Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties (Lecture Notes in Mathematics, 1959) 9783540705642, 3540705643

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1959

Yukiyoshi Nakkajima · Atsushi Shiho

Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties

ABC

Yukiyoshi Nakkajima

Atsushi Shiho

Department of Mathematics Tokyo Denki University 2-2 Nishiki-cho Kanda Chiyoda-ku Tokyo 101-8457 Japan [email protected]

Graduate School of Mathematical Sciences University of Tokyo 3-8-1, Komaba Meguro-ku Tokyo 153-8914 Japan [email protected]

ISBN: 978-3-540-70564-2 e-ISBN: 978-3-540-70565-9 DOI: 10.1007/978-3-540-70565-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008932186 Mathematics Subject Classification (2000): 14F30 (Primary), 14F40, 13K05 (Secondary) c 2008 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

The main goal of this book is to construct a theory of weights for the log crystalline cohomologies of families of open smooth varieties in characteristic p > 0. This is a p-adic analogue of the theory of the mixed Hodge structure on the cohomologies of open smooth varieties over C developed by Deligne in [23]. We also prove the fundamental properties of the weightfiltered log crystalline cohomologies such as the p-adic purity, the functoriality, the weight-filtered base change theorem, the weight-filtered K¨ unneth formula, the convergence of the weight filtration, the weight-filtered Poincar´e duality and the E2 -degeneration of p-adic weight spectral sequences. One can regard some of these results as the logarithmic and weight-filtered version of the corresponding results of Berthelot in [3] and K. Kato in [54]. Following the suggestion of one of the referees, we have decided to state some theorems on the weight filtration and the slope filtration on the rigid cohomology of separated schemes of finite type over a perfect field of characteristic p > 0. This is a p-adic analogue of the mixed Hodge structure on the cohomologies of separated schemes of finite type over C developped by Deligne in [24]. The detailed proof for them is given in another book [70] by the first-named author. We have to assume that the reader is familiar with the basic premises and properties of log schemes ([54], [55]) and (log) crystalline cohomologies ([3], [11], [54]). We hope that the findings in this book will serve as a role as a first step to understanding the rich structures which p-adic cohomology theory should have. Tokyo January 2008

Yukiyoshi Nakkajima Atsushi Shiho

v

Acknowledgements

We would like to thank K. Bannai, as the genesis of this work was a discussion with him in Azumino in 2000. Three lectures of P. Berthelot in Dwork trimester in 2001 [8] were also very helpful for us. Most key ideas in §1.1 and §1.2 can be attributed to him. We are deeply grateful to him for his permission to include some of his findings in this book. We would also like to thank to K. Sato for a useful discussion. We gave some proofs in this book during both authors’ stay at Padova University and during the firstnamed author’s stay at Universit´e de Rennes 1. Finally, we are very grateful to B. Chiarellotto, B. Le Stum and the universities for their hospitality.

vii

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Preliminaries on Filtered Derived Categories and Topoi . . 1.1 Filtered Derived Category. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Filtered Derived Category. II.–RHom and ⊗L . . . . . . . . . . . . . . 1.3 Filtered Derived Category. III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Some Remarks on Filtered Derived Categories . . . . . . . . . . . . . 1.5 The Topos Associated to a Diagram of Topoi. I . . . . . . . . . . . . 1.6 The Topos Associated to a Diagram of Topoi. II . . . . . . . . . . . .

15 15 26 35 37 41 45

2

Weight Filtrations on Log Crystalline Cohomologies . . . . . . 2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Log Linearization Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves . . . . . 2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Well-Definedness of the Preweight-Filtered Restricted Crystalline and Zariskian Complexes . . . . . . . . . . . . . . . . . . . . . . 2.6 The Preweight Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Vanishing Cycle Sheaf and the Preweight Filtration . . . . . 2.8 Boundary Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The Functoriality of the Preweight-Filtered Zariskian Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The Base Change Theorem and the K¨ unneth Formula . . . . . . . 2.11 Log Crystalline Cohomology with Compact Support . . . . . . . . 2.12 Filtered Log de Rham-Witt Complex . . . . . . . . . . . . . . . . . . . . . 2.13 Filtered Convergent F -isocrystal . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Specialization Argument in Log Crystalline Cohomology . . . . . 2.15 The E2 -degeneration of the p-adic Weight Spectral Sequence of an Open Smooth Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 64 84 93 102 112 116 127 136 143 156 175 192 196 199

ix

x

Contents

2.16 The Filtered Log Berthelot-Ogus Isomorphism . . . . . . . . . . . . . 2.17 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Family of Open Smooth Varieties . . . . . . . . . . . . . . . . . . . . . 2.18 Strict Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 The Weight-Filtered Poincar´e Duality . . . . . . . . . . . . . . . . . . . . . 2.20 l-adic Weight Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A

Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries for Later Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Rigid Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Weight Filtration on Rigid Cohomology . . . . . . . . . . . . . . . . . . . 3.5 Slope Filtration on Rigid Cohomology . . . . . . . . . . . . . . . . . . . . . 3.6 Weight Filtration on Rigid Cohomology with Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202 205 210 215 216

219 219 226 228 234 240 244

Relative SNCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Introduction

Though they are vague, we have the following dreams as in [37], [22] and [25]: (1) Cohomologies in characteristic 0 of algebro-geometric objects in any characteristic have remarkable increasing filtrations, which are called weight filtrations. (2) They are motivic. (3) They are constructible sheaves. Sometimes they are, in fact, smooth sheaves. (4) They are compatible with canonical operations, e.g., base change, K¨ unneth formula, Poincar´e duality. (5) They are functorial: certain classes of morphisms (e.g., the induced morphisms by the morphisms of algebro-geometric objects) are strictly compatible with them. In this book, for a family of open smooth varieties in characteristic p > 0, we discuss the p-adic aspects of (1), (3), (4), (5) and the following new aspect: (6) In some cases, they grow and they are rigid as Grothendieck said for crystalline sheaves. Here we assume that the family is the complement of a relative simple normal crossing divisor on a family of smooth varieties. Before explaining our results, we recall Deligne’s result on the weight filtration on the higher direct image of Q by a morphism from a family of open smooth algebraic varieties with good compactifications to a base scheme over the complex number field C ([23], [25]). Let U be a smooth variety over C. Let X be a smooth variety over C with ⊂ a simple normal crossing divisor D such that U = X \ D. Let j : U −→ X (0) be the natural open immersion. Set D := X and, for a positive integer k, let D(k) be the disjoint union of all k-fold intersections of the different irreducible components of D. Let P := {Pk }k∈Z be the weight filtration on the sheaf ΩiX/C (log D) (i ∈ N) of the logarithmic differential forms on X which Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties. Lecture Notes in Mathematics 1959, c Springer-Verlag Berlin Heidelberg 2008


1

2

Introduction

is obtained by counting the number of the logarithmic poles of local sections of ΩiX/C (log D). Let a(k) : D(k) −→ X (k ∈ N) be the natural morphism of schemes over C. Then we have the following isomorphism of complexes (the Poincar´e residue isomorphism): (0.0.0.1)

∼

•−k • (k) Res : grP (D/C)(−k)). k ΩX/C (log D) −→ a∗ (ΩD (k) /C ⊗Z
(k)

Here
(k) (D/C) is the orientation sheaf of D(k) /C;
(k) (D/C)(−k) := kZ in [23]. By using the isomorphism (0.0.0.1), we have the following spectral sequence (0.0.0.2) E1−k,h+k = H h−k (D(k) , Ω•D(k) /C ⊗Z
(k) (D/C))(−k) =⇒ H h (X, Ω•X/C (log D)). Moreover we have the following isomorphisms in the filtered derived category of bounded below filtered complexes of CXan -modules: (0.0.0.3)

∼

∼

(Ω•Xan /C (log Dan ), P ) ←− (Ω•Xan /C (log Dan ), τ ) −→ ∼

(jan∗ (Ω•Uan /C ), τ ) −→ (Rjan∗ (CUan ), τ ) = (Rjan∗ (ZUan ) ⊗Z C, τ ), where τ := {τk }k∈Z is the canonical filtration. By using the exponential sequence on Uan and the cup product, we have the purity isomorphism (0.0.0.4)

∼

(k)

Rk jan∗ (ZUan ) ←− aan∗ (
(k) (Dan /C))(−k)

(k ∈ N)

(cf. [58, (1.5.1)]). By the same calculation as that in [69, (3.3)], the following morphism (k)

aan∗ (C(Dan )(k) ⊗Z
(k) (Dan /C))(−k)

(0.0.0.5) (0.0.0.4)⊗C ∼

←− (0.0.0.3)

=

Res

Rk jan∗ (CUan ) •+k grP k ΩXan /C (log Dan )

∼

aan∗ (Ω•(Dan )(k) /C ⊗Z
(k) (Dan /C))(−k)

=

aan∗ (C(Dan )(k) ⊗Z
(k) (Dan /C))(−k)

−→

(k) (k)

is equal to the multiplication by (−1)k . Hence we use the following isomorphism (0.0.0.6) (−1)k

(0.0.0.4) ∼

(k)

∼

(k)

Rk jan∗ (ZUan ) −→ aan∗ (
(k) (Dan /C))(−k) −→ aan∗ (
(k) (Dan /C))(−k)

Introduction

3

instead of (0.0.0.4). The isomorphism (0.0.0.6) is equal to the isomorphism in [23, (3.1.9)]. Using the Leray spectral sequence for two functors jan∗ and Γ (Xan , ?), using the isomorphism (0.0.0.6) and renumbering Er−k,h+k := h−k,k Er+1 , we have the following spectral sequence (0.0.0.7)

E1−k,h+k = H h−k ((Dan )(k) ,
(k) (Dan /C))(−k) =⇒ H h (Uan , Z).

If X is proper, one obtains the weight filtration on H h (Uan , Z) by the spectral sequence (0.0.0.7); if X is proper, (0.0.0.2) is equal to (0.0.0.7)⊗Z C by GAGA and the Poincar´e lemma. In fact, the existence of the weight filtration above modulo torsion has been generalized to the case of a family in characteristic 0 by Deligne as follows. Let f : X −→ S be a proper smooth morphism of schemes of finite type over C. Let D be a relative simple normal crossing divisor on X over S. Set U := X \ D, and let f also denote the structural morphism f : U −→ S. Then Rh fan∗ (QUan ) is a local system ([21, II (6.14)]), and there exists a filtration P on Rh fan∗ (QUan ) by sub local systems such that the induced filtration Ps on the stalk Rh fan∗ (QUan )s = H h ((Uan )s , Q) (s ∈ San ) ([loc. cit.]) is obtained from the spectral sequence (0.0.0.7) ([25]). Now let us turn to the case of characteristic p > 0. Let (S, I, γ) be a PD-scheme with a quasi-coherent ideal sheaf I. Set S0 := SpecS (OS /I). Assume that p is locally nilpotent on S. Let f : X −→ S0 be a smooth scheme with a relative simple normal crossing divisor D on X over S0 (by abuse of notation, we denote by the same symbol f the composite f

⊂

morphism X −→ S0 −→ S). Then the pair (X, D) of S0 -schemes defines an fs(=fine and saturated) log scheme (X, M (D)) over S0 (§2.1 below, cf. [54]) in the sense of Fontaine-Illusie-Kato and (X, D)/S defines a log crystalline
topos ((X, M (D))/S)log ([54], cf. [29]). By abuse of notation, we often denote crys


(X, M (D)) by (X, D). Once we obtain the topos ((X, D)/S)log crys , we can use powerful techniques of [42] and many techniques of [3] (cf. [54]). Let O(X,D)/S
be the structure sheaf in ((X, D)/S)log and O (k) the structure sheaf crys

D

/S

(k) /S) in the classical crystalline topos (D
crys . (See (2.2.13.2) and (2.2.15) below for the precise definition of D(k) for a nonnegative integer k.) Let
(k) /S)


D)/S)log f(X,D)/S : ((X, crys −→ Szar be the crys −→ Szar and fD (k) /S : (D natural morphisms of topoi. Then one of our main results in this book is to show the existence of the following functorial spectral sequence:

(0.0.0.8)

(k) E1−k,h+k = Rh−k fD(k) /S∗ (OD(k) /S ⊗Z
crys (D/S))(−k)

=⇒ Rh f(X,D)/S∗ (O(X,D)/S ). (k) (k) /S) Here
crys (D/S) is the crystalline orientation sheaf of D/S in (D
crys which will be defined in §2.2. If S0 is of characteristic p > 0, then the relative

4

Introduction

Frobenius morphism F : (X, D) −→ (X  , D ) over S0 induces the relative  Frobenius morphism F (k) : D(k) −→ D(k) = D(k) . Let a(k) : D(k) −→ X   and a(k) : D(k) −→ X  be the natural morphisms. We define the relative Frobenius action (k)

(k) (k) Φ(k) : acrys∗
crys (D /S) −→ Fcrys∗ acrys∗
crys (D/S) (k)

as the identity under the natural identification ∼

(k) (k)
crys (D /S) −→ Fcrys∗
crys (D/S). (k)

Then (0.0.0.8) is compatible with the Frobenius action. We call (0.0.0.8) the preweight spectral sequence of (X, D)/(S, I, γ). Here, as noted in [68], we use the terminology “preweight” instead of the terminology “weight” since OS is a sheaf of torsion modules (and hence Rh f(X,D)/S∗ (O(X,D)/S ) is also). If S0 is the spectrum of a perfect field κ of characteristic p > 0 and if S is the spectrum of the Witt ring Wn of finite length n > 0 of κ, then (0.0.0.8) is canonically isomorphic to the following preweight spectral sequence (k) (k) /W ) E1−k,h+k = H h−k ((D
n crys , OD (k) /Wn ⊗Z
crys (D/Wn ))(−k) h ((X, D)/Wn ), =⇒ Hlog-crys

essentially constructed in [65] and [68]. Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field of characteristic p > 0. Then we can also construct the spectral sequence (0.0.0.8) when S is a p-adic formal V -scheme in the sense of [74]. In this case, we call (0.0.0.8) the p-adic weight spectral sequence of (X, D)/S and the induced filtration on Rh f(X,D)/S∗ (O(X,D)/S ) by (0.0.0.8) the weight filtration on Rh f(X,D)/S∗ (O(X,D)/S ). Let us return to the case where (S, I, γ) is a PD-scheme as above; especially, a prime number p is locally nilpotent on S. Let OX/S be  crys . Let the structure sheaf in the classical crystalline topos (X/S) 


D)/S)log u(X,D)/S : ((X, crys −→ Xzar (resp. uX/S : (X/S)crys −→ Xzar ) be the natural (resp. classical) projection. Then uX/S induces a mor crys , O
zar , f −1 (OS )) of ringed topoi. Let phism u : ((X/S) ) −→ (X X/S

X/S

 (k) /S) (k) uD(k) /S : (D
zar be also the classical projection. Let crys −→ D 
(X,D)/S : ((X, D)/S)log crys −→ (X/S)crys be the forgetting log morphism ∗ ) of log schemes. Then induced by the morphism (X, M (D)) −→ (X, OX   crys be a uX/S ◦ (X,D)/S = u(X,D)/S . Let QX/S : (X/S)Rcrys −→ (X/S) morphism of topoi defined in [3, IV (2.1.1)]. Then we have a morphism  Rcrys , Q∗ (OX/S )) −→ ((X/S)  crys , OX/S ) of ringed topoi. QX/S : ((X/S) X/S −1  Rcrys , Q∗ (O
Set uX/S := uX/S ◦ QX/S : ((X/S) (OS )) X/S )) −→ (Xzar , f X/S

as in [3].

Introduction

5

To construct (0.0.0.8) and to prove the functoriality of it, we define two filtered complexes (0.0.0.9) (Ecrys (O(X,D)/S ), P ) := (Ecrys (O(X,D)/S ), {Pk Ecrys (O(X,D)/S )}k∈Z ) ∈ D+ F(OX/S ), (0.0.0.10) (Ezar (O(X,D)/S ), P ) := (Ezar (O(X,D)/S ), {Pk Ezar (O(X,D)/S )}k∈Z ) ∈ D+ F(f −1 (OS )) and construct two other filtered complexes (0.0.0.11) (CRcrys (O(X,D)/S ), P ) := (CRcrys (O(X,D)/S ),{Pk CRcrys (O(X,D)/S )}k∈Z ) ∈ D+ F(Q∗X/S (OX/S )), (0.0.0.12) (Czar (O(X,D)/S ), P ) := (Czar (O(X,D)/S ), {Pk Czar (O(X,D)/S )}k∈Z ) ∈ D+ F(f −1 (OS )). Here D+ F(OX/S ), D+ F(Q∗X/S (OX/S )) and D+ F(f −1 (OS )) are the filtered derived categories of the bounded below filtered OX/S -modules, Q∗X/S (OX/S )modules and f −1 (OS )-modules, respectively. The definitions of (Ecrys (O(X,D)/S ), P ) and (Ezar (O(X,D)/S ), P ) are as follows: (Ecrys (O(X,D)/S ), P ) := (R(X,D)/S∗ (O(X,D)/S ), τ ), (Ezar (O(X,D)/S ), P ) := RuX/S∗ (Ecrys (O(X,D)/S ), P ). (Here τ denotes the canonical filtration (§2.7).) Note that they are functorial with respect to (X, D). (This is not the case for (CRcrys (O(X,D)/S ), P ).) In a simple case we soon give the definition of (CRcrys (O(X,D)/S ), P ) in (0.0.0.14) below. In the general case we give it in the text. The filtered complex (Czar (O(X,D)/S ), P ) is, by definition, RuX/S∗ (CRcrys (O(X,D)/S ), P ). We call (Ecrys (O(X,D)/S ), P ) and call (Ezar (O(X,D)/S ), P ) the preweightfiltered vanishing cycle crystalline complex and the preweight-filtered vanishing cycle zariskian complex of (X, D)/(S, I, γ), respectively. We also call (CRcrys (O(X,D)/S ), P ) the preweight-filtered restricted crystalline complex of (X, D)/(S, I, γ) and call (Czar (O(X,D)/S ), P ) the preweight-filtered zariskian complex of (X, D)/(S, I, γ), respectively. The main theme of this book is to investigate fundamental properties of (Ecrys (O(X,D)/S ), P ), (Ezar (O(X,D)/S ), P ), (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ). They enjoy the following properties:

6

Introduction ∼

(0.0.0.13): (CRcrys (O(X,D)/S ), P ) ←− Q∗X/S (Ecrys (O(X,D)/S ), P ); {Pk Ecrys (O (X,D)/S )}k∈Z is an “increasing filtration” on Ecrys (O(X,D)/S ) which is finite ∼ locally on X such that P−1 Ecrys (O(X,D)/S ) = 0, Q∗X/S P0 Ecrys (O(X,D)/S ) ←− ∼

Q∗X/S (OX/S ) and CRcrys (O(X,D)/S ) ←− Q∗X/S R(X,D)/S∗ (O(X,D)/S ). (0.0.0.14): Let ∆ :=  {Dλ }λ∈Λ be a decomposition of D by smooth components of D: D = λ∈Λ Dλ and each Dλ is smooth over S0 . If (X, D) has ⊂

an admissible immersion (X, D) −→ (X , D) over S with respect to ∆ (see (2.1.10) below for the definition of the admissible immersion), then (CRcrys (O(X,D)/S ), P )  (Q∗X/S LX/S (Ω•X /S (log D)), {Q∗X/S LX/S (Pk Ω•X /S (log D))}k∈Z ), where LX/S is the classical linearization functor for OX -modules ([3, IV 3], [11, §6]). ∼

(0.0.0.15): (Czar (O(X,D)/S ), P ) ←− (Ezar (O(X,D)/S ), P ). (Hence (Czar (O(X,D)/S ), P ) is functorial with respect to (X, D).) In particular, (0.0.0.16): {Pk Czar (O(X,D)/S )}k∈Z is an “increasing filtration” on Czar (O (X,D)/S ) which is finite locally on X such that P−1 Czar (O(X,D)/S ) = 0, ∼ ∼ P0 Czar (O(X,D)/S ) ←− RuX/S∗ (OX/S ), and Czar (O(X,D)/S ) ←− Ru(X,D)/S∗ (O(X,D)/S ), and ⊂

(0.0.0.17): If (X, D) has an admissible immersion (X, D) −→ (X , D) over S with respect to ∆ = {Dλ }λ∈Λ , (Czar (O(X,D)/S ), P )  (OD ⊗OX Ω•X /S (log D), {OD ⊗OX Pk Ω•X /S (log D)}k∈Z ), ⊂

where D is the PD-envelope of the immersion X −→ X over (S, I, γ). ∗ (0.0.0.18): grP k (CRcrys (O(X,D)/S )) = QX/S acrys∗ (OD (k) /S ⊗Z
crys (D/S)) {−k}, where {−k} is the shift which will be defined in the Convention (1) below (note that we do not consider the Tate twist (−k) on the right hand side of (0.0.0.18) because the functor Q∗X/S appears on the right hand side (In [3, IV (2.5)] Berthelot has noted that the restricted crystalline topos does not have the functoriality in general).). (k)

(k)

(k)

(k)

(0.0.0.19): grP k (Czar (O(X,D)/S )) = azar∗ (RuD (k) /S∗ (OD (k) /S ) ⊗Z
zar (D/S0 )) (k)

(−k){−k}, where
zar (D/S0 ) is the zariskian orientation sheaf of D/S0 which will be defined in §2.2. Here, see §2.9 for the meaning of the Tate twist (−k). ∼

(0.0.0.20): (CRcrys (O(X,D)/S ), τ ) −→ (CRcrys (O(X,D)/S ), P ), where τ is the canonical filtration.

Introduction

7

(0.0.0.21): If S0 is the spectrum of a perfect field κ of characteristic p and if S = Spec(Wn (κ)) (n > 0), then (Czar (O(X,D)/S ), P ) is canonically isomorphic to the filtered complex (Wn Ω•X (log D), P ) := (Wn Ω•X (log D), {Pk Wn Ω•X (log D)}k∈Z ) in [65]. Thus we obtain the following translation which let us recall Grothendieck’s project to unify algebra, geometry and analysis ([37]): (0.0.0.22) /C

crystal log

Uan , (Xan , Dan )

log ((X
an , Dan ))et ([51])


((X, D)/S)log crys

 Xan , X an


(X/S) crys ⊂

jan : Uan −→ Xan top : (Xan , Dan )log −→ Xan log  an : ((X
an , Dan ))et −→ Xan



(X,D)/S : ((X, D)/S)log crys −→ (X/S)crys

Rjan∗ (Z) = Rtop∗ (Z) ([58]), Q∗X/S R(X,D)/S∗ (O(X,D)/S )

Rtop∗ (Z/n) = Ran∗ (Z/n) (n ∈ Z) ([72]) Xan −→ X

 crys −→ X
zar uX/S : (X/S)

Z(Xan ,Dan )log (Z/n)(Xan ,Dan )log , (Z/n)((X
,D

O(X,D)/S

(Z/n)Xan (n ∈ Z) (Ω•Xan /C (log Dan ), τ )

OX/S (CRcrys (O(X,D)/S ), τ )

= (Ω•Xan /C (log Dan ), P )

= (CRcrys (O(X,D)/S ), P )

(Ω•X/C (log D), P )

(Czar (O(X,D)/S ), P )

(n ∈ Z) ZXan

an

log an ))et

Here (Xan , Dan )log is the real blow up of (Xan , Dan ) defined in [58] and top is the natural morphism of topological spaces which is denoted by τ in  [loc. cit.], and X an is the topos defined by the local isomorphisms to Xan and an is the natural morphism forgetting the log structure. To construct (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ), we use local admissible immersions of (X, D) over S, which are local exact closed immersions. On the other hand, in the case where S0 is the spectrum of a perfect field κ of characteristic p > 0 and where S = Spec(Wn (κ)), Mokrane has used local lifts of (X, D) over S in [64] and [65] in order to construct the filtered log de Rham-Witt complex (Wn Ω•X (log D), P ). Our guiding principle

8

Introduction

is: if we can do something by using local lifts, we can do something analogous and more by using admissible immersions. Since a standard exactification of the product of two local lifts of (X, D) is not a local lift of (X, D) at all, the notion “local lift” is not flexible for the construction of the spectral sequence (0.0.0.8). Moreover, because we do not take a local lift of (X, D), we can give a simple proof of the independence of (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ) of the choice of the open covering of (X, D) and that of the admissible immersion of each open log scheme: we do not need a concrete (slightly) laboring key calculation in [47]; in a future paper we shall develop analogous theory for a family of simple normal crossing log varieties over log points, and we shall show that a concrete key calculation in [48] and [64] is not necessary. Furthermore, we come to know that the filtered log de Rham-Witt complex of an open variety is not a necessary ingredient for the construction of (0.0.0.8) (in the special case S = Spec(Wn (κ))). We are sure that it is natural to capture something producing (0.0.0.8) as an object in a filtered derived category; to capture it as a real filtered complex is not flexible. However, because it is anyway possible to capture it as a real filtered complex in the case above, it gives us something deep in a special case. Indeed, in [70], we use the simplicial version of the filtered log de Rham-Witt complex above for the proof of a variant of the Serre-Grothendieck formula on the virtual Betti numbers of a separated scheme of finite type over κ, which has been conjectured in [37]. From the filtered object (Czar (O(X,D)/S ), P ), we immediately obtain the spectral sequence (0.0.0.8) by (0.0.0.19). If one wishes to obtain only (0.0.0.8), the object (Czar (O(X,D)/S ), P ) is enough; however, it is an important fact that something producing (0.0.0.8) exists not only in D+ F(f −1 (OS )) but also in a “higher stage” D+ F(Q∗X/S (OX/S )). By (0.0.0.17), it is well-known that the canonical filtration τ and the preweight filtration P on Czar (O(X,D)/S ) do not coincide in general; however, impressively, τ and P on CRcrys (O(X,D)/S ) coincide ((0.0.0.20)). The equality (0.0.0.20) follows from the following p-adic purity (0.0.0.23)

Q∗X/S Rk (X,D)/S∗ (O(X,D)/S ) (k) (D/S))(−k) (k ∈ N), = Q∗X/S acrys∗ (OD(k) /S ⊗Z
crys (k)

which will be proved in §2.7. The reason why we obtain the equality (0.0.0.20) is that (CRcrys (O(X,D)/S ), P ) exists in the world of the classical restricted crystalline topos; we can consider divided powers “a[n] = an /n!” (n ∈ N) in the restricted crystalline topos and we can use a Poincar´e lemma in it. By (0.0.0.23) and the Poincar´e lemma for R(X,D)/S∗ (O(X,D)/S ) which will be proved in the text, we obtain (0.0.0.13) and then an important property of (Czar (O(X,D)/S ), P ): it is functorial, that is, for another smooth scheme X  with a relative simple normal crossing divisor D over S0 and for a morphism g : (X, D) −→ (X  , D ) of log schemes over S0 in the sense of Fontaine-Illusie-

Introduction

9

Kato, we have a natural morphism g ∗ : (Czar (O(X  ,D )/S ), P ) −→ Rg∗ (Czar (O(X,D)/S ), P ). Finally in this rough explanation of the book, we remark that the following naive p-adic purity (k)

(k) Rk (X,D)/S∗ (O(X,D)/S ) = acrys∗ (OD(k) /S ⊗Z
crys (D/S))(−k)

(k ∈ N)

does not hold in general, which will be proved in Remark 2.7.11. For this reason, we have to consider the undesirable functor Q∗X/S . Now we outline the contents of this book. In Chapter 1, we prove preliminary results which we use in later chapters. From §1.1 to §1.4, we show some facts on filtered modules in a ringed topos which are necessary for later sections. Many key notions and many results are due to P. Berthelot. Especially, the notions (co)special modules and strictly injective resolutions are due to him. The notion strictly flat resolutions is also due to him. The filtered adjunction formula, which is also due to him, is a key ingredient for the proof of the filtered base change theorem of (Ecrys (O(X,D)/S ), P ). In §1.5 and §1.6 we review general facts on diagrams of topoi for this book and future papers. In Chapter 2, which is the main body of this book, we construct the theory of the weight filtration of the log crystalline cohmologies of families of open smooth varieties. In §2.1 we give the definition of a relative simple normal crossing divisor and a key notion admissible immersion and give a local description of the admissible immersion. In §2.2 we recall a (log) linearization functor and we calculate the graded pieces of Q∗X/S LX/S (Ω•X /S (log D)). In §2.3 we prove a crystalline Poincar´e lemma for RY /S∗ (E) for a morphism g : Y −→ S0 of fine log schemes which can be embedded into a fine log ◦

smooth scheme Y/S whose underlying scheme Y/S is also smooth and for a crystal E of OY /S -modules. In §2.4 we construct (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ) for an open covering of X and an admissible immersion of each open log subscheme of (X, D). In §2.5 we prove the independence of (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ) of the choice of the open covering of X and that of the admissible immersion of each open log subscheme of (X, D). In §2.6 we prove (0.0.0.18) and (0.0.0.19). In §2.7 we calculate the restriction of the vanishing cycle sheaves Q∗X/S Rk (X,D)/S∗ (O(X,D)/S ) (k ∈ Z) and we prove the p-adic purity (0.0.0.23). Then we prove (0.0.0.13) and (0.0.0.20); as a corollary, we immediately see that (CRcrys (O(X,D)/S ), P ) and

10

Introduction

(Czar (O(X,D)/S ), P ) are independent of the choice of the decomposition of D by smooth components of D. In §2.8 we give the description of the boundary morphism of the E1 -terms of the spectral sequence (0.0.0.8). In §2.9 we prove the functoriality of (Czar (O(X,D)/S ), P ). In §2.10 we prove the filtered base change theorem of (Ecrys (O(X,D)/S ), P ), and we prove the filtered K¨ unneth formula of (Ecrys (O(X,D)/S ), P ). As in [3] and [11], we have some important corollaries of the filtered base change theorem. In §2.11 we develop the analogous theory for a log crystalline cohomology with compact support, and, especially, we obtain the preweight spectral sequence of a log crystalline cohomology with compact support. In §2.12 we prove that, if S0 is the spectrum of a perfect field κ of characteristic p > 0 and if S = Spec(Wn (κ)), then (Czar (O(X,D)/S ), P ) is canonically isomorphic to (Wn Ω•X (log D), P ). In §2.13 we prove the convergence of the weight filtration as a corollary of the filtered base change theorem in §2.10. This is a filtered version of the convergence of a (log) crystalline cohomology in [74], [76] and [29]. In §2.14 we give a specialization argument of Deligne-Illusie ([49]) in our situation. In §2.15 we prove the E2 -degeneration of (0.0.0.8) modulo torsion in the case where S0 := Spec(κ) and S = Spf(W (κ)). The proof in this book is another, more natural proof than the proof in [68]; In §2.16 we prove that a filtered log Berthelot-Ogus isomorphism is strictly compatible with the weight filtration. In §2.17 we generalize the E2 -degeneration of (0.0.0.8) modulo torsion to the case where the base scheme S is a p-adic formal V -scheme in the sense of [74], by using the results in the previous two sections. Using this generalized result, we reprove the convergence of the weight filtration by reducing it to a result in [74]. In §2.18 we prove the strict compatibility of the weight filtration with respect to the induced morphism of log schemes by using the specialization argument in §2.14 and the convergence of the weight filtration. In §2.19 we prove the strict compatibility of the weight filtration with the Poincar´e duality. In §2.20 we give a remark on the corresponding l-adic weight spectral sequence. Following the suggestion of one of the referees, we state some results obtained in [70] in Chapter 3 to answer natural questions arising from results in Chapter 2. For a (separated) scheme of finite type U over C, P. Deligne has endowed H h (Uan , Q) (h ∈ Z) with the mixed Hodge structure in [24] by using results in [23]. In [70] the first-named author has succeeded in defining the weight filtration on the rigid cohomology of a separated scheme U of finite type over a perfect field κ of characteristic p > 0 by using de Jong’s alteration theorem

Introduction

11

([28]), Tsuzuki’s proper descent ([87]), Shiho’s comparison theorems ([82]) and results up to Chapter 2. In §3.1 we give preliminaries for later sections: the single complex of a complex in a multisimplicial topos in [24], the diagonal filtration in [loc. cit.], and the key functor Γ in [19], [87] and [88]. In §3.2 we give a reformulation of Tsuzuki’s proper descent on rigid cohomology. ⊂ Let U −→ U be an open immersion into a proper scheme over κ. In §3.3 we state comparison theorems between the rigid cohomology of U , the log convergent cohomology of a certain split proper hypercovering of (U, U ) and the log crystalline cohomology of the proper hypercovering. As an application of the comparison theorems, we can endow the rigid cohomology with closed support with the weight filtration (§3.4). As another application, we can calculate the slope filtration on the rigid cohomology by the log Hodge-Witt sheaves of the proper hypercovering (§3.5). In §3.6 we state the existence theorem of the weight filtration on the rigid cohomology with compact support. Finally we add the Appendix: we give some results on relative simple normal crossing divisors which are used in the text or related to it. In a future paper we shall construct an analogous theory for a family of semistable varieties, which is a generalization of [64] and [68]. ◦

Notations. (1) For a log scheme Y , Y denotes the underlying scheme of ◦

Y . For a morphism f : Y −→ Z of log schemes, f denotes the underlying ◦

◦

morphism Y −→ Z of schemes. (2) (S)NCD=(simple) normal crossing divisor. Conventions. We assume that the log structures on (formal) schemes are defined on Zariski site unless otherwise stated. Also, we make the following conventions about signs. Let A be an additive exact category. (1) For a complex (E • , d• ) of objects in A and for an integer n, , d•+n ) or (E • {n}, d• {n}) denotes the following complex: (E •+n

dq−1+n

dq+n

dq+1+n

· · · −→ E q−1+n −→ E q+n −→ E q+1+n −→ · · · . q−1

q

q+1

Here the numbers under the objects above in A mean the degrees. For a morphism f : (E • , d•E ) −→ (F • , d•F ) of complexes of objects of A, f {n} denotes a natural morphism (E • {n}, d•E {n}) −→ (F • {n}, d•F {n}) induced by f . A morphism f : (E • , d•E ) −→ (F • , d•F ) in the derived category D (A) ( = b, +, −, nothing) of the complexes of objects in A naturally induces a morphism f {n} : (E • {n}, d•E {n}) −→ (F • {n}, d•F {n}) in D (A).

12

Introduction

(2) For a complex (E • , d• ) of objects in A and for an integer n, (E • [n], d• [n]) denotes the following complex as usual: (E • [n])q := E q+n with boundary morphism d• [n] = (−1)n d•+n . For a morphism f : (E • , d•E ) −→ (F • , d•F ) of complexes of objects of A, f [n] denotes a natural morphism (E • [n], d•E [n]) −→ (F • [n], d•F [n]) induced by f without change of signs. This operation is well-defined in the derived category as in (1). (3) ([10, 0.3.2], [18, (1.3.2)]) For a short exact sequence f

g

0 −→ (E • , d•E ) −→ (F • , d•F ) −→ (G• , d•G ) −→ 0 of bounded below complexes of objects in A, let (E • [1], d•E [1]) ⊕ (F • , d•F ) be the mapping cone of f . We fix an isomorphism “(E • [1], d•E [1]) ⊕ (F • , d•F ) (x, y) −→ g(y) ∈ (G• , d•G )” in the derived category D+ (A). (4) ([10, 0.3.2], [18, (1.3.3)]) Under the situation (3), the boundary morphism (G• , d•G ) −→ (E • [1], d•E [1]) in D+ (A) is the following composite morphism ∼

proj.

(−1)×

(G• , d•G ) ←− (E • [1], d•E [1]) ⊕ (F • , d•F ) −→ (E • [1], d•E [1]) −→ (E • [1], d•E [1]). (5) Let A be an abelian category with enough injectives. Let F : A −→ B be a left exact functor of abelian categories. Then, in the situation (3), the boundary morphism ∂ : Rq F((G• , d•G )) −→ Rq+1 F((E • , d•E )) of cohomologies is, by definition, the induced morphism by the morphism (G• , d•G ) −→ (E • [1], d•E [1]) in (4). By taking injective resolutions (I • , d•I ), (J • , d•J ) and (K • , d•K ) of (E • , d•E ), (F • , d•F ) and (G• , d•G ), respectively, which fit into the following commutative diagram (0.0.0.24) 0 −−−−→ (I • , d•I ) −−−−→ (J • , d•J ) −−−−→ (K • , d•K ) −−−−→ 0    ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 0 −−−−→ (E • , d•E ) −−−−→ (F • , d•F ) −−−−→ (G• , d•G ) −−−−→ 0 of complexes of objects in A such that the upper horizontal sequence is exact, it is easy to check that the boundary morphism ∂ above is equal to the usual boundary morphism obtained from the upper short exact sequence of (0.0.0.24). (For a short exact sequence in (3), the existence of the commutative diagram (0.0.0.24) has been proved in, e.g., (1.1.7) below, as a very special case.) (6) For a complex (E • , d• ) of objects in A, the identity id : E q −→ E q ∼ (∀q ∈ Z) induces an isomorphism Hq ((E • , −d• )) −→ Hq ((E • , d• )) (∀q ∈ Z) of cohomologies. We sometimes use this convention. (7) We often denote a complex (E • , d• ) simply by (E • , d) or E • as usual when there is no risk of confusion.

Introduction

13

(8) Let r ≥ 2 be a positive integer. As usual, an r-uple complex of objects in A is, by definition, a pair (E •···• , {di }ri=1 ) such that E m1 ···mr (mi ∈ Z) is an object of A with morphisms di : E •···•,mi ,•···• −→ E •···•,mi +1,•···• satisfying the following relations d2i = 0 and di dj + dj di = 0 (i = j).

Chapter 1

Preliminaries on Filtered Derived Categories and Topoi

In this chapter, we prove some preliminary results on filtered derived categories and topoi which we need in later chapters. In the first four sections, we recall several notions and basic properties concerning filtered derived categories: filtered complex, filtered quasi-isomorphism, filtered derived category, filtered injective resolution, filtered flat resolution, filtered derived functor and filtered adjunction formula. It is important for us because our weight-filtered complexes will be defined as objects in certain filtered derived categories. In the last two sections, we recall the notion of topoi associated to diagrams of topoi and prove several basic properties of the topoi associated to diagrams of (restricted) log crystalline topoi.

1.1 Filtered Derived Category. I In this section we recall the definition of a filtered derived category and calculate derived functors for complexes of filtered modules in a ringed topos (cf. [24, §7], [60], [78], [8]). Many facts in this section are due to Berthelot ([8]). Though many facts in this section hold for abelian categories with enough injectives by using a standard technique reducing the general case to the case of the category of usual modules over a ring with unit element (e.g., [90, Remark on p. 12]), we are content with the case of a ringed topos. Let (T , A) be a ringed topos. Let E be an A-module in T . In this book, a filtration on E is, by definition, a family {Ek } = {Ek }k∈Z of A-submodules of E such that Ek ⊂ Ek+1 for all k ∈ Z; in this book we consider only increasing filtrations indexed by Z except stupid decreasing filtrations in later sections. As in [8], filtrations are not necessarily exhaustive nor separated. We denote by (E, {Ek }) an A-module E with filtration {Ek }. We sometimes denote (E, {Ek }) only by E if there is no risk of confusion. For an A-module E, we mean by the trivial filtration on E a filtration {Ek } such that E0 = E and E−1 = 0. In [8], A may have a nontrivial filtration and A is not necessarily Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties. Lecture Notes in Mathematics 1959, c Springer-Verlag Berlin Heidelberg 2008


15

16

1 Preliminaries on Filtered Derived Categories and Topoi

commutative; in this book, we consider only the case where the filtration on A is trivial and A is commutative; hence many facts in this section are special cases of results in [8]. Let MF(A) be the category of filtered A-modules. Let CF(A) be the category of filtered complexes of A-modules, and let C+ F(A), C− F(A) and Cb F(A) be the categories of the bounded below, bounded above and bounded filtered complexes of A-modules, respectively. Let KF(A) be the category of filtered complexes of A-modules modulo filtered homotopies, and let K+ F(A), K− F(A) and Kb F(A) be the categories of bounded below, bounded above and bounded filtered complexes of A-modules modulo filtered homotopies, respectively. We define the direct image and the inverse image of an object of CF(A) or KF(A) by a morphism of ringed topoi in an obvious way. Since MF(A) is an additive category, K F(A) ( = +, −, b, nothing) is a triangulated category. For a filtered complex E • = (E • , {Ek• }) ∈ CF(A) and for an integer l, we define the shift • ([23, (1.1)]). E • l = (E • , {Ek• })l := (E • , {E • lk }) by E • lk := El+k • • We say that a filtered complex (E , {Ek }) ∈ CF(A) is strictly exact if Hq (E • ) = Hq (Ek• ) = 0 (∀q, ∀k ∈ Z). It is easy to see that, if a filtered complex (E • , {Ek• }) is strictly exact and (E • , {Ek• })  (F • , {Fk• }) in KF(A), then (F • , {Fk• }) is also strictly exact (a special case of [78, Lemma 1.2.12, Remark 1.2.13]); we can define the notion of the strict exactness of a filtered complex (E • , {Ek• }) ∈ KF(A). We say that a filtered morphism f : (E • , {Ek• }) −→ (F • , {Fk• }) in CF(A) is a filtered quasi-isomorphism if the mapping cone of f is strictly exact. It is easy to check that f is a filtered quasi-isomorphism if and ∼ ∼ only if f induces isomorphisms Hq (E • ) −→ Hq (F • ) and Hq (Ek• ) −→ Hq (Fk• ) for all q, k ∈ Z; we can define the notion of filtered quasi-isomorphism in KF(A). Let us consider the set of morphisms (FQis) whose elements are the filtered quasi-isomorphisms in KF(A). Then (FQis) forms a saturated multiplicative system in the sense of [89, II (2.1.1)]: the conditions (SM1) and (SM5) are obviously satisfied; (SM2) and (SM3) can be proved as in [44, I (4.2)]; (SM6) follows from the five lemma; it is easy to check the saturatedness (SM4). Hence we can consider the localization of K F(A) ( = +, −, b, nothing) by the multiplicative system (FQis)|K
F(A) and we obtain a derived category D F(A) := K F(A)(FQis)|K
F(A) . Let K (Γ(T , A)) ( = +, −, b, nothing) be the category of complexes of Γ(T, A)-modules modulo homotopies for  = +, −, b, nothing. Let D (Γ(T , A)) := K (Γ(T , A))(Qis) be its derived category. Let MF(Γ(T , A)) and C F(Γ(T , A)) be the category of filtered Γ(T , A)-modules and that of filtered complexes of Γ(T , A)-modules, respectively. Let K F(Γ(T , A)) be the category of filtered complexes of Γ(T , A)-modules modulo filtered homotopies and let D F(Γ(T , A)) be the derived category K F(Γ(T , A))(FQis) localized by the set of the filtered quasi-isomorphisms in K F(Γ(T , A)). For two objects (E, {El }), (F, {Fl }) ∈ MF(A), we define HomA ((E, {El }), (F, {Fl })) ∈ MF(Γ(T , A))

1.1 Filtered Derived Category. I

17

in a well-known way: HomA ((E, {El }), (F, {Fl })) is, by definition, a Γ(T , A)module HomA (E, F ) with filtration HomA ((E, {El }), (F, {Fl }))k := {f ∈ HomA (E, F ) | f (El ) ⊂ Fl+k (∀l ∈ Z)} (k ∈ Z). An equality (1.1.0.1) HomA ((E, {El }), (F, {Fl }))k = HomMF(A) ((E, {El }), (F, {Fl })k) is immediate to verify. Let f : (E, {El }) −→ (F, {Fl }) be a morphism in MF(A). We say that f is a strict morphism if the inclusion morphism ⊂ f (El ) −→ (Im f ) ∩ Fl (∀l ∈ Z) is an isomorphism. We say that f is a strict monomorphism (resp. strict epimorphism) if f is strict and if f is a monomorphism (resp. epimorphism). We obtain a similar object Hom A ((E, {El }), (F, {Fl })) ∈ MF(A) in a similar way. Following [8], let us define the special module of (E, {El }) for AEl with filtration modules E, El (l ∈ Z): we set A (E, {El }) := E × l∈Z   ( A (E, {El }))k := E × El . Then we have a filtered A-module l≤k

   (E, {El }) = ( (E, {El }), ( (E, {El }))k ). A

A

A



In [8] Berthelot has called A(E, {El }) the special module of (E, {El }). A E −→ F morphism f : (E, {Ek }) −→ A (F, {Fk })  induces a morphism  and a composite morphism E/Ek−1 −→ A (F, {Fl })/( A (F, {Fl }))k−1 =  proj. Fl −→ Fk . The following formula in [8] is easily verified but very useful: l≥k

(1.1.0.2)   HomMF(A) ((E, {Ek }), (F, {Fk })) = HomA (E, F )× HomA (E/Ek−1 , Fk ). A

k∈Z

We recall the following definition due to Berthelot ([8]): Definition 1.1.1 ([8]). Let (J, {Jk }) be an object of MF(A). We say that (J, {Jk }) is strictly injective if J and Jk are injective A-modules and if, for a ⊂ strict monomorphism (E, {Ek }) −→ (F, {Fk }), the induced morphism HomA ((F, {Fk }), (J, {Jk })) −→ HomA ((E, {Ek }), (J, {Jk })) is a strict epimorphism.

18

1 Preliminaries on Filtered Derived Categories and Topoi

Next, we calculate the filtered derived functor of a morphism of ringed topoi. To do this, we consider several additive full subcategories of MF(A): Iflas (A) := {(J, {Jk }) | J and Jk (∀k ∈ Z) are flasque A-modules}, Iinj (A) := {(J, {Jk }) | J and Jk (∀k ∈ Z) are injective A-modules}, Istinj (A) := {(J, {Jk }) | (J, {Jk }) is a strictly injective A-module},  Ispinj (A) := { (I, {Ik }) | I and Ik are injective A-modules (∀k ∈ Z)}. A

We obviously see that Istinj (A) ⊂ Iinj (A) ⊂ Iflas (A). We also see that Ispinj (A) ⊂ Istinj (A), which is due to Berthelot:  Proposition 1.1.2 ([8]). Let A (I, {Ik }) be an object of Ispinj (A). Then  A (I, {Ik }) ∈ Istinj (A). ⊂

Proof. For the completeness of this book, we give  a proof. Let (E, {Ek }) −→ (F, {Fk }) be a strict monomorphism. Since I× k Ik is an injective A-module, the morphism   Ik ) −→ HomA (E, I × Ik ) HomA (F, I × k

k

is surjective. By (1.1.0.1), it suffices to prove that the morphism (1.1.2.1)   HomMF(A) ((F, {Fk }), (I, {Ik })) −→ HomMF(A) ((E, {Ek }), (I, {Ik })) A

A

is surjective. For each integer k, the induced morphism E/Ek−1 −→ F/Fk−1 is a monomorphism. Hence (1.1.2.1) is surjective by (1.1.0.2).   Definition 1.1.3. We say that a filtered module (J, {Jl }) ∈ MF(A) is filteredly flasque, filteredly injective, and specially injective if (J, {Jl }) ∈ Iflas (A), (J, {Jl }) ∈ Iinj (A) and (J, {Jl }) ∈ Ispinj (A), respectively. The category MF(A) has enough special injectives in the following sense: Proposition 1.1.4 ([8]). For a filtered module E = (E, {Ek }) ∈ MF(A),  ⊂  there exists a strict monomorphism E −→ A (I, {Il }) with A (I, {Il }) ∈ Ispinj (A). ⊂

⊂

Proof. Let E −→ I andE/El−1 −→ Il be  monomorphisms into injective Amodules. Set J := I × Il and Jk := I × Il . There are natural monomorl∈Z ⊂

l≤k ⊂

phisms Jk −→ Jk+1 and Jk −→ J. Since I and Ik (k ∈ Z) are injective A-modules, (J, {Jk }) is an object of Ispinj (A).

1.1 Filtered Derived Category. I ⊂

19 proj.

⊂

Two morphisms E −→ I and E −→ E/El−1 −→ Il induce a monomor⊂ ⊂ phism E −→ J; furthermore, two composite morphisms Ek −→ E −→ I and proj. ⊂ ⊂ Ek −→ E −→ E/El−1 −→ Il induce a monomorphism Ek −→ Jk . It remains to prove that the morphism (E, {Ek }) −→ (J, {Jk }) is strict. Set Nk := Im(E −→ J) ∩ Jk . Then  Nk is isomorphic to the kernel of a Il . This kernel is nothing but Ek by composite morphism E −→ J −→ l>k

the definition of Il (l > k). Hence the morphism (E, {Ek }) −→ (J, {Jk }) is strict.   Definition 1.1.5. Let (E • , {Ek• }) be an object of K+ F(A). (1) ([8]) We say that a filtered complex (J • , {Jk• }) ∈ K+ F(A) with a filtered morphism (E • , {Ek• })−→(J • , {Jk• }) is a strictly injective resolution of (E • , {Ek• }) if (J q , {Jkq }) ∈ Istinj (A) for all q ∈ Z and if the morphism (E • , {Ek• })−→(J • , {Jk• }) is a filtered quasi-isomorphism which is a strict monomorphism for all • = q ∈ Z. (2) We say that a filtered complex (J • , {Jk• }) ∈ K+ F(A) with a filtered morphism (E • , {Ek• })−→(J • , {Jk• }) is a filtered flasque resolution, filtered injective resolution and specially injective resolution of (E • , {Ek• }) if (J q , {Jkq }) ∈ Iflas (A), ∈ Iinj (A) and ∈ Ispinj (A), respectively, for all q ∈ Z and if the morphism (E • , {Ek• })−→(J • , {Jk• }) is a filtered quasi-isomorphism which is a strict monomorphism for all • = q ∈ Z. Remark 1.1.6. Let (T , A) be a ringed topos which has enough points. Let (F • , {Fk• }) be an object of C+ F(A). For an integer p, let (I pq , {Ikpq }; dpq )q∈Z≥0 be the Godement resolution of (F p , {Fkp }). Then the sequence 0 −→ (F p , {Fkp }) −→ (I p0 , {Ikp0 })

(−1)p dp0

−→

(I p1 , {Ikp1 })

(−1)p dp1

−→

···

(p ∈ Z)

gives a filtered flasque resolution s(I •• , {Ik•• }) of (F • , {Fk• }). We prove the following (cf. (1.1.4) and [78, Lemma 1.3.3]): Proposition 1.1.7. For a filtered complex (E • , {Ek• }) ∈ K+ F(A), there exists a specially injective resolution (J • , {Jk• }) of (E • , {Ek• }). Proof. (cf. [44, I (4.6) 1)]) We may assume that E q = 0 for q < 0. Assume that we are given (J 0 , {Jk0 }), (J 1 , {Jk1 }), . . . , (J q , {Jkq }) ∈ Ispinj (A). We consider A-modules J q ⊕E q E q+1 and Jkq ⊕Ekq Ekq+1 . Using the strictness of the morphism E q −→ J q , we can easily check that Jkq ⊕Ekq Ekq+1 −→ J q ⊕E q E q+1 is a monomorphism. Hence {Jkq ⊕Ekq Ekq+1 } is a filtration on J q ⊕E q E q+1 . The natural morphism E q+1 s −→ (0, s) ∈ J q ⊕E q E q+1 induces a filtered morphism (E q+1 , {Ekq+1 }) −→ (J q ⊕E q E q+1 , {Jkq ⊕Ekq Ekq+1 }). It is immediate to check that this filtered morphism is strict. Let I q+1 and Ikq+1 be injective A-modules such that there exist the following monomorphisms of A-modules:

20

(1.1.7.1)

1 Preliminaries on Filtered Derived Categories and Topoi ⊂

J q ⊕E q E q+1 −→ I q+1 ,

Set J q+1 := I q+1 ×

 l∈Z

⊂

q+1 J q ⊕E q E q+1 /Jkq ⊕Ekq Ekq+1 −→ Ik+1 .

Ilq+1 and Jkq+1 := I q+1 ×

 l≤k

Ilq+1 . Then (J q+1 , {Jkq+1 })

∈ Ispinj (A). By (1.1.0.2) and (1.1.7.1), we have a natural monomorphism (1.1.7.2)

⊂

(J q ⊕E q E q+1 , {Jkq ⊕Ekq Ekq+1 }) −→ (J q+1 , {Jkq+1 }).

Since E q −→ J q is a monomorphism, the morphism E q+1 −→ J q ⊕E q E q+1 is a monomorphism, and so is the following composite morphism ⊂

⊂

E q+1 −→ J q ⊕E q E q+1 −→ J q+1 . In fact, we have a morphism (E q+1 , {Ekq+1 }) −→ (J q+1 , {Jkq+1 }). Using a morphism J q s −→ (s, 0) ∈ J q ⊕E q E q+1 and the morphism (1.1.7.2), we have a morphism (J q , {Jkq }) −→ (J q+1 , {Jkq+1 }). By the proof of (1.1.4) and by the strictness of the morphism (E q+1 , {Ekq+1 }) −→ (J q ⊕E q E q+1 , {Jkq ⊕Ekq Ekq+1 }), the morphism (E q+1 , {Ekq+1 }) −→ (J q+1 , {Jkq+1 }) is strict. We obtain (J • , {Jk• }) inductively by the argument above. We claim that (J • , {Jk• }) is filteredly quasi-isomorphic to (E • , {Ek• }). Let ◦ be nothing or an integer k ∈ Z and let q be an integer. Because Ker(J◦q −→ J◦q+1 ) = Ker(J◦q −→ J◦q ⊕E◦q E◦q+1 ), the morphism Ker(E◦q −→ E◦q+1 ) −→ Ker(J◦q −→ J◦q+1 ) is an epimorphism; furthermore, the morphism J◦q−1 −→ J◦q factors through J◦q−1 −→ J◦q−1 ⊕E◦q−1 E◦q . Note that J◦q−1 ⊕E◦q−1 E◦q −→ J◦q is a monomorphism. Because the inverse image of Im(J◦q−1 −→ J◦q ) in E◦q is equal to the inverse image of Im(J◦q−1 −→ J◦q−1 ⊕E◦q−1 E◦q ), the morphism Ker(E◦q −→ E◦q+1 )/Im(E◦q−1 −→ E◦q ) −→ Hq (J◦• ) is an isomorphism. There∼   fore we have Hq (E◦• ) −→ Hq (J◦• ). Proposition 1.1.8. Let f • : (E • , {Ek• }) −→ (F • , {Fk• }) be a morphism in C+ F(A). Then there exists a morphism g • : (J • , {Jk• }) −→ (K • , {Kk• }) in C+ F(A) such that (J • , {Jk• }) (resp. (K • , {Kk• })) is a specially injective resolution of (E • , {Ek• }) (resp. (F • , {Fk• })), and such that g • makes the following diagram commutative: ⊂

(E • , {Ek• }) −−−−→ (J • , {Jk• }) ⏐ ⏐ ⏐ • ⏐ f • g ⊂

(F • , {Fk• }) −−−−→ (K • , {Kk• }). Proof. Assume that we are given morphisms g 0 , . . . , g q . Then we have the following commutative diagram

1.1 Filtered Derived Category. I

21

(E q+1 , {Ekq+1 }) −−−−→ (J q ⊕E q E q+1 , {Jkq ⊕Ekq Ekq+1 }) ⏐ ⏐ ⏐ q q q+1 ⏐ f q+1  g ⊕f f (F q+1 , {Fkq+1 }) −−−−→ (K q ⊕F q F q+1 , {Kkq ⊕Fkq Fkq+1 }). ⊂

q+1 Let J q ⊕E q E q+1 −→ IE be a monomorphism into an injective A-module.  q+1 be the push-out of the following diagram: Let IF ⊂

q+1 J q ⊕E q E q+1 −−−−→ IE ⏐ ⏐ g q ⊕f q f q+1 

K q ⊕F q F q+1

.

Note that the morphism K q ⊕F q F q+1 −→ IF q+1 is a monomorphism. Let ⊂ IF q+1 −→ IFq+1 be a monomorphism into an injective A-module. Obviously we have the following commutative diagram: ⊂

q+1 J q ⊕E q E q+1 −−−−→ IE ⏐ ⏐ ⏐ ⏐ g q ⊕f q f q+1  

K q ⊕F q F q+1 −−−−→ IFq+1 . q+1 Similarly there exists a morphism (IE )k+1 −→ (IFq+1 )k+1 of injective A-modules fitting into the following commutative diagram: ⊂

q+1 (J q ⊕E q E q+1 )/(Jkq ⊕Ekq Ekq+1 ) −−−−→ (IE )k+1 ⏐ ⏐ ⏐ ⏐ g q ⊕f q f q+1   ⊂

(K q ⊕F q F q+1 )/(Kkq ⊕Fkq Fkq+1 ) −−−−→ (IFq+1 )k+1 .  q+1  q+1 q+1 q+1 × (IE )k , Jkq+1 := IE × (IE )l , K q+1 := IFq+1 Set J q+1 := IE k∈Z l≤k  q+1  × (IF )k and Kkq+1 := IFq+1 × (IFq+1 )l . Obviously we have the following k∈Z

l≤k

commutative diagram in C+ F(A): (E q+1 , {Ekq+1 }) −−−−→ (J q+1 , {Jkq+1 }) ⏐ ⏐ ⏐ ⏐   (F q+1 , {Fkq+1 }) −−−−→ (K q+1 , {Kkq+1 }). ∼

∼

By the proof of (1.1.7), we have Hq (E◦• ) −→ Hq (J◦• ) and Hq (F◦• ) −→ Hq (K◦• ). Repeating these processes, we have the desired filtered complexes.  

22

1 Preliminaries on Filtered Derived Categories and Topoi

For an additive full subcategory I of MF(A), let K+ F(I) be the category of the bounded below filtered complexes whose components belong to I. We prove the following two lemmas for (1.1.12) below and for later sections. Lemma 1.1.9. Let (E • , {Ek• }) be a filtered complex of A-modules and let (I • , {Ik• }) be an object of K+ F(Istinj (A)). Assume that (E • , {Ek• }) is strictly exact. Let f : (E • , {Ek• }) −→ (I • , {Ik• }) be a morphism of filtered complexes. Then f is filteredly homotopic to zero. Proof. By the definition of the strict injectivity, the same argument as that in the classical case works.   Lemma 1.1.10. (1) Let (I • , {Ik• }) be an object of K+ F(Istinj (A)). Let s : (E • , {Ek• }) −→ (F • , {Fk• }) be a filtered quasi-isomorphism. Then s induces an isomorphism ∼

s∗ : HomKF(A) ((F • , {Fk• }), (I • , {Ik• })) −→ HomKF(A) ((E • , {Ek• }), (I • , {Ik• })). (2) If a morphism s : (I • , {Ik• }) −→ (E • , {Ek• }) is a filtered quasi-isomorphism from an object of K+ F(Istinj (A)) to a filtered complex of A-modules, then s has a filtered homotopy inverse. Proof. (1): Let (G• , {G•k }) be the mapping cone of s. Since (G• , {G•k })[n]  (G• , {G•k }){n}, we have (1.1.10.1)

HomKF(A) ((G• , G•k )[n], (I • , {Ik• })) HomKF(A) ((G• , G•k ){n}, (I • , {Ik• })) =HomKF(A) ((G• , G•k ), (I • , {Ik• }){−n})

for any integer n. By (1.1.9) we have HomKF(A) ((G• , G•k ), (I • , {Ik• }){−n}) = 0. By [44, I (1.1)], HomKF(A) (•, (I • , {Ik• })) is a cohomological functor. Hence s∗ is an isomorphism. (2): The proof is the same as that of [44, I (4.5)] by using (1.1.9) (cf. (1.1.11) below).   Remark 1.1.11. There is a mistake in signs in the proof of [44, I (4.5)]. Let the notations be as in [loc. cit.]. Then the homotopy operator (k, t) : T (I • ) ⊕ Y • −→ I • in [loc. cit.] is not right because the boundary operators of T (I • ) are twisted by −1; strictly speaking, we have a homotopy operator T (I • ) ⊕ Y • −→ T (I • ){−1}. Thus we have an equation v = (idI • , 0) = (k, t)dZ − dI (k, t) and two formulas idI • = −dk − kd + ts and dt = td. Anyway, ts is homotopic to idI • .

1.1 Filtered Derived Category. I

23

Corollary 1.1.12. (1) The following equalities hold: D+ F(A) = K+ F(Iflas (A))(FQis) = K+ F(Iinj (A))(FQis) = K+ F(Istinj (A)) = K+ F(Ispinj (A)). (2) Set I := Iflas (A), Iinj (A), Istinj (A) or Ispinj (A). Let f : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Then there exists the right derived functor Rf∗ : D+ F(A) −→ D+ F(A ) of f∗ such that Rf∗ [(I • , {Ik• })] = [(f∗ (I • ), f∗ (Ik• ))] for an object (I • , {Ik• }) ∈ K+ F(I). Here [ ] is the localization functor. (Because we shall use the symbol Q for an object of MF(A), we do not use Q as the localization functor. We sometimes omit the notation [ ] for simplicity.) (3) Let f : (T , A) −→ (T  , A ) and g : (T  , A ) −→ (T  , A ) be morphisms of ringed topoi. Then R(gf )∗ = Rg∗ Rf∗ . Proof. (1): The first two equalities follow from (1.1.7) and the proof of [44, I (5.1)]. The last two equalities follow from the proof of [44, I (4.7)] and (1.1.10) (2). (2): (2) follows from [44, I (5.1)].   (3): (3) follows by setting I := Iflas (A) in (2). We also need to recall the cospecial module and the strictly flat resolution ([8]). For two objects (M, {Mk }), (N, {Nk }) ∈ MF(A), we define the filtered tensor product (M ⊗A N, {(M ⊗A N )k }) of (M, {Mk }) and (N, {Nk }) as usual: Mj ⊗A Nl −→ we define the filtration by the following: (M ⊗A N )k := Im( j+l=k

M ⊗A N ). Following [8], for A-modules E, El (l ∈ Z), we

set ΣA (E, {El }) :=

El with filtration (ΣA (E, {El }))k := El , which Berthelot E ⊕ l∈Z

l≤k

has called the cospecial module of (E, {El }). Then ΣA (E, {El }) (ΣA (E, {El }), (ΣA (E, {El }))k ) is an object of MF(A). A formula

=

(1.1.12.1)  HomMF(A) (ΣA (E, {El })), (F, {Fl })) = HomA (E, F ) × HomA (El , Fl ) l∈Z

in [8] is easily verified. Definition 1.1.13 ([8]). We say that a filtered module (Q, {Qk }) ∈ MF(A) is strictly flat if it satisfies the following two conditions: (1) Q and Q/Qk are flat A-modules ⊂ (2) For a strict monomorphism (E, {Ek }) −→ (F, {Fk }), the induced morphism

24

1 Preliminaries on Filtered Derived Categories and Topoi

(Q⊗A E, {(Q⊗A E)k }) −→ (Q⊗A F, {(Q⊗A F )k }) is a strict monomorphism. Let us consider the following additive full subcategories of MF(A): (1.1.13.1) Qfl (A) := {(Q, {Qk }) | Q and Q/Qk (∀k ∈ Z) are flat A-modules }, (1.1.13.2) Qstfl (A) := {(Q, {Qk }) | (Q, {Qk }) is a strictly flat A-module }, (1.1.13.3) Qspfl (A) := {ΣA (Q, {Qk }) | Q and Qk are flat A-modules (∀k ∈ Z)}. We obviously have Qstfl (A) ⊂ Qfl (A). Definition 1.1.14. We say that a filtered module (Q, {Qk }) ∈ MF(A) is filteredly flat (resp. specially flat) if (Q, {Qk }) ∈ Qfl (A) (resp. (Q, {Qk }) ∈ Qspfl (A)). Lemma 1.1.15 ([8]). Qspfl (A) ⊂ Qstfl (A). ⊂

Proof. Let ΣA (Q, {Ql }) be an object of Qspfl (A) and let ι : (E, {El }) −→ (F, {Fl }) be a strict monomorphism. Let k be an integer. Denote by the same ⊂ symbol ι the induced morphism ΣA (Q, {Ql }) ⊗A E −→ ΣA (Q, {Ql }) ⊗A F . We have to prove that an A-submodule ι(ΣA (Q, {Ql }) ⊗A E) ∩ (ΣA (Q, {Ql }) ⊗A F )k is equal to ι((ΣA (Q, {Ql })⊗A E)k ). Let s be a local section of ι(ΣA (Q, {Ql }) ⊗A E) ∩ (ΣA (Q, {Ql }) ⊗A F )k . By the definition of the filtration on the filtered tensor product, s is a finite sum of local sections of Qi ⊗A Fj (i + j ≤ k). Because (Qi ⊗A Fj ) ∩ (Qi ⊗A ι(E)) = Qi ⊗A (Fj ∩ ι(E)) = Qi ⊗A ι(Ej ), s is a local section of ι((ΣA (Q, {Ql }) ⊗A E)k ).

 

Proposition 1.1.16 ([8]). For a filtered A-module E = (E, {Ek }), there exists a strict epimorphism ΣA (Q, {Ql }) −→ E with ΣA (Q, {Ql }) ∈ Qspfl (A). Proof. Recall the functor L0 : {A-modules} −→ {flat A-modules} ([11, §7]): for an A-module, L0 (E) is, by definition, the sheafification of the presheaf (U −→ free Γ(U, A)-module with basis Γ(U, E) \ {0}). The natural morphism L0 (E) −→ E is an epimorphism. Let Q −→

E and Ql −→ E

l be epimorphisms from flat A-modules. Ql and Rk := Ql ; (R, {Rk }) is an object of Qspfl (A). Set R := Q ⊕ l∈Z

l≤k

1.1 Filtered Derived Category. I

25 ⊂

The morphisms Q −→ E and Ql −→El −→ E induce an epimorphism ⊂ R −→ E. The morphism Ql −→ El −→ Ek (l ≤ k) induces an epimorphism Rk −→ Ek . It is trivial to check that the morphism R −→ E is strict. Thus (1.1.16) follows.   Definition 1.1.17. Let (E • , {Ek• }) be an object of K− F(A). (1) ([8]) We say that a filtered complex (Q• , {Q•k }) ∈ K− F(A) with a filtered morphism (Q• , {Q•k }) −→ (E • , {Ek• }) is a strictly flat resolution of (E • , {Ek• }) if (Qq , {Qqk }) ∈ Qstfl (A) for all q ∈ Z and if the morphism (Q• , {Q•k }) −→ (E • , {Ek• }) is a filtered quasi-isomorphism which is a strict epimorphism for all • = q ∈ Z. (2) We say that a filtered complex (Q• , {Q•k }) ∈ K− F(A) with a filtered morphism (Q• , {Q•k }) −→ (E • , {Ek• }) is a filtered flat resolution (resp. specially flat resolution) of (E • , {Ek• }) if (Qq , {Qqk }) ∈ Qfl (A) (resp. (Qq , {Qqk }) ∈ Qspfl (A)) for all q ∈ Z and if the morphism (Q• , {Q•k }) −→ (E • , {Ek• }) is a filtered quasi-isomorphism which is a strict epimorphism for all • = q ∈ Z. Proposition 1.1.18 ([8]). For a filtered complex (E • , {Ek• }) ∈ K− F(A), there exists a specially flat resolution (Q• , {Q•k }) of (E • , {Ek• }). Proof. (1.1.18) follows from (1.1.16) and [78, Lemma 1.3.3].

 

For an additive full subcategory Q of MF(A), let K− F(Q) be the category of the bounded above filtered complexes whose components belong to Q. Corollary 1.1.19. (1) The following equalities hold: D− F(A) = K− F(Qfl (A))(FQis) = K− F(Qstfl (A))(FQis) = K− F(Qspfl (A))(FQis) . (2) Set Q := Qfl (A ), Qstfl (A ) or Qspfl (A ). Let f : (T , A) −→ (T , A ) be a morphism of ringed topoi. Then there exists the left derived functor Lf ∗ : D− F(A ) −→ D− F(A) such that Lf ∗ [(Q• , {Q•k })] = [(f ∗ (Q• ), {f ∗ (Q•k )})] for an object (Q• , {Q•k }) ∈ K− F(Q ). (3) Let f : (T , A) −→ (T  , A ) and g : (T  , A ) −→ (T  , A ) be morphisms of ringed topoi. Then L(gf )∗ = Lf ∗ Lg ∗ . 

Proof. (1) and (2) are obvious; (3) follows by setting Q := Qfl (A) in (2).   Remark 1.1.20. In the case of the trivial filtration, a flat resolution of a complex E • ∈ K− (A) has been given in [11, §7] by the single complex of the double complex L• (E • ), though we have to change some signs of boundary morphisms of L• (E • ) in order to make L• (E • ) a double complex (see Convention (8)). We make the following convention:

26

1 Preliminaries on Filtered Derived Categories and Topoi

(1.1.20.1) · · · −−−−→

···  ⏐ −d⏐

−−−−→

···  ⏐ d⏐

−−−−→

···  ⏐ −d⏐

−−−−→

···  ⏐ d⏐

· · · −−−−→ L−3 (E 2 ) −−−−→ L−2 (E 2 ) −−−−→ L−1 (E 2 ) −−−−→ L0 (E 2 )     ⏐ ⏐ ⏐ ⏐ −d⏐ −d⏐ d⏐ d⏐ · · · −−−−→ L−3 (E 1 ) −−−−→ L−2 (E 1 ) −−−−→ L−1 (E 1 ) −−−−→ L0 (E 1 )     ⏐ ⏐ ⏐ ⏐ −d⏐ −d⏐ d⏐ d⏐ · · · −−−−→ L−3 (E 0 ) −−−−→ L−2 (E 0 ) −−−−→ L−1 (E 0 ) −−−−→ L0 (E 0 )     ⏐ ⏐ ⏐ ⏐ −d⏐ −d⏐ d⏐ d⏐ · · · −−−−→

···

−−−−→

···

−−−−→

···

−−−−→

··· .

1.2 Filtered Derived Category. II.–RHom and ⊗L In this section we prove the adjoint property of Lf ∗ and Rf∗ for a morphism of topoi f ([8]). The notion of strict injectivity in (1.1.1) due to Berthelot is a key notion for the definition of RHom and the proof for the adjoint property. We also define the derived functor of the filtered tensor product. (1) RHom. As in [44, p. 63], we set HomnA ((E • , {Ek• }), (F • , {Fk• })) :=



HomA ((E q , {Ekq }), (F q+n , {Fkq+n }))

q∈Z

for (E • , {Ek• }), (F • , {Fk• }) ∈ CF(A). Then we have a filtered complex of Γ(T , A)-modules Hom•A ((E • , {Ek• }), (F • , {Fk• })) ∈ CF(Γ(T , A)); the boundary morphism • • • • HomnA ((E • , {Ek• }), (F • , {Fk• })) −→ Homn+1 A ((E , {Ek }), (F , {Fk }))

is defined as in [10, p. 4] and [18, p. 10]:  dn := ((−1)n+1 dqE + dq+n F ). q∈Z

This boundary morphism is different from that in [44, p. 64]. As in the classical case, an n-cocycle of Hom•A ((E • , {El• }), (F • , {Fl• }))k corresponds to a filtered morphism E • −→ F • [n]k. (Recall that k is the shift of the filtration defined in §1.1.) An n-coboundary of Hom•A ((E • , {El• }),

1.2 Filtered Derived Category. II.–RHom and ⊗L

27

(F • , {Fl• }))k corresponds to a morphism E • −→ F • [n]k which is homotopic to zero: dn−1 (((−1)n f q )q ) = (f q+1 dqE + (−1)n dn+q−1 f q )q ((f q )q ∈ Homn−1 A F • • • • ((E , {El }), (F , {Fl }))k ). Hence (1.2.0.1) H n (Hom•A ((E • , {El• }), (F • ,{Fl• }))k ) = HomKF(A) ((E • , {El• }), (F • , {Fl• })[n]k). In particular, (1.2.0.2) H 0 (Hom•A ((E • , {El• }), (F • , {Fl• }))0 ) = HomKF(A) ((E • , {El• }), (F • , {Fl• })). Similarly we obtain a filtered complex Hom •A ((E • , {Ek• }), (F • , {Fk• })) of A-modules. To define the derived functor of the functor Hom•A (•, •) : KF(A)◦ × K+ F(A) −→ KF(Γ(T , A)), we have to check the following: Lemma 1.2.1. Let (E • , {Ek• }) ∈ KF(A) be a filtered complex of A-modules and let (I • , {Ik• }) be an object of K+ F(Istinj (A)). Assume that one of the following two conditions holds. (1) (I • , {Ik• }) is strictly exact. (2) (E • , {Ek• }) is strictly exact. Then Hom•A ((E • , {Ek• }), (I • , {Ik• })) is strictly exact. Proof. (1): By the definition of the strict injectivity, there exist A-modules J n and Jkn (n, k ∈ Z) satisfying the following conditions: n (i) Jkn ⊂ J n , Jk−1 ⊂ Jkn , n n (ii) (I , {Ik })  (J n−1 , {Jkn−1 }) ⊕ (J n , {Jkn }), (iii) the boundary morphism d : (I n , {Ikn }) −→ (I n+1 , {Ikn+1 }) is identified id

with the induced morphism by the morphisms J n−1 −→ 0 and J n −→ J n . By (1.1.0.1) and (1.2.0.1), we have only to construct a filtered homotopy for a morphism f ∈ HomCF(A) ((E • , {Ek• }), (I • , {Ik• })), which is easy. (2): By (1.2.0.1) and by the definition of the strict injectivity, the same argument as that in the classical case works.   By (1.2.1) we obtain the following derived functor RHom•A : DF(A)◦ × D+ F(A) −→ DF(Ab). As in [11, 7.7 Proposition, 7.8 Theorem], we shall need the following adjunction formula for the filtered base change theorem in §2.10.

28

1 Preliminaries on Filtered Derived Categories and Topoi

Theorem 1.2.2 ([8], Adjunction formula). Let f : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Let (E • , {Ek• }) (resp. (F • , {Fk• })) be an object of K− F(A ) (resp. K+ F(A)). Then there exists a canonical isomorphism RHomA (Lf ∗ (E • , {Ek• }), (F • ,{Fk• })) −→ RHomA ((E • , {Ek• }), Rf∗ (F • , {Fk• })) =

in D+ F(Γ(T  , A )). The isomorphism above satisfies the transitive condition (cf. [3, V Proposition 3.3.1]). Proof. (The proof below is a filtered version of that in [3, V Proposition 3.3.1].) Let (I • , {Ik• }) be a strictly injective resolution of (F • , {Fk• }). Let (Q• , {Q•k }) be a filtered flat resolution of (E • , {Ek• }). Let (J • , {Jk• }) ∈ K+ F (Istinj ) be a strictly injective resolution of f∗ (I • , {Ik• }). Then we have a morphism (1.2.2.1) Hom•A (f ∗ (Q• , {Q•k }), (I • , {Ik• })) = Hom•A ((Q• , {Q•k }), f∗ (I • , {Ik• })) ∼

−→ Hom•A ((Q• , {Q•k }), (J • , {Jk• })) ←− Hom•A ((E • , {Ek• }), (J • , {Jk• })). The last quasi-isomorphism follows from (1.2.1) (2). As in [3, V Proposition 3.3.1], by the transitive condition, we have only to prove that (1.2.2) holds for a morphism f : (T , A) −→ (T , B) of ringed topoi such that f = idT as a morphism of topoi. As in the trivial filtered case, consider the following functor f ! : f ! : MF(B) (K, {Kk }) −→ Hom B (f∗ (A), (K, {Kk })) ∈ MF(A). Here we endow f∗ (A) with the trivial filtration. The functor f ! is the right adjoint functor of f∗ : (1.2.2.2) HomA ((M, {Mk }), f ! (K, {Kk })) = HomB (f∗ (M,{Mk }), (K, {Kk })) ((M, {Mk }) ∈ MF(A)). By (1.2.2.2), we see that, if (K, {Kk }) is a strictly injective B-module, then f ! (K, {Kk }) is a strictly injective A-module. Moreover, for a strict ⊂ monomorphism f∗ (M, {Mk }) −→ (K, {Kk }) of B-modules, the correspond⊂ ing morphism (M, {Mk }) −→ f ! (K, {Kk }) is a strict monomorphism of A-modules, which is easily checked. Hence, by the same proof as that of (1.1.7) (especially, by noting that the functor f ! commutes with the direct product), we can take f ! (K • , {Kk• }) as (I • , {Ik• }), where (K • , {Kk• }) is a bounded below complex of strictly injective B-modules. Let R• be a flat resolution of f∗ (A) with the trivial filtration. Since the filtration on R• is trivial, it is obvious that the morphism (Qq , {Qqk })⊗B R• −→

1.2 Filtered Derived Category. II.–RHom and ⊗L

29

(Qq , {Qqk })⊗B f∗ (A) is a filtered quasi-isomorphism (q ∈ Z). By (1.2.1) (2), we have (1.2.2.3) ∼

Hom•B ((Qq , {Qqk })⊗B f∗ (A), (K • , {Kk• }))

−→Hom•B ((Qq , {Qqk })⊗B R• , (K • , {Kk• })). (1.2.2.3) is equal to the following: (1.2.2.4) ∼

Hom•B ((Qq , {Qqk }), Hom B (f∗ (A), (K • , {Kk• })))

−→Hom•B ((Qq , {Qqk }), Hom •B (R• , (K • , {Kk• }))). Here Hom B (f∗ (A), (K • , {Kk• })) is considered as a filtered B-module, which is nothing but f∗ f ! (K • , {Kk• }). It is easy to check that Hom B (Rq , (K q+n , {Kkq+n })) is a strictly injective B-module; so is Hom nB (R• , (K • , {Kk• })) (n ∈ Z). Therefore Hom •B (R• , (K • , {Kk• })) is a strictly injective resolution of f∗ f ! (K • , {Kk• }) by the sheafification of (1.2.1) (2). Hence we can take Hom •B (R• , (K • , {Kk• })) as (J • , {Jk• }), and we have a filtered quasi-isomorphism (1.2.2.5) ∼ Hom•B ((Qq , {Qqk }), f∗ f ! (K • , {Kk• })) −→ Hom•B ((Qq , {Qqk }), (J • , {Jk• })) by (1.2.2.4). Let (C • , {Ck• }) be the mapping cone of the morphism f∗ f ! (K • , {Kk• }) −→ • (J , {Jk• }). Then we have a triangle Hom•B ((Q• , {Q•k }), f∗ f ! (K • , {Kk• })) −→ Hom•B ((Q• , {Q•k }), (J • , {Jk• })) −→Hom•B ((Q• , {Q•k }), (C • , {Ck• })) −→ · · · . +1

By (1.2.2.5), the filtered complex Hom•B ((Qq , {Qqk }), (C • , {Ck• })) is strictly exact. As in [3, p. 327], by noting that (Q• , {Q•k }) is bounded above, one can easily check that Hom•B ((Q• , {Q•k }), (C • , {Ck• })) is also strictly exact. Therefore we obtain ∼

Hom•B ((Q• , {Q•k }), f∗ f ! (K • , {Kk• })) −→ Hom•B ((Q• , {Q•k }), (J • , {Jk• })), which enables us to finish the proof of (1.2.2).

 

Let (E • , {Ek• }) (resp. (F • , {Fk• })) be an object of KF(A) (resp. K+ F(A)). Set ExtqA ((E • , {Ek• }), (F • , {Fk• })) := HomDF(A) ((E • , {Ek• }), (F • , {Fk• })[q]). The following lemma is a filtered version of a classical lemma [44, I (6.4)], and it will be used in §2.10 below.

30

1 Preliminaries on Filtered Derived Categories and Topoi

Lemma 1.2.3. The following formula holds: (1.2.3.1) H q (RHom•A ((E • , {Ek• }), (F • , {Fk• }))0 ) = ExtqA ((E • , {Ek• }), (F • , {Fk• })). In particular, (1.2.3.2) H 0 (RHom•A ((E • , {Ek• }), (F • , {Fk• }))0 ) = HomDF(A) ((E • , {Ek• }), (F • , {Fk• })). Here RHom•A ((E • , {Ek• }), (F • , {Fk• }))0 is an object of D+ ((Ab)), which is, by definition, π0 (RHom•A ((E • , {Ek• }), (F • , {Fk• }))) in the case (T , A) = (1-point, Z) in (1.3.4.4) below. ⊂

Proof. Let ι : (F • , {Fk• }) −→ (I • , {Ik• }) be a filtered quasi-isomorphism into an object of K+ F(Istinj (A)). Then ExtqA ((E • , {Ek• }), (F • , {Fk• })) = ExtqA ((E • , {Ek• }), (I • , {Ik• })) = HomDF(A) ((E • , {Ek• }), (I • , {Ik• })[q]). Let f : (E • , {Ek• }) −→ (I • , {Ik• })[q] be a morphism in DF(A). Then f is g s represented by a diagram (E • , {Ek• }) −→ (G• , {G•k }) ←− (I • , {Ik• })[q] in KF(A), where s is a filtered quasi-isomorphism. By (1.1.10) (2), s has a homotopy inverse t : (G• , {G•k }) −→ (I • , {Ik• })[q] in KF(A). We prove that the morphism f −→ t ◦ g is well-defined. Let u : (G• , {G•k }) −→ (G• , {G•k }) be a filtered quasi-isomorphism. Let t : (G• , {G•k }) −→ (I • , {Ik• })[q] be a homotopy inverse of u ◦ s. We have to show that t ◦ u = t in KF(A). This equality immediately follows from the following commutative diagram HomKF(A) ((I • , {Ik• })[q], (I • , {Ik• })[q])

HomKF(A) ((I • , {Ik• })[q], (I • , {Ik• })[q])

 ⏐ s∗ ,=⏐

 ⏐(u◦s)∗ ,= ⏐

u∗ ,=

HomKF(A) ((G• , {G•k }), (I • , {Ik• })[q]) ←−−−−− HomKF(A) ((G• , {G•k }), (I • , {Ik• })[q]),

where the two vertical equalities and the lower horizontal one are obtained from (1.1.10) (1). Hence the morphism f −→ t ◦ g is well-defined. Moreover we also have a natural morphism HomKF(A) ((E •, {Ek• }), (I •, {Ik• })[q]) −→ HomDF(A) ((E •, {Ek• }), (I •, {Ik• })[q]). Therefore we have an isomorphism ∼

HomDF(A) ((E •, {Ek• }), (I •, {Ik• })[q]) −→ HomKF(A) ((E •, {Ek• }), (I •, {Ik• })[q]). Consequently we have

1.2 Filtered Derived Category. II.–RHom and ⊗L

31

ExtqA ((E • , {Ek• }), (F • , {Fk• })) = HomDF(A) ((E • , {Ek• }), (I • , {Ik• })[q]) = HomKF(A) ((E • , {Ek• }), (I • , {Ik• })[q]) = H q (Hom•A ((E • , {Ek• }), (I • , {Ik• }))0 ) = H q (RHom•A ((E • , {Ek• }), (F • , {Fk• }))0 ). (The third equality above follows from (1.2.0.1).)

 

(2) ⊗L . Next we define the filtered derived tensor product. First we need the following lemma: Lemma 1.2.4. (1) (cf. [13, I Proposition 7]) Let E and F be A-modules. Let E  and F  be A-submodules of E and F , respectively. Assume that F and F/F  are flat A-modules. Then (E  ⊗A F ) ∩ (E⊗A F  ) = E  ⊗A F  . (The intersection above is considered in E⊗A F.) (2) Let (E, {Ek }) and (F, {Fk }) be filtered A-modules. Assume that (F, {Fk }) ∈ Qfl (A) ((1.1.13.1)). Then the natural morphism  grl E⊗A grm F −→grk (E ⊗A F ) l+m=k

is an isomorphism. Proof. (1): The A-module F  is a flat A-module by the assumptions. (1) immediately follows from the following commutative diagram with exact rows 0 −−−−→ E  ⊗A F  −−−−→ E  ⊗A F −−−−→ E  ⊗A (F/F  ) −−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    0 −−−−→ E⊗A F  −−−−→ E⊗A F −−−−→ E⊗A (F/F  ) −−−−→ 0, where the three vertical morphisms are monomorphisms. (2): For two A-modules L and M and for two A-submodules L and M  of L and M , respectively, it is an elementary exercise to check that Im((L ⊗A M ⊕L⊗A M  ) −→ L⊗A M ) = Ker(L⊗A M −→ (L/L )⊗A (M/M  )). Hence we have a natural morphism  grl E⊗A grm F −→ grk (E⊗A F ), l+m=k

which we denote by f . The morphism f is an epimorphism by the definition of the tensor product of two filtered A-modules.

32

1 Preliminaries on Filtered Derived Categories and Topoi

Assume that f were not a monomorphism. Let (sl )l ∈ l+m=k grl E⊗A grm F be a non-zero local section such that f ((sl )l ) = 0. Let l0 be the minimum integer such that sl0 = 0. Set m0 := k − l0 . Let g be the following composite morphism ⊂

grl0 E⊗A grm0 F −→



f

grl E⊗A grm F −→ (E⊗A F )k /(E⊗A F )k−1

l+m=k

−→ (E⊗A F )k /((E⊗A F )k−1 + (E⊗A Fm0 −1 ) ∩ (E⊗A F )k ). Then we have g(sl0 ) = 0 since f ((sl )l ) = 0 and f ((sl )l>l0 ) ∈ (E⊗A Fm0 −1 ) ∩ (E⊗A F )k . Let s
∈ El0 ⊗A Fm0 be a local lift of sl0 . Then we have s
∈ (E⊗A F )k−1 + (E⊗A Fm0 −1 ) ∩ (E⊗A F )k ⊂ El0 −1 ⊗A F + E⊗A Fm0 −1 . Take two sections
t ∈ El0 −1 ⊗A F , u
∈ E⊗A Fm0 −1 such that s
=
t+u
. Then
t = s
− u
∈ E⊗A Fm0 . Hence we have (1.2.4.1)


t ∈ (El0 −1 ⊗A F ) ∩ (E⊗A Fm0 ) = El0 −1 ⊗A Fm0

by (1). Then u
= s
−
t ∈ El0 ⊗A Fm0 . Hence we have (1.2.4.2)

u
∈ (E⊗A Fm0 −1 ) ∩ (El0 ⊗A Fm0 ) = El0 ⊗A Fm0 −1

again by (1). By (1.2.4.1) and (1.2.4.2) we have s
=
t+u
∈ El0 −1 ⊗A Fm0 + El0 ⊗A Fm0 −1 . Hence we have sl0 = 0 and this contradicts the definition of l0 . Therefore f is a monomorphism.   As in the non-filtered case, we need a filtered derived tensor product in the preweight-filtered K¨ unneth formula in §2.10. Let (E • , {Ek• }) and (F • , {Fk• }) be two objects of CF(A). Set    q Elp ⊗A Fm −→ E p ⊗A F q ). (1.2.4.3) (E • ⊗A F • )nk := Im( p+q=n l+m=k

p+q=n

Then we have a filtered complex (E • , {Ek• }) ⊗A (F • , {Fk• }) := ((E • ⊗A F • )• , {(E • ⊗A F • )•k }) of A-modules; the boundary morphism is defined by the following formula (1.2.4.4)

d|E p ⊗A F q = (dpE ⊗ 1) + (−1)p (1 ⊗ dqF ).

