On the Maximal Unramified Quotients of

p-Adic Étale Cohomology Groups

and Logarithmic Hodge–Witt Sheaves

Dedicated to Professor K. Kato on his 50th birthday

Takeshi Tsuji

Received: November 11, 2002 Revised: November 9, 2003

Abstract. Let O_{K} be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. From the semi-stable conjecture (C_{st}) and the theory of slopes, we obtain isomorphisms between the maximal unramified quotients of certain Tate twists of p-adic étale cohomology groups and the cohomology groups of logarithmic Hodge-Witt sheaves for a proper semi-stable scheme over O_{K}. The object of this paper is to show that these isomorphisms are compatible with the symbol maps to the padic vanishing cycles and the logarithmic Hodge-Witt sheaves, and that they are compatible with the integral structures under certain restrictions. We also treats an open case and a proof of C_{st} in such a case is given for that purpose. The results are used in the work of U. Jannsen and S. Saito in this volume.

2000 Mathematics Subject Classification: 14F30, 14F20 Keywords and Phrases: p-adic étale cohomology, p-adic vanishing cycles, logarithmic Hodge-Witt sheaf, p-adic Hodge theory

We will study a description of the maximal unramified quotients of certain p-adic étale cohomology groups in terms of logarithmic Hodge-Witt sheaves. Let K be a complete discrete valuation field of mixed characteristic (0, p) whose residue field is perfect, let O_{K} be the ring of integers of K, and let \overline{K} be an algebraic closure of K. We consider a proper semi-stable scheme X over O_{K}, i.e. a regular scheme X proper and flat over O_{K} such that the special fiber Y of X is reduced and is a divisor with normal crossings on X. For such an X, we have a comparison theorem (the semi-stable conjecture by Fontaine-Jannsen) between the p-adic étale cohomology H_{ét}^{r}(X\overline{K}, ℚ_{p}) and the logarithmic crystalline cohomology of X with some additional structures (Theorem 3.2.2). Combining this with the description of the maximal slope part of the logarithmic crystalline cohomology in terms of logarithmic Hodge-Witt sheaves (see §2.3 for details), we obtain canonical isomorphisms (=(3.2.6), (3.2.7)):

(0.1)

H_{ét}^{r}(X\overline{K}, ℚ_{p}(r))_{I} ≅ ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{r})
H_{ét}^{r}(X\overline{K}, ℚ_{p}(d))_{I} ≅ ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{r−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{d})  if r ≥ d

by a simple argument. Here d = dim X_{K}, I denotes the inertia subgroup of Gal(\overline{K}/K) and the subscript I denotes the cofixed part by I, i.e. the maximal unramified quotient. The purpose of this paper is to answer the following two questions (partially for the second one) on these isomorphisms. First, if we denote by i and j the closed immersion and the open immersion Y → X and X_{K} := X ×_{Spec(O_{K})} Spec(K) → X respectively, then we have a unique surjective homomorphism of sheaves on the étale site Y_{ét} of Y:

(0.2)  i^{∗}R^{r}j_{∗}ℤ/p^{n}ℤ(r) ⟶ W_{n}ω_{Y/s,log}^{r}

p-Adic Étale Cohomology and Log Hodge–Witt Sheaves 835

compatible with the symbol maps (see §3.1 for details), from which we obtain homomorphisms from the LHS’s of (0.1) to the RHS’s of (0.1). Do these homomorphisms coincide with (0.1) constructed from the semi-stable conjecture? In [Sat], it is stated without proof that they coincide. We will give its precise proof. This second construction via (0.2) is necessary in the applications [Sat] and [J-Sai]. Secondly, the both sides of the isomorphisms (0.1) have natural integral structures coming from H_{ét}^{r}(X\overline{K}, ℤ_{p}(s)) and H_{ét}^{r−s}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{s}) for s = r, d. Do the two integral structures coincide? We will prove that it is true in the case r ≤ p−2 and the base field K is absolutely unramified by using the comparison theorem of C. Breuil (Theorem 3.2.4) between the p-torsion étale cohomology and certain log crystalline cohomology. By [Ts4], one can easily extend the theorem of C. Breuil to any fine and saturated smooth log scheme whose special fiber is reduced (including the case that the log structure on the generic fiber is non-trivial), and we will discuss on the second question under this more general setting. For the ℚ_{p} case, G. Yamashita [Y] recently proved the semi-stable conjecture in the open case (more precisely under the condition (3.1.2)) by the syntomic method. If we use his result, we can prove our result also in the open case. Considering the necessity in [J-Sai] of our result in the open case, we will give an alternative proof in §4 when the horizontal divisors at infinity do not have self-intersections by proving the compatibility of the comparison maps with the Gysin sequences. This paper is organized as follows. In §1, we will give a description of the maximal unramified quotients of semi-stable ℚ_{p}-representations and semi-stable ℤ_{p}-representations (in the sense of C. Breuil) in terms of the corresponding objects in \underline{MF}_{K}(φ, N) and in \underline{MF}_{W,tor}(φ, N) respectively. In §2, we study the relation between the maximal slope part of the log crystalline cohomology and the logarithmic Hodge-Witt sheaves taking care of their integral structures. In §3, we will state our main theorem, review the construction of the comparison map in the semi-stable conjecture and then prove the main theorem. I dedicate this paper to Professor K. Kato, who guided me to the p-adic world, especially to the p-adic Hodge theory. This paper is based on his work at many points. I would like to thank Professors U. Jannsen and S. Saito for fruitful discussions on the subject of this paper.

Notation. Let K be a complete discrete valuation field of mixed characteristic (0, p) whose residue field k is perfect, and let O_{K} denote the ring of integers of K. Let W be the ring of Witt vectors with coefficients in k, and let K_{0} denote the field of fractions of W. We denote by σ the Frobenius endomorphisms of k, W and K_{0}. Let \overline{K} be an algebraic closure of K, and let \overline{k} be the residue field of \overline{K}, which is an algebraic closure of k. Let G_{K} (resp. G_{k}) be the Galois group Gal(\overline{K}/K) (resp. Gal(\overline{k}/k)), and let I_{K} be the inertia group of G_{K}. We have G_{K}/I_{K} ≅ G_{k}. We denote by P_{0} the field of fractions of the ring of Witt vectors W (\overline{k}) with coefficients in \overline{k}.

§1. The maximal unramified quotients of semi-stable representations. In this section, we study the maximal unramified quotients (i.e. the coinvariant by the inertia group I_{K}) of semi-stable p-adic or ℤ_{p}-representations of G_{K}.

§1.1. Review on slopes. Let D be a finite dimensional P_{0}-vector space with a semi-linear automorphism φ. For a rational number α = sr^{−1} (s, r ∈ ℤ, r > 0, (s, r) = 1), we denote by D_{α} the P_{0}-subspace of D generated by the Frac(W (𝔽_{p^{r}}))-vector space D^{φ^{r}=p^{s}}. Then, the natural homomorphisms D^{φ^{r}=p^{s}} ⊗_{Frac(W (𝔽_{p^{r}}))} P_{0} → D_{α} and ⊕_{α∈ℚ}D_{α} → D are isomorphisms. We call D_{α} the subspace of D whose slope is α, and we say that α is a slope of D if D_{α} ≠ 0. Let D be a finite dimensional K_{0}-vector space with a σ-semi-linear automorphism φ. Then the above slope decomposition of (P_{0} ⊗_{K_{0}} D, φ ⊗ φ) descends to the decomposition D = ⊕_{α∈ℚ}D_{α} of D. We call D_{α} the subspace of D whose slope is α and we say that α is a slope of D if D_{α} ≠ 0. For a subset I ⊂ ℚ, we denote the sum ⊕_{α∈I}D_{α} by D_{I}.

§1.2. Review on semi-stable, crystalline and unramified p-adic representations ([Fo1], [Fo2]A1, [Fo3], [Fo4]). Let \underline{MF}_{K}(φ, N) denote the category of finite dimensional K_{0}-vector spaces endowed with σ-semilinear automorphisms φ, K_{0}-linear endomorphisms N satisfying Nφ = pφN, and exhaustive and separated descending filtrations F il^{·} on D_{K} := K ⊗_{K_{0}} D by K-subspaces. Let \underline{MF}_{K}(φ) denote the full subcategory of \underline{MF}_{K}(φ, N) consisting of the objects such that N = 0. We denote by \underline{M}_{K_{0}}(φ) the category of a finite dimensional K_{0}-vector space endowed with a σsemilinear automorphism φ whose slope is 0. We regard an object D of \underline{M}_{K_{0}}(φ) as an object of \underline{MF}_{K}(φ) by giving the filtration F il^{0}D_{K} = D_{K}, F il^{1}D_{K} = 0. By a p-adic representation of G_{K}, we mean a finite dimensional ℚ_{p}-vector space endowed with a continuous and linear action of G_{K}, and we denote by \underline{Rep}(G_{K}) the category of p-adic representations. We denote by \underline{Rep}_{st}(G_{K}) (resp. \underline{Rep}_{crys}(G_{K}), resp. \underline{Rep}_{ur}(G_{K})) be the full subcategory of \underline{Rep}(G_{K}) consisting of semi-stable (resp. crystalline, resp. unramified) p-adic representations. Choose and fix a uniformizer π of K. Then, by the theory of Fontaine, we have the following commutative diagram of categories and functors:

\underline{Rep}_{ur}(G_{K}) ⟶^{D_{ur}}  \underline{M}_{K_{0}}(φ)
∩  ∩
\underline{Rep}_{crys}(G_{K}) ⟶^{D_{crys}}  \underline{MF}_{K}(φ)
∩  ∩
\underline{Rep}_{st}(G_{K}) ⟶^{D_{st}}  \underline{MF}_{K}(φ, N)

The functors D_{crys} and D_{st} are fully faithful and the functor D_{ur} is an equivalence of categories; they are defined by

D_{•}(V) = (B_{•} ⊗_{ℚ_{p}} V)^{G_{K}} (• = st, crys, ur),

where B_{st} and B_{crys} are the rings of Fontaine and B_{ur} = P_{0}. A semi-stable representation V is crystalline if and only if N = 0 on D_{st}(V). Recall that the embedding B_{st} ↪ B_{dR} and hence the functor D_{st} depends on the choice of π. The quasi-inverse of D_{ur} is given by

V_{ur}(D) = (P_{0} ⊗_{K_{0}} D)^{φ⊗φ=1}

We say that an object D of \underline{MF}_{K}(φ, N) (resp. \underline{MF}_{K}(φ)) is admissible if there exists a semi-stable (resp. crystalline) representation V such that D_{st}(V) ≅ D (resp. D_{crys}(V) ≅ D). We denote by \underline{MF} _{K}^{ad}(φ, N) (resp. \underline{MF} _{K}^{ad}(φ)) the full subcategory of \underline{MF}_{K}(φ, N) (resp. \underline{MF}_{K}(φ)) consisting of admissible objects. Then the quasi-inverse V_{st} : \underline{MF} _{K}^{ad}(φ, N) → \underline{Rep}_{st}(G_{K}) (resp. V_{crys} : \underline{MF} _{K}^{ad}(φ) → \underline{Rep}_{crys}(G_{K})) of the functor D_{st} (resp. D_{crys}) is given by

V_{st}(D) = F il^{0}(B_{dR} ⊗_{K} D_{K}) ∩ (B_{st} ⊗_{K_{0}} D)^{φ⊗φ=1,N⊗1+1⊗N=0}
(resp. V_{crys}(D) = F il^{0}(B_{dR} ⊗_{K} D_{K}) ∩ (B_{crys} ⊗_{K_{0}} D)^{φ⊗φ=1})

For an object D of \underline{MF}_{K}(φ, N) and an integer r, we denote by D(r) the object of \underline{MF}_{K}(φ, N) whose underlying K_{0}-vector space and monodromy operator N are the same as D and whose Frobenius endomorphism (resp. filtration) is defined by φ_{D(r)} = p^{−r}φ_{D} (resp. F il^{i}D(r)_{K} = F il^{i+r}D_{K}). If D is admissible, then D(r) is also admissible and there is a canonical isomorphism V_{st}(D)(r) ≅ V_{st}(D(r)) induced by (B_{st} ⊗_{K_{0}} D) ⊗_{ℚ_{p}} ℚ_{p}(r) ≅ B_{st} ⊗_{K_{0}} D(r); (a ⊗ d) ⊗ t^{r} ↦ at^{r} ⊗ d. Here (r) in the left-hand sides means the usual Tate twist For an object D of \underline{MF}_{K}(φ, N), we define the integers t_{H}(D) and t_{N}(D) by

t_{H}(D) = ∑_{i∈ℤ}(dim_{K} gr_{F il}^{i}D_{K}) · i
t_{N}(D) = ∑_{α∈ℚ}(dim_{K_{0}} D_{α}) · α,

where D_{α} denotes the subspace of D whose slope is α. We say that an object D of \underline{MF}_{K}(φ, N) is weakly admissible if t_{H}(D) = t_{N}(D) and if, for any K_{0}-subspace D′ of D stable under φ and N, t_{H}(D′) ≤ t_{N}(D′). Here we endow D′_{K} with the filtration F il^{i}D′_{K} = F il^{i}D_{K} ∩ D′_{K} (i ∈ ℤ). Note that t_{H}(D′) ≤ t_{N}(D′) is equivalent to t_{H}(D/D′) ≥ t_{N}(D/D′) under the assumption t_{H}(D) = t_{N}(D). The admissibility implies the weak admissibility. P. Colmez and J.-M. Fontaine proved that the converse is also true ([C-Fo]).

§1.3. The maximal unramified quotients of semi-stable p-adic representations.

Lemma 1.3.1. Let D be a weakly admissible object of \underline{MF}_{K}(φ, N). If F il^{r}D_{K} = D_{K} and F il^{s+1}D_{K} = 0 for integers r ≤ s, then the slopes of φ on D are contained in [r, s].

Proof. We prove that the slopes of φ on D are not less than r. The proof of slopes≤ s is similar and is left to the reader. (Consider the projection D → D_{α} for the largest slope α.) Let α be the smallest slope of D, and let D_{α} be the subspace of D whose slope is α. Then, D_{α} is stable under φ. By the formula Nφ = pφN and the choice of α, we see N = 0 on D_{α}, especially D_{α} is stable under N. Hence t_{H}(D_{α}) ≤ t_{N}(D_{α}) = α · dim_{K_{0}} D_{α}. Since t_{H}(D_{α}) ≥ r · dim_{K_{0}} D_{α} by the assumption on D, we obtain α ≥ r. □

Let V be a semi-stable p-adic representation of G_{K}, and set D = D_{st}(V). Let s be an integer such that F il^{s+1}D_{K} = 0, and let V_{s} be the quotient V (s)_{I_{K}} (−s) of V. We will construct explicitly the corresponding admissible quotient of D. For α ∈ ℚ, let D_{α} denote the subspace of D whose slope is α. By Lemma 1.3.1, D_{α} = 0 if α > s and hence D = ⊕_{α∈ℚ,α≤s}D_{α}. We define the monodromy operator on D_{s} by N = 0 and the filtration on (D_{s})_{K} by F il^{s}(D_{s})_{K} = (D_{s})_{K} and F il^{s+1}(D_{s})_{K} = 0. Then, we see that D_{s}(s) is an object of \underline{M}_{K_{0}}(φ). Especially D_{s} is admissible. Using the relation Nφ = pφN on D and F il^{s+1}D_{K} = 0, we see that the projection D → D_{s} is a morphism in the category \underline{MF} _{K_{0}}^{ad}(φ, N). Especially it is strictly compatible with the filtrations ([Fo4]4.4.4. Proposition i)), that is, the image of F il^{i}D_{K} is F il^{i}(D_{s})_{K}.

