and we topologize the set U (Λ) by declaring this map to be a homeomorphism. We say that an action of a profinite group G on U is continuous just when G acts continuously on U (Λ) for all Qp -algebras Λ (equivalently, G acts continuously on Lie(U ) or O(U ) [7, Definition–Lemma 4.0.1]). We say that U is de Rham just when Lie(U ) is pro-de Rham (equivalently O(U ) is ind-de Rham [7, Definition–Lemma 4.2.2]). Though the period rings Bcris , Bst and BdR also carry topologies, none of our definitions or results here depend on these topologies.

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An argument of Kim establishes that the Bloch–Kato Selmer presheaf H1f (GK , U ) is representable when U is the maximal n-step unipotent quotient of the Qp -pro-unipotent étale fundamental group of a smooth projective curve. The argument only uses a condition on the weights of U . Definition 8 We say that U is mixed with only negative weights just when O(U ) is endowed with an exhaustive filtration Qp = W0 O(U ) ≤ W1 O(U ) ≤ · · · ≤ O(U ) by GK -stable subspaces, stable under the Hopf algebra structure maps, such that grW i O(U ) is pure of weight i for all i ≥ 0. Equivalently, Lie(U ) is endowed with a separated filtration · · · ≤ W−2 Lie(U ) ≤ W−1 Lie(U ) = Lie(U ) by GK -stable pro-finite-dimensional subspaces, stable under the Lie algebra structure maps, such that grW −i Lie(U ) is pure of weight −i for all i > 0. Lemma 5 (Kim [24, Lemma 5]) Assume that U is mixed with only negative weights. Then H1f (GK , U ) is representable by an affine scheme over Qp . In fact, [25, Proposition 1.4] gives a precise description of the representing scheme. Let DdR (U ) be the pro-unipotent group over K representing the presheaf Spec(Λ) → DdR (U )(Λ) := U (BdR ⊗K Λ)GK where BdR is the de Rham period ring (see [7, Lemma 4.2.1]). The argument of Kim establishes that in the setup of Lemma 5 there is an isomorphism   0 D H1f (GK , U ) ∼ (U )/F = ResK dR Qp of presheaves on AffQp (where the right-hand side denotes the presheaf quotient of DdR (U ) by the right-multiplication action of the subgroup F 0 DdR (U ) corresponding to the 0th step of the Hodge filtration on Lie(U )). In particular, the representing variety is an affine space: if one chooses a splitting DdR (Lie(U )) = V ⊕ F 0 of the Hodge filtration on the Lie algebra of U , then there is an isomorphism   K 0 ∼ ResK Qp V = ResQp DdR (U )/F of presheaves on AffQp , where by abuse of notation we also denote by V the associated affine space Spec(Λ) → VΛ . Our aim in this section is to extend this to descriptions of all three Bloch–Kato Selmer presheaves, and in particular to show that under the same assumptions on the weights, they are all also represented by affine spaces. In fact, by imitating the

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arguments in [4] we will obtain descriptions of the Bloch–Kato Selmer presheaves as (the affine spaces underlying) vector spaces; these descriptions are all canonical, up to the above choice of splitting of the Hodge filtration. Theorem 7 Let U be a W -filtered de Rham representation of GK on a finitely generated pro-unipotent group over Qp , which is mixed with negative weights. Then there are canonical natural isomorphisms   0 D H1e (GK , U ) ∼ (U )/F = ResK dR Qp   0 D (U )/F H1f (GK , U ) ∼ = ResK dR Qp   0 × V(U )ϕ=1 H1g (GK , U ) ∼ = ResK Qp DdR (U )/F of presheaves for a certain ϕ-module V(U ) functorially assigned to U (for a precise description, see below). These descriptions are compatible with the inclusions (1). In particular all three presheaves are representable by affine spaces, and the dimension of these spaces is given by dimQp H1e (GK , U ) = [K : Qp ]

  0 W dimK DdR (grW −i U ) − dimK F DdR (gr−i U ) i>0

  0 W dimK DdR (grW dimQp H1f (GK , U ) = [K : Qp ] −i U ) − dimK F DdR (gr−i U ) i>0

  0 W dimK DdR (grW dimQp H1g (GK , U ) = [K : Qp ] −i U ) − dimK F DdR (gr−i U ) +



i>0 ∗ dimQp Dcris ((grW −i U ) (1)) ϕ=1

i>0

where the right-hand side is the sum of the dimensions of the Bloch–Kato Selmer groups. If ∗ ∈ {e, f }, or if ∗ = g and U is Frobenius-semisimple, the same holds for the descending central series in place of the weight filtration. Example 5 Suppose that X/K is a smooth projective curve of genus g with semistable reduction, and that all irreducible components of the geometric special fibre of the minimal regular model of X are defined over the residue field k. Let Y = X \ {x} for a point x ∈ X(K) and let Un /Qp denote the maximal n-step unipotent quotient of the Qp -pro-unipotent étale fundamental group of YK (at a basepoint b ∈ Y (K)). Then we have  g n+1 − g0 dimQp H1g (GK , Un ) = [K : Qp ] · L≤n (2g) − L≤n (g) − L≤n (g0 ) + 0 g0 − 1

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+

n/2 (n − 1)g0n − ng0n−1 + 1  2 g0 −1 · (ν√q + ν−√q ) , · ν − λ 2 2(g0 − 1) 2(g0 − 1) λ

where g0 is the genus of the reduction graph of X, νλ are the multiplicities of the weight 1 eigenvalues of ϕK acting on H1et´ (XK , Qp ), and L≤n (T ) :=   μ(d) i/d is the summed necklace polynomial (the number of Lyndon 1≤i≤n d|i i T words of length ≤ n in an alphabet of T letters). ∗ Proof (Sketch) We will calculate the Qp -dimension of Dcris ((grW −i Un ) (1)) (for i ≤ n), leaving the remainder of the calculation to the reader. This Qp -dimension is ∗ equal to the K0 -dimension of the subspace of Dst (grW −i Un ) on which ϕK acts via q ∗ and N acts by 0. Since Un is a free n-step unipotent group, Dst (grW −i Un ) has a basis parametrised by Lyndon words in a basis of Dst (H1et´ (XK , Qp )). By semisimplicity, we may pick a basis of K ⊗K0 Dst (H1et´ (XK , Qp )) consisting of ϕK -eigenvectors. Exactly g0 of the corresponding eigenvalues are equal to 1 and g0 are equal to q; all the remaining eigenvalues are q-Weil numbers of weight 1. The monodromy operator N maps the q eigenspace isomorphically onto the 1 eigenspace, and acts as 0 on all other eigenspaces. The corresponding Lyndon basis ∗ of Dst (grW −i Un ) is also a basis of ϕK -eigenvectors, with the eigenvalue of (the basis element corresponding to) a Lyndon word w being the product of the eigenvalues of its letters. ∗ It follows from this description that the monodromy operator on grW −i Un maps the q eigenspace surjectively onto the 1 eigenspace, and so the desired dimension is equal to the number of Lyndon words of length i and eigenvalue q, minus the number of Lyndon words of length i and eigenvalue 1. This latter quantity is simply the number of Lyndon words in the g0 vectors of eigenvalue 1, and hence equal to  μ(d) i/d d|i i g0 [30, Theorem 7.1]. Summed over 1 ≤ i ≤ n, this yields the term −L≤n (g0 ) in the claimed formula. Now there are two types of Lyndon words w of eigenvalue q: the letters of w with eigenvalue not equal to 1 are either a single eigenvector with eigenvalue q, or two eigenvectors with eigenvalues λ, q/λ with λ of weight 1. To count words of the former type, suppose that x is an eigenvector with eigenvalue q. There are g0i−1 Lyndon words of length i containing x and i − 1 vectors of eigenvalue 0 by Reutenauer [30, Theorem 7.1(2)]. Summed over x and over 1 ≤ i ≤ n, this yields ϕ=1

g n+1 −g

the term 0g0 −1 0 in the claimed formula. To count words of the latter type, suppose that x and y are eigenvectors with eigenvalues λ and q/λ respectively. Using [30, Theorem 7.1(7.1.2)] again, we find that if x = y, then the number of Lyndon words containing x, y and i − 2 eigenvectors of eigenvalue 1 is (i − 1)g0i−2 . If x = y, then the number of Lyndon i−2 words containing two copies of x, y and i−2 eigenvectors of eigenvalue 1 is i−1 2 g0 i−2 1 i/2−1 if i is odd, and i−1 if i is even. Summed over x, y and over 1 ≤ i ≤ n, 2 g0 − 2 g0 this yields the quantity in the final line of the claimed formula. Here, we are using

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  2 − ν√ − ν √ that νλ = νq/λ by Poincaré duality, so that there are 12 ν q − q λ λ unordered pairs with x = y, and ν√q + ν−√q pairs with x = y.

4.1 Torsors in Tannakian Categories We will prove Theorem 7 via an alternative interpretation of the presheaves H1∗ (GK , U ) as classifying spaces of certain GK -equivariant torsors. For our purposes, it will be convenient to recall the definition of torsors in slightly greater generality. Let T be a Tannakian category—not necessarily neutral—over a characteristic 0 field E. As explained in [14, §5.2–5.4], one can make sense of many of the basic notions of affine algebraic geometry inside T, defining algebras in ind−T, affine schemes in T, and so on, all functorial with respect to faithfully exact ⊗-functors. We say that an affine group scheme U in T is pro-unipotent just when ω(T) is pro-unipotent for some fiber functor ω : T → Modfin E valued in finite-dimensional E -vector spaces for some extension field E ⊇ E. Since any two fiber functors become isomorphic over a common field extension, this notion is independent of ω. A torsor under an affine group scheme U in T over an affine scheme Spec(Λ) in T is a faithfully flat Spec(Λ)-scheme T → Spec(Λ) endowed with a fiberwise right action of U such that the induced map ∼

T ×Spec(1) U → T ×Spec(Λ) T is an isomorphism. Here, faithfully flat means5 that the map ω(T ) → ω(Spec(Λ)) is faithfully flat for some fiber functor ω—as above this is independent of ω. When U is pro-unipotent, the condition that ω(T ) → ω(Spec(Λ)) is faithfully flat is equivalent to it being split, i.e. ω(T )(ω(Λ)) being non-empty. There is a functor Λ → Λ ⊗ 1 from E-algebras to algebras in T [14, §5.6], and so we define the first cohomology presheaf of an affine group-scheme U in T to be the presheaf on AffE given by H1 (T, U ) : Spec(Λ) → {U -torsors over Spec(Λ ⊗ 1)}/iso . This is in fact a presheaf of pointed sets, with basepoint corresponding to the class of the trivial torsor U over Spec(1). Example 6 Let G be a profinite group, and T the category of continuous representations of G on finite-dimensional Q -vector spaces. Then ind−T is the category of continuous representations of G on general Q -vector spaces, where the topology

5 There

is an alternative internal definition of faithful flatness in [14, §5.3]. Although these presumably define the same notion, we will neither need nor prove this in what follows.

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on a Q -vector space is as in Remark 12. The category of affine schemes in T is equivalent to the category of affine Q -schemes X endowed with an action of G which is continuous in the sense that the action map G × X(Λ) → X(Λ) is continuous for every Q -algebra Λ. Here, the topology on X(Λ) is the subspace topology induced from any (possibly infinite-dimensional) affine embedding X(Λ) → Spec(Sym• (V ))(Λ) = VΛ∗ , where VΛ∗ is topologized as in Remark 12. A pro-unipotent group in T is a Q -pro-unipotent group endowed with a continuous action of G. A U -torsor T over Spec(Λ ⊗ 1) is determined up to isomorphism by the U (Λ)-valued cohomology class of the cocycle σ → γ −1 σ (γ ) for γ ∈ T (Λ), and hence there is an isomorphism H1 (T, U ) ∼ = H1 (G, U ) of presheaves on AffQ , where H1 (G, U ) denotes the continuous Galois cohomology presheaf. Example 7 Let T = Mod(ϕ,N,GK ) be the Qp -linear Tannakian category of discrete (ϕ, N, GK )-modules. There are two natural fiber functors on T: one given by taking the underlying Qnr p -vector space, and one defined over K given by taking the GK invariants in the base change to K [18, 4.3.1]. We denote these by (−)Qnrp and (−)K , respectively. If D is the (ϕ, N, GK )-module associated to a de Rham representation V , then DQnrp = Dpst (V ) and DK = DdR (V ). For an affine scheme T in T, TQnrp carries various extra structures, including ∼

a crystalline Frobenius ϕ : TQnrp → σ ∗ TQnrp , where σ ∗ denotes pullback along the arithmetic Frobenius in GQnrp /Qp , as well as a vector field determined by the monodromy operator. In particular, the crystalline Frobenius acts on Hom-sets between affine schemes in T. In what follows, we will often make use of the following well-known lemma, which we have already seen implicitly in Definition 6. Lemma 6 (Transport of Filtrations) Let U be a pro-unipotent group in T endowed with an exhaustive and increasing filtration 1 = W0 O(U ) ≤ W1 O(U ) ≤ . . . by objects in ind−T, compatible with the Hopf algebra structure. Then for every U -torsor T over Spec(Λ) there is a unique exhaustive filtration W• O(T ) of O(T ) by objects in ind−T compatible with the Λ-algebra structure and the coaction map ∼ W O(T ) → O(T )⊗O(U ). There is a unique isomorphism grW • O(T ) → Λ⊗gr• O(U ) of graded Λ-algebras in ind−T compatible with the graded coaction map. Proof (Sketch) When T = Modfin E , this is well-known: one chooses an element γ ∈ ∼ T (Λ), which provides an isomorphism T → Spec(Λ) ×Spec(1) U of U -torsors over ∼ Spec(Λ), and the W -filtration on O(T ) is defined so as to make O(T ) → Λ ⊗ O(U )

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a W -filtered isomorphism. This filtration is independent of γ , as is the induced W ∼ W graded isomorphism grW • O(T ) → Λ ⊗ gr• O(U ). Both the W -filtration and the induced graded isomorphism admit intrinsic descriptions. For the W -filtration, Wn O(T ) is the preimage of O(T ) ⊗ Wn O(U ) under the coaction map, while the graded isomorphism is the composite W W W grW • O(T ) → gr• O(T ) ⊗ gr• O(U ) → Λ ⊗ gr• O(U ), where the first map is the graded coaction and the second is the inverse of the unit isomorphism ∼ W Λ → grW 0 O(T ), tensored with gr• O(U ). This latter definition of W• O(T ) makes sense in any Tannakian category T; that this gives the unique filtration with the claimed properties can be checked after applying a fiber functor, where we use the previous case (over a field extension). ∼ This in particular ensures that the unit map Λ → grW 0 O(T ) is an isomorphism, so that the definition of the graded isomorphism also makes sense in T. Again applying a fiber functor proves that it is the unique isomorphism with the claimed properties.

Torsors in Mixed Weil–Deligne Representations The main example we will compute is the case when T is the category of mixed Weil–Deligne representations over a characteristic 0 field E. Throughout this subsection, we fix a pro-unipotent group U in Repmix E ( WK ) (equivalently, an Epro-unipotent group U with an action of the Tannaka group Gmix from Sect. 2.3), and assume that U has only negative weights in the sense that W0 O(U ) = Q (equivalently W−1 Lie(U ) = Lie(U )). We will describe the cohomology of U explicitly in terms of the following vector space (which was denoted by Lie(U )can in [4, Definition 2.2.2] in the context of -adic Galois representations). Definition 9 The Lie algebra Lie(U ), being the space of E-valued derivations on O(U ), is naturally a pro-object of Repmix E ( WK ). We define V(U ) := Lie(U )(−1)WK ,X to  bethe (pro-finite dimensional) subspace of Lie(U )(−1) fixed by WK and X = 11 ∈ SL2 (E) (see Sect. 2.3 for the definition of the SL2 -action). In other words, 01 V(U ) consists of those elements v ∈ Lie(U ) such that ρ(w)(x) = p−v(w) · v for all w ∈ WK and N r (v) ∈ W−r−2 Lie(U ) for all r ≥ 0. Since the weight torus is central in SL2  WKmix , it acts on V(U ) and endows it with a product grading6

6 The grading we actually use here is the grading on Lie(U ), which we identify with Lie(U )(−1) via the trivialization of E(1). In other words, we have shifted the grading on Lie(U )(−1) by two. This is because we will see that grW −r V is related to the operator δr from Definition 5, which is W -graded of degree −r.

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V(U ) =



grW −r V(U ) .

r>0

Note that grW −1 V(U ) = 0. Lemma 7 There is a canonical isomorphism ∼ H1 (Repmix E ( WK ), U ) = V(U )

of presheaves on AffE , where we identify the pro-finite-dimensional vector space V(U ) with its corresponding affine space in the usual way. The key to this theorem is the following version of the canonical paths from Definition 7 over a base. Proposition 6 For every U -torsor T over Spec(Λ), there is a unique element pT ∈ T (Λ) fixed under the action of the weight torus. This element is in fact invariant under the action of Gpure . Proof The action of the weight torus Gm on T corresponds to the canonical splitting of the weight filtration on O(T ), and the inclusion T Gm → T corresponds to the projection O(T ) → grW 0 O(T ) of the canonical splitting. According to Lemma 6, W we have gr0 O(T ) = Λ, and hence T Gm ∼ = Spec(Λ) consists of a single Λ-point pT ∈ T (Λ). Since Gm is central in Gpure , it follows that the action of Gpure preserves T Gm , so must fix pT . Proof of Lemma 7 Recall from Lemma 3 that the Tannaka group Gmix of Repmix E ( WK ) is canonically isomorphic to U  Gpure , where Gpure is the Tannaka group of pure Weil–Deligne representations and U is a pro-unipotent group containing certain elements exp(δr ) for r > 0. We define the map W WK ,X as follows. α : H1 (Repmix r>0 gr−r Lie(U )(−1) E ( WK ), U ) → V(U ) = Given a U -torsor T over some Spec(Λ) and an integer r > 0, by differentiating the action of the copy of Ga in U ≤ Gmix spanned by exp(δr ), we obtain a relative vector field Dr on T . Evaluating this vector field at the canonical path pT from Proposition 6 gives a relative tangent vector based at pT , and hence pT−1 Dr (pT ) is an element of Lie(U )(Λ). The commutation conditions for δr (see Remark 6) WK ,X , so we define the map α by ensure that in fact pT−1 Dr (pT ) ∈ grW −r Lie(U )(−1) −1 αSpec(Λ) ([T ]) := (pT Dr (pT ))r>0 . Next we show that each αSpec(Λ) is injective. Suppose we are given two torsors Dr (pT1 ) = pT−1 Dr (pT2 ) for all r > 0. There T1 , T2 over Spec(Λ) such that pT−1 1 2 ∼

is a unique isomorphism β : T1 → T2 of U -torsors taking pT1 to pT2 , and this is automatically Gpure -equivariant. To show that β is equivariant for the action of the copy of Ga in U generated by exp(δr ), it suffices to show that it is compatible with the relative vector fields Dr . But since the action map Ti × U → Ti is Ga equivariant, it is automatically compatible with the vector fields Dr on Ti and U , so the vector field Dr on Ti is determined by its value at any given Λ-point of Ti . Since

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by assumption β preserves Dr at pT1 , it thus preserves Dr everywhere, and so is Ga equivariant. Since Gmix is generated by Gpure and the elements exp(δr ) for r > 0, the isomorphism β is Gmix -equivariant, so [T1 ] = [T2 ]. Finally, we show that each αSpec(Λ) is surjective, by constructing an explicit  WK ,X W inverse. For a tuple v = (vr )r>0 ∈ , we let Uv r>0 gr−r Lie(U )(−1)Λ denote the Gmix -equivariant U -torsor over Spec(Λ) constructed as follows. We take Uv = Spec(Λ) ×Spec(E) U (as a right U -torsor) with the same action of Gpure . For r > 0, we define an action of Ga on Uv by ρr (t)(u) = exp(tvr ) · u. The choice of vr ensures that ρr (t) commutes with the action of X, and satisfies the commutation relations ρ(w) ◦ ρr (t) ◦ ρ(w) = ρr (p−v(w) t) ρ(λ) ◦ ρr (t) ◦ ρ(λ−1 ) = ρr (λ−r t) against elements w ∈ WK in the Weil group and λ ∈ Gm in the weight torus. It is easy to check that these actions extend uniquely to an action of U on Uv equivariant for the action of U and Gpure , and hence endow it with the structure of a Gmix equivariant U -torsor. It is then easy to check that α([Uv ]) = v, so that α is surjective, as claimed. Before we proceed to the proof of Theorem 7, we note that we already have proved its -adic counterpart, re-proving [4, Theorem 2.2.4] (in mildly more generality). Corollary 4 Let  = p be a prime and let U be a Q -pro-unipotent group endowed with a continuous action of GK , and suppose that U is mixed with only negative weights with respect to some filtration on O(U ). Then we have a canonical isomorphism H1 (GK , U ) ∼ = V(U ) of presheaves on AffQ . In particular, H1 (GK , U ) is representable by an affine space. Proof The filtration on Lie(U ) induces a corresponding filtration Q = W0 O(U ) ≤ W1 O(U ) ≤ . . . making O(U ) into an ind-mixed representation of GK . If T is a U torsor over Spec(Λ), then by Lemma 6 there is a unique GK -invariant W -filtration on O(P ) compatible with all the appropriate structures, which makes O(P ) into an ind-mixed representation of GK . We see from this construction and Example 6 that H1 (GK , U ) is isomorphic to H1 (Repmix Q (GK ), U ). It follows from the fullfaithfulness of the functor mix Repmix Q (GK ) → RepQ ( WK )

of Example 2 that the induced map

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1 mix H1 (Repmix Q (GK ), U ) → H (RepQ ( WK ), U )

is injective; surjectivity follows from the fact that the image of this fully faithful functor is closed under extensions. We conclude via Lemma 7. Proof We now come to the proof of Theorem 7. We proceed in several steps. Proposition 7 H1g (GK , U ) ⊆ H1 (GK , U ) consists of those GK -equivariant U torsors which are de Rham. Such a torsor T is automatically mixed (i.e. a torsor under U in Repmix Qp (GK )) when O(T ) is endowed with the unique weight filtration compatible with the weight filtration on O(U ), as in Lemma 6. Proof For the first part, if O(T ) is ind-de Rham then we have a GK -equivariant ∼ BdR ⊗ Λ-algebra isomorphism BdR ⊗Qp O(T ) → BdR ⊗K DdR (O(T )). Since DdR is a fiber functor on the category of de Rham representations, DdR (T ) := Spec(DdR (O(T ))) is automatically a DdR (U )-torsor over Spec(K ⊗ Λ), and hence DdR (T )(K ⊗ Λ) = ∅. In particular, T (BdR ⊗ Λ)GK = DdR (BdR ⊗ Λ)GK = ∅, and hence the class of T lies in H1g . In the other direction, let C• denote the conilpotency filtration on O(U ), and also C ∼ the induced filtration on O(T ) from Lemma 6, so that grC • O(P ) = Λ ⊗Qp gr• O(U ) 1 G K is ind-de Rham. When T lies in Hg , we have that T (BdR ⊗Qp Λ) = ∅, from which we deduce that there exists a GK -equivariant and C-filtered isomorphism BdR ⊗Qp O(T )  BdR ⊗Qp Λ ⊗Qp O(U ). Since O(U ) is ind-de Rham, the conilpotency filtration on BdR ⊗Qp O(U ) splits BdR -linearly and GK -equivariantly, and hence so too does the conilpotency filtration on BdR ⊗Qp O(T ). This implies that O(T ) is ind-de Rham, as desired. The second part follows by the same argument as in Corollary 4. Proposition 8 Let T be a de Rham U -torsor over Spec(Λ). Then: 1. the canonical path pT ∈ Dpst (T )(Λ ⊗Qp Qnr p ) is ϕ- and GK -invariant; and dR 2. there exists a “Hodge-filtered path” pT ∈ DdR (T )(Λ ⊗Qp K), such that the ∼

induced isomorphism O(DdR (T )) → Λ ⊗Qp O(DdR (U )) is a Hodge-filtered isomorphism. This choice of pTdR is unique up to the right-multiplication action of F 0 DdR (U ) (the pro-unipotent subgroup whose Lie algebra is F 0 Lie(DdR (U ))). In particular, pT ∈ DdR (T )(Λ ⊗Qp K) = TQp (Λ ⊗Qp K)GK . Proof For the first part, ϕ-invariance of pT follows from the fact that the canonical splitting of the weight filtration on a mixed Weil–Deligne representation is ϕequivariant (Remark 7). Since it is also invariant under the Weil group action, it is thus invariant under the GK -action. The second part follows by the argument in [7, Proposition 9.9].  0 × Using this, we define a morphism α : H1g (GK , U ) → ResK Qp DdR (U )/F V(U )ϕ=1 by declaring that

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α([T ]) := (pT−1 pTdR , (pT−1 Dr (pT ))r>0 ) , where Dr is as in the proof of Lemma 7. Proposition 9 α is an isomorphism. Proof Firstly, we show that α is injective as a morphism of presheaves. Suppose that T1 and T2 are de Rham U -torsors such that α([T1 ]) = α([T2 ]). We have ∼ ∼ isomorphisms βdR : DdR (T1 ) → DdR (T2 ) and βpst : Dpst (T1 ) → Dpst (T2 ) (of DdR (U )- and Dpst (U )-torsors, respectively) sending pT1 to pT2 . Our assumptions that α([T1 ]) = α([T2 ]) ensure firstly that βdR is compatible with the Hodge filtration (on the affine rings) and secondly that βpst is an isomorphism on the underlying mixed Weil–Deligne representations. Since βpst is also ϕ-equivariant (as pT1 and pT2 are ϕ-invariant), it follows that βpst is an isomorphism of (ϕ, N, GK )modules. Thus Dpst (T1 ) and Dpst (T2 ) are isomorphic as Dpst (U )-torsors in the category of weakly admissible Hodge-filtered discrete (ϕ, N, GK )-modules. Since ∼ w.a. 7 Dpst : RepdR Qp (GK ) → MF(ϕ,N,GK ) is an equivalence [8, Théorème A] [18, Théorème 5.6.7(v)], it follows that T1 and T2 are GK -equivariantly isomorphic as U -torsors. To conclude, we show that α is surjective. Let Λ be a Qp -algebra and take some (udR , (vr )r>0 ) ∈ DdR (U )(Λ ⊗Qp K) ×



W ,X,ϕ=1 nr . p Qp

K grW −r Lie(U )(−1)Λ⊗Q

r>0

We define a de Rham U -torsor TudR ,v as follows. We set DdR (TudR ,v ) := Spec(Λ ⊗Qp K) ×Spec(K) DdR (U ) , Dpst (TudR ,v ) := Spec(Λ ⊗Qp Qnr Dpst (U ) , p ) ×Spec(Qnr p) which are torsors under DdR (U ) and Dpst (U ) respectively. We define a comparison ∼ isomorphism cudR : Spec(K) ×Spec(K) DdR (TudR ,v ) → Spec(K) ×Spec(Qnrp ) D(Tu,v ) to be the map given by left-multiplication by udR on Spec(Λ ⊗Qp K) ×Spec(Λ⊗Qp K) D(U ) = Spec(Λ ⊗Qp K) ×Spec(Λ⊗Qp Qnrp ) Dpst (U ). Now we endow Dpst (TudR ,v ) with the semilinear crystalline Frobenius (on its affine ring) induced from the crystalline Frobenius on Dpst (U ). The proof of

7 Strictly speaking, [8, Théorème A] only shows that weakly admissible (ϕ, N )-modules are admissible. However, it implies the same result for (ϕ, N, GK )-modules by a straightforward argument. If D is a weakly admissible (ϕ, N, GK )-module, then there is a finite Galois extension L/K contained in K such that D = K 0 ⊗L0 D0 for some weakly admissible (ϕ, N, GL|K )-module D0 . Since D0 is weakly admissible as a (ϕ, N )-module by Fontaine [18, Proposition 4.4.9], it follows from the result of Colmez–Fontaine that Vst (D0 ) is a semistable representation of GL . Hence Vpst (D) = Vst (D0 ) is a potentially semistable representation of GK , and D ∼ = Dpst (Vpst (D)) → D is admissible.

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Lemma 7 explains also how to use the elements (vr )r>0 to endow Dpst (TudR ,v ) with the structure of a Qnr p -linear mixed Weil–Deligne representation, whose underlying Weil group action is the same as that induced from Dpst (U ). It thus follows, by reversing the construction in Example 3, that this crystalline Frobenius and mixed Weil–Deligne representation underlie a mixed (ϕ, N, GK )-module structure on Dpst (TudR ,v ). We also endow DdR (TudR ,v ) with the Hodge filtration arising from that on DdR (U ). Via the comparison isomorphism cudR , this endows Spec(Λ ⊗Qp K) ×Spec(Λ⊗Qp Qnrp ) Dpst (TudR ,v ) with a GK -invariant Hodge filtration. We claim that this Hodge filtration is weakly admissible. To see this, by the argument of Lemma 6, the W -graded isomorphism ∼

∗ W grW O(Dpst (TudR ,v )) → K ⊗K O(DdR (TudR ,v )) • cudR : gr• K ⊗Qnr p

∼ doesn’t depend on udR , and hence the canonical isomorphism grW • O(Dpst (TudR ,v )) = W O(D (U )) is Hodge-filtered (after base-changing to Λ ⊗ nr (Λ ⊗Qp Qnr ) ⊗ gr pst Qp Qp p • K). This implies that grW O(D (T )) is ind-weakly admissible, and hence pst u ,v • dR O(Dpst (TudR ,v )) is ind-weakly admissible by Fontaine [18, Proposition 4.4.4(iii)]. Now since Dpst (TudR ,v ) is weakly admissible, we obtain via Fontaine’s Vpst functor a de Rham U -torsor TudR ,v whose image under Dpst is Dpst (TudR ,v ). It follows by a simple calculation that α([TudR ,v ]) = (udR , v), so α is surjective as claimed. Proposition 9 proves Theorem 7 for H1g . The cases of H1e and H1f are covered by Kim [25, Proposition 1.4]. 

Appendix: A Canonical Presentation for the Weight-Graded Fundamental Group In this appendix, we outline another proof of Theorems 4 and 5 on mixedness and Frobenius-semisimplicity of the Q -pro-unipotent étale fundamental groups of smooth geometrically connected varieties Y /K. The aim here, following [21], is to write down an explicit and Galois-equivariant presentation of the associated W -graded of the fundamental group, from which mixedness and Frobeniussemisimplicity are immediate. For this section, we fix a smooth geometrically connected variety Y over a characteristic 0 field K with fixed algebraic closure K, and fix a simple normal crossings compactification Y of Y , with complementary divisor D. We write I1 for  the set of irreducible components of DK = i∈I1 Di , and I2 for the set of ordered pairs of distinct elements i, j ∈ I1 such that Di ∩ Dj = ∅. Both the sets I1 and I2 carry an action of GK = Gal(K/K). We write g = gb for the Lie algebra of the continuous Q -Mal˘cev completion of the profinite étale fundamental group π1et´ (YK , b) for some geometric basepoint b.

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´ 1 (Y , Q )∗ , which There is a canonical isomorphism gab ∼ = Het  1 (YK , Q ) := Het ´ K endows g with a weight filtration W• g [3, Definition 1.5], namely the unique ´ increasing filtration inducing the usual filtration on gab = Het 1 (YK , Q ) for which the  g → g is strict. This is the filtration described in Corollary 3. Lie bracket [·, ·] : g ⊗ Now the associated graded grW • g is independent of b up to canonical isomorphism, and hence inherits an action of GK from that on YK . Our main theorem of this appendix gives an explicit GK -equivariant presentation of this graded Lie algebra.

Theorem 8 Keep notation as above. Let f denote the free W -graded pro-nilpotent ´ ⊕I1 in degree Lie algebra over Q generated by Het 1 (Y K , Q ) in degree −1 and Q (1) −2. Let rf denote the (homogenous) ideal generated by the images of the following maps: – the map ´ W Het 2 (Y K , Q ) → gr−2 f =

2

´ ⊕I1 Het 1 (Y K , Q ) ⊕ Q (1)

induced by the map dual to cup product and the maps dual to the Gysin maps H0et´ (Di , Q )(−1) → H2et´ (Y K , Q ); – the map 

´ W Het 1 (Di , Q )(1) → gr−3 f

i∈I1 ´ ´ et given by taking the commutator of the maps Het 1 (Di , Q ) → H1 (Y K , Q ) and ⊕I the ith coproduct inclusion Q (1) → Q (1) 1 ; and – the map

Q (2)⊕I2 → grW −4 f given by taking the commutator of the ith and j th coproduct inclusions Q (1) → Q (1)⊕I1 whenever Di and Dj intersect. Then there is a canonical GK -equivariant and W -graded isomorphism ∼ grW • g = f/r . For us, the point of Theorem 8 is that it allows us to automatically deduce properties of étale fundamental groupoids of smooth varieties from the corresponding properties of low-dimensional (co)homology of smooth proper varieties. For instance, the cohomology of a smooth proper variety is pure in degrees ≤ 2 (Proposition 3 plus Chow’s Lemma and the Lefschetz Hyperplane Theorem)

Semisimplicity for π1

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and Frobenius-semisimple in degrees ≤ 1 (Proposition 4). This then implies the corresponding properties for the fundamental groupoid. Corollary 5 (Weight–Monodromy and Frobenius-Semisimplicity for the Fundamental Group) Suppose that Y is a smooth geometrically connected variety over a p-adic field K. Then O(π1Q (YK ; x, y)) and Lie(π1Q (YK , z)) are mixed and Frobenius-semisimple for all rational points x, y, z ∈ Y (K) (or tangential points). Here, the weight filtration on O(π1Q (YK ; x, y)) is dual to the weight filtration on O(π1Q (YK ; x, y))∗ = lim Z [[π1 (YK ; x, y)]] ⊗Z Q as defined in Definition 6, ← −n and the weight filtration on Lie(π1Q (YK ; z)) is the one defined above. Proof We will argue the  = p and  = p cases simultaneously, recalling in the latter case that O(π1Q (YK ; x, y))∗ is de Rham (Remark 9), and hence so too Q ∗ is Lie(π1Q (YK , z)). By Definition 4, it suffices to prove that grW • O(π1 (YK )) Q and grW • Lie(π1 (YK )) are pure and Frobenius-semisimple. Note that these objects are independent of the points x, y, z up to canonical GK -equivariant isomorQ ∗ phism, and that grW • O(π1 (YK )) is the completed universal enveloping algebra Q W of grW • Lie(π1 (YK )) = gr• g [4, Example A.3.8]. Since the class of pure and Frobenius-semisimple representations is closed under cokernels, it suffices to show that grW • g is pure and Frobenius-semisimple. ´ ⊕I1 , Now, for purity, it suffices by Theorem 8 to show that Het 1 (Y K , Q ), Q (1)

´ ´ et ⊕I2 are pure of weights −1, −2, −2, Het i∈I1 H1 (Di , Q )(1) and Q (2) 2 (Y K , Q ), −3 and −4, respectively. Purity of the second and fifth of these is immediate; purity of the first, third and fourth follows from Proposition 3. ´ ⊕I1 For Frobenius-semisimplicity, it suffices to prove that Het 1 (Y K , Q ) and Q (1) are Frobenius-semisimple. The second is obvious; the first is Proposition 4. The rest of this section is devoted to a proof of Theorem 8.

The Complex Case To begin with, suppose that K = C, so that g is the base change to Q of the Lie algebra of the Q-Mal˘cev completion of the Betti fundamental group of Y = Y (C). Henceforth, we will let g instead denote this Q-linear pro-nilpotent Lie algebra, and prove the Q-linear analogue of Theorem 8, using Betti homology in place of étale homology. We fix an orientation on C, which provides us with a generator of Q(1). Throughout the proof, we will use a certain natural topological compactification  of Y , whose construction is due to A’Campo [1]. Y i denote the real (oriented) blowup of Y Construction 1 For each i ∈ I1 , we let Y i  Y of manifolds with corners which along Di , so that there is a proper map Y  is an isomorphism away from Di and is an oriented S 1 -bundle over Di . We let Y

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i )i∈I1 over Y , so that the inclusion Y → Y factors denote the fibre product of (Y   Y . The first of these maps is a homotopy equivalence, so through Y → Y that g is canonically identified with the Lie algebra of the Q-pro-unipotent Betti . fundamental group of Y To begin the proof of Theorem 8, we first construct a graded Lie algebra W map f → grW • g, for which we need to construct maps H1 (Y , Q) → gr−1 g and Q(1)⊕I1 → grW −2 g. For the first of these, we take the inverse of the isomorphism ∼

W grW −1 g → gr−1 H1 (Y, Q) = H1 (Y , Q) arising from the definition of W• . For the second, we choose for each i ∈ I1 a point xi in Di not lying in any other component   Y over xi is an oriented circle γi , and we write log(γi ) ∈ of D. The fibre of Y W gr−2 g for the element determined by this loop. The desired map Q(1)⊕I1 → grW −2 g is then the map taking the ith basis vector to log(γi ). It is easy to see that the free homotopy class of the loop γi , and hence the map Q(1)⊕I1 → grW −2 g, is independent of the choice of xi . Changing notation slightly from Theorem 8, we write r for the kernel of the map f → grW • g defined above. We thus want to prove the following.

(a) The map f → grW • g is surjective.

W (b) The images of the maps H2 (Y , Q) → grW i∈I1 H1 (Di , Q)(1) → gr−3 f and −2 f, Q(2)⊕I2 → grW −4 f defined as in Theorem 8 are contained in r. (c) The images of these maps generate r as an ideal; equivalently they generate r/[f, r] as a vector space. Proof of (a) We may lift the W -graded map f → grW • g to a W -filtered map f → g. , Q) is surjective Now it is easy to see that the map fab → gab = H1 (Y, Q) = H1 (Y and W -strict: it surjects onto the quotient grW H (Y, Q) = H (Y , Q) and the kernel 1 1 −1 is generated by the classes of the loops γi , situated in degree −2. It follows that the map f → g is surjective and W -strict; passing to the associated W -gradeds shows that f  grW • g is surjective, as desired. The proof of (b) is purely topological, showing that the claimed relations in grW • g . are witnessed by certain immersed surfaces in Y Proof of (b) Weight −4: Suppose that Di ∩ Dj = ∅, and choose a point xij in their   Y over xij intersection not lying in any other component of D. The fibre of Y 1 1 is diffeomorphic to S × S , with the two product projections corresponding to the Y i , Y Y j , respectively. It is easy to see that the two standard projections Y 1 loops γ1 = S × {∗} and γ2 = {∗} × S 1 are freely homotopic to the loops γi and the fundamental group of S 1 × S 1 is commutative, we have γj respectively. Since  ⊕I2 → grW f is log(γi ), log(γj ) = 0 in grW −4 g. This says that the image of Q(2) −4 contained in r, as desired. Weight −3: Consider any element [γ ] ∈ H1 (Di , Z), which  we may choose to be represented by an immersed loop γ : S 1 → Di◦ := Di \ j =i Dj . The fibre of   Y over γ is an oriented S 1 -bundle over S 1 , and hence diffeomorphic to a torus Y  takes the standard loops γ1 and γ2 to (a lift S 1 × S 1 . The immersion S 1 × S 1 → Y

Semisimplicity for π1

59

  of) γ and γi , respectively. As before, this yields the relation log(γ ), log(γi ) = 0 W in grW −3 g, so that the image of H1 (Di , Q)(1) → gr−3 f is contained in r, as desired. Weight −2: Consider any class [Σ] ∈ H2 (Y , Z), represented by an immersion ι : Σ → Y with Σ a compact connected oriented surface which meets D transversely at smooth points. We let ι−1 D = {x1 , . . . , xn } denote the preimage of D in Σ and for each 1 ≤ j ≤ n write Dι(j ) for the component containing xi and j ∈ {±1} for the local intersection number of Σ and D at xj . The fundamental group of Σ := Σ \ {x1 , . . . , xn } has a presentation of the form  π1 (Σ) =

g g (aj )j =1 , (bj )j =1 , (cj )nj=1

:

g 

[aj , bj ] ·

j =1

n 

 cj = 1 ,

j =1

where the aj and bj form a symplectic basis of H1 (Σ, Z) and each cj is freely homotopic to a positively oriented loop around xj . It follows that we have g n     [ι∗ (aj )], [ι∗ (bj )] + j [γι(j ) ] = 0 j =1

j =1

g in grW j =1 [ι∗ (aj )] ∧ [ι∗ (bj )] is the image of [Σ] under the cup coproduct −2 g. But map, while  the image of [Σ] under the dual Gysin map associated to Di is [Di ∩ Σ] = ι(j )=i j . Thus, the above identity establishes that the image of [Σ] in grW g is zero, as desired. −2 The proof of (c) is somewhat more technical, and follows [21, §5]. We will control the quotient r/[f, r] using Lie algebra homology, specifically the exact sequence 0 → H2 (grW • g) →

 ab r → fab → grW g →0 • [f, r]

(2)

of low-degree terms in the Hochschild–Serre spectral sequence associated to the extension 0 → r → f → grW • g → 0 of pro-nilpotent Lie algebras. We control the leftmost term of this sequence using the following proposition, whose proof uses the Hodge theory of the pro-unipotent fundamental group in an essential way. Proposition 10 W (i) The natural map grW • H2 (g) → H2 (gr• g) is an isomorphism. (ii) The natural map H2 (Y, Q) → H2 (g) is surjective and W -strict.

Here, the W -filtration on H2 (g) is the natural one whereby W−k H2 (g) is spanned by the classes of “relations of weight −k”. Proof g carries a pro-mixed Hodge structure, compatible with the Lie bracket, whose weight filtration is W• [20]. The Deligne splitting provides a canonical

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splitting of the weight filtration on gC compatible with the Lie bracket. It follows W that the natural map grW • H2 (g) → H2 (gr• g) becomes an isomorphism after tensoring with C, and hence is already an isomorphism. This proves (i). The surjectivity in (ii) is well-known; W -strictness follows from the fact that it is a morphism of mixed Hodge structures [9, Theorem 11.7]. Proof of (c) It follows from Proposition 10 and sequence (2) that the grading on r/[f, r] is supported in degrees −2, −3 and −4. We thus want to show surjectivity of the three maps 

H2 (Y , Q) → grW −2 (r/[f, r]) ,

(∗2 )

W H1 (Di , Q)(1) → grW −3 (r/[f, r]) = gr−3 H2 (g) ,

(∗3 )

W Q(2)⊕I2 → grW −4 (r/[f, r]) = gr−4 H2 (g) ,

(∗4 )

i∈I1

which we address one by one. Weight −3: It suffices to prove that (∗3 ) is equal to the composite 

W H1 (Di , Q)(1)  grW −3 H2 (Y, Q)  gr−3 H2 (g)

(†3 )

i∈I1

where the first map is the map arising from the weight spectral sequence (the Leray  → Y ) and the second map is the map from spectral sequence associated to Y Proposition 10(ii). To do this, consider an element [γ ] ∈ H1 (Di , Z), which we represent by an immersed loop γ : S 1 → Di◦ as in the proof of (b). Pulling back  → Y along γ yields an oriented S 1 -bundle over S 1 , so that γ lifts to an immersion Y . Considering the map on Leray spectral sequences induced by the ι : S1 × S1 → Y commuting square

shows that the image of the class [γ ](1) under the first map of (†3 ) is the 1 1 pushforward ι∗ [S 1 × S 1 ] ∈ grW −3 H2 (Y, Q) of the orientation class on S × S . On the other hand, if we write gS 1 ×S 1 for the Lie algebra of the Q-Mal˘cev completion of the fundamental group of S 1 × S 1 (so a 2-dimensional vector group), then H2 (gS 1 ×S 1 ) is one-dimensional, spanned by the class of the relation [log(γ1 ), log(γ2 )] with γ1 and γ2 the standard generators of π1 (S 1 × S 1 ). Note that this generating class is the image of the orientation class on S 1 × S 1 under the natural map H2 (S 1 × S 1 , Q)  H2 (gS 1 ×S 1 ). Now the image of [γ ](1) under (∗3 ) is the pushforward ι∗ ([log(γ1 ), log(γ2 )]) ∈ grW −3 H2 (g), so that from the commuting square

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61

we see that [γ ](1) has the same image under (∗3 ) and (†3 ). Thus these two maps are equal, as desired. Weight −4: Let I2 denote the set of ordered pairs of distinct indices i, j ∈ I1 together with an irreducible component Dij of Di ∩ Dj . It suffices to prove that the  composite of (∗4 ), precomposed with the natural surjection Q(2)⊕I2  Q(2)⊕I2 , is equal to the composite 

W Q(2)⊕I2  grW −4 H2 (Y, Q)  gr−4 H2 (g)

(†4 )

where the first map is the map arising from the weight spectral sequence and the second map is the map from Proposition 10(ii). To do this, consider an element (i, j, Dij ) ∈ I2 , and choose a point xij ∈ Dij not lying in any other component of  → Y over xij is a torus S 1 ×S 1 , with the two standard loops γ1 and D. The fibre of Y γ2 being freely homotopic to the loops γi and γj , as in the proof of (b). As above, we find that the image of [xij ](2) under the first map of (†4 ) is the pushforward ι∗ ([S 1 × , while its image S 1 ]) of the orientation class under the inclusion ι : S 1 × S 1 → Y under (∗4 ) is ι∗ ([log(γ1 ), log(γ2 )]). The same argument establishes that ι∗ ([S 1 × W S 1 ]) maps to ι∗ ([log(γ1 ), log(γ2 )]) under the map grW −4 H2 (Y, Q) → gr−4 H2 (g), and hence we are done also in this case. Weight −2: There are two claims to prove here: that map (∗2 ) restricts to the W surjection grW −2 H2 (Y, Q)  gr−2 H2 (g) from Proposition 10(ii); and that it induces  ab  grW g ab . For a surjection from H2 (Y , Q)/grW • −2 H2 (Y, Q) to the kernel of f the first of these claims, we proceed as above. Pick a class [Σ] ∈ H2 (Y, Q), represented by a immersed closed oriented surface ι : Σ → Y , and write gΣ for the Lie algebra of the Q-Mal˘cev completion of the fundamental group   of Σ. Thus g H2 (gΣ ) is spanned by the class of the relation log j =1 [aj , bj ] , where the aj and bj are the standard generators of the fundamental group of a closed oriented surface of genus g. Again, this class is the image of the orientation class of Σ under the map H2 (Σ, Q)  H2 (gΣ ), and hence the image of [Σ] ∈ H2 (Y, Q) under (∗2 ) W is the same as its image under the surjection grW −2 H2 (Y, Q)  gr−2 H2 (g) from Proposition 10(ii). This establishes the first claim. For the second claim, we note that the low-degree terms of the Leray spectral sequence provides an exact sequence ⊕I1 0 → grW → grW −2 H2 (Y, Q) → H2 (Y , Q) → Q(1) −2 H1 (Y, Q) → 0

where the central map is dual to the sum of the Gysin maps and the right-hand map  ab . This directly establishes the second claim. is the weight −2 part of fab  grW • g

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The General Case Finally, we deduce Theorem 8 from the special case K = C. By the Lefschetz principle, it suffices to prove this when K admits an embedding K → C. Using this embedding, we obtain from the previous section a presentation of grW • g of the desired form, which we wish to show is GK -equivariant. The only part of this which is non-obvious is GK -equivariance of the map Q (1)⊕I1 → grW −2 g. To do this, we pick for each i a point xi ∈ Di (K) not lying in any other component, and write  OY ,xi for the completed local ring at xi . We choose a K local parameter ti ∈ OY ,xi cutting out Di , and denote the evident morphism K   alg Spec  OY ,xi [1/ti ] → Y by γi . Abhyankar’s Lemma [32, Proposition XIII.5.2] K   ensures that the étale fundamental group of Spec  OY ,xi [1/ti ] is canonically K Z(1), and hence pushforward yields an outer homomorphism  Z(1) → isomorphic to  π1et´ (YK ). The key result here states that the maps γi loops γi .

alg

are algebraic avatars of the

alg Lemma 8 For every embedding K → C, the outer homomorphism γi,∗ :  Z(1) →

π1et´ (YK ) is identified with the profinite completion of the outer homomorphism γi,∗ : Z(1) → π1 (Y (C)) under the usual identification π1et´ (YK ) ∼ π1 (Y (C)). = Corollary 6 The outer homomorphisms γi,∗ :  Z(1) → π1et´ (YK ) are independent of alg

the choice of points xi , and are Galois-equivariant in the sense that σ ◦γi,∗ ◦σ −1 = alg

alg

γσ (i),∗ for all σ ∈ GK . Proof of Corollary 6 If K admits an embedding in C, then the first assertion is immediate; in general use the Lefschetz Principle to reduce to this case. The second follows easily from the first. Proof of Lemma 8 It suffices to prove this in the case K = C. Denote by OY an ,xi (resp. OY an ,xi ) the ring of germs of holomorphic functions on Y OY an (U i )

OY an (U i

an

(resp. Y an ) at xi ,

(resp. ∩ Y an )) as U i runs over i.e. the direct limit of the rings open neighbourhoods of xi in Y i . Fix one such neighbourhood U i , and assume that U i is biholomorphic to a polydisc Dn in such a way that U i ∩ Di = {0} × Dn−1 and U i meets no other components of Di . We write Ui = U i ∩ Y an . There is thus a commuting diagram

Semisimplicity for π1

63 alg

of schemes, where the composite along the bottom row is γi . We will deduce the lemma by showing that applying the functor π1et´ to this diagram yields the diagram

of outer homomorphisms. We justify this in several steps. (1) Note that by construction Ui is homotopy equivalent to a complex punctured disc, and so has fundamental group Z(1); moreover the outer homomorphism Z(1) → π1 (Y an ) induced by the inclusion Ui → Y an is the pushforward map γi,∗ . But there is an equivalence of categories between finite coverings of Ui and finite étale algebras over OY an (Ui ), given by taking a finite covering π : V → Ui to the algebra O(V an ) where V an is endowed with the unique complex structure making π holomorphic. It thus follows that π1et´ (Spec (OY an (Ui ))) =  Z(1) → π1et´ (Y ) induced by the scheme Z(1), and that the outer homomorphism  morphism Spec (OY an (Ui )) → Y is equal to  γi,∗ . (2) A similar argument as for (1) establishes that Spec (OY an , xi ) is the inverse limit Z(1), of the profinite completions of the fundamental groups of U i ∩ Y an , i.e.  and that map (2) is the identity. (3) The ring OY an ,xi is strictly Henselian, so Abhyankar’s Lemma again implies that    π1et´ Spec OY an ,xi [1/ti ] =  Z(1), and that map (3) is the identity. (4) We know that the finite étale algebras over both OY an ,xi [1/ti ] and OY an ,xi are given by adjoining roots of ti : the former via Abhyankar’s Lemma and the latter via identifying these algebras with covers of Ui . It follows that (4) is the identity.

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Logarithmic Adic Spaces: Some Foundational Results Hansheng Diao, Kai-Wen Lan, Ruochuan Liu, and Xinwen Zhu

1 Introduction There are two main goals of this paper. Firstly, we would like to adapt many fundamental notions and features of the theory of log geometry for schemes, as in [19, 22, 23, 25], and [36], to the theory of adic spaces, as in [17] and [18]. For example, we would like to introduce the notion of log adic spaces, which allow us to study the de Rham and étale cohomology of nonproper adic spaces by introducing the log de Rham and Kummer étale cohomology of proper adic spaces equipped with suitable log structures. Secondly, we would like to adapt many foundational techniques in recent developments of p-adic geometry, as in [27, 38, 39, 41], and [43], to the context of log geometry. For example, we would like to introduce the pro-Kummer étale site, and show that log affinoid perfectoid objects form a basis for such a site, under suitable assumptions. In particular, we would like to establish the primitive comparison theorem and some related cohomological finiteness or vanishing results in this context.

H. Diao Yau Mathematical Sciences Center, Tsinghua University, Beijing, China e-mail: [email protected] K.-W. Lan University of Minnesota, Minneapolis, MN, USA e-mail: [email protected] R. Liu Beijing International Center for Mathematical Research, Peking University, Beijing, China e-mail: [email protected] X. Zhu () California Institute of Technology, Pasadena, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_3

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Although a general formalism of log topoi has been introduced in [11, Section 12.1], there are nevertheless several special features (such as the integral structure sheaves) or pathological issues (such as the lack of fiber products in general, or the necessary lack of noetherian property when working with perfectoid spaces) in the theory of adic spaces, which resulted in some complications in our adaption of many “well-known arguments”; and we have chosen to spell out the modifications of such arguments in some detail, for the sake of clarity. Moreover, this paper is intended to serve as the foundation for our development of a p-adic analogue of the Riemann–Hilbert correspondence in [8] (and forthcoming works such as [30]). Therefore, in addition to the above-mentioned goals, we have also included some foundational treatment of quasi-unipotent nearby cycles, following (and reformulating) Beilinson’s ideas in [3]. Here is an outline of this paper. In Sect. 2, we introduce log adic spaces and study their basic properties. In Sect. 2.1, we review some basic terminologies of monoids. In Sect. 2.2, we introduce the definition and some basic notions of log adic spaces, and study some important examples. In Sect. 2.3, we study the important notion of charts in the context of log adic spaces, which are useful for defining the categories of coherent, fine, and fs log adic spaces, and for constructing fiber products in them. In Sect. 3, we study log smooth morphisms of log adic spaces, and their associated sheaves of log differentials. In Sect. 3.1, we introduce the notion of log smooth and log étale morphisms, and show the existence of smooth toric charts for smooth fs log adic spaces. In Sects. 3.2 and 3.3, we develop a theory of log differentials for homomorphisms of log Huber rings and morphisms of coherent log adic spaces, and compare it with the theory in Sect. 3.1. In Sect. 4, we study the Kummer étale topology of locally noetherian fs log adic spaces. In Sect. 4.1, we introduce the Kummer étale site and study its basic properties. In Sect. 4.2, we establish an analogue of Abhyankar’s lemma for rigid analytic varieties, and record some related general facts. In Sect. 4.3, we study the structure sheaves and analytic coherent sheaves on the Kummer étale site, and show that their higher cohomology vanishes on affinoids. In Sect. 4.4, we show that Kummer étale surjective morphisms satisfy effective descent in the category of finite Kummer étale covers, and define Kummer étale fundamental groups with desired properties. In Sect. 4.5, we study certain direct and inverse images of abelian sheaves on Kummer étale sites. In Sect. 4.6, we establish some purity results for torsion Kummer étale local systems. In Sect. 5, we study the pro-Kummer étale topology of locally noetherian fs log adic spaces. In Sect. 5.1, based on the theory in Sect. 4, we introduce the proKummer étale site, by following Scholze’s ideas in [39] and [41]. In Sect. 5.2, we study certain direct and inverse images of abelian sheaves on pro-Kummer étale sites. In Sect. 5.3, we introduce the log affinoid perfectoid objects, and show that they form a basis for the pro-Kummer étale topology, for locally noetherian fs log adic spaces over Spa(Qp , Zp ). In Sect. 5.4, we introduce the completed structure sheaves and their integral and tilted variants on the pro-Kummer étale site, and prove various almost vanishing results for them.

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In Sect. 6, we study the Kummer étale cohomology of fs log adic spaces log smooth over a nonarchimedean base field k. In Sect. 6.1, we start with some preparations using the log affinoid perfectoid objects defined by towers over some associated toric charts. In Sect. 6.2, we establish the primitive comparison theorem, generalizing the strategy in [39, Section 5], and deduce from it some finiteness results for the cohomology of torsion Kummer étale local systems. In Sect. 6.3, we p -local systems, and record some Zp -, and Q introduce the notions of Zp -, Qp -,  finiteness results. In Sect. 6.4, as an application of the theory thus developed, we reformulate Beilinson’s ideas in [3] and define the unipotent and quasi-unipotent nearby cycles in the rigid analytic setting. In Appendix A, we state a version of Tate’s sheaf property and Kiehl’s gluing property for the analytic and étale sites of adic spaces that are either locally noetherian or analytic stably adic. This includes, in particular, a proof of Kiehl’s property for coherent sheaves on (possibly nonanalytic) noetherian adic spaces which (as far as we know) is not yet available in the literature.

1.1 Notation and Conventions By default, all monoids are assumed to be commutative, and the monoid operations are written additively (rather than multiplicatively), unless otherwise specified. For a monoid P , let P gp denote its group completion. For any commutative ring R with unit and any monoid P , we denote by R[P ] the monoid algebra over R associated with P . The image of a ∈ P in R[P ] will often be denoted by ea . Then we have ea+b = ea · eb in R[P ], for all a, b ∈ P . Group cohomology will always mean continuous group cohomology. For each site C, the category of sheaves (resp. abelian sheaves) on C is denoted by Sh(C) (resp. ShAb (C)), although the associated topos is denoted by C∼ . We shall follow [43, Lectures 2–7] for the general definitions and results of Huber rings and pairs, adic spaces, and perfectoid spaces. Unless otherwise specified, all Huber rings and pairs will be assumed to be complete. We say that an adic space is locally noetherian if it is locally isomorphic to Spa(R, R + ), where either R is analytic (see [43, Remark 2.2.7 and Proposition 4.3.1]) and strongly noetherian—i.e., the rings R#T1 , . . . , Tn $ =





 ai1 ,...,in T1i1 · · · Tnin ∈ R[[T1 , . . . , Tn ]] : ai1 ,...,in → 0

i1 ,...,in ≥0

are noetherian, for all n ≥ 0; or R is (complete, by our convention on Huber pairs, and) finitely generated over a noetherian ring of definition. We say that an adic space is noetherian if it is locally noetherian and qcqs (i.e., quasi-compact and quasiseparated). We shall follow [18, Definition 1.2.1] for the definition for morphisms of locally noetherian adic spaces to be locally of finite type (lft for short). A useful fact is

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that a fiber product Y ×X Z of locally noetherian adic spaces exists when the first morphism Y → X is lft, in which case its base change (i.e., the second projection) Y ×X Z → Z is also lft (see [18, (1.1.1), Proposition 1.2.2, and Corollary 1.2.3]). An affinoid field (k, k + ) is a Huber pair in which k is a (possibly trivial) nonarchimedean local field (i.e., a field complete with respect to a nonarchimedean multiplicative norm | · | : k → R≥0 ), and k + is an open valuation subring of Ok := {x ∈ k : |x| ≤ 1} (see [43, Definition 4.2.4]). When k is a nontrivial nonarchimedean field (i.e., a field that is complete with respect to a nontrivial nonarchimedean multiplicative norm), we shall regard rigid analytic varieties over k as adic spaces over (k, Ok ), by virtue of [18, (1.1.11)]. We shall follow [18, Sections 1.6 and 1.7] for the definition and basic properties of unramified, smooth, and étale morphisms of locally noetherian adic spaces. More generally, without the locally noetherian hypothesis, we say that a homomorphism (R, R + ) → (S, S + ) of Huber pairs is finite étale if R → S is finite étale as a ring homomorphism, and if S + is the integral closure of R + in S. We say that a morphism f : Y → X of adic spaces is finite étale if, for each x ∈ X, there exists an open affinoid neighborhood U of x in X such that V = f −1 (U if the ) is affinoid, and + induced homomorphism of Huber pairs OX (U ), O+ (U ) → O (V ), O (V ) is Y X Y finite étale. We say that a morphism f : Y → X of adic spaces is étale if, for each y ∈ Y , there exists open neighborhood V of y in Y such that the restriction of f to V factors as the composition of an open immersion, a finite étale morphism, and another open immersion. Given any adic space X, we denote by Xét the category of adic spaces étale over X. If fiber products exist in Xét , then Xét acquires a natural structure of a site. We say that X is étale sheafy if Xét is a site and if the étale structure presheaf OXét : U → OU (U ) is a sheaf. Étale sheafiness is known when X is either locally noetherian or a perfectoid space—see Appendix A for more information. A geometric point of an adic space X is a morphism η : ξ = Spa(l, l + ) → X, where l is a separably closed nonarchimedean field. For simplicity, we shall write ξ → X, or even ξ , when the context is clear. The image of the unique closed point ξ0 of ξ under η : ξ → X is called the support of ξ . Given any x ∈ X, we have a geometric point x = Spa(κ(x), κ(x)+ ) above x (i.e., x is the support of x), as in [18, (2.5.2)], where κ(x) is the completion of a separable closure of the residue field κ(x) of OX,x . An étale neighborhood of η is a lifting of η to a composition φ

ξ → U → X in which φ is étale. For any sheaf F on Xét , the stalk of F at η is Fξ := Γ ξ, η−1 (F) ∼ = lim F(V ), where the direct limit runs through all étale − → neighborhoods V of ξ . (Recall that, by [18, Proposition 2.5.5], when X is locally noetherian, geometric points form a conservative family for Xét .) An adic space X = Spa(R, R + ) is strictly local if R is a strictly local ring and if X contains a unique closed point x such that the support of the valuation | · (x)| is the maximal ideal of R. We shall denote by X(ξ ) = Spa(OX,ξ , O+ X,ξ ) the strict localization of a geometric point ξ → X of a locally noetherian adic space X, as in [18, (2.5.9) and Lemma 2.5.10]. By the explicit description of the completion of (OX,ξ , O+ X,ξ ) as in [18, Proposition 2.5.13], X(ξ ) is a noetherian adic space, which is canonically isomorphic to ξ when the support of ξ is analytic.

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As for almost mathematics, we shall adopt the following notation and conventions. We shall denote by M a the almost module associated with a usual module M, depending on the context. For usual modules M and N, we shall say “there is an ∼ ∼ almost isomorphism M → N” when there is an isomorphism M a → N a between the associated almost modules. We shall write interchangeably both “M a = 0” and “M is almost zero”, with exactly the same meaning.

2 Log Adic Spaces 2.1 Recollection on Monoids In this subsection, we recollect some basics in the theory of monoids. This is mainly to introduce the terminologies and fix the notation. For more details, we refer the readers to [36]. Definition 2.1.1 (1) A monoid P is called finitely generated if there exists a surjective homomorphism Zn≥0  P for some n. (2) A monoid P is called integral if the natural homomorphism P → P gp is injective. (3) A monoid P is called fine if it is integral and finitely generated. (4) A monoid P is called saturated if it is integral and, for every a ∈ P gp such that na ∈ P for some integer n ≥ 1, we have a ∈ P . A monoid that is both fine and saturated is called an fs monoid. (5) For any monoid P , we denote by P inv the subgroup of invertible elements in P , and write P := P /P inv . A monoid P is called sharp if P inv = {0}. (6) An sharp fs monoid is called a toric monoid. Remark 2.1.2 Arbitrary direct and inverse limits exist in the category of monoids (see [36, Section I.1.1]). In particular, for a homomorphism of monoids u : P → Q, we have ker(u) = u−1 (0), and coker(u) is determined by the conditions that Q → coker(u) is surjective and that two elements q1 , q2 ∈ Q have the same image in coker(u) if and only if there exist p1 , p2 ∈ P such that u(p1 ) + q1 = u(p2 ) + q2 . If P is a submonoid of Q, and if u : P → Q is the canonical inclusion, then we shall denote coker(u) by Q/P . Note that Q/P can be zero even when P = Q. In general, the induced map P / ker(u) → im(u) is surjective, but not necessarily injective. (For a typical example, consider the homomorphism u : Z2≥0 → Z≥0 : (x1 , x2 ) → x1 + x2 . Then ker(u) = 0 but u is not injective.) Therefore, the category of monoids is not abelian. Remark 2.1.3 It is not hard to show that a monoid P is finitely generated if and only if P inv is finitely generated (as a group) and P = P /P inv is finitely generated (as a monoid). (See [36, Proposition I.2.1.1].) A deeper fact is that a finitely generated (commutative) monoid P is always finitely presented; i.e., it is the coequalizer of n some homomorphisms Zm ≥0 ⇒ Z≥0 , for some m, n. (See [36, Theorem I.2.1.7].) As

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a result, if P is finitely generated and Q = lim Qi is a filtered direct limit of − →i∈I monoids, then any injective map P → Q lifts to an injective P → Qi , for some i ∈ I . (The opposite assertion, that any surjective map Q → P lifts to a surjective Qi → P , for some i ∈ I , only requires the finite generation of P .) Definition 2.1.4 Given any two homomorphisms of monoids u1 : P → Q1 and u2 : P → Q2 , the amalgamated sum Q1 ⊕P Q2 is the coequalizer of P ⇒ Q1 ⊕Q2 , with the two homomorphisms given by (u1 , 0) and (0, u2 ), respectively. Lemma 2.1.5 In Definition 2.1.4, suppose moreover that any of P , Q1 , or Q2 is a group. Then the natural map Q1 /P → (Q1 ⊕P Q2 )/Q2 is an isomorphism. Proof The surjectivity is clear. As for the injectivity, by assumption and by [36, Proposition I.1.1.5], two elements (q1 , q2 ), (q1 , q2 ) ∈ Q1 ⊕Q2 have the same image in Q1 ⊕P Q2 if and only if there exist a, b ∈ P such that q1 + u1 (a) = q1 + u1 (b) and q2 + u2 (b) = q2 + u2 (a). Therefore, for q1 , q1 ∈ Q1 , if they have the same image in (Q1 ⊕P Q2 )/Q2 —i.e., there exist q2 , q2 ∈ Q2 such that (q1 , q2 ) and (q1 , q2 ) have the same image in Q1 ⊕P Q2 —then there exist a, b ∈ P such that q1 + u1 (a) = q1 + u1 (b). Thus, q1 and q1 have the same image in Q1 /P .  Definition 2.1.6 For any monoid P , let P int denote the image of the canonical homomorphism P → P gp . For any integral monoid P , let P sat := {a ∈ P gp : na ∈ P , for some n ≥ 1}. For a general monoid P not necessarily integral, we write P sat for (P int )sat . Remark 2.1.7 The functor P → P int is the left adjoint of the inclusion from the category of integral monoids into the category of all monoids. Similarly, P → P sat is the left adjoint of the inclusion from the category of saturated monoids into the category of integral monoids. Lemma 2.1.8 Let P → Q1 and P → Q2 be homomorphisms of monoids. Then gp (Q1 ⊕P Q2 )int can be naturally identified with the image of Q1 ⊕P Q2 in Q1 ⊕P gp gp Q2 . Moreover, if P , Q1 , and Q2 are integral and if any of these monoids is a group, then Q1 ⊕P Q2 is also integral. Proof See [36, Proposition I.1.3.4].



Lemma 2.1.9 The quotient of an integral (resp. a saturated) monoid by a submonoid is also integral (resp. saturated). In particular, for any fs monoid P , the quotient P = P /P inv is a toric monoid. Proof Let Q be any submonoid of an integral monoid P . By [36, Proposition I.1.3.3], P /Q is also integral. Suppose moreover that P is saturated. For any a ∈ (P /Q)sat , by definition, there exists some n ≥ 1 such that na ∈ P /Q. That is, there exist b ∈ P and q1 , q2 ∈ Q such that na = b + (q1 − q2 ) in P gp . Then n(a + q2 ) = b + q1 + (n − 1)q2 , and hence a + q2 ∈ P and a ∈ P /Q. Thus, P /Q = (P /Q)sat is also saturated. 

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Lemma 2.1.10 Let P be an integral monoid, and u : P → Q a surjective homomorphism onto a toric monoid Q. Suppose that ker(ugp ) ⊂ P . Then u admits a (noncanonical) section. In particular, for any fs monoid P , the canonical homomorphism P → P admits a (noncanonical) section. Proof For a ∈ Qgp , if na = 0 for some n ≥ 1, then a = 0, as Q is saturated and sharp. Hence, Qgp is torsion-free, Qgp ∼ = Zr for some r, and the projection gp gp gp u : P → Q admits a section s : Qgp → P gp . It remains to show that s(Q) ⊂ P . For each q ∈ Q, choose any  q ∈ P lifting q. Then s(q) −  q ∈ P gp lies gp in ker(u ) ⊂ P , and therefore s(q) ∈  q + P ⊂ P , as desired.  Construction 2.1.11 Let P be a monoid, and S a subset of P . There exists a monoid S −1 P together with a homomorphism λ : P → S −1 P sending elements of S to invertible elements of S −1 P satisfying the universal property that any homomorphism of monoids u : P → Q with the property that u(S) ⊂ Qinv uniquely factors through S −1 P . The monoid S −1 P is called the localization of P with respect to S. Concretely, let T denote the submonoid of P generated by S. Then, as a set, S −1 P consists of equivalence classes of pairs (a, t) ∈ P × T , where two such pairs (a, t) and (a , t ) are equivalent if there exists some t ∈ T such that a + t + t = a + t + t . The monoid structure of this set is given by (a, t) + (a , t ) = (a + a , t + t ). The homomorphism λ is given by λ(a) = (a, 0). Remark 2.1.12 The localization of an integral (resp. saturated) monoid is still integral (resp. saturated). Remark 2.1.13 Let P → Q1 and P → Q2 be homomorphisms of monoids, and let S be a subset of P . Let S1 , S2 , and S3 denote the images of S in Q1 , Q2 , and Q1 ⊕P Q2 , respectively. Then the natural homomorphism Q1 ⊕P Q2 → (S1−1 Q1 ) ⊕S −1 P (S1−1 Q2 ) factors through an isomorphism ∼

S3−1 (Q1 ⊕P Q2 ) → (S1−1 Q1 ) ⊕S −1 P (S2−1 Q2 ), by the universal properties of the objects. Definition 2.1.14 Let u : P → Q be a homomorphism of monoids. (1) We say it is local if P inv = u−1 (Qinv ). (2) We say it is sharp if the induced homomorphism P inv → Qinv is an isomorphism. (3) We say it is strict if the induced homomorphism P → Q is an isomorphism. (4) We say it is exact if the induced homomorphism P → P gp ×Qgp Q is an isomorphism. (When P and Q are integral and canonically identified as submonoids of P gp and Qgp , respectively, we simply need P = (ugp )−1 (Q).)

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2.2 Log Adic Spaces In this subsection, we give the definition of log adic spaces, introduce some basic notions, and study some important examples. Convention 2.2.1 From now on, we shall only work with adic spaces that are étale sheafy. (They include locally noetherian adic spaces and perfectoid spaces.) Definition 2.2.2 Let X be an (étale sheafy) adic space. (1) A pre-log structure on X is a pair (MX , α), where MX is a sheaf of monoids on Xét and α : MX → OXét is a morphism of sheaves of monoids, called the structure morphism. (Here OXét is equipped with the natural multiplicative monoid structure.) (2) Let (M, α) and (N, β) be pre-log structures on X. A morphism from (M, α) to (N, β) is a morphism M → N of sheaves of monoids that is compatible with the structure morphisms α and β. (3) A pre-log structure (MX , α) on X is called a log structure if the morphism × α −1 (O× Xét ) → OXét induced by α is an isomorphism. In this case, we call the triple (X, MX , α) a log adic space. We shall simply write (X, MX ) or X when the context is clear. (4) We say that a sheaf of monoids M on Xét is integral (resp. saturated) if it is a sheaf of integral (resp. saturated) monoids. A pre-log structure (MX , α) on X is called integral (resp. saturated) if MX is. We say that a log adic space (X, MX , α) is integral (resp. saturated) if MX is. (5) For a log structure (MX , α) on X, we set MX := MX /α −1 (O× Xét ), called the characteristic of the log structure. (6) For a pre-log structure (MX , α) on X, we have the associated log structure × −1 (a MX , a α), where a MX is the pushout of O× Xét ← α (OXét ) → MX in a a the category of sheaves of monoids on Xét , and where α : MX → OXét is canonically induced by the natural morphism O× Xét → OXét and the structure morphism α : MX → OXét (cf. [11, Section 12.1.6]). Again, we shall simply write a MX when the context is clear. (7) A morphism f : (Y, MY , αY ) → (X, MX , αX ) of log adic spaces is a morphism f : Y → X of adic spaces together with a morphism of sheaves of monoids f % : f −1 (MX ) → MY compatible with f % : f −1 (OXét ) → OYét , f −1 (αX ) : f −1 (MX ) → f −1 (OXét ), and αY : MY → OYét . In this case, we have the log structure f ∗ (MX ) on Y associated with the pre-log structure f −1 (MX ) → f −1 (OXét ) → OYét . The morphism f is called strict if the induced morphism f ∗ (MX ) → MY is an isomorphism. (8) A morphism f : (Y, MY ) → (X,MX ) is called exact if, at each geometric y of Y , the induced homomorphism f ∗ (MX ) y → MY,y is exact. (9) A log adic space is called noetherian (resp. locally noetherian, resp. quasicompact, resp. quasi-separated, resp. affinoid, resp. perfectoid, resp. stably uniform, resp. analytic) if its underlying adic space is.

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(10) A morphism of log adic spaces is called lft (resp. quasi-compact, resp. quasiseparated, resp. separated, resp. proper, resp. finite, resp. surjective) if the underlying morphism of adic spaces is. As usual, a separated (resp. proper) log adic space over Spa(k, k + ) is a locally noetherian log adic space with a separated (resp. proper) structure morphism to Spa(k, k + ). Remark 2.2.3 As explained in [11, Section 12.1.6], the functor of taking associated log structures from the category of pre-log structures to the category of log structures on X is the left adjoint of the natural inclusion functor from the category of log structures to the category of pre-log structures on X. Lemma 2.2.4 A sheaf of monoids M on an adic space Xét is integral (resp. saturated) if and only if Mx is integral (resp. saturated) at each geometric point x of X. In particular, a log adic space (X, MX , α) is integral (resp. saturated) if and only if MX,x is integral (resp. saturated) at each geometric point x of X. Proof This follows from the proof of [11, Lemma 12.1.18(ii)].



Remark 2.2.5 For a log adic space (X, MX , α) and a geometric point x of X, it ∼ × × −1 follows that Minv X,x = α (OXét ,x ) → OXét ,x (i.e., the homomorphism MX,x → OX,x is local and sharp). Hence, MX,x ∼ = MX,x /α −1 (O× Xét ,x ) is a sharp monoid. If MX,x is integral (which is the case when (X, MX , α) is integral, by Lemma 2.2.4), gp gp then MX,x ∼ = MX,x /α −1 (O× Xét ,x ), and MX,x → MX,x is exact. Remark 2.2.6 Let f : (Y, MY , αY ) → (X, MX , αX ) be a morphism of log adic % spaces. At each geometric point y of Y , since fy : OXét ,f (y) → OYét ,y is local, % and since f : MX,f (y) ∼ = f −1 (MX )y → MY,y is by definition compatible with y

% % fy : OXét ,f (y) ∼ = f −1 (OXét )y → OYét ,y , we see that fy : MX,f (y) → MY,y is  local as in Definition 2.1.14. By Lemma 2.1.5, f ∗ (MX ) y ∼ = MX,f (y) . Therefore, ∼

%

f is strict if and only if MX,f (y) → MY,y , i.e., fy : MX,f (y) → MY,y is strict, at each geometric point y of Y . Here are some basic examples of log adic spaces. Example 2.2.7 Every (étale sheafy) adic space X has a natural log structure given can. by α : MX = O× Xét → OXét . We call it the trivial log structure on X. Example 2.2.8 A log point is a log adic space whose underlying adic space is Spa(l, l + ), where l is a nonarchimedean local field. We remark that the underlying topological space may not be a single point. Example 2.2.9 In Example 2.2.8, if l is separably closed, then the étale topos of Spa(l, l + ) is equivalent to the category of sets (see [18, Corollary 1.7.3 and Proposition 2.3.10, and the paragraph after (2.5.2)]). In this case, a log structure of Spa(l, l + ) is given by a homomorphism of monoids α : M → l inducing ∼ an isomorphism α −1 (l × ) → l × . For simplicity, by abuse of notation, we shall sometimes introduce a log point by writing s = (Spa(l, l + ), M). Also, we shall simply denote by s the underlying adic space Spa(l, l + ), when the context is clear.