It is a routine work to check that the functor ⊗ : CF(A) × CF(A) −→ CF(A)

1.2 Filtered Derived Category. II.–RHom and ⊗L

33

induces a functor ⊗ : KF(A) × KF(A) −→ KF(A). As in [44, II (4.1)], we need the following lemma to define a filtered derived functor ([8]) − − − ⊗L A : D F(A) × D F(A) −→ D F(A). Lemma 1.2.5. Let E • := (E • , {Ek• }) be a filtered complex of A-modules and let F • := (F • , {Fk• }) be an object of K− F(Qfl (A)). Assume that either (a) E • is strictly exact, or (b) F • is strictly exact, and assume also that either (c) E • is bounded above, or (d) F • is bounded below. Then E • ⊗A F • is strictly exact. Proof. By [44, II (4.1)], E • ⊗A F • is exact. We have only to prove that (E • ⊗A F • )k is exact for all k ∈ Z. Let G•• be a filtered complex defined by Gpq := E p ⊗A F q with double pq p q El ⊗A Fm in Gpq . Then we have the following two filtration Gk := l+m=k

spectral sequences p p+q E2pq = HII HIq (G•• ((E • ⊗A F • )k ). k ) =⇒ H q p+q E2pq = HIp HII (G•• ((E • ⊗A F • )k ). k ) =⇒ H

The assumptions (c), (d) imply that the two spectral sequences above are bounded and regular.   p q First, assume that (a) holds. Set E∞ := k∈Z Ekp and F∞ := k∈Z Fkq q q • (p, q ∈ Z). Then F∞ is a flat A-module since Fk (∀k ∈ Z) is so. Because E∞ • q • q is exact by (a), E∞ ⊗A F∞ is exact. We prove that (E ⊗A F )k is exact; we have only to prove that (1.2.5.1) Im((E p−1 ⊗A F q )k → (E p ⊗A F q )k ) ⊃ Ker((E p ⊗A F q )k → (E p+1 ⊗A F q )k ). Since the lower horizontal sequence of the following commutative diagram

(1.2.5.2)

(E p−1 ⊗A F q )k −−−−→ (E p ⊗A F q )k −−−−→ (E p+1 ⊗A F q )k ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    p−1 q E∞ ⊗A F∞

p q −−−−→ E∞ ⊗A F∞ −−−−→

p+1 q E∞ ⊗A F∞

• q is exact, we may assume that E • = E∞ , F q = F∞ . Set K◦p := Ker(E◦p −→ p+1 E◦ ) (◦ = k ∈ Z or nothing). Then the sequence 0 −→ K◦p −→ E◦p −→

34

1 Preliminaries on Filtered Derived Categories and Topoi

K◦p+1 −→ 0 is exact. Moreover, we have the following commutative diagram with lower exact row: (1.2.5.3) (K p ⊗A F q )k −−−−→ (E p ⊗A F q )k −−−−→ (K p+1 ⊗A F q )k ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    0 −−−−→ K p ⊗A F q

−−−−→ E p ⊗A F q

−−−−→ K p+1 ⊗A F q −−−−→ 0.

We claim that, to prove (1.2.5.1), we have only to prove that the following sequence (1.2.5.4) 0 −→ grk (K p ⊗A F q ) −→ grk (E p ⊗A F q ) −→ grk (K p+1 ⊗A F q )(−→ 0) is exact. Indeed, let s be a local section of the sheaf on the right hand side of (1.2.5.1). Then, by the lower exact sequence of (1.2.5.3) and by the as• q and F q = F∞ , there exists an integer k  ≥ k such that sumptions E • = E∞ p q   s ∈ (K ⊗A F )k . If k = k, there is nothing to prove. If k  > k, then (1.2.5.4) for k  implies that s ∈ (K p ⊗A F q )k −1 . Repeating this process k  − k times, we see that s ∈ (K p ⊗A F q )k ; thus we have shown the claim. Now let us prove the exactness of (1.2.5.4). By (1.2.4) (2), we have only to prove that the following sequence

(1.2.5.5)

0 −→



grl K p ⊗A grm F q −→

l+m=k

grl E p ⊗A grm F q

l+m=k



−→



grl K p+1 ⊗A grm F q (−→ 0)

l+m=k

is exact. Because grm F is a flat A-module, (1.2.5.5) is exact by the assumption (a). • p • is bounded above, E∞ ⊗A F∞ is Next, assume that (b) holds. Since F∞ q q q+1 p • exact. We prove that (E ⊗A F )k is exact. Set L◦ := Ker(F◦ −→ F◦ ) (◦ = k ∈ Z or nothing). Then Lq◦ is a flat A-module since F◦• is bounded above. As in the case (a), we have only to prove that the following sequence   grl E p ⊗A grm Lq −→ grl E p ⊗A grm F q (1.2.5.6) 0 −→ q

l+m=k

−→

l+m=k



grl E p ⊗A grm Lq+1 −→ 0

l+m=k q+1

is exact. Because grm L We finish the proof.

is a flat A-module, (1.2.5.6) is exact.

 

1.3 Filtered Derived Category. III

35

As in [44, II §4], we have a filtered derived functor ([8]) − − − ⊗L A : D F(A) × D F(A) −→ D F(A)

(1.2.5.7) by (1.2.5).

Remark 1.2.6. If (F q , Fkq ) ∈ Qstfl (A) (∀q ∈ Z) and if (a) in (1.2.5) holds, then the proof in (1.2.5) for the case (a) is simpler: indeed, let q be a fixed integer. Since F q is strictly flat, the filtered complex (E • ⊗A F q , (E • ⊗A F q )k ) is p • • strictly exact. Hence HIp (G•• k ) = 0 (∀p ∈ Z); therefore H ((E ⊗A F )k ) = 0 (∀p ∈ Z).

1.3 Filtered Derived Category. III This section is a complement of §1.1 and §1.2. Let the notations be as in §1.1. First we consider a larger category than Iflas (A), which is convenient in later sections. Let f : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Consider the following full subcategory If∗ -acyc (A) of MF(A) (cf. [23, (1.4.5)]): If∗ -acyc (A) := {(J, {Jl }) | J and Jl are f∗ -acyclic} Then the following holds by (1.1.7): Proposition 1.3.1. The canonical morphism K+ F(If∗ -acyc (A)) −→ D+ F(A) induces an equivalence ∼

K+ F(If∗ -acyc (A))(FQis) −→ D+ F(A) of categories and the right derived functor Rf∗ is calculated by the following formula Rf∗ [(I • , {Il• })] = [f∗ (I • , {Il• })] ((I • , {Il• }) ∈ K+ F(If∗ -acyc (A))). We can consider the dual notion of the above: consider the following full subcategory Qf ∗ -acyc (A) of MF(A ): Qf ∗ -acyc (A) := {(Q, {Ql }) | Q and Q/Ql are f ∗ -acyclic} Then the following holds by (1.1.18): Proposition 1.3.2. The canonical morphism K− F(Qf ∗ -acyc (A )) −→ D− F (A ) induces an equivalence ∼

K− F(Qf ∗ -acyc (A ))(FQis) −→ D− F(A ) of categories and the left derived functor Lf ∗ is calculated by the following formula Lf ∗ [(P • , {Pl• })] = [f ∗ (P • , {Pl• })] ((P • , {Pl• }) ∈ K− F(Qf ∗ -acyc (A ))).

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1 Preliminaries on Filtered Derived Categories and Topoi

Next, we define the gr-functor. For an integer k, there exists the following functor (1.3.2.1) • grk : K F(A) (E • , {Ek• }) −→ grk E • := Ek• /Ek−1 ∈ K (A) ( = +, −, b, nothing), which we call the gr-functor. Lemma 1.3.3. If f : (E • , {Ek• }) −→ (F • , {Fk• }) is a filtered quasi-isomorphism in KF(A), then grk f : grk E • −→ grk F • is a quasi-isomorphism.  

Proof. (1.3.3) follows from the five-lemma. By (1.3.3), the gr-functor (1.3.2.1) induces a functor (1.3.3.1)

grk : D F(A) −→ D (A).

We also call this functor the gr-functor. Lemma 1.3.4. For a morphism f : (T , A) −→ (T  , A ) of ringed topoi, the following diagrams are commutative: gr

(1.3.4.1)

k D+ F(A) −−−− → D+ (A) ⏐ ⏐ ⏐Rf ⏐ Rf∗   ∗

gr

k D+ F(A ) −−−− → D+ (A ),

gr

(1.3.4.2)

k → D− (A) D− F(A) −−−−   ⏐Lf ∗ ⏐ Lf ∗ ⏐ ⏐

gr

k D− F(A ) −−−− → D− (A ).

Proof. Let (E • , {Ek• }) be an object of K+ F(A) and let (I • , {Ik• }) be a filtered • is a flasque resolution flasque resolution of (E • , {Ek• }). Then grk I • = Ik• /Ik−1 q • • • ) = 0 of grk E by (1.3.3); hence Rf∗ (grk E ) = f∗ (grk I ). Since R1 f∗ (Ik−1 (∀q ∈ Z), the following sequence q 0 −→ f∗ (Ik−1 ) −→ f∗ (Ikq ) −→ f∗ (grk I q ) −→ 0

(∀q ∈ Z)

• is exact, and f∗ (Ik• )/f∗ (Ik−1 ) = f∗ (grk I • ). Hence grk Rf∗ (E • , {Ek• }) = • • f∗ (grk I ) = Rf∗ (grk E ). The commutativity of (1.3.4.2) follows from the same proof as that of the commutativity of (1.3.4.1).  

Lastly we give the definition of the functor forgetting the filtration π and the functor taking the filtration πk .

1.4 Some Remarks on Filtered Derived Categories

37

Let ◦ be an integer k or nothing. A morphism (1.3.4.3) π◦ : K F(A) (E • , {El• }) −→ E◦• ∈ K (A)

( = +, −, b, nothing)

induces a morphism (1.3.4.4)

π◦ : D F(A) −→ D (A) ( = +, −, b, nothing).

It is easy to check the following diagrams are commutative: π

(1.3.4.5)

D+ F(A) −−−◦−→ D+ (A) ⏐ ⏐ ⏐Rf ⏐ Rf∗   ∗ π

D+ F(A ) −−−◦−→ D+ (A ), π

(1.3.4.6)

D− F(A) −−−◦−→ D− (A)   ⏐Lf ∗ ⏐ Lf ∗ ⏐ ⏐ π

D− F(A ) −−−◦−→ D− (A ).

1.4 Some Remarks on Filtered Derived Categories In this section we prove some properties of filtered complexes of modules in a ringed topos with an emphasis on the relation with the formulation of derived categories in [78]. Let the notations be as in §1.1. Let I be an additive full subcategory of MF(A). First recall the conditions in our situation which are dual to [78, Definition 1.3.2 (a), (b), (c)]. (1.4.0.1): For any object (E, {Ek }) ∈ MF(A), there exists an object (I, {Ik }) ⊂ ∈ I with a strict monomorphism (E, {Ek }) −→ (I, {Ik }). (1.4.0.2): In any short strictly exact sequence 0 −→ (I, {Ik }) −→ (J, {Jk }) −→ (K, {Kk }) −→ 0 in MF(A), if (I, {Ik }) ∈ I and (J, {Jk }) ∈ I, then (K, {Kk }) ∈ I. (1.4.0.3): If 0 −→ (I, {Ik }) −→ (J, {Jk }) −→ (K, {Kk }) −→ 0

38

1 Preliminaries on Filtered Derived Categories and Topoi

in MF(A) is a short strictly exact sequence with (I, {Ik }), (J, {Jk }), (K, {Kk }) ∈ I, then, for any morphism f : (T , A) −→ (T  , A ) of ringed topoi, the sequence 0 −→ f∗ (I, {Ik }) −→ f∗ (J, {Jk }) −→ f∗ (K, {Kk }) −→ 0 is strictly exact. Proposition 1.4.1. Let I be Iflas (A), Iinj (A) or Istinj (A). Then I satisfies the conditions (1.4.0.1), (1.4.0.2) and (1.4.0.3). Proof. By (1.1.7), I satisfies the condition (1.4.0.1); I also satisfies the condition (1.4.0.3). It is easy to check that the categories Iflas (A) and Iinj (A) satisfy the condition (1.4.0.2). Consider the exact sequence in (1.4.0.2) with (I, {Ik }), (J, {Jk }) ∈ Istinj (A). Then, by the definition of Istinj (A), there ⊂ exists a splitting of the strict monomorphism (I, {Ik }) −→ (J, {Jk }). Hence (J, {Jk })  (I, {Ik }) ⊕ (K, {Kk }). Now it is easy to see that (K, {Kk }) ∈   Istinj (A). Remark 1.4.2. We do not know whether Ispinj (A) satisfies the condition (1.4.0.2). Let I be Iflas (A), Iinj (A) or Istinj (A). Let N+ F(I) be a full subcategory of K+ F(I) which consists of the strictly exact sequences of K+ F(I). We can prove the following as in the classical case ([78, (1.3.4)]): Corollary 1.4.3. The canonical functor K+ F(I)/N+ F(I) −→ D+ F(A) is an equivalence of categories. Let Q be an additive full subcategory of MF(A). Next, let us consider the conditions [78, Definition 1.3.2 (a), (b), (c)] in our situation: (1.4.3.1): For any object (E, {Ek }) ∈ MF(A), there exists an object (Q, {Qk }) ∈ Q with a strict epimorphism (Q, {Qk })−→(E, {Ek }). (1.4.3.2): In any short strictly exact sequence 0 −→ (Q, {Qk }) −→ (R, {Rk }) −→ (S, {Sk }) −→ 0 in MF(A), if (R, {Rk }) ∈ Q and (S, {Sk }) ∈ Q, then (Q, {Qk }) ∈ Q. (1.4.3.3): If a sequence 0 −→ (Q, {Qk }) −→ (R, {Rk }) −→ (S, {Sk }) −→ 0

1.4 Some Remarks on Filtered Derived Categories

39

in MF(A) is a short strictly exact sequence with (Q, {Qk }), (R, {Rk }), (S, {Sk }) ∈ Q, then, for any morphism f : (T  , A ) −→ (T , A) of ringed topoi, the sequence 0 −→ f ∗ (Q, {Qk }) −→ f ∗ (R, {Rk }) −→ f ∗ (S, {Sk }) −→ 0 is strictly exact. Proposition 1.4.4. The category Qfl (A) satisfies the conditions (1.4.3.1), (1.4.3.2) and (1.4.3.3). Proof. By (1.1.16), we obtain (1.4.4) immediately.

 

It is easy to check that Qstfl (A) satisfies the condition (1.4.3.3), and Qstfl (A) satisfies (1.4.3.1) by (1.1.16); however it is not so easy to show that Qstfl (A) satisfies (1.4.3.2). Theorem 1.4.5. The category Qstfl (A) satisfies the condition (1.4.3.2). Proof. Consider the strictly exact sequence in (1.4.3.2). Assume that (R, {Rk }), (S, {Sk }) ∈ Qstfl (A). Obviously (Q, {Qk }) satisfies the condition ⊂ in (1.1.13) (1). We check the condition in (1.1.13) (2). Let (E, {Ek }) −→ (F, {Fk }) be a strict monomorphism. Then we have a commutative diagram 0 −−−−→ E⊗A Q −−−−→ E⊗A R −−−−→ E⊗A S −−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    0 −−−−→ F ⊗A Q −−−−→ F ⊗A R −−−−→ F ⊗A S −−−−→ 0, where the two horizontal sequences are exact, the left vertical morphism is a monomorphism and the middle vertical morphism is a strict monomorphism. By this diagram we can reduce (1.4.5) to the following: Proposition 1.4.6. Let 0 −→ (Q, {Qk }) −→ (R, {Rk }) −→ (S, {Sk }) −→ 0 be a strictly exact sequence in MF(A). Assume that (R, {Rk }) ∈ Qfl (A) and that (S, {Sk }) ∈ Qfl (A). Then, for an object (E, {Ek }) ∈ MF(A), the induced monomorphism E⊗A Q −→ E⊗A R is strict.  Proof. For a filtered A-module (G, {Gk }), set G∞ := k∈Z Gk . Let s be a local section of (E⊗A Q) ∩ (E⊗A R)k . We have to prove that s ∈ (E⊗A Q)k . First we claim the following weaker statement: s ∈ E∞ ⊗A Q∞ . Indeed, the image of s by the following composite morphism E⊗A Q −→ E⊗A R −→ (E/E∞ )⊗A R

40

1 Preliminaries on Filtered Derived Categories and Topoi

is the zero. Obviously the morphism above is equal to the following composite morphism E⊗A Q −→ (E/E∞ )⊗A Q−→(E/E∞ )⊗A R. Because S is a flat A-module, the morphism (E/E∞ )⊗A Q−→(E/E∞ )⊗A R is a monomorphism. Hence the image of s in (E/E∞ )⊗A Q is the zero; thus s ∈ E∞ ⊗A Q. Moreover we obtain s ∈ E∞ ⊗A Q∞ similarly by noting that a composite morphism E∞ ⊗A Q −→ E∞ ⊗A R −→ E∞ ⊗A (R/R∞ ), is the zero, which is equal to a composite morphism E∞ ⊗A Q −→ E∞ ⊗A (Q/Q∞ )−→E∞ ⊗A (R/R∞ ), and by noting that S/S∞ is a flat A-module since S/S∞ = limk S/Sk . Thus −→ we may assume that E = E∞ , Q = Q∞ , R = R∞ . In this case, it suffices to prove that a canonical morphism grk (E⊗A Q) −→ grk (E⊗A R) is a monomorphism. Let T be Q or R. Then we have a natural isomorphism  ∼ grl E⊗A grm T −→ grk (E⊗A T ) fT : l+m=k

by (1.2.4) (2) and (1.4.4). Consider the following diagram:



grk (E⊗A Q) ⏐ ⏐ f Q ,  l+m=k

−−−−→

grl E⊗A grm Q −−−−→



grk (E⊗A R) ⏐ ⏐f , R l+m=k

grl E⊗A grm R.

Because grm S is flat, the lower horizontal morphism above is a monomorphism. Hence the upper horizontal morphism is a monomorphism. Now we have proved (1.4.6).   Let N− F(Q) be a full subcategory of K− F(Q) which consists of the strict exact sequences of K− F(Q). By (1.4.4) and (1.4.5), we have the following: Corollary 1.4.7. Let Q be Qfl (A) or Qstfl (A). Then the canonical functor K− F(Q)/N− F(Q) −→ D− F(A) is an equivalence of categories.

1.5 The Topos Associated to a Diagram of Topoi. I

41

1.5 The Topos Associated to a Diagram of Topoi. I We quickly review the general theory of the diagram of topoi in [3, V 3.4.1] (=the D-topos in [42, Vbis (1.2)]) (cf. [24, §5]), and we apply it to filtered derived categories. Let I be a small category. Assume that we are given a contravariant functor F : I o −→ (2-category of ringed topoi). The functor F is called a diagram of ringed topoi. To give F is equivalent to giving the I-ringed topos in the sense of [42, Vbis (1.2.1)]. Set (Ti , Ai ) := F(i) (i ∈ I) and α : (Ti , Ai ) −→ (Ti , Ai ) be the corresponding morphism to α : i −→ i. Then we have a ringed topos (T• , A• )•∈I ; an object of (T• , A• )•∈I is a system ((Ei )i∈I , (ρα )α∈Mor(I) ), where Ei is an Ai -module, α : i −→ i is a morphism in I and ρα : α−1 (Ei ) −→ Ei is a morphism of abelian sheaves which is compatible with the morphism α−1 (Ai ) −→ Ai such that ρidi = id and ρβ◦α = ρβ ◦β −1 (ρα ) for a morphism β : i −→ i . We call the ringed topos (T• , A• )•∈I the ringed topos associated to F. If no confusion seems likely to occur, we denote (T• , A• )•∈I simply by (T• , A• ). Let ei : (Ti , Ai ) −→ (T• , A• ) (1.2.8)] (cf. [3, V 3.4.2]): e−1 is the natbe the morphism defined in [42, Vbis i α∗ (E))i ∈I , canonical morphisms) ural restriction and ei∗ (E) = (( α∈HomI (i ,i)

for an Ai -module E. Let (1.5.0.1)

 I(T• ,A• ) := { ei∗ (Ii ) | each Ii is a flasque Ai -module} i∈I

be an additive full subcategory of (T• , A• ). Then, by [42, Vbis (1.3.10.1)], for any object F ∈ I(T• ,A• ) , the inverse image e−1 i (F ) is a flasque Ai -module. For an additive full subcategory I of the category of A• -modules, let K+ (I) (resp. N+ (I)) be the category of the bounded below (resp. exact) complexes whose components belong to I. By [42, Vbis (1.3.10)], (1.5.0.2)

∼

K+ (I(T• ,A• ) )(Qis) = K+ (I(T• ,A• ) )/N+ (I(T• ,A• ) ) −→ D+ (A• ).

By the proof of [3, V Proposition 3.4.4], if α is flat for all α, that is, α∗ is exact for all α, then we can replace the condition ‘each Ii is a flasque Ai -module’ in (1.5.0.1) by the condition ‘each Ii is an injective Ai -module’ without changing the property (1.5.0.2). Let f• : (T• , A• )•∈I −→ (T• , A• )•∈I be a morphism of the ringed topoi associated to diagrams of ringed topoi. Consider the following additive full subcategory: I(T• ,A• ;f• ) := {(Ei )i∈I | Ei : fi∗ -acyclic Ai -module (∀i ∈ I)}.

42

1 Preliminaries on Filtered Derived Categories and Topoi

Then we have ∼

K+ (I(T• ,A• ;f• ) )(Qis) = K+ (I(T• ,A• ;f• ) )/N+ (I(T• ,A• ;f• ) ) −→ D+ (A• ). By using the proof of (1.1.7), it is straightforward to generalize the above for filtered modules. Let (1.5.0.3)

 fil I(T := { ei∗ (Ii ,{(Ii )k }) • ,A• ) i∈I

| each (Ii , {(Ii )k }) is a filtered flasque Ai -module} be an additive full subcategory of MF(A• ). For an additive full subcategory I of MF(A• ), let K+ F(I) (resp. N+ F(I)) be the bounded below (resp. strictly exact) complexes whose components belong to I. Then we have an equivalence of categories (1.5.0.4) ∼ fil fil fil ) = K+ F(I(T )/N+ F(I(T ) −→ D+ F(A• ) K+ F(I(T • ,A• ) (FQis) • ,A• ) • ,A• ) by the proof of [3, V Proposition 3.4.4] and that of (1.1.7). By the proof of [3, V Proposition 3.4.4] and by the proof of (1.1.7), if the morphism α : (Ti , Ai ) −→ (Ti , Ai ) of ringed topoi is flat for all α : i −→ i, then we can replace the condition ‘each (Ii , {(Ii )k }) is a filtered flasque Ai -module’ in (1.5.0.3) by the condition ‘each (Ii , {(Ii )k }) is a strictly injective Ai -module’ without changing the property (1.5.0.4) because the product of strictly injective modules is strictly injective. Let f• : (T• , A• )•∈I −→ (T• , A• )•∈I be a morphism of the ringed topoi associated to diagrams of ringed topoi. Let fil I(T := {(Ei ,{(Ei )k })i∈I • ,A• ;f• )

| (Ei , {(Ei )k }) : filtered fi∗ -acyclic Ai -module (∀i ∈ I)}. Then we have the following by the proof of (1.1.7): ∼

fil fil fil K+ F(I(T ) = K+ F(I(T )/N+ F(I(T ) −→ D+ F(A• ). • ,A• ;f• ) Qis • ,A• ;f• ) • ,A• ;f• )

In particular, the following holds: let (E•• , {(E•• )k }) be an object of K+ F(A• ). Assume that (Eiq , {(Eiq )k }) is filtered fi∗ -acyclic for all i ∈ I and for all q ∈ Z. Then the following formula holds (cf. [24, (5.2.5)]): Rf•∗ ((E•• , {(E•• )k )}) = (fi∗ (Ei• , {(Ei• )k }))i∈I . Let (T , A) and (T  , A ) be two ringed topoi. Let g : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Let I0 be a (not necessarily finite) set. We fix a total order on I0 . Let {Ui0 }i0 ∈I0 and {Ui0 }i0 ∈I0 be open coverings of the final objects of T and T  , respectively. Let I be a category whose objects are

1.5 The Topos Associated to a Diagram of Topoi. I

43

i = (i0 , . . . , ir )’s (i0 < i1 < · · · < ir , r ∈ Z≥0 ). Set {i} := {i0 , . . . , ir }. For two objects i , i ∈ I, a morphism from i to i is, by definition, the inclusion ⊂ {i } −→ {i}. As in [3, V 3.4.6], we set Ui := Ui0 ×· · ·×Uir , Ui := Ui0 ×· · ·×Uir , Ti := T |Ui and Ti := T  |Ui . Let Ai (resp. Ai ) be the pull-back of A (resp. A ) to Ti (resp. Ti ). Let gi : (Ti , Ai ) −→ (Ti , Ai ) (i ∈ I) be morphisms of ringed topoi. Assume that the following two conditions hold: (1.5.0.5): If there exists a morphism i −→ i in I, then the following diagram is commutative: gi (Ti , Ai ) −−−−→ (Ti , Ai ) ⏐ ⏐ ⏐ ⏐   g



(Ti , Ai ) −−−i−→ (Ti , Ai ). (1.5.0.6): The following diagram is commutative for all i0 ∈ I0 : gi

0 → (Ti0 , Ai0 ) (Ti0 , Ai0 ) −−−− ⏐ ⏐ ⏐ ⏐  

(T , A)

g

−−−−→ (T  , A ).

Let (T• , A• ) be the ringed topos associated to a diagram of ringed topoi obtained from the open covering {Ui0 }i0 ∈I0 . Let π : (T• , A• ) −→ (T , A) be a morphism of ringed topoi characterized by the following: π −1 (E) (E: A-module) is obtained from restrictions of E to the ringed topos (Ti , Ai ) (i ∈ I) (e.g., [3, V 3.4.6]). We also have an analogous morphism π  : (T• , A• ) −→ (T  , A ) of ringed topoi. Then we have the following commutative diagram of ringed topoi by (1.5.0.5) and (1.5.0.6): g•

(1.5.0.7)

(T• , A• ) −−−−→ (T• , A• ) ⏐ ⏐ ⏐  ⏐ π π g

(T , A) −−−−→ (T  , A ). Thus we have the following two commutative diagrams by (1.1.12) (3) and (1.1.19) (3), respectively: Rg•∗

(1.5.0.8)

D+ F(A• ) −−−−→ D+ F(A• ) ⏐ ⏐ ⏐Rπ ⏐ Rπ∗   ∗ Rg∗

D+ F(A) −−−−→ D+ F(A ),

44

1 Preliminaries on Filtered Derived Categories and Topoi Lg ∗

(1.5.0.9)

D− F(A• ) ←−−•−− D− F(A• )   ⏐ ∗ ⏐ Lπ ∗ ⏐ ⏐Lπ Lg ∗

D− F(A) ←−−−− D− F(A ). Let (T , A) be a ringed topos. Let {Ui0 }i0 ∈I0 and {Uj0 }j0 ∈J0 be two open coverings of the final object of T . As above, we fix total orders on I0 and J0 . Then we have two sets I and J as above. For an object i × j ∈ I × J, set Uij = Ui × Uj , Tij := T |Uij and Aij := A|Uij . Then we have an I × J-ringed topos, which is denoted by (TIJ , AIJ ) by abuse of notation, and we have the ringed topos (T•• , A•• ) := (Tij , Aij )i×j∈I×J associated to the diagram of ringed topoi (TIJ , AIJ ). If we fix an object i ∈ I, we have a ringed topos (Ti• , Ai• ). Varying i ∈ I, we have an I-ringed topos, which is denoted by (TI• , AI• ) by abuse of notation. Similarly we have an analogous J-ringed topos (T•J , A•J ). We have a morphism of ringed topoi (1.5.0.10)

η : (T•• , A•• ) −→ (T• , A• )•∈I

which is characterized by the following: η −1 is the natural restriction. For each object i ∈ I, there also exists a natural morphism of ringed topoi (1.5.0.11)

ηi : (Ti• , Ai• ) −→ (Ti , Ai )

which is characterized as above. The morphisms η and ηi induce morphisms of filtered derived categories: (1.5.0.12)

Rη∗ : D+ F(A•• ) −→ D+ F((A• )•∈I ),

(1.5.0.13)

Rηi∗ : D+ F(Ai• ) −→ D+ F(Ai ).

The following (filtered) cohomological descent is often used in this book: Lemma 1.5.1 (cf. [3, V Proposition 3.4.8]). Let (T , A) be a ringed topos and let (T• , A• ) be the ringed topos associated to a diagram of ringed topoi which is obtained from a covering of the final object of T . Let π : (T• , A• ) −→ (T , A) be the natural morphism. Then the following hold: (1) Let F • be an object of K+ (A). Then the natural morphism F • −→ Rπ∗ π −1 (F • ) is an isomorphism in D+ (A). (2) Let (F • , {Fk• }) be an object of K+ F(A). Then the natural morphism (F • , {Fk• }) −→ Rπ∗ π −1 (F • , {Fk• })

1.6 The Topos Associated to a Diagram of Topoi. II

45

is an isomorphism in D+ F(A). Proof. (1): Let I •• be a double complex of flasque A• -modules such that there exists a flasque resolution π −1 (F • ) −→ s(I •• ), where s means the single complex, and such that π −1 (F i ) −→ I i• (∀i) is also a flasque resolution; for example we can take the Cartan-Eilenberg resolution of π −1 (F • ) as I •• . It suffices to prove that the following natural morphism Hi (F • ) −→ Ri π∗ π −1 (F • )

(1.5.1.1)

is an isomorphism. One can calculate the right hand side by the following spectral sequence E1ij = Hj (π∗ I i• ) =⇒ Hi+j (π∗ s(I •• )) = Ri+j π∗ π −1 (F • ). The usual cohomological descent in [3, V Proposition 3.4.8] tells us that E1i0 = F i and E1ij = 0 if j > 0. Hence (1.5.1.1) follows. (2): Let π −1 (F • , {Fk• }) −→ (I • , {Ik• }) be a filtered quasi-isomorphism into a complex of filtered flasque A• -modules. Since I • (resp. Ik• ) is a flasque resolution of π −1 (F • ) (resp. π −1 (Fk• )), we see that the morphism (F • , {Fk• }) −→ (π∗ (I • ), {π∗ (Ik• )}) = π∗ (I • , {Ik• }) = Rπ∗ π −1 (F • , {Fk• })  

is a filtered quasi-isomorphism by (1).

1.6 The Topos Associated to a Diagram of Topoi. II In this section we apply the theory in §1.5 to (restricted) log crystalline topoi. Let S be a fine log scheme with a PD-structure γ on a quasi-coherent ideal sheaf I of OS . Assume that a power of a prime number p kills OS . Let ◦

f : Y −→ S be a morphism of fine log schemes such that γ extends to Y .  log Then we have a log  crystalline topos (Y /S)crys associated to Y /(S, I, γ) ([54, Yi0 be an open covering, where I0 is a (not necessarily §5]). Let Y = i0 ∈I0

finite) set. We fix a total order on I0 . Let I be the category defined in §1.5. ◦

For i = (i0 , . . . , ir ) ∈ I (ij ∈ I0 ), set Yi := Yi0 ∩ · · · ∩ Yir (each Y i has a log structure induced by that of Y ). If there is a morphism α : i −→ i, the open immersion Yi −→ Yi induces a morphism log

log α : ((Y i /S)crys , OYi /S ) −→ ((Yi /S)crys , OYi /S ) log of ringed topoi. Then we have the ringed topos ((Y • /S)crys , OY• /S ) associated to the diagram of the ringed topoi {((Yi /S)log crys , OYi /S )}i∈I . Let

46

1 Preliminaries on Filtered Derived Categories and Topoi

log log  log πcrys : (Y • /S)crys −→ (Y /S)crys be the morphism of topoi defined in §1.5. log −1 log induces a morphism Note that (πcrys ) (OY /S ) = OY• /S ; πcrys log log  log πcrys : ((Y • /S)crys , OY• /S ) −→ ((Y /S)crys , OY /S ) log of ringed topoi. The morphism πcrys induces a morphism of filtered derived categories:

(1.6.0.1)

log Rπcrys∗ : D+ F(OY• /S ) −→ D+ F(OY /S ).

If we are given another open covering {Yj0 }j0 ∈J0 of Y , we have a ringed log topos ((Y
•• /S)crys , OY•• /S ) and we have two morphisms of filtered derived categories: (1.6.0.2)

(1.6.0.3)

log Rηcrys∗ : D+ F(OY•• /S ) −→ D+ F((OY• /S )•∈I ),

log Rηi,crys∗ : D+ F(OYi• /S ) −→ D+ F(OYi /S ).

We also have analogous objects for the Zariski topos Y
zar : by using the ◦

◦

⊂

◦

◦

f

◦

composite morphisms f i : Y i −→ Y −→ S (i ∈ I), we also have an analogous
◦ ringed topos (Y •zar , f•−1 (OS )) and an analogous morphism (1.6.0.4)


◦
◦ πzar : (Y •zar , f•−1 (OS )) −→ (Y zar , f −1 (OS ))

of ringed topoi. We have three morphisms of filtered derived categories: (1.6.0.5)

Rπzar∗ : D+ F(f•−1 (OS )) −→ D+ F(f −1 (OS )),

(1.6.0.6)

−1 (OS )) −→ D+ F((f•−1 (OS ))•∈I ), Rηzar∗ : D+ F(f••

(1.6.0.7)

−1 (OS )) −→ D+ F(fi−1 (OS )). Rηi,zar∗ : D+ F(fi•

Let (1.6.0.8)


◦ −1 uY /S : ((Y /S)log (OS )) crys , OY /S ) −→ (Y zar , f

be the projection. We also have natural projections (1.6.0.9)


◦ log −1 uY• /S : ((Y • /S)crys , OY• /S ) −→ (Y •zar , f• (OS ))

1.6 The Topos Associated to a Diagram of Topoi. II

47


◦ log −1 uY•• /S : ((Y
•• /S)crys , OY•• /S ) −→ (Y ••zar , f•• (OS )).

(1.6.0.10)

We have the following commutative diagrams: log Rπcrys∗

−−−−−→

D+ F(OY• /S ) ⏐ ⏐ RuY• /S∗ 

(1.6.0.11)

D+ F(OY /S ) ⏐ ⏐Ru  Y /S∗

Rπ

D+ F(f•−1 (OS )) −−−zar∗ −→ D+ F(f −1 (OS )) and log Rηcrys∗

D+ F(OY•• /S ) −−−−−→ ⏐ ⏐ RuY•• /S∗ 

(1.6.0.12)

D+ F(OY• /S ) ⏐ ⏐Ru  Y• /S∗

Rηzar∗

−1 D+ F(f•• (OS )) −−−−→ D+ F(f•−1 (OS )).

Let Y ⏐ ⏐ f

g

−−−−→

Y ⏐ ⏐  f

(S, I, γ) −−−−→ (S  , I  , γ  ) be a commutative diagram of fine log schemes such that the PD-structure ◦

γ  extends to Y  . Assume that we are given open coverings {Yi0 }i0 ∈I0 of Y and {Yi0 }i0 ∈I0 of Y  such that g induces a morphism gi0 : Yi0 −→ Yi0 of log schemes. Let πYlogcrys and πYlog crys be morphisms of (ringed) topoi defined above for Y /(S, I, γ) and Y  /(S  , I  , γ  ), respectively. Let πY zar and πY  zar be obvious analogous morphisms of (ringed) topoi. Then the family {gi0 }i0 ∈I0 induces morphisms satisfying the conditions (1.5.0.5) and (1.5.0.6) for the log crystalline topoi and the Zariski topoi of Y and Y  ; we obtain the following commutative diagrams: log Rg•crys∗

(1.6.0.13)

D+ F(OY• /S ) −−−−−→ D+ F(OY• /S  ) ⏐ ⏐ ⏐Rπlog ⏐ log RπY  Y  crys∗ crys∗  log Rgcrys∗

D+ F(OY /S ) −−−−−→ D+ F(OY  /S  ), Rg•zar∗

(1.6.0.14)

D+ F(f•−1 (OS )) −−−−−→ D+ F(f −1 • (OS  )) ⏐ ⏐ ⏐Rπ  ⏐ RπY zar∗   Y zar∗ Rgzar∗

D+ F(f −1 (OS )) −−−−→ D+ F(f −1 (OS  )).

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1 Preliminaries on Filtered Derived Categories and Topoi

We leave the analogous formulations of the left derived functors to the reader. ◦

Now, assume that f is locally of finite presentation. Let (Y /S)log Rcrys be the restricted log crystalline site of Y /S ([85, (6.2)], (cf. [3, IV Proposition 1.5.5, log D´efinition 1.7.1])): (Y /S)log Rcrys is a full subcategory of (Y /S)crys whose objects are isomorphic to triples (V, DV (V), [ ])’s,

(1.6.0.15)

⊂

where V is an open log subscheme of Y , ι : V −→ V is a locally closed immersion into a log smooth scheme over S and DV (V) is the log PD-envelope of ι over (S, I, γ) ([54, (5.3)]); the topology of (Y /S)log Rcrys is the induced log log  topology by that of (Y /S) . Let (Y /S) be the topos associated to crys

Rcrys

(Y /S)log Rcrys . Let (1.6.0.16)

 log QY /S : (Y /S)log Rcrys −→ (Y /S)crys

be the natural morphism of topoi which is the log version of the morphism in [3, IV (2.1.1)]. Then we have a morphism (1.6.0.17)

∗  log QY /S : ((Y /S)log Rcrys , QY /S (OY /S )) −→ ((Y /S)crys , OY /S ) ⊂

of ringed topoi. Let j : U −→ Y be an open immersion over S. For a sheaf E in (Y /S)log Rcrys and for an open log subscheme VU of U and a log smooth scheme VU over S which contains VU as a locally closed log scheme, log −1 (E))(VU ,DVU (VU ),[ ]) := E(VU ,DVU (VU ),[ ]) . Then we have a sheaf set (jRcrys

log −1 log −1 (E) and we see that the functor jRcrys commutes with finite inverse jRcrys limits (in fact, it commutes with inverse limits and direct limits). Hence we have a morphism

(1.6.0.18)

log  log −→ (Y jRcrys : (U/S) /S)log Rcrys Rcrys

of topoi and a morphism log ∗  log , Q∗ (O  log (1.6.0.19) jRcrys : ((U/S) U/S )) −→ ((Y /S)Rcrys , QY /S (OY /S )) U/S Rcrys

of ringed topoi. The morphisms (1.6.0.18) and (1.6.0.19) fit into the following commutative diagrams of (ringed) topoi, respectively: log jRcrys

(1.6.0.20)

 log  log (U/S) Rcrys −−−−→ (Y /S)Rcrys ⏐ ⏐ ⏐Q ⏐ QU/S   Y /S log jcrys

 log −−−−→ (Y (U/S) /S)log crys crys

1.6 The Topos Associated to a Diagram of Topoi. II

49

and log jRcrys

(1.6.0.21)

∗  log  log , Q∗ (O ((U/S) U/S )) −−−−→ ((Y /S)Rcrys , QY /S (OY /S )) Rcrys U/S ⏐ ⏐ ⏐Q ⏐ QU/S   Y /S log jcrys

 log , O ((U/S) U/S ) crys

−−−−→

((Y /S)log crys , OY /S ).

Let (V, DV (V), [ ]) be an object (1.6.0.15), which is considered as a reprelog −1 sentable sheaf in (Y /S)log Rcrys . By the definition of jRcrys , (1.6.0.22)

log −1 jRcrys ((V, DV (V), [ ])) = (V ∩ U, DV (V)|U , [ ]|U ).

Here, because DV (V)|U = DV ∩U (V \ (U \ V )), we can consider (V ∩ U, DV (V)|U , [ ]|U ) as an object of (U/S)log Rcrys . By using this equality, we have the following equality (1.6.0.23)

log log Q∗Y /S jcrys∗ = jRcrys∗ Q∗U/S .

 log , we have the following equalities: Indeed, for a sheaf E in (U/S) crys log (Q∗Y /S jcrys∗ (E))(V,DV (V),[

])

log = (jcrys∗ (E))(V,DV (V),[

= E(V ∩U,DV (V)|U ,[ =

])

]|U )

log (jRcrys∗ Q∗U/S (E))(V,DV (V),[ ]) .

Lemma 1.6.1 (cf. [3, IV Proposition 2.3.2]). Let E be a sheaf in (Y /S)log crys . ⊂

Let U be an open log subscheme of Y . Let U −→ P be an immersion into a log smooth scheme over S. Let DU (P 2 ) be the log PD-envelope for the diagonal ⊂ immersion U −→ P ×S P over (S, I, γ). Then the following sequence (1.6.1.1)

uY /S∗ (E)|U −→ EDU (P) ⇒ EDU (P 2 )

is exact. Proof. The proof is the same as that in the proof of [3, IV Proposition 2.3.2].   As in [3, IV (2.3.2)], set (1.6.1.2) ∗  log uY /S := uY /S ◦ QY /S : ((Y /S)log Rcrys , QY /S (OY /S )) −→ ((Y /S)crys , OY /S )

(1.6.1.3)


◦ −→ (Y zar , f −1 (OS )).

50

1 Preliminaries on Filtered Derived Categories and Topoi

Corollary 1.6.2. For a sheaf E in (Y /S)log crys , uY /S∗ (E) = uY /S∗ (Q∗Y /S (E)).

(1.6.2.1)

Proof. By using the exactness of the sequence (1.6.1.1), the proof is the same as that of [3, IV Corollaire 2.3.5].   Corollary 1.6.3. The morphism (1.6.0.17) defines a well-defined functor Q∗Y /S : D+ F(OY /S ) −→ D+ F(Q∗Y /S (OY /S )) fitting into the following commutative diagram D+ F(OY /S ) ⏐ ⏐ RuY /S∗ 

(1.6.3.1)

D+ F(f −1 (OS ))

Q∗ Y /S

−−−−→ D+ F(Q∗Y /S (OY /S )) ⏐ ⏐Ru  Y /S∗ D+ F(f −1 (OS )).

Proof. One can reduce (1.6.3) in the general case to (1.6.3) in the non-filtered case. In this non-filtered case, one can prove (1.6.3) in the same way as [3, V Corollaire 1.3.3], by using (1.6.1), (1.6.2), the analogue of [3, IV Proposition ˇ 2.2.5] and the Cech-Alexander complex computing RuY /S∗ and RuY /S∗ .   Let the notations be as in the beginning of this section. Using the morlog phisms (1.6.0.18) and (1.6.0.19), we have (ringed) topoi (Y • /S)Rcrys and log ((Y , Q∗ (OY /S )), and natural morphisms of (ringed) topoi • /S) Rcrys

Y• /S

•

log log  log πRcrys : (Y • /S)Rcrys −→ (Y /S)Rcrys

and log log ∗ ∗  log πRcrys : ((Y • /S)Rcrys , QY• /S (OY• /S )) −→ ((Y /S)Rcrys , QY /S (OY /S )).

Hence we have a morphism of filtered derived categories: (1.6.3.2)

log RπRcrys : D+ F(Q∗Y• /S (OY• /S )) −→ D+ F(Q∗Y /S (OY /S )).

Furthermore, we have morphisms log (1.6.3.3) RηRcrys∗ : D+ F(Q∗Y•• /S (OY•• /S )) −→ D+ F((Q∗Y• /S (OY• /S ))•∈I ),

(1.6.3.4)

log Rηi,Rcrys∗ : D+ F(Q∗Yi• /S (OYi• /S )) −→ D+ F(Q∗Yi /S (OYi /S )).

By the commutative diagram (1.6.0.21), we have the following commutative diagram:

1.6 The Topos Associated to a Diagram of Topoi. II

51

log RπRcrys∗

(1.6.3.5)

D+ F(Q∗Y• /S (OY• /S )) −−−−−−→ D+ F(Q∗Y /S (OY /S )) ⏐ ⏐ ⏐RQ ⏐ RQY• /S∗   Y /S∗ log Rπcrys∗

−−−−−→

D+ F(OY• /S )

D+ F(OY /S ).

We also have the following commutative diagrams log RηRcrys∗

(1.6.3.6)

D+ F(Q∗Y•• /S (OY•• /S )) −−−−−−→ D+ F((Q∗Y• /S (OY• /S )•∈I )) ⏐ ⏐ ⏐RQ ⏐ RQY•• /S∗   Y• /S∗ D+ F(OY•• /S )

log Rηcrys∗

−−−−−→

D+ F((OY• /S )•∈I )

and log RηiRcrys∗

(1.6.3.7)

D+ F(Q∗Yi• /S (OYi• /S )) −−−−−−→ D+ F(Q∗Yi /S (OYi /S )) ⏐ ⏐ ⏐RQ ⏐ RQYi• /S∗   Yi /S∗ log Rηicrys∗

−−−−−→

D+ F(OYi• /S )

D+ F(OYi /S ).

We also have the following equality of functors, whose proof is slightly complicated: Proposition 1.6.4. There exists the following equality of functors (1.6.4.1)

log log Q∗Y /S Rπcrys∗ −→ RπRcrys∗ Q∗Y• /S . =

Proof. Note that the morphism (1.6.4.1) factors as log log (1.6.4.2) Q∗Y /S Rπcrys∗ −→ Q∗Y /S Rπcrys∗ RQY• /S∗ Q∗Y• /S log log Q∗Y• /S −→ RπRcrys∗ Q∗Y• /S . = Q∗Y /S RQY /S∗ RπRcrys∗

Hence it suffices to prove that the two morphisms in (1.6.4.2) are isomorphisms. Moreover, it suffices to prove that these are isomorphisms as morphisms between functors on non-filtered derived categories. For an object I • in D+ (Q∗Y /S (OY /S )) consisting of flasque Q∗Y /S (OY /S )modules, we have Q∗Y /S RQY /S∗ (I • ) = Q∗Y /S QY /S∗ (I • ) = I • , where the second equality follows from the direct calculation. From this, we easily see that the second morphism in (1.6.4.2) is an isomorphism. We prove that the first morphism in (1.6.4.2) is also an isomorphism. Let us say that a sheaf F of OY /S -modules (resp. OY• /S -modules) is parasitic if

52

1 Preliminaries on Filtered Derived Categories and Topoi

Q∗Y /S (F ) = 0 (resp. Q∗Y• /S (F ) = 0). Then, to prove that the first morphism in (1.6.4.2) is an isomorphism, it suffices to prove that, for any sheaf F of OY• /S log modules and for any q ∈ Z, the sheaf Rq πcrys∗ Cone(F → RQY /S∗ Q∗Y /S (F )) is parasitic. Note that we have the spectral sequence log E2st = Rs πcrys∗ Ht (Cone(F → RQY /S∗ Q∗Y /S (F ))) log =⇒ Rs+t πcrys∗ Cone(F → RQY /S∗ Q∗Y /S (F )). log Hence we are reduced to proving that Rs πcrys∗ Ht (Cone(F → RQY /S∗ Q∗Y /S (F ))) is parasitic for any s, t ∈ Z. If we take a flasque resolution Q∗Y /S (F ) −→ I • of Q∗Y /S (F ), we have

Q∗Y /S Ht (Cone(F → RQY /S∗ Q∗Y /S (F ))) = Q∗Y /S Ht (Cone(F → QY /S∗ (I • ))) = Ht (Cone(Q∗Y /S (F ) → Q∗Y /S QY /S∗ (I • ))) = Ht (Cone(Q∗Y /S (F ) → I • )) = 0 (∀t), that is, Ht (Cone(F → RQY /S∗ Q∗Y /S (F ))) is parasitic for any t ∈ Z. Hence it suffices to prove that, for any parasitic sheaf F of OY• /S -modules and for log any s ∈ Z, Rs πcrys∗ (F ) is a parasitic sheaf of OY /S -modules. To prove this, it log suffices to prove the equality (Rq πcrys∗ (F ))D = 0 for any D := (V, DV (V), [ ]) in (1.6.0.15). Let us put D• := (V ×Y Y• , DV (V)|Y• , [ ]|Y• ) and denote the natural morphism D• −→ D by πD . Then we have the following commutative diagram

(1.6.4.3)

log
(Y • /S)crys −−−−→ D•zar ⏐ ⏐ ⏐π log ⏐ πcrys∗  D∗ 


(Y /S)log crys −−−−→ Dzar , where the horizontal morphisms are the canonical restrictions to Zariski topoi. Then the horizontal morphisms in (1.6.4.3) are exact and they send injective sheaves to injective sheaves (cf. the proof of [11, 7.21]). Hence we have, log for a parasitic sheaf F of OY• /S -modules, (Rs πcrys∗ (F ))D = Rs πD∗ (FD• ) = 0. Hence we have finished the proof of the proposition.   As in (1.6.1.2), we have natural morphisms (1.6.4.4)


◦ log ∗ −1 uY• /S : ((Y • /S)Rcrys , QY• /S (OY• /S )) −→ (Y •zar , f• (OS )),

(1.6.4.5)


◦ log ∗ −1 uY•• /S : ((Y
•• /S)Rcrys , QY•• /S (OY•• /S )) −→ (Y ••zar , f•• (OS )).

1.6 The Topos Associated to a Diagram of Topoi. II

53

We have the following commutative diagrams by (1.6.3.5) and (1.6.0.11), and by (1.6.3.6) and (1.6.0.12), respectively: log RπRcrys∗

(1.6.4.6)

D+ F(Q∗Y• /S (OY• /S )) −−−−−−→ D+ F(Q∗Y /S (OY /S )) ⏐ ⏐ ⏐Ru ⏐ RuY• /S∗   Y /S∗ D+ F(f•−1 (OS ))

Rπ

−−−zar∗ −→

D+ F(f −1 (OS ))

and log Rηcrys∗

(1.6.4.7)

D+ F(Q∗Y•• /S (OY•• /S )) −−−−−→ D+ F(Q∗Y• /S (OY• /S )) ⏐ ⏐ ⏐Ru ⏐ RuY•• /S∗   Y• /S∗ −1 D+ F(f•• (OS ))

Rηzar∗

−−−−→

D+ F(f•−1 (OS )).

Set Z := Y• or Y•• and g := f• or f•• , respectively. Then, by (1.6.2), uZ/S∗ (E) = uZ/S∗ (Q∗Z/S (E))  log . As in (1.6.3), we have the following commutative for a sheaf E in (Z/S) crys diagram

(1.6.4.8)

D+ F(OZ/S ) ⏐ ⏐ RuZ/S∗  D+ F(g −1 (OS ))

Q∗ Z/S

−−−−→ D+ F(Q∗Z/S (OZ/S )) ⏐ ⏐Ru  Z/S∗ D+ F(g −1 (OS )).

Chapter 2

Weight Filtrations on Log Crystalline Cohomologies

In this chapter, we construct a theory of weights of the log crystalline cohomologies of families of open smooth varieties in characteristic p > 0, by constructing four filtered complexes. We prove fundamental properties of these filtered complexes. Especially we prove the p-adic purity, the functoriality of three filtered complexes, the convergence of the weight filtration, the weight-filtered K¨ unneth formula, the weight-filtered Poincar´e duality and the E2 -degeneration of p-adic weight spectral sequences. We also prove that our weight filtration on log crystalline cohomology coincides with the one defined by Mokrane in the case where the base scheme is the spectrum of a perfect field of characteristic p > 0.

2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions In this section we give some results on exact closed immersions. After that, we define a relative simple normal crossing divisor (=:relative SNCD) and a key notion admissible immersion of a smooth scheme with a relative SNCD. (1) Let the notations be as in §1.6. Consider triples (2.1.0.1)

(V, DV (V), [ ])’s, ⊂

where V is an open log subscheme of Y , ι : V −→ V is an exact immersion into a log smooth scheme over S and DV (V) is the log PD-envelope of ι over log (S, I, γ). Let (Y /S)log ERcrys be a full subcategory of (Y /S)crys whose objects log are the triples (2.1.0.1). We define the topology of (Y /S)ERcrys as the induced topology by that of (Y /S)log crys .

Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties. Lecture Notes in Mathematics 1959, c Springer-Verlag Berlin Heidelberg 2008


55

56

2 Weight Filtrations on Log Crystalline Cohomologies

 log Definition 2.1.1. We call the site (Y /S)log ERcrys (resp. the topos (Y /S)ERcrys ) the exact restricted log crystalline site (resp. exact restricted log crystalline topos) of Y /(S, I, γ). Let  log  log QER Y /S : (Y /S)ERcrys −→ (Y /S)Rcrys

(2.1.1.1)

 log be a natural morphism of topoi: QER∗ Y /S (E) for an object E ∈ (Y /S)Rcrys is the natural restriction of E and QER∗ Y /S commutes with inverse limits. We also have a morphism (2.1.1.2) ER∗ ∗ ∗  log  log QER Y /S : ((Y /S)ERcrys , QY /S QY /S (OY /S )) −→ ((Y /S)Rcrys , QY /S (OY /S )) of ringed topoi. Proposition 2.1.2. The morphism (2.1.1.1) (resp. (2.1.1.2)) gives an equivalence of topoi (resp. ringed topoi). =

ER Proof. One can check easily the isomorphism F −→ QER∗ Y /S QY /S∗ F for any F ∈ (Y /S)log ERcrys . On the other hand, let D := (V, DV (V), [ ]) be an object of (Y /S)log Rcrys . By ⊂

[54, (5.6)], DV (V) is constructed locally by a local exactification V −→ V ex  ⊂ of V −→ V. Hence there exists a covering D = i Di such that each Di is an log object in (Y /S)log ERcrys . Note that Di ×D Di is also an object in (Y /S)ERcrys . log , we have Then, for any F ∈ (Y /S) Rcrys

  F (D) = Ker( F (Di ) −→ F (Di ×D Di )) i

i,i

  ER∗ ER∗ QER = Ker( QER Y /S∗ QY /S F (Di ) −→ Y /S∗ QY /S F (Di ×D Di )) i

=

i,i

ER∗ QER Y /S∗ QY /S F (D).

ER∗ Hence we have F = QER Y /S∗ QY /S F . Thus the equivalences follow.

 

Next we prove the second fundamental exact sequence for exact closed immersions of fine log schemes and using this, we give a local description of exact closed immersions of fine log schemes under certain assumption. Lemma 2.1.3 (Second fundamental exact sequence). ⊂ Let ι : Z −→ Y be an exact closed immersion of fine log schemes over a fine log scheme S defined by a coherent ideal J of OY . Then the following sequence

2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions

(2.1.3.1)

57

J /J 2 −→ ι∗ (Λ1Y /S ) −→ Λ1Z/S −→ 0 ∆

is exact. Here ∆ is the composite morphism ∆ : J /J 2 −→ι∗ (Ω1Y /S )−→ι∗ (Λ1Y /S ). ◦

If Z/S is log smooth, then ∆ is injective. If Z/S is log smooth and if Y is affine, then (2.1.3.1) is split. Proof. Let MY and MZ be the log structures of Y and Z with structural morphisms αY : MY −→ OY and αZ : MZ −→ OZ , respectively. Let MS be the log structure of S. Because the natural morphisms ι∗ (Ω1◦ ◦ ) −→ Ω1◦ ◦ Y /S

Z/S

∗ and ι−1 (MY /OY∗ ) −→ MZ /OZ are surjective, so is ι∗ (Λ1Y /S ) −→ Λ1Z/S . To prove the exactness of the middle term of (2.1.3.1), it suffices to prove that the following sequence

(2.1.3.2) HomOZ (J /J 2 , E) ←− HomOZ (ι∗ (Λ1Y /S ), E) ←− HomOZ (Λ1Z/S , E) is exact for any OZ -module E. The question is local. Assume that the restriction of an element of g ∈ HomOZ (ι∗ (Λ1Y /S ), E) to ∆(J ) is the zero. Let t be a section of J such that 1 + t ∈ OY∗ . Then g(d log(1 + t)) = g(dt/(1 + t)) = g(dt) = 0. Let β : ι−1 (MY ) −→ ι−1 (OY ) −→ OY /J be the natural morphism. Since MZ is the push-out of the following diagram β −1 ((OY /J )∗ ) −−−−→ ι−1 (MY ) ⏐ ⏐  (OY /J )∗

,

we may assume that a local section of MZ is represented by (u, m) (u ∈ (OY /J )∗ , m ∈ ι−1 (MY )). Let g  : Λ1Z/S −→ E be a morphism defined by ω ) (ω ∈ Ω1◦ ◦ ) and g  ([(u, m)]) = g(d log u) + g(d log m) ([(u, m)] ∈ g  (ω) = g(
Z/S

MZ ), where ω
denotes any lift of ω to ι∗ (Ω1◦

◦

). It is straightforward to check

Y /S

that g  is well-defined and that g  induces g. Thus (2.1.3.2) is exact. ◦

Next assume that Z/S is log smooth and that Y is affine. Let Y 1 be the ⊂ first log infinitesimal neighborhood of the exact closed immersion Z −→ Y . −1 −1 For two sections m ∈ ι (MY ) and a ∈ ι (OY ), let [m] and [a] be the images ◦

in MZ and OZ , respectively. Because Z/S is log smooth and Y is affine, there ⊂ exists a section s : Y 1 −→ Z of the exact closed immersion Z −→ Y 1 induced −1 by ι. In particular, there exist morphisms smo : s (MZ ) −→ MY |Y 1 and sri : s−1 (OZ ) −→ OY 1 such that smo ([m]) = m(1 + t) (∃t ∈ J /J 2 , 1 + t ∈ OY∗ 1 ) and sri ([a]) = a + t (∃t ∈ J /J 2 ); moreover, smo and sri fit into the following commutative diagram:

58

2 Weight Filtrations on Log Crystalline Cohomologies s

mo s−1 (MZ ) −−− −→ MY |Y 1 ⏐ ⏐ ⏐α | ⏐ s−1 (αZ )  Y Y1

s

s−1 (OZ ) −−−ri−→ OY 1 , where the vertical morphisms above are structural morphisms. To prove the existence of the local splitting of (2.1.3.1), we need the module of the log derivations, e.g., in [53, (5.1)]. Let F be an OY -module. Let f : Y −→ S be the structural morphism. ◦

Let DerS (Y, F) be a Γ (S, OS )-module whose elements are the pairs (δ, δ)’s satisfying the following conditions: ◦

(1) δ is a derivation OY −→ F over S, (2) δ is a morphism MY −→ F of monoids, ◦

(3) αY (m)δ(m) = δ(αY (m)) (m ∈ MY ), (4) δ(f −1 (n)) = 0 (n ∈ MS ). Then, by [53, (5.3)], we have an isomorphism HomOY (Λ1Y /S , F) h −→ (h ◦ d, h ◦ d log) ∈ DerS (Y, F). In particular, ∼

HomOZ (ι∗ (Λ1Y /S ), J /J 2 ) = HomOY (Λ1Y /S , J /J 2 ) −→ DerS (Y, J /J 2 ). Let β be the isomorphism (1+J )/(1+J 2 ) 1+t −→ t ∈ J /J 2 of abelian ◦

sheaves. It is easy to check that the morphisms δ : OY a −→ a − sri ([a]) ∈ J /J 2 and δ : MY m −→ β(m/smo ([m])) ∈ J /J 2 satisfy (1) ∼ (4) and give a local splitting of (2.1.3.1).   Lemma 2.1.4. Let the notations be as in (2.1.3) with Y, Z log smooth over S. Let AnS (n ∈ N) be a log scheme whose underlying scheme is An◦ and whose S

◦

log structure is the pull-back of that of S by the natural projection An◦ −→ S. S

◦

Let z be a point of Z and assume that there exists a chart (Q −→ MS , P −→ ρ MZ , Q −→ P ) of Z −→ S on a neighborhood of z such that ρ is injective, such that Coker(ρgp ) is torsion free and that the natural homomorphism OZ,z ⊗Z (P gp /Qgp ) −→ Λ1Z/S,z is an isomorphism. Then, on a neighborhood of z, there exist a nonnegative integer c and the following cartesian diagram:

(2.1.4.1)

Z ⏐ ⏐ 

ι

−−−−→ ⊂

Y ⏐ ⏐ 

(S ⊗Z[Q] Z[P ], P a ) −−−−→ (S ⊗Z[Q] Z[P ], P a ) ×S AcS .

2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions

59

Here the vertical morphisms are strict etale and the lower horizontal morphism is the base change of the zero section S −→ AcS . ◦

Proof. Assume that Y is affine. By (2.1.3) we have the following split exact sequence 0 −→ J /J 2 −→ ι∗ (Λ1Y /S ) −→ Λ1Z/S −→ 0. ∆

(2.1.4.2)

Let s be the image of z in S. Since Coker(ρgp ) is torsion free, there exgp which is compatible with the monoid ists a homomorphism P gp −→ MY,ι(z) homomorphisms Q −→ MS,s −→ MY,ι(z) and P −→ MZ,z . Since we have gp ∗ )z = (MY /OY∗ )ι(z) , the homomorphism P gp −→ MY,ι(z) induces the (MZ /OZ homomorphism P −→ MY,ι(z) , which induces a chart of Y −→ S on a neighborhood of ι(z). By the exact sequence (2.1.4.2), there exist local sections xr+1 , . . . , xr+c ∈ J and elements m1 , . . . , mr ∈ P such that {d log mi }ri=1 is 1 a basis of Λ1Z/S,z and {{d log mi }ri=1 , {dxj }d+c j=r+1 } is a basis of ΛY /S,ι(z) . By the same argument as that in [54, p. 205], we have compatible etale morphisms ◦

◦

◦

◦

Z −→ S ⊗Z[Q] Z[P ] and Y −→ (S ⊗Z[Q] Z[P ]) × ◦ Spec ◦ (O ◦ [xd+1 , . . . , xd+c ]) S S S in the classical sense.   ⊂

Corollary 2.1.5. Let S0 −→ S be a closed immersion of fine log schemes. Let Z0 (resp. Y ) be a log smooth scheme over S0 (resp. S), which can be ⊂ considered as a log scheme over S. Let ι : Z0 −→ Y be an exact closed im◦

mersion over S. Let z be a point of Z 0 and assume that there exists a chart ρ ⊂ (Q −→ MS , P −→ MZ0 , Q −→ P ) of Z0 −→ S0 −→ S on a neighborhood of gp z such that ρ is injective, such that Coker(ρ ) is torsion free and that the natural homomorphism OZ0 ,z ⊗Z (P gp /Qgp ) −→ Λ1Z/S0 ,z is an isomorphism. Then, on a neighborhood of z, there exist a nonnegative integer c and the following cartesian diagram (2.1.5.1) Z0

⏐ ⏐ 

−−−−−→

⊂

Y

⏐ ⏐ 

−−−−−→

Y

⏐ ⏐ 

⊂

(S0 ⊗Z[Q] Z[P ], P a ) −−−−−→ (S ⊗Z[Q] Z[P ], P a ) −−−−−→ (S ⊗Z[Q] Z[P ], P a ) ×S AcS ,

where the vertical morphisms are strict etale and the lower second horizontal ⊂ morphism is the base change of the zero section S −→ AcS and Y  := Y ×AcS S. ⊂

Proof. Set Y0 := Y ×S S0 and let ι0 : Z0 −→ Y0 be the closed immersion induced by ι. Apply (2.1.4) for ι0 . Then we have a cartesian diagram ◦

(2.1.4.1) for Z0 /S0 and Y0 /S0 around any point z ∈ Z 0 . By the same argument as in the proof of (2.1.4) using the isomorphism (MY /OY∗ )ι(z)  (MY0 /OY∗ 0 )ι0 (z) , we see that the chart P −→ MY0 extends to a chart

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2 Weight Filtrations on Log Crystalline Cohomologies

P −→ MY around ι(z). Let J0 (resp. J ) be the ideal sheaf of ι0 (resp. ι). (0) (0) Let {{d log mi }ri=1 , {dxj }r+c ∈ J0 ) be a basis of Λ1Y0 /S0 . j=r+1 } (mi ∈ P, xj (0)

Let xj be any lift of xj in J . Then, using [13, Corollaire to II §3 Propo1 sition 6], we see that {{d log mi }ri=1 , {dxj }r+c j=r+1 } is a basis of ΛY /S,ι(z) (cf.[40, 4 (17.12.2)]). Hence we have a strict etale morphism Y −→ (S ⊗Z[Q] Z[P ], P a ) ×S AcS . Now we obtain the diagram (2.1.5.1).   Remark 2.1.6. By a similar argument to the proof of (2.1.4) and (2.1.5) and using [54, (3.5), (3.13)], we see that the diagrams as in (2.1.4.1), (2.1.5.1) always exist etale locally (for some Q −→ P ) even if we drop the condition on the existence of a nice chart which we assumed in (2.1.4), (2.1.5). (2) Let Y be a scheme over a scheme T . Let Div(Y /T )≥0 be the integral monoid of effective Cartier divisors on Y over T (e.g., [56, (1.1.1)]). We say that a family {Eλ }λ∈Λ of non-zero elements in Div(Y /T )≥0 has a locally finite intersection if, for any point z ∈ Y , there exists a Zariski open neighborhood V of z such that ΛV := {λ ∈ Λ | Eλ |V = 0} is a finite set. If {Eλ }λ∈Λ has a locally finite intersection, then we can define a sum λ∈Λ nλ Eλ (nλ ∈ N) in Div(Y /T )≥0 . Let f : X −→ S0 be a smooth morphism of schemes. Definition 2.1.7. We call an effective Cartier divisor D on X/S0 is a relative simple normal crossing divisor (=:relative SNCD) on X/S0 if there exists a family ∆ := {Dλ }λ∈Λ of non-zero effective Cartier divisors on X/S0 of locally finite intersection which are smooth schemes over S0 such that

Dλ in Div(X/S0 )≥0 (2.1.7.1) D= λ∈Λ

and, for any point z of D, there exist a Zariski open neighborhood V of z in X and the following cartesian diagram: (2.1.7.2) D|V ⏐ ⏐ 

⊂

−−−−→

V ⏐ ⏐g 

SpecS (OS0 [y1 , . . . , yd ]/(y1 · · · ys )) −−−−→ SpecS (OS0 [y1 , . . . , yd ]) 0

0

(for some positive integers s and d such that s ≤ d), where the morphism g is etale. Note that we do not require a relation a priori between {Dλ |V }λ∈ΛV and the family {yi = 0}si=1 of closed subschemes in V in the diagram (2.1.7.2). However, by (A.0.1) below, we obtain {Dλ |V }λ∈ΛV = {{yi = 0}}si=1 in the diagram (2.1.7.2) if V is small.

2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions

61

Definition 2.1.8. We call a smooth divisor on X/S0 contained in D a smooth component of D. We call ∆ = {Dλ }λ∈Λ a decomposition of D by smooth components of D over S0 . Note that the decomposition of a relative SNCD by smooth components is not unique. Let DivD (X/S0 )≥0 be a submonoid of Div(X/S0 )≥0 consisting of effective Cartier divisors E’s on X/S0 such that there exists an open covering X =  i∈I Vi (depending on E) of X such that E|Vi is contained in the submonoid of Div(Vi /S0 )≥0 generated by Dλ |Vi (λ ∈ Λ). By (A.0.1) below, we see that the definition of DivD (X/S0 )≥0 is independent of the choice of ∆. The pair (X, D) gives a natural fs(=fine and saturated) log structure in
Xzar as follows (cf. [54, p. 222–223], [29, §2]).
zar defined as follows: for an open Let M (D) be a presheaf of monoids in X subscheme V of X, (2.1.8.1)

Γ (V, M (D) ) := {(E, a) ∈DivD|V (V /S0 )≥0 × Γ (V, OX )| a is a generator of Γ (V, OX (−E))}

with a monoid structure defined by (E, a) · (E  , a ) := (E + E  , aa ). The natural morphism M (D) −→ OX defined by the second projection (E, a) → a
zar . induces a morphism M (D) −→ (OX , ∗) of presheaves of monoids in X The log structure M (D) is, by definition, the associated log structure to the sheafification of M (D) . Because DivD|V (V /S0 )≥0 is independent of the choice of the decomposition of D|V by smooth components, M (D) is independent of the choice of the decomposition of D by smooth components of D. Proposition 2.1.9. Let the notations be as above. Let z be a point of D and let V be an open neighborhood of z in X which admits the diagram s (2.1.7.2). Assume that z ∈ i=1 {yi = 0}. If V is small, then the log struc⊂ ture M (D)|V −→ OV is isomorphic to OV∗ y1N · · · ysN −→ OV . Consequently M (D)|V is associated to the homomorphism NsV ei −→ yi ∈ M (D)|V (1 ≤ i ≤ s) of sheaves of monoids on V , where {ei }si=1 is the canonical basis of Ns . In particular, M (D) is fs. Proof. By the definition of M  (D) and by (A.0.1) below, the homomorphism M  (D)|V −→ OV factors through OV∗ y1N · · · ysN if V is small. Hence there exists a natural morphism M (D)|V −→ OV∗ y1N · · · ysN of log structures on V . By taking the stalks, one can easily check that the morphism above is an isomorphism.   By abuse of notation, we denote the log scheme (X, M (D)) by (X, D). ⊂ Set U := X \ D and let j : U −→ X be the natural open immersion. Set ∗ N (D) := OX ∩ j∗ (OU ). We remark that M (D)  N (D) in general; indeed, ∗ are not even finitely generated in general (see (A.0.9) the stalks of N (D)/OX below).

62

2 Weight Filtrations on Log Crystalline Cohomologies ⊂

Let S0 −→ S be a closed immersion of schemes defined by a quasi-coherent ideal sheaf I of OS . We can consider the scheme X as a scheme over S by the ⊂ closed immersion S0 −→ S. Let (X , D)(= (X , M (D))) be a smooth scheme with a relative SNCD over S. Let ι : X −→ X be a closed immersion over S defined by a quasi-coherent ideal sheaf of OX . Definition 2.1.10. Let ∆ := {Dλ }λ∈Λ be a decomposition of D by smooth ⊂ components of D. Let ι : (X, D) −→ (X , D) be an exact (closed) immersion into a smooth scheme with a relative SNCD over S. Then we call ι (or a pair (X , D)/S by abuse of terminology) an admissible (closed) immersion over S
:= {Dλ }λ∈Λ by smooth with respect to ∆ if D admits a decomposition ∆ ∼ components of D such that ι induces an isomorphism Dλ −→ Dλ ×X X
is compatible with ∆. of schemes over S0 for all λ ∈ Λ. We say that ∆ ⊂ We sometimes denote the admissible (closed) immersion by ι : (X, D; ∆) −→
(X , D; ∆). ◦

◦

Remark 2.1.11. If the underlying topological spaces of S 0 and S are the same and if (X , D) is a lift of (X, D) with a decomposition ∆ of D by smooth
of D by smooth components components of D, we obtain the decomposition ∆ of D canonically. ⊂


be an admissible immersion. Let V be an Let ι : (X, D; ∆) −→ (X , D; ∆) open subscheme of X. If we set V := X \ (X \ V ) (here X is the closure of X in X ), the restriction of ι to (V, D ∩ V )  ⊂ (2.1.11.1) ιV : (V, D ∩ V ) −→ (V, ( Dλ ) ∩ V) λ∈ΛV

is an admissible immersion with respect to {Dλ }λ∈ΛV . Definition 2.1.12. We call the admissible immersion ιV the restriction of ι to V , and ∆|V := {Dλ }λ∈ΛV the restriction of ∆ to V . By (2.1.5) and (A.0.1) below, we have the following: ⊂
be an admissible immersion. Lemma 2.1.13. Let ι : (X, D; ∆) −→ (X , D; ∆) Then, for any point z of X, there exist Zariski open neighborhoods V of z and V of ι(z), positive integers s ≤ d ≤ d and the following two cartesian diagrams:

(2.1.13.1) D|V ⏐ ⏐ 

⊂

−−−−→

V ⏐ ⏐g 

SpecS (OS [x1 , . . . , xd ]/(x1 · · · xs )) −−−−→ SpecS (OS [x1 , . . . , xd ]),

2.1 Exact Closed Immersions, SNCD’s and Admissible Immersions

63

(2.1.13.2) V ⏐ ⏐ 

⊂

−−−−→

V ⏐ ⏐g 

SpecS (OS0 [x1 , . . . , xd ]/(xd+1 , . . . , xd )) −−−−→ SpecS (OS [x1 , . . . , xd ]), 0

where g is etale and {Dλ |V }λ∈ΛV = {{xi = 0}}si=1 in the diagram (2.1.13.1). Let (S, I, γ) be a PD-scheme and let (X, D) be a smooth scheme with a relative SNCD over S0 := SpecS (OS /I). Let ∆ be a decomposition of D by smooth components of D. Consider triples ((U, D|U ), D(U,D|U ) ((U, D)), [ ])’s,

(2.1.13.3)

⊂

where U is an open subscheme of X, (U, D|U ) −→ (U, D) is an admissible immersion over S with respect to ∆U and D(U,D|U ) ((U, D)) is the log PDenvelope of the immersion above over (S, I, γ). Let ((X, D)/S)log ARcrys be a whose objects are the triples (2.1.13.3). full subcategory of ((X, D)/S)log crys log We define the topology of ((X, D)/S)ARcrys as the induced topology by
that of ((X, D)/S)log . Let ((X, D)/S)log be the topos associated to crys

ARcrys

((X, D)/S)log ARcrys . Definition 2.1.14. We call the site ((X, D)/S)log ARcrys (resp. the topos log
((X, D)/S)ARcrys ) the admissible restricted log crystalline site (resp. admissible restricted log crystalline topos) of (X, D)/(S, I, γ). Let
log
log QAR (X,D)/S : ((X, D)/S)ARcrys −→ ((X, D)/S)Rcrys

(2.1.14.1)


be a natural morphism of topoi: For an object E ∈ ((X, D)/S)log Rcrys , AR∗ QAR∗ (E) is the natural restriction of E and Q commutes with (X,D)/S (X,D)/S inverse limits. We also have a morphism (2.1.14.2)

AR∗ ∗
log QAR (X,D)/S : (((X, D)/S)ARcrys , Q(X,D)/S Q(X,D)/S (O(X,D)/S )) ∗
−→ (((X, D)/S)log Rcrys , Q(X,D)/S (O(X,D)/S ))

of ringed topoi. Proposition 2.1.15. The morphism (2.1.14.1) (resp. (2.1.14.2)) gives an equivalence of topoi (resp. ringed topoi). ⊂

Proof. Let ι : (X, D) −→ P be an exact closed immersion into a log smooth scheme over S. Let P  be an exact closed log subscheme of P locally obtained

64

2 Weight Filtrations on Log Crystalline Cohomologies

in (2.1.5) for ι. Then ι is locally an admissible immersion with respect to the restriction of ∆ to an open subscheme of X since P  is a local lift of (X, D). Hence we obtain (2.1.15) by (2.1.2) and by the proof of (2.1.2).  

2.2 The Log Linearization Functor In this section we recall the log version of the linearization functor in [11, §6] (cf. [54, (6.9)]) and the log HPD differential operators. After that, we show some properties of the log linearization functor for a smooth scheme with a relative SNCD. (1) Let (S, I, γ) and f : Y −→ S be as in §1.6. For an object (V, T, MT , ι, δ) of the log crystalline site (Y /S)log crys , we sometimes denote it simply by (V, T, MT , δ), (V, T, δ) or even T as usual. We also denote by T the rep log resentable sheaf in (Y /S)log crys defined by T . Let F be an object of (Y /S)crys . Let (Y /S)log |F be the localization of the topos (Y /S)log at F : the objects crys

crys

 log in (Y /S)log crys |F are the pairs (E, φ)’s, where E is an object in (Y /S)crys and φ is a morphism E −→ F in (Y /S)log . As usual, let crys

(2.2.0.1)

 log jF : (Y /S)log crys |F −→ (Y /S)crys

 be a morphism of topoi defined by the following: for an object E in (Y /S)log crys , proj. j ∗ (E) is a pair (E × F, E × F −→ F ); for an object (E, φ) in (Y /S)log |F , crys

F

jF ∗ ((E, φ)) is the sheaf of the sections of φ. Let (V, T, MT , δ) be an object of the log crystalline site (Y /S)log crys . Let  log jT : (Y /S)log crys |T −→ (Y /S)crys be the localization morphism in (2.2.0.1) for F = T . Let (2.2.0.2)


ϕ : ((Y /S)log crys |T , OY /S |T ) −→ (Tzar , OT )

be a morphism of ringed topoi defined by the following (cf. [11, 5.26 Proposition]): for an OT -module E, the sections of ϕ∗ (E) at (T  , φ) is Γ (T  , φ∗ (E));   /S)log for an OY /S -module E in (Y crys |T , ϕ∗ (E) is defined as follows: let T be ⊂

an open log subscheme of T . Let T  also denote the object (T  ×T V −→  (T  ×T T = T  )) in (Y /S)log crys . Then we have a natural morphism ι : T −→ T   in (Y /S)log crys ; the section of ϕ∗ (E) is, by definition, Γ (T , ϕ∗ (E)) := E((T , ι)). By the log version of the ringed topos version of [11, 5.26 Proposition], we have the following diagram of ringed topoi

2.2 The Log Linearization Functor

65

(2.2.0.3) uY /S jT −1  log
(OS )) ((Y /S)log crys |T , OY /S |T ) −−−−→ ((Y /S)crys , OY /S ) −−−−→ (Yzar , f  ⏐ ⏐ ⏐ ϕ ⏐ (T
zar , OT )

←−−−−

(V
zar , OV )

−−−−→ (V
zar , f −1 (OS )|V )

and the following commutative diagram of topoi  log (Y /S)log crys |T −−−−→ (Y /S)crys ⏐ ⏐ ⏐uY /S ⏐ ϕ  jT

(2.2.0.4)

T
zar = V
zar −−−−→

Y
zar ,

where ϕ is defined as follows: Γ((T  , φ), ϕ−1 (E)) := Γ(T  , φ−1 (E)) for E ∈ T
zar and (T  , φ) ∈ (Y /S)log crys |T . By the log version of [11, 5.27 Corollary], we have the following: Proposition 2.2.1. Let the notations be as above. Assume that V = Y . Then the following hold: (1) The functors jT ∗ is exact. (2) For an abelian sheaf E in (Y /S)log crys , jT ∗ (E) is uY /S∗ -acyclic. Now let us recall the log linearization functor briefly (cf. [11, 6.10 Proposition], [54, (6.9)]). ⊂ Let ι : Y −→ Y be a closed immersion into a log smooth scheme over ◦

S such that γ extends to Y. Let DY (Y) be the log PD-envelope of ι over (S, I, γ). Let (2.2.1.1)

◦
ϕ : ((Y /S)log crys |DY (Y) , OY /S |DY (Y) ) −→ (DY (Y)zar , ODY (Y) )

be the morphism (2.2.0.2) for T = DY (Y). For an ODY (Y) -module E, we define L(E) as follows: (2.2.1.2)

L(E) := jDY (Y)∗ ϕ∗ (E) ∈ (Y /S)log crys .

As in the classical crystalline case, L defines a functor: (2.2.1.3) {the category of ODY (Y) -modules and ODY (Y) -linear morphisms} −→ {OY /S -modules}. ⊂

For (U, T, δ) ∈ (Y /S)log crys , let DU (T ×S Y) be the PD-envelope of U −→ T ×S Y compatible with γ and δ and let pT : DU (T ×S Y) −→ T , pY : DU (T ×S Y) −→ DY (Y) be natural morphisms. Then the sheaf L(E)(U,T,δ) on Tzar induced by L(E) is given by L(E)(U,T,δ) = pT ∗ p∗Y E = ODU (T ×S Y) ⊗ODY (Y) E.

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2 Weight Filtrations on Log Crystalline Cohomologies

As in the classical crystalline case, another definition of the log linearization functor is known. To state it, we need to recall the definition of a log HPD stratification (cf. [11, 4.3H Definition]; however there is a mistype in [loc.cit., 1)]: “DX/S -linear” should be replaced by “DX/S (1)-linear”). ⊂

Let DY (Y 2 ) be the log PD-envelope of the locally closed immersion Y −→ Y ×S Y over (S, I, γ). Let J be the PD-ideal sheaf defining the exact locally ⊂ closed immersion Y −→ DY (Y 2 ). Definition 2.2.2. Let E and F be two ODY (Y) -modules. (1) An isomorphism  : ODY (Y 2 ) ⊗ODY (Y) E −→ E ⊗ODY (Y) ODY (Y 2 ) is called a log HPD stratification if  is ODY (Y 2 ) -linear, if  mod J is the identity and if the cocycle condition holds. (2) ([75, (1.1.3)]) An ODY (Y) -linear morphism u : ODY (Y 2 ) ⊗ODY (Y) E −→ F is called a log HPD differential operator. (3) ([75, (1.1.3)]) For a positive integer n, an ODY (Y) -linear morphism u : (ODY (Y 2 ) /J [n+1] ) ⊗ODY (Y) E −→ F is called a log PD differential operator of order ≤ n. Set L (E) := ODY (Y 2 ) ⊗ODY (Y) E. Then, as in the classical crystalline case, there is a canonical log HPD stratification ∼

ODY (Y 2 ) ⊗ODY (Y) L (E) −→ L (E) ⊗ODY (Y) ODY (Y 2 ) . Hence L (E) defines a crystal of OY /S -modules (cf. [54, (6.7)]), which we denote by the same symbol L (E). L defines a functor {the category of ODY (Y) -modules and log HPD differential operators} −→ {the category of crystals of OY /S -modules} : For a log HPD differential operator u : ODY (Y 2 ) ⊗ODY (Y) E −→ F, L (u) : L (E) −→ L (F) is given by the composite (2.2.2.1)

δ⊗id

ODY (Y 2 ) ⊗ODY (Y) E −→ ODY (Y 2 ) ⊗ODY (Y) ODY (Y 2 ) ⊗ODY (Y) E id⊗u

−→ ODY (Y 2 ) ⊗ODY (Y) F,

where δ : ODY (Y 2 ) −→ ODY (Y 2 ) ⊗ODY (Y) ODY (Y 2 ) = ODY (Y 3 ) is the map induced by the projection Y 3 −→ Y 2 to the first and the third factors. By the log version of [11, 6.10 Proposition], the following holds: Proposition 2.2.3. For an ODY (Y) -module E, there exists a canonical isomorphism ∼ L (E) −→ L(E).

2.2 The Log Linearization Functor

67

Hence L also defines the functor {the category of ODY (Y) -modules and log HPD differential operators} −→ {the category of crystals of OY /S -modules}. By (2.2.2.1) and (2.2.3), we see the following: For a log HPD differential operator u : ODY (Y 2 ) ⊗ODY (Y) E −→ F and (U, T, δ) ∈ (Y /S)log crys , L(u)(U,T,δ) : L(E)(U,T,δ) −→ L(F)(U,T,δ) is given by the composite (2.2.3.1) δ ⊗id

T ODY (T ×S Y) ⊗ODY (Y) ODY (Y 2 ) ⊗ODY (Y) E ODY (T ×S Y) ⊗ODY (Y) E −→

id⊗u

−→ ODY (T ×S Y) ⊗ODY (Y) F,

where δT : ODY (T ×S Y) −→ ODY (T ×S Y) ⊗ODY (Y) ODY (Y 2 ) = ODY (T ×S Y 2 ) (the equality follows from the log version of [11, 6.3, proof of 6.10]) is the map induced by the projection T ×S Y 2 −→ T ×S Y to the first and the third factors. It is easy to obtain the following lemma from the definition of L . Lemma 2.2.4. The functor L, regarded as the functor {the category of ODY (Y) -modules and ODY (Y) -linear morphisms} −→ {the category of crystals of OY /S -modules} is exact. Remark 2.2.5. (cf. [3, IV Remarque 1.7.8]) The functor L is not left exact as a functor (2.2.1.3) in general. Indeed, let κ be a perfect field of characteristic p > 0 and let Wn (n ∈ Z≥2 ) be the Witt ring of κ of length n. Set S := (Spec(Wn ), Wn∗ , pWn , [ ]), Y := (Spec(κ), κ∗ ), Y := S and E := Wn . Then, though a sequence pn−1 ×

0 −→ pE −→ E −→ E of Wn -modules is exact, the following sequence pn−1 ×

0 −→ L(pE) −→ L(E) −→ L(E) in OY /S -modules is not exact since the value of the sequence above at Y is 0

0

0 −→ pWn /p2 Wn −→ κ −→ κ. The following is analogous to [11, 6.2 Proposition]. Lemma 2.2.6. (1) Let Yi (i = 1, 2) and (S, I, γ) be as in §1.6. Let Ti = (Ui , Ti , δi ) = (Ui , Ti , MTi , δi ) (i = 1, 2) be an object of the log crystalline site  log (Yi /S)log crys , which is considered as a representable sheaf in the topos (Yi /S)crys .