Proposition 1.3.2. Under the notation and the assumption as above, the quotient V_{s} of V corresponds to the admissible quotient D_{s} of D.

Proof. Since D_{s}(s) is contained in \underline{M}_{K_{0}}(φ), V_{st}(D_{s})(s) ≅ V_{st}(D_{s}(s)) is unramified. Hence the natural surjection V → V_{st}(D_{s}) factors through V_{s}. On the other hand, since V_{s}(s) is unramified, D_{st}(V_{s})(s) ≅ D_{st}(V_{s}(s)) is contained in \underline{M}_{K_{0}}(φ). Hence the unique slope of D_{st}(V_{s}) is s and the natural projection D → D_{st}(V_{s}) factors as D → D_{s} → D_{st}(V_{s}). D_{s} → D_{st}(V_{s}) is strictly compatible with the filtrations because D → D_{s} and D → D_{st}(V_{s}) are strictly compatible with the filtrations. (Compatibility with φ and N is trivial). □

Corollary 1.3.3. Under the above notations and assumptions, there exists a canonical G_{k}-equivariant isomorphism:

V (s)_{I_{K}} ≅ (P_{0} ⊗_{K_{0}} D)^{φ=p^{s}}.

Proof. By Proposition 1.3.2, we have canonical G_{k}-equivariant isomorphisms:

V (s)_{I_{K}} = V_{s}(s) ≅ V_{st}(D_{s})(s) ≅ V_{st}(D_{s}(s)) = V_{ur}(D_{s}(s))
= (P_{0} ⊗_{K_{0}} D_{s})^{φ⊗p^{−s}φ=1} = (P_{0} ⊗_{K_{0}} D)^{φ⊗φ=p^{s}}.

□

§1.4. Review on semi-stable, crystalline and unramified p-torsion representations ([Fo-L], [Fo2]A1, [Br2], [Br3]§3.2.1). In this section, we assume K = K_{0}. Following [Br2], we denote by S the p-adic completion of the PD-polynomial ring in one variable W ⟨u⟩ = W ⟨u − p⟩, and by f_{0} (resp. f_{p}) the W -algebra homomorphism S → W defined by u^{[n]} ↦ 0 (resp. p^{[n]} = p^{n}/n!) for n ≥ 1. We define the filtration F il^{i}S (i ∈ ℤ, i > 0) to be the p-adic completion of the i-th divided power of the PD-ideal of W ⟨u − p⟩ generated by u − p. We set F il^{i}S = S for i ∈ ℤ, i ≤ 0. Let φ_{S}: S → S denote the lifting of Frobenius defined by σ on W and u^{[n]} ↦ (u^{p})^{[n]}. For an integer i such that 0 ≤ i ≤ p − 2, we have φ(F il^{i}S) ⊂ p^{i}S and we denote by φ_{i}: F il^{i}S → S the homomorphism p^{−i} · φ|_{F il^{i}S}. Finally let N denote the W -linear derivation N: S → S defined by N(u^{[n]}) = nu^{[n]} (n ∈ ℕ). Let \underline{MF}_{W,[0,p−2],tor}(φ) be the category of Fontaine-Laffaille of level within [0, p − 2], let \underline{MF}_{W,[0,p−2],tor}(φ, N) be the category \underline{M}^{p−2} of Breuil, and let \underline{M}_{W,tor}(φ) be the category of W -modules of finite length endowed with σsemilinear automorphisms. These categories are abelian and artinian. An object of \underline{MF}_{W,[0,p−2],tor}(φ) is a W -module M of finite length endowed with a descending filtration F il^{i}M (i ∈ ℤ) by W -submodules such that F il^{0}M = M, F il^{p−1}M = 0 and σ-semilinear homomorphisms φ_{i}: F il^{i}M → M (0 ≤ i ≤ p − 2) such that φ_{i}|_{F il^{i+1}M} = pφ_{i+1} (0 ≤ i ≤ p−3) and M = ∑_{0≤i≤p−2} φ_{i}(F il^{i}M). We can prove that F il^{i}M (i ∈ ℤ) are direct summands of M. For an integer 0 ≤ r ≤ p − 2, we say that M is of level within [0, r] if F il^{r+1}M = 0. The sequence M_{1} → M_{2} → M_{3} in \underline{MF}_{W,[0,p−2],tor}(φ) is exact if and only if it is exact as a sequence of W -modules. Furthermore, for an exact sequence M_{1} → M_{2} → M_{3} in \underline{MF}_{W,[0,p−2],tor}(φ), the sequence F il^{i}M_{1} → F il^{i}M_{2} → F il^{i}M_{3} is exact for any i ∈ ℤ. An object of \underline{MF}_{W,[0,p−2],tor}(φ, N) is an S-module M isomorphic to a finite sum of S-modules of the form S/p^{n}S (n ≥ 1) endowed with the following three structures: A submodule F il^{p−2}M such that F il^{p−2}S · M ⊂ F il^{p−2}M. A φ_{S}-semi-linear homomorphism φ_{p−2} : F il^{p−2}M → M such that (φ_{1}(u − p))^{p−2}φ_{p−2}(ax) = φ_{p−2}(a)φ_{p−2}((u − p)^{p−2}x) (a ∈ F il^{p−2}S, x ∈ M) and that M is generated by φ_{p−2}(F il^{p−2}M) as an S-module. A W -linear map N: M → M such that N(ax) = N(a)x + aN(x) (a ∈ S, x ∈ M), (u − p)N(F il^{p−2}M) ⊂ F il^{p−2}M and φ_{1}(u−p)Nφ_{p−2}(x) = φ_{p−2}((u−p)N(x)) (x ∈ F il^{p−2}M). Note that φ_{1}(u−p) = (p − 1)!u^{[p]} − 1 is invertible in S. For an object M of \underline{MF}_{W,[0,p−2],tor}(φ, N), we define the filtration F il^{i}M (0 ≤ i ≤ p − 2) by F il^{i}M := {x ∈ M| (u−p)^{p−2−i}x ∈ F il^{p−2}M} and the Frobenius φ_{i} : F il^{i}M → M (0 ≤ i ≤ p−2) by the formula φ_{i}(x) = φ_{1}(u − p)^{−(p−2−i)}φ_{p−2}((u − p)^{p−2−i}x). We have F il^{0}M = M and φ_{i}|F il^{i+1}M = pφ_{i+1} for 0 ≤ i ≤ p − 3. For an object M of \underline{MF}_{W,[0,p−2],tor}(φ, N) and an integer 0 ≤ r ≤ p − 2, we say that M is of level within [0, r] if F il^{p−2−r}S · M ⊃ F il^{p−2}M. The sequence M_{1} → M_{2} → M_{3} in \underline{MF}_{W,[0,p−2],tor}(φ, N) is exact if and only if it is exact as S-modules. Furthermore, for an exact sequence M_{1} → M_{2} → M_{3} in \underline{MF}_{W,[0,p−2],tor}(φ, N) and an integer 0 ≤ r ≤ p − 2, if M_{s} (s = 1, 2, 3) are of level within [0, r], then the sequence F il^{r}M_{1} → F il^{r}M_{2} → F il^{r}M_{3} is exact. We regard an object M of \underline{M}_{W,tor}(φ) as an object of \underline{MF}_{W,[0,p−2],tor}(φ) by setting F il^{0}M = M, F il^{1}M = 0 and φ_{0} = φ. We have a canonical fully faithful exact functor \underline{MF}_{W,[0,p−2],tor}(φ) → \underline{MF}_{W,[0,p−2],tor}(φ, N) defined as follows ([Br2]2.4.1): To an object M of \underline{MF}_{W,[0,p−2],tor}(φ), we associate the following object M. The underlying S-module is S ⊗_{W} M and F il^{p−2}M = ∑_{0≤i≤p−2} F il^{p−2−i}S ⊗_{W} F il^{i}M. The Frobenius φ_{p−2} : F il^{p−2}M → M is defined by the formula: φ_{p−2}(a ⊗ x) = φ_{p−2−i}(a) ⊗ φ_{i}(x) (0 ≤ i ≤ p − 2, a ∈ F il^{p−2−i}S, x ∈ F il^{i}M). The monodromy operator is defined by N(a ⊗ x) = N(a) ⊗ x (a ∈ S, x ∈ M). To prove that φ_{p−2} is well-defined, we use the fact that F il^{i}M (i ∈ ℤ) are direct summands of M. Note that, for an integer 0 ≤ r ≤ p − 2, M is of level within [0, r] if and only if M is of level within [0, r]. Let \underline{Rep}_{tor}(G_{K}) be the category of ℤ_{p}-modules of finite length endowed with continuous actions of G_{K}, and let \underline{Rep}_{tor,ur}(G_{K}) be the full subcategory of \underline{Rep}_{tor}(G_{K}) consisting of the objects such that the actions of G_{K} are trivial on I_{K}. First we have an equivalence of categories:

T_{ur} : \underline{M}_{W,tor}(φ) → \underline{Rep}_{tor,ur}(G_{K})

defined by T_{ur}(M) = (W (\overline{k}) ⊗_{W} M)^{φ⊗φ=1}. Its quasi-inverse is given by T ↦ (W (\overline{k}) ⊗_{ℤ_{p}} T)^{G_{K}} (cf. for example, [Fo2]A1).

J.-M. Fontaine and G. Laffaille constructed a covariant fully faithful exact functor ([Fo-L]):

T_{crys} : \underline{MF}_{W,[0,p−2],tor}(φ) → \underline{Rep}_{tor}(G_{K}).

(Strictly speaking, they constructed a contravariant functor \underline{U}_{S}. We define T_{crys} to be its dual: T_{crys}(M) := Hom(\underline{U}_{S}(M), ℚ_{p}/ℤ_{p}).) For an object M of \underline{MF}_{W,[0,p−2],tor}(φ) and an integer 0 ≤ r ≤ p − 2 such that M is of level within [0, r], we have the following exact sequence functorial on M (cf. [Ka1]II §3):

(1.4.1) 0 ⟶ T_{crys}(M)(r) ⟶ F il^{r}(A_{crys} ⊗_{W} M) ⟶^{1−φ_{r}} A_{crys} ⊗_{W} M ⟶ 0.

Here F il^{i}(A_{crys} ⊗_{W} M) = ∑_{0≤j≤i} F il^{i−j}A_{crys} ⊗_{W} F il^{j}M (0 ≤ i ≤ p − 2) and φ_{i} is defined by φ_{i}(a ⊗ x) = φ_{i−j}(a) ⊗ φ_{j}(x) (0 ≤ j ≤ i, a ∈ F il^{i−j}A_{crys}, x ∈ F il^{j}M). C. Breuil constructed a covariant fully faithful exact functor ([Br2]):

T_{st} : \underline{MF}_{W,[0,p−2],tor}(φ, N) → \underline{Rep}_{tor}(G_{K}).

(Strictly speaking, he constructed a contravariant functor V_{st}. We define T_{st} to be its dual: T_{st}(M) := Hom(V_{st}(M), ℚ_{p}/ℤ_{p}).) For an object M of \underline{MF}_{W,[0,p−2],tor}(φ, N) and an integer 0 ≤ r ≤ p − 2 such that M is of level within [0, r], we have the following exact sequence functorial on M ([Br3]§3.2.1):

(1.4.2) 0 ⟶ T_{st}(M)(r) ⟶ (F il^{r}(Â_{st}⊗_{S}M))^{N=0} ⟶^{1−φ_{r}} (Â_{st}⊗_{S}M)^{N=0} ⟶ 0

Here we define F il^{·}, φ_{r} and N on Â_{st} ⊗_{S} M as follows (see [Br1]§2 for the definition of Â_{st}): We define F il^{i}(Â_{st} ⊗_{S} M) (0 ≤ i ≤ p − 2) to be the sum of the images of F il^{i−j}Â_{st} ⊗_{S} F il^{j}M (0 ≤ j ≤ i). The homomorphism φ_{r}: F il^{r}(Â_{st} ⊗_{S} M) → Â_{st} ⊗_{S} M is defined by φ_{r}(a ⊗ x) = φ_{r−i}(a) ⊗ φ_{i}(x) (0 ≤ i ≤ r, a ∈ F il^{r−i}Â_{st}, x ∈ F il^{i}M). (The well-definedness is non-trivial). The monodromy operator N is defined by N(a ⊗ x) = N(a) ⊗ x + a ⊗ N(x) (a ∈ Â_{st}, x ∈ M). Now we have the following diagram of categories and functors commutative up to canonical isomorphisms:

(1.4.3)

\underline{M}_{W,tor}(φ)  ⟶^{T_{ur}}  \underline{Rep}_{tor,ur}(G_{K})
∩  ∩
\underline{MF}_{W,[0,p−2],tor}(φ) ⟶^{T_{crys}}  \underline{Rep}_{tor}(G_{K})
↓  ||
\underline{MF}_{W,[0,p−2],tor}(φ, N) ⟶^{T_{st}}  \underline{Rep}_{tor}(G_{K}).

For an object M of \underline{M}_{W,tor}(φ), the isomorphism T_{ur}(M) ≅ T_{crys}(M) is induced by the natural homomorphism (W (\overline{k}) ⊗_{W} M)(r) → F il^{r}(A_{crys} ⊗_{W} M) = (F il^{r}A_{crys}) ⊗_{W} M (0 ≤ r ≤ p − 2) defined by the natural inclusion W (\overline{k})(r) ⊂ F il^{r}A_{crys}. (The isomorphism is independent of the choice of r.) For an object M of \underline{MF}_{W,[0,p−2],tor}(φ), if we denote by M the corresponding object of \underline{MF}_{W,[0,p−2],tor}(φ, N), then the natural homomorphisms F il^{i}(A_{crys} ⊗_{W} M) → F il^{i}(Â_{st} ⊗_{S} M)^{N=0} (0 ≤ i ≤ p − 2) are isomorphisms and we obtain the natural isomorphism T_{crys}(M) ≅ T_{st}(M) from the two exact sequences (1.4.1) and (1.4.2).

§1.5. Review of the comparison between ℚ_{p} and p-torsion theories ([Fo-L], [Br1], [Br2]). We assume K = K_{0} and keep the notation of §1.4. We review the relation between the functor T_{•} (§1.4) and the functor V_{•} (§1.2) (• ∈ {ur, crys, st}). Let us begin with • = ur. Let (M_{n})_{n≥1} be a projective system of objects of \underline{M}_{W,tor}(φ) such that the underlying W -module of M_{n} is a free W/p^{n}W -module of finite rank and the morphism M_{n+1}/p^{n}M_{n+1} → M_{n} is an isomorphism. Associated to such a system, we define the object D of \underline{M}_{K_{0}}(φ) to be K_{0} ⊗_{W} \underleftarrow{lim}_{n} M_{n}. Then, we have a canonical isomorphism defined in an obvious way:

(1.5.1)  V_{ur}(D) ≅ ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} T_{ur}(M_{n}).