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Example 2.2.10 Let (X, MX , αX ) be a perfectoid log adic space; i.e., a log adic space whose underlying adic space X is a perfectoid space (see Definition 2.2.2). Let MX := lim MX , where the transition maps are given by sending a section ← − to its p-th multiple. Let X be the tilt of X. Then there is a natural morphism of sheaves of monoids αX : MX → OX making (X , MX , αX ) a perfectoid log ét

∼

−1 (O× adic space, called the tilt of (X, MX , αX ). Note that the isomorphism αX Xét ) → −1 × × O× Xét induces an isomorphism αX (O ) → O by taking inverse limit. Xét

Xét

We would like to study log adic spaces of the form Spa(R[P ], R + [P ]), whenever (R, R + ) is a Huber pair. Lemma 2.2.11 Suppose that P is a monoid and (R, R + ) is a Huber pair with a ring of definition R0 ⊂ R, which is adic with respect to a finitely generated ideal I . Let us equip R[P ] with the topology determined by the ring of definition R0 [P ] such that {I m R0 [P ]}m≥0 forms a basis of open neighborhoods of 0. Then (R[P ], R + [P ]) is also a Huber pair. Proof Note that R + [P ] is open in R[P ] because R + is open in R. We only need to check that R + [P ] is integrally closed in R[P ]. By writing P as the direct limit of its finitely generated submonoids, we may assume that P is finitely generated. But this case is standard (see, for example, [6, Theorem 4.42]).  Remark 2.2.12 Let (R#P $, R + #P $) denote the completion of (R[P ], R + [P ]). Since taking completions of Huber pairs does not alter the associated adic spectra, we can identify Spa(R[P ], R + [P ]) with Spa(R#P $, R + #P $) (not just as adic spaces, but also as log adic spaces) whenever it is convenient to do so. Lemma 2.2.13 Let P be a finitely generated monoid. Suppose that R is either (1) analytic and strongly noetherian; or (2) (complete, by our convention, and) finitely generated over a noetherian ring of definition. Then so is R#P $ (which is complete by definition). As a result, Spa(R#P $, R + #P $) is a noetherian adic space when Spa(R, R + ) is, and Spa(R#P $, R + #P $) is étale sheafy (see Corollary A.11). Moreover, the formation of the canonical morphism Spa(R#P $, R + #P $) → Spa(R, R + ) is compatible with rational localizations on the target Spa(R, R + ). Proof Suppose that R is analytic and strongly noetherian. Since P is finitely generated, there is some surjection Zr≥0  P , which induces a continuous surjection R#T1 , . . . , Tr $ ∼ = R#Zr≥0 $  R#P $. In this case, R#P $#T1 , . . . , Tn $ is r a quotient of R#Z≥0 $#T1 , . . . , Tn $ ∼ = R#T1 , . . . , Tr+n $, for each n ≥ 0, which is noetherian as R is strongly noetherian. Hence, R#P $ is also analytic and strongly noetherian. Alternatively, suppose that R is generated by some u1 , . . . , un over a noetherian ring of definition R0 , with an ideal of definition I ⊂ R0 . Since R0 [P ] is noetherian

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as P is finitely generated, its I R0 [P ]-adic completion R0 #P $ is also noetherian. Then the image of R0 #P $ is a noetherian ring of definition of R#P $, over which R#P $ is generated by the images of u1 , . . . , un , as desired. In both cases, the formation of Spa(R#P $, R + #P $) → Spa(R, R + ) is clearly compatible with rational localizations on Spa(R, R + ).  In a different direction, we would like to show that, under certain condition on P , if (R, R + ) is a perfectoid affinoid algebra, then (R#P $, R + #P $) also is. In this case, (R#P $, R + #P $) is étale sheafy (see Corollary A.11, again). Definition 2.2.14 For each integer n ≥ 1, a monoid P is called n-divisible (resp. uniquely n-divisible) if the n-th multiple map [n] : P → P is surjective (resp. bijective). Lemma 2.2.15 Suppose that (R, R + ) is a perfectoid Huber pair. If a monoid P is uniquely p-divisible, then (R#P $, R + #P $) is also a perfectoid Huber pair. Moreover, the formation of the canonical morphism Spa(R#P $, R + #P $) → Spa(R, R + ) is compatible with rational localizations on the target Spa(R, R + ). Proof If pR = 0, then the unique p-divisibility of P implies that R + [P ] is perfect, and so is its completion R + #P $. Also, it is clear that (R#P $)◦ = R ◦ #P $ in R#P $. Hence, R#P $ is uniform, and (R#P $, R + #P $) is a perfectoid Huber pair. In general, let (R , R + ) be the tilt of (R, R + ). Let  ∈ R be a pseudouniformizer of R satisfying  p |p in R ◦ and admitting a sequence of p-th power 1

1

roots  pn , so that  = (,  p , . . .) ∈ R ◦ is a pseudo-uniformizer of R , as in [43, Lemma 6.2.2]. Let ξ be a generator of ker(θ : W (R + ) → R + ), which can be written as ξ = p + [ ]a for some a ∈ W (R + ), by Scholze and Weinstein [43, Lemma 6.2.8]. By the first paragraph above and the tilting equivalence (see [43, Theorem 6.2.11]), it suffices to show that R + #P $ ∼ = W (R + #P $)/(ξ ). For this purpose, note that there is a natural homomorphism θ : W (R + #P $) → R + #P $ induced by the surjective homomorphism R + #P $ → (R + / )[P ] ∼ = (R + / )[P ] and the universal property of Witt vectors, and θ is surjective because both its source and target are complete. Since ξ = p + [ ]a, we have W (R + #P $)/(ξ, [ ]) = W (R + #P $)/(p, [ ]) ∼ = (R + / )[P ] ∼ = (R + / )[P ].

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Since θ ([ ]) =  , by induction, we see that the homomorphism W (R + #P $)/(ξ, [ ]n ) → (R + / n )[P ] induced by θ is an isomorphism, for each n ≥ 1. Thus, since W (R + #P $)/(ξ ) is [ ]-adically complete and separated, ker(θ ) is generated by ξ , as desired. Finally, the formation of Spa(R#P $, R + #P $) → Spa(R, R + ) is clearly compatible with rational localizations on Spa(R, R + ), as in Lemma 2.2.13.  Remark 2.2.16 In Lemma 2.2.15, perfectoid Huber pairs are Tate by our convention following [43, Lecture 6], but the statement of the lemma remains true for more general analytic perfectoid Huber pairs as in [26], by using [26, Lemma 2.7.9] instead of [43, Theorem 6.2.11]. ∼ Spa(R#P $, R + #P $) is étale Definition 2.2.17 When X = Spa(R[P ], R + [P ]) = sheafy, we denote by PX the constant sheaf on Xét defined by P . Then the natural homomorphism P → R#P $ of monoids induces a pre-log structure PX → OXét on X, whose associated log structure we simply denote by P log . Convention 2.2.18 From now on, when Spa(R#P $, R + #P $) is étale sheafy and regarded as a log adic space, we shall endow it with the log structure P log as in Definition 2.2.17, unless otherwise specified. Let us continue with some more examples of log adic spaces. Example 2.2.19 Given any locally noetherian adic space Y with trivial log structure as in Example 2.2.7, and given any finitely generated monoid P , by gluing the morphisms Spa(R#P $, R + #P $) → Spa(R, R + ) as in Lemma 2.2.13 over the noetherian affinoid open Spa(R, R + ) in Y , where each Spa(R#P $, R + #P $) is equipped with the structure of a log adic space as in Definition 2.2.17 and Convention 2.2.18, we obtain a morphism X → Y of log adic spaces, which we shall denote by Y #P $ → Y . In this case, we shall also denote the log structure of X = Y #P $ by P log . Example 2.2.20 If P is a toric monoid as in Definition 2.1.1(6), then we say that X = Spa(k#P $, k + #P $) is an affinoid toric log adic space. This is a special case of Example 2.2.19 with Y = Spa(k, k + ), and is closely related to the theory of toroidal embeddings and toric varieties (see, e.g., [28] and [9]). Roughly speaking, such affinoid toric log adic spaces provide affinoid open subspaces of the rigid analytification of toric varieties, which are then also useful for studying local properties of more general varieties or rigid analytic varieties which are locally modeled on toric varieties. Note that the underlying spaces of affinoid toric log adic spaces are always normal, by Bosch et al. [5, Section 7.3.2, Proposition 8], [13, IV-2, 7.8.3.1], and [16, Theorem 1] (cf. [24, Theorem 4.1]). Example 2.2.21 A special case of Example 2.2.20 is when P ∼ = Zn≥0 for some integer n ≥ 0. In this case, we obtain

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X = Spa(k#P $, k + #P $) ∼ = Dn := Spa(k#T1 , . . . , Tn $, k + #T1 , . . . , Tn $), the n-dimensional unit disc, with the log structure of Dn associated with the pre-log structure given by Zn≥0 → k#T1 , . . . , Tn $ : (a1 , . . . , an ) → T1a1 · · · Tnan . The following proposition provides many more examples of log adic spaces coming from locally noetherian log formal schemes. Proposition 2.2.22 The canonical fully faithful functor from the category of locally noetherian formal schemes to the category of locally noetherian adic spaces defined locally by Spf(A) → Spa(A, A) (as in [17, Section 4.1]) canonically extends to a fully faithful functor from the category of locally noetherian log formal schemes (as in [11, Section 12.1]) to the category of locally noetherian log adic spaces (introduced in this paper). Proof Given any locally noetherian log formal scheme (X, MX ), let X denote the adic spaces associated with the formal scheme X, with a canonical morphism of sites λ : Xét → Xét , as in [18, Lemma 3.5.1]. By construction, we have a canonical morphism λ−1 (OXét ) → O+ Xét . Let MX be the log structure of X associated with the pre-log structure λ−1 (MX ) → λ−1 (OXét ) → O+ Xét → OXét . Then the assignment (X, MX ) → (X, MX ) gives the desired functor, which is fully faithful by adjunction.  Definition 2.2.23 (cf. [36, Definition III.2.3.1]) We say that a morphism f : Y → X of log adic spaces is an open immersion (resp. a closed immersion) if the underlying morphism of adic spaces is an open immersion (resp. a closed immersion) and if the morphism f % : f −1 (MX ) → MY is an isomorphism (resp. a surjection). We say that f is an immersion if it is a composition of a closed immersion of log adic spaces followed by an open immersion of log adic spaces. We say that f is strict if it is a strict morphism of log adic spaces. Example 2.2.24 Let (X, MX , αX ) be a log adic space and ı : Z → X an immersion of adic spaces. Let (Z, MZ , αZ ) be the log adic space associated with the pre-log structure ıét−1 (MX ) → ıét−1 (OXét ) → OZét . Then the induced morphism (Z, MZ , αZ ) → (X, MX , αX ) of log adic spaces is a strict immersion. It is an open (resp. a closed) immersion exactly when the immersion ı of adic spaces is. Remark 2.2.25 More generally, by the same argument as in [36, the paragraph after Definition III.2.3.1], a closed immersion is strict when it is exact.

2.3 Charts and Fiber Products In this subsection, we introduce the notion of charts for log adic spaces. Compared with the corresponding notion for log schemes, a notable difference is that the definition of charts for a log adic space X involves not just OXét but also O+ Xét .

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Based on this notion, we also introduce the category of coherent (resp. fine, resp. fs) log adic spaces and study the fiber products in it. Definition 2.3.1 Let (X, MX , α) be a log adic space. Let P be a monoid, and let PX denote the associated constant sheaf of monoids on Xét . A (global) chart of Xmodeled on P is a morphism of sheaves of monoids θ : PX → MX such that α θ (PX ) ⊂ O+ Xét and such that θ canonically induces (by the universal property of ∼

pushouts) an isomorphism a PX → MX from the log structure a PX associated with the pre-log structure α ◦ θ : PX → OXét . We call the chart finitely generated (resp. fine, resp. fs) if P is finitely generated (resp. fine, resp. fs).  Remark 2.3.2 Giving a morphism θ : PX → MX such that α θ (PX ) ⊂ O+ Xét as in Definition 2.3.1 is equivalent to giving a homomorphism P → MX (X) of monoids whose composition with α(X) : MX (X) → OXét (X) factors through O+ Xét (X). If the monoid P is finitely generated, and if the underlying adic space X is over some affinoid adic space Spa(R, R + ), then giving such a homomorphism P → MX (X) whose composition with α(X) factors through O+ Xét (X) is equivalent to giving a morphism f : (X, MX ) → (Spa(R#P $, R + #P $), P log ) of log adic spaces, whenever Spa(R#P $, R + #P $) is an étale sheafy adic space. In this case, θ is a chart if and only if the morphism f is strict. We imposed the condition α θ (PX ) ⊂ O+ Xét in because we will make crucial use of morphisms f : (X, MX ) →  Definition 2.3.1 Spa(R#P $, R + #P $), P log as above in this paper. Remark 2.3.3 In Remark 2.3.2, if the underlying adic space X is over some locally noetherian  adic space Y , then giving a morphism θ : PX → MX such that α θ (PX ) ⊂ O+ Xét is also equivalent to giving a morphism g : X → Y #P $ as in Example 2.2.19, in which case θ is a chart if and only if the morphism g is strict. Moreover, if X is itself locally noetherian, then we can take Y = X, and obtain a closed immersion h : X → X#P $, in which case θ is a chart if and only if h is a strict closed immersion. Remark 2.3.4 Let θ : PX → MX be a chart of a log adic space (X, MX , α). By Lemma 2.1.5 and Remark 2.2.5, for each geometric point x of X, we obtain a ∼ × −1 ∼ canonical isomorphism P /(α ◦ θ )−1 (O× Xét ,x ) → MX,x /α (OXét ,x ) = MX,x . In θ

particular, the composition PX → MX → MX is surjective. Definition 2.3.5 A quasi-coherent (resp. coherent, resp. fine, resp. fs) log adic space is a log adic space X that étale locally admits some charts modeled on some monoids (resp. finitely generated monoids, resp. fine monoids, resp. fs monoids). (Quasi-coherent log adic spaces will not play any important role in this paper.) Lemma 2.3.6 Let (X, MX , α) be a log adic space, and θ : PX → MX a chart modeled on some monoid P . Suppose that there is a finitely generated monoid P θ

such that θ factors as PX → PX → MX and such that α ◦ θ : PX → OXét factors through O+ Xét . Then, étale locally on X, there exists a chart θ : PX → MX modeled on some finitely generated monoid P such that θ factors through θ .

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Proof The proof is similar to [36, Proposition II.2.2.1], except that, when compared with charts on log schemes, charts θ on log adic spaces (X, MX , α) are subject to the additional requirement that α ◦ θ factors through O+ Xét . Let {ai }i∈I be a finite set of generators of P . Since PX → MX is surjective (by Remark 2.3.4), étale locally on X, there exist some ai ∈ P and fi ∈ O× X (X) such that θ (ai ) = θ (ai ) fi , for all i ∈ I . By [18, (1) in the proof of Proposition 2.5.13], for each geometric point x of X, we have O+ Xét ,x = {f ∈ OXét ,x : |f (x)| ≤ 1} in OXét ,x . By Remark 2.1.3, up to further étale localization on X, we may assume that, for each i ∈ I , at least one of fi and fi−1 is in O+ X (X). Consider the homomorphism P ⊕ ZI≥0 → MX (X) sending (ai , 0) → θ (ai ) and sending 

(0, ei ) → fi ,

if fi ∈ O+ X (X);

−1 (0, ei ) → fi−1 , if fi ∈ O+ ∈ O+ X (X) but fi X (X),

where ei denotes the i-th standard basis element of ZI≥0 . Let β denote the homomorphism P → P , and let P be the quotient of P ⊕ ZI≥0 modulo the relations  (ai , 0) ∼ (β(ai ), ei ), if fi ∈ O+ X (X); (ai , ei ) ∼ (β(ai ), 0),

−1 if fi ∈ O+ ∈ O+ X (X) but fi X (X).

By construction, P ⊕ ZI≥0 → MX (X) factors through an induced homomorphism θ : P → MX (X), and α ◦ θ factors through O+ X , as desired. It remains to check that the log structure associated with the pre-log structure α ◦ θ : PX → MX → OXét coincides with MX ; i.e., the natural morphism a P → a P induced by P → P is an isomorphism. It is injective because the X X composition a PX → a PX → MX is an isomorphism. It is also surjective, because × −1 the induced morphism PX /(α ◦ θ )−1 (O× Xét ) → PX /(α ◦ θ ) (OXét ) is surjective, since the target is generated by the images of ai which lift to the images of ai in the source.  Lemma 2.3.7 Let (X, MX , α) be a locally noetherian coherent log adic space, P a monoid, and PX → MX a chart. Suppose that (X, MX , α) is integral (resp. saturated), in which case PX → MX factors through PXint → MX (resp. PXsat → MX ). Then PXint → MX (resp. PXsat → MX ) is also a chart. Proof Suppose that (X, MX , α) is integral. Since PX → MX is a chart, the composition a PX → a PXint → MX is an isomorphism, and hence the induced morphism a PX → a PXint is injective. Since P → P int is surjective, the composition a a int PXint → MX → OXét factors through O+ Xét , and the induced map PX → PX is surjective and hence is an isomorphism.

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Suppose that (X, MX , α) is saturated. Then the chart PX → MX factors as a composition of PX → PXsat → MX . Since O+ Xét is integrally closed in OXét , the composition PXsat → MX → OXét factors through O+ Xét . It remains to show that the sat a a induced morphism PX → PX is an isomorphism. By Remark 2.3.4, it suffices to show that, at each geometric point x of X, if we denote by β : P → MX,x and β : P sat → MX,x the induced homomorphisms, then the canonical homomorphism × sat −1 P /(α ◦ β)−1 (O× Xét ,x ) → P /(α ◦ β ) (OXét ,x )

(2.3.8)

is an isomorphism. Let β denote the composition of P → MX,x → MX,x . Since gp gp × −1 ker(MX,x → MX,x ) = Minv X,x = α (OXét ,x ), because MX,x is integral (see Remark 2.2.4), we obtain  gp ker(β ) = (β gp )−1 α −1 (O× Xét ,x ) .  gp gp gp Since P gp / ker(β ) ∼ , we obtain = MX,x ∼ = P gp / (α ◦ β)−1 (O× Xét ,x )   gp (α ◦ β)−1 (O× = (β gp )−1 α −1 (O× Xét ,x ) Xét ,x ) .  × −1 gp −1 α −1 (O× Since (α ◦ β)−1 (O× Xét ,x ) ⊂ (α ◦ β ) (OXét ,x ) ⊂ (β ) Xét ,x ) , we obtain   gp (α ◦ β )−1 (O× = (β gp )−1 α −1 (O× Xét ,x ) Xét ,x ) . By Lemma 2.1.9 and the above, we see that the natural homomorphism gp P sat /(α ◦ β )−1 (O× Xét ,x ) → P / ker(β ) gp

(2.3.9)

 sat is injective, whose image is contained in P /(α ◦ β)−1 (O× . Moreover, the Xét ,x ) composition of (2.3.8) and (2.3.9) induces the canonical homomorphism  sat × −1 P /(α ◦ β)−1 (O× . Xét ,x ) → P /(α ◦ β) (OXét ,x )

(2.3.10)

∼ By Lemmas 2.1.9 and 2.2.4, P /(α ◦ β)−1 (O× Xét ,x ) = MX,x is saturated. Thus, (2.3.10) is an isomorphism, and so is (2.3.8), as desired.  Proposition 2.3.11 Let (X, MX , α) be a locally noetherian coherent log adic space. Then it is fine (resp. fs) if and only if it is integral (resp. saturated). Proof If (X, MX , α) is integral (resp. saturated), then it is fine (resp. fs) by Lemma 2.3.7. Conversely, if (X, MX , α) is fine, then it is integral by Lemma 2.1.8. Suppose that (X, MX , α) admits an fs chart θ : PX → MX . By Remark 2.3.4, we ∼ × −1 ∼ have P /(α ◦ θ )−1 (O× Xét ,x ) → MX,x /α (OXét ,x ) = MX,x at each geometric point x of X. By Lemma 2.1.9, MX,x is saturated, because P is. By [36, Proposition

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I.1.3.5], MX,x is also saturated, because MX,x is integral and MX,x is saturated. Thus, (X, MX , α) is saturated, by Lemma 2.2.4.  Lemma 2.3.12 Let (X, MX , α) be a fine (resp. fs) log adic space. For any geometric point x of X, the monoid MX,x is sharp fine (resp. toric—i.e., sharp fs), and the canonical homomorphism MX,x → MX,x admits a splitting s that factors through the preimage of O+ Xét ,x in MX,x . Proof Let P := MX,x , which is finitely generated because X is fine. Under the assumption that X is fine (resp. fs), by Proposition 2.3.11 and Lemma 2.2.4, MX,x is integral (resp. saturated), and so its sharp quotient P is sharp fine (resp. toric). By Lemma 2.1.10, the surjective homomorphism f : MX,x → P admits a section s0 . We need to modify this into a section s : P → MX,x such that (α ◦ s)(P ) ⊂ O+ Xét ,x . By [18, (1) in the proof of Proposition 2.5.13], we have O+ Xét ,x = {f ∈ OXét ,x : |f (x)| ≤ 1} and {f ∈ OXét ,x : |f (x)| > 1} ⊂ O× Xét ,x in OXét ,x . Let {a1 , . . . , ar } be a finite set of generators of P . For each i, let γi := |α(s0 (ai ))(x)|. If γi ≤ 1 for all i, then we set f0 := 1 in OXét ,x . Otherwise, there exists some i0 such that γi0 > 1 and γi0 ≥ γi , for all i. Then α(s0 (ai0 )) admits . By [36, Corollary I.2.2.7], we can an inverse f0 in OXét ,x , so that |f0 (x)| = γi−1 0

identify P with a submonoid of Zr≥0 , for some r ≥ 0, so that we can describe elements of P by r -tuples of integers. Then the homomorphism n +···+nr

s : P → MX,x : (n1 , . . . , nr ) → f0 1 satisfies (α ◦ s)(P ) ⊂ O+ Xét ,x , as desired.

 s0 (n1 , . . . , nr ) 

Proposition 2.3.13 Let (X, MX , α) be an fine log adic space, and x any geometric point of X. Then X admits, étale locally at x, a chart modeled on MX,x . Proof By Lemma 2.3.12, we have a splitting s : P := MX,x → MX,x such that (αx ◦ s)(P ) ⊂ O+ Xét ,x . Since P is fine because X is fine, by Remark 2.1.3, up to étale localization on X, the splitting s lifts to a morphism  s : PX → MX such that (α ◦  s)(PX ) ⊂ O+ Xét (see Remark 2.1.3). Then the composition of PX → MX with α is a pre-log structure, whose associated log structure a PX → OXét factors through a s : a PX → MX (and α). The induced a sx : a PX,x → MX,x is an isomorphism, because the quotients of both sides by the isomorphic preimages of ∼ O× Xét ,x induce the canonical isomorphism P → MX,x , by construction. Hence, up to further étale localization on X, we may assume that a s : a PX → MX is an isomorphism, because the quotients of both sides by the isomorphic preimages of

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O× Xét induce the canonical morphism PX → MX (again, see Remark 2.1.3). As a s : PX → MX is a chart modeled on P = MX,x , as desired.  result,  Example 2.3.14 An fs log point is a log point (as in Example 2.2.8) that is an fs log adic space. In the setting of Example 2.2.9, by Remark 2.2.5, a log point s = (Spa(l, l + ), M) with l separably closed is an fs log point exactly when M/ l × is toric (i.e., sharp fs). In this case, by Lemmas 2.1.10 and 2.3.12, there always exists a homomorphism of monoids M/ l × → M splitting the canonical homomorphism M → M/ l × and defining a chart of s modeled on M. Example 2.3.15 A special case of Example 2.3.14 is a split fs log point i.e., a log point of the form s = (X, MX ) ∼ = (Spa(l, l + ), O× Xét ⊕ PX ) for some (necessarily) toric monoid P . This is equivalent to a log point (Spa(L, L+ ), M), where L is the completion of a separable closure l sep of l, with a Gal(l sep / l)-equivariant splitting of the homomorphism M → M/L× . We also remark that this is the same as a Gal(l sep / l)-equivariant splitting of the homomorphism M gp → M gp /L× . Example 2.3.16 Let D be an effective Cartier divisor on a normal rigid analytic variety X over a nonarchimedean field k, and let ı : D → X denote the associated closed immersion. By viewing X as a noetherian adic space, we equip X with the log structure α : MX → OXét defined by setting MX (V ) = {f ∈ OXét (V ) : f is invertible on the preimage of X − D}, for each object V → X in Xét , with α(V ) : MX (V ) → OXét (V ) given by the natural inclusion. This makes X a locally noetherian fs log adic space. (The normality of X is necessary for showing that the log structure MX is indeed saturated.) Then X − D is the maximal open subspace of X over which MX is trivial. Note that, in Example 2.2.21, the log structure of X ∼ = Dn can be defined alternatively as above by the closed immersion ı : D := {T1 · · · Tn = 0} → Dn . The following special case is useful in many applications: Example 2.3.17 Let X, D, and k be as in Example 2.3.16. Suppose moreover that X is smooth. We say that D is a (reduced) normal crossings divisor of X if, étale locally on X—or equivalently (by [21, Lemma 3.1.5]), analytic locally on X, up to replacing the base field k with a finite separable extension—X and D are of the form S × Dm and S × {T1 · · · Tm = 0}, where S is a smooth connected rigid analytic variety over k, and ι : D → X is the pullback of the canonical closed immersion {T1 · · · Tm = 0} → Dm . (This definition is justified by [29, Theorem 1.18].) Then we equip X with the fs log structure defined as in Example 2.3.16, which is compatible with the one of Dm as in Example 2.2.21 via pullback. The following example will be useful when studying the geometric monodromy and nearby cycles of étale local systems “along the boundary”: Example 2.3.18 Let X, D, and k be as in Example 2.3.17. Suppose that {Dj }j ∈I is the set of irreducible components of D (see [7]). For each J ⊂ I , as locally closed

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 subspaces of X, consider XJ := X ∩ ∩j ∈J Dj , DJ := ∪J J ⊂I XJ , and UJ := XJ − DJ . By pulling back the log structure from X to XJ and UJ , respectively, we obtain log adic spaces (XJ∂ , MX∂ ) and (UJ∂ , MU ∂ ) (with strict immersions to J J X). When XJ is also smooth and so DJ is a normal crossings divisor, we equip XJ with the fs log structure defined by DJ as in Example 2.3.16, whose restriction to UJ is then the trivial log structure. If we also consider D J := ∪j ∈I −J Dj , and let XJ denote the same adic space X but equipped with the fs log structure defined by D J as in Example 2.3.16, then MXJ and MUJ = O× UJ ,ét are nothing but the log structures pulled back from XJ . Moreover, since D J ⊂ D, there is a canonical morphism of log adic spaces X → XJ ; and since DJ = D J ∩ XJ , this morphism induces a canonical morphism of log adic spaces XJ∂ → XJ , whose underlying morphism of adic spaces is an isomorphism. Since X and D is étale locally of the form S ×Dm and S ×{T1 · · · Tm = 0} for some smooth S over k, it follows that XJ is étale locally of the form S×Dm−|J | , in which case the log structures MX∂ and MXJ m−|J |

are associated with the pre-log structures Zm ≥0 → OXJ∂ ,ét and Z≥0 ∼ MX ⊕ (ZJ )X . and we have a direct sum MX∂ = J ≥0 J

J

→ OXJ ,ét ,

J

Definition 2.3.19 Let f : (Y, MY , αY ) → (X, MX , αX ) be a morphism of log adic spaces. A chart of f consists of charts θX : PX → MX and θY : QY → MY and a homomorphism u : P → Q of monoids such that the diagram PY

u

QY

θX

f −1 (MX )

θY f

MY

commutes. We say that the chart is finitely generated (resp. fine, resp. fs) if both P and Q are finitely generated (resp. fine, resp. fs). When the context is clear, we shall simply say that u : P → Q is the chart of f . Example 2.3.20 Let P := Zn≥0 and let Q be a toric submonoid of m1 Zn≥0 containing P , for some m ≥ 1. Then the canonical homomorphism u : P → Q induces a morphism f : Y := Spa(k#Q$, k + #Q$) → X := Spa(k#P $, k + #P $) ∼ = Dn of normal adic spaces, whose source and target are equipped with canonical log structures as in Examples 2.2.20 and 2.2.21, making f : Y → X a morphism of fs log adic spaces. Moreover, these log structures on X and Y coincide with those on X and Y defined by D = {T1 · · · Tn = 0} → X and its pullback to Y , respectively, as in Example 2.3.16. A chart of f : Y → X is given by the canonical charts P → MX (X) and Q → MY (Y ) and the above u : P → Q. Proposition 2.3.21 Let f : Y → X be a morphism of coherent log adic spaces, and let P → MX (X) be a chart modeled on a finitely generated monoid P . Then, étale locally on Y , there exist a chart Q → MY (Y ) modeled on a finitely generated monoid Q and a homomorphism P → Q, which together provide a chart of f .

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Proof Up to étale localization on Y and X, we may assume that (X, MX ) and (Y, MY ) are modeled on some finitely generated monoids P and Q , respectively. Then the composition of PY ∼ = f −1 (PX ) → f −1 (MX ) → MY induces a morphism PY → (P ⊕ Q )Y → MY . Note that P ⊕ Q is finitely generated, and that the composition (P ⊕ Q )Y → MY → OYét factors through O+ Yét . By applying Lemma 2.3.6 to Q Y → (P ⊕ Q )Y → MY , we see that, étale locally, (P ⊕ Q )Y → MY factors as (P ⊕ Q )Y → QY → MY , where QY → MY is a chart modeled on a finitely generated monoid Q. Thus, the composition P → P ⊕ Q → Q gives a chart of f , as desired.  Proposition 2.3.22 Any morphism between fine (resp. fs) log adic spaces étale locally admits fine (resp. fs) charts. Proof By Proposition 2.3.13, étale locally, X admits a chart modeled on a fine (resp. fs) monoid P . By Proposition 2.3.21, f admits, étale locally on Y , a chart P → Q with finitely generated Q. By Lemma 2.3.7, the induced Qint Y → MY (resp. Qsat → M ) is also a chart of Y . Hence, the composition of P → Q → Qint Y Y sat (resp. P → Q → Q ) is a fine (resp. fs) chart of f .  Proposition 2.3.23 (1) The inclusion from the category of noetherian (resp. locally noetherian) fine log adic spaces to the category of noetherian (resp. locally noetherian) coherent log adic spaces admits a right adjoint X → Xint , and the corresponding morphism of underlying adic spaces is a closed immersion. (2) The inclusion from the category of noetherian (resp. locally noetherian) fs log adic spaces to the category of noetherian (resp. locally noetherian) fine log adic spaces admits a right adjoint X → Xsat , and the corresponding morphism of underlying adic spaces is finite and surjective. Proof In case (1) (resp. (2)), let ? = int (resp. sat) in the following. Suppose that X = Spa(R, R + ) is noetherian affinoid and admits a global chart modeled on a finitely generated (resp. fine) monoid P , so that we have a homomorphism P → R of monoids, inducing a homomorphism Z[P ] → R of rings. Let R ? := R ⊗Z[P ] Z[P ? ], and let R ?+ denote the integral closure of R + ⊗Z[P ] Z[P ? ] in R ? . Since P is finitely generated, Z[P ? ] is a finite Z[P ]-algebra, and (R ? , R ?+ ) is equipped with a unique topology extending that of (R, R + ), which is not necessarily complete. Let X? := Spa(R ? , R ?+ ) (which, as usual, depends only on the completion of (R ? , R ?+ )), with the log structure induced by P ? → OX? (X? ) = R ⊗Z[P ] Z[P ? ] : a → 1 ⊗ ea (where ea denotes the image of a ∈ P ? in Z[P ? ], by our convention). Clearly, the natural projection X? → X is a closed immersion (resp. finite and surjective morphism) of log adic spaces. We claim that, if (Y, MY ) is a fine (resp. fs) log adic space, then each morphism f : (Y, MY ) → (X, MX ) of log adic spaces factors through X? , yielding Mor(Y, X) ∼ = Mor(Y, X? ). Indeed, by Proposition 2.3.11, the induced −1 ∼ morphism PY = f (PX ) → f −1 (MX ) → MY factors through PY? , and hence Y → X factors through Y → X? , as desired.

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In general, there exists an étale covering of X by affinoids Xi = Spa(Ri , Ri+ ) such that each Xi admits a global chart modeled on a finitely generated (resp. fine)  = i Xi . By the affinoid case treated monoid (see Definition 2.3.5). Consider X  which is equipped ? → X, in the last paragraph, we obtain a finite morphism X with a descent datum. By étale descent of coherent sheaves (see Proposition A.10),  descends to a locally noetherian adic space X? → X. Also, the étale ? → X X sheaf of monoids descends (essentially by definition). Finally, by Proposition 2.3.22 and the local construction in the previous paragraph, the formation X → X? is functorial, as desired.  Remark 2.3.24 For a noetherian (resp. locally noetherian) coherent log adic space X, we shall simply denote by Xsat the fs log adic space (Xint )sat . By combining the two cases in Proposition 2.3.23, the functor X → Xsat from the category of noetherian (resp. locally noetherian) coherent log adic spaces to the category of noetherian (resp. locally noetherian) fs log adic spaces is the right adjoint of the inclusion from the category of noetherian (resp. locally noetherian) fs log adic spaces to the category of noetherian (resp. locally noetherian) coherent log adic spaces. Remark 2.3.25 By construction, both the functors X → Xint and X → Xsat send strict and finite (resp. étale) morphisms to strict and finite (resp. étale) morphisms. Remark 2.3.26 Again by construction, when X is a locally noetherian log adic space over a locally noetherian fs log adic space Y and admits a global chart modeled on a finitely generated (resp. fine) monoid P , for ? = int (resp. sat), we have X? ∼ = X ×Y #P $ Y #P ? $ as adic spaces, where Y #P $ and Y #P ? $ are as in Example 2.2.19. (Note that the fiber product X ×Y #P $ Y #P ? $ exists because the morphism Y #P ? $ → Y #P $ is lft when P is finitely generated.) Now, let us study fiber products in the category of locally noetherian coherent (resp. fine, resp. fs) log adic spaces: Proposition 2.3.27 (1) Finite fiber products exist in the category of locally noetherian log adic spaces when the corresponding fiber products of the underlying adic spaces exist. Moveover, finite fiber products of locally noetherian coherent log adic spaces over locally noetherian coherent log adic spaces are coherent (when defined). The forgetful functor from the category of locally noetherian log adic spaces to the category of locally noetherian adic spaces respects finite fiber products (when defined). (2) Finite fiber products exist in the category of locally noetherian fine (resp. fs) log adic spaces when the corresponding fiber products of the underlying adic spaces exist. Proof As for (1), let Y → X and Z → X be morphisms of locally noetherian log adic spaces such that the fiber product W := Y ×X Z of the underlying adic spaces is defined. Let prY , prZ , and prX denote the natural projections from W to Y , Z,

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and X, respectively, and equip W with the log structure associated with the pre-log structure pr−1 pr−1 Y (MY ) ⊕pr−1 Z (MZ ) → OWét . Then the log adic space thus X (M X ) obtained clearly satisfies the desired universal property. Suppose moreover that X, Y , and Z are all coherent. By Proposition 2.3.21, étale locally, Y → X and Z → X admit charts P → Q and P → R, respectively, where P , Q, and R are all finitely generated monoids, in which case W is (by construction) modeled on the finitely generated monoid S := Q ⊕P R, and hence is coherent. As for (2), let Y → X and Z → X be morphisms of locally noetherian fine (resp. fs) log adic spaces such that the fiber product Y ×X Z of the underlying adic spaces is defined, in which case we equip it with the structure of a coherent log int adic space as in (1). Then, by Proposition 2.3.23, Y ×fine X Z := (Y ×X Z) (resp. sat Y ×fs  X Z := (Y ×X Z) ) satisfies the desired universal property. Remark 2.3.28 Let P → Q and P → R be homomorphisms of finitely generated (resp. fine, resp. fs) monoids, and let S ? := (Q ⊕P R)? , where ? = ∅ (resp. int, resp. sat). Let Y be a locally noetherian fs log adic space. By Remark 2.3.2 and Proposition 2.3.27 (and the construction in its proof), Y #S ? $ is canonically isomorphic to the fiber product of Y #Q$ and Y #R$ over Y #P $ in the category of noetherian coherent (resp. fine, resp. fs) log adic spaces. Remark 2.3.29 Let P → Q and P → R be fine (resp. fs) charts of morphisms Y → X and Z → X, respectively, of locally noetherian fine (resp. fs) log adic fs spaces such that Y ×X Z is defined. Then Y ×fine X Z (resp. Y ×X Z) is modeled on int sat (Q ⊕P R) (resp. (Q ⊕P R) ). Remark 2.3.30 The forgetful functor from the category of locally noetherian fine (resp. fs) log adic spaces to the category of locally noetherian adic spaces does not respect fiber products (when defined), because the underlying adic spaces may change under the functor X → Xint (resp. X → Xsat ). Convention 2.3.31 From now on, all fiber products of locally noetherian fs log adic spaces are taken in the category of fs ones unless otherwise specified. For simplicity, we shall omit the superscript “fs” from “×”. We will need the following analogue of Nakayama’s Four Point Lemma [33, Proposition 2.2.2]: Proposition 2.3.32 Let f : Y → X and g : Z → X be two lft morphisms of locally noetherian fs log adic spaces, and assume that f is exact. Then, given any two points y ∈ Y and z ∈ Z that are mapped to the same point x ∈ X, there exists some point w ∈ W := Y ×X Z that is mapped to y ∈ Y and to z ∈ Z. In order to prove Proposition 2.3.32, it suffices to treat the case where X, Y , and Z are geometric points, and where x, y and z are the respective unique closed points. By [18, Lemma 1.1.10], it suffices to prove the following: Lemma 2.3.33 Let f : Y → X and g : Z → X be morphisms of fs log adic spaces such that the underlying adic spaces of X, Y , and Z are Spa(l, l + ) for the same

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complete separably closed nonarchimedean field l, and such that the underlying morphisms of adic spaces of f and g are the identity morphism. Assume that f is exact. Then W = Y ×X Z is nonempty. Proof Let x, y, and z be the unique closed points of X, Y , and Z, respectively, and let P = MX,x , Q = MY,y , and R = MZ,z . Let u : P → Q and v : P → R be the corresponding maps of monoids. By [36, Proposition I.4.2.1], u is exact. Consider the homomorphism φ : P gp → Qgp ⊕ R gp : a → ugp (a), −v gp (a) . Since u is exact and R is sharp, φ −1 (Q ⊕ R) is trivial. By [33, Lemma 2.2.6], the sharp monoid S := Q ⊕P R is quasi-integral (i.e., if a + b = a, then b = 0), and the natural homomorphism P → Q ⊕P R is injective. Note that f and g admit charts modeled on u : P → Q and v : P → R, respectively. This is because, by the proof of [33, Lemma 2.2.3], there exist compatible homomorphisms (MX (X))gp → l × , (MY (Y ))gp → l × , and (MZ (Z))gp → l × such that the compositions l × → (MX (X))gp → l × , l × → (MY (Y ))gp → l × , and l × → (MZ (Z))gp → l × are the identity homomorphisms. Therefore, the morphisms f ∗ (MX ) → MY and g ∗ (MX ) → MZ can be (noncanonically) identified with Id ⊕u : l × ⊕ P → l × ⊕ Q and Id ⊕v : l × ⊕ P → l × ⊕ R, respectively. Consequently, W ∼ = Spa(l, l + ) ×Spa(l#S$,l + #S$) Spa(l#S sat $, l + #S sat $). The image + of Spa(l, l ) → Spa(l#S$, l + #S$) consists of equivalence classes of valuations on l#S$ (bounded by 1 on l + ) whose support contains the ideal I of l#S$ generated by {ea : a ∈ S, a = 0}. On the other hand, the kernel of l#S$ → l#S sat $, which is generated by {ea − eb : a, b ∈ S, a = b in S int }, is contained in I because S is  quasi-integral. Thus, W is nonempty, as desired.

3 Log Smoothness and Log Differentials 3.1 Log Smooth Morphisms Definition 3.1.1 Let f : Y → X be a morphism between locally noetherian fs log adic spaces. We say that f is log smooth (resp. log étale) if, étale locally on Y and X, the morphism f admits an fs chart u : P → Q such that (1) the kernel and the torsion part of the cokernel (resp. the kernel and cokernel) of ugp : P gp → Qgp are finite groups of order invertible in OX ; and (2) f and u induce a morphism Y → X ×X#P $ X#Q$ of log adic spaces (cf. Remark 2.3.3) whose underlying morphism of adic spaces is étale. Remark 3.1.2 In Definition 3.1.1, the fiber product in (2) exists and the morphism f : Y → X is lft, because X#Q$ → X#P $ and hence the first projection X ×X#P $ X#Q$ → X is lft when Q is finitely generated. Hence, fiber products involving log smooth or log étale morphisms always exist.

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Proposition 3.1.3 Base changes of log smooth (resp. log étale) morphisms (by arbitrary morphisms between locally noetherian fs log adic spaces, which are justified by Remark 3.1.2) are still log smooth (resp. log étale). Proof Suppose that Y → X is a log smooth (resp. log étale) morphism of locally noetherian fs log adic spaces, with a chart P → Q satisfying the conditions in Definition 3.1.1. Let Z → X be any morphism of locally noetherian fs log adic spaces. By Proposition 2.3.22, up to étale localization, we may assume that Z → X admits an fs chart P → R. By Remark 2.3.29, Z ×X Y is modeled on S := (R ⊕P Q)sat . By Remark 2.3.26, Z ×X#P $ X#Q$ ∼ = Z ×Z#R$ Z#S$. By Remark 2.3.25, the morphism Z ×X Y → Z ×Z#R$ Z#S$ induces an étale morphism of underlying adic gp  spaces. It remains to note that R gp → (Q ⊕P R)sat satisfies the analogue of  Definition 3.1.1(1), by the assumption on P gp → Qgp and the fact that (Q ⊕P gp ∼ R)sat  = (Q ⊕P R)gp ∼ = Qgp ⊕P gp R gp . Proposition 3.1.4 Let f : Y → X be a log smooth (resp. log étale) morphism of locally noetherian fs log adic spaces. Suppose that X is modeled on a global fs chart P . Then, étale locally on Y and X, there exists an injective fs chart u : P → Q of f satisfying the conditions in Definition 3.1.1. Moreover, if P is torsion-free, we can choose Q to be torsion-free as well. Proof This is an analogue of the smooth and étale cases of [22, Lemma 3.1.6]. Suppose that, étale locally, f admits a chart P1 → Q1 satisfying the conditions in Definition 3.1.1. We may assume that X = Spa(R, R + ) is a noetherian affinoid log adic space. Let us begin with some preliminary reductions. Firstly, we may assume that PX → MX factors through (P1 )X → MX . Indeed, by Lemma 2.3.6, étale locally, X admits an fs chart P2 such that the canonically induced morphism (P ⊕ P1 )X → MX factors through (P2 )X → MX . Let Q2 be gp gp gp (P2 ⊕P1 Q1 )sat . Then Q2 ∼ = P2 ⊕P gp Q1 (cf. the proof of Proposition 3.1.3) and 1 hence P2 → Q2 is also an fs chart of f satisfying the conditions in Definition 3.1.1. gp gp Secondly, we may assume that P1 → Q1 is injective. Indeed, since P1 and Q1 gp gp are finitely generated, and since K := ker(P1 → Q1 ) is finite, there exists some finitely generated abelian group H1 fitting into a cartesian diagram P1gp

H1

P1gp /K

Qgp 1

such that gp gp gp coker(P1 → H1 ) ∼ = coker(P1 /K → Q1 ),

and so that

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gp gp gp K∼ = ker(P1  P1 /K) ∼ = ker(H1  Q1 ). gp

For any geometric point y of Y , let Q2 be the preimage of Q1 under H1 → Q1 . gp Note that Q2 is fs, Q2 = H1 , and P1 → Q2 is injective. We claim that P1 → Q2 is an fs chart of f , étale locally at y and f (y), satisfying the conditions in Definition 3.1.1. By Remark 2.3.4, Q1 → MY,y is surjective with kernel given by ∼ the preimage of O× Yét ,y . Since Q2 /K = Q1 , the induced homomorphism Q2 → MY,y satisfies the same properties, and P1 → Q2 is an fs chart of f , étale locally at y and f (y). By construction, it satisfies the condition (1) in Definition 3.1.1. It also satisfies the condition (2) in Definition 3.1.1, because Y → X ×X#P1 $ X#Q2 $ is the composition of Y → X ×X#P1 $ X#Q1 $ → Y → X ×X#P1 $ X#Q2 $, and X#Q1 $ → X#Q2 $ is étale, by [18, Proposition 1.7.1], as |K| is invertible in OX . Thus, the claim follows. Thirdly, we claim that, up to further modifying P1 → Q1 , we can find H fitting into a cartesian diagram of finitely generated abelian groups: P gp

H

P1gp

Qgp 1 .

(3.1.5) gp

Given such an H , let Q be the preimage of Q1 via H → Q1 . Since P and P1 gp are both fs charts of X, the homomorphism P gp → P1 induces an isomorphism after passing to quotients of the source and target by the preimages of O× Xét ,f (y) . Since (3.1.5) is Cartesian, and since Q1 is an fs chart of Y , the analogous statement for Q, Q1 , and O× Yét ,y is also true. Thus, u : P → Q is an injective fs chart of f , étale locally at y and f (y), satisfying the conditions of Definition 3.1.1. Let us verify the claim by modifying the arguments in the proofs of [22, Lemma 3.1.6] and [35, Lemma 2.8]. (We need to modify the arguments because + our requirement that charts induce morphisms to O+ Xét and OYét can be affected gp gp by localizations of monoids.) Let G := im(P gp → P1 ), G1 := P1 /G, and gp gp W := coker(P1 → Q1 ), and consider the pushout 0 → G1 → T1 → W → 0 gp gp gp of the extension 0 → P1 → Q1 → W → 0 via P1 → G. By assumption, there exists some integer n ≥ 1 invertible in OX which annihilates the torsion part gp gp of W . Since K1 := ker(P1 → MX,f (y) ) is finitely generated, there exists some finitely generated abelian group K2 such that K1 ∼ = nK2 = {nk : k ∈ K2 }. Let gp gp H2 denote the pushout of Q1 ← K1 → K2 , which contains Q1 as a finite index subgroup. Let P2 := {a ∈ H2 : na ∈ P1 } and Q2 := {b ∈ H2 : nb ∈ Q1 }, which are fs monoids because P1 and Q1 are. Note that P1 , P2 , Q1 , and Q2 are all gp submonoids of H2 . Let G2 := P2 /G, and let 0 → G2 → T2 → W → 0 be defined by pushout as before. Since n is invertible in OX , the induced homomorphism × K1 → O× Xét ,f (y) lifts to some homomorphism K2 → OXét ,f (y) . Hence, up to further étale localization, we may assume that P1 → Q1 lifts to an fs chart

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P2 → Q2 of f at y and f (y), which still satisfies the conditions in Definition 3.1.1. Given any torsion element w of W of order m (which necessarily divides n), let gp t1 ∈ T1 be any lifting of w. Then g1 := mt1 ∈ G1 = coker(P gp → P1 ). Since ∼ gp gp P gp → P1 /K1 → MX,f (y) is surjective (by Remark 2.3.4 again), g1 lifts to some k1 ∈ K1 , which is the m-th multiple of some k2 ∈ K2 with image g2 in G2 . Then t2 := t1 − g2 ∈ T2 is a lifting of w which satisfies mt2 = g1 − mg2 = 0. Hence, Zt2 ⊂ T2 defines a lifting of Zw ⊂ W . Since w is arbitrary, the homomorphism T2 → W of finitely generated abelian groups splits, and the preimage H of the split gp image of W in Q2 is an extension 0 → G → H → W → 0 whose pushout via gp gp gp G → P1 recovers 0 → P1 → Q1 → W → 0. Since H is finitely generated, there is some surjection F  H from a finitely generated free abelian group, and the preimage E of G is also finitely generated free and lifts to some E → P gp . Then the claim follows by taking H to be the pushout of P gp ← E → F . Finally, if P is torsion-free, let us show that we can take Q to be torsion-free as well. We learned the following argument from [34, Proposition A.2]. Consider the torsion submonoid Qtor of Q, which is necessarily contained in Qinv ; and choose any splitting s of π : Q  Q := Q/Qtor . Let n be any integer invertible in OX which annihilates the torsion in coker(ugp ). Since P is torsion-free, the composition u = π ◦u : P → Q is injective, and Qtor is also annihilated by n. Let S be the finite étale R-algebra obtained from R#Qtor $ by formally joining the n-th roots of ea , for all a ∈ Qtor ; and let S + be the integral closure of R + #Qtor $ in S. Then the morphism Z := Spa(S, S + ) → X#Qtor $ = Spa(R#Qtor $, R + #Qtor $) over X is finite étale and surjective, with base change Z#Q $ → X#Q$. Consider the composition v = s ◦ π ◦ u : P → Q. Then u−v : P → Q factors through P → Qtor , which extends to some × φ : Q → Stor ; and a → φ(a)a, for a ∈ Q , induces an isomorphism between the two compositions g, h : Z#Q $ → X#Q$ → X#P $ induced by u, v, respectively. Since u : P → Q is a chart of f , the induced morphism Y → X ×X#P $ X#Q$ is étale, whose pullback is an étale morphism Y ×X#Qtor $ Z → X ×X#P $,g Z#Q $. The target is isomorphic to X ×X#P $,h Z#Q $, and hence is étale over X ×X#P $ X#Q $, by the above explanation. Consequently, the morphism Y → X ×X#P $ X#Q $ induced by f and u : P → Q is étale, and so u is also an injective fs chart of f , as desired.  Proposition 3.1.6 Compositions of log smooth (resp. log étale) morphisms are still log smooth (resp. log étale). Proof This follows from Definition 3.1.1 and Proposition 3.1.4.



Proposition 3.1.7 If f : Y → X is log smooth (resp. log étale) and strict, then the underlying morphism of adic spaces is smooth (resp. étale). Proof Étale locally at geometric points y of Y and f (y) of X, by Propositions 2.3.13 and 3.1.4, we may assume that f : Y → X admits an injective fs chart u : P = MX,f (y) → Q as in Definition 3.1.1, where the torsion part Ktor of K := coker(ugp : P gp → Qgp ) is a finite group of order invertible in OX , and where K itself is finite when f is log étale. Since f is strict, by Remark 2.2.6,

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P = MX,f (y) ∼ = MY,y . Hence, we can identify K with ker(Qgp → MY,y ), so that ugp (P gp ) ∩ K = 0 in Qgp . Since K is a finitely generated abelian group, we have ∼  a decomposition Ktor ⊕ ⊕ri=1 Zai → K, for some elements ai ∈ K which are necessarily mapped to O× 1 ≤ i ≤ r, we Yét ,y . Up to replacing ai with −ai , for each  r + may assume that ai is mapped to OYét ,y . Let Q := u(P ) ⊕ Ktor ⊕ ⊕i=1 Z≥0 ai in gp Qgp . Then Qgp → MY,y maps Q to MY,y , the induced map Q → OYét ,y factors gp

through O+ Yét ,y , and the induced map Q → MY,y is surjective. In this case, up to further étale localization, u : P → Q is also an injective fs chart of f . Thus, it suffices to show that X#Q $ → X#P $ is smooth (resp. étale) at the image of y. Since X#Q $ ∼ = X#P $ ×X X#Ktor $ ×X X#Zr≥0 $ over X#P $, it remains to note that, by [18, Corollary 1.6.10 and Proposition 1.7.1], X#Ktor $ ×X X#Zr≥0 $ is smooth (resp. étale) over X, because Ktor is a finite groups of order invertible in OX , and because r = 0 when K itself is finite (i.e., when f is log étale). 

Definition 3.1.8 If f satisfies the condition in Proposition 3.1.7, we say that f is strictly smooth (resp. strictly étale), or simply smooth (resp. étale), when the context is clear. Definition 3.1.9 Let (k, k + ) be an affinoid field. A locally noetherian fs log adic space X is called log smooth over Spa(k, k + ) if there is a log smooth morphism X → Spa(k, k + ), where Spa(k, k + ) is endowed with the trivial log structure. When X is log smooth (resp. smooth) over Spa(k, Ok ), we simply say that X is log smooth (resp. smooth) over k. Local structures of log smooth log adic spaces can be described by toric charts, by the following proposition: Proposition 3.1.10 Let X be an fs log adic space log smooth over Spa(k, k + ), where (k, k + ) is an affinoid field. Then, étale locally on X, there exist a sharp fs monoid P and a strictly étale morphism X → Spa(k#P $, k + #P $) that is a composition of rational localizations and finite étale morphisms. Proof By Proposition 3.1.4, étale locally on X, there exists a torsion-free fs monoid Q and a strictly étale morphism X → Spa(k#Q$, k + #Q$). We may further assume that X → Spa(k#Q$, k + #Q$) is a composition of rational localizations and finite étale morphisms. By Lemma 2.1.10, Q → Q splits, and hence there is a ∼ ∼ decomposition Q → Q ⊕ Qinv → Q ⊕ Zr , for some r. Let P := Q ⊕ Zr≥0 , which is a sharp fs monoid. Since P → Q is a localization of monoids, as in Construction 2.1.11, Spa(k#Q$, k + #Q$) → Spa(k#P $, k + #P $) is a rational localization, whose pre-composition with X → Spa(k#Q$, k + #Q$) gives the  desired morphism. Corollary 3.1.11 Let X and (k, k + ) be as in Proposition 3.1.10. Suppose moreover that underlying adic space of X is smooth over Spa(k, k + ). Then, étale locally on X, there exists a strictly étale morphism X → Dn (see Example 2.2.21) that is a composition of rational localizations and finite étale morphisms.

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Proof As in the proof of Proposition 3.1.10, étale locally on X, there is a torsionfree fs monoid Q and a strictly étale morphism X → Y := Spa(k#Q$, k + #Q$) that is a composition of rational localizations and finite étale morphisms. Consider the canonical morphism Y → Z := Spec(k[Q]). Note that Z = Spec(k[Q]) admits a stratification by locally closed subschemes of the form ZF := Spec(k[F gp ]), where F are the faces of Q (see [36, Sections I.1.4 and I.3.4]), which is a closed subscheme of the open subscheme Z(F ) := Spec(k[QF ]) of Z, where QF denotes the localization of Q with respect to F (as in Construction 2.1.11). By [36, Section I.3.6], given any closed point z of ZF with residue field κ(z), the completion O∧ Z,z of the local ring OZ,z is isomorphic to κ(z)[[QF ]]. For each F , let YF := Y ×Z ZF and Y(F ) := Y ×Z Z(F ) . Then we have a canonical open immersion of adic spaces Y(F ) → Spa(k, k + ) ×Spec(k) Z(F ) , by comparing the construction of both sides using Lemma 2.2.11. (For the construction of such fiber products of adic spaces with schemes, see [17, Proposition 3.8 and its proof].) For any F such that YF meets the image of X → Y , we claim that QF ∼ = Zs≥0 ⊕ Zt , for some s and t. Assuming this claim, then Z(F ) admits an open immersion into (P1k )n , for n := s + t, and we obtain an open immersion Y(F ) → Spa(k, k + ) ×Spec(k) (P1k )n of log adic spaces, where the log structure on the target is defined (via fiber product) by naturally covering each factor Spa(k, k + ) ×Spec(k) P1k with two log adic spaces Spa(k#T $, k + #T $) and Spa(k#T −1 $, k + #T −1 $) isomorphic to D (see Example 2.2.21). Therefore, up to further localization on X, we may assume that X → Y extends to a strictly étale morphism X → Dn , which is still a composition of rational localizations and finite étale morphisms. Thus, the corollary follows from the claim. It remains to verify the claim. Since it only concerns monoids QF as above, we may base change to Spa(k, Ok ), and assume that k + = Ok , so that X is a smooth rigid analytic variety over k. For any F such that YF meets the image of the étale morphism X → Y , since X is smooth over k, and since the open an maps Y to Z an , we see immersion Y(F ) → Spa(k, k + ) ×Spec(k) Z(F ) ∼ = Z(F F F ) ∼ that Z is smooth over k at some closed point z of ZF , so that O∧ Z,z = κ(z)[[QF ]] (as explained above) is regular. Since the localization QF is torsionfree fs as Q is, ∼ ∼ t by decomposing QF → QF ⊕ Qinv F → QF ⊕ Z , for some t, as in the proof of ∼ Proposition 3.1.10, we obtain κ(z)[[QF ]] → κ(z)[[QF ]][[T1 , . . . , Tt ]]. Hence, the regularity of κ(z)[[QF ]] implies that of κ(z)[[QF ]]. Since QF is fine and sharp, by [36, Lemma I.1.11.7], we have QF ∼  = Zs≥0 , for some s, and the claim follows. Definition 3.1.12 A strictly étale morphism X → Spa(k#P $, k + #P $) as in Proposition 3.1.10 is called a toric chart. A strictly étale morphism X → Dn as in Corollary 3.1.11 is called a smooth toric chart. Example 3.1.13 Let X, D, and k be as in Example 2.3.17. We claim that, étale locally, X admits a smooth toric chart X → Dn , where n = dim(X). In order to see this, we may assume that, up étale localization, X is S × Dm as in Example 2.3.17, and that there is a morphism (of adic spaces with trivial log structures) S → ± ± Tn−m = Spa(k#T1± , . . . , Tn−m $, Ok #T1± , . . . , Tn−m $) that is a composition of finite

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étale morphisms and rational localizations. Then the composition of X = S×Dm → Tn−m × Dm → Dn−m × Dm ∼ = Dn is a desired smooth toric chart. In particular, X is log smooth over k.

3.2 Log Differentials In this subsection, we develop a theory of log differentials from scratch. We first introduce log structures and log differentials for Huber rings. Definition 3.2.1 (1) A pre-log Huber ring is a triple (A, M, α) consisting of a (not necessarily complete) Huber ring A, a monoid M, and a homomorphism α : M → A of multiplicative monoids called a pre-log structure. We sometimes denote a pre-log Huber ring just by (A, M), when the pre-log structure α is clear from the context. (2) A log Huber ring is a pre-log Huber ring (A, M, α) where A is complete and where the induced homomorphism α −1 (A× ) → A× is an isomorphism. In this case, α is called a log structure. (3) Given a pre-log Huber ring (A, M, α), let us still denote by α the composition α can.  where A  denotes the completion of A. Then we define of M → A → A,  a M,  the associated log Huber ring to be (A, α ), where a M is the pushout of × −1 ×  ) → M in the category of monoids, which is equipped with the  ← α (A A  In this case,  canonical homomorphism  α : a M → A. α is called the associated log structure. (4) A homomorphism f : (A, M, α) → (B, N, β) of pre-log Huber rings consists of a continuous homomorphism f : A → B of Huber rings and a homomorphism of monoids f% : M → N such that β ◦ f% = f ◦ α. In this case, we have a canonically induced morphism (B, M, β ◦ f% ) → (B, N, β) of prelog Huber rings, and we say that f is strict if the associated morphism of log Huber rings is an isomorphism. In general, any homomorphism f : (A, M) →  (B, N ) of log Huber rings factors as (A, M) → B, f∗ (M) → (B, N ), where f∗ (M) → B is associated with β ◦ f% : M → B as above. Definition 3.2.2 Let f : (A, M, α) → (B, N, β) be a homomorphism of prelog Huber rings. Given any complete topological B-module L, a derivation from (B, N, β) to L over (A, M, α) (or an (A, M, α)-derivation of (B, N, β) to L) consists of a continuous A-linear derivation  d : B →  L and a homomorphism of monoids δ : N → L such that δ f% (m) = 0 and d β(n) = β(n) δ(n), for all m ∈ M and n ∈ N. We denote the set of all (A, M, α)-derivations from (B, N, β) log to L by DerA (B, L). It has a natural B-module structure induced by that of L. If log M = α −1 (A× ) and N = β −1 (B × ), then DerA (B, L) is simply DerA (B, L), the usual B-module of continuous A-derivations from B to L, and we shall omit the superscript “log” from the notation.

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Remark 3.2.3 In Definition 3.2.2, (d, δ) naturally extends to a log derivation on log (B, a N, β), and the B-module DerA (B, L) remains unchanged if we replace a  N, β ). In addition, δ naturally extends to a group homomor(B, N, β) with (B, phism δ gp : (a N )gp → L. Definition 3.2.4 A homomorphism f : (A, M, α) → (B, N, β) of pre-log Huber rings is called topologically of finite type (or tft for short) if A and B are complete, f : A → B is topologically of finite type (as in [17, Section 3]), and  N gp / (f% (M))gp β −1 (B × ) is a finitely generated abelian group. Now, let f : (A, M, α) → (B, N, β) be tft, as in Definition 3.2.4. Consider the A B)[N] over B ⊗ A B associated with the monoid N, and for monoid algebra (B ⊗ each n ∈ N, its element en corresponding to n (by our convention). Let I be its ideal generated by {ef% (m) − 1}m∈M and {(β(n) ⊗ 1) − (1 ⊗ β(n)) en }n∈N . Note that, −1 × n −1 A B)[N] /I . Let J be the kernel if n ∈ β (B ), then e = β(n) ⊗ β(n) in (B ⊗ of the homomorphism  A B)[N] /I → B Δlog : (B ⊗

(3.2.5)

sending b1 ⊗ b2 to b1 b2 and all en to 1. We set log

ΩB/A := J /J 2 , log

(3.2.6) log

and define dB/A : B → ΩB/A and δB/A : N → ΩB/A by setting dB/A (b) = (b ⊗ 1) − (1 ⊗ b) and δB/A (n) = en − 1. A short computation shows that dB/A is an A-linear derivation, and that δB/A is a homomorphism of monoids satisfying the required properties in Definition 3.2.2. As observed in Remark 3.2.3, δB/A naturally extends to a group homomorphism gp log gp  gp log δB/A : N gp → ΩB/A such that δB/A f% (M gp ) = 0. Then ΩB/A is generated as gp A B → B) and {δB/A (n)}, where n runs through a set of a B-module by ker(B ⊗  representatives of generators of N gp / (f% (M))gp β −1 (B × ) . More precisely,   log ΩB/A ∼ = ΩB/A ⊕ (B ⊗Z N gp ) R,

(3.2.7)

where ΩB/A is the usual B-module of continuous differentials (see [18, Definition 1.6.1 and (1.6.2)]), and where R is the B-module generated by {(dβ(n), −β(n) ⊗ n) : n ∈ N} ∪ {(0, 1 ⊗ f% (m)) : m ∈ M}.

(3.2.8)

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95 log

In particular, ΩB/A is a finite B-module. Therefore, ΩB/A is complete with respect to its natural B-module topology, and dB/A is continuous. log

Proposition 3.2.9 Under the above assumption, (ΩB/A , dB/A , δB/A ) is a universal object among all (A, M, α)-derivations of (B, N, β). Proof Let (d, δ) be a derivation from (B, N, β) to some complete topological Bmodule L over (A, M, α). We turn the B-module B ⊕ L into a complete topological B-algebra, which we denote by B ∗ L, with the multiplicative structure defined by (b1 , x1 ) (b2 , x2 ) = (b1 b2 , b1 x2 +b2 x1 ). Note that the A-linear derivation d gives rise A B → B ∗ L sending to a continuous homomorphism of topological B-algebras B ⊗  A B)[N] → B ∗L b1 ⊗b2 to b1 b2 , b1 d(b2 ) . This extends to a homomorphism (B ⊗ by sending en to 1, δ(n) , for each n ∈ N. By the conditions in Definition 3.2.2, A B)[N] /I , inducing a homomorphism factors through (B ⊗ this homomorphism A B)[N ] /I → B ∗L which we denote by ϕ. By construction, the composition (B ⊗ of ϕ with the natural projection B ∗ L → B recovers (3.2.5). Therefore, ϕ induces log a continuous morphism of B-modules ϕ : ΩB/A = J /J 2 → L. Now, a careful chasing of definitions verifies that ϕ ◦ dB/A = d and ϕ ◦ δB/A = δ, as desired.  Given any complete topological B-module L, there is a natural forgetful functor log DerA (B, L) → DerA (B, L) defined by (d, δ) → d. The following lemma is obvious: Lemma 3.2.10 If f : (A, M, α) → (B, N, β) is a strict homomorphism of log log Huber rings, then the canonical morphism DerA (B, L) → DerA (B, L) is an isomorphism, for every complete topological B-module L. Consequently, the log canonical morphism ΩB/A → ΩB/A is an isomorphism. Definition 3.2.11 A homomorphism (D, T , μ) → (D , T , μ ) of log Huber rings is called a log thickening of first order if it satisfies the following conditions: (1) The underlying homomorphism D → D of Huber rings is surjective, whose kernel H is a closed ideal satisfying H 2 = 0. (2) The subgroup 1 + H of D × ∼ = T inv (via the log structure μ) acts freely on T , ∼ and induces an isomorphism T /(1 + H ) → T of monoids. Remark 3.2.12 The condition (2) in Definition 3.2.11 is automatic when T inv acts freely on T ; or, equivalently, when T inv → T gp is injective. (In this case, T is uintegral, as in [36, Definition I.1.3.1].) In particular, the condition (2) is satisfied when T is integral (i.e., T → T gp is injective).

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Consider the following commutative diagram (A, M, α) f

(B, N, β)

(D, T, μ) g g

(D , T , μ )

(3.2.13)

of solid arrows given by homomorphisms of log Huber rings, in which the arrow (D, T , μ) → (D , T , μ ) is a log thickening of first order as in Definition 3.2.11. Definition 3.2.14 A homomorphism f : (A, M, α) → (B, N, β) of log Huber rings is called formally log smooth (resp. formally log unramified, resp. formally log étale) if, for any diagram as in (3.2.13), there exists at least one (resp. at most one, resp. exactly one) lifting  g : (B, N, β) → (D, T , μ) of g, as the dotted arrow in (3.2.13), making the whole diagram commute. If M = α −1 (A× ) and N = β −1 (B × ), then we simply say that f : A → B (the underlying ring homomorphism of Huber rings) is formally smooth (resp. formally unramified, resp. formally étale) (cf. the formal lifting conditions in [18, Definition 1.6.5]). Remark 3.2.15 Let k be a nontrivial nonarchimedean field. By [18, Proposition 1.7.11], a tft homomorphism f : A → B of Tate k-algebras is formally smooth (resp. formally unramified, resp. formally étale) if and only if the induced morphism Spa(B, B ◦ ) → Spa(A, A◦ ) is smooth (resp. unramified, resp. étale) in the sense of classical rigid analytic geometry. Remark 3.2.16 It follows easily from the definition that we have the following: (1) Formally log smooth (resp. formally log unramified, resp. formally log étale) homomorphisms are stable under compositions and completed base changes. (2) If a homomorphism f : (A, M, α) → (B, N, β) is formally log étale, then a homomorphism g : (B, N, β) → (C, O, γ ) is formally log smooth (resp. formally log étale) if and only if g ◦ f : (A, M, α) → (C, O, γ ) is. Lemma 3.2.17 If the homomorphism f : (A, M, α) → (B, N, β) is a strict homomorphism of log Huber rings, then it is formally log smooth (resp. formally log unramified, resp. formally log étale) if the underlying homomorphism of Huber rings f : A → B is formally smooth (resp. formally unramified, resp. formally étale). If A× ∼ = M inv → M gp is injective, then the converse is true. Proof Suppose we are given any diagram as in (3.2.13), with the top horizontal row denoted by h : (A, M, α) → (D, T , μ) in this proof. Since the homomorphism f : (A, M, α) → (B, N, β) is strict, each n ∈ N is of the form n = b + f% (m) for some b ∈ B × and m ∈ M, where m is uniquely determined by n up to an element of A× . Hence, any homomorphism  g : B → D of Huber rings lifting g : B → D uniquely extends to a homomorphism  g : (B, N, β) → (D, T , μ) of log Huber rings lifting g : (B, N, β) → (D , T , μ ), by setting  g% (n) =  g (b) + h% (m), for

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each n = b + f% (m) ∈ N as above. Hence, the formal lifting properties without log structures imply those with log structures. Conversely, suppose we are given a diagram as in (3.2.13), but without any log structures. Nevertheless, we can define μ : T → D and μ : T → D to be the log f

α

g

β

structures associated with M → A → D and N → B → D , respectively. Since A× ∼ = M inv → M gp is injective, A× acts freely on M. Therefore, by choosing any set-theoretic section of M → M, we obtain a bijection A× × M → M compatible with the actions of A× (on A× and M), which induces a bijection D × × M → T compatible with the actions of D × (on D × and T ). As a result, 1 + H ⊂ D × ∼ = T inv (via μ) acts freely on T , and (D, T , μ) → (D , T , μ ) is a log thickening of first order, as in Definition 3.2.11. Thus, we obtain a full diagram as in (3.2.13), with log structures, and the formal lifting properties with log structures imply those without, as desired.  We have the first fundamental exact sequence for log differentials, as follows: Theorem 3.2.18 f

g

(1) A composition (A, M, α) → (B, N, β) → (C, O, γ ) of tft homomorphisms of log Huber rings leads to an exact sequence log

log

log

C ⊗B ΩB/A → ΩC/A → ΩC/B → 0 of finite topological C-modules (cf. [18, Proposition where the first  1.6.3]), map sends c ⊗ dB/A (b) and c ⊗ δB/A (n) to c dC/A g(b) and c δC/A g% (n) , respectively, and the second map sends dC/A (c) and δC/A (l) to dC/B (c) and δC/B (l), respectively. (2) If the homomorphism g : (B, N, α) → (C, O, γ ) is formally log smooth, then log log C ⊗B ΩB/A → ΩC/A is injective, and the short exact sequence log

log

log

0 → C ⊗B ΩB/A → ΩC/A → ΩC/B → 0 is split in the category of topological C-modules. log (3) If g is formally log unramified, then ΩC/B = 0. log log (4) If g is formally log étale, then ΩC/A ∼ = C ⊗B ΩB/A . (5) If g ◦ f is formally log smooth, then the converses of (2), (3), and (4) hold. log

log

Proof Since the homomorphisms of log Huber rings are all tft, C ⊗B ΩB/A , ΩC/A , log

and ΩC/B are finite C-modules. Thus, to prove the exactness in (1), it suffices to show that, for any complete topological C-module H , the induced sequence log

log

log

0 → HomC (ΩC/B , H ) → HomC (ΩC/A , H ) → HomC (C ⊗B ΩB/A , H ) (3.2.19)

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is exact. By Proposition 3.2.9, this sequence is nothing but log

log

log

0 → DerB (C, H ) → DerA (C, H ) → DerA (B, H ), whose exactness is obvious. Therefore, (1) follows. In the rest of the proof, let H be a complete topological C-module, and let (d, δ) : (B, N, β) → H be an (A, M, α)-derivation. Let C ∗ H be the C-algebra defined as in the proof of Proposition 3.2.9, equipped with the log structure (γ ∗Id) : O ⊕H → C ∗ H : (a, b) → (γ (a), γ (a) b), and denote by (C, O, γ ) ∗ H the log Huber ring thus obtained. Note that (C, O, γ ) ∗ H → (C, O, γ ) is a log thickening of first order, as in Definition 3.2.11, because the action of 1 + H on O ⊕ H is free. We claim that there is a natural bijection between the set of extensions of (d, δ)  to (A, M, α)-derivations (d, δ ) : (C, O, γ ) → H and the set of homomorphisms of log Huber rings h : (C, O, γ ) → (C, O, γ ) ∗ H making the diagram (B, N, β) g

(C, O, γ) ∗ H h Id

(C, O, γ)

(C, O, γ)

(3.2.20)

commute. Here the upper horizontal map is a homomorphism of log Huber rings sending (b, n) to (g(b), d(b)), (g% (n), δ(n)) , and the right vertical one is the natural projection. To justify the claim, for each map h : (C, O) → (C ∗H, O ⊕H )  (Id,  lifting the projection (C ∗ H, O ⊕ H ) → (C, O), let us write h = (Id, d), δ) . Then a short computation shows that h is a homomorphism of log Huber rings δ is a homomorphism of monoids such that if and only if d is a derivation and   (x)) = γ (x)  d(γ δ (x) for all x ∈ O, and the claim follows. Thus, if (C, O, γ ) is formally log unramified over (B, N, β), then the natural log log map DerA (C, H ) → DerA (B, H ) is surjective for each finite C-module H . (Since log log log C ⊗B ΩB/A , ΩC/A , and ΩC/B are finite C-modules, it suffices to consider finite Clog

modules H in this paragraph.) In other words, the natural map HomC (ΩC/A , H ) → log

log

log

HomC (C ⊗B ΩB/A , H ) is surjective, and therefore C ⊗B ΩB/A → ΩC/A is injective, yielding (3). Similarly, if (C, O, γ ) is formally log smooth over (B, N, β), log log then HomC (ΩC/A , H ) → HomC (C ⊗B ΩB/A , H ) is surjective, and we can log

justify (2) by taking H = C ⊗B ΩB/A , which shows that the natural map log

log

HomC (ΩC/B , H ) → HomC (ΩC/A , H ) is injective and admits a left inverse splitting (3.2.19). By combining (2) and (3), we obtain (4). Finally, let us prove (5). Suppose we are given a commutative diagram u

(B, N, β) g

(C, O, γ)

v

(D, T, μ) i

v

(D , T , μ )

(3.2.21)

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of solid arrows, with i a log thickening of first order. If g ◦ f is formally log smooth, then there exists some lifting w : (C, O, γ ) → (D, T , μ) of v making the diagram u◦f

(A, M, α)

(D, T, μ)

w

g◦f

(C, O, γ)

i v

(D , T , μ )

commute. Note that this w might not satisfy u = g ◦ w, but u − g ◦ w defines a derivation (d, δ) : (B, N, β) → H = ker(D → D ). Hence, we obtain a homomorphism  g : (B, N, β) → (C, O, γ ) ∗ H extending g : (B, N, β) → (C, O, γ ), as before. Since H ⊂ D by definition, w canonically extends to a homomorphism w  : (C, O, γ ) ∗ H → (D, T , μ) sending H canonically into D. By combining these, we can extend (3.2.21) to a commutative diagram (A, M, α)

u◦f

f

u

(B, N, β) g

(C, O, γ)

g

(C, O, γ) ∗ H

h

w v

Id

(C, O, γ)

(D, T, μ) i

v

(D , T , μ )

(3.2.22)

of solid arrows, and any h : (C, O, γ ) → (C, O, γ ) ∗ H making the diagram (3.2.22) commute canonically induces  v := w  ◦ h : (C, O, γ ) → (D, T , μ) making the diagrams (3.2.21) and (3.2.22) commute. Moreover, h is uniquely determined by  v=w  ◦ h, because if h is another such map such that  v=w ◦h = w  ◦ h , then w  ◦ (h − h ) = 0, but (h − h )(C) ⊂ H and w |H : H → D is (by definition) the canonical injection. Thus, if the conclusion in (2) (resp. (3), log log resp. (4)) holds, then C ⊗B ΩB/A → ΩC/A is injective and splits (resp. is surjective, log

log

resp. is bijective), and so HomC (ΩC/A , H ) → HomC (C ⊗B ΩB/A , H ) is surjective (resp. injective, resp. bijective). By the first three paragraphs of this proof, and by the relation between h and  v explained above, there exists at least one (resp. at most one, resp. exactly one)  v making the diagrams (3.2.21) and (3.2.22) commute. Since (3.2.21) is arbitrary, g is formally log smooth (resp. formally log unramified, resp. formally log étale), as desired.  Lemma 3.2.23 In order to verify that a tft homomorphism of log Huber pairs f : (A, M, α) → (B, N, β) is formally smooth (resp. formally unramified, resp. formally étale), it suffices to verify the corresponding lifting condition in Definition 3.2.14 only for all commutative diagrams (3.2.13) in which the underlying homomorphism of Huber rings A → D is tft and in which H = ker(D → D ) is a finite D -module.

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Proof Given any diagram (3.2.13), since f : (A, M, α) → (B, N, β) is tft, the homomorphism B → D factors through a complete topological B-subalgebra D˘ of D that is tft over A. By lifting the topological generators of D over A, there exists some complete topological A-subalgebra D˘ of D such that the composition of the homomorphisms D˘ → D → D factors through a surjection D˘ → D˘ such that H˘ := ker(D˘ → D˘ ) ⊂ H is a finite D˘ -module. Let μ˘ : T˘ → D˘ and μ˘ : T˘ → D˘ denote the pullbacks of μ : T → D and μ : T → D , respectively. Note that H˘ 2 = 0 because H 2 = 0, and 1 + H˘ acts freely on T˘ because 1 + H acts freely on T . Then (3.2.13) extends to a commutative diagram (A, M, α) f

(B, N, β)

˘ T˘, μ (D, ˘)

(D, T, μ)

˘ , T˘ , μ (D ˘)

(D , T , μ )

g ˘ g ˘

g

˘ T˘ , μ) of solid arrows, in which (D, ˘ → (D˘ , T˘ , μ˘ ) is a log thickening of first order.  Since any dotted arrow g˘ lifting g˘ in the above diagram induces a dotted arrow  g lifting g in (3.2.13), we obtain the formally log smooth case of this lemma. It remains to establish the formally log unramified case of this lemma. Given any liftings  g and  g of g in (3.2.13), their difference  g − g defines a derivation (d, δ) : (B, N, β) → H = ker(D → D ) over A, which corresponds to a morphism log log ΩB/A → H of B-modules, by Proposition 3.2.9. Since f is tft, ΩB/A is a finite Bmodule. Thus, in order to show that f is formally log unramified, it suffices to show log that, when H is a finite B-module, all morphisms ΩB/A → H as above are zero. As in the proof of Theorem 3.2.18, this can be verified using only diagrams (3.2.13) in which (D, T , μ) → (D , T , μ ) is (B, N, β) ∗ H → (B, N, β), where the underlying homomorphism A → B is tft and H is a finite B-module.  Definition 3.2.24 Let u : P → Q be a homomorphism of fine monoids, and let R be a Huber ring. Then we have the pre-log Huber ring P → R#P $ : a → ea (resp. Q → R#Q$ : a → ea ), with the topology given in Lemma 2.2.11. In this case, we say that R#P $ is a pre-log Huber R-algebra. By abuse of notation, we shall still denote by R#P $ (resp. R#Q$) the log Huber R-algebras thus obtained. Proposition 3.2.25 Let u : P → Q and R be as in Definition 3.2.24. If the kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of ugp : P gp → Qgp are finite groups of orders invertible in R, then R#Q$ is formally log smooth (resp. formally log étale) over R#P $. In this case, the map δ : Qgp → log ΩR#Q$/R#P $ induces an isomorphism of finite free R#Q$-modules  log ΩR#Q$/R#P $ ∼ = R#Q$ ⊗Z Qgp /ugp (P gp ) .

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Proof Consider the commutative diagram RP

(D, T, μ)

RQ

(D , T , μ )

(3.2.26)

as in (3.2.13), in which (D, T , μ) → (D , T , μ ) is a log thickening of first order as in Definition 3.2.11. This gives rise to a commutative diagram of monoids P

T

Q

T ,

(3.2.27)

which in turn induces a commutative diagram of abelian groups P gp

T gp

Qgp

(T )gp .

(3.2.28)

Note that there is a natural bijection between the set of homomorphisms of log Huber R-algebras R#Q$ → (D, T , μ) extending (3.2.26) and the set of homomorphisms of monoids Q → T extending (3.2.27). By using the cartesian diagram T

T gp

T

(T )gp ,

and the fact that P and Q are fine monoids, we see that there is also a bijection between the set of desired homomorphisms R#Q$ → (D, T , μ) and the set of group homomorphisms Qgp → T gp extending (3.2.28). Since (D, T , μ) → (D , T , μ ) is a log thickening of first order as in Definition 3.2.11, we have ker(T gp → (T )gp ) = μ−1 (1 + H ) ∼ = H . Let G = Qgp /ugp (P gp ). Since the kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of ugp are finite groups of orders invertible in R, the set of desired homomorphisms Qgp → T gp is a torsor under Hom(G, H ) ∼ = Hom(G/Gtor , H ), where Gtor is the torsion subgroup of G. This proves the first statement of the proposition.