68

2 Weight Filtrations on Log Crystalline Cohomologies ⊂

Let Ji be the defining ideal sheaf of the closed immersion Ui −→ Ti . Let ⊂ Y1 −→ Y2 be an exact closed immersion which induces an exact closed im⊂ mersion U1 −→ U2 . Let g : T1 −→ T2 be an exact closed immersion of fine log PD-schemes over S fitting into the following commutative diagram ⊂

U1 −−−−→ ⏐ ⏐ 

U2 ⏐ ⏐ 

g

T1 −−−−→ T2 . ⊂

Assume that g ∗ induces a surjective morphism g ∗ : g ∗ (J2 ) −→ J1 . Let log  log ι : (Y 1 /S)crys |T1 −→ (Y2 /S)crys |T2

be the induced morphism of topoi. Let (U, T, δ, φ) = (U, T, MT , δ, φ) be a log representable object in (Y 2 /S)crys |T2 . Let J be the defining ideal sheaf of ⊂

the closed immersion U −→ T . Set J := J + IOT and let δ be the extension of δ and γ on J . Let Dδ (T ×T2 T1 ) be the log PD-envelope of the ⊂ closed immersion U ×U2 U1 −→ T ×T2 T1 over (T, J , δ) with natural morphism q : (U ×U2 U1 , Dδ (T ×T2 T1 ), [ ]) −→ (U1 , T1 , δ1 ) in (Y1 /S)log crys . Then ι∗ ((U, T, δ, φ)) is representable by an object (U ×U2 U1 , Dδ (T ×T2 T1 ), [ ], q) ∈ (Y1 /S)log crys |T1 ; the functor ι∗ is exact. (2) Let the notations and the assumptions be as in (1). Then Dδ (T ×T2 T1 ) = T ×T2 T1 . Proof. (1): We have to check that (U ×U2 U1 , Dδ (T ×T2 T1 ), [ ], q) is actually an object of (Y1 /S)log crys |T1 . Since U ×U2 U1 is an open subscheme of U1 , γ extends to OU ×U2 U1 . Since the image J in OU ×U2 U1 is IOU ×U2 U1 , δ actually extends to OU ×U2 U1 (cf. [11, 6.2.1 Lemma]). Since δ extends to OU ×U2 U1 , the exact closed immersion U ×U2 ⊂

U1 −→ Dδ (T ×T2 T1 ) is a PD closed immersion by [11, 3.20 Remarks 4)]. Set J i := Ji +IOTi (i = 1, 2) and let δ i be the extension of δi and γ on J i .  Set J12,T := Ker(OT ×T2 T1 −→ OU ×U2 U1 ). Let J 12,T be the PD-ideal sheaf 

of ODδ (T ×T2 T1 ) obtained from J12,T . Set J 12,T := J 12,T + IODδ (T ×T2 T1 ) . Consider the following commutative diagram ODδ (T ×T2 T1 )  ⏐ ⏐

ODδ (T ×T2 T1 )  ⏐ ⏐

OT ×T2 T1  ⏐ ⏐

OT  ⏐ ⏐

OT1

←−−−−

OT2 .

2.2 The Log Linearization Functor

69

Here we omit to write the direct images. We claim that the left vertical composite morphism induces a PD-morphism (OT1 , J 1 ) −→ (ODδ (T ×T2 T1 ) , J 12,T ). Indeed, by the definition of δ, the composite morphism (OT2 , J 2 ) −→ (OT , J ) −→ (ODδ (T ×T2 T1 ) , J 12,T ) is a PD-morphism. Let s be a local section of Ker(g ∗ : J2 −→ J1 ). Then the image of s in ODδ (T ×T2 T1 ) by the right vertical composite morphism is the zero. Hence the claim follows because g ∗ : g ∗ (J 2 ) −→ J 1 is surjective by the assumption. Consequently we actually have a natural morphism q : (U ×U2 U1 , Dδ (T ×T2 T1 ), [ ]) −→ (U1 , T1 , δ1 ) of log PD-schemes over (S, I, γ). By using the universality of the log PD-envelope, it is straightforward to see that (2.2.6.1)

ι∗ ((U, T, δ, φ)) = (U ×U2 U1 , Dδ (T ×T2 T1 ), [ ], q).

log Therefore, for an object E in (Y 2 /S)crys |T2 , we have

(2.2.6.2)

ι∗ E((U, T, δ, φ)) = Hom(Y /S)log 1

crys |T1

(ι∗ ((U, T, δ, φ)), E)

= E((U ×U2 U1 , Dδ (T ×T2 T1 ), [ ], q)). Using the formula (2.2.6.2) and noting that Dδ (T ×T2 T1 ) ≈ T ×T2 T1 is a closed set of T as a topological space, we can easily see that the functor ι∗ is exact. (2): Set J12 := Ker(OT2 −→ g∗ (OT1 )). The structure sheaf of T ×T2 T1 is equal to OT /J12 OT . By the following commutative diagram OT /J  ⏐ ⏐

OU   

φ−1 (OT2 /J2 )

φ−1 (OU2 ),

we have J ∩ φ−1 (J12 )OT = φ−1 (J2 ∩ J12 )OT . It is easy to see that the ideal sheaf J12 ∩ J2 is a sub PD-ideal sheaf of J2 . Hence, by the same proof of [11, 3.5 Lemma], the PD-structure δ defines a unique PD-structure δ12 on J (OT /J12 OT ). Moreover, it is easy to see that γ extends to OT ×T2 T1 . Hence (OT /J12 OT , J (OT /J12 OT ), δ12 ) is a sheaf of the universal PD-algebras of   (OT ×T2 T1 , J12,T ) over (OT , J , δ), that is, we have (2). Following [31], let us denote by ΛiY /S the sheaf of log differential forms of Y /S of degree i (i ∈ N). The following is a log version of [11, 6.12 Theorem]: ⊂

Proposition 2.2.7. Let ι : Y −→ Y be a closed immersion of fine log schemes over S. Assume that Y is log smooth over S and that γ extends

70

2 Weight Filtrations on Log Crystalline Cohomologies ◦

to Y . Let DY (Y) be the log PD-envelope of ι over (S, I, γ). Then the natural morphism OY /S −→ L(ODY (Y) ⊗OY Λ•Y/S )

(2.2.7.1)

is a quasi-isomorphism. Proof. Let DY (Y i ) (i ∈ Z>0 ) be the log PD-envelope of the composite im⊂ ⊂ ⊂ mersion Y −→ Y −→ Y i over S, where Y −→ Y i is the diagonal immersion. i-th proj. Let pi : DY (Y 2 ) −→ Y 2 −→ Y (i = 1, 2) be a natural morphism and let J be the ideal sheaf of the locally exact closed immersion Y −→ DY (Y 2 ). The problem is local as in [11, 6.12 Theorem]; we may assume that Λ1Y/S has a basis {d log tj }nj=1 , where tj is a local section of the log structure of Y. Let ∗ ∗ −→ OY ) such that p∗2 (tj ) = p∗1 (tj )uj . uj be a local section of Ker(OD 2 Y (Y ) Then, by [54, (6.5)], the following morphism [n]

OY s1 , . . . , sn  sj −→ (uj − 1)[n] ∈ ODY (Y 2 ) is an isomorphism, where sj ’s are independent indeterminates. We identify ODY (Y 2 ) with OY s1 , . . . , sn  by this isomorphism. By the log version of [11, 6.2 Proposition], ιlog crys∗ (OY /S ) is a crystal of OY/S -modules. Hence, as in [11, ∼ 6.3 Corollary], we obtain a canonical isomorphism ODY (Y) ⊗OY ODY (Y 2 ) −→ ODY (Y 2 ) . Consequently we can identify ODY (Y 2 ) with ODY (Y) s1 , . . . , sn  (cf. [54, (6.5)]). Moreover, by [54, (5.8.1)] and [81, Proposition 3.2.5], there ∼ exists an isomorphism Λ1Y/S d log tj −→ uj − 1 ∈ J /J 2 = J /J [2] of OY -modules. Let p13 : DY (Y 3 ) −→ DY (Y 2 ) be the induced morphism by the product of the first and the third projections Y 3 −→ Y 2 . Let p∗

∼

13 δ : ODY (Y 2 ) −→ ODY (Y 3 ) −→ ODY (Y 2 ) ⊗ODY (Y) ODY (Y 2 )

be the morphism in [75, p. 14]. Then, by the formula [75, (1.1.4.2)], δ(uj ) = uj ⊗ uj . Hence δ(sj ) = sj ⊗ sj + sj ⊗ 1 + 1 ⊗ sj (the last formula in [75, p. 16]). Hence the natural connection ∇ : ODY (Y 2 ) ⊗OY ΛqY/S −→ ODY (Y 2 ) ⊗OY Λq+1 Y/S is given by (2.2.7.2)

[i ] ∇(as1 1

n] · · · s[i n ⊗ω)

n

[i −1] [i ] n] = a( s1 1 · · · sj j · · · s[i n (sj +1)d log tj ∧ω j=1

[i ]

q n] +s1 1 · · · s[i n ⊗ dω) (a ∈ ODY (Y) , i1 , . . . , in ∈ N, ω ∈ ΛY/S )

2.2 The Log Linearization Functor

71

as in [11, 6.11 Lemma]. Let (U, T, δ) be an object of (Y /S)log crys . Because the problem is local, we may assume that there exists the following commutative diagram: ⊂ U −−−−→ T ⏐ ⏐ ⏐ ⏐   ι

Y −−−−→ Y. Then we have a natural morphism (U, T, δ) −→ (Y, DY (Y), [ ]) in (Y /S)log crys and a natural complex OT s1 , . . . , sn  ⊗OY Λ•Y/S , which is equal to the complex L(ODY (Y) ⊗OY Λ•Y/S )(U,T,δ) . Now, consider the case n = 1 and set s1 = s and t1 = t. Then the ∇T complex OT s1 , . . . , sn ⊗OY Λ•Y/S is equal to OT s −→ OT sd log t. Because [n] [n−1] [n] [n−1] ∇T (s ) = s (s + 1)d log t = (ns + s )d log t for a positive integer n, we have the following formula (2.2.7.3) m m

an s[n] ) = (an + (n − 1)an−1 )s[n−1] d log t + mam s[m] d log t ∇T ( n=0

n=1

(m ∈ N, an ∈ OT (0 ≤ n ≤ m)). ◦

Hence Ker(∇T ) = OT . Because p is locally nilpotent on S, we may assume that pN apN = 0 if N is sufficiently large. Hence we see that Coker(∇T ) = 0 by the formula (2.2.7.3). Therefore we have checked that the morphism (2.2.7.1) is a quasi-isomorphism for the case n = 1. The rest of the proof is the same as that of [11, 6.12 Theorem].   Proposition 2.2.8 ([54, the proof of (6.9)]). ⊂ Let ι : Y −→ Y, Y and DY (Y) be as in (2.2.7). Let E be a crystal of OY /S modules. Let (E, ∇) be the corresponding ODY (Y) -module with integrable connection. Then there exists a natural quasi-isomorphism E −→ L(E ⊗OY Λ•Y/S ).

(2.2.8.1)

Proof. The proof is the same as that in [11, 6.14 Theorem]: we have the following equalities in D+ (OY /S ): E = E ⊗OY /S L(ODY (Y) ⊗OY Λ•Y/S ) = L(E ⊗OY Λ•Y/S ).   ⊂

Let ι : Z −→ Y be an exact closed immersion of fine log schemes over S to which γ extends. Assume that there exists the following cartesian diagram

72

2 Weight Filtrations on Log Crystalline Cohomologies ι

Z −−−−→ ⊂ ⏐ ⏐ ∩

(2.2.8.2)

Y ⏐ ⏐∩ 

ιY,Z

Z −−−−→ Y, ⊂

where ιY,Z is an exact closed immersion of fine log schemes over S and the vertical two morphisms are closed immersions. Let DZ (Z) and DY (Y) be ⊂ ⊂ the log PD-envelopes of the closed immersions Z −→ Z and Y −→ Y over (S, I, γ), respectively. Then we have the following diagram of ringed topoi: (2.2.8.3) ϕ

D (Z) g  log
(Z
zar , OZ ) ←−−Z−−− (D −−Z −−−− ((Z/S) crys |D Z (Z) , OZ/S |D Z (Z) ) Z (Z)zar , OD Z (Z) ) ←

⏐

⏐ ⏐

ιY,Z ⏐ 

⏐ ⏐

ιPD Y,Z 

ιlog,loc  crys ϕ

gY D (Y)

zar , OY ) ←−− (Y −−− (D −−Y−−−− ((Y /S)log crys |D Y (Y) , OY /S |D Y (Y) ) Y (Y)zar , OD Y (Y) ) ← jD (Z)

Z − −−− −− →

 log ((Z/S) crys , OZ/S ) ⏐ ⏐

ιlog crys  jD (Y)

Y − −−− −− →

((Y /S)log crys , OY /S ).

Let J Z (resp. J Y ) be the PD-ideal sheaf of DZ (Z) (resp. DY (Y)). Let JY,Z be the ideal sheaf of the closed immersion ιY,Z . Lemma 2.2.9. Assume that DZ (Z) = Z ×Y DY (Y). Then the diagram g∗

(Z
zar , OZ ) −−−Z−→ (D
Z (Z)zar , ODZ (Z) ) ⏐ ⏐ ⏐ιPD ⏐ ιY,Z∗   Y,Z∗

(2.2.9.1)

∗

gY (Y
zar , OY ) −−−−→ (D
Y (Y)zar , ODY (Y) ).

is commutative for a quasi-coherent OZ -module E, that is, the natural mor∗ phism gY∗ ιY,Z∗ (E)−→ιPD Y,Z∗ gZ (E) is an isomorphism. ◦

Proof. Since ιY,Z is affine, (2.2.9) immediately follows from the affine base change theorem ([39, (1.5.2)]).   PD∗ Lemma 2.2.10. Assume that ιPD Y,Z induces a surjection ιY,Z (J Y ) −→ J Z . Then the diagram ϕ∗

(2.2.10.1)

D (Z)  log | (D
−−Z−−→ ((Z/S) Z (Z)zar , ODZ (Z) ) − crys DZ (Z) , OZ/S |DZ (Z) ) ⏐ ⏐ ⏐ιlog,loc ⏐ ιPD  crys∗ Y,Z∗ 

ϕ∗

D (Y) (D
−−Y−−→ ((Y /S)log Y (Y)zar , ODY (Y) ) − crys |DY (Y) , OY /S |DY (Y) )

2.2 The Log Linearization Functor

73

is commutative for a quasi-coherent ODZ (Z) -module E, that is, the natural log,loc ∗ morphism ϕ∗DY (Y) ιPD Y,Z∗ (E)−→ιcrys∗ ϕDZ (Z) (E) is an isomorphism. Proof. Let E be a quasi-coherent ODZ (Z) -module. Let (T, φ) = (U, T, MT , δ, φ) be an object of (Y /S)log crys |DY (Y) . Then, by (2.2.6) (1) and (2), (2.2.10.2)

∗ ∗ ιlog,loc crys∗ ϕDZ (Z) (E)(T, φ) = Γ (T ×DY (Y) DZ (Z), p2 (E)),

where p2 : T ×DY (Y) DZ (Z) −→ DZ (Z) is the second projection. On the other hand, (2.2.10.3)

∗ PD ϕ∗DY (Y) ιPD Y,Z∗ (E)(T, φ) = Γ (T, φ ιY,Z∗ (E)).

Since DZ (Z) −→ DY (Y) is a closed immersion, in particular, an affine morphism, the affine base change theorem tells us that both right hand sides of (2.2.10.2) and (2.2.10.3) are the same. This completes the proof of (2.2.10).   PD∗ Lemma 2.2.11. Assume that ιPD Y,Z induces a surjection ιY,Z (J Y ) −→ J Z . Then the following diagram of topoi

 log |  log (Z/S) −−Z−−→ (Z/S) crys DZ (Z) − crys ⏐ ⏐ ⏐ log,loc ⏐ ιcrys  ιlog crys jD

(2.2.11.1)

(Z)

(Y /S)log −−Y−−→ (Y /S)log crys |DY (Y) − crys . jD

(Y)

is commutative. Proof. Let T = (U, T, MT , δ) be an object of (Y /S)log crys . Let δ be the PDstructure of Ker(OT −→ OU ) + IOT which is an extension of δ and γ. Let D(T ) := DU ∩Z,δ (T ) be the log PD-envelope of the closed immersion U ∩ Z −→ T over (T, MT , δ). By the log version of [11, 6.2.1 Lemma], ∗ log∗ ιlog∗ crys (T ) = (U ∩ Z, D(T )). Hence jDZ (Z) ιcrys (T ) = (D(T ) × DZ (Z), p2,Z ) as a sheaf, where p2,Z : D(T ) × DZ (Z) −→ DZ (Z) is the second projection. Analogously, let p2,Y : T × DY (Y) −→ DY (Y) be the second projection. Let δZ be the PD-structure of DZ (Z) and let δ Z be the extension of the δ and γ on Ker(ODZ (Z) −→ OZ ) + IODZ (Z) . Let D(T ×S DY (Y)) be the double log PD-envelope of T and DY (Y) (cf. [11, 5.12 Lemma]) over (S, I, γ). Let D(δ) be the PD-structure of D(T ×S DY (Y)) and D(δ) the extension of D(δ) and γ. Then we have log,loc∗ ∗ log,loc∗ ιcrys jDY (Y) (T ) = ιcrys (T × DY (Y), p2,Y ) log,loc∗ = ιcrys (D(T ×S DY (Y)), p2,Y )

= DD(δ) (D(T ×S DY (Y)) ×DY (Y) DZ (Z)) = D(T ) × DZ (Z)

(2.2.6)

(the universality of D(T ×S DY (Y))).

74

2 Weight Filtrations on Log Crystalline Cohomologies

 log | Here we consider the last equality as sheaves in (Z/S) crys DZ (Z) . Hence (2.2.11.1) is commutative.   Corollary 2.2.12. Assume that DZ (Z) = Z ×Y DY (Y). Let LPD Z/S (resp. ) be the linearization functor of O -modules (resp. O LPD DZ (Z) DY (Y) -modY /S ules). Then there exists a canonical isomorphism of functors (2.2.12.1)

PD log PD LPD Y /S ◦ ιY,Z∗ −→ ιcrys∗ ◦ LZ/S

∗ for quasi-coherent ODZ (Z) -modules. Set LY /S := LPD Y /S ◦ gY and LZ/S := ∗ LPD Z/S ◦ gZ . Then there also exists a canonical isomorphism of functors

(2.2.12.2)

LY /S ◦ ιY,Z∗ −→ ιlog crys∗ ◦ LZ/S

for quasi-coherent OZ -modules. Moreover, the isomorphism (2.2.12.1) is functorial with respect to log HPD differential operators of quasi-coherent ODZ (Z) -modules. Proof. Because DZ (Z) = Z ×Y DY (Y) and because the diagram (2.2.8.2) is PD∗ cartesian, the natural morphism ιPD Y,Z : ιY,Z (J Y ) −→ J Z is surjective. The first statement of (2.2.12) immediately follows from (2.2.1.2), (2.2.10) and (2.2.11). The second statement follows from the former and (2.2.9). Let us prove the last statement. For a quasi-coherent ODZ (Z) -module E and (U, T, δ) ∈ (Y /S)log crys , the isomorphism PD log PD LPD Y /S ◦ ιY,Z∗ (E)T −→ ιcrys∗ ◦ LZ/S (E)T

induced by (2.2.12.1) is given by the natural homomorphism (2.2.12.3)

ODU (T ×S Y) ⊗ODY (Y) ιPD Y,Z∗ (E) −→ ODU ×Y Z (T ×S Z) ⊗ODZ (Z) E.

If we are given a log HPD differential operator u : ODZ (Z 2 ) ⊗ODZ (Z) E −→ F of ODZ (Z) -modules, the composite morphism u

u
: ODY (Y 2 ) ⊗ODY (Y) ιPD Y,Z∗ (E) −→ ODZ (Z 2 ) ⊗ODZ (Z) E −→ F is a log HPD differential operator of ODY (Y) -modules and we see easily that the diagram PD ODU (T ×S Y) ⊗ODY (Y) ιPD Y,Z∗ (E) −−−−→ ODU (T ×S Y) ⊗ODY (Y) ιY,Z∗ (F) ⏐ ⏐ ⏐ ⏐  

ODU ×Y Z (T ×S Z) ⊗ODZ (Z) E

−−−−→

ODU ×Y Z (T ×S Z) ⊗ODZ (Z) F

is commutative for any T = (U, T, δ) ∈ (Y /S)log crys , where the upper horizontal morphism (resp. the lower horizontal morphism) is the homomorphism

2.2 The Log Linearization Functor

75

induced by u
(resp. u) in the way described in (2.2.3.1) and the vertical morphisms are the homomorphism (2.2.12.3) for E and F. Therefore we see the compatibility of (2.2.12.1) with log HPD differential operators.   Remark 2.2.13. In the case where Y , Z are trivial log smooth schemes over a trivial log scheme S, we can also prove (2.2.12) by an analogous proof of [3, IV Proposition 3.1.7]. In the case where Y , Z are fine log (not necessarily smooth) schemes over a fine log scheme S, we can also prove (2.2.12) by the second fundamental exact sequence of log differential forms on fine log smooth schemes ((2.1.3)) and by the log version of an analogous proof of [3, IV Proposition 3.1.7]. (2) Now let us study some properties of log linearization functors for a smooth scheme with a relative SNCD. ⊂ Let S0 −→ S be a closed immersion of schemes(=trivial log schemes) defined by a quasi-coherent ideal sheaf. Let f : X −→ S0 be a smooth scheme with a relative SNCD D on X over S0 . Let Z be a relative SNCD on X over S0 which intersects D transversally over S0 . Let ∆D := {Dλ }λ (resp. ∆Z := {Zµ }µ ) be a decomposition of D (resp. Z) by smooth components of D (resp. Z). Then ∆ := {Dλ , Zµ }λ,µ is a decomposition of D ∪ Z by ⊂ smooth components of D ∪ Z. Let (X, D ∪ Z) −→ (X , D ∪ Z) be an admissi
:= {Dλ , Zµ }λ,µ be the ble closed immersion over S with respect to ∆. Let ∆ decomposition of D ∪ Z which is compatible with ∆. Set (2.2.13.1)

D{λ1 ,λ2 ,...λk } := Dλ1 ∩ Dλ2 ∩ · · · ∩ Dλk

for a positive integer k, and set ⎧ ⎨ (2.2.13.2) D(k) = ⎩

X 

(λi = λj if i = j)

D{λ1 ,λ2 ,...,λk }

{λ1 ,...,λk | λi =λj (i=j)}

(k = 0), (k ≥ 1)

for a nonnegative integer k. Set (2.2.13.3)

D∅ := X

for later convenience. The following proposition says that a decomposition of a relative SNCD by smooth components is locally unique: Proposition 2.2.14. Let ∆ and ∆ be decompositions of D by smooth components. Then, for any z ∈ X, there exists an open neighborhood V of z in X such that ∆V = ∆V . Proof. If V is small enough, we can take the diagram (2.1.7.2) such that  (A.0.1) below holds for both ∆ and ∆ . Then ∆V = {yi = 0}si=1 = ∆V . 

76

2 Weight Filtrations on Log Crystalline Cohomologies

Proposition 2.2.15. D(k) is independent of the choice of the decomposition of D by smooth components of D. Proof. Obviously we may assume that k is positive. First we prove (2.2.15) for the case k = 1. Let ∆D = {Dλ }λ and ∆D = {Dλ  }λ be two decompositions of D by smooth components of D. By (2.2.14) there exists an open covering {Xi }i of X such that ∆D |Xi = ∆D |Xi .   ∼ Hence we have an isomorphism ( λ Dλ ) ×X Xi −→ ( λ Dλ  ) ×X Xi . This ⊂ local isomorphism is compatible with the open Xi ∩ Xi −→ Xi ;  immersions ∼  therefore we have the global isomorphism λ Dλ −→ λ Dλ  . Let D[k] be the k-fold fiber product of D(1) over X; D[k] admits the action of the symmetric group Sk of degree k. For a positive integer k, denote the set {1, 2, . . . , k} by [1, k]. For a surjective map α : [1, k] −→ [1, l], we have the corresponding morphism D[l] −→ D[k] , which we denote by sα . Let Sk be the set of surjective morphisms [1, k] −→ [1, k − 1]. Set D{k} := D[k] \ α∈Sk sα (D[k−1] ); D{k} is an open subscheme of D[k] . The scheme D{k} also admits the action of Sk . Then we can check D(k) = D{k} /Sk by the construction of D{k} . Consequently D(k) is independent of the choice of the decomposition of D by smooth components of D.   Set Z|D(k) := Z ×X D(k) .

(2.2.15.1)

The scheme Z|D(k) is a relative SNCD on D(k) . We use analogous notations
Let a(k) : (D(k) , Z|D(k) ) −→ (X, Z) D(k) and Z|D(k) (k ∈ N) for D∪Z with ∆. (k) (k) and b : (D , Z|D(k) ) −→ (X , Z) be morphisms induced by natural closed immersions. As usual, we define the preweight filtration P•D on the sheaf of the log differential forms ΩiX /S (log(D ∪ Z)) (i ∈ N) in X
zar with respect to D as follows: PkD ΩiX /S (log(D ∪ Z)) =

(2.2.15.2) ⎧ ⎪ ⎨0

Im(ΩkX /S (log(D ∪ Z))⊗OX Ωi−k (log Z) −→ ΩiX /S (log(D ∪ Z))) X /S

⎪ ⎩Ω i

X /S

(log(D ∪ Z))

(k < 0), (0 ≤ k ≤ i), (k > i).

Now, assume that the defining ideal sheaf I of the closed immersion ⊂ S0 −→ S is a PD-ideal sheaf with a PD-structure γ. Let the right objects in the following table be the log PD-envelopes of the left exact closed immersions over (S, I, γ):

2.2 The Log Linearization Functor

77

⊂

(X, D ∪ Z) −→ (X , D ∪ Z) DD ⊂ (X, Z) −→ (X , Z) D ⊂ (D(k) , Z|D(k) ) −→ (D(k) , Z|D(k) ) D(k) Let gD : DD −→ (X , D∪Z), g : D −→ (X , Z) and g (k) : D(k) −→ (D(k) , Z|D(k) ) be natural morphisms. Note that the underlying schemes of the log schemes DD and D are the same. Let c(k) : D(k) −→ D be a morphism induced by b(k) : (D(k) , Z|D(k) ) −→ (X , Z). Lemma 2.2.16. (1) The natural morphism (D(k) , Z|D(k) ) −→ (D(k) , Z|D(k) ) ×(X ,Z) (X, Z) is an isomorphism. (2) The natural morphism D(k) −→ D ×(X ,Z) (D(k) , Z|D(k) ) is an isomorphism. (k) (3) Let J (resp. J ) be the PD-ideal sheaf of OD (resp. OD(k) ). Then the natural morphism c(k)∗ : c(k)∗ (J ) −→ J

(k)

is surjective.

Proof. Apply (2.1.13) to the SNCD D ∪ Z and assume that D (resp. Z) is defined by an equation x1 = · · · = xt = 0 (resp. xt+1 = · · · = xs = 0) (1 ≤ t ≤ s). (1): (1) is obvious. (2): By the universality of the log PD-envelope, this is a local question. We may have two cartesian diagrams in (2.1.13) for D ∪ Z; we may assume that k ≤ t. Let D1···k be a closed subscheme defined by an equation x1 = · · · = xk = 0. Then OD ⊗OX OD1···k = OX xd+1 , . . . , xd  ⊗OX (OX /(x1 , . . . , xk )) = OX xd+1 , . . . , xd /(x1 , . . . , xk ). Set D1···k := D1···k ×X X. Then the structure sheaf of the PD-envelope of ⊂ the closed immersion D1···k −→ D1···k is OD1···k xd+1 , . . . , xd  = OX xd+1 , . . . , xd /(x1 , . . . , xk ). Furthermore it is immediate to see that there exists a natural isomorphism D(k)  D ×(X ,Z) (D(k) , Z|D(k) ) as log schemes. Thus (2) follows. (3): The proof of (3) is evident by the local description of OD and OD(k) .   As usual, we denote the left objects in the following table by the right ones for simplicity of notation: ⊂

((X, D ∪ Z) −→ DD ) ∈ ((X, D ∪ Z)/S)log crys

DD

((X, Z) −→ D) ∈ ((X, Z)/S)log crys

D

⊂

((D

(k)

⊂

, Z|D(k) ) −→ D

(k)

) ∈ ((D

(k)

, Z|D(k) )/S)log crys

D(k)

Furthermore, as usual, we identify the representable sheaf by DD in ((X, log

D ∪ Z)/S)log crys with DD . Let ((X, D ∪ Z)/S)crys |DD be the localization of

78

2 Weight Filtrations on Log Crystalline Cohomologies

(k) ,


log ((X, D ∪ Z)/S)log Z|D(k) )/S)log crys at DD . Let ((X, Z)/S)crys |D and ((D crys (k)log
| (k) be obvious analogues. Let acrys : ((D(k) ,
Z| (k) )/S)log −→ ((X, Z)/S) D log crys

D

crys

be a morphism of topoi induced by the morphism a(k) . By the log version (k)log

of [11, 6.2 Proposition], the functor acrys∗ is exact. Let the right objects in the following table be the log PD-envelope of the locally closed immersion of the left ones: ⊂

(X, D ∪ Z) −→ (X , D ∪ Z) ×S (X , D ∪ Z) DD (1) ⊂ (X, Z) −→ (X , Z) ×S (X , Z) D(1) ⊂ (k) (k) (k) (D , Z|D(k) ) −→ (D , Z|D(k) ) ×S (D , Z|D(k) ) D(k) (1) Let log

jDD : ((X, D ∪ Z)/S)log crys |DD −→ ((X, D ∪ Z)/S)crys

jD : ((X, Z)/S)log |D −→ ((X, Z)/S)log crys

jD(k) :

((D(k) ,
Z|D(k) )/S)log crys |D(k)

crys

−→ ((D(k) ,
Z|D(k) )/S)log crys

be localization functors (2.2.0.1) and let
◦


ϕD : (((X, D ∪ Z)/S)log crys |D D , O(X,D∪Z)/S |D D ) −→ (Dzar , OD )


◦


ϕ : (((X, Z)/S)log crys |D , O(X,Z)/S |D ) −→ (Dzar , OD )  (k) ϕ(k) : (((D(k) ,
Z|D(k) )/S)log zar , OD (k) ) crys |D (k) , O(D (k) ,Z| )/S |D (k) ) −→ (D D (k)

be morphisms of ringed topoi defined in (2.2.1.1) and let
◦
◦ g : (Dzar , OD ) −→ (X zar , OX ) ◦   (k) g (k) : (D(k) zar , OD ) −→ (D zar , OD (k) ) be natural morphisms. For an OX -module E, set
∪ Z)/S)log L(X,D∪Z)/S (E) := jDD ∗ ϕ∗D g ∗ (E) ∈ ((X, D crys and


Z)/S)log L(X,Z)/S (E) := jD∗ ϕ∗ g ∗ (E) ∈ ((X, crys .

For an OD(k) -module E, set also L(k) (E) := jD(k) ∗ ϕ(k)∗ g (k)∗ (E) ∈ ((D(k) ,
Z|D(k) )/S)log crys .

2.2 The Log Linearization Functor

79

As usual, we have a complex L(X,D∪Z)/S (Ω•X /S (log(D ∪ Z))) of O(X,D∪Z)/ S -modules. By (2.2.7) we have a natural quasi-isomorphism (2.2.16.1)

∼

O(X,D∪Z)/S −→ L(X,D∪Z)/S (Ω•X /S (log(D ∪ Z))).

Similarly we have two quasi-isomorphisms: ∼

(2.2.16.2)

O(X,Z)/S −→ L(X,Z)/S (Ω•X /S (log Z)),

(2.2.16.3)

O(D(k) ,Z|

∼

D (k)

)/S

−→ L(k) (Ω•D(k) /S (log Z|D(k) )).

Let {PkD Ω•X /S (log(D ∪ Z))}k∈Z be the filtration on Ω•X /S (log(D ∪ Z)) defined in (2.2.15.2). Then PkD Ω•X /S (log(D ∪ Z)) forms a subcomplex of Ω•X /S (log(D ∪ Z)) and the boundary morphisms of PkD Ω•X /S (log(D ∪ Z)) are log HPD differential operators of order ≤ 1 with respect to (X , Z)/S. Set PkD L(X,Z)/S (Ω•X /S (log(D∪Z))) := L(X,Z)/S (PkD Ω•X /S (log(D∪Z))) (k ∈ Z). Lemma 2.2.17. (1) The natural morphism (2.2.17.1) OD ⊗OX PkD Ω•X /S (log(D ∪ Z)) −→ OD ⊗OX Ω•X /S (log(D ∪ Z)) is injective. (2) The natural morphism (2.2.17.2) Q∗(X,Z)/S PkD L(X,Z)/S (Ω•X /S (log(D ∪ Z))) −→ Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))) is injective. Proof. (1): The question is local. We may have cartesian diagrams (2.1.13.1) and (2.1.13.2) for SNCD D∪Z on X ; we assume that D (resp. Z) is defined by an equation x1 · · · xt = 0 (resp. xt+1 · · · xs = 0). Set J := (xd+1 , . . . , xd )OX , X  := SpecX (OX /J ) and X  := SpecS (OS [xd+1 , . . . , xd ]). Then X  is smooth over S. Let D (resp. Z  ) be a closed subscheme of X  defined by an equation x1 · · · xt = 0 (resp. xt+1 · · · xs = 0). Because p is locally nilpotent on S, we may assume that there exists a positive integer N such that J N OD = 0. Since X  is smooth over S, there exists a section of the surjection OX /J N −→ OX  . Hence, as in [11, 3.32 Proposition], we have a morphism OX  [xd+1 , . . . , xd ] −→ OX /J N

80

2 Weight Filtrations on Log Crystalline Cohomologies

such that the induced morphism OX  [xd+1 , . . . , xd ]/J0N −→ OX /J N is an isomorphism, where J0 := (xd+1 , . . . , xd ). By [11, 3.32 Proposition], OD is isomorphic to the PD-polynomial algebra OX  xd+1 , . . . , xd . Hence we have the following isomorphisms ∼

OD ⊗OX Ω•X /S (log(D ∪ Z)) −→s(Ω•X  /S (log(D ∪ Z  ))⊗OS OS xd+1 , . . . , xd  ⊗OX  Ω•X  /S ) and ∼



OD ⊗OX PkD Ω•X /S (log(D ∪ Z)) −→s(PkD Ω•X  /S (log(D ∪ Z  ))⊗OS OS xd+1 , . . . , xd  ⊗OX  Ω•X  /S ). Since the complex OS xd+1 , . . . , xd  ⊗OX  Ω•X  /S consists of free OS modules, we obtain the desired injectivity. (2): By (1) and (2.2.4), the natural morphism (2.2.17.3) PkD L(X,Z)/S (Ω•X /S (log(D ∪Z))) −→ L(X,Z)/S (Ω•X /S (log(D ∪Z))) is injective in the category of crystals of O(X,Z)/S -modules. As in [3, IV Proposition 2.1.3], the functor {the category of crystals of O(X,Z)/S -modules} −→ {the category of Q∗(X,Z)/S (O(X,Z)/S )-modules} is exact. Hence (2.2.17.2) is injective.

 

By (2.2.17) (2), a family {Q∗(X,Z)/S PkD L(X,Z)/S (Ω•X /S (log(D ∪ Z)))}k∈Z of complexes of Q∗(X,Z)/S (O(X,Z)/S )-modules defines a filtration on the complex Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))). Hence we obtain an object (Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))), {Q∗(X,Z)/S PkD L(X,Z)/S (Ω•X /S (log(D ∪ Z)))}k∈Z ) in C+ F(Q∗(X,Z)/S (O(X,Z)/S )). Now we consider the Poincar´e residue isomorphism with respect to D. Though a relative divisor in this book is a union of smooth divisors, we consider the orientation sheaf of it for showing that our theory in this book is independent of the choice of the numbering of the smooth components of a relative SNCD. First, let us recall the orientation sheaf in [23, (3.1.4)]. k E Z if k ≥ 1 Let E be a finite set with cardinality k ≥ 0. Set
E := and
E := Z if k = 0.

2.2 The Log Linearization Functor

81

Let k be a positive integer. Let P be a point of D(k) . Let Dλ0 , . . . , Dλk−1 be different smooth components of D such that Dλ0 ∩ · · · ∩ Dλk−1 contains P . Then the set E := {Dλ0 , . . . , Dλk−1 } gives an abelian sheaf
λ0 ···λk−1 zar (D/S0 ) :=

k 

ZE Dλ

0

∩···∩Dλk−1

on a local neighborhood of P in D(k) . The sheaf
λ0 ···λk−1 zar (D/S0 ) is globalized on D(k) ; we denote this globalized abelian sheaf by the same symbol
λ0 ···λk−1 zar (D/S0 ). We denote a local section of
λ0 ···λk−1 zar (D/S0 ) by the following way: m(λ0 · · · λk−1 ) (m ∈ Z). Set  (k) (D/S0 ) :=
λ0 ···λk−1 zar (D/S0 ).
zar {λ0 ,...λk−1 } (k)

(k)

(k)

By abuse of notation, we often denote a∗
zar (D/S0 ) simply by
zar (D/S0 ). (0) (k) Set
zar (D/S0 ) := ZX . The sheaves
λ0 ···λk−1 zar (D/S0 ) and
zar (D/S0 ) (k)log are extended to abelian sheaves
λlog (D/S; Z) and
crys (D/S; Z), 0 ···λk−1 crys Z| (k) )/S)log since, for respectively, in the log crystalline topos ((D(k) ,
D

an object (U, T, MT , ι, δ) ∈ ⊂

((D(k) ,
Z|D(k) )/S)log crys ,

crys

the closed immersion

ι : U −→ T is a homeomorphism of topological spaces. If Z = ∅, then de(k)log note
λlog (D/S; Z) and
crys (D/S; Z) by
λ0 ···λk−1 crys (D/S) and 0 ···λk−1 crys (k)


crys (D/S), respectively. Definition 2.2.18. We call (k) (k) (k)log
zar (D/S0 ) (resp.
crys (D/S),
crys (D/S; Z))

the zariskian orientation sheaf (resp. crystalline orientation sheaf , log crystalline orientation sheaf) of D(k)/S0 (resp. D(k) /(S, I, γ), (D(k), Z|D(k) )/(S, I, γ)). (k)

(k)

(k)log

Remark 2.2.19. The sheaves
zar (D/S0 ),
crys (D/S) and
crys (D/S; Z) are defined by the local nature of D; they are independent of the choice of the decomposition by smooth components of D. Lemma 2.2.20. Let E be an OD(k) -module. Then there exists a canonical isomorphism (2.2.20.1)

∼

(k) (k)log L(k) (E ⊗Z
zar (D/S)) −→ L(k) (E) ⊗Z
crys (D/S; Z). (k)log

Proof. (2.2.20) immediately follows from the definition of
crys (D/S; Z).  

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2 Weight Filtrations on Log Crystalline Cohomologies

Proposition 2.2.21. (1) There exists the following exact sequence: (2.2.21.1) D 0 −→ OD ⊗OX Pk−1 Ω•X /S (log(D ∪ Z)) −→ OD ⊗OX PkD Ω•X /S (log(D ∪ Z)) (k) (D/S){−k}) −→ 0. −→ OD ⊗OX b∗ (Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (k)

(2) There exist quasi-isomorphisms (2.2.21.2) Q∗

grk (X,Z)/S

PD

Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z)))

∼

(k)log

∼

(k)log

(k) −→ Q∗(X,Z)/S acrys∗ L(k) (Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S)){−k}

←− Q∗(X,Z)/S acrys∗ (O(D(k) ,Z|

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)){−k}.

Proof. (1): By the Poincar´e residue isomorphism with respect to D (cf. [21, 3.6]), we have the following isomorphism (2.2.21.3) D

• ResD : grP k ΩX /S (log(D ∪ Z)) ∼

(k) −→ b∗ (Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S){−k}). (k)

Hence (1) follows from (2.2.17) (1). (2): By the isomorphism (2.2.21.3), (2.2.17) (1) and (2.2.4), we have Q∗

grk (X,Z)/S

PD

Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))) D

• = Q∗(X,Z)/S L(X,Z)/S (grP k ΩX /S (log(D ∪ Z))) D Q∗ (X,Z)/S L(X,Z)/S (Res )

=

Q∗(X,Z)/S L(X,Z)/S (b∗ (Ω•D(k) /S (log Z|D(k) ) (k)

(k) (D/S))){−k}. ⊗Z
zar

By (2.2.12) and (2.2.16) (1), (2), this complex is equal to (k) Q∗(X,Z)/S acrys∗ L(k) (Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S)){−k}, (k)log

which is equal to (k)log Q∗(X,Z)/S acrys∗ (L(k) (Ω•D(k) /S (log Z|D(k) )) ⊗Z
crys (D/S; Z)){−k} (k)log

by (2.2.20). By (2.2.7) we obtain the second quasi-isomorphism in (2.2.21.2).

 

2.2 The Log Linearization Functor

83

For simplicity of notation, set (Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))), Q∗(X,Z)/S P D ) := (Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))), {Q∗(X,Z)/S PkD L(X,Z)/S (Ω•X /S (log(D ∪ Z)))}k∈Z )

and

(OD ⊗OX Ω•X /S (log(D ∪ Z)), P D ) := (OD ⊗OX Ω•X /S (log(D ∪ Z)), {OD ⊗OX PkD Ω•X /S (log(D ∪ Z))}k∈Z ).

Proposition 2.2.22. Let (2.2.22.1)
u(X,Z)/S : (((X, Z)/S)log

∗ Rcrys , Q(X,Z)/S (O(X,Z)/S ))


zar , f −1 (OS )) −→ (X

be the morphism in (1.6.1.2). Then (2.2.22.2) Ru(X,Z)/S∗ (Q∗(X,Z)/S L(X,Z)/S (Ω•X /S (log(D ∪ Z))), Q∗(X,Z)/S P D ) = (OD ⊗OX Ω•X /S (log(D ∪ Z)), P D ) in D+ F(f −1 (OS )). Proof. By (1.6.3.1), by (2.2.1) (2) and by (1.3.1), the left hand side of (2.2.22.2) is equal to (u(X,Z)/S∗ L(X,Z)/S (Ω•X /S (log(D ∪ Z))), u(X,Z)/S∗ L(X,Z)/S (PkD Ω•X /S (log(D ∪ Z)))).

For an OX -module F, we have u(X,Z)/S∗ L(X,Z)/S (F) = u(X,Z)/S∗ jD∗ ϕ∗ g ∗ (F) = ϕ∗ ϕ∗ g ∗ (F) = OD ⊗OX F by (2.2.0.4). Hence u(X,Z)/S∗ (L(X,Z)/S (PkD Ω•X /S (log(D ∪ Z)))) = OD ⊗OX PkD Ω•X /S (log(D ∪ Z)). Thus (2.2.22) follows.

 

Remark 2.2.23. For simplicity, we assume that Z = ∅ in this remark. By the proof of (2.2.7), the differential operator of OD ⊗X PkD Ω•X /S (log D) is not a log HPD differential operator in general since the log HPD differential operator ODD (1) ⊗OX Ω•X /S (log D) −→ OD ⊗X Ω•+1 X /S (log D) induces a morphism D • D ODD (1) ⊗OX Pk ΩX /S (log D) −→ OD ⊗X Pk+1 Ω•+1 X /S (log D), but does not induce a morphism ODD (1) ⊗OX PkD Ω•X /S (log D) −→ OD ⊗X PkD Ω•+1 X /S (log D) D • in general; there does not exist a complex L(X,D)/S (Pk ΩX /S (log D)) in C+ (O(X,D)/S ) in general.

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2 Weight Filtrations on Log Crystalline Cohomologies

2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves In this section we investigate some properties of the forgetting log morphism of log crystalline topoi. Let the notations be as in §1.6. However, in this section, we denote the underlying scheme of the log scheme Y also by Y by abuse of notation. Let M be the log structure of Y . Let N ⊂ M be also a fine log structure on Yzar . Then we have a natural morphism  = (Y,M,N )/S : (Y, M ) −→ (Y, N )

(2.3.0.1)

of log schemes over S. The morphism  induces a morphism of topoi which is denoted by the same notation: (2.3.0.2)


log  = (Y,M,N )/S : ((Y,
M )/S)log crys −→ ((Y, N )/S)crys .

When N is trivial, we denote (Y,M,N )/S by Y /S ; the morphism Y /S is a p-adic analogue of the l-adic forgetting log morphism in [30] and [67, (1.1.2)]. In this section, let us assume the following condition on the log structure N unless otherwise stated: (2.3.0.3) Locally on Y , there exists a chart P −→ N such that P gp has no p-torsion. Then we have the following lemma: Lemma 2.3.1. Let the notation be as above and let (U, T, MT , ι, δ) be an inv ∗ be the inverse image of N |U /OU by the object of ((Y, M )/S)log crys , let NT proj.

ι∗

∼

∗ following morphism: MT −→ MT /OT∗ −→ M |U /OU . Then NTinv is a fine log structure on T (under the assumption (2.3.0.3)).

Proof. It is easy to see that NTinv is a log structure on T such that NTinv /OT∗ = ∗ . Set IT := Ker(OT −→ OU ). Then we have the exact sequence N |U /OU 0 −→ 1 + IT −→ NTinv,gp −→ N gp |U −→ 0. Shrink U and take a chart α : P −→ N |U such that P gp has no p-torsion. Then, since any element of 1 + IT is killed by some power of p, we have
: P gp −→ NTinv,gp Ext1 (P gp , 1 + IT ) = 0. Hence we have a homomorphism α lifting αgp locally on T and it induces the homomorphism of monoids P −→ NTinv , which we also denote by α
. If we denote the log structure α



induces a homomorphism of associated to P −→ NTinv −→ OT by P a , α log structures α
a : P a −→ NTinv such that the induced homomorphism ∗ α
a : P a /OT∗ −→ NTinv /OT∗ is nothing but the identity on N |U /OU . Hence a inv inv
is a chart of NT . Therefore NT is a fine α
is an isomorphism, that is, α log structure.  

2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves

85

Under the assumption (2.3.0.3), the explicit description of  = (∗ , ∗ ) is given as follows: for an object F of ((Y,
N )/S)log crys and an object log (U, T, MT , ι, δ) ∈ ((Y, M )/S)crys , ∗ (F )((U, T, MT , ι, δ)) = F ((U, T, NTinv , ι, δ)); for an object G of ((Y,
M )/S)crys and an object (U, T, NT , ι, δ)∈((Y, N )/S)log crys , (∗ (T ), G). ∗ (G)((U, T, NT , ι, δ)) = Hom((Y,M
)/S)log crys

Definition 2.3.2. We call the morphism (Y,M,N )/S in (2.3.0.1) and the morphism (Y,M,N )/S in (2.3.0.2) the forgetting log morphism of log schemes over S along M \ N and the forgetting log morphism of log crystalline topoi along M \ N , respectively. When N is trivial, we call the two (Y,M,N )/S ’s the forgetting log morphisms of Y /S. When Y is a smooth scheme X over S0 := SpecS (OS /I), M = M (D ∪ Z) and N = M (Z), where D and Z are transversal relative SNCD’s on X/S0 , we call the two (Y,M,N )/S ’s the forgetting log morphisms along D and denote them by (X,D∪Z,Z)/S . Let {Yi }i∈I be an open covering of Y . Let Mi (resp. Ni ) be the pull-back of M (resp. N ) to Yi . Then we also have an analogous morphism of topoi (2.3.2.1)

log
• : ((Y•
, M• )/S)log crys −→ ((Y• , N• )/S)crys ,

and we have the following commutative diagram • log
((Y•
, M• )/S)log crys −−−−→ ((Y• , N• )/S)crys ⏐ ⏐ ⏐πlog ⏐ log πM  N crys crys 



(2.3.2.2)


log ((Y,
M )/S)log crys −−−−→ ((Y, N )/S)crys . log log Here πM crys and πN crys are morphisms of topoi defined in §1.6; we have written the symbols M and N in subscripts for clarity. Let u(Y,L)/S , u(Y• ,L• )/S and u(Y•• ,L•• )/S (L := M, N ) be the projections in (1.6.0.8), (1.6.0.9) and (1.6.0.10) for (Y, L), respectively. Since ∗ ◦ u∗(Y,N )/S = u∗(Y,M )/S and ∗• ◦ u∗(Y• ,N• )/S = u∗(Y• ,M• )/S , we have the following two equations

(2.3.2.3)

u(Y,N )/S ◦  = u(Y,M )/S ,

u(Y• ,N• )/S ◦ • = u(Y• ,M• )/S

as morphisms of topoi. Let the notations be as in §1.6. Then we have the following commutative diagram:

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2 Weight Filtrations on Log Crystalline Cohomologies

(2.3.2.4)

•• log
((Y••
, M•• )/S)log crys −−−−→ ((Y•• , N•• )/S)crys ⏐ ⏐ ⏐ηlog ⏐ log ηM  N crys crys  • log
((Y•
, M• )/S)log crys −−−−→ ((Y• , N• )/S)crys .




L)/S)log Let O(Y,L)/S (L := M, N ) be the structure sheaf in ((Y, crys . Since ∗ there is a morphism  (O(Y,N )/S ) −→ O(Y,M )/S , there is a morphism (2.3.2.5)

O(Y,N )/S −→ ∗ (O(Y,M )/S ).

The morphism  also induces a morphism (2.3.2.6)


log  : (((Y,
M )/S)log crys , O(Y,M )/S ) −→ (((Y, N )/S)crys , O(Y,N )/S )

of ringed topoi. We have the analogues of the commutative diagrams (2.3.2.2) and (2.3.2.4) for the ringed topoi: (2.3.2.7) • log
(((Y•
, M• )/S)log crys , O(Y• ,M• )/S ) −−−−→ (((Y• , N• )/S)crys , O(Y• ,N• )/S ) ⏐ ⏐ ⏐πlog ⏐ log πM  N crys crys 

(((Y,
M )/S)log crys , O(Y,M )/S )

 −−−−→ (((Y,
N )/S)log crys , O(Y,N )/S )),

(2.3.2.8) •• log
(((Y••
, M•• )/S)log crys , O(Y•• ,M•• )/S ) −−−−→ (((Y•• , N•• )/S)crys , O(Y•• ,N•• )/S ) ⏐ ⏐ ⏐ηlog ⏐ log ηM  N crys crys  (((Y•
, M• )/S)log crys , O(Y• ,M• )/S )



−−−•−→

(((Y•
, N• )/S)log crys , O(Y• ,N• )/S ).

The morphism (2.3.2.5) gives a morphism (2.3.2.9)

O(Y,N )/S −→ R∗ (O(Y,M )/S ).

Using (2.3.2.3), we have a morphism (2.3.2.10)

Ru(Y,N )/S∗ (O(Y,N )/S ) −→ Ru(Y,M )/S∗ (O(Y,M )/S ).

Next we define the localization of . Let FM = (UF , TF , MF , ιF , δF ) be a representable sheaf in ((Y,
M )/S)log . Set FN := (UF , TF , N inv , ιF , δF ), crys

F

where NFinv is the inverse image of N |UF by the morphism MF −→ ∗ as before. Then we have a morphism M |UF /OU F (2.3.2.11)


log M )/S)log |F : ((Y,
crys |FM −→ ((Y, N )/S)crys |FN

of topoi and a morphism

2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves

87

(2.3.2.12)
log |F : (((Y,
M )/S)log crys |FM , O(Y,M )/S |FM ) −→ (((Y, N )/S)crys |FN , O(Y,N )/S |FN ). of ringed topoi. Lemma 2.3.3. Let the notations be as above. Then the functor |F ∗ is exact. Proof. Let (U, T, NT , ι, δ, φN ) be an object in ((Y, N )/S)log crys |FN . Let φ : T −→ TF be the underlying morphism of schemes of φN . Set MT := φ∗ (MF ). Let φM : (U, T, MT , ι, δ) −→ (UF , TF , MF , ιF , δF ) be the natural morphism. Then (U, T, MT , ι, δ, φM ) is an object in ((Y, M )/S) Let (TF , MF )×(TF ,NFinv ) (T, NT ) be the fiber product of (TF , MF ) and (T, NT ) over (TF , NFinv ) in the category of fine log schemes. We claim that

log crys |FM .

(2.3.3.1)

(TF , MF ) ×(TF ,NFinv ) (T, NT ) = (T, MT ). ⊂

Indeed, let φU : U −→ UF be the open immersion. Then we have the following: (φ∗ (MF ) ⊕φ∗ (NFinv ) NT )/OT∗ = φ∗ (MF )/OT∗ ⊕φ∗ (NFinv )/OT∗ NT /OT∗ = φ−1 (MF /OT∗ F ) ⊕φ−1 (NFinv /OT∗ ∗  φ−1 U (M |UF /OUF ) ⊕φ−1 (N |U U

F

∗ = M |U /OU  φ∗ (MF )/OT∗ .

F

)

NT /OT∗

∗ ) /OU F

∗ N |U /OU

Hence the natural morphism φ∗ (MF ) −→ φ∗ (MF ) ⊕φ∗ (NFinv ) NT is an isomorphism and we have shown the claim. Denote (U, T, LT , ι, δ, φL ) (L := M, N ) by (T, LT , φL ) for simplicity of notation. By the formula (2.3.3.1), (|F )∗ ((T, NT , φN )) is represented by (T, MT , φM ). Therefore, for an object E in ((Y,
M )/S)log |F , we have crys

M

(2.3.3.2) Γ ((T, NT , φN ), (|F )∗ (E)) = Hom((Y,M
)/S)log

crys |FM

((|F )∗ ((T, NT , φN )), E)

= E((T, MT , φM )). Using this formula, we see that the functor |F ∗ is exact.

 

Lemma 2.3.4. Let the notations be as above. Then the following diagram of topoi is commutative: M
log ((Y,
M )/S)log crys |FM −−−−→ ((Y, M )/S)crys ⏐ ⏐ ⏐ ⏐ |F  

jF

(2.3.4.1)

jFN
log ((Y,
N )/S)log crys |FN −−−−→ ((Y, N )/S)crys .

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2 Weight Filtrations on Log Crystalline Cohomologies

The obvious analogue of (2.3.4.1) for ringed topoi also holds. ∗ Proof. Let G be an object of ((Y,
N )/S)log crys . By the proof of (2.3.3), (|F ) ∗ ∗ ∗ ∗ (FN ) = FM . Hence (|F ) jFN (G) = (|F ) (G × FN ) =  (G) × FM = jF∗ M ∗ (G). Hence the former statement follows. The latter statement immediately follows.  


M )/ Lemma 2.3.5. Let FM = (Y, T, MT , ι, δ) be a representable sheaf in ((Y, log log
S)crys . Let E ∈ ((Y, M )/S)crys |FM be an O(Y,M )/S |FM -module. Then the canonical morphism ∗ jFM ∗ (E) −→ R∗ jFM ∗ (E) is an isomorphism in the derived category D+ (O(Y,N )/S ). Proof. Indeed, we have ∗ jFM ∗ (E)

(2.3.4)

=

jFN ∗ (|F )∗ (E)

(2.2.1) (1)

=

(2.3.4)

=

(2.3.3)

=

jFN ∗ R(|F )∗ (E)

RjFN ∗ R(|F )∗ (E) = R(jFN |F )∗ (E)

R(jFM )∗ (E) = R∗ RjFM ∗ (E)

(2.2.1) (1)

=

R∗ jFM ∗ (E).  

Though ∗ is not exact in general (see (2.7.1) below), the following holds: ⊂

Corollary 2.3.6. Let ι : (Y, M ) −→ (Y, M) be a closed immersion into a log smooth scheme over S. Let DY (Y) be the log PD-envelope of ι over (S, I, γ). Let E be an ODY (Y) -module. Let LPD (Y,M )/S (E) be the linearization of E with respect to ι. Then the canonical morphism (2.3.6.1)

PD ∗ LPD (Y,M )/S (E) −→ R∗ L(Y,M )/S (E)

is an isomorphism in the derived category D+ (O(Y,N )/S ). Proof. (2.3.6) immediately follows from (2.2.1.2) and (2.3.5).

 

Lemma 2.3.7. Let (Y, M) be a log smooth scheme over S. Let N be a fine sub-log structure of M on Y such that (Y, N ) is also log smooth over S. Let ι

(2.3.7.1)

M (Y, M ) −−− −→ (Y, M) ⏐ ⏐ ⏐(Y,M,N )/S (Y,M,N )/S ⏐  

ι

(Y, N ) −−−N−→ (Y, N ) be a commutative diagram whose horizontal morphisms are closed immersions. Let DM and DN be the log PD-envelopes of ιM and ιN over (S, I, γ), respectively, with the natural following commutative diagram:

2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves

89

gM

DM −−−−→ (Y, M) ⏐ ⏐ ⏐ ⏐ h 

(2.3.7.2)

gN

DN −−−−→ (Y, N ) ◦

Assume that the underlying morphism h of schemes is the identity. Then there exist natural isomorphisms (2.3.7.3)

∼

PD LPD (Y,N )/S −→ (Y,M,N )/S∗ L(Y,M )/S

and (2.3.7.4)

◦

◦

∼

∗ PD ∗ LPD (Y,N )/S g N −→ (Y,M,N )/S∗ L(Y,M )/S g M

of functors. Moreover, the functor (2.3.7.3) is functorial with respect to log HPD differential operators. ◦  M )/S)log Proof. Let ϕM : (((Y,
crys |DM , O(Y,M )/S |DM ) −→ (DM zar , ODM ) be the morphism of ringed topoi in (2.2.1.1). Let ϕN be the analogue of ϕM for (Y, N ). Let


log |D : (((Y,
M )/S)log crys |D M , O(Y,M )/S |D M ) −→ (((Y, N )/S)crys |D N , O(Y,N )/S |D N )

be the natural morphism. Then, using the formula (2.3.3.2), we can immediately check that (|D )∗ ϕ∗M = ϕ∗N . Hence we have the following commutative diagram

(2.3.7.5)

◦
◦ g∗ ϕ∗ −→ (DMzar , ODM ) −−−M −→ (Y
zar , OY ) −−−M       ◦
◦ g∗ ϕ∗ (Y
zar , OY ) −−−N−→ (DN zar , ODN ) −−−N−→

M
log (((Y,
M )/S)log crys |DM , O(Y,M )/S |DM ) −−−−→ (((Y, M )/S)crys , O(Y,M )/S ) ⏐ ⏐ ⏐ (Y,M,N )/S∗ ⏐ |D∗  

jD

∗

jDN ∗
log (((Y,
N )/S)log crys |DN , O(Y,N )/S |DN ) −−−−→ (((Y, N )/S)crys O(Y,N )/S ),

and this implies the isomorphisms (2.3.7.3), (2.3.7.4). Finally we check the functoriality of the isomorphism (2.3.7.3) with respect to log HPD differential operators. To show this, it suffices to prove the required functoriality for the morphism (2.3.7.6)

PD ∗(Y,M,N )/S LPD (Y,N )/S −→ L(Y,M )/S .

90

2 Weight Filtrations on Log Crystalline Cohomologies

inv For TM := (U, T, MT , ι, δ) in ((Y, M )/S)log crys , let TN := (U, T, NT , ι, δ) be as above. Then, for an ODN -module E, the homomorphism PD (∗(Y,M,N )/S LPD (Y,N )/S (E))TM −→ (L(Y,M )/S (E))TN

induced by (2.3.7.6) is given by the canonical homomorphism ODU (TN ×S (Y,N )) ⊗ODN E −→ ODU (TM ×S (Y,M)) ⊗ODM E, and it is easy to see that this homomorphism is functorial with respect to log HPD differential operators (see (2.2.3.1)). Hence we finish the proof of the lemma.   Remark 2.3.8. In (2.3.7), we do not have to assume the condition (2.3.0.3) on the log structure N . The reason why we imposed the condition (2.3.0.3) was to assure that the log structure NTinv is always fine. However, in the situation in (2.3.7), the fineness of NTinv for any T = (U, T, MT ) follows from the assumption. Indeed, we have a morphism ψ : (T, MT ) −→ DM etale locally on T and one can see that NTinv is isomorphic to the pull-back of the log structure of DN by ψ. Definition 2.3.9. For an OY /S -module E, we call R(Y,M,N )/S∗ (E) the vanishing cycle sheaf of E along M \ N . We call R(Y,M,N )/S∗ (O(Y,M )/S ) the vanishing cycle sheaf of (Y, M )/(S, I, γ) along M \ N . If N is trivial, we omit the word “along M \ N ”. The following theorem is the crystalline Poincar´e lemma of a vanishing cycle sheaf: Theorem 2.3.10 (Poincar´ e lemma of a vanishing cycle sheaf ). Let MS be the log structure of S. Let E be a crystal of O(Y,N )/S -modules and let (E, ∇) be the ODM -module with integrable log connection corresponding to ∗(Y,M,N )/S (E). Assume that we are given the commutative diagram (2.3.7.1) ◦

and that h in (2.3.7) is the identity. Then there exists a canonical isomorphism (2.3.10.1) ∼ • R(Y,M,N )/S∗ ∗(Y,M,N )/S (E) −→ LPD (Y,N )/S (E ⊗OY ΩY/S (log M/MS )) in D+ (O(Y,N )/S ). Proof. By (2.2.8.1), we have an isomorphism (2.3.10.2)

∼

• ∗(Y,M,N )/S (E) −→ LPD (Y,M )/S (E ⊗OY ΩY/S (log M/MS )).

Applying R(Y,M,N )/S∗ to both hands of (2.3.10.2) and using (2.3.6) and (2.3.7), we obtain

2.3 Forgetting Log Morphisms and Vanishing Cycle Sheaves

91

R(Y,M,N )/S∗ ∗(Y,M,N )/S (E) ∼

• −→ R(Y,M,N )/S∗ LPD (Y,M )/S (E ⊗OY ΩY/S (log M/MS )) ∼

• ←− (Y,M,N )/S∗ LPD (Y,M )/S (E ⊗OY ΩY/S (log M/MS )) • = LPD (Y,N )/S (E ⊗OY ΩY/S (log M/MS )).

  We prove the boundedness of log crystalline cohomology in a general situation. Proposition 2.3.11. Let (S, I, γ) be the log PD-scheme in §1.6. Set S0 := SpecS (OS /I). Let f : X −→ Y be a morphism of fine log schemes over S0 . ◦

◦

◦

◦

◦

Assume that X and Y are quasi-compact and that f : X −→ Y is quasiseparated morphism of finite type. Let E be a quasi-coherent crystal of OY /S log modules. Then Rfcrys∗ (E) is bounded. Proof. For (U, T, δ) ∈ (Y /S)log crys , put XU := X×Y U and denote the morphism of topoi (f |XU ) ◦ uXU /T by fXU /T . By the same argument as [3, V Th´eor`eme 3.2.4], we are reduced to proving the following claim: there exists a positive integer r such that, for any (U, T, δ) ∈ (Y /S)log crys and for any quasi-coherent log i crystal E on (XU /T )crys , we have R fXU /T ∗ (E) = 0 for i > r. Again, by the same argument as [3, V Th´eor`eme 3.2.4, Proposition 3.2.5], we are reduced ◦

◦

to showing the above claim in the case where X and Y are sufficiently small affine schemes. Hence we may assume that X admits a chart α : P −→ MX . (Note that, in this book, log structures are defined on a Zariski site.) Let us take surjections ϕ1 : OY [Na ] −→ OX and ϕ2 : Nb −→ P (a, b ∈ N). For (U, T,

MT , ι, δ) ∈ (Y /S)log crys , let us define T := (T , MT
) by T
:= SpecO (OT [Na ⊕ Nb ]), T

MT
:= the log structure associated to MT ⊕ Nb −→ OT [Na ⊕ Nb ], where the map MT ⊕ Nb −→ OT [Na ⊕ Nb ] is induced by MT −→ OT and the ⊂ natural inclusion Nb −→ OT [Na ⊕ Nb ]. Then we have the canonical affine log smooth morphism g : T
−→ T . Let ψ1 be the morphism OT [Na ⊕Nb ] −→ OXU ⊂

ϕ1 | U

ϕ2

α|X

induced by Na −→ OU [Na ] −→ OXU and Nb −→ P −→U MXU −→ OXU . Let ψ2 be the morphism MT ⊕ Nb −→ MXU induced by MT −→ MU −→ MXU α|X ϕ2 ⊂ and Nb −→ P −→U MXU . Then we have the closed immersion ψ : XU −→ T
of log schemes induced by ψ1 , ψ2 and we have the commutative diagram of log schemes

92

2 Weight Filtrations on Log Crystalline Cohomologies ψ

XU −−−−→ ⏐ ⏐ 

T
⏐ ⏐g 

ι

U −−−−→ T. Let D be the log PD-envelope of ψ and let h : D −→ T be the composite morphism D −→ T
−→ T . Then we have Ri fXU /T ∗ (E) = Hi (h∗ (E(XU ,D) ⊗OT Λ•T
/T )) = 0 for i > a + b. Hence we have proved the claim and consequently

 

we finish the proof of (2.3.11).

Corollary 2.3.12. Let (S, I, γ) and S0 be as in (2.3.11). Let (Y, M ) be a fine log smooth scheme over S0 such that Y is quasi-compact. Let N be a fine sub log structure of M on Y . Then, for a quasi-coherent crystal E of O(Y,M )/S -modules, the complex R(Y,M,N )/S∗ (E) is bounded. Remark 2.3.13. In the proof of (2.3.11), we used the convention that the log structures in this book are defined on a Zariski site. However, if we assume that f is log smooth, we can prove the statement of (2.3.11) also in the case where the log structures are defined on an etale site. Indeed, in this case, by ◦

◦

(2.3.14) below, if we assume that X and Y are affine, then we have always a log smooth lift g : T
−→ T of XU −→ U for any (U, T, δ) ∈ (Y /S)log crys such ◦

that T is affine. Then we have Ri fXU /T ∗ E = Hi (g∗ (E(XU ,T
) ⊗OT Λ•T
/T )) = 0 for i > r, where r is the maximum of the rank of Λ1X/Y,x (x ∈ X). We give a proof of a lemma which has been used in (2.3.13), which is useful also in later sections. Lemma 2.3.14. Let S be a fine log scheme and let I be a quasi-coherent nil-ideal sheaf of OS . Let S0 be an exact closed log subscheme of S defined ◦

by I. Assume that S is affine. Let Z be a log smooth scheme over S0 . Then, ◦

if Z is affine, there exists a unique log smooth lift Z (up to an isomorphism) of Z over S and Z is also affine. Proof. Let (P) be a property of a scheme or a morphism of schemes. In this proof, for simplicity, we say that a log scheme W (resp. a morphism ◦

◦

f : W −→ W  of log schemes) has the property (P) if W (resp. f ) has the property (P). Though the unique existence of Z seems more or less wellknown, we give a proof as follows (cf. [54, (3.14) (1)], [11, N.B. in 5.28]). Express I as the inductive limit of the inductive system {Iλ } of quasicoherent nilpotent ideal sheaves of OS : I = limλ Iλ . Let Sλ be an exact −→ closed log subscheme of S defined by Iλ . Since S0 = limλ Sλ and since Z is of ←− finite presentation over S0 , there exists a fine log smooth scheme Zλ over Sλ

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes

93

such that Z = Zλ ×Sλ S0 for a large λ (cf. [40, 3 (8.8.2) (ii)], [40, 4 (17.7.8)], [86, 4.11]). By [40, 3 (8.10.5) (viii)], we may assume that Zλ is affine. Since Iλ is nilpotent, the existence of Z follows from [54, (3.14) (1)]. Let Z  be another lift of Z over S. Since the structural morphism Z −→ S0 is quasi-separated, the structural morphisms Z −→ S and Z  −→ S are quasiseparated by [40, 1 (1.2.5)]. Set Zλ := Z ×S Sλ and Zλ := Z  ×S Sλ . Then Zλ and Zλ are quasi-compact, quasi-separated and of finite presentation over ∼ Sλ . Because limλ Zλ = Z = limλ Zλ , there exists an isomorphism Zλ −→ Zλ ←− ←− over Sλ for a large λ which induces the identity of Z (cf. [40, 3 (8.8.2) (i)], ∼ [86, 4.11.3]). Since Iλ is nilpotent, there exists an isomorphism Z −→ Z  over ∼  S which induces the isomorphism Zλ −→ Zλ ([54, (3.14) (1)]). The rest we have to prove is that Z is affine. Let Zλ be the affine fine log scheme above. Because Iλ is nilpotent, we may assume that Iλ2 = 0. Let J be a coherent ideal sheaf of OZ . By the proof in [45, III (3.7)] of Serre’s theorem on the criterion of the affineness of a scheme, we have only ◦

to prove that H 1 (Z, J ) = 0 (the assumption “noetherianness” in [loc. cit.] is unnecessary). Consider the following exact sequence 0 −→ Iλ J −→ J −→ J /Iλ J −→ 0. ◦

◦

Because Zλ is affine, H 1 (Z, J /Iλ J ) = H 1 (Z λ , J /Iλ J ) = 0. Similarly, ◦

◦

H 1 (Z, Iλ J ) = 0. Hence H 1 (Z, J ) = 0. Hence we finish the proof.

 

Let X be a smooth scheme over S0 and let D and Z be relative SNCD’s on X/S0 which meets transversally over S0 . In §2.7 below, we investigate important properties of R(Y,M,N )/S∗ (O(Y,M )/S ) for the case where (Y, M ) = (X, D ∪ Z) and (Y, N ) = (X, Z).

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes Let (S, I, γ) be a PD-scheme such that OS is killed by a power of a prime number p and such that I is quasi-coherent. Set S0 := SpecS (OS /I). Let ◦

f : X −→ S0 be a smooth morphism and D a relative SNCD on X over S0 . Let f : (X, D) −→ S0 be the natural morphism of log schemes. By abuse of ⊂ notation, we also denote by f the composite morphism (X, D) −→ S0 −→ S. The aim in this section is to construct two fundamental objects in D+ F(Q∗X/S (OX/S )) and in D+ F(f −1 (OS )) which we call the preweightfiltered restricted crystalline complex of (X, D)/(S, I, γ) and preweight-filtered zariskian complex of (X, D)/(S, I, γ), respectively. In fact, we construct these complexes in a more general setting.

94

2 Weight Filtrations on Log Crystalline Cohomologies

As explained in §2.1, X has the fs log structure M (D) defined by D. As in §2.1, we denote this log scheme by (X, D). Let ∆ = {Dλ }λ∈Λ be a decomposition of D by smooth components of D over S0 . Let X = Xi0 be an open covering, where I0 is a set. Set Di0 := D ∩ Xi0 i0 ∈I0

and D(λ;i0 ) := Dλ ∩ Xi0 . Fix a total order on I0 and let I be the category defined in §1.5. For an object i = (i0 , . . . , ir ) ∈ I, set Xi := r r r X s=0 is , Di := s=0 Dis and D(λ;i) := s=0 D(λ;is ) . As explained in §1.6, log we have two ringed topoi (((X•
, D• )/S) , Q∗ (O(X ,D )/S )) and • zar , f•−1 (OS )). (X Thus we have the following datum:

Rcrys

(X• ,D• )/S

•

•

 (2.4.0.1): An open covering X = i0 ∈I0 Xi0 and the family {(Xi , Di )}i∈I of log schemes which form a diagram of log schemes over the log scheme (X, D), which we denote by (X• , D• ). That is, (X• , D• ) is nothing but a contravariant functor I o −→ {smooth schemes with relative SNCD’s over S0 which are augmented to (X, D)} Assume that, for any element i0 of I0 , there exists a smooth scheme Xi0 with a relative SNCD Di0 on Xi0 over S such that there exists an admissible immersion ⊂ (Xi0 , Di0 ) −→ (Xi0 , Di0 ) with respect to ∆i0 := {D(λ;i0 ) }λ∈Λ . By (2.3.14), if {Xi0 }i0 ∈I0 is an affine open covering of X, we can assume that (Xi0 , Di0 ) is, in fact, a lift of (Xi0 , Di0 ): (Xi0 , Di0 ) ×S S0 = (Xi0 , Di0 ). We wish to construct the following object: ⊂

(2.4.0.2): A diagram (X• , D• ) −→ (X• , D• ) (• ∈ I) of admissible immersions into a diagram of smooth schemes with relative SNCD’s over S with respect to ∆• , where ∆i := {D(λ;i) }λ∈ΛXi (i ∈ I).
i = {D(λ;i ) }λ∈Λ Let ∆ be a decomposition of Di0 which is compatible 0 Xi 0 0 with ∆i0 : Di0 = λ∈ΛX D(λ;i0 ) and D(λ;i0 ) ×Xi0 Xi0 = D(λ;i0 ) (∀λ ∈ ΛXi0 ). i0

Let i = (i0 , . . . , ir ) be an object of I. Set X(iα ,i) := Xiα \ (X iα \ Xi ) (0 ≤ α ≤ r), where X iα denotes the closure of Xiα in Xiα . Since X iα \ Xi is a closed subscheme of Xiα , X(iα ,i) is an open subscheme of Xiα . It is easy ⊂

to see that the morphism Xi −→ X(iα ,i) is a closed immersion. Denote by D(λ;iα ,i) (resp. D(iα ,i) ) the closed subscheme D(λ;iα ) ∩ X(iα ,i) (resp. Diα ∩  ⊂ X(iα ,i) ) of X(iα ,i) . Set Xi := rα=0 X(iα ,i) . The closed immersions Xi −→ S

⊂

X(iα ,i) (α = 0, . . . , r) induce an immersion Xi −→ Xi . Blow up Xi along

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes



r λ∈Λ

S

α=0 D(λ;iα ,i) .

95

Denote this scheme by Xi . We consider the complement

Xi of the strict transform of r  

(X(i0 ,i) ×S · · · ×S X(iβ−1 ,i) ×S D(λ;iβ ,i) ×S X(iβ+1 ,i) ×S · · · ×S X(ir ,i) )

λ∈Λ β=0

in Xi . Let Di be the exceptional divisor on Xi . Then Di is a relative SNCD on Xi by (2.4.2) below. Considering the strict transform of the image of Xi ⊂ of the diagonal embedding in Xi , we have an immersion Xi −→ Xi , in fact, ⊂ an admissible immersion (Xi , Di ) −→ (Xi , Di ) with respect to ∆i by (2.4.2) ⊂ below. Let Di be the log PD-envelope of the immersion (Xi , Di ) −→ (Xi , Di ) over (S, I, γ) with structural morphism fi : Di −→ S. First we give the local description of OXi at a point of Di (cf. [47, 2], [48, (1.7)], [64, 3.4]) for the warm up for the general description of OXi in (2.4.2) below. Lemma 2.4.1. Let i = (i0 , . . . , ir ) be an element of I. Then, Zariski locally at the image of a point of Di in Xi , the structure sheaf OXi of Xi is etale over the following sheaf of rings (i )

(i )

(i i0 )±1

OS [x1 α , . . . , xdiα | 0 ≤ α ≤ r][ut α α

(i )

(i i0 ) (i0 ) xt

(xt α − ut α

| 1 ≤ α ≤ r, 1 ≤ t ≤ s]/

| 1 ≤ α ≤ r, 1 ≤ t ≤ s),

(i ) (i ) x1 α , . . . , xdiα α

(i i )

(i i )

(0 ≤ α ≤ r) and u1 α 0 , . . . , us α 0 (1 ≤ α ≤ r) are where independent variables over OS and s is a positive integer. The exceptional (i ) (i ) divisor Di is defined by an equation x1 0 · · · xs 0 = 0. Proof. The problem is etale local. We may assume that there exists an isomor∼ (i ) (i ) phism Xiα −→ SpecS (OS [x1 α , . . . , xdiα ]). Assume, furthermore, that D(iα ,i) α

(i )

(i )

is defined by an equation x1 α · · · xs α = 0 (1 ≤ s ≤ min{diα | 0 ≤ α ≤ r}). Here a positive integer s is independent of α. (i ) (i ) Set A := OS [x1 α , . . . , xdiα | 0 ≤ α ≤ r]. Let It ⊂ A (1 ≤ t ≤ s) be the α ideal sheaf of a closed subscheme (i )

(i )

(i )

(xt 0 = 0) ∩ (xt 1 = 0) ∩ · · · ∩ (xt r = 0). Set U0 := Xi and let Ut (1 ≤ t ≤ s) be a scheme defined inductively as follows: Ut is the blowing up of Ut−1 with respect to the ideal sheaf It OUt−1 . Then, by [77, (5.1.2) (v)], Us = Xi . By the construction of Us , Us is covered by the following spectrums over S of the following sheaves of rings: (iβ1 )

A[u1 (iβ1 )

(x1

(iβ1 )

− (u1

(iα1 )

/u1

(iα1 )

/u1

(iα1 )

)x1

(i

)

(i

)

, . . . , us βs /us αs | 0 ≤ β1 , . . . , βs ≤ r]/ (i

)

(iβ1 )

, . . . , xs βs − (us

(i

)

(i

)

/us αs )xs αs )

(0 ≤ α1 , . . . , αs ≤ r).

96

2 Weight Filtrations on Log Crystalline Cohomologies

Since the following equations (i )

xt β = 0

(1 ≤ t ≤ s, 0 ≤ ∀β ≤ r)

are equivalent to (i )

ut β = 0,

(i )

xt α = 0

(1 ≤ t ≤ s, 0 ≤ ∀β ≤ r),

OXi is isomorphic to (i )

(i iα )±1

(i )

OS [x1 α , . . . , xdiα | 0 ≤ α ≤ r][ut β α

(i )

(i iα ) (iα ) (i i ) (i i ) xt , ut β α ut α β

(xt β − ut β

| 0 ≤ α = β ≤ r, 1 ≤ t ≤ s]/ (i iα )

− 1, ut γ

(i iβ ) (iβ iα ) ut

− ut γ

| 0 ≤ α = β = γ = α ≤ r, 1 ≤ t ≤ s). The last sheaf of rings is isomorphic to (i )

(i )

(i i0 )±1

OS [x1 α , . . . , xdiα | 0 ≤ α ≤ r][ut α α

(i )

(i i0 ) (i0 ) xt

(xt α − ut α

| 1 ≤ α ≤ r, 1 ≤ t ≤ s]/

| 1 ≤ α ≤ r, 1 ≤ t ≤ s).

Now the claim on the exceptional divisor is obvious.

 

We think that the reader is ready to read the following theorem which ⊂ tells us that (Xi , Di ) −→ (Xi , Di ) is, indeed, an admissible immersion with respect to ∆i . Theorem 2.4.2. Fix i = (i0 , . . . , ir ) ∈ I. Let A := rα=0 OX(iα ,i) be the structure sheaf of Xi . Set Λi := {λ ∈ Λ | D(λ;iα ,i) = ∅ (0 ≤ ∀α ≤ r)}. (Then we have Λi = ΛXi .) Let J(λ;iα ,i) (λ ∈ Λi ) be the ideal sheaf of OX(iα ,i) defining the closed im⊂

mersion D(λ;iα ,i) −→ X(iα ,i) . Let X(iα ,i) = ∪µ(iα ,i) Xµ(iα ,i) be an open covering of X(iα ,i) such that the restriction of J(λ;iα ,i) to Xµ(iα ,i) is generated by a local (µ

)

section xλ (iα ,i) for all λ ∈ Λi (such open covering exists by the commutative diagram (2.1.7.2) for (X(iα ,i) , D(iα ,i) )). Set (r)

Λi

(r)

:= Λi (µ(i0 ,i) , . . . , µ(ir ,i) ) := {λ ∈ Λi | D(λ;iα ,i) ∩ Xµ(iα ,i) = ∅, (0 ≤ ∀α ≤ r)}.

Then Xi is covered by the spectrums over S of the following sheaves of rings (µ(iα ,i) µ(i0 ,i) ) ±1

A[(uλ (µ(iα ,i) )

(xλ

)

(µ(iα ,i) µ(i0 ,i) ) (µ(i0 ,i) ) xλ

− uλ

(r)

| λ ∈ Λi , 1 ≤ α ≤ r]/ (r)

| λ ∈ Λi ) (µ(i0 ,i) , . . . , µ(ir ,i) ).

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes (µ

97

)

µ

Here uλ (iα ,i) (i0 ,i) ’s are independent variables. The exact locally closed im⊂ mersion (Xi , Di ) −→ (Xi , Di ) is an admissible immersion with respect to {D(λ;i) }λ∈Λi . Proof. We have the restriction 

⊂

(Xi , Di ) −→ (X(iα ,i) ,

D(λ;iα ,i) )

λ∈Λi ⊂

of the admissible immersion (Xiα , Diα ) −→ (Xiα , Diα ) with respect to ∆iα (0 ≤ α ≤ r). Set M (λ) := {(µ(i0 ,i) , . . . , µ(ir ,i) ) | D(λ;iα ) ∩ Xµ(iα ,i) = ∅ (0 ≤ ∀α ≤ r)} and let M1 (λ) be the set of the µ(is ,i) ’s (0 ≤ s ≤ r) appearing in an element of M (λ). Then, by [77, (5.1.2) (v)], Xi is covered by the spectrums over S of the following sheaves of rings: (µ(iβ

A[uλ (µ(iβ

(xλ

λ

,i) )

λ

,i) )

(µ(iβ

−(uλ

(µ(iα

/uλ

λ

,i) )

λ

,i) )

(µ(iα

/uλ

(r)

| 0 ≤ βλ ≤ r, λ ∈ Λi , µ(iβλ ,i) ∈ M1 (λ)]/

λ

,i) )

(µ(iα

)xλ

λ

,i) )

)

(0 ≤ αλ ≤ r, µ(iαλ ,i) ∈ M1 (λ)).

Since the following equations (µ(iα ,i) )

xλ

= 0

(0 ≤ ∀α ≤ r)

are equivalent to (µ(iα ,i) )

uλ

= 0,

(µ(iα ,i) )

xλ

= 0

(0 ≤ ∀α ≤ r),

Xi is covered by the spectrums of the quotient sheaves of (µ(iβ ,i) µ(iα ,i) ) ±1

A[(uλ

(r)

| 0 ≤ α = β ≤ r, λ ∈ Λi , µ(iα ,i) , µ(iβ ,i) ∈ M1 (λ)]

)

divided by ideal sheaves generated by (µ(iβ ,i) )

xλ

(µ(iβ ,i) µ(iα ,i) ) (µ(i ,i) ) xλ α ,

− uλ

(µ(iα ,i) µ(iβ ,i) ) (µ(iβ ,i) ,µ(iα ,i) )

uλ

uλ

− 1,

and (µ(iγ ,i) µ(iα ,i) )

uλ

(µ(iγ ,i) µ(iβ ,i) ) (µ(iβ ,i) µ(iα ,i) )

− uλ

(0 ≤ α = β = γ = α ≤ r, λ ∈

uλ

(r) Λi , µ(iα ,i) , µ(iβ ,i) , µ(iγ ,i)

∈ M1 (λ)).

98

2 Weight Filtrations on Log Crystalline Cohomologies

This quotient sheaf is isomorphic to (µ(iα ,i) µ(i0 ,i) ) ±1

A[(uλ

)

(r)

| λ ∈ Λi ]/

(µ(iα ,i) µ(i0 ,i) ) (µ(i0 ,i) ) (r) xλ | λ ∈ Λi ). r Let D(λ;i) be the strict transform of α=0 D(λ;iα ,i) in Xi . Now we see that,  (r) for λ ∈ Λi , the intersection of D(λ;i) and the inverse image of rα=0 Xµ(iα ,i) in S (µ ) Xi is defined by an equation xλ (i0 ,i) = 0. Hence D(λ;i) is a smooth divisor on (µ(iα ,i) )

(xλ

− uλ

Xi over S and Di is a relative SNCD on Xi over S, and D(λ;i) ×Xi Xi = D(λ;i) . Therefore we obtain (2.4.2).  

Now we change notations. Let X be a smooth scheme and let D and Z be transversal relative SNCD’s on X/S0 . Let ∆D := {Dλ }λ (resp. ∆Z := {Zµ }µ ) be a decomposition of D (resp. Z) by smooth components of D (resp. Z). Then ∆D and ∆Z give a decomposition ∆ := {Dλ , Zµ }λ,µ of D∪Z by smooth components of D ∪ Z. We can construct the objects in (2.4.0.1) and (2.4.0.2) ⊂ for D ∪ Z and ∆: (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ). Let Di be the log PD⊂ envelope of the admissible immersion (Xi , Zi ) −→ (Xi , Zi ) with respect to (k) (k) (k) ∆Z |Xi . Set Zi |D(k) := Zi ∩ Di and Zi |D(k) := Zi ∩ Di (k ∈ N), where Di i i is a scheme over S defined in (2.2.13.2) for Di . (k)

Lemma 2.4.3. The log scheme Di ×(Xi ,Zi ) (Di , Zi |D(k) ) is the log PD-envei

(k)

(k)

lope of the locally closed immersion (Di , Zi |D(k) ) −→ (Di , Zi |D(k) ). i

i

 

Proof. (2.4.3) is a special case of (2.2.16) (2). {PkDi }k∈Z

be the filtration on Ω•Xi /S (log(Di ∪Zi )) defined in (2.2.15.2). Let As in §2.2, we set PkDi L(Xi ,Zi )/S (Ω•Xi /S (log(Di ∪ Zi ))) := L(Xi ,Zi )/S (PkDi Ω•Xi /S (log(Di ∪ Zi ))), PkDi (ODi ⊗OXi Ω•Xi /S (log Di )) := ODi ⊗OXi PkDi Ω•Xi /S (log(Di ∪ Zi )). By (2.2.17) (1) and (2), we have two filtered complexes (Q∗(Xi ,Zi )/S L(Xi ,Zi )/S (Ω•Xi /S (log(Di ∪ Zi ))), Q∗(Xi ,Zi )/S P Di ) ∈ C+ F(O(Xi ,Zi )/S ), (ODi ⊗OXi Ω•Xi /S (log(Di ∪ Zi )), P Di ) ∈ C+ F(fi−1 (OS )). Lemma 2.4.4. For a morphism α : i −→ j in I, let α : (Xj , Dj ∪ Zj ) −→ (Xi , Di ∪ Zi ) be the natural morphism. Then {(Xi , Di ∪ Zi ), α}i∈I,α∈Mor(I) defines a diagram of smooth schemes with relative SNCD’s over S : I o −→ {smooth schemes with relative SNCD’s over S},

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes

99

that is, for another morphism β : j −→ l in I, α ◦ β = β ◦ α, and idi = id. Moreover, {Di }i∈I is a diagram of log schemes. In particular, there are natural morphisms ρα : α−1 (Q∗(Xi ,Zi )/S PkDi L(Xi ,Zi )/S (Ω•Xi /S (log(Di ∪ Zi )))) −→ Q∗(Xj ,Zj )/S Pk j L(Xj ,Zj )/S (Ω•Xj /S (log(Dj ∪ Zj ))), D

ρα : α−1 (PkDi (ODi ⊗OXi Ω•Xi /S (log(Di ∪ Zi )))) D

−→ Pk j (ODj ⊗OXj Ω•Xj /S (log(Dj ∪ Zj ))) such that ρidi = id and ρβ◦α = ρβ ◦ β −1 (ρα ). ⊂

Proof. The open immersion Xj −→ Xi induces a morphism Xj −→ Xi . By the universality of the blow ups, we have a morphism Xj −→ Xi and this morphism induces morphisms (Xj , Dj ∪ Zj ) −→ (Xi , Di ∪ Zi ), (Xj , Dj ) −→ (Xi , Di ) and (Xj , Zj ) −→ (Xi , Zi ). The universality of the log PD-envelope   induces a morphism Dj −→ Di . Thus (2.4.4) follows. By (2.4.4), we obtain a complex (Q∗(Xi ,Zi )/S PkDi L(Xi ,Zi )/S (Ω•Xi /S (log(Di ∪ Zi ))))i∈I ∈ C+ (Q∗(X• ,Z• )/S (O(X• ,Z• )/S )) of Q∗(X• ,Z• )/S (O(X• ,Z• )/S )-modules and a complex (PkDi (ODi ⊗OXi Ω•Xi /S (log(Di ∪ Zi ))))i∈I ∈ C+ (f•−1 (OS )) of f•−1 (OS )-modules. Now we have the following filtered complex of Q∗(X• ,Z• ) −1 /S (O(X• ,Z• )/S )-modules and the following filtered complex of f• (OS )-modules: log,Z• (O(X• ,D• ∪Z• )/S ), P D• ) := (CRcrys (Q∗(Xi ,Zi )/S L(Xi ,Zi )/S (Ω•Xi /S (log(Di ∪ Zi ))), Q∗(Xi ,Zi )/S P Di )i∈I and log,Z• (Czar (O(X• ,D• ∪Z• )/S ), P D• ) := (ODi ⊗OXi Ω•Xi /S (log Di ), P Di )i∈I .