Next consider the case • = crys. Let (M_{n})_{n≥1} be a projective system of objects of \underline{MF}_{W,[0,p−2],tor}(φ) such that the underlying W -module of M_{n} is a free W/p^{n}W -module of finite rank and the morphism M_{n+1}/p^{n}M_{n+1} → M_{n} is an isomorphism. We define the object D of \underline{MF}_{K}(φ) associated to this system as follows: The underling vector space is K_{0} ⊗_{W} (\underleftarrow{lim}_{n} M_{n}), the filtration is defined by K_{0} ⊗_{W} (\underleftarrow{lim}_{n} F il^{i}M_{n}) and the Frobenius endomorphism is defined to be the projective limit of φ_{0} of M_{n}. Then D is admissible, and we have a canonical isomorphism ([Fo-L]§7, §8):

(1.5.2)  V_{crys}(D) ≅ ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n} T_{crys}(M_{n})).

The homomorphism from the RHS to the LHS is constructed as follows: By taking the projective limit of the exact sequence (1.4.1) for M = M_{n} and 0 ≤ r ≤ p − 2 such that M_{n} are of level within [0, r] for all n ≥ 1 and tensoring with ℚ_{p}, we obtain an isomorphism

ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n} T_{crys}(M_{n}))(r) ≅ F il^{r}(B_{crys}^{+} ⊗_{K_{0}} D)^{φ=p^{r}},

whose RHS is contained in

V_{crys}(D)(r) ≅ F il^{r}(B_{crys} ⊗_{K_{0}} D)^{φ=p^{r}}
v ⊗ t^{⊗r} ↦ t^{r}v (v ∈ V_{crys}(D), t ∈ ℚ_{p}(1)).

Finally let us explain the case • = st. In this case, we need to introduce another category \underline{MF}_{K}(φ, N) defined by C. Breuil [Br1]§6 (\underline{MF}_{S⊗_{W} K_{0}}(Φ, N) in his notation), which is categorically equivalent to \underline{MF}_{K}(φ, N). Set S_{K_{0}} := K_{0}⊗_{W} S, and let f_{0} (resp. f_{p}) also denote the surjective homomorphism S_{K_{0}} → K_{0} induced by f_{0} (resp. f_{p}) : S → W. The filtration F il^{·}, the Frobenius endomorphism φ and the monodromy operator N on S naturally induce those on S_{K_{0}}, which we also denote by F il^{·}, φ and N. An object of \underline{MF}_{K}(φ, N) is a free S_{K_{0}}-module D of finite rank endowed with the following three structures: A descending filtration F il^{i}D (i ∈ ℤ) by S_{K_{0}}-submodules such that F il^{i}S_{K_{0}} · F il^{j}D ⊂ F il^{i+j}D (i, j ∈ ℤ), F il^{i}D = D (i << 0) and F il^{i}D ⊂ F il^{1}S_{K_{0}} · D (i >> 0). A φ_{S_{K_{0}}} -semilinear endomorphism D → D whose linearization D ⊗_{S_{K_{0}} ,φ} S_{K_{0}} → D is an isomorphism. A homomorphism N: D → D such that N(ax) = N(a)x + aN(x) (a ∈ S_{K_{0}}, x ∈ D), N(F il^{i}D) ⊂ F il^{i−1}D (i ∈ ℤ) and Nφ = pφN. We can construct a functor \underline{MF}_{K}(φ, N) → \underline{MF}_{K}(φ, N) easily as follows: Let D be an object of \underline{MF}_{K}(φ, N). The corresponding object D is the S_{K_{0}}-module S_{K_{0}} ⊗_{K_{0}} D with the Frobenius φ⊗φ and the monodromy operator defined by N(a⊗x) = N(a)⊗x+a⊗N(x). The filtration is defined inductively by the following formula, where i_{0} is an integer such that F il^{i_{0}}D = D.

F il^{i}D = D (i ≤ i_{0})
F il^{i}D = {x ∈ F il^{i−1}D|f_{p}(x) ∈ F il^{i}D, N(x) ∈ F il^{i−1}D} (i > i_{0})

Here f_{p} denotes the natural projection D → D ⊗_{S_{K_{0}} ,f_{p}} K_{0} and we identify D⊗_{S_{K_{0}} ,f_{p}} K_{0} with D by the natural isomorphism (D⊗_{S_{K_{0}}} S_{K_{0}})⊗_{S_{K_{0}} ,f_{p}} K_{0} ≅ D. To construct the quasi-inverse of the above functor, we need the following

proposition. (Compare with B_{st}^{+} = B̂_{st}^{+}^{N-nilp}.)

Proposition 1.5.3. ([Br1]§6.2.1). Let D be an object of \underline{MF}_{K}(φ, N). Then D := D^{N-nilp} is a finite dimensional vector space over K_{0} and the natural homomorphism D ⊗_{K_{0}} S_{K_{0}} → D is an isomorphism. Here N-nilp denotes the part where N is nilpotent.

With the notation of Proposition 1.5.3, we can verify easily that D is stable under N and φ and we can define an exhaustive and separated filtration on D by the image of F il^{i}D under the homomorphism D → D ⊗_{S_{K_{0}} ,f_{p}} K_{0} ≅ D. Thus we obtain an object D of \underline{MF}_{K}(φ, N). The functor associating D to D is the quasi-inverse of the above functor. Let (M_{n})_{n≥1} be a projective system of objects of \underline{MF}_{W,[0,p−2],tor}(φ, N) such that the underlying S-module of M_{n} is a free S/p^{n}S-module and the morphism M_{n+1}/p^{n}M_{n+1} → M_{n} is an isomorphism for n ≥ 1. Set M := \underleftarrow{lim}_{n} M_{n}, F il^{p−2}M := \underleftarrow{lim}_{n} F il^{p−2}M_{n}, and define the Frobenius φ_{p−2} : F il^{p−2}M → M and the monodromy operator N: M → M by taking the projective limit of those on M_{n} (n ∈ ℕ). Then M is a free S-module of finite rank and the additional three structures satisfy the same conditions required in the definition of the category \underline{MF}_{W,[0,p−2],tor}(φ, N). Furthermore M/F il^{p−2}M is p-torsion free. Indeed, we have the following injective morphism between two short exact sequences:

0 ⟶ M ⟶^{p} M ⟶ M_{1}  ⟶ 0
↑  ↑  ↑
0 ⟶ F il^{p−2}M ⟶^{p} F il^{p−2}M ⟶ F il^{p−2}M_{1} ⟶ 0.

(With the terminology of [Br2]Définition 4.1.1.1, M with the three additional structures is a strongly divisible S-module.) For each integer i such that 0 ≤ i ≤ p − 2, we define the S-submodule F il^{i}M of M by {x ∈ M|(u − p)^{p−2−i}x ∈ F il^{p−2}M} and the Frobenius φ_{i}: F il^{i}M → M by φ_{i}(x) = φ_{1}(u − p)^{−(p−2−i)}φ_{p−2}((u − p)^{p−2−i}x). We have F il^{0}M = M, φ_{i}|F il^{i+1}M = pφ_{i+1} for 0 ≤ i ≤ p − 3 and F il^{i}M = \underleftarrow{lim}_{n} F il^{i}M_{n} (0 ≤ i ≤ p − 2). If M_{n} are of level within [0, r] for an integer 0 ≤ r ≤ p − 2, then we see that M/F il^{r}M is p-torsion free. Now we define the object D of \underline{MF}_{K}(φ, N) associated to the projective system (M_{n})_{n} as follows ([Br2]§4.1.1): The underlying S_{K_{0}}-module is ℚ_{p}⊗_{ℤ_{p}} M. The filtration is defined by F il^{i}D = ℚ_{p}⊗_{ℤ_{p}} F il^{i}M (0 ≤ i ≤ p−2) and F il^{i}D = ∑_{0≤j≤p−2} F il^{i−j}S_{K_{0}} · F il^{j}D (i ≥ p − 1). The Frobenius and the monodromy operator on D are defined to be the endomorphisms induced by φ_{0} and N on M. Finally let D be the object of \underline{MF}_{K}(φ, N) corresponding to D. (If M_{n} are of level within [0, r] for an integer 0 ≤ r ≤ p − 2, then so is D, that is, F il^{0}D = D and F il^{r+1}D = 0.) Then D is admissible and there is a canonical isomorphism

(1.5.4)  V_{st}(D) ≅ ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} T_{st}(M_{n})

functorial on (M_{n})_{n} ([Br2]4.2). This isomorphism is constructed as follows (cf. [Br3]4.3.2). First, since M is a free S-module of finite rank and M_{n} ≅ M/p^{n}M, we have

(1.5.5)  ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n}(Â_{st} ⊗_{S} M_{n})) ≅ B̂_{st}^{+} ⊗_{S_{K_{0}}} D.

See [Br1]§2 for the definition of B̂_{st}^{+}.

Lemma 1.5.6. Let r be an integer such that 0 ≤ r ≤ p − 2 and M_{n} (n ≥ 1) are of level within [0, r]. Then the above isomorphism induces an isomorphism:

ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n}(F il^{r}(Â_{st} ⊗_{S} M_{n})^{N=0})) ≅ F il^{r}(B̂_{st}^{+} ⊗_{S_{K_{0}}} D)^{N=0}.

Here F il^{r}(Â_{st} ⊗_{S} M_{n}) is the Â_{st}-submodule of Â_{st} ⊗_{S} M_{n} defined after (1.4.2), and F il^{r}(B̂_{st}^{+} ⊗_{S_{K_{0}}} D) is the sum of the images of F il^{r−j}B̂_{st}^{+} ⊗_{S_{K_{0}}} F il^{j}D (0 ≤

j ≤ r) in B̂_{st}^{+} ⊗_{S_{K_{0}}} D.

This is stated in [Br3] Lemme 4.3.2.2 in the special case that (M_{n}) comes from crystalline cohomology with only an outline of a proof, and his proof seems to work for a general (M_{n}). To make our argument certain, I will give a proof, which seems a bit different from his.

Proof. We use the terminology and the notation in [Br3]3.2.1 freely. We define another filtration \overline{F il}^{i}M (0 ≤ i ≤ r) of M by \overline{F il}^{i}M := F il^{i}D ∩ M. We see easily that this filtration satisfies the three conditions of [Br3] Définition 3.2.1.1, and \overline{F il}^{i}M ⊃ F il^{i}M. Note that M/F il^{r}M is p-torsion free. Define another filtration \overline{F il}^{i}M_{n} (0 ≤ i ≤ r) of M_{n} to be the images of \overline{F il}^{i}M. Then this filtration is admissible in the sense of [Br3] Définition 3.2.1.1. Define \overline{F il}^{r}(Â_{st} ⊗_{S} M_{n}) using \overline{F il}^{·}M_{n} instead of F il^{·}M_{n}, and \overline{F il}^{r}(Â_{st} ⊗_{S} M) similarly using \overline{F il}^{·}M. Then, by [Br3] Proposition 3.2.1.4, we have

(\overline{F il}^{r}(Â_{st} ⊗_{S} M_{n}))^{N=0} = (F il^{r}(Â_{st} ⊗_{S} M_{n}))^{N=0}.

Hence it suffices to prove that (1.5.5) induces an isomorphism

ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n} \overline{F il}^{r}(Â_{st} ⊗_{S} M_{n})) ≅ F il^{r}(B̂_{st}^{+} ⊗_{S_{K_{0}}} D).

To prove this isomorphism, we choose a basis e_{λ} (λ ∈ Λ) of D over S_{K_{0}} and integers 0 ≤ r_{λ} ≤ r such that

F il^{i}D = ⊕_{λ}F il^{i−r_{λ}}S_{K_{0}}e_{λ} (0 ≤ i ≤ r)

([Br1] A). Let M′ be the free S-module generated by e_{λ}. By multiplying p^{−m} for some m > 0 if necessary, we may assume that there exists an integer ν ≥ 0 such that p^{ν}M′ ⊂ M ⊂ M′. We define the filtration \overline{F il}^{i}M′ (0 ≤ i ≤ r) and \overline{F il}^{r}(Â_{st} ⊗_{S} M′) in the same way as M. Then, for 0 ≤ i ≤ r, we have

\overline{F il}^{i}M′ = ⊕_{λ}F il^{i−r_{λ}}S · e_{λ}

and hence

\overline{F il}^{r}(Â_{st} ⊗_{S} M′) = ⊕_{λ}F il^{r−r_{λ}} Â_{st} · e_{λ}.

Especially \overline{F il}^{r}(Â_{st} ⊗_{S} M′) and (Â_{st} ⊗_{S} M′)/(\overline{F il}^{r}(Â_{st} ⊗_{S} M′)) are p-adically complete and separated, and p-torsion free. On the other hand, we have p^{ν} \overline{F il}^{i}M′ ⊂ \overline{F il}^{i}M ⊂ \overline{F il}^{i}M′ and hence

p^{ν} \overline{F il}^{r}(Â_{st} ⊗_{S} M′) ⊂ \overline{F il}^{r}(Â_{st} ⊗_{S} M) ⊂ \overline{F il}^{r}(Â_{st} ⊗_{S} M′).

Since \overline{F il}^{r}(Â_{st} ⊗_{S} M_{n}) is the image of \overline{F il}^{r}(Â_{st} ⊗_{S} M) and (Â_{st} ⊗_{S} M)/p^{n} = Â_{st} ⊗_{S} M_{n}, we have the following commutative diagram:

\overline{F il}^{r}(Â_{st} ⊗_{S} M′)/p^{n} ⟶^{“p^{ν}”} \overline{F il}^{r}(Â_{st} ⊗_{S} M_{n}) ⟶ \overline{F il}^{r}(Â_{st} ⊗_{S} M′)/p^{n}
∩  ∩  ∩
(Â_{st} ⊗_{S} M′)/p^{n}  ⟶^{“p^{ν}”}  Â_{st} ⊗_{S} M_{n}  ⟶ (Â_{st} ⊗_{S} M′)/p^{n}.

By taking the projective limit with respect to n, we obtain the following commutative diagram:

\overline{F il}^{r}(Â_{st} ⊗_{S} M′) ⟶^{p^{ν}}  \underleftarrow{lim}_{n}(\overline{F il}^{r}(Â_{st} ⊗_{S} M_{n})) ⟶ \overline{F il}^{r}(Â_{st} ⊗_{S} M′)
∩  ∩  ∩
Â_{st} ⊗_{S} M′  ⟶^{p^{ν}}  Â_{st} ⊗_{S} M  ⟶  Â_{st} ⊗_{S} M′.