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On the other hand, for any finite R#Q$-module L, by the same argument as in the log proof of Theorem 3.2.18, there is a bijection between DerR#P $ (R#Q$, L) and the set of h : R#Q$ → R#Q$ ∗ L extending the following commutative diagram RP

RQ

L

h

RQ

Id

RQ.

log log Then HomR#Q$ (ΩR#Q$/R#P $ , L) ∼ = DerR#P $ (R#Q$, L) ∼ = Hom(G/Gtor , L), by the previous paragraph. The second statement of the proposition follows. 

Corollary 3.2.29 Let u : P → Q and R as in Definition 3.2.24 such that the kernel and the torsion part of the cokernel of ugp : P gp → Qgp are finite groups of orders invertible in R. Let Q be any fine monoid, and let S denote the log Huber ring  := Q ⊕ Q → R#Q$ (mapping Q − {0} associated with the pre-log Huber ring Q  → S of log Huber rings is strict. to 0), so that the surjective homomorphism R#Q$ log gp  Then the map δ : Q → ΩS/R#P $ induces an isomorphism  gp gp gp log  /u (P ) . ΩS/R#P $ ∼ = R#Q$ ⊗Z Q If, in addition, the torsion part of (Q )gp is a finite group whose order is invertible in R, then we also have log log ΩS/R#P $ ∼ . = R#Q$ ⊗R#Q$  ΩR#Q$/R#P  $ log

log

Proof By comparing the definitions of ΩS/R#P $ and ΩR#Q$/R#P $ as in (3.2.6), we log log  → S maps Q − {0} obtain ΩS/R#P $ ∼ = ΩR#Q$/R#P $ ⊕ (R#Q$ ⊗Z (Q )gp ), because Q to zero. Since this isomorphism is compatible with the canonical maps Qgp → log log gp → Ω log gp denoted by δ, we can finish ΩR#Q$/R#P $ , Q  S/R#P $ , and Q → ΩR#Q$/R#P $   the proof by applying Proposition 3.2.25 to P → Q and P → Q.

3.3 Sheaves of Log Differentials Our next step is to define sheaves of log differentials for locally noetherian coherent log adic spaces, and show that their formation is compatible with fiber products in the category of locally noetherian coherent, fine, and fs log adic spaces. Then we shall globalize several results in Sect. 3.2 and relate them to the definitions we made in Sect. 3.1.

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Definition 3.3.1 Let f : (Y, MY , αY ) → (X, MX , αX ) be a morphism of locally noetherian log adic spaces, and let F be a sheaf of complete topological OYét modules. By a derivation of (Y, MY , αY ) over (X, MX , αX ) valued in F, we mean derivation a pair (d, δ), where d : OYét → F is a continuous OXét -linear  −1 and δ : M Y → F is a morphism of sheaves of monoids such that δ f (M ) =0 X and d αY (m) = αY (m) δ(m), for all sections m of MY . Construction 3.3.2 Let f : (Y, MY , αY ) → (X, MX , αX ) be a morphism of noetherian coherent log adic spaces, where X = Spa(A, A+ ) and Y = Spa(B, B + ) are affinoid. Suppose that f induces a tft homomorphism of log Huber rings (A, M, α) → (B, N, β), where M := MX (X) and N := MY (Y ), and where log α := αX (X) : M → A and β := αY (Y ) : N → B. Let ΩY /X denote the coherent sheaf on Yét associated with ΩB/A (see (3.2.6)). For each Spa(C, C + ) ∈ Yét , by log

log

Theorem 3.2.18, the log differential ΩC/A for (A, M, α) → (C, N, (B → C) ◦ β) log log  is naturally isomorphic to C ⊗B ΩB/A = ΩY /X Spa(C, C + ) ; and the maps log

log

dC/A : C → ΩC/A and δC/A : N → ΩC/A naturally and compatibly extend log

to a continuous OXét -linear derivation dY /X : OYét → ΩY /X and a morphism  log δY /X : NY → ΩY /X of sheaves of monoids satisfying δY /X f −1 (MX ) = 0 and  dY /X αY (n) = αY (n) δY /X (n), for sections n of NY over objects of Yét . We may log further extend δY /X to a morphism δY /X : MY → ΩY /X of sheaves of monoids  −1  satisfying δY /X f (MX ) = 0 and dY /X αY (m) = αY (m) δY /X (m), for sections m of MY over objects of Yét . log

Lemma 3.3.3 In Construction 3.3.2, the triple (ΩY /X , dY /X , δY /X ) is a universal object among all derivations of (Y, MY , αY ) over (X, MX , αX ). Moreover, for any affinoid objects V ∈ Yét and U ∈ Xét fitting into a commutative diagram V

Y

U

X,

log log we have a canonical isomorphism (ΩY /X , dY /X , δY /X )|V ∼ = (ΩV /U , dV /U , δV /U ).

Proof Let (d, δ) be a derivation of (Y, MY , αY ) over (X, MX , αX ) valued in some complete topological OYét -module F. At each Spa(C, C + ) ∈ Yét , the evaluation of the derivation (d, δ) defines a derivation   C, MY (Spa(C, C + )), αY (Spa(C, C + )) → F Spa(C, C + )  over A, MX (X), αX (X) , which restricts to a derivation

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 C, N, (B → C) ◦ β → F Spa(C, C + )

over (A, M, α). By the universal property of log differentials, it factors through a  log continuous C-linear morphism ΩC/A → F Spa(C, C + ) . Moreover, we deduce from the universal property of log differentials a commutative diagram log ΩC 2 /A

Spa(C2 , C2+ )

log ΩC 1 /A

Spa(C1 , C1+ ) ,

for any morphism Spa(C1 , C1+ ) → Spa(C2 , C2+ ) in Yét . As a result, the morphisms log ΩC/A → F(Spa(C, C + )), for Spa(C, C + ) ∈ Yét , are compatible with each other log

and define a continuous OXét -linear morphism ΩY /X → F, whose compositions with dY /X and δY /X are equal to d and δ, respectively. This proves the first assertion of the lemma. The second then follows from Theorem 3.2.18.  Construction 3.3.4 Given any lft morphism f : Y → X of noetherian coherent log adic spaces, by Proposition 2.3.21, there exist a finite index set I and étale coverings {Xi → X}i∈I and {Yi → Y }i∈I , respectively, by affinoid log adic spaces such that f induces a morphism Yi → Xi which fits into the setting of Construction 3.3.2, for log each i ∈ I . By Lemma 3.3.3, the pullbacks of the triples (ΩYi /Xi , dYi /Xi , δYi /Xi ) are canonically isomorphic to each other over the fiber products of Yi over Y . Thus, by log Proposition A.10, we obtain a triple (ΩY /X , dY /X , δY /X ) on Yét , where (dY /X , δY /X ) log

gives a derivation of Y over X valued in ΩY /X . By Lemma 3.3.3, we immediately obtain the following: log

Lemma 3.3.5 In Construction 3.3.4, (ΩY /X , dY /X , δY /X ) is a universal object log

among all derivations of Y over X. As a result, (ΩY /X , dY /X , δY /X ) is well defined, i.e., independent of the choice of étale coverings; and its definition extends to all lft morphisms f : Y → X of locally noetherian coherent log adic spaces. log

Definition 3.3.6 We call the ΩY /X in Lemma 3.3.5 the sheaf of log differentials of f and (dY /X , δY /X ) the associated universal log derivations. By abuse of notation, log we shall denote the pushforward of ΩY /X to Yan by the same symbols. (When there is any risk of confusion, we shall denote the sheaves of log differentials on Yét and log log Yan more precisely by ΩY /X,ét and ΩY /X,an , respectively.) If X = Spa(k, Ok ), for log

log

simplicity, we shall write ΩY instead of ΩY /X .

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Proposition 3.3.7 Let Y

X

f

Y

X

be a cartesian diagram in the category of locally noetherian coherent (resp. fine, log log resp. fs) log adic spaces in which Y → X is lft. Then f ∗ (ΩY /X ) ∼ = ΩY /X . Proof By the étale local construction of sheaves of log differentials, we may assume that Y = Spa(B, B + ), X = Spa(A, A+ ), X = Spa(A , A + ), and Y = Spa(B , B + ) are affinoid, and that Y → X and X → X admit charts P → Q and P → P , respectively, given by finitely generated (resp. fine, resp. fs) monoids. Let Q := Q ⊕P P . By the proofs of Propositions 2.3.23 and 2.3.27, B is isomorphic A A (resp. (B ⊗ A A )⊗ Z[Q ] Z[(Q )int ], resp. (B ⊗ A A )⊗ Z[Q ] Z[(Q )sat ]), and to B ⊗ int sat Y is modeled on Q (resp. (Q ) , resp. (Q ) ). We need to show that log log ΩB /A ∼ = ΩB/A ⊗B B .

Since we have log log HomB (ΩB /A , L) ∼ = DerA (B , L)

and log log log HomB (ΩB/A ⊗B B , L) ∼ = HomB (ΩB/A , L) ⊗B B ∼ = DerA (B, L) ⊗B B ,

for each complete topological B -module L, it suffices to show that DerA (B , L) ∼ = DerA (B, L) ⊗B B . log

log

In the case of coherent log adic spaces, by Remark 3.2.3, this follows from essentially the same argument as in the proof of [36, Proposition IV.1.1.3] (in the log scheme case). Since (Q )gp ∼ = ((Q )int )gp ∼ = ((Q )sat )gp , by essentially the same argument as in the proof of [36, Proposition IV.1.1.9], we also have log log log ∼ A A , L) ⊗B ⊗ DerA (B , L) ∼ = DerA (B ⊗ A A B = DerA (B, L) ⊗B B ,

yielding the desired isomorphism in the cases of fine and fs log adic spaces. Z



→ Z of log adic spaces Definition 3.3.8 (cf. Definition 3.2.11) A morphism i : is called a log thickening of first order if it satisfies the following conditions:

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(1) It is a strict closed immersion (see Definition 2.2.23 and Example 2.2.24) defined by an OZ -ideal I satisfying I2 = 0. (2) The subsheaf 1 + Iét of O× Zét , where Iét denotes the natural pullback of I to Zét as a coherent ideal, acts freely on MZ , and induces an isomorphism  ∼ . i −1 MZ /(1 + Iét ) → MZ (of sheaves of monoids) over Zét Remark 3.3.9 (cf. Remark 3.2.12) The condition (2) in Definition 3.3.8 is automatic when O× Zét acts freely on MZ ; or, equivalently, when the canonically induced morphism αZ−1 (O× Zét ) → MZ is injective. Hence, the condition is satisfied when Z is integral. gp

Definition 3.3.10 (cf. Definition 3.2.14) A morphism f : Y → X of log adic spaces is called formally log smooth (resp. formally unramified, resp. formally log étale) if, for each commutative diagram Z i

Z

Y g

f g

X

(3.3.11)

of solid arrows in which i is a log thickening of first order as in Definition 3.3.8, there exists, up to (strictly) étale localization on Z, at least one (resp. at most one, resp. exactly one) lifting  g : Z → Y of g, as the dotted arrow in (3.2.13), making the whole diagram commute. Remark 3.3.12 It follows easily from the definition that we have the following: (1) Formally log smooth (resp. formally log unramified, resp. formally log étale) morphisms are stable under compositions and base changes (when defined). (2) If a morphism g : X → S of log adic spaces is formally log étale, then a morphism f : Y → X of log adic spaces is formally log smooth (resp. formally log étale) if and only if g ◦ f : Y → S is. Remark 3.3.13 By Lemmas 3.2.17 and 3.2.23, and by [18, Proposition 1.7.1], in order to show that an lft morphism f : Y → X of locally noetherian fine log adic spaces is formally log smooth (resp. formally log étale), it suffices to take locally finite (strictly) étale coverings {Xi → X}i∈I and {Yi → Y }i∈I by affinoid log adic spaces such that f induces lft morphisms fi : Yi → Xi , and verify for each such fi the corresponding formal lifting condition in Definition 3.3.10 only for all commutative diagrams (3.3.11) with affinoid log adic spaces Z. Lemma 3.3.14 If f : Y → X is a strict lft morphism of locally noetherian log adic spaces, then it is formally log smooth (resp. formally log unramified, resp. formally log étale) if the underlying morphism f : Y → X of adic spaces is formally smooth (resp. formally unramified, resp. formally étale) in the sense that it satisfies the formal lifting conditions in [18, Definition 1.6.5]. If the canonical morphism gp O× Xét → MX is injective, then the converse is true.

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Proof In the notation of (3.3.11), when f satisfies the assumptions of this lemma, and when Z is noetherian and affinoid, the formal lifting conditions in [18, Definition 1.6.5] of the underlying adic spaces can be verified étale locally on Z, covering as in the theory for schemes in [12, III], because  any liftings over an étale of Z defines a cohomology class in H 1 Zét , HomOZ ((Z → Y )∗ (ΩY /X ), I) (by working locally as in the proof of Theorem 3.2.18 and in Construction 3.3.2, ignoring all log structures), which vanishes exactly when the liftings can be modified (upto further étale localization) to descend to a global lifting on Z ; and because 1 ∗ H Zét , HomOZ ((Z → Y ) (ΩY /X ), I) = 0, by Proposition A.10, since Z and hence Z are noetherian and affinoid, since the OZ -ideal I can be identified with a coherent OZ -module in this case, and since ΩY /X is a coherent OY -module when f is lft. Thus, this lemma follows from essentially the same arguments as in the proof of Lemma 3.2.17 (by working with sheaves of monoids and their stalks instead).  Lemma 3.3.15 If f : Y → X is a formally log smooth lft morphism of locally log noetherian fs log adic spaces, then ΩY /X is a locally free OY -module of finite rank. Proof Since X and Y are fs, up to étale localization, we may assume that X = Spa(A, A+ ) and Y = Spa(B, B + ) are affinoid, with log structures induced by some homomorphisms P → A+ → A and Q → B + → B from fs monoids P and Q, respectively, and that there exists a surjection A#T1 , . . . , Tn $  B, for some n ≥ 0. Moreover, since Q is fs, it contains the torsion part Qtor of Qgp , which we may assume to be embedded into B × . Hence, we may assume that Qtor is a finite subgroup of B × annihilated by an integer invertible in B, so that B ⊗Z Qgp is a log finite free B-module. It suffices to show that ΩB/A is a finite projective B-module. Let us equip the Huber ring D := A#T1 , . . . , Tn $#Q$ with the log structure induced by the pre-hog structure P ⊕ Q → D given by P → A+ → A and Q → A#T1 , . . . , Tn $#Q$ : q → eq , for all q ∈ Q. Then we obtain a strict surjection D  B of log Huber rings over A, whose kernel we denote by H , which factors as a composition of strict surjections D  D := D/H 2  B ∼ = D/H . Note that D → B is a log thickening, as in Definition 3.2.11, because the log structure of D is integral (see Remark 3.2.12), as P and Q are fine. Since A → B is formally log smooth, there exists some splitting B → D of log Huber rings over A. By Theorem 3.2.18 (applied to A → A#T1 , . . . , Tn $ → D), Lemma 3.2.10 (applied to the strict homomorphism A → A#T1 , . . . , Tn $), Proposition 3.2.25 (applied to A#T1 , . . . , Tn $ → D with u : P → P ⊕ Q), and Proposition 3.3.7, we log obtain a split short exact sequence 0 → D n → ΩD/A → D ⊗Z Qgp → 0 of finite D-modules, which remains exact after base change to B. Therefore, since B ⊗Z Qgp log is a finite free B-module, so is B ⊗D ΩD/A . By construction (see (3.2.6)), we have log

log

log

canonical surjections B ⊗D ΩD/A → B ⊗D ΩD /A → ΩB/A of finite B-modules. (This assertion can be interpreted as a comparison between the second fundamental exact sequences associated with the strict surjections D  B and D  B via the log log strict surjection D  D .) The first morphism B ⊗D ΩD/A → B ⊗D ΩD /A is an isomorphism, because its kernel is generated over B ∼ = D/H by 1 ⊗ d(xy) =

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x ⊗ dy + y ⊗ dx = 0 in B ⊗D ΩD/A , for all x, y ∈ H , by (3.2.7) and (3.2.8). On the other hand, any splitting B → D above induces a splitting of the second morphism log log log log B ⊗D ΩD /A → ΩB/A , which embeds ΩB/A as a direct summand of B ⊗D ΩD /A . log

log

Thus, ΩB/A is a finite projective B-module as B ⊗D ΩD/A is, as desired.



Proposition 3.3.16 Let f : Y → X be a lft morphism of locally noetherian fs log adic spaces. Then f is formally log smooth (resp. formally log étale) as in Definition 3.3.10 if and only if it is log smooth (resp. log étale) as in Definition 3.1.1. Proof Suppose f is log smooth (resp. log étale). Then f admits étale locally an fs chart u : P → Q as in Definition 3.1.1, and hence is formally log smooth (resp. log étale) by Remarks 3.3.12(1) and 3.3.13, and Proposition 3.2.25. Conversely, suppose f is formally log smooth (resp. formally log étale). Let y = Spa(l, l + ) be a geometric point of Y , which is mapped to a geometric point x = f (y) of X. By Proposition 2.3.13, up to étale localization at x, we may assume that X admits an fs chart θX : PX → MX , with P = MX,x . We need to show that, up to further étale localization at x and y, there exists some fs chart u : P → Q satisfying the conditions in Definition 3.1.1. (Note that f remains formally log smooth (resp. formally log étale), by Remarks 3.3.12 and 3.3.13.) log Consider δ := δY /X,y : MY,y → ΩY /X,y (see Constructions 3.3.2 and 3.3.4). Since δ(t) = t −1 dt for every t ∈ MY,y that is mapped to O× Yét ,y , by (3.2.7) gp

log

and (3.2.8), δ induces a surjection OYét ,y ⊗Z MY,y → ΩY /X,y . Since f is formally log

log smooth, by Lemma 3.3.15, ΩY /X is locally free of finite rank. Take t1 , . . . , tr log

in MY,y whose images in ΩY /X,y form a basis over OYét ,y . Consider the homomorphism of monoids Zr≥0 ⊕ P → MY,y induced by sending the i-th basis element of  Zr≥0 to δ(ti ), and by the composition of P → MX,x ∼ = f −1 (MX ) y → MY,y . ∼ MY,y /α −1 (O× ) is a sharp fs monoid. Also, recall By assumption, MY,y = Y,y

Yét ,y

that y = Spa(l, l + ). By (3.2.7) and (3.2.8) again, the canonical homomorphism  gp gp log MY,y → MY,y / im(P gp ) induces a surjection ΩY /X,y  l ⊗Z MY,y / im(P gp ) . gp

Consequently, Zr≥0 ⊕P → MY,y induces a surjection l⊗Z (Zr ⊕P gp )  l⊗Z MY,y . gp

gp

Since MY,y is a free abelian group of finite rank, the cokernel of Zr ⊕P gp → MY,y is a finite group annihilated by some integer n invertible in l, and hence in OYét ,y . r gp → Mgp to Since O× Y,y Yét ,y is n-divisible, we can (noncanonically) extend Z ⊕ P gp some h : G → MY,y , where G is some free abelian group of finite rank containing Zr ⊕ P gp such that G/(Zr ⊕ P gp ) is annihilated by n, and such that the induced gp map G → MY,y is surjective. Let Q1 := h−1 (MY,y ). By construction, P is a submonoid of Q1 , the induced gp map Q1 → MY,y is strict, and the torsion part of Q1 /P gp is annihilated by n. By the same argument as in the proof of Lemma 2.3.12, there is a finitely generated gp gp submonoid Q2 of Q1 such that Q2 = Q1 , the induced map Q2 → MY,y is still strict, and the composition of Q2 → MY,y → OYét ,y factors through O+ Yét ,y . Let

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Q be the saturation of the submonoid of Q1 generated by Q2 and P , which is an fs submonoid of Q1 with the same properties. Let u : P → Q denote the induced map. By construction, u is injective and compatible with P → MX,x → MY,y and Q → MY,y , and the torsion part of Qgp /ugp (P gp ) is annihilated by n. Since these are all finitely generated monoids, because of the explanation in Remark 2.1.3, up to étale localization at y we may assume that Q → MY,y extends to a chart θY : QY → MY ; that θX , θY , and u : P → Q form an fs chart of f , as in Definition 2.3.19;  log that n is invertible in OY ; and that OY,y ⊗Z Qgp /ugp (P gp ) ∼ = ΩY /X,y . It remains to show that u : P → Q satisfies the conditions in Definition 3.1.1 after these étale localizations. We already know that ker(ugp ) = 0 and the torsion part of Qgp /ugp (P gp ) is annihilated by n. If f is formally log étale (and hence formally log unramified), log by Theorem 3.2.18(3) (and the construction of ΩY /X over affinoid coverings), we log

have ΩY /X,y = 0, in which case the whole Qgp /ugp (P gp ) is torsion and therefore annihilated by n. Thus, u satisfies the condition (1) of Definition 3.1.1. Let g : Y → Y := X ×X#P $ X#Q$ ∼ = X ×Spa(k#P $,k + #P $) Spa(k#Q$, k + #Q$) be log

the morphism induced by the chart θX , θY , and u. Note that g is strict. Since ΩY /X,y is locally free of finite rank, up to further localization at y, we may assume that θY  log and δY /X induce OY ⊗Z Qgp /ugp (P gp ) ∼ = ΩY /X . By Proposition 3.3.7 (and the construction of sheaves of log differentials over affinoid coverings), the canonical log log morphism g ∗ (ΩY /X ) → ΩY /X is an isomorphism. Since f : Y → X is formally smooth, by Theorem 3.2.18(5) and Remark 3.3.13, g : Y → Y is formally log étale. Since Y is integral, by Lemma 3.3.14, the underlying lft morphism of g is formally étale, and hence étale (see the definition and the equivalent formulations in [18, Sections 1.6 and 1.7]). Thus, u also satisfies the condition (2) of Definition 3.1.1.  Theorem 3.3.17 f

g

(1) A composition of lft morphisms Y → X → S of locally noetherian coherent log log log adic spaces naturally induces an exact sequence f ∗ (ΩX/S ) → ΩY /S → log

ΩY /X → 0 of coherent OY -modules. (2) If f is a log smooth morphism of locally noetherian fs log adic spaces, then log log log f ∗ (ΩX/S ) → ΩY /S is injective, and ΩY /X is a locally free OY -module of finite rank. (Étale locally on X and Y , when f admits an fs chart u : P → Q as in log Definition 3.1.1, the rank of ΩY /X as a locally free OY -module is equal to the rank of Qgp /ugp (P gp ) as a finitely generated abelian group.) (3) If f is a log étale morphism of locally noetherian fs log adic spaces, then log log log f ∗ (ΩX/S ) ∼ = ΩY /S and ΩY /X = 0. (4) If X, Y , and S are locally noetherian fs log adic spaces, and if g ◦ f is log smooth, then the converses of (2) and (3) hold.

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(5) If X, Y , and S are locally noetherian fs log adic spaces, and if g is log étale, log ∼ log then ΩY /S → ΩY /X , and f is log smooth (resp. log étale) if and only if g ◦ f is log smooth (resp. log étale). Proof By the construction of sheaves of log differentials, the assertion (1) follows from Theorem 3.2.18(1). The assertions (2) and (3) follow from Theorem 3.2.18 (2) and (4), and Propositions 3.2.25 and 3.3.7. The assertion (4) follows from Theorem 3.2.18(5), Remark 3.3.13, and Proposition 3.3.16. The assertion (5) follows from the assertion (1), Remark 3.3.12(2), and Proposition 3.3.16.  f

g

 → X → S are morphisms of locally noetherian Corollary 3.3.18 Suppose that X fs log adic spaces such that g is log smooth; such that the underlying morphism of adic spaces of f is an isomorphism; and such that the canonical homomorphism MX,x → MX,x  of fs monoids splits as a direct summand, with αX,x  mapping the ∼ split image of (MX,x /M ) − {0} to 0 in O , at each geometric point O =  ét ,x X,x Xét ,x X  log log gp gp ∼ x of X. Then Ω ⊕ OX ,x ⊗Z (M /M ) at each x. Moreover, =Ω  X/S,x

X/S,x

ét

 X,x

X,x

 → Y to a log adic space Y log smooth if there is a strict closed immersion ı : X log ∼ ∗ log over S, then we also have ΩX/S (Ω ı  = Y /S ). Proof This follows from Theorem 3.3.17 and Corollary 3.2.29.



Definition 3.3.19 Let X → S be a log smooth morphism of locally noetherian log fs log adic spaces. Then ΩX/S is a locally free OX -module of finite rank, by log,a

log

Theorem 3.3.17(2), and we set ΩX/S := ∧a ΩX/S , for each integer a ≥ 0. More  over X such that X  → X → S is as in Corollary 3.3.18, and generally, for any X  admits a strict closed immersion to a log adic space Y log smooth over such that X log,a log := ∧a ΩX/S S, we also set ΩX/S   , which is canonically isomorphic to the pullback of ΩY /S , for each integer a ≥ 0. When S = Spa(k, k + ), for some nonarchimedean field k with k + = Ok , and when there is no risk of confusion in the context, we shall often omit S and k from the notation, for the sake of simplicity. In particular, log when X is log smooth over k as in Definition 3.1.9, we shall simply write ΩX and log,• ΩX . log,a

Example 3.3.20 In Example 2.3.18, the morphisms X ← XJ∂ → XJ satisfy the  → X in the second half of Corollary 3.3.18, requirements of the morphisms Y ← X log,• log,• and hence we have a canonical isomorphism Ω ∂ ∼ = (XJ → X)∗ (ΩX ) and XJ ∼ étale locally on X∂ = XJ,an (depending on the choices of coordinates) some J,an log ∼ log = ΩXJ XJ∂

isomorphisms Ω

⊕ OJX of vector bundles.

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4 Kummer étale Topology 4.1 The Kummer étale Site Definition 4.1.1 A homomorphism u : P → Q of saturated monoids is called Kummer if it is injective and if the following conditions hold: (1) For any a ∈ Q, there exists some integer n ≥ 1 such that na ∈ u(P ). (2) The quotient Qgp /ugp (P gp ) is a finite group. Definition 4.1.2 (1) A morphism (resp. finite morphism) f : Y → X of locally noetherian fs log adic spaces is called Kummer (resp. finite Kummer) if it admits, étale locally on X and Y (resp. étale locally on X), an fs chart u : P → Q that is Kummer as in Definition 4.1.1. Such a chart u is called a Kummer chart of f . (2) An f as above is called Kummer étale (resp. finite Kummer étale) if the Kummer chart u above can be chosen such that |Qgp /ugp (P gp )| is invertible in OY , and such that f and u induce a morphism Y → X ×X#P $ X#Q$ of log adic spaces (cf. Remark 2.3.3) whose underlying morphism of adic spaces is étale (resp. finite étale). (3) A Kummer morphism is called a Kummer cover if it is surjective. Remark 4.1.3 Definition 4.1.2 can be extended beyond the case of locally noetherian fs log adic spaces, with suitable P and Q, when all adic spaces involved are étale sheafy. However, we will not pursue this generality in this paper. Remark 4.1.4 Any Kummer homomorphism u : P → Q as in Definition 4.1.1 is exact. Accordingly, as we shall see in Lemma 4.1.11, any Kummer morphism f : Y → X as in Definition 4.1.2 is exact. In particular, Proposition 2.3.32 is applicable to Kummer morphisms. (See also Lemma 4.1.13.) Definition 4.1.5 (1) For any saturated torsion-free monoid P and any positive integer n, let n1 P be the saturated torsion-free monoid such that the inclusion P → n1 P is isomorphic to the n-th multiple map [n] : P → P . (2) Let X be a locally noetherian log adic space modeled on a torsion-free fs monoid P , and n any positive integer. Then we have the log adic space 1 X n := X ×X#P $ X# n1 P $, with a natural chart modeled on n1 P . 1

The structure morphism X n → X is a finite Kummer cover with a Kummer chart given by the natural inclusion P → n1 P , which is finite Kummer étale when n is invertible in OX . Such morphisms will play an important role in Sects. 4.3 and 4.4. More generally, we have the following: Proposition 4.1.6 Let X be a locally noetherian log adic space with a chart modeled on an fs monoid P . Let u : P → Q be a Kummer homomorphism of

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fs monoids such that G := Qgp /ugp (P gp ) is a finite group. Consider Y := X ×X#P $ X#Q$, which is equipped with a canonical action of the group object GD X := X#G$ over X, which is an analogue of a diagonalizable group scheme that is Cartier dual to the constant group scheme G. Then the following hold: (1) The natural projection f : Y → X is a finite Kummer cover, which is finite (see [18, (1.4.4)]) and surjective. (2) When X and therefore Y are affinoid, we have a canonical exact sequence 0 → OX (X) → OY (Y ) → OY ×X Y (Y ×X Y ). (3) The morphism GD X ×X Y → Y ×X Y induced by the action and the second projection is an isomorphism. (4) When G is annihilated by an integer m ≥ 1 invertible in OX , the group GD X is étale over X (which is simply the constant group Hom(G, OX (X)× )X when OX (X) contains all the m-th roots of unity); and the cover f : Y → X is a Galois finite Kummer étale cover with Galois group GD X , which is open. For the proof of Proposition 4.1.6, we need the following general construction, which will also be useful later in Sect. 4.4. Lemma 4.1.7 Let Y = Spa(S, S + ) → X = Spa(R, R + ) be a finite morphism of noetherian adic spaces, and let Γ be a finite group which acts on Y by morphisms over X. Then (T , T + ) := (S Γ , (S + )Γ ) is a Huber pair, and Z := Spa(T , T + ) is a noetherian adic space finite over X. Moreover, the canonical morphism Y → X factors through a finite, open, and surjective morphism Y → Z, which induces a ∼ homeomorphism Y /Γ → Z of underlying topological spaces and identifies Z as the categorical quotient Y /Γ in the category of adic spaces. Proof For analytic adic spaces, and for any finite group Γ such that |Γ | is invertible in S, this essentially follows from [14, Theorem 1.2] without the noetherian hypothesis. Nevertheless, we have the noetherian hypothesis, but not the analytic or invertibility assumptions here. Moreover, we have a base space X over which Y is finite. Hence, we can resort to the following more direct arguments. Since R is noetherian, and since S is a finite R-module, T = S Γ is also a finite R-module, and T + = (S + )Γ is the integral closure of R + in T . Therefore, (T , T + ) has a canonical structure of a Huber pair such that Z := Spa(T , T + ) is a noetherian adic space finite over X = Spa(R, R + ). Moreover, Y → Z is also finite. By [18, (1.4.2) and (1.4.4)] and [17, Section 2], if {s1 , . . . , sr } is any set of generators of S as an T -module, then the topology of S is generated by ri=1 Ui si , where Ui runs through a basis of the topology of T , for all i. Suppose that w : T → Γw is any continuous valuation, and that v : S → Γv is any valuation extending w.

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Note that v and w factor through the domains S := S/v −1 (0) and T := T /w −1 (0), respectively, and Frac(S) is a finite extension of Frac(T ). Therefore, we may assume that Γv and Γw are generated by v(S) and w(T ), respectively, and that Γw is a finite index subgroup of Γv . Foreach γ ∈ Γv , the subgroup {s ∈ S : v(s) < γ } of S is open because it contains ri=1 Ui si , where Ui := {t ∈ T : w(t) < γ − v(si )} is open by the continuity of w. Consequently, Cont(S) → Cont(T ) is surjective. This replaces the main argument in Step 1 of the proof of [14, Theorem 3.1] where the Tate assumption is used. After this step, the remaining arguments in the proof of [14, Theorem 3.1] work verbatim and show that Spa(S, S + ) → Spa(T , T + ) induces a homeomorphism Spa(S, S + )/Γ → Spa(T , T + ). Since Y and Z are finite over X, and since T = S Γ , by [18, (1.4.4)] and Proposition A.9, the canonical morphism OZ → (Y → Z)∗ (OY ) factors through Γ ∼  an isomorphism OZ → (Y → Z)∗ (OY ) . (This provides a replacement of [14, Theorem 3.2].) Thus, the canonical morphism Y → Z factors through an ∼ isomorphism Y /Γ → Z of adic spaces, as in [14, Theorem 1.2], as desired.  Now we are ready for the following: Proof (of Proposition 4.1.6) Let us identify P as a submonoid of Q via the injection u : P → Q. Since the assertions are local in nature on X, we may assume that X = Spa(R, R + ) and hence Y is affinoid. Then OX (X) → OY (Y ) is injective, because it is the base change of Z[P ] → Z[Q] from Z[P ] to R, and Z[P ] is a direct summand of Z[Q] as Z[P ]-modules, as explained in the proof of [20, Lemma 2.1]. Moreover, since Z[P ] → Z[Q] is finite because Q is finitely generated and u is Kummer, its base change OX (X) → OY (Y ) is also finite, and therefore (1) follows. Since the canonical sequence OX (X) → OY (Y ) → OY ×X Y (Y ×X Y ) is exact by [35, Lemma 3.28], (2) also follows. By [19, Lemma 3.3], (Q ⊕P Q)sat ∼ = Q ⊕ G. Accordingly, by Remark 2.3.28, X#G$ ×X X#Q$ ∼ = X#Q$ ×X#P $ X#Q$, and the action of GD X = X#G$ on Y induces ∼ X#G$ ×X Y → Y ×X Y . This verifies (3). As for (4), since it can be verified étale locally on X, we may assume that OX (X) = R contains all |G|-th roots of unity. In this case, X#G$ ∼ = ΓX , where Γ := Hom(G, OX (X)× ), and the action of GD is induced by the canonical actions X of Γ on X#Q$ and X#Qgp $, by sending q to γ (q)q, for each q ∈ Qgp and γ ∈ Γ . Note that (R#Q$)Γ = (R#Qgp $)Γ ∩ R#Q$ = R#P gp $ ∩ R#Q$ = R#P $, where the last one follows from the assumptions that u : P → Q is Kummer and that P is saturated; and the formation of Γ -invariants commutes with the base change from R#P $ to R, because |Γ | is invertible in R. Thus, if Y = Spa(S, S + ), then the morphism Y → X = Spa(R, R + ) ∼ = Spa(S Γ , (S + )Γ ) is open and induces ∼ an isomorphism Y /Γ → X, by Lemma 4.1.7. Moreover, for any subgroup Γ of Γ = Hom(G, R × ), which is of the form Hom(G , R × ) for some quotient G of G = Qgp /ugp (P gp ), we have (R#Q$)Γ ∼ = R#Q $ and (R + #Q$)Γ ∼ = R + #Q $, where Q is the preimage of ker(G → G ) under the canonical homomorphism Q → G = Qgp /ugp (P gp ), so that Y /Γ ∼ = X ×X#P $ X#Q $ → X is a finite

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Kummer étale. Consequently, f : Y → X is a Galois finite Kummer étale cover with Galois group Γ , as desired.  Definition 4.1.8 Kummer (resp. Kummer étale) covers f : Y → X as in Proposition 4.1.6 are called standard Kummer (resp. standard Kummer étale) covers. Corollary 4.1.9 Kummer étale morphisms are open. Proof This is because, by definition, Kummer étale morphisms are, étale locally on the source and target, compositions of standard Kummer étale covers and strictly étale morphisms, both of which are open.  In the remainder of this subsection, let us study some general properties of Kummer étale morphisms. Our goal is to introduce the Kummer étale site. Lemma 4.1.10 Let f : Y → X be a Kummer étale morphism of locally noetherian fs log adic spaces. Suppose that X is modeled on an fs monoid P . Then f admits, étale locally on Y and X, a Kummer chart P → Q as in Definition 4.1.2(2), with the same prescribed P as above. Moreover, if P is torsion-free (resp. sharp), then we can choose Q to be torsion-free (resp. sharp). Proof Étale locally on Y and X, let u1 : P1 → Q1 be a Kummer chart of f as in Definition 4.1.2. (A priori, P1 might be different from P .) As in the proof of Proposition 3.1.4, up to further étale localization and modifying P1 → Q1 , we can find some group H fitting into the cartesian diagram (3.1.5). Let Q := {a ∈ H : na ∈ P , for some n ≥ 1 invertible in OY }, so that u : P → Q is Kummer, as in Definition 4.1.1. Let Q be the preimage of Q1 gp via H → Q1 , so that u : P → Q is an fs chart of f satisfying the conditions of Definition 3.1.1, as explained in the paragraph after (3.1.5). Since gp

Q1 = {a ∈ Q1 : na ∈ u1 (P1 ), for some n ≥ 1 invertible in OY }, by the assumption on u1 , and since (3.1.5) is cartesian, we can identify Q with the localization of Q with respect to ker(Q → Q1 ). Therefore, u : P → Q is an fs chart of f as u is, which also satisfies the conditions of Definition 3.1.1, or rather of Definition 4.1.2(2). By the proof of the last assertion of Proposition 3.1.4, if P is torsion-free, then we may assume that Q is torsion-free as well. Finally, suppose that P is sharp and Q is torsion-free. For any q ∈ Qinv , there is some n ≥ 1 such that nq and −nq are both in u(P ) and hence in u(P inv ) = {0}. Since Q is torsionfree, this forces q = 0. Thus, Q is also sharp, as desired.  Lemma 4.1.11 Let f : Y → X be a Kummer morphism of locally noetherian fs log adic spaces. Then:

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(1) The morphism f is exact. (2) For any geometric point y of Y , the induced homomorphism of sharp fs monoids % fy : MX,f (y) → MY,y is Kummer. Moreover, if f is Kummer étale, then  % coker (fy )gp is invertible in OY,y . (3) Suppose that f admits a Kummer chart u : P → Q. For any geometric point y of Y , if Kf (y) := ker(P → MX,f (y) ) and Ky := ker(Q → MY,y ), then Kf (y) = u−1 (Ky ), and the induced homomorphism w : Kf (y) → Ky is Kummer. Thus, if Kf (y) = 0, then Ky is torsion, and is zero if Q is sharp. %

Proof Let us start with (3). By Remark 2.3.4, u and v := fy are compatible with surjective homomorphisms θf (y) : P → MX,f (y) and θy : Q → MY,y , with × kernels Kf (y) and Ky given by preimages of O× Xét ,f (y) and OYét ,y , respectively.

Since fy : OXét ,f (y) → OYét ,y is local, Kf (y) = u−1 (Ky ). If a ∈ MX,f (y) and v(a) = 0, then a = θf (y) (a), for some a ∈ P such that u(a) ∈ Ky . Hence, a ∈ Kf (y) , and a = θf (y) (a) = 0. This shows that v is injective. Since u : P → Q is Kummer, if b ∈ Ky ⊂ Q, then nb = u(a) for some integer n ≥ 1 and a ∈ P , and v maps θf (y) (a) to θy (nb) = 0, and so θf (y) (a) = 0 by the injectivity of v. Therefore, a ∈ Kf (y) . It follows that the induced homomorphism w : Kf (y) → Ky is Kummer, with coker(w gp ) given by a subgroup of coker(ugp ), and (3) follows. Next, let us verify (2). By assumption, up to étale localization on Y and X, the morphism f admits a Kummer chart u : P → Q. If b ∈ MY,y , then b = θy (b), for some b ∈ Q. Since u is Kummer, nb = u(a), for some integer n ≥ 1 and a ∈ P . Then v maps a := θf (y) (a) to θy (nb) = nb. Furthermore, coker(v gp ) is a finite group, because it is a quotient of coker(ugp ). Thus, v is Kummer. By Lemma 4.1.10, if f is Kummer étale, then we may assume that the order of coker(ugp ) is invertible in OY,y , and the same is true for its quotient coker(v gp ). Finally, since any Kummer homomorphism of monoids is exact (see Remark 4.1.4), (1) follows from (2), [36, Proposition I.4.2.1], and Remark 2.2.5.  %

Definition 4.1.12 In Lemma 4.1.11, the ramification index of f at y is defined to gp gp be the smallest positive integer n that annihilates coker(MX,f (y) → MY,y ). The ramification index of a Kummer étale morphism f is the least common multiple, when defined, of the ramification indices among the geometric points y of Y . (The ramification index is not always defined.) The ramification index of a Kummer étale morphism is 1 if and only if f is strictly étale. Lemma 4.1.13 A morphism f : Y → X of locally noetherian fs log adic spaces is Kummer étale if and only if it is log étale and Kummer, and if and only if it is log étale and exact. It is finite Kummer étale if and only it is log étale and finite Kummer. Proof If f is Kummer étale (resp. finite Kummer étale), then it is log étale and Kummer (resp. finite Kummer) by definition, and it is exact by Lemma 4.1.11. Conversely, assume that f is log étale and exact. By Propositions 2.3.13 and 3.1.4, f admits, étale locally at geometric points y of Y and x = f (y) of X, an injective

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fs chart u : P = MX,x → Q satisfy the conditions in Definition 3.1.1(2), in which case coker(ugp ) is a finite group whose order is invertible in OY . Since f is % exact, by [36, Proposition I.4.2.1] and Remark 2.2.5, f y : MX,x → MY,y is exact. Given any b ∈ Q, since coker(ugp ) is annihilated by n, we have nb = ugp (a) for some a ∈ P gp . Since a is mapped to the image of nb in MY,y , by the exactness %

of f y , we have a ∈ P = MX,x . Hence, u : P → Q is a Kummer chart as in Definition 4.1.2(2). Since y is arbitrary, f is Kummer étale, as desired. Alternatively, assume that f is log étale and finite Kummer. Up to étale localization on X, we may assume that it admits a Kummer chart u : P → Q, and that X has at most one positive residue characteristic . When no such  exists, we set  = 0, for simplicity. Then we have a Kummer homomorphism u : P → Q := {b ∈ Q : nb ∈ u(P ), for some n ≥ 1 s.t.   n}  such that   coker (u )gp . We claim that the morphism of adic spaces Y → X ×X#P $ X#Q $ induced by u is étale. Note that this can be verified up to étale localization on Y . By the previous paragraph, f is Kummer étale. By Lemmas 4.1.10 and 4.1.11, f admits, étale locally at geometric points y of Y and f (y) of X, another Kummer ∼ ∼ gp chart u1 : P1 → Q1 , with P1 → MX,f (y) , Q1 → MY,y , and   | coker(u1 )|, such that the morphism Y → X ×X#P1 $ X#Q1 $ induced by u1 is étale. Note that   | coker(u1 )| implies that Q → MY,y is surjective as Q → MY,y is, and so u is an fs chart as u is. Then u : P → Q and u1 : P1 → Q1 compatibly extend to a Kummer homomorphism u2 : P2 → Q2 of fs monoids, where P2 (resp. Q2 ) is the localization of P ⊕ P1 (resp. Q ⊕ Q1 ) with respect to the kernel of P ⊕ P1 → MX,f (y) (resp. Q ⊕ Q1 → MY,y ). Since ∼ ∼ P1 → MX,f (y) and Q1 → MY,y , we have P2 = P1 ⊕ P2inv , Q2 = Q1 ⊕ Qinv 2 , and inv inv inv inv u2 = u1 ⊕ u2 , for some Kummer homomorphism u2 : P1 → H1 such that   | coker(uinv 2 )|. In this case, the above morphism Y → X ×X#P $ X#Q $ induced by u is the composition of the morphisms gp

Y → X ×X#P2 $ X#Q2 $ → X ×X#P2 $ (X#P2 $ ×X#P $ X#Q $) ∼ = X ×X#P $ X#Q $, induced by u2 and the canonical homomorphisms among P , Q , P2 , and Q2 . Note that the second morphism is the pullback of the canonical morphism X#Q2 $ → X#P2 $ ×X#P $ X#Q $.

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Let P2 + Q be the submonoid of Q2 generated by the images of P2 and Q , and G := coker (u )gp . Then we have an isomorphism of monoids gp

∼

(P2 ⊕P Q )sat → (P2 + Q ) ⊕ G : (a, b) → (a + b, b), where a ∈ P2 and b ∈ Q , and b ∈ G denotes the image of b, as in the proof of [19, Lemma 3.3]. Moreover, we have an induced isomorphism of adic spaces ∼

X#P2 $ ×X#P $ X#Q $ → X#P2 + Q $ ×X X#G $, where X#G $ → X is étale, as in Proposition 4.1.6. Since Q → MY,y is surjective ∼ and Q1 → MY,y , we have P2 + Q = P2 + Q + P2inv ⊂ P2 + Q + Qinv 2 = gp Q2 in Q2 , and the monoid Q2 is generated by P2 + Q and some finitely many invertible elements of Q2 whose | coker(uinv 2 )|-th multiples are in P2 + Q . Since inv   | coker(u2 )|, the induced morphism X#Q2 $ → X#P2 + Q $ is étale, by [18, Proposition 1.7.1], and so is the above X#Q2 $ → X#P2 $ ×X#P $ X#Q $. Therefore, in order to verify the above claim, it suffices to show that the morphism Y → X ×X#P2 $ X#Q2 $ inv induced by u2 = u1 ⊕ uinv 2 is étale. Again since   | coker(u2 )|, this follows from the known exactness of the morphism of adic spaces Y → X ×X#P1 $ X#Q1 $ induced by u1 . Thus, f is finite Kummer étale because it admits, étale locally on X, an fs chart u : P → Q satisfying the conditions in Definition 4.1.2(2). 

Proposition 4.1.14 Kummer étale (resp. finite Kummer étale) morphisms as in Definition 4.1.2 are stable under compositions and base changes under arbitrary morphisms between locally noetherian fs log adic spaces (which are justified by Remark 3.1.2 and Proposition 3.1.3). Proof The stability under compositions follows from Proposition 3.1.6, Lemma 4.1.13, and the stability of exactness under compositions (by definition). As for the stability under base changes, it suffices to note that, if P → Q is a Kummer homomorphism (of fs monoids), and if P → R is any homomorphism of fs monoids, then the induced homomorphism R → (R ⊕P Q)sat is also Kummer, because it is injective as the composition R → (R ⊕P Q)sat → R gp ⊕P gp Qgp is, and because it satisfies the conditions in Definition 4.1.1 as P → Q does.  Proposition 4.1.15 Suppose that f : Y → X and g : Z → X are Kummer étale morphisms of locally noetherian fs log adic spaces. Then any morphism h : Y → Z such that f = g ◦ h is also Kummer étale. Proof By Lemma 4.1.13, it suffices to show that h is log étale and exact. By Theorem 3.3.17(5), h is log étale because f = g ◦ h and g are. By Lemma 4.1.11, étale locally at each geometric point y of Y , with x = f (y) and z = h(y), the

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%

homomorphisms fy : MX,x → MY,y and gz : MX,x → MZ,z are both Kummer. %

Consequently, the homomorphism hy : MZ,z → MY,y is also Kummer, and hence exact. Thus, h is exact, by [36, Proposition I.4.2.1] and Remark 2.2.5.  By Proposition 2.3.32 and Remark 4.1.4, and by Propositions 4.1.14 and 4.1.15, we are now ready for the following: Definition 4.1.16 Let X be a locally noetherian fs log adic space. The Kummer étale site Xkét has as underlying category the full subcategory of the category of locally noetherian fs log adic spaces consisting of objects that are Kummer étale over X, and has coverings given by the topological coverings. Remark 4.1.17 Let X be as in Definition 4.1.16. (1) For each U ∈ Xét , we can view U as a log adic space by restricting the log structure α : MX → OXét to Uét . This gives rise to a strictly étale morphism U → X of log adic spaces, which is Kummer étale by definition. Therefore, we obtain a natural projection of sites εét : Xkét → Xét , which is an isomorphism when the log structure of X is trivial. (2) For any morphism f : Y → X of locally noetherian fs log adic spaces, we have a natural morphism of sites fkét : Ykét → Xkét , because base changes of Kummer étale morphisms are still Kummer étale, by Proposition 4.1.14. Remark 4.1.18 By definition, the Kummer étale topology on X is generated by surjective (strictly) étale morphisms and standard Kummer étale covers.

4.2 Abhyankar’s Lemma An important class of finite Kummer étale covers arise in the following way: Proposition 4.2.1 (Rigid Abhyankar’s Lemma) Let X be a smooth rigid analytic variety over a nonarchimedean field k of characteristic zero, and let D be a normal crossings divisor of X. We equip X with the fs log structure induced by D as in Example 2.3.17. Suppose that h : V → U := X − D is a finite étale surjective morphism of rigid analytic varieties over k. Then it extends to a finite surjective and Kummer étale morphism of log adic spaces f : Y → X, where Y is a normal rigid analytic variety with its log structures defined by the preimage of D. Consequently, Yan has a basis consisting of affinoid W satisfying π0 W ∩ f −1 (U ) = π0 (W ). Proof By [15, Theorem 1.6] (which was based on [32, Theorem 3.1 and its proof]), h : V → U extends to a finite ramified cover f : Y → X, for some normal rigid analytic variety Y (viewed as a noetherian adic space). Then Yan has a basis  consisting of affinoid open subspaces W satisfying π0 W ∩ f −1 (U ) = π0 (W ), by the unique existence of extensions of bounded functions (which include locally constant functions, in particular) from W ∩ f −1 (U ) to W , for any affinoid open subspaces W of Y , by [2, Section 3] (see also [15, Theorem 2.6]). Just as X is

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equipped with the log structure defined by D, we equip Y with the log structure defined by the preimage of D. The question is whether the map f is Kummer étale (with respect to the log structures on X and Y ), and such a question can be answered analytic locally on X, up to replacing k with a finite extension. As in Example 2.3.17, we may assume that there is an affinoid smooth rigid analytic variety S over k such that X = S × Dr ∼ = S#Zr≥0 $ (see Example 2.2.19) for some r ∈ Z≥0 , with D = S × {T1 · · · Tr = 0}. Thus, we can finish the proof of this proposition by combining the following Lemmas 4.2.2 and 4.2.3.  For simplicity, let us introduce some notation for the following two lemmas. We write P := Zr≥0 and identify Dr with Spa(k#P $, k + #P $) as in Example 2.2.21. For each m ∈ Z≥1 , we also write m1 P = m1 Zr≥0 . For each power ρ of p, we denote by Dρ the (one-dimensional) disc of radius ρ, so that D = Dρ when ρ = 1. We also denote × by D× ρ the punctured disc of radius ρ, and by D the punctured unit disc. For any rigid analytic variety with a canonical morphism to Dr , we denote with a subscript “ρ” (resp. superscript “×”) its pullback under Drρ → Dr (resp. (D× )r → Dr ). Lemma 4.2.2 Suppose that X = S × Dr ∼ = S#P $× = S#P $, D, and U = X − D ∼ are as in the proof of Proposition 4.2.1. Assume there is some ρ ≤ 1 such that, for each connected component Y of Yρ , there exist d1 , . . . , dr ∈ Z≥1 such that induced cover Y → X := Xρ is refined (i.e., admits a further cover) by some finite ramified  cover Z := S#P $ρ → X , where P = ⊕1≤i≤r d1i Z≥0 . Then, up to replacing k with a finite extension, we have Y ∼ = S#Q$ρ , for some sharp fs monoid Q such that P ⊂ Q ⊂ P . Consequently, Yρ → X = Xρ is finite Kummer étale. Moreover, 1 if mQ ⊂ P for some m ∈ Z≥1 , and if X m := S# m1 P $, then the finite (a priori 1

1

ramified) cover Y ×X Xρm → Xρm splits completely (i.e., the source is a disjoint union of sections) and is therefore strictly étale. Proof Let V (resp. W ) be the preimage of U := Uρ ∼ = S#P $× ρ in Y (resp. Z). Up to replacing k with a finite extension containing all dj -th roots of unity for all j , by Proposition 4.1.6, the finite étale cover W → U is Galois with  gp gp × Galois group G := Hom (P ) /P , k , and V is (by the usual arguments, as in [12, V]) the quotient of W by some subgroup G of G (as in Lemma 4.1.7), which is isomorphic to S#Q$× ρ for some monoid Q such that P ⊂ Q ⊂ P and gp Q = Q ∩ P . These conditions imply that Q is toric, and hence S#Q$ is normal because Spa(k#Q$, k + #Q$) is (see Example 2.2.20 and the references given there). Since Y and S#Q$ρ are both normal and are both finite ramified covers of X extending the same finite étale cover V of U , they are canonically isomorphic by [15, Theorem 1.6], as desired. Finally, for the last assertion of the lemma, it suffices to note that, for any Q as above satisfying mQ ⊂ P , up to replacing k with a finite extension containing all m-th roots of unity, the connected components sat of S#Q$ ×S#P $ S# m1 P $ are all of the form S# m1 P $, because Q ⊕P ( m1 P ) is the 1  product of m P with a finite group annihilated by m.

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Lemma 4.2.3 The (cover-refinement) assumption in Lemma 4.2.2 holds up to replacing k with a finite extension and S with a strictly finite étale cover; and we may assume that the positive integers d1 , . . . , dr there (for various Y ) are no greater than the degree d of f : Y → X. Moreover, we can take ρ = p−b(d,p) , where b(d, p) is defined as in [32, Theorem 2.2], which depends on d and p but not on r; and we can take m = d! in the last assertion of Lemma 4.2.2. Proof We shall proceed by induction on r. When r = 0, the assumption in Lemma 4.2.2 means, for each connected component Y of Y , the strictly étale cover Y → X = S splits completely. This can always be achieved up to replacing S with a Galois strictly finite étale cover refining Y → S for all Y . In the remainder of this proof, suppose that r ≥ 1, and that the lemma has been proved for all strictly smaller r. Let ρ = p−b(d,p) be as above. Fix some a ∈ k such ˘ (instead of ×). that |a| = ρ. We shall denote normalizations of fiber products by × r−1 Let P1 be the submonoid Z≥0 ⊕ {0} of P = Zr≥0 . Let X1 := S × Dr−1 ∼ = S#P1 $, r−1 r ∼ × {a} of X = S × D = S#P $. Let which we identify with the subspace S × D ˘ X X1 . Note that the degree of Y1 → X1 is also d. By induction, up to Y1 := Y × replacing k with a finite extension and S with a strictly finite étale cover, for each connected component Y1 of (Y1 )ρ , there exist 1 ≤ d1 , . . . , dr−1 ≤ d such that the induced finite ramified cover Y1 → (X1 )ρ is refined by S#P1 $ρ → (X1 )ρ , where 1

1 P1 := ⊕1≤i≤r−1 ( d1i Z≥0 ). Let X1d! := S# d! P1 $, with 1

1

1 d! P1 1

=

1 r−1 d! Z≥0 .

 := X d! ×X1 X ∼  := X d! ×  → X  ˘ X1 Y . Let f : Y Let X = X1d! × D and Y 1 1 denote the induced finite ramified cover, which is also of degree d. Then the (strictly 1 finite étale) pullback of f to (X d! )× ρ × {a} can be identified with the pullback of 1

1 d!

(X1 )× ρ,

Y1 → X1 to which splits completely, by the induction hypothesis and the last assertion in Lemma 4.2.2. Hence, since ρ = p−b(d,p) , for each connected  )× → X ρ×  of Y ρ , by applying [32, Lemma 3.2] to the morphism (Y component Y × d r     induced by f , we obtain a rigid analytic function T on (Y ) such that T = Tr , where Tr is the coordinate on the r-th factor of Dr . By [2, Section 3] (see also [15,  , which still Theorem 2.6]), T extends to a rigid analytic function on the normal Y 1 1 ∼ (X d! × D)ρ ρ =  → X satisfies Tdr = Tr . Hence, we can view T as Trdr , and Y 1

1

$ρ → (X d! × D)ρ , where P  := ( 1 P1 ) ⊕ ( 1 Z≥0 ). Since these factors through S#P 1 d! dr are finite ramified covers of the same degree dr from connected and normal rigid $ρ .  ∼ analytic varieties, we obtain an induced isomorphism Y = S#P Since each connected component Y of Yρ is covered by some connected  of Y ρ , by Lemma 4.2.2, up to replacing k with a finite extension, component Y ∼  = ( 1 P1 ) ⊕ ( 1 Z≥0 ), Y = S#Q$ρ for some monoid Q satisfying P ⊂ Q ⊂ P d! dr  determined by Y  as above. By the construction d and P for some 1 ≤ dr ≤   , the monoid ( 1 P1 ) ⊕P1 Q sat is the product of P  with a finite group, of Y d! 1 r−1 ⊕ ( dr Z≥0 ) ⊂ Q. By the construction of Y1 and the induction which forces {0}

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 → hypothesis, the projection P

1 d! P1

121

maps Q into ⊕1≤i≤r−1 ( d1i Z≥0 ) for some

1 ≤ d1 , . . . , dr−1 ≤ d. Thus, P ⊂ Q ⊂ P := ⊕1≤i≤r ( d1i Z≥0 ), as desired.



Remark 4.2.4 Proposition 4.2.1 can be regarded as the Abhyankar’s lemma (cf. [12, XIII, 5.2]) in the rigid analytic setting, because of the last assertion in Lemma 4.2.2. More generally, we have the following basic but useful facts: Lemma 4.2.5 Let X be a noetherian fs log adic space modeled on a sharp fs monoid P , and let f : Y → X be a Kummer étale (resp. finite Kummer étale) 1 1 morphism. Then Y ×X X n → X n is étale (resp. finite étale) for some positive integer n. If X has at most one positive residue characteristic, then we can take n to be invertible on all of X. Proof Since X is noetherian, by taking the least common multiple of the positive integers obtained on finitely many members in an étale covering, it suffices to work étale locally on X. By Lemma 4.1.10, up to étale localization on X, there exists an étale covering {Yi → Y }i∈I indexed by a finite set I such that each induced Kummer étale morphism Yi → X admits a Kummer chart P → Qi with a sharp Qi . Then there exists some positive integer n, which we may assume to be invertible on all of X when X has at most one positive residue characteristic, such that P → n1 P ui

factors as P → Qi → 1 n

1 nP

for some injective ui , for all i ∈ I . The induced

1

for each i ∈ I , because morphism Yi ×X X → X n is étale, sat it admits1a Kummer sat ∼  1 ∼ chart n1 P → Qi ⊕P ( n1 P ) ⊕ Q ) ⊕ ( P ) (Q = = Gi ⊕ ( n P ), where i P i Qi n gp gp gp Gi := (Qi ) /ui (P ) has order invertible in OYi by assumption. (Since Gi is a group, n1 P → Gi ⊕ ( n1 P ) is strict, by definition. See also Remark 2.2.6.) By  1 1 Proposition 2.3.32, the étale map i∈I Yi ×X X n → Y ×X X n is surjective as  1 1 n n i∈I Yi → Y is. Hence, by étale descent, Y ×X X → X is also étale. Finally, by 1

1

[18, Lemma 1.4.5 i)], Y ×X X n → X n is finite when Y → X is.



Lemma 4.2.6 Let X be a noetherian fs log adic space modeled on a sharp fs monoid P . Let {Ui → X}i∈I be a Kummer étale covering indexed by a finite set I . Then there exists a Kummer étale covering {Vj → X}j ∈J indexed by a finite set J refining {Ui → X}i∈I such that each Vj → X admits a chart P → n1j P for 1

1

some integer nj invertible in OVj , and such that {Vj ×X X n → X n }j ∈J is an étale 1

covering of X n for some integer n divisible by all nj . If X has at most one positive residue characteristic, then we may take n to be invertible in OX . Proof Since X is noetherian, by Lemma 4.1.10, we may replace {Ui → X}i∈I with a finite refinement {Uj → X}j ∈J such that each Uj → X admits a Kummer chart P → Qj with a sharp Qj . Then there exists some positive integer nj invertible in OUj such that P →

1 nj

uj

P factors as P → Qj →

1 nj

P for some injective uj ,

for each j ∈ J . Therefore, each Vj := Uj ×Uj #Qj $ Uj # n1j P $ → X is Kummer

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étale with a Kummer chart P →

1 nj

1

P , and the induced morphism Vj → X nj =

X ×X#P $ X# n1j P $ is étale (as in Definition 4.1.2). In this case, if n is divisible by 1

1

all nj , then Vj ×X X n → X n = X ×X#P $ X# n1 P $ is also étale, and we can take n to be invertible on X when X has at most one positive residue characteristic (cf. the proof of Lemma 4.2.5). Since j ∈J Uj → X is surjective by assumption, by 1

1

1

Proposition 2.3.32, {Vj ×X X n }j ∈J → X n is an étale covering of X n , as desired.  The following two propositions show that the properties of morphisms being Kummer étale, log smooth, and log étale can be verified up to Kummer étale localization on either the source or the target: f

g

Proposition 4.2.7 Let Y → X → S be lft morphisms of locally noetherian fs log adic spaces such that f is Kummer étale and surjective. Then g is log smooth (resp. log étale, resp. Kummer étale) if and only if g ◦ f is. Proof Since f is Kummer étale (and hence log étale), by Propositions 3.1.6 and 4.1.14, if g is log smooth (resp. log étale, resp. Kummer étale), then so is g ◦ f . (The surjectivity of f is not needed in this direction of implication.) Conversely, suppose that g◦f is log smooth (resp. log étale, resp. Kummer étale). It suffices to show that X → S is log smooth (resp. log étale, resp. Kummer étale), étale locally at geometric points x of X and s = g(x) of S. Up to étale localization at x, we may assume that X has at most one positive residue characteristic. By Proposition 2.3.22, up to étale localization at x and s, we may assume that X → S admits an fs chart u : L := MS,s → P , inducing a % strict morphism g : X → X := S ×S#L$ S#P $. Let u := gx : L → P := MX,x . By Remark 2.3.4, P → P is surjective with kernel given by the preimage of O× L and P are sharp fs, we may assume that the order of the torsion Xét ,x . Since  part of ker (P )gp → P gp is invertible in OXét ,x . Since f is Kummer étale, by Definition 4.1.2 and Lemma 4.2.6, and by the first paragraph above, we are reduced 1 to the case where f : Y → X is of the form X m → X, for some integer m invertible in OX , which admits a global fs chart v : P → m1 P . Let y be any geometric point of Y such that f (y) = x, which exists because % f is surjective. By Lemma 4.1.11, v := fy : MX,x → MY,y is also given by v : P → m1 P . By the same argument as above, up to étale localization at y, we may assume that Y → S admits an fs global chart w : L → Q, and we have a surjection Q → MY,y ∼ = m1 P such that the order of the torsion part of ker Qgp → m1 P gp %

is invertible in OXét ,x . By definition, w := (g ◦ f )y = v ◦ u as homomorphisms from L to m1 P . If g ◦ f is log smooth (resp. log étale), then the kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of w gp have orders invertible in OXét ,x , and so are ugp and ugp (cf. Definition 3.1.1). If g ◦ f is Kummer étale, then w is Kummer, and so are u and u (see Definitions 4.1.1 and 4.1.2, and

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Lemmas 4.1.11 and 4.1.13). Therefore, by the first paragraph above, X → S is log smooth (resp. log étale, resp. Kummer étale) when g ◦ f is. Thus, up to replacing g with g : X → X , we are reduced to the case where g is strict, and it remains to 1 show that g : X → S is (strictly) étale when g ◦ f : Y = X m → S is log étale. Up to étale localization on S, we may assume that OS (S)× and hence OX (X)× contain 1 all m-th roots of unity. Then Y = X m → X is a Galois finite Kummer étale cover with Galois group Γ := Hom ( m1 P gp )/P gp , OS (S)× , by Proposition 4.1.6.    Let us write OS (S), MS (S) = (A, M), OX (X), MX (X) = (B, N ), and OY (Y ), MY (Y ) = (C, O), for simplicity. Since X → S is strict, up to further étale localization on X and S, we may assume that (A, M) → (B, N ) is also strict. By Lemma 3.3.14, Remark 3.3.13, and Proposition 3.3.16, it suffices to show that (A, M) → (B, N ) is formally log étale (as in Definition  3.2.14) when sat (A, M) → (C, O) is. Since Y ∼ = X ×X#P $ X# m1 P $, we have O ∼ = N ⊕P ( m1 P )  Z[P ] Z[ m1 P ])⊗ Z[N ⊕ ( 1 P )] Z[ N ⊕P ( m1 P ) sat ] ∼ Z[N ] Z[O]. and C ∼ = (B ⊗ = B⊗ P m Consider any commutative diagram as in (3.2.13) for (A, M) → (B, N ) (with sat α, β, etc omitted here). For ? = ∅ and , consider T? := T ? ⊕P ( m1 P ) and ? ? ?      D := D ⊗Z[T ? ] Z[T ], so that (D , T ) → (D , T ) is the completion of the  T) in the category of log common pullback of (B, N ) → (C, O) and (D, T ) → (D, ∼  ? )Γ , (T? )Γ , Huber rings with fs log structures. Moreover, we have (D ? , T ? ) → (D for ? = ∅ and , because the formation of Γ -invariants is compatible with arbitrary base changes and completions when |Γ | is invertible (as m is), and because of Remark 2.1.7. Thus, we have obtained an extended commutative diagram for  T) → (D  , T ) of (D, T ) → (D , T ). (A, M) → (C, O) and the base change (D,  , T ) uniquely lifts to Since (A, M) → (C, O) is formally log étale, (C, O) → (D   (C, O) → (D, T ), whose pre-composition with (B, N ) → (C, O) is Γ -invariant and hence factors through (D, T ). This shows that (A, M) → (B, N ) is also formally log étale, as desired.  Proposition 4.2.8 Let f : Y → X and g : X → X be lft morphisms of locally noetherian fs log adic spaces such that g is Kummer étale and surjective. Then f is log smooth (resp. log étale, resp. Kummer étale) if and only if its pullback f : Y := Y ×X X → X under g is. Proof By definition, we have the following commutative diagram Y

f

X g

g

Y

f

X

in which g is the pullback of g under f . If f is log smooth (resp. log étale, resp. Kummer étale), then so is f , by Proposition 3.1.3 and 4.1.14. Conversely, suppose f is log smooth (resp. log étale, resp. Kummer étale). Since g is Kummer étale (and hence log étale), by Propositions 3.1.6 and 4.1.14, g ◦ f = f ◦ g is also log

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smooth (resp. log étale, resp. Kummer étale). By Propositions 4.1.14 and 2.3.32, g is Kummer étale and surjective as g is. Thus, by Propositions 4.2.7, f is log smooth (resp. log étale, resp. Kummer étale), as desired. 

4.3 Coherent Sheaves In this subsection, we show that, when X is a locally noetherian fs log adic space, the presheaf OXkét (resp. O+ Xkét ) on Xkét defined by U → OU (U ) (resp. U → O+ (U )) is indeed a sheaf, generalizing a well-known result of Kato’s [25] for log U schemes. We also study some problems related to the Kummer étale descent of coherent sheaves. Theorem 4.3.1 Let X be a locally noetherian fs log adic space. (1) The presheaves OXkét and O+ Xkét are sheaves. i (2) If X is affinoid, then H (Xkét , OXkét ) = 0, for all i > 0. A key input is the following: Lemma 4.3.2 Let X be an affinoid noetherian fs log adic spaces, endowed with a chart modeled on a sharp fs monoid P . Let Y → X be a standard Kummer cover ˇ (see Definition 4.1.8). Then the Cech complex C • (Y /X) : 0 → O(X) → O(Y ) → O(Y ×X Y ) → O(Y ×X Y ×X Y ) → · · · (where we omit the subscripts of the structure sheaf O for simplicity) is exact. Proof This is essentially [35, Lemma 3.28], based on the idea in [25, Lemma 3.4.1]. Suppose that Y → X = Spa(R, R + ) is associated with a Kummer homomorphism u : P → Q as in Proposition 4.1.6. Then C • (Y /X) is already known to be exact at the first three terms; and O(Y ×X Y ×X · · · ×X Y ) ∼ = O(Y ) ⊗R R[G] ⊗R R[G] · · · ⊗R R[G], where G = Qgp /ugp (P gp ), in which case we can write the differentials of C • (Y /X) explicitly and construct a contracting homotopy for C • (Y /X), by the same argument as in the proof of [35, Lemma 3.28].  We emphasize that Lemma 4.3.2 also works for standard Kummer covers that are not necessarily Kummer étale. Proof (of Theorem 4.3.1) (1) It suffices to prove that OXkét is a sheaf, in which case O+ Xkét also is, because + O+ Xkét (U ) = OU (U ) = {f ∈ OXkét (U ) = OU (U ) : |f (x)| ≤ 1, for all x ∈ U },

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exactly as in [43, Proposition 3.1.7]. Since the sheafiness for the étale topology is known for all locally noetherian adic spaces, by Lemma 4.2.6, the statement is reduced to Lemma 4.3.2. (2) By Propositions 2.3.13 and A.10, we may reduce to the case where X is affinoid with a global sharp fs chart P . By Lemma 4.2.6, any Kummer étale covering {Ui → X}i∈I of X admits some refinement {Vj → X}j ∈J as finite Kummer 1

1

étale covering such that {Vj ×X X m → X m }j ∈J is an étale covering, for some 1

m, and such that each Vj ×X X m → Vj is a composition of étale morphisms ˇ and standard Kummer étale covers. Thus, by Lemma 4.3.2, the Cech complex 1

1

1

O(X) → O(X m ) → O(X m ×X X m ) → · · · ˇ is exact. As a result, by Proposition A.10, the Cech complex O(X) → ⊕j O(Vj ) → ⊕j,j O(Vj ×X Vj ) → · · · is also exact, as desired.



Corollary 4.3.3 Let X be a locally noetherian fs log adic space. Consider the natural projections of sites εan : Xkét → Xan and εét : Xkét → Xét . Then we ∼ ∼ have canonical isomorphisms OXan → Rεan,∗ (OXkét ) and OXét → Rεét,∗ (OXkét ). As a result, the pullback functor from the category of vector bundles on Xan (resp. Xét ) to the category of OXkét -modules is fully faithful (cf. Proposition A.10). Proposition 4.3.4 Let X be a locally noetherian fs log adic space. Then the presheaf MXkét assigning U → MU (U ) is a sheaf on Xkét . In particular, we also ∼ have a canonical isomorphism εét,∗ (MXkét ) → MX . Proof The proof is similar to [25, Lemma 3.5.1]. Since MX is already a sheaf on the étale topology, by replacing X with its strict localization at a geometric point x, it suffices to show the exactness of 0 → MX (X) → MY (Y ) ⇒ MY ×X Y (Y ×X Y ), ∼ MX,x , where X = Spa(R, R + ) admits a chart modeled on a sharp fs monoid P = for some strictly local ring R (see Proposition 2.3.13); and where Y → X is a standard Kummer étale cover with a Kummer chart u : P → Q with a sharp Q such that the order of G := coker(ugp ) is invertible in R (see Lemma 4.1.10). Note that P is sharp by assumption, and u is injective (see Definition 4.1.1). Let R := OY (Y ) and R := OY ×X Y (Y ×X Y ). By Definition 4.1.8 and f1 ,R#P $,f2 R#Q$, where f1 : R#P $ → R and Proposition 4.1.6, we have R ∼ = R⊗ f2 : R#P $ → R#Q$ are induced by the charts. By Proposition 4.1.6, we have R#P $ R#Q ⊕ G$ ∼ R#P $ R#(Q ⊕P Q)sat $ ∼ R ∼ = R⊗ = R [G]. = R⊗

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Let I (resp. I , resp. I ) be the ideal of R (resp. R , resp. R ) generated by the image of P \ {0} (resp. Q \ {0}, resp. Q \ {0}), which is a proper ideal because P and Q are sharp. Since I is contained in the maximal ideal of the strictly local ring R, and since u : P → Q is Kummer, I must be contained in all maximal ideals of R . The canonical morphism R/I → R /I is an isomorphism, because it is induced by compatibly completing both sides of the canonical isomorphism ∼ R ⊗f1 ,R#P $,f3 R → (R ⊗f1 ,R#P $,f2 R#Q$) ⊗f5 ,R#Q$,f4 R, where: • f1 : R#P $ → R and f2 : R#P $ → R#Q$ are given by the charts, as above; • f3 : R#P $ → R and f4 : R#Q$ → R are R-algebra homomorphisms defined by sending nonzero elements of P and Q to 0, respectively; and • f5 : R#Q$ → R ⊗f1 ,R#P $,f2 R#Q$ is the pullback of f1 under f2 . Since I is contained in all maximal ideals of R , this forces R to be local. Since R → R is finite (by Proposition 4.1.6(1)), R is strictly local as R is. Let V (resp. V , resp. V ) be the subgroup of elements in R × (resp. (R )× , resp. (R )× ) congruent to 1 modulo I (resp. I , resp. I ). Since R and R are strictly local, and since R ∼ = R [G], where the order of G is invertible in R and hence in R , we have compatible canonical isomorphisms ∼

R × /V → (R/I )× , ∼ (R )× /V → (R /I )× ∼ = (R/I )× ,

and ∼ (R )× /V → (R /I )× ∼ = ((R /I )[G])× ∼ = ((R/I )[G])× .

By Lemma 4.3.2, we know that 0 → R → R ⇒ R is exact. Since the injection R → R is finite, we can identify R as a subring of R over which R is integral. Hence, it is elementary that R × = (R )× ∩ R, and 0 → R × → (R )× ⇒ (R )× is exact. Moreover, we have I = I ∩ R and V = V ∩ R × , and 0 → V → V ⇒ V is also exact. By some diagram chasing, it suffices to show the exactness of 0 → MX (X)/V → MY (Y )/V ⇒ MY ×X Y (Y ×X Y )/V .

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Since MX (X) → R and MY (Y ) → R are associated with the pre-log structures P → R and Q → R , since u : P → Q is a Kummer chart, and since P and Q are sharp, we have compatible isomorphisms MX (X)/V ∼ = (R × /V ) ⊕ P ∼ = (R/I )× ⊕ P and MY (Y )/V ∼ = (R/I )× ⊕ Q. = ((R )× /V ) ⊕ Q ∼ Since the log structure MY ×X Y (Y ×X Y ) → R is associated with the pre-log structure Q ⊕ G → R ∼ = R [G] induced by the same Q → R as above and by the identity map G → G, we have MY ×X Y (Y ×X Y )/V ∼ = ((R/I )[G])× ⊕ Q. Accordingly, the above sequence can be identified with 0 → (R/I )× ⊕ P → (R/I )× ⊕ Q ⇒ ((R/I )[G])× ⊕ Q, where the double arrows are (x, q) → (x, q) and (x, q) → (xeq , q), with q denoting the image of q in G = Qgp /ugp (P gp ). Thus, it suffices to note that the sequence 0 → P → Q ⇒ Q ⊕ G, where the double arrows are q → (q, 0) and q → (q, q), is exact.