Remark 2.4.5. Once we are given the data (2.4.0.1) and (2.4.0.2) for (X, D ∪ Z) with respect to ∆ = {Dλ , Zµ }λ,µ , we can obtain two filtered complexes log,Z• log,Z• (CRcrys (O(X• ,D• ∪Z• )/S ), P D• ) and (Czar (O(X• ,D• ∪Z• )/S ), P D• ).

Let (2.4.5.1)

log ∗ π(X,Z)/SRcrys : (((X•
, Z• )/S)log Rcrys , Q(X• ,Z• )/S (O(X• ,Z• )/S )) −→

100

2 Weight Filtrations on Log Crystalline Cohomologies ∗
(((X, Z)/S)log Rcrys , Q(X,Z)/S (O(X,Z)/S ))

and (2.4.5.2)

•zar , f•−1 (OS )) −→ (X
zar , f −1 (OS )) πzar : (X

be natural morphisms of ringed topoi defined in §1.5 and §1.6. Definition 2.4.6. Assume that we are given the data (2.4.0.1) and (2.4.0.2) for (X, D ∪ Z) with respect to ∆ = {Dλ , Zµ }λ,µ . (1) We call (2.4.6.1) log log,Z• Rπ(X,Z)/SRcrys∗ (CRcrys (O(X• ,D• ∪Z• )/S ), P D• ) ∈ D+ F(Q∗(X,Z)/S (O(X,Z)/S )) the preweight-filtered restricted crystalline complex of O(X,D∪Z)/S (or (X, D∪ log,Z Z)/S) with respect to D. We denote it by (CRcrys (O(X,D∪Z)/S ), P D ). If log,Z D Z = ∅, then we call (CRcrys (O(X,D∪Z)/S ), P ) the preweight-filtered restricted crystalline complex of O(X,D)/S or (X, D)/S and we denote it by (CRcrys (O(X,D)/S ), P ). (2) We call (2.4.6.2)

log,Z• (O(X• ,D• ∪Z• )/S ), P D ) ∈ D+ F(f −1 (OS )) Rπzar∗ (Czar

the preweight-filtered zariskian complex of O(X,D∪Z)/S (or (X, D ∪ Z)/S) log,Z (O(X,D∪Z)/S ), P D ). If Z = ∅, then with respect to D. We denote it by (Czar log,Z D we call (Czar (O(X,D∪Z)/S ), P ) the preweight-filtered zariskian complex of O(X,D)/S or (X, D)/S and we denote it by (Czar (O(X,D)/S ), P ). Let
(2.4.6.3) (X,D∪Z,Z)/S : (((X, D ∪ Z)/S)log crys ,O(X,D∪Z)/S )
−→ (((X, Z)/S)log crys , O(X,Z)/S ) be the forgetting log morphism along D ((2.3.2)) and let (2.4.6.4) −1

u(X,D∪Z)/S : (((X, D ∪ Z)/S)log (OS )) crys , O(X,D∪Z)/S ) −→ (Xzar , f

be the canonical projection ((1.6.0.8)). Proposition 2.4.7. There exists the following canonical isomorphisms (2.4.7.1)

∼

log,Z (O(X,D∪Z)/S ), Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) −→ CRcrys

(2.4.7.2) ∼ log,Z Ru(X,Z)/S∗ (CRcrys (O(X,D∪Z)/S )) −→ Ru(X,D∪Z)/S∗ (O(X,D∪Z)/S ),

2.4 Preweight-Filtered Restricted Crystalline and Zariskian Complexes

(2.4.7.3)

101

∼

log,Z Ru(X,D∪Z)/S∗ (O(X,D∪Z)/S ) −→ Czar (O(X,D∪Z)/S ).

Proof. Let (2.4.7.4)

log log log

π(X,D∪Z)/Scrys : ((X• , D • ∪ Z• )/S)crys −→ ((X, D ∪ Z)/S)crys

and log log log

(2.4.7.5) π(X,D∪Z)/SRcrys : ((X• , D • ∪ Z• )/S)Rcrys −→ ((X, D ∪ Z)/S)Rcrys

be natural morphisms of topoi defined in §1.6. The isomorphism (2.4.7.1) follows from the cohomological descent [42, log,Z Vbis ], (2.3.2.2), (2.3.10.1), (1.6.4.1) and the definition of CRcrys (O(X,D∪Z)/S ). Indeed, the left hand side of (2.4.7.1) is equal to log log,−1 π(X,D∪Z)/Scrys (O(X,D∪Z)/S ) Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ Rπ(X,D∪Z)/Scrys∗ log R(X• ,D• ∪Z• ,Z• )/S∗ (O(X• ,D• ∪Z• )/S ) =Q∗(X,Z)/S Rπ(X,Z)/Scrys∗ log L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))) =Q∗(X,Z)/S Rπ(X,Z)/Scrys∗ log Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))) =Rπ(X,Z)/SRcrys∗ log,Z (O(X,D∪Z)/S ). =CRcrys

By the trivially filtered case of (1.6.3.1) and by (2.4.7.1), log,Z (O(X,D∪Z)/S ) = Ru(X,Z)/S∗ R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) Ru(X,Z)/S∗ (CRcrys

= Ru(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ). (2.4.7.3) is a special case of [46, (2.20)], which follows from the cohomological descent.   Remark 2.4.8. (1) In the next section we shall prove that log,Z (O(X,D∪Z)/S ), P D ) ∈ D+ F(Q∗(X,Z)/S (O(X,Z)/S )) (CRcrys

and

log,Z (Czar (O(X,D∪Z)/S ), P D ) ∈ D+ F(f −1 (OS ))

are independent of the data (2.4.0.1) and (2.4.0.2) for (X, D ∪ Z) if we fix a decomposition of D and Z by their smooth components, and then, in §2.7, we shall prove that they are independent of the choice of the decompositions of D and Z by their smooth components. Once we know that the definitions log,Z log,Z (O(X,D∪Z)/S ), P D ) and (Czar (O(X,D∪Z)/S ), P D ) are well-defined, of (CRcrys we know that

102

2 Weight Filtrations on Log Crystalline Cohomologies

(2.4.8.1) log,Z log,Z Ru(X,Z)/S∗ (CRcrys (O(X,D∪Z)/S ), P D ) = (Czar (O(X,D∪Z)/S ), P D ) by the constructions of them. (2) The complex Czar (O(X,D)/S ) in (2.4.6) is different from that defined in [46, (2.19)]. Because the latter depends on an embedding system of (X, D), it should be called a crystalline complex with respect to an embedding system.

2.5 Well-Definedness of the Preweight-Filtered Restricted Crystalline and Zariskian Complexes In this section we prove that the preweight-filtered restricted crystalline complex log,Z (O(X,D∪Z)/S ), P D ) ∈ D+ F(Q∗(X,Z)/S (O(X,Z)/S )) (CRcrys

in (2.4.6.1) and the preweight-filtered zariskian complex log,Z (O(X,D∪Z)/S ), P D ) ∈ D+ F(f −1 (OS )) (Czar

in (2.4.6.2) are independent of the data (2.4.0.1) and (2.4.0.2). To prove this independence, we need not make local explicit calculations of PD-envelopes; the notion of the admissible immersion enables us to use the classical crystalline Poincar´e lemma implicitly; see (2.5.1), (2.5.2) and (2.5.3) below for the detail. ⊂ Let S0 −→ S be a PD-closed immersion defined by a quasi-coherent ideal sheaf I. Let (X, D∪Z), ∆D , ∆Z and ∆ be as in the previous section. Consider the following commutative diagram ⊂

(X, D ∪ Z) −−−−→ (X1 , D1 ∪ Z1 ) ⏐  ⏐    ⊂

(X, D ∪ Z) −−−−→ (X2 , D2 ∪ Z2 ), where the horizontal morphisms above are admissible immersions with respect to a decomposition ∆; assume that the horizontal morphisms in⊂ duce admissible immersions (X, D) −→ (Xi , Di ) with respect to ∆D and ⊂ (X, Z) −→ (Xi , Zi ) with respect to ∆Z (i = 1, 2). Let Di (i = 1, 2) be the ⊂ log PD-envelope of the admissible immersion (X, Z) −→ (Xi , Zi ). Then the following holds: Lemma 2.5.1. The induced morphisms (2.5.1.1)

(Q∗(X,Z)/S L(X,Z)/S (Ω•X2 /S (log(D2 ∪ Z2 ))), Q∗(X,Z)/S P D )

−→ (Q∗(X,Z)/S L(X,Z)/S (Ω•X1 /S (log(D1 ∪ Z1 ))), Q∗(X,Z)/S P D ),

2.5 Well-Definedness of the Preweight-Filtered Complexes

103

(OD2 ⊗OX2 Ω•X2 /S (log(D2 ∪ Z2 )), P D )

(2.5.1.2)

−→ (OD1 ⊗OX1 Ω•X1 /S (log(D1 ∪ Z1 )), P D ) are filtered quasi-isomorphisms. Q∗

PD

Proof. Apply the gr-functor grk (X,Z)/S (k ∈ N) to (2.5.1.1). Then, by (2.2.21.2), we obtain the following morphism: Q∗

grk (X,Z)/S

PD

{(2.5.1.1)} :

(k)log Q∗(X,Z)/S acrys∗ L(k) (Ω•D(k) /S (log Z2 |D(k) )){−k} 2 2

(k)log ⊗Z
crys (D/S; Z)

(k)log −→ Q∗(X,Z)/S acrys∗ L(k) (Ω•D(k) /S (log Z1 |D(k) )){−k} ⊗Z
crys (D/S; Z). (k)log

1

1

Q∗

Then gk := grk (X,Z)/S gram:

PD

{(2.5.1.1)} fits into the following commutative dia-

Q∗(X,Z)/S acrys∗ L(k) (Ω• (k)

(log Z2 |D(k) )){−k} ⊗Z
crys (D/S; Z) ⏐ 2 ⏐ gk 

Q∗(X,Z)/S acrys∗ L(k) (Ω• (k)

(log Z1 |D(k) )){−k} ⊗Z
crys (D/S; Z)

(k)log

D2 /S

(k)log

D1 /S

(k)log

(k)log

1

←−−−− Q∗(X,Z)/S acrys∗ O(D(k) ,Z| (k)log

←−−−− Q∗(X,Z)/S acrys∗ O(D(k) ,Z| (k)log

D (k)

(k)log

)/S {−k}

⊗Z
crys (D/S; Z)

)/S {−k}

⊗Z
crys (D/S; Z),

D (k)

  

(k)log

where the horizontal morphisms are quasi-isomorphisms. Hence gk is also a quasi-isomorphism and so is (2.5.1.1). Applying the filtered direct image Ru(X,Z)/S∗ to (2.5.1.1), we immediately see that (2.5.1.2) is a filtered quasi-isomorphism by the log version of [11, 5.27.2, (7.1.2)].   Remark 2.5.2. To compare our straight method with previous works, assume ⊂ that Z = ∅ and consider two admissible immersions (X, D) −→ (Xi , Di ) (i = 1, 2) with respect to a decomposition ∆ = {Dλ }λ∈Λ of D by smooth  components of D. As  in §2.4, we make the following operation. Set X12 := D(λ;i) (i = 1, 2) be the union of smooth components X1 ×S X2 . Let Di = λ∈Λ   along λ∈Λ (D(λ;1) ×S D(λ;2) ). Let X12 be the complement of Di . Blow up X12 of the strict transform of  {(D(λ;1) ×S X2 ) ∪ (X1 ×S D(λ;2) )} λ∈Λ

104

2 Weight Filtrations on Log Crystalline Cohomologies

in this blow up. Let D12 be the exceptional divisor on X12 . By considering the strict transform of X in X12 , we have an admissible immersion ⊂ (X, D) −→ (X12 , D12 ) with respect to ∆, and we have the following commutative diagram: ⊂

(2.5.2.1)

(X, D) −−−−→ (X12 , D12 ) ⏐  ⏐    ⊂

(X, D) −−−−→ (Xi , Di ), Let Di (i = 1, 2) and D12 be the log PD-envelope of the admissible immersions ⊂ ⊂ (X, D) −→ (Xi , Di ) and (X, D) −→ (X12 , D12 ), respectively. Then the induced morphisms (X12 , D12 ) −→ (Xi , Di ) (i = 1, 2) induce morphisms of filtered complexes (2.5.2.2)

(Q∗X/S LX/S (Ω•Xi /S (log Di )), Q∗X/S P ) −→ (Q∗X/S LX/S (Ω•X12 /S (log D12 )), Q∗X/S P ),

and (2.5.2.3) (ODi ⊗OXi Ω•Xi /S (log Di ), P ) −→ (OD12 ⊗OX12 Ω•X12 /S (log D12 ), P ), which are filtered quasi-isomorphisms by (2.5.1). Thus the proof for (2.5.2.3) gives a simpler proof of a filtered version of the last lemma in [47] (cf. [48, (1.7)], [64, 3.4]). Because we allow not only local lifts of (X, D) but also local admissible immersions in the constructions of (CRcrys (O(X,D)/S ), P ) and (Czar (O(X,D)/S ), P ), we can use the Poincar´e lemma implicitly for the proof of the quasi-isomorphism (2.5.2.3). We can also use a complicated version of [64, 3.4] to prove that (2.5.2.3) is a filtered quasi-isomorphism; however we omit this proof because this proof is lengthy. log,Z log,Z Next we prove that (CRcrys (O(X,D∪Z)/S ), P D ) and (Czar (O(X,D∪Z)/S ), P ) are independent of the data (2.4.0.1) and (2.4.0.2) for D ∪ Z and ∆. Let the notations be as in §2.4. Let {Xi0 }i0 ∈I0 and {Xj0 }j0 ∈J0 be two open coverings of X, where I0 and J0 are two sets. Let I and J be two sets in §1.5. By §1.6 we have a diagram of ringed topoi (((X••
, Z•• )/S)log crys , O(X•• ,Z•• )/S ) −1  and (X••zar , f•• (OS )). Let i and j be arbitrary elements of I and J, respectively. For simplicity of notation, set E := D ∪ Z. Let {Dλ }λ and {Zµ }µ be decompositions of D and Z by smooth components of D and Z, respectively. Set ∆ := {Eν }ν := {Dλ , Zµ }λ,µ . Then ∆ is a decomposition of E by smooth components of E. Assume that there exist two diagrams of admissible im⊂ ⊂
i )i∈I and (Xj , Ej ; ∆|X )j∈J −→ mersions (Xi , Ei ; ∆|Xi )i∈I −→ (Xi , Ei ; ∆ j
j )j∈J over S. Set Xij := Xi ∩ Xj and Eij := Ei ∩ Ej . Let X(i,ij) := (Xj , Ej ; ∆ D

2.5 Well-Definedness of the Preweight-Filtered Complexes

105

Xi\(X i \ Xij ) (resp. X(j,ij) := Xj \ (X j \ Xij )) and set Xij := X(i,ij) ×S X(j,ij) . ⊂
i Then we have a locally closed immersion Xij −→ Xij . Set {E(ν;i) }ν := ∆
and {E(ν;j) }ν := ∆j . Set also E(i,ij) := Ei ∩ X(i,ij) , E(j,ij) := Ej ∩ X(j,ij) , E(ν;i,ij) := E(ν;i) ∩ X(i,ij) and E(ν;j,ij) := E(ν;j) ∩ X(j,ij) . Blow up Xij along  (E(ν;i,ij) ×S E(ν;j,ij) ). Let Xij be the resulting scheme. Let Xij be the comν

plement of the strict transform of   [ E(ν;i,ij) ×S X(j,ij) )] ∪ [ X(i,ij) ×S E(ν;j,ij) ] ν

ν

in Xij . Let Eij be the exceptional divisor on Xij . Then Eij is a relative SNCD on Xij by (2.4.2). Considering the strict transform of the image of Xij in ⊂ Xij , we have a locally closed immersion Xij −→ Xij , in fact, an admissible ⊂ immersion (Xij , Eij ) −→ (Xij , Eij ) by (2.4.2). Let {E(ν;ij) }ν be the resulting decomposition of Eij by smooth components of Eij . We also have a relative SNCD Zij on Xij /S by using Z instead of E. Let Dij be the log PD-envelope ⊂ of the locally closed immersion (Xij , Zij ) −→ (Xij , Zij ). Let log RηRcrys∗ : D+ F(Q∗(X•• ,Z•• )/S (O(X•• ,Z•• )/S )) −→ D+ F((Q∗(X• ,Z• )/S (O(X• ,Z• )/S ))•∈I )

and log Rηi,Rcrys∗ : D+ F(Q∗(Xi• ,Zi• )/S (O(Xi• ,Zi• )/S )) −→ D+ F(Q∗(Xi ,Zi )/S (O(Xi ,Zi )/S ))

be the natural morphisms defined in (1.6.0.2) and (1.6.0.3), respectively. Let −1 (OS )) −→ D+ F((f•−1 (OS ))•∈I ), Rηzar∗ : D+ F(f•• −1 Rηi,zar∗ : D+ F(fi• (OS )) −→ D+ F(fi−1 (OS ))

be the natural morphisms defined in (1.6.0.6) and (1.6.0.7), respectively. Then we have the following: Theorem 2.5.3. (2.5.3.1) log RηRcrys∗ (Q∗(X•• ,Z•• )/S L(X•• ,Z•• )/S (Ω•X•• /S (log E•• )), Q∗(X•• ,Z•• )/S P D•• ) =(Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log E• )), Q∗(X• ,Z• )/S P D• )•∈I . (2.5.3.2)

Rηzar∗ (OD•• ⊗OX•• Ω•X•• /S (log E•• ), P D•• ) =(OD• ⊗OX• Ω•X• /S (log E• ), P D• )•∈I .

Proof. Because (2.5.3.2) follows from (2.5.3.1) by (2.2.22) and by the commutative diagram (1.6.4.7), we have only to prove (2.5.3.1).

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2 Weight Filtrations on Log Crystalline Cohomologies

Let γij : Xij −→ Xi (i ∈ I, j ∈ J) be the natural morphism. Then ∗

log

•

∗

ηRcrys∗ (Q(X•• ,Z•• )/S L(X•• ,Z•• )/S (ΩX•• /S (log E•• )), Q(X•• ,Z•• )/S P Ker{



∗

γ•j0 Rcrys∗ (Q(X

•j0 ,Z•j0 )/S

j0

−→



L(X•j

0

∗

•j0 j1 ,Z•j0 j1 )/S

∗

Q(X

)=

D•j • ∗ 0 ,Z•j )/S (ΩX•j /S (log E•j0 )), Q(X•j ,Z•j )/S P 0 0 0 0

γ•j0 j1 Rcrys∗ (Q(X

j0 0 ) be the induced decomposition of En of smooth components of En . Let log ∗
π  log Rcrys : (((Xn , Zn )/S)Rcrys , Q(Xn ,Zn )/S (O(Xn ,Zn )/S ))n∈N ∗
−→ (((X, Z)/S)log Rcrys , Q(X,Z)/S (O(X,Z)/S ))

be a natural morphism of ringed topoi. Then there exists an admissible immer⊂ sion (Xn , En )n∈N −→ (Xn , En )n∈N of simplicial smooth schemes with simplicial relative SNCD’s over S with respect to (∆n )n∈N . Moreover,

2.5 Well-Definedness of the Preweight-Filtered Complexes

111

(2.5.7.1) log,Z (CRcrys (O(X,E)/S ), P D ) = ∗ • ∗ Dn )n∈N ). Rπ  log (X,Z)Rcrys∗ ((Q(Xn ,Zn )/S L(Xn ,Zn )/S (ΩXn /S (log En )), Q(Xn ,Zn )/S P

Proof. Let I  be a category whose objects are (i0 , . . . , ir )’s (r ∈ N, i0 , . . . , ir ∈ I0 ) and the morphism from i := (i0 , . . . , ir ) −→ j := (j0 , . . . , js ) is one point if {i0 , . . . , ir } ⊂ {j r0 , . . . , js } and empty r otherwise. For an object i = (i0 , . . . , ir ), set Xi := s=0 Xis , Ei := s=0 (E ∩ Xis ). Then we have the following contravariant functor: (X• , E• ) : I o −→ {smooth schemes with relative SNCD’s over S0 }. The construction in §2.4 shows the existence of a diagram of admissible immersions into a diagram of smooth schemes with relative SNCD’s over S: ⊂ (X• , E• ) −→ (X• , E• ) (• ∈ I  ) with respect to ∆• , where ∆• is the induced decomposition of E by ∆ ((2.1.12)). For an element j1 , j2 ∈ I0 , there exists two natural morphisms δk : (X(j1 ,j2 ) , E(j1 ,j2 ) ) −→ (Xjk , Ejk ) (k = 1, 2). Using these morphisms, we have natural face morphisms δm : (Xn , En ) −→ (Xn−1 , En−1 ) (m = 0, . . . , n). Moreover, note that X(i,i) (i ∈ I0 ) is an open scheme of the blow up of Xi ×S Xi by a closed subscheme of it. By considering ⊂ the strict transform of the diagonal immersion Xi −→ Xi ×S Xi , we have a natural morphism s : Xi −→ X(i,i) . Using this morphism, we have natural degeneracy morphisms sm : (Xn−1 , En−1 ) −→ (Xn , En ) (m = 0, . . . , n − 1). The morphisms sm and δm (m ∈ N) satisfy the standard relations in [90, (8.1.3)]. Hence we have a desired simplicial log scheme (Xn , En )n∈N . Fix a total order < on I0 . Let I be a subcategory of I  whose objects are (i0 , . . . , ir )’s (r ∈ N, i0 < · · · < ir , ij ∈ I0 ). Let log ∗ π(X,Z)/SRcrys : (((X•
, Z• )/S)log Rcrys , Q(X• ,Z• )/S (O(X• ,Z• )/S ))•∈I ∗
−→ (((X, Z)/S)log Rcrys , Q(X,Z)/S (O(X,Z)/S ))

be a natural morphism of ringed topoi. Then we have log,Z (CRcrys (O(X,E)/S ), P D ) log =Rπ(X,Z)/Scrys∗ ((Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log E• )), Q∗(X• ,Z• )/S P D• )•∈I ) log,Z ˇ by the definition of (CRcrys (O(X,E)/S ), P D ). Because Cech complexes are calculated by alternating cochains as in [80, §3], the right hand side is canonically isomorphic to

Rπ  log ((Q∗(Xn ,Zn )/S L(Xn ,Zn )/S (Ω•Xn /S (log En )), Q∗(Xn ,Zn )/S P Dn )n∈N ). (X,Z)/SRcrys∗

 

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2 Weight Filtrations on Log Crystalline Cohomologies

Corollary 2.5.8. With the notation of (2.5.7), let Dn be the log PD-envelope ⊂ of the locally closed immersion (Xn , Zn ) −→ (Xn , Zn ) over (S, I, γ). Let 
n )n∈N −→ X
be a natural morphism of topoi. Then the following πzar : (X holds: (2.5.8.1) log,Z  (O(X,E)/S ), P D ) = Rπzar∗ ((ODn ⊗OXn Ω•Xn /S (log En ), P Dn )n∈N ). (Czar Proof. We immediately have (2.5.8) since we have the analogue of (2.5.4.1) for Rπ  log ((Q∗(Xn ,Zn )/S L(Xn ,Zn )/S (Ω•Xn /S (log En )), Q∗(Xn ,Zn )/S P Dn )n∈N ). (X,Z)/SRcrys∗

 

2.6 The Preweight Spectral Sequence Let the notations be as in §2.4 and §2.5. Recall the projections u(X,Z)/S and u(X,D∪Z)/S ((2.2.22.1), (2.4.6.4)). Set f(X,Z)/S := f ◦ u(X,Z)/S and f(X,D∪Z)/S := f ◦ u(X,D∪Z)/S . Then we have the log crystalline cohomology sheaf Rh f(X,D∪Z)/S∗ (O(X,D∪Z)/S ) (h ∈ Z). We also have the log crystalline cohomology sheaf Rh f(D(k) ,Z| (k) )/S∗ (O(D(k) ,Z| (k) )/S ) of D

D

(D(k) , Z|D(k) )/(S, I, γ). In this section we construct the following spectral sequence of OS -modules: (2.6.0.1) E1−k,h+k = Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S

(k)log ⊗Z
crys (D/S; Z))

=⇒ Rh f(X,D∪Z)/S∗ (O(X,D∪Z)/S ). Theorem 2.6.1. Let a(k) : (D(k) , Z|D(k) ) −→ (X, Z) (k ∈ N) be the natural morphism. Let (k)log

acrys∗ : D+ (O(D(k) ,Z| and

D (k)

)/S )

−→ D+ (O(X,Z)/S )

azar∗ : D+ ((f ◦ a(k) )−1 (OS )) −→ D+ (f −1 (OS )) (k)

be the induced morphisms by a(k) . Fix decompositions of D and Z by their smooth components. Then there exist the following canonical isomorphisms (2.6.1.1)

D

log,Z grP k (CRcrys (O(X,D∪Z)/S ))

=Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k)log

and

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)){−k}

2.6 The Preweight Spectral Sequence

113

D

log,Z grP k (Czar (O(X,D∪Z)/S ))

(2.6.1.2)

(k)

=azar∗ Ru(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S

(k) ⊗Z
zar (D/S0 )){−k}.

Proof. Let the notations be as in §2.4. By applying Ru(X,Z)/S∗ to both hands of (2.6.1.1), we immediately have (2.6.1.2) by (1.3.4.1) and (2.5.4.1); hence we have only to prove (2.6.1.1). Let log (2.6.1.3) π(D (k) ,Z|

D (k)

)/Scrys

(k)
: (((D• , Z• |D(k) )/S)log crys , O(D (k) ,Z• | •

•

(k) )/S D•

−→ (((D(k) ,
Z|D(k) )/S)log crys , O(D (k) ,Z|

)

D (k)

)/S )

be the natural morphism of ringed topoi (§1.6). Then we have the following equalities: (2.6.1.4) D

log,Z grP k (CRcrys (O(X,D∪Z)/S )) log ∗ • =grP k Rπ(X,Z)/SRcrys∗ (Q(X• ,Z• )/S L(X• ,Z• )/S (ΩX• /S (log(D• ∪ Z• )))) D

Q∗

log =Rπ(X,Z)/SRcrys∗ grk (X• ,Z• )/S

P D•

(Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))))

log Q∗(X• ,Z• )/S a•crys∗ (O(D(k) ,Z• | =Rπ(X,Z)/SRcrys∗ (k)log

•

(k) )/S D•

(k)log ⊗Z
crys (D• /S; Z• )){−k} log =Q∗(X,Z)/S Rπ(X,Z)/Scrys∗ a•crys∗ (O(D(k) ,Z• | (k)log

•

(k) )/S D•

(k)log ⊗Z
crys (D• /S; Z• )){−k} log =Q∗(X,Z)/S acrys∗ Rπ(D (k) ,Z| (k)log

D (k)

)/Scrys∗

(O(D(k) ,Z• | •

(k) )/S D•

(k)log ⊗Z
crys (D• /S; Z• )){−k}

=Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k)log

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)){−k}.

Here the second, the third, the fourth and the fifth equalities follow from (1.3.4.1), (2.2.21.2), (1.6.4.1) and (1.6.0.13), respectively. The last equality follows from the cohomological descent. Next we prove that the isomorphism (2.6.1.4) is independent of the choice of the data (2.4.0.1) and (2.4.0.2). Assume that we are given the other data (2.4.0.1) and (2.4.0.2) as in §2.5. By the trivially filtered version of (2.5.3), we have

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2 Weight Filtrations on Log Crystalline Cohomologies

log RηRcrys∗ (Q∗(X•• ,Z•• )/S a••crys∗ L(X•• ,Z•• )/S (Ω•D(k) /S ( log Z•• |D(k) ) (k)log

••

••

(k)log (D•• /S; Z•• ))) ⊗Z
crys (k)log = Q∗(X• ,Z• )/S a•crys∗ (L(X• ,Z• )/S (Ω•D(k) /S (log Z• |D(k) )) ⊗Z
crys (D• /S; Z• )). (k)log

•

•

P D••

P D•

log grk = grk Since RηRcrys∗ commutative diagram Q∗

grk (X• ,Z• )/S

P D•

log RηRcrys∗ by (1.3.4.1), we have the following

(Q∗(X• ,Z• )/S L(X• ,Z• )/S   

Q∗

log RηRcrys∗ grk (X•• ,Z•• )/S

P D••

∼

(Ω•X• /S (log(D• ∪ Z• ))))

−−−−→

(Q∗(X•• ,Z•• )/S L(X•• ,Z•• )/S

∼

(Ω•X•• /S (log(D•• ∪ Z•• )))) Q∗(X• ,Z• )/S a•crys∗ (L(D(k) ,Z• | (k)log

•

(k) )/S D•

(Ω•D(k) /S (log Z• |D(k) )){−k}

   log Q∗(X• ,Z• )/S Rηcrys∗ a••crys∗ (L(D(k) ,Z•• | (k)log

••

−−−−→

(k) )/S D••

•

•

(k)log ⊗Z
crys (D• /S; Z• ))

(Ω•D(k) /S (log Z•• |D(k) )){−k} ••

••

(k)log ⊗Z
crys (D•• /S; Z•• )).

Hence we see that the isomorphism (2.6.1.1) (and hence (2.6.1.2)) is independent of the choice of the data (2.4.0.1) and (2.4.0.2).   Corollary 2.6.2. Let k  be a nonnegative integer. For integers k and h, set E1−k,h+k ((X, D ∪ Z)/S; k  )  (k)log Rh−k f(D(k) ,Z| (k) )/S∗ (O(D(k) ,Z| (k) )/S ⊗Z
crys (D/S; Z)) (k ≤ k  ), D D := 0 (k > k ). Set f (X,Z)/S := f ◦ u(X,Z)/S . Then there exists the following spectral sequence (2.6.2.1)

E1−k,h+k = E1−k,h+k ((X, D ∪ Z)/S; k  ) log,Z (O(X,D∪Z)/S )). =⇒ Rh f (X,Z)/S∗ (PkD CRcrys

In particular, there exists the following spectral sequence

2.6 The Preweight Spectral Sequence

115

(2.6.2.2) E1−k,h+k = E1−k,h+k ((X, D ∪ Z)/S) = Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S

(k)log ⊗Z
crys (D/S; Z))

=⇒ Rh f(X,D∪Z)/S∗ (O(X,D∪Z)/S ). Proof. Let (Ik• , {Il• }l≤k ) ∈ K+ F(Q∗(X,Z)/S (O(X,Z)/S )) be a filtered flasque log,Z log,Z resolution of a representative of (PkD CRcrys (O(X,D∪Z)/S ), {PlD CRcrys (O(X, + ∗  )} ) ∈ D F(Q (O )). Consider the following spectral D∪Z)/S l≤k (X,Z)/S (X,Z)/S sequence

E1−k,h+k = Hh (f (X,Z)/S∗ grk (Ik• )) =⇒ Hh (f (X,Z)/S∗ Ik• ). log,Z (O(X,D∪Z)/S )). Obviously we have Hh (f (X,Z)/S∗ Ik ) = Rh f (X,Z)/S∗ (PkD CRcrys log,Z By the proof of (1.3.4.1), grk (Ik• ) is a flasque resolution of grP k (CRcrys (O(X,   D∪Z)/S )) for k ≤ k . Hence, for k ≤ k , we have D

log,Z E1−k,h+k =Rh f (X,Z)/S∗ (grP k (CRcrys (O(X,D∪Z)/S ))) D

(k)log

=Rh f (X,Z)/S∗ (Q∗(X,Z)/S (acrys∗ (O(D(k) ,Z|

D (k)

=Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z| (k) )/S D

)/S {−k}

(k)log

⊗Z crys (D/S; Z))))

(k)log

⊗Z crys (D/S; Z)).

Here, in the last equality, we have used the commutativity of the diagram (1.6.3.1) for the trivially filtered case. Therefore we obtain (2.6.2.1). By using (2.4.7.2), we obtain (2.6.2.2) similarly.   Corollary 2.6.3. Fix decompositions ∆D and ∆Z of D and Z by their ⊂ smooth components, respectively. Let ι : (X, D ∪ Z) −→ (X , D ∪ Z) be an admissible immersion over S with respect to ∆D and ∆Z . Let f : (X, D∪Z) −→ S0 and fS : (X , D ∪ Z) −→ S be the structural morphisms. Let D be the log ⊂ PD-envelope of the locally closed immersion (X, Z) −→ (X , Z) over (S, I, γ). (k) Let fS : D(k) −→ S be the PD-envelope of the locally closed immersion ⊂ D(k) −→ D(k) over (S, I, γ). Let k  be a nonnegative integer. For integers k and h, set E1−k,h+k ((X , D ∪ Z)/S; k  )  (k) (k) Rh−k fS∗ (OD(k) ⊗OD(k) Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S)) (k ≤ k  ), := 0 (k > k ). Then the following spectral sequence (2.6.3.1)

E1−k,h+k := E1−k,h+k ((X , D ∪ Z)/S; k  ) =⇒ Rh fS∗ (OD ⊗OX PkD Ω•X /S (log(D ∪ Z)))

116

2 Weight Filtrations on Log Crystalline Cohomologies

is isomorphic to (2.6.2.1), and hence it is independent of the choice of the admissible immersion fS . In particular, if fS : (X , D ∪ Z) −→ S is a lift of f : (X, D ∪ Z) −→ S0 , then the following spectral sequence (2.6.3.2) E1−k,h+k = E1−k,h+k ((X , D ∪ Z)/S; k  ) =⇒ Rh fS∗ (PkD Ω•X /S (log(D ∪ Z))) is independent of the choice of the lift. Here E1−k,h+k ((X , D ∪ Z)/S; k  )  (k) (k) Rh−k fS∗ (Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S)) = 0

(k ≤ k  ), (k > k ).  

Proof. (2.6.3) immediately follows from (2.5.4.1) and (2.6.2.1).

Remark 2.6.4. In §2.9 below, we consider the functoriality of (2.6.2.2); in particular, in the case where S0 is of characteristic p, we shall consider the compatibility of (2.6.2.2) with the relative Frobenius F : (X, D) −→ (X  , D ) over S0 .

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration Let S, S0 and f : (X, D ∪Z) −→ S0 be as in §2.4. Let a(k) : (D(k) , Z|D(k) ) −→ (X, Z) be as in §2.2 (2). In §2.4 and §2.5, we have constructed the preweightfiltered restricted crystalline complex log,Z (CRcrys (O(X,D∪Z)/S ), P D ) ∈ D+ F(Q∗(X,Z)/S (O(X,Z)/S ))

such that log,Z CRcrys (O(X,D∪Z)/S ) = Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S )

in D+ (Q∗(X,Z)/S (O(X,Z)/S )). Here

log (X,D∪Z,Z)/S : ((X, D ∪ Z)/S)log crys −→ ((X, Z)/S)crys ⊂

is the forgetting log morphism along D ((2.3.2)). Let j : U := X \ D −→ X be the natural open immersion. Let n be a positive integer. Let (X, D ∪ Z) be as above or an analogous log scheme over C or an algebraically closed field of characteristic p > 0. Then we have the following translation if Z = ∅:

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration

(2.7.0.1) /C Uan

l-adic
et U
(X, D)log

log (Xan , Dan )log , (X
an , Dan )et
et  X Xan , X an ⊂

jan : Uan −→ Xan jet : top : (Xan , Dan )log −→ Xan log  an : (X
an , Dan )et −→ Xan et :

et

117

crystal ?
((X, D)/S)log
(X/S) crys

crys


et −→ X
et ? U


(X, D)log et
et −→ X


(X,D)/S : ((X, D)/S)log crys
−→ (X/S)

Rjan∗ (Z) = Rtop∗ (Z) Rtop∗ (Z/n) = Ran∗ (Z/n) Xan −→ X Z(Xan ,Dan )log (Z/n)(Xan ,Dan )log

Rjet∗ (Z/ln ) = Ret∗ (Z/ln )
et −→ X
zar X

? R(X,D)/S∗ (O(X,D)/S )  crys −→ X
zar uX/S : (X/S)

(Z/n)
log

O(X,D)/S

(Z/n)(X
,D

(p  n)

an

log an )et

ZXan (Z/n)Xan (n ∈ Z) (Ω•X/C (log D), P ) (Ω•Xan /C (log Dan ), P ) (Ω•Xan /C (log Dan ), τ )

(X,D)et

(Z/n)X et (p  n) ? ? ?

crys

OX/S (Czar (O(X,D)/S ), P ) (CRcrys (O(X,D)/S ), P ) (CRcrys (O(X,D)/S ), τ )

Here (Xan , Dan )log is the real blow up of (Xan , Dan ) ([58, (1.2)]) and top is log the natural morphism of topological spaces, (X
an , Dan )et is the analytic log etale topos of (Xan , Dan ) ([51]) and an is the forgetting log morphism to
an defined by the local isomorphisms to Xan ; the morphism et the topos X in the middle column is the forgetting log morphism ([30], cf. [67, (1.1.2)]); the upper (resp. lower) equality in the left column has been obtained in [58, (1.5.1)] (resp. [72]), and the equality in the middle column ([30, (3.6)]) follows from the following composite equality (2.7.0.2) Rh et∗ (Z/ln ) =

h 

gp ∗ /OX ) ⊗Z Z/ln (−h) = Rh jet∗ (Z/ln ) (h ∈ Z, n ∈ Z>0 ). (MD

Here the first equality follows from [58, (2.4)] and the second equality is Gabber’s purity ([33]) which has solved Grothendieck’s purity conjecture. Recall that, in the crystalline case, Rjcrys∗ (OU/S ) is not a good object ([3, VI Lemme 1.2.2]). The purpose of this section is to give another intrinsic description of the log,Z (O(X,D∪Z)/S ), P D ) preweight-filtered restricted crystalline complex (CRcrys and, as a corollary, to obtain the spectral sequence (2.6.2.2) in a different way.

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2 Weight Filtrations on Log Crystalline Cohomologies

We start with the following, which includes a crystalline analogue of Gabber’s purity. Theorem 2.7.1 (p-adic purity). Let k be a nonnegative integer. Then (2.7.1.1) Q∗(X,Z)/S Rk (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) = Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k)log

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)).

log,Z log,Z Proof. The “increasing filtration” {PkD CRcrys (O(X,D∪Z)/S )}k∈Z on CRcrys (O (X,D∪Z)/S ) gives us the following spectral sequence

(2.7.1.2) D log,Z log,Z h E1−k,h+k = Hh (grP k CRcrys (O(X,D∪Z)/S )) =⇒ H (CRcrys (O(X,D∪Z)/S )). Let I • be a flasque resolution of O(X,D∪Z)/S . By (2.4.7.1) and by the exactness of Q∗(X,Z)/S , we have log,Z (O(X,D∪Z)/S )) = Hh (Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S )) Hh (CRcrys

= Hh (Q∗(X,Z)/S (X,D∪Z,Z)/S∗ (I • )) = Q∗(X,Z)/S Hh ((X,D∪Z,Z)/S∗ (I • )) = Q∗(X,Z)/S Rh (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ), and by (2.6.1.1) we have D

log,Z Hh (grP k CRcrys (O(X,D∪Z)/S ))

=Hh−k (Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k)log

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)));

this is equal to Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k) )/S ⊗Z
crys (D/S; Z)), 0 for D k = h and k = h, respectively. Hence (2.7.1.2) degenerates at E1 ; thus we have a canonical isomorphism (k)log

(k)log

Q∗(X,Z)/S Rk (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) =Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k)log

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)).

 
By the Leray spectral sequence for the functor (X,D∪Z,Z)/S∗ : ((X, D ∪ Z)/S) log
log
log
crys −→ ((X, Z)/S)crys and f(X,Z)/S∗ : ((X, Z)/S)crys −→ Xzar , we obtain the following spectral sequence (2.7.1.3)

E2st := Rs f(X,Z)/S∗ Rt (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) =⇒

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration

119

Rs+t f(X,D∪Z)/S∗ (O(X,D∪Z)/S ).

log Z)/S)log Set f (X,Z)/S := f(X,Z)/S ◦ Q(X,Z)/S : ((X, Rcrys −→ ((X, Z)/S)crys −→
zar . Because Rf (X,Z)/S∗ ◦ Q∗ X (X,Z)/S = Rf(X,Z)/S∗ , (2.7.1.3) is equal to the following spectral sequence (2.7.1.4) E2st = Rs f(D(t) ,Z|

D (t)

)/S∗ (O(D (t) ,Z|D(t) )/S

(t)log ⊗Z
crys (D/S; Z)) =⇒

Rs+t f(X,D∪Z)/S∗ (O(X,D∪Z)/S ) by (2.7.1). log,Z Using (2.7.1), we can give another simpler expression of (CRcrys (O(X,D∪Z) D /S ), P ). To do this, let us recall the canonical filtration of a complex. Let (T , A) be a ringed topos and let E • be an object in C(A). Then the canonical filtration τ := {τk E • }k∈Z of E • is defined as follows: τk E i := E i (i < k), := Ker(E k −→ E k+1 ) (i = k), := 0 (i > k). Let E • and F • be objects in C+ (A). Then a homotopy h between two morphisms f, g : E • −→ F • also gives a filtered homotopy between two morphisms f, g : (E • , τ ) −→ (F • , τ ) of filtered complexes. Furthermore, a quasi-isomorphism f : E • −→ F • induces a filtered quasi-isomorphism f : (E • , τ ) −→ (F • , τ ); thus a functor C+ (A) E • −→ (E • , τ ) ∈ C+ F(A) induces a functor D+ (A) −→ D+ F(A), which is also denoted by E • → (E • , τ ). We prove the following lemma for a main result (2.7.3) below in this section: Lemma 2.7.2. Let f : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Then, for an object E • in D+ (A), there exists a canonical morphism (2.7.2.1)

(Rf∗ (E • ), τ ) −→ Rf∗ ((E • , τ ))

in D+ F(A ). Proof. Let E • −→ I • be a quasi-isomorphism into a complex of flasque A-modules. Let (I • , τ ) −→ (J • , {Jk• }) be a filtered flasque resolution of (I • , τ ). Then, by applying the functor f∗ to the morphism of this resolution, we obtain a morphism (2.7.2.2)

(f∗ (I • ), {f∗ (τk I • )}) −→ (f∗ (J • ), {f∗ (Jk• )}).

By (1.1.12) (2), the right hand side of (2.7.2.2) is equal to Rf∗ ((E • , τ )). On the other hand, there exists a natural morphism f∗ (τk I • ) −→ τk f∗ (I • ); in ∼ fact, by the left exactness of f∗ , we have f∗ (τk I • ) −→ τk f∗ (I • ). Hence the left hand side of (2.7.2.2) is equal to (f∗ (I • ), {τk f∗ (I • )}) = (Rf∗ (E • ), τ ). It is easy to check that the induced morphism in D+ F(A ) by the morphism (2.7.2.2) is independent of the choice of I • and (J • , {Jk• }). Therefore we have a canonical morphism (2.7.2.1).   log,Z (O(X,D∪Z)/S ), P D ). Now we give another description of (CRcrys

120

2 Weight Filtrations on Log Crystalline Cohomologies

Theorem 2.7.3 (Comparison theorem). Let S0 , S, X, D and Z be as in §2.4. Set (2.7.3.1)

log,Z (Ecrys (O(X,D∪Z)/S ), P D )

:=(R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ), τ ) ∈ D+ F(O(X,Z)/S ). Then there exists a canonical isomorphism (2.7.3.2) ∼ log,Z log,Z Q∗(X,Z)/S (Ecrys (O(X,D∪Z)/S ), P D ) −→ (CRcrys (O(X,D∪Z)/S ), P D ). In particular, (2.7.3.3)

log,Z log,Z (CRcrys (O(X,D∪Z)/S ), τ ) = (CRcrys (O(X,D∪Z)/S ), P D ).

Proof. Fix the data (2.4.0.1) and (2.4.0.2) for D ∪ Z. Then, as usual, there exists a natural morphism of filtered O(X• ,Z• )/S -modules: (2.7.3.4)

Q∗(X• ,Z• )/S (L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))), τ ) −→

(Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))), Q∗(X• ,Z• )/S P D• ). By (2.7.2) there exists a canonical morphism (2.7.3.5) log (Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))), τ ) −→ log Rπ(X,Z)/SRcrys∗ (Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))), τ ). log ((2.7.3.4)), we obBy composing (2.7.3.5) with the morphism Rπ(X,Z)/SRcrys∗ tain a morphism

(2.7.3.6) log (Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))), τ ) −→ log Rπ(X,Z)/SRcrys∗ (Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪Z• ))), Q∗(X• ,Z• )/S P D• )

which is nothing but a morphism (2.7.3.7) log,Z (Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ), τ ) −→ (CRcrys (O(X,D∪Z)/S ), P D ) by (1.6.4.1). (We have not yet claimed that the morphism (2.7.3.7) is independent of the data (2.4.0.1) and (2.4.0.2).) To prove that the morphism (2.7.3.7) is a filtered quasi-isomorphism, it suffices to prove that the induced morphism (2.7.3.8) D log,Z grτk Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) −→ grP k CRcrys (O(X,D∪Z)/S )

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration

121

is a quasi-isomorphism for each k ∈ Z. By the definition of the canonical filtration τ and by the proof of (2.7.1), we have (2.7.3.9) Hi (grτk Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ))  Q∗(X,Z)/S Rk (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) (i = k) = 0 (i = k)  (k)log (k)log Q∗(X,Z)/S acrys∗ (O(D(k) ,Z| (k) )/S ⊗Z
crys (D/S; Z)) D = 0

(i = k), (i = k).

D

log,Z By the proof of (2.7.1) again, Hi (grP k CRcrys (O(X,D∪Z)/S )) is also equal to the last formulas in (2.7.3.9). Hence the morphism (2.7.3.7) is a quasiisomorphism. Finally we show that the morphism (2.7.3.7) is independent of the data (2.4.0.1) and (2.4.0.2). Indeed, let the notations be as in §2.5. Using (2.5.3.1), we have the following commutative diagram: log Q∗(X• ,Z• )/S L(X• ,Z• )/S (Rπ(X,Z)/SRcrys∗

(Ω•X• /S (log(D•   

∼

−−−−→

∪ Z• ))), τ )

log log (Rπ(X,Z)/SRcrys∗ RηRcrys∗ Q∗(X•• ,Z•• )/S L(X•• ,Z•• )/S

(Ω•X•• /S (log(D•• ∪ Z•• ))), τ )

−−−−→

log (Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))),

  

Q∗(X• ,Z• )/S P D• )

log log (Rπ(X,Z)/SRcrys∗ RηRcrys∗ Q∗(X•• ,Z•• )/S L(X•• ,Z•• )/S (Ω•X•• /S (log(D•• ∪ Z•• ))),

Q∗(X•• ,Z•• )/S P D•• ). Thus the independence in question follows.

 

log,Z Definition 2.7.4. We call (Ecrys (O(X,D∪Z)/S ), P D ) ∈ D+ F(O(X,Z)/S ) the preweight-filtered vanishing cycle crystalline complex of (X, D ∪ Z)/S with respect to D. Set log,Z log,Z (Ezar (O(X,D∪Z)/S ), P D ) := Ru(X,Z)/S∗ (Ecrys (O(X,D∪Z)/S ), P D )

122

2 Weight Filtrations on Log Crystalline Cohomologies

and we call it the preweight-filtered vanishing cycle zariskian complex of (X, D ∪ Z)/S with respect to D. Corollary 2.7.5. There exists a canonical isomorphism (2.7.5.1)

∼

log,Z log,Z (Ezar (O(X,D∪Z)/S ), P D ) −→ (Czar (O(X,D∪Z)/S ), P D ).

Proof. The left hand side of (2.7.5.1) is equal to log,Z Ru(X,Z)/S∗ Q∗(X,Z)/S (Ecrys (O(X,D∪Z)/S ), P D ) log,Z log,Z (O(X,D∪Z)/S ), P D ) = (Czar (O(X,D∪Z)/S ), P D ). =Ru(X,Z)/S∗ (CRcrys

Here we have used (2.5.4.1) for the last equality.

 

Corollary 2.7.6. The spectral sequence (2.7.1.4) is equal to (2.6.2.2) if we h−k,k make the renumbering Er−k,h+k = Er+1 (r ≥ 1). Proof. By [23, (1.4.8)], the spectral sequence (2.7.1.4) is obtained from the log,Z increasing filtration {τk CRcrys (O(X,D∪Z)/S )}k∈Z ; this filtration is equal to D log,Z {Pk CRcrys (O(X,D∪Z)/S )}k∈Z by (2.7.3). Hence (2.7.6) follows.   log,Z (O(X,D∪Z)/S ), P D ) is inCorollary 2.7.7. (1) The filtered complex (CRcrys dependent of the choice of the decompositions of D and Z by their smooth components. The spectral sequence (2.6.2.2) is also independent of the choice of them. (2) Let the assumptions be as in (2.5.6). Then the right hand sides of (2.5.6.1) and (2.5.6.2) are independent of the choice of the decompositions of D and Z by their smooth components.

Proof. The proof is obvious.

 

Corollary 2.7.8. The isomorphism (2.6.1.1) is independent of the choice of the decompositions of D and Z by their smooth components. Consequently the isomorphism (2.6.1.2) and the spectral sequences (2.6.2.1), (2.6.2.2), (2.6.3.1) and (2.6.3.2) are also independent of the choice. Proof. Since both hands of (2.6.1.1) is independent of the choice by (2.7.3) and (2.2.15), the problem is local. By (A.0.1) below, we may assume that two choices of the decompositions of D and Z by their smooth components are the same. Now the independence follows from the proof of (2.5.1) and the argument in (2.5.3).   The following is another proof of (2.5.7): Corollary 2.7.9. (2.5.7) and (2.5.8) hold.

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration

123

Proof. By (1.6.4.1), (2.3.10.1) and the cohomological descent, we have ∗ • Rπ  log (X,Z)/SRcrys∗ (Q(Xn ,Zn )/S L(Xn ,Zn )/S (ΩXn /S (log En ))n∈N )

=Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ). By the same proof as that for the formula (2.7.3.2), we also have ∗ • Rπ  log (X,Z)/SRcrys∗ ((Q(Xn ,Zn )/S L(Xn ,Zn )/S (ΩXn /S (log En )),

Q∗(Xn ,Zn )/S PkDn )n∈N ) • =(Rπ  log (X,Z)/SRcrys∗ (L(Xn ,Zn )/S (ΩXn /S (log En ))n∈N ), τ ).

 

Hence we have (2.5.7) and (2.5.8). We shall use the following for the preweight-filtered K¨ unneth formula:

Proposition 2.7.10. Assume that X is quasi-compact. Then the filtered log,Z complex (Ecrys (O(X,D∪Z)/S ), P D ) is bounded. Proof. By (2.3.11), R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) is bounded. Hence (2.7.10) immediately follows.   Remark 2.7.11. In this remark we show an unexpected nonequality (2.7.11.1) (k) (k) Rk (X,D)/S∗ (O(X,D)/S ) = acrys∗ (OD(k) /S ⊗Z
crys (D/S))

(k ∈ N)

in general. More specially, in this remark, we prove that the natural morphism (2.7.11.2)

OX/S −→ (X,D)/S∗ (O(X,D)/S ) = R0 (X,D)/S∗ (O(X,D)/S )

is not surjective in general if p = 0 on S0 . ⊂ Let (X, D) −→ (X , D) be an exact closed immersion into a smooth scheme with a relative SNCD over S. Let ι : LX/S (Ω1X /S ) −→ LX/S (Ω1X /S (log D)) be a natural morphism of OX/S -modules, and let d : LX/S (OX ) −→ LX/S (Ω1X /S ) be the natural boundary morphism. By the crystalline Poincar´e lemma and the Poincar´e lemma of a vanishing cycle sheaf ((2.3.10)), we have the following: (2.7.11.3) OX/S ⏐ ⏐ 

=

−−−−→

Ker(d : LX/S (OX ) −→ LX/S (Ω1X /S )) ⏐ ⏐ 

=

(X,D)/S∗ (O(X,D)/S ) −−−−→ Ker(ι ◦ d : LX/S (OX ) −→ LX/S (Ω1X /S (log D))) Consider the following commutative diagram of exact sequences:

124

2 Weight Filtrations on Log Crystalline Cohomologies

0 −−−−→

0 ⏐ ⏐ 

−−−−→ LX/S (OX ) ⏐ ⏐ d

LX/S (OX ) −−−−→ 0 ⏐ ⏐ ι◦d

0 −−−−→ Ker ι −−−−→ LX/S (Ω1X /S ) −−−−→

Im ι

−−−−→ 0.

Hence, by the snake lemma and (2.7.11.3), we obtain the following exact sequence (2.7.11.4) 0 −→ OX/S −→ (X,D)/S∗ (O(X,D)/S ) −→ Ker(ι) −→ Coker(d). Now set X := SpecS (OS [x]) and let D be a relative smooth divisor on X defined by an equation x = 0. Set (X, D) := (X , D) ×S S0 . In this case, Coker(d) = 0 by the crystalline Poincar´e lemma. Hence, to prove that (2.7.11.2) is not an isomorphism in this case, it suffices to prove that Ker(ι) = 0. Set A0 := OS0 [x, y]/(xy). Let f : A0 −→ OS0 [x] be a morphism of sheaves of rings over OS0 defined by equations f (x) = x and f (y) = 0. be the PD-envelope of A0 with respect to Ker(f ). Let δ be the PDLet APD 0 −→ OS0 [x] be the induced morphism structure on Ker(f ) and let f PD : APD 0 of sheaves of rings over OS0 by f . Set T := SpecS (APD 0 ). Then f induces a ⊂

0

PD closed immersion X −→ T ; the triple (X, T, δ) is an object of (X/S)crys . Let g : APD 0 ⊗OS0 OS0 [x] −→ OS0 [x] be a morphism of sheaves of rings over OS0 defined by g(s ⊗ t) := f PD (s)t (s ∈ APD 0 , t ∈ OS0 [x]) and let B be the ⊗ O [x] with respect to Ker(g). Then, by the proof PD-envelope of APD OS0 S0 0 of [11, (6.10)], the value LX/S (Ω1X /S )T of LX/S (Ω1X /S ) at T is given by the following formula LX/S (Ω1X /S )T = B ⊗OS [x] OS [x]dx = Bdx, while the value LX/S (Ω1X (log D))T is given by the following formula LX/S (Ω1X /S (log D))T = Bd log x. Let ιT : LX/S (Ω1X /S )T −→ LX/S (Ω1X /S (log D))T be the value of ι at T . Then ιT (dx) = (1 ⊗ x)d log x. To prove that ιT is not injective, it suffices to prove that a morphism B −→ B given by multiplication by 1 ⊗ x is not injective. Here we denote the image of a local section s of APD 0 ⊗OS0 OS0 [x] in B by the same symbol s by abuse of notation. We check (A) y ⊗ 1 = 0 in B and (B) (1 ⊗ x)p (y ⊗ 1) = 0 in B. First we check (A). Consider the following commutative diagram

2.7 The Vanishing Cycle Sheaf and the Preweight Filtration

A0 ⏐ ⏐ 

125

f

−−−−→ OS0 [x] ⏐ ⏐ 

OS0 [y] −−−−→ OS0 , where the vertical morphisms are defined by sending x to 0 and the lower horizontal morphism is defined by sending y to 0. By taking the PD-envelopes with respect to the kernels of the horizontal morphisms, we obtain the following commutative diagram:

(2.7.11.5)

APD 0 ⏐ ⏐ 

f PD

−−−−→ OS0 [x] ⏐ ⏐ 

OS0 y −−−−→ OS0 . ⊗OS0 Denote by ϕ the left vertical morphism in (2.7.11.5) and let ψ : APD 0 OS0 [x] −→ OS0 y be a morphism defined by ψ(s⊗t) := ϕ(s)·(t mod xOS0 [x]) (s ∈ APD 0 , t ∈ OS0 [x]). Then the diagram (2.7.11.5) gives the following commutative diagram g

APD 0 ⊗OS0 OS0 [x] −−−−→ OS0 [x] ⏐ ⏐ ⏐ ⏐ ψ  OS0 y

−−−−→

OS0

and then the following commutative diagram:

(2.7.11.6)

B ⏐ ⏐ 

−−−−→ OS0 [x] ⏐ ⏐ 

OS0 y −−−−→ OS0 . Since the image of y ⊗ 1 ∈ B by the left vertical morphism in (2.7.11.6) is equal to y ∈ OS0 y, y ⊗ 1 = 0 in B. Next we check (B). It is clear that 1 ⊗ x − x ⊗ 1 ∈ B is a local section of the PD-ideal sheaf of B. Hence we have the following equalities in B (1 ⊗ x)p (y ⊗ 1) = xp y ⊗ 1 + (1 ⊗ xp − xp ⊗ 1)(y ⊗ 1) = 0 + (1 ⊗ x − x ⊗ 1)p (y ⊗ 1) = p!(1 ⊗ x − x ⊗ 1)[p] (y ⊗ 1) = 0 because p = 0 in B. Now we have proved that the morphism (2.7.11.2) is not an isomorphism in general.

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2 Weight Filtrations on Log Crystalline Cohomologies

We also remark the following.  Rcrys ((2.7.1)), (O By the p-adic purity in (X/S) X/S )X = ((X,D)/S∗ (O(X,D) /S ))X . Hence the exact sequence (2.7.11.4) tells us that (X,D)/S∗ (O(X,D)/S ) is not a crystal of OX/S -modules in general. Remark 2.7.12. (1) Let (X, D) be a smooth analytic variety with (not necessarily simple) NCD over the complex number field. Let U be the complement ⊂
(0) be the norof D in X and let j be the natural inclusion U −→ X. Let D (k)
malization of D and for a positive integer k, define D in the way described
(k) −→ X be the natural morphism. Then,
(0) . Let
a(k) : D in (2.2.15) from D in [23, (3.1.8)], Deligne has proved that (2.7.12.1)

(Ω•X/C (log D), τ ) −→ (Ω•X/C (log D), P )

is a quasi-isomorphism by using the Poincar´e residue isomorphism and the Poincar´e lemma Res

• grP a∗ (Ω•D
(k) /C {−k} ⊗Z

(k) (D/C)(−k)) k ΩX/C (log D) −→
∼

(k)

(k)


(k) (D/C)(−k)), =
a∗ (CD
(k) {−k} ⊗Z

(k) (Since we have used the where

(k) (D/C) is the orientation sheaf of D notation  as a forgetting log morphism, we cannot use the notation  in [23]). Note that, in (2.7.1), (2.7.3) and (2.7.12.1), the graded pieces grP k is isomorphic to the complex which consists of one component; this property is a key point for (2.7.1) and the quasi-isomorphism (2.7.12.1). It is reasonable to expect that, if D is an SNCD, if we use the log infinitesimal topos and if we develop analogous theory for this topos by the same method as that in this book, we will be able to prove that (2.7.12.2)

(k)

Rk ∗ (OX/C ) = a∗ (OD(k) /C ⊗Z
(k) (D/C)(−k)),

 inf is the forgetting log morphism of infin log −→ (X/C) where  : (X/C) inf  log itesimal topoi, OX/C (resp. O (k) ) is the structure sheaf in (X/C) D

/C

inf

(k) /C) ), a(k) :=

(k) −→ X and
(k) (D/C) := (resp. (D
a(k) : D(k) = D inf (k)

(D/C). (2) The morphism (2.7.2.1) is not a filtered isomorphism in general. Indeed, if it were so, we would have the following contradiction. Assume that Z = ∅ and that it were an isomorphism. Then, by applying RuX/S∗ to (2.7.3.2), we would have

2.8 Boundary Morphisms

(2.7.12.3)

127

(Czar (O(X,D)/S ), P ) = RuX/S∗ (Ecrys (O(X,D)/S ), P ) = RuX/S∗ (R(X,D)/S∗ (O(X,D)/S ), τ ) = (Ru(X,D)/S∗ (O(X,D)/S ), τ ) = (Czar (O(X,D)/S ), τ ).

Here the first equality follows from (2.7.5.1). The third equality follows from our assumption. The fourth equality follows from (2.4.7.3). However it is practically well-known that the equality (2.7.12.3) does not hold in general. Indeed, let κ be a field of characteristic p > 0 and let (X, D) be a smooth scheme with an SNCD over κ. Assume that S = S0 = Spec(κ). Then (2.7.12.3) is an isomorphism (Ω•X/κ (log D), τ ) = (Ω•X/κ (log D), P ). If we take X := A1κ , D: the origin of X and k = 0, we have a contradiction. Hence (2.7.2.1) is not a filtered isomorphism in general.

2.8 Boundary Morphisms In this section we define the log cycle class of a smooth divisor which intersects the log locus transversally (cf. [29, §2]). As an application, we give the description of the boundary morphism between the E1 -terms of the spectral sequence (2.6.2.2). Let f : (X, Z) −→ S0 be a smooth scheme with a relative SNCD over a scheme S0 . Let D be a smooth divisor on X which intersects Z transversally over S0 ; for a decomposition ∆ = {Zµ }µ of Z by smooth components of Z, ∆(D) := {D, Zµ }µ is a decomposition of D ∪ Z by smooth components of D ∪ Z. The closed subscheme Z|D := Z ∩ D in D is a relative SNCD on D/S0 ; ∆|D := {Zµ |D }µ be a decomposition of Z|D by smooth ⊂ components of Z|D . Let a : (D, Z|D ) −→ (X, Z) be the natural closed im
zar , OD ) −→ (X
zar , OX ) be the induced mormersion over S0 . Let azar : (D log phism of Zariski ringed topoi. Let a : (((D,
Z|D )/S)log , O(D,Z| )/S ) −→ crys

crys

D


(((X, Z)/S)log crys , O(X,Z)/S ) be also the induced morphism of log crystalline ringed topoi. Let (2.8.0.1) (1) ResD : Ω•X/S0 (log(D ∪ Z)) −→ azar∗ (Ω•D/S0 (log(Z|D )) ⊗Z
zar (D/S0 )){−1} be the Poincar´e residue morphism with respect to D/S0 . Then we have the following exact sequence: (2.8.0.2)

0 −→ Ω•X/S0 (log Z) −→ Ω•X/S0 (log(D ∪ Z))

128

2 Weight Filtrations on Log Crystalline Cohomologies ResD

(1) −→ azar∗ (Ω•D/S (log(Z|D )) ⊗Z
zar (D/S0 )){−1} −→ 0.

Let (S, I, γ) and S0 be as in §2.4. As in §2.4, by abuse of notation, we also ⊂ denote by f the composite morphism (X, Z) −→ S0 −→ S. As in §2.4, we have the following data:  r (2.8.0.3): An open covering X = i0 ∈I0 Xi0 with Xi = s=0 Xis (i = (i0 , . . . , ir )). The family {(Xi , Di ∪ Zi )}i∈I (Di := D ∩ Xi , Zi := Z ∩ Xi ) of log schemes form a diagram of log schemes over (X, D ∪ Z), which we denote by (X• , D• ∪ Z• ). That is, (X• , D• ∪ Z• ) is a contravariant functor I o −→ {smooth schemes with relative SNCD’s over S0 which are augmented to (X, D ∪ Z)}. We have a diagram ∆• (D• ) of a decomposition of D• ∪ Z• by a diagram of smooth components of D• ∪ Z• . ⊂

(2.8.0.4): A family (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ) (• ∈ I) of admissible immersions into a diagram of smooth schemes with relative SNCD’s over S with respect to ∆• (D• ). Let b• : D• −→ X• be a diagram of the natural closed immersions. By using the Poincar´e residue isomorphism with respect to D• , we have the following exact sequence ([29, §2]): (2.8.0.5)

0 −→ Ω•X• /S (log Z• ) −→ Ω•X• /S (log(D• ∪ Z• )) −→ Res

(1) (D• /S)){−1} −→ 0. b•zar∗ (Ω•D• /S (log(Z• |D• )) ⊗Z
zar

Let L(X• ,Z• )/S (resp. L(D• ,Z• |D• )/S ) be the log linearization functor with ⊂

respect to the diagram of the locally closed immersions (X• , Z• ) −→ ⊂ (X• , Z• ) (resp. (D• , Z• |D• ) −→ (D• , Z• |D• )). By (2.2.12) and (2.2.16), L(X• ,Z• )/S b•zar∗ = alog crys•∗ L(D• ,Z• |D• )/S . Hence we have the following exact sequence by (2.2.17) (2) and (2.2.21.2): (2.8.0.6) 0 −→ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log Z• )) −→ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))) • −→ Q∗(X• ,Z• )/S alog crys•∗ (L(D• ,Z• |D• )/S (ΩD• /S (log(Z• |D• ))) (1)log (D• /S; Z• )){−1}) −→ 0. ⊗Z
crys log log and π(D,Z| of ringed topoi in Recall the morphisms π(X,Z)/Scrys D )/Scrys (2.4.7.4) for the case D = φ and (2.6.1.3). By (1.6.0.23) we have the following triangle

2.8 Boundary Morphisms

129

(2.8.0.7) log −→Q∗(X,Z)/S Rπ(X,Z)/Scrys∗ L(X• ,Z• )/S (Ω•X• /S (log Z• )) −→ log Q∗(X,Z)/S Rπ(X,Z)/Scrys∗ L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))) −→ log • Q∗(X,Z)/S alog crys∗ Rπ(D,Z|D )/Scrys∗ L(D• ,Z• |D• )/S (ΩD• /S (log(Z• |D• )) +1

(1)log ⊗Z
crys (D• /S; Z• )){−1} −→ · · · .

By (2.2.7), (2.3.10.1) and by the cohomological descent, we have the following triangle: (2.8.0.8) −→ Q∗(X,Z)/S (O(X,Z)/S ) −→ Q∗(X,Z)/S R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) −→ (1)log Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z)){−1} −→ · · · . +1

Using the Convention (4), we have the boundary morphism (2.8.0.9) (1)log d : Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z))){−1} −→ Q∗(X,Z)/S (O(X,Z)/S )[1] in D+ (Q∗(X,Z)/S (O(X,Z)/S )). Equivalently, we have the following morphism (2.8.0.10) (1)log d : Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z))) −→ Q∗(X,Z)/S (O(X,Z)/S )[1]{1} in D+ (Q∗(X,Z)/S (O(X,Z)/S )). Set (2.8.0.11)

GD/(X,Z) := −d.

and call GD/(X,Z) the Gysin morphism of D. Then we have a cohomology class (2.8.0.12) c(X,Z)/S (D) := GD/(X,Z) ∈Ext 0Q∗

(X,Z)/S (O(X,Z)/S )

(Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S

(1)log ⊗Z
crys (D/S; Z)), Q∗(X,Z)/S (O(X,Z)/S )[1]{1}). (1)log

Since
crys (D/S; Z) is canonically isomorphic to Z and since there exists a natural morphism Q∗(X,Z)/S (O(X,Z)/S ) −→ Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ), we have a cohomology class

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2 Weight Filtrations on Log Crystalline Cohomologies

c(X,Z)/S (D) ∈ Ext 0Q∗

(X,Z)/S

∗ ∗ (O(X,Z)/S ) (Q(X,Z)/S (O(X,Z)/S ), Q(X,Z)/S (O(X,Z)/S )[1]{1})

=: Q∗(X,Z)/S H2log-crys ((X, Z)/S).

As usual, if Z = ∅, we denote GD/(X,Z) and c(X,Z)/S (D) simply by GD/X and cX/S (D), respectively. Remark 2.8.1. (cf. [35, (1.6)]) Let t• = 0 be a local equation of D• in X• . If we use a Poincar´e residue morphism Ω•X• /S (log(D• ∪ Z• )) d log t• ∧ ω• −→ ω• |D• ∈b•zar∗ (Ω•D• /S (log(Z• |D• )) (1) (D• /S; Z• ))[−1] ⊗Z
zar

instead of the Poincar´e residue morphism in (2.8.0.5), then we have a Gysin morphism (1)log GD/(X,Z) : Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z))[−1]

−→ Q∗(X,Z)/S (O(X,Z)/S )[1]. Here we have used the Convention (4). Hence, by the Convention (2), we have a Gysin morphism (1)log (2.8.1.1) GD/(X,Z) : Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z))[−2]

−→ Q∗(X,Z)/S (O(X,Z)/S ). However we do not use this Gysin morphism in this book. Proposition 2.8.2. The morphism GD/(X,Z) and the class c(X,Z)/S (D) are independent of the data (2.8.0.3) and (2.8.0.4). Proof. Use notations in §2.5. Assume that we are given two data in (2.8.0.3) and two data in (2.8.0.4). Because the question is local, we may assume that the two admissible immersions are admissible immersions with respect to the same decompositions of D and Z by their smooth components. As in §2.5 we have two morphisms , Z•• )/S)log η(X,Z)/S : (((X••
crys , O(X•• ,Z•• )/S ) , Z• )/S)log −→ (((X•
crys , O(X• ,Z• )/S ), and log
η(D,Z|D )/S : (((D•• , Z •• |D•• )/S)crys , O(D•• ,Z•• |D•• )/S )

Z• |D• )/S)log −→ (((D• ,
crys , O(D• ,Z• |D• )/S ) of ringed topoi. Then we have the following commutative diagram of triangles:

2.8 Boundary Morphisms

131

(2.8.2.1) −−−−→ Q∗(X• ,Z• )/S Rη(X,Z)/S∗ L(X•• ,Z•• )/S (Ω•X•• /S (log Z•• )) −−−−→  ⏐ ⏐ −−−−→

Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log Z• ))

−−−−→

Q∗(X• ,Z• )/S Rη(X,Z)/S∗ L(X•• ,Z•• )/S (Ω•X•• /S (log(D•• ∪ Z•• ))) −−−−→  ⏐ ⏐ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))) Q∗(X• ,Z• )/S alog crys•∗ Rη(D,Z|D )/S∗ L(D•• ,Z•• |D•• )/S  ⏐ ⏐

(Ω•D•• /S (log(Z•• |D•• )){−1})

• Q∗(X• ,Z• )/S alog crys•∗ L(D• ,Z• |D• )/S (ΩD• /S (log(Z• |D• )){−1})

−−−−→ +1

−−−−→

+1

−−−−→ .

By the proof of (2.5.3), the three vertical morphisms above are isomorphisms. Hence (2.8.2) follows.   Remark 2.8.3. We can also construct c(X,Z)/S (D) by using the vanishing cycle sheaf as follows.

log ∪ Z)/S)log Let (X,D∪Z,Z)/S : ((X, D crys −→ ((X, Z)/S)crys be the forgetting log morphism along D ((2.3.2)). By (2.3.2.9), there exists a natural morphism (2.8.3.1)

O(X,Z)/S −→ R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S )

in D+ (O(X,Z)/S ). Let RΓD (O(X,Z)/S ) be the mapping fiber of (2.8.3.1). Then we have a triangle (2.8.3.2) +1 −→ RΓD (O(X,Z)/S ) −→ O(X,Z)/S −→ R(X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) −→ . i Set HD (O(X,Z)/S ) := Hi (RΓD (O(X,Z)/S )) (i ∈ Z). Then we have the following exact sequence i (2.8.3.3) · · · −→ HD (O(X,Z)/S ) −→ Hi (O(X,Z)/S )

−→ Ri (X,D∪Z,Z)/S∗ (O(X,D∪Z)/S ) −→ · · · . Here we have used the Convention (4) and (5). By (2.7.1), we have (2.8.3.4)

i Q∗(X,Z)/S HD (O(X,Z)/S )  (1)log Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z)) = 0

(i = 2), (i = 2).

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2 Weight Filtrations on Log Crystalline Cohomologies

Let E • be a representative of Q∗(X,Z)/S RΓD (O(X,Z)/S ). Then we have an isomorphism ∼ τ2 E • −→ E • and we can take an isomorphism (2.8.3.5)

∼

2 τ2 E • −→ Q∗(X,Z)/S HD (O(X,Z)/S ){−1}[−1].

Therefore we have a canonical isomorphism Q∗(X,Z)/S RΓD (O(X,Z)/S ) (1)log =Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z)){−1}[−1].

Since there exists a natural morphism RΓD (O(X,Z)/S ) −→ O(X,Z)/S by the definition of RΓD (O(X,Z)/S ), we have a canonical morphism (2.8.3.6)

(1)log Q∗(X,Z)/S alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z)){−1}[−1]

−→ Q∗(X,Z)/S (O(X,Z)/S ). By (2.8.0.8), we see that the morphism (2.8.3.6) is equal to −GD/(X,Z) . If we take the canonical isomorphism (2.8.3.7)

∼

2 τ2 E • −→ Q∗(X,Z)/S HD (O(X,Z)/S )[−2].

instead of (2.8.3.5), we obtain the Gysin morphism (2.8.1.1) again. Proposition 2.8.4. Let u : (S  , I  , γ  ) −→ (S, I, γ) be a morphism of PDschemes. Set S0 := SpecS  (OS  /I  ). Let h : Y −→ S0 be a smooth morphism of schemes fitting into the following commutative diagram g

Y −−−−→ ⏐ ⏐ h

X ⏐ ⏐f 

S0 −−−−→ S0 . Set E := D ×X Y and W := Z ×X Y . Assume that E ∪ W is a relative SNCD ⊂ on Y over S0 . Let b : (E, W |E ) −→ (Y, W ) be a natural closed immersion of −1 Ru(X,Z)/S∗ (c(X,Z)/S (D)) in (2.8.0.12) by log schemes. Then the image of gzar the natural morphism −1 (1)log gzar Ext 0f −1 (OS ) (Ru(X,Z)/S∗ alog crys∗ (O(D,Z|D )/S ⊗Z
crys (D/S; Z)),

Ru(X,Z)/S∗ (O(X,Z)/S )[1]{1}) −→

Ext 0h−1 (OS ) (Ru(Y,W )/S∗ blog crys∗ (O(E,W |E )/S 

(1)log ⊗Z
crys (E/S  ; W )),

Ru(Y,W )/S∗ (O(Y,W )/S )[1]{1})

2.8 Boundary Morphisms

133

is equal to Ru(Y,W )/S∗ (c(Y,W )/S  (E)). Proof. (2.8.4) immediately follows from the functoriality of the construction given in (2.8.3).   Finally we prove that the boundary morphism d•• 1 of (2.6.2.2) is expressed by summation of Gysin morphisms with signs. Henceforth D denotes a (not necessarily smooth) relative SNCD on X over S0 which meets Z transversally. First, fix a decomposition {Dλ }λ∈Λ of D by smooth components of D over S0 . Assume that D{λ0 ,...,λk−1 } = ∅. j , . . . , λk−1 }, Dλ := D{λ ,...,λ } , Set λ := {λ0 , . . . , λk−1 }, λj := {λ0 , . . . , λ 0 k−1 and Dλj := D{λ0 ,...,λ j ,...,λk−1 } for k ≥ 2 and Dλ0 := X. Here  means the elimination. Then Dλ is a smooth divisor on Dλj over S0 . Let ⊂

λ

ιλj : (Dλ , Z|Dλ ) −→ (Dλj , Z|Dλj ) be the closed immersion. Set log
λcrys (D/S; Z) :=
λlog (D/S; Z) 0 ···λk−1 crys

and
λlog (D/S; Z) :=
log j crys

 j ···λk−1 crys λ0 ···λ

(D/S; Z).

By (2.8.0.11) we have a morphism λ

Gλj := GDλ /(Dλj ,Z|Dλ ) :

(2.8.4.1)

j

λj log Q∗(Dλ ,Z|D )/S ιλcrys∗ (O(Dλ ,Z|Dλ )/S j λ j

−→ Q∗(Dλ

j

(1)log

⊗Z
λj crys (D/S; Z))

,Z|Dλ )/S (O(Dλj ,Z|Dλj )/S )[1]{1}. j

We fix an isomorphism ∼

log
λlog (D/S; Z) ⊗Z
λlog (D/S; Z) −→
λcrys (D/S; Z) j crys j crys

(2.8.4.2)

by the following morphism j · · · λk−1 ) −→ (−1)j (λ0 · · · λk−1 ). (λj ) ⊗ (λ0 · · · λ log (D/S; Z)⊗Z
λlog (D/S; Z) with
λcrys (D/S; Z) by this We identify
λlog j crys j crys isomorphism. We also have the following composite morphism

(2.8.4.3) λ (−1)j Gλj : Q∗(Dλ

j

Q∗(Dλ

j

λj log ,Z|Dλ )/S ιλcrys∗ (O(Dλ ,Z|Dλ )/S j

λj log ,Z|Dλ )/S ιλcrys∗ (O(Dλ ,Z|Dλ )/S j

⊗Z
λlog (D/S; Z) ⊗Z
λlog (D/S; Z)) j crys j crys

λ

Gλj ⊗1

−→ Q∗(Dλ

j

,Z|Dλ )/S (O(Dλj ,Z|Dλj )/S j

∼

log ⊗Z
λcrys (D/S; Z)) −→

⊗Z
λlog (D/S; Z))[1]{1} j crys

134

2 Weight Filtrations on Log Crystalline Cohomologies

defined by λ j · · · λk−1 )”. “x ⊗ (λ0 · · · λk−1 ) −→ (−1)j Gλj (x) ⊗ (λ0 · · · λ

(2.8.4.4)

The morphism (2.8.4.3) induces a morphism of log crystalline cohomologies: (2.8.4.5) λ log (−1)j Gλj : Rh−k f(Dλ ,Z|Dλ )/S∗ (O(Dλ ,Z|Dλ )/S ⊗Z
λcrys (D/S; Z)) −→ Rh−k+2 f(Dλj ,Z|Dλ

j

)/S∗ (O(Dλj ,Z|Dλ )/S j

⊗Z
λlog (D/S; Z)). j crys λ

Here we have used the Convention (6). If D{λ0 ,...,λk−1 } = ∅, set (−1)j Gλj : = 0. Denote by aλ (resp. aλj ) the natural exact closed immersion (Dλ , Z|Dλ ) ⊂

⊂

−→ (X, Z) (resp. (Dλj , Z|Dλj ) −→ (X, Z)). : E1−k,h+k −→ E1−k+1,h+k be the boundary Proposition 2.8.5. Let d−k,h+k 1 k−1 λ morphism of (2.6.2.2). Set G := {λ0 ,...,λk−1 | λi =λj (i=j)} j=0 (−1)j Gλj . Then d−k,h+k = −G. 1

Proof. (cf. [64, 4.3]) Assume that we are given the data (2.4.0.1) and (2.4.0.2) for D ∪ Z. Consider the following exact sequence D• Q∗ (X ,D• )/S P (Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D•

0 −→ grk−1 •

∪ Z• )))) −→

D• (Q∗(X• ,D• )/S PkD• /Q∗(X• ,D• )/S Pk−2 )(Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• )))) D• Q∗ (X• ,D• )S P (Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D•

−→ grk

∪ Z• )))) −→ 0.

Then the boundary morphism d−k,h+k is induced by the boundary morphism 1 of the following triangle D• Q∗ (X ,D• )/S P

log −→ Rπ(X,Z)/SRcrys∗ grk−1 •

(Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))))

log D• −→ Rπ(X,Z)/SRcrys∗ ((Q∗(X• ,D• )/S PkD• /Q∗(X• ,D• )/S Pk−2 )

(Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• ))))) −→ D• Q∗ (X• ,D• )S P

log grk Rπ(X,Z)/SRcrys∗

+1

(Q∗(X• ,D• )/S L(X• ,Z• )/S (Ω•X• /S (log(D• ∪ Z• )))) −→ .

Here we have used the Convention (4). Assume that D(λ;•) := D(λ0 ;•) ∩ · · · ∩ D(λk−1 ;•) = ∅. Set D(λj ;•) := D(λ0 ;•) ∩ · · · ∩ D(λj−1 ;•) ∩ D(λj+1 ;•) ∩ · · · ∩ D(λk−1 ;•) . We use a shorter notation
λzar (D• /S; Z• ) for a zariskian orientation sheaf
λ0 ···λk−1 zar (D• /S; Z• ) and so on as for crystalline orientation sheaves. The Poincar´e residue morphisms with respect to Dλj (0 ≤ j ≤ k − 1) and Dλ induce the following morphisms

2.8 Boundary Morphisms

135 D•

P • • ResD λj : grk−1 ΩX• /S (log(D• ∪ Z• )) −→

Ω•D(λ

j ;•)

and

/S (log Z• |D(λj ;•) ){−(k P • ResD λ : grk

D•

− 1)} ⊗Z
λj zar (D• /S; Z• )

Ω•X• /S (log(D• ∪ Z• )) −→

Ω•D(λ;•) /S (log Z• |D(λj ;•) ){−k} ⊗Z
λzar (D• /S; Z• ). As in (2.8.4.2), we fix an isomorphism (2.8.5.1)

∼


λj zar (D• /S; Z• ) ⊗Z
λj zar (D• /S; Z• ) −→
λzar (D• /S; Z• )

by the following morphism j · · · λk−1 ) −→ (−1)j (λ0 · · · λk−1 ). (λj ) ⊗ (λ0 · · · λ We identify
λj zar (D• /S; Z• ) ⊗Z
λj zar (D• /S; Z• ) with
λzar (D• /S; Z• ) by this isomorphism. Let Resj be the Poincar´e residue morphism Ω•D(λ

(2.8.5.2)

j ;•)

/S (log(D(λ;•)

∪ Z• |D(λj ;•) )) −→

Ω•D(λ;•) /S (log Z• |D(λ;•) ){−1} ⊗Z
λj zar (D• /S; Z• ) with respect to the divisor D(λ;•) on D(λj ;•) . Then we have a composite morphism (−1)j Resj : Ω•D(λ

j ;•)

/S (log(D(λ;•)

∪ Z• |D(λj ;•) )) ⊗Z
λj zar (D• /S; Z• )

−→ Ω•D(λ;•) /S (log Z• |D(λ;•) ){−1} ⊗Z
λj zar (D• /S; Z• ) ⊗Z
λj zar (D• /S; Z• ) ∼

−→ Ω•D(λ;•) /S (log Z• |D(λ;•) ){−1} ⊗Z
λzar (D• /S; Z• ). defined by (2.8.5.3)

j · · · λk−1 ) −→ (−1)j ResD• (x) ⊗ (λ0 · · · λk−1 ). x ⊗ (λ0 · · · λ λj

It is easy to check that (−1)j Resj is well-defined. The morphism (−1)j Resj induces a morphism (2.8.5.4) L(X• ,Z• )/S ((−1)j Resj ) : L(X• ,Z• )/S (Ω•D(λ

j ;•)

/S (log(D(λ;•)

∪ Z• |D(λj ;•) )) ⊗Z
λlog (D/S; Z• )) j crys

log −→ L(X• ,Z• )/S (Ω•D(λ;•) /S (log Z• |D(λ;•) ){−1} ⊗Z
λcrys (D• /S; Z• )). • As in [64, 4.3], the morphism Q∗(X• ,D• )/S L(X• ,Z• )/S (ResD λj ) uniquely extends

• to a morphism Q∗(X• ,D• )/S L(X• ,Z• )/S (ResD λj ,λ ) fitting into the following commutative diagram:

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2 Weight Filtrations on Log Crystalline Cohomologies

(2.8.5.5) 0 − −−−−−− →

Q∗ P D• (X• ,D• )/S gr Q∗ L (Ω• (log(D• ∪ Z• ))) (X• ,D• )/S (X• ,Z• )/S X• /S k−1 D ⏐ Q∗ L (Res • )⏐ (X• ,D• )/S (X• ,Z• )/S λj 

log )){−(k − 1)} ⊗Z  0 − −−−−−− → Q∗ L (Ω• (D• /S; Z• ) (log Z• |D (X• ,D• )/S (X• ,Z• )/S D(λ ;•) /S λj crys (λj ;•) j − −−−−−− →

− −−−−−− → D D (Q∗ P • /Q∗ P • )Q∗ L (Ω• (D• ∪ Z• )))) (X• ,D• )/S k (X• ,D• )/S k−2 (X• ,D• )/S (X• ,Z• )/S X• /S ⏐ D (Res • )⏐ Q∗ L (X• ,D• )/S (X• ,Z• )/S λj ,λ  log Q∗ L (Ω• ))){−(k − 1)} ⊗Z  (log(D(λ;•) ∪ Z• |D (D• /S; Z• ) (X• ,D• )/S (X• ,Z• )/S D(λ ;•) /S λj crys (λj ;•) j − −−−−−− →

Q∗ L ((−1)j Resj ) (X• ,D• )/S (X• ,Z• )/S − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− → Q∗ P D• (X• ,D• )/S gr (Ω• (log D• )) Q∗ L X• /S (X• ,D• )/S (X• ,Z• )/S k

− −−−−−− → 0

⏐ D ⏐ Q∗ L (Res • ) (X• ,D• )/S (X• ,Z• )/S λ log Q∗ L (Ω• (log Z• |D ;•) )){−k} ⊗Z  (D• /S; Z• ) − −−−−−− → 0. (X• ,D• )/S (X• ,Z• )/S D(λ;•) /S λcrys (λ • Here the morphism ResD λj ,λ is defined by a formula

j •  ResD λj ,λ (yd log xλ0 · · · d log xλk−1 ) = (−1) yd log xλj ⊗ (λ0 · · · λj · · · λk−1 ),

where xλi = 0 (xλi ∈ OX• ) is a local equation of D(λi ;•) in X• and y is a local section of Ω•X• (log Z• ) (the formula R´esIIq (ω) = α ∧ dxiq /xiq |DIq in [64, p. 323, l. -9] have to be replaced by R´esIIq (ω) = (−1)q−1 α ∧ dxiq /xiq |DIq ). By the formulas (2.8.4.4) and (2.8.5.3), by the definition of the Gysin morphism for smooth divisors ((2.8.0.11)) and by the Convention (4) and (5), we see λ that (−1)j (−Gλj ) is the boundary morphism of the lower exact sequence. Hence we obtain (2.8.5).  