By tensoring with ℚ_{p}, we obtain the claim. □

Let r be an integer as in Lemma 1.5.6. Then by taking the projective limit of the exact sequence (1.4.2) for M_{n} and r and using the isomorphism (1.5.5) and Lemma 1.5.6, we obtain an isomorphism:

ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n}(T_{st}(M_{n})(r)) ≅ F il^{r}(B̂_{st}^{+} ⊗_{S_{K_{0}}} D)^{N=0,φ=p^{r}},

where F il^{r}(B̂_{st}^{+} ⊗_{S_{K_{0}}} D) is as in Lemma 1.5.6. Recall that D denotes the object of \underline{MF}_{K}(φ, N) corresponding to D. By definition, D = D^{N-Nilp}, the canonical homomorphism D ⊗_{K_{0}} S_{K_{0}} → D is an isomorphism, φ and N on D are induced from those on D, and the filtration F il^{i}D is the image of F il^{i}D by f_{p}: D → D ⊗_{S_{K_{0}} ,f_{p}} K_{0} ≅ D. (Recall that we assume K = K_{0}.) On the

other hand, B_{st}^{+} = B̂_{st}^{+}^{N-Nilp} ([Ka3]Theorem (3.7)) and we have a canonical homomorphism B̂_{st}^{+} → B_{dR}^{+} ([Br1]§7, see also [Ts2]§4.6) compatible with the

filtrations and f_{p}: S_{K_{0}} → K_{0} such that the composite with B_{st} ⊂ B̂_{st}^{+} is the

inclusion B_{st}^{+} ⊂ B_{dR}^{+} (associated to p). Hence we have (B̂_{st}^{+} ⊗_{S_{K_{0}}} D)^{N-Nilp} =

(B̂_{st}^{+}⊗_{K_{0}} D)^{N-Nilp} = B_{st}^{+}⊗_{K_{0}} D and the image of F il^{i}(B̂_{st}^{+}⊗_{S_{K_{0}}} D)∩(B_{st}^{+}⊗_{K_{0}} D) by the homomorphism B_{st}^{+} ⊗_{K_{0}} D → B_{dR}^{+} ⊗_{K_{0}} D is contained in F il^{i}(B_{dR}^{+} ⊗_{K_{0}} D). Thus we obtain an injective homomorphism

ℚ_{p} ⊗_{ℤ_{p}} (\underleftarrow{lim}_{n} T_{st}(M_{n})(r)) ↪ (B_{st}^{+} ⊗_{K_{0}} D)^{N=0,φ=p^{r}} ∩ F il^{r}(B_{dR}^{+} ⊗_{K_{0}} D)

and the RHS is contained in

V_{st}(D)(r) ≅ (B_{st} ⊗_{K_{0}} D)^{N=0,φ=p^{r}} ∩ F il^{r}(B_{dR} ⊗_{K_{0}} D).

§1.6. Unramified quotients of semi-stable ℤ_{p}-representations. We assume K = K_{0} and keep the notation of §1.4 and §1.5. Let (M_{n})_{n≥1} be a projective system of objects of \underline{MF}_{W,[0,p−2],tor}(φ, N) such that the underlying S-module M_{n} is a free S/p^{n}S-module of finite rank and the morphism M_{n+1}/p^{n}M_{n+1} → M_{n} in \underline{MF}_{W,[0,p−2],tor}(φ, N) is an isomorphism for every integer n ≥ 1. Let D and D be the objects of \underline{MF}_{K}(φ, N) and \underline{MF}_{K}(φ, N) associated to (M_{n})_{n≥1}. If we denote by M the projective limit of M_{n} with respect to n as S-modules, then D = ℚ_{p} ⊗_{ℤ_{p}} M, D = D^{N-Nilp} and D ⊗_{K_{0}} S_{K_{0}} ≅ D. Set T_{n} := T_{st}(M_{n}), T := \underleftarrow{lim}_{n} T_{n}, and V := V_{st}(D). Recall that D is admissible. We have a canonical isomorphism ℚ_{p} ⊗_{ℤ_{p}} T ≅ V (1.5.4) and we will regard T as a lattice of V by this isomorphism in the following. Let r be an integer such that 0 ≤ r ≤ p−2 and let V′ be a quotient of the representation V such that V′(r) is unramified. Since V′(r) is unramified and hence crystalline, V′ is also crystalline, and if we denote by D′ the corresponding quotient of D in the category \underline{MF}_{K}(φ, N), then F il^{r}D′ = D′, F il^{r+1}D′ = 0, and the slope of the Frobenius φ is r. Furthermore, we have G_{K}-equivariant isomorphisms (1.6.1) V′ ≅ V_{crys}(D′) ≅ V_{crys}(D′(r))(−r) ≅ V_{ur}(D′(r))(−r) = ((P_{0}⊗_{K_{0}} D′)^{φ=p^{r}})(−r).

Let T′ be the image of T in V′ and let M′ be the image of M under the composite

D → D ⊗_{S_{K_{0}} ,f_{0}} K_{0} ≅ D → D′.

Theorem 1.6.2. Let the notation and the assumption be as above. Then the composite of the isomorphisms (1.6.1) induces an isomorphism

T′ ≅ ((W (\overline{k}) ⊗_{W} M′)^{φ⊗φ=p^{r}})(−r).

Proof. Set T′_{n} := T′/p^{n}T′. Since T′_{n}(r) is unramified, T′_{n} is crystalline. If we denote by M′_{n} the corresponding quotient of M_{n}, the projective system (M′_{n})_{n≥0} comes from the projective system (M′_{n})_{n≥1} in \underline{MF}_{W,[0,p−2],tor}(φ) such that, for every integer n ≥ 1, F il^{r}M′_{n} = M′_{n} and F il^{r+1}M′_{n} = 0, the underlying W_{n}-module of M′_{n} is free of rank rank_{ℤ/p^{n}ℤ}T′_{n} = dim_{ℚ_{p}} V′, and the morphism M′_{n+1}/p^{n}M′_{n+1} → M′_{n} in \underline{MF}_{W,[0,p−2],tor}(φ) is an isomorphism. We define M′_{n}(r) to be the object of \underline{M}_{W,tor}(φ) ⊂ \underline{MF}_{W,[0,p−2],tor}(φ) whose underlying W_{n}-module is the same as M′_{n} and whose Frobenius φ is φ_{r}. Then, we have isomorphisms:

T′_{n} ≅ T_{st}(M′_{n}) ≅ T_{crys}(M′_{n}) ≅ T_{crys}(M′_{n}(r))(−r)
≅ T_{ur}(M′_{n}(r))(−r) = ((W (\overline{k}) ⊗_{W} M′_{n})^{φ⊗φ_{r}=1})(−r).

The third isomorphism follows from the exact sequence (1.4.1). By taking the projective limit, we obtain an isomorphism

T′ ≅ (W (\overline{k}) ⊗_{W} (\underleftarrow{lim}_{n} M′_{n}))^{φ⊗φ_{r}=1})(−r).

Let M′ be the projective limit of M′_{n} as S-modules. By taking the projective limit of the surjective homomorphisms M_{n} → M′_{n} of S/p^{n}S-modules, we obtain a surjective homomorphisms M → M′ of S-modules such that the following diagram is commutative:

M ⟶ M′
∩↓  ^{∩}↓
D ⟶ D′,

where D′ denotes the quotient of D corresponding to the quotient D′ of D. Note that D′ and D′ are the objects of \underline{MF}_{K_{0}}(φ, N) and \underline{MF}_{K_{0}}(φ, N) associated to (M′_{n})_{n}. On the other hand, since M′_{n} ⊗_{S,f_{0}} W ≅ M′_{n}, we have M′ ⊗_{S,f_{0}} W ≅ \underleftarrow{lim}_{n} M′_{n}. Hence there exists an injective homomorphism \underleftarrow{lim}_{n} M′_{n} → D′ which makes the following diagram commutative:

M′ ⟶ M′ ⊗_{S_{K_{0}} ,f_{0}} W ≅ \underleftarrow{lim}_{n} M′_{n}
↓  ↓
D′ ⟶ D′ ⊗_{S_{K_{0}} ,f_{0}} K_{0} ≅  D′.

By the definition of M′, the image of \underleftarrow{lim}_{n} M′_{n} in D′ is M′, and φ_{r} on \underleftarrow{lim}_{n} M′_{n} is induced by p^{−r}φ on D′. Thus we obtain an isomorphism:

T′ ≅ ((W (\overline{k}) ⊗_{W} M′)^{φ⊗φ=p^{r}})(−r).

Now it remains to prove that this is compatible with (1.6.1), which is straightforward. □

§2. The maximal slope of log crystalline cohomology. Set s := Spec(k) and \overline{s} := Spec(\overline{k}). Let L be any fine log structure on s, and let \overline{L} denote its inverse image on \overline{s}. We consider a fine log scheme (Y, M_{Y}) smooth and of Cartier type over (s, L) such that Y is proper over s. Let H^{q}((Y, M_{Y})/(W, W (L))) be the crystalline cohomology \underleftarrow{lim}_{n} H_{crys}^{q}((Y, M_{Y})/(W_{n}, W_{n}(L))) defined by Hyodo and Kato in [H-Ka] (3.2), which is a finitely generated W -module endowed with a σ-semi-linear endomorphism φ called the Frobenius. The Frobenius φ becomes bijective after ⊗_{W} K_{0}. In this section, we will construct a canonical decomposition M_{1}⊕M_{2} of H^{q}((Y, M_{Y})/(W, W (L))) stable under φ such that K_{0}⊗_{W} M_{1} (resp. K_{0}⊗_{W} M_{2}) is the direct factor of slope q (resp. slopes< q), and a canonical isomorphism

W (\overline{k}) ⊗_{W} M_{1} ≅ W (\overline{k}) ⊗_{ℤ_{p}} H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{q}).

We will also construct a canonical decomposition M′_{1} ⊕M′_{2} stable under φ such that K_{0}⊗_{W} M′_{1} (resp. K_{0}⊗_{W} M′_{2}) is the direct factor of slope d (resp. slopes< d), and a canonical isomorphism

W (\overline{k}) ⊗_{W} M′_{1} ≅ W (\overline{k}) ⊗_{ℤ_{p}} H_{ét}^{q−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{d}).

Here (\overline{Y} , M\overline{Y}) := (Y, M_{Y})×_{(s,L)} (\overline{s}, \overline{L}). See §2.2 for the definition of the RHS’s.

§2.1. De Rham-Witt complex. First, we will review the de Rham-Witt complex (with log poles) W_{•}ω_{Y/s}^{•} associated to (Y, M_{Y})/(s, L). We don’t assume that Y is proper over s in §2.1. Noting the crystalline description of the de Rham-Witt complex given in [I-R] III (1.5) for a usual smooth scheme over s, we define the sheaf of W_{n}-modules W_{n}ω_{Y/s}^{i} on Y_{ét} to be

σ_{∗}^{n}R^{i}u_{(Y,M_{Y} )/(W_{n},W_{n}(L))∗}O_{(Y,M_{Y} )/(W_{n},W_{n}(L))}

for integers i ≥ 0 and n > 0 ([H-Ka] (4.1)). Here u_{(Y,M_{Y} )/(W_{n},W_{n}(L))} denotes the canonical morphism of topoi: ((Y, M_{Y})/(W_{n}, W_{n}(L)))_{crys}^{∼} → Y_{ét}^{∼}. These sheaves are endowed with the canonical projections π: W_{n+1}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} ([H-Ka] (4.2)) and the differentials d: W_{n}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i+1} ([H-Ka] (4.1)) such that πd = dπ and ((W_{n}ω_{Y/s}^{•}, d)_{n≥1}, π)_{n} becomes a projective system of graded differential algebras over W. The projections π are surjective ([H-Ka] Theorem (4.4)). Furthermore, we have the operators F: W_{n+1}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} and V : W_{n}ω_{Y/s}^{i} → W_{n+1}ω_{Y/s}^{i} ([H-Ka] (4.1)) compatible with the projections π. As in the classical case (cf. [I1] I Introduction), one checks easily that we have (1) F V = V F = p, F dV = d.

(2) F x.F y = F (xy) (x ∈ W_{n}ω_{Y/s}^{i}, y ∈ W_{n}ω_{Y/s}^{j}), xV y = V (F x.y) (x ∈ W_{n+1}ω_{Y/s}^{i}, y ∈ W_{n}ω_{Y/s}^{j}).

(3) V (xdy) = V x.dV y (x ∈ W_{n}ω_{Y/s}^{i}, y ∈ W_{n}ω_{Y/s}^{j}). (The property (3) follows from (1) and (2): V (xdy) = V (x.F dV y) = V x.dV y.) We have W -algebra isomorphisms τ: W_{n}(O_{Y}) →^{∼} W_{n}ω_{Y/s}^{0} (see [H-Ka] (4.9) for the construction of τ and the proof of the isomorphism) compatible with π, F and V, where the projections π and the operators F and V on W_{n}(O_{Y}) are defined in the usual manner. With the notation in [H-Ka] (4.9), C_{Y/W_{n}}^{•} is a complex of quasi-coherent W_{n}(O_{Y})-modules and hence W_{n}ω_{Y/s}^{i} = H^{i}(C_{Y/W_{n}}^{•}) are quasi-coherent W_{n}(O_{Y})-modules. In the special case n = 1, we have an isomorphism W_{1}ω_{Y/s}^{i} ≅ H^{i}(ω_{Y/s}^{•}) ←^{∼}_{C^{−1}}

ω_{Y/s}^{i} ([Ka2] (6.4), (4.12)), which is O_{Y} -linear and compatible with the differentials. Recall that we regard H^{i}(ω_{Y/s}^{•}) as an O_{Y} -module by the action via O_{Y} → O_{Y}; x ↦ x^{p}. We will identify (W_{1}ω_{Y/s}^{•}, d) with (ω_{Y/s}^{•}, d) in the following by this isomorphism. With the notation of [H-Ka] Definition (4.3) and Theorem (4.4) B_{n+1}ω_{Y/s}^{i} ⊕ Z_{n}ω_{Y/s}^{i−1} (resp. B_{1}ω_{Y/s}^{i}) are coherent subsheaves of the coherent O_{Y} -module F_{∗}^{n+1}(ω_{Y/s}^{i} ⊕ω_{Y/s}^{i−1}) (resp. F_{∗}ω_{Y/s}^{i}), the homomorphism (C^{n}, dC^{n}): B_{n+1}ω_{Y/s}^{i} ⊕ Z_{n}ω_{Y/s}^{i−1} → B_{1}ω_{Y/s}^{i} is O_{Y} -linear and the isomorphism ([H-Ka] Theorem (4.4)): (2.1.1)

(V^{n}, dV^{n}):F_{∗}^{n+1}(ω_{Y/s}^{i}⊕ω_{Y/s}^{i−1})/Ker(C^{n}, dC^{n}) →^{∼} Ker(π: W_{n+1}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i})

is O_{Y} -linear, where we regard the right-hand side as an O_{Y} -module via F: O_{Y} = W_{n+1}O_{Y} /V W_{n+1}O_{Y} → W_{n+1}O_{Y} /pW_{n+1}O_{Y}. Especially this implies, by induction on n, that W_{n}ω_{Y/s}^{i} is a coherent W_{n}(O_{Y})-module.