As a byproduct, let us show that representable presheaves are sheaves on Xkét . The log scheme version can be found in [19, Theorem 2.6], which can be further traced back to [25, Theorem 3.1]. Proposition 4.3.5 Let Y → X be a morphism of locally noetherian fs log adic spaces. Then the presheaf MorX ( · , Y ) on Xkét is a sheaf. Proof We follow the idea of [25, Theorem 3.1]. It suffices to show that the presheaf Mor( · , Y ) on Xkét is a sheaf, because MorX ( · , Y ) is just the sub-presheaf of sections of Mor( · , Y ) with compatible morphisms to X. We may assume that Y = Spa(R, R + ) is affinoid with a chart modeled on a sharp fs monoid P . + + We claim that the  presheaves F : T → Hom (R, R , (O T (T ), OT (T )) , G : T → Hom P , MT (T ) , H : T → Hom P , OT (T ) on Xkét , where the first Hom is in the category of Huber pairs, and where the latter two are in the category of monoids, are As for the case of F, it suffices to  all sheaves. show that F : T → Homcont R, OT (T ) is a sheaf, where the homomorphisms  are continuous ring homomorphisms; or that F : T → Hom R, OT (T ) is a sheaf, where we consider all ring homomorphisms. Consider any presentation

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∼ ker(OT (T )I → OT (T )J ). R∼ = Z[Ti ]i∈I /(fj )j ∈J of the ring R, so that F (T ) = Then F is a sheaf on Xkét as T → OT (T ) is (see Theorem 4.3.1). As for the cases of G and H, consider any presentation Zr≥0 ⇒ Zs≥0 → P → 0 of the finitely generated monoid P , which exists by [36, Theorem I.2.1.7]. Then G(T ) (resp. H(T )) is the equalizer of MT (T )s ⇒ MT (T )r (resp. OT (T )s ⇒ OT (T )r ). Hence, both presheaves are sheaves on Xkét as T → MT (T ) and T → OT (T ) are (see Theorem 4.3.1 and Proposition 4.3.4). By the claim just established, since Mor( · , Y ) (when Y = Spa(R, R + ) is modeled on P as above) is the fiber product of the morphisms F → H and G → H induced by P → R and MT (T ) → OT (T ), respectively, it is also a sheaf, as desired.  In the remainder of this subsection, we study coherent sheaves on the Kummer étale site. Definition 4.3.6 Let X be a locally noetherian fs log adic space. (1) An OXkét -module F is called an analytic coherent sheaf if it is isomorphic to the inverse image of a coherent sheaf on the analytic site of X. (2) An OXkét -module F is called a coherent sheaf if there exists a Kummer étale covering {Ui → X}i such that each F|Ui is an analytic coherent sheaf. The following results are analogues of [25, Proposition 6.5], the proof of which is completed in [35, Proposition 3.27]. Theorem 4.3.7 Suppose that X is an affinoid noetherian fs log adic space. Then H i (Xkét , F) = 0, for all i > 0, in the following two situations: (1) F is an analytic coherent OXkét -module. (2) F is a coherent OXkét -module, and X is over an affinoid field (k, k + ). Proof (1) As in the proof of Theorem 4.3.1, by Lemma 4.2.6 and Proposition A.10, it ˇ suffices to show the exactness of the Cech complex • CF (Y /X) : 0 → F(X) → F(Y ) → F(Y ×X Y ) → · · · ,

where X is affinoid with a sharp fs chart P , and where Y → X is a standard Kummer cover. By Proposition 4.1.6, the morphisms Y → X, Y ×X Y → X, Y ×X Y ×X Y → X, etc are finite, and hence • CF (Y /X) ∼ = C • (Y /X) ⊗OX (X) F(X),

where C • (Y /X) is as in Lemma 4.3.2. Since the contracting homotopy used in the proof of Lemma 4.3.2 (based on the proof of [35, Lemma 3.28]) is OX (X)• (Y /X) is also exact, as desired. linear, CF

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(2) First assume that X is modeled on a sharp fs monoid P . By definition, there exists a Kummer étale covering {Ui → X}i such that each F|Ui is analytic coherent. By Lemma 4.2.6 and Proposition A.10, we may assume that F|U 1 is analytic coherent, where U = X n for some n invertible in k. Let G :=  1 gp gp j ( n P ) /P . Since H (U ×X · · · ×X U )két , F = 0, for all j > 0, by (1),  • it suffices to show that H i CF (U/X) = 0, for all i > 0. As in the proof of Lemma 4.3.2, by Proposition 4.1.6, OXkét (U ×X · · · ×X U ) ∼ = OXkét (U ) ⊗k k[G] ⊗k · · · ⊗k k[G], and we can identify the complex F(U ) → F(U ×X U ) → F(U ×X U ×X U ) → · · ·  with the complex computing the group cohomology H i G, F(U ) . Since |G| is invertible  in k as n is, and since F(U ) is a k-vector space, we have H i G, F(U ) = 0, for all i > 0. Let εét : Xkét → Xét denote the natural projection of sites. Then the argument above shows that R j εét,∗ (F) = 0, for all j > 0, and that εét,∗ (F) is a coherent sheaf on Xét . Since these statements are étale local in nature, they extend to all X considered in the statement of the theorem, by Proposition 2.3.13. Thus, we  have H i (Xkét , F) ∼ = H i Xét , εét,∗ (F) = 0, as desired, by Proposition A.10.  Kummer étale descent of objects (coherent sheaves, log adic spaces, etc) are usually not effective, mainly because fiber products of Kummer étale covers do not correspond to fiber products of structure rings. Here is a standard counterexample. Example 4.3.8 Let k be a nonarchimedean field. As in Example 2.2.21, consider the unit disc D = Spa(k#T $, Ok #T $) equipped with the log structure modeled on the chart Z≥0 → k#T $ : 1 → T . By Proposition 4.1.6, we have a Galois standard Kummer étale cover fn : D → D corresponding to the chart Z≥0 → Z≥0 : 1 → n, where n is invertible in k, with Galois group μn . Then the ideal sheaf I of the origin, a μn -invariant invertible sheaf on D, does not descend via fn . Kummer étale descent of morphisms are more satisfactory. Proposition 4.3.9 Let X be a locally noetherian fs log adic space, and let f : Y → X be a Kummer étale cover. Let pr1 , pr2 : Y ×X Y → Y denote the two projections. Suppose that E and F are analytic coherent OXkét -modules; and that g : f ∗ (E) → f ∗ (F) is a morphism on Y such that pr∗1 (g ) = pr∗2 (g ) on Y ×X Y . Then there exists a unique morphism g : E → F such that f ∗ (g) = g . Proof By Lemma 4.2.6 and Proposition A.10, we may assume that X is affinoid and that Y → X is a standard Kummer cover. Let A := OX (X), B := OXkét (Y ), C := OXkét (Y ×X Y ), M := E(X), and N := F(X). We need to show that

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0 → HomA (M, N ) → HomB (B ⊗A M, B ⊗A N) → HomC (C ⊗A M, C ⊗A N) is exact, and where the third arrow is the difference between two pullbacks as usual. Equivalently, we need to show that 0 → HomA (M, N ) → HomA (M, B ⊗A N) → HomA (M, C ⊗A N) is exact. By the left exactness of HomA (M, · ), we are reduced to showing that the sequence 0 → N → B ⊗A N → C ⊗A N is exact. But this is just the first three • (Y /X) in the proof of Theorem 4.3.7(1). terms in the complex CF  To wrap up the subsection, let us introduce a convenient basis for the Kummer étale topology. Lemma 4.3.10 Let X be a locally noetherian fs log adic space. Let B be the full subcategory of Xkét consisting of affinoid adic spaces V with fs global charts. Then ∼ ∼ → B is a basis for Xkét , and we have an isomorphism of topoi Xkét B∼ . Proof By Artin et al. [1, III, 4.1], it suffices to show that every object in Xkét has a covering by objects in B. But this is clear.  Lemma 4.3.11 Let X be a locally noetherian fs log adic space, and let B be as in Lemma 4.3.10. Suppose that F is a rule that functorially assigns to each V ∈ B ∼ a finite OXkét (V )-module F(V ) such that F(V ) ⊗OV (V ) OV (V ) → F(V ) for all V → V that are either étale morphisms or standard Kummer étale covers. Then F defines an analytic coherent sheaf on Xkét . Proof This follows from Propositions 4.3.9 and A.10, and Lemma 4.3.10.



4.4 Descent of Kummer étale Covers Definition 4.4.1 Let X be a locally noetherian fs log adic space X. Let Xfkét denote the full subcategory of Xkét consisting of log adic spaces that are finite Kummer étale over X. Let Fkét denote the fibered category over the category of locally noetherian fs log adic spaces such that Fkét(X) = Xfkét . The goal of this subsection is to show that Kummer étale covers satisfy effective descent in Fkét. We first study Xfkét when X is as in Examples 2.2.8 and 2.2.9. Definition 4.4.2 (1) A log geometric point is a log point ζ = (Spa(l, l + ), M, α) (cf. Examples 2.2.8 and 2.2.9) such that: (a) l is a complete separably closed nonarchimedean field; and (b) if M := Γ (Spa(l, l + ), M), then M = M/ l × is uniquely n-divisible (see Definition 2.2.14) for all positive integers n invertible in l.

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(2) Let X be a locally noetherian fs log adic space. A log geometric point of X is a morphism of log adic spaces η : ζ → X from a log geometric point ζ . (3) Let X be a locally noetherian fs log adic space. A Kummer étale neighborhood of a log geometric point η : ζ → X is a lifting of η to some composition φ

ζ → U → X in which φ is Kummer étale. Construction 4.4.3 For each geometric point ξ : Spa(l, l + ) → X, let us construct ξ above it (i.e., the morphism  ξ → X of underlying some log geometric point  adic spaces factors through ξ → X) as follows. By Proposition 2.3.13, up to étale localization on X, we may assume that X admits a chart modeled on a sharp fs monoid P , so that we have a strict closed immersion X → X#P $ as in Remark 2.3.3. We equip Spa(l, l + ) with the log structure P log associated with the pre-log structure given by the composition of P → OX (X) → l, so that (Spa(l, l + ), P log ) is an fs log point with a chart given by P → l. We shall still denote this fs log point by ξ . For each positive integer m, let P → m1 P be as in Definition 4.1.5. Consider 1

ξ ( m ) := (Spa(l, l + ) ×X#P $ X# m1 P $)red , equipped with the natural log structure modeled on 1 m

1 1 (m ) m P . Note that ξ

might differ

from the ξ in Definition 4.1.5, because we are taking the reduced subspace, so that 1 the underlying adic space of ξ ( m ) is still isomorphic to Spa(l, l + ). Then  ξ := lim ξ ( m ) ← − 1

m

where the inverse limit runs through all positive integers m invertible in l, is a log ξ is isomorphic to Spa(l, l + ), geometric point above ξ . The underlying adic space of  endowed with the natural log structure modeled on PQ≥0 := lim − →m

1 mP,

where the direct limit runs through all positive integers m invertible in l. Lemma 4.4.4 Let ζ → X be a log geometric point of a locally noetherian fs log adic space. Then the functor Sh(Xkét ) → Sets : F → Fζ := lim F(U ) from the − → category of sheaves on Xkét to the category of sets, where the direct limit is over Kummer étale neighborhoods U of ζ , is a fiber functor. The fiber functors defined by log geometric points form a conservative system. Proof By Proposition 2.3.32 and Remark 4.1.4, the category of Kummer étale neighborhoods of ζ is filtered, and hence the first statement follows. Since every point of X admits some geometric point and hence some log geometric point above it (see Construction 4.4.3), and since every object U in Xkét is covered by liftings of log geometric points of X, the second statement also follows. 

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Definition 4.4.5 For each profinite group G, let G-FSets denote the category of finite sets (with discrete topology) with continuous actions of G. Definition 4.4.6 Let l be a separably closed field. For each positive integer m, let μm (l) denote the group of m-th roots of unity in l. Let μ∞ (l) := lim μm (l) − →m Z (1)(l) := lim μm (l), where the limits run through all positive integers m and  ← −m Z(1)(l) instead of  Z (1)(l). When invertible in l. When char(l) = 0, we shall write  there is no risk of confusion in the context, we shall simply write μm , μ∞ , and  Z (1), without the symbols (l). Proposition 4.4.7 Let ξ = (Spa(l, l + ), M) be an fs log point with l complete (by our convention) and separably closed. Let M := Mξ and so M ∼ = M/ l × . Let  ξ be a log geometric point constructed as in Construction 4.4.3. Then the functor Fξ : Y → Homξ ( ξ , Y ) induces an equivalence of categories  gp ξfkét ∼ Z (1)(l) −FSets. = Hom M ,  Proof For simplicity, we shall omit the symbols (l) as in Definition 4.4.6. Let P := ∼ M, a sharp and fs monoid. By Lemma 2.1.10, we have some splitting M → l × ⊕ P ∼

1⊕Id

such that P → l × ⊕ P → M defines a chart for ξ . For each m invertible in l, the ∼

1 (m )

Id ⊕[m]

∼

cover ξ → ξ is given by M → l × ⊕ P → l × ⊕ P → M. Note that any finite Kummer étale cover of ξ is a finite disjoint union of fs log adic spaces of the form ξQ := ξ ×ξ #P $ ξ #Q$, where P → Q is a Kummer homomorphism of sharp fs monoids whose cokernel is annihilated by some integer invertible in l. We have Fξ (ξQ ) = Morξ ( ξ , ξQ ) ∼ = Homl × ⊕P (l × ⊕ Q, l × ⊕ PQ≥0 ) ∼ = Hom(Qgp /P gp , μ∞ ). The last group has a natural transitive action of  Autξ ( ξ) ∼ = HomP (PQ≥0 , l × ) ∼ = Hom (PQ≥0 )gp /P gp , μ∞   ∼ Z (1) . = lim Hom P gp ⊗Z ( m1 Z/Z), μm ∼ = Hom P gp ,  ← − m

 gp Hence, Fξ is indeed a functor from ξfkét to Hom M ,  Z (1) −FSets. Let us verify that Fξ is fully faithful. By working with connected components, it suffices to show that, for any Q1 and Q2 , the natural map  gp gp Morξ (ξQ1 , ξQ2 ) → Hom Hom(Q1 /P gp , μ∞ ), Hom(Q2 /P gp , μ∞ ) (4.4.8)

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is bijective. Note that Morξ (ξQ1 , ξQ2 ) ∼ = HomP (Q2 , l × ⊕ Q1 ). Since PQ≥0 is uniquely divisible, the sharp fs Q1 and Q2 monoids can be viewed as submonoids of PQ≥0 . If Q2 ⊂ Q1 , then both sides of (4.4.8) are zero. Otherwise, Q2 ⊂ Q1 , and (4.4.8) sends the morphism induced by Q2 → Q1 in HomP (Q2 , l × ⊕ gp gp Q1 ) to the homomorphism induced by restriction from Q1 /P gp to Q2 /P gp . Consequently, (4.4.8) is bijective, because both sides of (4.4.8)  gp are principally homogeneous under compatible actions of Autξ (ξQ2 ) ∼ = Hom Q2 /P gp , μ∞ . Finally, let us verify that Fξ is essentially surjective. Since any discrete finite   Z (1) ∼ set S with a continuous action of Hom P gp ,  = Hom (PQ≥0 )gp /P gp , μ∞ is a disjoint union of orbits, action on S is transitive. Then S  we may assume the is a quotient space of Hom (PQ≥0 )gp /P gp , μ∞ , which corresponds by Pontryagin duality to a finite subgroup G ⊂ (PQ≥0 )gp /P gp . Let Q denote the preimage of G in PQ≥0 . Then Qgp /P gp ∼  = S, as desired. = G and Fξ (ξQ ) ∼ Proposition 4.4.9 Let (X, MX ) be a locally noetherian fs log adic space. Let ξ = Spa(l, l + ) be a geometric point of X, and let X(ξ ) be the strict localization of X at ξ , with its log structure pulled back from X. Without loss of generality, let us assume that l ∼ = κ(x), the completion of a separable closure of the residue field κ(x) of OX,x , for some x ∈ X. Let M := MX,ξ and so M ∼ = M/ l × . Let us view ξ and X(ξ ) as log adic spaces by equipping them with the log structures pulled back ξ be the log geometric point over ξ constructed as in Construction 4.4.3. from X. Let  Then the functor Hξ : Y → MorX ( ξ , Y ) induces an equivalence of categories  gp X(ξ )fkét ∼ Z (1)(l) − FSets. = Hom M ,  In addition, we have Hξ = Fξ ◦ ι−1 , where Fξ is as in Proposition 4.4.7, and ι−1 : X(ξ )fkét → ξfkét is the natural pullback functor defined by ι : ξ → X(ξ ). Note that, if x is an analytic point of X, then Proposition 4.4.9 follows immediately from Proposition 4.4.7, because ξ ∼ = X(ξ ) in this case. Nevertheless, the proof below works for non-analytic points as well. Proof (of Proposition 4.4.9) It suffices to show that ι−1 is an equivalence of categories. Write P = M and X(ξ ) = Spa(R, R + ). By Lemma 2.1.10, we can ∼

1⊕Id

∼

choose a splitting R × ⊕ P → M such that P → R × ⊕ P → M defines a chart for X(ξ ). Note that objects in X(ξ )fkét (resp. ξfkét ) are finite disjoint unions of fs log adic spaces of the form X(ξ )Q := X(ξ ) ×X(ξ )#P $ X(ξ )#Q$

(4.4.10)

(resp. ξQ ), where P → Q is a Kummer homomorphism of sharp monoids. Then ι−1 sends X(ξ )Q to ξQ , and hence is essentially surjective. To see that ι−1 is fully faithful, it suffices to show that the canonical map MorX(ξ ) (X(ξ )Q1 , X(ξ )Q2 ) → Morξ (ξQ1 , ξQ2 )

(4.4.11)

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∼ Spa(RQ , R + ), for i = 1, 2, where is bijective. By definition, we have X(ξ )Qi = i Qi RQi = R ⊗R#P $ R#Q$ ∼ = R ⊗R[P ] R[Q] are also strictly local rings with residue field l. Therefore, × MorX(ξ ) (X(ξ )Q1 , X(ξ )Q2 ) ∼ ⊕ Q1 ) = HomP (Q2 , RQ 1

∼ = HomP (Q2 , l × ⊕ Q1 ) ∼ = Morξ (ξQ1 , ξQ2 ), and hence the map (4.4.11) is bijective, as desired.



Now, we are ready to prove the main result of this subsection; i.e., Kummer étale covers satisfy effective descent in the fibered category Fkét. Theorem 4.4.12 Let X be a locally noetherian fs log adic space, and let f : Y → X be a Kummer étale cover. Let pr1 , pr2 : Y ×X Y → Y denote the two projections. ˘ ∼ −1 ˘ Suppose that Y˘ ∈ Yfkét and that there exists an isomorphism pr−1 1 (Y ) → pr2 (Y ) satisfying the usual cocycle condition. Then there exists a unique X˘ ∈ Xfkét (up to isomorphism) such that Y˘ ∼ = X˘ ×X Y . Proof By étale descent (see Proposition A.10), by étale localization on X, it suffices to prove the theorem in the case where X is affinoid with a sharp fs chart P , and where Y → X is a standard Kummer étale cover induced by a Kummer homomorphism of sharp monoids u : P → Q, with G := Qgp /ugp (P gp ) a finite group of order invertible in OX . By Proposition 4.1.6, up to further étale localization on X, we may assume that the morphism Y → X is a Galois cover with Galois group Γ := Hom(G, OX (X)× ); that |G| is invertible in OX , and OX (X)× contains all the |G|-th roots of unity; and that Y ×X Y ∼ = ΓX ×X Y . In this case, the descent datum is equivalent to an action of Γ on Y˘ over X lifting the action of Γ on Y over X. Let ˘ S˘ + ), M ˘ ). By Lemma 4.1.7, us write X = (Spa(R, R + ), MX ) and Y˘ = (Spa(S, Y ˘ R˘ + ) := (S˘ Γ , (S˘ + )Γ ) is a Huber pair, and X˘ := Spa(R, ˘ R˘ + ) is a noetherian (R, adic space finite over X. Moreover, the morphism Y˘ → X˘ is finite, open, surjective, and invariant under the Γ -action on Y˘ . The étale sheaf of monoids MX˘ defined by  Γ MX˘ (U ) := MY˘ (Y˘ ×X˘ U ) , for each U ∈ X˘ ét , is fine and saturated, and defines ˘ We claim that the log adic space X˘ thus obtained gives the a log structure of X. desired descent. Let us first verify that the canonical morphism Y˘ → X˘ ×X Y induced by the structure morphisms Y˘ → X˘ and Y˘ → Y is an isomorphism. Since the morphism is between spaces that are finite over X, and since the formation of Γ -invariants is compatible with (strict) base change (as |Γ | is invertible in OX ), we may assume that X = X(ξ ) is strictly local, and so is Y ∼ = X(ξ )Q = X(ξ ) ×X(ξ )#P $ X(ξ )#Q$ (as in the proof of Proposition 4.4.9). Without loss of generality, we may assume that ˘ for some Kummer homomorphism of fs Y˘ ∼ = X(ξ )Q˘ := X(ξ ) ×X(ξ )#P $ X(ξ )#Q$ ˘ (as in (4.4.10), but without assuming that Q ˘ is sharp), which monoids u˘ : P → Q ˘ is the composition of u : P → Q with some homomorphism  gp Q → Q. Under ∼  the equivalence of categories Hξ : X(ξ )fkét = Hom P , Z (1)(l) −FSets as in

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˘ gp /u˘ gp (P gp ), μ∞ ) Proposition 4.4.9, Y˘ → X corresponds to the set Γ˘ := Hom(Q with a Γ -action; Y → X corresponds to Γ itself (with its canonical Γ -action); and Y˘ → Y corresponds to a surjective Γ -equivariant map Γ˘  Γ . Since Y˘ → gp X is Kummer, we have (MΓ˘ )gp = (M ˘ )Γ , and hence X˘ ∼ = X(ξ )P˘ for the fs Y Y gp gp gp ∼ Γ˘ /Γ , by explicitly ˘ ∩ P˘ such that Hom(P˘ /u˘ (P ), μ∞ ) = monoid P˘ = Q ˘ Γ using the identifications in the proof computing R˘ = S˘ Γ ∼ = (R ⊗R[P ] R[Q]) of Proposition 4.4.9. In particular, X˘ → X is finite Kummer étale, and Y˘ → X˘ corresponds to the quotient Γ˘ → Γ˘ /Γ under Hξ . Since Γ˘ is an abelian group, the canonical map Γ˘ → (Γ˘ /Γ )×Γ is bijective, and hence the corresponding canonical morphism Y˘ → X˘ ×X Y is indeed an isomorphism. ˘ ˘ Consequently, Y˘ ∼ = X˘ ×X#P = X˘ ×X Y ∼ ˘ $ X#Q$ → X is finite Kummer étale. By construction, X˘ → X is also finite Kummer (firstly by assuming that X is strictly local as above, and then by extending the identifications of charts over étale neighborhoods of X in general). By Lemma 4.1.13, it remains to show that X˘ → X is log étale. Since Y˘ → Y is log étale, and since Y → X is a Kummer étale covering, this follows from Proposition 4.2.8, as desired.  Corollary 4.4.13 Let X be a locally noetherian fs log adic space, and let f : Y → X be a finite Kummer étale cover. Let Γ be a finite group which acts on Y by morphisms over X. Then the canonical morphisms Y → Z := Y /Γ → X induced by f (by Lemma 4.1.7) are both finite Kummer étale covers. Proof By Lemma 4.1.7, both morphisms Y → Z and Z → X are finite, and Y → Z is finite Kummer. By Lemma 4.1.13 and Proposition 4.1.15, it suffices to show that Z → X is finite Kummer étale. Then the first projection f˘ : Y˘ := Y ×X Y → Y is a pullback of Y → X, which inherits an action of Γ . By [18, Lemma 1.7.6], under the noetherian hypothesis, the formation of quotients by Γ as in Lemma 4.1.7 is compatible with base changes under étale morphisms of affinoid adic spaces. By Proposition 4.1.6 and Remark 4.1.18, up to étale localization on X, we may assume that Y → X is a composition of a finite étale morphism f1 and a standard Kummer étale cover f2 as in Proposition 4.1.6, in which case Y˘ is a disjoint union of sections of f˘. Then Γ acts on Y˘ by permuting such sections, and we have a quotient Z˘ := Y˘ /Γ → Y , which is clearly finite Kummer étale. Moreover, the pullbacks of Z˘ → Y to Y˘ along the two projections are isomorphic to each other by interchanging the factors, and hence Z˘ → Y descends to a finite Kummer étale cover of X, by Theorem 4.4.12. We claim that this cover is canonically isomorphic to Z → X. Since the set of sections of f˘ : Y˘ → Y is a disjoint union of subsets formed by the sections of pullback of the finite étale morphism f1 , we can reduce the claim to the extremal cases where either f = f1 is finite étale or f = f2 is standard Kummer étale. In the former case, the claim follows from the usual theory for finite étale covers of schemes, as in [12, V]. In the latter case, the claim follows from Proposition 4.1.6(4). 

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Definition 4.4.14 Let X be a locally noetherian fs log adic space, and let Λ be a commutative ring. (1) A sheaf F on Xkét is called a constant sheaf of sets (resp. constant sheaf of Λmodules) if it is the sheafification of a constant presheaf given by some set S (resp. some Λ-module M). (2) A sheaf F on Xkét is called locally constant if there exists a Kummer étale covering {Ui }i∈I → X such that all F|Ui are constant sheaves. We denote by Loc(Xkét ) the category of locally constant sheaves of finite sets on Xkét . Theorem 4.4.15 Let X be a locally noetherian fs log adic space. The functor φ : Xfkét → Loc(Xkét ) : Y → MorX ( · , Y ) is an equivalence of categories. Moreover: (1) Fiber products exist in Xfkét and Loc(Xkét ), and φ preserves fiber products. (2) Categorical quotients by finite groups exist in Xfkét and Loc(Xkét ), and φ preserves such quotients. Proof By Proposition 4.3.5, representable presheaves on Xkét are sheaves. By Proposition 4.1.6 and Remark 4.1.18, any Y ∈ Xfkét is Kummer étale locally (on X) a disjoint union of finitely many copies of X. Hence, MorX ( · , Y ) is indeed a locally constant sheaf of finite sets, and the functor φ is defined. The functor φ is fully faithful for formal reasons. Since any locally constant sheaf of finite set is Kummer étale locally represented by objects in Xfkét , these objects glue to a global object Y by Theorem 4.4.12 and the full faithfulness of φ. This shows that φ is also essentially surjective, as desired. As for the statements (1) and (2), by Kummer étale localization, we just need to note that the statements become trivial after replacing the source and target of the functor φ with the categories of finite disjoint unions of copies of X and of constant sheaves of finite sets, respectively.  Next, let us define the Kummer étale fundamental groups. Lemma 4.4.16 Let X be a connected locally noetherian fs log adic space, and η : ζ → X a log geometric point. Let FSets denote the category of finite sets. Consider the fiber functor F : Xfkét → FSets : Y → Yζ := MorX (ζ, Y ).

(4.4.17)

Then Xfkét together with the fiber functor F is a Galois category. Proof We already know that the final object, fiber products (see Proposition 4.1.14), categorical quotients by finite groups (see Corollary 4.4.13), and finite coproducts exist in Xfkét (and FSets). It remains to verify the following conditions: (1) F preserves fiber products, finite coproducts, and quotients by finite groups. (2) F reflexes isomorphisms (i.e., F (f ) being an isomorphism implies f also being an isomorphism).

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(We refer to [12, V, 4] for the basics on Galois categories.) As for condition (1), since F is defined Kummer étale locally at the log geometric point ζ , it suffices to verify the condition after restricting F to the category of finite disjoint unions of X, in which case the condition clearly holds. As for condition (2), note that F factors through the equivalence of categories φ in Theorem 4.4.15 and induces the stalk functor Loc(Xkét ) → FSets : F → Fζ := lim F(U ), − → where the direct limit is over Kummer étale neighborhoods U of ζ . Since X is connected, the stalk functors at any two log geometric points are isomorphic. Thus, whether f is an isomorphism can be checked at just one stalk.  Corollary 4.4.18 Let X and ζ → X be as in Lemma 4.4.16. Then the fiber functor F in (4.4.17) is prorepresentable. Let π1két (X, ζ ) be the automorphism group of F . Then F induces an equivalence of categories ∼

Xfkét → π1két (X, ζ )−FSets,

(4.4.19)

which is the composition of the equivalence of categories φ in Theorem 4.4.15 with the equivalence of categories ∼

Loc(Xkét ) → π1két (X, ζ )−FSets

(4.4.20)

induced by the stalk functor F → Fζ . Remark 4.4.21 In Corollary 4.4.18, since stalk functors at any two log geometric points ζ and ζ are isomorphic, the fundamental groups π1két (X, ζ ) and π1két (X, ζ ) are isomorphic. We shall omit ζ from the notation when the context is clear. Corollary 4.4.22 Let (X, MX ), ξ = Spa(l, l + ), X(ξ ), and M := MX,ξ be as in Proposition 4.4.9. In particular, the underlying adic spaces of ξ (resp. X(ξ )) is a geometric point (resp. a strictly local adic space). Then we have   gp π1két X(ξ ) ∼ Z (1)(l) . = π1két (ξ ) ∼ = Hom M ,  gp ∼ Since M is sharp and fs, we have M = Zr for some r, and we obtain a  r két ∼  noncanonical isomorphism π1 (ξ ) = Z (1)(l) .

Remark 4.4.23 For any connected locally noetherian fs log adic space X and any log geometric point ξ of X, the natural inclusion from the category of finite étale covers to that of finite Kummer étale covers is fully faithful, and hence induces a canonical surjective homomorphism π1két (X, ξ ) → π1ét (X, ξ ) (see [12, V, 6.9]). Example 4.4.24 Let (k, k + ) be an affinoid field, and let s = (Spa(k, k + ), M) be an fs log point as in Example 2.3.14. Let s = (Spa(K, K + ), M) be a geometric point over s, where K is the completion of a separable closure k sep of k, and let  s

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be a log geometric point over s. Then we have a canonical short exact sequence 1 → π1két (s, s) → π1két (s, s) → π1ét (s, s) → 1,  gp where π1két (s, s) ∼ Z (1)(k sep ) (as we have seen in Corollary 4.4.22) = Hom M ,  and π1ét (s, s) ∼ = Gal(k sep /k). If s is a split fs log point as in Example 2.3.15, then any choice of a Gal(k sep /k)-equivariant splitting of M → M ∼ = M/K × also splits this exact sequence, inducing an isomorphism  gp ∼ π1két (s, s) → Hom M ,  Z (1)(k sep )  Gal(k sep /k). Example 4.4.25 Let (k, k + ) be an affinoid field. Consider 0 = Spa(k, k + ), the point of Spa(k#Z≥0 $, k + #Z≥0 $) ∼ = Spa(k#T $, k + #T $) defined by T = 0. Let us ∂ denote by 0 the log adic space with underlying adic space 0 and with its log structure pulled back from Spa(k#Z≥0 $, k + #Z≥0 $), which is the split fs log point ∂ ∂ (X, MX ) = (Spa(k, k + ), O× Xét ⊕ (Z≥0 )X ) as in Example 2.3.15. Let 0 and 0 be defined over 0∂ as in Example 4.4.24 (with s = 0∂ there). Then π1két (0∂ ,  0∂ ) ∼ Z (1)(k sep )  Gal(k sep /k). = For each n invertible in k, and each r ≥ 1, we have a Z/n-local system J∂r,n on 0∂két defined by the representation of π1két (0∂ ,  0∂ ) on (Z/n)r such that a topological generator of  Z (1)(k sep ) acts as the standard upper triangular principal unipotent matrix Jr and Gal(k sep /k) acts diagonally on (Z/n)r and trivially on ker(Jr − 1). (The local system thus defined is independent of the choice of the generator of  Z (1)(k sep ) up to isomorphism.) Moreover, for each m ≥ 1 with m invertible in k, we also have the Z/n-local system K∂m,n defined by the representation of π1két (0∂ ,  0∂ ) sep sep  induced from the trivial representation of mZ (1)(k )  Gal(k /k) on Z/n. (These local systems will be useful for defining quasi-unipotent nearby cycles in Sect. 6.4.) Example 4.4.26 Let s, s, and  s be as in Example 4.4.24, and let f : X → s be any strict lft morphism of log adic spaces. Let ξ be a geometric point of X above ξ be a log geometric point above ξ and  s, and let  s. Let X(ξ ) denote the strict localization of X at ξ . Then, by Proposition 4.4.9 and  gp Corollary 4.4.22, we have két (s, sep ) . ∼  π1két (X(ξ ),  ξ) ∼ π s) Hom M , Z (1)(k = 1 = Lemma 4.4.27 Let (X, MX ), ξ = Spa(l, l + ), X(ξ ), x, and M := MX,ξ be as in Proposition 4.4.9. Let εét : Xkét → Xét be the natural projection of sites, as before. Then, for each sheaf F of finite abelian groups on Xkét , we have  i  ξ ), Fξ . R εét,∗ (F) ξ ∼ = H i π1két (ξ, 

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 Proof By definition, we have R i εét,∗ (F) ξ ∼ = lim H i (Ukét , F), where the direct − → limit is over the filtered category of étale neighborhoods iU : ξ → U in X. By Proposition 2.3.13, up to étale localization, we may assume that X admits a chart modeled on P := M. Consider the morphism i −1 : lim Ukét → xkét , − → where the direct limit of sites lim Ukét is as in [1, VI, 8.2.3], induced by the − → morphisms iU−1 : Ukét → xkét . Since each Kummer étale covering of ξ can be further covered by some standard Kummer étale covers induced by n-th multiple maps [n] : P → P , for some integers n ≥ 1 invertible in l, and since coverings of the latter kind are in the essential image of i −1 , by Proposition 4.4.7, we have ∼ ∼ lim U ∼ of the associated topoi. Consequently, we also have an equivalence ξkét = − két ←  i ∼ lim H (Ukét , F) = H i ξkét , ξ −1 (F) ∼ ξ ), Fξ , as desired.  = H i π1két (ξ,  − → Let (X, MX ) be a locally noetherian fs log adic space, and let MXkét be as in Proposition 4.3.4. For each positive integer n invertible in OX , let [n]

× μn,két := ker(O× Xkét → OXkét )

and [n]

× μn,ét := ker(O× Xét → OXét ).

By Corollary 4.3.3, we have canonical isomorphisms ∼

∗ εét (μn,ét ) → μn,két

and ∼

μn,ét → εét,∗ (μn,két ). We shall omit the subscripts “két” and “ét”, and write simply μn , when there is no risk of confusion. Consider the sequence gp

[n]

gp

1 → μn → MXkét → MXkét → 1 on Xkét , which is exact by comparing stalks at log geometric points of X (cf. Construction 4.4.3), whose pushforward under εét : Xkét → Xét induces a long exact sequence gp [n]

gp

1 → μn → MXét → MXét → R 1 εét,∗ (μn ), which is compatible with the Kummer exact sequence

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× 1 → μn → O × Xét → OXét → 1

and induces a canonical morphism gp  gp MX nMX → R 1 εét,∗ (μn )

(4.4.28)

on Xét , which is nothing but the inverse of the isomorphism in Lemma 4.4.27, by comparing stalks at geometric points of X. Therefore, we obtain the following: Lemma 4.4.29 The above morphism (4.4.28) is an isomorphism and, for each i, the canonical morphism ∧i R 1 εét,∗ (μn ) → R i εét,∗ (μn ) is an isomorphism.

4.5 Localization and Base Change Functors In this subsection, we study the behavior of sheaves on Kummer étale sites under certain direct image and inverse image functors. (The readers are referred to [1, IV] for general notions concerning sites, topoi, and the functors and morphisms among them.) For any morphism f : Y → X of locally noetherian fs log adic spaces, since pulling back by f respects fiber products, we have a morphism of topoi −1 ∼ ∼ (fkét , fkét,∗ ) : Ykét → Xkét .

Concretely, we have the direct image (or pushforward) functor  fkét,∗ : Sh(Ykét ) → Sh(Xkét ) : F → U → fkét,∗ (F)(U ) := F(U ×X Y ) , and the inverse image (or pullback) functor −1 fkét : Sh(Xkét ) → Sh(Ykét )

sending G ∈ Sh(Xkét ) to the sheafification of V → lim G(U ), where U runs − →U through the objects in Ykét such that V → X factors through f −1 (U ) → Y . It is −1 −1 formal that fkét,∗ is the right adjoint of fkét . Moreover, fkét is exact, and fkét,∗ is left exact. For any Kummer étale morphism f : Y → X, the functor fkét,∗ is also called the −1 −1 base change functor, while the functor fkét is simply fkét (F)(U ) := F(U ), because any object U of Ykét gives an object in Xkét by composition with f . Moreover, we have the localization functor fkét,! : Sh(Ykét ) → Sh(Xkét ).

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141

sending F ∈ Sh(Ykét ) to the sheafification of the presheaf p

fkét,! (F) : U →

!

F(U, h),

h∈MorX (U,Y ) h

where F(U, h) means the value of F on the object U → Y of Xkét . We shall also denote by fkét,! : ShAb (Ykét ) → ShAb (Xkét ) the induced functor between the categories of abelian sheaves, in which case the above coproduct becomes a direct −1 sum. It is also formal that fkét,! is left adjoint to fkét , and that fkét,! is right exact. Lemma 4.5.1 Let f : V → W be a finite Kummer étale morphism in Xkét . If f has a section g : W → V , then there exists a finite Kummer étale morphism W → W  ∼ h : V → W W such that the composition h◦g is the natural and an isomorphism inclusion W → W W . Proof We may assume that W is connected. Let G := π1két (W ) (see Remark 4.4.21). Via the equivalence (4.4.19), the finite Kummer étale cover ∼ V → W (resp. W → W ) corresponds to a finite set S (resp. a singleton S0 ) with a continuous G-action (resp. the trivial action), and the section g : W → V corresponds to a G-equivariant map  g∗ : S0 → S. This gives rise to a G-equivariant decomposition S = g∗ (S0 ) S , and hence to the desired decomposition  ∼ h : V → W W , by Corollary 4.4.18.  Proposition 4.5.2 Given any finite Kummer étale morphism f : Y → X of locally noetherian fs log adic spaces, we have a natural isomorphism ∼

fkét,! → fkét,∗ : ShAb (Ykét ) → ShAb (Xkét ). Consequently, both functors are exact. Proof Let F be an abelian sheaf on Ykét . For any U ∈ Xkét , each morphism h in MorX (U, Y ) induces a section U → U ×X Y of the natural projection  U ×X Y → ∼ U . By Lemma 4.5.1, we obtain a decomposition U × Y U U identifying = X  U → U ×X Y with U → U U , which gives rise to a canonical map F(U ) → F(U ×X Y ) because F is a sheaf. By combining such maps, we obtain a map of presheaves 

 p fkét,! (F) (U ) = ⊕h∈MorX (U,Y ) F(U, h) → fkét,∗ (F) (U ) = F(U ×X Y ),

which induces a canonical morphism fkét,! → fkét,∗ by sheafification. By the above construction, it remains to show that, étale locally on U , there exists a finite Kummer étale cover V → U such that MorX (V , Y ) is a finite set and such that the sections V → V ×X Y given by h ∈ MorX (V , Y ) induces  ∼ h∈MorX (V ,Y ) V → V ×X Y . Note that this is true in the special case where Y → X is strictly finite étale, because Y is étale locally on X a finite disjoint

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union of copies of X. In general, up to étale localization on X, we may assume that X is affinoid and modeled on a sharp fs monoid P ; and that Y → X factors as a composition Y → XQ := X ×X#P $ X#Q$ → X, where the first morphism is strictly finite étale, and where the second morphism is the standard Kummer étale cover induced by a Kummer homomorphism u : P → Q of sharp fs monoids such that the order of G = Qgp /ugp (P gp ) is invertible in OY . By Proposition 4.1.6, Y ×X XQ ∼ = Y ×X X#G$ → XQ is = Y ×XQ (XQ ×X XQ ) ∼ strictly finite étale. Hence, as explained above, there exists a finite Kummer étale  cover V → U ×X XQ such that h∈MorX (V ,Y ×X XQ ) V ∼ = V ×XQ (Y ×X XQ ). Q   V ∼ V ∼ Since = = V ×X (Y ×X XQ ) ∼ = V ×X Y , h∈MorX (V ,Y )

h∈MorXQ (V ,Y ×X XQ )

Q

the composition of the finite Kummer étale covers V → U ×X XQ → U gives the desired finite Kummer étale cover V → U .  Lemma 4.5.3 Let X be a locally noetherian fs log adic space. Let ı : Z → X be a strict closed immersion of log adic spaces, and j : W → X an open immersion of log adic spaces, as in Definition 2.2.23, such that W = X − Z. For ? = an, ét, or két, let (ı?−1 , ı?,∗ ) and (j?−1 , j?,∗ ) denote the associated morphisms of topoi, and let j?,! denote the left adjoint of j?−1 (which is defined as explained above). (1) For each abelian sheaf F on X? , we have the excision short exact sequence 0 → j?,! j?−1 (F) → F → ı?,∗ ı?−1 (F) → 0 in ShAb (X? ). (2) For each abelian sheaf G on Z? , the adjunction morphism ı?−1 ı?,∗ (G) → G is an isomorphism in ShAb (Z? ), and hence ı?,∗ is exact and fully faithful. ∼ (3) For each abelian sheaf H on W? , the adjunction morphism H → j?−1 j?,! (H) is an isomorphism in ShAb (W? ), and hence j!,∗ is exact and fully faithful. Proof These follow easily from the definitions of the objects involved, by evaluating them at points (resp. geometric points, resp. log geometric points) when ? = an (resp. ét, resp. két). (See [18, Proposition 2.5.5] and Lemma 4.4.4.)  Lemma 4.5.4 Let f : X → X˘ and g : Z → Z˘ be morphisms of locally noetherian fs log adic spaces whose underlying morphisms of adic spaces are isomorphisms, and let ı : Z → X and ı˘ : Z˘ → X˘ be strict immersions compatible with f and g. Then, for any abelian sheaf F on Xkét , and for each i ≥ 0, we have −1 −1 i R i gkét,∗ ıkét (F) ∼ R fkét,∗ (F). = ı˘két

(4.5.5)

This applies, in particular, to the case where X˘ and Z˘ are the underlying adic spaces of X and Z, respectively, equipped with their trivial log structures, in which case X˘ két ∼ = Xét and Z˘ két ∼ = Zét , and therefore fkét : Xkét → X˘ két and gkét : Zkét → Z˘ két can be identified with the natural morphisms Xkét → Xét and Zkét → Zét , respectively. Proof Up to compatibly replacing X and X˘ with open subspaces, we may assume that ı and ı˘ are compatible strict closed immersions. By Lemma 4.5.3(2), and by applying ı˘két,∗ to (4.5.5), it suffices to show that we have

Logarithmic Adic Spaces: Some Foundational Results −1 −1 i R i fkét,∗ ıkét,∗ ıkét (F) ∼ R fkét,∗ (F). = ı˘két,∗ ı˘két

143

(4.5.6)

Let j : W → X and j˘ : W˘ → X˘ denote the complementary open immersions. By Lemma 4.5.3(1), we have a long exact sequence −1 −1 · · · → R i fkét,∗ jkét,! jkét (F) → R i fkét,∗ (F) → R i fkét,∗ ıkét,∗ ıkét (F) → · · · .

By the definition of jkét,! and j˘két,! , and by comparing stalks at log geometric points −1 −1 i (F) ∼ R fkét,∗ (F), which as in Lemma 4.4.4, we obtain R i fkét,∗ jkét,! jkét = j˘két,! j˘két induces the desired (4.5.6), by Lemma 4.5.3(1) again.  Lemma 4.5.7 Let ı : Z → X be a strict closed immersion of locally noetherian fs log adic spaces over Spa(Qp , Zp ). −1 + (1) For ? = an, ét, or két, the canonical morphism ıZ,? (O+ X? /p) → OZ? /p is an isomorphism. (2) For any Fp -sheaf F on Zkét , the canonical morphism

  + ıZ,két,∗ (F) ⊗ (O+ Xkét /p) → ıZ,két,∗ F ⊗ (OZkét /p) is an isomorphism. Proof The case where ? = an or ét is already in [40, Lemma 3.14]. As for ? = két, the proof is similar, which we explain as follows. At each log geometric point  + ζ = Spa(l, l + ), M → X, the stalk of O+ Xkét /p at ζ is isomorphic to l /p by + construction, because ker(OXkét ,ζ → l + ), which is the same as ker(OXζ ,ζ → l), is p-divisible (as X is defined over Spa(Qp , Zp )). The analogous statement for O+ Zkét /p is true. Thus, we can finish the proof by comparing stalks, by Lemma 4.4.4.  Lemma 4.5.8 Let g : Y → Z be a morphism of locally noetherian fs log adic spaces over Spa(Qp , Zp ) such that its underlying morphism of adic spaces is an % isomorphism. Suppose moreover that gy : MZ,g(y) → MY,y is injective and splits, for each geometric point y of Y . Then, for any Fp -sheaf F on Ykét , the  + ig F ⊗ canonical morphism R i gkét,∗ (F) ⊗Fp (O+ /p) → R két,∗ Fp (OYkét /p) is Zkét an isomorphism. Proof By Lemma 4.4.4, it suffices to show that, for each log geometric point  z : (Spa(l, l + ), M) → Z as in Construction 4.4.3, the induced morphism  i  i + R gkét,∗ (Fp ) z ⊗Fp (O+ z → R gkét,∗ (OYkét /p)  Zkét /p) z

(4.5.9)

is an isomorphism. Since g induces an isomorphism of the underlying adic spaces, the underlying geometric point z of  z uniquely lifts to a geometric point y. By % assumption, gy : MZ,z → MY,y is injective and splits, in which case we have MY,y ∼ z to = MZ,z ⊕ N, for some fs monoid N . Therefore, we can lift 

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 some (saturated by not necessarily fine) log point Spa(l, l + ), M → Y such that M ∼ = M ⊕ N. By taking fiber products with Spa(l# m1 N$, l + # m1 N$) over Spa(l#N$, l + #N$), by taking reduced subspaces, and by taking the limit with respect to m (cf. Construction 4.4.3), we can further lift this log point to a log geometric point  y of Y above y. Thus, by a limit argument similar to the one in the proof of Lemma 4.4.27, and by Proposition 4.4.7 and Lemma 4.5.7, we may identify (4.5.9) with H i (Γ, F) ⊗Fp (l + /p) → H i (Γ, l + /p), where   gp Γ := ker π1két (y,  y ) → π1két (z, z) ∼ Z(1)(l) . = Hom N ,  Since H i (Γ, F) is computed by some bounded complex of free Fp -modules, and since H i (Γ, l + /p) is computed by the tensor product of this complex with the flat Fp -module l + /p, we see that (4.5.9) is an isomorphism, as desired.  Proposition 4.5.10 Let f : Y → X be a morphism of locally noetherian fs log adic spaces over Spa(Qp , Zp ) such that its underlying morphism of adic spaces  is a closed immersion. Suppose moreover that f ∗ (MX ) y → MY,y is injective and splits, for each geometric point y of Y . Then, for any Fp-sheaf F on Ykét , the + i canonical morphism R i fkét,∗ (F) ⊗Fp (O+ Xkét /p) → R fkét,∗ F ⊗Fp (OYkét /p) is an isomorphism. Proof In this case, let Z denote the underlying adic space of Y equipped with the log structure pulled back from X. Then f : Y → X factors as the composition of a morphism g : Y → Z as in Lemma 4.5.8 and a strict closed immersion ı : Z → X as in Lemma 4.5.7, and we can combine Lemmas 4.5.3, 4.5.7, and 4.5.8. 

4.6 Purity of Torsion Local Systems We have the following purity result for torsion Kummer étale local systems. Theorem 4.6.1 Let X, D, and k be as in Example 2.3.17. Let U := X − D, and let j : U → X denote the canonical open immersion. Suppose moreover that char(k) = 0 and k + = Ok . Let L be a torsion local system on Uét . Then jkét,∗ (L) is a torsion local system on Xkét , and R i jkét,∗ (L) = 0 for all i > 0. Let us start with some preparations. Lemma 4.6.2 In the setting of Theorem 4.6.1, consider the commutative diagram:

Logarithmic Adic Spaces: Some Foundational Results

Uk´et

jk´et

∼ =

U´et

145

Xk´et ε´et

j´et

X´et .

(4.6.3)

Then the canonical morphism Rεét,∗ (Z/n) → Rjét,∗ (Z/n)

(4.6.4)

 gp gp is an isomorphism; and R i jét,∗ (Z/n) ∼ = ∧i (MX /nMX ) (−i), for every i ≥ 0. Proof By Lemma 4.4.29, it suffices to show that the composition 

gp gp ∧i (MX /nMX ) (−i) → R i εét,∗ (Z/n) → R i jét,∗ (Z/n)

(induced by (4.4.28) and (4.6.4)) is an isomorphism. Since this assertion is étale local on X, we may assume that D ⊂ X is the analytification of a normal crossings divisor on a smooth scheme over k, and further reduce the assertion to its classical analogue for schemes by [18, Proposition 2.1.4 and Theorem 3.8.1], which is known (see, for example, [19, Theorem 7.2]).  Lemma 4.6.5 In the setting of Lemma 4.6.2, the canonical morphism Z/n → Rjkét,∗ (Z/n)

(4.6.6)

is an isomorphism. Proof Let C be the cone of (4.6.6) (in the derived category). It suffices to show that H • (Wkét , C) = 0, for each W → X that is the composition of an étale covering and a standard Kummer étale cover of X. Note that the complement of U ×X W in W is a normal crossings divisor, which induces the fs log structure of W as in Example 2.3.17. Consider the diagram (4.6.3), with U → X replaced with U ×X W → W . Since the corresponding morphism (4.6.4) for this new diagram is an isomorphism by Lemma 4.6.2, and since (4.6.4) is obtained from (4.6.6) by applying εét to both sides, we have Rεét,∗ (C|Wkét ) = 0 on Wét , and so H • (Wkét , C) = 0, as desired.  Proof (of Theorem 4.6.1) Let V → U be a finite étale cover trivializing L. By Proposition 4.2.1, it extends to a finite Kummer étale cover f : Y → X, where Y is a normal rigid analytic variety with its log structures defined by the preimage of D. Moreover, if Y → Y is Kummer étale, then Y is locally a normal rigid analytic variety, and any section of a finite torsion constant sheaf over the preimage of U uniquely extends to a section of the constant sheaf with the same coefficients over Y , by Example 2.2.20, Proposition 4.1.6, Corollary 4.1.9, and Proposition 4.2.1. Thus, jkét,∗ (L)|Ykét is constant, and jkét,∗ (L) is a torsion local system on Xkét . Given 1 this f : Y → X, up to étale localization on X, we have some X m → X as in

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Lemma 4.2.5. Then the underlying adic space of Z := Y ×X X m is a smooth rigid analytic variety, its fs log structure is defined by some normal crossings divisor as in Example 2.3.17, and the induced morphism Z → X is Kummer étale, because 1 these are true for X m . (Alternatively, we can construct Z → X, as in the proof of Proposition 4.2.1, by using Lemma 4.2.3 and the last assertion of Lemma 4.2.2.) Thus, in order to show that R i jkét,∗ (L) = 0, for all i > 0, up to replacing X with Z, we may assume that L = Z/n is constant, in which case Lemma 4.6.5 applies.  Corollary 4.6.7 Let k and j : U → X be as in Theorem 4.6.1. Let L be an étale Fp -local system on Uét . Then L := jkét,∗ (L) is a Kummer étale Fp -local system extending L. Conversely, any étale Fp -local system L on Xkét is of this form. In either case, H i (Uét , L) ∼ = H i (Xkét , L), for all i ≥ 0. Proof This follows from Theorem 4.6.1 and the fact that, for any Kummer étale −1 Fp -local system L on X, the canonical adjunction morphism L → Rjkét,∗ jkét (L) is an isomorphism because it is a morphism between local systems whose restriction to the open dense U is the identity morphism of L. 

5 Pro-Kummer étale Topology 5.1 The Pro-Kummer étale Site In this subsection, we define the pro-Kummer étale site on log adic spaces, a log analogue of Scholze’s pro-étale site in [39]. For any category C, by [39, Proposition 3.2], the category pro−C is equivalent to the category whose objects are functors F : I → C from small cofiltered  index cat egories and whose morphisms are Mor(F, G) = lim lim Mor F (i), G(j ) , ← −j ∈J − →i∈I for each F : I → C and G : J → C. We shall use this equivalent description in what follows. For each F : I → C as above, we shall denote F (i) by Fi , for each i ∈ I , and denote the corresponding object in pro−C as lim Fi . ← −i∈I Let X be a locally noetherian fs log adic space, with the category pro−Xkét as above. Then any object in pro-Xkét is of the form U = lim Ui , where each Ui → ← −i∈I X is Kummer étale, with underlying topological space |U | := lim |Ui |. ← −i∈I Definition 5.1.1 (1) We say that a morphism U → V in pro-Xkét is Kummer étale (resp. finite Kummer étale, resp. étale, resp. finite étale) if it is the pullback under some morphism V → V0 in pro-Xkét of some Kummer étale (resp. finite Kummer étale, resp. strictly étale, resp. strictly finite étale) morphism U0 → V0 in Xkét . (2) We say that a morphism U → V in pro-Xkét is pro-Kummer étale if it can be written as a cofiltered inverse limit U = lim Ui of objects Ui → V ← −i∈I Kummer étale over V such that Uj → Ui is finite Kummer étale and surjective

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for all sufficiently large i (i.e., all i ≥ i0 , for some i0 ∈ I ). Such a presentation U = lim Ui → V is called a pro-Kummer étale presentation. ← −i∈I (3) We say that a morphism U → V as in (2) is pro-finite Kummer étale if all Ui → V there are finite Kummer étale. Definition 5.1.2 The pro-Kummer étale site Xprokét has as underlying category the full subcategory of pro−Xkét consisting of objects that are pro-Kummer étale over X, and each covering of an object U ∈ Xprokét is given by a family of pro-Kummer étale morphisms {fi : Ui → U }i∈I such that |U | = ∪i∈I fi (|Ui |) and such that fi : Ui → U can be written as an inverse limit Ui = lim U → U satisfying ← −μ 0, and there are almost isomorphisms O Xprokét (U ) =  (lim G)(U ) ∼ = lim (R + /p) ∼ = lim G(U ) ∼ = R + . By [39, Lemma 3.18] again, ← −Φ ← −Φ ← −Φ

+ X H i (Uprokét , O )a ∼ = H i (Uprokét , lim G)a = 0, for all i > 0. prokét ← −Φ Finally, let us prove (4). Consider the sheaf associated with the presheaf H on  Xprokét determined by H(U ) = O+  (U ), for each U ∈ B. It suffices to show that H U satisfies the sheaf property for coverings by objects in B. Let U and V be log affinoid perfectoid objects in Xprokét , and let V → U be a pro-Kummer étale cover. Let R, S, and T be the perfectoid algebras associated with U , V and U ×V U , respectively. Then it suffices to show the exactness of 0 → R + → S + → T + . Note that this is the inverse limit (along Frobenius) of 0 → R + /p → S + /p → T + /p, and this

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last sequence is exact by the fact that O+ két is a sheaf and p-torsion free. Thus, the U  desired exactness follows. The following proposition is an analogue of [27, Theorem 9.2.15]. Theorem 5.4.4 Suppose that X is a locally noetherian fs log adic space over Spa(Qp , Zp ). Let U be a log affinoid perfectoid object of Xprokét . The functor H → H := H(U ) Xprokét |U -modules on is an equivalence from the category of finite locally free O ∼ Xprokét (U )-modules, with a quasiXprokét to the category of finite projective O /U inverse given by  H → H(V ) := H ⊗O X (U ) OXprokét (V ), prokét for each log affinoid perfectoid object V in Xprokét over U . Moreover, for each finite Xprokét |U -module H, for all i > 0, we have locally free O H i (Xprokét /U , H) = 0. Proof By Lemma 5.3.8, the first statement follows from [27, Theorem 9.2.15]. The proof of the second statement is similar to that of [31, Proposition 2.3], with the input of [27, Lemma 2.6.5(a)] replaced with Lemma 5.1.4(3) here.  By combining Lemma 5.3.7 and Theorem 5.4.3, we obtain the following: Proposition 5.4.5 Let ı : Z → X be a strict closed immersion of locally noetherian Xprokét → fs log adic spaces over Spa(Qp , Zp ). Then the natural morphism O  ıprokét,∗ (OZprokét ) is surjective. More precisely, its evaluation at every log affinoid perfectoid object U in Xprokét is surjective.

6 Kummer étale Cohomology 6.1 Toric Charts Revisited Let V = Spa(S1 , S1+ ) be a log smooth affinoid fs log adic space over Spa(k, k + ), where (k, k + ) is as in Definition 3.1.9 and where k + = Ok , with a toric chart V → E = Spa(k#P $, k + #P $) = Spa(R1 , R1+ ), as in Proposition 3.1.10 and Definition 3.1.12, where P is a sharp fs monoid. The goal of this subsection is to prove the following:

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Proposition 6.1.1 In the above setting, assume moreover that k is characteristic zero and contains all roots of unity. Let V → E be a toric chart as above, and let L be an Fp -local system on Vkét . Then we have the following:  (1) H i Vkét , L ⊗Fp (O+ V /p) is almost zero, for all i > n = dim(V ). (2) Let V ⊂ V be a rational subset such that V is strictly contained in V (i.e., the closure V of V is contained the image of the  in Vi). Then canonical morphism + H i Vkét , L ⊗Fp (O+ V /p) → H Vkét , L ⊗Fp (OV /p) is an almost finitely generated k + -module, for each i ≥ 0. In order to prove Proposition 6.1.1, we need some preparations. Let us first introduce an explicit pro-finite Kummer étale cover of E. For each m ≥ 1, consider + Em := Spa(k# m1 P $, k + # m1 P $) = Spa(Rm , Rm )

and the log affinoid perfectoid object  E := lim Em ∈ Eprokét , ← − m

where the transition maps Em → Em (for m|m ) are induced by the natural 1 1 1 inclusions m P → m P . Let PQ≥0 := lim m P as before. Then the associated − →m perfectoid space is

  E := Spa(k#PQ≥0 $, k + #PQ≥0 $) = Spa(R, R + ). For each m ≥ 1, let us write + Vm := V ×E Em = Spa(Sm , Sm )

and  := V ×E  E ∈ Vprokét . V ∼ Then V = lim Vm is also a log affinoid perfectoid object in Vprokét , with associated ← −m   +) . ∼ perfectoid space V = Spa(S, S + ), where (S, S + ) = lim (Sm , Sm − →m Definition 6.1.2 Suppose that X is a locally noetherian fs log adic space over Spa(k, k + ), where (k, k + ) is an affinoid field, and where k is of characteristic zero and contains all roots of unity. Let G be a profinite group. A pro-Kummer étale cover U → X is a Galois cover with (profinite) Galois group G if there exists a pro-Kummer étale presentation U = lim Ui → X such that each Ui → X is a ← −i Galois finite Kummer étale cover with Galois group Gi (as in Proposition 4.1.6, where Gi is a constant group object because contains all roots of unity), and such that G ∼ = lim Gi , in which case the group action and the second projection induces ← −i a canonical isomorphism G × U ∼ = U ×X U over X.

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Since P is a sharp fs monoid, P gp is a finitely generated free abelian group. Let := (PQ≥0 )gp ∼ = P gp ⊗Z Q. Then Em → E and therefore Vm → V are finite Kummer étale covers with Galois group

gp PQ

 Γ/m := Hom ( m1 P )gp /P gp , μ∞ ∼ = Hom(P gp /mP gp , μm ) ∼ = Hom(P gp , μm ),

(6.1.3)

 → V is a Galois pro-finite Kummer étale cover with Galois group and V   Γ := lim Γ/m ∼ Z(1) = Hom P gp ,  = Hom P gp , lim μm ∼ ← − ← − m

m

∼ Hom(P gp /P gp , μ∞ ), ∼ = Hom lim ( m1 P )gp /P gp , μ∞ = Q − →m (6.1.4) Z(1) are as in Definition 4.4.6 (with the symbols (k) omitted). where μm , μ∞ , and  Consider the k + [P ]-module decomposition 

 k + [PQ≥0 ] = ⊕χ k + [PQ≥0 ]χ

(6.1.5)

according to the action of Γ , where the direct sum is over all finite-order characters χ of Γ . Note that the set of finite-order characters of Γ can be naturally identified gp gp with PQ /P gp , via (6.1.4). If we denote by π the natural map π : PQ≥0 → PQ /P gp , then we have the k + -module decomposition  k + [PQ≥0 ]χ = ⊕a∈PQ≥0 , π(a)=χ k + ea . Lemma 6.1.6 (1) k + [PQ≥0 ]1 = k + [P ] for the trivial character χ = 1. (2) Each direct summand k + [PQ≥0 ]χ is a finite k + [P ]-module. Proof The assertion (1) follows from the observation that PQ≥0 ∩ P gp = P as gp subsets of PQ . As for the assertion (2), it suffices to show that, for each χ in gp PQ /P gp , if χ ∈ ( m1 P )gp /P gp for some m ≥ 1, and if P is generated as a monoid by some finite subset {a1 , . . . , ar }, then there exists some integer m ≥ m (depending on m) such that π −1 (χ ) ⊂ m1 P , so that π −1 (χ ) = Sχ + P for the finite subset  Sχ := { ri=1 mci ai ∈ π −1 (χ ) : 0 ≤ ci < m } of π −1 (χ ). Concretely, since P is sharp by assumption, σ := R≥0 a1 + · · · + R≥0 ar is a convex subset of P gp ⊗Z R of the form σ = {a ∈ P gp ⊗Z R : bj (a) ≥ 0, for all j = 1, . . . , s} for some homomorphisms bj : P gp → Z (cf. [28, Chapter I, Section 1, pages 6–7]) such that   HomZ (P gp , Q) = sj =1 Q bj . It follows that HomZ (P gp , Z) ⊂ sj =1 N1 Z bj , for some N ≥ 1, and hence {a ∈ P gp ⊗Z Q : bj (a) ∈ Z, for all j = 1, . . . , s }⊂ N1 P gp by duality. Thus, π −1 (χ ) ⊂ PQ≥0 ∩ m1 P gp = m1 P , for m = mN, as desired. 

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Lemma 6.1.7 Fix n ≥ 0. Let M be any k + /pn -module on which Γ acts via a primitive character χ : Γ → μm . Then H i (Γ, M) is annihilated by ζm − 1, where ζm ∈ μm is any primitive m-th root of unity, for each i ≥ 0. Moreover, if we have a finite extension k0 of Qp (μm ) in k with ring of integers k0+ , a finitely generated k0+ /pn -algebra T0 , and a finite (and therefore finitely presented) T0 -module M0 such that M ∼ = M0 ⊗k + /pn (k + /pn ) as Γ -modules over T := T0 ⊗k + /pn (k + /pn ), then 0

H i (Γ, M) is a finitely presented T -module, for each i ≥ 0.

0

Proof By choosing a Z-basis of P gp , we have Γ ∼ Z(1)n , where n = rkZ (P gp ) = (see (6.1.4)). Then the lemma follows from a direct computation using the Koszul complex of Γ (as in the proof of [39, Lemma 5.5]) and (for the last assertion of the lemma) using the flatness of k + /pn over k0+ /pn (and the compatibility with flat base change in the formation of Koszul complexes).  ∼ (k + /p)[P ] and R + /p ∼ Remark 6.1.8 Since R1+ /p = = (k + /p)[PQ≥0 ], by + 6.1.6 and injection R1 /p → R + /p induces an injection Lemmas   natural 6.1.7,ithe + i + H Γ, (R1 /p) → H Γ, (R /p) , with cokernel annihilated by ζp − 1, for  each i ≥ 0. Moreover, the R1+ /p-module H i Γ, (R + /p) is almost finitely presented, because, for each  > 0 such that p -torsion makes sense, there are only finitely χ such that the finitely presented R + /p-module direct  many i + summand H Γ, (k /p)[PQ≥0 ]χ is nonzero and not p -torsion. By the use of Koszul complexes as in the proof of Lemma 6.1.7, for any composition of rational localizations and finite étale morphisms Spa(S1 , S1+ ) → Spa(R1 , R1+ ), we have  + + i + ∼ H Γ, (S1 /p) ⊗R + /p (R /p) = (S1 /p) ⊗R + /p H i (Γ, R + /p). 1

1

By the same argument as in the proof of [39, Lemma 4.5], we obtain the following: Lemma 6.1.9 Let X be a locally noetherian fs log adic space over Spa(k, k + ). Let U = lim Ui = lim (Spa(Ri , Ri+ ), Mi ) ← − ← − i∈I

i∈I

 ∧ be a log affinoid perfectoid object in Xprokét , and let (R, R + ) := lim (Ri , Ri+ ) , − →i  = Spa(R, R + ) is the associated affinoid perfectoid space. so that U Suppose that, for some i ∈ I , there exists a strictly étale morphism Vi = Spa(Si , Si+ ) → Ui that is a composition of rational localizations and strictly finite étale morphisms. For each j ≥ i, let Vj := Vi ×Ui Uj = Spa(Sj , Sj+ ), and let V := Vi ×Ui U ∼ = lim Vj ∈ Xprokét . ← − j

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 ∧ Let (S, S + ) := lim (Sj , Sj+ ) . Let Tj be the p-adic completion of the p-torsion − →j free quotient of Sj+ ⊗R + R + . Then we have the following: j

is a perfectoid affinoid (k, k + )-algebra, and V is a log affinoid  = Spa(S, S + ). perfectoid object in Xprokét with associated perfectoid space V  = Vj ×Uj U  in the category of adic spaces. Moreover, V (2) For each j ≥ i, we have S ∼ = Tj [ p1 ], and the cokernel of Tj → S + is annihilated by some power of p. (3) For each ε ∈ Q>0 , there exists some j ≥ i such that the cokernel of Tj → S + is annihilated by pε . (1)

(S, S + )

Remark 6.1.10 Lemma 6.1.9 is applicable, in particular, to the log affinoid perfectoid object U = lim E in Eprokét and any strictly étale morphism V = ← −m≥1 m + Spa(S1 , S1 ) → E (for m = 1) giving a toric chart. Lemma 6.1.11 Let X be a locally noetherian fs log adic space over Spa(k, k + ). Suppose that U is a log affinoid perfectoid object of Xprokét , with associated  = Spa(R, R + ). Let L be an Fp -local system on Ukét . Then: perfectoid space U  (1) H i Ukét , L ⊗Fp (O+ /p) is almost zero, for all i > 0. X  (2) L(U ) := H 0 Ukét , L ⊗Fp (O+ X /p) is an almost finitely generated projective R + /p-module (see [10, Definition 2.4.4]). In addition, for any morphism U → U in Xprokét , where U is a log affinoid perfectoid object in Xprokét , with  = Spa(R , R + ), we have a canonical almost associated perfectoid space U ∼ isomorphism L(U ) = L(U ) ⊗R + /p (R + /p). Proof By replacing X with its connected components, we may assume that X is connected. Choose any Galois finite Kummer étale cover Y → X trivializing ∼ L → Frp . By Lemma 5.3.8, W := U ×X Y → U is finite étale, and W  = Spa(T , T + ). is log affinoid perfectoid, with associated perfectoid space W j/U For each j ≥ 1, let W denote the j -fold fiber product of W over U . By  j/U Proposition 5.1.7 and Theorem 5.4.3, H i Wkét , L⊗Fp (O+ W /p) is almost zero, for  j/U all i > 0 and j , and H 0 Wkét , L ⊗Fp (O+ V /p) is canonically almost isomorphic + j/U r to (OW j/U (W )/p) . By the faithful flatness of T +a /p → R +a /p, the desired results follow from almost faithfully flat descent (see [10, Section 3.4]).  Now we are ready for the following:  → V = Spa(S1 , S + ) Proof (of Proposition 6.1.1) Consider the Galois cover V 1   = Spa(S, S + ), as above. Since V  ×Γ j −1 j/V ∼ with Galois group Γ , and with V =V is a log affinoid perfectoid object in Vprokét , for each j ≥ 1, we have  j/V   , L ⊗Fp (O+ /p) ∼ Hi V = Homcont Γ j −1 , Li , V két

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 két , L ⊗Fp (O+ /p) a is an almost finitely generated projective where Li := H i V V S +a /p-module, equipped with the discrete topology, which vanishes when i > 0, by Lemma 6.1.11 and by Propositions 5.1.6 and 5.1.7. By Proposition 5.1.7 again, and by Lemma 6.1.11 and the Cartan–Leray spectral sequence (see [1, V, 3.3]), we have an almost isomorphism   + 0 ∼ ˇi  ∼ i H i Vkét , L ⊗Fp (O+ V /p) = H {V → V }, L ⊗Fp (OV /p) = H (Γ, L ), where the last isomorphism follows from Proposition 5.1.12 and [39, Proposition 3.7(iii)] (and the correction in [41]). Hence, the statement (1) of Proposition 6.1.1 follows from the fact that Γ ∼ Z(1)n has cohomological dimension n. =   = Spa(S , S + ). We As for the statement (2), write V = Spa(S1 , S1 + ) and V  i 0 i 0 need to show that the image of H (Γ, L ) → H Γ, L ⊗S + /p (S + /p) is an almost finitely generated k + -module. Since L0 is an almost finitely generated projective S +a /p-module, it suffices to show that the image of H i (Γ, S + /p) → H i (Γ, S + /p) is an almost finitely generated k + -module. Choose rational subsets {V (j ) }1≤j ≤n+2 such that V (n+2) = V , V (1) = V , and V (j +1) is strictly contained in V (j ) , for (j ) (j ) (j )+ 1 ≤ j ≤ n+1. Write Vm := V (j ) ×E Em = Spa(Sm , Sm ), for all 1 ≤ j ≤ n+2 (j ) := lim Vm(j ) is a log affinoid perfectoid object in Vprokét , and m ≥ 1. Then V ← −m  (j ) = Spa(S (j ) , S (j )+ ). By Lemma 6.1.9 and with associated perfectoid space V Remark 6.1.10, it suffices to show that the image of (1)+ (n+2)+ H i (Γ, (Sm ⊗Rm+ R + )/p) → H i (Γ, (Sm ⊗Rm+ R + )/p) (j )+

is almost finitely generated, for all m ∈ Z≥1 . Note that mΓ acts trivially on Sm and we have the Hochschild–Serre spectral sequence

,

 (j )+ (j )+ H i1 Γ/m , H i2 (mΓ, (Sm ⊗Rm+ R + )/p) ⇒ H i1 +i2 (Γ, (Sm ⊗Rm+ R + )/p). By [39, Lemma 5.4] and Remark 6.1.8, it suffices to show that the image of (j )+

(Sm

(j +1)+

/p) ⊗Rm+ /p H i (mΓ, R + /p) → (Sm

/p) ⊗Rm+ /p H i (mΓ, R + /p)

is almost finitely generated, for all j = 1, . . . , n + 1 and m ≥ 1. Since the image (j )+ (j +1)+ /p is an almost finitely generated k + -module, it suffices to of Sm /p → Sm i + + /p, by Remark 6.1.8 note that H (mΓ, R /p) is almost finitely generated over Rm + + (up to replacing (R1 , R1 ), Γ , etc with (Rm , Rm ), Γm , etc). 

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6.2 Primitive Comparison Theorem The main goal of this subsection is to prove the following primitive comparison theorem, with the finiteness of cohomology as a byproduct, generalizing the strategy in [39, Section 5]: Theorem 6.2.1 Let (k, k + ) be an affinoid field, where k is algebraically closed and of characteristic zero, and let X be a proper log smooth fs log adic space over Spa(k, k + ) (see Definitions 2.2.2 and 3.1.1). Let L be an Fp -local system on Xkét . Then we have the following:  + (1) H i Xkét , L ⊗Fp (O+ X /p) is an almost finitely generated k -module (see [10, Definition 2.3.8]) for each i ≥ 0, and is almost zero for i ' 0. (2) There is a canonical almost isomorphism  ∼ H i (Xkét , L) ⊗Fp (k + /p) → H i Xkét , L ⊗Fp (O+ X /p) of k + -modules, for each i ≥ 0. Consequently, H i (Xkét , L) is a finite-dimensional Fp -vector space for each i ≥ 0, and H i (Xkét , L) = 0 for i ' 0. In addition, if X is as in Example 2.3.17, then H i (Xkét , L) = 0 for i > 2 dim(X). Remark 6.2.2 Recall that there is no general finiteness results for the étale cohomology of Fp -local systems on non-proper rigid analytic varieties over k, as is well known (via Artin–Schreier theory) that H 1 (D, Fp ) is infinite. Nevertheless, we have the following: Corollary 6.2.3 Let U be a smooth rigid analytic variety that is Zariski open in a proper rigid analytic variety over k. Then H i (Uét , L) is a finite-dimensional Fp -vector space, for each Fp -local system L on Uét and each i ≥ 0. Moreover, H i (Uét , L) = 0 for i > 2 dim(U ). Proof By resolution of singularities (as in [4]), we may assume that we have a smooth compactification U → X such that U = X − D for some normal crossings  divisor D of X. Now apply Theorems 4.6.1 and 6.2.1. Lemma 6.2.4 Let X be a proper log smooth fs log adic space over Spa(k, Ok ). For each integer N ≥ 2, we can find N affinoid étale coverings of X (N ) m (1) }h=1 , . . . , {Vh }m h=1

{Vh

satisfying the following properties: (N )

• Vh

(j +1)

• Vh

(1)

⊂ · · · ⊂ Vh (j +1)

⊂ Vh

is a chain of rational subsets, for each h = 1, . . . , m. (j )

⊂ Vh , for all h = 1, . . . , m and j = 1, . . . , N − 1.

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(1)

(1)

• Vh1 ×X Vh2 → Vh1 is a composition of rational localizations and finite étale morphisms, for 1 ≤ h1 , h2 ≤ m. • Each Vh(1) admits a toric chart Vh(1) → Spa(k#Ph $, Ok #Ph $), for some sharp fs monoid Ph . Proof By Proposition 3.1.10 and the same argument as in the proof of [39, Lemma 5.3], there exist N affinoid analytic open coverings of X (N )

(1)

m {Uh }m h=1 , . . . , {Uh }h=1

satisfying the following properties: (N )

• Uh

(1)

⊂ · · · ⊂ Uh is a chain of rational subsets, for each h = 1, . . . , m.

(j +1)

(j +1)

(j )

• Uh ⊂ Uh ⊂ Uh , for all h = 1, . . . m and j = 1, . . . , N − 1. (1) (1) (1) • Uh1 ∩ Uh2 ⊂ Uh1 is a rational subset, for 1 ≤ h1 , h2 ≤ m. (1)

(1)

(1)

• There exist finite étale covers Vh → Uh such that each Vh admits a toric (1) chart Vh → Ej = Spa(k#Ph $, Ok #Ph $) (which is, in particular, a composition of rational localizations and finite étale morphisms) for some sharp fs monoid Ph . (j )

Then it suffices to take Vh

(j )

(1)

:= Vh ×U (1) Uh , for all h and j . h



Proof (of Theorem 6.2.1(1)) Consider X := X ×Spa(k,k + ) Spa(k, Ok ) ⊂ X. Consider any covering {Uh }h of X by log affinoid perfectoid objects in Xprokét , whose pullback {Uh ×X X }h is a covering of X by log affinoid perfectoid objects in Xprokét . By Lemma 6.1.11, we have a canonical almost isomorphism  ∼ i + H i Uh,két , L ⊗Fp (O+ X /p) → H Uk,két ×X X , L ⊗Fp (OX /p) , for all i ≥ 0 and all h. By Proposition 5.1.7 and by comparing the spectral sequences associated with the coverings, we obtain an almost isomorphism  ∼ i + H i Xkét , L ⊗Fp (O+ X /p) → H Xkét , L ⊗Fp (OX /p) , for each i ≥ 0. Hence, for the purpose of this proof, up to replacing X with X , we may assume that k + = Ok in what follows. (N ) (1) m Let {Vh }m h=1 , . . . , {Vh }h=1 be affinoid étale coverings of X satisfying the same properties as in Lemma 6.2.4. For each subset H = {h1 , . . . , hs } of (j ) (j ) (j ) {1, . . . , m}, let VH := Vh1 ×X · · · ×X Vhs . For each j = 1, . . . , N , we have a spectral sequence  (j )  i1 ,i2 i2 i1 +i2 E1,(j VH,két , L ⊗Fp (O+ Xkét , L ⊗Fp (O+ X /p) ⇒ H X /p) . ) = ⊕|H |=i1 +1 H

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For j = 1, . . . , N − 1, we also have natural morphisms between spectral sequences i1 ,i2 i1 ,i2 E∗,(j ) → E∗,(j +1) . Then the desired finiteness result follows from Proposition 6.1.1 and [39, Lemma 5.4]. 6.1.1 and the spectral sequence for  Moreover, by Proposition a j = 1, we have H i Xkét , L ⊗Fp (O+ /p) = 0 for i ' 0.  X Proof (of Theorem 6.2.1(2)) Consider the Artin–Schreier sequence σ  X  X 0 → L → L ⊗Fp O → L ⊗Fp O → 0, prokét prokét

(6.2.5)

where σ = Id ⊗(Φ − Id) and Φ is the Frobenius morphism (induced by x → x p ). The exactness of (6.2.5) can be checked locally on log affinoid perfectoid objects U ∈ Xprokét over which L is trivial, which then follows (by using Lemma 5.3.8) from the same argument in the proof of [39, Theorem 5.1]. Choose any  ∈ k such that  % = p. By Theorem 6.2.1(1) and [39, Lemma 2.12], there exists some r ≥ 0 such that we have  m a ∼  + H i Xprokét , L ⊗Fp (O = (Oak / m )r , Xprokét / ) for all m, which are compatible with each other and with the Frobenius morphism. By [39, Lemma 3.18], we have  a a  +  + R lim L ⊗Fp (O / m ) ∼ = (L ⊗Fp O X X ) , ← − m

and so a ∼ a r  + H i (Xprokét , L ⊗Fp O Xprokét ) = (Ok )

and  X H i (Xprokét , L ⊗Fp O )∼ = (k )r prokét (by inverting  ), which are still compatible with the Frobenius morphisms. Thus, by considering the long exact sequence associated with (6.2.5), and by Proposition 5.1.7, we see that  X H i (Xkét , L) ∼ )Φ−Id ∼ = H i (Xprokét , L ⊗Fp O = Frp prokét and  a H i (Xkét , L) ⊗Fp (k +a /p) ∼ = H i Xkét , L ⊗Fp (O+ Xkét /p) , as desired.



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Proof (of the Remaining Statements of Theorem 6.2.1) It remains to show that, if X is as in Example 2.3.17, then H i (Xkét, L) = 0 for i > 2 dim(X). By a Theorem 6.2.1(2), it suffices to show that H i Xkét , L ⊗Fp (O+ /p) = 0, for Xkét i > 2 dim(X). Note that, in Example 3.1.13, since k is algebraically closed, X analytic locally admits smooth toric charts X → Dn . Hence, by the same argument (1) as in the proof of [39, Lemma 5.3], all the étale coverings {Vj }m j =1 in Lemma 6.2.4 can be chosen to be analytic coverings. Let λ : Xkét → Xan denote a the natural projection of sites. By Proposition 6.1.1, R j λ∗ L ⊗Fp (O+ /p) = 0, for all Xkét j > dim(X). Since the cohomological dimension of Xan is bounded by dim(X), by [21, Proposition 2.5.8], the desired vanishing follows. (This is essentially the same argument as in the proof of [39, Lemma 5.9].) 