2.9 The Functoriality of the Preweight-Filtered Zariskian Complex Let S0 , S and (X, D ∪ Z) be as in §2.4. In this section we prove the functolog,Z (O(X,D∪Z)/S ), P D ); (2.7.3) is indispensable for the proof of riality of (Czar the functoriality.

2.9 The Functoriality of the Preweight-Filtered Zariskian Complex

137

Let (S  , I  , γ  ) be another PD-scheme satisfying the same conditions in the beginning of §2.4. Set S0 := SpecS  (OS  /I  ). Let u : (S, I, γ) −→ (S  , I  , γ  ) be a morphism of PD-schemes. Let u0 : S0 −→ S0 be the induced morphism by u. Let (X  , D ∪ Z  ) be a smooth scheme with a relative SNCD over S0 . Let g

(2.9.0.1)

(X, D ∪ Z) −−−−→ (X  , D ∪ Z  ) ⏐ ⏐ ⏐ ⏐   u

S0

−−−0−→

S0

be a commutative diagram of log schemes. Assume that the morphism g induces g(X,D) : (X, D) −→ (X  , D ) and g(X,Z) : (X, Z) −→ (X  , Z  ) over u0 : S0 −→ S0 . Let
log
 : ((X, D ∪ Z)/S)log crys −→ ((X, Z)/S)crys and

 ∪ Z  )/S  )log −→ ((X
 , Z  )/S  )log  : ((X  , D
crys crys

be the forgetting log morphisms along D and D , respectively. Theorem 2.9.1 (Functoriality). Let the notations be as above. Then the following hold: (1) There exists a canonical morphism (2.9.1.1)





log∗ log,Z g(X,Z)crys : (Ecrys (O(X  ,D ∪Z  )/S  ), P D ) log log,Z −→ Rg(X,Z)crys∗ (Ecrys (O(X,D∪Z)/S ), P D ).

(2) There exists a canonical morphism (2.9.1.2)  ∗ log,Z  log,Z gzar : (Czar (O(X  ,D ∪Z  )/S  ), P D ) −→ Rgzar∗ (Czar (O(X,D∪Z)/S ), P D ). Proof. (1): (1) is clear. (2): Let log
: (((X, D ∪ Z)/S)log gcrys crys ,O(X,D∪Z)/S )  ∪ Z  )/S  )log , O   −→ (((X  , D
(X ,D ∪Z  )/S  ) crys

be the morphism of log crystalline ringed topoi induced by g. Then we construct a desired morphism in the following way: 



log,Z (Czar (O(X  ,D ∪Z  )/S  ), P D ) 



log,Z (O(X  ,D ∪Z  )/S  ), P D ) = Ru(X  ,Z  )/S  ∗ (Ecrys

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2 Weight Filtrations on Log Crystalline Cohomologies

= Ru(X  ,Z  )/S  ∗ (R∗ (O(X  ,D ∪Z  )/S  ), τ ) log −→ Ru(X  ,Z  )/S  ∗ (R∗ Rgcrys∗ (O(X,D∪Z)/S ), τ ) log = Ru(X  ,Z  )/S  ∗ (Rg(X,Z)crys∗ R∗ (O(X,D∪Z)/S ), τ ) log (R∗ (O(X,D∪Z)/S ), τ ) −→ Ru(X  ,Z  )/S  ∗ Rg(X,Z)crys∗ log log,Z (Ecrys (O(X,D∪Z)/S ), P D ) = Ru(X  ,Z  )/S  ∗ Rg(X,Z)crys∗ log,Z = Rgzar∗ Ru(X,Z)/S∗ (Ecrys (O(X,D∪Z)/S ), P D ) log,Z = Rgzar∗ (Czar (O(X,D∪Z)/S ), P D ).

Here the first and the last equalities follow from (2.7.5.1); the first arrow is log ∗ induced by gcrys and the second arrow is obtained from (2.7.2).   Corollary 2.9.2. Let Ess ((X, D ∪ Z)/S) (resp. Ess ((X  , D ∪ Z  )/S  )) be the spectral sequence (2.6.2.2) (resp. (2.6.2.2) for (X  , D ∪ Z  )/S  ). Then the log∗ morphism gcrys induces a morphism (2.9.2.1)

log∗ : Ess ((X  , D ∪ Z  )/S  ) −→ Ess ((X, D ∪ Z)/S) gcrys

of spectral sequences.  

Proof. The proof is straightforward.

Let a(k) : (D(k) , Z  |D (k) ) −→ (X  , Z  ) be a natural morphism. Assume that g induces a morphism gD(k) : (D(k) , Z|D(k) ) −→ (D(k) , Z  |D (k) ) for any log∗ k ∈ N. By (2.6.1.1), (2.9.1) and (1.3.4.1), the morphism g(X,Z)crys induces the following morphism (2.9.2.2) log∗ grP k (g(X,Z)crys ) :

Ru(X  ,Z  )/S  ∗ a crys∗ (O(D (k) ,Z  | (k)log

D  (k)

)/S 

(k)log ⊗Z
crys (D /S  ; Z  )){−k} −→

log Ru(X  ,Z  )/S  ∗ a crys∗ RgD (k) crys∗ (O(D (k) ,Z| (k)log

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)){−k}.

log∗ In the following, we make the morphism grP k (g(X,Z)crys ) in (2.9.2.2) explicit in certain cases by using a notion which is analogous to the D-twist in [71]. Assume that the following two conditions hold:

(2.9.2.3): there exists the same cardinality of D and  of smooth components  D over S0 and S0 , respectively: D = λ∈Λ Dλ , D = λ∈Λ Dλ , where Dλ and Dλ are smooth divisors over S0 and S0 , respectively. (2.9.2.4): there exist positive integers eλ (λ ∈ Λ) such that eλ Dλ = g ∗ (Dλ ).

2.9 The Functoriality of the Preweight-Filtered Zariskian Complex

139

As in the previous section, set λ := {λ1 , . . . , λk } (λj ∈ Λ, (λi = λj (i = j))). Let aλ : (Dλ , Z|Dλ ) −→ (X, Z) and aλ : (Dλ , Z  |Dλ ) −→ (X  , Z  ) be natural morphisms. Consider the following direct factor of the morphism (2.9.2.2): (2.9.2.5) log∗ Ru(X  ,Z  )/S  ∗ a log λcrys∗ (gλcrys ) : log     Ru(X  ,Z  )/S  ∗ a log λcrys∗ (O(Dλ ,Z|D )/S  ⊗Z
λcrys (D /S ; Z )){−k} λ

log log −→ Ru(X  ,Z  )/S  ∗ a log λcrys∗ Rgλcrys∗ (O(Dλ ,Z|Dλ )/S ⊗Z
λcrys (D/S; Z)){−k}.

Proposition 2.9.3. Let the notations and the assumptions be as above. Let g(Dλ ,Z|Dλ ) : (Dλ , Z|Dλ ) −→ (Dλ , Z  |Dλ ) log∗ be the induced morphism by g. Then the morphism Ru(X  ,Z  )/S  ∗ (gλcrys ) in k log log∗  (2.9.2.5) is equal to ( j=1 eλj )Ru(X  ,Z  )/S  ∗ a λcrys∗ (g(Dλ ,Z|D )crys ) for k ≥ 0. λ k Here we define j=1 eλj as 1 for k = 0.

Proof. We may assume that k ≥ 1. Let us take affine open coverings X =      i0 ∈I0 Xi0 , X = i0 ∈I0 Xi0 of X, X by the same index set I0 satisfying  g(Xi0 ) ⊆ Xi0 (i0 ∈ I0 ) and let us form diagrams of log schemes (X• , D• ∪ Z• ) and (X• , D• ∪ Z• ) indexed by I as in (2.4.0.1). Then we have a morphism g• : (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ) of diagrams of log schemes over g. Next let us take log smooth lifts ⊂

⊂

(Xi0 , Di0 ∪ Zi0 ) −→ (Xi0 , Di0 ∪ Zi0 ), (Xi0 , Di0 ∪ Zi0 ) −→ (Xi0 , Di0 ∪ Zi0 ) for each i0 ∈ I0 and from these data, let us construct the diagrams of admissible immersions ⊂

⊂

(X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ), (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ) by the method explained in §2.4 before (2.4.1). Let g(X• ,Z• ) : (X• , Z• ) −→ (X• , Z• ) be the morphism induced by g• , which exists by assumption on g and let πzar be the morphism defined in (2.4.5.2). Then we have (2.9.3.1) D log,Z P D• • ΩX• /S (log(D• ∪ Z• ))), grP k Czar (O(X,D∪Z)/S ) = Rπzar∗ (OD• ⊗OX• grk (2.9.3.2)

grP k

D



log,Z Czar (O(X  ,D ∪Z  )/S )

=Rπzar∗ Rg(X• ,Z• )zar∗ (OD• ⊗OX  grP k •

 D•

Ω•X• /S (log(D• ∪ Z• ))),

140

2 Weight Filtrations on Log Crystalline Cohomologies ⊂

where D• (resp. D• ) denotes the log PD-envelope of (X• , Z• ) −→ (X• , Z• ) ⊂ (resp. (X• , Z• ) −→ (X• , Z• )). Because (Xi0 , Di0 ∪ Zi0 ) is log smooth over S  ⊂

and the exact closed immersion (Xi0 , Di0 ∪Zi0 ) −→ (Xi0 , Di0 ∪Zi0 ) is defined by the nil-ideal sheaf IOXi0 , there exists a morphism g
i0 : (Xi0 , Di0 ∪Zi0 ) −→ (Xi0 , Di0 ∪ Zi0 ) which is a lift of g|(Xi0 ,Di0 ∪Zi0 ) (cf. [11, N.B. in 5.27]). The family {
gi0 }i0 ∈I0 induces a morphism (2.9.3.3)

g
• : (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• )

of diagrams of log schemes by the universality of blow-up. Let 
 h(λ;•) : (D(λ;•) , Z• |D(λ;•) ) −→ (D(λ;•) , Z  |D(λ;•) )

be the induced morphism. (Here k   i=1 D(λi ;•) , where D(λi ;•) , D(λi ;•) For i0 ∈ I0 , Let x(j;i0 ) = 0  of D(λj ;i0 ) in Xi0 (resp. D(λ j ;i0 ) eλ

we put D(λ;•) :=

k i=1

 D(λi ;•) , D(λ;•) :=

are as in §2.4 before (2.4.1).) (resp. x(j;i0 ) = 0) be a local equation in Xi0 ) (1 ≤ j ≤ k). Then we have

g
i∗0 (x(j;i0 ) ) = u(j;i0 ) x(j;ij 0 ) for some unit u(j;i0 ) . For i = (i0 , ..., ir ) ∈ I, let us put x(j;i) := x(j;i0 ) , x(j;i) := x(j;i0 ) , u(j;i) := u(j;i0 ) . Then, by definition  of D(λj ;i) , D(λ (via the blow-up construction), x(j;i) = 0 (resp. x(j;i) = 0) j ;i)  is a local equation of D(λj ;i) in Xi (resp. D(λ in Xi ) (1 ≤ j ≤ k) and j ;i) eλ

j . So, for a local section ω = we have the equality g
i∗ (x(j;i) ) = u(j;i) x(j;i) 

 ad log x(1;i) · · · d log x(k;i) of PkD Ω•X  /S  (log(D ∪ Z  )) (a ∈ Ω•−k X  /S  (log Z )),  k we have g
i∗ (ω) = ( j=1 eλj )
gi∗ (a)d log x(1;i) · · · d log x(k;i) + ω  , where ω  ∈ Di Pk−1 Ω•Xi /S (log(Di ∪ Zi )). So, if we put

Ω•(λ;•) := Ω•D(λ;•) /S (log Z• |D(λ;•) ) ⊗Z
λzar (D• /S), Ω•(λ;•) := Ω•D

(λ;•) /S

 (log Z• |D(λ;•) ) ⊗Z
λzar (D• /S),

we have the following commutative diagram (the vertical arrows are Poincar´e residue morphisms with respect to Dλ and Dλ ): grP k

(2.9.3.4)

D

log,Z•

(Czar

grP

D•

(
g∗ )

k • −−− −−−− →

(O(X• ,D• ∪Z• )/S ))   

 D•

Ω•X  /S (log(D• ∪ Z• ))) ⏐•  ⏐ D• Resλ 

(OD• ⊗OX• grP k

(

k


∗ eλ j ) h (λ;•)

(OD• ⊗OX  (aλ |(D• ,Z  |D ) )zar∗ Ω•(λ;•) ){−k} −−−−−−−−−−−→ •

•

j=1

2.9 The Functoriality of the Preweight-Filtered Zariskian Complex

141

D

log,Z• g(X• ,Z• )zar∗ grP (O(X• ,D• ∪Z• )/S )) k (Czar   

g(X• ,Z• )zar∗ (OD• ⊗OX• grP k ⏐ D• ⏐ Resλ 

D•

Ω•X• /S (log(D• ∪ Z• )))

g(X• ,Z• )zar∗ (OD• ⊗OX• (aλ |(D• ,Z|D• ) )zar∗ Ω•(λ;•) ){−k}. Now, by (2.9.3.1), (2.9.3.2), (2.9.3.4) and log crystalline Poincar´e lemma for (D• , Z|D• ), (D• , Z  |D• ), (2.9.3) is reduced to the following obvious lemma.   Lemma 2.9.4. Let F : A −→ B be a left exact functor of abelian categories. Let M • and M • (resp. N • and N • ) be objects of K+ (B) (resp. K+ (A)). Let f

M • −−−−→ F (N • ) ⏐ ⏐ ⏐ ⏐   f

M • −−−−→ F (N • ) be the commutative diagram in K+ (B). Assume that A has enough injectives. Then the following diagram is commutative: f

M • −−−−→ RF (N • ) ⏐ ⏐ ⏐ ⏐   f

M • −−−−→ RF (N • ). Proof. The proof is obvious.

 

Definition 2.9.5. (1) We call {eλ }λ∈Λ ∈ ZΛ >0 the multi-degree of g with respect to a decomposition ∆ := {Dλ }λ and ∆ := {Dλ }λ of D and D , respectively. We denote it by deg∆,∆ (g) ∈ ZΛ >0 . If eλ ’s for all λ’s are equal, we also denote eλ ∈ Z>0 by deg∆,∆ (g) ∈ Z>0 . (2) Assume that eλ ’s for all λ’s are equal. Let u : E −→ F be a morphism of OS -modules. Let k be a nonnegative integer. The k-twist u(−k) : E(−k; g; ∆, ∆ ) −→ F(−k; g; ∆, ∆ ) of u with respect to g, ∆ and ∆ is, by definition, the morphism deg∆,∆ (g)k u : E −→ F. Corollary 2.9.6. Assume that eλ ’s for all λ’s are equal. Let Ess ((X, D ∪ Z)/S) be the following spectral sequence

142

2 Weight Filtrations on Log Crystalline Cohomologies

E1−k,h+k ((X, D ∪ Z)/S) = Rh−k f(D(k) ,Z|

D (k)

 )/S∗ (O(D (k) ,Z|D(k) )/S )(−k; g; ∆, ∆ )

=⇒ Rh f(X,D∪Z)/S∗ (O(X,D∪Z)/S ) and let Ess ((X  , D ∪Z  )/S  ) be the obvious analogue of the above for (X  , D ∪ Z  )/S  . Then there exists a morphism log∗ gcrys : Ess ((X  , D ∪ Z  )/S  ) −→ Ess ((X, D ∪ Z)/S)

(2.9.6.1)

of spectral sequences.  

Proof. (2.9.6) immediately follows from (2.9.3).

Assume that S0 is a scheme of characteristic p > 0. Let FS0 : S0 −→ S0 be the p-th power endomorphism. Let (X  , D ∪ Z  ) be the base change of (X, D ∪ Z) by FS0 . The relative Frobenius morphism F : (X, D ∪ Z) −→ (X  , D ∪ Z  ) over S0 induces the relative Frobenius morphisms F(X,Z) : (X, Z) −→ (X  , Z  ) and



F (k) : (D(k) , Z|D(k) ) −→ (D(k) , Z  |D(k) ).

Let a(k) : (D(k) , Z|D(k) ) −→ (X, D ∪ Z) and





a(k) : (D(k) , Z  |D(k) ) −→ (X  , D ∪ Z  )

be the natural morphisms. We define the relative Frobenius action Φ(D(k) ,Z|

(k) log

)/S D (k)

(k)log log (k)log : acrys∗
crys (D /S; Z  ) −→ Fcrys∗ acrys∗
crys (D/S; Z) (k)log

as the identity under the natural identification ∼

(k)log (k)log
crys (D /S; Z  ) −→ Fcrys∗
crys (D/S; Z). (k)log

When g is the relative Frobenius F : (X, D ∪ Z) −→ (X  , D ∪ Z  ), we denote (2.9.6.1) by (2.9.6.2) E1−k,h+k ((X, D ∪ Z)/S) = Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S (k)log ⊗Z
crys (D/S; Z))(−k)

=⇒ Rh f(X,D∪Z)/S∗ (O(X,D∪Z)/S ).

2.10 The Base Change Theorem and the K¨ unneth Formula

143

((2.9.6.2) is equal to (2.6.2.2)+(the compatibility with Frobenius).) (2.9.6.2) is generalized to the following spectral sequence (2.9.6.3)

E1−k,h+k = E1−k,h+k ((X, D ∪ Z)/S; k  )(−k) log,Z (O(X,D∪Z)/S )) =⇒ Rh f (X,D∪Z)/S∗ (PkD CRcrys log,Z (O(X,D∪Z)/S )) = Rh f(X,D∪Z)/S∗ (PkD Ecrys

by (2.6.2.1) and (2.7.3.2). Definition 2.9.7. We call the sequence (2.9.6.2) the preweight spectral sequence of (X, D ∪ Z)/(S, I, γ) with respect to D. If Z = ∅, then we call it the preweight spectral sequence of (X, D)/(S, I, γ). By the proof of (2.8.5) and (2.9.3), the morphism G in (2.8.5) is a morphism (2.9.7.1) G : Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S

−→ Rh−k+2 f(D(k−1) ,Z|

D (k−1)

(k)log ⊗Z
crys (D/S; Z)))(−k)

)/S∗ (O(D (k−1) ,Z|D(k−1) )/S (k−1)log ⊗Z
crys (D/S; Z))(−(k − 1)).

By (2.7.6) we also have the following Leray spectral sequence (2.9.7.2) E2st := Rs f(D(t) ,Z|

D (t)

)/S∗ (O(D (t) ,Z|D(t) )/S

(t)log ⊗Z
crys (D/S; Z))(−t)

=⇒ Rs+t f(X,D∪Z)/S∗ (O(X,D∪Z)/S ).

2.10 The Base Change Theorem and the K¨ unneth Formula In this section we prove the base change theorem of a preweight-filtered vanishing cycle crystalline complex and the K¨ unneth formula of it. (2.7.5) plays an important role in this section. We keep the notations in §2.4. In this section we assume that X is quasicompact. Hence we can assume that the cardinality of the family {Xi0 }i0 ∈I0 of an open covering of X is finite.

(1) Base change theorem. Proposition 2.10.1. Let

144

2 Weight Filtrations on Log Crystalline Cohomologies

Y ⏐ ⏐ f 

(2.10.1.1)

g

−−−−→

Y ⏐ ⏐f 

u

(T  , J  , γ  ) −−−−→ (T, J , γ) be a commutative diagram of fine log schemes, where a PD-structure γ (resp. γ  ) on a PD-ideal sheaf J (resp. J  ) of OT (resp. OT  ) extends to Y (resp. Y  ) and u is a PD-morphism of PD-log schemes. Let (E • , {Ek• }) be a bounded below filtered complex of OY /T -modules. Assume that RfY /T ∗ (E • , {Ek• }) is bounded above. Then there exists a canonical morphism log−1 Lu∗ RfY /T ∗ (E • , {Ek• }) −→ RfY  /T  ∗ gcrys (E • , {Ek• })

(2.10.1.2) in DF(OT  ).

Proof. By (1.2.3.2) we have only to find an element in log−1 H0 [{RHomOT  (Lu∗ RfY /T ∗ (E • , {Ek• }), RfY  /T  ∗ gcrys (E • , {Ek• })}0 ].

Using (1.2.2), we have the following formula log−1 RHomOT  (Lu∗ RfY /T ∗ (E • , {Ek• }), RfY  /T  ∗ gcrys (E • , {Ek• }))

(2.10.1.3)

log−1 = RHomOT (RfY /T ∗ (E • , {Ek• }), Ru∗ RfY  /T  ∗ gcrys (E • , {Ek• })) log log−1 = RHomOT (RfY /T ∗ (E • , {Ek• }), RfY /T ∗ Rgcrys∗ gcrys (E • , {Ek• })). log log−1 gcrys (E • , {Ek• }) induces a The adjunction morphism (E • , {Ek• }) −→ gcrys∗ log • • log−1 • • morphism (E , {Ek }) −→ Rgcrys∗ gcrys (E , {Ek }). This morphism induces a morphism log log−1 RfY /T ∗ (E • , {Ek• }) −→ RfY /T ∗ Rgcrys∗ gcrys (E • , {Ek• })

 

in DF(OT ). ⊂

Proposition 2.10.2. (1) Let f : (X, D ∪ Z) −→ S0 (−→ S) and (S, I, γ) be ◦

as in §2.4. Assume moreover that S is quasi-compact and that f : X −→ S0 is ⊂ quasi-separated and quasi-compact. Let f(X,Z) : (X, Z) −→ S0 (−→ S) be the log,Z (O(X,D∪Z)/S ) (h, k ∈ induced morphism by f . Then Rh f(X,Z)/S∗ PkD (Ecrys log,Z (O(X,D∪Z)/S ), P D ) Z) are quasi-coherent OS -modules and Rf(X,Z)/S∗ (Ecrys is isomorphic to a bounded filtered complex of OS -modules. (2) Let (S, I, γ) and S0 be as in §2.4. Let Y be a quasi-compact smooth scheme over S0 (with trivial log structure). Let f : (X, D ∪ Z) −→ Y be a ◦

morphism of log schemes such that f : X −→ Y is smooth, quasi-compact and quasi-separated and such that D∪Z is a relative SNCD over Y . (In particular, D ∪ Z is also a relative SNCD on X over S0 .) Let f(X,Z) : (X, Z) −→ Y be

2.10 The Base Change Theorem and the K¨ unneth Formula

145

log log,Z the induced morphism by f . Then Rf(X,Z)crys∗ (Ecrys (O(X,D∪Z)/S ), P D ) is isomorphic to a bounded filtered complex of OY /S -modules. log,Z (O(X,D∪Z) Proof. (1): Let (I • , {Ik• }) be a filtered flasque resolution of (Ecrys D log,Z D • ), P ). Then Rf (E (O ), P ) = (f ◦ u /S (X,Z)/S∗ (X,D∪Z)/S (X,Z)/S )∗ (I , crys • {Ik }). Now, fix a decomposition {Dλ }λ of D by its smooth components and give a ∼ (k)log total order on λ’s. Then there exists an isomorphism Z −→
crys (D/S; Z). Furthermore, for each k, fix a decomposition {(Z|D(k) )µ } of Z|D(k) by its smooth components and give a total order on µ’s. Because X is quasicompact, the sets λ’s and µ’s are finite. By (2.6.2.2) we have the following spectral sequence

(2.10.2.1)

E1−l,h+l = Rh−l fZ (l) |

D (k)

=⇒ Rh f(D(k) ,Z|

/S∗ (OZ (l) |D(k) /S )

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S ).

By [11, 7.6 Theorem] and by the spectral sequences (2.6.2.2) and (2.10.2.1), Hh ((f(X,Z) ◦ u(X,Z)/S )∗ (Ik• )) (h, k ∈ Z) are quasi-coherent OS -modules and there exists an integer h0 such that, for all h ≥ h0 and for all k ∈ Z, log,Z Hh ((f(X,Z) ◦u(X,Z)/S )∗ (Ik• )) = 0. Hence Rh f(X,Z)/S∗ PkD (Ecrys (O(X,D∪Z)/S ) log,Z (O(X,D∪Z)/S ), (h, k ∈ Z) are quasi-coherent OS -modules and Rf(X,Z)/S∗ (Ecrys P D ) = ((f(X,Z) ◦ u(X,Z)/S )∗ (I • ), (f ◦ u(X,Z)/S )∗ (Ik• )) is isomorphic to a bounded filtered complex of OS -modules. (2): (2) immediately follows from (1) and from the proof of [3, V Corollaire 3.2.3] (cf. the proof of [11, 7.11 Corollary]).   ⊂

Theorem 2.10.3 (Base change theorem). Let f : (X, D ∪ Z) −→ S0 (−→ S) and (S, I, γ) be as in (2.10.2). Let u : (S  , I  , γ  ) −→ (S, I, γ) be a morphism of PD-schemes. Assume that I  is a quasi-coherent ideal sheaf of OS  . Set S0 := SpecS  (OS  /I  ). Let f  : (X  , D ∪ Z  ) := (X ×S0 S0 , (D ∪ Z) ×S0 S0 ) −→ S0 be the base change morphism of f with respect to u|S0 . Then there exists a canonical isomorphism (2.10.3.1)

∼

log,Z Lu∗ Rf(X,Z)/S∗ (Ecrys (O(X,D∪Z)/S ), P D ) −→ 



 log,Z Rf(X (O(X  ,D ∪Z  )/S  ), P D )  ,Z  )/S  ∗ (Ecrys

in the filtered derived category DF(f −1 (OS  )). Proof. Let g(X,Z) : (X  , Z  ) −→ (X, Z) and g(X,D∪Z) : (X  , D ∪ Z  ) −→ (X, D ∪ Z) be the natural morphisms of log schemes. First we use the general theory in §1.5 as follows. Consider a small category I := {i, i } consisting of two elements. The morphisms in I, by definition, consist of three elements idi , idi and a morphism i −→ i . By corresponding the natural morphism

146

2 Weight Filtrations on Log Crystalline Cohomologies log  ∪ Z  )/S  )log , O   g(X,D∪Z)crys : (((X  , D
(X ,D ∪Z  )/S  ) crys


−→ (((X, D ∪ Z)/S)log crys , O(X,D∪Z)/S ) log
to the morphism i −→ i , we have a ringed topos (((Xj , D j ∪Zj )/Sj )crys , O(Xj , • Dj ∪Zj )/Sj )j∈I . Let (Ij )j∈I be a flasque resolution of (O(Xj ,Dj ∪Zj )/Sj )j∈I  
 

log ((1.5.0.2)). Let  : ((X, D ∪ Z)/S)log crys −→ ((X, Z)/S)crys and  : ((X , D ∪ Z    log
)/S  )log crys −→ ((X , Z )/S )crys be the forgetting log morphisms along D and log,Z log,Z  D , respectively. Then (Ecrys (O(X,D∪Z)/S ), P D ) and (Ecrys (O(X  ,D ∪Z  ) D •  • ), P ) are represented by ( (I ), τ ) and ( (I ), τ ), respectively. Since  ∗ i /S ∗ i log −1 log −1 log −1 • • g(X,Z)crys is exact, g(X,Z)crys (∗ (Ii ), τ ) = (g(X,Z)crys ∗ (Ii ), τ ). By the following commutative diagram g(X,D∪Z)

(X  , D ∪ Z  ) −−−−−−→ (X, D ∪ Z) ⏐ ⏐ ⏐ ⏐    (X  , Z  )

g(X,Z)

−−−−→

(X, Z),

log −1 log −1 ∗ (Ii• ), τ ) −→ (∗ g(X,D∪Z)crys (Ii• ), τ ). we have a natural morphism (g(X,Z)crys

log −1 By the definition of (Ij• )j∈I , we have the morphism g(X,D∪Z)crys (Ii• ) −→ Ii• . Hence we have a composite morphism log −1 (g(X,Z)crys ∗ (Ii• ), τ ) −→ (∗ (Ii• ), τ ).

Therefore we have a canonical morphism (2.10.3.1) by (2.10.1) and (2.10.2) (1). We prove that (2.10.3.1) is an isomorphism. By the filtered cohomological descent (1.5.1) (2) and by the same argument as that in the proof of [3, V Proposition 3.5.2] ([11, 7.8 Theorem]), we may assume that S is affine and that X is an affine scheme over S0 . Then (X, D ∪ Z) has a lift (X , D ∪ Z)/S (D = D ×X X, Z = Z ×X X) by (2.3.14). In this case, we may assume
that the morphism (2.4.5.1) is the identity of (((X, Z)/S)log , O(X,Z)/S ). Let crys

f : (X , D ∪ Z) −→ S be the lift of f . Set f∗ (PkD ) := f∗ (PkD Ω•X /S (log(D ∪ Z))) (k ∈ Z) and f∗ (P D ) := {f∗ (PkD )}k∈Z for simplicity of notation. Then, by (2.7.5), we have log,Z Rf(X,Z)/S∗ (Ecrys (O(X,D∪Z)/S ), P D ) = (f∗ (Ω•X /S (log(D ∪ Z))), f∗ (P D )).

and we have the same formula for (X  , D ∪Z  )/S  . We claim that f∗ (Ω•X /S (log (D ∪ Z)))/f∗ (PkD ) is a flat OS -module for any k. Indeed, the filtration PkD on Ω•X /S (log(D ∪ Z)) is finite and f∗ (Ω•X /S (log(D ∪ Z))) is a flat OS -module. Because X is affine over S, we have the following exact sequence

2.10 The Base Change Theorem and the K¨ unneth Formula

147

(2.10.3.2) D

• • D 0 −→ f∗ (grP k ΩX /S (log(D ∪ Z))) −→ f∗ (ΩX /S (log(D ∪ Z)))/f∗ (Pk−1 )

−→ f∗ (Ω•X /S (log(D ∪ Z)))/f∗ (PkD ) −→ 0. By the Poincar´e residue isomorphism, the left term of (2.10.3.2) is isomorphic (k) (k) to f∗ (b∗ Ω•D(k) /S (log Z|D(k) ) ⊗Z
zar (D/S)){−k}, where b(k) : D(k) −→ X is the natural morphism. Hence, the descending induction on k shows the claim. Therefore the left hand side of (2.10.3.1) is equal to u∗ f∗ (Ω•X /S (log(D ∪ Z)), P D ). Since f : X −→ S is an affine morphism, we obtain (2.10.3) by the affine base change theorem ([39, (1.5.2)]) as in the classical case ([11, 7.8 Theorem]).   As in [3, V] and [11, §7], we have some important consequences of (2.10.3). Corollary 2.10.4. Let f : (X, D ∪ Z) −→ Y be as in (2.10.2) (2). Then log log,Z Rf(X,Z)crys∗ (Ecrys (O(X,D∪Z)/S ), P D )

is a filtered crystal in DF(OY /S ). That is, for a morphism v : (U  , T  , δ  ) −→ (U, T, δ) of the crystalline site (Y /S)crys , the canonical morphism log log,Z (Ecrys (O(X,D∪Z)/S ), P D ))T ) −→ Lv ∗ ((Rf(X,Z)crys∗ 



log log,Z Rf(X (O(X  ,D ∪Z  )/S ), P D )T   ,Z  )crys∗ (Ecrys

is an isomorphism, where (X  , D ∪ Z  ) := (X  , D ∪ Z  ) ×U U  . Corollary 2.10.5. Let f : (X, D ∪ Z) −→ Y be as in (2.10.2) (2). Assume that Y has a smooth lift Y over S. Let h be an integer. Then the following holds: (1) There exists a quasi-nilpotent integrable connection ∇

k log,Z Rh f(X,Z)/Y∗ (PkD Ecrys (O(X,D∪Z)/S )) −→

(2.10.5.1)

(k ∈ Z)

log,Z Rh f(X,Z)/Y∗ (PkD Ecrys (O(X,D∪Z)/S ))⊗OY Ω1Y/S

making the following diagram commutative for any two nonnegative integers k≤l: (2.10.5.2) ∇k D E log,Z (O D E log,Z (O Rh f(X,Z)/Y∗ (Pk −−−−−− → Rh f(X,Z)/Y∗ (Pk crys crys (X,D∪Z)/S )) − (X,D∪Z)/S ))⊗O ⏐ ⏐ 

Y

Ω1 Y/S

Y

Ω1 . Y/S

⏐ ⏐ 

∇l log,Z (O log,Z (O Rh f(X,Z)/Y∗ (PlD Ecrys −−−−−− → Rh f(X,Z)/Y∗ (PlD Ecrys (X,D∪Z)/S )) − (X,D∪Z)/S ))⊗O

148

2 Weight Filtrations on Log Crystalline Cohomologies

(2) For k ∈ Z, set PkD Rh f(X,D∪Z)/Y∗ (O(X,D∪Z)/S ) := log,Z Im(Rh f(X,Z)/Y∗ (PkD Ecrys (O(X,D∪Z)/S )) −→ Rh f(X,D∪Z)/Y∗ (O(X,D∪Z)/S )).

Then there exists a quasi-nilpotent connection PkD Rh f(X,D∪Z)/Y∗ (O(X,D∪Z)/S ) −→ PkD Rh f(X,D∪Z)/Y∗ (O(X,D∪Z)/S )⊗OY Ω1Y/S . Corollary 2.10.6. Let f : (X, D ∪ Z) −→ Y be as in (2.10.2) (2). Let g

(X  , D ∪ Z  ) −−−−→ (X, D ∪ Z) ⏐ ⏐ ⏐f ⏐ f   Y ⏐ ⏐ 

−−−−→

h

Y ⏐ ⏐ 

(S  , I  , γ  )

−−−−→

(S, I, γ)

be a commutative diagram such that the upper rectangle is cartesian. Assume that Y  is a quasi-compact smooth scheme over S  . Then the natural morphism log log,Z Lh∗crys Rf(X,Z)crys∗ (Ecrys (O(X,D∪Z)/S ), P D ) −→ 



log,Z D     Rf  log ) (X  ,Z  )crys∗ (Ecrys (O(X ,D ∪Z )/S ), P

is an isomorphism. Corollary 2.10.7. Let the notations and the assumptions be as in (2.10.2) log log,Z (1). Then Rf(X,Z)/Scrys∗ (PkD Ecrys (O(X,D∪Z)/S )) (k ∈ N) has finite tordimension. Moreover, if S is noetherian and if f is proper, then Rf(X,Z)/S∗ log,Z (PkD Ecrys (O(X,D∪Z)/S )) is a perfect complex of OS -module. Definition 2.10.8. Let A be a noetherian commutative ring. Let (E • , {Ek• }) ∈ CF(A) be a filtered complex of A-modules. We say that (E • , {Ek• }) is filteredly strictly perfect if it is bounded, if the filtration {Ekq } is finite for any q and if all Ekq ’s are finitely generated projective A-modules. Definition 2.10.9. Let A be a commutative ring with unit element. For a filtered A-module (E, {Ek }) whose filtration is finite and for a family {Tl }l∈Z of

A-modules, we say that (E, {Ek }) is the direct sum of {Tl }l∈Z if Ek = l≤k Tl (∀k ∈ Z). The following is a nontrivial filtered version of [11, 7.15 Lemma]:

2.10 The Base Change Theorem and the K¨ unneth Formula

149

Theorem 2.10.10. Let A be a noetherian commutative ring. Let (E • , {Ek• }) be a filtered complex of A-modules. Assume that there exist integers k0 ≤ k1 such that Ekq1 = E q and Ekq0 = 0 for all q ∈ Z. Then (E • , {Ek• }) is quasiisomorphic to a filteredly strictly perfect complex if and only if Ek• (∀k) has finite tor-dimension and finitely generated cohomologies. Proof. Roughly speaking, the proof is dual to that of (1.1.7) with some additional calculations. We have only to prove the “if” part. Let k be an integer such that k0 < k ≤ k1 . By the assumption, we may assume that E q = 0 (q > 0). Since H 0 (Ek• ) is finitely generated, there exists a free A-module Tk0 of finite rank with a morphism Tk0 −→ Ek0 such that the induced morphism Tk0 −→ H 0 (Ek• ) is surjective. Set Tk0 := 0 for k ≤ k0 or k > k1 . Let (Q0 , {Q0k }) be the direct sum of {Tk0 }. Then we have a natural filtered morphism (Q0 , {Q0k }) −→ (E 0 , {Ek0 }). Assume that, for a nonpositive integer q, we are given a morphism (Q•≥q , {Q•≥q }) −→ (E •≥q , {Ek•≥q }) k of (≥ q)-truncated filtered complexes such that the induced morphism H ∗ (Q•k ) −→ H ∗ (Ek• ) is an isomorphism for ∗ > q, Ker(Qqk −→ Qq+1 k ) −→ H q (Ek• ) is surjective, Q• = 0 for • ≥ 0, Q• = Q•k1 , Q•k0 = 0 (q ≤ • ≤ 0) and that (Qr , {Qrk }) (∀r ≥ q) is the direct sum of some family {Tkr }k∈Z of free A-modules of finite rank. For an integer k0 < k ≤ k1 , consider the fiber product Ekq−1 ×Ekq q Ker(Qqk −→ Qq+1 k ). Let Ik be the image of the following composite morphism ⊂

q q+1 q Ekq−1 ×Ekq Ker(Qqk −→ Qq+1 k ) −→ Ker(Qk −→ Qk ) −→ Qk .

Since A is noetherian, Ikq is finitely generated. Let {yi }i∈I be a system of finite generators of Ikq . Take an element (xi , yi ) ∈ Ekq−1 ×Ekq Ker(Qqk −→ Qq+1 k ). q−1 • Because H (Ek ) is finitely generated, we can take a family {zj }j∈J of finite elements of Ker(Ekq−1 −→ Ekq ) whose images in H q−1 (Ek• ) form a system of generators of H q−1 (Ek• ). Now consider a finitely generated A-module Skq−1 generated by {(xi , yi )}i∈I q−1 and {(zj , 0)}j∈J in Ekq−1 ×Ekq Ker(Qqk −→ Qq+1 be a free Ak ). Let Tk module of finite rank such that there exists a surjection Tkq−1 −→ Skq−1 . Set Tkq−1 := 0 for k ≤ k0 or k > k1 . Let (Qq−1 , {Qq−1 k }) be the direct sum of {Tkq−1 }k∈Z . Then we have a natural filtered morphism q−1 , {Ekq−1 }). (Qq−1 , {Qq−1 k }) −→ (E q q • By assumption, Ker(Qk −→ Qq+1 k ) −→ H (Ek ) is a surjection. Moreover, q q+1 if the image of an element of Ker(Qk −→ Qk ) vanishes in H q (Ek• ), then this element belongs to Im(Tkq−1 −→ Qqk ) by the definition of Tkq−1 . In partic−→ Qqk ). Hence the natural morphism ular, this element belongs to Im(Qq−1 k

150

2 Weight Filtrations on Log Crystalline Cohomologies ∼

q • q • Ker(Qqk −→ Qq+1 k ) −→ H (Ek ) induces an isomorphism H (Qk ) −→ q−1 q q • q−1 (Ek• ) H (Ek ). Moreover, it is easy to see that Ker(Qk −→ Qk ) −→ H is surjective. Hence the induction works well and so we have constructed a filtered complex (Q• , {Q•k }) such that Qq = 0 (q > 0), such that Q•k0 = 0 and Q•k1 = Q• , such that (Qq , {Qqk }) (q ∈ Z) is the direct sum of a family {Tkq }k∈Z of free A-modules of finite rank and such that there exists a filtered quasi-isomorphism (Q• , {Q•k }) −→ (E • , {Ek• }). Because Ek• (∀k) has finite tor-dimension, grk E • (∀k) also has it. Since (Q• , {Q•k }) is filteredly quasi-isomorphic to (E • , {Ek• }), grk Q• (∀k) also has it. Since the filtration on Q• is finite, there exists a nonpositive integer r and a complex Fk• of flat A-modules for each k ∈ Z satisfying the following properties: (a) Fk• is quasi-isomorphic to grk Q• , (b) Fk• = 0 for • > 0 or • ≤ r. Set Bkq := Im(Qq−1 −→ Qqk ). Let l ≤ k1 − k0 be a positive integer. Set k ⎧ ⎪ (q < r − l + 1 or q > 0), ⎨0 Rkq 0 +l = Qqk0 +l /(Qqk0 +r−q + Bkq0 +r−q+1 ) (r − l + 1 ≤ q ≤ r), ⎪ ⎩ q Qk0 +l (r < q ≤ 0).

Then we claim that Rkq 0 +l is a flat A-module. We proceed on induction on l. Unusually we assume that the initial case l = 1 holds and that l ≥ 2. Consider the following exact sequence 0 −→ Rkq 0 +l−1 −→ Rkq 0 +l −→ grk0 +l Qq −→ 0

(r − l + 1 < q ≤ r).

By the induction hypothesis, we may assume that Rkq 0 +l−1 (r − l + 1 < q ≤ r) is a flat A-module. Since grk0 +l Qq is a flat A-module, so is Rkq 0 +l (r − l + 1 < q ≤ r). Now we show that Rkr−l+1 is a flat A-module. By the properties (a) 0 +l and (b), we have the following exact sequence · · · −→ grk0 +l Qr−l −→ grk0 +l Qr−l+1 −→ Rkr−l+1 −→ 0. 0 +l For a positive integer i and for any A-module M , r−l+1 TorA i (Rk0 +l , M )

=H −i (· · · −→ grk0 +l Qr−l ⊗A M −→ grk0 +l Qr−l+1 ⊗A M −→ 0) =H r−l+1−i (grk0 +l Q• ⊗A M ) = H r−l+1−i (Fk•0 +l ⊗A M ) = 0. is a flat A-module. The rest for showing the claim is to prove Hence Rkr−l+1 0 +l that Rkr 0 +1 is a flat A-module. As above, we can prove this using the following resolution r r · · · −→ Qr−1 k0 +1 −→ Qk0 +1 −→ Rk0 +1 −→ 0.

2.10 The Base Change Theorem and the K¨ unneth Formula

151

Set R• := Rk• := Rk•1 for k ≥ k1 and Rk• := 0 for k ≤ k0 . Then {Rk• }k∈Z is an increasing filtration on R• since the natural morphism Rk•0 +l−1 −→ Rk•0 +l is injective. Note that R• is a bounded complex of projective A-modules. Finally we claim that the natural morphism (Q• , {Q•k }) −→ (R• , {Rk• }) is a filtered quasi-isomorphism. Indeed, for a positive integer l ≤ k1 − k0 , grk0 +l R• is the following complex 0 −→ grk0 +l Qr−l+1 /Im(grk0 +l Qr−l −→ grk0 +l Qr−l+1 ) r−l+1

−→ grk0 +l Qr−l+2 −→ · · · . r−l+2

This complex is isomorphic to grk0 +l Q• by the properties (a) and (b). Hence we have finished the proof of (2.10.10).

 

Corollary 2.10.11. Let the notations and the assumptions be as in (2.10.7). log,Z Then the filtered complex Rf(X,Z)/S∗ (Ecrys (O(X,D∪Z)/S ), P D ) is a filtered perfect complex of OS -modules, that is, locally on Szar , filteredly quasiisomorphic to a filtered strictly perfect complex.  

Proof. (2.10.11) immediately follows from (2.10.7) and (2.10.10). (2) K¨ unneth formula.

Next, we give the K¨ unneth formula of preweight-filtered vanishing cycle crystalline complexes. Let Xj /S0 (j = 1, 2) be a smooth scheme with transversal relative SNCD’s Dj and Zj over S0 . Set X3 := X1 ×S0 X2 , D3 = (D1 ×S0 X2 ) ∪ (X1 ×S0 D2 ) and Z3 = (Z1 ×S0 X2 ) ∪ (X1 ×S0 Z2 ). Let fj : (Xj , Dj ∪ Zj ) −→ S0 (j = 1, 2, 3) be the structural morphism. Assume that S is quasi-compact ◦

and that f j (j = 1, 2) is quasi-compact and quasi-separated. We denote log,Z

log,Z

Rfj(Xj ,Zj )/S∗ (Ecrys j (O(Xj ,Dj ∪Zj )/S ), P Dj ) simply by Rf(Xj ,Zj )/S∗ (Ecrys j (O(Xj ,Dj ∪Zj )/S ), P Dj ). We have the following commutative diagram of ringed topoi for j = 1, 2: (2.10.11.1) log

qjcrys

log log

(((Xj , D j ∪ Zj )/S)crys , O(Xj ,Dj ∪Zj )/S ) ←−−−−− (((X3 , D3 ∪ Z3 )/S)crys , O(X3 ,D3 ∪Z3 )/S )

⏐

⏐ ⏐(X ,D ∪Z ,Z )/S  3 3 3 3

(X ,D ∪Z ,Z )/S ⏐ j j j j 

(((Xj
, Zj )/S)log crys , O(Xj ,Zj )/S )

⏐

f(X ,Z )/S ⏐  j j


zar , OS ) (S

log

pjcrys

←−−−−−

(((X3
, Z3 )/S)log crys , O(X3 ,Z3 )/S )

⏐ ⏐f  (X3 ,Z3 )/S


zar , OS ), (S

152

2 Weight Filtrations on Log Crystalline Cohomologies

where qj : (X3 , D3 ∪ Z3 ) −→ (Xj , Dj ∪ Zj ) and pj : (X3 , Z3 ) −→ (Xj , Zj ) are the projections. We shall construct a canonical morphism (2.10.11.2)

log,Z1 Rf(X1 ,Z1 )/S∗ (Ecrys (O(X1 ,D1 ∪Z1 )/S ), P D1 )⊗L OS log,Z2 Rf(X2 ,Z2 )/S∗ (Ecrys (O(X2 ,D2 ∪Z2 )/S ), P D2 ) log,Z3 −→ Rf(X3 ,Z3 )/S∗ (Ecrys (O(X3 ,D3 ∪Z3 )/S ), P D3 ).

For simplicity of notation, set j := (Xj ,Dj ∪Zj ,Zj )/S (j = 1, 2, 3). We have to construct a morphism (2.10.11.3) Rf(X1 ,Z1 )/S∗ (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗L OS Rf(X2 ,Z2 )/S∗ (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ ) −→ Rf(X3 ,Z3 )/S∗ (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ).

To construct it, we need the following two lemmas: Lemma 2.10.12 (cf. (2.7.2)). Let f : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Then, for an object E • in D− (A ), there exists a canonical morphism (2.10.12.1)

Lf ∗ ((E • , τ )) −→ (Lf ∗ (E • ), τ )

in D− F(A). Proof. Let Q• −→ E • be a quasi-isomorphism from a complex of flat A modules. Let (R• , {Rk• }) −→ (Q• , τ ) be a filtered flat resolution of (Q• , τ ). Then, by applying the functor f ∗ to the morphism of this resolution, we obtain a diagram

(2.10.12.2)

f ∗ (Rk• ) −−−−→ f ∗ (τk Q• ) ⏐ ⏐ ⏐ ⏐   f ∗ (R• ) −−−−→ f ∗ (Q• ).

By (1.1.19) (2), the left hand side of (2.10.12.2) is equal to Lf ∗ ((E • , τ )). On the other hand, there exists a natural morphism f ∗ (τk Q• ) −→ τk f ∗ (Q• ). Hence there exists a natural diagram

(2.10.12.3)

f ∗ (τk Q• ) −−−−→ τk f ∗ (Q• ) ⏐ ⏐ ⏐ ⏐   f ∗ (Q• )

f ∗ (Q• ).

Composing (2.10.12.2) with (2.10.12.3), we have a morphism (2.10.12.1).   Lemma 2.10.13. Let (T , A) be a ringed topos. Let E • and F • be two complexes of A-modules. Assume that E • is bounded above. Then there exists a canonical morphism

2.10 The Base Change Theorem and the K¨ unneth Formula

153

• • L • (E • , τ ) ⊗L A (F , τ ) −→ (E ⊗A F , τ ).

(2.10.13.1)

Proof. Let P • −→ E • be a flat resolution of E • . Let (Q• , {Q•k }) −→ (P • , τ ) be a filtered flat resolution of (P • , τ ). Then we have the following: • (E • , τ ) ⊗L A (F , τ ) = (Q• , {Q•k }) ⊗A (F • , τ )

Q•l ⊗A τm F • −→ Q• ⊗A F • )}k∈Z ) = (Q• ⊗A F • , {Im( l+m=k •

•

−→ (P ⊗A F , {Im(

τl P • ⊗A τm F • −→

l+m=k

P • ⊗A F • )}k∈Z )

l+m=k

• −→ (E • ⊗L A F , τ ).

  Now we construct the canonical morphism (2.10). We need a canonical element in (2.10.13.2) ∗ H0 [RHomO(X1 ,Z1 )/S (Lf(X {Rf(X1 ,Z1 )/S∗ (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗L OS 3 ,Z3 )/S Rf(X2 ,Z2 )/S∗ (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ )}, (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ))]. First we have the following morphism ∗ Lf(X {Rf(X1 ,Z1 )/S∗ (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗L OS 3 ,Z3 )/S

Rf(X2 ,Z2 )/S∗ (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ )} ∗ Rf(X1 ,Z1 )/S∗ (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗L =Lf(X O(X3 ,Z3 )/S 3 ,Z3 )/S ∗ Lf(X Rf(X2 ,Z2 )/S∗ (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ ) 3 ,Z3 )/S ∗ L =Lplog∗ 1crys Lf(X1 ,Z1 )/S Rf(X1 ,Z1 )/S∗ (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗O(X

3 ,Z3 )/S

∗ Lplog∗ 2crys Lf(X2 ,Z2 )/S Rf(X2 ,Z2 )/S∗ (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ ) L −→Lplog∗ 1crys (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗O(X

3 ,Z3 )/S

Lplog∗ 2crys (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ ). Note that Rj∗ (O(Xj ,Dj ∪Zj )/S ) (j = 1, 2, 3) is bounded above by (2.7.10). Therefore it suffices to construct a canonical morphism (2.10.13.3) L Lplog∗ 1crys (R1∗ (O(X1 ,D1 ∪Z1 )/S ), τ )⊗O(X

3 ,Z3 )/S

Lplog∗ 2crys (R2∗ (O(X2 ,D2 ∪Z2 )/S ), τ )

−→ (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ). We also have the following composite morphism

154

2 Weight Filtrations on Log Crystalline Cohomologies log∗ Lplog∗ jcrys (Rj∗ (O(Xj ,Dj ∪Zj )/S ), τ ) −→ (Lpjcrys Rj∗ (O(Xj ,Dj ∪Zj )/S ), τ ) log∗ (O(Xj ,Dj ∪Zj )/S ), τ ) −→ (R3∗ Lqjcrys

= (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ) Here we have obtained the first morphism by (2.10.12), and the second morphism by the commutative diagram (2.10.11.1) and the adjunction morphism. Thus we have only to construct a canonical morphism (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ )⊗L O(X

3 ,Z3 )/S

(R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ) −→ (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ).

By (2.10.13), it suffices to construct a canonical morphism (R3∗ (O(X3 ,D3 ∪Z3 )/S )⊗L O(X

3 ,Z3 )/S

R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ ) −→ (R3∗ (O(X3 ,D3 ∪Z3 )/S ), τ )

and, furthermore, to construct a canonical morphism R3∗ (O(X3 ,D3 ∪Z3 )/S ) ⊗L O(X

3 ,Z3 )/S

R3∗ (O(X3 ,D3 ∪Z3 )/S ) −→ R3∗ (O(X3 ,D3 ∪Z3 )/S ).

Hence we have only to have a canonical element of (2.10.13.4)

H0 [RHomO(X3 ,D3 ∪Z3 )/S (L∗3 {R3∗ (O(X3 ,D3 ∪Z3 )/S )⊗L O(X

3 ,Z3 )/S

R3∗ (O(X3 ,D3 ∪Z3 )/S )}, O(X3 ,D3 ∪Z3 )/S )]. The source of [ ] in (2.10.13.4) is L∗3 R3∗ (O(X3 ,D3 ∪Z3 )/S ) ⊗L O(X

3 ,D3 ∪Z3 )/S

L∗3 R3∗ (O(X3 ,D3 ∪Z3 )/S ).

Using the adjunction, we have a composite morphism L∗3 R3∗ (O(X3 ,D3 ∪Z3 )/S ) ⊗L O(X

3 ,D3 ∪Z3 )/S

O(X3 ,D3 ∪Z3 )/S ⊗L O(X

3 ,D3 ∪Z3 )/S

L∗3 R3∗ (O(X3 ,D3 ∪Z3 )/S ) −→

O(X3 ,D3 ∪Z3 )/S = O(X3 ,D3 ∪Z3 )/S .

Thus we have a morphism (2.10.11.2). Theorem 2.10.14 (K¨ unneth formula). (1) Let the notation be as above. Then there exists a canonical isomorphism (2.10.14.1)

log,Z1 Rf(X1 ,Z1 )/S∗ (Ecrys (O(X1 ,D1 ∪Z1 )/S ), P D1 )⊗L OS log,Z2 Rf(X2 ,Z2 )/S∗ (Ecrys (O(X2 ,D2 ∪Z2 )/S ), P D2 )

2.10 The Base Change Theorem and the K¨ unneth Formula

155

∼

log,Z3 −→ Rf(X3 ,Z3 )/S∗ (Ecrys (O(X3 ,D3 ∪Z3 )/S ), P D3 ).

(2) Let Y and fj : (Xj , Dj ∪ Zj ) −→ Y (j = 1, 2) be as in (2.10.2) (2). Set f3 := f1 ×Y f2 . Then there exists a canonical isomorphism log log,Z1 Rf(X (Ecrys (O(X1 ,D1 ∪Z1 )/S ), P D1 )⊗L OY /S 1 ,Z1 )crys∗

(2.10.14.2)

log log,Z2 Rf(X (Ecrys (O(X2 ,D2 ∪Z2 )/S ), P D2 ) 2 ,Z2 )crys∗ ∼

log log,Z3 −→ Rf(X (Ecrys (O(X3 ,D3 ∪Z3 )/S ), P D3 ). 3 ,Z3 )crys∗

Proof. (1): By virtue of the filtered cohomological descent (1.5.1) (2), we may assume that Xj (j = 1, 2) and S are affine as in the proof of [3, V Corollary 4.2.2], and hence that (Xj , Dj ∪ Zj ) (j = 1, 2) has a log smooth lift (Xj , Dj ∪ Zj ) over S. Let (X3 , D3 ∪ Z3 ) be the fiber product of (X1 , D1 ∪ Z1 ) and (X2 , D2 ∪ Z2 ) over S. Let gj : (Xj , Dj ∪ Zj ) −→ S (j = 1, 2, 3) be the structural morphism. In this case, by (2.7.5), the proof of (1) is reduced to showing an isomorphism (g1∗ Ω•X1 /S (log(D1 ∪ Z1 )) ⊗OS g2∗ Ω•X2 /S (log(D2 ∪ Z2 )), {

D2 • g1∗ PlD1 Ω•X1 /S (log(D1 ∪ Z1 )) ⊗OS g2∗ Pm ΩX2 /S (log(D2 ∪ Z2 ))}k∈Z )

l+m=k

−→ g3∗ (Ω•X3 /S (log(D3 ∪ Z3 )), P D3 ), which is easily verified. (2): (2) follows from (1) as in [3, V Theorem 4.2.1].

 

The following is the compatibility of the preweight-filtered K¨ unneth formula with the base change formula. Proposition 2.10.15. Let u be the morphism in (2.10.3). Let  mean the base unneth change of an object over S by u|S0 . Let K be the preweight-filtered K¨ unneth isomorphism isomorphism (2.10.14.1) and K  the preweight-filtered K¨ for (Xi , Di ∪ Zi ) (i = 1, 2, 3). Set log,Zi Hi := Rf(Xi ,Zi )/S∗ (Ecrys (O(Xi ,Di ∪Zi )/S ), P Di )

and

log,Z 



Hi := Rf(Xi ,Zi )/S  ∗ (Ecrys i (O(Xi ,Di ∪Zi )/S  ), P Di )

(i = 1, 2, 3). Then the following diagram is commutative: Lu∗ (K)

(2.10.15.1)

∗ −−−−→ Lu∗ H3 Lu∗ H1 ⊗L OS  Lu H2 − ⏐ ⏐ ⏐ ⏐    H1 ⊗L OS  H2

K

−−−−→

H3 .

156

2 Weight Filtrations on Log Crystalline Cohomologies

Proof. We leave the proof of (2.10.15) to the reader because the proof is a straightforward (but long) exercise by recalling the constructions of the base change isomorphism and the K¨ unneth isomorphism (cf. [3, V Proposition 4.1.3]).  

2.11 Log Crystalline Cohomology with Compact Support Let the notations be as in §2.4. Let us define a variant of a special case of the definition of the log crystalline cohomology sheaf with compact support in [85, §5] briefly (cf. [29, §2]). Let (U, T, ι, MT , δ) be an object of the log crystalline log site ((X, D ∪ Z)/S)log crys = ((X, M (D ∪ Z))/S)crys . Set MU := M (D ∪ Z)|U . Because ι : (U, MU ) −→ (T, MT ) is an exact closed immersion, MT /OT∗ = ∗ on Uzar = Tzar . Hence the defining local equation of the relative MU /OU SNCD D ∩ U on U lifts to a local section t of MT . We define an ideal D D ⊂ O(X,D∪Z)/S by the following: I(X,D∪Z)/S (T )= the ideal sheaf I(X,D∪Z)/S generated by the image of t by the structural morphism MT −→ OT . One D ) is a crystal on the restricted log can prove that Q∗(X,D∪Z)/S (I(X,D∪Z)/S crystalline site ((X, D ∪ Z)/S)log Rcrys in the same way as [85, (5.3)]. D Definition 2.11.1. We call the higher direct image sheaf Rh f(X,D∪Z)/S∗ (I(X,
D∪Z)/S ) in Szar the log crystalline cohomology sheaf with compact support

with respect to D and denote it by Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ). The local description of Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) is as follows; as⊂

sume that there exists an exact closed immersion ι : (X, D∪Z) −→ (X , D∪Z) into a smooth scheme with a relative SNCD over S such that ι induces exact ⊂ ⊂ closed immersions (X, D) −→ (X , D) and (X, Z) −→ (X , Z). Let D be the log ⊂ PD-envelope of the exact closed immersion (X, Z) −→ (X , Z) over (S, I, γ)
with structural morphism fS : D −→ S. Let u(X,D∪Z)/S : ((X, D ∪ Z)/S)log crys
zar be the canonical projection. Let F be the crystal on ((X, D −→ X ∪ Z)/S)log crys corresponding to the integrable log connection OD ⊗OX OX (−D) −→ OD ⊗OX OX (−D)Ω1X /S (log(D ∪ Z)). Then there exists a natural morD and it induces an isomorphism Q∗(X,D∪Z)/S (F) phism F −→ I(X,D∪Z)/S =

D −→ Q∗(X,D∪Z)/S (I(X,D∪Z)/S ) by [85, (5.3)]. Hence we have the following formula:

(2.11.1.1)

D Ru(X,D∪Z)/S∗ (I(X,D∪Z)/S ) D = Ru(X,D∪Z)/S∗ Q∗(X,D∪Z)/S (I(X,D∪Z)/S )

= Ru(X,D∪Z)/S∗ Q∗(X,D∪Z)/S (F) = Ru(X,D∪Z)/S∗ (F) = OD ⊗OX Ω•X /S (log(Z − D)),

2.11 Log Crystalline Cohomology with Compact Support

157

where Ω•X /S (log(Z − D)) := OX (−D)Ω•X /S (log(D ∪ Z)). As a result, we have Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) = Rh fS∗ (OD ⊗OX Ω•X /S (log(Z − D))). Let {Dλ }λ be a decomposition of D by smooth components of D. Let λ ⊂ the notations be as in §2.8. The exact closed immersion ιλj : (Dλ , Z|Dλ ) −→ (Dλj , Z|Dλj ) induces the morphism λ log∗

j (−1)j ιλcrys : O(Dλj ,Z|Dλ

(2.11.1.2)

j

)/S

⊗Z
λlog (D/S; Z) −→ j crys

λ log

log j (O(Dλ ,Z|Dλ )/S ) ⊗Z
λcrys (D/S; Z) ιλcrys∗

j · · · λk−1 ) −→ (−1)j ιλj log∗ (x) ⊗ (λ0 · · · λk−1 ). It is defined by x ⊗ (λ0 · · · λ λcrys λ log∗

j easy to check that the morphism (−1)j ιλcrys is well-defined. Set

(2.11.1.3)

:= ι(k−1)log∗ crys

k−1

{λ0 ,λ1 ,··· ,λk−1 | λi =λl (i=l)} j=0 (k−1)log

acrys∗

(O(D(k−1) ,Z|

D (k−1)

)/S

(k−1)log ⊗Z
crys (D/S; Z)) −→

)/S

(k)log ⊗Z
crys (D/S; Z)).

(k)log

acrys∗ (O(D(k) ,Z|

D (k)

(k)log∗

λ log∗

j j alog λj crys∗ ◦ ((−1) ιλcrys ) :

(k−1)log∗

is the zero. Indeed, the question The composite morphism ιcrys ◦ ιcrys is local. By taking trivializations of orientation sheaves, we can reduce this vanishing to the usual well-known case. In this section we start with the following: (k−1)log∗

is independent of the choice of the Lemma 2.11.2. The morphism ιcrys decomposition of {Dλ }λ by smooth components of D/S0 . Proof. The question is local. Let ∆ and ∆ be two decompositions of D by smooth components of D. Let x be a point of X. By (A.0.1) below, there exists an open neighborhood U of x such that ∆ |U = ∆|U . Thus we have (2.11.2).  

log Theorem 2.11.3. Let  : ((X, D ∪ Z)/S)log crys −→ ((X, Z)/S)crys be the forgetting log morphism along D ((2.3.2)). Set log,Z (2.11.3.1) Ecrys,c (O(X,D∪Z)/S ) ι(0)log∗ crys

(0)log := (O(X,Z)/S ⊗Z
crys (D/S; Z) −→

ι(1)log∗ crys

(1)log

acrys∗ (O(D(1) ,Z|

D (1)

)/S

(1)log ⊗Z
crys (D/S; Z)) −→

158

2 Weight Filtrations on Log Crystalline Cohomologies ι(2)log∗ crys

(2)log

acrys∗ (O(D(2) ,Z|

)/S D (2)

(2)log ⊗Z
crys (D/S; Z)) −→ · · · ).

Then there exists the following canonical isomorphism in D+ (Q∗(X,Z)/S (O(X,Z)/S )) : (2.11.3.2)

∼

D log,Z Q∗(X,Z)/S R∗ (I(X,D∪Z)/S ) −→ Q∗(X,Z)/S Ecrys,c (O(X,D∪Z)/S ).

Before the proof of (2.11.3), we prove two lemmas. Lemma 2.11.4. There exists a morphism of topoi

log ∪ Z)/S)log Rcrys : ((X, D Rcrys −→ ((X, Z)/S)Rcrys fitting into the following commutative diagram of topoi:

log ((X, D ∪ Z)/S)log Rcrys −−−−→ ((X, Z)/S)Rcrys ⏐ ⏐ ⏐Q ⏐ Q(X,D∪Z)/S   (X,Z)/S Rcrys

(2.11.4.1)



log ((X, D ∪ Z)/S)log crys −−−−→ ((X, Z)/S)crys .

Proof. First we show the existence of Rcrys . To show this, it suffices to see that, for an object T := (U, T, MT , ι, δ) ∈ ((X, D ∪ Z)/S)log Rcrys , the object log inv (U, T, NT , ι, δ) constructed in §2.3 belongs to ((X, Z)/S)Rcrys Zariski locally on T . (Then we can define the exact functor ∗Rcrys in the same way as ∗ in §2.3.) Let us assume that T is the log PD-envelope of the closed immersion ⊂ i : (U, (D ∪Z)|U ) −→ (U, MU ), where (U, MU ) is log smooth over S. Since the log structure MU is defined on the Zariski site of U, we have a factorization ⊂

(U, (D ∪ Z)|U ) −→ (U  , MU  ) −→ (U, MU ) of i Zariski locally on U such that the first morphism is an exact closed immersion and that the second morphism is log etale. Then T is the log PD-envelope of the first morphism. Hence we may suppose that i is an exact closed immersion. Then, by (2.1.5), we may assume that i is an admissible ⊂ closed immersion (U, (D ∪ Z)|U ) −→ (U, D ∪ Z). In this case, the log structure NTinv on T is nothing but the pull-back of the log structure on U defined by Z. Hence (T, NTinv ) is the log PD-envelope of the exact closed immer⊂ sion (U, Z|U ) −→ (U, Z). Hence (U, T, NTinv , ι, δ) belongs to ((X, Z)/S)log Rcrys Zariski locally on T . Now it is clear that we have the morphism Rcrys of topoi. It is easy to see that we have the commutative diagram (2.11.4.1).   Lemma 2.11.5. Let the notations be as in (2.11.4). Then the following natural morphism of functors Q∗(X,Z)/S R∗ −→ RRcrys∗ Q∗(X,D∪Z)/S

2.11 Log Crystalline Cohomology with Compact Support

159

for O(X,D∪Z)/S -modules is an isomorphism. Proof. By the same argument as that in the proof of (1.6.4), we are reduced to showing that, for any parasitic O(X,D∪Z)/S -module F of ((X, D ∪ Z)/S)log crys , Rq ∗ (F ) is also parasitic for any q ≥ 0. To see this, it suffices to prove that, for any object T := (U, T, MT , ι, δ) ∈ ((X, Z)/S)log Rcrys with T sufficiently small, the sheaf (Rq ∗ (F ))T on Tzar induced by Rq ∗ (F ) is equal to zero. Hence we ⊂ may assume that there exists a closed immersion i : (U, Z|U ) −→ X into an affine log smooth scheme over S such that (T, MT ) is the log PD-envelope of ⊂ i. On the other hand, let us take a closed immersion i : (U, (D ∪ Z)|U ) −→ Y into an affine log scheme which is log smooth over S. Then, for any n ∈ Z≥1 , ⊂ we have the closed immersion in : (U, (D ∪ Z)|U ) −→ X ×S Y n induced by  i ◦ (|U ) and i . Let D(n) be the log PD-envelope of the closed immersion in over (S, I, γ). Then it is isomorphic to the log PD-envelope of the closed ⊂ immersion (U, (D ∪ Z)|U ) −→ (T, MT ) ×S Y n (induced by the composite ι◦(|U ) and i ) compatible with δ, where δ is the PD-structure on Ker(OT −→ OU ) + IOT extending γ and δ. By the log version of [3, V 1.2.5], we have ˇ ), (Rq ∗ (F ))T = Rq f(U,(D∪Z)|U )/T ∗ F = Rq (ι ◦ (|U ))∗ CA(F ˇ ˇ where CA(F ) = FD(•) is the log version of the Cech-Alexander complex of F ([3, V 1.2.3]). Since F is parasitic, we have FD(n) = 0 for any n. Now we   have (Rq ∗ (F ))T = 0. Proof (of Theorem 2.11.3). Assume that we are given the data (2.4.0.1) and (k) (k) (2.4.0.2) for (X, D ∪ Z). Let b• : D• −→ X• be the natural morphism. Let log log π(X,D∪Z)/Scrys be the morphism of topoi defined in (2.4.7.4). Let π(X,Z)/Scrys be the morphism of topoi defined in (2.4.7.4) for the case D = φ. Let F• be the crystal on (X• , D• ∪Z• )/S corresponding to the integrable log connection OD• ⊗OX• OX• (−D• ) −→ OD• ⊗OX• OX• (−D• )Ω1X• /S (log(D• ∪ Z• )), where D• denotes the log PD-envelope of (X• , D• ∪ Z• ) in (X• , D• ∪ Z• ). Then we have D Q∗(X,Z)/S R∗ (I(X,D∪Z)/S ) log log,−1 D −→ Q∗(X,Z)/S R∗ Rπ(X,D∪Z)/Scrys∗ π(X,D∪Z)/Scrys (I(X,D∪Z)/S ) =

log log,−1 D −→ RRcrys∗ Rπ(X,D∪Z)/SRcrys∗ Q∗(X• ,D• ∪Z• )/S π(X,D∪Z)/Scrys (I(X,D∪Z)/S ) =

log ←− RRcrys∗ Rπ(X,D∪Z)/SRcrys∗ Q∗(X• ,D• ∪Z• )/S (F• ) =

log ←− Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S R•∗ (F• ) =

log −→ Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S R•∗ L(X• ,D• ∪Z• )/S (ΩX• /S (log(Z• − D• ))) =

log ←− Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (ΩX• /S (log(Z• − D• ))). =

160

2 Weight Filtrations on Log Crystalline Cohomologies

By the same argument as that in [27, (4.2.2) (a), (c)], the following sequence (2.11.5.1)

(0) 0 −→ Ω•X• /S (log(Z• − D• )) −→ Ω•X• /S (log Z• ) ⊗Z
zar (D• /S) ι(0)∗ •,zar

ι(1)∗ •,zar

(1) −→ b•∗ (Ω•D(1) /S (log Z• |D(1) ) ⊗Z
zar (D• /S)) −→ · · · (1)

•

•

is exact. Here we define similarly as for ιcrys . Hence Ω•X• /S (log(Z• − D• )) is quasi-isomorphic to the single complex of the following double complex (2.11.5.2) (k)∗ ι•,zar

···

(k)log∗

−−−−−→

 ⏐ d⏐

···

 ⏐

−d⏐ (0)∗

ι•,zar

(0)

(1)

Ω2X

(log Z• ) ⊗Z zar (D• /S) −−−−−→ b•∗ (Ω2 (1)

Ω1X

(log Z• ) ⊗Z zar (D• /S) −−−−−→ b•∗ (Ω1 (1)

• /S

D• /S

 ⏐ d⏐

(0)∗

ι•,zar

(0)

• /S

(1)

D• /S

 ⏐

(1)

) ⊗Z zar (D• /S))

(log Z• |

(1)

) ⊗Z zar (D• /S))

D•

 ⏐ −d⏐

 ⏐

−d⏐ (0)∗

ι•,zar

(0)

OX• ⊗Z zar (D• /S)

(1)

−−−−−→

−−−−−→

(1)∗

(1)

D•

d⏐

ι•,zar

(1)

(log Z• |

b•∗ (O

(1)

(1)

D•

⊗Z zar (D• /S))

···

−−−−−→ · · ·

 ⏐ d⏐ (2)

−−−−−→ b•∗ (Ω2 (2)

D• /S

(log Z• |

(2)∗

(2)

ι•,zar

(2)

ι•,zar

(2)

) ⊗Z zar (D• /S)) −−−−−→ · · ·

(2)

) ⊗Z zar (D• /S)) −−−−−→ · · ·

D•

 ⏐

d⏐ (1)∗

ι•,zar

(2)

−−−−−→ b•∗ (Ω1 (2)

D• /S

(log Z• |

D•

 ⏐

(2)∗

d⏐ (1)∗

ι•,zar

−−−−−→

(2)

b•∗ (O

(2) D•

(2)

⊗Z zar (D• /S))

(2)∗

ι•,zar

−−−−−→ · · · .

We claim that the following sequence (2.11.5.3) 0 −→Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(Z• − D• ))) −→ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log Z• ) (0) (D• /S)) ⊗Z
zar

(0)∗ Q∗ (X• ,Z• )/S (ι•,zar )

−→

2.11 Log Crystalline Cohomology with Compact Support

161

Q∗(X• ,Z• )/S L(X• ,Z• )/S (b•∗ (Ω•D(1) /S (log Z• |D(1) ) (1)

•

•

(1) (D• /S))) ⊗Z
zar

(1)∗ Q∗ (X• ,Z• )/S (ι•,zar )

−→

···

is exact. Indeed, the question is local and we have only to prove that the sequence (2.11.5.3) for • = i is exact for a fixed i ∈ I. As in (2.2.17), we have only to prove that the following sequence (2.11.5.4) 0 −→ ODi ⊗OXi Ω•Xi /S (log(Zi − Di )) (0) −→ ODi ⊗OXi Ω•Xi /S (log Zi ) ⊗Z
zar (Di /S) (0)∗

(1)∗

ιi,zar

ιi,zar

(1) −→ ODi ⊗OXi bi∗ (Ω•D(1) /S (log Zi |D(1) ) ⊗Z
zar (Di /S)) −→ · · · (1)

i

i

is exact. The following argument is the same as that in the proof of (2.2.17) (1). We may have cartesian diagrams (2.1.13.1) and (2.1.13.2) for SNCD Di ∪ Zi on Xi ; we assume that Di (resp. Zi ) is defined by an equation x1 · · · xt = 0 (resp. xt+1 · · · xs = 0). Set Ji := (xd+1 , . . . , xd )OXi . We may assume that there exists a positive integer N such that JiN ODi = 0. Set Xi := SpecX (OXi /Ji ) and X  := SpecS (OS [xd+1 , . . . , xd ]). Let Di i (resp. Zi ) be the closed subscheme of Xi defined by an equation x1 · · · xt = 0 (resp. xt+1 · · · xs = 0). As in [11, 3.32 Proposition], we may assume that there exists a morphism OXi [xd+1 , . . . , xd ] −→ OXi /JiN such that the induced morphism OXi [xd+1 , . . . , xd ]/J0iN −→ OXi /JiN is an isomorphism, where J0i := (xd+1 , . . . , xd ). By [11, 3.32 Proposition], ODi is locally isomorphic to the PD-polynomial algebra OXi xd+1 , . . . , xd . Let b i

(k)

(k ∈ Z>0 ) and ι i,zar (k ∈ Z≥0 ) be analogous morphisms to bi (k)∗

(k)

and

ιi,zar , respectively, for Xi , Di and Zi . Then we have an exact sequence (k)∗

(2.11.5.5)

(0) (Di /S) 0 −→ Ω•Xi /S (log(Zi − Di )) −→ Ω•Xi /S (log Zi ) ⊗Z
zar ι i,zar

ι i,zar

(0)∗

(1)∗

(1) −→ b i∗ (Ω•D (1) /S (log Zi |D (1) ) ⊗Z
zar (Di /S)) −→ · · · (1)

i

i

ΩqX  /S

(q ∈ N) is a free OS -module, applying Since OS xd+1 , . . . , xd  ⊗OX  the tensor product ⊗OS OS xd+1 , . . . , xd  ⊗OX  ΩqX  /S (q ∈ N) to the exact sequence (2.11.5.5) preserves the exactness. Because ODi ⊗OXi Ω•Xi /S (log(Zi − Di )) Ω•Xi /S (log(Zi − Di ))⊗OS

OS xd+1 , . . . , xd  ⊗OX  Ω•X  /S

162

2 Weight Filtrations on Log Crystalline Cohomologies

and because the similar formulas for ODi ⊗Xi bi∗ (Ω• (k) (k)

Di

(k)
zar (Di /S))

/S

(log Zi |D(k) ) ⊗Z i

(k ∈ N) hold, we have the exactness of (2.11.5.4). By (2.2.12) and (2.11.5.3), we have the following quasi-isomorphism

(2.11.5.6) Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(Z• − D• ))) ∼

(0) −→{Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log Z• ) ⊗Z
zar (D• /S))

−→ (Q∗(X• ,Z• )/S a•crys∗ L(D(1) ,Z• | (1)log

•

(1) )/S D•

(Ω•D(1) /S (log Z• |D(1) )) •

•

(1) (D• /S)), −d) −→ · · · }. ⊗Z
zar log to (2.11.5.6), we have Applying the direct image Rπ(X,Z)/SRcrys∗ ∼

log (2.11.5.7) Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log(Z• − D• ))) −→ log (0) {Rπ(X,Z)/SRcrys∗ Q∗(X• ,Z• )/S L(X• ,Z• )/S (Ω•X• /S (log Z• ) ⊗Z
zar (D• /S)) −→ log Q∗(X• ,Z• )/S a•crys∗ L(D(1) ,Z• | (Rπ(X,Z)/SRcrys∗ (1)log

•

D (1)

)/S

(Ω•D(1) /S (log(Z• |D(1) )) •

•

(1) ⊗Z
zar (D• /S), −d)

−→ · · · }.

(See (2.11.8) below.) By (1.6.4.1) and (2.2.20.1), the isomorphism (2.11.5.7) is nothing but an isomorphism (2.11.3.2). Now we show that the isomorphism (2.11.3.2) is independent of the data (2.4.0.1) and (2.4.0.2). Let the notations be as in the proof of (2.5.3). Let log : D+ F(Q∗(X•• ,Z•• )/S (O(X•• ,Z•• )/S )) RηRcrys∗

−→ D+ F((Q∗(X• ,Z• )/S (O(X• ,Z• )/S ))•∈I ) be a morphism of filtered derived categories in §2.5. Then we have the following commutative diagram by the cohomological descent: Rπ

log Q∗ L (Ω• (log(Z• − D• ))) X• /S (X,Z)/SRcrys∗ (X• ,Z• )/S (X• ,Z• )/S

∼ − −−−−−− →

⏐ ⏐ 

Rπ

∼ log log RηRcrys∗ Q∗ L (Ω• (log(Z•• − D•• ))) − −−−−−− → (X•• ,Z•• )/S (X•• ,Z•• )/S X•• /S (X,Z)/SRcrys∗ {Rπ

(0) log Q∗ L (Ω• (log Z• ) ⊗Z zar (D• /S)) −→ X• /S (X,Z)/SRcrys∗ (X• ,Z• )/S (X• ,Z• )/S ⏐ ⏐ 

{Rπ

(0) log log RηRcrys∗ Q∗ L (Ω• (log Z•• ) ⊗Z zar (D•• /S)) −→ (X•• ,Z•• )/S (X•• ,Z•• )/S X•• /S (X,Z)/SRcrys∗

2.11 Log Crystalline Cohomology with Compact Support (R(π

log ∗ (1)log • Q a ) L (Ω (1) (log Z• | (1) ) (X,Z)/SRcrys (X• ,Z• )/S •crys ∗ (D (1) ,Z | )/S D• D• /S • • D (1) (1) ⊗Z zar (D• /S)), −d) −→ · · · }

⏐ ⏐  (Rπ

163

log log • ∗ (1)log (Ω (1) RηRcrys∗ Q(X a L (log Z•• | (1) ) (X,Z)/SRcrys∗ •• ,Z•• )/S ••crys∗ (D (1) ,Z D•• /S D•• •• | (1) )/S •• D (1) ⊗Z zar (D•• /S)), −d) −→ · · · }.

Hence the isomorphism (2.11.3.2) is independent of the data (2.4.0.1) and (2.4.0.2).   Remark 2.11.6. Let the notation be as in the proof of (2.11.3) and let L•(X• ,Z• )/S be the complex (0) (D• /S)) {L(X• ,Z• )/S (Ω•X• /S (log Z• ) ⊗Z
zar (1)log

(a•crys∗ L(D(1) ,Z• | •

(1) )/S D•

(1) (Ω•D(1) /S (log Z• |D(1) )) ⊗Z
zar (D• /S)), −d) −→ · · · }. •

•

Then, by the proof of (2.11.3), we see that the isomorphism (2.11.3.2) is obtained by applying Q∗(X,Z)/S to the following diagram: D R∗ (I(X,D∪Z)/S )

(2.11.6.1) =

log log,−1 D −→ R∗ Rπ(X,D∪Z)/Scrys∗ π(X,D∪Z)/Scrys (I(X,D∪Z)/S ) log ←− R∗ Rπ(X,D∪Z)/Scrys∗ (F• ) =

log ←− Rπ(X,Z)/Scrys∗ R•∗ (F• ) =

log −→ Rπ(X,Z)/Scrys∗ R•∗ L(X• ,D• ∪Z• )/S (ΩX• /S (log(Z• − D• ))) =

log ←− Rπ(X,Z)/Scrys∗ L(X• ,Z• )/S (ΩX• /S (log(Z• − D• ))) log −→ Rπ(X,Z)/Scrys∗ L•(X• ,Z• )/S =

log log,−1 log,Z ←− Rπ(X,Z)/Scrys∗ π(X,Z)/Scrys Ecrys,c (O(X,D∪Z)/S ) =

log,Z ←− Ecrys,c (O(X,D∪Z)/S ).

Note that the arrows in the above diagram without = are not necessarily isomorphisms: they become isomorphic only after we apply Q∗(X,Z)/S . Note also that they become isomorphic if we apply Ru(X,Z)/S∗ or Rf(X,Z)/S∗ because Ru(X,Z)/S∗ = Ru(X,Z)/S ◦ Q∗(X,Z)/S and Rf(X,Z)/S∗ = Rf (X,Z)/S ◦ Q∗(X,Z)/S . log,Z Let PcD := {PcD,k }k∈Z be the stupid filtration on Ecrys,c (O(X,D∪Z)/S ). log,Z Then, by (2.11.3), we have a filtered complex (Ecrys,c (O(X,D∪Z)/S ), PcD ) ∈ D+ F(O(X,Z)/S ).

164

2 Weight Filtrations on Log Crystalline Cohomologies

log,Z Definition 2.11.7. We call (Ecrys,c (O(X,D∪Z)/S ), PcD ) the preweight-filtered vanishing cycle crystalline complex with compact support of O(X,D∪Z)/S (or (X, D ∪ Z)/S) with respect to D. Set log,Z log,Z (Ccrys,c (O(X,D∪Z)/S ), PcD ) := Q∗(X,Z)/S (Ecrys,c (O(X,D∪Z)/S ), PcD ). log,Z (O(X,D∪Z)/S ), PcD ) the preweight-filtered crystalline complex We call (Ccrys,c with compact support of O(X,D∪Z)/S (or (X, D ∪ Z)/S) with respect to D. Set log,Z log,Z (Ezar,c (O(X,D∪Z)/S ), PcD ) := Ru(X,Z)/S∗ (Ecrys,c (O(X,D∪Z)/S ), PcD ). log,Z We call (Ezar,c (O(X,D∪Z)/S ), PcD ) the preweight-filtered vanishing cycle zariskian complex with compact support of O(X,D∪Z)/S (or (X, D ∪ Z)/S) with respect to D. log,Z By the definition of (Ezar,c (O(X,D∪Z)/S ), PcD ), there exists the following canonical isomorphism in D+ (f −1 (OS )) :

(2.11.7.1) log,Z Ezar,c (O(X,D∪Z)/S ) ∼

(0)log −→ {Ru(X,Z)/S∗ (O(X,Z)/S ⊗Z
crys (D/S; Z)) −→ (1)

azar∗ (Ru(D(1) ,Z|

D (1)

)/S∗ (O(D (1) ,Z|D(1) )/S

(1)log ⊗Z
crys (D/S; Z)), −d)

−→ · · · }. Remark-Definition 2.11.8. Because the notation for the right hand side of (2.11.7.1) is only suggestive, we have to give the strict definition of it. Let I •• be a double complex of O(X,Z)/S -modules such that, for each nonnegative (k)log

integer k, I k• is a u(X,Z)/S∗ -acyclic resolution of (acrys∗ (O(D(k) ,Z| (k)log
crys (D/S; Z)), (−1)k d). +

D (k)

)/S

⊗Z

Then the right hand side of (2.11.7.1) is, by definition, an object in D (f −1 (OS )) which is given by the single complex of u(X,Z)/S∗ (I •• ). Let PcD := {PcD,k }k∈Z be the stupid filtration with relog,Z (O(X,D∪Z)/S ), PcD ) = spect to the first degree of u(X,Z)/S∗ (I •• ). Then (Ezar,c •• D + −1 (u(X,Z)/S∗ (I ), Pc ) in D F(f (OS )). log,Z D Corollary 2.11.9. Ezar,c (O(X,D∪Z)/S ) = Ru(X,D∪Z)/S∗ (I(X,D∪Z)/S ).

Proof. We have only to apply the direct image Ru(X,Z)/S∗ to (2.11.3.2) and to use the commutative diagram (1.6.3.1) for the case of the trivial filtration.   By applying Rf∗ to both hands of (2.11.7.1) (cf. (2.11.8)), we have a canonical isomorphism

2.11 Log Crystalline Cohomology with Compact Support

165

(2.11.9.1) ∼ (0) (D/S; Z)) Rf(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) −→ {Rf(X,Z)/S∗ (O(X,Z)/S ⊗Z
crys −→ (Rf(D(1) ,Z|

D (1)

)/S∗ (O(D (1) ,Z|D(1) )/S

(1) ⊗Z
crys (D/S; Z)), −d) −→ · · · }.

log,Z Next we prove the base change theorem of (Ecrys,c (O(X,D∪Z)/S ), PcD ).

Proposition 2.11.10. Let the notations and the assumptions be as in log,Z (2.10.2) (1). Then Rh f(X,Z)/S∗ (PcD,k Ecrys,c (O(X,D∪Z)/S )) (h, k ∈ Z) is a log,Z (O(X,D∪Z)/S )) (k ∈ Z) quasi-coherent OS -module and Rf(X,Z)/S∗ (PcD,k Ecrys,c has finite tor-dimension. Proof. This immediately follows from the spectral sequence (2.10.2.1) and [11, 7.6 Theorem], [11, 7.13 Corollary].   Theorem 2.11.11 (Base change theorem). Let the notations and the assumptions be as in (2.10.3). Then there exists the following canonical isomorphism log,Z Lu∗ Rf(X,Z)/S∗ (Ecrys,c (O(X,D∪Z)/S ), PcD )

(2.11.11.1) ∼





log,Z −→Rf(X  ,Z  )/S  ∗ (Ecrys,c (O(X  ,D ∪Z  )/S  ), PcD ).

Proof. Let I •• be a double complex of O(X,Z)/S -modules such that, for (k)log

each k ∈ N, I k• is an injective resolution of (acrys∗ (O(D(k) ,Z|

)/S ⊗Z D (k) (k)log
crys (D/S; Z)), (−1)k d). Then we have a double complex ((f u(X,Z)/S )∗ (I 0• ) −→ (f u(X,Z)/S )∗ (I 1• ) −→ · · · ). This double complex is a representative log,Z (O(X,D∪Z)/S )). For a nonnegative integer r, let τr (f of Rf(X,D∪Z)/S∗ (Ecrys,c k• u(X,Z)/S )∗ (I ) be the canonical filtration of the complex (f u(X,Z)/S )∗ (I k• ).