We set W ω_{Y/s}^{i} := \underleftarrow{lim}_{n} W_{n}ω_{Y/s}^{i} and denote also by d, F and V the projective limits of d, F and V for W_{•}ω_{Y/s}^{•}. Then, by the same argument as in the proof of [I1] II Proposition 2.1 (a) using the above coherence, we see that, if Y is proper over k, then H^{j}(Y, W_{n}ω_{Y/s}^{i}) and H^{j}(Y, W_{n}ω_{Y/s}^{•}) are finitely generated W_{n}-modules and the canonical homomorphisms

(2.1.2)  H^{j}(Y, W ω_{Y/s}^{i}) → \underleftarrow{lim}_{n} H^{j}(Y, W_{n}ω_{Y/s}^{i})
(2.1.3)  H^{j}(Y, W ω_{Y/s}^{•}) → \underleftarrow{lim}_{n} H^{j}(Y, W_{n}ω_{Y/s}^{•})

are isomorphisms.

Theorem 2.1.4. ([H-Ka] Theorem (4.19), cf. [I1] II Théorème 1.4). There exists a canonical isomorphism in D^{+}(Y_{ét}, W_{n}):

Ru_{(Y,M_{Y} )/(W_{n},W_{n}(L))∗}O_{(Y,M_{Y} )/(W_{n},W_{n}(L))} ≅ W_{n}ω_{Y/s}^{•}

functorial on (Y, M_{Y}) and compatible with the products, the Frobenius and the transition maps. Here the Frobenius on the RHS is defined by p^{i}F in degree i.

Recall that p: W_{n}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} factors through the canonical projection π: W_{n}ω_{Y/s}^{i} → W_{n−1}ω_{Y/s}^{i} and that the induced homomorphism \underline{p}: W_{n−1}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} is injective ([H-Ka] Corollary (4.5) (1), cf. [I1] I Proposition 3.4). Hence p^{i}F: W_{n}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} is well-defined for i ≥ 1. For i = 0, it is defined by the usual F on W_{n}O_{Y}.

Remark. Strictly speaking, the compatibility with the products is not mentioned in [H-Ka] Theorem (4.19), but one can verify it simply by looking at the construction of the map carefully as follows: We use the notation of the proof in [H-Ka]. We have a PD-homomorphism O_{D_{n}^{·}} → W_{n}(O_{Y^{·}}) and ω_{Z_{n}^{·}/(W_{n},W_{n}(L))}^{1}|_{Y^{·}} → ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L))}^{1}, which induce a morphism of graded algebras C_{n} = O_{D_{n}^{·}} ⊗ ω_{Z_{n}^{·}/(W_{n},W_{n}(L))}^{•} → ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L))}^{•}. By taking the quotient ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L),[ ])}^{•} of the target, we obtain a morphism C_{n} → ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L),[ ])}^{•} of differential graded algebras. The homomorphism W_{n}(O_{Y^{·}}) → W_{n}ω_{Y^{·}/s}^{0} is extended uniquely to a W_{n}(O_{Y^{·}})-linear morphism of differential graded algebras: ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L))}^{•} → W_{n}ω_{Y^{·}}^{•} compatible with d log’s from W_{n}(M^{·}) = M ⊕ Ker(W_{n}(O_{Y^{·}})^{∗} → O_{Y^{·}}^{∗}). It factors through the quotient ω_{W_{n}(Y^{·})/(W_{n},W_{n}(L),[ ])}^{•}. Thus we see that the morphism C_{n} → W_{n}ω_{Y^{·}/s}^{•} constructed in the proof of [H-Ka] Theorem (4.19) is a morphism of differential graded algebras. From the definition, we immediately obtain the following exact sequences (cf. [I1] Remarques 3.21.1):

(2.1.5)  \begin{matrix}W_{2n+k}ω_{Y/s}^{i} ⟶^{F^{n+k}} W_{n}ω_{Y/s}^{i} ⟶^{dV^{k}} W_{n+k}ω_{Y/s}^{i+1}, \\ W_{k}ω_{Y/s}^{i−1} ⟶^{F^{k}dV^{n}} W_{n}ω_{Y/s}^{i} ⟶^{V^{k}} W_{n+k}ω_{Y/s}^{i}, \\ W_{2n+k}ω_{Y/s}^{i} ⟶^{F^{n}} W_{n+k}ω_{Y/s}^{i} ⟶^{F^{k}d} W_{n}ω_{Y/s}^{i+1}, \\ W_{k}ω_{Y/s}^{i} ⟶^{V^{n}} W_{n+k}ω_{Y/s}^{i} ⟶^{F^{k}} W_{n}ω_{Y/s}^{i}. \end{matrix}

For example, the first exact sequence is obtained by considering the following morphism of short exact sequences:

0 ⟶ C_{n+k}^{•} ⟶_{\underline{p}^{n+k}} C_{2n+2k}^{•} ⟶ C_{n+k}^{•} ⟶ 0
∥  ↑^{\underline{p}^{k}}  ↑^{\underline{p}^{k}}
0 ⟶ C_{n+k}^{•} ⟶_{\underline{p}^{n}} C_{2n+k}^{•} ⟶ C_{n}^{•} ⟶ 0,

where C_{n}^{•} is the same complex as in [H-Ka] (4.1). Finally we review a generalization of the Cartier isomorphism to the de Rham-Witt complex (cf. [I-R] III Proposition (1.4)) and the operator V′ (cf. [I-R] III (1.3.2)), which will be used in the proof of our main result in §2. By the exact sequences (2.1.5), F^{n}: W_{2n}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i} induces an isomor-

phism F^{n}: W_{2n}ω_{Y/s}^{i}/V^{n}W_{n}ω_{Y/s}^{i} →^{∼} ZW_{n}ω_{Y/s}^{i}. We define the operator V′ on

ZW_{•}ω_{Y/s}^{i} by the following commutative diagram:

(2.1.6)  \begin{matrix}W_{2n+2}ω_{Y/s}^{i}/V^{n+1}W_{n+1}ω_{Y/s}^{i} & ⟶^{π^{2}} & W_{2n}ω_{Y/s}^{i}/V^{n}W_{n}ω_{Y/s}^{i} \\ ^{≀}↓_{F^{n+1}} & & ^{≀}↓_{F^{n}} \\ ZW_{n+1}ω_{Y/s}^{i} & ⟶^{V′} & ZW_{n}ω_{Y/s}^{i}. \end{matrix}

We see easily that this operator satisfies the relations:

(2.1.7)  V′π = πV′, F V′ = V′F = π^{2}, V′d = dV π^{2}.

The last relation implies V′(BW_{n+1}ω_{Y/s}^{i}) ⊂ BW_{n}ω_{Y/s}^{i} and hence V′ induces a morphism H^{i}(W_{n+1}ω_{Y/s}^{•}) → H^{i}(W_{n}ω_{Y/s}^{•}), which we will also denote by V′. Using the property F^{n}dV^{n} = d, we also see that F^{n} induces an isomorphism

W_{2n}ω_{Y/s}^{i}/(V^{n}W_{n}ω_{Y/s}^{i}+dV^{n}W_{n}ω_{Y/s}^{i−1}) →^{∼} H^{i}(W_{n}ω_{Y/s}^{•}). On the other hand, from

[H-Ka] Theorem (4.4), we can easily derive

(2.1.8) Ker(π^{m}: W_{n+m}ω_{Y/s}^{i} → W_{n}ω_{Y/s}^{i}) = V^{n}W_{m}ω_{Y/s}^{i} + dV^{n}W_{m}ω_{Y/s}^{i−1}

by induction on m (cf. [I1] I Proposition 3.2). Hence F^{n} induces an isomorphism:

(2.1.9)  C^{−n}: W_{n}ω_{Y/s}^{i} →^{∼} H^{i}(W_{n}ω_{Y/s}^{•}),

which we can regard as a generalization of the Cartier isomorphism. Indeed, if n = 1, this coincides with the Cartier isomorphism by the identification W_{1}ω_{Y/s}^{i} = ω_{Y/s}^{i}. (To prove this coincidence, we need d log which will be explained in the next subsection.) We need the following lemma in the proof of Lemma 3.4.4.

Lemma 2.1.10. The composite of the following homomorphisms is the identity.

W_{n}ω_{Y/s}^{r} ⟶_{C^{−n}}^{∼} H^{r}(W_{n}ω_{Y/s}^{•}) ≅^{2.1.4} R^{r}u_{(Y,M_{Y} )/(W_{n},W_{n}(L))∗}O = W_{n}ω_{Y/s}^{r}.

Proof. Denote by α the homomorphism in question. Then α is compatible with the products. Since W_{n}ω_{Y/s}^{r} (r ≥ 2) is generated by x_{1}.x_{2} … x_{r} (x_{i} ∈ W_{n}ω_{Y/s}^{1}) as a sheaf of modules ([H-Ka] Proposition (4.6)), it suffices to prove the lemma in the case r = 0, 1. We use the notation in the proof of [H-Ka] Theorem (4.19). The lemma for r = 0 follows from the fact that the composite of W_{n}O_{Y^{•}} →^{τ} O_{D^{•}} → W_{n}O_{Y^{•}} coincides with F^{n}, where τ(x_{0}, …, x_{n−1}) = ∑_{i=0}^{n−1} p^{i}x̃_{i}^{p^{n−i}}. By [H-Ka] Proposition (4.6), W_{n}ω_{Y/s}^{1} is generated by dx (x ∈ W_{n}O_{Y}) and d log(a) (a ∈ M_{Y}^{gp}) as a W_{n}O_{Y} - module. Using the quasi-isomorphism W_{2n}ω_{Y/s}^{•}/\underline{p}^{n}W_{n}ω_{Y/s}^{•} → W_{n}ω_{Y/s}^{•} ([H-Ka] Corollary (4.5)), we see that α commutes with the differentials. Hence α(dx) = d(α(x)) = dx. For a ∈ M_{Y}^{gp}, we have C^{−n}(d log(a)) = the class of d log(a). On the other hand, by the construction of the quasi-isomorphism of [H-Ka] Theorem (4.19), the following diagram is commutative:

M_{D^{•}}^{gp} ⟶ W_{n}(M_{Y^{•}})^{gp}
d log↓  ^{d log}↓
C_{n}^{1} ⟶ W_{n}ω_{Y^{•}/s}^{1}.

Choose a lifting ã ∈ M_{D^{•}}^{gp} of a and let au, u ∈ Ker(W_{n}O_{Y^{•}}^{∗} → O_{Y^{•}}^{∗}) be its image in W_{n}(M^{•})^{gp}. Then the image of d log(ã) ∈ C_{n}^{1} in W_{n}ω_{Y^{•}/s}^{1} is d log(a) + d(log(u)), which is congruent to d log(a) modulo dW_{n}(O_{Y^{•}}). □

§2.2. Logarithmic Hodge-Witt sheaves. We will review the logarithmic Hodge-Witt sheaves W_{•}ω_{Y/s, log}^{•} associated to (Y, M_{Y})/(s, L) (See [I1] for usual smooth schemes and [H], [L] for log smooth schemes). We still don’t assume that Y is proper over s. As in [H-Ka] (4.9), we have natural homomorphisms

d log: M_{Y}^{gp} → W_{n}ω_{Y}^{1} (n ≥ 1),

which satisfy d log(g^{−1}(L^{gp})) = 0, [b]d log(a) = d([b]) for a ∈ M_{Y}, its image b in O_{Y} and [b] = (b, 0, 0, ⋯) ∈ W_{n}(O_{Y}), πd log = d log, F d log = d log, and d(d log(M_{Y}^{gp})) = 0. For n = 1, d log coincides with the usual d log: M_{Y}^{gp} → ω_{Y/s}^{1}. We define the logarithmic Hodge-Witt sheaves W_{n}ω_{Y/s, log}^{i} to be the subsheaves of abelian groups of W_{n}ω_{Y/s}^{i} generated by local sections of the form d log(a_{1}) ∧ ⋯ ∧ d log(a_{i}) (a_{1}, …, a_{i} ∈ M_{Y}^{gp}).

Theorem 2.2.1. ([I1] 0 Théorème 2.4.2, [Ts1] Theorem (6.1.1)). The following sequence is exact for any integer i ≥ 0:

0 ⟶ ω_{Y/s, log}^{i} ⟶ Zω_{Y/s}^{i} ⟶^{1−C} ω_{Y/s}^{i} ⟶ 0.

Now, by the same argument as the proof of [I1] I (3.26) and [I1] I (5.7.2) plus some additional calculation, we can derive the following theorem from Theorem 2.2.1, the exact sequences (2.1.5) and the isomorphism (2.1.1) (cf. [L] 1.5.2).

Theorem 2.2.2. (cf. [I1] I (3.26), (5.7.2)). For any integers n ≥ 1 and i ≥ 0, the following sequence is exact:

0 ⟶ W_{n}ω_{Y/s, log}^{i} + V^{n−1}W_{1}ω_{Y/s}^{i} ⟶ W_{n}ω_{Y/s}^{i} ⟶^{π−F} W_{n−1}ω_{Y/s}^{i} ⟶ 0.

Note that we easily obtain

(2.2.3)  \begin{matrix}Ker(V^{n}: ω_{Y/s}^{i} → W_{n+1}ω_{Y/s}^{i}) = B_{n}ω_{Y/s}^{i}, \\ Ker(V^{n}: ω_{Y/s}^{i} → W_{n+1}ω_{Y/s}^{i}/dV^{n}ω_{Y/s}^{i−1}) = B_{n+1}ω_{Y/s}^{i} \end{matrix}

(cf. [I1] I (3.8)) from the isomorphism (2.1.1) ([L] Proposition 1.2.7).

Corollary 2.2.4. (cf. [H] (2.6), [L] (1.5.4)). The following sequence is exact for any integers i ≥ 0, n, m ≥ 1:

0 ⟶ W_{n}ω_{Y/s, log}^{i} ⟶^{\underline{p}^{m}} W_{n+m}ω_{Y/s, log}^{i} ⟶ W_{m}ω_{Y/s, log}^{i} ⟶ 0.

Corollary 2.2.5. (cf. [I-R] IV §3). The homomorphism W_{n}ω_{Y/s, log}^{i} → H^{i}(W_{n}ω_{Y/s}^{•}) is injective and the following sequence is exact:

0 ⟶ K_{n}^{i} ⟶ H^{i}(W_{n}ω_{Y/s}^{•}) ⟶^{V′−π} H^{i}(W_{n−1}ω_{Y/s}^{•}) ⟶ 0,

where K_{n}^{i} denotes the image of W_{n}ω_{Y/s, log}^{i} + p^{n−1}F (W_{n+1}ω_{Y/s}^{i}) in H^{i}(W_{n}ω_{Y/s}^{•}).

Proof. This immediately follows from Theorem 2.2.2 using the following commutative diagram:

W_{n}ω_{Y/s}^{i}  ⟶^{π−F} W_{n−1}ω_{Y/s}^{i}
≀↓_{C^{−n}}  ^{≀}↓_{C^{−(n−1)}}
H^{i}(W_{n}ω_{Y/s}^{•}) ⟶^{V′−π} H^{i}(W_{n−1}ω_{Y/s}^{•}).