6.3 p-Adic Local Systems Definition 6.3.1 Let X be a locally noetherian fs log adic space. (1) A Zp -local system on Xkét , also called a lisse Zp -sheaf on Xkét , is an inverse system of Z/pn -modules L = (Ln )n≥1 on Xkét such that each Ln is a locally constant sheaf which are locally (on Xkét ) associated with finitely generated Z/pn -modules, and such that the inverse system is isomorphic in the procategory to an inverse system in which Ln+1 /pn ∼ = Ln . (2) A Qp -local system (or lisse Qp -sheaf ) on Xkét is an object of the stack associated with the fibered category of isogeny lisse Zp -sheaves. Definition 6.3.2 Let X be a locally noetherian fs log adic space. Let  Zp := lim(Z/pn ) ← − n

as a sheaf of rings on Xprokét , and let p :=  Zp [ p1 ]. Q Zp -local system on Xprokét is a sheaf of  Zp -modules on Xprokét that is locally (on A Zp for some finitely generated Zp -modules L. The Xprokét ) isomorphic to L ⊗Zp  p -local system on Xprokét is defined similarly. notion of Q Lemma 6.3.3 Let X be a locally noetherian fs log adic space over Spa(Qp , Zp ). Let υ : Xprokét → Xkét denote the natural projection of sites. (1) The functor L = (Ln )n≥1 →  L := lim υ −1 (Ln ) ← − n

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is an equivalence of categories from the category of Zp -local systems on Xkét to   Zp -local systems on Xprokét . Moreover,  L ⊗ the category of  Zp Qp is a Qp -local system. (2) For all i > 0, we have R i lim υ −1 (Ln ) = 0. ← −n Proof Apply Proposition 5.3.13 and [39, Lemma 3.18].  Corollary 6.3.4 Let k, X, and U be as in Theorem 4.6.1. Let L be an étale Zp -local system on Uét . Then L := jkét,∗ (L) is a Kummer étale Zp -local system extending L. Conversely, any étale Zp -local system L on Xkét is of this form. In either case, there are canonical isomorphisms  H i (Uk,ét , L) ∼ = H i (Xk,két , L) ∼ = H i (Xk,prokét , L) of finite Zp -modules, for each i ≥ 0, where k denotes any algebraic closure of k. Proof The assertions on L and L in the first four sentences, together with the first isomorphism (displayed above), follow from Corollary 4.6.7 by taking limits of Zp /pm -local systems over m ∈ Z≥1 , which is justified by the finite-dimensionality of the cohomology of Fp -local systems on Xk,két shown in Theorem 6.2.1. The second isomorphism follows from Proposition 5.1.7 and Lemma 6.3.3(2). The finiteness of these isomorphic Zp -modules follows, again, from Theorem 6.2.1.  Corollary 6.3.5 Let f : X → Y be a log smooth morphism of log adic spaces whose log structures are defined by normal crossings divisors D and E of smooth rigid analytic varieties X and Y , respectively, as in Example 2.3.17. Assume that the underlying morphisms of adic spaces of f and f |X−D : X − D → Y − E are both proper. Let L be any Zp -local system L on Xkét . Then R i fkét,∗ (L) is a Zp -local system on Ykét , for each i. Proof This follows from [43, Theorem 10.5.1] and Corollary 6.3.4.



p -local systems and completed structure The combination of pullbacks of Q sheaves under strict closed immersions can be described as follows: Lemma 6.3.6 Let ı : Z → X be a strict closed immersion of locally noetherian fs p -local system on Xprokét . Then we log adic spaces over Spa(Qp , Zp ). Let  L be a Q have a canonical isomorphism Xprokét )(U ) ⊗  ( L ⊗Q p O OX

prokét

(U )

Zprokét (U ×X Z) O

∼  −1 Zprokét (U ×X Z), → ıprokét ( L) ⊗Q p O for each log affinoid perfectoid object U of Xprokét . Proof By Lemma 5.3.7, U ×X Z is a log affinoid perfectoid object of Zprokét , and the Xprokét → ıprokét,∗ (O Zprokét ) induces a surjective homomorphism natural morphism O

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Xprokét (U ) → O Zprokét (U ×X Z). By Theorem 5.4.4, it suffices to prove the lemma O by replacing U with some log affinoid perfectoid object V of Xprokét over U such L|V is trivial, in which case the assertion is clear. that   Finally, let us define and study the notion of unipotent and quasi-unipotent geometric monodromy actions along a normal crossings divisor. Let ı : D → X and k be as in Example 2.3.17, with j : U := X − D → X the complementary open immersion. Let L be a Qp -local system on Xkét . Definition 6.3.7 With k, X, D, and L as above, we say that L|Uét has unipotent ξ acts unipo(resp. quasi-unipotent) geometric monodromy along D if π1két X(ξ ),  ξ tently (resp. quasi-unipotently) on the stalk Lξ , for each log geometric points  of X lying above each geometric point ξ of D, where the log structure of the strict localization X(ξ ) is pulled back from X, as in Proposition 4.4.9. By abuse of language, when there is no risk of confusion, we shall also say that L has unipotent (resp. quasi-unipotent) geometric monodromy along D, without writing L|Uét . Example 6.3.8 Suppose that {Dj }j ∈I is the set of irreducible components of D  (see [7]). For each J ⊂ I , suppose moreover that XJ := X ∩ ∩j ∈J Dj is smooth and geometrically connected, and consider the fs log adic spaces UJ and UJ∂ introduced in Example 2.3.18, together with a canonical morphism εJ∂ : UJ∂ → UJ (whose underlying morphism of adic spaces is a canonical isomorphism) and a strict immersion ıJ∂ : UJ∂ → X. Note that the log structure of UJ is trivial, while the one of UJ∂ is pulled back from XJ . We shall simply denote the underlying adic space of UJ∂ by UJ . By construction, X is set-theoretically the disjoint union of such locally closed subspaces UJ . At each geometric point ξ = Spa(l, l + ) of UJ (and hence also of UJ∂ ), by projection to factors of polydiscs as in Examples 2.3.17 and 2.3.18, we have locally a strict morphism from UJ∂ to s = (Spa(k, Ok ), ZJ≥0 ) as in Example 4.4.26, which is the restriction of a strict morphism from a neighborhood of ξ in X to a neighborhood of s in D|J | (with its canonical log structure defined as in Example 2.2.21). As result, by Corollary 4.4.22, we have compatible isomorphisms ∼

∼

ZJ≥0 → MX,ξ → MU ∂ ,ξ J

(6.3.9)

and   gp π1két UJ∂ (ξ ) ∼ Z (1) = Hom MU ∂ ,ξ ,  J

  gp ∼  J → π1két X(ξ ) ∼ Z (1) → Γ J :=  Z (1) = Hom MX,ξ ,  ∼

(6.3.10)

Z (1)(l), whose operations will be denoted (with (l) omitted from the notation of  multiplicatively). Therefore, any Zp -local system on X(ξ )két is equivalent to a Zp -local system on UJ∂ (ξ )két , which is in turn equivalent to a (trivial) Zp -local system on UJ (ξ )ét with Γ J -action. (The analogous statement for Qp -local systems follows.) Thus, in Definition 6.3.7, the local system L on Xkét has unipotent (resp.

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quasi-unipotent) geometric monodromy along D if and only if, J ⊂ I for each and each geometric point ξ of UJ , the action of π1két X(ξ ) ∼ = Γ J on Lξ is unipotent (resp. quasi-unipotent), and this property depends only the pullback of L to UJ∂ (ξ )két . Lemma 6.3.11 In Definition 6.3.7, it suffices to verify the condition for geometric points ξ of X lying above the smooth locus of D. (That is, ξ does not lie on the intersections, including self-intersections, of irreducible components of D.) Proof Since Definition 6.3.7 requires only strict localizations of X, we may replace k with a complete algebraic closed extension. Moreover, up to étale localization, we may assume that X is affinoid and admits a smooth toric chart X → Dn as in Example 3.1.13, with the log structure induced by maps Zn≥0 → MX (X) → OX (X) sending the i-th standard basis element ei to the images of the i-th coordinate Ti of Dn . Consider the tower · · · → Xm → · · · → X defined bythe toric n chart X → Dn as in Sect. 6.1 (with P = Zn≥0 ), with Galois group Γ ∼ Z (1) . =  Up to further étale localization, we may assume that the subspace of X defined by Ti = 0 is either empty or irreducible. Then, in the setting of Example 6.3.8, we may identify I with a subset of {1, . . . , n}, with irreducible components Dj of D defined by Tj = 0, for j ∈ I . In this case, if J ⊂ J ⊂ I , then we have canonical projections Zn≥0  ZJ≥0  ZJ≥0 which induce inclusions Γ J → Γ J → Γ , by (6.3.9) and (6.3.10). Let ξ and ξ be any geometric points of UJ and UJ , respectively. By pulling back the tower above to X(ξ ) and X(ξ ), respectively, and J J by Proposition 4.4.9, we can identify Γ the above inclusions  → Γ and Γ → Γ két két with homomorphisms π1 X(ξ ) → Γ and π1 X(ξ ) → Γ . As a result, we   obtain an inclusion π1két X(ξ ) → π1két X(ξ ) , which can be canonically identified with the above inclusion Γ J → Γ J above inclusion inside Γ , for any ξ and ξ as above. Since UJ is contained in the closure XJ of UJ , every Kummer étale neighborhood of ξ admits the lifting of some geometric point ξ of XJ . Thus, since each Zp -local system is trivialized by some inverse system of such neighborhoods, and since each Qp -local system is (by definition) an isogeny class of Zp -local systems, if π1két X(ξ ) acts unipotently (resp. quasi-unipotently) on L|ξ , for all  geometric points ξ of UJ , then the subgroup Γ J of Γ J ∼ = π1két X(ξ ) acts  unipotently (resp. quasi-unipotently) on Lξ . Since Γ J ∼ = j ∈J Γ {j } is generated {j } by Γ , for j ∈ J ; and since the smooth locus of D is (set-theoretically) ∪j ∈I U{j } , the lemma follows.  Remark 6.3.12 Suppose that X, D, and L is the analytification of X0 , D0 , and L0 , respectively, where D0 is a normal crossings divisor on a smooth algebraic variety X0 over k, and where L0 is an étale Qp -local system on X0 . Since the construction of standard Kummer étale covers are compatible with analytification, by comparing the constructions in Proposition 4.1.6 and [19, Proposition 3.2], we have obvious analogues of Definition 6.3.7, Example 6.3.8, and Lemma 6.3.11 in the algebraic setting, which are all compatible with analytification.

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Remark 6.3.13 In Remark 6.3.12, for each irreducible component of D0 , its generic point is a point of codimension one, and hence the strict localization X0 (ξ0 ) at any geometric point ξ0 = Spec(l) above such a generic point is the spectrum of a strictly local ring R with residue field l. Let K := Frac(R), let K be any algebraic closure of K, and let K tr be the maximal tamely ramified extension of K in K. Let η0 := Spec(K). Then L0 |η0 is naturally a representation of Gal(K/K).  As explained in [19, Example 4.7(b)], π1két X0 (ξ0 ) is canonically isomorphic tr /K) ∼  to the tame inertia group, which is = Z (1) in this case; and the Gal(K két ∼ induced isomorphism π1 X0 (ξ0 ) =  Z (1) can be canonically identified with   gp két ∼ ∼  π1 X0 (ξ0 ) = Hom MX0 ,ξ0 , Z (1) =  Z (1), with the last isomorphism induced by MX0 ,ξ0 ∼ = Z≥0 . Since L0 |U0,ét extends over X0,két , the action of Gal(K/K) on L0 |η0 factors through Gal(K tr /K). Note that, in the algebraic setting, if ξ0 specializes to some geometric point ξ0 of X0 , then we have a canonical morphism  X0 (ξ0 ) → X0 (ξ0 ), and hence a canonical homomorphism π1két X0 (ξ0 ) →  π1két X0 (ξ0 ) . When ξ0 does not lie on any other irreducible component of D0 , this last homomorphism can be canonically identified with the identity homomorphism Z (1). Since every geometric point of the smooth locus of D0 is some such ξ0 , by of  the algebraic analogue of Lemma 6.3.11, we see that L0 has unipotent (resp. quasiunipotent) monodromy along D0 (by the algebraic analogue of Definition 6.3.7) if and only if, for each ξ0 as above, the induced action of Gal(K tr /K) ∼ Z (1) =  is unipotent (resp. quasi-unipotent). In fact, this last condition is a more classical definition for schemes, whose formulation does not rely on log geometry at all. Nevertheless, our Definition 6.3.7 has the advantage of not relying on the notion of generic points (or specializations).

6.4 Quasi-Unipotent Nearby Cycles In this subsection, as an application of our results, we reformulate Beilinson’s ideas (see [3]; cf. [37]) and define the unipotent and quasi-unipotent nearby cycles in the rigid analytic setting. Let k be any field of characteristic zero, and let k be any algebraic closure of k. Let Gm := Spec(k[z, z−1 ]) be the multiplicative group scheme over k. Let k be any fixed algebraic closure of k. Then π1 (Gm , 1) ∼ = π1 (Gm,k , 1)  Gal(k/k), and π1 (Gm,k , 1) ∼ Z(1) as Gal(k/k)-modules. For each r ≥ 1, let Jr denote =  the rank r unipotent étale Zp -local system on Gm defined by the representation of π1 (Gm,k , 1) on Zrp such that a topological generator γ ∈ π1 (Gm,k , 1) acts as a principal unipotent matrix Jr and such that Gal(k/k) acts diagonally on Zrp and trivially on ker(Jr − 1). (As in Example 4.4.25, the local system thus defined is independent of the choice of γ up to isomorphism.) There is a natural inclusion Jr → Jr+1 , together with a projection Jr+1 → Jr (−1) such that the composition Jr → Jr (−1) is given by the monodromy action. For each m ≥ 1, let [m] denote the m-th power homomorphism [m] of Gm , and let Km := [m]∗ (Zp ). When m | m , there is a natural inclusion Km → Km (defined by adjunction).

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Now let k be a nontrivial nonarchimedean field, and let k + = Ok . We shall denote the analytifications of the above objects and morphisms to Gan m , and their further pullbacks to D× = D − {0}, by the same symbols. Let X be a rigid analytic variety over k. Let f : X → D be a morphism over k that induces an open immersion j : U := f −1 (D× ) → X and a closed immersion ı : f −1 (0) → X such that D := f −1 (0)red (the reduced subspace) is a normal crossings divisor, so that X is equipped with  the fs log structure defined by D → X, as in Example 2.3.17. Note that f −1 (0) ét ∼ = Dét . Let U be equipped with the trivial log structure, with an open immersion j : U → X. Let D ∂ be the adic space D equipped with the log structure pulled back from X, with a canonical morphism ε∂ : D ∂ → D and a strict closed immersion ı ∂ : D ∂ → X. Definition 6.4.1 In the above setting, for any given Qp -local system L on Uét ∼ = Ukét , its sheaf of unipotent nearby cycles (with respect to f ) is    ∂,−1 ∂ lim ıkét RΨfu (L) := Rεét,∗ jkét,∗ L ⊗Zp fét−1 (Jr ) , − →r and its sheaf of quasi-unipotent nearby cycles is    qu ∂,−1 ∂ lim RΨf (L) := Rεét,∗ ıkét jkét,∗ L ⊗Zp fét−1 (Km ) ⊗Zp fét−1 (Jr ) . − →m,r Suppose that {Dj }j ∈I is the set of irreducible components of D (see [7]), so that f −1 (0) = j ∈I nj Dj (as Cartier divisors on X; see [43, Lecture 5.3, especially Proposition 5.3.4]), for some integers nj ≥ 1 giving the multiplicities of Dj . For each J ⊂ I , let XJ , εJ∂ : UJ∂ → UJ , and ıJ∂ : UJ∂ → X be as in Example 6.3.8. ∂ : Given any geometric point ξ = Spa(l, l + ) of UJ (and hence also of UJ∂ ), let εJ,ξ  J ∼  UJ∂ (ξ ) → UJ (ξ ) denote the pullback of εJ∂ to UJ (ξ ). Let Γ J = Z(1) be as in (6.3.10). Then, by Lemma 4.4.27 and the explanations in Example 6.3.8, we have ∂ a canonical isomorphism R i εJ,ξ,ét,∗ (M) ∼ = H i (Γ J , M), for each i ≥ 0. Let 0∂ and  0∂ , and the (Z/n)-local systems J∂r,n and K∂m,n on 0∂két defined by representations of π1két (0∂ ,  0∂ ) ∼ Z(1)  Gal(k/k), be as in Example 4.4.25. By = Z ≥1 taking limits over n ∈ p , we obtain Zp -local systems J∂r and K∂m on 0∂két , which can be identified with the pullbacks of Jr := jkét,∗ (Jr ) and Km := jkét,∗ (Km ), respectively. By pulling back f : X → D (as a morphism of fs log adic spaces), ξ as in the we obtain a canonical morphism fξ : UJ∂ (ξ ) → 0∂ for any ξ and  ξ ) → π1két (0∂ ,  0∂ ) last paragraph, and the induced homomorphism π1két (UJ∂ (ξ ),   J can be identified with the composition of Γ J ∼ Z(1) : (xj )j ∈J → Z(1) →  =     Z(1) → Z(1)  Gal(k/k). Let γj n x with the canonical homomorphism j ∈J j j  J be any element of the j -th factor of Γ J ∼ Z(1). Z(1) that is mapped to nj γ in  =   −1 nj −1 ∂ ∼ Then γj acts by Jr on the rank r local system f (Jr ) |U ∂ (ξ ) = fξ (Jr ). J

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For each Qp -local system M on UJ∂ (ξ )két , let us denote by W a formal variable  on which γj−1 acts by W → W + nj , and write M[W ] = lim M[W ]≤r−1 , where − →r the superscript “≤ r − 1” means “up to degree r − 1”. Note that, by matching a standard basis of J∂r with binomial monomials up to degree r − 1 in W , as in the proof of [31, Lemma 2.10], we have M[W ]≤r−1 ∼ = M ⊗Zp fξ−1 (J∂r ). Lemma 6.4.2 Suppose there exists some j0 ∈ J such that γj0 acts quasiunipotently (i.e., a positive power of γj0 acts unipotently) on M. Then the local systems H i Γ J , M[W ]≤r−1 stabilize as r → ∞, and hence the direct limit  H i Γ J , M[W ]) exists as a Qp -local system, for each i ≥ 0. When J = {j0 } is  a singleton, H i Γ {j0 } , M[W ]) vanishes when i = 0, 1; is canonically isomorphic to the maximal subsheaf of M on which Γ {j0 } acts unipotently, when i = 0; is finite-dimensional when i = 1; and is zero when i = 1 and γj acts unipotently. Proof By the Hochschild–Serre spectral sequence, by first considering the action of [m](Γ {j0 } ) for some m, and then the induced action of the finite quotient Γ {j0 } /[m](Γ {j0 } ) ∼ = Z/mZ, and then the induced action of Γ J /Γ {j0 } ∼ = Γ J −{j0 } , it suffices to treat the special case where J = {j0 } is a singleton and γj0 acts unipotently. Then the lemma is reduced to its last statement, which follows from the same argument as in the proof of [31, Lemma 2.10], by matching a basis of Jr with binomial monomials up to degree r − 1 in W .  Lemma 6.4.3 Let L be a Qp -local system on Xkét such that L|Uét has quasiunipotent geometric monodromy along D (as in Definition 6.3.7). For each integer m ≥ 1, consider the canonical morphism [m] : D → D induced by sending the standard coordinate of D to its m-th power, whose pullback under f : X → D is gm

f

a finite Kummer étale cover gm : Xm → X, which induces fm : Xm → X → D by composition. Let Dm denote the reduced subspace of Xm ×X D (in the category of adic spaces), which is canonical isomorphic to D via the second projection, and −1 (U ) = X − D . Then there exists m ≥ 1 such that let Um := fm−1 (D× ) = gm m 0  −1 m qu u RΨf (L|U ) ∼ = RΨfm gm (L)|Um over Dét , whenever m0 |m. ∂ denote the adic space D equipped with the log Proof For each m ≥ 1, let Dm m ∂ : D ∂ → X , and ε ∂ : structure pulled back from Xm . Let jm : Um → Xm , ım m m m ∂ → D denote the canonical morphisms. Then Dm m

L|U ⊗Zp fét−1 (Km ) ∼ = (gm |Um )ét,∗ (L|Um ) over Uét , and  ∂,−1 ∂ Rεét,∗ ıkét jkét,∗ L|U ⊗Zp fét−1 (Km ) ⊗Zp fét−1 (Jr ) ∂,−1  −1 ∂ ∼ ım,két (Jr ) (L|Xm ) ⊗Zp fkét = Rεm,ét,∗

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over Dét , by Proposition 4.5.2 and Lemma 4.5.4. Since L has quasi-unipotent geometric monodromy along D, there exists some m0 ≥ 1 such that L|Um has unipotent geometric monodromy along Dm , whenever m0 |m. We claim that, when m0 |m, the canonical morphism  ∂,−1  −1 −1 ∂ ∂ L|Xm ) ⊗Zp fkét Rεm ı ∂,−1 L|Xm0 ) ⊗Zp fkét (Jr ) → Rεm,ét,∗ ım,két (Jr ) 0 ,ét,∗ m0 ,két induced by Km0 → Km is an isomorphism for all sufficiently large r (depending on m0 and m). Given this claim, for all m divisible by m0 , we have  qu ∂,−1 ∂ RΨf (L) ∼ ıkét jkét,∗ (L|U ) ⊗Zp fét−1 (Km ) ⊗Zp fét−1 (Jr ) = Rεét,∗   −1 ∂,−1 −1 ∂ ∼ ım,két jm,két,∗ (L|Um ) ⊗Zp fm,ét (Jr ) ∼ (L) = Rεm,ét,∗ = RΨfum gm for all sufficiently large r, and the lemma follows. It remains to verify the claim. For this purpose, by (6.3.10), we may pullback to UJ∂ (ξ ), for all nonempty J ⊂ I and all geometric point ξ of UJ∂ . By Lemmas 4.5.4 and 6.4.2, and by (6.3.10) again, it suffices to show that the canonical morphism   H i [m0 ](Γ J ), L|U ∂ (ξ ) [W ] → H i [m](Γ J ), L|U ∂ (ξ ) [W ] J

J

is an isomorphism. By the Hochschild–Serre spectral sequence, we may first compare the cohomology of [m0 ](Γ {j0 } ) and [m](Γ {j0 } ), for some j0 ∈ J , which is concentrated in degree zero and gives the full L|U ∂ (ξ ) in both cases, because J

γjm0 and γjm act unipotently, by assumption. Then we compare the cohomology groups of [m0 ](Γ J −{j0 } ) and [m](Γ J −{j0 } ), which coincide as they are related by a Hochschild–Serre spectral sequence in terms of the cohomology of Qp -modules with unipotent actions of the finite group [m0 ](Γ J −{j0 } )/[m](Γ J −{j0 } ), and the claim follows.  Proposition 6.4.4 Let L be a Qp -local system on Xkét such that L|Uét has quasiunipotent geometric monodromy along D (as in Definition 6.3.7). Consider any integer m ≥ 1 such that L|Um has unipotent geometric monodromy along Dm , where Um and Dm are as in Lemma 6.4.3. Then, for each nonempty J ⊂ I and each i ≥ 0, and for each geometric point ξ of UJ∂ , we have  R i Ψfu (L|U )|U ∂ (ξ ) ∼ = H i Γ J , L|U ∂ (ξ ) [W ] J

J

and  qu R i Ψf (L|U )|U ∂ (ξ ) ∼ = H i [m](Γ J ), L|U ∂ (ξ ) [W ] J

as Qp -local systems on UJ (ξ )ét .

J

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 Consequently, if D = f −1 (0) red is smooth over k and if L|U has quasiunipotent monodromy along D, then RΨfu (L|U ) is concentrated in degree zero unip

and can be identified with the subsheaf L|D ∂ of L|D ∂ whose pullback to D ∂ (ξ ) is the maximal subsheaf on which π1két (D ∂ (ξ ),  ξ) ∼ Z(1) (as in Example 4.4.26) = ξ of D ∂ above each geometric point acts unipotently, for each log geometric point  qu ξ of D; and RΨf (L|U ) (which is the same RΨfu (L|U ) as above when L|U has unipotent monodromy along D) is also concentrated in degree zero and can be identified with the whole L|D ∂ . Proof Combine Lemmas 4.5.4, 6.4.2, and 6.4.3.



A Kiehl’s Property for Coherent Sheaves In this appendix, by adapting the gluing argument in [27, Section 2.7] and by using [17, Theorem 2.5], we establish Kiehl’s property for coherent sheaves on (possibly nonanalytic) noetherian adic spaces. By combining this with results in [27, Section 8.2] and [26, Sections 1.3–1.4], we also state some versions of Tate’s sheaf property and Kiehl’s gluing property for adic spaces that are either locally noetherian, or analytic and stably adic. (We will review the definition below.) Recall the following definition from [27, Definition 1.3.7]: Definition A.1 By a gluing diagram, we will mean a commuting diagram of ring homomorphisms R

R1

R2

R12

such that the R-module sequence 0 → R → R1 ⊕ R2 → R12 → 0, in which the last nontrivial arrow is the difference between the given homomorphisms, is exact. By a gluing datum over this diagram, we mean a datum consisting of modules M1 , M2 , and M12 over R1 , R2 , and R12 , respectively, equipped with ∼ ∼ isomorphisms ψ1 : M1 ⊗R1 R12 → M12 and ψ2 : M2 ⊗R2 R12 → M12 . We say such a gluing datum is finite if the modules are finite over the respective rings. Given a gluing datum as above, let M := ker(ψ1 − ψ2 : M1 ⊕ M2 → M12 ). There are natural morphisms M → M1 and M → M2 of R-modules, which induce maps M ⊗R R1 → M1 and M ⊗R R2 → M2 , respectively. The following is [27, Lemma 1.3.8]:

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Lemma A.2 Consider a finite gluing datum for which M ⊗R R1 → M1 is surjective. Then the following are true. (1) The morphism ψ1 − ψ2 : M1 ⊕ M2 → M12 is surjective. (2) The morphism M ⊗R R2 → M2 is also surjective. (3) There exists a finitely generated R-submodule M0 of M such that, for i = 1, 2, the morphism M0 ⊗R Ri → Mi is surjective. Lemma A.3 In the above setting, suppose in addition that Ri is noetherian and that Ri → R12 is flat, for i = 1, 2. Suppose that, for every finite gluing datum, the map M ⊗R R1 → M1 is surjective. Then, for any finite gluing datum, M is a finitely presented R-module, and M ⊗R R1 → M1 and M ⊗R R2 → M2 are bijective. Proof Let M0 be as in Lemma A.2. Choose a surjection F → M0 of R-modules, with F finite free. Let F1 := F ⊗R R1 , F2 := F ⊗R R2 , and F12 := F ⊗R R12 . Let N := ker(F → M), N1 := ker(F1 → M1 ), N2 := ker(F2 → M2 ), and N12 := ker(F12 → M12 ). By Lemma A.2, we have a commutative diagram 0

0

0

0

N

N1 ⊕ N 2

N12

0

0

F

F1 ⊕ F2

F12

0

0

M

M 1 ⊕ M2

M12

0

0

0

0

(A.4)

with exact rows and columns, excluding the dotted arrows. Since R12 is flat over Ri , the sequence 0 → Ni ⊗Ri R12 → F12 → M12 → 0 is exact, and hence Ni ⊗Ri R12 ∼ = N12 . By hypothesis, Ri is noetherian, and so Ni is finite over Ri . Consequently, N1 , N2 , and N12 form a finite gluing datum as well. By Lemma A.2 again, the dotted horizontal arrow in (A.4) is surjective. By diagram chasing, the dotted vertical arrow in (A.4) is also surjective; that is, we may add the dotted arrows to (A.4) while preserving exactness of the rows and columns. In particular, M is a finitely generated R-module. It follows that N is finitely generated. This implies that M is finitely presented.

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For i = 1, 2, we obtain a commutative diagram

0

N ⊗R Ri

Fi

M ⊗R Ri

0

Ni

Fi

Mi

0

with exact rows—the first one is derived from the left column of (A.4) by tensoring with Ri over R, while the second one is derived from the middle column of (A.4). Since the left vertical arrow is surjective, by the five lemma, the right vertical arrow is injective. It follows that the map M ⊗R Ri → Mi is a bijection, as desired.  Definition A.5 We call a homomorphism of Huber rings f : A → B strict adic if, for one (and hence every) choice of an ideal of definition I ⊂ A, the image f (I ) is an ideal of definition of B. It is clear that a strict adic morphism is adic. The following is modeled on [27, Lemma 2.7.2]. Lemma A.6 Let R1 → S and R2 → S be homomorphisms of complete Huber rings such that their sum ψ : R1 ⊕ R2 → S is strict adic. Then, for any ideal of definition IS of S, there exists some integer l ≥ 1 such that, for every n > 0, every U ∈ GLn (S) with U − 1 ∈ Mn (ISl ) is of the form ψ(U1 ) ψ(U2 ) for some Ui ∈ GLn (Ri ), for i = 1, 2. Proof Since ψ is strict adic, for any ideals of definition I1 ⊂ R1 and I2 ⊂ R2 , we have an ideal of definition IS := ψ(I1 ⊕ I2 ) ⊂ S. Choose l > 0 such that ISl ⊂ IS . Then it suffices to show that every U ∈ GLn (S) with U − 1 ∈ Mn (IS ) is of the form ψ(U1 ) ψ(U2 ) for some Ui ∈ GLn (Ri ), for i = 1, 2. Given U ∈ GLn (S) with U − 1 ∈ Mn (IS m ) for some m > 0, put V = U − 1. By assumption, we may lift V to a pair (X, Y ) ∈ Mn (I1m ) × Mn (I2m ). Then it is straightforward that the matrix U = ψ(1 − X) U ψ(1 − Y ) satisfies U − 1 ∈ Mn (IS 2m ). Hence, we may construct the desired matrices by iterating this construction.  The following is modeled on [27, Lemma 2.7.4]. Lemma A.7 In the context of Definition A.1 and the paragraph following it, suppose in addition that (1) the Huber rings R1 , R2 and R12 are complete; (2) R1 ⊕ R2 → R12 is strict adic; and (3) the map R2 → R12 has a dense image. Then, for i = 1, 2, the natural map M ⊗R Ri → Mi is surjective. Proof Choose sets of generators {m1,1 , . . . , mn,1 } and {m1,2 , . . . , mn,2 } of M1 and M2 , respectively, of the same cardinality. Then there exist  A, B ∈ Mn (R12 ) such that ψ2 (mj,1 ) = A ψ (m ) and ψ (m ) = ij 1 i,2 1 j,2 i i Bij ψ2 (mi,1 ), for all j . Since R2 → R12 has a dense image, by Lemma A.6, there exists B ∈ Mn (R2 )

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such that 1 + A(B − B) = C1 C2−1 for some Ci ∈ GLn (Ri ), for i = 1, 2. For j = 1, . . . , n, let xj := (xj,1 , xj,2 ) =



(C1 )ij mi,1 ,

i



 (B C2 )ij mi,2 ∈ M1 × M2 .

i

Then xj ∈ M, because ψ1 (xj,1 ) − ψ2 (xj,2 ) =



(C1 − AB C2 )ij ψ1 (mi,1 )

i

 = (1 − AB)C2 ij ψ1 (mi,2 ) = 0. i

For i = 1, 2, since Ci ∈ GLn (Ri ), we see that {xj,i }nj=1 generates Mi over Ri as well. Thus, M ⊗R Ri → Mi is surjective, as desired.  Theorem A.8 Let X = Spa(R, R + ) be a noetherian affinoid adic space. The categories of coherent sheaves on X and finitely generated R-modules are equivalent via the global sections functor. Proof By Kedlaya and Liu [27, Lemma 2.4.20], it suffices to verify Kiehl’s gluing property for any simple Laurent covering {Spa(Ri , Ri+ ) → X}i=1,2 . In this case, + ) = Spa(R1 , R1+ ) ×X Spa(R2 , R2+ ), with all Huber pairs let us write Spa(R12 , R12 completed by our convention. By the noetherian hypothesis, and by Huber [17, Theorem 2.5], R, Ri , and R12 form a gluing diagram. Also, Ri → R12 is flat with dense image, for i = 1, 2. Hence, we can finish the proof by applying Lemmas A.3 and A.7.  Thus, we have the following version of Tate’s sheaf property and Kiehl’s gluing property (see [27, Definition 2.7.6]) over certain affinoid adic spaces: Proposition A.9 Let X = Spa(R, R + ) be a noetherian (resp. analytic) affinoid  denote adic space, and let M be a finite (resp. finite projective) R-module. Let M  ) = M ⊗R OX (U ), for each open subset the presheaf on X defined by setting M(U U ⊂ X. Then the following are true:  is a sheaf. Moreover, the sheaf M  is acyclic in the sense that (1) The presheaf M i  H (U, M) = 0 for every rational subset U ⊂ X and every i > 0.  defines an equivalence of categories from the category (2) The functor M → M of finite (resp. finite projective) R-modules to the category of coherent sheaves (resp. vector bundles) on X, with a quasi-inverse given by F → F(X). Proof When X is noetherian, (1) is [17, Theorem 2.5], while (2) is Theorem A.8. When X is analytic, these follow from [26, Theorems 1.4.2 and 1.3.4].  Recall that an adic space X is called stably adic (as in [27, Definition 8.2.19]) if Xét is a site with a stable basis B; i.e., a basis stable under fiber products such that,

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for any morphism Y → Y in Xét that is either finite étale or a rational localization, if Y ∈ B, then Y ∈ B as well. We know X is stably adic if X is locally noetherian (see [18, (1.1.1) and Section 1.7]) or a perfectoid space (see [43, Lecture 7]). By Kedlaya and Liu [27, Proposition 8.2.20], we have the following analogue of Proposition A.9 for the étale topology: Proposition A.10 Let X = Spa(R, R + ) be a noetherian (resp. analytic stably adic) affinoid adic space. Let B be a stable basis of Xét as above, which exists because X  denote the is stably adic. Let M be a finite (resp. finite projective) R-module. Let M  ) = M ⊗R OX (U ), for each U ∈ Xét . Then presheaf on Xét defined by setting M(U the following are true:  is a sheaf. Moreover, M  is acyclic on B; i.e., for every Y ∈ B, (1) The presheaf M 0 i    = 0, for all i > 0. we have H (Y, M) = M(U ) and H (Y, M)  (2) The functor M → M defines an equivalence of categories from the category of finite (resp. finite projective) R-modules to the category of coherent sheaves (resp. vector bundles) on Xét , with a quasi-inverse given by F → F(X). Corollary A.11 For any X in Proposition A.10, the presheaf OXét is a sheaf. Therefore, X is étale sheafy. Acknowledgments This paper was initially based on a paper written by the first author, and we would like to thank Christian Johansson, Kiran Kedlaya, Teruhisa Koshikawa, Martin Olsson, Fucheng Tan, and Jilong Tong for helpful correspondences and conversations during the preparation of that paper. We would also like to thank David Sherman and an anonymous referee for their careful reading and many helpful corrections, questions, and suggestions. Moreover, we would like to thank the Beijing International Center for Mathematical Research and the California Institute of Technology for their hospitality. Finally, we would like to thank the following organizations for their financial support during the preparation of this paper. K.-W. Lan was partially supported by the National Science Foundation under agreement No. DMS-1352216, by an Alfred P. Sloan Research Fellowship, and by a Simons Fellowship in Mathematics. R. Liu was partially supported by the National Natural Science Foundation of China under agreement Nos. NSFC-11571017 and NSFC-11725101, and by the Tencent Foundation. X. Zhu was partially supported by the National Science Foundation under agreement Nos. DMS-1602092 and DMS-1902239, by an Alfred P. Sloan Research Fellowship, and by a Simons Fellowship in Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this writing are those of the authors, and do not necessarily reflect the views of the funding organizations.

References 1. Artin, M., Grothendieck, A., Verdier, J.L. (eds.): Théorie des topos et cohomologie étale des schémas (SGA 4), Lecture Notes in Mathematics, vol. 269, 270, 305. Springer-Verlag, Berlin, Heidelberg, New York (1972, 1972, 1973) 2. Bartenwerfer, W.: Der erste Riemannsche Hebbarkeitssatz im nichtarchimedischen Fall. J. Reine Angew. Math. 286/287, 144–163 (1976) 3. Beilinson, A.A.: How to glue perverse sheaves. In: K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Mathematics, vol. 1289, pp. 42–51. Springer-Verlag, Berlin, Heidelberg, New York (1987)

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Twisted Differential Operators and q-Crystals Michel Gros, Bernard Le Stum, and Adolfo Quirós

Introduction In their recent article [3], Bhargav Bhatt and Peter Scholze have introduced two new cohomological theories with a strong crystalline flavor, the prismatic and the qcrystalline cohomologies, allowing them to generalize some of their former results obtained with Matthew Morrow on p-adic integral cohomology. As explained in [6, section 6], once a theory of coefficients for these new cohomologies is developed, these tools could also be a way for us to get rid of some non-canonical choices in our construction of the twisted Simpson correspondence [7, corollary 8.9]. At the end, this correspondence should hopefully appear just as an avatar of a deeper canonical equivalence of crystals on the prismatic and the q-crystalline sites, whose general pattern should look like a “q-deformation” of Hidetoshi Oyama’s reinterpretation [14, theorem 1.4.3] of Ogus-Vologodsky’s correspondence as an equivalence between categories of crystals (see [6, section 6]) for a general overview. In this note, we start elaborating on this project by showing Theorem 7.5 how to construct a functor from the category of crystals on Bhatt-Scholze’s q-crystalline site (that we call for short q-crystals) to the category of modules over the ring Dq of twisted differential operators of [7, section 5]. The construction of this functor has also the independent interest of describing explicitly, at least locally, the kind of structure hiding behind a q-crystal. Indeed, the ring Dq = DA/R,q is defined

M. Gros () · B. Le Stum IRMAR, Université de Rennes, Rennes Cedex, France e-mail: [email protected]; [email protected] A. Quirós Departamento de Matématicas, Universidad Autónoma de Madrid, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_4

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only for a very special class of algebras A over a base ring R and some element q ∈ R (let’s call them in this introduction, forgetting additional data, simply twisted R-algebras; see [11] or [7] for a precise definition) and it is obviously only for them that the functor will be constructed. The construction consists then of two main steps. The first one, certainly the less standard, is to describe concretely the q-PD-envelope introduced a bit abstractly in [3, lemma 16.10], of the algebra of polynomials in one indeterminate ξ over these twisted R-algebras and to relate it to the algebra appearing in the construction of Dq . The second step is to develop the q-analogs of the usual calculus underlying the theory of classical crystals: hyperstratification, connection. . . for the q-crystalline site associated to a twisted R-algebra. Let us now describe briefly the organization of this article. In Sect. 1, we recall for the reader’s convenience some basic vocabulary and properties of δ-rings used in [3]. In Sect. 2, we essentially rephrase some of our considerations about twisted powers [7], using this time δ-structures instead of relative Frobenius. Following [3], we introduce in Sect. 3 the notion of q-PD-envelope, first ignoring for it some hypotheses of completeness in loc. cit, and prove (Theorem 3.5) that our twisted divided power algebra in the indeterminate ξ coincides exactly with this a priori completely different object. Section 4 then addresses the general question of completion of q-PD-envelopes. In Sect. 5 we start the elaboration of twisted calculus and develop the language of hyper-q-stratifications (Definition 5.7), that is to say, the twisted variant of hyperstratifications, where new phenomena like non-trivial flip maps (Proposition 5.4) appear. Once we have obtained all these materials, we explain in Sect. 6 how they can be reinterpreted in the language of Dq -modules and we establish (Propositions 6.16 and 6.18) the twisted version of the classical equivalences of categories of usual calculus. We have then all the ingredients to come back, in Sect. 7, to the question of describing the complete q-PD-envelope of the diagonal embedding (Theorem 7.3) and to give the construction of the functor (Theorem 7.5) going from the category of q-crystals to that of Dq -modules in our particular geometric setting. Everything below depends on a prime p, fixed once for all. We stick to dimension one, leaving for another occasion the consideration of the higher dimensional case, which requires additional inputs. Also, since we are mainly interested in local questions, we concentrate on the affine case. We are extremely grateful and indebted to the referee for her/his very meticulous and constructive reading of the successive versions, pointing out in many places the need for further explanations or the existence of several issues, as well as for providing us with precious hints on how to fix them and suggestions to improve the whole presentation. After the release of the first version of this article, two preprints on related matters have appeared and could be of great interest for the readers: one by Andre Chatzistamatiou [5], the other by Matthew Morrow and Takeshi Tsuji [13].

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The first author (M.G.) heartily thanks the organizers, Bhargav Bhatt and Martin Olsson, for their invitation to the Simons Symposium on p-adic Hodge Theory (April 28-May 4, 2019), allowing him to follow the progress on topics related to those treated in this note.

1 δ-Structures In this section, we briefly review the notion of a δ-ring. We start from the Witt vectors point of view since it automatically provides all the standard properties. As a set, the ring of (p-typical) Witt vectors of length two1 on a commutative ring A is W1 (A) = A × A. It is endowed with the unique natural ring structure such that both the projection map W1 (A) → A,

(f, g) → f

and the ghost map W1 (A) → A,

(f, g) → f p + pg

are ring homomorphisms. A δ-structure on A is a section A → W1 (A),

f → (f, δ(f ))

of the projection map in the category of rings. This is equivalent to giving a pderivation: a map δ : A → A such that δ(0) = δ(1) = 0, ∀f, g ∈ A,

δ(f + g) = δ(f ) + δ(g) −

p−1  k=1

  1 p p−k k f g p k

(1)

and ∀f, g ∈ A,

δ(fg) = f p δ(g) + δ(f )g p + pδ(f )δ(g).

(2)

Note that a δ-structure is uniquely determined by the action of δ on the generators of the ring. A δ-ring is a commutative ring endowed with a δ-structure and δ-rings make a category in the obvious way. We refer the reader to André Joyal’s note [8] for a short but beautiful introduction to the theory. When R is a fixed δ-ring, a δ-Ralgebra is an R-algebra endowed with a compatible δ-structure.

1 We

use the more recent index convention for truncated Witt vectors.

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Examples 1. There exists only one δ-structure on the ring Z of rational integers which is given by δ(n) =

n − np ∈ Z. p

2. If f ∈ Z[x], then there exists a unique δ-structure on Z[x] such that δ(x) = f . 3. There exists no δ-structure at all when pk A = 0, unless A = 0 (show that vp (δ(n)) = vp (n) − 1 when vp (n) > 0 and deduce that vp (δ k (pk )) = 0). 4. It may also happen that there √ exists no δ-structure at all even when A is p-torsionfree: take p = 2 and A = Z[ −1]. A Frobenius on a commutative ring A is a (ring) morphism φ : A → A that satisfies ∀f ∈ A,

φ(f ) ≡ f p mod p.

If A is a δ-ring, then the map φ : A → A,

f → f p + pδ(f ),

obtained by composition of the section with the ghost map, is a Frobenius on A and this construction is functorial. Moreover, it is easy to verify that φ ◦ δ = δ ◦ φ. Note that the multiplicative condition (2) may then be rewritten in the asymmetric but sometimes more convenient way ∀f, g ∈ A,

δ(fg) = δ(f )g p + φ(f )δ(g) = f p δ(g) + δ(f )φ(g).

(3)

Conversely, when A is p-torsion-free, any Frobenius of A comes from a unique δ-structure through the formula δ(f ) =

φ(f ) − f p ∈ A. p

We will systematically use the fact that the category of δ-rings has all limits and colimits and that they both preserve the underlying rings: actually the forgetful functor is conservative and has both an adjoint (δ-envelope) and a coadjoint (Witt vectors). We refer the reader to the article of James Borger and Ben Wieland [4] for a plethystic interpretation of these phenomena. Definition 1.1 Let R be a δ-ring. If A is an R-algebra, then its δ-envelope Aδ is a δ-R-algebra which is universal for morphisms of R-algebras into δ-R-algebras. Actually, it follows from the above discussion that the forgetful functor from δR-algebras to R-algebras has an adjoint A → Aδ (so that δ-envelopes always exist

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and preserve colimits). Be careful that, even if it does not show up in the notation, the δ-envelope does depend on the base δ-ring R. However, if R → S is a morphism of δ-rings, we have (S ⊗R A)δ  S ⊗R Aδ .

(4)

Of course, it is always possible to define the δ-envelope of a ring A by considering the case R = Z. Example We have2 R[x]δ = R[{xi }i∈N ] (polynomial ring with infinitely many variables) with the unique δ-structure such that δ(xi ) = xi+1 for i ∈ N. More generally, since δ-envelope preserves colimits, we have R[{xk }k∈F ]δ = R[{xk,i }k∈F,i∈N ], with the unique δ-structure such that δ(xk,i ) = xk,i+1 for k ∈ F, i ∈ N. Definition 1.2 A δ-ideal in a δ-ring A is an ideal I which is stable under δ. In general, the δ-closure Iδ of an ideal I is the smallest δ-ideal containing I . Equivalently, a δ-ideal is the kernel of a morphism of δ-rings. It immediately follows from formulas (1) and (2) above that the condition for an ideal to be a δideal may be checked on generators. As a consequence, if I = {fi }i∈E , then Iδ =



j

δ (fi )





i∈E,j ∈N

.

Also, if R is a δ-ring, A an R-algebra and I ⊂ A an ideal, then we have (A/I )δ = Aδ /(I Aδ )δ . This provides a convenient tool to describe δ-envelopes by choosing a presentation:     A = R[{xk }k∈F ]/ {fi }i∈E ⇒ Aδ = R[{xk,i }k∈F,i∈N ]/ δ j (fi ) i∈E,j ∈N

(with xk → xk,0 ). Example We usually endow the polynomial ring Z[q] with the unique δ-structure such that δ(q) = 0. Then the principal ideal generated by q − 1 is a δ-ideal: since q p − 1 ≡ (q − 1)p both modulo p and modulo q − 1, we can write q p − 1 − (q − 1)p = p(q − 1)c

R = Z, this is in fact the same as the free δ-ring Z{e} of [3], lemma 2.11 or the ring of p-typical symmetric functions Λp of [4].

2 For

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in the unique factorization domain Z[q], and then we have δ(q − 1) = (q − 1)c. We will need later the following property: Lemma 1.3 Assume that p lies in the Jacobson radical of both A and B. If A × B is endowed with a δ-structure, then both A and B are δ-ideals. Equivalently, there exists a (unique) quotient δ-structure on both A and B. Proof It is enough to prove that A has a unique quotient δ-structure, and we only need to check that δ(1, 0) = (0, 0) because everything else is automatic. If we write δ(1, 0) =: (f, g), then we have δ(1, 0) = δ((1, 0)2 ) = 2(f, g)(1, 0)p + p(f, g)2 = (2f + pf 2 , pg 2 ). It follows that f = 2f + pf 2 and g = pg 2 or, in other words, that (1 + pf )f = 0 and (1 − pg)g = 0. Since p lies in the Jacobson radical of A (resp. B) then 1 + pf (resp. 1−pg) is invertible in A (resp. B) and it follows that f = 0 (resp. g = 0).  Note that the condition will be satisfied when A and B are both (p)-adically complete. Note also that some condition is necessary because the result does not hold for example when A = B = Q. It is important for us to recall also that a p-derivation δ on a ring A is systematically I -adically continuous with respect to any finitely generated ideal I containing p. In particular, δ will then automatically extend in a unique way to a δ-structure on the I -adic completion of A. Finally, we will use the convenient terminology of f being a rank one element to mean δ(f ) = 0 (and consequently φ(f ) = f p ) and f being a distinguished element to mean δ(f ) ∈ A× (and therefore p ∈ (f p , φ(f )) ⊂ (f, φ(f ))).

2 δ-Rings and Twisted Divided Powers We fix a δ-ring R as well as a rank one element q in R. Alternatively, we may consider R as a δ-Z[q]-algebra, where q is then seen as a parameter with δ(q) = 0, in which case we would still write q instead of q1R ∈ R. As a δ-ring, R is endowed with a Frobenius endomorphism φ. Note that φ(q) = q p and the action on qanalogs of natural numbers3 is given by φ((n)q ) = (n)q p . We let A be a δ-R-algebra and we fix some rank one element x in A. The Ralgebra A is automatically endowed with a Frobenius φ which is semilinear (with respect to the Frobenius φ of R) and satisfies φ(x) = x p . Although we could consider as well the relative Frobenius, which is the R-linear map4

q n −1 q−1 .

3 We

write (n)q :=

4 We

∗ used to put a star and write FA/R in [7].

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F : A := R φ (⊗R A → A,

a φ (⊗ f → af p + paδ(f ),

so that φ(f ) = F (1⊗f ), we will stick here to the absolute Frobenius φ and modify our formulas from [7] accordingly. Let us remark, however, that A has a natural δstructure and that F is then a morphism of δ-R-algebras. There exists a unique structure of δ-A-algebra on the polynomial ring A[ξ ] such that x + ξ has rank one and we call it the symmetric δ-structure. It is given by δ(ξ ) =

p−1  i=1

  1 p p−i i x ξ p i

(5)

(which depends on x but not on q or δ). Recall that we introduced in section 4 of [9] the twisted powers ξ (n)q := ξ(ξ + x − qx) · · · (ξ + x − q n−1 x) ∈ A[ξ ] (that clearly depend on both the choice of q and x). We will usually drop the index q and simply write ξ (n) . They form an alternative basis for A[ξ ] (as an A-module) which is better adapted to working with q-analogs: if φ denotes the Frobenius of A[ξ ] attached to the symmetric δ-structure, then we have the fundamental congruence φ(ξ ) ≡ ξ (p) mod (p)q .

(6)

This follows from corollary 7.6 of [7] and lemma 2.12 of [9], but may also be checked directly. There exist more general explicit formulas for higher twisted powers: we showed in proposition 7.5 of [7] that5 φ(ξ (n) ) =

pn 

an,i x pn−i ξ (i)

i=n

with an,i :=

n  j =0

(−1)n−j q

p(n−j )(n−j −1) 2

    n pj ∈ Z[q] j qp i q

(7)

(in which we use the q-analogs of the binomial coefficients, as in section 2 of [9] for example).

5 In

[7] we actually used An,i and Bn,i instead of an,i and bn,i .

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In section 2 of [7], we also introduced the ring of twisted divided polynomials A#ξ $q , which depends on the choice6 of both q and x (but does not require any δ-structure on A). This is a commutative A-algebra. As an A-module, it is free on some generators ξ [n]q (called the twisted divided powers) indexed by n ∈ N. The multiplication rule is quite involved: ξ [n]q ξ [m]q =



q

i(i−1) 2



0≤i≤n,m

n+m−i n

   n (q − 1)i x i ξ [n+m−i]q . i q q

(8)

Again, we will usually drop the index q and simply write ξ [n] . Heuristically,7 we have ∀n ∈ N,

ξ [n] =

ξ (n) . (n)q !

In section 7 of [7], we showed that there exists a divided Frobenius map [F ] : A #ω$q p → A#ξ $q (we denote by ω the variable on the left hand side in order to avoid confusion) such that, heuristically again, ∀n ∈ N,

 φ ξ [n]  [n] = [F ] ω . (p)nq

(9)

More precisely, we have the following formula pn    [n] = [F ] ω bn,i x pn−i ξ [i]

(10)

i=n

with bn,i =

(i)q ! an,i ∈ Z[q] (n)q p !(p)nq

and an,i as in (7). We may then define the absolute Frobenius of A#ξ $q by composing the “blowing up”

it depends on q and y := (1 − q)x. precisely, there exists a unique natural homomorphism A[ξ ] → A#ξ $q sending ξ (n) → (n)q !ξ [n] .

6 Actually, 7 More

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A#ξ $q → A #ω$q p ,

ξ [n] → (p)nq ω[n]

with the divided Frobenius [F ]. Note that this blowing up is a morphism of rings: this is actually a generic question that may be checked on polynomial rings. One must also check that this is a lifting of the absolute Frobenius. We may clearly assume that R = Z[q] and A = R[x] and we are reduced to showing that the map induced on Z[q, x]#ξ $/p is the absolute Frobenius. Since this ring is (n)q -torsion free for all n > 0, this boils down to the same assertion on Fp [q, x, ξ ]. It is therefore sufficient to recall that q, x and x + ξ map respectively to q p , x p and (x + ξ )p . When A is p-torsion-free, the Frobenius of A#ξ $q corresponds to a unique δstructure on A#ξ $q and this provides a natural δ-structure in general using the isomorphism A#ξ $q  A ⊗Z[q,x] Z[q, x]#ξ $q

(11)

(where q and x are seen as parameters). Remark The rank one condition on x is crucial for our results to hold. Otherwise, the Frobenius of A[ξ ] will not extend to A#ξ $q . This is easily checked when p = 2. Let us denote by δc the δ structure given by δc (x) = c ∈ R and by φc the corresponding Frobenius, so that φc (x) = x 2 + 2c = φ0 (x) + 2c

and

φc (ξ ) = ξ 2 + 2xξ = φ0 (ξ ).

Recall now from the first remark after definition 7.10 in [7] that φ0 (ξ ) = (1 + q)(ξ [2] + xξ ).  Then, the following computation shows that φc ξ (2) is not divisible by  (2)q 2 = 1 + q 2 in general (so that φc ξ [2] does not exist in A#ξ $q ):   φc ξ (2) = φc (ξ(ξ + (1 − q)x)) = φ0 (ξ )(φ0 (ξ ) + (1 − q 2 )(φ0 (x) + 2c))   = φ0 ξ (2) + 2c(1 − q 2 )φ0 (ξ )   = (1 + q 2 )φ0 ξ [2] + 2c(1 − q)(1 + q)2 (ξ [2] + xξ ).

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3 q-Divided Powers and Twisted Divided Powers As before, we let R be a δ-ring with fixed rank one element q. We consider the maximal ideal (p, q − 1) ⊂ Z[q] and we assume from now on that R is actually a Z[q](p,q−1) -algebra. Note that, under this new hypothesis, we have (k)q ∈ R × as long as p  k so that R/(p)q is a q-divisible ring of q-characteristic p in the sense of [9], which was a necessary condition for the main results of [7] to hold. We have the following congruences when k ∈ N: (p)q k ≡ p mod q − 1 and

p−1

(p)q k ≡ (k)q

(q − 1)p−1 mod p,

(12)

which both imply that (p)q k ∈ (p, q −1). It also follows that (p)q k is a distinguished element in R since δ((p)q k ) ≡ δ(p) = 1 − pp−1 ≡ 1 mod (p, q − 1) (the first congruence follows from the fact that (q − 1) is a δ-ideal). We recall now the following notion from [3, definition 16.2]: Definition 3.1 1. If B is a δ-R-algebra and J ⊂ B is an ideal, then (B, J ) is a δ-pair. We may also say that the surjective map B  B := B/J is a δ-thickening. 2. If, moreover, B is (p)q -torsion-free and ∀f ∈ J,

φ(f ) − (p)q δ(f ) ∈ (p)q J,

(13)

then (B, J ) is a q-PD-pair.8 We may also say that J is a q-PD-ideal or has q-divided powers or that the map B  B is a q-PD-thickening. Remarks 1. Condition (13) may be split in two, as Bhatt and Scholze do in loc. cit.: one may first require that φ(J ) ⊂ (p)q B, then introduce the map γ : J → B, f →

φ(f ) − δ(f ) (p)q

and also require that γ (J ) ⊂ J . 2. In the special case where J is a δ-ideal, condition (13) simply reads φ(J ) ⊂ (p)q J , and this implies that φ k (J ) ⊂ (pk )q J for all k ∈ N.

8 There

are a lot more requirements, that we may ignore at this moment, in definition 16.2 of [3].

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3. In general, the elements f ∈ J that satisfy the property in condition (13) form an ideal and the condition can therefore be checked on generators. Examples 1. In the case q = 1, condition (13) simply reads ∀f ∈ J,

f p ∈ pJ

and (B, J ) is a q-PD-pair if and only if B is p-torsion-free and J has usual divided powers (use lemma 2.35 of [3]). In other words, a 1-PD-pair is a ptorsion-free PD-pair endowed with a lifting of Frobenius (the δ-structure). 2. If B is a (p)q -torsion-free δ-R-algebra, then the Nygaard ideal N := φ −1 ((p)q B) has q-divided powers (this is shown in [3, lemma 16.7]). This is the maximal q-PD-ideal in B. Note that N is the first piece of the Nygaard filtration. 3. If B is a (p)q -torsion-free δ-R-algebra, then J := (q−1)B has q-divided powers. This is also shown in [3], but actually simply follows from the equalities φ(q − 1) = q p − 1 = (p)q (q − 1), since (q − 1) is a δ-ideal. 4. With the notations of the previous section, we may endow A#ξ $q with the augmentation ideal I [1] generated by all the ξ [n] for n ≥ 1. When A is (p)q torsion-free, this is a q-PD pair as equality (9) shows (since I [1] is clearly a δ-ideal). For later use, let us also mention the following: Lemma 3.2 1. If (B, J ) is a q-PD-pair, B is a (p)q -torsion-free δ-ring and B → B is a morphism of δ-rings, then (B , J B ) is a q-PD-pair. 2. The category of δ-pairs has all colimits and they are given by

with B = lim Be and J = − → are (p)q -torsion free.



lim(Be , Je ) = (B, J ) − → Je B. Colimits preserve q-PD-pairs as long as they

Proof Both assertions concerning q-PD-pairs follow from the fact that the property in condition (13) may be checked on generators. The first part of the second assertion is a consequence of the fact that the category of δ-rings has all colimits and that they preserve the underlying rings. 

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Remarks 1. As a consequence of the first assertion, we see that if (B, J ) is a q-PD-pair and b ⊂ B is a δ-ideal such that B/b is (p)q -torsion-free, then (B/b, J + b/b) is also a q-PD-pair. 2. As a particular case of the second assertion, we see that fibered coproducts in the category of δ-pairs are given by (B1 , J1 ) ⊗(B,J ) (B2 , J2 ) = (B1 ⊗B B2 , im(B1 ⊗B J2 + J1 ⊗B B2 )). Note that if B1 is B-flat and B2 is (p)q torsion-free, then B1 ⊗B B2 is automatically (p)q -torsion-free. Definition 3.3  [] Let (B, J ) be a δ-pair. Then (if it exists) its q-PD envelope B q , J [ ]q is a q-PD-pair that is universal for morphisms to q-PD-pairs: there  exists a morphism of δ-pairs (B, J ) → B [ ]q , J [ ]q such that any morphism  (B, J ) → (B , J ) to a q-PD-pair extends uniquely to B [ ]q , J [ ]q . Remarks 1. Note that the q-PD-envelope only depends on the underlying δ-Z[q]-algebra structure: we may as well take R = B. 2. In Definition 3.3, it might be necessary/useful for later developments to add the condition B  B [ ]q , but this is not clear yet. Examples 1. When q = 1, B is p-torsion-free and J is generated by a regular sequence modulo p, then the q-PD-envelope of B is its usual PD-envelope (corollary 2.39 of [3]). 2. If B is (p)q -torsion-free and J already has q-divided powers, then B [ ]q = B. 3. If J = B, then B [ ]q = B[{1/(n)q }n∈N ] (= 0 in general). As a consequence of Lemma 3.2, let us mention the following: Lemma 3.4 1. Let (B, J ) be a δ-pair and b ⊂ B a δ-ideal. Assume that (B, J ) has a q-PDenvelope and B [ ]q /bB [ ]q is (p)q -torsion-free. Then,   B [ ]q /bB [ ]q , (J [ ]q + bB [ ]q )/bB [ ]q is the q-PD-envelope of (B/b, J + b/b). 2. Let {(Be , Je )}e∈E be a commutative diagram of δ-pairs all having a q-PD [] Je B. If B := lim Be q is (p)q -torsion free, envelope, B := lim Be and J = − → − →    [] then B , Je q B is the q-PD-envelope of (B, J ).

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Proof Concerning both assertions, we already know from Lemma 3.2 (and the remarks thereafter) that the given pair is a q-PD-pair and the universal property follows from the universal properties of quotient and colimit respectively.  The next example is fundamental for us (we use the notations of the previous section) but we first want to make precise some general notations. If A is any commutative ring and d, g ∈ A, we define A[g/d] := A⊗A[X] A[Y ], with structural maps X → g on the left and X → dY on the right. Then, if A → B is any morphism, we have B ⊗A A[g/d]  B[g/d]. When A is a δ-ring, we denote by A[g/d]δ the δ-envelope of the A-algebra A[g/d]. When A → B is a morphism of δ-rings, it follows from (4) that B ⊗A A[g/d]δ  B[g/d]δ . Theorem 3.5 If A is a (p)q -torsion-free δ-R-algebra with fixed rank one element x and A[ξ ] is endowed with the symmetric δ-structure, then (A#ξ $q , I [1] ) is the qPD-envelope of (A[ξ ], (ξ )). Moreover, we have an isomorphism of δ-R-algebras A#ξ $q  A[ξ ][φ(ξ )/(p)q ]δ . Proof We already know from (11) that A#ξ $q is stable under the base change of A and, as we just saw, the same holds for A[ξ ][φ(ξ )/(p)q ]δ . We may therefore assume that R = Z[q](p,q−1) and A = R[x] and we first construct some other basis of A#ξ $q as an A-module that will be useful later. We endow A#ξ $q with the degree filtration by the A-submodules Fn generated by ξ [k] for k ≤ n. Formula (10) shows that Frobenius sends Fn into Fpn . Since this is clearly also true for the pth power map and A is p-torsion-free, we see that the same also holds for δ (recall that 1 δ(u) = p1 (φ(u) − up ) and for γ (recall that γ (u) = (p) φ(u) − δ(u)). At this point, q we may actually notice that, although δ is not an additive map, formula (1) shows that it satisfies ∀u, v ∈ Fn ,

(u ≡ v mod Fn−1 ) ⇒ (δ(u) ≡ δ(v) mod Fpn−1 ),

(14)

and the analogous property is also satisfied by γ when u, v ∈ I [1] . Now, it is not difficult to compute the following leading terms  (np) !  q [np] ξ φ ξ [n] ≡ mod Fnp−1 (n)q p ! and

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 p (np)q ! [np] ξ [n] ≡ ξ mod Fnp−1 . ((n)q !)p It follows that   (((n)q !)p − (n)q p !)(np)q ! δ ξ [n] ≡ dn ξ [np] mod Fnp−1 , with dn = ∈ R. p(n)q p !((n)q !)p (15) We claim that dpr ∈ R × when r ∈ N \ {0}. Since R is a local ring and q − 1 belongs to the maximal ideal, we may assume that q = 1 and in this case, dpr =

((pr !)p − pr !)(pr+1 !) ((pr !)p−1 − 1)(pr+1 !) = . r r p pp !(p !) p(pr !)p

We have vp (pr+1 !) − pvp (pr !) − 1 =

pr − 1 pr+1 − 1 −p −1=0 p−1 p−1

(16)

and therefore vp (dpr ) = 0. We can now show by induction that ∀r ∈ N, ∃cr ∈ R × ,

γ r (ξ ) ≡ cr ξ [p

r]

mod Fpr −1 .

The formula trivially holds for r = 0. Moreover, since d1 = 0 in (15), we have δ(ξ ) ≡ 0 mod Fp−1 and therefore γ (ξ ) ≡

φ(ξ ) ≡ (p − 1)q !ξ [p] mod Fp−1 . (p)q

Thus the formula also holds for r = 1 and, assuming that it holds for some r ≥ 1, we will have, thanks to property (14) and the asymmetric formula (3),   r γ r+1 (ξ ) ≡ γ cr ξ [p ] mod Fpr+1 −1     1 r r φ cr ξ [p ] − δ cr ξ [p ] mod Fpr+1 −1 (p)q  r   r   r  1 p ≡ φ(cr )φ ξ [p ] − δ(cr )φ ξ [p ] − cr δ ξ [p ] mod Fpr+1 −1 (p)q  r+1  pr −1 p ≡ (p)q (φ(cr ) − (p)q δ(cr ))bpr ,pr+1 − cr dpr ξ [p ] mod Fpr+1 −1 ≡

with bn,i ∈ R as in (10). Since cr dpr ∈ R × , it is then sufficient to recall that (p)q belongs to the maximal ideal (p, q − 1) and the coefficient is therefore necessarily invertible, as asserted. p

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Now, if n ∈ N has p-adic expansion n = vn :=



γ r (ξ )kr ≡

r≥0



crkr

r≥0





r≥0 kr p

r,

then

(ξ [p ] )kr ≡ c(n)ξ [n] mod Fn−1 r

r≥0

with c(n) ∈ R × . It follows that {vn }n∈N is a basis for the free A-module A#ξ $q . At this point, we should recall that I [1] is a q-PD-ideal because it is a δ-ideal and φ(I [1] ) ⊂ (p)q A#ξ $q . In particular, I [1] is stable under γ . It follows that {vn }n>0 is a basis for the free A-module I [1] . One can also show, exactly as we did above (but with a shift in the indices) that ∀r ∈

N, ∃cr

×

∈R ,

 δ

r

φ(ξ ) (p)q

Then, if n has p-adic expansion n = vn := ξ k0







≡ cr ξ [p

r≥0 kr p

r

r+1 ]

mod Fpr+1 −1 .

(17)

and we set

δ r (φ(ξ )/(p)q )kr+1 ,

r≥0

we obtain another basis {vn }n∈N for the free A-module A#ξ $q . Assume now that (B, J ) is a q-PD-pair and that we are given a morphism of δpairs u : (A[ξ ], (ξ )) → (B, J ). Then g := u(ξ ) ∈ J and therefore φ(g)/(p)q ∈ B. It follows that u extends uniquely to a morphism of R-algebras A[ξ ][φ(ξ )/(p)q ]  A[ξ, z]/((p)q z − φ(ξ )) → B, and since B is a δ-R-algebra, this morphism then extends uniquely to a morphism of δ-R-algebras U : A[ξ ][φ(ξ )/(p)q ]δ  A[ξ ][z]δ /((p)q z − φ(ξ ))δ → B (the δ-envelopes are taken over A[ξ ], which is already a δ-ring). As a particular case, we can apply this construction to the canonical map   λ : (A[ξ ], (ξ )) → A#ξ $q , I [1] , and we get a morphism of δ-R-algebras Λ : A[ξ ][φ(ξ )/(p)q ]δ → A#ξ $q . We will show below that Λ is an isomorphism. As a consequence, the map u : A[ξ ] → B will extend uniquely to a morphism of δ-rings that we may still call U : A#ξ $q → B. In particular, U commutes with γ . Since {vn }n>0 is a basis for the free A-module I [1] , it follows that U sends I [1] into J . In other words, U is

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a morphism of δ-pairs that extends uniquely our original morphism u. This shows that (A#ξ $q , I [1] ) has the expected universal property. It remains to show that Λ is an isomorphism. The δ-envelope A[ξ ][z]δ (over A[ξ ]) of the polynomial ring A[ξ, z] is the polynomial ring over A on infinitely many variables {zr }r∈N where z0 := ξ

and

zr := δ r−1 (z) if r > 0.

 In particular, this is a free A-module with basis zk := r≥0 zrkr where kr = 0 for r >> 0. We will endow this free A-module with the weighted degree defined by deg(zk ) = r≥0 kr pr (and deg(0) = −∞). In order to lighten the notations (and only in this proof), we will set E := A[ξ ][φ(ξ )/(p)q ]δ . We define the degree of u ∈ E as the minimum (weighted) degree of all its representations in A[ξ ][z]δ so that ∀u, v ∈ E,

deg(u + v) ≤ max{deg(u), deg(v)}

and

deg(uv) ≤ deg(u) + deg(v).

Using the multiplicative formula ∀u, v ∈ E,

δ(uv) = up δ(v) + δ(u)v p + pδ(u)δ(v),

one easily checks by induction on n that ∀u ∈ E,

deg(u) < n ⇒ deg(δ(u)) < pn.

(18)

Note that we have an analogous property for u → up , and therefore also for the map u → φ(u). We consider now the filtration of E by the A-submodules En defined by the condition deg(u) ≤ n for n ∈ N. Let us then improve on (18) and show that ∀u, v ∈ En ,

(u ≡ v mod En−1 ) ⇒ (δ(u) ≡ δ(v) mod Epn−1 ).

If we write v = u + w with deg(w) < n, we have δ(v) = δ(u) + δ(w) −

p−1  k=1

  1 p p−k k u w . p k

It follows from (18) that deg(δ(w)) < pn and we also have for all 0 < k < p, deg(up−k w k ) ≤ (p − k) deg(u) + k deg(w) ≤ (p − k)n + k(n − 1)

(19)

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= pn − k ≤ pn − 1. We will now prove by induction that, if we denote by zr the class of zr in E, then ∀r ≥ 0, ∃er ∈ R ×

p

er zr ≡ (p)q pr zr+1 mod Epr+1 −1 .

(20)

The formula holds when r = 0 with e0 = 1 because φ(ξ ) = (p)q z and φ(ξ ) ≡ ξ p mod Ep−1 thanks to Eq. (5), which shows that deg(δ(ξ )) < p. For the induction process, we also need the case r = 1 and we use the formula δ(φ(ξ )) = δ((p)q z). On the one hand, we have δ(φ(ξ )) = φ(δ(ξ )) ≡ 0 mod Ep2 −1 thanks to property (18) (or more precisely its φ analog). On the other hand, δ((p)q z) = δ((p)q )zp + (p)q p δ(z). Property (20) therefore holds when r = 1 with e1 = −δ((p)q ) wich is invertible because (p)q is distinguished. Now we fix r ≥ 2 and we assume that for all s < r, there exists es ∈ R × such that p

es zs ≡ (p)q ps zs+1 mod Eps+1 −1 .

(21)

From the case s = r − 1 and property (19), we get p

δ(er−1 zr−1 ) ≡ δ((p)q pr−1 zr ) mod Epr+1 −1 , which can be rewritten as p2

p

p

p

p

δ(er−1 )zr−1 + φ(er−1 )δ(zr−1 ) ≡ δ((p)q pr−1 )zr + (p)q pr zr+1 mod Epr+1 −1 , or, for later use, p2

δ(er−1 )zr−1 + φ(er−1 )δ(zr−1 ) − δ((p)q pr−1 )zr ≡ (p)q pr zr+1 mod Epr+1 −1 . (22) At this point, it is necessary to introduce the R-submodule Wr+1 of E generated by all zk such that deg(zk ) ≤ pr+1

and

kr+1 = 0

(23)

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(so that zr+1 is ruled out). We will now show that p

∀u ∈ Wr+1 , ∃e ∈ R,

u ≡ ezr mod Epr+1 −1 .

(24)

We may clearly assume that u = zk and that conditions (23) are satisfied. If deg(u) < pr+1 , then we are done (put e = 0). Otherwise, since kr+1 = 0, p there exists s ∈ {0, . . . , r} such that ks ≥ p. We can then write u = vzs with l v = z (ls = ks − p and lr = kr otherwise) and proceed by infinite descent on |k| := ri=0 ki ≥ 0. In the case s = r, we have v = 1 and we are done. So we may assume now that s < r and use our induction hypothesis (21) so that u≡

(p)q ps es

vzs+1 mod Epr+1 −1

and vzs+1 = zm (ms+1 = 1 and mr = lr otherwise) with |m| < |k|. Property (24) is now proved and we can apply it to p2

p

p

u := δ(er−1 )zr−1 + φ(er−1 )δ(zr−1 ) − δ((p)q pr−1 )zr . Using (22), we obtain formula (20) but we still need to prove that er is invertible. In order to show that, we can assume as above that q = 1. Then, we can use corollary 2.39 of [3] and identify E with the divided polynomial ring A#ξ $. Using formula (17), we also see that En then identifies with Fn . In this new setting, since we already showed property (20) up to the fact that er is invertible, we can write    p  r−1 φ(ξ ) r φ(ξ ) mod Fpr+1 −1 . er δ ≡ pδ p p But we know from (17) that there exists cr , cr−1 ∈ R × such that

 δr

φ(ξ ) p



≡ cr ξ [p

r+1 ]



 and

δ r−1

φ(ξ ) p

p

 r p p ≡ cr−1 ξ [p ] mod Fpr+1 −1 .

Moreover, it follows from computation (16) that ∃f ∈ R × ,

 r p r+1 ξ [p ] ≡ fpξ [p ] mod Fpr+1 −1 . −p

Therefore, since R is p-torsion free, er = f −1 cr cr−1 ∈ R × and property (20) is now completely proved. In the same way as we showed property (24), one can now prove that E is generated as an A-module by {zk , kr < p}. In other words, we want to show that any u ∈ E is an A-linear combination of zk with all kr < p. We proceed by induction on n := deg(u), the result being trivial for n < p. We may clearly

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 assume that u = zk with r kr pr ≤ n. Now, we proceed by infinite descent on p |k| := i≥0 ki ≥ 0. Assume that some ks ≥ p and write u = vzs , where v = zl with ls = ks − p and lr = kr for r = s. Thanks to (21), we have p

zs ≡ azs+1 mod Eps+1 −1 , with a ∈ R. Since deg(v) ≤ n−ps+1 , it follows that u ≡ avzs+1 mod En−1 . Since vzs+1 ∈ En and vzs+1 = zm with |m| < |k|, we are done. Let us come back to our morphism Λ : E → A#ξ $q . We can define an A-linear section of Λ by sending the basis vector vn to zk if n  r has p-adic expansion n = r≥0 kr p . Since E is generated as an A-module by k {z , kr < p}, this section is surjective so that Λ is bijective and we can identify both δ-R-algebras as we claimed.  Remarks 1. In the proof of Theorem 3.5, it is actually not obvious at all from the explicit formulas that the various coefficients are indeed invertible. For example, in the simplest non trivial case p = 2 and r = 1, we have d2 = q + q 2 + q 3 , which is a non-trivial unit. 2. Be careful that the isomorphism in Theorem 3.5 is very specific to our situation and it is not true for example that, if X is an indeterminate, then R[X, φ(X)/(p)q ]δ is the q-PD-envelope of R[X]δ with respect to the augmentation ideal unless q = 1. 3. Our result may be seen as a decompletion in our situation of lemma 16.10 of [3]. Note however that it is not even clear a priori that A[ξ ][φ(ξ )/(p)q ]δ is (p)q torsion free before completion. 4. Theorem 3.5 is closely related to Jonathan Pridham’s work in [15]. For example, his lemma 1.3 shows that (when A is not merely a δ-ring but actually a λ-ring)  λ

n

ξ q −1



1 = ξ [n] (q − 1)n

# =



ξ q −1

[n] $

when q − 1 ∈ R × (in which case A[ξ ] = A#ξ $q ). 5. Note also that, as a corollary of our theorem, one recovers the existence of the q-logarithm (after completion, but see below) as in lemma 2.2.2 of [1].

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4 Complete q-PD-Envelopes As before, R denotes a δ-ring with fixed rank one element q and we assume that R is actually a Z[q](p,q−1) -algebra. Any R-algebra B will be implicitly endowed with  or B ∧ its completion for this its (p, q − 1)-adic topology and we will denote by B topology, which is automatically a Zp [[q − 1]]-algebra (as in [3]). We recall that, if B is a δ-R-algebra, then δ is necessarily continuous and extends therefore uniquely  We also want to mention that a complete ring is automatically (p)q -complete. to B. Actually, the (p, q − 1)-adic topology and the (p, (p)q )-adic topology coincide thanks to congruences (12), and we will often simply call this the adic topology.  of B, one may also consider its derived Remark Besides the usual completion B • completion [16, Tag 091N] R lim Bn , where Bn• denotes the Koszul complex ← −

More generally, one can consider the derived completion functor • %• := R lim(K • ⊗L K • → K B Bn ) ← −

on complexes of B-modules. If M is a B-module and M[0] denotes the corresponding complex concentrated in degree zero, then there exists a canonical map  → M[0]  which is not an isomorphism in general (but see below). M[0] We recall that an abelian group M has bounded p∞ -torsion if ∃l ∈ N, ∀s ∈ M, ∀m ∈ N,

pm s = 0 ⇒ pl s = 0.

We will need the following result: Lemma 4.1 If M has bounded p∞ -torsion, so does its (p)-adic completion. Proof We use the notations of the definition and we assume that {sn }n∈N is a sequence in M such that pm sn → 0 when n → ∞ for some m ≥ l. Thus, given k ∈ N with k ≥ l, if we write k = k + m − l, there exists N ∈ N such that for all n ≥ N , we can write pm sn = pk t with t ∈ M. But then pm (sn − pk −m t) = 0 and −m l k l k therefore already p (sn − p t) = 0 or p sn = p t. This shows that pl sn → 0.  Definition 4.2 An R-algebra B is bounded9 if B is (p)q -torsion free and B/(p)q has bounded p∞ -torsion.

9 We

should perhaps say q-bounded or even p-q-bounded, but we prefer the simpler name.

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Remarks 1. If B is bounded, then the computations carried out in the proof of lemma 3.7 of & = B[0].  [3] show that B[0] In other words, for our purpose, there is no reason in this situation to introduce the notion of derived completion. 2. It is equivalent to say that B is a complete bounded δ-ring or that (B, (p)q ) is a bounded prism in the sense of [3, definition 1.4]. Proposition 4.3 If B is a bounded R-algebra, then the adic topology on (p)q B is identical to the topology induced by the adic topology of B. Proof Since the topology is defined by the ideal (p, (p)q ), it is actually sufficient to show that the (p)-adic topology on (p)q B is identical to the topology induced by the (p)-adic topology of B: ∀n ∈ N, ∃m ∈ N,

(p)q B ∩ p m B ⊂ p n (p)q B.

In other words, we have to show that ∀n ∈ N, ∃m ∈ N, ∀f ∈ B,

(∃g ∈ B, pm f = (p)q g) ⇒ (∃h ∈ B, pm f = (p)q pn h).

But since B/(p)q has bounded p∞ -torsion, we know that ∃l ∈ N, ∀f ∈ B, ∀m ∈ N,

(∃g ∈ B, pm f = (p)q g) ⇒ (∃h ∈ B, pl f = (p)q h).

It is therefore sufficient to set m = n + l.



As a consequence, we see that the q-PD-condition is closed: Corollary 4.4 If B is a bounded δ-R-algebra and J ⊂ B is a q-PD- ideal, then its cl closure J for the adic topology is also a q-PD-ideal. Proof Recall that the condition means that (φ −(p)q δ)(J ) ⊂ (p)q J . By continuity, cl

cl

we will have (φ − (p)q δ)(J ) ⊂ (p)q J . Now, it follows from Proposition 4.3 that multiplication by (p)q induces a homeomorphism B  (p)q B and therefore (p)q J

cl

cl

= (p)q J .



We also obtain that boundedness is preserved by completion:  also is bounded and the ideal Corollary 4.5 If B is a bounded R-algebra, then B  is closed in B  (or equivalently B/(p)   (p)q B is complete). B q Proof It follows from Proposition 4.3 that the sequence (p)q

0 −→ B −→ B −→ B/(p)q −→ 0 is strict exact, and then the sequence

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q q −→ 0  −→  −→ B/(p) 0 −→ B B

 is (p)q -torsion free and the ideal (p)q B  is therefore also strict exact. In particular, B ∞ q has bounded p -torsion thanks to  Moreover, B/(p)  is closed in B. q = B/(p) Lemma 4.1.  The next result is basic but very useful in practice: Lemma 4.6 If R is bounded and B is flat (over R) then B is also bounded. Proof Upon tensoring with B over R, the exact sequence (p)q

0 −→ R −→ R −→ R/(p)q −→ 0 provides an exact sequence (p)q

0 −→ B −→ B −→ B/(p)q −→ 0. This shows that B is (p)q -torsion free. Also, upon tensoring with B over R, the exact sequence pn

0 −→ R/(p)q [pn ] −→ R/(p)q −→ R/(p)q −→ R/((p)q , pn ) −→ 0 provides another exact sequence pn

0 −→ B ⊗R (R/(p)q )[pn ] −→ B/(p)q −→ B/(p)q −→ B/((p)q , pn ) −→ 0. This shows that B ⊗R (R/(p)q )[pn ] = (B/(p)q )[pn ]. It follows that B/(p)q has bounded p∞ -torsion (with the same bound as R/(p)q ).  Definition 4.7 A δ-pair (B, J ) is complete if B is complete and J is closed (or equivalently B := B/J is also complete). More generally, the completion of a δ J cl ) where J cl denotes the closure of J B  in B.  pair (B, J ) is the δ-pair (B, Remarks  = J unless B is noetherian. 1. We usually have J B 2. Completion is clearly universal for morphisms to complete δ-pairs. cl

 0) and (B,  φ −1 ((p)q B))  are complete Example If B is a bounded δ-ring, then (B,   bounded q-PD-pairs. This is also the case for (B, (q − 1)B) if we assume for example that B/(q − 1) is p-torsion free as in definition 16.2 of [3] (in which case the ideal is closed).