Because Rf(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S ) (k)

is “bounded” by (2.10.2.1) and

[11, 7.6 Theorem], and because D = ∅ if k  0 (since X is quasi⊂ compact), if r is large enough, the natural inclusions τr (f u(X,Z)/S )∗ (I k• ) −→ (f u(X,Z)/S )∗ (I k• ) are quasi-isomorphisms for all k. Hence the natural morphism s(τr (f u(X,Z)/S )∗ (I •• )) −→ s((f u(X,Z)/S )∗ (I •• )) is a quasi-isomorphism. Let d : (f u(X,Z)/S )∗ (I •l ) −→ (f u(X,Z)/S )∗ (I •+1,l ) and

d : (f u(X,Z)/S )∗ (I k• ) −→ (f u(X,Z)/S )∗ (I k,•+1 )

be the boundary morphisms. Using the functor L0 in [11, §7], we have a flat resolution Q•k∗ of τr (f u(X,Z)/S )∗ (I k∗ ) for a fixed r  0. The morphisms d and d induce morphisms dQ : Qj•l −→ Qj,•+1,l and dQ : Qjk• −→ Qjk,•+1 , respectively. We fix the boundary morphisms as follows: (−1)j dQ : Qj•l −→

166

2 Weight Filtrations on Log Crystalline Cohomologies

Qj,•+1,l and (−1)j dQ : Qjk• −→ Qjk,•+1 . We also have a natural boundary morphism Q•kl −→ Q•+1,k,l . By these three boundary morphisms, we have log,Z (O(X,D∪Z)/S )) = a triple complex Q••• . Then Lu∗ PcD,k Rf(X,D∪Z)/S∗ (Ecrys,c ∗ •k• ∗ •,k+1,• −→ u Q −→ · · · ){−k}. By the base change theorem of Kato (u Q ([54, (6.10)]), this complex is isomorphic to {(Rf(D(k) ,Z  |

D  (k)

)/S  ∗ (O(D (k) ,Z  |  (k) )/S  D

· · · } = Rf(X  ,Z  )/S  ∗ PcD



,k

(k)log

⊗Z crys (D /S  ; Z  ))){−k}, (−1)k d) −→ 

log,Z (Ecrys,c (O(X  ,D ∪Z  )/S  )).

  Proposition 2.11.12. Let the notations and the assumptions be as those in (2.10.2) (1). Assume moreover that f : X −→ S0 is proper. Then log,Z Rf(X,Z)/S∗ (Ecrys,c (O(X,D∪Z)/S ), PcD )

is a filteredly strictly perfect complex. Proof. We use the criterion (2.10.10); we have checked the condition as to the tor-dimension in (2.11.10) and we obtain the finiteness from the spectral sequence (2.10.2.1) and [11, 7.16 Theorem].   We prove the boundedness property of the log crystalline cohomology for the coefficient I(X,D∪Z)/S : Proposition 2.11.13. Let f : (X, D ∪ Z) −→ Y be as in (2.10.2) (2) and let  : (X, D ∪ Z) −→ (X, Z) be the forgetting log morphism along D. Then log D D Rfcrys∗ (I(X,D∪Z)/S ) and R∗ (I(X,D∪Z)/S ) are bounded. log D Proof. Let us first prove that Rfcrys∗ (I(X,D∪Z)/S ) is bounded. Let the notations be as that in the proof of (2.3.11). By the same argument as [3, V Th´eor`eme 3.2.4], we are reduced to proving the following claim: there exists a positive integer r such that, for any (U, T, δ) ∈ (Y /S)log crys , we have i D R fXU /T ∗ (I(X,D∪Z)/S ) = 0 for i > r. Again by the same argument as that in the proof of [3, V Th´eor`eme 3.2.4, Proposition 3.2.5], we are reduced to showing the above claim in the case where X and Y are sufficiently small affine schemes. Hence we may assume that the log structure M (D ∪ Z) associated to D ∪ Z admits a chart of the form Nb −→ M (D ∪ Z). If we take = a surjection ϕ1 : OY [Na ] −→ OX and if we set ϕ2 := id : Nb −→ Nb , we can construct a commutative diagram ψ

(2.11.13.1)

XU −−−−→ ⏐ ⏐  ι

T
⏐ ⏐g 

U −−−−→ T

2.11 Log Crystalline Cohomology with Compact Support

167

in the same way as in the proof of (2.3.11) such that ψ is an exact closed immersion and that g is log smooth. Then we can form a crystal F on ∗ D ∗ (XU /T )log crys satisfying the equality QXU /S (I(X,D∪Z)/S ) = QXU /S (F). Then i D i we have R fXU /T ∗ (I(X,D∪Z)/S ) = R fXU /T ∗ (F) and it vanishes for i > a + b log D by (2.3.11). Now we have proved that Rfcrys∗ (I(X,D∪Z)/S ) is bounded. D Let us prove that R∗ (I(X,D∪Z)/S ) is bounded. It suffices to prove that there exists a positive integer r such that, for any (U, T, δ) ∈ ((X, Z)/S)log crys , D Ri fXU /T ∗ (I(X,D∪Z)/S ) = 0 (where XU := U ×(X,Z) (X, D ∪ Z) = (U, (D ∪ Z)|U )) for i > r. We may also assume that X is sufficiently small. Hence we may assume that the log structure M (D) associated to D admits a chart of the form α : Nb −→ M (D). Let us denote the log structure on X associated to D ∪ Z by MX . If we put ϕ1 := idOX and ϕ2 := idNb , we can construct the commutative diagram (2.11.13.1) in the same way as (2.3.11) and then we can form a crystal F on (XU /T )log crys which satisfies the equality D D Q∗XU /S (I(X,D∪Z)/S ) = Q∗XU /S (F). Then we have Ri fXU /T ∗ (I(X,D∪Z)/S ) = i R fXU /T ∗ (F) and it vanishes for i > b by (2.3.11). Hence we have also proved D ) is bounded.   that R∗ (I(X,D∪Z)/S

Using (2.11.11) and (2.11.13), we can prove the following: Proposition 2.11.14. Let the notations and the assumptions as in (2.10.6). (1) The natural morphism (2.11.14.1)

log log,Z Lh∗crys Rf(X,Z)crys∗ (Ecrys,c (O(X,D∪Z)/S ), PcD ) −→ 



log,Z Rf  (X  ,Z  )crys∗ (Ecrys,c (O(X  ,D ∪Z  )/S ), PcD ) log

is an isomorphism. (2) There exists a natural isomorphism (2.11.14.2)



log D D Lh∗crys Rfcrys∗ (I(X,D∪Z)/S ) −→ Rf  crys∗ (I(X  ,D  ∪Z  )/S  ) log

which is compatible with the isomorphism (2.11.14.1). Proof. (1) follows from (2.11.12) in the same way as [3, V], [11, §7] (see also §17). Let us prove (2). One can construct the morphism (2.11.14.2) in usual log D way ([3, V Th´eor`eme 3.5.1]), using the boundedness of Rfcrys∗ (I(X,D∪Z)/S ) ⊂

which has been proved in (2.11.13). We can take data (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ) as (2.4.0.1), (2.4.0.2) for (X, D ∪ Z). If we put (X• , D• ∪ Z• ) := (X• , D• ∪ Z• ) ×S S  and (X• , D• ∪ Z• ) := (X• , D• ∪ Z• ) ×S S  , we obtain the ⊂ data (X• , D• ∪ Z• ) −→ (X• , D• ∪ Z• ) as (2.4.0.1), (2.4.0.2) for (X  , D ∪ Z  ). Then we see from the diagram (2.11.6.1) that there exists a diagram of base change morphisms

168

2 Weight Filtrations on Log Crystalline Cohomologies log log D Lh∗crys Rfcrys∗ (I(X,D∪Z)/S ) ←−−−− Lh∗crys Rf(X (F• ) • ,D• ∪Z• )crys∗ ⏐ ⏐ ⏐ ⏐   

D Rf  crys∗ (I(X  ,D  ∪Z  )/S  ) log

Rf  (X  ,D ∪Z  )crys∗ (F• ) log

←−−−−

•

•

•

log log,Z −−−−→ Lh∗crys Rf(X,Z)crys∗ (Ecrys,c (O(X,D∪Z)/S )) ⏐ ⏐  

log,Z −−−−→ Rf  (X  ,Z  )crys∗ (Ecrys,c (O(X  ,D ∪Z  )/S )), log

 where f(X• ,D• ∪Z• ) (resp. f(X    ) denotes the composite morphism of • ,D• ∪Z• ) (X• , D• ∪Z• ) −→ (X, D∪Z) with f (resp. (X• , D• ∪Z• ) −→ (X  , D ∪Z  ) with f  ) and F• is the crystal on (X• , D• ∪Z• )/S  defined in the same way as F. To prove (2), it suffices to prove that the horizontal arrows are isomorphisms. We are reduced to showing (as in [3, V 3.5.5]) that, in the situation in (2.10.3), the horizontal arrows in the following diagram of base change morphisms D ) ←−−−− Lu∗ Rf(X• ,D• ∪Z• )/S∗ (F• ) Lu∗ Rf(X,D∪Z)/S∗ (I(X,D∪Z)/S ⏐ ⏐ ⏐ ⏐   

D Rf(X  ,D ∪Z  )/S  ∗ (I(X −−−−  ,D  ∪Z  )/S  ) ←

Rf(X• ,D• ∪Z• )/S  ∗ (F• )

log,Z −−−−→ Lu∗ Rf(X,Z)/S∗ (Ecrys,c (O(X,D∪Z)/S )) ⏐ ⏐  

log,Z −−−−→ Rf(X  ,Z  )/S  ∗ (Ecrys,c (O(X  ,D ∪Z  )/S  ))

are isomorphisms. This follows from (2.11.6) because the arrows in (2.11.6.1)   become isomorphic if we apply Rf(X,Z)/S∗ . Hence we have proved (2). log,Z By using the filtered complex (Ecrys,c (O(X,D∪Z)/S ), PcD ), by (2.11.9) and by the Convention (6), we have the following spectral sequence

(2.11.14.3)

k,h−k ((X, D ∪ Z)/S) E1,c

= Rh−k f(D(k) ,Z|

D (k)

)/S∗ (O(D (k) ,Z|D(k) )/S

(k)log ⊗Z
crys (D/S; Z)))

=⇒ Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ). Let k be a fixed integer. Set 



k ,h−k E1,c ((X, D ∪ Z)/S) =

  Rh−k f(D(k ) ,Z| 0

D

(k ) )/S∗

(O(D(k ) ,Z|

(k )log

D

(k ) )/S

⊗Z
crys

(D/S; Z))

(k  ≥ k), (k  < k).

2.11 Log Crystalline Cohomology with Compact Support

169

We shall also need the following spectral sequence later 







k ,h−k ((X, D ∪ Z)/S) =⇒ E1k ,h−k = E1,c

(2.11.14.4)

(k)log

Rh−k f(X,D∪Z)/S∗ ((acrys∗ (O(D(k) ,Z| (k+1)log

acrys∗

(O(D(k+1) ,Z|

D (k+1)

)/S

D (k)

)/S

(k)log ⊗Z
crys (D/S; Z)), (−1)k d) −→

(k)log ⊗Z
crys (D/S; Z)), (−1)k+1 d) −→ · · · ).

Definition 2.11.15. We call the spectral sequence (2.11.14.3) the preweight spectral sequence of (X, D ∪ Z)/(S, I, γ) with respect to D for the log crystalline cohomology with compact support. If Z = ∅, then we call (2.11.14.3) the preweight spectral sequence of (X, D)/(S, I, γ) for the log crystalline cohomology with compact support. Let PcD,• be the filtration on Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) induced from the spectral sequence (2.11.14.3). Since PcD,• is the decreasing filtration, we D : also consider the following increasing filtration P•,c (2.11.15.1) D Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) Ph−•,c = PcD,• Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ). Proposition 2.11.16. Let the notations be as in (2.10.3). There exists a canonical morphism of spectral sequences (2.11.16.1) −k,h+k {E1,c ((X, D ∪ Z)/S) ⊗OS OS  =⇒ Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ) ⊗OS OS  } −k,h+k −→{(E1,c ((X  , D ∪ Z  )/S  )

=⇒ Rh f(X  ,D ∪Z  )/S  ∗,c (O(X  ,D ∪Z  ;Z  )/S  )} of OS  -modules. Proof. (2.11.16) immediately follows from the construction of (2.11.14.3).   k,h−k Proposition 2.11.17. The boundary morphism dk,h−k : E1,c ((X, D ∪ Z) 1 (k)log∗

k+1,h−k ((X, D∪Z)/S) is equal to the morphism induced by ιcrys /S) −→ E1,c

.

Proof. Though the proof is the same as that of [68, (5.1)], we give the proof here. We have the following triangle (2.11.17.1) −→ Rf(D(k+1) ,Z|

D (k+1)

)/S∗ (O(D (k+1) ,Z|D(k+1) )/S )[−(k

PcD,k /PcD,k+2 ((2.11.9.1)) −→ Rf(D(k) ,Z|

D (k)

+ 1)] −→

)/S∗ (O(D (k) ,Z|D(k) )/S )[−k]

+1

−→ .

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2 Weight Filtrations on Log Crystalline Cohomologies

Hence the boundary morphism dk,h−k is equal to the boundary morphism 1 Rf(D(k) ,Z|

D (k)

Rf(D(k+1) ,Z|

)/S∗ (O(D (k) ,Z|D(k) )/S )[−k]

D (k+1)

−→

)/S∗ (O(D (k+1) ,Z|D(k+1) )/S )[−k]

by the Convention (4) and (5). By the Convention (3), (4), (6) and by (l)log taking the Godement resolution of the complex acrys∗ (O(D(l) ,Z| (l) )/S∗ ⊗Z D

(l)log


crys (D/S; Z)) (l = k, k + 1) , we can check that dk,h−k is equal to the 1 (k)log∗   morphism induced by ιcrys . Proposition 2.11.18. Let u be as in (2.10.3). Let u0 : S0 −→ S0 be the induced morphism by u. Let (Y, E ∪ W ) and (X, D ∪ Z) be smooth schemes with relative SNCD’s over S0 and S0 , respectively. Let g

(Y, E ∪ W ) −−−−→ (X, D ∪ Z) ⏐ ⏐ ⏐ ⏐  

(2.11.18.1)

S0

u

−−−0−→

S0

be a commutative diagram of log schemes such that the morphism g induces morphisms g (k) : (E (k) , W |E (k) ) −→ (D(k) , Z|D(k) ) of log schemes over u0 : S0 −→ S0 for all k ∈ N. Then the isomorphism in (2.11.11.1) and the log∗ . spectral sequence (2.11.14.3) are functorial with respect to gcrys  

Proof. The proof is obvious.

The following is the K¨ unneth formula for the log crystalline cohomology sheaf with compact support Rh f(X,D∪Z)/S∗,c (O(X,D∪Z;Z)/S ). Theorem 2.11.19 (K¨ unneth formula). Let the notations be as in those in (2.10.14) (2). Then the following hold: log log,Zi (Ecrys,c (O(Xi ,Di ∪Zi )/S ), PcDi ) (i = 1, 2, 3). (1) Set Hi.c := Rf(X i ,Zi )crys∗ Then there exists a canonical isomorphism ∼

H1,c ⊗L OY /S H2,c −→ H3,c .

(2.11.19.1)

(2) There exists a canonical isomorphism (2.11.19.2) log log D1 D2 Rf(X (I(X ) ⊗L OY /S Rf(X2 ,D2 ∪Z2 )crys∗ (I(X2 ,D2 ∪Z2 )/S ) 1 ,D1 ∪Z1 )crys∗ 1 ,D1 ∪Z1 )/S ∼

log D3 −→ Rf(X (I(X ) 3 ,D3 ∪Z3 )crys∗ 3 ,D3 ∪Z3 )/S

which is compatible with the isomorphism (2.11.19.1). (3) The isomorphisms (2.11.19.1), (2.11.19.2) are compatible with the base change isomorphism (cf. (2.10.15)).

2.11 Log Crystalline Cohomology with Compact Support

171

 (k) (i) (j) Proof. (1): Let k be a nonnegative integer. Then D3 = i+j=k D1 ×S0 D2 . Hence (1) follows from the usual K¨ unneth formula. (2): We have to check that the diagram (2.11.6.1) is compatible with log K¨ unneth morphisms. ⊂ Let (Xj• , Dj• ∪ Zj• )•∈I −→ (Xj• , Dj• ∪ Zj• )•∈I (j = 1, 2) be the data (2.4.0.1) and (2.4.0.2) with ∆j• for (Xj , Dj ∪ Zj )/S0 /S. Here note that we may assume that the index set I is independent of j = 1, 2 since I0 in §2.4 can be assumed to be independent of j = 1, 2 by considering the product of two index sets. Set X3• := X1• ×S X2• , D3• := (D1• ×S X2• ) ∪ (X1• ×S D2• ) and Z3• := (Z1• ×S X2• )∪(X1• ×S Z2• ). Then we have a natural datum (X3• , D3• ∪ ⊂ Z3• )•∈I −→ (X3• , D3• ∪ Z3• )•∈I with ∆3• . Set j• := (Xj• ,Dj• ∪Zj• ,Zj• )/S (j = 1, 2, 3). Then we have the following diagram (2.11.19.3) log
(((Xj• , D j•∪Zj• )/S)crys , O(X

j• ,Dj• ∪Zj• )/S

log qj•crys log
)← −−−−−− −(((X3• , D 3•∪Z3• )/S)crys , O(X

3• ,D3• ∪Z3• )/S

)

⏐ ⏐  3•

⏐ j• ⏐  (((Xj•
, Zj• )/S)log crys , O(Xj• ,Zj• )/S )

log pj•crys ← −−−−−− −

(((X3•
, Z3• )/S)log crys , O(X3• ,Z3• )/S ) ⏐ ⏐f  (X

⏐ ⏐ j• ,Zj• )/S 

f(X

3• ,Z3• )/S


zar , O ) (S S


zar , O ) (S S

as (2.10.11.1) (j = 1, 2). Let Fj• (j = 1, 2, 3) be the crystal F• in the proof of ⊂ (2.11.3) for the admissible immersion (Xj• , Dj• ∪ Zj• ) −→ (Xj• , Dj• ∪ Zj• ). Then we have a natural morphism log ∗ log ∗ q1•crys (F1• ) ⊗O(X3• ,D3• ∪Z3• )/S q2•crys (F2• ) −→ F3•

and hence a natural morphism (2.11.19.4)

log ∗ (F1• ) ⊗L Lq1•crys O(X

3• ,D3• ∪Z3• )/S

log ∗ Lq2•crys (F2• ) −→ F3• .

Using the adjunction formula, we have a natural morphism (2.11.19.5)

∗ L Lplog 1•crys R1•∗ (F1• ) ⊗O(X

3• ,D3• ∪Z3• )/S

log ∗ −→ R3•∗ (Lq1•crys (F1• ) ⊗L O(X

3• ,D3• ∪Z3• )/S

∗ Lplog 2•crys R2•∗ (F2• ) log ∗ Lq2•crys (F2• )).

Here, note that Rj•∗ (Fj• ) (j = 1, 2) is bounded above by (2.3.12). Composing (2.11.19.5) with (2.11.19.4), we have a morphism (2.11.19.6) ∗ L Lplog 1•crys R1•∗ (F1• ) ⊗O(X

3• ,Z3• )/S

∗ Lplog 2•crys R2•∗ (F2• ) −→ R3•∗ (F3• ).

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2 Weight Filtrations on Log Crystalline Cohomologies

Now let us set • Llog j• := L(Xj• ,Dj• ∪Zj• )/S (ΩXj• /S (log(Zj• − Dj• ))),

Lj• := L(Xj• ,Zj• )/S (Ω•Xj• /S (log(Zj• − Dj• ))) and (k) log

Lkj• := aj•crys∗ L(D(k) ,Zj• | j•

(k) )/S Dj•

(k)log (Ω•D(k) /S (log Zj• |D(k) )⊗Z
crys (Dj• /S; Zj• )) j•

j•

for k ∈ N, where (k) log

aj•

(k)

: (Dj• , Zj• |D(k) ) −→ (Xj• , Zj• ) j•

is a natural morphism. Then we have a morphism (2.11.19.7) ∗ log L Lplog 1•crys R1•∗ (L1• ) ⊗O(X

3• ,Z3• )/S

∗ log log Lplog 2•crys R2•∗ (L2• ) −→ R3•∗ (L3• )

which is constructed in the same way as (2.11.19.6) and we also have natural morphisms (2.11.19.8)

∗ L Lplog 1•crys (L1• ) ⊗O(X

(2.11.19.9)

∗ log ∗ • • • Lplog 1•crys (L1• ) ⊗O(X3• ,Z3• )/S Lp2•crys (L2• ) −→ L3• .

3• ,Z3• )/S

∗ Lplog 2•crys (L2• ) −→ L3• ,

We can check that the canonical morphism Fj• −→ Llog j• induces the morphism (2.11.19.6) −→ (2.11.19.7), =

the isomorphism Rj•∗ (Llog j• ) −→ Lj• induces the isomorphism =

(2.11.19.7) −→ (2.11.19.8) and the morphism Lj• −→ L•j• induces the morphism (2.11.19.8) −→ (2.11.19.9). Hence we have the commutative diagram ∗ L Lplog 1•crys R1•∗ (F1• ) ⊗O(X

⏐ ⏐ 

3• ,Z3• )/S

(2.11.19.10)

∗ Lplog 2•crys R2•∗ (F2• ) −−−−→

R3•∗ (F3• ) ∗ log ∗ • • Lplog 1•crys (L1• ) ⊗O(X3• ,Z3• )/S Lp2•crys (L2• ) ⏐ ⏐ 

L•3• .

−−−−→

2.11 Log Crystalline Cohomology with Compact Support

173

log By applying Rπ(X to the diagram (2.11.19.10) and by using the 3 ,Z3 )/Scrys∗ adjunction formula, we obtain the commutative diagram

(2.11.19.11) ∗ log Lplog 1crys R1∗ Rπ(X

1 ,D1 ∪Z1 )/Scrys∗

(F1• ) ⊗L O

(X3• ,Z3• )/S

∗ log Lplog 2crys R2∗ Rπ(X

2 ,D2 ∪Z2 )/Scrys∗

(F2• )

⏐ ⏐ 

log R3∗ Rπ(X

3 ,D3 ∪Z3 )/Scrys∗

∗ log − −−−−− → Lplog 1crys Rπ(X

1 ,Z1 )/Scrys∗

(F3• )

(L• 1• ) ⊗O(X

3• ,Z3• )/S

∗ log Lplog 2crys Rπ(X

2 ,Z2 )/Scrys∗

(L• 2• )

⏐ ⏐ 

log Rπ(X

− −−−−− →

3 ,Z3 )/Scrys∗

(L• 3• ).

log log (Note that, by (2.3.11), Rj∗ Rπ(X (Fj• ) and Rπ(X j ,Dj ∪Zj )/Scrys∗ j ,Zj )/Scrys∗ (L•j• ) are bounded.) Let us put (k)log

Ojk := ajcrys∗ (O(D(k) ,Zj | j

(k) )/S Dj

(k)log ⊗Z
crys (Dj /S; Zj )),

where (k) log

aj

(k)

: (Dj , Zj |D(k) ) −→ (Xj , Zj ) j

is a natural morphism. Then we have a natural morphism ∗ • L Lplog 1crys (O1 ) ⊗O(X

(2.11.19.12)

3 ,Z3 )/S

∗ • • Lplog 2crys (O2 ) −→ O3 ,

=

log and the isomorphism Rπ(X (L•j• ) ←− Oj• induces the isomorphism j ,Zj )/Scrys∗ =

(the right column of (2.11.19.11)) ←− (2.11.19.12). On the other hand, we have a natural morphism (2.11.19.13) ∗ D1 L Lplog 1crys R1∗ (I(X1 ,D1 ∪Z1 )/S ) ⊗O(X

3 ,Z3 )/S

∗ D2 Lplog 2crys R2∗ (I(X2 ,D2 ∪Z2 )/S ) D3 ), −→ R3∗ (I(X 3 ,D3 ∪Z3 )/S

D

(note that Rj∗ (I(Xjj ,Dj ∪Zj )/S ) is bounded by (2.11.13)) and the morphism D

log (Fj• ) induces the morphism I(Xjj ,Dj ∪Zj ) ←− Rπ(X j ,Dj ∪Zj )/Scrys∗

(2.11.19.13) ←− (the left column of (2.11.19.11)).

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2 Weight Filtrations on Log Crystalline Cohomologies

Hence we obtain the diagram (2.11.19.14) ∗ 1 Lplog 1crys R1∗ I(X D

1 ,D1 ∪Z1 )/S

⊗L O

∗ 2 Lplog 2crys R2∗ I(X D

(X3 ,Z3 )/S

2 ,D2 ∪Z2 )/S

← −−−−− −

⏐ ⏐ 

D

R3∗ (I(X3 ∗ log Lplog 1crys R1∗ Rπ(X

1 ,D1 ∪Z1 )/Scrys∗

3 ,D3 ∪Z3 )/S

(F1• ) ⊗L O

← −−−−− −

)

(X3• ,Z3• )/S

∗ log Lplog 2crys R2∗ Rπ(X

2 ,D2 ∪Z2 )/Scrys∗

(F2• )

⏐ ⏐ 

log R3∗ Rπ(X

3 ,D3 ∪Z3 )/Scrys∗

− −−−−− →

∗ • Lplog 1crys (O1 )

(F3• )

⊗L O(X ,Z )/S 3 3

∗ • Lplog 2crys (O2 )

⏐ ⏐ 

O3• .

− −−−−− →

log By applying Rf(X to the diagram (2.11.19.14) and by using the adjunc3 ,Z3 ) tion formula, we obtain the diagram

(2.11.19.15) Rf

D D2 log log (I 1 ) ⊗L −−−−−− − OY /S Rf(X2 ,D2 ∪Z2 )crys∗ (I(X2 ,D2 ∪Z2 )/S ) ← (X1 ,D1 ∪Z1 )crys∗ (X1 ,D1 ∪Z1 )/S ⏐ ⏐ 

Rf

Rf

D log (I 3 ) (X3 ,D3 ∪Z3 )crys∗ (X3 ,D3 ∪Z3 )/S

← −−−−−− −

log log log log Rπ (F ) ⊗L OY /S Rf(X2 ,D2 ∪Z2 )crys∗Rπ(X2 ,D2 ∪Z2 )/Scrys∗(F2• ) (X1 ,D1 ∪Z1 )crys∗ (X1 ,D1 ∪Z1 )/Scrys∗ 1• ⏐ ⏐ 

Rf

log log Rπ (F3• ) (X3 ,D3 ∪Z3 )crys∗ (X3 ,D3 ∪Z3 )/Scrys∗

− −−−−−− → Rf

log log • • ) ⊗L (O1 OY /S Rf(X2 ,Z2 )crys∗ (O2 ) (X1 ,Z1 )crys∗ ⏐ ⏐ 

− −−−−−− →

Rf

log • ). (O3 (X3 ,Z3 )crys∗

The left vertical morphism in (2.11.19.15) is the morphism in the statement of (2) and the right vertical morphism is (the non-filtered version of) the morphism (2.11.19.1). Therefore, to prove (2), it suffices to prove that the horizontal morphisms in (2.11.19.15) are isomorphisms. We can check this in the same way as (2.11.14). (3): (3) immediately follows from [3, V Corollary 4.1.4], (2.11.14) and (2).  

2.12 Filtered Log de Rham-Witt Complex

175

2.12 Filtered Log de Rham-Witt Complex Let κ be a perfect field of characteristic p > 0. Let W (resp. Wn ) be the Witt ring of κ (resp. the Witt ring of length n ∈ Z>0 ). Let K0 be the fraction field of W . Let (X, D) be a smooth scheme with an SNCD over κ. In this section, as a special case, we prove that (Czar (O(X,D)/S ), P ) in the case S = Spec(Wn ) is canonically isomorphic to the filtered log de RhamWitt complex (Wn Ω•X (log D), P ) := (Wn Ω•X (log D), {Pk Wn Ω•X (log D)}k∈Z ) constructed by Mokrane ([64, 1.4]). Before proceeding on our way, we have to give the following remarks. Let ◦

s = (Spec(κ), L) be a fine log scheme. Let g : Y := (Y , M ) −→ s be a log smooth morphism of Cartier type. Let Wn Λ•Y be the “reverse” log de RhamWitt complex defined in [46, (4.1)] and denoted by Wn ωY• in [loc. cit.]. Then, in [46, (4.19)], we find the following statements: (1) There exists a canonical isomorphism ∼

RuY /Wn ∗ (OY /Wn ) −→ Wn Λ•Y

(n ∈ Z>0 ).

(2) These isomorphisms for various n ∈ Z>0 are compatible with transition morphisms with respect to n. However, as pointed out in [68, §7], the proofs of these two claims have gaps: especially we cannot find a proof of (2) in the proof of [46, (4.19)]; in [68, (7.19)], we have completed the proof of [46, (4.19)]. Hence we can use [46, (4.19)]. In addition, we have to note one more point as in [68, (7.20)] for the completeness of this book; in the definition of the embedding system in [46, p. 237], we allow the (not necessarily closed) immersion as in [82, Definition 2.2.10]. Now we come back to our situation. We keep the notations in §2.4. For example, the morphism f : X −→ Spec(κ) is smooth and D ∪ Z is a transversal SNCD on X; by abuse of notation, we also denote by f the composite ⊂ morphism X −→ Spec(κ) −→ Spec(Wn ) (n ∈ Z>0 ). Because the morphism (X, D ∪ Z) −→ (Spec(κ), κ∗ ) of log schemes is of Cartier type, we can apply the general theory of the log de Rham-Witt complexes in [46, §4] and [68, §6, §7] (cf. [48]) to our situation above. In particular, we have a canonical isomorphism (2.12.0.1)

∼

Ru(X,D∪Z)/Wn ∗ (O(X,D∪Z)/Wn ) −→ Wn Ω•X (log(D ∪ Z))

by the Zariski analogue of [46, (4.19)]=[68, (7.19)]. In other words, we have a canonical isomorphism (2.12.0.2)

∼

log,Z Czar (O(X,D∪Z)/Wn ) −→ Wn Ω•X (log(D ∪ Z))

176

2 Weight Filtrations on Log Crystalline Cohomologies

Let (Yn , En ∪ Wn ) be a lift of (X, D ∪ Z) over Wn . Then we have Wn ΩiX (log(D ∪ Z)) = Hi (Ω•Yn /Wn (log(En ∪ Wn ))). Set (2.12.0.3)

PkD Wn ΩiX (log(D ∪ Z)) = Hi (PkEn Ω•Yn /Wn (log(En ∪ Wn ))).

Definition 2.12.1. We call the filtration P D := {PkD Wn ΩiX (log(D∪Z))}k∈Z the preweight filtration on Wn ΩiX (log(D ∪ Z)) with respect to D. We shall prove, in (2.12.4) below, that PkD Wn ΩiX (log(D ∪ Z)) is independent of the choice of the lift (Yn , En ∪ Wn ). If Z = ∅, PkD Wn ΩiX (log(D ∪ Z)) is the preweight filtration defined in [64, (1.4.1)]. Here, as noted in [68, (4.3)], we use the terminology “preweight filtration” instead of the terminology “weight
zar . filtration” since Wn ΩiX (log(D ∪ Z)) is a sheaf of torsion W -modules in X To prove a filtered version of (2.12.0.2), we need some lemmas (cf. [64, 1.2, 1.4.3]). Let ∆D := {Dλ }λ (resp. ∆Z := {Zµ }µ ) be a decomposition of D (resp. Z) by smooth components of D (resp. Z). Set ∆ := {Dλ , Zµ }λ,µ . ⊂ Let ι : (X, D ∪ Z) −→ (Xn , Dn ∪ Zn ) be an admissible immersion over ⊂ Wn with respect to ∆ which induces an admissible immersion (X, D) −→ ⊂ (Xn , Dn ) (resp. (X, Z) −→ (Xn , Zn )) with respect to ∆D (resp. ∆Z ). Let ⊂ ι : (X, D ∪ Z) −→ (Yn , En ∪ Wn ) be a lift of (X, D ∪ Z) over Wn such ⊂ ⊂ that ι induces a lift (X, D) −→ (Yn , En ) (resp. (X, Z) −→ (Yn , Wn )). Assume that (Yn , En ∪ Wn ) and (Xn , Dn ∪ Zn ) are affine log schemes. Because (Xn , Dn ∪ Zn ) is log smooth over Wn , there exists a morphism of log schemes f : (Yn , En ∪ Wn ) −→ (Xn , Dn ∪ Zn ) over Wn such that f induces morphisms (Yn , En ) −→ (Xn , Dn ) and (Yn , Wn ) −→ (Xn , Zn ) and such that f ◦ ι = ι. Let DX (Xn ) be the PD-envelope of the closed immersion ι : X −→ Xn over (Spec(Wn ), pWn , [ ]). The morphism f also induces a morphism f : (Yn , pOYn ) −→ (DX (Xn ), Ker(OXn −→ OX )ODX (Xn ) ) of PDschemes. Hence f induces a morphism f∗ : ODX (Xn ) ⊗OXn Ω•Xn /Wn (log(Dn ∪ Zn )) −→ Ω•Yn /Wn (log(En ∪ Wn )) of complexes. By (2.2.17) (1), we have the following exact sequence (2.12.1.1)

Dn • 0 −→ ODX (Xn ) ⊗OXn Pk−1 ΩXn /Wn (log(Dn ∪ Zn ))

−→ ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )) −→ (k) (log Zn |Dn(k) ) ⊗Z
zar (Dn /Wn )(−k) −→ 0 ODX (Xn ) ⊗OXn Ω•−k D (k) /W n

n

by using the Poincar´e residue isomorphism with respect to Dn ((2.2.21.3)). (The compatibility of the Poincar´e residue isomorphism with the Frobenius can be checked as in [68, (9.3) (1)].) Note that the derivative d : PkDn Ω•Xn /Wn (log(Dn ∪ Zn )) −→ PkDn Ω•+1 Xn /Wn (log(Dn ∪ Zn ))

2.12 Filtered Log de Rham-Witt Complex

177

extends to a derivative of ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )) (cf. [50, 0 (3.1.4)], [54, (6.7)]). Lemma 2.12.2. The long exact sequence associated to (2.12.1.1) is decomposed into the following short exact sequences: (2.12.2.1)

Dn • 0 −→ Hq (ODX (Xn ) ⊗OXn Pk−1 ΩXn /Wn (log(Dn ∪ Zn )))

−→ Hq (ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn ))) −→ Hq−k (ODX (Xn ) ⊗OXn Ω•D(k) /W (log Zn |Dn(k) ) n

n

(k) (Dn /Wn )(−k)) ⊗Z
zar

−→ 0

(q ∈ Z).

Proof. (cf. [64, 1.2]) The problem is Zariski local. In the following, we fix an ∼ (k) isomorphism
zar (Dn /Wn ) −→ Z.
zar be a canonical morphism of topoi. For a coherent
et −→ X Let u : X ODX (Xn ) -module (resp. a coherent OYn -module) F on DX (Xn )zar  Xzar (resp. Ynzar  Xzar ), let Fet be the corresponding coherent ODX (Xn ) -module (resp. a coherent OYn -module) on DX (Xn )et  Xet (resp. Yn et  Xet ). Let us consider the following diagram (2.12.2.2)

Dn • 0 −→ Hq ((ODX (Xn ) ⊗OXn Pk−1 ΩXn /Wn (log(Dn ∪ Zn )))et )

−→ Hq ((ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )))et ) −→ Hq−k ((ODX (Xn ) ⊗OXn Ω•D(k) /W (log Zn |Dn(k) ))et ) −→ 0 n

n

(q ∈ Z), which is the etale analogue of the diagram (2.12.2.1). We prove that the diagram (2.12.2.1) is exact for any k, q ∈ Z if and only if the diagram (2.12.2.2) is exact for any k, q ∈ Z. By the Zariski analogue of [54, (6.4)], both Hq (ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) ))) n

and

n

Hq (Ω•E (k) /W (log(Wn(k) |En(k) ))) = Wn ΩqD(k) (log(Z|D(k) )) n

n

calculate Rq u(D(k) ,Z|

D (k)

)/Wn ∗ (O(D (k) ,Z|D(k) )/Wn ).

Hence we have

Hq (ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) ))) = Wn ΩqD(k) (log(Z|D(k) )) n

n

and it is a quasi-coherent Wn (OX )-module on Xzar . On the other hand, let (k) ((D(k) , Z|D(k) )/Wn )log , Z|D(k) ) over Wn crys,et be the log crystalline site of (D with respect to the etale topology and let u(D(k) ,Z|

D (k)

)/Wn ,et


: ((D(k) ,
Z|D(k) )/Wn )log crys,et −→ Xet

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2 Weight Filtrations on Log Crystalline Cohomologies

be the morphism of topoi which is defined in the same way as the morphism u(D(k) ,Z| (k) )/Wn . Then, by [54, (6.4)], both D

Hq ((ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) )))et ) n

and

n

Hq ((Ω•E (k) /W (log(Wn(k) |En(k) )))et ) = Wn ΩqD(k) (log(Z|D(k) )) n

n

calculate R u(D(k) ,Z| (k) )/Wn ,et∗ (O(D(k) ,Z| (k) )/Wn ). Hence we have Hq ((ODX D D • (log(Zn |Dn(k) )))et ) = Wn ΩqD(k) (log(Z|D(k) )) on Xet and it (Xn ) ⊗OXn Ω (k) q

Dn /Wn

is the quasi-coherent Wn (OX )-module on Xet corresponding to Hq (ODX (Xn ) ⊗OXn Ω• (k) (log(Zn |Dn(k) ))). Hence there exists the canonical isomorphism Dn /Wn

Hq (ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) )))

(2.12.2.3)

n

n

−→ Ru∗ Hq ((ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) )))et ) =

n

n



and for any etale morphism ϕ : X −→ X, there exists the following canonical isomorphism (2.12.2.4) Wn (OX  ) ⊗ϕ−1 (Wn (OX )) Hq (ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) ))) n

n

 . −→ Hq ((ODX (Xn ) ⊗OXn Ω•D(k) /W (log(Zn |Dn(k) )))et )|Xzar

=

n

n

Now let us assume that the diagram (2.12.2.2) is exact for any k, q ∈ Z. Then, by (2.12.2.3) and the induction on k, we see that each term of (2.12.2.2) is u∗ -acyclic and that u∗ ((2.12.2.2)) gives the exact sequence (2.12.2.1). On the other hand, assume that the diagram (2.12.2.1) is exact for any k, q ∈ Z. In this case, note that the morphisms in the diagram (2.12.2.1) and the long exact sequence Dn • ΩXn /Wn (log(Dn ∪ Zn )))et ) · · · −→ Hq ((ODX (Xn ) ⊗OXn Pk−1

−→ Hq ((ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )))et ) −→ Hq−k ((ODX (Xn ) ⊗OXn Ω•D(k) /W (log Zn |Dn(k) ))et ) −→ · · · n

n

are Wn (OX )-linear with respect to the natural action of Wn (OX ) = H0 (ODX (Xn ) ⊗OXn Ω•Xn /Wn ). Then, by (2.12.2.4) and the induction on k, we see that there exists the canonical isomorphism Wn (OX  ) ⊗ϕ−1 (Wn (OX )) Hq (ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )))  −→ Hq ((ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )))et )|Xzar

=

2.12 Filtered Log de Rham-Witt Complex

179

 for any etale morphism ϕ : X  −→ X. Hence the diagram (2.12.2.2)|Xzar is exact for any ϕ : X  −→ X as above, and this implies the exactness of (2.12.2.2) for any k, q ∈ Z. Hence the exactness of (2.12.2.1) for any k, q ∈ Z is equivalent to the exactness of (2.12.2.2) for any k, q ∈ Z. By the claim we have shown in the previous paragraph, we may work etale locally to prove the lemma. Hence we may assume that Xn is the scheme ⊂ Spec(Wn [x1 , . . . , xd ]) and that Dn −→ Xn is the closed immersion defined by the ideal (x1 · · · xs ) for some 0 ≤ s ≤ d. In this case, by the proof of [64, 1.2], the morphism

Z(ODX (Xn ) ⊗OXn PkDn ΩqXn /Wn (log(Dn ∪ Zn ))) −→ Z(ODX (Xn ) ⊗OXn Ωq−k (k)

Dn /Wn

(log Zn |Dn(k) ))  

is surjective on Xzar . Hence we obtain the exactness of (2.12.2.1).

By the Zariski analogue of [54, (6.4)] we have the following commutative diagram: (2.12.2.5)

∼

Rq u(X,D∪Z)/Wn ∗ (O(X,D∪Z)/Wn ) −−−−−→ Hq (OD X (Xn ) ⊗OXn Ω•X

n /Wn

  

(log(Dn ∪ Zn )))

⏐ ⏐Hq (f ∗ ) 

∼

Hq (Ω•Y

Rq u(X,D∪Z)/Wn ∗ (O(X,D∪Z)/Wn ) −−−−−→

n /Wn

(log(En ∪ Wn ))).

Lemma 2.12.3. Let k be a nonnegative integer. Then Hq (f∗ ) induces an isomorphism Hq (ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn ( log(Dn ∪ Zn ))) ∼

−→ Hq (PkEn Ω•Yn /Wn (log(En ∪ Wn ))). Proof. We have two proofs. First proof: The morphism f induces a morphism ODX (Xn ) ⊗OXn PkDn Ω•Xn /Wn (log(Dn ∪ Zn )) −→ PkEn Ω•Yn /Wn (log(En ∪ Wn )). By using the Poincar´e residue isomorphisms with respect Dn , by (2.12.2) and by induction on k, it suffices to prove that f∗ induces an isomorphism ∼

(k) Hq−k (ODX (Xn ) ⊗OXn Ω•D(k) /W (log Zn |Dn(k) ) ⊗Z
zar (Dn /Wn )(−k)) −→ n

n

(k) Hq−k (Ω•E (k) /W (log Wn |En(k) ) ⊗Z
zar (En /Wn )(−k)). n

n

(k)

By noting that DX (Xn )×Xn Dn is the PD-envelope of the closed immersion (k) D(k) −→ Dn ((2.2.16) (2)), we see that the morphism above is an isomorphism by [11, 7.1 Theorem].

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2 Weight Filtrations on Log Crystalline Cohomologies

 

Second proof: (2.12.3) immediately follows from (2.5.4) (2).

The following is a generalization of the preweight filtration on Wn ΩiX (log ⊂ D) ([64, (1.4.1)]) for an admissible closed immersion (X, D∪Z) −→ (Xn , Dn ∪ Zn ) over (Spec(Wn ), pWn , [ ]) even if Z = ∅: Corollary 2.12.4. (1) The preweight filtration on Wn Ω•X (log(D ∪ Z)) with respect to D is well-defined. More generally, {H• (ODX (Xn ) ⊗OXn PkDn Ω∗Xn /Wn (log(Dn ∪ Zn )))}k∈N induces the preweight filtration on Wn Ω•X (log(D ∪ Z)). (2) Let i be a nonnegative integer. Then (2.12.4.1)

log,Z PkD Wn ΩiX (log(D ∪ Z)) = Hi (PkD Czar (O(X,D∪Z)/Wn ))

(3) There exists the following canonical isomorphism (2.12.4.2) D ∼ • • (k) ResD : grP k Wn ΩX (log(D ∪ Z)) −→ Wn ΩD (k) (log Z|D (k) ) ⊗Z
zar (D/κ)(−k) which is compatible with the Frobenius endomorphisms. Proof. (1): We can show the well-definedness by the standard method in, e.g., [64, 3.4] and by (2.12.3). The latter statement is obvious by (2.12.3). (2): (2) is obvious by the definition (2.12.0.3). (3): (2.12.4.2) is an isomorphism of complexes of Wn -modules by (2.6.1.1) and the definition of the boundary morphism of the two log de Rham Witt complexes in (2.12.4.2). The compatibility with the Frobenius endomorphisms is obtained by the same argument as that in [68, (9.3) (1)].   ◦

Let g : Y := (Y , M ) −→ s be as in the beginning of this section. ◦

⊂

By abuse of notation, we denote Y by Y . Let ι : (Y, M ) −→ (Y, M) be a closed immersion into a fine formally log smooth scheme over (Spf(W ), W (L)), where W (L) is the canonical lift of L over Spf(W ) (cf. [46, (3.1)]). Let g
: (Y, M) −→ (Spf(W ), W (L)) be the structural morphism. Let (DY (Y), MDY (Y) ) be the log PD-envelope of the closed ⊂

immersion (Y, M ) −→ (Y, M). Set (Yn , Mn ) := (Y, M) ⊗W Wn , (DY (Yn ), MDY (Yn ) ) := (DY (Y), MDY (Y) ) ⊗W Wn and gn := g
⊗W Wn ⊂

(n ∈ Z>0 ). Let ιn : (Y, M ) −→ (Yn , Mn ) be the induced natural closed immersion. Let Wn (M ) be the canonical lift of M over Wn (Y ). Assume that there exists an endomorphism Φ of (Y, M) which is a lift of the Frobenius morphism of (Y1 , M1 ). Then there exists a morphism (2.12.4.3)

Wn (ι) : (Wn (Y ), Wn (M )) −→ (Yn , Mn )

of log schemes which has been constructed in ([68, (7.17)]) by using a log version of a lemma of Dieudonn´e-Cartier ([68, (7.10)]). In this book we only review the definition of the morphism Wn (ι). As a morphism of schemes,

2.12 Filtered Log de Rham-Witt Complex

181

Wn (ι) is well-known (e.g., [50, 0 (1.3.21), II (1.1.4)]). Let m
be a local section of M with image m ∈ Mn . Let zj (1 ≤ j ≤ n − 1) be a unique local section j
=m
p zj . Let {
sj }n−1 of 1 + pOY satisfying an equality Φ∗j (m) j=1 be a family of local sections of OY satisfying the following equalities j−1

1 + p
sp1

+ · · · + pj s
j = zj .

(The existence of {
sj }n−1 j=1 has been proved in the proof of the log version of a lemma of Dieudonn´e-Cartier ([68, (7.10)]) by using the argument in [61, VII sj ) (1 ≤ j ≤ n − 1) and s0 := 1. Then Wn (ι) as a morphism 4].) Set sj := ι∗ (
of log structures is, by definition, the following morphism: (2.12.4.4)

Wn (ι)∗ (Mn ) m −→(ι∗n (m), (s0 , . . . , sn−1 )) ∈ M ⊕ (1 + V Wn−1 (OY )) = Wn (M ).

Here we denote Wn (ι)∗ (m) simply by m. By the universality of the log PD-envelope, Φ induces a natural morphism ΦDY (Y) : (DY (Y), MDY (Y) ) −→ (DY (Y), MDY (Y) ). Following [31], let us denote by ΛiYn /Wn the sheaf of log differential forms of degree i on (Yn , Mn )/(Spec(Wn ), Wn (L)), and by Wn ΛiY the Hodge-Witt sheaf of log differential forms of degree i on (Y, M )/s. The morphism Wn (ι) induces a morphism (2.12.4.5)

ODY (Yn ) ⊗OYn Λ•Yn /Wn −→ Λ•Wn (Y )/(Wn ,Wn (L)),[

]

of complexes of gn−1 (Wn )-modules, where Λ•Wn (Y )/(Wn ,Wn (L)),[ ] is defined in • the proof of [46, (4.19)] and denoted by ωW in [loc. cit.]. By n (Y )/(Wn ,Wn (L)),[ ] [46, (4.9)] there exists a canonical morphism (2.12.4.6) Λ•Wn (Y )/(Wn ,Wn (L)),[ ] −→ H• (ODY (Yn ) ⊗OYn Λ∗Yn /Wn )(= Wn Λ•Y ). Composing (2.12.4.5) with (2.12.4.6), we have a morphism (2.12.4.7)

ODY (Yn ) ⊗OYn Λ•Yn /Wn −→ H• (ODY (Yn ) ⊗OYn Λ∗Yn /Wn ).

As usual ([50, II (1.1)]), the induced morphism by (2.12.4.7) in the derived category is independent of the choice of Y and Φ. Lemma 2.12.5. Set ϕ := Φ∗DY (Y) . Then the morphism (2.12.4.7) is equal to the morphism (ϕ/p• )n mod pn . Proof. First consider the case • = 0. Because OY is p-torsion-free, the following morphism sϕ is well-defined: (2.12.5.1)

sϕ : OY x −→ (s0 , s1 , . . . , sn−1 , . . .) ∈ W (OY ),

182

2 Weight Filtrations on Log Crystalline Cohomologies m−1

m−2

where si ’s satisfy the following equations sp0 + psp1 + · · · + pm−1 sm−1 = m−1 (x) (m ∈ Z>0 ) (e.g., [50, 0 (1.3.16)]). The morphism (2.12.4.5) for • = 0 ϕ is induced by the following composite morphism OYn

sϕ mod V n W (OY1 )

−→

Wn (OY1 ) −→ Wn (OY ).

The morphism (2.12.4.6) for • = 0 is defined by Wn (OY ) (t0 , t1 , . . . , tn−1 ) −→

(2.12.5.2) n

n−1


tp0 + p
tp1

+ · · · + pn−1
tpn−1 ∈ H0 (ODY (Yn ) ⊗OYn Λ•Yn /Wn ),

where
tj ∈ ODY (Yn ) (1 ≤ j ≤ n − 1) is a lift of tj ∈ OY ([46, (4.9)]). Since n−i n−i+1 ϕ(sp ) ≡ sp mod pn−i+1 OYn (s ∈ OYn , i ∈ {0, 1, . . . , n}), (2.12.4.7) for • = 0 is induced by the morphism x −→ ϕn (x) (x ∈ OYn ). Next, consider the case • = 1. Because the image of (2.12.4.5) is contained in the image of Wn (OY )⊗Z Wn (M )gp in Λ1Wn (Y )/(Wn ,Wn (L)),[ ] , consider the following composite morphism (2.12.5.3)

Wn (OY )⊗Z Wn (M )gp −→ Λ1Wn (Y )/(Wn ,Wn (L)),[ (2.12.4.6)

−→

]

H1 (ODY (Yn ) ⊗OYn Λ•Yn /Wn ).

The morphism (2.12.5.3) is defined by the morphisms (2.12.5.2) and d log m

−→ d log m
mod pn (m ∈ M ), where m
∈ MDY (Y) is a lift of m. Since ϕ : MDY (Y) −→ MDY (Y) is a lift of the Frobenius endomorphism, there
=m
p (1 + pa). Then exists a section a of ODY (Y) such that ϕ(m)
= p−1 d log(m
p (1 + pa)) = d log m
+ p−1 d log(1 + pa) p−1 d log ϕ(m) ∞

= d log m
+ d( (−1)i−1 (pi−1 /i)ai ). i=1

Hence d log m
mod pn = p−1 d log ϕ(m)
mod pn = ··· = p−n d log ϕn (m)
mod pn . in H1 (ODY (Yn ) ⊗OYn Λ•Yn ). Furthermore the image of 1 ⊗ (1 + V Wn−1 (OY )) by the morphism (2.12.5.3) is the zero.
Let m and {sj }n−1 j=1 be local sections in (2.12.4.4). Then the image of d log m n−1 j pn−j by the morphism (2.12.4.7) is the class of d log m
+ d log(1 + j=1 p s
j ), where s
j is a lift of sj in ODY (Yn ) . As in the argument above, the second form is exact. Hence the morphism (2.12.4.7) for • = 1 is induced from (ϕ/p)n mod pn .

2.12 Filtered Log de Rham-Witt Complex

183

When • ≥ 2, (2.12.5) follows from the definition of (2.12.4.6), from [46, (4.9)] and from the calculation above.   ⊂

Corollary 2.12.6. Let ι : (X, D ∪Z) −→ (X , D ∪Z) be an admissible immersion into a formally smooth scheme over Spf(W ) with a relative transversal SNCD over Spf(W ). Set (Xn , Dn ∪ Zn ) := (X , D ∪ Z) ⊗W Wn . Assume that there exists a lift Φ : (X , D ∪ Z) −→ (X , D ∪ Z) of the Frobenius endomorphism of (X1 , D1 ∪ Z1 ). Let DX (X ) be the PD-envelope of the closed ⊂ immersion X −→ X over (Spf(W ), pW, [ ]). Set DX (Xn ) := DX (X ) ⊗W Wn . Then the morphism (2.12.6.1)

ODX (Xn ) ⊗OXn Ω•Xn /Wn (log(Dn ∪ Zn )) −→ Wn Ω•X (log(D ∪ Z))

defined in (2.12.4.7) induces an isomorphism (2.12.6.2) (ODX (Xn ) ⊗OXn Ω•Xn /Wn (log(Dn ∪ Zn )), P Dn ) −→ (Wn Ω•X (log(D ∪ Z)), P D ) in D+ F(f −1 (Wn )). Proof. The endomorphism Φ induces an endomorphism ΦDX (X ) : DX (X ) −→ DX (X ). Set ϕ := Φ∗DX (X ) . By the definition of Wn Ω•X (log(D ∪ Z)) ([46, (4.1)]), we have Wn Ω•X (log(D ∪ Z)) = H• (ODX (Xn ) ⊗OXn Ω∗Xn /Wn (log(Dn ∪ Zn ))). By (2.12.5), the morphism (2.12.6.1) is induced by ϕn := (ϕ/p• )mod pn . By a calculation in [68, (8.1), (8.4)], ϕn preserves the preweight filtrations with respect to Dn : ϕn (ODX (X ) ⊗OX PkD Ω•X /W (log(D ∪ Z))/pn ) ⊂ H• (ODX (X ) ⊗OX PkD Ω∗X /W (log(D ∪ Z))/pn ). Hence, by using the Poincar´e residue isomorphism and by (2.12.2), it suffices to prove that ϕn induces an isomorphism ∼

(k) (2.12.6.3) ODX (Xn ) ⊗OXn Ω•D(k) /W (log Zn |Dn(k) ) ⊗Z
zar (Dn /Wn )(−k) −→ n

n

(k) H• (ODX (Xn ) ⊗OXn Ω∗D(k) /W (log Zn |Dn(k) ) ⊗Z
zar (Dn /Wn )(−k)). n

n

This immediately follows from (2.2.16) (2) and [46, (4.19)]=[68, (7.19)].

 

Lemma 2.12.7. Let Y be a scheme over Fp with a closed subscheme E. ⊂ Let U be the complement of E in Y and j : U −→ Y the open immersion. ∗ ) ∩ OY ). Let (Wn (Y ), Wn (E)) Denote by (Y, E) a log scheme (Y, j∗ (OU be a similar log scheme over Wn (Fp ) = Z/pn : (Wn (Y ), Wn (E)) := (Wn (Y ), Wn (j)∗ (Wn (OU )∗ ) ∩ Wn (OY )). Assume that the natural morphism OY −→ j∗ (OU ) is injective. Then (Wn (Y ), Wn (E)) is the canonical lift of (Y, E) in the sense of [46, (3.1)].

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2 Weight Filtrations on Log Crystalline Cohomologies

Proof. Let [ ] : OY a −→ (a, 0, . . . , 0) ∈ Wn (OY ) be the Teichm¨ uller lift. By noting that V Wn (OU ) is a nilpotent ideal sheaf of Wn (OU ) ([50, 0 (1.3.13)]), ∗ ∗ ] ⊕ Ker(Wn (OU )∗ −→ OU ). We claim we have a formula Wn (OU )∗ = [OU that ∗ Ker{Wn (j)∗ (Wn (OU )∗ ) ∩ Wn (OY ) −→ j∗ (OU )} = Ker(Wn (OY )∗ −→ OY∗ ).

The inclusion ⊃ is obvious. Let a be a local section on the left hand side. Then the image of a in OY is 1 since OY −→ j∗ (OU ) is injective. Hence we have a ∈ Wn (OY )∗ since V Wn (OY ) is a nilpotent ideal sheaf of Wn (OY ). Therefore ∗ Wn (j)∗ (Wn (OU )∗ ) ∩ Wn (OY ) = [j∗ (OU ) ∩ OY ] ⊕ Ker(Wn (OY )∗ −→ OY∗ ).

 

This equality shows (2.12.7).

Let us also consider the case of the log crystalline cohomology with compact support. Assume that Z = ∅ for the time being. Fix a total order on λ’s only in (2.12.7.1) below. In [64, Lemma 3.15.1], it is claimed that the following sequence (2.12.7.1)

0 −→ Wn Ω•X (− log D) −→ Wn Ω•X −→ Wn Ω•D(1) −→ · · ·

is exact. Let R be the Cartier-Dieudonn´e-Raynaud algebra over κ ([52, I (1.1)]). Set Rn := R/(V n R + dV n R). The second isomorphism ∼

• • Rn ⊗L R W ΩX (− log D) −→ Wn ΩX (− log D)

in [64, 1.3.3] (we have to say that the turn of the tensor product in [64, 1.3.3] is not desirable) is necessary for the proof of [64, Lemma 3.15.1]. However the proof of the second isomorphism in [64, 1.3.3] is too sketchy. In [68, §6] we have given a precise proof of the second isomorphism in [64, 1.3.3]. Hence we can use [64, Lemma 3.15.1] without anxiety, and we identify Wn Ω•X (− log D) with the following complex (2.12.7.2)

(1) Wn Ω•X −→ (Wn Ω•D(1) ⊗Z
zar (D/κ), −d) (2) −→ Wn Ω•D(2) ⊗Z
zar (D/κ) −→ · · ·

in D+ (f −1 (Wn )). We generalize the exact sequence (2.12.7.2) to the case Z = ∅ as follows. First assume that X is affine. Let (X , D ∪ Z) be a formal lift of (X, D ∪ Z) over Spf(W ) with a lift Φ : (X , D ∪ Z) −→ (X , D ∪ Z) of the Frobenius of (X, D ∪ Z). Let f
: X −→ Spf(W ) be the structural morphism. Set Ω• := Ω•X /W (log(Z − D)), Ω•1 := Ω• /pΩ• and φ = Φ∗ : Ω• −→ Ω• . Then (Ω• , φ) satisfies the axioms of (6.0.1) ∼ (6.0.5) in [68], that is,

2.12 Filtered Log de Rham-Witt Complex

185

(2.12.7.3): Ωi = 0 for i < 0. (2.12.7.4): Ωi (i ∈ N) are p-torsion-free, p-adically complete Zp -modules in C+ (f
−1 (Zp )). (2.12.7.5): φ(Ωi ) ⊂ {ω ∈ pi Ωi | dω ∈ pi+1 Ωi+1 } (i ∈ N). (2.12.7.6): There exists an Fp -linear isomorphism ∼

C −1 : Ωi1 −→ Hi (Ω•1 )

(i ∈ N).

((19.7.6) is an isomorphism in [27, (4.2.1.3)].) (2.12.7.7): A composite morphism (mod p) ◦ p−i φ : Ωi −→ Ωi −→ Ωi1 factors through Ker(d : Ωi1 −→ Ωi+1 1 ), and the following diagram is commutative: mod p

Ωi −−−−−→ ⏐ ⏐ p−i φ

Ωi1 ⏐ ⏐ −1 C

mod p

Ωi −−−−−→ Hi (Ω•1 ). Theorem 2.12.8 ([68, (6.2), (6.3), (6.4)]). (1) For a gauge  : Z −→ N ([11, 8.7 Definition]), let η : Z −→ N be the associated cogauge to  defined by  (i) + i (i ≥ 0), η(i) := (0) (i ≤ 0). Let Ω• (resp. Ω•η ) be the largest complex of Ω• whose i-th degree is contained in p(i) Ωi (resp. pη(i) Ωi ). Then the morphism φ : Ω• −→ Ω• induces a quasiisomorphism φ : Ω• −→ Ω•η . (2) Assume that Ω• and Ω•η are bounded above and that they consist of flat Zp -modules. Let M be an f
−1 (Zp ) = Zp -module. Then the morphism φ ⊗Zp idM : Ω• ⊗Zp M −→ Ω•η ⊗Zp M

(2.12.8.1)

is a quasi-isomorphism. (3) (cf. [52, III (1.5)]) Let i (resp. n) be a nonnegative (resp. positive) integer. Then (2.12.8.2) φ pi {ω ∈ Ωi | dω ∈ pn+1 Ωi+1 } {ω ∈ Ωi | dω ∈ pn Ωi+1 } ∼ ←− . i i+1 i i−1 ∈ Ω | dω ∈ pΩ } + p dΩ pn Ωi + pdΩi−1

pi+n {ω

Proof. (1): We only remark that the proof is the same as that in [11, 8.8 Theorem]. (2): By the assumption, the complex MC(φ ) ⊗Zp M is acyclic. (3): Set M := Z/pn in (2). Let  be any gauge such that (i − 1) = 1 and (i) = 0. Then (2.12.8.1) at the degree i is equal to (2.12.8.2).  

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2 Weight Filtrations on Log Crystalline Cohomologies

Set (2.12.8.3) Zni := {ω ∈ Ωi | dω ∈ pn Ωi+1 },

Bni := pn Ωi + dΩi−1 ,

Wn Ωi := Zni /Bni .

As usual (e.g., [68, §6]), we can define the following operators: F : Wn+1 Ωi −→ Wn Ωi ,

V : Wn Ωi −→ Wn+1 Ωi ,

p : Wn Ωi −→ Wn+1 Ωi

d : Wn Ωi −→ Wn Ωi+1 ,

and π : Wn+1 Ωi −→ Wn Ωi .

We only remark that p is an injective morphism induced by p−(i−1) φ : Ωi −→ Ωi (note that −(i−1) is positive if i = 0) and that π is the following composite surjective morphism ([68, (6.5)]): (2.12.8.4) proj.

i i i Wn+1 Ωi = Zn+1 /Bn+1 −→Zn+1 /(pn Z1i + dΩi−1 ) (p−i φ)−1 ∼

−→

pn Ωi

Zni proj. i −→ Zn /Bni = Wn Ωi . + pdΩi−1

Here the isomorphism p−i φ in (2.12.8.4) is given by (2.12.8.2). As usual, one can endow Wn Ωi with a natural Wn (OX )-module structure, and the following formulas hold: d2 = 0, F dV = d, F V = V F = p , F p = pF, V p = pV, dp = pd, pπ = πp = p. Set WΩ• = lim Wn Ω• . Then WΩ• is a complex of sheaves of W (OX )-modules ← − π and torsion-free W -modules in C+ (f
−1 (W )). In fact, WΩ• (resp. Wn Ω• ) is naturally an R-module (resp. Rn -module). Set  Ker(WΩi −→ Wr Ωi ) (r > 0), r i Fil WΩ := (r ≤ 0). WΩi We recall the following (cf. [50, I (3.31)], [50, I (3.21.1.5)], [62, (1.20)], [52, II (1.2)], [62, (2.16)]): Proposition 2.12.9 ([68, §6, (A), (B), (C)]). The following formulas hold: (1) Filr WΩi = V r WΩi + dV r WΩi−1 (i ∈ Z, r ∈ Z≥0 ). (2) d−1 (pn WΩ• ) = F n WΩ• . • • (3) Rn ⊗L R WΩ = Wn Ω (n ∈ Z>0 ). Proof. Here we only remark that (3) is a formal consequence of (1) and (2) (see [52, II (1.2)]).  

2.12 Filtered Log de Rham-Witt Complex

187

Let us come back to the general case. D of O(X,D∪Z)/S in §2.11. Set Recall the ideal sheaf I(X,D∪Z)/S D ) (i ∈ N). (2.12.9.1) Wn ΩiX (log(Z − D)) := Ri u(X,D∪Z)/Wn ∗ (I(X,D∪Z)/W n ∼

Zariski locally on X, we have an isomorphism Wn ΩiX (log(Z −D)) −→ Wn Ωi . It is a routine work to check that the family {Wn Ω•X (log(Z − D))}n∈Z>0 of complexes has the operators F , V , d, p and π (cf. [46, (4.1), (4.2)]) (especially one can check that p and π are well-defined by considering embedding systems of (X, D ∪ Z) over W ); in fact, W Ω•X (log(Z − D)) is naturally an R-module. Then the following holds: Proposition 2.12.10. The complex Wn Ω•X (log(Z − D)) (n ∈ Z>0 ) is quasiisomorphic to the single complex of the following double complex: ···  ⏐ d⏐

(2.12.10.1)

−−−−→

···  ⏐ −d⏐

ι(0)∗

(1)

ι(0)∗

(1)

ι(0)∗

(1)

Wn Ω2X (log Z) −−−−→ Wn Ω2D(1) (log Z|D(1) ) ⊗Z
zar (D/κ)   ⏐ ⏐ −d⏐ d⏐ Wn Ω1X (log Z) −−−−→ Wn Ω1D(1) (log Z|D(1) ) ⊗Z
zar (D/κ)   ⏐ ⏐ −d⏐ d⏐ Wn Ω0X (log Z) −−−−→ Wn Ω0D(1) (log Z|D(1) ) ⊗Z
zar (D/κ) −−−−→

···  ⏐ d⏐

−−−−→ · · ·

ι(1)∗

(2)

ι(2)∗

ι(1)∗

(2)

ι(2)∗

ι(1)∗

(2)

ι(2)∗

−−−−→ Wn Ω2D(2) (log Z|D(2) ) ⊗Z
zar (D/κ) −−−−→ · · ·  ⏐ d⏐ −−−−→ Wn Ω1D(2) (log Z|D(2) ) ⊗Z
zar (D/κ) −−−−→ · · ·  ⏐ d⏐ −−−−→ Wn Ω0D(2) (log Z|D(2) ) ⊗Z
zar (D/κ) −−−−→ · · · . Proof. The proof is the same as that of [64, Lemma 3.15.1]: by using (2.12.9) (3) and Ekedahl’s Nakayama duality, we can reduce the exactness to that for the case n = 1, and in this case, we obtain the exactness by the argument of [27, (4.2.2) (a), (c)] (cf. (2.11.5.1)).   The complex (2.12.10.1) has a stupid filtration σ k (k ∈ Z) with respect to the columns and we set PcD,k := σ k . Hence we obtain a filtered complex in

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2 Weight Filtrations on Log Crystalline Cohomologies

C+ F(f −1 (Wn )), K+ F(f −1 (Wn )) and D+ F(f −1 (Wn )), and we denote it by (2.12.10.2)

(Wn Ω•X (log(Z − D)), PcD ).

The following is the main result in this section: Theorem 2.12.11 (Comparison theorem). (1) In D+ F(f −1 (Wn )), there exists the following canonical isomorphism: (2.12.11.1)

∼

log,Z (Czar (O(X,D∪Z)/Wn ), P D ) −→ (Wn Ω•X (log(D ∪ Z)), P D ).

The isomorphisms (2.12.11.1) for n’s are compatible with two projections of both hands of (2.12.11.1). (2) In D+ F(f −1 (Wn )), there exists the following canonical isomorphism: (2.12.11.2)

∼

log (Ezar,c (O(X,D∪Z)/Wn ), PcD ) −→ (Wn Ω•X (log(Z − D)), PcD ).

If one forgets the filtrations of both hands of (2.12.11.2), one can identify the isomorphism (2.12.11.2) with the isomorphism (2.12.11.3)

∼

D Ru(X,D∪Z)/Wn ∗ (I(X,D∪Z)/W ) −→ Wn Ω•X (log(Z − D)) n

induced by the isomorphism (2.12.0.2). The isomorphisms (2.12.11.2) for n’s are compatible with two projections of both hands of (2.12.11.2). The isomorphism (2.12.11.2) is functorial for the commutative diagram (2.11.18.1) for the case S0 = Spec(κ), S = Spec(Wn ), S0 = Spec(κ ) and S  = Spec(Wn (κ )), where κ is a perfect field of characteristic p. Proof. (1): Let {Xi0 }i0 ∈I0 be an affine open covering of X. Set Di0 := D∩Xi0 and Zi0 := Z ∩ Xi0 . Then there exists an affine formal log scheme (Xi0 , Di0 ∪ Zi0 )i0 ∈I0 over Spf(W ) such that each (Xi0 , Di0 ∪Zi0 ) is a lift of the log scheme (Xi0 , Di0 ∪Zi0 ). The Frobenius morphism (Xi0 , Di0 ∪Zi0 ) −→ (Xi0 , Di0 ∪Zi0 ) lifts to a morphism Φi0 : (Xi0 , Di0 ∪ Zi0 ) −→ (Xi0 , Di0 ∪ Zi0 ). Then, using {(Xi0 , Di0 ∪Zi0 )}i0 ∈I0 , we have a diagram of log schemes (X• , D• ∪Z• )•∈I over Spf(W ) as in §2.4. Using {Φi0 }i0 ∈I0 , we have an endomorphism Φ• : (X• , D• ∪ Z• ) −→ (X• , D• ∪Z• ) of a diagram of log schemes; Φi is a lift of the Frobenius of (Xi , Di ∪ Zi ) ⊗W κ (i ∈ I). Let DX• (X• ) be the PD-envelope of the locally ⊂ closed immersion X• −→ X• over (Spec(Wn ), pWn , [ ]). Then the morphism Φ• induces a natural morphism DX• (X• ) −→ DX• (X• ). Set (X•,n , D•,n ∪ Z•,n )•∈I := (X• ⊗W Wn , (D• ⊗W Wn ) ∪ (Z• ⊗W Wn ))•∈I and set Φ•,n := Φ• mod pn . Then there exists a morphism (Wn (X• ), Wn (D• )∪ Wn (Z• )) −→ (X•,n , D•,n ∪ Z•,n ) of diagrams of log schemes, where (Wn (X• ), Wn (D• ) ∪ Wn (Z• )) is a log scheme defined in (2.12.7). By (2.12.4.7), this morphism induces a morphism (2.12.11.4) ODX• (X•,n ) ⊗OX•,n Ω•X•,n /Wn (log(D•,n ∪ Z•,n )) −→ Wn Ω•X• (log(D• ∪ Z• )).

2.12 Filtered Log de Rham-Witt Complex

189

(Note that (Wn (Xi ), Wn (Di ) ∪ Wn (Zi )) is the canonical lift of (Xi , Di ∪ Zi ) over Wn by (2.12.7); thus, by applying the filtered higher direct image of
•zar , f −1 (Wn )) −→ (X
zar , f −1 (Wn )) to the the natural morphism πzar : (X • morphism in (2.12.11.4), we obtain a morphism which is equal to a special case of a morphism defined in [46, (4.19)].) The morphism (2.12.11.4) induces a filtered quasi-isomorphism with respect to preweight filtrations. Indeed, the problem is local; in this case, it follows from (2.12.6). By applying the filtered higher direct image of πzar to (2.12.11.4), we have an isomorphism (2.12.11.1). As in the proof of (2.6.1), we can check that the morphism (2.12.11.1) is independent of the choice of the open covering of X and the lift of each open scheme. Let g : (X1 , D1 ∪ Z1 ) −→ (X2 , D2 ∪ Z2 ) be a morphism of smooth schemes with SNCD’s over κ which induces morphisms (X1 , D1 ) −→ (X2 , D2 ) and (X1 , Z1 ) −→ (X2 , Z2 ). Then, by the proof of [68, (9.3) (2)], g induces a morphism g ∗ : (Wn ΩiX2 (log(D2 ∪ Z2 )), P D2 ) −→ (Wn ΩiX1 (log(D1 ∪ Z1 )), P D1 ). Using the diagram of log schemes, we see that the proof of the functoriality of (2.12.11.1) is reduced to the local question on (Xi , Di ∪ Zi ) (i = 1, 2). In this case, by the functoriality of the morphisms (2.12.4.3) and (2.12.4.6) and by (2.12.6), we obtain the functoriality of (2.12.11.1). In [68, (7.18)] we have proved that the morphism (2.12.4.6) is compatible with two projections; as a result, the morphism (2.12.4.7) is also compatible with them. In particular, we have the following commutative diagram (2.12.11.4)

(ODX• (X•,n+1 ) ⊗OX•,n+1 Ω•X•,n+1 /Wn+1 (log(D•,n+1 ∪ Z•,n+1 )), P D ) −−−−−−−→ ⏐ ⏐ proj. (ODX• (X•,n ) ⊗OX•,n Ω•X•,n /Wn (log(D•,n ∪ Z•,n )), P D )

(2.12.11.4)

−−−−−−−→

(Wn+1 Ω•X• (log(D• ∪ Z• )), P D ) ⏐ ⏐π  (Wn Ω•X• (log(D• ∪ Z• )), P D ). Applying the direct image Rπzar∗ , we obtain the compatibility with two projections. (2): The morphism (2.12.11.4) induces a morphism (2.12.11.5) ODX• (X•,n ) ⊗OX•,n Ω•X•,n /Wn (log(Z•,n − D•,n )) −→ Wn Ω•X• (log(Z• − D• )). By (2.2.16) (2), DX• (X• ) ×X• D(k) is the PD-envelope of the locally closed (k) ⊂ (k) (k) (k) immersion D• −→ D• . Set DD(k) (D•,n ) := (DX• (X• ) ×X• D• ) ⊗W Wn . • By (2.11.9) and (2.12.10), we have the following commutative diagram:

190

2 Weight Filtrations on Log Crystalline Cohomologies ∼

Rπzar∗ (ODX• (X•,n ) ⊗OX•,n Ω•X•,n /Wn (log(Z•,n − D•,n ))) −−−−→ ⏐ ⏐ (2.12.11.6)  ∼

Rπzar∗ (Wn Ω•X• (log(Z• − D• ))) (Rπzar∗ (OD

(0) (0) (D•,n ) D•

⊗O

(0) D•,n

Ω• (0)

−−−−→ (0)

(log Z•,n |D(0) )) ⊗Z
zar (D/κ) −→ · · · ) •,n ⏐ ⏐ 

D•,n /Wn

(Rπzar∗ (Wn Ω•X• (log Z• )) ⊗Z
zar (D/κ) −→ · · · ). (0)

By the cohomological descent, the lower vertical morphism in (2.12.11.6) is equal to (0) {Ru(X,Z)/Wn ∗ (O(X,Z)/Wn ) ⊗Z
zar (D/κ) −→ · · · } −→ (0) {Wn Ω•X (log Z) ⊗Z
zar (D/κ) −→ · · · }.

By [46, (4.19)]=[68, (7.19)], this is an isomorphism. The claim as to the compatibility of the filtrations is obvious by the definitions. As usual (cf. [50, II (1.1)], §2.5), we see that the lower vertical morphism in (2.12.11.6) is independent of the choice of the open covering of X, that of the lift of each open subscheme and that of the lift of the Frobenius. The compatibility with respect to two projections follows from the following commutative diagram: Ru(D(•) ,Z|

D (•)

Ru(D(•) ,Z|

D (•)

∼

proj.

−−−−→ Wn+1 Ω•D(•) (log Z|D(•) ) ⏐ ⏐π 

)/Wn (O(D (•) ,Z|D(•) )/Wn )

−−−−→ Wn Ω•D(•) (log Z|D(•) ),

)/Wn+1 (O(D (•) ,Z|D(•) )/Wn+1 )

⏐ ⏐

∼

which we can prove in the same way as [46, (4.19)]=[68, (7.19)]. The functoriality claimed in (2) is obvious by the proof above.

 

Let i be a nonnegative integer. We conclude this section by constructing the preweight spectral sequences of Wn ΩiX (log(D∪Z)) and Wn ΩiX (log(Z−D)) with respect to D and describing the boundary morphisms between the E1 terms of the spectral sequences. The following is a generalization of [68, (5.7.1;n)]: Proposition 2.12.12. Let i be a nonnegative integer. Then there exists the following spectral sequence (k) (log Z|D(k) )⊗Z
zar (D/κ))(−k) (2.12.12.1) E1−k,h+k = H h−i (D(k) , Wn Ωi−k D (k)

=⇒ H h−i (X, Wn ΩiX (log(D ∪ Z))). The spectral sequences (2.12.12.1) for n’s are compatible with the projections.

2.12 Filtered Log de Rham-Witt Complex

191

Proof. (2.12.12.1) immediately follows from (2.12.4.2). The compatibility with the projection immediately follows from the same proof as that of [68, (8.4) (2)].   Next we describe the boundary morphism between the E1 -terms of the spectral sequence (2.12.12.1). Let the notations be before (2.8.5). Consider the following exact sequence (2.12.12.2)

0 −→ Wn ΩiDλ (log Z|Dλj ) −→ Wn ΩiDλ (log(Z ∪ Dλ )) j

Dλ

Res

j

λ

−→ ιλ∗j (Wn Ωi−1 Dλ (log Z|Dλ ))(−1) −→ 0.

We have the boundary morphism λ

λ log

(2.12.12.3) −Gλj : ιλ∗j

i Wn Ωi−1 Dλ (log ZDλ )(−1) −→ Wn ΩDλ (log Z|Dλj )[1]. j

of (2.12.12.2). Here we have used the Convention (4). As in (2.8.4.5), the morphism (2.12.12.3) induces the following morphism (2.12.12.4) λ log (−1)j Gλj : H h−i (Dλ , Wn Ωi−k (log Z|Dλ ) ⊗Z
λ,zar (D/κ))(−k) −→ H h−i+1 (Dλj , Wn Ωi+1−k (log Z|Dλj ) ⊗Z
λlog (D/κ))(−(k − 1)). j ,zar Definition 2.12.13. We call the morphism (2.12.12.4) the Gysin morphism in log Hodge-Witt cohomologies associated to the closed immersion ⊂ (Dλ , Z|Dλ ) −→ (Dλj , Z|Dλj ). Proposition 2.12.14. Set G := Then the boundary morphism (2.12.12.1) is equal to −G.

{λ0 ,...,λk−1 | λi =λj (i=j)} d−k,h+k : E1−k,h+k −→ 1

k−1

j λj j=0 (−1) Gλ . E1−k+1,h+k of

Proof. The proof is the same as that of (2.8.5).

 

Proposition 2.12.15. Let i be a nonnegative integer. Then there exists the following spectral sequence (2.12.15.1)

(k) E1k,h−k = H h−i−k (D(k) , Wn ΩiD(k) (log Z|D(k) ) ⊗Z
zar (D/κ))

=⇒ H h−i (X, Wn ΩiX (log(Z − D))). The spectral sequences (2.12.15.1) for n’s are compatible with the projections. The boundary morphism

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2 Weight Filtrations on Log Crystalline Cohomologies

(k) (2.12.15.2) dk,h−k : H h−i−k (D(k) , Wn ΩiD(k) (log Z|D(k) ) ⊗Z
zar (D/κ)) −→ 1 (k+1) H h−i−k (D(k+1) , Wn ΩiD(k+1) (log Z|D(k+1) ) ⊗Z
zar (D/κ))

is equal to ι(k)∗ . Proof. (2.12.15) immediately follows from (2.12.10.1). (The compatibility with the projection is easy to check.)   Remark 2.12.16. If X is proper over κ and if Z = ∅, the first-named author has proved the E2 -degeneration of the following spectral sequences modulo torsion ([68, (5.9)]): (k) ⊗Z
zar (D/κ))(−k) E1−k,h+k = H h−i (D(k) , W Ωi−k D (k)

=⇒ H h−i (X, W ΩiX (log D)), (k) E1k,h−k = H h−i−k (D(k) , W ΩiD(k) ⊗Z
zar (D/κ))

=⇒ H h−i (X, W ΩiX (− log D)).

2.13 Filtered Convergent F -isocrystal So far we have worked over a base scheme whose structure sheaf is killed by a power of p. We can also work over a (not necessarily affine) P -adic base in the sense of [11, 7.17 Definition], and the analogues of results in previous sections hold in this case. Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field κ of characteristics p > 0. Let W be the Witt ring of κ with fraction field K0 . Let K be the fraction field of V . For a V module M , MK denotes the tensor product M ⊗V K. Unless otherwise stated, from this section to §2.19, S denotes a p-adic formal V -scheme in the sense of [74, §1], i.e., S is a noetherian formal scheme over V with the p-adic topology such that, for any affine open formal subscheme U , there exists a surjective morphism V {x1 , . . . , xn } −→ Γ(U, OU ) of topological rings for some n. Let f : (X, D ∪ Z) −→ S denote a proper smooth morphism of p-adic formal V -schemes (e.g., V /p-schemes) of finite type with relative transversal SNCD. Following [74], for a p-adic formal scheme T /Spf(V ), set T1 := SpecT (OT /pOT ). By virtue of results in previous sections, we can give the compatibility of the weight filtrations on log crystalline cohomologies as convergent F -isocrystals with some canonical operations, e.g., the base change, the K¨ unneth formula, the functoriality. Later, in §2.19, we shall give the compatibility of them with the Poincar´e duality.

2.13 Filtered Convergent F -isocrystal

193

(1) Base change theorem Theorem 2.13.1. Let k, h be two nonnegative integers. Then there exists a convergent F -isocrystal Ekh on S/V such that DT 1

(Ekh )T = Rh f(XT1 ,ZT1 )/T ∗ (Pk

log,ZT1

Ecrys

(O(XT1 ,DT1 ∪ZT1 )/T ))K

for any p-adic enlargement T of S/V . In particular, there exists a convergent F -isocrystal Rh f∗ (O(X,D∪Z)/K ) on S/V such that Rh f∗ (O(X,D∪Z)/K )T = Rh f(XT1 ,DT1 ∪ZT1 )/T ∗ (O(XT1 ,DT1 ∪ZT1 )/T )K for any p-adic enlargement T of S/V . Proof. The base change theorem (2.10.3) and the argument in [74, (3.1)] show the existence of a p-adically convergent isocrystal Ekh . As in the proof of [74, (3.7)], we may assume that V = W ; furthermore, by the log version of [74, (3.4)], we may assume that pOS = 0. The spectral sequence in (2.9.6.3) for DT 1

Rh f(XT1 ,ZT1 )/T ∗ (Pk

log,ZT1

Ecrys

(O(XT1 ,DT1 ∪ZT1 )/T ))

shows that the Frobenius action FS∗ (Ekh ) −→ Ekh is an isomorphism. Thus   Ekh prolongs to a convergent F -isocrystal as in [74, (3.7)]. Remark 2.13.2. The existence of the convergent F -isocrystal Rh f∗ (O(X,D∪Z) /K ) is a special case of [76, Theorem 4] and [29, §2 (e), (f)]. This existence also follows from the log base change theorem ([54, (6.10)]), the bijectivity of the Frobenius [46, (2.24)], and the same proof of [74, (3.1), (3.7)]. Corollary 2.13.3. The weight filtration on Rh f∗ (O(X,D∪Z)/K ) with respect to D is a convergent F -isocrystal on S/V . That is, the image PkD Rh f∗ (O(X,D∪ h h Z)/K ) := Im(Ek −→ R f∗ (O(X,D∪Z)/K )) (k ∈ N) is a convergent F -isocrystal. Proof. The category of the convergent isocrystals on S/V is abelian ([74, (2.10)]); hence the image Im(Ekh −→ Rh f∗ (O(X,D∪Z)/K )) is a convergent isocrystal. Now, by [74, (2.18), (2.21)], we have only to prove that PkD Rh f∗ (O(X,D∪Z) ) /K gives a p-adically convergent F -isocrystal for the case V = W . The existence of the Frobenius on PkD Rh f∗ (O(X,D∪Z)/K ) is clear by the functoriality which will be stated in (2.13.9) below soon. Because the Frobenius F ’s on the E1 -terms of (2.9.6.3)⊗V K for a p-adic formal V -scheme T are isomorphisms, the Frobenius on PkD Rh f∗ (O(X,D∪Z)/K ) is also an isomorphism. This completes the proof of (2.13.3).   Remark 2.13.4. We can also develop theory of weight filtrations by virtue of theory of log convergent topoi ([82]). See [73] for details.

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2 Weight Filtrations on Log Crystalline Cohomologies

Corollary 2.13.5. Let k, h be two nonnegative integers. For any p-adic enlargement T of S/V , (2.13.5.1)

DT 1

Pk

Rh f(XT1 ,DT1 ∪ZT1 )/T ∗ (O(XT1 ,DT1 ∪ZT1 )/T )K := DT 1

Im(Rh f(XT1 ,ZT1 )/T ∗ (Pk

log,ZT1

Ecrys

(O(XT1 ,DT1 ∪ZT1 )/T ))K −→

Rh f(XT1 ,DT1 ∪ZT1 )/T ∗ (O(XT1 ,DT1 ∪ZT1 )/T )K ) is a flat OT ⊗V K-module. Proof. (2.13.5) follows from [74, (2.9)] and (2.13.3).

 

Remark 2.13.6. The flatness of Rh f(XT1 ,DT1 ∪ZT1 )/T ∗ (O(XT1 ,DT1 ∪ZT1 )/T )K is a special case of [76, Lemma 36] and [29, §2 (e), (f)]. (2) K¨ unneth formula Theorem 2.13.7. Let (Xj , Dj ∪ Zj ) (j = 1, 2) be a log scheme stated in the beginning of this section. Let (X3 , D3 ∪ Z3 ) be the product (X1 , D1 ∪ Z1 ) ×S (X2 , D2 ∪ Z2 ) in the category of fine log schemes. Then the there exists the following canonical isomorphism  (2.13.7.1) Ri f∗ (O(X1 ,D1 ∪Z1 )/K )⊗OS/K Rj f∗ (O(X2 ,D2 ∪Z2 )/K ) i+j=h

−→ Rh f∗ (O(X3 ,D3 ∪Z3 )/K ) of convergent F -isocrystals on S/V which is compatible with the weight filtrations with respect to D1 , D2 and D3 . Proof. The existence of the canonical isomorphism in (2.13.7.1) as weightfiltered convergent F -isocrystals immediately follows from (2.10.15).   (3) Log crystalline cohomology sheaf with compact support Using (2.11.11) and (2.11.19), we obtain the following as in (1) and (2). Theorem 2.13.8. Let k, h be two nonnegative integers. h (1) There exists a convergent F -isocrystal Ek,c on S/V such that DT 1

h (Ek,c )T = Pk

Rh f(XT1 ,DT1 ∪ZT1 )/T ∗,c (O(XT1 ,DT1 ∪ZT1 ;ZT1 )/T )K

for any p-adic enlargement T of S/V . In particular, there exists a convergent F -isocrystal Rh f∗,c (O(X,D∪Z;Z)/K ) on S/V such that Rh f∗,c (O(X,D∪Z;Z)/K )T = Rh f(XT1 ,DT1 ∪ZT1 )/T ∗,c (O(XT1 ,DT1 ∪ZT1 ;ZT1 )/T )K for any p-adic enlargement T of S/V .