□ Corollary 2.2.6. (cf. [I-R] IV §3). The homomorphism V′ −π: ZW_{n}ω_{Y/s}^{i} → ZW_{n−1}ω_{Y/s}^{i} is surjective and, if we denote its kernel by L_{n}^{i}, then W_{n}ω_{Y/s, log}^{i} ⊂ L_{n}^{i} and L_{n}^{i}/W_{n}ω_{Y/s, log}^{i} is killed by π^{3}.

Proof. The surjectivity follows from Theorem 2.2.2 using the commutative diagram:

W_{2n}ω_{Y/s}^{i} ⟶^{π^{2}−πF} W_{2n−2}ω_{Y/s}^{i}
F^{n}↓  _{F^{n−1}}↓
ZW_{n}ω_{Y/s}^{i} ⟶^{V′−π} ZW_{n−1}ω_{Y/s}^{i}.

The assertion on the kernel follows from Theorem 2.2.2 and Lemma 2.2.7 below by considering the commutative diagram:

ZW_{n+1}ω_{Y/s}^{i} ⟶^{π−F} ZW_{n}ω_{Y/s}^{i}
∥  ↓_{V′}
ZW_{n+1}ω_{Y/s}^{i} ⟶^{πV′−π^{2}} ZW_{n−1}ω_{Y/s}^{i}.

□

Lemma 2.2.7. The homomorphism V′: ZW_{n+1}ω_{Y/s}^{i} → ZW_{n}ω_{Y/s}^{i} is surjective and its kernel is killed by π^{2}.

Proof. In the diagram (2.1.6), the upper horizontal map is a surjection with its kernel (V^{n}W_{n+2}ω_{Y/s}^{i} + dV^{2n}W_{2}ω_{Y/s}^{i−1})/V^{n+1}W_{n+1}ω_{Y/s}^{i} (2.1.8). Hence V′ is surjective and its kernel is p^{n}F W_{n+2}ω_{Y/s}^{i} + dV^{n−1}W_{2}ω_{Y/s}^{i−1}. □

§2.3. The maximal slope. In §2.3, we always assume that Y is proper over s. For i ∈ ℕ, we define the projective system of morphisms of complexes (cf. [I-R] III (1.7)):

V′_{≤i}: {τ_{≤i}W_{n}ω_{Y/s}^{•} → τ_{≤i}W_{n−1}ω_{Y/s}^{•}}_{n≥1}

by the morphism p^{i−j−1}π^{2}V in degree j ≤ i − 1 and V′: ZW_{n}ω_{Y/s}^{i} → ZW_{n−1}ω_{Y/s}^{i} in degree i. If j ≤ i or i = d := dim Y, then the natural homomorphism H^{j}(Y, τ_{≤i}W_{n}ω_{Y/s}^{•}) → H^{j}(Y, W_{n}ω_{Y/s}^{•}) is an isomorphism and hence V′_{≤i} induces an endomorphism on H^{j}(Y, W ω_{Y/s}^{•}), which we will also denote by V′_{≤i}. We need the following lemma, which is well-known for a σ-semilinear endomorphism and is proven in the same way as in the σ-semilinear case.

Lemma 2.3.1. Let M be a finitely generated W -module and let V be a σ^{−1}- semilinear endomorphism of M.

(1) There exists a unique decomposition M_{bij} ⊕ M_{nil} of M stable under V such that V is bijective on M_{bij} and p-adically nilpotent on M_{nil}.

(2) If k is algebraically closed, V − 1 is surjective on M and bijective on M_{nil}. Furthermore, the natural homomorphism:

W ⊗_{ℤ_{p}} M^{V =1} = W ⊗_{ℤ_{p}} (M_{bij})^{V =1} → M_{bij}

is bijective.

Recall that we have canonical isomorphisms:

H^{j}(Y, W_{n}ω_{Y/s}^{•}) ≅ H_{crys}^{j}((Y, M_{Y})/(W_{n}, W_{n}(L)))
H^{j}(Y, W ω_{Y}^{•}) ≅ H_{crys}^{j}((Y, M_{Y})/(W, W (L)))

by Theorem 2.1.4 and the right hand sides are finitely generated modules over W_{n} and over W respectively. By applying Lemma 2.3.1 to H^{j}(Y, W ω_{Y/s}^{•}) and V′_{≤i} for i ≥ j or i = d, we obtain a decomposition

(2.3.2)  H^{j}(Y, W ω_{Y/s}^{•}) = H^{j}(Y, W ω_{Y/s}^{•})_{V′_{≤i}−bij} ⊕ H^{j}(Y, W ω_{Y/s}^{•})_{V′_{≤i}−nil}

and a natural isomorphism

(2.3.3)  W ⊗_{ℤ_{p}} H^{j}(Y, W ω_{Y/s}^{•})^{V′_{≤i}=1} →^{∼} H^{j}(Y, W ω_{Y/s}^{•})_{V′_{≤i}−bij}

if k is algebraically closed.

Lemma 2.3.4. For any integers i and j such that i ≥ j or i = d, we have

K_{0} ⊗_{W} H^{j}(Y, W ω_{Y/s}^{•})_{V′_{≤i}−bij} = (K_{0} ⊗_{W} H^{j}(Y, W ω_{Y/s}^{•}))_{[i]},
K_{0} ⊗_{W} H^{j}(Y, W ω_{Y/s}^{•})_{V′_{≤i}−nil} = (K_{0} ⊗_{W} H^{j}(Y, W ω_{Y/s}^{•}))_{[0,i[}.

(See §1.1 for the definition of D_{I} (I ⊂ ℚ) for an F -isocrystal D over k.)

Proof. Let F denote the morphism τ_{≤i}W_{n}ω_{Y/s}^{•} → τ_{≤i}W_{n−1}ω_{Y/s}^{•} whose degree q-part is p^{q}F, which induces the Frobenius endomorphism φ on H^{j}(Y, W ω_{Y/s}^{•}). Then FV′_{≤i} = V′_{≤i}F = p^{i}π^{2} : τ_{≤i}W_{n}ω_{Y/s}^{•} → τ_{≤i}W_{n−2}ω_{Y/s}^{•}. Hence, we have φV′_{≤i} = V′_{≤i}φ = p^{i} on H^{j}(Y, W ω_{Y/s}^{•}), which implies the lemma. □

We set H^{j}(Y, W ω_{Y/s, log}^{i}) := \underleftarrow{lim}_{n} H^{j}(Y, W_{n}ω_{Y/s, log}^{i}).

Proposition 2.3.5. Assume that k is algebraically closed. Then, for any integers i and j, we have

H^{0}(Y, W ω_{Y/s, log}^{i}) ⟶^{∼} H^{i}(Y, W ω_{Y/s}^{•})^{V′_{≤i}=1},
H^{j}(Y, W ω_{Y/s, log}^{d}) ⟶^{∼} H^{j+d}(Y, W ω_{Y/s}^{•})^{V′_{≤d}=1}.

Proof. First note that V π is a nilpotent endomorphism of W_{n}ω_{Y/s}^{i}. By Corollary 2.2.6, the morphism of complexes

V′_{≤i} − π: τ_{≤i}W_{n}ω_{Y/s}^{•} ⟶ τ_{≤i}W_{n−1}ω_{Y/s}^{•}

is surjective, and if we denote its kernel by K_{i,n}^{•}, K_{i,n}^{j} = Ker(π: W_{n}ω_{Y/s}^{j} → W_{n−1}ω_{Y/s}^{j}) if j ≤ i−1, W_{n}ω_{Y/s, log}^{i} ⊂ K_{i,n}^{i} and K_{i,n}^{i}/W_{n}ω_{Y/s, log}^{i} is annihilated by π^{3}. Hence we have a long exact sequence

⋯ → H^{j}(Y, K_{n,i}^{•}) ⟶ H^{j}(Y, τ_{≤i}W_{n}ω_{Y/s}^{•}) ⟶^{V′_{≤i}−π} H^{j}(Y, τ_{≤i}W_{n−1}ω_{Y/s}^{•}) ⟶ ⋯

and the natural homomorphism

H^{j−i}(Y, W ω_{Y/s, log}^{i}) ⟶ \underleftarrow{lim}_{n} H^{j}(Y, K_{n,i}^{•})

is bijective. By Lemma 2.3.1 (2), if j ≤ i or i = d, the endomorphism V′_{≤i} − 1 on H^{j}(Y, W ω_{Y/s}^{•}) is surjective. On the other hand, we have the following morphism of short exact sequences:

0 ⟶ H^{j}(Y/W)/p^{n} ⟶ H^{j}(Y/W_{n}) ⟶ H^{j+1}(Y/W)_{p^{n}} ⟶ 0
↑^{proj}  ↑^{proj}  ↑^{p}
0 ⟶ H^{j}(Y/W)/p^{n+1} ⟶ H^{j}(Y/W_{n+1}) ⟶ H^{j+1}(Y/W)_{p^{n+1}} ⟶ 0,

where we abbreviate (Y, M_{Y})/(W, W (L)) or (W_{n}, W_{n}(L)) to Y/W or W_{n}. Hence, if the torsion part of H^{j+1}((Y, M_{Y})/(W, W (L))) is killed by p^{ν_{j+1}}, then the cokernel of the homomorphism

V′_{≤i} − π: H^{j}(Y, W_{n}ω_{Y/s}^{•}) → H^{j}(Y, W_{n−1}ω_{Y/s}^{•})

is killed by π^{ν_{j+1}} if j ≤ i or i = d. By taking \underleftarrow{lim}_{n} of the above long exact sequences, we obtain

\underleftarrow{lim}_{n} H^{i}(Y, K_{n,i}^{•}) →^{∼} H^{i}(Y, W ω_{Y/s}^{•})^{V′_{≤i}=1}
\underleftarrow{lim}_{n} H^{j}(Y, K_{n,d}^{•}) →^{∼} H^{j}(Y, W ω_{Y/s}^{•})^{V′_{≤d}=1}.

□

We also need the following lemma (in (3.4.5)):

Lemma 2.3.6. Assume that k is algebraically closed. Then, for any integers i ≥ 0 and j ≥ 0, we have an isomorphism

H^{j}(Y, W ω_{Y/s, log}^{i}) ⟶^{∼} \underleftarrow{lim}_{n} H^{j}(Y, H^{i}(W_{n}ω_{Y/s}^{•}))^{V′=1}.

Proof. Set M_{n}^{j} := H^{j}(Y, H^{i}(W_{n}ω_{Y/s}^{•})), M^{j} := \underleftarrow{lim}_{n} M_{n}^{j} and let M_{n}^{j′} be the image of M^{j} in M_{n}^{j}. By (2.1.9), M_{n}^{j} are finitely generated W_{n}-modules and hence {(V′ −π)(M_{n+1}^{j})}_{n≥1} satisfies the Mittag-Leffler condition. On the other hand, by Lemma 2.3.7 below, (V′ − π)(M_{n+1}^{j′}) = M_{n}^{j′}, which implies

\underleftarrow{lim}_{n}((V′ − π)(M_{n+1}^{j})) ⊃ \underleftarrow{lim}_{n}((V′ − π)(M_{n+1}^{j′})) = \underleftarrow{lim}_{n} M_{n}^{j′} = \underleftarrow{lim}_{n} M_{n}^{j}.

Hence \underleftarrow{lim}_{n}(M_{n}^{j}/(V′ − π)(M_{n+1}^{j})) = 0. Since \{M_{n}^{j−1}/(V′ − π)(M_{n+1}^{j−1})\}_{n≥1} satisfies the Mittag-Leffler condition, Corollary 2.2.5 implies \underleftarrow{lim} H^{j}(Y, K_{n}^{i}) ≅ (M^{j})^{V′=1}. Since π(p^{n−1}F W_{n+1}ω_{Y/s}^{i}) = 0, the LHS is isomorphic to H^{j}(Y, W ω_{Y/s, log}^{i}). □

Lemma 2.3.7. Let M_{1} and M_{2} be W -modules of finite length, let π: M_{1} → M_{2} be a surjective W -linear homomorphism and let V′: M_{1} → M_{2} be a σ^{−1}- linear homomorphism. If k is algebraically closed, then V′ − π: M_{1} → M_{2} is surjective.

Proof. Using the short exact sequences 0 → pM_{i} → M_{i} → M_{i}/pM_{i} → 0 (i = 1, 2), we are easily reduced to the case pM_{i} = 0 (i = 1, 2). In this case, π has a W -linear section s: M_{2} → M_{1} and V′ ∘ s − π ∘ s = V′ ∘ s − 1 is surjective by Lemma 2.3.1 (2). □

§3. The maximal unramified quotient of p-adic étale cohomology.

§3.1. Statement of the main theorem. Let (S, N) be the scheme Spec(O_{K}) endowed with the canonical log structure (i.e. the log structure defined by its closed point). Let f: (X, M) → (S, N) be a smooth fs(=fine and saturated) log scheme and let g: (Y, M_{Y}) → (s, L) be the reduction of f modulo the maximal ideal of O_{K}. We assume that X is proper over S and that f is universally saturated, which is equivalent to saying that g is of Cartier type, or also to saying that Y is reduced ([Ts3]). Let X_{triv} denote the locus where the log structure M is trivial, which is open and contained in the generic fiber of X. Let (\overline{s}, \overline{L}) be the scheme Spec(\overline{k}) endowed with the inverse image of L, and set (\overline{Y} , M\overline{Y}) := (Y, M_{Y}) ×_{(s,L)} (\overline{s}, \overline{L}). Set (X_{triv})\overline{K} := X_{triv} ×_{Spec(K)} Spec(\overline{K}). We will describe the maximal unramified quotients

H_{ét}^{r}((X_{triv})\overline{K}, ℚ_{p}(r))_{I_{K}} , H_{ét}^{r}((X_{triv})\overline{K}, ℚ_{p}(d))_{I_{K}} (r ≥ d) of p-adic étale cohomology groups and the images of H_{ét}^{r}((X_{triv})\overline{K}, ℤ_{p}(r′)) (r′ = r or d) in them in terms of the logarithmic Hodge-Witt sheaves of (\overline{Y} , M\overline{Y})/(\overline{s}, \overline{L}) (Theorem 3.1.11). In the rest of §3.1, we choose and fix an integer r ≥ 0, and assume that (X, M)/(S, N) and r satisfy one of the following conditions: (3.1.1) r ≤ p − 2. (3.1.2) Étale locally on X, there exists an étale morphism over S:

X → Spec((O_{K}[T_{1}, …, T_{u}]/(T_{1} ⋯ T_{u} − π^{e}))[U_{1}, …, U_{s}, V_{1}, …, V_{t}])

for some integers u ≥ 1, s, t ≥ 0 and e ≥ 1 such that e | [K : K_{0}] and X_{triv} = X − (Y ∪ D), where D is the inverse image of {U_{1} ⋯ U_{s} = 0}. Let i and j denote the immersions Y → X and X_{triv} → X respectively. By [Ka4]Theorem (11.6), we have

(3.1.3)  M = O_{X} ∩ j_{∗}O_{X_{triv}}^{∗} and M^{gp} = j_{∗}O_{X_{triv}}^{∗}

and, hence, from the Kummer sequence on (X_{triv})_{ét}, we obtain a symbol map:

(3.1.4)  (i^{∗}M^{gp})^{⊗r} ⟶ i^{∗}R^{r}j_{∗}ℤ/p^{n}ℤ(r).