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We may now make the following definition: Definition 4.8   If (B, J ) is a δ-pair, then (if it exists) its complete q-PD envelope cl  B [ ]q , J [ ]q is a complete q-PD-pair which is universal for morphisms to complete q-PD-pairs. Proposition 4.9 Let (B, J ) be a δ-pair. If the (non complete) q-PD-envelope exists and is bounded, then its completion is the complete q-PD-envelope of (B, J ). [ ]q is bounded and in particular (p) Proof It follows from Corollary 4.5 that B& q & [ ] q is a q-PD-ideal. Finally, we showed in Corollary 4.4 torsion free. Moreover, J B   cl & & [ ] [ ] q q that the q-PD-condition is closed thus we see that B , J B is a complete



q-PD-pair. The universal property is then automatic. Remark This proposition will apply when R itself is bounded and then, thanks to Lemma 4.6, B [ ]q will be bounded.

B [ ]q

is flat since

There exists a complete analog to Lemma 3.4 but we will actually only need the following consequence: Lemma 4.10 If, for i = 1, 2, (B, J ) → (Bi , Ji ) is a morphism of δ-pairs such that each (Bi , Ji ) admits a complete q-PD-envelope and []

[]q

 B B2 B1 q ⊗

is bounded, then this is the complete q-PD-envelope of (B1 , J1 ) ⊗(B,J ) (B2 , J2 ) when it is endowed with the closure of the image of cl

[ ]q

J1

[]q

⊗ B B2

[]q

+ B1

cl

[ ]q

⊗B J2

.

Proof Corollary 4.4 implies that this is a complete q-PD-pair and the universal property then follows automatically from the universal properties of tensor product and completion.  Examples   1. If B is bounded and J is a q-PD-ideal, then B [ ]q = B. []q 2. If J = B, then B = 0. This fundamental property does not rely on Proposition 4.9. It follows from Example 3 after Definition 3.3, which also shows that it is not true before completion. 3. If A is a bounded complete unramified δ-R-algebra and if we look at the δ-pair (P , I ), where P := A ⊗R A and I is the kernel of multiplication P → A, then  P [ ]q = A. This reduces to the previous cases since

(P , I )  (N, N ) × (A, 0).

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4. If A is a complete bounded δ-R-algebra withrank one element  x, then it follows from Theorem 3.5 and Proposition 4.9 that

 A#ξ $q , I [1]

cl

is the complete q-

PD-envelope of (A[ξ ], ξ ).

5 Hyper q-Stratifications We let R be a Z[q](p,q−1) -algebra. Any R-module will be implicitly endowed with its (p, q − 1)-adic topology and completion always means (p, q − 1)-adic completion. We let A be a complete R-algebra with a fixed topologically étale (that is, formally étale and topologically finitely presented) coordinate x. We let P := A ⊗R A and we denote by I the kernel of multiplication e : P → A. We denote by p1 , p2 : A → P the canonical maps and we let ξ := p2 (x) − p1 (x) ∈ I . Unless otherwise specified, we will use the “left” action f → p1 (f ) of A on P in order to turn P into an A-algebra. Since q − 1 is topologically nilpotent on A, which is complete and has x as a topologically étale coordinate, there exists a unique endomorphism σ of the Ralgebra A such that σ (x) = qx and σ ≡ IdA mod q − 1. In particular, A is canonically a twisted algebra in the sense of [11]. We extend σ to P in an asymmetric way by setting σP := σ ⊗ IdA (we will simply write σ when no confusion can arise). For n ∈ N, we let I (n+1) := I σ (I ) · · · σ n (I ) cl

be the (n + 1)th twisted power of the ideal I and denote by I (n+1) the closure of cl (n+1)  I (n+1) in general, but this is the case for . It is not clear if I its image in P n = 0 because the sequence 0 → I → P → A → 0 is split exact as a sequence of A-modules (and A is complete). One can show that x is a topological q-coordinate on A in the following sense (we denote by the same letter ξ both the indeterminate and the element of P ): Lemma 5.1 The canonical maps are bijective for all n ∈ N: cl

/I (n+1) . A[ξ ]/(ξ (n+1) )  P Proof We fix some n ∈ N. We have ξ n+1 ≡ ξ (n+1) mod q − 1 and ξ is therefore topologically nilpotent in A[ξ ]/(ξ (n+1) ). Since ξ is a topological coordinate for P and A[ξ ]/(ξ (n+1) ) is complete, there exists a unique morphism U below making the diagram commute:

Twisted Differential Operators and q-Crystals

P

207

e

A

U

A[ξ]

A[ξ]/(ξ (n+1) ).

This morphism U is necessarily surjective and extends by continuity to a surjective morphism :P  → A[ξ ]/(ξ (n+1) ). U Byconstruction, we have U (I ) ⊂ (ξ ) and therefore U (I (n+1) ) = 0 so that  cl  I (n+1)  induces a surjective map = 0. Thus, U U cl

/I (n+1) → A[ξ ]/(ξ (n+1) ). u:P Now, if we compose U with the canonical map /I (n+1) v : A[ξ ]/(ξ (n+1) ) → P

cl

and the projection onto A, we recover e : P → A. Using again the fact that ξ is a cl /I (n+1) is complete and Iconsists of topological coordinate for P over A, since P cl

topologically nilpotent elements modulo I (n+1) , we see that v ◦ U is necessarily cl /I (n+1) . It follows that u is an inverse for v. the projection onto P   Let us consider now the canonical map A[ξ ] → A#ξ $q from Sect. 2. Since the [k] (k) image (k)q !ξ of ξ goes to zero when k goes to infinity, we see that this map factors uniquely through the twisted power series ring A[[ξ ]]q := lim A[ξ ]/(ξ (n+1) ). ← − Definition 5.2 The q-Taylor map (of level zero) is the composite θ :A

p2

P

limP /I (n+1)

cl

A[[ξ ]]q

A ξ q.

We may still denote by the letter θ the map so defined, but with the arrow cl &$q , if no confusion could result. Notice /I (n+1) , A[[ξ ]]q or A#ξ stopping at lim P ← − that we always have θ (x) = x + ξ .

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&$q We will need to be able to move between the canonical (left) structure of A#ξ as an A-module and its (right) structure through the map θ and we first prove the following technical result: Lemma 5.3 If we denote by ' τ : A[ξ ] → A[[ξ ]]q ,

f → θ (f ) for f ∈ A ξ → −ξ

the flip map (which is a morphism of R-algebras), then we have for all n > 0,     n    n n−1 (n) k k(k−1) n−k = τ ξ (−1) q 2 (1 − q) (n − k)q ! x n−k ξ (k) . k−1 q k q k=1

Proof Recall first that we have ξ (n) =

n−1 

(ξ + (1 − q k )x).

k=0

Since τ (x) = x + ξ and τ (ξ ) = −ξ , we will have for all k = 1, . . . , n − 1, τ (ξ + (1 − q k )x) = −ξ + (1 − q k )(x + ξ ) = x − q k (x + ξ ), and it follows that  n−1   (n) = τ ξ (x − q k (x + ξ )). k=0

We may then apply the twisted binomial formula n−1 

n 

k=0

m=0

(q k X + Y ) =

q

m(m−1) 2

  n Xm Y n−m m q

from proposition 2.14 of [9] with X = −(x + ξ ) and Y = x and obtain   n    n (n) m m(m−1) 2 = τ ξ (−1) q (x + ξ )m x n−m . m q m=0

Using our formula

(25)

Twisted Differential Operators and q-Crystals

(x + ξ )m =

209

m    m

k

k=0

x m−k ξ (k)

q

from lemma 7.1 of [7], we obtain $   # n m      m(m−1) n m τ ξ (n) = (−1)m q 2 x m−k ξ (k) x n−m . m q k q m=0

k=0

Making the change of indices m = k + l, we have k(k − 1) l(l − 1) m(m − 1) = + + kl 2 2 2 and         n m n n−k = . m q k q l q k q It follows that $   #   n n−k    l(l−1) n − k n (n) k k(k−1) l kl = τ ξ (−1) q 2 (−1) q 2 q x n−k ξ (k) . k q l q k=0

l=0

We may call again our twisted binomial formula (25) in the case X = q k and Y = −1 (and also n replaced with n − k) which gives n−k−1 

(1 − q k+i ) = (−1)n−k

i=0

n−k−1 

(q i q k − 1)

i=0

= (−1)

n−k

n−k 

q

l(l−1) 2

l=0

=

n−k 

(−1)l q

l=0

l(l−1) 2



n−k l

 (q k )l (−1)n−k−l q

  n−k q kl . l q

On the other hand, we also have n−k−1 

n−k−1 

i=0

i=0

(1 − q k+i ) =

(1 − q)(k + i)q = (1 − q)n−k (n − k)q !

so finally (since this is 0 when k = 0)

  n−1 , k−1 q

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    n    k(k−1) n n−1 τ ξ (n) = (−1)k q 2 (1 − q)n−k (n − k)q ! x n−k ξ (k) . k−1 q k q k=1

 &$q → A#ξ &$q defined by Proposition 5.4 The R-linear flip map τ : A#ξ f ξ [n] → θ (f )

  n  k(k−1) n−1 (−1)k q 2 (1 − q)n−k x n−k ξ [k] k−1 q k=1

&$q such that τ ◦ τ = IdA#ξ is an R-algebra automorphism of A#ξ &$q . Proof In order to show that this is a morphism of rings, it is sufficient to check that ∀m, n ∈ N,

      τ ξ [n] ξ [m] = τ ξ [n] τ ξ [m] .

& so that A has By functoriality, we may assume that R = Z[q](p,q−1) and A = R[x] no torsion and it is then sufficient to show the equality ∀m, n ∈ N,

      τ ξ (n) ξ (m) = τ ξ (n) τ ξ (m) .

We are therefore led to prove the similar equality for the map τ of Lemma 5.3, but this is a morphism of rings by definition. Let us now show that the composite map τ ◦ τ is A-linear, or, equivalently, that &$q is the canonical map. In order to do so, we the composite map τ ◦ θ : A → A#ξ may first remark that, the map τ of Lemma 5.3 sends the ideal (ξ (n) ) into the ideal (ξ (n−k) , (q − 1)k ) when n ≥ k. It follows that τ induces, for all k ∈ N, a morphism A[[ξ ]]q = lim A[ξ ]/(ξ (n) ) → lim A[ξ ]/(ξ (n−k) , (q − 1)k ) = A[[ξ ]]q /(q − 1)k ← − ← − n≥k

n≥k

and, taking the limit on k, a ring endomorphism of A[[ξ ]]q that we will still denote by τ . This allows us to decompose τ ◦ θ as the composite τ θ &$q . A → A[[ξ ]]q → A[[ξ ]]q → A#ξ

Since ξ is topologically nilpotent on A[[ξ ]]q and x is a topologically étale coordinate on A, our claim then follows from the fact that τ (θ (x)) = τ (x + ξ ) = (x + ξ ) − ξ = x. It orderto prove that τ ◦ τ is the identity, it is now sufficient to show the equality (τ ◦ τ ) ξ [n] = ξ [n] . As above, we may assume that A has no torsion and we are

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211

reduced to showing that (τ ◦ τ )(ξ (n) ) = ξ (n) . This follows again from Lemma 5.3 because (τ ◦ τ )(ξ ) = τ (−ξ ) = ξ and τ ◦ τ is A-linear.  Remarks 1. The flip map τ exchanges the canonical map with θ in the sense that τ ◦ θ = can and τ ◦ can = θ . 2. The flip map also extends to an automorphism of [n+1]  A##ξ $$q := lim A#ξ $q /I [n+1] = lim A#ξ $q /I ← − ← −

(26)

[n+k] into I& [n] + (q − 1)k A#ξ &$q . because τ sends I 3. Actually, the same holds with A[[ξ ]]q and there exists a commutative diagram

f ⊗g

P

A[[ξ]]q

Aξ

q

A ξ

q

g⊗f

P

A[[ξ]]q

Aξ

q

A ξ

q

of flip maps, all of which we may denote by τ . Note that the left hand square commutes thanks to Remark 1.  We will also let e : A#ξ $q → A be the augmentation map and     A A#ξ p1 , p2 : A#ξ $q → A#ξ $q ⊗ $q be the obvious maps (we use the notation ⊗ to indicate that we use the q-Taylor map θ for the A-structure on the left hand side). Proposition 5.5 The diagonal map, which is the A-linear map (for the left Amodules structures)

is a morphism of A-algebras. Proof We have to prove that ∀m, n ∈ N,

      Δ ξ [n] ξ [m] = Δ ξ [n] Δ ξ [m] .

& Since By functoriality, we may clearly assume that R = Z[q](p,q−1) and A = R[x]. now A has no torsion, it is sufficient to prove that

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  Δ ξ (n) ξ (m) = Δ(ξ (n) )Δ(ξ (m) ).

∀m, n ∈ N,

This follows from theorem 3.5 of [10]: the diagonal map

(where ⊗ A is built using x → x + ξ on the left) is a ring homomorphism.



Remarks 1. The same formula also defines a diagonal map  A A##ξ $$q Δ : A##ξ $$q → A##ξ $$q ⊗ with A##ξ $$q as in (26). 2. Actually, this also holds with A[[ξ ]]q and the diagram P

A[[ξ]]q Δ

P ⊗A P

Aξ

Δ

A ξ

q

Δ

A[[ξ]]q ⊗A A[[ξ]]q

A ξ q ⊗A A ξ

q

Δ

A ξ

q

q ⊗A A

ξ

where the left vertical map is given by f ⊗ g → f ⊗ 1 ⊗ g, is commutative. For the left hand square, this follows from the formula given in theorem 3.5 of [10] & In general, it is sufficient, since x is a topologically étale in the case A = R[x]. coordinate, to check the commutativity modulo (p, q − 1). But then ξ becomes topologically nilpotent and we can work modulo ξ (or more precisely modulo (1 ⊗ ξ, ξ ⊗ 1)) in which case the assertion is trivial.  &$q , θ, can, e, Δ and τ satisfy properties similar We now show that A, A#ξ (because we use completions) to those of a formal groupoid in the sense of definition 1.1.3 in chapter II of [2] (we follow the same order for the statements): Proposition 5.6 The following diagrams are commutative:

1.

A

Aξ e

A

q

and

A

θ

Aξ e

A

q

q,

Twisted Differential Operators and q-Crystals

2.

A

Aξ

213

and

q

Δ

3.

p1

Aξ

q

Aξ

q

Δ

θ

A

Aξ

Δ

θ

A ξ q ⊗A A ξ

q

A ξ q ⊗A A ξ

q

and

p2

Aξ

q

Aξ

q

Δ

A ξ q ⊗A A ξ

q

A ξ q ⊗A A ξ

q

e⊗Id

Aξ

4.

Aξ

Id⊗e

Aξ

q

Δ

q

A ξ q ⊗A A ξ

A ξ q ⊗A A ξ

id⊗Δ q

A

and

τ

q

6.

q

A ξ q ⊗A A ξ q ⊗A A ξ

q

A

θ

Aξ

q

Δ⊗Id

Δ

5.

q

θ

Aξ

Aξ

q

Aξ

q

τ

Aξ e

τ

q

Aξ

q

q

e

A

7.

Aξ

e

q

A

and

Δ

A ξ q ⊗A A ξ

Aξ

e

q

Δ Id×τ q

Aξ

q

A ξ q ⊗A A ξ

A θ

τ ×Id q

A ξ q.

Proof Only the last assertion needs a proof, the other ones following either directly from the definitions or from what we already proved. For example, the commutativity of the right hand square in assertion (2) follows from the second remark before the proposition. Now, we consider the last assertion and since both proofs are similar, we only do the first diagram. We have to show that

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∀n > 0,

   (Id × τ ) Δ ξ [n] = 0.

As usual, we can reduce by functoriality to showing that (Id × τ )(Δ(ξ (n) )) = 0 and this results from the commutativity of the diagram e

P

A,

f ⊗g

Δ

e

fg

Id×τ

f g ⊗ 1.

Δ

P ⊗A P

Id×τ

P

f ⊗1⊗g

 There exists a notion of stratification in this setting that reads as follows: Definition 5.7 A hyper-q-stratification (of level zero)10 on an A-module M is an  A#ξ $q -linear isomorphism &$q ⊗ A M  M ⊗A A#ξ &$q  : A#ξ such that e∗ () = IdM

and

Δ∗ () = p1∗ () ◦ p2∗ ().

(27)

There also exists an apparently weaker notion which is defined as follows : Definition 5.8 A hyper-q-Taylor structure (of level zero) on an A-module M is an A-linear map  θ : M → M ⊗A A#ξ $q , where we consider the target as an A-module via the q-Taylor map of A, such that (IdM ⊗ e) ◦ θ = IdM and (θ ⊗ IdA#ξ ) ◦ θ = (IdM ⊗ Δ) ◦ θ.  $ q

10 We

could also say hyper-q-PD-stratification.

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215

Proposition 5.9 Restriction and scalar extension along the q-Taylor map θ : A → &$q provide an equivalence between hyper-q-stratifications and hyper-q-Taylor A#ξ structures. In other words, A-modules endowed with a hyper-q-stratification (resp. a hyperq-Taylor structure) form a category in an obvious way and both categories are isomorphic. Proof This is analogous to corollary 1.4.4 in chapter II of [2]. It is easy to see that any hyper-q-stratification induces a hyper-q-Taylor structure and that any hyper-q&$q -linear map Taylor structure extends to an A#ξ &$q ⊗ A M → M ⊗A A#ξ &$q ,  : A#ξ

1 ⊗ s → θ (s)

that satisfies the normalization and cocycle conditions. It only remains to show that  is bijective and it is then that the flip map enters the picture: if we pull back the cocycle condition along Id × τ (resp. τ × Id), we get Id =  ◦ τ ∗ () (resp. Id = τ ∗ () ◦ ). 

6 q-Calculus As before, we let R be a Z[q](p,q−1) -algebra and endow all modules with their (p, q − 1)-adic topology. Completion is always meant with respect to this topology. We let A be a complete R-algebra with a fixed topologically étale coordinate x and we keep the notations from the previous section. We want to extend here some notions from our previous articles [10], [11] and [7] by taking into account the topology (see also [12], where we did the affinoid case). Recall that there exists a unique endomorphism σ of the R-algebra A such that σ (x) = qx and σ ≡ IdA mod q − 1. Then, the basic notion in q-calculus is the following: Definition 6.1 1. A q-derivation of A with values in an A-module M is an R-linear map D : A → M that satisfies the twisted Leibniz rule ∀f, g ∈ A,

D(fg) = D(f )g + σ (f )D(g).

2. A q-derivation on an A-module M with respect to some q-derivation DA : A → A is a map DM : M → M that satisfies the twisted Leibniz rule ∀f ∈ A, ∀s ∈ M,

DM (f s) = DA (f )s + σ (f )DM (s).

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3. We will denote by TA/R,q the module of q-derivations on A with values in A. An action of TA/R,q by q-derivations on M is an A-linear map TA/R,q → EndR (M) such that the image of D ∈ TA/R,q is a q-derivation of M with respect to D. The q-derivation DA plays a secondary role in the definition of DM and we will not mention it in the future (we may even use the same letter D for both). Note also that both definitions of a q-derivation coincide when A = M (in which case DM = DA ). In order to study q-derivations, it is necessary to introduce the (complete) module of q-differential forms. Let us first recall that we extend in an asymmetric way the endomorphism σ to P := A ⊗R A and that we denote by I the kernel of multiplication and by I (n+1) its twisted powers. Definition 6.2 The (complete) module of q-differential forms11 of A/R is cl

ΩA/R,q := I/I (2) . The R-linear map p2 − p1 : A → P takes values inside I and induces therefore an R-linear map dq : A → ΩA/R,q . It has the following universal property: Proposition 6.3 The map dq : A → ΩA/R,q is a q-derivation which is universal among all q-derivations with value in complete A-modules. Moreover, ΩA/R,q is a free A-module of rank one with basis dq x. Proof We already know from proposition 2.4 of [10] that I /I (2) is universal with respect to q-derivations into any A-module and ΩA/R,q is by definition the completion of I /I (2) . The second assertion follows from Lemma 5.1.  As an immediate consequence, we see that TA/R,q is a free A-module on one generator ∂A,q determined by the condition ∂A,q (x) = 1. In particular, giving an action of TA/R,q by q-derivations of M amounts to specifying a q-derivation ∂M,q on M with respect to ∂A,q . There exists an equivalent notion which is more natural in some sense: Definition 6.4 A q-connection on an A-module M is an R-linear map ∇ : M → M ⊗A ΩA/R,q that satisfies the twisted Leibniz rule ∀f ∈ A, ∀s ∈ M,

11 There

∇(f s) = s ⊗ dq (f ) + σ (f )∇(s).

is no natural isomorphism between ΩA/R,q and the usual (complete) module of derivations ΩA/R .

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A q-connection on M is equivalent to an action of TA/R,q by q-derivations via the formula ∇(s) = ∂M,q (s) ⊗ dq (x). We shall soon discuss the general notion of a q-differential operator but we can already introduce the following: Definition 6.5 The ring of q-differential operators (of level zero) of A/R is the non-commutative polynomial ring DA/R,q in one generator ∂q over A with the commutation rule ∂q ◦ f = σ (f )∂q + ∂A,q (f ). Clearly, a q-connection is equivalent to a structure of a DA/R,q -module through the formula ∂q s = ∂M,q (s). The twisted polynomial ring will now enter the picture: Definition 6.6 For each n ∈ N, the q-Taylor map of (level zero and) order n is the composite θ [n+1]  A#ξ $ /I [n+1] . &$q  A#ξ  θn : A → A#ξ $q /I q

This is a morphism of R-algebras that endows A#ξ $q /I [n+1] with a new structure of A-module that we may informally call the right structure (by symmetry with the left structure corresponding to the canonical action of A). As we already did with &$q , we will use the notation ⊗ to indicate that we consider A#ξ $q /I [n+1] as an A#ξ A-module via the q-Taylor map θn and not with respect to the canonical map. We will prove general formulas below but we can already notice that the q-Taylor map is closely related to q-derivations because, by definition, ∀f ∈ A,

θ1 (f ) − f = ∂A,q (f )ξ.

In order to make explicit computations later, it will be necessary to have a better grasp on the right A-module structure: Lemma 6.7 The action of A via the q-Taylor map on the ideal I [n] /I [n+1] is identical to the canonical action of A twisted by σ n . Proof Since θ is a morphism of R-algebras and I [n] /I [n+1] is the free A-module of rank one generated by (the image of) ξ [n] , there exists a unique morphism of R-algebras σn : A → A such that ∀f ∈ A,

θ (f )ξ [n] ≡ σn (f )ξ [n] mod I [n+1] .

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In the case q = 1, we have σn = Id = σ n . Moreover, since σ (x + ξ ) = x + ξ and ξ (n+1) = σ n (ξ )ξ (n) , we have θ (x)ξ [n] = (x + ξ )ξ [n] = σ n (x + ξ )ξ [n] = (σ n (x) + σ n (ξ ))ξ [n] = σ n (x)ξ [n] + (n + 1)q ξ [n+1] so that σn (x) = σ n (x). Since x is a topological coordinate on A, which is complete, we can conclude that σn = σ n .  As a consequence, the map θn is finite free. More precisely: Lemma 6.8 1. The ideal I [n] /I [n+1] is a free A-module of rank one on ξ [n] for both the left and the right A-module structures. ( )n 2. The ring A#ξ $q /I [n+1] is a free A-module on ξ [k] k=0 for both the left and the right A-module structures. ( )n,m 3. The ring A#ξ $q /I [n+1] ⊗ A A#ξ $q /I [m+1] is a free A-module on ξ [k] ⊗ ξ [l] k,l=0 for the left, middle and right A-module structures. Proof The first assertion follows from Lemma 6.7 since σ is an automorphism of A and the other ones are then obtained by induction on n (and m).  As we announced, there exists a more general (and more formal) approach to q-differential operators that we explain now (see section 5 of [7] for the analogous situation when there is no topology). Definition 6.9 Let M and N be two A-modules. A q-differential operator of order at most n (and level zero) from M to N is an A-linear map u : A#ξ $q /I [n+1] ⊗ A M → N.

(28)

We insist on the fact that we have ϕ ⊗ f s = θn (f )ϕ ⊗ s on the left hand side and linearity means that u(f ϕ ⊗ s) = f u(ϕ ⊗ s). We will write   Diffq,n (M, N ) := HomA A#ξ $q /I [n+1] ⊗ A M, N and Diffq (M, N ) := lim Diffq,n (M, N ) − →n∈N

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(or just Diffq,n (M) and Diffq (M) when N = M). Note that Diffq,n (M, N ) (resp. Diffq (M, N )) is naturally an A#ξ $q /I [n+1] -module (resp. an A#ξ $q -module). By composition, Diffq,n (M, N ) (resp. Diffq (M, N )) inherits both a left and a right structure of A-module. We may now consider for each m, n ∈ N the diagonal map of finite order (a morphism of A-algebras)

which is induced by the diagonal map Δ of Proposition 5.5. We may compose qdifferential operators in the usual way. More precisely, if we are given u as in (28) and v : A#ξ $q /I [m+1] ⊗ A L → M, then we define u ◦v as the composite u◦v

A ξ q /I [n+m+1] ⊗A L

N u

Δn,m ⊗ Id

A ξ q /I [n+1] ⊗A A ξ q /I [m+1] ⊗A L

Id⊗ v

A ξ q /I [n+1] ⊗A M.

Composition of q-differential operators is somehow tricky because we have for example Diffq,0 (M, N ) = HomA (M, N), and in particular Diffq,0 (A)  A, so that any f ∈ A may be considered as a q-differential operator of order 0. But, if u ∈ Diffq,n (A), then u ◦f = u ◦ θn (f ). Associativity of composition of q-differential operators follows from the commutativity of diagram (4) in Proposition 5.6 and, if M is any A-module, we obtain a structure of R-algebra on Diffq (M). It should also be noticed that composition of q-differential operators is compatible with composition of R-linear maps under the restriction map / HomR (M, N )

Diffq,n (M, N ) u

(A#ξ $q /I [n+1] ⊗ A M → N)

u / (M → A#ξ $q /I [n+1] ⊗ M → N) A

(which is not injective in general). In particular, there exits a canonical A-linear morphism of R-algebras Diffq (M) → EndR (M)

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when M is an A-module. In the case M = A, we recover our previous ring of q-differential operators: Lemma 6.10 There exists an A-linear isomorphism of R-algebras DA/R,q  Diffq (A),

∂qk

 '  1 if n = k [n]  k → ∂q : ξ → ∈ Diffq,k (A). 0 otherwise

Proof This map is clearly bijective. Thus, we have to check that ∂q ◦f = σ (f )∂q + ∂A,q (f ) and

∀f ∈ A,

∀k ≥ 1,

k+1 ∂* = ∂q ◦∂qk . q

First of all, we have (∂q ◦f )(1) = ∂q (θ1 (f )) = ∂q (f + ∂A,q (f )ξ ) = ∂A,q (f ), and (∂q ◦f )(ξ ) = ∂q (θ1 (f )ξ ) = ∂q (σ (f )ξ ) = σ (f ). Since, in general, we have



fk ∂q

k



· ξ [n] = fn , the first equality holds. Also,

(∂q ◦∂qk )(1) = (∂q ◦ Id ⊗ ∂qk ◦ Δ1,k )(1) = (∂q ◦ Id ⊗ ∂qk )(1 ⊗ 1) = ∂q (0) = 0, for 1 ≤ n ≤ k, (∂q ◦∂qk )(ξ [n] ) = (∂q ◦ Id ⊗ ∂qk ◦ Δ1,k )(ξ [n] ) = (∂q ◦ Id ⊗ ∂qk )(1 ⊗ ξ [n] + ξ ⊗ ξ [n−1] )   = ∂q θ (∂qk (ξ [n] )) + θ (∂qk (ξ [n−1] ))ξ ' =

∂q (0) = 0 if k < n ∂q (1) = 0 if k = n,

and finally (∂q ◦∂qk )(ξ [k+1] ) = (∂q ◦ Id ⊗ ∂qk ◦ Δ1,k )(ξ [k+1] )

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= (∂q ◦ Id ⊗ ∂qk )(ξ ⊗ ξ [k] ) = ∂q (ξ ) = 1.  Remarks 1. In other words, {∂qk }k∈N and {ξ [n] }n∈N are “dual basis” for DA/R,q and A#ξ $q . , 2. If we denote the image of P under the isomorphism DA/R,q  Diffq (A) by P then   (ξ [k] )∂qk and ∀ϕ ∈ A#ξ $q , ϕ =  ∀P ∈ DA/R,q , P = P ∂qn (ϕ)ξ [n] . k≥0

n≥0

As a consequence of Lemma 6.10, we see that DA/R,q is naturally an A#ξ $q module and we can make the structure explicit: Lemma 6.11 We have ∀k, n ∈ N,

ξ

[n]

· ∂qk

=

n 

q

i(i−1) 2

i=0

    k n (q − 1)i x i ∂qk−n+i n q i q

(for n ≤ k and 0 otherwise). Proof From formula (8), we deduce that, for m ∈ N,     ξ [n] · ∂qk (ξ [m] ) = ∂qk ξ [n] ξ [m] ⎛ ⎞      i(i−1) n + m − i n = ∂qk ⎝ q 2 (q − 1)i x i ξ [n+m−i] ⎠ n i q q 0≤i≤n,m

 =

q

i(i−1) 2

n+m−i n n

0

(q q i q

− 1)i x i if n + m − i = k, otherwise.

Note that the condition 0 ≤ i ≤ m, n is only meant to insist on the fact that the q-binomial coefficients are zero otherwise and this is why it could disappear from the final formula. Therefore, ξ [n] · ∂qk =

n  i=0

for n ≤ k, and 0 otherwise.

q

i(i−1) 2

    k n k−n+i (q − 1)i x i ∂ q n q i q 

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Remarks 1. The formula in the lemma may also be obtained in two steps: ξ [n] ·∂qk =

  k (ξ [n] ·∂qn )◦∂qk−n n q

with ξ [n] ·∂qn =

n 

q

i(i−1) 2

i=0

  n (q−1)i x i ∂qi . i q

2. Alternatively, if P ∈ DA/R,q and ϕ ∈ A#ξ $q , we have ϕ·P =

 0≤i≤n≤k

q

i(i−1) 2

    k n (ξ [k] ) (q − 1)i x i P ∂qn (ϕ)∂qk−n+i . n q i q

Lemma 6.12 A q-derivation D : M → M extends uniquely to a q-differential operator of order one  : A#ξ $q /I [2] ⊗ A M → M D and we have ∀s ∈ M,

 ⊗ s) = D(x)s + (q − 1)xD(s). D(ξ

Proof First of all, since D is R-linear, it extends uniquely to an A-linear map (for the left structure) P ⊗ A M → M. Since the canonical map P → A#ξ $q /I [2] induces an isomorphism A[ξ ]/ξ (2)  A#ξ $q /I [2] , it is surjective and uniqueness follows. As  we first show that the map an intermediate step, in order to prove the existence of D, θD : M → M ⊗A A#ξ $q /I [2] ,

s → s ⊗ D(x) + D(s) ⊗ ξ

is A-linear for the right A-module structure. If f ∈ A and s ∈ M, we have θD (f s) = f s ⊗ D(x) + D(f s) ⊗ ξ = f s ⊗ D(x) + (D(f )s + σ (f )D(s)) ⊗ ξ = s ⊗ (D(x)f + D(f )ξ ) + D(s) ⊗ σ (f )ξ = s ⊗ D(x)θ (f ) + D(s) ⊗ θ (f )ξ = θ (f )(s ⊗ D(x) + D(s) ⊗ ξ ) = θ (f )θD (s). It follows that θD extends uniquely to an A#ξ $q /I [2] -linear map D : A#ξ $q /I [2] ⊗ A M → M ⊗A A#ξ $q /I [2]

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that we can compose with the canonical map Id ⊗A  ∂q : M ⊗A A#ξ $q /I [2] → M  of order one. Now, we compute in order to obtain a differential operator D  ⊗ s) = (Id ⊗A  D(1 ∂q )(D (1 ⊗ s)) = (Id ⊗A  ∂q )(s ⊗ D(x) + D(s) ⊗ ξ ) = D(s) and  ⊗ s) = (Id ⊗A  D(ξ ∂q )(D (ξ ⊗ s)) = (Id ⊗A  ∂q )(s ⊗ D(x)ξ + D(s) ⊗ ξ 2 ) = (Id ⊗A  ∂q )(s ⊗ D(x)ξ + D(s) ⊗ (q − 1)xξ ) = D(x)s + (q − 1)xD(s).  Remark Since we have a split short exact sequence 0 → ΩA,q → A#ξ $q /I [2] → A → 0, given an A-module M, it is equivalent to give an R-linear map ∇ : M → M ⊗A ΩA,q or an R-linear map θ1 : M → M ⊗A A#ξ $q /I [2] such that the composite map θ1

M → M ⊗A A#ξ $q /I [2] → M is the identity. This equivalence is given by ∀s ∈ M,

θ1 (s) = s ⊗ 1 + ∇(s).

Moreover, ∇ is a q-connection if and only if θ1 is semilinear in the sense that ∀f ∈ A, s ∈ M,

θ1 (f s) = θ1 (f )θ1 (s).

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More precisely, since θ1 acts as σ on ΩA,q , we have for f ∈ A and s ∈ M, θ1 (f s) = θ1 (f )θ1 (s) ⇔ f s ⊗ 1 + ∇(f s) = θ1 (f )(s ⊗ 1 + ∇(s)) ⇔ ∇(f s) = s ⊗ θ1 (f ) − s ⊗ f + σ (f )∇(s) ⇔ ∇(f s) = s ⊗ dq (f ) + σ (f )∇(s). The map θ1 then extends uniquely to an A#ξ $q /I [2] -linear map 1 : A#ξ $q /I [2] ⊗ A M → M ⊗A A#ξ $q /I [2] . If ∂M,q denotes the q-derivation associated to ∇, then we have Id⊗∂*

1 A,q [2] ⊗A M −→ M ⊗A A#ξ $q /I [2] −→ M. ∂* M,q : A#ξ $q /I

(29)

We will need below the following technical result: Lemma 6.13 If M is a DA/R,q -module and s ∈ M, then ∀k, n ∈ N,

k  ∂M,q (ξ [n] ⊗ s) =

n 

q

i(i−1) 2

i=0

    k n k−n+i (q − 1)i x i ∂M,q (s) n q i q

(for n ≤ k and 0 otherwise). Proof We proceed by induction on k (the case k = 0 being trivial). Note that the case k = 1 is essentially Lemma 6.12. By definition, k+1 [n] k  (ξ ⊗ s) = ( ∂M,q ◦ Id ⊗  ◦ Δ1,k ⊗ Id)(ξ [n] ⊗ s) ∂M,q ∂M,q k )(1 ⊗ ξ [n] ⊗ s + ξ ⊗ ξ [n−1] ⊗ s) ∂M,q = ( ∂M,q ◦ Id ⊗  k k = ∂M,q (1 ⊗  ∂M,q (ξ [n] ⊗ s)) +  ∂M,q (ξ ⊗  ∂M,q (ξ [n−1] ⊗ s))

and we will compute separately k Xn :=  ∂M,q (1 ⊗  ∂M,q (ξ [n] ⊗ s))

and

k Yn :=  ∂M,q (ξ ⊗  ∂M,q (ξ [n−1] ⊗ s)).

First of all, we have for 0 ≤ n ≤ k, Xn =

n 

q

i(i−1) 2

    k n k−n+i (q − 1)i ∂M,q (x i ∂M,q (s)) n q i q

q

i(i−1) 2

    k n k−n+i (q − 1)i (i)q x i−1 ∂M,q (s) n q i q

i=0

=

n  i=1

Twisted Differential Operators and q-Crystals n 

+

q

i(i−1) 2

    k n k−n+i+1 (q − 1)i q i x i ∂M,q (s) n q i q

q

i(i+1) 2

    k n k−n+i+1 (q − 1)i+1 (i + 1)q x i ∂M,q (s) n q i+1 q

q

i(i−1) 2

    k n k−n+i+1 (q − 1)i q i x i ∂M,q (s). n q i q

i=0 n−1 

=

i=0

+

n 

225

i=0

We can rearrange the first sum: q

i(i+1) 2



n i+1



  n (n − i)q (q − 1)i+1 i q   i(i−1) i n 2 =q q (q n−i − 1)(q − 1)i i q   i(i−1) n 2 =q (q n − q i )(q − 1)i . i q

(q − 1)i+1 (i + 1)q = q q

i(i−1) 2

qi

It follows that Xn =

n 

q

i(i−1) 2

i=0

qn

    k n k−n+i+1 (q − 1)i x i ∂M,q (s). n q i q

We turn now to the second term and it follows from Lemma 6.12 that, for 0 ≤ n − 1 ≤ k, Yn =

n−1 

q

i(i−1) 2



i=0

+ (q − 1)x

k n−1

n−1 

q

i(i−1) 2

i=0

=

n−1 

q

i(i−1) 2

q

i(i−1) 2

q

i(i−1) 2



i=0

+

n−1 



i=0

+

n−1  i=0



   n−1 k−n+i+1 (q − 1)i x i ∂M,q (s) i q q

k n−1



k n−1

   n−1 k−n+i+1 (q − 1)i ∂M,q (x i ∂M,q (s)) i q q

   n−1 k−n+i+1 (q − 1)i x i ∂M,q (s) i q q

k n−1

   n−1 k−n+i+1 (q − 1)i+1 (i)q x i ∂M,q (s) i q q

k n−1

   n−1 k−n+i+2 (q − 1)i+1 q i x i+1 ∂M,q (s) i q q

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=

n−1 



q

q

i(i−1) 2

q

(i−2)(i−1) 2

i=0

+

n−1  i=0

+

n 

k n−1

i(i−1) 2



k n−1

   n−1 k−n+i+1 (q − 1)i x i ∂M,q (s) i q q

   n−1 k−n+i+1 (q − 1)i (q i − 1)x i ∂M,q (s) i q q



   k n−1 k−n+i+1 (q − 1)i q i−1 x i ∂M,q (s). n−1 q i−1 q

i=1

Since q

(i−2)(i−1) 2

q i−1 = q

i(i−1) 2

and

      n−1 n−1 n qi + = , i i−1 q i q q

we finally get Yn =

n 

q

(i−1)i 2



   k n k−n+i+1 (q − 1)i x i ∂M,q (s). n−1 q i q

i=0

Now, we use       k k k+1 q + = n q n−1 q n q n

in order to obtain k+1 [n]  (ξ ⊗ s) = ∂M,q

n  i=0

q

i(i−1) 2



   k+1 n k−n+i+1 (q − 1)i x i ∂M,q (s). n q i q 

Lemma 6.14 If M is a DA/R,q -module, then the structural map DA/R,q → EndR (M) lifts uniquely to an A#ξ $q -linear morphism of R-algebras ρ : Diffq (A)  DA/R,q → Diffq (M). Proof The image of the generator ∂q ∈ DA/R,q in EndR (M) is a q-derivation ∂M,q of M. Moreover, we may observe that ∀f ∈ A,

 ∂M,q ◦f = σ (f )  ◦ ∂M,q + ∂A,q (f ).

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To verify this formula, it is sufficient to check it on ξ ⊗ s for s ∈ M, but we have ∂q (f )(ξ ⊗ s) = 0 and ( ∂M,q  ◦ f )(ξ ⊗ s) =  ∂M,q (θ1 (f )ξ ⊗ s) = ∂M,q (σ (f )ξ ⊗ s) = σ (f ) ∂M,q (ξ ⊗ s). It therefore follows from Lemma 6.12 that the structural map DA/R,q → EndR (M) lifts uniquely to an A-linear morphism of R-algebras ρ : DA/R,q → Diffq (M) sending ∂q to  ∂M,q . It remains to show that ρ is A#ξ $q -linear. It follows from Lemmas 6.13 and 6.11 that, for s ∈ M, we have   ∀k, n ∈ N, ρ(∂qk )(ξ [n] ⊗ s) = ρ ξ [n] ·  ∂qk (1 ⊗ s). By linearity, we see that ∀P ∈ DA/R,q , ∀ϕ ∈ A#ξ $q ,

 )(ϕ ⊗ s) = ρ ϕ · P  (1 ⊗ s). ρ(P

Now, if P ∈ DA/R,q and ϕ, ψ ∈ A#ξ $q , we will have ))(ψ ⊗ s) = ρ(P )(ϕψ ⊗ s) (ϕ · ρ(P   (1 ⊗ s) = ρ ϕψ · P  ) (1 ⊗ s) = ρ ψ · (ϕ · P   (ψ ⊗ s). =ρ ϕ·P  As a consequence, we see that ρ induces for all n ∈ N an A#ξ $q map

/I [n+1] -linear

ρn : Diffq,n (A) → Diffq,n (M). The next notion is that of a q-Taylor structure whose infinite level analog is studied in section 5 of [10]: Definition 6.15 A q-Taylor structure (of level zero) on an A-module M is a compatible family of semilinear maps (A-linear if we consider the target as an Amodule via the q-Taylor map) θn : M → M ⊗A A#ξ $q /I [n+1] ,

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such that ∀s ∈ M,

θ0 (s) = s ⊗ 1

(30)

(θn ⊗ IdA#ξ $q /I [m+1] ) ◦ θm = (IdM ⊗ Δn,m ) ◦ θm+n .

(31)

and ∀m, n ∈ Z≥0 ,

We can make explicit the semilinear condition, which sounds very natural: ∀f ∈ A, ∀s ∈ M,

θn (f s) = θn (f )θn (s).

Proposition 6.16 A q-Taylor structure on an A-module M is equivalent to a DA/R,q -module structure through the formula ∀s ∈ M,

θn (s) =

n 

k ∂M,q (s) ⊗ ξ [k] .

(32)

k=0

Proof In general, if X and Y are two A-modules, then we have an adjunction X → HomA (Y, M)

⇔

X ⊗A Y → M

⇔

Y → HomA (X, M)

⇔

x ⊗ y → #x, y$

⇔

y → (x → #x, y$).

given by x → (y → #x, y$)

If B is an A-module and B is an A-bimodule, then we can apply this to X = B ⊗A M (in which we use the right structure for the tensor product and the left structure in order to turn the tensor product into an A-module) and Y = HomA (B, A) and we get B ⊗A M → HomA (HomA (B, A), M)

⇔

HomA (B, A) → HomA (B ⊗A M, M)

(33) given by ϕ ⊗ s → (P → #ϕ ⊗ s, P $)

⇔

P → (ϕ ⊗ s → #ϕ ⊗ s, P $).

When B is free over A, we also have an isomorphism M ⊗A B  HomA (HomA (B, A), M), from which we deduce an adjunction

s ⊗ ϕ → (P → P (ϕ)s)

Twisted Differential Operators and q-Crystals 

B ⊗A M → M ⊗A B

229 ρ

HomA (B, A) → HomA (B ⊗A M, M).

⇔

One can make this explicit: if (ϕ ⊗ s) =    ϕ ⊗ s → P → P (ϕk )sk )



sk ⊗ ϕk , then adjunction (33) reads

⇔

P → (ϕ → ρ(P )(ϕ ⊗ s)).

In other words (ϕ ⊗ s) =



sk ⊗ ϕk

⇔

ρ(P )(ϕ ⊗ s) =



P (ϕk )sk .

(34)

We now apply this to the case where B (resp. B ) is equal to A#ξ $q /I [n+1] with its canonical A-module structure (resp. the A-bimodule structure given by the canonical structure on the left and the Taylor structure on the right). In this situation, we have B ⊗A M = A#ξ $q /I [n+1] ⊗ A M and we write n and ρn instead of  and ρ. One easily sees that n is A#ξ $q /I [n+1] linear if and only if ρn is A#ξ $q /I [n+1] -linear. Therefore, an A-linear map θn : M → M ⊗A A#ξ $q /I [n+1] extends uniquely to an A#ξ $q /I [n+1] -linear map n : A#ξ $q /I [n+1] ⊗ A M → M ⊗A A#ξ $q /I [n+1] , which in turn corresponds by adjunction to a unique A#ξ $q /I [n+1] -linear map

This shows that the compatible families {θn }n∈N of A-linear maps and {ρn }n∈N of A#ξ $q /I [n+1] -linear maps determine each other uniquely. Moreover, it follows from equivalence (34) that, for s ∈ M and r ≤ n, n (ξ [r] ⊗ s) =

n  k=0

In other words,

srk ⊗ ξ [k]

⇔

∀m ≤ n ∈ N, ρn ( ∂qm )(ξ [r] ⊗ s) = srm .

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∀s ∈ M, ∀r ≤ n,

n (ξ [r] ⊗ s) =

n 

ρn (∂qk )(ξ [r] ⊗ s) ⊗ ξ [k]

(35)

k=0

(and the formula extends trivially to the case r > n). Now, condition (30) translates into ρ0 (1) = IdM and condition (31) translates into n,m 

(ρn (∂qk ) ◦ρm (∂ql ))(ξ [r] ⊗s)⊗ξ [k] ⊗ξ [l] =

k,l=0

n,m 

k+l [r] [k] [l] ρn+m (∂* q )(ξ ⊗s)⊗ξ ⊗ξ .

k,l=0

( )n,m We know from Lemma 6.8 that ξ [k] ⊗ ξ [l] k,l=0 is a basis for A#ξ $q /I [n+1] ⊗ A A#ξ $q /I [m+1] , and condition (31) is therefore equivalent to ∀k ≤ n, l ≤ m ∈ N,

k+l ρn (∂qk ) ◦ρm (∂ql ) = ρn+m (∂* q ).

(36)

Taking the limit on the family {ρn }n∈N , we obtain an R-linear morphism ρ

DA/R,q  Diffq (A) → Diffq (M). Since, by construction, ρ is linear for both the left and right structures of A-module, condition (36) therefore means that it is a morphism of rings. Our assertion therefore follows from Lemma 6.14. Finally, it follows from the case r = 0 of equality (35) that formula (32) holds.  One can also define a q-stratification (of level zero) as a compatible family of A#ξ $q /I [n+1] -linear isomorphisms n : A#ξ $q /I [n+1] ⊗ A M  M ⊗A A#ξ $q /I [n+1] satisfying a normalization and a cocycle condition of the same type as (27). Clearly, a q-stratification induces a q-Taylor structure. Moreover applying equivalence (34) to n , we see that n (ϕ ⊗ s) =



sk ⊗ ξ [k]

⇔

l ∀l ≤ n,  ∂M,q (ϕ ⊗ s) =



 ∂ql (ξ [k] )sk = sl

and it follows that ∀ϕ ∈ A#ξ $q /I [n+1] , ∀s ∈ M,

n (ϕ ⊗ s) =

n 

k  (ϕ ⊗ s) ⊗ ξ [k] . ∂M,q

k=0

Said differently, we obtain the following improvement on (29):

Twisted Differential Operators and q-Crystals

231 Id⊗ ∂n

n A,q n  : A#ξ $q /I [n+1] ⊗ A M −→ M ⊗A A#ξ $q /I [n+1] −→ M. ∂M,q

However, there is no flip map on A#ξ $q /I [n+1] (or on A[ξ ]/ξ (n+1) either) because τ (I [n+1] ) ⊂ I [n+1] , and it follows that a q-Taylor structure will not extend to a qstratification in general. Actually, if we are given a q-stratification on an A-module M, then one can verify that 1−1 (s ⊗ ξ ) ≡

N  (−1)k (q − 1)k x k ξ ⊗ ∂qk (s) mod (q − 1)N +1 . k=0

Clearly, if M is a general DA/R,q -module and (q − 1) is not nilpotent, then this sum could be infinite. It is therefore necessary to consider the following classical topological condition: Definition 6.17 A DA/R,q -module M is topologically quasi-nilpotent if ∀s ∈ M,

k lim ∂M,q (s) = 0.

k→+∞

We may also say that the corresponding connection, q-Taylor structure, etc. is topologically quasi-nilpotent. Recall that we introduced the notion of hyper-q-stratification in definition 5.7. Then, we have: Proposition 6.18 If M is a finitely presented flat A-module, then it is equivalent to give M the structure of a topologically quasi-nilpotent DA/R,q -module or to endow it with a hyper-q-stratification. Proof Thanks to Propositions 6.16 and 5.9, it is sufficient to show that a topologically nilpotent q-Taylor structure is equivalent to a hyper-q-Taylor structure. By reduction modulo I [n+1] for all n ∈ N, the latter will automatically induce the former. Conversely, since M is finitely presented over A, a q-Taylor structure on M provides a formal q-Taylor map θˆ : M → lim(M ⊗A A#ξ $q /I [n+1] )  M ⊗A A##ξ $$q , ← −

s →

+∞ 

k ∂M,q (s) ⊗ ξ [k] ,

k=0

where the last isomorphism results from [16,Tag 059K] since we have an isomorphism of free A-modules A#ξ $q /I [n+1]  nk=0 Aξ [k] . Since we also assume that M is flat, it is a direct summand of a finite free module [16, Tag 00NX] and we have

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k k &$q ⇔ lim ∂M,q ∂M,q (s) ⊗ ξ [k] ∈ M ⊗A A#ξ (s) = 0. k→+∞

k=0

&$q if and only Thus, we see that the formal q-Taylor map factors through M ⊗A A#ξ if the q-Taylor structure is topologically quasi-nilpotent and it remains to check &$q is a hyper-q-Taylor structure. Since that the induced map θ : M → M ⊗A A#ξ (IdM ⊗ e) ◦ θ = IdM , we only have to check that &$q ⊗ &$q A A#ξ (θ ⊗ IdA#ξ ) ◦ θ = (IdM ⊗ Δ) ◦ θ : M → M ⊗A A#ξ  $ q

and it is sufficient to make sure that the canonical map    A A#ξ  A A##ξ $$q M ⊗A A#ξ $q ⊗ $q → M ⊗A A##ξ $$q ⊗ is injective. Since M is flat, this follows from the fact that, in the following diagram, the bottom map is injective and the flip maps are bijective. A ξ q ⊗A A ξ

q

A ξ

τ ⊗Id

A ξ q ⊗A A ξ

i,j∈N A(ξ

[i]

q ⊗A A

ξ

q

ξ

q

τ ⊗Id q

⊗ ξ [j] )

A ξ

i,j∈N

q ⊗A A

A(ξ [i] ⊗ ξ [j] ).



7 q-Crystals All δ-rings are assumed to live over the local ring Z[q](p,q−1) (with q of rank one) and they are endowed with their (p, q − 1)-adic topology. We fix a morphism of bounded q-PD-pairs (R, r) → (A, a) with A complete with fixed topologically étale coordinate x. We also assume that a is closed in A so that A is also complete. The absolute (big) q-crystalline site q−CRIS is the category opposite12 to the category of complete bounded13 q-PD-pairs (B, J ) (or equivalently complete qPD-thickenings B  B with B bounded). We may actually consider the slice

12 We will always write morphisms of pairs in the usual way and call it an opposite morphism if we consider it as a morphism in the q-crystalline site. 13 As in [3], we could add some other technical assumptions: see their definition 16.2.

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233

 r): an object is a morphism (R, r) → (B, J ) to a category q−CRIS/R over (R, complete bounded q-PD-pair (and a morphism is the opposite of a morphism of qPD-pairs which is compatible with the structural maps). Now,  if we denote by FSch the category of (p, q − 1)-adic formal schemes (over Spf Zp [[q − 1]] ), then we may consider the functor q−CRIS → FSch,

(B, J ) → Spf(B)

(with B = B/J ).

If X is a (p, q − 1)-adic formal scheme, then the absolute q-crystalline site q−CRIS(X) of X will be the corresponding fibered site over X: an object is a pair made of a complete bounded q-PD-pair (B, J ) and a morphism Spf(B) → X of formal schemes. We will actually mix both constructions and consider, if X is a (p, q − 1)-adic formal R-scheme, the q-crystalline site q−CRIS(X/R) of X/R: an object is a pair made of a morphism (R, r) → (B, J ) to a complete bounded q-PD-pair together with a morphism Spf(B) → X of formal R-schemes. We will write q−CRIS(A/R) := q−CRIS(Spf(A)/R). For the moment, we endow the category q−CRIS(X/R) with the coarse topology, so that a sheaf is simply a presheaf, and we denote by (X/R)q−CRIS the corresponding topos.14 A sheaf (of sets) E on q−CRIS(X/R) is thus a family of sets EB together with a compatible family of maps EB → EB for any morphism opposite to (B, J ) → (B , J ) in q−CRIS(X/R). In particular, we may consider the sheaf of rings OX/R,q that sends B to itself (we will write OA/R,q when X = Spf(A) as above). A sheaf of OX/R,q -modules is a family of B-modules EB endowed with a compatible family of semilinear maps EB → EB , or better, of linear maps B ⊗B EB → EB , called the transition maps. Definition 7.1 If X is a (p, q − 1)-adic formal R-scheme, then a q-crystal on X/R is a sheaf of OX/R,q -modules whose transition maps are all bijective. Remarks 1. An OX/R,q -module E is finitely presented if and only if it is a q-crystal and all EB ’s are finitely presented B-modules. 2. An OX/R,q -module E is flat if and only all EB ’s are flat B-modules. Since x is a topologically étale coordinate on A, there exists a unique δ-structure on A such that x has rank one (use lemma 2.18 of [3] for example). We endow the polynomial ring A[ξ ] with the symmetric δ-structure so that x + ξ also has rank one. The ring P := A ⊗R A comes with its tensor product δ-structure and the canonical map A[ξ ] → P is a morphism of δ-rings. It follows that P has a Frobenius φ, but, as in Sect. 2, if we set

14 Unlike

in [3], we follow the standard notations for sites and topos.

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R A A := R φ (⊗

and

P := A ⊗R A = R φ (⊗R P ,

we could as well consider the relative Frobenius F : P → P (which is a morphism of δ-algebras) as we did in section 7 of [7]. Lemma 7.2 If two morphisms of δ-pairs u1 , u2 : (P , I ) → (B, J ) to a complete q-PD-pair coincide when restricted to A[ξ ], then they must be equal. Proof The statement means that any morphism of δ-pairs (P , I ) ⊗(A[ξ ],ξ ) (P , I ) → (B, J ) will factor (necessarily uniquely) through the multiplication map (P , I ) ⊗(A[ξ ],ξ ) (P , I ) → (P , I ). Since x is a topologically étale coordinate on A, if we denote by N the kernel of multiplication from Q := A ⊗R[x] A to A, then there exists a split exact sequence →Q →A→0 0→N of rings (see [16, Tag 02FL] in the algebraic setting), and therefore also of δ-rings thanks to Lemma 1.3. Therefore, letting M := A ⊗R N, there exists a sequence of isomorphisms ×P . A[ξ ] P  A⊗ R Q  A⊗ R (N × A)  M P⊗ Actually, there even exists an isomorphism of exact sequences 0

P ⊗A[ξ] I + I⊗A[ξ] P

P ⊗A[ξ] P

A

0

0

M ×I

M ×P

A

0,

where the upper right map is total multiplication and the lower right map is composition of projection and multiplication. In particular, there exists an isomorphism of δ-pairs ×P , M  × I) = (M,  M)  × (P , I).  (A[ξ ],ξ ) (P , I ) = (P ⊗ A[ξ ] P , P ⊗ I + I ⊗  P )  (M (P , I )⊗

Let us also note that there exists a commutative diagram

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235

(P, I)⊗(A[ξ],ξ) (P, I)

(P , I)

(M , M ) × (P , I)

(P , I),

where the upper map is multiplication and the lower map is the projection. To finish the proof, it is therefore sufficient to show that any morphism of δ-pairs  M)  × (P , I) → (B, J ) u : (M, , I). But, for any such u, there exists a decomposition factors through (P  M)  → (B , J ) and u : (P , I) → (B , J ) as a u = u × u with u : (M, product of morphisms of δ-rings (use Lemma 1.3 again) and u factors necessarily  M),  which is the zero ring (because through the complete q-PD-envelope of (M, we use completions).  The morphism     θ : (P , I ) → A#ξ $q , I[1] obtained by linear extension from the q-Taylor map of Definition 5.2 is clearly a morphism of δ-pairs and we can show that it is universal:    Theorem 7.3 The q-PD-pair A#ξ $q , I[1] is the complete q-PD-envelope of (P , I ). Proof We give ourselves a map u : (P , I ) → (B, J ) to a complete q-PD-pair. Thanks to Theorem 3.5 and Proposition 4.9, the restriction of u to A[ξ ] extends  uniquely to a morphism v : A#ξ $q , I[1] → (B, J ). Moreover, the diagram

is commutative. We may then apply Lemma 7.2 to the maps u1 = u and



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We will need a slight generalization of Theorem 7.3 and we consider now P (r) := A ⊗R · · · ⊗R A  P ⊗ A · · · ⊗ A P 01 2 / / 01 2 r+1 times

r times

and denote by I (r) the kernel of multiplication P (r) → A. We also define &    A#ξ1 , . . . , ξr $∧ q := A#ξ1 $q ⊗A · · · ⊗A A#ξr $q ∧   A#ξ1 , . . . , ξr−1 $∧ . q #ξr $q Corollary 7.4 A#ξ1 , . . . , ξr $∧ q is the complete q-PD-envelope of I (r) in P (r). Proof This is a direct consequence of Lemma 4.10. The only thing to check is that A#ξ1 , . . . , ξr $∧ q is bounded and this is shown by induction: since A#ξr $q is flat over A, the map ∧ A#ξ1 , . . . , ξr−1 $∧ q → A#ξ1 , . . . , ξr−1 $q ⊗A A#ξr $q

is flat and we may then apply successively Lemma 4.6 and Corollary 4.5.



Remarks cl [1] is a q-PD-ideal in A#ξ  1. If $q and  a is a q-PD-ideal  in A, then a ⊕ I cl [1]  is the complete q-PD-envelope of (P , a ⊕ I ). This follows A#ξ $q , a ⊕ I

from Lemma 4.10 applied to (A, 0) → (A, a) and (A, 0) → (P , I ), or can be proved directly. 2. For the same reason, il follows from Corollary 7.4 that 

     A A#ξ  A I [1] + I [1] ⊗  A A#ξ $q ⊗ $q , acl ⊕ A#ξ $q A#ξ $q ⊗



is the complete q-PD-envelope of (P ⊗ A P , a ⊕ P ⊗ A I + I ⊗ A P ). Theorem 7.5 There exists a functor E → EA from finitely presented flat q-crystals on A/R to finitely presented flat topologically quasi-nilpotent DA/R,q -modules. Proof There exists an obvious functor from q-crystals on A/R to hyper-q-stratified modules on A/R obtained by composing the transition maps    EA ⊗A A#ξ A#ξ $q ⊗ A EA  EA#ξ $q .  $ q

If E is flat finitely presented, then EA may be seen by Proposition 6.18 as a finitely presented flat topologically quasi-nilpotent DA/R,q -module. 

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237

Remarks 1. In a forthcoming article, we will explain what happens when we endow the qcrystalline site with the flat topology. We are confident that (A, a) will then cover q−CRIS(A/R) for this topology and, as a consequence, the functor in Theorem 7.5 will be then an equivalence. Note that this result could also be deduced from the recent work of Chatzistamatiou in [5] and partly recovered from the preprint [13] of Morrow and Tsuji. 2. There should also exist a comparison theorem in cohomology. One approach to try to prove such a result would be to investigate more closely the relation ˇ between the Cech-Alexander complexes used in [3, 16.13], and our constructions. A second possibility would be to develop a theory of linearization of q-differential operators. Acknowledgments Adolfo (MCIU/AEI/FEDER, UE).

Quirós

supported

by

grant

PGC2018-095392-B-I00

References 1. Anschütz, J., Le Bras, A.-C.: Prismatic Dieudonné theory. Eprint (2019): arXiv:1907.10525[math.AG]. 2. Berthelot, P.: Cohomologie cristalline des schémas de caractéristique p > 0. Lecture Notes in Mathematics, Vol. 407. Springer-Verlag, Berlin (1974). 3. Bhatt, B., Scholze, P.: Prisms and prismatic cohomology. Ann. of Math. (2) 196 (3), 1135–1275 (2022). https://doi.org/10.4007/annals.2022.196.3.5 4. Borger, J., Wieland, B.: Plethystic algebra, Adv. Math. 194 (2), 246–283 (2005). https://doi. org/10.1016/j.aim.2004.06.006 5. Chatzistamatiou, A.: q-crystals and q-connections. Eprint (2020): arXiv:2010.02504[math.AG]. 6. Gros, M.: Sur une q-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive. In: Bhatt, B., Olsson, M. (eds) p-adic Hodge Theory, 143–160. Simons Symposia. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-43844-9_5 7. Gros, M., Le Stum, B., Quirós, A.: Twisted divided powers and applications. J. Number Theory 237, 285–331 (2022). https://doi.org/10.1016/j.jnt.2019.02.009 8. Joyal, A.: δ-anneaux et vecteurs de Witt. C. R. Math. Acad. Sci. Soc. R. Can. 7 (3), 177–182 (1985). 9. Le Stum, B., Quirós, A.: On quantum integers and rationals. In: Chamizo, F., Guàrdia, J., RojasLeón, A., Tornero, J.-M. (eds) Trends in Number Theory–Sevilla 2013, 107–130, Contemp. Math. 649, Amer. Math. Soc., Providence, RI (2015). https://doi.org/10.1090/conm/649 10. Le Stum, B., Quirós, A.: Formal confluence of quantum differential operators. Pacific J. Math. 292 (2), 427–478 (2018). https://doi.org/10.2140/pjm.2018.292.427 11. Le Stum, B., Quirós, A.: Twisted calculus. Comm. Algebra 46 (12), 5290–5319 (2018). https:// doi.org/10.1080/00927872.2018.1464168 12. Le Stum, B., Quirós, A.: Twisted calculus on affinoid algebras. Pacific J. Math. 304 (2), 523– 560 (2020). https://doi.org/10.2140/pjm.2020.304.523

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13. Morrow, M., Tsuji, T.: Generalised representations as q-connections in integral p-adic Hodge theory. Eprint (2020): arXiv:2010.04059[math.NT]. 14. Oyama, H.: PD Higgs crystals and Higgs cohomology in characteristic p. J. Algebraic Geom. 26 (4), 735–802 (2017). https://doi.org/10.1090/jag/699 15. Pridham, J. P.: On q-de Rham cohomology via Λ-rings. Math. Ann.375 (1–2), 425–452 (2019). https://doi.org/10.1007/s00208-019-01806-7 16. The Stacks Project Authors: The Stacks project. https://stacks.math.columbia.edu

Full Level Structures on Elliptic Curves Jacob Lurie

1 Overview Let p be a prime number, which we regard as fixed throughout this paper. For each n > 0, let X(pn ) denote the modular curve parametrizing (generalized) elliptic curves equipped with a full level-pn structure, which we regard as a scheme defined over the cyclotomic field Q[ζpn ]. Each X(pn ) determines a rigid-analytic curve X(pn )an over the local field Qp [ζpn ]. These rigid-analytic curves can be organized into an inverse system · · · → X(p4 )an → X(p3 )an → X(p2 )an → X(p)an → X(1)an . The starting point of this paper is the following result (which is a special case of Theorem III.1.2 of [5]): Theorem 1 (Scholze) There exists a perfectoid space X(p∞ )an over the perfectoid cyc field Qp , characterized up to unique isomorphism by the requirement X(p∞ )an ∼ lim X(pn )an (in the sense of [6], Definition 2.4.1). ← − The primary goal of this paper is to prove an integral version of Theorem 1. For pn = 2, we can identify X(pn ) with the generic fiber of a Deligne-Mumford stack Ell(pn ) over the ring of integers Z[ζpn ] ⊆ Q[ζpn ], which parametrizes (generalized) elliptic curves equipped with a full level-pn structure in the sense of Drinfeld (see [3] and [2]). These stacks can be organized into an inverse system · · · → Ell(p4 ) → Ell(p3 ) → Ell(p2 ) → Ell(p) → Ell(1) J. Lurie () Institute for Advanced Study, Princeton, NJ, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_5

239

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with affine transition maps, so that the inverse limit Ell(p∞ ) is a Deligne-Mumford stack defined over the ring defined over the ring Z[ζp∞ ] = lim Z[ζpn ]. Let − →n Ell(p∞ )p=0 denote the closed substack of Ell(p∞ ) given by the vanishing locus of p, and let Fp [ζp∞ ] denote the quotient ring Z[ζp∞ ]/(p). The main result of this paper is the following: Theorem 2 The structure map Ell(p∞ )p=0 → Spec(Fp [ζp∞ ]) is relatively perfect. That is, the commutative diagram of Deligne-Mumford stacks Ell(p∞ )p=0

Spec(Fp [ζp∞ ])

ϕ

ϕ

Ell(p∞ )p=0

Spec(Fp [ζp∞ ])

(in which the horizontal maps are given by the Frobenius) is a pullback square. Note that the Frobenius map ϕ : Fp [ζp∞ ] → Fp [ζp∞ ] is a surjection, whose kernel is the ideal generated by the image of the element π = (ζp2 − 1)p−1 ∈ Z[ζp∞ ]. We can therefore reformulate Theorem 2 more concretely as follows: Theorem 3 The absolute Frobenius map induces an isomorphism from Ell(p∞ )p=0 to the closed substack Ell(p∞ )π =0 ⊆ Ell(p∞ ) given by the vanishing locus of the element π = (ζp2 − 1)p−1 . It follows from Theorem 3 that the moduli stack Ell(p∞ ) is integral perfectoid (after p-adic completion). More precisely, we have the following: Corollary 1 For every étale map Spec(R) → Ell(p∞ ), there exists a regular element π ∈ R such that π p is a unit multiple of p, and the Frobenius map R/π R → R/π p R is an isomorphism. Remark 1 The conclusion of Corollary 1 is satisfied more generally for maps f : Spec(R) → Ell(p∞ ) which are “log étale at infinity” (in particular, our result can be applied to the study of elliptic curves equipped with auxiliary “prime to p” level structures). Remark 2 In [8], Weinstein supplies an explicit description of the coordinate ring for Lubin–Tate space at infinite level (see Theorem 2.17 of [8]). From this description, one can immediately deduce that Corollary 1 holds after formal completion along the locus of supersingular elliptic curves. Warning 1 For pn > 2, the generic fiber of Ell(pn ) is the modular curve X(pn ), which is a scheme. However, the stack Ell(pn ) itself is never a scheme: over a field of characteristic p, any supersingular elliptic curve E admits a unique full levelpn structure, which is preserved by any automorphism of E. Consequently, there is a slight mismatch between the statements of Theorem 1 and Corollary 1: the first concerns the local structure of the inverse system {X(pn )an } with respect to

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the analytic topology, while the second concerns the local structure of the inverse system {Ell(pn )} with respect to the étale topology. Nevertheless, it is not difficult to deduce Theorem 1 formally from Corollary 1; we leave details to the interested reader. Remark 3 Theorem 2 provides a moduli-theoretic interpretation of the tilt X(p∞ )an, of the perfectoid space of Theorem 1: it can be realized as the “generic fiber” of the formal Deligne-Mumford stack given by the direct limit of the system ϕ

ϕ

ϕ

Ell(p∞ )p=0 − → Ell(p∞ )p=0 − → Ell(p∞ )p=0 − → ··· , where the transition maps are given by the absolute Frobenius. Let us now outline the contents of this paper. We begin in Sect. 2 by reviewing the definition of the moduli stacks Ell(pn ) (following Katz–Mazur [3]) and formulating a “finite-level” analogue of Theorem 3, which asserts the existence of a commutative diagram of Deligne-Mumford stacks Ell(pn )p=0

ϕ

Ell(pn )π=0

Θ

Ell(pn−1 )p=0

ϕ

Ell(pn−1 )π=0

(1)

for n ≥ 3 (see Theorem 4 and Remark 8), where the horizontal maps are given by the Frobenius and the vertical maps by “forgetting” level structure. The difficulty is then to prove the existence of the morphism Θ in (1). Working away from the cusps, we can think of points of Ell(pn )π =0 as elliptic curves E equipped with a full levelpn structure (x, y) for which the Weil pairing epn (x, y) is a primitive pn−1 st root of unity. The heuristic idea is that this property of the Weil pairing ensures that pn−1 x and pn−1 y “generate” a subgroup S ⊆ E of order p. The morphism Θ then carries (E, x, y) → (E/S, x , y ), where x and y denote the images of x and y in the quotient elliptic curve E/S. In Sect. 3, we translate this heuristic into a precise mathematical construction in the case where E is an ordinary elliptic curve, and use n−1 ) this to construct a morphism Ell(pn )ord p=0 on the open substack π =0 → Ell(p ord n n Ell(p )π =0 ⊆ Ell(p )π =0 parametrizing ordinary elliptic curves (Proposition 1). We extend this construction to the supersingular locus (and to the cusps) using a descent argument together with the fact that Frobenius map ϕ : Ell(pn )p=0 → Ell(pn )π =0 is (faithfully) flat (Proposition 2). This follows from the regularity of the moduli stack Ell(pn ) (Theorem 5.1.1 and Corollary 10.9.2 of [3]) together with a mixed-characteristic analogue of Kunz’s characterization of regular Fp -algebras, which we prove in Sect. 4 (Theorem 6). Remark 4 Many of the results of this paper can be extended to a more general setting, where the (algebraic) moduli stack Ell of elliptic curves is replaced by the (non-algebraic) moduli stack of 1-dimensional p-divisible groups.

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2 Level Structures on Elliptic Curves In this section, we briefly review the theory of Drinfeld level structures on elliptic curves and give a more detailed outline of our proof of Theorem 3. For a more comprehensive account, we refer the reader to [3]. Notation 1 Let E be an elliptic curve over a commutative ring R and let x ∈ E(R) be an R-valued point of E. Then x determines a closed immersion of schemes Spec(R) → E, whose image is an effective Cartier divisor in E. We will denote this effective Cartier divisor by [x]. Definition 1 (Drinfeld, Katz–Mazur) Let E be an elliptic curve over a commutative ring R. A full level-pn structure on E is a group homomorphism γ : (Z/pn Z)2 → E(R) for which there is an equality 

[γ (v)] = E[pn ]

v∈(Z/pZ)2

of effective Cartier divisors in E. Here E[pn ] denotes the kernel of the map pn : E → E. Remark 5 Let E be an elliptic curve over a commutative ring R. We will generally abuse notation by identifying group homomorphisms (Z/pn Z)2 → E(R) with pairs of pn -torsion points x, y ∈ E(R). We will say that a pair of pn -torsion points (x, y) is a full level-pn structure if, under this identification, it corresponds to a full levelpn structure (Z/pn Z)2 → E(R) in the sense of Definition 1. Notation 2 Let R be a commutative ring. We let Ell(R) denote the groupoid whose objects are elliptic curves over R and whose morphisms are isomorphisms of elliptic curves. If n is a positive integer, we let Ell(pn )(R) denote the groupoid whose objects are pairs (E, γ ), where E is an elliptic curve over R and γ : (Z/pn Z)2 → E(R) is a full level-pn structure on E; a morphism from (E, γ ) to (E , γ ) is an isomorphism of elliptic curves f : E → E which carries γ to γ . We regard the constructions R → Ell(R) and R → Ell(pn )(R) as functors from the category of commutative rings to the 2-category of groupoids. We will refer to Ell as the moduli stack of elliptic curves, to Ell(pn ) as the moduli stack of elliptic curves with a full level-pn structure. Remark 6 Let E → Ell denote the universal elliptic curve, and let E[pn ] denote its p-torsion subgroup (so that E[pn ] → Ell is a finite flat map of degree p2n ). Then the construction (E, x, y) → ((E, x), (E, y)) determines a closed immersion of Deligne-Mumford stacks Ell(pn ) → E[pn ] ×Ell E[pn ]. In particular, the projection map Ell(pn ) → Ell is finite.

(E, x, y) → E

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Notation 3 Let Ell denote the Deligne-Mumford compactification of the moduli stack Ell and let j : Ell → Ell denote the inclusion map. Let n be a nonnegative integer, and let q : Ell(pn ) → Ell denote the projection map. Then q is finite (Remark 6), and j is an affine open immersion (it is the inclusion of the complement of an effective Cartier divisor). Consequently, the composite map (j ◦ q) : Ell(pn ) → Ell is affine, determined by a quasi-coherent sheaf of algebras (j ◦ q)∗ OEll(pn ) on the moduli stack Ell. Let A denote the integral closure of OEll in (j ◦ q)∗ OEll(pn ) , and let Ell(pn ) denote the relative spectrum of A. By construction, we have a pullback diagram of Deligne-Mumford stacks Ell(pn )

Ell(pn )

q

Ell

q j

Ell,

where the vertical maps are finite and the horizontal maps are open immersions. Remark 7 The construction of Notation 3 is somewhat unsatisfying, because it does not a priori give a concrete description of the functor represented by the DeligneMumford stack Ell(pn ). For a moduli-theoretic perspective, we refer the reader to [2]. Notation 4 (The Weil Pairing) Let E be an elliptic curve over a commutative ring R and let x, y ∈ E(R) be a pair of pn -torsion points of E. We let epn (x, y) denote the Weil pairing of x and y, which we regard as an element of the group n

μpn (R) = {u ∈ R : up = 1} of pn th roots of unity in R. If (x, y) is a full level pn -structure on E, then epn (x, y) is a primitive pn th root of unity: that is, it is a root of the cyclotomic polynomial n up −1 . n−1 p u

−1

pn

Let Z[ζpn ]  Z[u]/( upn−1−1 ) denote the ring of integers in the cyclotomic field u −1 Q[ζpn ], so that the construction (E, x, y) → epn (x, y) induces a morphism of Deligne-Mumford stacks Ell(pn ) → Spec(Z[ζpn ]). Since Z[ζpn ] is integral over Z, this extends uniquely to a morphism Ell(pn ) → Spec(Z[ζpn ]). For n ≥ 2, we let Ell(pn )π =0 denote the vanishing locus of the element π = (ζp2 −1)p−1 ∈ Z[ζp2 ] ⊆ Z[ζpn ]. Note that π p is a unit multiple of p, so the Frobenius induces a morphism of Deligne-Mumford stacks ϕ : Ell(pn )p=0 → Ell(pn )π =0 .

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Theorem 4 For each n ≥ 2, there is an essentially unique commutative diagram of Ell(pn )p=0

ϕ

Ell(pn )π=0

Θ

Ell(pn−1 )p=0

ϕ

Ell(pn−1 )p=0 ,

where the horizontal maps are given by the absolute Frobenius and the vertical maps by forgetting level structure. Remark 8 If n ≥ 3, then the absolute Frobenius of Ell(pn−1 )p=0 factors through the closed substack Ell(pn−1 )π =0 ⊆ Ell(pn−1 ). In this case, Theorem 4 supplies a commutative diagram Ell(pn )p=0

ϕ

Ell(pn )π=0

Θ

Ell(pn−1 )p=0

ϕ

Ell(pn−1 )π=0 .

Proof By virtue of Theorem 4 (and Remark 8), we have a commutative diagram of Deligne-Mumford stacks ···

···

Ell(p3 )p=0

ϕ

Ell(p3 )π=0

Ell(p2 )p=0

ϕ

Ell(p2 )π=0

Ell(p)p=0

ϕ

Ell(p)p=0 .