2.13 Filtered Convergent F -isocrystal

195

(2) The OT ⊗V K-module DT 1

Pk

Rh f(XT1 ,DT1 ∪ZT1 )/T ∗,c (O(XT1 ,DT1 ∪ZT1 ;ZT1 )/T )K

is flat for any p-adic enlargement T of S/V . (3) Let (Xj , Dj ∪ Zj ) (j = 1, 2) be as in (2.13.7). Then there exists the following canonical isomorphism  Ri f∗,c (O(X1 ,D1 ∪Z1 ;Z1 )/K ) ⊗OS/K Rj f∗,c (O(X2 ,D2 ∪Z2 ;Z2 )/K ) i+j=h ∼

−→ Rh f∗,c (O(X3 ,D3 ∪Z3 ;Z3 )/K ) of convergent F -isocrystals on S/V which is compatible with the weight filtrations with respect to D1 , D2 and D3 . (4) Functoriality Theorem 2.13.9. Let f : (X, D ∪ Z) −→ S be as in the beginning of this section. Let k, h be nonnegative integers. Then the following hold: (1) The convergent F -isocrystal PkD Rh f∗ (O(X,D∪Z)/K ) (k ∈ Z) is functorial. (2) The convergent F -isocrystal PkD Rh fc∗ (O(X,D∪Z;Z)/K ) (k ∈ Z) is functorial with respect to the obvious analogue of the morphism in (2.11.18). Proof. (1) and (2) immediately follow from (2.9.1) and (2.11.18), respectively.   (5) Gysin morphisms Proposition 2.13.10. The Gysin morphism (2.8.4.5) induces the following morphism λ

(2.13.10.1) (−1)j Gλj : Rh−k f∗ (O(Dλ ,Z|Dλ )/K ⊗Z
λlog (D/K; Z))(−k) −→ Rh−k+2 f∗ (O(Dλj ,Z|Dλ

j

)/K

⊗Z
λlog (D/K; Z))(−(k − 1)). j

of convergent F -isocrystals on S/V . Here Rh f∗ (O(Dλ ,Zλ |Dλ )/K ⊗Z
λlog (D/K; Z)) is a convergent F -isocrystal on S/V such that Rh f∗ (O(Dλ ,Zλ |Dλ )/K ⊗Z

log
λlog (D/K; Z))T = Rh fXT1 /T ∗ (O((Dλ )T1 ,(Zλ )T1 |(Dλ )T /T ) ⊗Z
λcrys (DT1 /T ; 1 ZT1 )) for a p-adic enlargement T of S/V .

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2 Weight Filtrations on Log Crystalline Cohomologies

Proof. (2.13.10) immediately follows from (2.8.4).

 

Using (1), (3), (4) and (5), we obtain the following: Theorem 2.13.11. Let Rh f∗ (O(D(k) ,Z| (k) )/K ⊗Z
(k)log (D/K; Z)) be a conD vergent F -isocrystal on S/V such that Rh f∗ (OD(k) /K ⊗Z
(k)log (D/K; Z))T = Rh fXT1 /T ∗ (O(D(k) ,Z| T1

(k) )/T DT 1

(k)log ⊗Z
crys (DT1 /T ; ZT1 ))

for any p-adic enlargement T of S/V . Then the following hold: (1) There exist the following weight spectral sequences of convergent F -isocrystals

(2.13.11.1)

E1−k,h+k ((X, D ∪ Z)/K) = Rh−k f∗ (O(D(k) ,Z|

D (k)

)/K

⊗Z
(k)log (D/K; Z))(−k)

=⇒ Rh f∗ (O(X,D∪Z)/K ), (2.13.11.2)

k,h−k E1,c ((X, D ∪ Z)/K)

= Rh−k f∗ (O(D(k) ,Z|

D (k)

)/K

⊗Z
(k)log (D/K; Z))

=⇒ Rh f∗,c (O(X,D∪Z;Z)/K ). The boundary morphism of (2.13.11.1) (resp. (2.13.11.2)) is given by −G (resp. ι(k)∗ ) induced by the morphism in (2.8.5) (resp. (2.11.1.3)). (2) The spectral sequences (2.13.11.1) and (2.13.11.2) are functorial with respect to the obvious analogue of the morphism in (2.9.0.1) and (2.11.18), respectively. Proof. (1): (1) follows from (2.9.6.2) and (2.11.14.3). (2): Obvious.

 

Definition 2.13.12. In the case Z = ∅, we call (2.13.11.1) (resp. (2.13.11.2)) the p-adic weight spectral sequence of Rh f∗ (O(X,D)/K ) (resp. Rh f∗,c (O(X,D)/K )).

2.14 Specialization Argument in Log Crystalline Cohomology Let us recall a specialization argument of Deligne-Illusie in log crystalline cohomologies (cf. [49, (3.10)], [68, §3]) for later sections §2.15 and §2.18.

2.14 Specialization Argument in Log Crystalline Cohomology

197

◦

Let p be a prime number. Let T be a noetherian formal scheme with ◦

an ideal sheaf of definition aOT , where a is a global section of Γ (T , OT ). Assume that there exists a positive integer n such that pOT = an OT . Let ◦

T be a fine formal log scheme with underlying formal scheme T . Assume that OT is a-torsion-free, that is, the endomorphism a × idOT ∈ EndOT (OT ) is injective, and that the ideal sheaf aOT has a PD-structure γ. We call T = (T, aOT , γ) above an adic fine formal log PD-scheme. We define the notion of a morphism g  : T  −→ T of adic fine formal log PD-schemes in the following way: the morphism g  is nothing but a morphism of formal fine log PD-schemes, and T  is a -adically complete and separated and a -torsion-free, where a := g ∗ (a). In this section we assume that, for each affine open set Spf(R) of T , aR is a prime ideal and that the localization ring Ra at the ideal aR is a discrete valuation ring. Let H be an OT -module of finite type. Since Ra is a PID, there exists a non-empty open log formal subscheme T  of T such that there exists an isomorphism H|T   OTr  ⊕ Htor , where Htor is a direct sum of OT  -modules OT  /ae (e ∈ Z>0 ) (Deligne’s remark ([49, (3.10)])). Let E be an a-torsion-free OT -module. Then, as in [68, (3.1)], it is easy to see that (2.14.0.1)

T  (H|  , E|  ) = 0 Tor O T T r

(∀r ∈ Z>0 )

and (2.14.0.2)

Tor gr

−1

(OT  )

(g −1 (H|T  ), OT  ) = 0

(∀r ∈ Z>0 )

for any morphism g : T  −→ T  of adic fine formal log PD-schemes. Set T1 := SpecT (OT /a), and set T1 := SpecT  (OT  /a) for an open log formal subscheme T  of T . Let f : X −→ T1 be a proper log smooth integral morphism. By the finiteness of log crystalline cohomologies (cf. [11, 7.24 Theorem]), there exists a non-empty open log formal subscheme T  of T such that (2.14.0.3)

T  (Rh f Tor O XT  /T  ∗ (OXT  /T  ), E|T  ) = 0 (∀r ∈ Z>0 ) r 1

1

for any a-torsion-free OT -module E and for any h ∈ Z. Assume furthermore that the log structures on X, T are fs. Let IX/T be the ideal sheaf on OX/T ◦

defined in [85, §5]. (In [85, §5], IX/T is defined under the condition that T is equal to Spec Wm (κ) (κ is a perfect field of characteristic p > 0), the log structure on T is associated to the morphism N 1 → b ∈ Wm (κ) for some b and that the morphism f is universally saturated. However, for the definition of IX/T , we do not need these assumptions.) Set Rh fX/T ∗,c (OX/T ) := Rh fX/T ∗ (IX/T ). One can see that IX/S is a crystal on the restricted log crystalline site (X/T )log Rcrys as in [85, (5.3)] and that, for any log smooth integral lift X −→ T of f , the sheaf (IX/T )X is flat over OT by [85, (2.22)]. By using

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2 Weight Filtrations on Log Crystalline Cohomologies

these facts, we see that the log version of the proofs of [11, (7.8), (7.13), (7.16), (7.24)] and [74, (3.3)] work for the coefficient IX/S . Hence Rh fX/T ∗,c (OX/T ) is a perfect complex of OT -modules and it satisfies the base change property. Therefore, if T  is sufficiently small, we have T  (Rh f (2.14.0.4) Tor O XT  /T  ∗,c (OXT  /T  ), E|T  ) = 0 r 1

(∀r ∈ Z>0 , ∀h ∈ Z).

1

Proposition 2.14.1. Let T = (T, aOT , γ) be as above. Let g : T  −→ T  be a morphism from an adic fine formal log scheme into an open log formal subscheme of T . If T  is small enough, then the following hold: (1) The canonical morphism g ∗ Rh fXT  /T  ∗ (OXT  /T  )−→Rh fXT  /T  ∗ (OXT  /T  ) 1

1

1

1

is an isomorphism of OT  -modules. (2) The canonical morphism g ∗ Rh fXT  /T  ∗,c (OXT  /T  )−→Rh fXT  /T  ∗,c (OXT  /T  ) 1

1

1

1

is an isomorphism of OT  -modules. Proof. We may assume that (2.14.0.2), (2.14.0.3) and (2.14.0.4) hold. (1): As in [68, (3.2)], we immediately obtain (1) using the existence of a strictly perfect complex of OT  -modules representing RΓ (XT1 /T  , OXT  /T  ) 1 (cf. [11, 7.14 Definition, 7.24.3 Theorem]), using (2.14.0.2) and (2.14.0.3), and using the log base change theorem ([54, (6.10)], cf. [74, (3.3)]). (2): By the facts described before (2.14.0.4), the same proof as that of (1) works.   We will use the following proposition in §2.18 below. Proposition 2.14.2. Let T be an adic formal scheme. Let g : T  −→ T  be a morphism from an adic scheme into an open formal subscheme of T . Let f : (X, D ∪ Z) −→ T1 be a proper smooth scheme with a relative SNCD over T1 . If T  is small enough, then the following hold: (1) The canonical morphism DT 

g ∗ Pk

1

Rh f(X,D∪Z)T  /T  ∗ (O(X,D∪Z)T  /T  ) −→ 1

DT 

Pk

1

1

Rh f(X,D∪Z)T  /T  ∗ (O(X,D∪Z)T  /T  ) 1

1

is an isomorphism. (2) The canonical morphism DT 

g ∗ Pk

1

Rh f(X,D∪Z)T  /T  ∗,c (O(X,D∪Z;Z)T  /T  ) −→ 1

1

2.15 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Variety DT 

Pk

1

199

Rh f(X,D∪Z)T  /T  ∗,c (O(X,D∪Z;Z)T  /T  ) 1

1

is an isomorphism Proof. By (2.9.6.2), there exist the following two spectral sequences (2.14.2.1) E1−k,h+k = Rh−k f(D(k) ,Z|

)  /T D (k) T1

∗

(O(D(k) ,Z|

)  /T D (k) T1



(k)log ⊗Z
crys (DT1 /T  ; ZT1 ))(−k)

=⇒ Rh f(X,D∪Z)T  /T  ∗ (O(X,D∪Z)T  /T  ), 1

(2.14.2.2) E1−k,h+k = Rh−k f(D(k) ,Z|

1

)  /T  ∗ D (k) T1

(O(D(k) ,Z|

)  /T  D (k) T1

(k)log ⊗Z
crys (DT1 /T  ; ZT  ))(−k)

=⇒ Rh f(X,D∪Z)T  /T  ∗ (O(X,D∪Z)T  /T  ). 1

1

By (2.9.1) (2), there exists a canonical morphism g −1 ((2.14.2.1)) ⊗g−1 (OT  ) OT  −→ (2.14.2.2). Then, by (2.14.0.3), there exists a non-empty open formal subscheme T  such that T  (Rh f T or O (D (k) ,Z| r

)  /T D (k) T1

∗

(O(D(k) ,Z|

)  /T D (k) T1



), E|T  ) = 0

for any OT -module E without a-torsion and for all r ∈ Z>0 . Hence we have an isomorphism ∼

g −1 E1−k,h+k ((X, D∪Z)T  1 /T  )⊗g−1 (OT  ) OT  −→ E1−k,h+k ((X, D∪Z)T1 /T  ) as in the proof of (2.14.1) (1), and therefore the morphism in (1) is an isomorphism. The proof of (2) is the same as that of (1).  

2.15 The E2 -degeneration of the p-adic Weight Spectral Sequence of an Open Smooth Variety Let κ be a perfect field of characteristic p > 0. Let W be the Witt ring of κ. Let K0 be the fraction field of W . In [68, (5.2)] we have proved the E2 degenerations modulo torsion of the weight spectral sequences (2.9.6.2) and (2.11.14.3) when Z = ∅ and S = Spf(W ). To prove the degenerations, we

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2 Weight Filtrations on Log Crystalline Cohomologies

have used a somewhat tricky argument in [68, (5.2)] (cf. [68, (3.2), (3.4), (3.5), (3.6)]) based on Deligne’s remark ([49, 3.10]). Though we also use Deligne’s remark in this book, the proof in this section is not tricky by virtue of the existence of the weight spectral sequences (2.9.6.2) and (2.11.14.3) over a general base (cf. [68, (3.7)]). Let (X, D) be a proper smooth scheme with an SNCD over κ. By [40, 3, (8.9.1) (iii), (8.10.5)] and [40, 4, (17.7.8)] , there exist a smooth affine scheme S1 over a finite field Fq and a model (X , D) of (X, D) over S1 . By a standard deformation theory ([41, III (6.10)]), there exists a formally smooth scheme S such that S ⊗W (Fq ) Fq = S1 . Let T be an affine open subscheme of S, and set T1 := T ⊗W (Fq ) Fq . Take a closed point t of T1 . The point t is the spectrum of a finite field κt . We fix a lift FT : T −→ T of the Frobenius(=p-th power morphism) FT1 of T1 . Then we have the Teichm¨ uller lift Γ (T, OT ) −→ W (κt ) (resp. Γ (T, OT ) −→ W ) of the morphism Γ (T1 , OT1 ) −→ κt (resp. Γ (T1 , OT1 ) −→ κ) (e.g., [50, 0 1.3]). The rings W (κt ) and W become Γ (T, OT )-algebras by these lifts. To prove the E2 -degenerations, we prove some elementary lemmas. Let A be a p-adically complete and separated p-torsion-free ring with a lift f of the Frobenius endomorphism of A1 := A/p. Then there exists a unique section τ
: A −→ W (A) of the projection W (A) −→ A such that τ
◦ f = F ◦ τ
, where F is the Frobenius of W (A) (e.g., [50, 0 (1.3.16)]). This morphism induces morphisms τ : A −→ W (A1 ) and τn : A/pn −→ Wn (A1 ). Then the following holds: Lemma 2.15.1. If A1 is reduced, then the morphism τ : A −→ W (A1 ) is injective. Proof. Let F∗n (A1 ) be the restriction of scalars of A1 by the n-th power of the Frobenius endomorphism of A1 . By the assumption, the morphism F n : A1 −→ F∗n (A1 ) is injective. (2.15.1) follows from the following commutative diagram in [50, 0 (1.3.22)]: A1 ⏐ ⏐ pn 

Fn

−−−−→

F∗n (A1 ) ⏐ ⏐ V n 

(∀n ∈ N)

n

gr τn+1

pn A/pn+1 A −−−−−→ V n W (A1 )/V n+1 W (A1 ).   Lemma 2.15.2. (1) Let B be a commutative ring whose Jacobson radical rad(B) is the zero. Let M(B)  be the set of the maximal ideals of B. Then the morphism W (B) −→ W (B/m) is injective. m∈M(B)

(2) Let C be a commutative ring with unit element and let D be a smooth C-algebra. If rad(C) = 0, then rad(D) = 0.

2.15 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Variety

201

 Proof. (1): By the assumption, the natural morphism B −→ B/m is m∈M(B)   injective. Thus W (B) −→ W ( B/m) = W (B/m) is injective. m∈M(B) m∈M(B)  (2): Let {fi }i be a family of elements of D  such that Spec(D) = i Spec ). Then the natural morphism D −→ i Dfi is injective since D −→ (Df i Dm is injective. Thus the problem is local; we may assume that there m∈M(D) √ √ exists a finite etale morphism C[X1 , . . . , Xm ] −→ D. Let √ ( 0)C and ( 0)D be the √ nilpotent radicals of C and D, respectively. Since ( 0)C ⊂ rad(C) = 0, ( 0)C = 0. Hence C is√a Jacobson ring and D is also by [13, V §3, n◦ 4, Since C[X1 , . . . , Xm ] is reduced, D Theorem 3]. Therefore ( 0)D = rad(D). √   is also by [41, I Proposition 9.2]. Hence ( 0)D = 0. Corollary 2.15.3. Let κ be a perfect field of characteristic p > 0. Let A be a p-adically complete and separated formally smooth algebra over W (κ ) witha lift of the Frobenius morphism of A1 . Then the morphism A −→ W (A1 /m) is injective. m∈M(A1 )

 

Proof. (2.15.3) follows from (2.15.1) and (2.15.2).

Theorem 2.15.4 ([68, (5.2)]). If Z = ∅ and S = Spf(W ), then (2.9.6.2) and (2.11.14.3) degenerate at E2 modulo torsion. Proof. For a W (Fq )-module M , MK0 (Fq ) denotes M ⊗W (Fq ) K0 (Fq ). First we prove (2.15.4) for (2.9.6.2). Replace T by a sufficiently small affine open sub log formal scheme in order that, for any h, k ∈ Z, r ∈ Z>0 , Er−k,h+k ((XT1 , DT1 )/T ) has the form OT⊕n ⊕ N (n ∈ N), where N is a direct sum of modules of type OT /pe (e ∈ Z>0 ). Then we have Tor gs

−1

(OT )

(g −1 Er−k,h+k ((XT1 , DT1 )/T ), OT  ) = 0

(∀s ∈ Z>0 )

for any morphism g : T  −→ T of p-adic fine log PD-schemes and for any h, k ∈ Z, r ∈ Z>0 . Then we have g ∗ Er−k,h+k ((XT1 , DT1 )/T ) = Er−k,h+k ((XT1 , DT1 )/T  ) for any morphism g : T  −→ T of p-adic fine log PD-schemes and for any h, k ∈ Z, r ∈ Z>0 . Indeed, for r = 1, it is nothing but (2.14.1) (1); for general r, it follows from the functoriality of the spectral sequence (2.9.6.2) and induction. Hence, to prove the theorem for the spectral sequence (2.9.6.2), we have to only to prove that the morphism ((XT1 , DT1 )/T )K0 (Fq ) : Er−k,h+k ((XT1 , DT1 )/T )K0 (Fq ) −→ d−k,h+k r Er−k+r,h+k−r+1 ((XT1 , DT1 )/T )K0 (Fq ) is zero for any r ≥ 2. Let us express

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2 Weight Filtrations on Log Crystalline Cohomologies

Er−k,h+k ((XT1 , DT1 )/T ) = OT⊕n ⊕ N , 

Er−k+r,h+k−r+1 ((XT1 , DT1 )/T ) = OT⊕n ⊕ N  , where N , N  are direct sums of modules of type OT /pe (e ∈ Z>0 ). Then we have 

d−k,h+k ((XT1 , DT1 )/T ) ∈ HomOT (OT⊕n ⊕ N , OT⊕n ⊕ N  ) r 

= HomOT (OT⊕n , OT⊕n ) ⊕ N, where N is a direct sum of modules of type Γ (T, OT )/pe (e ∈ Z>0 ). Then, for any closed point t of T1 , we have d−k,h+k ((Xt , Dt )/W (κt )) r =d−k,h+k ((XT1 , DT1 )/T ) ⊗OT W (κt ) r 

∈HomW (κt ) (W (κt )⊕n , W (κt )⊕n ) ⊕ (N ⊗Γ (T,OT ) W (κt )). By the purity of the weight [15, (1.2)] or [68, (2.2) (4)], we have d−k,h+k ((Xt , Dt )/W (κt ))K0 (Fq ) = 0 for any closed point t of T1 , that is, r ((Xt , Dt )/W (κt )) is contained in N ⊗Γ (T,OT ) W (κt ). From this d−k,h+k r ((XT1 , DT1 )/T ) is contained in N . Hence and (2.15.3), we see that d−k,h+k r ((X , D )/T ) = 0. d−k,h+k T1 T1 K0 (Fq ) r The proof of the degeneration of (2.11.14.3) is the same as the above. (One may use the duality between (2.9.6.2) ⊗W K0 and (2.11.14.3) ⊗W K0 for the case Z = ∅ and S = Spf(W ).)  

2.16 The Filtered Log Berthelot-Ogus Isomorphism In this section we prove a filtered version of Berthelot-Ogus isomorphism. Because the proof of this isomorphism is almost the same as that in [12] and [74], we give only the sketch of the proof. Proposition 2.16.1. Let S be a scheme of characteristic p > 0 and let ⊂ ⊂ S0 −→ S be a nilpotent immersion. Let S −→ T be a PD-closed immersion into a formal scheme with p-adic topology such that OT is p-torsion-free. Let f : (X, D ∪ Z) −→ S and f  : (X  , D ∪ Z  ) −→ S be smooth schemes with relative transversal SNCD’s. Assume that X, X  , S and T are noetherian. Set (X0 , D0 ∪ Z0 ) := (X, D ∪ Z) ×S S0 and (X0 , D0 ∪ Z0 ) := (X  , D ∪ Z  ) ×S S0 . Let g : (X0 , D0 ∪ Z0 ) −→ (X0 , D0 ∪ Z0 ) be a morphism of log schemes over S0 which induces morphisms (X0 , D0 ) −→ (X0 , D0 ) and (X0 , Z0 ) −→ (X0 , Z0 ). Then the following hold: (1) There exists a canonical filtered morphism

2.16 The Filtered Log Berthelot-Ogus Isomorphism

203 

 L D g ∗ : (Rf(X ) −→  ,D  ∪Z  )/T ∗ (O(X  ,D  ∪Z  )/T ) ⊗Z Q, P

(2.16.1.1)

D (Rf(X,D∪Z)/T ∗ (O(X,D∪Z)/T ) ⊗L Z Q, P ),

which is compatible with compositions. If g has a lift g
: (X, D ∪ Z) −→ log ∗ (X  , D ∪ Z  ), then g ∗ = g
crys . (2) Assume that g induces a morphism g (k) : (D0 , Z0 |D(k) ) −→ (D 0 , Z0 0 |D (k) ) for all k ∈ N. Then there exists a canonical filtered morphism (k)

(k)

0



 L D gc∗ : (Rf(X ) −→  ,D  ∪Z  )/T ∗,c (O(X  ,D  ∪Z  ;Z  )/T ) ⊗Z Q, Pc

(2.16.1.2)

D (Rf(X,D∪Z)/T ∗,c (O(X,D∪Z;Z)/T ) ⊗L Z Q, Pc ),

which is compatible with compositions. If g has a lift g
: (X, D ∪ Z) −→ log ∗ (X  , D ∪ Z  ), then gc∗ = g
crys . Proof. (1): The relative Frobenius F(X,D∪Z)/S : (X, D ∪ Z) −→ (X (p) , D(p) ∪ Z (p) ) over S induces an isomorphism PkD

(p)

Rf(X (p) ,D(p) ∪Z (p) )/T ∗ (O(X (p) ,D(p) ∪Z (p) )/T ) ⊗L Z Q ∼

−→ PkD Rf(X,D∪Z)/T ∗ (O(X,D∪Z)/T ) ⊗L Z Q (k ∈ Z) by (2.9.6.3) and (2.10.2.1) because the relative Frobenius induces an isomorphism of the classical iso-crystalline cohomology of a smooth scheme over S ([12, (1.3)]). Hence the same proof as that in [12, (2.1)] shows that we have the morphism (2.16.1.1). (2): The proof for (2.16.1.2) is the same as that for (2.16.1.1) by using (2.11.14.4) instead of (2.9.6.3) and using (2.11.18).   Corollary 2.16.2. If (X0 , D0 ∪ Z0 ) = (X0 , D0 ∪ Z0 ), then D (Rf(X,D∪Z)/T ∗ (O(X,D∪Z)/T ) ⊗L Z Q, P ) = 

D (Rf(X  ,D ∪Z  )/T ∗ (O(X  ,D ∪Z  )/T ) ⊗L ) Z Q, P

and D (Rf(X,D∪Z)/T ∗,c (O(X,D∪Z;Z)/T ) ⊗L Z Q, Pc ) = 

D (Rf(X  ,D ∪Z  )/T ∗,c (O(X  ,D ∪Z  ;Z  )/T ) ⊗L Z Q, Pc ).

Proof. Obvious (cf. [12, (2.2)]).

 

Theorem 2.16.3 (Filtered log Berthelot-Ogus isomorphism). Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field κ. Let p be the characteristic of κ. Set K := Frac(V ). Let S be a p-adic formal V -scheme in the sense of [74, §1]. Let (X, D ∪ Z) −→ S be a proper formally smooth scheme with a relative transversal SNCD over S. Let

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2 Weight Filtrations on Log Crystalline Cohomologies

T be an enlargement of S with morphism z : T0 := (SpecT (OT /p))red −→ S. Set T1 := SpecT (OT /p). Let f0 : (X0 , D0 ∪ Z0 ) := (X, D ∪ Z) ×S,z T0 −→ T0 be the base change of f : (X, D ∪ Z) −→ S. Then the following hold: (1) If there exists a log smooth lift f1 : (X1 , D1 ∪ Z1 ) −→ T1 of f0 , then there exist the following canonical filtered isomorphisms σT : (Rh f(X1 ,D1 ∪Z1 )/T ∗ (O(X1 ,D1 ∪Z1 )/T )K , P D1 ) ∼

−→ (Rh f∗ (O(X,D∪Z)/K )T , P D ), σT,c : (Rh f(X1 ,D1 ∪Z1 )/T ∗,c (O(X1 ,D1 ∪Z1 ;Z1 )/T )K , PcD1 ) ∼

−→ (Rh f∗,c (O(X,D∪Z;Z)/K )T , PcD ). (2) If there exists a log smooth lift f : (X , D ∪ Z) −→ T of f0 , then there exist the following canonical filtered isomorphisms ∼

log σcrys,T : (Rh f∗ (Ω•X /T (log(D ∪ Z)))K , P D ) −→ (Rh f∗ (O(X,D∪Z)/K )T , P D ), ∼

log : (Rh f∗ (Ω•X /T (log(Z−D)))K , PcD ) −→ (Rh f∗,c (O(X,D∪Z;Z)/K )T , PcD ). σcrys,T,c

Proof. The proof is the same as that of [74, (3.8)].

 

Remark 2.16.4. Let V , κ and p be as in (2.16.3). Then V /p is a κ-algebra by [79, II Proposition 8]. (1) Let (X , D ∪Z) a proper smooth scheme over Spec(V ) with an (S)NCD. Set UK := XK \(DK ∪ZK ). Then, by (2.16.3) and (2.16.2) and the base change theorem of the log crystalline cohomology ([54, (6.10)]), there are canonical isomorphisms: (2.16.4.1) ∼ h ((Xκ , Dκ ∪ Zκ )/W (κ))K −→ H h (XK , Ω•XK /K (log(DK ∪ ZK ))) Hlog-crys h = HdR (UK /K),

(2.16.4.2) ∼ h Hlog-crys,c ((Xκ , Dκ ∪ Zκ ; Zκ )/W (κ))K −→ H h (XK , Ω•XK /K (log(ZK − DK ))) which are compatible with the weight filtrations with respect to Dκ and DK . See also [17] for analogous statements by the rigid analytic method in the case Z = ∅. (2) Let (X, D) be a proper smooth scheme with a relative SNCD over κ. Set U := X \ D. By the finite base change theorem ([5, (1.8)]) and by Shiho’s comparison theorems [82, Theorem 2.4.4, Corollary 2.3.9, ∼ h (U/K) −→ Theorem 3.1.1]), there exists a canonical isomorphism Hrig h h ((X, D)/W ) ⊗W K. As a result, Hrig (U/K) has a weight filtration. Hlog-crys

2.17 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Family

205

By [85], [82, Theorem 2.4.4, Corollary 2.3.9, Theorem 3.1.1] and [6, (2.4)], h h ((X, D)/W ) ⊗W K = Hrig,c (U/K). In particular, we obtain Hlog-crys,c h Hrig,c (U/K) has a weight filtration. If (X, D) is the special fiber of (X , D) in (1), there exists a weight-filtered ∼ h h (U/K) −→ HdR (UK /K). An analogous statement can be isomorphism Hrig found in [17]. (3) Let U be a separated scheme of finite type over κ. Let Z/κ be a closed subscheme of U . In [70] the first-named author has defined a finite increasing h (U/K) which deserves the name “weight filtration”. In filtration on Hrig,Z h (U/K) defined in (2) is independent particular, the weight filtration on Hrig of the choice of (X, D). See §3.4 below for more details. In [loc. cit.] he has h (U/K) which deserves the also defined a finite increasing filtration on Hrig,c name “weight filtration” in the case where U is embeddable into a smooth scheme over κ as a closed subscheme. See also §3.6 below for more details.

2.17 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Family of Open Smooth Varieties Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field κ of characteristic p > 0. Let B be a topologically finitely generated ring over V . For a V -module M , MK denotes the tensor product M ⊗V K. In particular, BK = B ⊗V K. Let m be a maximal ideal of BK . By the proof of [84, (4.5)], BK /m is a finite extension of K. Set K  := BK /m. Let C be the image of B in BK /m = K  . Let V  be the integer ring of K  . Then the following is well-known (cf. [74, the proof of (4.2)]): Lemma 2.17.1. V ⊂ C ⊂ V  . Proof. The inclusion V ⊂ C is obvious. Let π be a uniformizer of V . Let v be a normalized valuation of V  . Let e be the ramification index of V  /V . By the definition of B, there exists a surjection V {x1 , . . . , xr } −→ B. It suffices to show that the image yi (1 ≤ i ≤ r) of xi in K  belongs to V  . If not, v(yi ) < 0 for some i. Set  π n/(e+1) (n ∈ N, e + 1|n), an = 0 (n ∈ N, e + 1  n). ∞ Then the image of an element n=0 an xni ∈ V {x1 , . . . , xr } in K  does not   converge in K  . This is a contradiction. We keep the notations in §2.4 except that S is a p-adic formal V -scheme in the sense of [74, §1] and that X is a proper smooth scheme with a relative SNCD D over S1 := SpecS (OS /p). The main result in this section is the following:

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2 Weight Filtrations on Log Crystalline Cohomologies

Theorem 2.17.2 (E2 -degeneration). Assume that S is a p-adic formal V -scheme and that X is a proper smooth scheme over S1 . Then (2.9.6.2)⊗V K and (2.11.14.3) ⊗V K degenerate at E2 in the case Z = ∅ and S0 = S1 . Proof. (Compare the following proof with [20, (5.5)].) We first prove the theorem for (2.9.6.2)⊗V K for the case Z = ∅ and S0 = S1 . We may assume that S is a p-adic affine flat formal scheme Spf(B) over Spf(V ). Consider the following boundary morphism: (2.17.2.1) d−k,h+k : Er−k,h+k ((X, D)/S)K −→ Er−k+r,h+k−r+1 ((X, D)/S)K r

(r ≥ 2).

We prove that d−k,h+k = 0 (r ≥ 2). r Case I: First we consider a case where B is a topologically finitely generated ring over V such that BK is an artinian local ring. Let m be the maximal ideal of BK . Then m is nilpotent. Set K  := BK /m. Consider the following ideal of B: I := Ker(B −→ BK /m). Then C = B/I, CK = K  and ⊂ V ⊂ C ⊂ V  ((2.17.1)). Let ι : Spf(C) −→ Spf(B) be the nilpotent closed immersion. Since the characteristic of K is 0, the morphism Spec(CK ) −→ Spec(K) is smooth and hence there exists a section sK : Spec(BK ) −→ Spec(CK ) of the nilpotent closed immersion Spec(CK ) −→ Spec(BK ). By [74, (1.17)], there exists a finite modification π : Spf(B  ) −→ Spf(B), a nilpo⊂ tent closed immersion ι : Spf(C) −→ Spf(B  ) with π ◦ ι = ι and a morphism s : Spf(B  ) −→ Spf(C) such that s induces sK and that s ◦ ι = id. Set S  := Spf(B  ). Because the boundary morphisms {d−k,h+k } are summations 1 of Gysin morphisms (with signs) ((2.8.5)), the E2 -terms of (2.9.6.2) ⊗V K are convergent F -isocrystals by [74, (3.7), (3.13), (2.10)]. Hence we have  E2−k,h+k ((X, D)/S)K = E2−k,h+k ((XS1 , DS1 )/S  )K since BK = BK . Let  •• {dr } (r ≥ 1) be the boundary morphism of (2.9.6.2)⊗V K for (XS1 , DS1 )/S  . Because {d•• r } (r ≥ 2) are functorial with respect to a morphism of p-adic enlargements, we have the following commutative diagram for r ≥ 2: Er−k,h+k ((X, D)/S)K ⏐ ⏐ d−k,h+k  r

−−−−→

Er−k,h+k ((XS1 , DS1 )/S  )K ⏐ ⏐  −k,h+k dr

Er−k+r,h+k−r+1 ((X, D)/S)K −−−−→ Er−k+r,h+k−r+1 ((XS1 , DS1 )/S  )K . Here, if r = 2, then the two horizontal morphisms above are isomorphisms.  •• does. Hence it suffices By induction on r ≥ 2, we see that d•• r vanishes if dr to prove that the boundary morphism (2.17.2.2)

dr −k,h+k : Er−k,h+k ((XS1 , DS1 )/S  )K −→ Er−k+r,h+k−r+1 ((XS1 , DS1 )/S  )K

(r ≥ 2)

 is the zero. Let l(M ) be the length of a finitely generated BK = BK -module  •• M . Furthermore, to prove the vanishing of dr , it suffices to prove that

2.17 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Family

207

(2.17.2.3)  −k,h+k l(Rh f(XS ,DS )/S  ∗ (O(XS ,DS )/S  )K ) = l( E2 ((XS1 , DS1 )/S  )K ). 1

1

1

k

1

Set S  := Spf(C). Then we have the morphism (XS1 , DS1 ) −→ S  . Let us denote the pull-back of the morphism (XS1 , DS1 ) −→ S  by s : S  −→ S  by (XS  , DS  ) −→ S  . Then, since we have π ◦ ι = ι and s ◦ ι = id, both 1 1 (XS1 , DS1 ) and (XS  , DS  ) are deformations of (XS1 , DS1 ) to S1 . Hence, by 1 1 (2.16.2), the spectral sequence (16.6.2)⊗V K for (XS1 , DS1 )/S  and that for (XS  , DS  )/S  are isomorphic. Therefore we have 1

1

E2−k,h+k ((XS1 , DS1 )/S  )K = E2−k,h+k ((XS 1 , DS 1 )/S  )K = B  ⊗C E2−k,h+k ((XS1 , DS1 )/S  )K . Hence, to prove (2.17.2.3), it suffices to prove that (2.17.2.4) dimK  (Rh f(XS ,DS )/S  ∗ (O(XS ,DS )/S  )K ) 1 1 1 1  −k,h+k E2 ((XS1 , DS1 )/S  )K ). = dimK  ( k

Set V1 := V  /p. Because there exists a morphism Spf(V  ) −→ Spf(C) of p-adic enlargements of S, it suffices to prove that (2.17.2.5) dimK  (Rh f(XV  ,DV  )/V  ∗ (O(XV  ,DV  )/V  )K ) 1 1 1 1  −k,h+k E2 ((XV1 , DV1 )/V  )K ). = dimK  ( k

We reduce (2.17.2.5) to a result of [68, (5.2) (1)](=(2.15.4) for (2.9.6.2) in this book) by using (a log version of) a result of Berthelot-Ogus ([12, §2]) as follows. Let κ be the residue field of V  . Since κ is perfect and since κ is a finite extension of κ, κ is also perfect. Let W  be the Witt ring of κ . The ring V1 is an artinian local κ -algebra with residue field κ ([79, II Proposition 8]). Set X  := XV1 ⊗V1 κ and D := DV1 ⊗V1 κ . Then (X  ⊗κ V1 , D ⊗κ V1 ) and (XV1 , DV1 ) are two log deformations of (X  , D ). Therefore, by (2.16.2), the spectral sequence (2.9.6.2)⊗V K for (X  ⊗κ V1 , D ⊗κ V1 )/V  and that for (XV1 , DV1 )/V  are isomorphic. From this fact, the log base change theorem ([54, (6.10)]) and the compatibility of Gysin morphisms with base change ([3, VI Th´eor`eme 4.3.12]), we have (2.17.2.6)

Rh f(XV  ,DV  )/V  ∗ (O(XV  ,DV  )/V  ) ⊗V  K  1

1

1

∼

1

−→ Rh f(X  ,D )/W  ∗ (O(X  ,D )/W  )⊗W  K  ,

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2 Weight Filtrations on Log Crystalline Cohomologies

(2.17.2.7) ∼ E2−k,h+k ((XV1 , DV1 )/V  ) ⊗V  K  −→ E2−k,h+k ((X  , D )/W  )⊗W  K  . Hence it suffices to prove that −k,h+k E2−k,h+k ((X  , D )/W  )⊗W  K  = E∞ ((X  , D )/W  )⊗W  K  .

We have already proved this in [68, (5.2) (1)](=(2.15.4)). Case II: Next, we consider the general case. Let m be a maximal ideal of BK . Consider the following ideal I (n) and the following ring B(n) in [74, p. 780]: I (n) := Ker(B −→ BK /mn ),

B(n) := B/I (n)

(n ∈ N).

The ring B(n) defines a p-adic enlargement S(n) of S. Let : Er−k,h+k ((X(S(n) )1 , D(S(n) )1 )/S(n) )K d−k,h+k r,(n) −→ Er−k+r,h+k−r+1 ((X(S(n) )1 , D(S(n) )1 )/S(n) )K be the boundary morphism. Because {d•• r } is functorial, we have the following commutative diagram: −k,h+k ((X, D)/S)⊗ (B Er B (n) )K ⏐ ⏐ d−k,h+k ⊗ (B )  BK r (n) K

− −−−−−− →

−k,h+k ((X Er (S(n) )1 , D(S(n) )1 )/S(n) )K ⏐ ⏐ −k,h+k dr,(n)

−k+r,h+k−r+1 ((X −k+r,h+k−r+1 ((X, D)/S)⊗ (B −−−−−− → Er Er B (n) )K − (S(n) )1 , D(S(n) )1 )/S(n) )K .

Because E2−k,h+k ((X, D)/S)K is a convergent F -isocrystal, the two horizontal morphisms are isomorphisms if r = 2. By induction on r and by the proof for the Case I, the boundary morphism d•• r ⊗BK (B(n) )K (r ≥ 2) vann ⊗ B /m ) = 0. Because B ishes. Thus limn (d•• K is a noetherian ring and ←− r BK K −k,h+k E2 ((X, D)/S)K is a finitely generated BK -module, we have d•• B /mn ) = lim(d•• ⊗ B /mn ) = 0. r ⊗BK (lim ←−n K ← − r BK K n Since (BK )m is a Zariski ring, limn (BK )m /mn (BK )m is faithfully flat over ←− (BK )m ([13, III §3 Proposition 9]). Therefore d•• r ⊗BK (BK )m = 0. Since m is = 0 (r ≥ 2). Hence we have proved an arbitrary maximal ideal of BK , d•• r (2.17.2) for (2.9.6.2) ⊗V K. Next we prove (2.17.2) for (2.11.14.3)⊗V K for the case Z = ∅ and S0 = S1 . As we remarked before (2.14.0.4), we have the base change property for Rq f(X,D)/S∗,c (O(X,D)/S )K = (Rq f(X,D)/S∗ I(X,D)/S ) ⊗V K. Hence the proof is analogous to the proof of (2.17.2) for (2.9.6.2)⊗V K for the case Z = ∅ and S0 = S1 : we have only to use (2.16.2) for Rf(X,D)/S∗,c (O(X,D)/S )K , (2.11.17) and use [68, (5.2) (2)] (=(2.15.4) for (2.11.14.3)).  

2.17 The E2 -degeneration of the p-adic Weight Spectral Sequence of a Family

209

We can reprove (2.13.3) in the case Z = ∅ and more: Corollary 2.17.3. Let k be a nonnegative integer. Then the following hold: (1) There exists a convergent F -isocrystal E2−k,h+k ((X, D)/K) such that h E2−k,h+k ((X, D)/K)T = grP h+k R f(XT1 ,DT1 )/T ∗ (O(XT1 ,DT1 )/T )K

for any p-adic enlargement T of S over Spf(V ). (2) There exists a convergent F -isocrystal Pk Rh f∗ (O(X,D)/K ) such that Pk Rh f∗ (O(X,D)/K )T = Pk Rh f(XT1 ,DT1 )/T ∗ (O(XT1 ,DT1 )/T )K for any p-adic enlargement T of S over Spf(V ). (3) There exists a spectral sequence of convergent F -isocrystals on (X, D) /S over Spf(V ) : (2.17.3.1)

E1−k,h+k ((X, D)/K) = Rh−k f∗ (OD(k) /K ⊗Z
(k) (D/K))(−k) =⇒ Rh f∗ (O(X,D)/K ).

This spectral sequence degenerates at E2 . Proof. (1): By (2.8.5), the boundary morphism d•• 1 of (2.9.6.2) ⊗V K is a summation (with signs) of Gysin morphisms, and thus d•• 1 is a morphism of convergent F -isocrystals by [74, (3.13)]. By [74, (3.1)] and by (2.17.2), we obtain (1). (2): By (1), for a morphism g : T  −→ T of p-adic affine enlargements of S over Spf(V ), Pk Rh f(XT  ,DT  )/T  ∗ (O(XT  ,DT  )/T  )K = g ∗ Pk Rh f(XT1 ,DT1 )/T ∗ (O 1 1 1 1 (XT ,DT )/T )K . The claim on the F -isocrystal follows as in [74, (3.7)]. (3): (3) immediately follows from (2.17.2).   We can reprove (2.13.8) (1) and (2) in the case Z = ∅ and more: Corollary 2.17.4. Let k be a nonnegative integer. Then the following hold: k,h−k (1) There exists a convergent F -isocrystal E2,c ((X, D)/K) such that k,h−k h E2,c ((X, D)/K)T = grP h−k R f(XT1 ,DT1 )/T ∗,c (O(XT1 ,DT1 )/T )K

for any p-adic enlargement T of S over Spf(V ). (2) There exists a convergent F -isocrystal Pk Rh f∗,c (O(X,D)/K ) on S/ Spf(V ) such that (Pk Rh f∗,c (O(X,D)/K ))T = Pk Rh f(XT1 ,DT1 )/T ∗,c (O(XT1 ,DT1 )/T )K for any p-adic enlargement T of S/Spf(V ). (3) There exists a spectral sequence of convergent F -isocrystals on X/S over Spf(V ) :

210

(2.17.4.1)

2 Weight Filtrations on Log Crystalline Cohomologies k,h−k E1,c ((X, D)/K) = Rh−k f∗ (OD(k) /K ⊗Z
(k) (D/K))

=⇒ Rh f∗,c (O(X,D)/K ). This spectral sequence degenerates at E2 . Proof. (1), (2), (3): We obtain (1), (2) and (3) as in (2.17.3).

 

As in [11, §7], for a p-adic formal V -scheme S, we have a log crystalline

topos ((X, D)/S)log crys and the forgetting log morphism (X,D)/S : ((X, D)/S) ◦  log crys −→ (X/S)crys . The following is nothing but a restatement of a part of (2.17.2) by the p-adic version of (2.7.6): Corollary 2.17.5. The following Leray spectral sequence (2.17.5.1)

◦

E2k,h−k = Rh−k f (X,D)/S∗ Rk (X,D)/S∗ (O(X,D)/S )K =⇒ Rh f(X,D)/S∗ (O(X,D)/S )K

degenerates at E3 .

2.18 Strict Compatibility In this section, using a specialization argument of Deligne-Illusie (§2.14) and by using the convergence of the weight filtration (§2.13, §2.17), we prove the strictness of the induced morphism of log crystalline cohomologies by a morphism of log schemes with respect to the weight filtration. Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field κ of characteristic p > 0 and with fraction field K. Let g : (X  , D ) −→ (X, D) be a morphism of two proper smooth schemes with SNCD’s over κ. Let W be the Witt ring of κ and K0 the fraction field of W . Then the following holds: Theorem 2.18.1. Let h be an integer. Then the following hold: (1) The induced morphism (2.18.1.1)

log∗ h h gcrys : Hlog-crys (X/W )K −→ Hlog-crys (X  /W )K

is strictly compatible with the weight filtration. (2) Assume that g induces morphisms g (k) : D(k) −→ D(k) for all k ∈ N. Then the induced morphism (2.18.1.2)

log∗ h h gcrys,c : Hlog-crys,c (X/W )K −→ Hlog-crys,c (X  /W )K

is strictly compatible with the weight filtration.

2.18 Strict Compatibility

211

Proof. (1): In this proof, for the sake of clarity, denote by P and P  the weight h h (X/W )K0 and Hlog-crys (X  /W )K0 , respectively. filtrations on Hlog-crys h h Since Pk Hlog-crys (X/W )K0 ⊗K0 K = (Pk Hlog-crys (X/W ))K (k ∈ Z ∪ {∞}), we may assume that V = W . By (2.9.1) the morphism g induces a morphism (2.18.1.3) log∗ h h gcrys : Pk Hlog-crys (X/W )K0 −→ Pk Hlog-crys (X  /W )K0

(k ∈ Z ∪ {∞}).

h h log ∗ Let Pk Hlog-crys (X  /W )K0 be the image of Pk Hlog-crys (X/W )K0 by gcrys . Then we prove that

(2.18.1.4)

 Pk ∩ P∞ = Pk .

By [40, 3, (8.9.1) (iii), (8.10.5)] and [40, 4, (17.7.8)], there exists a model of g, that is, there exists a morphism g : (X  , D ) −→ (X , D) of proper smooth schemes with relative SNCD’s over the spectrum S1 := Spec(A1 ) of a smooth algebra A1 (⊂ κ) over a finite field Fq such that g ⊗A1 κ = g. By a standard deformation theory ([41, III (6.10)]), there exists a formally smooth scheme S = Spf(A) over Spf(W (Fq )) such that S ⊗W (Fq ) Fq = S1 . We fix a lift F : S −→ S of the Frobenius of S1 . Then, as in §2.15, W is an A-algebra. Let P  and P  be the analogous filtrations on Rh fX  /S∗ (OX  /S ) ⊗W (Fq ) K0 (Fq ), where K0 (Fq ) is the fraction field of W (Fq ). By (2.14.2), in order to prove (2.18.1.4), it suffices to prove that (2.18.1.5)

 Pk ∩ P∞ = Pk

by shrinking S. Here, note that the extension κ/Frac(A1 ) of fields may be  are convergent isocrystals infinite and transcendental. Because Pk and P∞   ((2.13.3) or (2.17.3)), so is Pk ∩ P∞ by [74, (2.10)]. Since two inclusions    ) ∩ Pk −→ Pk and (Pk ∩ P∞ ) ∩ Pk −→ Pk ∩ P∞ are morphisms of (Pk ∩ P∞ convergent isocrystals, it suffice to prove that (2.18.1.6)

 (Pk ∩ P∞ )s = (Pk )s

for any closed point s ∈ S by [74, (3.17)]. In this case, (2.18.1.6) immediately follows from the purity of the weight of the crystalline cohomologies ([15, (1.2)] or [68, (2.2) (4))]) and by the spectral sequence (2.9.6.2). Thus we have proved (1). (2): By the assumption of g, the analogue of (2.18.1.3) for the log crystalline cohomology with compact support holds. Using (2.13.8) instead of (2.13.3), we obtain (2) in a similar way.   Theorem 2.18.2 (Strict compatibility). Let S be a p-adic formal V scheme. Let f : (X, D) −→ S1 and f  : (X  , D ) −→ S1 be proper smooth schemes with relative SNCD’s over S1 . Let g : (X  , D ) −→ (X, D) be a morphism of log schemes over S1 . Let h be an integer. Then the following hold: (1) The induced morphism

212

2 Weight Filtrations on Log Crystalline Cohomologies

(2.18.2.1)  g ∗ : Rh f(X,D)/S∗ (O(X,D)/S )K −→ Rh f(X  ,D  )/S∗ (O(X  ,D  )/S )K

(h ∈ Z)

is strictly compatible with the weight filtration. (2) Assume that g induces morphisms g (k) : D(k) −→ D(k) for all k ∈ N. Then the induced morphism (2.18.2.2)  gc∗ : Rh f(X,D)/S∗,c (O(X,D)/S )K −→ Rh f(X  ,D  )/S∗,c (O(X  ,D  )/S )K

(h ∈ Z)

is strictly compatible with the weight filtration. Proof. Since the proofs of (1) and (2) are similar, we give only the proof of (1). By (2.13.3) (or (2.17.3)) and by the proof of [74, (3.17)], we may assume that S is the formal spectrum of a finite extension V  of V . Let κ be the residue field of V  . As mentioned in the proof of (2.17.2), V  /p is an κ -algebra; the two pairs (X, D) and ((X, D) ⊗V  κ ) ⊗κ V  /p are two deformations of (X, D) ⊗V  κ ; the obvious analogue for (X  , D ) also holds. Hence, by the deformation invariance of log crystalline cohomologies with weight filtrations ((2.16.2)), we may assume that S = Spf(W (κ )) and that (X, D) and (X  , D ) are smooth schemes with SNCD’s over a perfect field κ of characteristic p > 0. Hence (1) follows from (2.18.1) (1).   Corollary 2.18.3. Let the notations be as in (2.18.2). Let g : (X  , D ) −→ (X, D) be a log etale morphism such that Rg∗ (OX  ) = OX (e.g., the blowing up along center a smooth component of D(k) ). Then g ∗ in (2.18.2.1) is a filtered isomorphism. Proof. We may assume that S is flat over Spf(V ). By the second proof of [65, (2.2)] and by [loc. cit., (2.4)], the induced morphism Rf∗ (Ω•X/S1 (log D)) −→ Rf∗ (Ω•X  /S1 (log D )) is an isomorphism (cf. [43, VII (3.5)], (2.18.7) below). By the log version of a triangle in the proof of [11, 7.16 Theorem] and by the log version of [11, 7.22.2], the induced morphism  g ∗ : RfX/S∗ (O(X,D)/S ) −→ Rf(X  ,D  )/S∗ (O(X  ,D  )/S )

is an isomorphism; in particular, g ∗ : Rh f(X,D)/S∗ (O(X,D)/S )K −→ Rh f(X  ,D )   /S∗ (O(X  ,D  )/S )K is an isomorphism. (2.18.3) follows from (2.18.2) (1). Remark 2.18.4. Let the notations be as in (2.18.2). We do not know an example such that the induced morphism g ∗ : (Rh f(X,D)/S∗ (O(X,D)/S ), P ) −→  (Rh f(X  ,D  )/S∗ (O(X  ,D  )/S ), P ) is not strictly compatible with the weight filtration.

2.18 Strict Compatibility

213

Theorem 2.18.5. Let the notations be as in (2.18.2). Assume that g induces morphisms g (k) : D(k) −→ D(k) for all k ∈ N. Assume, moreover, that g is log etale, that Rg∗ (OX  ) = OX and that g ∗ (OX (−D)) = OX  (−D ). Then gc∗ in (2.18.2.2) is a filtered isomorphism. Proof. We may assume that S is flat over Spf(V ). Because g is log etale, we have g ∗ (ΩiX/S1 (log D)) = ΩiX  /S1 (log D ) (i ∈ N). Hence, by the assumption, we have g ∗ (ΩiX/S1 (− log D)) = ΩiX  /S1 (− log D ). By using the projection formula as in [65, p. 168], we have ΩiX/S1 (− log D) = Rg∗ (ΩiX  /S1 (− log D )). Consequently, as in [65, (2.4)], we have Ω•X/S1 (− log D) = Rg∗ (Ω•X  /S1 (− log D )) by using the spectral sequence E1ij = Rj g∗ (ΩiX  /S1 (− log D )) =⇒ Ri+j g∗ (Ω•X  /S1 (− log D )). Let n be a positive integer, and set Sn := SpecS (OS /pn ). Then we have an exact sequence 0 −→ pn OS /pn+1 OS −→ OSn+1 −→ OSn −→ 0. By using the base change theorem of the log crystalline cohomology sheaf with compact support ((2.11.11.1)), we have the following triangle as in [11, 7.16 Theorem]: (2.18.5.1)

n n+1 −→ Rf(X,D)/S1 ∗,c (O(X,D)/S ) ⊗L OS OS1 p OS /p

−→ Rf(X,D)/Sn+1 ∗,c (O(X,D)/Sn+1 ) +1

−→ Rf(X,D)/Sn ∗,c (O(X,D)/Sn ) −→ · · · . Hence, by induction on n and by (2.11.7.1) and [11, 7.22.2], we have Rh f(X  ,D )/S∗,c (O(X  ,D )/S ) = Rh f(X,D)/S∗,c (O(X,D)/S ). In particular, gc∗ is an isomorphism of OS ⊗V K-modules. Moreover, by   (2.18.2) (2), gc∗ is a filtered isomorphism. Remark 2.18.6. It is straightforward to generalize (2.18.2), (2.18.3), (2.18.5) into the framework of convergent F -isocrystals. Remark 2.18.7. The following example (=a very special case of [65, (2.3)]) shows that the strictness of the induced morphism on sheaves of log differential forms by a morphism of smooth schemes with relative SNCD’s does not hold. Let S be a scheme and let X be an affine plane A2S = SpecS (OS [x, y]). Let D be a relative SNCD on X/S defined by xy = 0. Let g : X  −→ X be the blow up of X along the center (0, 0). Let D be the union of the strict transform of D and the exceptional divisor of g; then D is a relative SNCD on X  /S. Let i be an integer. Then Mokrane has proved that

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2 Weight Filtrations on Log Crystalline Cohomologies

Rj g∗ (ΩiX  /S (log D )) = 0 (j ∈ Z>0 ) and g∗ (ΩiX  /S (log D )) = ΩiX/S (log D) (a very special case of [65, (2.2)]; however, note that in the notations in [loc. cit.], ⊂ the condition that the closed immersion Y −→ X is a regular embedding is necessary for [loc. cit.] because the fact Rf∗ (OX  ) = OX in [43, VII (3.5)] has been shown under this assumption.). The pull-back morphism g ∗ : (Ω2X/S (log D), P ) −→ g∗ (Ω2X  /S (log D ), P ) is a morphism of filtered sheaves; however, as remarked in [loc. cit.], g ∗ is not strict. (Consequently g ∗ does not induce an isomorphism of filtered sheaves of log differential forms.) Note that the number of smooth components of D is more than those of D; the log structure of (X  , D ) is “bigger” than that of (X, D). Remark 2.18.8. The following remark is the crystalline analogue of a part of results in [24, (9.2)]. Let (S, I, γ) and S0 be as in §2.4. Let f : (X, D) −→ S0 be a smooth scheme ⊂ with a smooth relative divisor over S0 . Let a : D −→ X be the natural closed immersion. Then, by (2.6.1.1), we have the following exact sequence (2.18.8.1)

0 −→ Q∗X/S (OX/S ) −→ Q∗X/S CRcrys (O(X,D)/S ) −→ Q∗X/S acrys∗ (OD/S )(−1){−1} −→ 0.

Applying the higher direct image functor R• f X/S∗ to (2.18.8.1), we have the following exact sequence (2.18.8.2) · · · −→ Rh−2 fD/S∗ (OD/S )(−1) −→ Rh fX/S∗ (OX/S ) −→ Rh f(X,D)/S∗ (O(X,D)/S ) −→ · · · . The spectral sequence (2.9.6.2) degenerates at E2 in this case since E2ij = 0 if i = 0 or i = −1. It is easy to check that the exact sequence (2.18.8.2) is strictly compatible with the preweight filtration. Using (2.11.7.1), we also have the following exact sequence which is strictly compatible with the preweight filtration (2.18.8.3) · · · −→ Rh fX/S∗ (OX/S ) −→ Rh fD/S∗ (OD/S ) −→ Rh+1 f(X,D)/S∗,c (O(X,D)/S ) −→ · · · . Now assume that S is a p-adic formal V -scheme (in the sense of [74, §1]) over a complete discrete valuation ring V of mixed characteristics with perfect residue field. Assume also that S0 = SpecS (OS /p), that X is projective over S0 of pure relative dimension d and that D is a smooth hypersurface section. Let K be the fraction field of V . Then the induced morphism

2.19 The Weight-Filtered Poincar´e Duality

(2.18.8.4)

215

Rh fX/S∗ (OX/S )K −→ Rh fD/S∗ (OD/S )K ⊂

by the closed immersion D −→ X is an isomorphism for h ≤ d − 2 and an injection for h = d − 1 (cf. [2, Th´eor`eme]). Indeed, first consider the case h ≤ d − 2. Then we can assume that S is the formal spectrum of a finite extension of V by [74, (3.17)]. In this case, the argument in the proof of (2.18.2) and the specialization argument of Deligne-Illusie ([49, 3.10], cf. the argument in (2.18.1)) show that the hard Lefschetz theorem holds for Rh fX/S∗ (OX/S )K (cf. [49, 3.8]). Hence the proof of [57, p. 76 Corollary] shows that (2.18.8.4) is an isomorphism for h ≤ d − 2. As to the case h = d − 1, the same proof works by considering the image of Rd−1 fX/S∗ (OX/S )K in Rd−1 fD/S∗ (OD/S )K . By the Poincar´e duality ([74, (3.12)]), the Gysin morphism Gh : Rh−2 fD/S∗ (OD/S )K (−1) −→ Rh fX/S∗ (OX/S )K is an isomorphism for h ≥ d + 2 and a surjection for h ≥ d + 1. Set Rd−1 fD/S∗,ev (OD/S )K (−1) := Ker Gd+1 . Then Rd−1 fD/S∗,ev (OD/S )K (−1) is the orthogonal part of the image of the injective morphism Rd−1 fX/S∗ (OX/S )K −→ Rd−1 fD/S∗ (OD/S )K . Therefore we have the following direct decomposition: (2.18.8.5) Rd−1 fD/S∗ (OD/S )K = Rd−1 fD/S∗,ev (OD/S )K ⊕ Rd−1 fX/S∗ (OX/S )K .

2.19 The Weight-Filtered Poincar´ e Duality The following is the Poincar´e duality: Theorem 2.19.1 (Weight-filtered Poincar´ e duality). Let V be a complete discrete valuation ring of mixed characteristics with perfect residue field of characteristic p > 0. Let S be a p-adic formal V -scheme. Let (X, D) be a formally smooth scheme with a relative SNCD over S. Assume that X/S is projective and that the relative dimension of X/S is of pure dimension d. Then there exists a perfect pairing of convergent F -isocrystal on S/Spf(V ) (2.19.1.1)

Rh f∗,c (O(X,D)/K ) ⊗ R2d−h f∗ (O(X,D)/K ) −→ OS/K (−d),

which is strictly compatible with the weight filtration. That is, the natural morphism (2.19.1.2) Rh f∗,c (O(X,D)/K ) −→ Hom OS/K (R2d−h f∗ (O(X,D)/K ), OS/K (−d)) is an isomorphism of weight-filtered convergent F -isocrystals on S/V .

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2 Weight Filtrations on Log Crystalline Cohomologies

Proof. By (2.11.3), there exists a canonical morphism R2d f∗,c (O(X,D)/K ) −→ R2d f∗ (OX/K ) of convergent isocrystals on S/Spf(V ), which is constructed from natural morphisms R2d f∗,c (O(XT1 ,DT1 )/T ) −→ R2d f∗ (OXT1 /T ) for padic enlargements T of S/Spf(V ). Using the cup product, we have the following composite morphism (2.19.1.3) ∪

Rh f∗,c (O(X,D)/K ) ⊗ R2d−h f∗ (O(X,D)/K ) −→ R2d f∗,c (O(X,D)/K ) Tr

◦ f

−→ R2d f∗ (OX/K ) −→ OS/K (−d). by [74, (3.12.1)]. The morphism (2.19.1.2) is an isomorphism. Indeed, by [74, (3.17)], we may assume that S is the spectrum of a perfect field κ of finite characteristic. In this case Tr ◦ is the classical trace map ([74, pp. 809–810]), f

Therefore (2.19.1.2) for S = Spec(κ) is an isomorphism by [85, (5.6)], and hence we have an isomorphism (2.19.1.2). By using the arguments in (2.18.1) and (2.18.2), we obtain the strict compatibility of the isomorphism (2.19.1.2) with the weight filtration.  

2.20 l-adic Weight Spectral Sequence Let S be a scheme. Let (X, D)/S be a proper smooth scheme with a relative SNCD. Set U := X \ D and let f : U −→ S be the structural morphism. Let f (k) : D(k) −→ S (k ∈ Z≥0 ) be the structural morphism and a(k) : D(k) −→ X also the natural morphism. Let l be a prime number (k) which is invertible on S. Let
et (D/S)(−k) (k ∈ N) be the etale orienk  (k) ∗ tation sheaf of D(k) :
et (D/S)(−k) := {u−1 ( (M (D)/OX ))}|D(k) , where et

u is the canonical morphism Xet −→ Xzar of topoi. Here note that we do (k)

not define “
et (D/S)”. If S is of characteristic p > 0, then the Frobenius (k) of (X, D) acts on
et (D/S)(−k) by the multiplication by pk . Almost all the results in the previous sections have l-adic analogues. For example, the excision spectral sequence (k)

(k)

(2.20.0.1) E1k,h−k = Rh−k f∗ (Ql (k) ⊗Z
et (D/S)(−k)) =⇒ Rh f∗,c (Ql ). calculates Rh f∗,c (Ql ). ⊂ Let j : U −→ X be the open immersion. By Grothendieck’s absolute pu∼ rity, which has been solved by O. Gabber ([33]), we obtain Rk j∗ (Ql ) −→ (k) (k) a∗ (Ql,D(k) ⊗Z
et (D/S)(−k)). As in the Introduction, we use the following isomorphism

2.20 l-adic Weight Spectral Sequence

(2.20.0.2)

∼

217 (k)

(k)

Rk j∗ (Ql ) −→ a∗ (Ql,D(k) ⊗Z
et (D/S)(−k)) (−1)k ∼

(k)

(k)

−→ a∗ (Ql,D(k) ⊗Z
et (D/S)(−k)). Then we have the following spectral sequence: (2.20.0.3)

(k)

(k)

E2k,h−k = Rh−k f∗ (Ql ⊗Z
et (D/S)(−k)) =⇒ Rk f∗ (Ql ).

The spectral sequence (2.20.0.1) (resp. (2.20.0.3)) degenerates at E2 (resp. E3 ) by the standard specialization argument (e.g., [34]) and the Weil conjecture ([26, (3.3.9)]).

Chapter 3

Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

In this chapter, we give an outline of the construction of the weight filtration and the calculation of the slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic p > 0. They are constructed and calculated by using de Jong’s alteration theorem, Tsuzuki’s proper cohomological descent for rigid cohomology, the comparison theorem between log crystalline cohomology and rigid cohomology by the secondnamed author and the results in the previous chapters in this book. It is analogous to the fact that the mixed Hodge structure on the cohomology of a separated scheme of finite type over C is constructed from the mixed Hodge complex on open smooth schemes over C via the technique of proper hypercovering. The detailed proof of the results in this chapter is given by another book [70] by the first-named author.

3.1 Preliminaries for Later Sections In the rest of this book, following the suggestion of one of the referees, we state some results to answer natural questions arising from results in previous sections: we outline the existence of the weight filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic p > 0 without proofs. In this section we gather general objects needed in the later sections. First we have to fix a convention on the sign of the boundary morphism of the single complex of a multicosimplicial complex in [24] and [70]. We follow the convention in [70], which is different from that in [24]. Let (T , A) be a ringed topos. For a positive integer r, let (Tt1 ···tr , At1 ···tr )t1 , ...,tr ∈N be a constant r-simplicial ringed topos defined by (T , A): Tt1 ···tr = T , At1 ···tr = A. Let M be an object of the category C(A•···• ) (rpoints) of complexes of A•···• -modules. For nonnegative integers t1 , . . . , tj (1 ≤ j ≤ r), set tj := t1 + · · · + tj and t0 := 0. Set also t := (t1 , . . . , tr ) Y. Nakkajima, A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties. Lecture Notes in Mathematics 1959, c Springer-Verlag Berlin Heidelberg 2008


219

220

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

and • := • · · · • (r-points). The object M defines an (r + 1)-uple complex M •• = (M t1 ···tr • )t1 ,...,tr ∈N of A-modules whose boundary morphisms will be fixed in (3.1.0.1) below. Let us consider the following single complex s(M ) with the following boundary morphism (cf. [24, (5.1.9.1), (5.1.9.2)]):   M ts = M t1 ···tr s ; (3.1.0.1) s(M )n = tr +s=n

d(xts ) =

t 1 +1

t1 +···+tr +s=n

(−1)i δ1i (xts ) + (−1)t1

t 2 +1

i=0

(−1)i δ2i (xts ) + · · ·

i=0

+ (−1)tr−1

t r +1

(−1)i δri (xts ) + (−1)tr dM (xts ) (xts ∈ M ts ),

i=0

where dM : M ts −→ M t,s+1 is the boundary morphism arising from the boundary morphism of the complex M and δji : M t1 ···tj ···tr s −→ M t1 ···tj−1 , tj +1,tj+1 ···tr s (1 ≤ j ≤ r, 0 ≤ i ≤ tj + 1) is a standard coface morphism. We can check that the functor s : C(A• ) −→ C(A)

(3.1.0.2)

induces the following functor s : D+ (A• ) −→ D+ (A)

(3.1.0.3)

(cf.[42, Vbis (2.3.2.2)]). Next we consider the filtered version of the above. of

CF(A• ). Then (M, P ) defines Let (M, P ) := (M, {Pk M }) be

an object ts an (r+1)-uple filtered complex ( t≥0,s M , { t≥0,s Pk M ts }) of A-modules. Here t ≥ 0 means that tj ≥ 0 (1 ≤ ∀j ≤ r). Then we have the following functors (3.1.0.4)

s : C+ F(A• ) (M, P ) −→ (s(M ), {s(Pk M )}) ∈ C+ F(A),

(3.1.0.5)

s : D+ F(A• ) [(M, P )] −→ [(s(M ), {s(Pk M )})] ∈ D+ F(A).

Let δ(P ) = {δ(P )k }k∈Z be the diagonal filtration on s(M ) defined by the following formula (For the case r ≤ 2, this is equal to [24, (7.1.6.1), (8.1.22)].):  Ptr +k M ts . (3.1.0.6) δ(P )k (s(M )) = t≥0,s

Then we have (3.1.0.7)

δ(P )

grk

(s(M )) =

 t≥0

t• grP tr +k M [−tr ].

3.1 Preliminaries for Later Sections

221

As in [24, (7.1.6.3)], we have the following functor (3.1.0.8)

(s, δ) : D+ F(A• ) [(M, P )] −→ [(s(M ), δ(P ))] ∈ D+ F(A).

Next we generalize the above to the mapping fiber of a morphism in D+ F(A• ). Let ρ : (M, P ) −→ (N, P ) be a morphism in CF(A• ). Let MF(ρ) := (M, P )⊕ (N, P )[−1] be the mapping fiber of ρ. Then s(MF(ρ)) = s(M, P ) ⊕ (s(N, P )[−1]). We define the diagonal filtration δMF (P ) on s(MF(ρ)) as follows:   (3.1.0.9) δMF (P )k s(MF(ρ)) := Ptr +k M ts ⊕ Ptr +k+1 N t,s−1 . t≥0,s

t≥0,s

By the definition of δMF (P ), we have   δ (P ) ts t,s−1 grP ⊕ grP . (3.1.0.10) grkMF s(MF(ρ)) = tr +k M tr +k+1 N t≥0,s

t≥0,s

t• t• Since the morphism ρ : grP −→ grP is the zero, tr +k M tr +k+1 N (3.1.0.11)   δ (P ) t• t• grkMF s(MF(ρ)) = grP grP tr +k M [−tr ] ⊕ tr +k+1 N [−tr − 1]. t≥0

t≥0

Assume that the two filtrations δ(P )’s on s(M ) and s(N ) are exhaustive and t• t• and grP are quasi-isomorphic to objects of complete, that grP tr +k M tr +k N + • C (A ) and that the spectral sequence arising from the two filtrations δ(P )’s on s(M ) and s(N ) are regular and bounded below. Here we say that δ(P ) is complete if s(M ) = limk s(M )/(δ(P )−k (s(M ))) (cf. [90, (5.4.4)]). Then ←− we have the following convergent spectral sequence by the Convention (6) (cf. [90, (5.5.10)], [24, (8.1.15)]): (3.1.0.12) E1−k,h+k =

 t≥0

t• Hh−tr (grP tr +k M ) ⊕



t• Hh−tr −1 (grP tr +k+1 N )

t≥0

=⇒ Hh (s(MF(ρ))). Let d(M ) :

 t≥0

t• Hh−tr (grP tr +k M ) −→



t• Hh−tr +1 (grP tr +k−1 M )

t≥0

be the boundary morphism arising from the following exact sequence δ(P )

δ(P )

0 −→ grk−1 s(M ) −→ (δ(P )k /δ(P )k−2 )s(M ) −→ grk

s(M ) −→ 0.

222

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

Let d(N ) :



t• Hh−tr −1 (grP tr +k+1 N ) −→

t≥0



t• Hh−tr (grP tr +k N )

t≥0

be the analogous boundary morphism for (N, P ). The boundary morphism d : E1−k,h+k −→ E1−k+1,h+k between the E1 -terms in (3.1.0.12) is equal to (3.1.0.13)     d(M ) ρ t• t• Hh−tr (grP Hh−tr −1 (grP : tr +k M ) ⊕ tr +k+1 N ) −→ 0 −d(N ) t≥0



t≥0

t• Hh−tr +1 (grP tr +k−1 M ) ⊕

t≥0



t• Hh−tr (grP tr +k N ).

t≥0

Let [ρ] be the morphism in D F(A ) induced by ρ. Let Mor(D+ F(A• )) be the set of the morphisms in D+ F(A• ). Then we have the following functor +

•

(3.1.0.14) (s(MF), δMF ) : Mor(D+ F(A• )) [(ρ : (M, P ) −→ (N, P ))]

−→ [(s(MF(ρ)), δMF (P ))] ∈ D+ F(A). Next we consider the decreasing stupid filtration on the mapping fiber. Let ρ : M −→ N be a morphism in C+ (A• ). Then ρ induces a morphism ρ : (M, σ) −→ (N, σ) of filtered complexes, where σ’s are decreasing stupid filtrations. Then we define a decreasing stupid filtration σ on s(MF(ρ)) as follows:   M ts ⊕ N t,s−1 (i ∈ N). (3.1.0.15) σ i s(MF(ρ)) = t≥0,s≥i

Then (3.1.0.16)

griσ s(MF(ρ)) =



t≥0,s−1≥i

M ti [−tr ]{−i} ⊕

t≥0



N ti [−tr − 1]{−i}

t≥0

= sMF(ρ {−i} : M {−i} −→ N i {−i}), i

i

where ρi : M i −→ N i is the degree i-part of ρ. Hence we have the following spectral sequence: (3.1.0.17)

E1i,h−i = Hh−i (s(MF(ρi ))) =⇒ Hh (s(MF(ρ))).

Let (T• , A• ) be an r-simplicial ringed topos. For a family {nj }uj=1 (1 ≤ u ≤ r) of nonnegative integers, we also need the following natural restriction functor in later sections:

3.1 Preliminaries for Later Sections

223

(3.1.0.18) + • •• e−1 •···•n1 •···•n2 •······•nu−1 •···•nu •···• : D (A ) [M ] −→ [M •···•n1 •···•n2 •······•nu−1 •···•nu •···• ] ∈ D+ (A•···•n1 •···•n2 •······•nu−1 •···•nu •···• ). Let N be a nonnegative integer. Next we recall the key functor ΓN in [19], [87] and [88]. Let ∆ in this section be the standard simplicial category: an object of ∆ is denoted by [n] := {0, . . . , n} (n ∈ N); a morphism in ∆ is a non-decreasing function [n] −→ [m] (n, m ∈ N). Let C be a category in which there exist finite inverse limits. Let l be a nonnegative integer. Following [19, §11], define a set Hom≤l ∆ ([n], [m]) ([n], [m]), by definition, consists of the morphisms (n, m ∈ N): the set Hom≤l ∆ γ : [n] −→ [m] in ∆ such that γ([n]) ≤ l. Let us recall the definition of a simplicial object Γ := ΓCN (X)≤l in C for an object X ∈ C ([19, §11], [88, (7.3.1)]): for an object [m], set  Xγ Γm := ≤l

γ∈Hom∆ ([N ],[m])

with Xγ = X; for a morphism α : [m ] −→ [m] in ∆, αΓ : Γm −→ Γm is defined to fit the following commutative diagram for any γ : [N ] −→ [m ] such that γ([N ]) ≤ l: α

Γm −−−Γ−→ ⏐ ⏐ proj.

Γm  ⏐ ⏐proj. 

id

Xαγ −−−X −→ Xγ . In fact, we have a functor (3.1.0.19)

ΓCN (?)≤l : C −→ Hom(∆o , C) =: {simplicial objects in C}.

Set ΓCN (X) = ΓCN (X)≤N . Then ΓCN (X)≤l = ΓCN (X) for l ≥ N . Let X•≤N be an N -truncated simplicial object of C and let f : XN −→ Y be a morphism in C. Then we have a morphism (3.1.0.20)

X•≤N −→ ΓCN (Y )•≤N

defined by the following commutative diagram:  Xm −−−−→ γ∈Hom∆ ([N ],[m]) Yγ ⏐ ⏐ ⏐proj ⏐ X(γ)  (3.1.0.21) f

XN −−−−→

Yγ = Y.

224

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

Assume, furthermore, that C has finite disjoint sums. Let ∆N be a full subcategory of ∆ consisting of [k]’s (0 ≤ k ≤ N ). An object of Hom(∆oN , C) is, by definition, an N -truncated simplicial object in C. Let us recall the skelton: skCN : Hom(∆o , C) X• −→ X•≤N ∈ Hom(∆oN , C). The coskelton

coskCN : Hom(∆oN , C) −→ Hom(∆o , C)

is, by definition, the right adjoint functor of skCN . Let ∆[r] (r ∈ N) be the category of objects of ∆ augmented to [r] and let ∆N [r] be a full subcategory of ∆[r] whose objects are [q] −→ [r]’s such that q ≤ N . The explicit description of the coskelton is the following ([42, Vbis (3.0.1.2)]): coskCN (X•≤N )r = lim∆ Xq ←− N [r]

(r ∈ N).

As in the proof of [19, (11.2.5)], we have the following by the adjointness of the skelton and the coskelton: Lemma 3.1.1 ([19, (11.2.5)], [88, (7.3.2)]). coskCl (ΓCN (X)•≤l ) = ΓCN (X)≤l . Let S be a (formal) scheme (resp. a fine log (formal) scheme). Let CS be the category of (formal) schemes (resp. fine log (formal) schemes) over S. The inverse limit of a finite inverse system exists in the category of fine log (formal) schemes (over a fine log (formal) schemes) (cf. [54, (1.6), (2.8)]). Set ΓSN (?) := ΓCNS (?) and coskSN (?) := coskCNS (?). The following are immediate generalizations of [19, (11.2.4), (11.2.6)]: Lemma 3.1.2 ([70, (6.5)]). Let N be a nonnegative integer. Let S be a fine log (formal) scheme. Then the following hold: (1) Let X•≤N be a fine N -truncated simplicial log (formal) scheme over S. If XN −→ Y is a closed immersion of fine log schemes over S, then the morphism X•≤N −→ ΓSN (Y )•≤N in (3.1.0.20) is an immersion of N -truncated ◦

◦

fine log (formal) schemes over S. Moreover, if X •≤N and Y are separated ◦

over S, then the morphism X•≤N −→ ΓSN (Y )•≤N is a closed immersion. (2) Let X −→ S be a morphism of fine log (formal) schemes. Assume that the morphism X −→ S satisfies a condition (P) which is stable under the base change X ×S Y −→ Y for any Y ∈ CS . Then the natural morphism coskSl (ΓSN (X)•≤l )m −→ coskSl (ΓSN (X)•≤l )m satisfies (P) for l < l and for any m. Let U be a separated scheme of finite type over a field κ. We recall the following definition: Definition 3.1.3 ([24, (5.3.8)]). Let N be a nonnegative integer (resp. ∞). Let U•≤N be a separated N -truncated simplicial scheme (resp. a separated

3.1 Preliminaries for Later Sections

225

simplicial scheme) of finite type over κ. The N -truncated simplicial scheme U•≤N is said to be an N -truncated proper hypercovering (resp. a proper hypercovering) of U if the natural morphism Un+1 −→ coskU n (U•≤n )n+1 (−1 ≤ n ≤ N −1) is proper and surjective. Here we set coskU (U )0 := U . •≤−1 −1 ⊂

Let U −→ U be an open immersion of separated schemes of finite type over κ. As in [19] and [87], we call (U, U ) a pair over κ. Let n be a nonnegative integer. We recall the following definition due to Tsuzuki: Definition 3.1.4 ([87, (2.1.1) (2)]). Let N be a nonnegative integer (resp. ∞). Let ⊂ U•≤N −−−−→ U •≤N ⏐ ⏐ ⏐ ⏐   ⊂

U −−−−→ U be a commutative diagram of (N -truncated simplicial) separated schemes of finite type over κ, where the horizontal morphisms are open immersions. (As in [19] and [87], we call (U•≤N , U •≤N ) a pair over (U, U ).) The pair (U•≤N , U •≤N ) is said to be an N -truncated proper hypercovering (resp. a proper hypercovering) of (U, U ) if U•≤N is an N -truncated proper hypercovering (resp. a proper hypercovering) of U , if U •≤N is proper over U and if the morphism (U•≤N , U •≤N ) −→ (U, U ) is strict, that is, U•≤N = U •≤N ×U U . Lemma 3.1.5 ([70, (10.5)]). Let N be a nonnegative integer and let (U, U ) be a pair. Let U•≤N be an N -truncated proper hypercovering of U . Then there exists an N -truncated proper hypercovering (U•≤N , U •≤N ) of (U, U ). Finally we note the following, which is an immediate generalization of the technique used in the proof of (2.10.3). Let F : (T , A) −→ (T  , A ) be a morphism of ringed topoi. Consider a small category I := {i, i } consisting of two elements. The morphisms in I, by definition, consist of three elements idi , idi and a morphism i −→ i. Set (Ti , Ai ) := (T , A) and (Ti , Ai ) := (T  , A ). By corresponding the natural morphism F to the morphism i −→ i, we have a ringed topos (Tj , Aj )j∈I . Let (F • , P ) and (F • , P  ) be a filtered complex of A-modules and a filtered complex of A -modules, respectively, with a morphism (F • , P  ) −→ F∗ ((F • , P )) of filtered complexes of A -modules. Set (Fi• , Pi ) := (F • , P ) and (Fi• , Pi ) := (F • , P  ). Then we have a filtered complex (Fj• , Pj )j∈I of (Aj )j∈I -modules. Let (Fj•• , Pj )j∈I be a filtered flasque resolution of (Fj• , Pj )j∈I . Then the complex s(Fj•• , Pj )j∈I gives us flasque resolutions of s(F •• , P  ) and s(F •• , P ) of (F • , P  ) and (F • , P ), respectively, fitting into the following commutative diagram

(3.1.5.1)

(F • , P  ) −−−−→ ⏐ ⏐ 

F∗ ((F • , P )) ⏐ ⏐ 

s(F •• , P  ) −−−−→ F∗ (s(F •• , P )).

226

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

3.2 Rigid Cohomology Let V be a complete discrete valuation ring of mixed characteristics with (not necessarily perfect) residue field κ. Let W be a Cohen ring of κ in V. Let K (resp. K0 ) be the fraction field of V (resp. W ). Let U be a separated scheme of finite type over κ. By Nagata’s embedding theorem ([66]), there ⊂ exists an open immersion j : U −→ U into a proper scheme over κ. Let C be an overconvergent m isocrystal on U/K ([7, (2.2), (2.3)], [19, (10.3.1)]). Let Z := i=1 Z i (m ∈ Z>0 ) be the disjoint union of affine open subschemes which cover U . Set Z := Z ×U U . The scheme Z can be embedded into a separated formally smooth p-adic formal V-scheme Z. Then we have U V ˇ a Cech diagram Z := (Z• , Z • , Z• ) := (coskU 0 (Z), cosk0 (Z), cosk0 (Z)). Let π : (Z• , Z • ) −→ (U, U ) be the natural augmentation. Denote the open im⊂ ˇ mersion Z• −→ Z • by j• . Then, by [19, (10.1.4)], the Cech diagram is a universally de Rham descendable hypercovering of (U, U ) over Spf(V). Then we have, by definition, (3.2.0.1)

RΓrig (U/K, C) := RΓ(]Z • [Z• , DR(π ∗ (C)))

([19, (10.4)]). Here DR(π ∗ (C)) is the complex π ∗ (C)Z• ⊗j•† O ∗

]Z • [Z

•

j•† Ω•]Z

• [ Z•

on ]Z • [Z• . For simplicity of notation, we denote DR(π (C)) only by DR(C). The complex RΓrig (U/K, C) is a well-defined object in the derived category D+ (K) of bounded below complexes of K-vector spaces ([19, (10.4.3)]). We call RΓrig (U/K, C) the rigid cohomological complex of C (in [19, p. 185] Chiarellotto and Tsuzuki have called it the rigid cohomology complex of C). When C is the trivial coefficient, we call RΓrig (U/K, C) the rigid cohomological complex of U/K and we denote it by RΓrig (U/K). Set (3.2.0.2)

h Hrig (U/K, C) := H h (RΓrig (U/K, C))

and (3.2.0.3)

h Hrig (U/K) := H h (RΓrig (U/K)).

h h We call Hrig (U/K, C) and Hrig (U/K) the rigid cohomology of C and the rigid cohomology of U/K, respectively. By [88, (6.4.1)] there exists an integer c such that

(3.2.0.4)

h Hrig (U/K, C) = 0

for all h > c.

The following is a reformulation of the proper cohomological descent of rigid cohomology in Tsuzuki’s theory ([87] and [88]): Theorem 3.2.1 ([70, (10.9)]). Let U• be a proper hypercovering of U . Let C t be the pull-back of C by the structural morphism Ut −→ U (t ∈ N). Let c

3.2 Rigid Cohomology

227

be an integer in (3.2.0.4). Let N be a positive integer satisfying the inequality N > 2−1 (c + 1)(c + 2). Let D+ (K •≤N ) be the derived category of N -truncated (0 ≤ t ≤ N )) cosimplicial complexes of K-vector spaces. Let s (resp. e−1 t be the N -truncated cosimplicial version of the functor (resp. the cosimplicial version of the functor) defined in (3.1.0.3) (resp. (3.1.0.18)) for D+ (K •≤N ) : e−1

s

t D+ (K) ←− D+ (K •≤N ) −→ D+ (K).

•≤N ((U•≤N /U )/K, C) in D+ (K •≤N ) such that Then there exists an object Crig there exist canonical isomorphisms

(3.2.1.1)

∼

•≤N RΓrig (U/K, C) −→ τc s(Crig ((U•≤N /U )/K, C))

and (3.2.1.2) ∼ •≤N RΓrig (Ut /K, C t ) −→ e−1 t (Crig ((U•≤N /U )/K, C))

(0 ≤ ∀t ≤ N ).

•≤N The object Crig ((U•≤N /U )/K, C) is functorial for a morphism of augmented simplicial schemes U•≤N −→ U ’s and a morphism of overconvergent isocrystal C’s. Furthermore, for N ≤ N  , there exists a canonical isomorphism

(3.2.1.3) ∼ •≤N  •≤N s{(Crig ((U•≤N  /U )/K, C))•≤N } −→ s(Crig ((U•≤N /U )/K, C)). Here • ≤ N on the left hand side of (3.2.1.3) means the N -truncation of the first cosimplicial degree. •≤N The definition of Crig ((U•≤N /U )/K, C) is as follows. By (3.1.5) there exists an N -truncated proper hypercovering (U•≤N , U •≤N ) of (U, U ). The pair (U•≤N ×U Z, U •≤N ×U Z) over (Z, Z) is an N -truncated proper hypercovering of (Z, Z). Let (V•≤N , V •≤N ) be a refinement ([87, (4.2.1)]) of the proper hypercovering (U•≤N ×U Z, U •≤N ×U Z) of (Z, Z) ⊂ such that there exists a closed immersion V N −→ PN into a separated formally smooth p-adic formal V-scheme ([87, (4.2.3)]). Consider the N truncated triple

(3.2.1.4)

(V•≤N , V •≤N , ΓV N (PN )•≤N ).

The simplicial formal V-scheme ΓV N (PN )•≤N contains V •≤N as an N -truncated simplicial closed subscheme. Set Q• := ΓV N (PN ). Following the idea in [87, (4.4)], we consider the triple (V•≤N,• , V •≤N,• , Q•≤N,• ) of (N, ∞)-truncated bisimplicial (formal) schemes defined by

228

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

(3.2.1.5) Um V m  (Vmn , V mn , Qmn ) :=(coskU 0 (Vm )n , cosk0 (V m )n , cosk0 (Qm ×V Z)n )

(0 ≤ m ≤ N, n ∈ N) with the natural morphisms which make (V•≤N,• , V •≤N,• , Q•≤N,• ) a triple of (N, ∞)-truncated bisimplicial schemes. Set (3.2.1.6)

•≤N ((U•≤N /U )/K, C) := RΓ•≤N (]V •≤N,• [Q•≤N,• , DR(C)). Crig

Here the N -truncated cosimplicial degree in RΓ•≤N means the first N -truncated cosimplicial degree. The natural morphism (V•≤N,• , V •≤N,• , Q•≤N,• ) −→ (Z• , Z • , Z• ) of triples is shown to induce an isomorphism (3.2.1.1) ([87, pp. 126–127], [70, (10.9)]). For a nonnegative integer t ≤ N , (Vt• , V t• , Qt• ) is a universally de Rham descendable covering of (Ut , U t ) by the proof of [87, (4.4.1) (2)]. Hence (3.2.1.7)

RΓrig (Ut /K) = RΓ(]V t• [Qt• , DR(C)),

which is nothing but the isomorphism (3.2.1.2).

3.3 Comparison Theorems ⊂

Let V, κ, K, K0 , W and j : U −→ U be as in §3.2. In this section we state comparison theorems between RΓrig (U/K), the log convergent cohomologial complex of a certain split proper hypercovering of (U, U ) and the log crystalline cohomological complex of it. The following is an easy corollary of de Jong’s alteration theorem ([28, (4.1)]): Theorem 3.3.1 ([28, Introduction]). There exists the following cartesian diagram ⊂

(3.3.1.1)

U0 −−−−→ ⏐ ⏐ 

X0 ⏐ ⏐ 

⊂

U −−−−→ U such that the left vertical morphism is proper and surjective and such that U0 is the complement of an SNCD D0 in a projective regular scheme X0 over κ. Using (3.3.1) and a general formalism in [42, Vbis ], we have the following:

3.3 Comparison Theorems

229

Proposition 3.3.2 ([70, (9.2)]). There exists a proper hypercovering (U• , U • )•∈N of (U, U ) ((3.1.4)). Moreover one can take the pair (U• , U • )•∈N over κ satisfying the following conditions: (3.3.2.1): X• := U • is regular, (3.3.2.2): D• := X• \ U• is a simplicial SNCD on X• , ⊂ (3.3.2.3): (U• , X• ) is split, that is, there exists an open immersion N (U )m −→ N (X)m (m ∈ N) of schemes such that     N (U )m , N (X)m ) (Un , Xn ) = ( 0≤m≤n [n]
[m]

0≤m≤n [n]
[m]

([42, Vbis (5.1.1)], [24, (6.2.2)]). Here [n]  [m] means a surjection in ∆. Definition 3.3.3. We say that a proper hypercovering of (U, U ) satisfying (3.3.2.1) and (3.3.2.2) is good. We say that a proper hypercovering of (U, U ) satisfying (3.3.2.1) ∼ (3.3.2.3) is gs(=good and split). We can prove the following without difficulty. ⊂

Lemma 3.3.4 ([70, (9.6)]). Let U• −→ Y• be an open immersion of simpli⊂ cial schemes over an open immersion U −→ Y of schemes over κ. Assume that (U• , Y• ) is split. Then there exists a pair (U• , Y• ) of split simplicial schemes over (U, Y ) with a natural morphism (U• , Y• ) −→ (U• , Y• ) of the pairs of the simplicial schemes satisfying the following conditions:  (3.3.4.1): Um (m ∈ N) is the disjoint union of open subschemes of Um which are open subschemes of affine open subschemes of Ym ; these open subschemes cover Um .

(3.3.4.2): Ym (m ∈ N) is the disjoint union of affine open subschemes which cover Ym and whose images in Y are contained in affine open subschemes of Y .  (3.3.4.3): Um = Um ×Ym Ym

(m ∈ N).

(3.3.4.4): If (U• , Y• ) is strict over (U, Y ), that is, U• = Y• ×Y U , then (U• , Y• ) is strict over (U, Y ). (3.3.4.5): If Ym (m ∈ N) is quasi-compact, then the number of the open subschemes in (3.3.4.1) and (3.3.4.2) can be assumed to be finite. In particular, there exists a natural morphism (U•• , Y•• ) −→ (U• , Y• ) by setYm   m ting (Umn , Ymn ) := (coskU 0 (Um )n , cosk0 (Ym )n ). Moreover, for each n ∈ N, (U•n , Y•n ) is split. Let Y be a proper smooth scheme over κ and E an SNCD on Y /κ. Set
V := Y \E. Let K(Y,E)/V be the isostructure sheaf in ((Y, E)/V)log conv . Then, by [82, Corollary 2.3.9] and [82, Theorem 2.4.4], we have a canonical isomorphism

230

(3.3.4.6)

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary) ∼
RΓ(((Y, E)/V)log conv , K(Y,E)/V ) −→ RΓrig (V /K).

By [82, Theorem 3.1.1] we also have a canonical isomorphism (3.3.4.7)

∼ RΓ(((Y,
E)/W )log conv , K(Y,E)/W ) −→ RΓlog-crys ((Y, E)/W )K0 .

Let (U• , X• ) be a gs proper hypercovering of (U, U ). Assume that X• is smooth over κ. Set D• := X• \ U• . Now we state the following comparison theorems. ((1) and (2) in (3.3.5) are generalizations of (3.3.4.6) and (3.3.4.7), respectively. The main part of the proof of (3.3.5) is the reduction to the facts (3.3.4.6) and (3.3.4.7).) Theorem 3.3.5 (Comparison theorems ([70, (11.6)])). (1) There exists an isomorphism (3.3.5.1)

∼ , D• )/V)log RΓ(((X•
conv , K(X• ,D• )/V ) −→ RΓrig (U/K).

The isomorphism (3.3.5.1) is functorial: for a morphism (V, V ) −→ (U, U ) of pairs over κ and for gs proper hypercoverings (U• , X• ) of (U, U ) and (V• , Y• ) of (V, V ), respectively, fitting into the following commutative diagram (V• , Y• ) −−−−→ (U• , X• ) ⏐ ⏐ ⏐ ⏐   (V, V ) −−−−→ (U, U ), the following diagram (3.3.5.2) log
RΓ(((Y•
, E• )/V)log conv , K(Y• ,E• )/V ) ←−−−− RΓ(((X• , D• )/V)conv , K(X• ,D• )/V ) ⏐ ⏐ ⏐ ⏐   ←−−−−

RΓrig (V /K)

RΓrig (U/K)

is commutative, where E• := Y• \ V• . In particular, RΓ(((X•
, D• )/V)log conv , K (X• ,D• )/V ) depends only on U/κ and K. (2) There exists a functorial isomorphism (3.3.5.3) ∼ RΓ(((X•
, D• )/W )log , K(X ,D )/W ) −→ RΓlog-crys (((X• , D• )/W )K . conv

•

•

0

The construction of the morphism (3.3.5.1) is as follows. Let h be a nonnegative integer. Let N be a sufficiently large integer (e.g., N > 2−1 (h + 1)(h + 2); the reader does not need to mind this inequality). Let (U•• , X•• ) be a pair of bisimplicial schemes over κ in (3.3.4) for the pair (U• , X• ). Let (Z, Z, Z) be the triple in the beginning of §3.2. Consider the

3.3 Comparison Theorems

231

fiber product (U•• ×U Z, X•• ×U Z) and a pair of schemes defined by the following: lm lm (Xlmn , Dlmn ) := (coskX (Xlm ×U Z)n , coskD (Dlm ×U Z)n ) 0 0

(l, m, n ∈ N). Then we have the pair (X••• , D••• ) of a smooth trisimplicial scheme with trisimplicial SNCD over κ. Set U••• := X••• \ D••• . Because XN 0 is affine, there exists a lift (XN 0 , DN 0 ) of (XN 0 , DN 0 ) over Spf(V) such that XN 0 is a formally smooth scheme over Spf(V) and such that DN 0 is a relative SNCD on XN 0 /Spf(V). By (3.1.2) (1) we have a closed immersion (X•≤N,0 , D•≤N,0 ) −→ ΓV N ((XN 0 , DN 0 ))•≤N into a formally log smooth N -truncated simplicial scheme over Spf(V). Set Γl := ΓV N ((XN 0 , DN 0 ))l (l ∈ N). Then we have a closed immersion ⊂

Dl V l (Xlm , Dlm ) = (coskX 0 (Xl0 )m , cosk0 (Dl0 )m ) −→ cosk0 (Γl )m

(0 ≤ l ≤ N ).

Set Γlm := coskV 0 (Γl )m . Then we have a closed immersion lm lm (Xlm• , Dlm• ) = (coskX (Xlm ×U Z), coskD (Dlm ×U Z)) 0 0

⊂  −→ coskV 0 (Γlm ×V Z) (m, n ∈ N).

In this way we have a closed immersion (3.3.5.4)

⊂

(X•≤N,•• , D•≤N,•• ) −→ (P•≤N,•• , M•≤N,•• )

into a formally log smooth (N, ∞, ∞)-truncated trisimplicial p-adic fine log formal V-scheme such that the underlying trisimplicial formal scheme P•≤N,•• is formally smooth over Spf(V); moreover there exists a morphism P•≤N,•• −→ Z• over V fitting into the following commutative diagram ⊂

X•≤N,•• −−−−→ P•≤N,•• ⏐ ⏐ ⏐ ⏐   Z•

⊂

−−−−→

Z• .

⊂

Let j•≤N,•• : U•≤N,•• −→ X•≤N,•• be the open immersion. The morphism (3.3.5.4) induces the following morphism (3.3.5.5) RΓrig (U/K) = RΓ(]Z • [Z• , j•† Ω•]Z • [Z ) •

† −→ RΓ(]X•≤N,•• [P•≤N,•• , j•≤N,•• Ω•]X•≤N,•• [P

•≤N,••

).

Using the triple (V•≤N,• , V •≤N,• , Q•≤N,• ) of (N, ∞)-truncated bisimplicial (formal) schemes in the previous section, we can prove that H h ((3.3.5.5)) is an isomorphism ([70, (11.6)]).

232

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

Let RΓ((X•≤N,•• , D•≤N,•• )/K, LDR(O]X•≤N,•• [log

P•≤N,••

)) be the complex

which produces log naive rigid cohomologies which are trisimplicial version of the log naive rigid cohomologies defined in [82, Definition 2.2.12] (cf. [4, p. 14]) (the terminology “naive rigid cohomology” has been used in [4, p. 14], while the terminology “log analytic cohomology” has been used in [82] instead of the terminology “log naive rigid cohomology”). Then, by the cohomological descent and by the proof of [82, Corollary 2.3.9] and [82, Theorem 2.4.4], there exists a natural morphism (3.3.5.6)

, D•≤N )/V)log RΓ(((X•≤N
conv , K(X•≤N ,D•≤N )/V ) , D•≤N,•• )/V)log = RΓ(((X•≤N,••
conv , K(X•≤N,•• ,D•≤N,•• )/V ) = RΓ((X•≤N,•• , D•≤N,•• )/K, LDR(O]X•≤N,•• [log

P•≤N,••

))

† −→ RΓ((X•≤N,•• , D•≤N,•• )/K, LDR(j•≤N,•• O]X•≤N,•• [P•≤N,•• )) † Ω•]X•≤N,•• [P := RΓ(]X•≤N,•• [P•≤N,•• , j•≤N,••

•≤N,••

).

We can check that the induced morphism H h ((3.3.5.6)) is an isomorphism by Shiho’s comparison theorems ([82, Corollary 2.3.9, Theorem 2.4.4]) ([70, (11.6)]). Thus, for any nonnegative integer h, there exists a sufficiently large positive integer N such that there exists an isomorphism (3.3.5.7)

, D• )/V)log τh RΓ(((X•
conv , K(X• ,D• )/V )

∼ , D•≤N )/V)log = τh RΓ(((X•≤N
conv , K(X•≤N ,D•≤N )/V ) −→ τh RΓrig (U/K).

By (3.2.0.4) for the trivial coefficient, there exists an integer c such that i Hrig (U/K) = 0 for i > c. Hence, by (3.3.5.7), (3.3.5.8) log
RΓ(((X•
, D• )/V)log conv , K(X• ,D• )/V ) = τc RΓ(((X• , D• )/V)conv , K(X• ,D• )/V ).

Using (3.3.5.7) for h = c, we have the isomorphism (3.3.5.1). We can check that the isomorphism (3.3.5.1) is independent of the choice of the lift (XN 0 , DN 0 ) and the choice of (U•• , X•• ). We can also check that the isomorphism (3.3.5.1) is independent of the choice of (Z• , Z • , Z• ) and the choice of N satisfying N > 2−1 (c + 1)(c + 2). Moreover we can prove the functoriality of the isomorphism (3.3.5.1). See [70, (11.6)] for details. The construction of the morphism (3.3.5.3) is as follows. Let A•≤N,•• be the structure sheaf of the log PD-envelope of the closed ⊂ immersion (X•≤N,•• , D•≤N,•• ) −→ (P•≤N,•• , M•≤N,•• ). By the proof of [82, Theorem 3.1.1], we have a morphism

3.3 Comparison Theorems

(3.3.5.9)

233

RΓ((X•≤N,•• , D•≤N,•• )/K0 , LDR(O]X•≤N,•• [log

P•≤N,••

))

−→ RΓ(X•≤N,•• , A•≤N,•• ⊗OP•≤N,•• Ω•P•≤N,•• /W (log M•≤N,•• ))K0 . By the usual cohomological descent, the canonical filtration τh on the right hand side is nothing but τh RΓlog-crys ((X• , D• )/W )K0 . As in the case of (3.3.5.6), by Shiho’s comparison theorem ([82, Theorem 3.1.1]), we see that the induced morphism H h ((3.3.5.9)) is an isomorphism ([70, (11.6)]). Now, by (3.3.5.8), we have the isomorphism (3.3.5.3). We can prove that the isomorphism (3.3.5.3) is independent of the choice of the lift (XN 0 , DN 0 ). We can also check that the isomorphism (3.3.5.3) is independent of the choice of N satisfying N > 2−1 (c + 1)(c + 2). Moreover we can prove the functoriality of the isomorphism (3.3.5.3) ([70, (11.6)]). Corollary 3.3.6. The following hold: (1) Let the notations and assumptions be as in (3.3.5) (1). Then there exists a canonical isomorphism (3.3.6.1)

∼

RΓrig (U/K) −→ RΓlog-crys ((X• , D• )/W )K .

In particular, RΓlog-crys ((X• , D• )/W )K depends only on U and K, and there exists a canonical isomorphism (3.3.6.2)

∼

h h Hrig (U/K) −→ Hlog-crys ((X• , D• )/W )K

(h ∈ Z).

h In particular, Hrig (U/K) is finite dimensional (because we have the following spectral sequence i+j j E1ij = Hcrys ((Xi , Di )/W ) =⇒ Hlog-crys ((X• , D• )/W )). h Remark 3.3.7. (1) In [5] Berthelot has proved that Hrig (U/K) is finite dimensional if U is smooth over κ. In [36] Große-Kl¨ onne has first proved that h (U/K) is finite dimensional. In [87] Tsuzuki has given an alternative proof Hrig h (U/K) by using his proper cohomological of the finite dimensionality of Hrig descent in rigid cohomology and by reducing the finiteness to Berthelot’s result above. (2) The right hand side of (3.3.6.1) depends only on U/κ and K; this solves a problem raised in [28, Introduction] for the split case in a stronger form. (3) In [1], in a different method from ours, Andreatta and Barbieri-Viale 1 ((X• , D• )/W ) depends only on U/κ for the case have proved that Hlog-crys where p ≥ 3 and where the augmentation morphism X0 \ D0 −→ U is generically etale, even for the non-split case.

We can generalize (3.3.6) as follows. Let ρ : V −→ U be a morphism of separated schemes of finite type over κ. ⊂ It is easy to see that there exists an open immersion V −→ V into a proper

234

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

scheme fitting into the following commutative diagram ⊂

(3.3.7.1)

V −−−−→ ⏐ ⏐ 

V ⏐ ⏐ 

⊂

U −−−−→ U over κ (e.g., [5, p. 340]). Using the general formalism in [42, Vbis ], we have gs proper hypercoverings (V• , Y• ) and (U• , X• ) of (V, V ) and (U, U ), respectively, fitting into the following commutative diagram: ρ•

(3.3.7.2)

(V• , Y• ) −−−−→ (U• , X• ) ⏐ ⏐ ⏐ ⏐   ρ

(V, V ) −−−−→ (U, U ). Set D• := X• \ U• and E• := Y• \ V• . For a morphism f : (F • , d•F ) −→ (G• , d•G ) of complexes in an additive category A, let MF(f ) := (F • , d•F ) ⊕ (G• [−1], d•G [−1]) be the mapping fiber of f . This operation is well-defined in the derived category D+ (A). Then we have the following by the functoriality in (3.3.5) (3). Corollary 3.3.8. There exists the following canonical isomorphism ∼

(3.3.8.1) MF(ρ∗rig : RΓrig (U/K) −→ RΓrig (V /K)) −→ MF(ρ∗• : RΓlog-crys ((X• , D• )/W )K −→ RΓlog-crys ((Y• , E• )/W )K ). This isomorphism is independent of the choice of the commutative diagram (3.3.7.2).

3.4 Weight Filtration on Rigid Cohomology Let (S, I, γ) be a PD-scheme on which a prime number p is locally nilpotent or a (not necessarily affine) P -adic base in the sense of [11, 7.17 Definition]. Set S0 := SpecS (OS /I). Let X• be a smooth simplicial scheme over S0 and let D• and Z• be transversal simplicial relative SNCD’s on X• /S0 . Let f : X• −→ S be the structural morphism. As in §2.3, we have the following forgetting log morphisms of simplicial log schemes over S and log crystalline ringed topoi: (3.4.0.1)

(X• ,D• ∪Z• ,Z• )/S : (X• , D• ∪ Z• ) −→ (X• , Z• ),

(3.4.0.2)

log
(X• ,D• ∪Z• ,Z• )/S : (((X• , D • ∪ Z• )/S)crys , O(X• ,D• ∪Z• )/S )

, Z• )/S)log −→ (((X•
crys , O(X• ,Z• )/S ).

3.4 Weight Filtration on Rigid Cohomology

235

Let I •• be a flasque resolution of O(X• ,D• ∪Z• )/S , which exists by [42, Vbis (1.3.10)]. Then, set (3.4.0.3)

log,Z• (Ecrys (O(X• ,D• ∪Z• )/S ), P D• ) := ((X• ,D• ∪Z• ,Z• )/S∗ (I •• ), τ )

in D+ F(O(X• ,Z• )/S ). Let (I •• , P D• ) be the filtered flasque resolution of the right hand side above. Set (3.4.0.4)

log,Z• (Ezar (O(X• ,D• ∪Z• )/S ), P D• ) := u(X• ,Z• )/S∗ ((I •• , P D• ))

in D+ F(f −1 (OS )). It is clear that log,Zt (Ecrys (O(Xt ,Dt ∪Zt )/S ), P Dt )

and

log,Zt (Ezar (O(Xt ,Dt ∪Zt )/S ), P Dt )

(t ∈ N)

in this section are isomorphic to log,Zt (Ecrys (O(Xt ,Dt ∪Zt )/S ), P Dt )

and

log,Zt (Ezar (O(Xt ,Dt ∪Zt )/S ), P Dt )

in (2.7.3.1) and (2.7.4) in D+ F(O(Xt ,Zt )/S ) and D+ F(ft−1 (OS )), respectively, where ft : Xt −→ S is the structural morphism. In the case Z• = ∅, we denote log,Z• (Ecrys (O(X• ,D• ∪Z• )/S ), P D• )

and

log,Z• (Ezar (O(X• ,D• ∪Z• )/S ), P D• )

(Ecrys (O(X• ,D• )/S ), P )

and

(Ezar (O(X• ,D• )/S ), P ),

by respectively. Now, let the notations be as in (3.3.7.2). Let f : X• −→ Spf(W ) and g : Y• −→ Spf(W ) be the structural morphisms. From this section, assume that κ is perfect unless stated otherwise. Consider the natural morphism O(X• ,D• )/W −→ ρlog •crys∗ (O(Y• ,E• )/W ). By the final note in §3.1, we have •• •• and I(Y of O(X• ,D• )/W and O(Y• ,E• )/W , flasque resolutions I(X • ,D• )/W • ,E• )/W respectively, with a natural morphism •• •• I(X −→ ρlog •crys∗ (I(Y• ,E• )/W ). • ,D• )/W •• Let (J •• , P ) be a filtered flasque resolution of ((Y• ,E• )/W ∗ (I(Y ), τ ). • ,E• )/W Then, by using the morphism (2.7.2.1), we have the following composite morphism

(3.4.0.5) •• •• ((X• ,D• )/W ∗ (I(X ), τ ) −→ ((X• ,D• )/W ∗ ρlog •crys∗ (I(Y• ,E• )/W ), τ ) • ,D• )/W •• −→ (ρ•crys∗ (Y• ,E• )/W ∗ (I(Y ), τ ) • ,E• )/W •• −→ ρ•crys∗ ((Y• ,E• )/W ∗ (I(Y ), τ ) • ,E• )/W

−→ ρ•crys∗ ((J •• , P )).

236

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

The morphism (3.4.0.5) in C+ F(OX• /W ) gives us a morphism (Ecrys (O(X• ,D• )/W ), P ) −→ Rρ•crys∗ (Ecrys (O(Y• ,E• )/W ), P )

(3.4.0.6)

in D+ F(f −1 (W )). As a result, we have a natural morphism (Ezar (O(X• ,D• )/W ), P ) −→ Rρ•∗ (Ezar (O(Y• ,E• )/W ), P )

(3.4.0.7)

in D+ F(f −1 (W )). We can also take a representative of the morphism (3.4.0.7) in C+ F(f −1 (W )). Consider the following morphism •
ρ∗• • : RΓ (X•zar , (Ezar (O(X• ,D• )/W ), P ))

−→ RΓ• (Y
•zar , (Ezar (O(Y• ,E• )/W ), P )) of filtered complexes of W • -modules induced by (3.4.0.7) and the following filtered mapping fiber of ρ∗• • : (s(MF(ρ∗• • )), δMF (P )). (k)

(k)

(k)

(k)

Let at : Dt −→ Xt (k, t ∈ N) and bt : Et −→ Yt (k, t ∈ N) be the natural morphisms. Then (3.1.0.11) for (s(MF(ρ∗• • )), δMF (P )) is (3.4.0.8) grkMF s(MF(ρ∗• • ))   grP grP = t+k Ezar (O(Xt ,Dt )/W )[−t] ⊕ t+k+1 Ezar (O(Yt ,Et )/W )[−t − 1] (P )

δ

t≥0

=

 t≥0

 t≥0

t≥0 (t+k) at,zar∗ RuD(t+k) /W ∗ (OD(t+k) /W t t

(t+k) ⊗Z
crys (Dt /W ))

{−t − k}[−t](−(t + k))⊕ (t+k+1) bt,zar∗ RuE (t+k+1) /W ∗ (OE (t+k+1) /W t t

(t+k+1) ⊗Z
crys (Et /W ))

{−t − k − 1}[−t − 1](−(t + k + 1)) in D+ (f −1 (W )) by (2.7.5.1) and (2.6.1.2). By (3.1.0.12) and (3.3.8.1), we have the following spectral sequences: Theorem 3.4.1 ([70, (16.1)]). Denote MF(ρ∗rig : RΓrig (U/K) −→ RΓrig (V /K)) simply by MF(ρ∗rig ). Then there exists the following spectral sequence: (3.4.1.1) E1−k,h+k =

 t≥0


(t+k) H h−2t−k ((Dt /W )crys , OD(t+k) /W t

(t+k) (Dt /W ))(−(t + k))K ⊕ ⊗Z
crys

3.4 Weight Filtration on Rigid Cohomology



237


(t+k+1)

H h−2t−k−2 ((Et

/W )crys , OE (t+k+1) /W t

t≥0

(t+k+1) ⊗Z
crys (Et /W ))(−(t + k + 1))K

=⇒H h (MF(ρ∗rig )). The spectral sequence (3.4.1.1) degenerates at E2 . The induced filtration P on H h (MF(ρ∗rig )) is independent of the choice of the commutative diagram (3.3.7.2). Definition 3.4.2. We call (3.4.1.1) the weight spectral sequence of MF(ρ∗rig ). We call the filtration P the weight filtration on H h (MF(ρ∗rig )). We can prove the E2 -degeneration of the spectral sequence (3.4.1.1) as in (2.17.2). We can prove the independence of P by using Grothendieck’s idea for the reduction of geometric problems to arithmetic problems ([40, 3], [37]) in the following way: (3.4.2.1): for two commutative diagrams (3.3.7.2)’s over ρ, we may assume that there exists a morphism between them (a general formalism in [42, Vbis ]), (3.4.2.2): for any nonnegative integer N , the existence of a model of ρ•≤N over the spectrum of a smooth ring of finite type over a finite field (the standard log deformation theory (cf. the proof of (2.18.1)), (3.4.2.3): the reduction to the case of finite fields (the specialization argument of Deligne-Illusie ([49], [68], §2.14)), (3.4.2.4): the purity of the eigenvalues of the Frobenius on the classical crystalline cohomology of a proper smooth variety over a finite field ([57], [15, (1.2)], [68, (2.2) (4)]). See [70, (12.5), (15.2)] for the more detailed proof of the independence of P . Remark 3.4.3. By using (3.1.0.13), we can give the explicit description of the boundary morphism d−k,h+k of the spectral sequence (3.4.1.1). 1 Theorem 3.4.4 (Strict compatibility ([70, (15.5)])). Let ρ

V  −−−−→ ⏐ ⏐ v

U ⏐ ⏐u 

ρ

V −−−−→ U be a commutative diagram of separated schemes of finite type over κ. Then the induced morphism

238

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

(u∗ , v ∗ ) : H h (MF(ρ∗rig )) −→ H h (MF(ρ∗rig ))

(h ∈ Z)

is strictly compatible with the weight filtration. Proposition 3.4.5 ([70, (15.4)]). The weight filtration on H h (MF(ρ∗rig )) is compatible with the extension of complete discrete valuation rings of mixed characteristics with perfect residue fields. Let Z be a closed subscheme of U . Let V be the complement of Z in U ⊂ and let ρ : V −→ U be the open immersion. Set (3.4.5.1)

RΓrig,Z (U/K) := MF(ρ∗rig : RΓrig (U/K) −→ RΓrig (V /K))

and h Hrig,Z (U/K) := H h (RΓrig,Z (U/K)).

(3.4.5.2)

We call RΓrig,Z (U/K) the rigid cohomological complex of U/K with support h (U/K) the rigid cohomology of U/K with support in Z in Z, and call Hrig,Z ([5, (2.3)]). As an immediate application of (3.4.1) and (3.4.4), we obtain the following: ⊂

Corollary 3.4.6 ([70, (16.1)]). (1) Let U −→ U be an open immersion into a proper scheme over κ. Let V be the closure of V in U . Let the notations be as in (3.3.7.2). Then there exists the following spectral sequence (3.4.6.1) E1−k,h+k =




(t+k) H h−2t−k ((Dt /W )crys , OD(t+k) /W t

t≥0



(t+k) (Dt /W ))(−(t + k))K ⊕ ⊗Z
crys


(t+k+1)

H h−2t−k−2 ((Et

t≥0

/W )crys , OE (t+k+1) /W t

(t+k+1) ⊗Z
crys (Et /W ))(−(t + k + 1))K h =⇒Hrig,Z (U/K).

The spectral sequence (3.4.6.1) degenerates at E2 . The induced filtration P h (U/K) is independent of the choice of the commutative diagram on Hrig,Z (3.3.7.2). (2) Let u : (U  , Z  ) −→ (U, Z) be a morphism of separated schemes of finite type over κ with closed subschemes. Then the induced morphism (3.4.6.2)

h h  u∗ : Hrig,Z (U/K) −→ Hrig,Z  (U /K)

is strictly compatible with P ’s.

3.4 Weight Filtration on Rigid Cohomology

239

h Definition 3.4.7. (1) We call (3.4.6.1) the weight spectral sequence of Hrig,Z h (U/K). We call the filtration P the weight filtration on Hrig,Z (U/K). h (U/K) is of weight ≥ k (resp. ≤ (2) Let k be an integer. We say that Hrig,Z h h h (U/K)). We say k) if Pk−1 Hrig,Z (U/K) = 0 (resp. Hrig,Z (U/K) = Pk Hrig,Z h h that Hrig,Z (U/K) is of pure weight k if Hrig,Z (U/K) is of weight ≥ k and ≤ k. h h (U/K) is of weight ≤ k if v ∈ Pk Hrig,Z (U/K). We call a vector v of Hrig,Z

If κ is a finite field, then the definitions in (3.4.7) are usual ones using the eigenvalues of the Frobenius endomorphism by the spectral sequence (3.4.6.1) and by the purity of the weight ([57], [15, (1.2)], [68, (2.2) (4)]). We list some fundamental properties of the weight filtration on rigid cohomology. h Proposition 3.4.8 ([70, (16.1)]). The weight filtration on Hrig,Z (U/K) is compatible with the extension of complete discrete valuation rings of mixed characteristics with perfect residue fields.

Theorem 3.4.9 ([70, (16.6)]). The following hold: (1) The exact sequence (3.4.9.1)

h h h · · · −→ Hrig,Z (U/K) −→ Hrig (U/K) −→ Hrig (V /K) −→ · · ·

(cf. [5, (2.3.1)]) is strictly exact with respect to the weight filtration. (2) Let U  be an open subscheme of U which contains Z as a closed subscheme. Then the isomorphism (3.4.9.2)

∼

h h Hrig,Z (U/K) −→ Hrig,Z (U  /K)

(cf. [5, (2.4.1)]) is an isomorphism of weight-filtered K-vector spaces. (3) Let T be a closed subscheme of Z. Then the natural morphism (3.4.9.3)

h h Hrig,T (U/K) −→ Hrig,Z (U/K)

(cf. [5, (2.5.1)]) is strictly compatible with respect to the weight filtration. (4) If Z = Z1  Z2 , then the isomorphism (3.4.9.4)

∼

h h h Hrig,Z (U/K) ⊕ Hrig,Z (U/K) −→ Hrig,Z (U/K) 1 2

(cf. [5, (2.4.2)]) is an isomorphism of weight-filtered K-vector spaces. The following (1) is a generalization of [14, (2.3)] (We reduce the proof of the following to [loc. cit.] as in (3.4.2.2), (3.4.2.3) and (3.4.2.4).). Theorem 3.4.10 ([70, (16.7)]). Set d := dim U , c := codim(Z, U ) and dZ := dim Z. Assume that U and Z are of pure dimensions and that U is smooth over κ. (Then d = c + dZ ([45, II Exercise, (3.20) (d)]).) Then the following hold:

240

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

h (1) The weights of Hrig,Z (U/K) lie in [h, 2(h − c)]. h (U/K) is less than or equal to 2d. (2) The weights of Hrig,Z

The following is a generalization of Berthelot’s K¨ unneth formula ([6, (3.2)]) and Kedlaya’s K¨ unneth formula ([59, (1.2.4)]): Theorem 3.4.11 (cf. [24, (8.1.25)]). Let U i (i = 1, 2) be a separated scheme of finite type over a not necessarily perfect field κ of characteristic ⊂ p > 0. Let Z i be a closed subscheme of U i . Let U i −→ U i be an open immersion into a proper scheme over κ. Set U 12 := U 1 ×κ U 2 and Z 12 := Z 1 ×κ Z 2 . Then the canonical morphism (3.4.11.1) RΓrig,Z 1 (U 1 /K0 ) ⊗K0 RΓrig,Z 2 (U 2 /K0 ) −→ RΓrig,Z 12 (U 12 /K0 ) is an isomorphism. Assume, furthermore, that κ is perfect. Then the induced isomorphism on the cohomologies by the isomorphism (3.4.11.1) is compatible with the weight filtration. h From the definition of the weight filtration on Hrig (U/K), we obtain the following:

Proposition 3.4.12 ([70, (12.9) (2)]). If U is the complement of an SNCD D on a proper smooth scheme X over κ, then the weight filtration P on h h Hrig (U/K) = Hlog-crys ((X, D)/W )K coincides with the weight filtration defined by the filtered zariskian complex (Czar (O(X,D)/W ), P ).

3.5 Slope Filtration on Rigid Cohomology Let the notations be as in §3.3. In this section, assume that κ is perfect and that V = W . By using [68, (9.1), (9.3) (2)], we have a complex Wn Ω•X• (log D• ) (n ∈ Z≥1 ) on X• , which is the cosimplicial version of the log de Rham-Witt complex Wn Ω•X (log D) in §2.12. We have a projection π : Wn+1 Ω•X• (log D• ) −→ Wn Ω•X• (log D• ). The family {Wn Ω•X• (log D• )}∞ n=1 forms an inverse system. Set W Ω•X• (log D• ) := limn Wn Ω•X• (log D• ). ←− As in (3.3.4), in [70] we have proved that there exists a split simplicial log scheme (X• , D• ) over Spec(κ) with a natural morphism  (m ∈ N) (X• , D• ) −→ (X• , D• ) of simplicial log schemes such that Xm is the disjoint union of affine open subschemes which cover Xm . Set Dm   m (Xmn , Dmn ) := (coskX 0 (Xm )n , cosk0 (Dm )n ). Then we have a natural morphism (X•• , D•• ) −→ (X• , D• ). Let g : X•• −→ Spec(Wn ) (n ∈ Z≥1 ) be the structural morphism. By using [68, (9.1), (9.3) (2)] again, we have a complex Wn Ω•X•• (log D•• ) on X•• in C+ (g −1 (Wn )), which is the bicosimplicial version of Wn Ω•X (log D) in §2.12. We have

3.5 Slope Filtration on Rigid Cohomology

241

a projection π : Wn+1 Ω•X•• (log D•• ) −→ Wn Ω•X•• (log D•• ). The family • {Wn Ω•X•• (log D•• )}∞ n=1 forms an inverse system. Set W ΩX•• (log D•• ) := limn Wn Ω•X•• (log D•• ). ←− Let N be a nonnegative integer. Then we have a closed immersion ⊂    , DN ) −→ (XN , DN ) into a formally smooth scheme over Spf(W ) (XN with a relative SNCD. We can assume that there exists an endomorphism    ) −→ (XN , DN ) lifting the Frobenius of (XN , DN ) mod p. Φ : (XN , DN By using the functor (3.1.0.19) and (3.1.2) (1), we have an immersion ⊂    (X•≤N , D•≤N ) −→ Y•≤N into a formally log smooth scheme over Spf(W ).   Note that Y•≤N also has an endomorphism lifting the Frobenius of Y•≤N   modp. Set Y•≤N mod pn := Y•≤N ⊗W Wn . Set Ylm := cosk0 (Yl )m n (0 ≤ l ≤ N ) and Ylm mod p := Ylm ⊗W Wn . By the obvious bicosimplicial version of (3.4.0.3) (resp. (3.4.0.4)), we have a complex Spf(W )

(3.5.0.1)

Ecrys (O(X•• ,D•• )/Wn ) (resp. Ezar (O(X•• ,D•• )/Wn ))

in C+ (OX•• /Wn ) or D+ (OX•• /Wn ) (resp. C+ (g −1 (Wn )) or D+ (g −1 (Wn ))). Let D•≤N,• be the log PD envelope of the (N, ∞)-truncated simplicial immersion ⊂ (X•≤N,• , D•≤N,• ) −→ Y•≤N,• over (Spf(W ), pW, [ ]). Set D•≤N,• mod pn := D•≤N,• ⊗W Wn . Then, in D+ (g −1 (Wn )), we have the following explicit description (3.5.0.2)

Ezar (O(X•≤N,• ,D•≤N,• )/Wn ) ∼

←− RuX•≤N,• /Wn ∗ (Ecrys (O(X•≤N,• ,D•≤N,• )/Wn )) ∼

←− uX•≤N,• /Wn ∗ LX•≤N,• /Wn (Λ•Y•≤N,• mod = OD•≤N,• mod

pn

⊗OY•≤N,• mod

pn

pn /Wn )

Λ•Y•≤N,• mod

pn /Wn

by the Poincar´e lemma of a vanishing cycle sheaf (2.3.10) and by (2.2.1) (cf. the proof of (2.2.22)). Because Wn ΩiX•≤N,• (log D•≤N,• ) = Hi (OD•≤N,• mod

pn

⊗OY•≤N,• mod

pn

Λ•Y•≤N,• /Wn ),

we have a natural morphism (3.5.0.3)

Ezar (O(X•≤N,• ,D•≤N,• )/Wn ) −→ Wn Ω•X•≤N,• (log D•≤N,• )

in D+ (g −1 (Wn )) as in §2.12. Indeed, this morphism is an isomorphism by (2.12.11.1). The isomorphisms (3.5.0.3) for n’s are compatible with two projections of both hands of (3.5.0.3). As a corollary, we have the following composite canonical isomorphism

242

3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

(3.5.0.4) ∼ RΓlog-crys ((X•≤N , D•≤N )/Wn ) −→ RΓlog-crys ((X•≤N,• , D•≤N,• )/Wn ) ∼

−→ RΓ(X•≤N,• , Wn Ω•X•≤N,• (log D•≤N,• )) ∼

←− RΓ(X•≤N , Wn Ω•X•≤N (log D•≤N )), which is the N -truncated split simplicial version of [46, (4.19)] and [68, (7.19)]. The isomorphism (3.5.0.4) is compatible with respect to n’s. Let c be an integer in (3.2.0.4). Let N be a sufficiently large integer (e.g., N > 2−1 (c + 1)(c + 2)). By (3.3.6.1) and (3.2.0.4), we have RΓlog-crys ((X• , D• )/W )K0 = τc RΓlog-crys ((X• , D• )/W )K0 . Because τc RΓlog-crys ((X• , D• )/Wn ) = τc RΓlog-crys ((X•≤N , D•≤N )/Wn ) and τc RΓ(X• , Wn Ω•X• (log D• )) = τc RΓ(X•≤N , Wn Ω•X•≤N (log D•≤N )), we have the following canonical isomorphism by (3.5.0.4): (3.5.0.5)

∼

RΓlog-crys ((X• , D• )/W )K0 −→ RΓ(X• , W Ω•X• (log D• ))K0 .

(The isomorphism (3.5.0.5) is independent of the choice of N .) Hence we have the following isomorphism by (3.3.8.1): ∼

∗ • (3.5.0.6) MF(ρ∗rig ) −→ MF(ρlog •dRW : RΓ(X• ,W ΩX• (log D• ))K0

−→ RΓ(Y• , W Ω•Y• (log E• ))K0 ).

For each nonnegative integer i, we have the following morphism by the functoriality of log Hodge-Witt sheaves ([68, (9.3) (2)]): (3.5.0.7)

∗i i i ρlog •dRW : RΓ(X• , W ΩX• (log D• )) −→ RΓ(Y• , W ΩY• (log E• )).

Here we calculate RΓ(X• , W ΩiX• (log D• )) and RΓ(Y• , W ΩiY• (log E• )) by the Godement resolutions of W ΩiX• (log D• ) and W ΩiY• (log E• ), respectively. By (3.4.1.1) we know that H h (MF(ρ∗rig )) is an F -isocrystal, which is a generalization of [87, (5.1.1)]. As a consequence, H h (MF(ρ∗rig )) has the slope filtration. We have the following spectral sequence (3.5.1.1) by (3.1.0.17) and (3.5.0.6): Theorem 3.5.1 (Slope decomposition ([70, (15.7)])). hold: (1) There exists the following spectral sequence

The following

3.5 Slope Filtration on Rigid Cohomology

(3.5.1.1)

243

∗i h ∗ E1i,h−i = H h−i (MF(ρlog dRW ))K0 =⇒ H (MF(ρrig )).

This spectral sequence degenerates at E1 . (2) There exists the following isomorphism: (3.5.1.2)

∗i h ∗ H h−i (MF(ρlog •dRW ))K0  (H (MF(ρ )))[i,i+1) .

(3) There exists the following slope decomposition (3.5.1.3)

H h (MF(ρ∗rig )) =

h 

∗i H h−i (MF(ρlog •dRW ))K0 .

i=0

Definition 3.5.2. We call the spectral sequence (3.5.1.1) the slope spectral sequence of H h (MF(ρ∗rig )). We call the direct sum decomposition (3.5.1.3) the slope decomposition of H h (MF(ρ∗rig )). By (3.5.1.1) we obtain the following: Proposition 3.5.3 ([70, (15.8)]). The slopes of H h (MF(ρ∗rig )) lie in [0, h]. Now consider the case Y• = E• = ∅. Then (3.5.1) is the following: Theorem 3.5.4 (Slope decomposition). (1) There exists the following spectral sequence (3.5.4.1)

i+j E1ij = H j (X• , W ΩiX• (log D• ))K0 =⇒ Hrig (U/K0 ).

This spectral sequence degenerates at E1 . (2) There exists the following canonical isomorphism: (3.5.4.2)

∼

i+j H j (X• , W ΩiX• (log D• ))K0 −→ Hrig (U/K0 )[i,i+1) .

In particular, H j (X• , W ΩiX• (log D• ))K0 depends only on U/κ. (3) There exists the following slope decomposition (3.5.4.3)

h (U/K0 ) = Hrig

h 

H h−i (X• , W ΩiX• (log D• ))K0 .

i=0

P. Berthelot, S. Bloch and H. Esnault have given an impressive description h (X/K0 )[0,1) by the cohomology of Witt rings on a proper scheme X of Hrig over κ: Theorem 3.5.5 ([9]). Let X be a proper scheme over κ. Then there exists a functorial morphism h Hrig (X/K0 ) −→ H h (X, W (OX ))K0 ∼

(h ∈ N)

h which induces an isomorphism Hrig (X/K0 )[0,1) −→ H h (X, W (OX ))K0 .

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3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

By (3.5.4) and (3.5.5), we have the following corollary: Corollary 3.5.6 ([70, (13.7)]). Let X be a proper scheme over κ. Let X• −→ X be a gs proper hypercovering of X. Then the natural morphism H h (X, W (OX ))K0 −→ H h (X• , W (OX• ))K0

(h ∈ N)

is a functorial isomorphism. We conclude this section by stating ranges of slopes on rigid cohomology with closed support (The following (3) is a generalization of the latter part of [16, (3.1.2)].): Theorem 3.5.7 ([70, (16.11)]). Let the notations be as in (3.4.10). Then the following hold: h (U/K0 ) lie in [0, h]. (1) The slopes of Hrig,Z h (U/K0 ) lie in [c, h − c]. (2) If U is smooth over κ, the slopes of Hrig,Z h (U/K0 ) lie in [h−d, d]. (3) Under the assumption in (2), the slopes of Hrig,Z h Consequently, the slopes of Hrig,Z (U/K0 ) lie in [max{c, h − d}, min{h − c, d}].

3.6 Weight Filtration on Rigid Cohomology with Compact Support Let the notations be as in §3.3. In this section, assume that κ is perfect. First we note that there is an announcement in [4, pp. 21–23] that the rigid cohomology with compact support of a separated scheme U of finite type over κ is defined. Let U be a separated smooth scheme of finite type over κ of pure dimension d. Then, by the same proof as that of [6, (2.4)], we have a canonical isomorphism (3.6.0.1)

∼

2d−h h (U/K) −→ HomK (Hrig (U/K), K). Hrig,c

2d The trace morphism TrU : Hrig,c (U/K) −→ K ([6]) is an underlying morphism of the following morphism

(3.6.0.2)

2d TrU : Hrig,c (U/K) −→ K(−d)

by the proof of [16, (2.1.1)] ([70, (16.9)]). Endow K(−d) with an increasing 2d−h filtration P : P2d K(−d) = K(−d), P2d−1 K(−d) = 0. Because Hrig (U/K) has the weight filtration, under the following identification (3.6.0.3)

∼

2d−h h Hrig,c (U/K) −→ HomK (Hrig (U/K), K(−d)),

h (U/K) has the weight filtration P . Hrig,c

3.6 Weight Filtration on Rigid Cohomology with Compact Support

245

The following are relative versions of [82, Theorem 2.4.4, Corollary 2.3.9, Theorem 3.1.1] for the trivial coefficients. In [83] and [73], we have proved that the morphism (3.6.1.1) below is a functorial isomorphism; in [83] the secondnamed author has also proved that the morphism (3.6.1.2) is a functorial isomorphism. Theorem 3.6.1 ([83] ([73])). Let S be a p-adic formally smooth formal W -scheme. Set S1 := S ⊗W κ. Endow S with the canonical PD structure (pOS , [ ]). Let f : (X, D) −→ S1 be a proper smooth scheme with a relative SNCD over S1 . Set U := X \ D. By abuse of notation, denote also by f the structural morphism U −→ S1 . Let Rh f(X,D)/SK0 ∗ (K(X,D)/SK0 ) and Rh f(X,D)/S∗ (O(X,D)/S ) (h ∈ Z) be the relative log convergent cohomology of (X, D)/S and the relative log crystalline cohomology of (X, D)/S, respectively. Let Rh frig∗ (U/SK0 ) be the relative rigid cohomology of U/SK0 ([19, (10.6)]). (It is an OSK0 -module, where SK0 denotes the rigid analytic space associated to S.) Let sp : SK0 −→ S be the specialization map defined in [8, (0.2.3)]. Then the canonical morphisms (3.6.1.1)

Rh f(X,D)/SK0 ∗ (K(X,D)/SK0 ) −→ Rh f(X,D)/S∗ (O(X,D)/S )K0

and (3.6.1.2)

Rh f(X,D)/SK0 ∗ (K(X,D)/SK0 ) −→ sp∗ Rh frig∗ (U/SK0 )

are functorial isomorphisms. Using (3.6.1) and similar arguments to (3.4.2.2), (3.4.2.3) and (3.4.2.4), we can obtain the following: Theorem 3.6.2 ([70, (17.6)]). Let U be a separated smooth scheme of finite type over κ of pure dimension d. Let Z be a smooth closed subscheme of U . Then the following composite morphism of the cup product and the trace morphism (3.6.2.1)

Tr

U 2d−h h 2d Hrig,c (Z/K) ⊗K Hrig,Z (U/K) −→ Hrig,c (U/K) −→ K(−d)

induces an isomorphism (3.6.2.2)

∼

2d−h h (U/K) −→ HomK (Hrig,c (Z/K), K(−d)) Hrig,Z

of weight-filtered vector spaces over K. Using Berthelot’s duality (3.6.0.3) and (3.6.2), we obtain the following: Corollary 3.6.3 ([70, (17.7)]). Assume that Z is of pure codimension c. Then the Gysin isomorphism (3.6.3.1)

∼

h−2c h GZ/U : Hrig (Z/K)(−c) −→ Hrig,Z (U/K)

is an isomorphism of weight-filtered K-vector spaces.

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3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

By (3.4.6) and (3.6.2), we obtain the following: Corollary 3.6.4 ([70, (17.9)]). Let f : U −→ V be a finite etale morphism of smooth affine schemes over κ. Then the trace morphism h h Trf : Hrig,c (U/K) −→ Hrig,c (V /K)

[6, (1.4)] is strictly compatible with the weight filtration. Let Z be a separated scheme of finite type over κ. Until (3.6.6), assume that there exists a closed immersion ⊂

Z −→ U

(3.6.4.1)

into a separated smooth scheme of finite type over κ of pure dimension d and that Z is of pure codimension c in U . By the proof of [6, (2.4)], we have the following isomorphism (3.6.4.2)

∼

2d−h h Hrig,c (Z/K) −→ HomK (Hrig,Z (U/K), K).

2d−h 2d−h Because Hrig,Z (U/K) has the weight filtration, HomK (Hrig,Z (U/K), K(−d)) has the weight filtration. Thus, under the following identification 2d−h h Hrig,c (Z/K) = HomK (Hrig,Z (U/K), K(−d)),

(3.6.4.3)

h (Z/K). we have a weight filtration P on Hrig,c If U are Z are reduced, then there exist dense open smooth subschemes U sm and Z sm of U and Z, respectively. Using induction on dim Z and using (3.6.2), we can prove the following: h Theorem 3.6.5 ([70, (17.11)]). The weight filtration P on Hrig,c (Z/K) by the formula (3.6.4.3) is independent of the choice of U . h Definition 3.6.6. We call the weight filtration on Hrig,c (Z/K) given by the h formula (3.6.4.3) the weight filtration on Hrig,c (Z/K). h We list some properties of the weight filtration on Hrig,c (Z/K).

Proposition 3.6.7 ([70, (17.1)]). Let U i (i = 1, 2) be a separated smooth scheme of finite type over κ. Then Berthelot’s K¨ unneth isomorphism  ∼ h1 h2 h Hrig,c (U 1 /K) ⊗K Hrig,c (U 2 /K) −→ Hrig,c (U 1 ×κ U 2 /K) (3.6.7.1) h1 +h2 =h

is compatible with the weight filtration. Let (X, D) be a proper smooth scheme with an SNCD over κ. Set U := X \ D. By [85], [82, Theorem 2.4.4, Corollary 2.3.9, Theorem 3.1.1] and [6, (2.4)], we obtain

3.6 Weight Filtration on Rigid Cohomology with Compact Support

(3.6.7.2)

247

h h Hlog -crys,c ((X, D)/W )K = Hrig,c (U/K).

h Proposition 3.6.8 ([70, (17.4)]). The weight filtration on Hrig,c (U/K) in h this section is equal to the weight filtration on Hlog -crys,c ((X, D)/W )K in (2.11.15.1) under the canonical isomorphism (3.6.7.2).

Proposition 3.6.9 ([70, (17.13)]). Let Z and U be as in (3.6.4.1). Set V := U \ Z. Then the following sequence (3.6.9.1)

h h h · · · −→ Hrig,c (V /K) −→ Hrig,c (U/K) −→ Hrig,c (Z/K) −→ · · ·

([4, (3.1) (iii)]) is strictly exact with respect to the weight filtration. Proposition 3.6.10 ([70, (17.19)]). Let f : U −→ V be a proper morphism of separated schemes of finite type over κ. Assume that U and V are closed subschemes of separated smooth schemes of finite type over κ. Then the pull-back h h f ∗ : Hrig,c (V /K) −→ Hrig,c (U/K) is strictly compatible with the weight filtration. The following immediately follows from [6, (2.2) (ii)] and (3.6.10). Corollary 3.6.11 ([70, (17.20)]). Let f : U −→ V be a finite etale morphism of affine smooth schemes over κ. Then the trace morphism h h Trf : Hrig (U/K) −→ Hrig (V /K)

([6, (1.4)]) is strictly compatible with the weight filtration. Proposition 3.6.12 ([70, (17.21)]). Let Z be as in (3.6.4.1). Then the canonical morphism (3.6.12.1)

h h Hrig,c (Z/K) −→ Hrig (Z/K)

([4, (3.1) (i)]) is strictly compatible with the weight filtration. Let Z be as in (3.6.4.1). Finally we give ranges of weights and slopes of h Hrig,c (Z/K). The following is a p-adic version of a special case of [26, III (3.3.4)]; see also [37, Partie II, III 3]. Proposition 3.6.13 ([70, (17.14)]). Let Z be a separated scheme of finite type over κ of pure dimension dZ . Assume that there exists a closed immerh (Z/K) sion (3.6.4.1). Let h ∈ [0, 2dZ ] be an integer. Then the weights of Hrig,c lie in [0, h]. The following is [16, (3.1.2)] (see also [26, III (3.3.8)]).

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3 Weight Filtrations and Slope Filtrations on Rigid Cohomologies (Summary)

Proposition 3.6.14 ([16, (3.1.2)], [70, (17.3)]). Let h ∈ [0, 2d] be an integer. Let U be a separated scheme of finite type h (U/K) lie in [max{0, h − d}, over κ of dimension d. Then the slopes of Hrig,c min{h, d}].

Appendix A

Relative SNCD

Let f : X −→ S be a smooth morphism of schemes and let D be a relative ⊂ SNCD on X/S. Set U := X \ D and let j : U −→ X be the natural open ∗ immersion. Set N (D) := OX ∩ j∗ (OU ) as in §2.1. In this appendix we prove basic properties of D, M (D) and N (D): (A) We determine the local behavior of the decomposition of D by smooth components ((A.0.1)). (B) We refine decompositions of D by smooth components as possible ((A.0.3), (A.0.7)). (C) We prove the equality N (D) = M (D) if S is reduced at any point of f (D) ((A.0.8) (1)). (D) We show the inequality N (D) = M (D) if S is not reduced at f (x) for ∗ )x is a point x ∈ X ((A.0.8) (2)). In fact, we prove that the stalk (N (D)/OX not finitely generated ((A.0.9)). For a morphism Y −→ T of schemes, recall Div(Y /T )≥0 in §2.1. If T is the spectrum of a commutative ring A with unit element, we often denote Div(Y /T )≥0 simply by Div(Y /A)≥0 in this section. For a non-zero divisor section y ∈ Γ (Y, OY ) such that SpecY (OY /yOY ) is flat over T , we denote by div(y) the effective Cartier divisor defined by the ideal sheaf yOY of OY . Let us take a diagram (2.1.7.2) (with S 0 replaced by S) and a point z in s V and V ⊆ X. Then, we may assume that z ∈ i=1 {yi = 0} by shrinking s replacing s if necessary. Henceforth we always assume that z ∈ i=1 {yi = 0} when we take a diagram like (2.1.7.2) and a point z in V . In (2.1.7), there is no relation a priori between a decomposition of D by smooth components and the diagram (2.1.7.2). Though the uniqueness of the decomposition does not necessarily hold, the local decomposition is determined by the diagram (2.1.7.2): Proposition A.0.1. Let f : (X, D) −→ S be as in the beginning of this section. Let ∆ := {Dλ }λ∈Λ be a decomposition of D by smooth components. Let z be a point of D and assume that we are given s a cartesian diagram (2.1.7.2) (with S0 replaced by S, such that z ∈ i=1 {yi = 0}). Then, by

249

250

A Relative SNCD

shrinking V , for any 1 ≤ i ≤ s, there exists a unique element λi ∈ Λ satisfying Dλi |V = div(yi ) in Div(V /S)≥0 . Proof. Set B := OX,z and Xz := Spec(B). Let D1 , . . . , Dm be the elements in ∆ which contain z and let di (1 ≤ i ≤ m) be elements of B such that Di ∩ Xz = div(di ). Then we have the following equality m

(A.0.1.1)

div(di ) =

i=1

s

div(yi )

i=1

in Div(Xz /S)≥0 by the diagram (2.1.7.2). We have to prove that s = m and that yi B = di B (1 ≤ i ≤ s) up to some renumbering of the indexes. Case I: First consider the case S = Spec(κ), where κ is a field. In this case, B is a regular local ring. By Auslander-Buchsbaum’s theorem [63, §19, Theorem 48], B is a UFD. Since ODi ,z = B/di B is also a regular local ring, the ideal di B of B is prime. Since yi ∈ B ∗ (1 ≤ i ≤ s) (because s z ∈ i=1 {yi = 0}), the ideal yi B of B is also prime. Since B is a UFD, the equality (A.0.1.1) implies the equalities s = m and di B = yi B (1 ≤ i ≤ s) up to some renumbering of the indexes. This completes the proof in the Case I. Case II: Next consider the case S = Spec(A), where A is an integral domain. Set K := Frac(A) and let η be the generic point of S. For a scheme T over S, set Tη := T ×S η. Set BK := B ⊗A K and for a point w in Xz,η := Spec(BK ), denote by (BK )w the localization of BK at the prime ideal corresponding to w. Then Spec((BK )w ) is nothing but the localization of Xη at w, where we identify w with its image in Xη . Fix an index i (1 ≤ i ≤ m) for the moment. Since Di ∩ Xz is flat over S, there exists a point w ∈ Di ∩ Xz such that w is sent to η by the structural morphism Xz −→ S (Indeed, we may assume that S is local; in this case the morphism Di ∩ Xz −→ S is faithfully flat.). Then w ∈ Di,η . By the result in the Case I, there exists an index j (1 ≤ j ≤ s) such that (A.0.1.2)

di (BK )w = yj (BK )w .

Denote by ϕ the composite morphism B −→ BK −→ (BK )w and by ϕ the induced morphism B/yj B −→ (BK )w /yj (BK )w by ϕ. Then ϕ is flat. Consider the following commutative diagram (A.0.1.3) 0 −−−−→

B ⏐ ⏐ ϕ

yj ×

−−−−→ yj ×

B ⏐ ⏐ ϕ

−−−−→

B/yj B ⏐ ⏐ ϕ

−−−−→ 0

0 −−−−→ (BK )w −−−−→ (BK )w −−−−→ (BK )w /yj (BK )w −−−−→ 0.

A Relative SNCD

251

Because the upper horizontal sequence in (A.0.1.3) is exact and because ϕ is flat, the lower horizontal sequence is also exact. The morphisms B −→ BK and B/yj B −→ BK /yj BK are injective. Because BK and BK /yj BK are regular, they are integral domains. Hence the morphisms BK −→ (BK )w and BK /yj B −→ (BK )w /yj (BK )w are also injective. Thus the vertical morphisms in the diagram (A.0.1.3) are injective. By an easy diagram-chasing, we have an equality yj B = ϕ−1 (yj (BK )w ). Similarly we have an equality di B = ϕ−1 (di (BK )w ). Hence di B = yj B by (A.0.1.2). For each i (1 ≤ i ≤ m), we may take j(i) with 1 ≤ j(i) ≤ m such that di B = yj(i) B. Then we obtain the following equality s

div(yj ) =

j=1

m

div(yj(i) )

in

Div(Xz /S)≥0 .

i=1

m Set Y := i=1 div(yj(i) ) ⊂ Xz . Since z is contained in Y and since Y is flat over S, there exists a point w ∈ Y such that w is sent to η by the structural morphism Xz −→ S. Hence w ∈ Yη . If we define (BK )w as before, we have the s m equality j=1 div(yj ) = i=1 div(yj(i) ) in Div(Spec((BK )w )/K)≥0 . Since (BK )w is a UFD and since yj ∈ (BK )∗w (1 ≤ j ≤ m), we obtain s = m and div(di ) = div(yi ) up to some renumbering of the indexes. Case III: Next consider the case where S is a noetherian scheme. Let t be the image of z by f . Then we may assume that S is the spectrum of a noetherian local ring A with closed point t. Let m be the maximal ideal of A. Let p be a prime ideal of A. By the result in the Case II, we have s = m and di (B/pB) = yi (B/pB) (1 ≤ i ≤ s) up to some renumbering of the indexes. Note that the correspondence di ↔ yi is independent of the choice of p. Indeed, di (B/pB) = yi (B/pB) implies di (B/mB) = yi (B/mB), and hence yi is uniquely determined by the result in the Case I. Now let ψ : B −→ B/pB be the projection and by ψ the induced morphism B/yi B −→ (B/pB)/yi (B/pB) by ψ. Consider the following diagram (A.0.1.4) 0 −−−−→

B ⏐ ⏐ ψ

yi ×

−−−−→

B ⏐ ⏐ ψ

−−−−→

B/yi B ⏐ ⏐ ψ

−−−−→ 0

yi ×

0 −−−−→ B/pB −−−−→ B/pB −−−−→ (B/pB)/yi (B/pB) −−−−→ 0. Since yi is a non-zero divisor in B and in B/pB, the horizontal sequences are exact. Because Ker(ψ) = (pB + yi B)/yi B, an easy diagram-chasing gives an equality ψ −1 (yi (B/pB)) = yi B + pB. Similarly we have an equality ψ −1 (di (B/pB)) = di B +pB, and hence we have an inclusion di B ⊂ yi B +pB, that is, (A.0.1.5)

di (B/yi B) ⊂ p(B/yi B)

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A Relative SNCD

for any 1 ≤ i ≤ s and for any prime ideal p of A. Let n be the nilpotent radical q of A and let n = k=1 pk (q ∈ Z>0 ) be the primary decomposition of n, where pk is a prime ideal q of A. Then, by (A.0.1.5) for p = pk (1 ≤ k ≤ q), we obtain di (B/yi B) ⊂ k=1 pk (B/yi B) = n(B/yi B), where the last equality follows from the flatness of B/yi B over A. Hence we obtain di B ⊂ yi B + nB for 1 ≤ i ≤ s. Now we prove the inclusion di B ⊂ yi B (1 ≤ i ≤ s). To prove this, it suffices to prove the inclusion di B ⊂ yi B + ne B (1 ≤ i ≤ s) for any positive integer e since ne = 0 for some e. We prove this inclusion by induction on e. We have already proved the inclusion for the case e = 1. Assume that we have the inclusion di B ⊂ yi B + ne B (1 ≤ i ≤ s) for a positive integer e. Then there exists an element ui (resp. i ) of B (resp. ne B) such that di = ui yi + i . By the diagram (2.1.7.2), we have (A.0.1.6)

(

s 

yi )B = (

i=1

s 

di )B = (

i=1

s 

(ui yi + i ))B.

i=1

By reducing the equality (A.0.1.6) modulo n, we obtain (

s 

yi )(B/nB) = (

i=1

s 

ui )(

i=1

s 

yi )(B/nB).

i=1

s Since i=1 yi is a non-zero divisor in B/nB, each ui is invertible in B/nB, and hence so is in B. Replacing di by u−1 i di , we may assume that ui = 1 for equation (A.0.1.6) (with any i. Fix an index i0 (1 ≤ i0 ≤ s) and consider the ui = 1) modulo yi0 B + ne+1 B. Then we see that i0 i=i0 yi ∈ yi0 B + ne+1 B. Because the following diagram e+1

(A.0.1.7)

B/n ⏐ ⏐ yi0 ×

(

 i =i

yi )×

−−−−−0−−−→ B/ne+1 ⏐ ⏐yi ×  0 (



i =i

yi )×

B/ne+1 −−−−−0−−−→ B/ne+1 is cartesian (because B/ne+1 B is flat over (A/ne+1 )[y1 , . . . , yd ]), i0 ∈ yi0 B + ne+1 B. Hence, for any i (1 ≤ i ≤ s), i ∈ yi B + ne+1 B and consequently di ∈ yi B + ne+1 B. Therefore we obtain the inclusion di B ⊂ yi B + ne+1 B, infact, the inclusion di B ⊂ yi B. By the last inclusion and the equality s s ( i=1 di )B = ( i=1 yi )B, we can easily deduce the equality di B = yi B. We complete the proof in the Case III. Case IV: Finally we prove the proposition in the general case. By shrinking V , the structural morphism V −→ S fits into the following cartesian diagram

A Relative SNCD

253

V −−−−→ ⏐ ⏐ 

V ⏐ ⏐ 

S −−−−→ S satisfying the following conditions: (1) S is a noetherian scheme and V −→ S is smooth. (2) There exists an effective Cartier divisor Di = div(di ) (1 ≤ i ≤ m) on m V which is smooth over S satisfying Di ×S S = Di |V . Set D := i=1 Di . (3) There exists a cartesian diagram D ⏐ ⏐ 

⊂

−−−−→

V ⏐ ⏐g 

SpecS (OS [y1 , . . . , yd ]/(y1 · · · ys )) −−−−→ SpecS (OS [y1 , . . . , yd ]), over S which is compatible with the diagram (2.1.7.2), where g is an etale morphism. Let z be the image of z in V . Then, by the result in the Case III to (V , D), s = m and div(di ) = div(yi ) (1 ≤ i ≤ s) in Div(Spec(OV ,z )/S) up to some renumbering of the indexes. Then div(di ) = div(yi ) (1 ≤ i ≤ s) in   Div(Xz /S). This equality completes the proof of the proposition. To study the global behavior of decompositions of a relative SNCD by smooth components, we introduce the notion of the refinement of a decomposition by smooth components. Definition A.0.2. (1) Let f : (X, D) −→ S be as in the beginning of this section and let ∆ := {Dλ }λ∈Λ and ∆ := {Dσ }σ∈Σ be decompositions of of ∆ if, for D by smooth components. Then we say that ∆ is a refinement  any λ ∈ Λ, there exists a subset Σλ of Σ with Σ = λ∈Λ Σλ such that Dλ = σ∈Σλ Dσ . (2) We say that a closed and open set of a topological space is clopen. If X is locally noetherian as a topological space, we can prove the existence of the finest (hence canonical) decomposition of a relative SNCD by smooth components: Proposition A.0.3. Let f : (X, D) −→ S be as in the beginning of this section. If X is locally noetherian as a topological space, then there exists a unique decomposition of D by smooth components which is a refinement of any decomposition of D by smooth components.  Proof. Let T be a locally noetherian topological space and let T = i∈I Ti (Ti ∩ Tj = ∅ for i = j) be the decomposition into the connected components of T . Then we claim that Ti is open. Let O beany noetherian open subset of T . Set I  := {i ∈ I | O ∩ Ti = ∅}. Then O = i∈I  (O ∩ Ti ). We claim that I  is finite. Indeed, if not, we have a union O = O1 ∪ · · · ∪ Or (r ∈ Z>0 ) of the

254

A Relative SNCD

irreducible components of O since O is noetherian. Then there exist a positive integer s (s ≤ r) and i = i (i, i ∈ I  ) such that Os ∩ Ti = ∅ and Os ∩ Ti = ∅. Since Os is irreducible, Os is connected. Hence Os ∪ Ti ∪ Ti is also connected. the definition However, this union contains both Ti and Ti ; this contradicts  of Ti . Thus we see that I  is finite and that Ti ∩O = O\( i ∈I  \{i} Ti ) is open in T . Now we see thatTi is open since T is locally noetherian. In conclusion, the disjoint sum T = i∈I Ti is the disjoint sum of clopen subspaces of T . Take a decomposition ∆ = {Dλ }λ∈Λ of D by smooth components and let Dλ = σ∈Σλ Dλ,σ be the decomposition of Dλ into the connected components. Since X is locally noetherian, so is  Dλ . Hence, by the previous paragraph, we have the decomposition Dλ = σ∈Σλ Dλ,σ into clopen subschemes. In particular, each Dλ,σ is smooth over S and we have an equality Dλ = σ∈Σλ Dλ,σ in Div(X/S)≥0 . Therefore ∆0 := {Dλ,σ }λ∈Λ,σ∈Σλ is a refinement of ∆. We say that a decomposition of D by smooth components is very fine if each member of the decomposition is connected as a topological space. It is clear that ∆0 is very fine. Hence we have shown that any decomposition of D by smooth components admits a refinement by a very fine one. To prove (A.0.3), it suffices to prove that there exists only one very fine decomposition of D by smooth components. Let ∆ = {Dλ }λ∈Λ and ∆ = {Dλ  }λ ∈Λ be very fine decompositions of D by smooth components. Fix an index λ ∈ Λ. For each point z in Dλ , take an open neighborhood Uz of z in X such that ∆Uz = ∆Uz ((A.0.1)). Then there exists a unique λ (λ, z) ∈ Λ such that Dλ ∩ Uz = Dλ  (λ,z) ∩ Uz . We claim that λ (λ, z) does not depend on z. Let V be a subset of Dλ defined as follows:     ∃ sequence of points z = z0 , z1 , . . . , zn = w in Dλ  V := w ∈ Dλ  . satisfying Dλ ∩ Uzj ∩ Uzj+1 = ∅ (0 ≤ j ≤ n − 1) Then, if w is a point of V , Dλ ∩ Uw is contained in V ; V is an open set of Dλ . On the other hand, if w ∈ Dλ is not contained in V , the points in Uw are not contained in V ; V is a closed set of Dλ . Since Dλ is connected and V is non-empty (since z ∈ V ), Dλ = V . Let w be a point of Dλ . Then there exists a sequence of points z = z0 , z1 , . . . , zn = w in Dλ satisfying Dλ ∩ Uzj ∩ Uzj+1 = ∅ for all 0 ≤ j ≤ n − 1. Since Dλ ∩ Uzj = Dλ  (λ,zj ) ∩ Uzj and Dλ ∩ Uzj+1 = Dλ  (λ,zj+1 ) ∩ Uzj+1 , Dλ ∩ Uzj ∩ Uzj+1 = Dλ  (λ,zj ) ∩ Uzj ∩ Uzj+1 = Dλ  (λ,zj+1 ) ∩ Uzj ∩ Uzj+1 . This implies the equality λ (λ, zj ) = λ (λ, zj+1 ) since Uzj ∩Uzj+1 is non-empty and that ∆Uzj ∩Uzj+1 = ∆Uz ∩Uz . Hence we have j

j+1

λ (λ, z) = λ (λ, z0 ) = · · · = λ (λ, zn ) = λ (λ, w).

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255

Thus we have proved the claim. We denote this index by λ (λ). Then we have   (Dλ ∩ Uz ) = (Dλ  (λ) ∩ Uz ) ⊂ Dλ  (λ) , Dλ = z∈Dλ

z∈Dλ

and we see that Dλ is an open subscheme of Dλ  (λ) . On the other hand, Dλ is a closed subscheme of Dλ  (λ) since it is closed in X. Since Dλ  (λ) is connected, Dλ = Dλ  (λ) .  map λ  → λ (λ) is injective. This fact and the equality It is clear that the λ∈Λ Dλ = D = λ ∈Λ Dλ imply the bijectivity of the correspondence   λ → λ (λ). In the case where X is not locally noetherian as a topological space, there exists an example of a relative SNCD which does not have the finest decomposition by smooth components:  Example A.0.4. Let k be a field and let A := n∈N k be the countable product of k. Set S := Spec(A), X := Spec(A[x]) and let D be a relative SNCD on X ∼ over S defined by the ideal (x) ⊂ A[x]. Then D −→ S. Let ∆ := {Dλ }λ∈Λ be any decomposition of D by smooth components. Then each Dλ is closed ⊂ ∼ in D and it is open in D because the composite morphism Dλ −→ D −→ S is smooth. Since Dλ is closed in D  Spec(A), Dλ is quasi-compact. For a subset T of N, define eT := (eT,n )n∈N ∈ A by eT,n = 1 (resp. 0) if / p} (T ⊂ N) n ∈ T (resp. n ∈ / T ). Then the sets UT := {p ∈ Spec(A) | eT ∈ forms an open basis of Spec(A)(= S  D). It is easy to see that, for a l finite number of subsets T1 , T2 , . . . , Tl of N, we have i=1 UTi = Ul Ti ; if i=1 T1 ∩ T2 = ∅, then UT1 ∩ UT2 = ∅. Because Dλ is open in D and quasi-compact, Dλ is a union of a finite numbers of open sets of the form UT . By the fact in the previous paragraph, ∼ Dλ −→ UTλ for a non-empty subset Tλ of N.  If  Tλ = 1 (∀λ ∈ Λ), then we can deduce an equality Spec(A) = n∈N U{n} . However, a maximal ideal of A containing the ideal {(xn )n∈N ∈ A | xn = 0 for except finite numbers of n’s}  does not belong to n∈N U{n} and it is a contradiction. Thus, for some λ ∈ Λ, thecardinality of Tλ is greater than 1. Then we have a disjoint sum Tλ = T  T  of Tλ with T  = ∅ and T  = ∅. Hence we have a refinement of ∆ by factorizing Dλ as Dλ = UT  + UT  . Thus ∆ is not the finest decomposition. Hence, in this case, there does not exist the finest decomposition of D by smooth components. As we have seen in the above, we cannot have the finest decomposition of D in general. However, we can prove that any two decompositions admit a common refinement in the following way.

256

A Relative SNCD

Let ≤ be a partial order in Div(X/S)≥0 such that E1 ≤ E2 if and only if there exists an effective Cartier divisor F on X over S such that E2 = E1 +F . Then we have the following lemma: Lemma A.0.5. Let f : X −→ S be as in the beginning of this section and let ∆ = {Dλ }λ∈Λ be a family of smooth effective Cartier divisors on X/S. For nonnegative integers n1,λ and n2,λ (λ ∈ Λ), λ∈Λ n1,λ Dλ ≤ λ∈Λ n2,λ Dλ if and only if n1,λ ≤ n2,λ for all λ ∈ Λ. Proof. It suffices to prove the ‘only if’ part. For a point t ∈ S and for a scheme T over S, set Tt := T ×S t. Then λ∈Λt n1,λ Dλ,t ≤ λ∈Λt n2,λ Dλ,t , where Λt := {λ ∈ Λ | Dλ,t = ∅}. Since the assertion is well-known in the case where S is the spectrum of a field, n1,λ ≤ n2,λ for λ ∈ Λt . Since any λ ∈ Λ   belongs to Λt for some point t ∈ S, n1,λ ≤ n2,λ for all λ ∈ Λ. Using (A.0.5), we define the operation ∧ for the elements in DivD (X/S)≥0 as follows: Proposition A.0.6. Let the notations be as above. Then the following hold: (1) For two elements E1 and E2 in DivD (X/S)≥0 , there exists a unique element E1 ∧ E2 ∈ DivD (X/S)≥0 satisfying the following equality: E1 ∧ E2 = max{F ∈ DivD (X/S)≥0 | F ≤ E1 and F ≤ E2 }. (2) Let E and F be elements in DivD (X/S)≥0 . If E is smooth over S, then E ∧ F is clopen in E. In particular, E ∧ F is smooth over S. (3) Let E be an element in DivD (X/S)≥0 . Let {Fσ }σ∈Σ be a family of elements in DivD (X/S)≥0 of locally finite intersection such that Fσ ∧ Fσ = 0 for any σ, σ  ∈ Σ with σ = σ  . Then

Fσ ) = (E ∧ Fσ ). E∧( σ∈Σ

σ∈Σ

Proof. (1): Take a decomposition {Dλ }λ∈Λ of D by smooth components. Then, by the definition of DivD (X/S) ≥0 , there exists an open covering X =  X of X such that E | = j i X j j∈J λ∈ΛX ni,j,λ (Dλ |Xj ) (i = 1, 2) for some j

nonnegative integers ni,j,λ , where ΛXj is the set {λ ∈ Λ | Dλ |Xj = ∅}. Define Fj ∈ DivD|Xj (Xj /S)≥0 by Fj := λ∈ΛX min{n1,j,λ , n2,j,λ }(Dλ |Xj ). Then it j

is easy to see (by using (A.0.5)) that Fj ≤ Ei |Xj (i = 1, 2) and that Fj is the maximum element among the elements F ’s in DivD|Xj (Xj /S)≥0 satisfying F ≤ Ei |Xj (i = 1, 2). By using this characterization, one can see that there exists a unique element F ∈ DivD (X/S)≥0 such that F |Xj = Fj for any j ∈ J. Set E1 ∧ E2 := F . Then E1 ∧ E2 satisfies the equality in (1). (2): Since E ∧ F is closed in E, it suffices to prove that E ∧ F is open in E. Since the problem is local, we may assume that F = λ∈Λ nλ Dλ (nλ ∈ N). Moreover, we may assume, by the smoothness of E, that there exists an

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257

element λ0 ∈ Λ satisfying E = Dλ0 . Then, E ∧ F = E (resp. 0) if nλ0 ≥ 1 (resp. nλ0 = 0). In both cases, E ∧ F is open in E. (3): Since the problem is local, we may assume that Σ is a finite set and that

nλ Dλ , Fσ = mσ,λ Dλ , E= λ∈Λ

λ∈Λσ

where nλ , mσ,λ are nonnegative integers and Λσ is a subset of Λ satisfying Λσ ∩ Λσ = ∅ for σ = σ  . Then



Fσ ) = min{nλ , mσ,λ }Dλ = (E ∧ Fσ ). E∧( σ∈Σ

σ∈Σ λ∈Λσ

σ∈Σ

  Using (A.0.6), we can prove that any two decompositions of a relative SNCD by smooth components have a common refinement: Proposition A.0.7. Let f : (X, D) −→ S be as in the beginning of this section. Let ∆ = {Dλ }λ∈Λ and ∆ = {Dλ  }λ ∈Λ be two decompositions of D by smooth components. Then there exists a decomposition ∆ of D by smooth components which is a refinement of ∆ and ∆ . Proof. By (A.0.6), Dλ ∧ Dλ  is smooth over S (possibly empty) and we have the following equalities

Dλ ∧ Dλ  , Dλ  = Dλ ∧ Dλ  . Dλ =  =0 λ ∈Λ ,Dλ ∧Dλ 

 =0 λ∈Λ,Dλ ∧Dλ 

Set Λ := {(λ, λ ) ∈ Λ × Λ | Dλ ∧ Dλ  = 0}. Then it is easy to see from the  above equalities that ∆ := {Dλ ∧ Dλ  }(λ,λ )∈Λ has a desired property.  Lastly we discuss log structures on X associated to D. Though we always consider log structures in the Zariski topos of X, (A.0.8) and (A.0.9) below remain valid if we consider log structures in the etale topos of X. By (2.1.9), (X, M (D)) is a fine log scheme; in fact, it is log smooth over S. ⊂ ∗ Let us recall N (D) := OX ∩ j∗ (OU ) with structural morphism N (D) −→ OX . From the local expression of M (D) given in (2.1.9), there exists a natural ⊂ inclusion M (D) −→ N (D). Both M (D) and N (D) have U as the maximal open subscheme where they are trivial. By [55, (8.2), (11.6)], they coincide in the case where S is regular. However we prove M (D)  N (D) in general, in fact, we give an equivalent condition for the equality M (D) = N (D) ((A.0.8) below) and show that N (D) is not a fine log structure in general ((A.0.9) below): Proposition A.0.8. Let f : (X, D) −→ S be in the beginning of this section. Then the following hold: (1) If S is reduced at the points of f (D), then M (D) = N (D).

258

A Relative SNCD

(2) If S is not reduced at some point of f (D), then M (D) = N (D). Consequently, the log structures M (D) and N (D) coincide if and only if S is reduced at the points of f (D). Proof. Let z be a point of X. Set B := OX,z and Xz := Spec(B). Take an open neighborhood V of z which admits the diagram (2.1.7.2). ⊂ (1): It suffices to prove the surjectivity of the homomorphism M (D) −→ N (D). To prove this, we may work locally. By (2.1.9), it suffices to prove Γ(Xz , N (D)) = B ∗ y1N · · · ysN . Here we denote the pull-back of N (D) to Xz by the same symbol. We may assume that S is noetherian because we can reduce the general case to the noetherian case by the well-known technique similar to that of Case IV in the proof of (A.0.1). Furthermore, we may assume that S is the spectrum of a noetherian local ring A and that t = f (z) is the closed point of S. By the assumption, A is reduced. Let g be an element of Γ(Xz , N (D)) = B ∩ B[(y1 · · · ys )−1 ]∗ . Because yi is a non-zero divisor of B (1 ≤ i ≤ s), there exist an element h ∈ B and a nonnegative integer a satisfying gh = (y1 · · · ys )a in B. For a prime ideal p of A, consider the equation gh = (y1 · · · ys )a in the ring B/pB. Then, since yi (B/pB)’s (1 ≤ i ≤ s) are different prime ideals of an integral domain B/pB, there exist unique integers bp,i (1 ≤ i ≤ s) with b a−b 0 ≤ bp,i ≤ a satisfying g ∈ yi p,i B + pB and h ∈ yi p,i B + pB for any prime b ideal p of A. Let m be the maximal ideal of A. Then g ∈ yi p,i B + mB and h ∈ a−bp,i B + mB. By the uniqueness of bm,i , we can conclude that bp,i is indeyi pendent of p. Set bi := bp,i . Then g ∈ p∈Spec(A) (yibi B+pB). Because B/yibi B is flat over A and because A is a reduced noetherian ring, p∈Spec(A) (yibi B + s pB) is equal to yibi B. Because B is flat over A[y1 , . . . , ys ], i=1 (yibi B) = s s s ( i=1 yibi )B. Hence g ∈ ( i=1 yibi )B. Similarly, h ∈ ( i=1 yia−bi )B. Define   s s g  := g/( i=1 yibi ) and h := h/( i=1 yia−bi ). Then g  h = 1. Consequently ∗ N N Γ(Xz , N (D)) = B y1 · · · ys . (2): Let t be a point of f (D) where S is not reduced at t. To prove (2), we may assume that S is the spectrum of a nonreduced local ring A with the closed point t. Denote by m the maximal ideal of A and by κ the residue field A/m. Consider a point z in Dt such that D is smooth at z. Then there exists an open neighborhood V of z which admits the cartesian diagram (2.1.7.2) for the case where s = 1, S0 = S = Spec(A) and g(z) is equal to the maximal ideal mA[y1 , . . . , yd ] + (y1 , . . . , yd ). The induced morphism g ∗ : A[y1 , . . . , yd ] −→ B by g is ind-etale. In the following, we denote the image g ∗ (a) of an element a ∈ A[y1 , . . . , yd ] by a by abuse of notation. To prove (2), it suffices to prove that Γ(Xz , N (D)) = B ∗ y1N . Set y := y1 for simplicity of notation. Take an element  of A satisfying  = 0 but 2 = 0. Then we have (y + )(y − ) = y 2 in B. Hence y +  ∈ Γ(Xz , N (D)). Assume that y +  ∈ B ∗ y N . Then there exists a nonnegative integer n and an element u ∈ B ∗ such that

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259

y +  = uy n .

(A.0.8.1)

Apply the projection B −→ B/(y) = OD,z to the equality (A.0.8.1). If n = 0,  = u in B/(y); this is a contradiction. If n ≥ 1,  = 0 in B/(y); this contradicts the choice of  since the natural ring homomorphism A −→ B/(y) is faithfully flat. In conclusion, y +  ∈ Γ(Xz , N (D)) does not belong to   Γ(Xz , M (D)). Proposition A.0.9. Let the notations be as in the proof of (A.0.8) above. ∗ is not a finitely generated monoid. Then Γ(Xz , N (D))/OX,z Proof. Keep the notation in the proof of (A.0.8) (2). Assume that Γ(Xz , N (D))/B ∗ were finitely generated. Since (y n + )(y n − ) = y 2n (n ∈ Z>0 ), y n +  is an element in Γ(Xz , N (D)). Fix a surjective homomorphism ϕ : Nr −→ Γ(Xz , N (D))/B ∗ of monoids for some r and for each positive integer n, take an element hn ∈ Nr satisfying ϕ(hn ) = y n + . Then there exist elements an ∈ Q (1 ≤ n ≤ r + 1) with (a1 , a2 , . . . , ar+1 ) = (0, 0, . . . , 0) r+1 such that n=1 an hn = 0 in Qr . By multiplying an ’s by some integers, we may assume that an ’s are integers with gcd{ai | ai = 0, (1 ≤ i ≤ r + 1)} = 1. Set {n | 1 ≤ n ≤ r + 1, an > 0} =: {n1 , n2 , . . . , nq }, {n | 1  ≤ n ≤ r + 1, an < 0} =: {nq+1 , nq+2 , . . . , nq+s }, ani , (1 ≤ i ≤ q), mi := −ani , (q + 1 ≤ i ≤ q + s). q+s q Then we obtain the equality i=1 mi hni = i=q+1 mi hni in Nr . By applying ϕ to this equality, we see that there exists an element v ∈ B ∗ such that (A.0.9.1)

v

q 

(y

ni

i=1

mi

+ )

=

q+s 

(y ni + )mi

i=q+1

in B. q+s q Define integers P and Q by P := i=1 mi ni and Q := i=q+1 mi ni , respectively. Since 2 = 0, we obtain the following equality (A.0.9.2)

q q+s

mi y P −ni )} = y Q + ( mi y Q−ni ). v{y P + ( i=1

i=q+1

Let κ[y]loc be the localization of the polynomial ring κ[y] at the prime ideal (y). Let β  be the composite morphism A[y, y2 , . . . , yd ] −→ κ[y] −→ κ[y]loc , where the first homomorphism sends a ∈ A (resp. y, yi (2 ≤ i ≤ d)) to a mod m (resp. y,0) and the second homomorphism is the localization. Then the ring C := B ⊗A[y,y2 ,...,yd ],β  κ[y]loc is a local ring which is ind-etale over κ[y]loc . Hence C is a discrete valuation ring with uniformizer y. Denote the natural homomorphism B −→ C by β and apply β to the equality (A.0.9.2). Then we obtain P = Q since β(v) ∈ C ∗ . Hence we obtain the equality

260

(A.0.9.3)

A Relative SNCD q q+s

P −ni P v{y + ( mi y )} = y + ( mi y P −ni ). P

i=1

i=q+1

Let p ≥ 0 be the characteristic of κ. For a positive integer L, let (C)L be the claim that mi is a multiple of p for i’s with ni ≥ L. We prove that (C)L holds for any L by descending induction. If L > ni for any 1 ≤ i ≤ q +s, (C)L obviously holds. Assume that (C)L+1 is true. Then, if there is no integer i with ni = L, (C)L obviously holds since (C)L is equivalent to (C)L+1 . Assume that there exists an integer i0 with ni0 = L. We may assume that 1 ≤ i0 ≤ q. Let γ  be the homomorphism A[y, y2 , . . . , yd ] −→ (A/pA)[y]/(y P −L+1 ) which sends a ∈ A (resp. y, yi (2 ≤ i ≤ d)) to the class of a (resp. y, 0). Then the ring R := B ⊗A[y,y2 ,...,yd ],γ  (A/pA)[y]/(y P −L+1 ) is a local ring which is indetale over the local ring (A/pA)[y]/(y P −L+1 ). Denote the homomorphism B −→ R by γ and apply γ to the equality (A.0.9.3). Then we obtain an equality mi0 y P −L  = 0 in R by the induction hypothesis. Since R is faithfully flat over (A/pA)[y]/(y P −L+1 ), the equality mi0 y P −L  = 0 in R implies that in (A/pA)[y]/(y P −L+1 ). Hence we have mi0  = 0 in A/pA. This implies that there exists an element a ∈ A such that (mi0 − pa) = 0 in A. Hence mi0 is equal to the zero in κ, and consequently mi0 is a multiple of p. Therefore the claim (C)L holds. Now, by the descending induction, (C)1 holds and all mi ’s (1 ≤ i ≤ r) are multiples of p. This contradicts gcd{mi | mi = 0, (1 ≤ ∗ is not a finitely generated i ≤ r)} = 1. In conclusion, Γ(Xz , N (D))/OX,z monoid.   The proposition (A.0.9) tells us that N (D) is far from nice in general. This is the reason why we have considered only M (D) in the text.

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Index

adic fine formal log PD-scheme, 197 adjunction formula, 28 admissible (closed) immersion, 62

Gysin morphism, 129, 130, 195 Gysin morphism in log Hodge-Witt cohomologies, 191

base change theorem, 145, 165, 193

K¨ unneth formula, 154, 170, 194

comparison theorem, 188, 230 compatible, 62 cospecial module, 23 crystalline orientation sheaf, 81

locally finite intersection, 60 log crystalline cohomology sheaf with compact support, 156, 194 log crystalline orientation sheaf, 81 log HPD differential operator, 66 log HPD stratification, 66 log PD differential operator of finite order, 66

decomposition of an SNCD by its smooth components, 61 diagonal filtration, 220 direct sum, 148

multi-degree, 141 E2 -degeneration, 206 filtered flasque resolution, 19 filtered flat resolution, 25 filtered injective resolution, 19 filtered log Berthelot-Ogus isomorphism, 203 filtered perfect complex, 151 filtered quasi-isomorphism, 16 filteredly flasque, 18 filteredly flat, 24 filteredly injective, 18 filteredly strictly perfect, 148 forgetting log morphism, 85 functor forgetting the filtration, 36 functor taking the filtration, 36 functoriality, 137, 195 good proper hypercovering, 229 gr-functor, 36 gs proper hypercovering, 229

p-adic purity, 118 p-adic weight spectral sequence, 196 pair, 225 Poincar´ e duality, 215 Poincar´ e lemma of a vanishing cycle sheaf, 90 preweight filtration, 76, 176 preweight spectral sequence, 143, 169 preweight-filtered restricted crystalline complex, 100 preweight-filtered vanishing cycle crystalline complex, 121 preweight-filtered vanishing cycle crystalline complex with compact support, 164 preweight-filtered vanishing cycle zariskian complex, 122 preweight-filtered zariskian complex, 100 proper hypercovering, 225 pure weight, 239

265

266

Index

refinement, 253 relative simple normal crossing divisor(=:relative SNCD), 60 restriction, 62 rigid cohomological complex, 226 rigid cohomological complex with support, 238 rigid cohomology, 226 rigid cohomology with compact support, 244 rigid cohomology with support, 238 ringed topos associated to a diagram of ringed topoi, 41

strict compatibility, 211, 237 strict epimorphism, 17 strict monomorphism, 17 strict morphism, 17 strictly exact, 16 strictly flat, 23 strictly flat resolution, 25 strictly injective, 17 strictly injective resolution, 19

slope decomposition, 242, 243 slope spectral sequence, 243 smooth component, 61 special module, 17 specially flat, 24 specially flat resolution, 25 specially injective, 18 specially injective resolution, 19

vanishing cycle sheaf, 90 very fine, 254

truncated proper hypercovering, 225 twist, 141

weight, 239 weight filtration, 237, 239, 246 weight spectral sequence, 237, 239 zariskian orientation sheaf, 81

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