Passing to the inverse limit in the vertical directions, we conclude that the Frobenius map ϕ : Ell(p∞ )p=0 → Ell(p∞ )π =0 is an isomorphism. Let Ellord p=0 denote the open substack of Ellp=0 classifying ordinary elliptic curves (over commutative rings of characteristic p). For each n ≥ 2, we let Ell(pn )ord p=0 and ord ord n n Ell(p )π =0 denote the inverse image of Ellp=0 in the moduli stacks Ell(p )p=0 and Ell(pn )π =0 , respectively. In Sect. 3, we will prove the following weaker version of Theorem 4:

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Proposition 1 For each n ≥ 2, there exists an essentially unique commutative diagram of Deligne-Mumford stacks Ell(pn )ord p=0

ϕ

Ell(pn )ord π=0

Θ ord

Ell(pn−1 )p=0

ϕ

Ell(pn−1 )p=0 ,

where the horizontal maps are given by the absolute Frobenius and the vertical maps by forgetting level structure. We will deduce Theorem 4 from Proposition 1 together with the following result: Proposition 2 For n ≥ 2, the Frobenius morphism ϕ : Ell(pn )p=0 → Ell(pn )π =0 is flat. Our proof of Proposition 2 will use a mixed-characteristic analogue of Kunz’s characterization of regular rings of characteristic p (Theorem 6), which we establish in Sect. 4. Proof of Proposition 2 Fix an étale morphism Spec(R) → Ell(pn ), and let us abuse notation by identifying the element π = (ζp2 − 1)p−1 ∈ Z[ζp2 ] with its image in R. We wish to show that the Frobenius map ϕ : R/π R → R/π p R = R/pR is flat. Since R is a regular Noetherian ring (Theorem 5.1.1 and Corollary 10.9.2 of [3]), this is a special case of Theorem 6. Proof of Theorem 4 Fix an integer n ≥ 2, so we have a commutative diagram Ell(pn )p=0

ϕ

Ell(pn )π=0

ϕ

Ell(pn−1 )p=0

θ

Ell(pn−1 )p=0

in the category C of Deligne-Mumford stacks equipped with an affine morphism to Ell(pn−1 )p=0 . We wish to show that, in the category C, the morphism θ factors uniquely as a composition ϕ

Θ

Ell(pn )p=0 − → Ell(pn )π =0 − → Ell(pn−1 )p=0 . Note that the Frobenius map ϕ : Ell(pn )p=0 → Ell(pn )π =0 is a flat surjection (Proposition 2), and is therefore an epimorphism in the category C; this proves the uniqueness of Θ. To prove existence, it will suffice (by faithfully flat descent) to

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show that θ ◦ q+ = θ ◦ q− , where q− , q+ : Ell(pn )p=0 ×Ell(pn )π =0 Ell(pn )p=0 → Ell(pn )p=0 are the two projection maps. Note that the fiber product Ell(pn )p=0 ×Ell(pn )π =0 Ell(pn )p=0 is flat over the moduli stack Ell(pn )p=0 (by Proposition 2), and therefore also over Ellp=0 (since Ell(pn ) is finite flat over Ell). It follows that the ordinary locus Ell(pn )ord Ell(pn )ord p=0 ×Ell(pn )ord p=0 is schematically dense in the fiber product π =0 n n Ell(p )p=0 ×Ell(pn )π =0 Ell(p )p=0 , so it suffices to prove that θ ◦ q+ and θ ◦ q− agree on the ordinary locus. This follows from the existence of the morphism n−1 ) Θ ord : Ell(pn )ord p=0 supplied by Proposition 1. π =0 → Ell(p

3 The Ordinary Locus Let E be an elliptic curve over a commutative Fp -algebra R. We say that E is ordinary if each fiber of the map E → Spec(R) is an ordinary elliptic curve. In this case, the subgroup scheme E[pn ] ⊆ E fits into a short exact sequence of finite flat group schemes 0 → E[pn ]conn → E[pn ] → E[pn ]ét → 0; here E[pn ]conn is the connected component of the identity in E[pn ] (given by the kernel of the nth power of the Frobenius morphism on E), and E[pn ]ét is an étale group scheme which is Cartier dual to E[pn ]conn (via the Weil pairing) Remark 9 (Level Structures on Ordinary Elliptic Curves) Let E → Spec(R) be an ordinary elliptic curve over an Fp -algebra R. Then a pair of pn -torsion points x, y ∈ E(R) determines a full level-pn structure on E if and only if the following two conditions are satisfied: – The induced map of finite flat group schemes (Z/pn Z)2 → E[pn ] induces an epimorphism of étale group schemes (Z/pn Z)2  E[pn ]ét . – The Weil pairing epn (x, y) ∈ μpn (R) is a primitive pn th root of unity: that is, it n

satisfies the cyclotomic polynomial

up −1 . n−1 up −1

See Proposition 1.11.2 of [3]. Proposition 3 Let R be a commutative Fp -algebra, let E be an ordinary elliptic curve over Spec(R), and suppose we are given a pair of p-torsion points x, y ∈ E(R) which determine a full level-p structure on E, which we identify with a morphism of group schemes γ : (Z/pZ)2 → E. If the Weil pairing ep (x, y) is equal to 1, then γ factors uniquely as a composition (Z/pZ)2  S → E, where S ⊆ E is an étale subgroup of degree p.

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Proof Our assumption that x and y determine a level structure guarantees that the γ composite map (Z/pZ)2 − → E[p] → E[p]ét is an epimorphism of étale group schemes over R. Let K denote its kernel and set S = (Z/pZ)2 /K, so that we have a diagram of short exact sequences 0

(Z /p Z)2

K

γ

γ0

0

S

E[p]conn

0 ∼

E[p]

E[p]´et

0.

Since the Weil pairing on E[p] induces a perfect pairing of E[p]conn with E[p]ét , the equation ep (x, y) = 1 guarantees that the morphism γ0 vanishes. It follows that γ factors through a unique homomorphism S → E[p], which splits the epimorphism E[p]  E[p]ét and therefore identifies S with a closed subgroup of E[p] ⊂ E. Remark 10 In the situation of Proposition 3, let E/S denote the quotient of E by the subgroup S ⊂ E, let (E/S)(p) denote its pullback along the Frobenius map Spec(R) → Spec(R), and let F : E/S → (E/S)(p) be the relative Frobenius map. F

Then the composite map E  E/S − → (E/S)(p) is an isogeny of degree p2 , whose kernel is the p-torsion subgroup E[p] ⊂ E. It follows that there exists a unique isomorphism of elliptic curves α : E → (E/S)(p) for which the diagram E

E/S

p

F α ∼

E

(E/S)(p)

is commutative. Corollary 2 Let R be a commutative Fp -algebra, let E be an ordinary elliptic curve over Spec(R), and suppose we are given a pair of p-torsion points x, y ∈ E(R) which determine a full level-pn structure on E for some n ≥ 1, which we identify with a morphism of group schemes γ : (Z/pn Z)2 → E. (a) If the Weil pairing epn (x, y) is a pn−1 st root of unity, then there exists a diagram of group schemes (which is unique up to unique isomorphism) (Z /pn Z)2

(Z /pn−1 Z)2

γ

E

γ f

E,

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where f is an étale isogeny of degree p. (b) If n ≥ 2 and the Weil pairing epn (x, y) is a primitive pn−1 st root of unity, then the map γ : (Z/pn−1 Z)2 → E is a full level-pn−1 structure on the elliptic curve E . Proof If epn (x, y) is a pn−1 st root of unity, then ep (pn−1 x, pn−1 y) = 1. Applying Proposition 3, we see that there is a unique étale subgroup S ⊆ E of degree p containing the points pn−1 x and pn−1 y. To prove (a), we take E = E/S and f : E  E to be the tautological map. Note that we have a commutative diagram of étale group schemes (Z /pn Z)2

(Z /pn−1 Z)2

E[pn ]´et

E [pn−1 ]´et ,

where the left vertical map and bottom horizontal map are epimorphisms; it follows that the right vertical map is also an epimorphism. Invoking Remark 9, we deduce that γ is a full level-pn−1 structure on the elliptic curve E if and only if epn−1 (f (x), f (y)) = epn (x, y) is a primitive pn−1 st root of unity, which proves (b). Example 1 Let E be an ordinary elliptic curve over an Fp -algebra R, let E (p) denote the pullback of E along the Frobenius map ϕ : Spec(R) → Spec(R), and let F : E → E (p) be the relative Frobenius map. Let x, y ∈ E(R) be a pair of points which supply a full level-pn structure on E, for some n ≥ 2. Then (F x, F y) is a full level-pn structure on E (p) , and the Weil pairing epn (F x, F y) = epn (x, y)p is a primitive pn−1 st root of unity (since epn (x, y) is a primitive pn th root of unity). Applying Corollary 2 to the triple (E (p) , F x, F y), we obtain the commutative diagram (Z /pn Z)2

(Z /pn−1 Z)2

(F x,F y)

E (p)

(px,py) V

E,

where V : E (p) → E is the Verschiebung morphism. Proof of Proposition 1 Fix n ≥ 2 and let R be a commutative Fp -algebra, so that we can identify R-valued points of Ell(pn )ord p=0 with triples (E, x, y) where E is an ordinary elliptic curve over Spec(R) and (x, y) is a full level-pn structure on E. Set π = (ζp2 − 1)p−1 ∈ Z[ζpn ]. Using the identity p−1

π ≡ 1 + ζp2 + · · · + ζp2

pn−2

= 1 + ζpn

(p−1)pn−2

+ · · · + ζp n

(mod p).

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249

It follows that (E, x, y) is an R-valued point of the closed substack Ell(pn )ord π =0 ⊆ ord n n−1 n Ell(p )p=0 if and only if the Weil pairing ep (x, y) is a primitive p st root of unity. In this case, Corollary 2 supplies an étale isogeny f : E  E of degree p such that (f (x), f (y)) is a full level-pn−1 structure on E . This construction depends functorially on R and therefore determines a map of moduli stacks Θ ord : n−1 ) Ell(pn )ord p=0 . It follows from Remark 10 and Example 1 that this π =0 → Ell(p map fits into a commutative diagram of Deligne-Mumford stacks Ell(pn )ord p=0

ϕ

Ell(pn )ord π=0

Θ ord

Ell(pn−1 )p=0

ϕ

Ell(pn−1 )p=0 .

4 Kunz’s Theorem in Mixed Characteristic We refer the reader to [4] for a proof of the following classical result: Theorem 5 (Kunz) Let R be a Noetherian Fp -algebra. Then R is regular if and only if the Frobenius morphism ϕR : R → R is flat. Our goal in this section is to prove the following mixed-characteristic variant of Theorem 5, which was suggested to us by Gabber (see [1] for a closely related result): Theorem 6 Let R be a Noetherian ring and let π ∈ R be a regular element for which π p divides p. The following conditions are equivalent: (1) For every maximal ideal m ⊆ R containing π , the local ring Rm is regular. (2) The Frobenius morphism ϕ : R/π R → R/π p R is flat. Remark 11 For the purpose of proving Theorem 4, we will need only the “easy” implication (1) ⇒ (2) of Theorem 6. However, since the converse implication may be of independent interest, we include a proof here. Warning 2 In the statement of Theorem 6, the assumption that π is regular cannot be omitted (note that the ring Fp [π ]/(π p ) satisfies condition (2) of Theorem 6, but does not satisfy (1)). Proof of (1) ⇒ (2) Let R be a Noetherian ring containing a regular element π for which π p divides p. To show that the Frobenius map ϕ : R/π R → R/π p R is flat, it will suffice to show that it becomes flat after localizing with respect to every prime ideal of R/(π ). We may therefore assume without loss of generality that R is a local ring whose maximal ideal m contains π . Choose a faithfully flat map R → S,

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where S is a complete regular local ring with perfect residue field. We then have a commutative diagram R/πR

S/πS

ϕ

R/π p R

ϕ

S/π p S,

where the vertical maps are faithfully flat. Consequently, to show that the upper horizontal map is flat, it will suffice to show that the lower horizontal map is flat. We may therefore replace R by S and thereby reduce to the case where R is a complete regular local ring with perfect residue field k = R/m. Choose a regular system of parameters x1 , x2 , . . . , xn ∈ m, and let R denote the power series ring W (k)[[X1 , . . . , Xn ]], so that there is a unique surjective ring homomorphism ρ : R → R lifting the identity map idk on residue fields and satisfying ρ(Xi ) = xi . Choose a ring homomorphism ϕR : R → R lifting the Frobenius map on R/pR (for example, we can choose ϕR to carry each generator Xi to its pth power); note that ϕR automatically restricts to the Witt vector Frobenius on W (k). We then have a commutative diagram R ϕR

R

R/πR ϕ

R/π p R,

(2)

where the left vertical map is flat. We will complete the proof by showing that this diagram is a pushout square of commutative rings. Since the element p ∈ R belongs to m, we can choose a power series f = f (X1 , . . . , Xn ) with vanishing constant term satisfying ρ(f ) = p, so that ρ induces an isomorphism R/(p − f )  R. Similarly, we can choose a power series π = π (X1 , . . . , Xn ) with vanishing constant term which satisfies ρ(π ) = π . It follows ρ → R  R/π R is a surjection with kernel ideal (p − that the homomorphism R − ρ → R  R/π p R is a surjection with kernel f, π ), and that the homomorphism R − ideal (p − f, π p ). Let I denote the ideal of R generated by the elements ϕR (p − f ) and ϕR (π ), so that the commutativity of diagram (2) guarantees that we have an inclusion I ⊆ (p − f, π p ). To complete the proof, it will suffice to show that the reverse inclusion holds: that is, that p − f and π p belong to I . Since the quotient ring R/(p − f, π )  R/π R is an Fp -algebra, the ideal (p − f, π ) ⊆ R contains p. Applying the ring homomorphism ϕR , we deduce that the ideal I also contains p. The congruence π p ≡ ϕR (π ) (mod p) shows that π p also belongs to I . Our assumption that π p divides p in R guarantees that we can write p = aπ p + b(p − f ) for some elements a, b ∈ R. Evaluating at X1 = X2 = · · · = Xn = 0, we see that the power series b must have constant term equal to 1, and is

Level Structures

251

therefore an invertible element of R. We then have p − f = b−1 (p − aπ p ) ∈ (p, π p ) ⊆ I, as desired. Proof of (2) ⇒ (1) Fix a Noetherian ring R, a regular element π ∈ R such that π p divides p, and assume that the Frobenius homomorphism ϕ : R/π R → R/π p R is flat. We wish to show that, for every maximal ideal m ⊆ R containing π , the local ring Rm is regular. Replacing R by its localization Rm , we may assume that R is a local ring whose maximal ideal m contains π . Let n denote the maximal ideal of the quotient ring R/π R, and let x1 , . . . , xd ∈ m be a collection of elements whose images form a basis of the quotient n/n2 (as a vector space over the residue p p field R/m). Let c = (R/(π p , x1 , . . . , xd )) denote the length of the Artinian ring p p R/(π p , x1 , . . . , xd ). p Let I denote the ideal of R/π p R generated by the images of the elements xi . Invoking the flatness of the Frobenius map ϕ : R/π R → R/π p R, we see that I /I 2 p p is a free module of rank d over the quotient ring R/(π p , x1 , . . . , xd ), with basis p given by the images of the elements xi . Applying [7, Tag 0EBY], we conclude that c = pd (R/(π p , x1 , . . . , xd )). In particular, we have c ≥ pd , and equality holds if and only if m = (π p , x1 , . . . , xd ). Let χ : Z≥0 → Z≥0 be the Hilbert-Samuel function χ (t) = (R/(π R + mt )). For t ' 0, χ (t) is a polynomial function of n, whose degree D is equal to the Krull dimension of R/π R (so that D + 1 is the Krull dimension of R, since the element π is not a zero divisor). Using flatness of the Frobenius map ϕ : R/π R → R/π p R, we obtain χ (t) = (R/(π R + (x1 , . . . , xd )t )) 1 p p (R/(π p R + (x1 , . . . , xd )t )) c 1 ≤ (R/(π p R + (x1 , . . . , xd )pt+(p−1)d )) c p ≤ (R/(π R + (x1 , . . . , xd )pt+(p−1)d )) c p = χ (pt + (p − 1)d). c =

Evaluating at t ' 0, we deduce that pD+1 ≥ c. We now consider two cases: – Suppose that c > pd . It follows that the Krull dimension dim(R) = D + 1 is strictly larger than d. Then the maximal ideal m = (π, x1 , . . . , xd ) can be generated by d + 1 ≤ dim(R) elements, so R is regular.

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– Suppose that c = pd , so that m = (π p , x1 , . . . , xd ). Applying Nakayama’s lemma, we deduce that m = (x1 , . . . , xd ) can be generated by d ≤ D + 1 = dim(R) elements, so R is regular.

Acknowledgments I would like to thank Bhargav Bhatt, Johan de Jong, Barry Mazur, and Peter Scholze for useful discussions related to the subject of this paper, Preston Wake for offering corrections on an earlier version, and the anonymous referee for numerous corrections and recommendations. Particular thanks are due to Ofer Gabber, who suggested Theorem 6 (thereby substantially simplifying the proof of Theorem 2). I offer thanks also to the National Science Foundation, for supporting this work under grant number 1510417.

References 1. Bhatt, B., Iyengar, S. and L. Ma. Regular rings and perfect(oid) algebras. Comm. Algebra 47 (2019), no. 6, 2367–2383. 2. Conrad, B. Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu 6 (2007), no. 2, 209–278. 3. Katz, N., and B. Mazur. Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. 4. Kunz, E. On Noetherian rings of characteristic p. Amer. J. Math. 98 (1976), no. 4, 999–1013. 5. Scholze, P. On the torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2) 182 (2015), no. 3, 945–1066. 6. Scholze, P., and J. Weinstein. Moduli of p-divisible groups. Camb. J. Math. 1 (2013), no. 2, 145–237. 7. The Stacks Project. https://stacks.math.columbia.edu. 8. Weinstein, J. Semistable models for modular curves of arbitrary level. Invent. Math. 205 (2016), no. 2, 459–526.

The Saturated de Rham–Witt Complex for Schemes with Toroidal Singularities Arthur Ogus

Introduction Crystalline cohomology [1, 3], conceived and developed by Grothendieck and Berthelot, provides a very satisfactory p-adic cohomology theory for smooth proper schemes X over a perfect field k. In particular, the crystalline cohomology of such a scheme consists of finitely generated modules over the Witt ring W of k, and the corresponding integral structure carries important geometric information. This theory does not behave well for schemes with even the simplest singularities, but further work by Berthelot and others has developed a theory of “rigid cohomology [2]”, which seem well behaved quite generally. However, these theories have coefficients in Q ⊗ W (k) and thus provide no information about p-adic lattice structures or torsion. One of the key applications of the integral version of crystalline cohomology has been to the study of the action of Frobenius, which has revealed subtle connections between zeta-functions and Hodge numbers [17]. An especially powerful tool for this study has been the “de Rham–Witt complex” [11], which is a canonical complex of abelian sheaves on the Zariski topology of a smooth scheme X/k and which calculates its crystalline cohomology. Recently, Bhatt, Lurie, and Mathew have found a new construction [4] of this complex, called the saturated de Rham–Witt complex, which is in some ways simpler and more general. It is the aim of this note to show how this new construction provides reasonable answers for a wide class of singular schemes, including schemes with “ideally toroidal” singularities. These results apply to fine saturated and smooth idealized log schemes X/k. It turns out that the saturated de Rham Witt complex of X/k is quasi-isomorphic to certain de

A. Ogus () Department of Mathematics, University of California, Berkeley, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Bhatt, M. Olsson (eds.), p-adic Hodge Theory, Singular Varieties, and Non-Abelian Aspects, Simons Symposia, https://doi.org/10.1007/978-3-031-21550-6_6

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Rham complexes on suitable PD thickenings of X, and in particular to Danilov’s de Rham complex of a smooth lifting Y /W . Applications include a version of the theorems of Deligne-Illusie (on Hodge to de Rham degeneration) and Mazur (on the Katz conjectures) for such a log scheme. Here is a summary of the contents of this paper. The first section contains a review of the main constructions of [4], as well as a few additional perspectives, generalizations, and results that we will need later. Generalizing two key ideas from [4], we introduce the notions of “quasi-saturated” and “quasi-Cartier type.” Corollary 1.9 shows that a quasi-isomorphism between quasi-saturated Dieudonné complexes induces an isomorphism between their strict saturations, a result which will be used later to prove that the saturated de Rham–Witt complex calculates crystalline cohomology (Theorem 6.8). Proposition 1.13 gives another such criterion, which will be used in the following section devoted to de Rham and de Rham–Witt complexes of monoid algebras. Sections 2 and 3 contain our main technical results about Dieudonné complexes of monoid algebras. A monoid algebra A over the Witt ring has a natural Frobenius lifting φ, and its saturated de Rham–Witt complex can be obtained by applying the saturation functor to the Dieudonné complex deduced from the de Rham complex · ΩA/W together with the action of φ. The singularities of A make this complex difficult to handle, and the technical key to our paper is the fact that the saturation · of ΩA/W is the same as the saturation of the complex Ω ·A/W of “Danilov” (or “Zariski”) differentials, which is much better behaved. In Sect. 3 we generalize slightly by considering algebras which are quotients of monoid algebras by ideals generated by monoidal ideals. This additional flexibility allows us to handle some reducible schemes: those obtained by gluing monoidal schemes along faces in a suitable way. Theorem 3.4 summarizes the conclusions we can draw about the saturated de Rham–Witt complex of such a k-algebra A, including an explicit description of W1 ΩA· . We also show in Theorem 3.5 that the saturated de Rham– Witt complex of such an algebra satisfies simplicial descent with respect to the normalization mapping. Section 4 applies the results of the previous two sections to a global setting, answering several questions asked by Illusie [12, §4.3]. Theorem 4.1 shows that, for a scheme X/k which looks étale locally like an ideally toric scheme, the components of Wn ΩX· are coherent over Wn OX . On the other hand, Wn ΩX· is is not always obtained as the pushforward of the classical construction on the regular locus of X; instead one must pushforward along a locally toric resolution of singularities. We have not addressed the possible comparison between the saturated de Rham–Witt complex and rigid cohomology, an important question raised in [12, §6]. Section 5 discusses the relation between the saturated de Rham–Witt complex and crystalline cohomology. In particular, it describes how to construct WΩX· from the PD-envelope of X in a suitable embedding in a smooth Y /W endowed with a Frobenius lifting. This gives a very direct proof of the comparison between crystalline and de Rham–Witt cohomology in the smooth case. (We should note that [4, Theorem 10.1.2] gives a a very general existence and uniqueness result for such an isomorphism.)

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In Sect. 6 we discuss logarithmic schemes. The scheme underlying a fine and saturated and smooth idealized log scheme X/k has ideally toroidal singularities, and the version of the de Rham–Witt complex we are discussing here does not see the log structure. However, the additional information provided by a log structure allows us to compare this complex to the de Rham cohomology of a (log) smooth lifting Y /W and to the singular cohomology of its generic fiber (see Corollary 6.9). The proof of this comparison requires Theorem 6.7, a crystalline Poincaré lemma for the complexes of Danilov differentials in mixed characteristic. Results in this section also include a second proof of the crystalline to de Rham–Witt comparison theorem (Theorem 6.8) and also Theorem 6.10, a version of the Deligne-Illusie theorem for the saturated de Rham–Witt complex, in a logarithmic context. In the last section we discuss the Nygaard filtration. We begin with a general definition, based on the “abstract” construction of Mazur [17], which is easy to explicate for Dieudonné complexes which are of Cartier type or are saturated (see Proposition 7.4). Theorem 7.5 is a filtered version of the key quasi-isomorphism theorem [4, 2.7.3], which turns out to also be a generalization of Nygaard’s key theorem [19, 1.5]. Finally, we show in Proposition 7.9 that formation of Nygaard filtrations commutes with passage to hypercohomology under certain conditions which are often satisfied for smooth log schemes over W . This paper also includes a technical appendix explaining the difference between Danilov differentials and W1 ΩX· in small characteristics. This paper owes a huge debt to Luc Illusie, whose enormous generosity with time, ideas, conversations, and guidance helped shape and motivate this project, and whose foundational work on the original construction of the de Rham–Witt complex remains a classic inspiration. I first learned about the saturated de Rham– Witt complex directly from him, and his paper [12] remains an invaluable guide. I am also grateful to Bhatt, Lurie, and Mathew, the three authors of [4], who patiently listened to some of my inchoate ideas at the early stages of the research and who provided useful feedback. Thanks also go to the anonymous referee, who pointed out many egregious errors and misprints in an earlier version of this manuscript.

1 Dieudonné Complexes and Dieudonné Algebras Let us briefly review the main definitions of [4] and gather the basic facts that we will need about them. Definition 1.1 A Dieudonné complex is a triple (M · , d, F ), where (M · , d) is a cochain complex of abelian groups and F : M · → M · is an endomorphism of the underlying graded abelian group such that dF = pF d. A Dieudonné algebra is a Dieudonné complex (A· , d, F ) endowed with a structure of a commutative differential graded algebra such that F (a) ≡ a p (mod p) for every a ∈ A0 and such that An = 0 if n < 0.

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If R is a ring endowed with an endomorphism σ , then a Dieudonné complex (resp. algebra) over R is a Dieudonné complex (resp. algebra) in which (M · , d) is complex (resp. differential graded algebra) of R-modules and F is σ -linear. The category of Dieudonné complexes admits kernels and cokernels in the obvious way and in fact is an abelian category. One can also consider Dieudonné complexes of sheaves on a topological space or topos, and we shall often do so without comment. If (M · , d, F ) is a Dieudonné complex, the endomorphism F is not a morphism of complexes, but we can adjust for this in several ways. For example, F induces a morphism of complexes: F : (M · , pd) → (M · , d)

(1.1)

and hence, after reduction modulo p, a morphism of graded abelian groups; γ : M · /pM · → H · (M · /pM · , d)

(1.2)

Alternatively, let Φ i : M i → M i denote pi F . Then Φ · : (M · , d) → (M · , d) is a morphism of complexes. If the terms of M · are p-torsion free, let η(M)i := {ω ∈ pi M i : dω ∈ pi+1 M i+1 } ⊆ M i [1/p]. Then Φ · factors through a morphism of complexes α : (M · , d) → (η(M · ), d).

(1.3)

The following proposition-definition is the basis of the new approach to the de Rham–Witt complex proposed in [4]. Its proof is immediate. Proposition 1.2 Let (M · , d, F ) be a Dieudonné complex each of whose terms is p-torsion free. Then the following conditions are equivalent. 1. The endomorphism F is injective, and an element x of M · lies in its image if and only if dx ∈ pM · . 2. The morphism α is an isomorphism. If these conditions are satisfied, (M · , d, F ) is said to be saturated.



Formation of the complex η(M · ) can be understood as a special case of Deligne’s construction of the “filtration décalée” [8, 1.3.3]. Let P denote the p-adic filtration of M[1/p]: P k M i := pk M i for i ∈ Z. Then η(M)i = P˜ 0 M i , where P˜ is the décalée of the filtration P . Let us recall for convenience the definition and essential points of this construction.

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Proposition 1.3 ([8, 1.3.3]) If (M · , P ) is a filtered complex, let P˜ k M i := {ω ∈ P i+k M i : dω ∈ P i+k+1 M i+1 }.

1. The obvious map  i P˜ k M i → Ker Gri+k P M

d-

i+1 Gri+k P M



induces a map π which fits in an exact sequence of complexes: 0

· , d) (KP,k

(GrkP˜ M · , Grk d)

π

(H · (Gr·P+k M · ), β · )

=

0

=

(E0k,·−k (M · , P˜ ), d0 )

(E1·+k,−k (M · , P ), d1 ).

Here KPi ,k := (P i+k+1 M i + dP i+k M i−1 )/P˜ k+1 M i , j +k

β j : H j (GrP

M · ) → H j +1 (GrP

j +k+1

M ·)

is the Bockstein map, and E··,· (M, F ) denotes the spectral sequence of a filtered complex (M, F ). 2. The complex (KP· ,k , d) is acyclic, so π is a quasi-isomorphism. 3. If a morphism θ : (M · , P ) → (N · , Q) of filtered complexes induces a quasiisomorphism GrP (M · ) → GrQ (N · ), then it induces a quasi-isomorphism GrP˜ (M · ) → GrQ˜ (N · ). Proof Statements (1) and (2) are just unraveling the definitions. For (3), which comes from [4, 2.4.5], observe that if Gr(θ ) is a quasi-isomorphism, then the induced map of complexes H · (Gr·P+k M, β) → H · (Gr·Q+k N, β) is an isomorphism, hence a quasi-isomorphism for every k, and because the map π in the diagram is a quasi-isomorphism, the map (GrkP˜ M · , d) → (GriQ˜ N · , d) 

is also a quasi-isomorphism. pi

In our case, since M is p-torsion free, multiplication by induces an iso- (H · (Gri M · , d1 ). Furthermore, one checks morphism (H · (Gr0P M · , d1 ) P immediately that P˜ 1 M · = pP˜ 0 M · . Thus we find a commutative diagram:

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(M · /pM · , d) α 0

· , d) (KP,0

(η(M · )/pη(M · ), d)

γ π

(H · (M · /pM · ), d1 )

0 (1.4)

Here π is the composition of the quasi-isomorphism π with the isomorphism induced by p−i , and γ is induced by F . This is the map of Eq. 1.2, now promoted to a morphism of complexes. Let us consider the following conditions. Definition 1.4 A p-torsion free Dieudonné complex is: 1. 2. 3. 4.

saturated if α is an isomorphism, quasi-saturated if α is a quasi-isomorphism, of Cartier type if γ is an isomorphism, of quasi-Cartier type if γ is a quasi-isomorphism.

Since π is always a quasi-isomorphism, we see that (M · , d, F ) is of quasi-Cartier type if and only if α is a quasi-isomorphism. Since M · and η(M · ) are p-torsion free, this is true if (M · , d, F ) is quasi-saturated. If M · is p-adically separated and complete, the converse also holds. Thus a p-torsion free and p-adically separated and complete Dieudonné complex is quasi-saturated if and only if it is of quasiCartier type. If Y /W is a formally smooth formal scheme over the Witt ring of a perfect field, with a Frobenius lifting φY , the associated Dieudonné complex (ΩY· /W , d, F ) is of Cartier type. Remark 1.5 Illusie and Mathew have pointed out that if a Dieudonné complex is saturated, of Cartier type, and p-adically separated, then F is an isomorphism and d = 0. Indeed, if (M · , d, F ) is saturated, then α is an isomorphism, and if it is of Cartier type, then γ is an isomorphism. It follows from diagram (1.4) that π is an isomorphism and hence that KP· ,0 = 0. Now if x ∈ M i , then pi+1 x ∈ P i+1 M i ⊆ pP˜ 0 M i , so dx ∈ pM i+1 , and, since M is saturated, x belongs to the image of F . Thus F is surjective. Since dF n is divisible by pn and M is p-adically separated, it follows that d = 0. The inclusion functor from the category of saturated Dieudonné complexes to the category of all Dieudonné complexes has a left adjoint M → Sat(M), which can be described in several convenient ways. If M is a Dieudonné complex, let M[F −1 ] := lim(M, F ), − → i.e., the localization of M by the endomorphism F of the graded abelian group underlying M. The differential d does not extend to M[F −1 ], but it does extend to M[F −1 , p−1 ], with dF −n (ω) := p−n F −n (ω).

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Proposition 1.6 If M is a Dieudonné complex, let Mtf be the quotient of M by its p-torsion submodule. Then Mtf is a Dieudonné complex, and Sat(M) = Sat(Mtf ). When M is p-torsion free, Sat(M) can be described in the following ways:  α α 1. Sat(M) = lim M - η(M) - η2 (M) · · · . − → 2. Sat(M) = {ω ∈ M[F −1 ] : dF n ω ∈ pn M for some (equivalently for all) n ' 0}. 3. Sat(M) := {ω ∈ M[F −1 ] : dω ∈ M[F −1 ]}. 4. Sat(M) is the largest graded subgroup of M[F −1 ] closed under d. Proof The proofs of (1), (2), and (3) can be found in [4, 2.3.1, 2.3.3, 2.3.4]. For (4), it will suffice to show that any graded subgroup N of M[F −1 ] which is stable under d is contained in Sat(M) (as defined by condition (2)). Indeed, if N is such a complex and x ∈ N, then also dx ∈ N and consequently there exists some n > 0 such that F n x and F n dx both belong to M. Then dF n x = pn F n dx ∈ pn M, and hence x ∈ Sat(M).  If M is a saturated Dieudonné complex and if x ∈ pM, then dx ∈ pM, so there is a unique x such that F x = x. Thus, there is a unique additive homomorphism V : M → M such that F V = p; moreover V F = F V = p and V d = pdV . One checks immediately that F il r M := dV r M + V r M is stable under d and that V F il r M ⊆ F il r+1 M and F il r M ⊆ F il r−1 M. Let Wr M := M/F il r M and WM := lim Wr M, which inherits a natural structure of a ← − Dieudonné complex. Here is a summary of the key results about saturated Dieudonné complexes. Proposition 1.7 (Higher Cartier Isomorphisms) Let (M · , d, F ) be a saturated Dieudonné complex. 1. For each r, the map F r induces an isomorphism of complexes:  r  F : Wr M · , d → H · (M · /p r M · ), β , where β is the Bockstein differential. 2. For each r, the natural projection induces a quasi-isomorphism πr : (M · /pr M · , d) → (Wr M · , d). 3. For each r, the composition of F and H · (πr ) defines an isomorphism of graded abelian groups: r

ψr : Wr M · → H · (Wr M · , d).

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Proof See [4, 2.7.2,2.7.3] and [12, 5.1.3] for the statements about the underlying abelian groups, as well as Theorem 7.5 for a refined version of statement (2). The compatibility of the map in (1) with the differentials is the commutativity of the diagram i

r

M /p M

F

i

r

H i (M · /pr M · )

d M

i+1

β r

/p M

i+1

F

r

H i+1 (M · /pr M · ).

Let us explain the straightforward calculation (up to sign). This Bockstein differential is the boundary map of the long exact sequence coming from the short exact sequence 0

- M · /pr M ·

r] [p-

M · /p2r M ·

- M · /pr M ·

- 0.

Thus, if y ∈ M i lifts the class y of an element of H i (M · /pr M · ), then dy is divisible by pr and the Bockstein of y is given by the class of p−r dy. If x ∈ M · and y = F r x, then p−r dy = p−r dF r x = F r dx, as required.  The following key result is proved in [4, 2.4.2] when n = 1 under the stronger hypothesis that (M · , d, F ) be of Cartier type. That proof is easily adopted to cover this more general statement. Theorem 1.8 If (M · , d, F ) is a Dieudonné complex of quasi-Cartier type, then for every n > 0, the natural maps (M · /pn M · , d) → (Sat(M · , d)/pn Sat(M · , d) → (Wn Sat(M · , d)) and the maps lim(M · /pn M · , d) → lim(Sat(M · )/pn Sat(M · ), d) → lim(Wn Sat(M · ), d) ← − ← − ← − are quasi-isomorphisms1 . If (M · , d, F ) is of Cartier type, then in addition the natural map of complexes: (M · /pM · , d)

∼-

W1 Sat(M · , d).

is an isomorphism. 1 If we are working on a topological space, we should assume that each point admits a neighborhood basis on which the transition maps are surjective on global sections. This will be the case in our applications, because, as we shall see, our sheaves will be quasi-coherent.

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Proof As we observed after Definition 1.4, if (M · , d, F ) is of quasi-Cartier type, then the map α in diagram (1.4) is a quasi-isomorphism. As explained in (3) of Proposition 1.3, it then follows that η(α) induces a quasi-isomorphism η(M · )/pη(M · ) → η2 M · /pη2 (M · ), and so on for every ηk . Passing to the limit, we conclude that the map (M · /pM · , d)) → (Sat(M · )/p Sat(M · ), d) is a quasi-isomorphism. One deduces the analogous statement with pn in place of p using induction on n, since M · and Sat(M · ) are p-torsion free. Statement (2) of Proposition 1.7 tells us that the map (Sat(M · , d)/pn Sat(M · , d) → Wn Sat(M · , d) are quasi-isomorphism. The same holds in the limit because the transition maps are surjective. This concludes the proof of the first statement of the theorem. If (M · , d, F ) is of Cartier type, then it is also of quasi-Cartier type, so the first statement holds again. Furthermore, the natural map (M · , d) → Sat(M · , d) → W1 (Sat M · , d) factors through M · /pM · and so induces the map in the second statement. To see that map is an isomorphism, consider the commutative diagram of graded groups: M · /pM · γ H · (M · /pM · )

W1 Sat(M · ) ψ1 H · (Sat(M · )/p Sat(M · )).

The left vertical arrow (induced by F ) is an isomorphism because (M · , d, F ) is of Cartier type, the bottom horizontal arrow is an isomorphism because (M · , d, F ) is of quasi-Cartier type, and the right vertical arrow is an isomorphism by Proposition 1.7. We conclude that the top horizontal arrow is also an isomorphism, as desired.  Corollary 1.9 Let θ : (M · , d, F ) → (M · , d, F ) be a morphism of p-torsion free Dieudonné complexes. Suppose that θ is a quasi-isomorphism and that either M · or M · is quasi-saturated (resp. of quasi-Cartier type). Then both complexes are quasi-saturated (resp. of quasi-Carier type), and the maps

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WSat (M · , d, F ) → WSat (M · , d, F ) and Wn Sat (M · , d, F ) → Wn Sat (M · , d, F )

are isomorphisms for all n ≥ 0. Proof We have a commutative diagram: (M · , d)

θ

(M · , d) α

α η(M · , d)

η(θ)

η(M · , d).

Since θ is a quasi-isomorphism, so is η(θ ). If either complex is quasi-saturated, then α or α is a quasi-isomorphism, and hence both are. Thus both complexes are quasi-saturated, hence of quasi-Cartier type. We also have a commutative diagram: (M · /pM · , d) γ (H · (M · /pM · ), d1 )

(M · /pM · , d) γ (H · (M · /pM · d), d1 ).

Since θ is a quasi-isomorphism and the complexes M · and M · are p-torsion free, the top arrow is a quasi-isomorphism, which implies that the bottom arrow is actually an isomorphism. If either complex is of quasi-Cartier type, then one of the two vertical arrows is a quasi-isomorphism, and it follows that both must be, i.e., that both complexes are of quasi-Cartier type. Under either hypothesis, both complexes are of quasi-Cartier type, so Theorem 1.8 implies that the vertical maps in the following commutative diagram are quasi-isomorphisms. (M · /pn M · , d)

(M · /pn M · , d)

(Wn SatM · , d)

(Wn SatM · , d).

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Since θ is a quasi-isomorphism and M · and M · are p-torsion free, the top horizontal arrow is also a quasi-isomorphism, and hence so is the bottom arrow. Then it follows  from (3) of Proposition 1.7 that the bottom maps are actually isomorphisms. Definition 1.10 A Dieudonné complex (M · , d, F ) or algebra is strict if it is saturated and the natural map M → WM · := lim Wn M is an isomorphism. ← − · It is proved in [4, 2.7.6] that if (M , d, F ) is saturated, then in fact (WM · , d, F ) is strict. We should also point out that statement (3) of Proposition 1.7 implies that any quasi-isomorphism of strict Dieudonné complexes is in fact an isomorphism. If M · is a saturated Dieudonné complex, then pM · ⊆ V M · ⊆ F il 1 M · , and hence W1 M · is annihilated by p. If A· is a saturated Dieudonné algebra, then F il 1 A· is an ideal of A· and hence W1 A0 is an Fp -algebra. Moreover, if M· is a saturated Dieudonné complex or algebra, then the same is true of WM· , which is in fact strict [4, 2.7.6]. We can now state the new construction (and version) of the de Rham–Witt complex described in [4, 4.1.5]. Theorem 1.11 ([4]) The functor A· → W A0 , from the category of strict 1

Dieudonné algebras to the category of Fp -algebras, admits a left adjoint R → WΩR· . If R is an Fp -algebra, WΩR· is called the saturated or strict de Rham–Witt complex of R. There are several constructions of WΩ · presented in [4]. Here we describe the R

one most useful for our present purposes. For another, more general construction, see Theorem 5.2. Proposition 1.12 ([4, 4.2.3]) Let R be an Fp -algebra which is the reduction modulo p of a p-torsion free ring R˜ admitting an endomorphism φ lifting the absolute Frobenius endomorphism of R. 1. The de Rham complex Ω · and its p-adic completion Ωˆ · admit a canonical R˜

structure F of a Dieudonné algebra. 2. The map

R˜

(WΩR· , d, F ) → W Sat(ΩR·˜ , d, F ) ˜ R˜ → W1 Sat(Ω · )0 is an isomorphism, and adjoint to the map R = R/p R˜ similarly for Ωˆ · .2 R˜

2 Actually

cases.

only the statement for the completion appears in [4], but the proof is the same in both

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We next explain a few technical results which will help us with the computation of the saturation of Dieudonné complexes in some cases. · Proposition 1.13 Let θ : M → M · be a morphism of Dieudonné complexes on a topological space, and suppose that M · is p-torsion free. Then the following conditions are equivalent. 1. The action of F on the kernel and cokernel of θ is locally nilpotent. · 2. The induced map M [F −1 ] → M · [F −1 ] is an isomorphism. · 3. The induced map Sat(θ ) : Sat(M ) → Sat(M · ) is an isomorphism. Proof Note first that if F is an endomorphism of an abelian group or sheaf M, then the kernel of the map M → M[F −1 ] consists of the elements (or sections) of M which are annihilated by some power of F . In particular, M[F −1 ] vanishes if and only if F is locally nilpotent on M. Now let A· (resp. B · ) be the kernel (resp. cokernel) of θ , so that we have an exact sequence: · 0 → A· → M → M · → B · → 0. Since the localization functor is exact, the sequence · 0 → A· [F −1 ] → M [F −1 ] → M · [F −1 ] → B · [F −1 ] → 0 is again exact. Thus the localization of θ is an isomorphism if and only if F is locally nilpotent on A· and on B · . This proves that (1) and (2) are equivalent. Suppose that (1) and (2) are verified. Since M · is p-torsion free, it follows that · the p-torsion of M is contained in the kernel of θ , hence that F is locally nilpotent · on this p-torsion, and hence that the map from M to its p-torsion free quotient induces an isomorphism after localization by F . Since this map also induces an · isomorphism on saturations, we may replace M by this quotient, and thus we may · and shall assume that M is p-torsion free. We now have a commutative diagram in which the horizontal arrows are isomorphisms: · M [F −1 , p−1 ]

M · [F −1 , p−1 ]

· M [F −1 ]

M · [F −1 ].

The objects in the top row are endowed with the operator d, and by the characterization of the saturation functor Sat in (3) of Proposition 1.6, we see that Sat(θ ) is an isomorphism also.

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Finally, observe that for any p-torsion free M, the maps M → Sat(M) → M[F −1 ] induce isomorphisms after localization by F , and so (3) implies (2).  Corollary 1.14 Suppose we are given a morphism of short exact sequences of ptorsion free Dieudonné complexes on a noetherian topological space 0

A

B

C

0

0

A

B

C

0,

and suppose that any two of the vertical arrows induce an isomorphism after saturation. Then the same is true of the third vertical arrow. Proof The two given sequences remain exact after localization by F . If any two of the vertical arrows become isomorphisms on saturation, then Proposition 1.13 implies that they remain isomorphisms after localization, and then it follows that the third is also an isomorphism after localization. Applying the proposition again, we see that that arrow also becomes an isomorphism after saturation.  Proposition 1.15 If 0 → A· → B · → C · is an exact sequence of p-torsion free Dieudonné complexes on a noetherian topological space, then 0 → Sat(A· ) → Sat(B · ) → Sat(C · ) is also exact. Proof Since C · is p-torsion free, pn B · ∩ A· = pn A· for all n. Thus the p-adic filtration of B · induces the p-adic filtration A· . Suppose a ∈ An and da ∈ pn+1 B n+1 . Then da maps to zero in C n+1 and hence da ∈ An+1 ∩ pn+1 B n = pn+1 An+1 . It follows that η(B · ) ∩ A· = η(A· ), and hence that the sequence 0 → η(A· ) → η(B · ) → η(C · ) is exact. The same is true with ηn in place of η, by induction. Taking the direct limit, we find the conclusion.  Remark 1.16 Let DC be the category of Dieudonné complexes and DCsat the full subcategory of saturated ones. It follows from Proposition 1.15 that the kernel of a homomorphism θ of saturated Dieudonné complexes is again saturated, but this is not true for its cokernel. However, since Sat is left adjoint to the inclusion functor, it is true that Sat(Cok(θ )) is the cokernel of θ in the category DCsat . Thus DCsat admits kernels and cokernels, although it is not abelian. The composite functor inc ◦ Sat : DC → DCsat → DC is left exact, but not right exact. For an example, consider the Dieudonné complex B · , whose component in degree 0 is freely generated by elements bn in degree zero and by bn , an in degree 1, for n ∈ N, where F (xn ) = xn+1 for x = a, b, b , and where dan = dbn = 0 and dbn = pn+1 bn + pn an . Then {an : n ∈ N} forms a sub-Dieudonné complex A of B, and the quotient C of B by the subcomplex A is freely generated in degree 0 by the images cn of bn and in degree 1 by the images cn of bn . The localization of these Dieudonné complexes by F is exhibited by writing the same formulas with n ∈ Z.

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One sees then that c−1 ∈ Sat(C) but is not in the image of Sat(B). On the other hand, it is true that the functor Sat ⊗Q preserves surjectivity. Indeed, if B · → C · is surjective, then so is its localization by F , and Sat(C · ) ⊗ Q = C · [F −1 ].

2 Dieudonné Complexes of Monoid Algebras Recall that a commutative monoid Q is said to be fine if it is finitely generated and integral and that Q is said to be toric if it fine and saturated and in addition the associated abelian group Qgp is free. If Q is a commutative integral monoid and R is a commutative ring, we denote by R[Q] the monoid algebra of Q over R. Thus R[Q] is the free R-module with basis β : Q → R[Q] : q → eq , and β is a homomorphism from Q to the multiplicative monoid underlying R[Q]. Endow R[Q] with the natural Qgp -grading in which eq has degree q. For each q ∈ Q, the degree q component of R[Q] is a free R-module of rank one, with · , d) basis eq , and is zero if q ∈ Qgp \ Q. Then the de Rham complexes (ΩR[Q] · gp and (ΩR[Qgp ] , d) inherit a natural Q -grading for which the differential preserves degrees. Exterior multiplication on these de Rham complexes gives them the structure of a strictly commutative differential graded algebra. To avoid confusing the two different gradings, we will say “Qgp -grading” for the grading induced by the Qgp -grading of the ring R[Q] when necessary. We fix a prime number p and assume that R is endowed with an endomorphism σ such that σ (r) ≡ r p (mod p) for every r ∈ R. For example, we could take R = Z and σ = idZ , or R could be the Witt ring of a perfect field k and σ its Frobenius endomorphism. Proposition 2.1 Let Q be an integral monoid and let R be a p-torsion free commutative ring endowed with an endomorphism σ as above. Then there is a unique σ -homomorphism of R-algebras: · · F : ΩR[Q]/R → ΩR[Q]/R with the following properties. q 1. For each q ∈ Q, we have F (eq ) = epq and F (deq ) = e(p−1)q de  . p 2. For each f ∈ R[Q], we have F (df ) = f p−1 df + d F (f p)−f .

In degree zero, F is the homomorphism: φ:



rq eq →



σ (rq )epq .

· The triple (ΩR[Q]/R , d, F ) is a Dieudonné algebra.

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Proof The map Q → Q sending q to pq is a monoid homomorphism. Thus there is a unique homomorphism of R-algebras R[Q] → R[Q] sending eq to epq for every q ∈ Q. The composition of this with the homomorphism   homomorphism R[Q] → R[Q] sending rq eq to σ (rq )eq is the homomorphism φ shown above; furthermore φ is the unique σ -homomorphism such that F (eq ) = epq for all q. Moreover, φ(f ) is congruent to f p mod p for every f ∈ R[Q]. Since R[Q] is p-torsion free, we can conclude from [4, 3.2.1] that there is a unique endomorphism · F of the ring ΩR[Q]/R such that F (f ) = φ(f ) for all f ∈ R[Q] and such that condition (2) above holds. If f = eq for some q ∈ Q, then F (f ) = f p , and so formula (2) reduces to the second formula in (1). Formula (2) implies that dF (f ) = pF d(f ) for every f ∈ R[Q] and · consequently that dF = pF d on all of ΩR[Q]/R . In fact, [4, 3.2.1] shows that ·  (ΩR[Q] , d, F ) is a Dieudonné algebra. · The complex ΩR[Q]/R seems hard to compute in general. We shall find it useful to consider some variants, which appear in various guises in the literature. First recall [20, V§2.2] that the map R[Q] → R[Q] ⊗ Qgp : eq → eq ⊗ q is a derivation and therefore induces a homomorphism of Q-graded R[Q]-modules 1 ΩR[Q]/R → R[Q] ⊗ Qgp .

Here the grading on the right is inherited from the grading on R[Q]; the elements of Qgp are viewed in degree zero. The above map induces a homomorphism of differential graded algebras: · ΩR[Q]/R → R[Q] ⊗ Λ· Qgp ,

(2.1)

where the differential on the right is wedge product with q in degree q: d(eq ⊗ ω) := eq ⊗ q ∧ ω

(2.2)

Define: F : R[Q] ⊗ Λi Qgp → R[Q] ⊗ Λi Qgp by  q

aq eq ⊗ ω →

 q

σ (aq )epq ⊗ ω.

(2.3)

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Note that for q ∈ Q and ω ∈ Λi Qgp , dF (eq ⊗ ω) = = = =

d(epq ⊗ ω) epq ⊗ (pq ∧ ω) pepq ⊗ (q ∧ ω) pF d(eq ⊗ ω).

Thus d and F endow R[Q] ⊗ Λ· R[Q] with the structure of a Dieudonné complex. Since F is compatible with multiplication and induces the Frobenius endomorphism of R/pR[Q], this Dieudonné complex is in fact a Dieudonné algebra. The following proposition justifies Definition 2.3. Proposition 2.2 The homomorphism (2.1) · ΩR[Q]/R → R[Q] ⊗ Λ· Qgp is a homomorphism of Dieudonné algebras and is an isomorphism if Q is an abelian group. Proof If q ∈ Q, statement (1) of Proposition 2.1 says that F (deq ) = e(p−1)q deq , which the homomorphism (2.1) takes to e(p−1)q eq ⊗ q = epq ⊗ q. On the other · hand, (2.1) maps deq to eq ⊗ q, which (2.3) takes to epq ⊗ q. Since F on ΩR[Q]/R and on R[Q] ⊗ Λ· Q are algebra homomorphisms over σ , it follows that (2.1) is compatible with F , hence is a homomorphism of Dieudonné algebras. If Q is an abelian group, it is well known that this homomorphism is an isomorphism of differential graded algebras; see for example [20, IV,1.1.5].  If Q is an integral monoid, the complex · gp R[Q] ⊗ Λ· Qgp ⊆ R[Qgp ] ⊗ Λ· Qgp ∼ = ΩR[Q ]/R corresponds to the so-called “logarithmic differentials” [20, IV,§2.2]. It has several variants, many of which have appeared in various forms in the literature [6, 7, 14, 20]. However, see the Appendix for some subtle technicalities concerning these constructions. Definition 2.3 Let Q be an integral commutative monoid, let R be a ring, and let G be a face of Q. For q ∈ Q, let #G, q$ denote the face of Q generated by G and q and let #q$ := #Q∗ , q$.  i i i gp 1. ΩR[Q]/R (log) := q∈Q ΩR[Q gp ]/R,q = R[Q] ⊗ Λ Q . i 2. Ω iR[Q]/R ⊆ ΩR[Q]/R (log) is the Qgp -graded submodule whose component in i degree q is R ⊗ Λ #q$gp if q ∈ Q and is zero if q ∈ Q. i 3. Ω iR[Q]/R (G) ⊆ ΩR[Q]/R (log) is the Qgp -graded submodule whose component in degree q ∈ Q is R ⊗ Λi #G, q$gp .

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These submodules are all stable under d, F , and exterior multiplication, and so · gp , d, F ). Note that Ω · ∗ define sub-Dieudonné algebras of (ΩR[Q ]/R R[Q]/R (Q ) = · · · Ω R[Q]/R and that Ω R[Q]/R (Q) = ΩR[Q]/R (log). Furthermore, formation of these complexes commutes with arbitrary base change R → R , as follows from the construction. In particular, they are defined over Z[Q]. If Q is fine, this ring is noetherian and Z[Q]⊗Λi Qgp is a noetherian Z[Q]-module, and it follows that each of the modules defined above is finitely generated over Z[Q]. Since their formation is compatible with base change, this conclusion holds for every R. Proposition 2.4 If Q is an integral monoid, there is a functorial commutative diagram of Dieudonné algebras: · (ΩR[Q]/R , d, F )

θ

· (Ω R[Q]/R , d, F )

· gp , d, F ) (ΩR[Q ]/R

∼

· gp , d, F ). (Ω R[Q ]/R

Proof The left vertical arrow comes from the functoriality of the construction in Proposition 2.1 and the natural map Q → Qgp , the right vertical arrow exists from the definitions, and the bottom horizontal isomorphism comes from Proposition 2.2. i → Ω iR[Qgp ]/R To see the existence of θ , it suffices to check that each map ΩR[Q]/R i factors through Ω R[Q]/R , and it suffices to do this when i = 1. This is clear: for 1 1 q  each q ∈ Q, the image of deq in ΩR[Q gp ]/R is e ⊗ q, which lies in Ω R[Q]/R . The following proposition reveals an important advantage of the complex Ω ·R[Q]/R . It is proved in [20, V, 2.3.17], but we repeat the proof here for the reader’s convenience. It is modeled on Kato’s proof in [14] of the log Cartier isomorphism, · which implies that (ΩR[Q]/R (log), d, F ) is of Cartier type. We note also that Blickle has also established a version of the Cartier isomorphism for “Danilov differentials” [5], but a technical subtlety prevents it from applying here (see the Appendix). Proposition 3.3 gives a generalization to the idealized case. Proposition 2.5 If R/pR is perfect and Q is toric, the Dieudonné complex (Ω ·Q/R , F, d) is of Cartier type. Proof We must prove that for every j , the homomorphism (1.2) j γ : Ω R[Q]/R ⊗ Fp → H j (Ω ·R[Q]/R ⊗ Fp , d)

induced by F is an isomorphism. As we have seen, formation of these complexes commutes with base change, so it suffices to treat the case when R is itself a perfect ring of characteristic p. Since the endomorphism φ of R is an isomorphism, we

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are reduced by base change to the case in which R = Fp , in which case σ = id. Note that F maps the component of degree q to the component of degree pq. Using formula (2.3), we can identify the map γ as the map ⊕q Fp ⊗ Λj #q$gp → ⊕q H j (Fp ⊗ Λ· #q $gp , d) induced by the identity map on #q$ = #pq$. Formula (2.2) identifies the differential

Ω jFp [Q]/Fp ,q

d

Ω j+1 Fp [Q]/Fp ,q

∼ =

∼ =

Fp ⊗ Λj q

gp

Fp ⊗ Λj+1 q

gp

as wedge product by q , which vanishes if q = pq, so γ is indeed an isomorphism for every q. It remains only to show that H j (Ω ·Fp [Q]/Fp ,q ) vanishes if q ∈ pQ. Since Q is saturated, in fact such a q does not belong to pQgp , and hence its image x in Fp ⊗ Qgp is not zero and belongs to a basis of the vector space Fp ⊗ #q $gp . As is well known, it follows that the complex (Fp ⊗ Λ· #q $gp , ∧x) 

is acyclic.

Remark 2.6 The saturation hypothesis on Q is not superfluous. For example, let Q be the monoid given by generators a, b with relation a n = bn . Then Qgp ∼ = Z ⊕ Z/nZ, and the homomorphism sending a to (1, 0) and b to (1, 1) identifies Q with {(x, t) ∈ Z ⊕ Z/nZ : x ≥ min(t ∩ N)}. Then Ω ·Q/k need not be of Cartier type even if k is of characteristic p relatively prime to n. The Cartier condition will fail gp unless whenever q ∈ Q \ pQ, the image of q in Qgp /pQgp ∼ = Qgp /(pQgp + Qtors ) is not zero. For example, if n = 3 and p = 5, let q = (5, 1) ∈ Q. Then q = 5(1, 2) = (5, 1) ∈ Q, while (1, 2) ∈ Qgp \ Q. The following theorem, which is our main computational tool, shows how the operation of saturation “cleans” the pathologies of the de Rham complex of a toroidal monoid. This result is enough to show that the saturated de Rham–Witt complex of schemes with toric singularities in the sense of [15] is well-behaved. We shall make this explicit in a more general context in Theorem 3.4 in the next section. Theorem 2.7 If Q is a toric monoid, the map · θ : (ΩR[Q]/R , d, F ) → (Ω ·R[Q]/R , d, F ) induces isomorphisms:

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· Sat(ΩR[Q]/R , d, F ) → Sat(Ω ·R[Q]/R , d, F ) and · · Sat(Ωˆ R[Q]/R , d, F ) → Sat(Ωˆ R[Q]/R , d, F ), where Mˆ means the p-adic completion of M. Proof It does not seem to be known whether or not the terms of the complex · ΩR[Q]/R contain any p-torsion, but it is clear that Ω ·R[Q]/R does not. In any case, by Proposition 1.13, it will suffice to show that the action of F on the kernel and cokernel of θ is nilpotent. To handle the cokernel, we will use the following explicit description of the image of θ . · · Lemma 2.8 Let Ω R[Q]/R be the image of ΩR[Q]/R in Ω ·[Q]/R . Then with the Qgp grading described in Definition 2.3, for each q ∈ Q, there is a natural identification i Ω R[Q]/R,q ∼ = R ⊗ Li,q ⊆ R ⊗ Λi #q$gp ∼ = Ω iR[Q]/R,q ,

where Li,q is the subgroup of Λi Qgp generated by {q1 ∧ · · · ∧ qi : q1 + · · · qi ≤ q}. 1 Proof Recall that ΩR[Q]/R is generated as an abelian group by elements of the i is generated by elements form f dg for f, g ∈ R[Q], and consequently ΩR[Q]/R  aq eq , we of the form f dg1 ∧ · · · ∧ dgi . Writing f and each gi as a sum i see that in fact ΩR[Q]/R is generated as an R-module by elements of the form ωq := eq0 deq1 ∧ · · · ∧ deqi , where q := (q0 , q1 , . . . , qi ). The degree of such i an ωq is q := q0 + q1 + · · · qi , and so every element of ΩR[Q]/R,q is a linear 1 1 combination of such elements. The map ΩR[Q]/R → Ω R[Q]/R ⊆ R[Q] ⊗ Qgp takes deq to eq ⊗ q and hence ωq to eq0 +q1 +···+qi ⊗ (q1 ∧ · · · ∧ qi ), which lies in i

Li,q . This shows that Ω R[Q]/R,q ⊆ Li,q . On the other hand, if q1 + · · · + qi ≤ q, then q0 := q − (q1 + · · · + qi ) ∈ Q, and we can let ω := eq0 deq1 ∧ · · · ∧ deqi , whose image in Ω iR[Q]/R,q is eq0 +q1 +···qi ⊗ q1 ∧ · · · ∧ qi .  · Proposition 2.9 If Q is a toric monoid, let (Ω R[Q]/R , d, F ) and (Ω ·R[Q]/R , d, F ) be the associated Dieudonné complexes as described in Lemma 2.8 and Definition 2.3. Then the following statements hold. 1. For each i ≥ 0 and each q ∈ Q, the map i

Ω R[Q],mq → Ω iR[Q]/R,mq is an isomorphism for m sufficiently large. 2. There is an n > 0 such that i

F n Ω iR[Q]/R ⊆ Ω R[Q]/R .

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· 3. The action of F on the quotient Ω ·R[Q]/R /Ω R[Q]/R is nilpotent. Proof Note first that for q ∈ Q and m ∈ Z+ , #q$ = #mq$. Lemma 2.8 identifies i Ω R[Q]/R with Li,mq ⊆ Λi #q$gp . Every element of Λi #q$gp is a linear combination of elements of the form q1 ∧ · · · ∧ qi with each qi ∈ #q$, so each qi ≤ mi q for some mi ∈ N. But then q1 ∧ · · · ∧ qi ∈ Li,mq for m ≥ m1 + · · · + mi . Thus Λi #q$gp = ∪{Li,mq : m ∈ N}, and since Λi #q$gp is a noetherian abelian group, it follows that Li,mq = Λi #q$gp for m ' 0, proving statement (1). Suppose ω ∈ Ω iR[Q]/R has degree q. Then F n (ω) has degree pn q and so statement (1) i

implies that it belongs to Ω R[Q]/R for n ' 0. Any ω ∈ Ω iR[Q]/R can be written as a sum of homogenous elements, and it follows that F is locally nilpotent on i Ω iR[Q]/R /Ω R[Q]/R . As we have seen in the discussion after Definition 2.3, this R[Q]-module is finitely generated, and since F is φ-linear, its action is nilpotent, proving statements (2) and (3).  To finish the proof of Theorem 2.7, it will now suffice to prove the following proposition. · Proposition 2.10 The kernel A·R[Q]/R of the natural map ΩR[Q]/R → Ω ·R[Q]/R is · stable under F , and the action of F on AR[Q]/R is nilpotent. We shall need some preparatory lemmas. If Q is a monoid, we let Q+ denote its maximal ideal, i.e., Q+ = Q \ Q∗ . i i Lemma 2.11 If Q is sharp and i ≥ 1, then F ΩR[Q]/R ⊆ Q+ ΩR[Q]/R . i 1 = Λi (ΩR[Q]/R ) and F is compatible with exterior multipliProof Since ΩR[Q]/R cation, it suffices to prove this when i = 1. Since Q is sharp, R[Q] is generated by 1 R[Q+ ], and hence ΩR[Q]/R is generated as an R[Q]-module by {deq : q ∈ Q+ }. As we saw in Proposition 2.1, 1 F (deq ) = (eq )p−1 deq ∈ Q+ ΩR[Q]/R



· Lemma 2.12 If Q is any toric monoid, then F A·R[Q]/R ⊆ Q+ ΩR[Q]/R , and · · n + F AR[Q]/R ⊆ Q AR[Q]/R for n sufficiently large. Proof Since Q is saturated, we can write Q ∼ = Q ⊕ Q∗ , and we get a corresponding ∼ product decomposition R[Q] = R[Q] ⊗R R[Q∗ ]. If q ∈ Q, q ∗ ∈ Q∗ , and q = q + q ∗ , then #q$gp = #q$gp ⊕ Q∗ , so Ω 1R[Q]/R,q = R ⊗ #q$gp ∼ = R ⊗ #q$gp ⊕ R ⊗ #q ∗ $gp   1 1 1 ∼ ∼ ∗] Ω ⊕ R[Q] ⊗ . = Ω 1R[Q]/R,q ⊕ΩR[Q ∗ ]/R,q ∗ = R[Q] ⊗R[Q] Ω ∗ R[Q R[Q ] R[Q]/R q

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It follows that 1 Ω 1R[Q]/R ∼ = R[Q] ⊗R[Q] Ω 1R[Q]/R ⊕ R[Q] ⊗R[Q∗ ] ΩR[Q ∗ ]/R .

Similarly, 1 1 1 ∼ ΩR[Q]/R ⊕ R[Q] ⊗R[Q∗ ] ΩR[Q = R[Q] ⊗R[Q] ΩR[Q]/R ∗ ]/R



i ∼ ΩR[Q]/R =

a b (R[Q] ⊗R[Q] ΩR[Q]/R ) ⊗R (R[Q] ⊗R[Q∗ ] ⊗ΩR[Q ∗ ]/R )

a+b=i

Ω iR[Q]/R ∼ =



b (R[Q] ⊗R[Q] Ω aR[Q]/R ) ⊗R (R[Q] ⊗R[Q∗ ] ΩR[Q ∗ ]/R )

a+b=i b b Since ΩR[Q ∗ ]/R = Ω R[Q∗ ]/R is free, since R[Q] → R[Q] is flat, and since A0R[Q]/R = 0, we conclude that

AiR[Q]/R ∼ =



b R[Q] ⊗R[Q] AaR[Q]/R ⊗R R[Q] ⊗R[Q∗ ] ΩR[Q ∗ ]/R .

a+b=i,a≥1

The splitting Q → Q∗ , although not canonical, is compatible with F . It now follows i from Lemma 2.11 that F AiR[Q]/R ⊆ Q+ ΩR[Q]/R . Then i i F n+1 AiR[Q]/R ⊆ F n (Q+ ΩR[Q]/R ) ⊆ (Q+ )p ΩR[Q]/R . n

Since AiR[Q]/R is stable under F , in fact i F n+1 AiR[Q]/R ⊆ (Q+ )p ΩR[Q]/R ∩ AiR[Q]/R . n

If R is noetherian, the Artin-Rees lemma implies that there exists an n0 such that i (Q+ )m+n0 ΩR[Q]/R ∩ AiR[Q]/R ⊆ (Q+ )m AiR[Q]/R for all m > 0. In particular this holds when R is the localization Z(p) of Z at p. Since our general R is by assumption flat over Z(p) , formation of AiR[Q]/R and of these intersections commutes with base extension Z(p) → R, and hence the same containment holds in general. We conclude that that F n A·R[Q]/R ⊆ Q+ A·R[Q]/R for n sufficiently large.  If G is a face of Q, we let QG denote the localization of Q by G, which is toric if Q is. The next lemma shows that formation of the Dieudonné complexes we are considering is compatible with localization. Lemma 2.13 If G is a face of Q, the natural maps of Dieudonné complexes · · (ΩR[Q]/R )G → ΩR[Q , G ]/R

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(Ω ·R[Q]/R )G → Ω ·R[QG ]/R , · (A·R[Q]/R )G → AR[Q G ]/R are isomorphisms. ∼ R[Q]G , and since formation of de Rham complexes Proof It is clear that R[QG ] = is compatible with localization, the first of the above maps is an isomorphism. In general, if M is any Qgp -graded R[Q]-module and G is a face of M, then MG is again Qgp -graded, and if x ∈ Qgp , there is a natural identification (MG )x ∼ = lim{Mx → Mgx : g ∈ G} [20, I,3.2.8]. Moreover, for each q ∈ Q, #q, G$gp is the − → group envelope of the face of QG generated by q, and is also lim{#q$gp → #gq$gp : − → g ∈ G}. Taking exterior powers, we deduce the second isomorphism, and the third  then follows by a diagram chase. Proof of Proposition 2.10 We proceed by induction on the dimension of Q. If this dimension is zero, then Q∗ = Q, so A·R[Q]/R = 0 and there is nothing to prove. If G is a nontrivial face of Q, then the dimension of Q/G is less than that of Q, and · so the induction hypothesis implies that the action of F on AR[Q is nilpotent: G ]/R · n there is an n such that F AR[QG ]/R = 0. Since Q has only finitely many faces, we can choose n independent of G. By Lemma 2.13, we conclude that 

F n A·R[Q]/R

 G

    = F n (A·R[Q]/R )G = F n A·R[QG ]/R

vanishes, for every nontrivial face of G. Each nontrivial face G of Q defines an open subset Spec R[Q], and the union of these open sets is the complement of the closed subset defined by the ideal Q+ . (This is just because every element of Q+ generates a nontrivial face of Q.) Thus F n A·R[Q]/R is supported at the ideal R[Q+ ]: every element of F n A·R[Q]/R is annihilated by some power of Q+ . Since this R[Q]-module is finitely generated, it is annihilated by (Q+ )n for some n > 0. On the other hand, by Lemma 2.12, there · m is an m such that F AR[Q]/R ⊆ Q+ A·R[Q]/R . Then for every i > 0,       F mi+n A·R[Q]/R ⊆ F n (Q+ )i A·R[Q]/R ⊆ (Q+ )i F n A·R[Q]/R , which vanishes for i large enough.



We have now proved that the kernel and cokernel of the map · θ : (ΩR[Q]/R ) → Ω ·R[Q]/R are annihilated by some power of F . The same holds after p-adic completion. Thus Proposition 1.13 implies that Sat(θ ) is an isomorphism, and similarly for the padically completed complexes. 

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3 Idealized Monoid Algebras Theorem 2.7 can be extended to apply to some reducible toroidal schemes. Continuing with the hypotheses of Proposition 2.1, let K be an ideal of Q, let R[K] ⊆ R[Q] denote the free R-module spanned by K, which forms an ideal of R[Q], and let R[Q, K] denote the quotient R[Q]/R[K]. Our aim is to compare the saturation of the de Rham complex of Spec R[Q, K] with a corresponding quotient of Ω ·R[Q]/R . Let Ω ·R[K]/R := ⊕{Ω ·R[Q]/R,k : k ∈ K}, which forms a differential ideal in · Ω R[Q]/R , and let Ω ·R[Q,K]/R denote the quotient complex, which can be identified with ⊕{Ω ·R[Q]/R,q : q ∈ Q \ K}. Finally, let   · · · , ΩR[K]/R := Ker ΩR[Q]/R → ΩR[Q,K]/R · a differential ideal in ΩR[Q]/R . Theorem 3.1 The obvious maps fit into a diagram of short exact sequences of Dieudonné complexes: 0

· ΩR[K]/R

· ΩR[Q]/R

φ 0

· ΩR[Q,K]/R

θ

Ω ·R[K]/R

0

ψ

· Ω R[Q]/R

Ω ·R[Q,K]/R

0.

The action of F on the kernel and cokernel of each of the vertical arrows is nilpotent. Consequently, the maps φ, θ , and ψ induce isomorphisms · Sat(ΩR[K]/R ) · ) Sat(Ω

∼∼-

Sat(Ω ·R[K]/R ) Sat(Ω · )

· Sat(ΩR[Q,K]/R )

∼-

Sat(Ω ·R[Q,K]/R ).

R[Q]/R

R[Q]/R

Proof We have already treated the arrow θ . To establish the existence of the arrows φ and ψ, begin by observing that the kernel of the map 1 1 ΩR[Q]/R → ΩR[Q,K] 1 is generated as an R[Q]-module by KΩR[Q]/R and dK. It follows that the · k differential ideal ΩR[K]/R is generated by {e : k ∈ K}. Then, since θ is a homomorphism of differential algebras and takes each ek to an element of the differential ideal Ω ·R[K]/R , it induces the arrows φ and ψ.

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We have already seen in Propositions 2.9 and 2.10 that F is nilpotent on the kernel and cokernel of θ , and it follows that it is also nilpotent on the kernel of φ and on the cokernel of ψ. The snake lemma yields an exact sequence: Ker(θ ) → Ker(ψ) → Cok(φ) → Cok(θ ). It will follow that F is nilpotent on the kernel of ψ and on the cokernel of φ provided we can prove that it is nilpotent on the cokernel of the map Ker(θ ) → Ker(ψ), which is isomorphic to the kernel of the map Cok(φ) → Cok(θ ). The following lemma will allow us to conclude the argument. · · Lemma 3.2 Let Ω R[K]/R denote the image of the map ΩR[K]/R → Ω ·R[K]/R . Then   · · F Ω R[Q]/R ∩ Ω ·R[K]/R ⊆ Ω R[K]/R . i

Proof Suppose that ω ∈ Ω R[Q]/R,k with k ∈ K. Lemma 2.8 shows that ω is a linear combination of elements of the form ek ⊗ q1 ∧ · · · ∧ qi , where q1 + · · · + qi ≤ k, and we may assume that ω itself has this form. Let q0 := k − (q1 + · · · + qi ). Then i k := q0 + (p − 1)k ∈ K, and so ek deq1 ∧ · · · ∧ deqi ∈ ΩR[K]/R . Then F (ω) = epk ⊗ q1 ∧ · · · ∧ qi = ek +q1 +···+qi ⊗ q1 ∧ · · · ∧ qi = φ(ek deq1 ∧ · · · ∧ deqi ), i

and so lies in Ω R[K]/R .



Now suppose that ω ∈ Ω iR[K]/R represents an element of the kernel of the i map Cok(φ) → Cok(θ ). Then its image in Ω iR[Q]/R lies in Ω R[Q]/R ∩ Ω ·R[K]/R . By Lemma 3.2, F (ω) lies in the image of φ and hence vanishes in Cok(φ). This concludes the proof that F is nilpotent on the kernel and cokernel of the vertical arrows in the diagram. The “consequence” then follows from Proposition 1.13.  We next show that, if K is a radical ideal in Q, then the Dieudonné complex (Ω ·R[Q,K]/R , d, F ) is of Cartier type. Proposition 3.3 Suppose that R is p-torsion free and that R/pR is perfect. If K is a radical ideal in a toric monoid Q, the Dieudonné complexes (Ω ·R[Q]/R , d, F ), (Ω ·R[K]/R , d, F ), and (Ω ·R[Q,K]/R , d, F ) are of Cartier type. Proof Recall that we have an exact sequence of Dieudonné complexes: · 0 → Ω ·R[K]/R → Ω ·R[Q]/R → ΩR[Q,K]/R → 0.

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The complexes are Qgp -graded, with the proviso that F multiplies degrees by p. Note that Ω ·R[K]/R is just the submodule of Ω ·R[Q]/R consisting of those terms whose degree q belongs to K. There is a natural splitting of this sequence obtained by identifying the quotient with the submodule of Ω ·R[Q]/R whose degree q does not belong to K. This submodule is evidently closed under d, and it is also closed under F because K is a radical ideal. The map γ (1.2) is then compatible with the direct sum decomposition, and since it is an isomorphism for Ω ·R[Q]/R , it must also be an isomorphism on each factor.  Theorem 3.4 Let K be a radical ideal in a toric monoid Q, let k be a perfect ring, and let R be a p-torsion free lift of k endowed with a lift σ of the Frobenius endomorphism of k. 1. The natural maps · · (WΩk[Q,K] , d, F ) → W Sat(ΩR[Q,K]/R , d, F ) → W Sat(Ω ·R[Q,K]/R , d, F ) and · · · (WΩk[Q,K] , d, F ) → W Sat(Ωˆ R[Q,K]/R , d, F ) → W Sat(Ωˆ R[Q,K]/R , d, F ) are isomorphisms. 2. The natural map · (Ωˆ R[Q,K] , d)

· ∼ - (W Sat(Ω · R[Q,K] ), d) = (WΩk[Q,K] , d)

is a quasi-isomorphism. · - W1 Ω · 3. The natural map Ωk[Q,K]/k k[Q,K]/k factors through an isomorphism     - W1 Ω · Ω ·k[Q,K]/k , d , d . k[Q,K]/k 4. For every n ∈ N, let Rn := R/pn R. Then there are isomorphisms of complexes:  · Wn Ωk[Q,K] ,d   H · (Ω · , d), β Rn [Q,]/Rn

  H · (Sat Ω ·Rn [Q,K]/Rn , d), β   - H · (Sat Ω · , d), β -

Rn [Q,K]/Rn

and a quasi-isomorphism   Ω ·Rn [Q,K]/Rn , d

-



 · Wn Ωk[Q,K]/k ,d .

Proof The ring R[Q, K] is a flat lift of k[Q] and φ lifts the absolute Frobenius endomorphism of k[Q, K], so statement (2) of Proposition 1.12 asserts that the · · natural map WΩk[Q,K] → W Sat(ΩR[Q,K]/R ) is an isomorphism of Dieudonné

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· algebras. Theorem 3.1 tells us that W Sat(ΩR[Q,K]/R ) → W Sat(Ω ·R[Q,K]/R ) is also an isomorphism, and similarly for the p-adic completions. This proves statement (1). By Proposition 3.3, (Ω ·R[Q,K]/R , d, F ) is of Cartier type, so Theorem 1.8 implies statements (2) and (3). Since (Ω ·R[Q,K]/R , d, F ) is of quasi-Cartier type, the maps (Ω ·Rn [Q,K]/Rn , d) → Sat(Ω ·Rn [Q,K]/Rn , d) are quasi-isomorphisms, and so statement (4) follows from Proposition 1.7.  The algebras R[Q, K] and the corresponding complexes Ω ·R[Q,K]/R and · have an appealing geometric interpretation. Recall first that if K is a WΩk[Q,K] prime ideal, its complement G is a face of Q, and the natural map R[G] → R[Q, K] is an isomorphism. In fact, it follows immediately from the definitions that Ω iR[G] =



Λi #g$gp = Ω iR[Q,K] ,

g∈G

so the map Ω ·R[G] → Ω ·R[Q,K] is also an isomorphism. More generally, every radical ideal K is the intersection of a finite number of primes, and Spec R[Q, K] is a union of the spectra of the monoid algebras of the · corresponding faces. We shall see that Ω ·R[Q,K]/R and WΩk[Q,K] satisfy descent with respect to the gluing of these faces. First note that if K1 and K2 are prime ideals of Q, then so is their union K12 , and R[K12 ] = R[K1 ] + R[K2 ] ⊆ R[Q]. Thus if Gi := Q \ Ki and G12 = G1 ∩ G2 , we have an exact sequence: 0 → R[Q, K] → R[G1 ] ⊕ R[G2 ] → R[G12 ] → 0. Indeed, if q ∈ Q, the degree q part of the sequence looks like: 0→0→0→0→0 0→Z→Z→0→0 0→Z→Z⊕Z→Z→0

if q ∈ K if q ∈ Gi \ G12 if q ∈ G12 .

In any of these cases, the sequence remains exact when tensored with Λi #q$gp , and consequently the sequence 0 → Ω iR[Q,K] → Ω iR[G1 ] ⊕ Ω iR[G2 ] → Ω iR[G12 ] → 0 is also exact. More generally, suppose that K = K1 ∩ · · · ∩ Km , where each Ki is a prime ideal with complementary face Gi . The normalization of R[Q, K] is the homomorphism

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˜ R[Q, K] → R[Q, K] := ⊕i R[Q, Ki ] ∼ = ⊕i R[Gi ]. For any multi-index I := (I0 , . . . In ), let KI := KI0 ∪· · ·∪KIn and GI := Q\KI = GI0 ∩ · · · GIn . Consider the cosimplicial ring: → R• [Q, K] := R0 [Q, K]→ →R1 [Q, K]→ →···

(3.1)

whose nth term is the n + 1-fold product ˜ ˜ ˜ Rn [Q, K] := R[Q, K] · · · ⊗R[Q,K] R[Q, K]   K] ⊗R[Q,K] R[Q, ∼ R[Q, KI ] ∼ R[GI ]. = = |I |=n+1

|I |=n+1

The face and degeneracy maps are the obvious ones, and in particular the map R[Q, K] → ⊕i R[Gi ] sends eq to ⊕i {eq ∈ R[Gi ] : q ∈ Gi }. The following theorem shows that the saturated de Rham–Witt complex behaves as expected for idealized monoid algebras. It generalizes a result of Illusie [12, §4.1]. Theorem 3.5 If K is a radical ideal of a toric monoid Q and i ∈ N, let C • (Ω iR[Q,K] , ∂) denote the cochain complex associated to the cosimplicial ring R• [Q, K] (3.1). Then the augmentation map Ω iR[Q,K]/R → C • (Ω iR[Q,K]/R , ∂) is a quasi-isomorphism. Similarly, the maps i i Wn Ωk[Q,K] → C • (Wn Ωk[Q,K] , ∂)

and i i WΩk[Q,K] → C • (WΩk[Q,K] , ∂)

are quasi-isomorphisms. Proof Let C −1 (Ω iR[Q,K]/R ) := Ω iR[Q,K]/R and let ∂ −1 : Ω iR[Q,K]/R → C 0 (Ω iR[Q,K]/R ) be the augmentation map. Our claim is that the augmented complex C˜ • (Ω iR[Q,K ) obtained by inserting the terms in degree −1 is acyclic. This complex is again Q-graded, where C˜ n (Ω iR[Q,K]/R,q ) =

 |I |=n+1

Ω iR[GI ]/R,q

 = {Λi #q$gp : |I | = n + 1 : q ∈ GI }, if n ≥ 0, and

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C˜ −1 (Ω iR[Q,K],q ) = Λi #q$gp . With the convention that G∅ = Q, this also holds for n = −1. For ω ∈ C˜ n (Ω iR[GI ],q ), the differential of this complex is given by: n+1  (∂ω)I := (−1)k ωk (I ) , k=0

where k is the kth face map. To show that the complex is acyclic, we construct homotopy operators as follows. For each face F of Q which does not meet K, there is some iF ∈ [1, m] such that F ⊆ GiF . If q ∈ Q \ K, then #q$ ∩ K = ∅, and we let iq := i#q$ . Thus q ∈ Giq , and iq = inq for every n > 0. If I := (I0 , . . . , In ), let sq (I ) := (iq , I0 , . . . , In ). Now if n ∈ N, define ρn,q : C˜ n (Ω iR[Q,K] )q → C˜ n−1 (Ω iR[Q,K] )q : (ρn,q (ω· ))I := ωsq (I ) . This makes sense because ωsq (I ) ∈ Λi #q$gp and q ∈ Gsq (I ) = Giq ∩ GI ⊆ GI . The following lemma shows that ρ is a homotopy operator with respect to the boundary operator ∂ and that it is compatible with the Dieudonné module structure of Ω ·R[Q,K]/R and of each Ω ·R[GI ]/R . Lemma 3.6 The following identities hold: 1. ∂ρ + ρ∂ = id : C˜ n (Ω iR[Q,K] ) → C˜ n (Ω iR[Q,K] ); 2. d∂ = ∂d : C˜ n (Ω i ) → C˜ n+1 (Ω i+1 ); R[Q,K]

R[Q,K]

3. dρ = ρd : C˜ n (Ω iR[Q,K] ) → C˜ n−1 (Ω i+1 R[Q,K] ); 4. Fρ = ρF : C˜ n (Ω iR[Q,K] ) → C˜ n−1 (Ω iR[Q,K] ).

Proof The proof of (1) is a straightforward and standard calculation we shall not repeat, and (2) is a consequence of the naturality of the constructions. For (3), observe that, for every I , we have: (ρn,q (dω· ))I = (ρn,q (q ∧ ω· ))I = q ∧ ωsq (I ) = dωsq (I ) = (dρn,q (ω))I . Statement (4) holds because iq = ipq for every q ∈ Q \ K.



Statement (1) of Lemma 3.6 implies the acyclicity of the augmented complex C˜ • (Ω iR[Q,K]/R ) and hence also statement (1) of Theorem 3.5. Statements (3) and (4) of the lemma imply that each ρn is fact a morphism of Dieudonné complexes and therefore extends to define homotopy operators C˜ n Sat(Ω ·R[Q,K] ) → C˜ n−1 Sat(Ω ·R[Q,K] )

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for all n. These homotopy operators necessarily commute with d, F , and V , and hence pass to the quotient complexes Wn Ω ·R[Q,K]/R . It follows that the maps i i Wn Ωk[Q,K] → C • (Wn Ωk[Q,K] , ∂)

are quasi-isomorphisms. Since the transition maps in these inverse systems are surjective, the same is true after taking the inverse limit. 

4 Ideally Toroidal Schemes Let k be a perfect field and let X/k be a k-scheme locally of finite type. We shall say that X/k is toroidal if étale locally on X, there exist a toric monoid Q and an étale map X → Spec(k[Q]). Note that such a scheme is necessarily normal. More generally, we shall say that X/k is ideally toroidal if, étale locally on X, there exist a toric monoid Q, an ideal K in Q, and an étale map X → Spec k[Q, K]. Our aim is to explain that the strict de Rham–Witt complex of such a scheme is well-behaved in various senses. For simplicity, we assume henceforth that our schemes are reduced; recall however that, in general, the saturated de Rham Witt complexes associated to a scheme and its reduced subscheme are the same. (This fact is a consequence of [4, 6.5.2] and also of the easier [4, 3.6.1].) Theorem 4.1 Let X/k be a reduced ideally toroidal scheme, locally of finite type over a perfect field k of characteristic p > 0. 1. The natural map OX → W1 ΩX0 is an isomorphism, and the sheaf W1 ΩXi is a coherent sheaf of OX -modules. It is torsion free if X is in fact toroidal. 2. More generally, for n > 0, there is a natural isomorphism Wn OX → Wn ΩX0 , j and the sheaf Wn ΩX is a coherent sheaf of Wn OX -modules. 3. Let X → X be the normalization mapping and let C • (Wn ΩXi ) denote the cochain complex associated to the simplicial scheme associated to the morphism X → X and the functor Wn Ω i . The natural map Wn ΩXi → C • (Wn ΩXi ) is a quasi-isomorphism. The analogous result also holds with W in place of Wn . Proof Since the sheaves W Ω · are compatible with étale localization [4, 5.3.5], n

X

these statements can be verified étale locally on X, and we may assume that X = Spec k[Q, K], where Q is a toric monoid and K is an ideal in Q. Then statement (1) follows from (3) of Theorem 3.4. (The k[Q]-modules in the complex Ω ·k[Q] are evidently torsion free.)

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Lemma 4.2 Let K be a radical ideal in a toric monoid Q and let k be a field. Then k[Q, K] is seminormal. Proof Let A := k[Q, K]. According to Swan’s characterization of semi-normality, it is enough to prove that if (x, y) is a pair of elements of A satisfying x 2 = y 3 , then there exists a t ∈ A such that x = t 3 and y = t 2 . Since K is reduced, its complement is a union of faces G1 , . . . , Gn of Q, and the normalization A of A can be identified with the direct product of the monoid algebras Ai := k[Gi ]. Each of these is a normal integral domain, and hence for each i there is a ti such that xi = ti3 and yi = ti2 , where (xi , yi ) is the image of (x, y) in Ai . The tensor product A ⊗A A can be identified with the direct product of the rings Ai,j := Ai ⊗A Aj ∼ = k[Gi ∩Gj ], each of which is also an integral domain. It follows easily that the images of ti and tj in Ai,j agree, hence, by the descent property proved in Theorem 3.5, that the element t· of A descends to A. It follows from Lemma 4.2 and [4, 3.6.2 and 6.5.2] that the map Wn OX → Wn ΩX0 is an isomorphism. Statement (2) then follows; see [12, 8.8(c)]. Statement (3) follows from Theorem 3.5.  Proposition 4.3 If k is perfect and X/k is a proper k-scheme with ideally toroidal singularities, then the hypercohomology groups H i (X, WΩX· ) and, for each n, H i (X, Wn ΩX· ) are finitely generated W -modules. Proof Recall from Proposition 1.7 that, for every n, the natural map (WΩX· /pn WΩX· , d) → (Wn Ω · , d) is a quasi-isomorphism. Theorem 4.1 shows that the terms of the complex W1 ΩX· are coherent sheaves of OX -modules. It follows that its hypercohomology, as well as that of WΩ · /pWΩ · , is finite dimensional. The exact sequences 0 → WΩX· /pWΩX· → WΩX· /pn+1 WΩX· → WΩX· /pn WΩX· → 0 then allow us to conclude by induction that the cohomology of each WΩ · /pn WΩX· is a W -module of finite length. The same is true for Wn ΩX· by another application of Proposition 1.7. Then a Mittag-Leffler argument shows that the natural maps H ∗ (X, WΩX· ) → lim H ∗ (X, Wn ΩX· ) ← − are isomorphisms. It follows that the cohomology modules are separated and complete for the p-adic topology. Since the terms of WΩX· are p-torsion free, we find an inclusion H ∗ (X, WΩX· )/pH ∗ (X, WΩX· ) → H ∗ (X, WΩX· /pWΩX· ).

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Since the latter is finitely generated, so is the former, and it follows that the same is true of H ∗ (X, WΩX· ).  Remark 4.4 In fact, as explained in [12, §6.2], when X/k is proper, the coherence j of the sheaves W1 ΩX is enough to establish that RΓ (X, WΩX· ) is finitely generated over the Raynaud ring. If X/k is of finite type and normal, then its smooth locus Xsm is open and its complement has codimension at least two. If j : Xsm → X is the inclusion, then the “Zariski differentials” of X/k are by definition the sheaves · := j Ω · Ω˜ X/k ∗ Xsm /k . These sheaves have been extensively studied [5, 7]; in particular Danilov has shown that if k = C and X has at most toroidal singularities, then the hypercohomology of j∗ ΩX· sm /C is isomorphic to the singular cohomology of the analytic space associated to X. Thus it is natural to ask whether, when k is perfect of characteristic p and X has at most toroidal singularities, the natural map Wn ΩX· → j∗ Wn ΩX· sm /C is an isomorphism. Unfortunately, this is not always the case, as explained in the Appendix. On the other hand, as Danilov has explained in [6, Lemma 1.5], if X/k has at most toroidal singularities, and if f : X → X is any resolution of singularities, then there is a natural isomorphism f∗ ΩX· /C → j∗ ΩXsm /C It turns out the natural analog of this statement for the saturated de Rham–Witt complex does hold. Since we do not know resolution of singularities in general, our statement is somewhat ad hoc. To make sense of it, let us say that a morphism f : X → X is a “toroidal blowup” if, étale locally on X, there exists a toric monoid Q and an ideal K of Q such that X = Spec k[Q] and X → X is the normalized blowup of X along the ideal k[K]. In this case both X and X have at most toroidal singularities. Theorem 4.5 Suppose that X/k has at most toroidal singularities and that f : X → X is a toroidal blowup. 1. The natural map W Ω · → f (W Ω · ) is an isomorphism for all n. n

X

∗

n

2. If X /k is smooth, the natural maps

X

Wn ΩX· → f∗ (Wn ΩX· ) and W1 ΩX· → f∗ (ΩX· /k ) are isomorphisms.

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Proof Statement (1) implies statement (2), since Wn ΩX· is naturally isomorphic to Wn ΩX· when X /k is smooth. Statement (1) can be verified étale locally on X, so we may and shall assume that Q is a toric monoid, that X = Spec k[Q], and that X → X is the normalized blowup of X along the ideal of k[Q] generated by an ideal K of Q. Statement (3) of Theorem 3.4 provides natural isomorphisms i Ω iX/k ∼ and Ω iX /k ∼ = W1 ΩX/k = W1 ΩXi /k . Then Proposition A.2 of the Appendix shows that the theorem is true when n = 1. We proceed by induction on n, using some results of Illusie and the following lemma, whose proof is immediate. Lemma 4.6 Let 0 → A → B → C (resp. 0 → A → B → C ) be an exact sequence of abelian sheaves on X (resp. X ) and that there exists a commutative diagram as follows. A

0

B

a

C

b

f∗ A

0

π

f∗ B

c π

f∗ C .

1. If b and c are isomorphisms, so is a. 2. If a and c are isomorphisms and π is an epimorphism, then π is an epimorphism and b is an isomorphism.  Applying the induction hypothesis and statement (2) of this lemma to the exact sequence: 0

- Grn WΩ i F il X

- Wn+1 Ω i X

π-

Wn ΩXi

- 0,

we see that it will be enough to prove that the natural map gn : GrnF il WΩXi → GrnF il WΩXi is an isomorphism for every n. To do this, we use Illusie’s exact sequences [12, §6.2]: 0

- W1 Ω i /Bn W1 Ω i X X

- Grn WΩ i F il X

- W1 Ω i /Z n W1 Ω i X X

Applying (2) of Lemma 4.6, we see that it will suffice to show that the maps zn : W1 ΩXi /Z n

- f∗ (W1 Ω i /Z n ) X

bn : W1 ΩXi /Bn

- f∗ (W1 Ω i /Bn ) X

- 0.

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are isomorphisms. We recall from [11, 2.2] that Z · and B· are, respectively, the “iterated cycle” and “iterated boundary” filtrations defined inductively using the Cartier isomorphisms (1.7) ψ1 : W1 ΩXi ∼ = H i (W1 ΩX· ) by: B0 W1 ΩXi B1 W1 ΩXi Bn W1 ΩXi Z 0 W1 ΩXi Z 1 W1 ΩXi Z n W1 ΩXi

:= 0 := Im(d : W1 ΩXi−1 → W1 ΩXi ) βi,n - Bn+1 W1 Ω i /B1 W1 Ω i X X ∼ =

:= W1 ΩXi := Ker(d : W1 ΩXi → W1 ΩXi+1 ) ζi,n - Z n+1 W1 Ω i /B1 W1 Ω i . X X ∼ =

Note for future reference that for n ≥ 0, ζi,n induces isomorphisms: i GrnZ W1 ΩXi → Grn+1 Z W1 ΩX

and hence (by induction) GrnZ W1 ΩXi ∼ = Gr0Z W1 ΩXi ∼ = B1 W1 ΩXi+1

(4.1)

Similarly, combining ζi,n and βi,n , we find isomorphisms: Z n W1 ΩXi /Bn W1 ΩXi ∼ = Z n+1 W1 ΩXi /Bn+1 W1 ΩXi and hence (by induction) Z n W1 ΩXi /Bn W1 ΩXi ∼ = W1 ΩXi .

(4.2)

We have already seen that the map z0 : W1 ΩXi → f∗ (W1 ΩXi ) is an isomorphism. Applying statement (1) of Lemma 4.6 to the exact sequence 0 → Z 1 W1 ΩXi

- W1 Ω i X

d-

W1 ΩXi+1 ,

we see that the map Z 1 W1 ΩXi → f∗ (Z 1 W1 ΩXi )

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is an isomorphism. Thanks to the Cartier isomorphism on X and X and the case n = 1, we also know that the map i ∼ WΩX/k = f∗ (W1 ΩXi /k ) = Hi (W1 ΩX· ) → f∗ (Hi (W1 ΩXi )) ∼

is an isomorphism. Applying (1) of Lemma 4.6 to the exact sequence 0

- B1 W 1 Ω i → Z 1 W 1 Ω i X X

- Hi (W1 Ω i ) X

- 0,

we deduce that the map B1 W1 ΩXi → f∗ (B1 W1 ΩXi ) is an isomorphism for all i. Equation (4.1) then implies that the map GrnZ W1 ΩXi → f∗ (GrnZ W1 ΩXi ) is also an isomorphism. Applying (2) of Lemma 4.6 to the exact sequence 0

- Grn W1 Ω i Z X

- W1 Ω i /Z n+1 W1 Ω i X X

- W1 Ω i /Z n W1 Ω i X X

- 0,

we see by induction on n that zn : GrnZ W1 ΩXi → f∗ (GrnZ W1 ΩXi ) is an isomorphism for all n. It remains only to prove that bn is an isomorphism. Using Eq. (4.2), we find an exact sequence: 0

- Bn W 1 Ω i X

- Z n W1 Ω i X

- W1 Ω i X

- 0.

Applying (2) of Lemma 4.6, we can conclude that the map bn : W1 ΩXi /Bn → f∗ (W1 ΩXi /Bn ) is indeed an isomorphism, concluding the proof.



5 Crystalline Cohomology The construction of the saturated de Rham–Witt complex of an Fp -scheme X explained in Proposition 1.12 depends on the existence of a lifting of X together with a lifting of its Frobenius endomorphism. The existence of such liftings is rare,

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but often one can find an embedding of X as a closed subscheme of a scheme which does admit such a lifting. We shall see that applying the construction in Proposition 1.12 to the PD-envelope of X in such a lifting gives another construction of WΩX· and provides a direct way to compare de Rham–Witt and crystalline cohomology. Before explaining how this works, we need to control the torsion in such PD-envelopes. Lemma 5.1 If X is a reduced scheme of finite type over a perfect field k, embedded as a closed subscheme of a smooth formal scheme Y /W endowed with a Frobenius lift, let DX (Y ) denote the (p-adically completed) PD-envelope of X in Y and let OD denote the structure sheaf of DX (Y ). Then the p-torsion of OD forms a sub PD-ideal of the PD-ideal ID of X in DX (Y ), as does its closure in the p-adic topology. Proof Let us use affine notation for simplicity. We suppose X = Spec A and Y = Spf B. Let Aperf denote the perfection of A. Since A is reduced, the map A → Aperf is injective, and since Aperf is perfect, its Witt ring W (Aperf ) is p-torsion free. Hence the same is true of W (A). The lift φ of Frobenius gives B the structure of a δ-ring, and by Joyal’s characterization [13] of the functor W , there is a unique homomorphism B → W (A) of δ-rings which is compatible with the given map B → A. Since the ideal of A in W (A) has a canonical divided power structure, this map extends to a PD-homomorphism DI (B) → W (A). Since W (A) is p-torsion free, the p-torsion of DI (B) maps to zero in W (A), hence also in A, and hence is contained in I . Say x ∈ I and pr x = 0. Then pir γi (x) = γi (pr x) = 0, so γi (x) is also a p-torsion element. This shows that the p-torsion of OD forms a sub PDideal J of I . Since I is p-adically closed, the closure of J is also contained in I , and since the divided power operations γi are p-adically continuous, this closure is stable under their action.  Suppose now that X/k is of finite type and reduced, embedded as a locally closed subscheme of a smooth formal scheme Y /W endowed with a Frobenius lift φY , Let D˜ denote the closed subscheme of D := DX (Y ) defined by the ideal of p-torsion elements of the structure sheaf OD of DX (Y ). It follows from Lemma 5.1 that the ˜ and that the ideal of X in D˜ (which we embedding X → DX (Y ) factors through D, denote by I˜) is again a PD-ideal. The OY -module ODX (Y ) admits an integrable connection [3, 6.4] which induces a connection on OD˜ . The endomorphism φY of Y extends uniquely to a PD-morphism φD of DX (Y ), which in turn induces ˜ of the module with connection (O ˜ , ∇), and of its de endomorphism, φD˜ of D, D · Rham complex OD˜ ⊗ΩY /W . This endomorphism is divisible by pi on OD˜ ⊗ΩYi /W , so φ i˜ = pi F for a unique OD˜ -linear endomorphism F of OD˜ ⊗ ΩYi /W . Thus D (OD˜ ⊗ Ω ·˜ , d, F ) is a Dieudonné complex. Y /W

Theorem 5.2 With the notation of the previous paragraph, let D˜ 1 denote the reduction of D˜ modulo p.

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1. The Dieudonné complex (OD˜ ⊗ ΩY· /W , d, F ) is in fact a Dieudonné algebra. 2. The natural map OD˜ 1 → W1 Sat (OD˜ ⊗ ΩY· /W , d, F )0 factors through a map OX → W1 Sat (OD˜ ⊗ ΩY· /W , d, F )0 . 3. The adjoint to the map in (2) is an isomorphism (WΩX· , d, F ) → WSat (OD˜ ⊗ ΩY· /W , d, F ) Proof To see that (OD˜ ⊗ ΩY· /W , d, F ) is a Dieudonné algebra, we must show that φD˜ : D˜ → D˜ reduces to the Frobenius endomorphism of D˜ 1 [4, 3.1.2]. Since D˜ 1 ⊆ D1 , It will suffice to show that φD reduces to the Frobenius endomorphism FD1 of D1 . By definition, φD is the unique PD morphism D → D extending FY , and so it will suffice to show that FD1 is in fact a PD-morphism. But if t is an element of the PD-ideal I of X in D1 , then FD∗ 1 (t) = t p = p!t [p] = 0, and hence for any n ≥ 1, FD1 ◦ γn and γn ◦ FD1 both vanish. ˜ the latter Note that OD˜ ⊗ ΩY· /W is not the same as the de Rham complex of D; has a lot of p-torsion. The comparison of these two complexes is the key to our proof. Lemma 5.3 In the following diagram, the top horizontal arrow is induced by adjunction and the natural map ΩY1 /W → Ω 1˜ , and ∇˜ := π ◦ t ◦ ∇. The lower D/Y triangle commutes, but the upper one does not. OD˜ ⊗ ΩY1 /W ∇ OD˜

t

1 ΩD/W ˜

d ˜ ∇

π 1 ΩD/W /(p-torsion)− ˜

Furthermore, the composite t˜ := π ◦ t : OD˜ ⊗ ΩY1 /W

- Ω 1 /(p-torsion)− ˜ D/W

is an isomorphism. (NB: here we always mean the p-adically completed de Rham complexes; and in particular we are dividing by the p-adic closure of the p-torsion in the lower right hand corner.) Proof The algebra OD˜ is topologically generated over OY by the divided powers f [n] of elements f of the ideal of X in Y , for n ≥ 1. For any such f , we have

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∇f [n] = f [n−1] ⊗ df in OD˜ ⊗ ΩY1 /W [3, 6.4]. On the other hand, since n!f [n] = f n and n!f [n−1] = nf n−1 , we have n!df [n] = d(n!f [n] ) = df n = nf n−1 df = n(n − 1)!f [n−1] df = n!f [n−1] df in Ω 1˜ . Thus ∇f [n] and df [n] have the same image in Ω 1˜ /(p-torsion)− , so D/W D/W the lower triangle commutes. Since d : OD˜ → Ω 1˜ is the universal derivation to a p-adically complete sheaf D/W

of OD˜ -modules, there is a unique map s : Ω 1˜

D/W

→ OD˜ ⊗ ΩY1 such that s ◦ d = ∇.

The map s factors through a map s˜ : Ω 1˜ /(p-torsion)− → OD˜ ⊗ ΩY1 . We find D/W the diagram: 1 ΩD/W /(p-torsion)− ˜

s˜

π ΩY1 /W

1 ΩD/W /(p-torsion)− ˜

π

id s

1 ΩD/W ˜

OD˜ ⊗ ΩY1 /W

i

t

1 ΩD/W ˜

d

∇

d OY

t˜

OD˜ ⊗ ΩY1

˜ ∇

OD˜

π 1 ΩD/W /(p-torsion)− , ˜

in which all triangles commute, except for the upper one in the right-hand bottom square. Then π ◦ t ◦ s ◦ d = π ◦ t ◦ ∇ = ∇˜ = π ◦ d, and it follows that π ◦ t ◦ s = π and hence that t˜ ◦ s˜ = id. On the other hand, if f is a local section of OY , then i(f ) is a section of OD˜ , and ∇(i(f )) = 1 ⊗ df in OD˜ ⊗ ΩY1 /W . Thus the problematic triangle does commute when restricted to OY ; that is, t ◦ ∇ ◦ i = d ◦ i. It follows that s ◦ t : OD˜ ⊗ ΩY1 /W → OD˜ ⊗ ΩY1 /W = id, and hence the same is true of s˜ ◦ t˜.  ˜ φ ˜ ) is a p-torsion free lifting of (D˜ 1 , F ˜ ), by Bhatt et al. [4, 3.2.1], Since (D, D D1 there is an endomorphism F of the graded abelian sheaf Ω ·˜ which gives it D/W the structure of a Dieudonné algebra, and [4, 4.2.3] constructs an isomorphism of Dieudonné algebras: · , d, F ) → (WΩ · , d.F ). W Sat(ΩD/W ˜ D˜ 1

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A. Ogus

Thus we find a commutative diagram OD˜ ⊗ ΩY· /W

π◦t

· ΩD/W /(p-torsion)− ˜

W Sat(OD˜ ⊗ ΩY· /W )

w

· W Sat(ΩD/W ) ˜

∼ =

· WΩD ˜1

g · . ΩX We have seen in Lemma 5.3 that π ◦ t is an isomorphism, and hence the same is true of w. Since X is the reduced subscheme of D˜ 1 , the map g is also an isomorphism [4, 6.5.2]. We conclude that the natural map W Sat(OD˜ ⊗ ΩY· /W ) → WΩX· is an isomorphism, as asserted in the last statement of the theorem.  When X/k is smooth, we know from [4, 4.4.12] that WΩX· agrees with the classical de Rham–Witt complex W ΩX· , which is known to compute crystalline cohomology [11, II.1.4]. The previous result gives a new and direct proof of this fact. Corollary 5.4 Suppose that X/k is smooth and embedded as a locally closed subscheme of a smooth formal scheme Y /W which is endowed with a lifting φY of the Frobenius endomorphism of its reduction p. Then the map c : (OD˜ ⊗ ΩY· /W , d) → WSat (OOD˜ ⊗ ΩY· /W , d) ∼ = WΩX· is a quasi-isomorphism. Thus, (WΩX· , d) is a representative of RuX/W ∗ (OX/W ). Proof By [3, 8.20], applied to the constant gauge  = 0, the morphism φ · factors through a quasi-isomorphism

D˜

α : OD˜ ⊗ ΩY· /W → η(OD˜ ⊗ ΩY· /W ). Thus the complex (OD˜ ⊗ΩY· /W , d, F ) is quasi-saturated and of Cartier of type (Definition 1.4), and so Theorem 1.8 implies that the map (OD˜ ⊗ ΩY· /W , d, F ) → W Sat(OD˜ ⊗ ΩY· /W , d, F ) is a quasi-isomorphism.



Remark 5.5 In fact, there is a more direct proof of Corollary 5.4, which does not refer to [3] or to statement (3) of Theorem 5.2. We explain this in a more general context in the proof of Theorem 6.8 in the next section.

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6 Log Schemes In the context of log geometry, one can define, in a somewhat ad hoc way, a variant of crystalline cohomology that coincides with the saturated de Rham–Witt cohomology we have been considering. This construction will allow us to obtain more precise information about the action of Frobenius, about the behaviour of the Hodge and conjugate spectral sequences, and about the relationship between the de Rham–Witt complex and the de Rham cohomology of a lifting. The log structures do not play a role in the construction of the de Rham–Witt complex we are considering here, but they seem to be important in the construction the crystalline complexes and in controlling the liftings from characteristic p to characteristic zero. Let us first explain why the results of Sect. 4 will be relevant to our constructions here. If Q is a monoid and R is a ring (understood from the context), we denote by AQ the log scheme Spec(Q → R[Q]), and if K is an ideal Q, we denote by AQ,K the idealized log scheme Spec((Q, K) → R[Q, K]) [20, III,§1.3]. If X is a log scheme, we denote by X the underlying scheme, which can also be viewed as a log scheme with trivial log structure. Proposition 6.1 Let R be a ring (with no log structure) and let Y /R be a fine saturated and smooth idealized log scheme over R. Then étale locally on Y , there exist a toric monoid Q, an ideal K in Q, and a strict étale morphism Y → AQ ,K . In particular, if R is a field, then Y is ideally toroidal in the sense of Sect. 4. Proof If y is a geometric point of Y , the stalk MY,y is a fine, saturated, and sharp, hence toric monoid [20, I,1.3.5], and in some neighborhood X of y there exists a chart (Q, K) → (MY , KY ) inducing an isomorphism (Q, K) → (My , Ky ) such that the associated morphism X → Spec(R[Q]/R[K]) is étale [20, IV,3.3.4,3.3.5].  We begin with a discussion of de Rham cohomology. Let T be a scheme (with trivial log structure), and let Y /T be a smooth, fine, and saturated idealized log scheme over T . We denote by ΩY· /T the logarithmic de Rham complex of Y /T [20]; in particular, (d, dlog) : (OY , MY ) → ΩY1 /T is the universal log derivation. When T = Spec(C) and K = ∅, this complex calculates the cohomology of Y ∗ , the open subset of Y where its log structure is trivial [20, V,4.2.5]. As explained in [20, V,2.3.21], the complex ΩY· /T has a canonical subcomplex Ω ·Y /T with the following properties. 1. If K is an ideal in a toric monoid Q, if T = Spec(R), and if Y is the log scheme AQ,K , then Ω ·Y /T is the complex of sheaves corresponding to the complex Ω ·R[Q,K] from Definition 2.3 (and the discussion at the beginning of Sect. 3 for the idealized case). 2. If f : Y → Y is a strict and étale morphism of idealized fs log schemes, the natural map f ∗ Ω ·Y /T → Ω ·Y /T is an isomorphism. (A morphism f : Y → Y of idealized log schemes is said to be strict if the map f : f ∗ MY → MY is an isomorphism and the ideal KY generates the ideal KY .) 3. Formation of Ω ·Y /T is compatible with arbitrary base change T → T .

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A. Ogus

We do not know how to define such complexes for general schemes with toroidal singularities without the additional information provided by a global log structure.3 Remark 6.2 Before proceeding, a word of warning. If f : Y → Z is a (log) étale · morphism of log schemes, the natural map f ∗ ΩZ/T → ΩY· /T is an isomorphism, but this need not be true for the map f ∗ Ω ·Z/T → Ω ·Y /T unless f is also strict. In particular, recall that, if X → Y is a closed immersion of log schemes the strict formal completion Yˆ of Y along X is constructed as follows. First one forms the usual formal completion Yˆ of Y along X, then the exactification Yˆ → Yˆ with respect to the morphism X → Yˆ , so that X → Yˆ is strict and M Yˆ → M X is an isomorphism. Then the ideal KX ⊆ MX generates an ideal K ⊆ MYˆ , and Yˆ is the closed formal subscheme of Yˆ defined by αYˆ (K). The complexes Ω ·ˆ , Ω ·ˆ , Y /T Y /T and Ω · can be identified with the respective pullbacks of Ω · , but this is not the Yˆ /T

Y /T

case for their underlined counterparts. We shall need to be wary of this fact when forming (strict) PD envelopes. In this rest of this section, we let X/k be a fine, saturated, reduced, and smooth idealized log scheme over a perfect field k of characteristic p. The following result generalizes the comparison between the de Rham and the de Rham–Witt complexes of a smooth scheme to the logarithmic case. It implies that for the log schemes we are considering, the complex Ω ·X/k does not depend on the log structure. Theorem 6.3 If X/k is a reduced and smooth idealized fs log scheme, there is a natural isomorphism Ω ·X/k ∼ = W1 ΩX· , uniquely determined by its naturality and compatibility with the classical isomorphism in the case of schemes with trivial log structure. Proof We first consider the case in which the underlying scheme X is also smooth. · → W Ω · is an isomorphism. In fact the same is true Then the natural map ΩX/k 1 X · → Ω · , as can be checked locally by reducing to the for the natural map ΩX/k X/k case in which X = AQ,K . In this case, if X is smooth, then K is a prime ideal and its complement Q \ K is a face G of Q, and necessarily G is a free monoid. It follows from the definitions that Ω ·X/k ∼ = Ω ·AG /k , in which case the result follows from [20, V,2.3.11]. Then there is a unique isomorphism Ω ·X/k → W1 ΩX· making the following diagram commute:

more general construction appears in [20], where a subcomplex of ΩY· /T is constructed associated to any relatively coherent sheaf F of faces of MY . Here we only consider the case in which F = M∗Y .

3A

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· ΩX/k ∼ =

· W1 Ω X ∼ =

Ω ·X/k . In general, if X/k is a smooth saturated and reduced idealized log scheme, the set Xsm where X is smooth is open and dense, and we have the solid arrows in the diagram: W1 Ω ·X

· ) j∗ (W1 ΩX sm ∼ =

· ΩX/k

· j∗ (Ω Xsm/k ).

If X admits an étale chart to some AQ,K , statement (3) of Theorem 3.4 shows that the dashed arrow exists and is an isomorphism. This isomorphism is compatible with localization, and we see from Theorem 3.5 and the Definition 2.3 that the horizontal maps are injective. Thus, if there exists a global left vertical isomorphism making the diagram commute, it is unique. Since such an arrow does exist locally, the uniqueness guarantees that it also exists globally.  We now turn to our construction of a crystalline incarnation of the cohomology of the complexes Ω ·Y we have been considering. Suppose that we are given a closed immersion of X into a fine, saturated, reduced, and smooth idealized log scheme Y /W . Let Yˆ denote the strict formal completion of Y along X and let (DX (Y ), J Y ) be the (p-adically completed) strict PD-envelope of X in Y , or equivalently in Yˆ , and let OD denote its structure sheaf. As Kato explained in [14], OD is p-torsion free and has a canonical (logarithmic) integrable connection ∇ whose de Rham complex OD ⊗ ΩY· /W calculates the cohomology of the structure sheaf OX/W of the (log) crystalline site of X/W . Since X → Yˆ is strict, ∇ factors through Ω · Yˆ /W

and thus defines a subcomplex OD ⊗Ω ·ˆ of OD ⊗Ω ·ˆ . These complexes can be Y /W Y /W endowed with an analog of the Hodge filtration, and we shall see that the resulting cohomology is crystalline in nature. We need to prepare with some technicalities concerning these filtered complexes. Proposition 6.4 If X → Y is a closed immersion as described in the previous paragraph, let

294

A. Ogus k F ilX Ω iˆ

[k−i]

Y /W k F ilX Ω iˆ Y /W

:= J Yˆ :=

[k−i] J Yˆ

⊗ Ω iˆ

Y /W ⊗ Ω iˆ Y /W

Then in fact k F ilX Ω iˆ

Y /W

so that (Ω ·ˆ

Y /W

k = F ilX ΩY·ˆ /W ∩ Ω iˆ

Y /W

,

· ) is a strict filtered subcomplex of (Ω · , F ilX ˆ

Y /W

· ). , F ilX

Proof The proof will rely on two lemmas, the second of which is formulated somewhat more generally than we will need here. Lemma 6.5 Let f : Y → T be a smooth morphism of idealized fs log formal schemes, where T has trivial log structure. Then the sheaves Ω iY /T , as well as the quotients ΩYi /T /Ω iY /T , are flat over T , and their formation commutes with base change T → T . If f admits a factorization f = p ◦ h, where h : Y → Z is smooth and p : Z → T is smooth and strict, then the sheaves Ω iY /T and ΩYi /T /Ω iY /T are also flat over Z. Proof We can check these statements étale locally, and so, by Proposition 6.1, we may assume that T = Spec(R) and that Y = Spec((Q, K) → R[Q, K]), where (Q, K) is a fine saturated idealized monoid. For each q ∈ Q, the group #q$gp is free abelian and a direct summand of the free abelian Qgp , and hence each R ⊗ Λi #q$gp is free and a direct summand of the free R-module R ⊗ Λi Qgp . Since Ω iR[Q,K]/R ∼ = i gp ⊕q∈K R ⊗ Λ #q$ , it is a free R-module. Similarly, i ΩR[Q,K]/R /Ω iR[Q,K]/R =



R ⊗ Λi Qgp /Λi #q$gp

q∈K

is a direct sum of free R-modules, hence free. This proves the flatness, and since both ΩYi /T and Ω iY /T commute with base change, the same is true of their quotient. As a first step toward the second statement, we shall show that Ω iR[Q,K]/R and i ΩR[Q,K]/R /Ω iR[Q/K]/R are free over R[Q∗ ]. Since Q is saturated, it can be written as a product Q = Q∗ ⊕ Q. Having chosen a section Q → Q, we can thus write every element of q uniquely as a sum q = q + u, with u ∈ Q∗ and q ∈ Q ⊆ Q. Then #q$ = #q$ ⊆ Q, and we see that there are isomorphisms of R[Q∗ ]-modules: Ω iR[Q,K]/R ∼ = R[Q∗ ] ⊗



Λi #q$gp

q∈Q\K i ΩR[Q,K]/R /Ω iR[Q,K]/R ∼ = R[Q∗ ] ⊗



q∈Q\K

Λi Qgp /Λi #q$gp .

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Thus both of these R[Q∗ ]-modules are free. Now to prove the second statement, working étale locally, we may assume that T = Spec R and that p is projection from affine n-space to T . After a further adjustment, we may in fact assume that p is the projection Gnm × T → T . Thus Z = Spec R[Γ ], where Γ is a finitely generated free abelian group. By Proposition 6.1 applied to Y /R[Γ ], we may also assume that h admits a strict étale chart subordinate to an idealized toric monoid (P , K), and even that Y = Spec((P , K) → R[Γ ][P , K]). Now let Q := P ⊕ Γ , so that Q∗ = P ∗ ⊕ Γ and Y = Spec((Q, K ⊕ Γ ) → R[Q, K ⊕ Γ ). The previous paragraph tells us that the modules under consideration are free over R[Q∗ ], and it follows that they are flat over R[Γ ], as claimed.  Lemma 6.6 Consider a commutative diagram j

X

Y g

f i

S

T,

where f and g are smooth integral morphisms of fs formal idealized log schemes and i and j are strict closed immersions. Then locally on Y and T , the morphisms g and i factor: X

j

Y h

f S

i

Z := ArT i

p T

such that p ◦ h = g, the square is Cartesian, the morphisms p and h are smooth and integral, and i is a strict closed immersion. The analogous statement holds if Y is a formal idealized log scheme and X is a subscheme of definition, with Z replaced by formal affine space relative to T . Proof Let J be the ideal of X in Y , let I be the ideal of S in T , and let YS := Y ×T S. Let JS denote the ideal of X in YS , and consider the sequence of OX -modules:

296

A. Ogus 1 0 → (JS /J2S ) → ΩY1S /S | → ΩX/S → 0. X

Since YS /S and X/S are smooth, this sequence is exact and locally split, and (JS /J2S ) is locally free [20, IV, 3.2.2].4 Working locally, we may choose sections f1 , . . . , fr of J whose images form a basis for (JS /J2S ). Let Z := ArT , , let p be the structure map, let h be the map defined by f1 , . . . , fr , and let i be the composition of i with the zero section of ArT . Then the diagram shown commutes, and p◦h = g. Since the images of (f1 , . . . , fr ) generate JS , the ideal J is generated by (f1 , . . . , fr ) and I, and thus the square is Cartesian. The morphism i is a strict closed immersion, and the morphism p is smooth, by construction. To see that h is smooth, consider the exact sequence 1 h∗ ΩZ/T → ΩY1 /T → ΩY1 /Z → 0. 1 The elements (dt1 , . . . , dtr ) of ΩZ/T map to the elements (df1 , . . . , dfr ) of ΩY1 /T , and at each point x of X, these form part of a basis of the free OY,x -module ΩY1 /T ,x . Thus the first map in the sequence is injective and locally split at x. Since g is smooth, it follows that h is also smooth at x [20, IV, 3.2.3], and hence in some neighborhood of x. Since p is strict, it is of course integral, and since p ◦ h is integral and p is strict, the morphism h is also integral [20, III,2.5.3]. The formal case is proved in the same way. 

We can now explain the proof of Proposition 6.4. Working locally, we apply Lemma 6.6, with f the map X → S := Spec k and g the map Y → T := Spec W . The resulting map h is smooth and integral, so the underlying morphism h is flat [20, IV,4.3.5]. Since formation of divided power envelopes commutes with flat base change, DX (Y ) ∼ = DX (Z) ×Z Y . We have an exact sequence of OZ -modules 0 → h∗ Ω iY /T → h∗ ΩYi /T → h∗ (ΩYi /T /Ω iY /T ) → 0 and by Lemma 6.5, these are all flat. Then the sequence 0 → h∗ Ω iY /T → h∗ ΩYi /T → h∗ (ΩYi /T /Ω iY /T ) → 0 [k]

remains exact when tensored with ODX (Z) /J Z , and the resulting sequence identifies with [k]

[k]

ODX (Y ) /J Y ⊗ Ω iY /T → ODX (Y ) /J Y ⊗ ΩYi /T → ODX (Y ) /JY[k] ⊗ (ΩYi /T /Ω iY /T ). The claim follows from the injectivity on the left.



4 Unfortunately this reference does not explicitly mention formal or idealized log schemes; however these variants present no difficulty.

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The following theorem explains the crystalline property of the filtered complexes we have been considering. Theorem 6.7 With the notation of the paragraph above, suppose that g : Y → Y is a morphism of smooth idealized log schemes over W , and that i : X → Y ; and i := g ◦ Y are closed immersions, so that g induces morphisms of strict formal completions Yˆ → Yˆ and strict PD-envelopes D := DX (Y ) → D := DX (Y ),

1. The morphism g induces quasi-isomorphisms: OD ⊗ ΩY·ˆ /W → OD ⊗ ΩY·ˆ /W OD ⊗ Ω Y·ˆ /W → OD ⊗ Ω Y·ˆ /W . 2. The natural maps · ) → (O ⊗ Ω · ·) g ∗ : (OD ⊗ ΩY·ˆ /W , F ilX , F ilX D Yˆ /W · ) → (O ⊗ Ω · ·) , F ilX g ∗ : (OD ⊗ Ω Y·ˆ /W , F ilX D Yˆ /W are filtered quasi-isomorphisms. Proof Statement (1) of the theorem is just a special case of statement (2). We begin the proof of statement (2) with the case in which Y = ArY and g : Y → Y is the zero section. Let p : Y → Y be the projection. We shall verify that g induces quasi-isomorphisms between the filtered complexes described in the theorem; since p ◦g = idY , it will follow that the same is true for p. An induction argument reduces us to the case in which r = 1. Working locally, we assume that Y = Spf((Q, K) → B) and let J be the ideal of X in Y . Then Y = Spf(Q, K) → B{t}), and the ideal of X in Y is (J , t). Let C be the completed PD-envelope J in B. The PD-envelope of (J , t) in B{t} can be identified with the completion of the PD-polynomial algebra C#t$. Thus every element of OD ⊗ ΩYi /W can be written uniquely as a formal sum ω=



t [j ] (αj + dt ∧ βj )

with αj ∈ OD ⊗ ΩYi /W , βj ∈ OD ⊗ ΩYi−1 /W , and lim αj = lim βj = 0. We have a commutative diagram of filtered complexes:

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A. Ogus

· ) (Ker· , F ilX

· ) (OD ⊗ ΩY· /W , F ilX

(OD ⊗ ΩY·

/W , F ilX )

· ) (Ker· , F ilX

· ) (OD ⊗ Ω ·Y /W , F ilX

(OD ⊗ Ω ·Y

/W , F ilX )

·

·

The rows are strictly short exact, and Proposition 6.4 shows that the vertical arrows are strict inclusions. The element ω lies in OD ⊗ Ω ·Y /W if and only if each αj and βj do, and ω lies in Keri if and only if α0 = 0. Now define ρ : OD ⊗ ΩYi /W → OD ⊗ ΩYi−1 /W by 

t [j ] (αj + dt ∧ βj ) →

j ≥0



t [j +1] βj ,

j ≥0

noting that ρ preserves the subcomplexes Ker· and OD ⊗ Ω ·Y /W as well as the · . We calculate: filtration F ilX ⎛



(dρ + ρd)(ω) = d ⎝

⎞ t [j +1] βj ⎠ +

j ≥0

⎛ ρ⎝

=





(t

[j −1]

dt ∧ αj +

j ≥1

(t [j ] dt ∧ βj + t [j +1] dβj ) +

j ≥0



t

[j ]

dαj −

j ≥0



⎞ t

[j ]

dt ∧ dβj )⎠

j ≥0

  (t [j ] αj − t [j +1] dβj ). j ≥1

j ≥0

= ω − α0 .

It follows that dρ − ρd is the identity on Ker· and its filtered subcomplexes and hence that g ∗ and p∗ are indeed filtered quasi-isomorphisms. The general case follows easily. First note that the map gˆ : Yˆ → Yˆ is necessarily strict, since X → Yˆ and X → Yˆ are strict. Suppose that g is smooth. Then gˆ is strict and smooth, and hence gˆ is smooth. Working étale locally, we may suppose that gˆ looks like projection from formal affine space over Y , so that the previous argument applies. Next suppose that g is a closed immersion. Then since Y is smooth and we are working locally, we may assume that g admits a smooth retraction r. Since the result is true for r and for idY and since r ◦ g = idY , the result also holds for g. For the general case, recall that every morphism can locally factored as a composition of a closed immersion and a smooth morphism. 

De Rham–Witt for Toroidal Singularites

299

In order to relate these crystalline constructions to the de Rham–Witt complex, we begin by supposing that X admits an embedding into a smooth Y /W which is endowed with a Frobenius lifting φY : Y → Y . Then φY induces a Frobenius lifting φD of the strict PD-envelope DX (Y ) of X in Y , which in turn induces an · of O ⊗ Ω · . Since these complexes are p-torsion free and endomorphism φD D Y /W · is visibly divisible by pi in degree i, these data define a Dieudonné complex φD (OD ⊗ Ω ·Y /W , d, F ). Arguing as in Steps 1 and 2 of Theorem 5.2, we see that this complex has the structure of a Dieudonné algebra, and that there is a pair of adjoint maps: OX → W1 Sat (OD ⊗ Ω ·Y /W )0 ,

cY : WΩX· → WSat (OD ⊗ Ω ·Y /W )

(6.1)

Theorem 6.8 Let X/k be a fine, saturated, smooth, and reduced idealized log scheme. 1. If i : X → Y is an embedding into a fine, saturated and smooth formal idealized log scheme Y over W endowed with a Frobenius lift, then the associated Dieudonné algebra (OD ⊗ Ω ·ˆ , d, F ) is quasi-saturated, and the natural map Y /W c : WΩ · → WSat (O ⊗ Ω · ) is an isomorphism. Y

D

X

Y /W

2. If Y /W is a smooth formal lifting of X (not necessarily endowed with a Frobenius lift), there is a natural derived isomorphism (WΩX· , d) ∼ (Ω Y·ˆ /W , d). Proof To prove statement (1), we may work locally on X. Thus we may assume that (X, FX ) admits a smooth formal lifting (Y , φY ). By Proposition 1.12, the map cY is an isomorphism. Since (Ω ·Y /W , d, F ) is of Cartier type and p-adically separated and complete, it is quasi-saturated. Let (Y , φY ) := (Y × Y , φY × φY ). By Theorem 6.7, the map (Y , φY ) → (Y , φY ) induces a quasi-isomorphism (Ω ·ˆ , d, F ) → (OD ⊗ Ω ·ˆ , d, F ), and hence by Corollary 1.9, the complex Y /W Y /W (O ⊗ Ω · , d, F ) is also quasi-saturated and the map D

Yˆ /W

WSat (OD ⊗ Ω ·ˆ

Y /W

, d, F ) → WSat (OD ⊗ Ω ·ˆ , d, F ) Y

is an isomorphism. The same corollary applied to the map (Y , φY ) → (Y, φY ) shows that (OD ⊗ Ω ·ˆ , d, F ) is quasi-saturated and that the map Y /W

WSat (OD ⊗ Ω Y·ˆ /W , d, F ) → WSat (O D ⊗ Ω Y·ˆ /W , d, F ) is an isomorphism. It follows that cY is also an isomorphism.

300

A. Ogus

To deduce statement (2), let Y 0 → Y be an open affine cover of Y , let Y n := ×Y · · · ×Y Y 0 (n + 1-times), and let Xn be its reduction modulo p. Now let Z n be the strict formal completion of Y 0 ×W · · · ×W Y 0 along Xn and let D n be the strict PD envelope of Xn in Z n .5 We find immersions of simplicial formal idealized log schemes: Y0

X• → Y • → D • . Since Y 0 is formally smooth and affine, it admits a Frobenius lift φY 0 , which induces Frobenius lifts on Z • and D • (but not Y • ). We find a diagram: · • ) b C · (WSat(O • ⊗ Ω · • )) c C · (O • ⊗ Ω · • ) C · (WΩX D D Z /W Z /W e

a · WΩX

Ω ·Y /W

f

C · (Ω ·Y /W ).

The arrows a and f are quasi-isomorphisms by descent, and arrow b is a quasi-isomorphism by statement (1). Arrow c is a quasi-isomorphism because OD ⊗ Ω ·Z/W is quasi-saturated, and arrow e is a quasi-isomorphism by Theorem 6.7. A standard simplicial argument, which we will not write out, shows that the resulting derived isomorphism is independent of the choices made and, in fact, is natural.  Corollary 6.9 Let Y /W be a fine saturated smooth and proper log scheme over W and let X/k be its reduction modulo p. Choose an embedding W → C. Then there are natural isomorphisms: H · (X, WΩX· ) ⊗W C → H · (Y, Ω ·Y /W ) ⊗W C → H · (Yan , Ω ·Y /C ) ← H · (Yan , C). The filtration on H · (Y, Ω ·Y /W ) coming from the “filtration bête” of ΩY· /W coincides with the Hodge filtration of the mixed Hodge structure on H · (Yan , C). Proof These results follow from the compatibility of formation of cohomology with flat base change, GAGA, and Danilov’s theorems [6, Theorem 3.4].  Theorem 6.10 Let Y /W be a fine saturated smooth and proper idealized log scheme over W and let X/k be its reduction modulo p. Assume the dimension of X/k is less than p. note that Z n is endowed with the idealized structure coming from any of the maps pi : Z n →

Y 0 ; these are all the same because, after the exactification used in the construction, pi (m) and

pj (m) differ by a unit.

5 We

De Rham–Witt for Toroidal Singularites

301

1. There is a derived isomorphism: (Ω ·Y /W , pd) ∼ (Ω ·Y /W , d). 2. The Hodge and conjugate spectral sequences of the hypercohomology of the complex Ω ·X/k ∼ = W1 ΩX· degenerate at E1 and E2 respectively. Proof Let us first assume that there exists a locally closed strict embedding Y → Y , where Y /W is a fine saturated and smooth idealized log scheme over W which is endowed with a Frobenius lifting φY . Let D := DX (Y ), and observe that the [m] endomorphism φY∗ of OD induced by φY takes J Y into pOD , hence J Y into p[m] OD for every m ∈ N. Since φY∗ is also divisible by pi on Ω iˆ , we find a Y /W morphism of complexes: m Φm : F ilX (OD ⊗ Ω ·ˆ

Y /W

) → (p[m] )OD ⊗ Ω ·ˆ

Y /W

.

Lemma 6.11 With the notations above, suppose that dim X ≤ m < p. 1. Multiplication by pm−i in degree i defines an isomorphism of complexes: m · Ψm : (Ω ·Y /W , pd) → (F ilX Ω Y /W , d).

2. The morphism Φm is a quasi-isomorphism. Proof The ideal of X in Y is just the ideal pOY , so m i F ilX Ω Y /W := (p)[m−i] Ω iY /W = pm−i Ω iY /W ,

since each m − i < p. Since n := dim X ≤ m, multiplication by pm−i in degree i defines an isomorphism of complexes: (Ω Y· /W , pd)

=

Ψm m · Ω Y /W , d) = (F ilX

OY

pd

pm pm OY

Ω 1Y /W pm−1

d

pm−1 Ω 1Y /W

pd ··· d

···

Ω nY /W pm−n

· · ·pm−n Ω nY /W .

This proves statement (1). Statement (2) can be checked locally, so we may assume that Y is affine and endowed with a Frobenius lift φY . Let Yˆ be the strict formal completion of X in Y × Y , with the Frobenius lift φY induced by φY × φY . Since DX (Y ) = Y , we have a commutative diagram:

302

A. Ogus m Φm : F ilX (OD ⊗ Ω ·Yˆ

/W

m Φm : F ilX (OD ⊗ Ω ·Yˆ

)

(p[m] )OD ⊗ Ω Y·ˆ

)

(p[m] )OD ⊗ Ω Y·ˆ

/W

m Φm : F ilX (Ω Y· /W )

(p[m] )Ω ·Yˆ

/W

/W

/W

.

The vertical arrows are quasi-isomorphisms by Theorem 6.7 so it will suffice to prove that Φm is a quasi-isomorphism. Composing Φm with multiplication by p−m , we find the morphism F : (Ω ·Y /W , pd) → Ω ·Y /W associated to the Dieudonné complex (Ω ·Y /W , d, F ), which we encountered earlier (1.1). Since this complex is of Cartier type, the reduction of F modulo p is a quasi-isomorphism,. Since the complex is p-torsion free and p-adically separated and complete, it follows that F is also a quasi-isomorphism, and then so is Φm .  To prove statement (1) of the theorem, observe that, if there exists an embedding of Y into a Y admitting a Frobenius lift, we have a diagram m (F ilX (OD ⊗ Ω Y·ˆ

/W

, d))

Φm

a (Ω ·Y /W , pd)

Ψm

m · (F ilX Ω Y /W , d)

(pm OD ⊗ Ω Y·ˆ

/W

, d)

b (pm Ω ·Y /W , d).

The morphisms Φm and Ψm are quasi-isomorphisms by Lemma 6.11, and the morphisms a and b are quasi-isomorphisms by Theorem 6.7. Inverting the morphism a in the derived category and composing with the other morphisms gives the desired derived isomorphism, when Y → Y exists. A standard simplicial argument will cover the general case. The reduction modulo p of the derived isomorphism of statement (1) gives a quasi-isomorphism (Ω ·X/k , 0) ∼ (Ω ·X/k , d).

De Rham–Witt for Toroidal Singularites

303

In other words, the complex (Ω ·X/k , d) is “completely decomposed” in the sense of [9]. The argument there, using a dimension count and the Cartier isomorphism, then applies to prove the theorem. 

7 The Hodge and Nygaard Filtrations Our aim here is to give a brief account of some of the essential features of the construction of the Nygaard filtration as discussed in [4]. We also explain its application to the proof of Katz’s conjecture, following Nygaard’s method in [19], but adapted to the language of [4]. We begin with a general construction, going back to Mazur’s original article [17]. Let p be a fixed natural number, typically a prime. By a p-span in an abelian category we mean a monomorphism Φ : M → M of p-torsion free objects. A pspan is a p-isogeny if there exist a natural number  and a morphism Ψ : M → M such that Φ ◦ Ψ = p idM and Ψ ◦ Φ = p idM . The smallest such  is called the level of the isogeny. Definition 7.1 If Φ : M → M is a p-span, let M := M/pM, and define, for i ≥ 0, NiM Ni M NiM Ni M

:= := := :=

Φ −1 (pi M) I m(p−i Φ : N i M → M) I m(N i M → M /pM ) I m(Ni M → M/pM).

The verification of the following proposition is immediate. Proposition 7.2 With the definitions above, N · is a descending filtration of M , and N· is an ascending filtration of M. Furthermore pN i−1 M = N i M ∩ pM pNi+1 M = Ni M ∩ pM. The map p−i Φ induces isomorphisms of pairs (N i M , N i+1 M ) (N i M , pN i−1 M ) (N i M , N i+1 M + pN i−1 ) and hence isomorphisms:

∼ =-

(Ni M, pNi+1 M) (Ni M, Ni−1 M) ∼ =(Ni M, Ni−1 M + pNi+1 M), ∼ =-

304

A. Ogus

GriN M i

N M GriN

M



∼ =∼ =∼ =-

Ni M GrN i M GrN i M.







It follows from the definitions that N 0 M = M and that N−1 M = 0. If Φ is a p-isogeny of level , then N +1 M = 0 and N M = M. Formation of these filtrations is natural: a morphism of p-spans induces morphisms of filtered objects in the obvious way. Example 7.3 Mazur’s proof of the Katz conjectures in [17] is based on an analysis of p-isogenies in the category of finitely generated W -modules. For example, let i be a natural number and let Φ : M → M denote multiplication by pi on W . Then N · M (resp. N· M) is the unique filtration on k such that Gri k (resp. Gri k) is nonzero. It is a standard fact that every p-isogeny in the category of finitely generated W -modules is a direct sum of objects of this type, and consequently is determined up to isomorphism by its “abstract Hodge numbers” hi (Φ) := dimk GriN M = dimk GrN i M. Let (M · , d, F ) be a p-torsion free Dieudonné complex with M i = 0 for i < 0. Then the morphism Φ : (M · , d, F ) → (M · , d, F ), given by pi F in degree i, defines a p-span in the category of Dieudonné complexes, and hence filtrations N · and N· of (M · , d, F ), Proposition 7.4 Let (M · , d, F ) be a p-torsion free Dieudonné complex such that M n = 0 for n < 0, and let N · and N· be the filtrations on M · defined by Φ as in Definition 7.1. 1. If (M · , d, F ) is of Cartier type, then N i M · = pi M 0 → pi−1 M 1 → · · · → pM i−1 → M i → M i+1 · · · · NiM =

0

→ 0 → ··· →

0

i

→M →M

i+1

··· .

2. If (M · , d.F ) is saturated, then N i M · = pi−1 V M 0 → pi−2 V M 1 → · · · → V M i−1 → M i → M i+1 · · · . Ni M · =

M0 →

M 1 → · · · → M i−1 → F M i → pF M i+1 · · · .

Furthermore, the inverse of the isomorphism p−i Φ : N i M · → Ni M · is given by pi−n−1 V in degree n. If M n = 0 for n ≥ , then Φ is a p-isogeny of level at most .

De Rham–Witt for Toroidal Singularites

305

Proof An element x of M n lies in N i M n if and only if pn F x = pi y for some y ∈ M n . If i ≤ n this condition is vacuous, so N i M n = M n when i ≤ n. Suppose that x ∈ N i M n , that n < i, and that M · is of Cartier type. It follows that F x ∈ pM n , hence x is killed by the isomorphism γ : M n /pM n → H n (M · /pM · ), hence x ∈ pM. Repeating the argument with p−1 x, we eventually see that x ∈ pi−n M n . The reverse inclusion is trivial. Now suppose that M · is saturated. If n < i, we see that x ∈ N i M n if and only if Φ(x) ∈ pi M n , i.e., if and only if F x = pi−n y for some y ∈ M n . Then F x = pi−n−1 py = pi−n−1 F V y, that is, if and only if x = pi−n−1 V y. Then p−i Φx = p−i Φpi−n−1 V y = pn−i Fpi−n−1 V y = y, so Ni M n = M n when n < i. On the other hand, if i ≤ n, then Ni M n := p−i ΦN i M n = pn−i F M n . If also M n = 0 for n > , then N M · = M · and Φ is a p-isogeny of level ≤ .  If (M · , d, F ) is a saturated Dieudonné complex and (WM · , d, F ) is its completion, we find a natural map of filtered complexes: (M · , d, N · ) → (WM · , d, N · ). I do not know if this map is strictly compatible with the filtrations. However it is easy to see that, for every r > 0, the filtrations N · M · and N · WM · induce the same filtration of Wr M · . Indeed, if x ∈ pj V WM n , then we can find sequences (ym ) ∈ M n , (zm ) ∈ M n−1 so that x = pj V = pj V

∞  m=0 r−1 

(V m ym + dV m zm ) (V m ym + dV m zm ) +

∞ 

(V m pj V ym + dV m pj +1 V zm ),

m=r

m=0

r−1 m so the element pj V m=0 (V ym + dV m zm ) of pj V M n has the same image in n Wr M as does x. The analogous statement is true for N· : if x ∈ pj F WM n , then x = pj F = pj F

r  m=0 r  m=0

∞ 

(V m ym + dV m zm ) + pj F (V m ym + dV m zm ) +

∞ 

(V m ym + dV m zm )

m=r+1

(V m pj F ym + dV m−1 pj zm )

m=r+1

The following result is the promised filtered version of statement (2) of Proposition 1.7 and [4, 2.7.3]. It is related to Nygaard’s [19, Theorem 1.5], which is essentially this result except applied to the rth powers of Φ and p and the corresponding filtrations.

306

A. Ogus

Theorem 7.5 If (M · , d, F ) is a saturated Dieudonné complex, then for every r > 0, the natural maps πr : (M · /pr M · , N · ) → (Wr M · , N · ) πr : (M · /pr M · , N· ) → (Wr M · , N· ) are filtered quasi-isomorphisms. Proof Let us write Mr· for M · /pr M · and Kr· for the kernel of πr , i.e., Krn = F il r M n /pr M n . By definition, N i Mrn is the image of the map N i M n → Mrn and N i Krn := Krn ∩ N i Mrn , so we have an exact sequence of complexes: 0 → N i Kr· → N i Mr· → N i Wr M · → 0. There is an analogous sequence with Ni in place of N i . It will suffice to show that the complexes N i Kr· and Ni Kr· are acyclic. Let us first check that Kr· is acyclic. An element of Krn is the image of an element x of F il r M n , say x = V r x + dV r x , so dx = dV r x . If x lifts a cycle, then dx = pr z for some z. Then dx = F r dV r x = F r dx = F r pr z = pr F r z. Since M · is saturated, it follows that x = F r x for some x . Then V r x = pr x , so in fact x ≡ dV r x (mod pr M n ). Since V r x ∈ F il r M n−1 , we see that Krn is indeed acyclic. To see that N i Kr· is acyclic, we must show that if x as above belongs to N i M + pr M n , then V r x ∈ N i M n−1 + pr M n−1 . If r = 0 or i < n, there is nothing to check. If r > 0 and i = n, then N i M n−1 = V M n−1 which contains V r x since r ≥ 1. Suppose r > 0 and i = n + j with j > 0. Since x ∈ N i M n + pr M n , we can write x = pj −1 V z + pr z , and since x ≡ dV r x (mod pr M n ), we find that dx = F r dV r x ≡ F r x ≡ F r pj −1 V z ≡ pj F r−1 z (mod pr M n ). If j ≥ r, then x ∈ pr M n and there is nothing to prove, so we may assume that j < r. Then dx ∈ pj M n , and since M is saturated, we can write x = F j x , and then V r x = V r−j V j F j x = pj V V r−j −1 x ∈ N i M n−1 , as required. The proof of the second part is similar. If x ∈ M n lifts a cycle of Ni Krn , then as before we can write x = dV r x + pr z and x = pj F z + pr z ; without loss of generality j < r. Then dx = F r dV r x ≡ F r x ≡ pj F r+1 z, so there exists x such that x = F j x . Then V r x = pj −1 F V r−j +1 x ∈ Ni M n−1 .  The second part of the following result is contained in [4, 8.2.1]. Corollary 7.6 If (M · , d, F ) is a saturated Dieudonné complex, there are natural quasi-isomorphisms:

De Rham–Witt for Toroidal Singularites

N iM

·

307 i

=

···0

V M i−1 /pM i−1

M

β ≥i W1 M · =

···0

0

W1 M i

M

i+1

···

W1 M i+1 · · ·

and Ni M

·

=

τ ≤i W 1 M · =

M

0

···

W1 M 0

M

i−1

· · · W1 M i−1

F M i /pM i

0···

Z i (W1 M · )

0···

Proof The first statement is just the special case of Theorem 7.5 when r = 1. Since · · M · is saturated, we can identify F M i /pM i with Z i (M ), and so Ni M identifies · · with τ ≤i M . Since M → W1 M · is a quasi-isomorphism, the same holds after ≤i applying τ , and the result follows.  The next result is an easy consequence of the previous one, but it is just as easy to check it directly. Corollary 7.7 If (M · , d, F ) is a saturated Dieudonné complex, there is a commutative diagram of quasi-isomorphisms: GriN M

·

GrN i M

·

W1 M i [−i]

H i (W 1 M · )[−i],

where the horizontal arrows are induced by the arrows of Theorem 7.5, the left vertical arrow is the one appearing in the last line of Proposition 7.2 and the right vertical arrow is the Cartier isomorphism ψ1 of Proposition 1.7.  The following result shows that the map from a Dieudonné complex of Cartier type to its saturation is a filtered quasi-isomorphism, mod powers of p. (Compare with [4, 8.3.4 and 8.3.5].)

308

A. Ogus

Proposition 7.8 If (M · , d, F ) is a p-torsion free Dieudonné complex and r ∈ N, let Mr· := M · /pr M · and let N i Mr· denote the image of N i M · in Mr· . Then if (M · , d, F ) is of Cartier type and M n = 0 for n < 0, then for all i and all r, the natural maps: · (7.8.1) GriN M · NiM (7.8.2) i · (7.8.3) N Mr (7.8.4) M · /N i M ·

→ → → →

· GriN Sat (M ) · N i Sat (M ) i · N Sat(Mr ) Sat M · /N i (Sat (M · ))

are quasi-isomorphisms. i

Proof Statement (1) of Proposition 7.4 shows that GriN M is just M [−i]. Compos· ing the map (7.8.1) with the quasi-isomorphism GriN SatM → W1 SatM i [−i] of i

Corollary 7.7, we find a map M [−i] → W1 SatM i [−i], which is nothing but the isomorphism in the last statement of Theorem 1.8. It follows that the map (7.8.1) is a also a quasi-isomorphism, and then induction shows that the same is true of (7.8.2). Since N i M n ∩ pr M n = pr N i−r M n (and similarly for SatM), we have a commutative diagram with exact rows: ·

0

N i−r M

0

N i−r (SatM · )

· N i Mr+1

N i Mr·

0

N i Sat(M · )r+1

N i Sat(M )r·

0.

Then another induction proves that (7.8.3) is also a quasi-isomorphism. Since pr M n ⊆ N i M n for r ' 0, (and similarly for Sat(M)) it follows that (7.8.4) is a quasi-isomorphism as well.  The following result shows that, under suitable hypotheses, formation of the filtrations N · and N· commutes with passage to hypercohomology. Proposition 7.9 Let (M · , F, d) be a strict Dieudonné complex on a topological space (or topos) X. Suppose that the following hypotheses are satisfied. 1. The groups H · (X, M · ) are p-torsion free. 2. The two spectral sequences of hypercohomology associated to the complex W1 M · degenerate, at E1 and at E2 respectively. That is: (a) For all i, the maps H · (X, β ≥i W1 M · ) → H · (X, W1 M · ) are injective. (b) For all i, the maps H · (X, τ ≤i W M · ) → H · (X, W M · ) are injective. 1

1

Let N · and N· be the filtrations on H · (X, M · ) defined by the map

De Rham–Witt for Toroidal Singularites

309

H · (Φ) : H · (X, M · ) → H · (X, M · ) as in Definition 7.1. Then the following conclusions hold. 1. For all i, the natural maps H · (X, M · )/p i H · (X, M · ) → H · (X, M · /pi M · ) are isomorphisms. In particular, the natural maps · H · (X, M · )/pH · (X, M · ) → H · (X, M ) → H (X, W1 M · ) are isomorphisms. 2. The natural maps H · (X, N i M · ) H · (X, Ni M · )

- N i H · (X, M · ) - Ni H · (X, M · )

are isomorphisms. 3. The natural maps - H · (X, β ≥i W1 M · ) - H · (X, τ ≤i W1 M · )

H · (X, N i M · ) H · (X, Ni M · ) are surjective.

Proof Conclusion (1) follows from the long exact cohomology sequence associated to the short exact sequence 0 → M·

i p-

M · → M · /pi M · → 0,

· hypothesis (1), and the fact that M → W1 M · is a quasi-isomorphism (see Proposition 1.7). The proof of the following lemma depends on the degeneration of the first hypercohomology spectral sequence. Lemma 7.10 For every i, the map H · (X, N i M · ) → H · (X, M · ) is injective. Proof We use induction on i, the case i = 0 being trivial. Thanks to Proposition 7.2, we have an exact sequence 0 → N i−1 M ·

[p] -

· NiM· → NiM → 0

and hence a commutative diagram in which the rows are exact:

(7.1)

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A. Ogus

H · (X, N i−1 M · )

[p]

H · (X, N i M · )

· H · (X, N i M )

ai

ai−1 p

H · (X, M · )

bi · H · (X, M ).

H · (X, M · )

The map ai−1 is injective by the induction hypothesis, the map p in the lower left is injective because H · (M) is torsion free, and by Theorem 7.5 the map bi identifies with the map H · (β ≥i W1 M · ) → H · (W1 M · ), which is injective by hypothesis (2a). It follows that ai is injective.  Since N i M · is the kernel of the map M·

Φ-

M · → M · /pi M · ,

we find a map φi : M · /N i M · → M · /pi M · . The next lemma uses the hypothesis that the second hypercohomology sequence degenerates. Lemma 7.11 For every i, the map H · (X, M · /N i M · ) → H · (X, M · /pi M · ) induced by φi is injective. Proof We argue by induction on i, the case i = 0 being trivial. Let ρi be the composition ρi : GriN M ·

αi

· · Ni M → M ,

where the first arrow is the isomorphism from Proposition 7.2 and the second is the evident inclusion. We have a commutative diagram: 0

GriN M ·

M · /N i+1 M ·

ρi 0

M · /pM ·

φi+1 [pi ]

M · /pi+1 M ·

with exact rows. This yields the diagram:

M · /N i

0

φi M · /pi M ·

0

De Rham–Witt for Toroidal Singularites

311

H · (X, GriN M · )

H · (X, M · /N i+1 M · )

H · (ρi )

H · (φi+1 )

· H · (X, M )

[pi ]

H · (X, M · /N i M · ) H · (φi )

H · (X, M · /pi+1 M · )

H · (X, M · /pi M · ).

The rows in the diagram are exact, the map labeled [pi ] is injective by hypothesis (1), and the map H · (φi ) is injective by the induction hypothesis. The map H · (ρi ) factors as a composite H · (X, GriN M · )

H · (αi)

· H · (X, Ni M )

βi

· H · (M ).

The first map is an isomorphism since αi is, and by Theorem 7.5, the map βi identifies with the map H · (X, τ ≤i W1 M · ) → H · (X, W1 M · ), which is injective by hypothesis (2b). It follows that H · (ρi ) is injective and then that H · (φi+1 ) is injective.  · Lemma 7.12 The map H · (X, N i M · ) → H · (X, N i M ) is surjective. Proof The exact sequence (7.1) yields a long exact sequence · H · (X, N i M · ) → H · (X, N i M ) → H ·+1 (X, N i−1 M · )

[p] -

H ·+1 (X, N i M · ).

Thus it suffices to show that the map [p] is injective. This follows from the commutative diagram H · (X, N i−1 M · )

[p]

H · (X, N i M · )

H · (X, M · )

p

H · (X, M · ),

the torsion freeness of H · (X, M · ), and Lemma 7.10.  · i Now to prove the theorem, recall that N H is by definition the kernel of the composition ci : H · (X, M · )

H · (Φ) -

H · (X, M · )

- H · (X, M · )/p i H · (X, M · ).

The top row of the following commutative diagram is exact:

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H · (X, N i M · )

ai

H · (X, M · )

H · (X, M · /N i M · )

ci H · (X, M · )/pi H · (X, M · )

φi ∼ =

H · (X, M · /pi M · ).

As we have seen, ai and φi are injective, and it follows that H · (X, N i M · ) identifies with the kernel of ci .  Let us sketch how Proposition 7.9 implies Katz’s conjecture for smooth proper log schemes, generalizing Mazur’s classic theorem [17]. Note that, thanks to Theorem 6.10, the hypothesis of Hodge degeneration is automatically satisfied if the dimension of X is less than p and if it admits a log structure such that the associated log scheme lifts smoothly to W . Theorem 7.13 Let X/k be a smooth proper ideally toroidal scheme over a perfect · (X) := H · (X, WΩ · ). Assume that field k of characteristic p > 0. and let HdRW X · (X) is torsion free and that the Hodge and/or conjugate spectral sequence of HdRW · (X) induced W1 ΩX· degenerates at E1 . Let Φ denote the endomorphism of HdRW · (X) as in · by FX and let N and N· be the corresponding filtrations of HdRW Definition 7.1. · · (X)/pH · (X) → H · (X, W Ω · ) is an 1. The natural map H := HdRW 1 X dRW isomorphism. 2. The filtration induced by N · on H (X, W1 ΩX· ) is the Hodge filtration. 3. The filtration induced by N· on H (X, W1 ΩX· ) is the conjugate filtration. n i ). 4. The dimension of GriN H is equal to the dimension of H n−i (X, W1 ΩX/k · 5. The Newton polygon of the action of Φ on HdRW (X) lies on or above the Hodge polygon of X/k in degree n. Proof We know from Proposition 4.3 that the cohomology groups H q (X, W1 ΩXi ) are finite dimensional. The Cartier isomorphism (see (3) of Proposition 1.7) implies that H q (X, W1 ΩXi ) ∼ = H q (X, Hi (W1 ΩX· )). Thus the dimensions of the E1 terms of the “Hodge” spectral sequence match the dimensions of the E2 terms of the “conjugate” spectral sequence, so if one of these degenerates, so does the other. Then Statements (1)–(4) follow from Proposition 7.9. Statement (5) follows, since the Newton polygon of an F-crystal always lies on or above the polygon formed from the numbers dim GriN H [17].  Finally, we give a give a relatively computation free proof of the theorem of Langer-Zink [16, 4.7], comparing the Nygaard filtration of the de Rham–Witt complex with the Hodge filtration on crystalline cohomology, generalized here to the logarithmic case.

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Theorem 7.14 Let X/k be a fine saturated and smooth idealized log scheme, strictly embedded in a smooth formal Y /W . Then if i < p, there is a natural derived isomorphism i F ilX (OD ⊗ ΩY· /W ) ∼ N i WΩX· .

Proof Using the standard simplicial argument, we reduce to the case in which Y is endowed with a Frobenius lift ΦY . Then OD ⊗ ΩY· /W has the structure of a Dieudonné algebra, and as we saw in Theorem 6.8, WΩX· can be identified with its completed saturation. Thus there is a natural map of Dieudonné algebras: c : OD ⊗ ΩY· /W → WΩX· . The endomorphism ΦY∗ of OD ⊗ ΩY· /W induced by ΦY is divisible by pi on i (O ⊗ Ω · F ilX D Y /W ) and hence c induces a morphism i cYi : F ilX (OD ⊗ ΩY·˜ /W → N i WΩX· := ΦY∗ −1 (pi WΩX· ).

To see that cYi is a quasi-isomorphism, we can work locally, with the aid of a lifting ˜ Φ ˜ ) := (Y × X, ˜ ΦY × Φ ˜ ), and let E˜ denote the ˜ Φ ˜ ) of (X, FX ). Let (Z, (X, X Z X ˜ divided power envelope of X in Z. Then we have morphisms: i F ilX (OD ⊗ ΩY·˜ )

i F ilX (OE˜ ⊗ ΩZ·˜ )

ciY

ciZ˜

i · ) F ilX (OX˜ ⊗ ΩX ˜

ciX˜

· . N i WΩX The horizontal arrows are quasi-isomorphisms by Theorem 6.7, and ci˜ is a quasiX

isomorphism by Propositions 3.3 and 7.8. It follows that ci˜ and cYi are also quasiZ isomorphisms, as claimed. 

A Technicalities of Toric Differentials Let Q be a fine saturated monoid, let R be a regular ring, and let X := Spec R[Q]. Then X is normal, so the complement of its regular locus Xreg has codimension at least two. Since the geometric fibers of Xreg → Spec R are also regular and X/R is flat, in fact Xreg /R is smooth, and the sheaves in the complex ΩX· reg /R are

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locally free. The pushforward j∗ ΩX· reg to all of X is called the complex of Zariski or Danilov differentials and has been extensively studied. In particular, if R = C, then Danilov [7] showed that the hypercohomology of this complex calculates the singular cohomology of the analytic space associated to X. If R is flat over Z, these Danilov differentials are the quasi-coherent sheaves associated to the modules Ω ·R[Q]/R defined in Definition 2.3, as explained in [20, V.2.1.1.2] and [20, V.2.3.15]. The flatness hypothesis was unfortunately neglected in these assertions, and an example due to Simon Felten [10, Example 7.5] shows that it is not superfluous. (We give a slightly simpler example below.) Although the complex of Danilov differentials also satisfies a Cartier isomorphism [5], we are forced to use instead the complex Ω ·R[Q]/R , since the notion of “Cartier type,” requires commutation with base change, which is not always the case for the Danilov differentials. For more details about this issue, we refer to the errata pages associated to [20], currently available at https://math.berkeley.edu/ ogus/loggeometryerrata.pdf. Example A.1 If X = Spec R[Q], (with R regular and Q fine and saturated), then its module of Danilov differentials Ω˜ R[Q]/R is the Qgp -graded submodule of R[Q] ⊗ Qgp which in degree q is the intersection of the submodules R ⊗ F gp as F ranges over the facets of Q containing q, as explained in the course of the proof of [20, V.2.3.13] and in [7, 4.3]. For example, let p be a prime and let Q be the monoid given by generators a, b, c satisfying the relation a + b = pc. This monoid is fine and saturated, and its facets F1 and F2 are the submonoids generated by a and b gp gp respectively. Thus F1 ∩ F2 = {0}, but a and −b become equal in Qgp ⊗ Fp , so gp gp gp (F1 ⊗ Fp ) ∩ (F2 ⊗ Fp ) = F1 ⊗ Fp . On the other hand, recall from Definition 2.3 that Ω 1Q/Z is the Q-graded submodule of Z[Q] ⊗ Qgp which in degree q is #q$gp . ∼ Fp . = 0 while Ω˜ 1 Thus we find that, in degree 0, Ω 1 = Fp [Q]/Fp

Fp [Q]/Fp

Although the complex Ω ·R[Q]/R cannot be computed as the pushforward of the de Rham complex on the regular locus of Spec R[Q], it can be viewed as the pushforward of the de Rham complex on a toric resolution of singularities. This will follow from the following result, inspired by ideas of Danilov [6, 1.5]. Proposition A.2 Let R be a ring, let K be an ideal of a toric monoid Q, and let f : XK → X be the (normalized) blowup of X := Spec R[Q] along the ideal R[K]. Then the natural map Ω ·X/R → f∗ Ω ·XK /R is an isomorphism. In particular, K can be chosen so that XK is smooth, in which case Ω ·XK /R ∼ = ΩX· /R . K

Proof Choose generators (k1 , . . . , km ) for K, and for each i, let Qi be the saturation of the submonoid of Qgp generated by Q and {kj − ki : j = 1, . . . , m}. Then {Xi := Spec R[Qi ] : i = 1, . . . , m} is an open affine cover of XK , and for each j

De Rham–Witt for Toroidal Singularites j

315 j

Γ (XK , Ω XK /R ) = ∩i {Ω R[Qi ]/R : i = 1, . . . , m}. All these modules are Qgp -graded, and the intersection formula above holds for j each degree. In particular, for each y ∈ Qgp , the degree y part of Γ (XK , Ω XK /R ) j

vanishes unless Ω R[Qi ]/R,y is nonzero for every i, and, in particular, unless y ∈ ∩{Qi : i = 1, . . . , m}. Assume henceforth that this is the case. It follows from [20, II,1.7.7] that the map ∪{Spec Qi : i = 1, . . . , m} → Spec Q is surjective. Thus, for every face F of Q, there exist some i and some face Fi of Qi such that Fi ∩ Q = F . Then the homomorphism QF → Qi Fi is local, and if F is a facet, also exact [20, 4.2.1]. Since y ∈ Qi ⊆ Qi Fi , it follows that y ∈ QF , and since this is true for every facet of Q and Q is saturated, it follows that y ∈ Q [20, I,2.4.5]. Now write q for y and let F := #q$. The strong surjectivity of log blowups [20, II,1.7.7] implies that i and Fi can be chosen so that the map (Q/F )gp → (Qi /Fi )gp gp is an isomorphism. This implies that F gp = Fi , and so if qi is the image of q in Qi and #qi $ is the face of Qi it generates, that R ⊗ Λj #q$gp = R ⊗ Λj #qi $gp . We j j conclude that Ω R[Q/R],q = Ω R[Qi /R],q and hence that j

j

Ω R[Q/R],q = Γ (XK , Ω XK /R )q . The fact that K can be chosen to make XK smooth follows from [18, 5.8], and then Ω ·XK /R ∼  = ΩX· /R by [20, V,2.3.11]. K

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