Theorem 3.1.5. ([Bl-Ka]Theorem (1.4), [H](1.6.1), [Ts2], [Ts4]). The homomorphism (3.1.4) is surjective.

Proof. In the case (3.1.2) with s = 0, this is [H](1.6.1) (=[Bl-Ka]Theorem (1.4) in the good reduction case). The case (3.1.2) for a general s is reduced to the case s = 0 as in the proof of [Ts2] Lemma 3.4.7. (The proof of Lemma 3.4.7 (1) works without the assumption μ_{p^{n}} ⊂ K). In the case (3.1.1), we are easily reduced to the case n = 1, and the theorem follows from [Ts3]Theorem 5.1 and Proposition A15 with r = q. □

We have a surjective homomorphism (§2.2):

(3.1.6) (M_{Y}^{gp})^{⊗r} ⟶ W_{n}ω_{Y/s, log}^{r}; a_{1} ⊗ ⋯ ⊗ a_{r} ↦ d log(a_{1}) ∧ ⋯ ∧ d log(a_{r}).

Proposition 3.1.7. (cf. [Bl-Ka]Theorem (1.4) (i), [H](1.6.2)). There exists a unique surjective homomorphism:

(3.1.8)  i^{∗}R^{r}j_{∗}ℤ/p^{n}ℤ(r) ⟶ W_{n}ω_{Y/s, log}^{r}

such that the following diagram commutes:

(i^{∗}M^{gp})^{⊗r} ⟶ i^{∗}R^{r}j_{∗}ℤ/p^{n}ℤ(r)
↓  ↓
(M_{Y}^{gp})^{⊗r} ⟶ W_{n}ω_{Y/s, log}^{r}.

Proof. We have the required map ν^{∗}(i^{∗}R^{r}j_{∗}ℤ/p^{n}ℤ(r)) → ν^{∗}(W_{n}ω_{Y/s, log}^{r}) for each generic point ν: η → Y of Y ([Bl-Ka] (6.6)). Note that Y is reduced. Since the homomorphism ω_{Y/s}^{r} → ⊕_{ν}ν_{∗}ν^{∗}ω_{Y/s}^{r} is injective, the homomorphism W_{n}ω_{Y/s, log}^{r} → ⊕_{ν}ν_{∗}ν^{∗}W_{n}ω_{Y/s, log}^{r} is also injective by Corollary 2.2.4. Now the proposition follows from Theorem 3.1.5 (cf. [Bl-Ka] (6.6)). □

The condition (3.1.1) or (3.1.2) still holds for the base change of (X, M)/(S, N) by any finite extension of K contained in \overline{K}. Hence, by taking the inductive limit, we obtain:

(3.1.9)  \overline{i}^{∗}R^{r}\overline{j}_{∗}ℤ/p^{n}ℤ(r) ⟶ W_{n}ω_{\overline{Y} /\overline{s},log}^{r}

Here \overline{X} = X ×_{Spec(O_{K})} Spec(O\overline{K}), and \overline{i} and \overline{j} denote the morphisms \overline{Y} → \overline{X} and (X_{triv})\overline{K} → \overline{X} respectively. Note that, for any fs log structure L′ on s and a morphism (s, L′) → (s, L), W_{n}ω^{•} and W_{n}ω_{log}^{•} associated to (Y, M_{Y})/(s, L) and (Y, M_{Y}) ×_{(s,L)} (s, L′)/(s, L′) coincide by the base change theorem [H-Ka]Proposition (2.23) (or [Ts2] Proposition 4.3.1). Let d = dim X_{K}. Then, for any affine scheme U étale over (X_{triv})\overline{K}, we have H^{i}(U, ℤ/p^{n}ℤ(d)) = 0 (i > d). Hence we have \overline{i}^{∗}R^{i}\overline{j}_{∗}ℤ/p^{n}ℤ(d) = 0 (i > d). By the proper base change theorem, we obtain from (3.1.9) homomorphisms:

(3.1.10)

H_{ét}^{r}((X_{triv})\overline{K}, ℤ/p^{n}ℤ(r)) ⟶ H_{ét}^{0}(\overline{Y} , W_{n}ω_{\overline{Y} /\overline{s},log}^{r}),
H_{ét}^{r}((X_{triv})\overline{K}, ℤ/p^{n}ℤ(d)) ⟶ H_{ét}^{r−d}(\overline{Y} , W_{n}ω_{\overline{Y} /\overline{s},log}^{d})

Theorem 3.1.11. (1) (cf. [Sat] Lemma 3.3). Assume K = K_{0} in the case (3.1.1) and s = 0 in the case (3.1.2). Then the homomorphisms (3.1.10) induce isomorphisms: (3.1.12)

H_{ét}^{r}((X_{triv})\overline{K}, ℚ_{p}(r))_{I_{K}} ⟶^{∼} ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s} ,log}^{r})
H_{ét}^{r}((X_{triv})\overline{K}, ℚ_{p}(d))_{I_{K}} ⟶^{∼} ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{r−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s} ,log}^{d})  if r ≥ d

(2) In the case (3.1.1), if K = K_{0}, (3.1.10) induce isomorphisms: (3.1.13)

H_{ét}^{r}((X_{triv})\overline{K}, ℤ_{p}(r))_{I_{K}} /tor ⟶^{∼} H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s}, log}^{r})
H_{ét}^{r}((X_{triv})\overline{K}, ℤ_{p}(d))_{I_{K}} /tor ⟶^{∼} H_{ét}^{r−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s}, log}^{d})/tor  if r ≥ d

Remark 3.1.14. For (1) in the case (3.1.2), if we use [Y] (resp. Theorem 4.1.2), the proof of Theorem 3.1.11 in 3.2–3.4 works without the assumption s = 0 (resp. under the assumption (4.1.1)). See Remark 3.2.3.

§3.2. Review of the semi-stable conjecture. We will review comparison theorems between p-adic étale cohomology and crystalline cohomology.

Let f: (X, M) → (S, N) and g: (Y, M_{Y}) → (s, L) be the same as in §3.1. Then the crystalline cohomology D^{q} := K_{0} ⊗_{W} \underleftarrow{lim}_{n} H^{q}((Y, M_{Y})/(W_{n}, W_{n}(L)) ([H-Ka] (3.2)) is a finite dimensional K_{0}-vector space endowed with a σ-semi-linear automorphism φ and a linear endomorphism N ([H-Ka] (3.4), (3.5), (3.6)) satisfying the relation Nφ = pφN. We choose and fix a uniformizer π of K. Then there exists a canonical isomorphism ([H-Ka] Theorem (5.1)):

(3.2.1)  ρ_{π}: K ⊗_{K_{0}} D^{q} ≅ H^{q}(X_{K}, Ω_{X_{K}}^{•} (log M_{K})).

Using the Hodge filtration on the RHS, the crystalline cohomology D^{q} becomes an object of \underline{MF}_{K}(φ, N) (§1.2). Set V^{q} := H_{ét}^{q}((X_{triv})\overline{K}, ℚ_{p}), which is a finite dimensional ℚ_{p}-vector space endowed with a continuous and linear action of G_{K}.

Theorem 3.2.2. (The semi-stable conjecture by Fontaine-Jannsen:[Ka3], [Ts2]). Assume that (X, M)/(S, N) satisfies the condition (3.1.2) with s = 0. Then, for any integer q ≥ 0, V^{q} is a semi-stable p-adic representation and there exists a canonical isomorphism D_{st}(V^{q}) ≅ D^{q} in \underline{MF}_{K}(φ, N). Here we define D_{st} using the same uniformizer π as (3.2.1).

Remark 3.2.3. G. Faltings ([Fa]) proved the theorem without the assumption on (X, M)/(S, N). However his construction of the comparison map is different from that in [Ka3], [Ts2] via syntomic cohomology, and we will use the latter construction in the proof of Theorem 3.1.11. Recently, G. Yamashita [Y] proved that the comparison map via syntomic cohomology is an isomorphism for any (X, M) satisfying (3.1.2). We give an alternative proof in §4 when the horizontal divisors at infinity do not have self-intersections. To prove Theorem 3.1.11 (2), we need the following refinement by C. Breuil for the integral p-adic étale cohomology T^{q} := H_{ét}^{q}((X_{triv})\overline{K}, ℤ_{p})/tor. We assume K = K_{0} and q ≤ p − 2. Let (E_{n}, M_{E_{n}}) be the scheme Spec(W_{n}⟨u⟩) = Spec(W_{n}⟨u − p⟩) endowed with the log structure associated to ℕ → W_{n}⟨u⟩; 1 ↦ u. We have a closed immersion i_{p}: (S_{n}, N_{n}) ↪ (E_{n}, M_{E_{n}}) defined by u ↦ p. Then the crystalline cohomology M_{n}^{q} := H^{q}((X_{n}, M_{n})/(E_{n}, M_{E_{n}})) ≅ H^{q}((X_{1}, M_{1})/(E_{n}, M_{E_{n}})) is naturally regarded as an object of \underline{MF}_{W,[0,q],tor}(φ, N) ([Br3] Théorème 2.3.2.1). Set M^{q} := (\underleftarrow{lim}_{n} M_{n}^{q})/tor and M_{n}^{′q} := M^{q}/p^{n}M^{q}. Then \{M_{n}^{′q}\}_{n} becomes a projective system of objects of \underline{MF}_{W,[0,p−2],tor}(φ, N) satisfying the condition in the beginning of §1.6 ([Br3] 4.1).

Theorem 3.2.4. ([Br3] Théorème 3.2.4.7, §4.2, [Ts4]). Assume K = K_{0} and let q be any integer such that 0 ≤ q ≤ p − 2. Then there exist canonical G_{K}-equivariant isomorphisms:

T_{st}(M_{n}^{q}) ≅ H_{ét}^{q}((X_{triv})\overline{K}, ℤ/p^{n}ℤ), T_{st}(M_{n}^{′q}) ≅ T^{q}/p^{n}T^{q}

The object of \underline{MF}_{K}(φ, N) associated to the projective system \{M_{n}^{′q}\}_{n≥0} (§1.6) is canonically isomorphic to D^{q} ([Br3] Proposition 4.3.2.3 and the remark after Corollaire 4.3.2.4). Hence, Theorem 3.2.4 implies (§1.5):

Theorem 3.2.5. ([Br3]Corollaire 4.3.2.4 and the following remark). If K = K_{0}, for any integer 0 ≤ q ≤ p − 2, the p-adic étale cohomology V^{q} is a semi-stable p-adic representation and there exists a canonical isomorphism D_{st}(V^{q}) ≅ D^{q} in \underline{MF}_{K}(φ, N)

By Corollary 1.3.3, Lemma 2.3.4 and Proposition 2.3.5, we obtain the following isomorphisms form Theorem 3.2.2 (resp. 3.2.5) for an integer r ≥ 0 such that the condition (3.1.2) with s = 0 is satisfied (resp. the condition (3.1.1) is satisfied and K = K_{0}):

(3.2.6)  V^{r}(r)_{I_{K}} ≅ ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{r})
(3.2.7)  V^{r}(d)_{I_{K}} ≅ ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{r−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{d})  if r ≥ d.

Here d = dim(X_{K}). See the proof of Lemma 2.3.4 for the relation between φ and V′_{≤i}. In the case (3.1.2), if we use the result in [Y] (resp. Theorem 4.1.2), we obtain the isomorphisms without the assumption s = 0 (resp. under the condition (4.1.1)). In the case (3.1.1) and K = K_{0}, the image of M^{r} under the projection M^{r} → D^{r} given by u ↦ 0 coincides with the image of H^{r}((Y, M)/(W, W (L)))(≅ H^{r}(Y, W ω_{Y/s}^{•})) ([Br3] Proposition 4.3.1.3). Hence, by Corollary 1.3.3, Theorem 1.6.2, (2.3.2), Lemma 2.3.4, Proposition 2.3.5 and Theorem 3.2.4, we see that the isomorphisms (3.2.6) and (3.2.7) induce isomorphisms:

(3.2.8)  T^{r}(r)_{I_{K}} /tor ≅ H_{ét}^{0}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{r})
(3.2.9)  T^{r}(d)_{I_{K}} /tor ≅ H_{ét}^{r−d}(\overline{Y} , W ω_{\overline{Y} /\overline{s},log}^{d})/tor if r ≥ d.

To prove Theorem 3.1.11, it remains to prove that (3.2.6) and (3.2.7) are induced by (3.1.10).

§3.3. Review of the construction of the comparison map. We will review the construction of the comparison map in Theorems 3.2.2 and 3.2.5. First recall that to give an isomorphism D_{st}(V^{q}) ≅ D^{q} is equivalent to give a B_{st}-linear isomorphism:

(3.3.1)  B_{st} ⊗_{ℚ_{p}} V^{q} ⟶^{∼} B_{st} ⊗_{K_{0}} D^{q}

compatible with φ, N, the actions of G_{K} and F il^{·} after B_{dR}⊗_{B_{st}}. Recall also that we fixed a uniformizer π of K (in the case K = K_{0}, we choose p as π) in order to define the functor D_{st} (or equivalently the embedding B_{st} ↪ B_{dR}) and to define the filtration on K ⊗_{K_{0}} D^{q}. Let (E_{n}, M_{E_{n}}) be the PD-envelope of the exact closed immersion (S_{n}, N_{n}) ↪ (Spec(W_{n}[u]), L(u)) defined by u ↦ π compatible with the canonical PDstructure on pW_{n}. Here L(u) denotes the log structure defined by ℕ → W [u]; 1 ↦ u. Since (E_{n}, M_{E_{n}}) is isomorphic to the PD-envelope of (S_{1}, N_{1}) ↪ (Spec(W_{n}[u]), L(u)), the lifting of Frobenius of (Spec(W_{n}[u]), L(u)) defined by σ: W_{n} → W_{n} and u ↦ u^{p} induces that of (E_{n}, M_{E_{n}}), which is compatible with the PD-structure \overline{δ} on the ideal \overline{J}_{E_{n}} of O_{E_{n}} defining S_{1}. We denote by i_{E_{n},π} the canonical exact closed immersion (S_{n}, N_{n}) ↪ (E_{n}, M_{E_{n}}). We have Γ(E_{n}, O_{E_{n}}) = W [u, \frac{u^{en}}{n!} (n ∈ ℕ)]/p^{n} (e = [K : K_{0}]) and (E_{n}, M_{E_{n}}) coincides with the log scheme appearing before Theorem 3.2.4 when K = K_{0} and π = p. Let W_{n}(L) be the “Teichmüller lifting” ([H-Ka] Definition (3.1)) of the log structure L on s to Spec(W_{n}), which already appeared in the definition of D^{q}. Then, we have a closed immersion i_{E_{n},0} : (Spec(W_{n}), W_{n}(L)) ↪ (E_{n}, M_{E_{n}}) defined by u ↦ 0, which is compatible with the lifting of Frobenius. First we review the crystalline interpretation of (B_{st}^{+} ⊗_{K_{0}} D)^{N=0}. We set R_{E,ℚ_{p}} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} Γ(E_{n}, O_{E_{n}}). We define the crystalline cohomology D^{q} to be

ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} H_{crys}^{q}((X_{n}, M_{n})/(E_{n}, M_{E_{n}}, \overline{J}_{E_{n}}, \overline{δ}))

≅ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} H_{crys}^{q}((X_{1}, M_{1})/(E_{n}, M_{E_{n}}, \overline{J}_{E_{n}}, \overline{δ})),

which is an R_{E,ℚ_{p}}-module endowed with φ and N satisfying Nφ = pφN ([Ts2] 4.3). The projection pr_{0} : D^{q} → D^{q} induced by the exact closed immersions {i_{E_{n},0}}, which is compatible with φ, N, has a unique K_{0}-linear section s: D^{q} → D^{q} compatible with φ and N, and it induces an isomorphism R_{E,ℚ_{p}} ⊗_{K_{0}} D^{q} →^{∼} D^{q} ([H-Ka] Lemma (5.2), [Ts2] Propositions 4.4.6, 4.4.9). We define H_{crys}^{q}((\overline{X}_{n}, \overline{M}_{n})/(E_{n}, M_{E_{n}})) to be the inductive limit of H_{crys}^{q}((X′_{n}, M′_{n})/(E_{n}, M_{E_{n}})), where (X′, M′) ranges over the base changes of (X, M) by all finite extensions K′ of K contained in \overline{K}. We define \overline{D}^{q} to be ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} H_{crys}^{q}((\overline{X}_{n}, \overline{M}_{n})/(E_{n}, M_{E_{n}})), which is also endowed with φ and N ([Ts2] 4.3). In the special case (X, M) = (S, N), we have D^{q} = 0 (q > 0) ([Ka3] Proposition (3.1), [Ts2] Lemma 1.6.7) and B̂_{st}^{+} = D^{0} by definition ([Br1]§2). Furthermore, there exists a canonical isomorphism (the crystalline

interpretation of B_{st}^{+}) ι: B_{st}^{+} ≅ B̂_{st}^{+}^{N-nil} compatible with the actions of G_{K}, φ and N ([Ka3] Theorem (3.7)). Here N-nil denotes the part where N is nilpotent. Let us return to a general (X, M). Then we have a Künneth isomorphism B̂_{st}^{+} ⊗_{R_{E,ℚp}} D^{q} →^{∼} \overline{D}^{q} ([Ka3]§4, [Ts2]Proposition 4.5.4). By taking N = 0, we obtain the following crystalline interpretation of (B_{st}^{+} ⊗_{K_{0}} D^{q})^{N=0} ([Ka3]§4, [Ts2]§4.5).

(3.3.2) (B_{st}^{+} ⊗_{K_{0}} D^{q})^{N=0} ⟶_{ι⊗s}^{∼} (B̂_{st}^{+} ⊗_{R_{E,ℚp}} D^{q})^{N=0} ⟶^{∼} (\overline{D}^{q})^{N=0}.

To compare \overline{D}^{q} with V^{q}, we use syntomic cohomology. The syntomic complex S_{n}^{∼}(r)_{(X,M)} ([Ts2]§2.1) is an object of D^{+}(Y_{ét}, ℤ/p^{n}ℤ) such that there exists a canonical distinguished triangle:

(3.3.3) → S_{n}^{∼}(r)_{(X,M)} → Ru_{(X_{n},M_{n})/W_{n}∗}J_{(X_{n},M_{n})/W_{n}}^{[r]}

⟶^{p^{r}−φ} Ru_{(X_{n},M_{n})/W_{n}∗}O_{(X_{n},M_{n})/W_{n}},

where u_{(X_{n},M_{n})/W_{n}} denotes the canonical morphism of topoi ((X_{n}, M_{n})/W_{n})_{crys}^{∼} → (X_{n})_{ét}^{∼}. We define the syntomic cohomology H^{q}(\overline{Y} , S_{ℚ_{p}}(r)_{(\overline{X},\overline{M})}) to be ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n}(\underrightarrow{lim}_{K′} H_{ét}^{q}(Y′, S_{n}^{∼}(r)_{(X′,M′)})), where K′ ranges over all finite extensions of K contained in \overline{K} and (X′, M′) denotes the base change of (X, M) by O_{K} → O_{K′}. From the above distinguished triangle (3.3.3), we obtain a natural homomorphism

(3.3.4) H^{q}(\overline{Y} , S_{ℚ_{p}}(r)_{(\overline{X},\overline{M})}) →

(ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n} H_{crys}^{q}((\overline{X}_{n}, \overline{M}_{n})/W_{n}))^{φ=p^{r}} → (\overline{D}^{q})^{N=0,φ=p^{r}}.

On the other hand, if we set ℤ/p^{n}ℤ(r)′ = (\frac{1}{p^{a}a!}ℤ_{p}(r))/p^{n} (r = a(p − 1) + b, a ≥ 0, 0 ≤ b ≤ p − 2), we have a canonical map

(3.3.5)  S_{n}^{∼}(r)_{(X,M)} ⟶ i^{∗}Rj_{∗}ℤ/p^{n}ℤ(r)′

in D^{+}(Y_{ét}, ℤ/p^{n}ℤ) ([Ts2] §3.1), which induces a homomorphism

(3.3.6)  H^{q}(\overline{Y} , S_{ℚ_{p}}(r)_{(\overline{X},\overline{M})}) ⟶ V^{q}(r)

by taking H_{ét}^{q}(Y, −) and the inductive limit with respect to all finite base changes of (X, M). We do not identify ℤ/p^{n}ℤ(r)′ with ℤ/p^{n}ℤ(r) by the multiplication by p^{a}a! because (3.3.5) is compatible with the products and the Tate twist if we use the natural maps ℤ/p^{n}ℤ(r)′ ⊗ ℤ/p^{n}ℤ(s)′ → ℤ/p^{n}ℤ(r + s)′ and ℤ/p^{n}ℤ(r)′(1) → ℤ/p^{n}ℤ(r + 1)′.

Theorem 3.3.7. ([Ku], [Ka3]Theorem (5.4), [Ts2]Theorem 3.4.4, [Ts4]Theorem 5.1). Assume (X, M)/(S, N) and r satisfy one of the conditions (3.1.1) and (3.1.2). Then the homomorphism (3.3.6) is an isomorphism if q ≤ r or r = d = dim X_{K}.

For the case r = d, note \overline{i}^{∗}R^{i}\overline{j}_{∗}ℤ/p^{n}ℤ(d) = 0 (i > d), H^{i}(S_{n}^{∼}(d)_{(\overline{X},\overline{M})}) = 0 (i > d + 1) ([Ts2] (2.3.3) and Lemma 2.3.4), and p^{N}H^{d+1}(S_{n}^{∼}(d)_{(\overline{X},\overline{M})}) = 0 for some N > 0 independent of n ([Ts2] Lemma 2.3.19: H^{d+1}(C_{n}(d)) = 0 and the proof of [Ts2] Theorem 2.3.2). Thus, under the condition (3.1.1) or (3.1.2), we obtain a homomorphism

(3.3.8) V^{q}(r) ⟵_{p^{−r}·(3.3.6)}^{∼} H^{q}(\overline{Y} , S_{ℚ_{p}}(r)_{(\overline{X},\overline{M})})

⟶_{(3.3.4)} (\overline{D}^{q})^{N=0,φ=p^{r}} ⟵_{(3.3.2)}^{∼} (B_{st}^{+} ⊗_{K_{0}} D^{q})^{N=0,φ=p^{r}}.

if q ≤ r or r = d. By using ℚ_{p}(−r) ⊂ B_{st}, we obtain a homomorphism

(3.3.9)  B_{st} ⊗_{ℚ_{p}} V^{q} ⟶ B_{st} ⊗_{K_{0}} D^{q},

which is independent of r ([Ts2] Corollary 4.8.8 and Lemma 4.9.1). If K = K_{0} in the case (3.1.1) and s = 0 in the case (3.1.2), then (3.3.9) gives the required isomorphism D_{st}(V^{q}) ≅ D^{q}. In the case (3.1.2), the result of [Y] (resp. Theorem 4.1.2) says that (3.3.9) still gives the isomorphism without the condition s = 0 (resp. under (4.1.1)).

§3.4. Proof of the main theorem. Let us prove the main theorem: Theorem 3.1.11. Let (X, M)/(S, N) be as in the beginning of §3.1. We fix an integer r ≥ 0 and assume that (X, M)/(S, N) and r satisfy (3.1.1) and K = K_{0}, or (3.1.2) with s = 0. (In the case (3.1.2), we can remove (resp. replace) the condition s = 0 (resp. by (4.1.1)) if we use [Y] (resp. Theorem 4.1.2).) The remaining problem is only to prove that the isomorphisms (3.2.6) and (3.2.7) induced by the comparison isomorphism D_{st}(V^{r}) ≅ D^{r} (Theorems 3.2.2 and 3.2.5) coincide with the homomorphisms induced by (3.1.10). First we define canonical projections \overline{f}_{0} : B̂_{st}^{+} → P_{0} and \overline{pr}_{0} : \overline{D}^{q} → P_{0} ⊗_{K_{0}} D^{q} compatible with φ. Let K′ be any finite extension of K contained in \overline{K}, let (S′, N′) be the scheme Spec(O_{K′}) endowed with the canonical log structure, and set (X′, M′) := (X, M) ×_{(S,N)} (S′, N′). Then we have a commutative diagram:

(3.4.1)  \xymatrix{(X′_{n}, M′_{n}) \ar[d] &  & (Y′, M_{Y′}) \ar[ll] \ar[d] \\ (S′_{n}, N′_{n}) \ar[d] &  & (s′, L′) \ar[ll] \ar[dd] \ar[dl] \\ (S_{n}, N_{n}) \ar[d]_{i_{En,π}} & (s, L) \ar[d] \ar[l] \\ (E_{n}, M_{E_{n}}) & (W_{n}(s), W_{n}(L)) \ar[l]_{i_{En,0}} & (W_{n}(s′), W_{n}(L′)), \ar[l]}

which induces a homomorphism

H^{q}((X′_{n}, M′_{n})/(E_{n}, M_{E_{n}})) ⟶ H^{q}((Y′, M_{Y′})/(W_{n}(s′), W_{n}(L′)))

⟵^{∼} W_{n}(k′) ⊗_{W_{n}} H^{q}((Y, M_{Y})/(W_{n}, W_{n}(L))).

By taking the inductive limit with respect to K′ and ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n}, we obtain

\overline{pr}_{0} : \overline{D}^{q} ⟶ P_{0} ⊗_{W} D^{q}.

This is compatible with φ and the action of G_{K}, but not with N. In the special case (X, M) = (S, N) and q = 0, this becomes a ring homomorphism

\overline{f}_{0} : B̂_{st}^{+} ⟶ P_{0}

and the above homomorphism \overline{pr}_{0} is compatible with \overline{f}_{0}. By definition, \overline{pr}_{0} and \overline{f}_{0} are compatible with pr_{0} : D^{q} → D^{q} and f_{0} : R_{E,ℚ_{p}} → K_{0} induced by {i_{E_{n},0}}.

Lemma 3.4.2. Let V be a semi-stable p-adic representation of G_{K} and set D := D_{st}(V). Let s be a positive integer such that F il^{s+1}D_{K} = 0 and let D′ be the quotient of D corresponding to the quotient V (s)_{I_{K}} (−s) of V (§1.3). Then the following diagram is commutative:

F il^{s}(B_{st}^{+} ⊗_{K_{0}} D)^{N=0,φ=p^{s}} ⟶_{\overline{f}_{0}⊗1} (P_{0} ⊗_{K_{0}} D)^{φ=p^{s}}
↓  ↓^{≀}
F il^{s}(B_{st}^{+} ⊗_{K_{0}} D′)^{N=0,φ=p^{s}} ⟵^{∼} (P_{0} ⊗_{K_{0}} D′)^{φ=p^{s}}.

Recall D′ = D_{[s]}, F il^{s}D′_{K} = D′_{K}, F il^{s+1}D′_{K} = 0 and N = 0 on D′.

Proof. We are reduced to the case D = D′, in which case the claim is trivial because \overline{f}_{0} is a P_{0}-algebra homomorphism. □

Note that, with the notation of Lemma 3.4.2, the isomorphism in Corollary 1.3.3 is characterized by the following commutative diagram: (3.4.3)

V (s)  F il^{s}(B_{st} ⊗_{K_{0}} D)^{N=0,φ=p^{s}}
↓  ↓
V (s)_{I_{K}}  ≅_{Cor. 1.3.3} (P_{0} ⊗_{K_{0}} D′)^{φ=p^{s}}  →^{∼} F il^{s}(B_{st} ⊗_{K_{0}} D′)^{N=0,φ=p^{s}}.

Lemma 3.4.4. Let K′, (S′, N′) and (X′, M′) be as in the definition of \overline{pr}_{0} above. Then the composites of the following two sequences of homomorphisms coincide:

H^{r}(S_{n}^{∼}(r)_{(X′,M′)}) ⟶^{p^{r}·(A)} R^{r}u_{(X′_{n},M′_{n})/W_{n}∗}O ⟶^{(B)} R^{r}u_{(X′_{n},M′_{n})/(E_{n},M_{En})∗}O

⟶^{(C)} R^{r}u_{(Y′,M_{Y ′})/(W_{n}(s′),W_{n}(L′))∗}O ≅^{(D)} H^{r}(W_{n}ω_{Y′/s′}^{•}) ⟶^{p^{m}} H^{r}(W_{n}ω_{Y′/s′}^{•}),

H^{r}(S_{n}^{∼}(r)_{(X′,M′)}) ⟶^{(E)} i^{′∗}R^{r}j′_{∗}ℤ/p^{n}ℤ(r)′ ⟶^{“p^{m}”} i^{′∗}R^{r}j′_{∗}ℤ/p^{n}ℤ(r)

⟶^{(F)} W_{n}ω_{Y′/s′, log}^{r} ⟶ H^{r}(W_{n}ω_{Y′/s′}^{•}).

Here the integer m is defined by p^{a}a! = p^{m}·unit in ℤ_{p}, r = a(p − 1) + b, a, b ∈ ℤ, 0 ≤ b ≤ p − 2. The homomorphisms (A), (B), (C), (D), (E) and (F) are induced by the distinguished triangle (3.3.3), the morphism (E_{n}, M_{E_{n}}) → Spec(W_{n}), the diagram (3.4.1), Theorem 2.1.4, (3.3.5) and (3.1.8) respectively.

Proof. The question is étale local on X′. Étale locally on X′, by choosing a closed immersion of (X′, M′) into a smooth fine log scheme (Z, M_{Z}) over W with liftings of Frobenius {F_{Z_{n}}} of {(Z_{n}, M_{Z_{n}})} satisfying the condition [Ts2] (2.1.1), we can define a complex S′_{n}(r) ([Ts2]§2.1) and a natural homomorphism S_{n}^{∼}(r) → S′_{n}(r) ([Ts2](2.1.2)). By the definition of the morphisms [Ts2](2.1.2) and [Ts2](3.1.1) (=(3.3.5)), the homomorphisms p^{r}· (A) and (E) canonically factor through H^{r}(S′_{n}(r)). Hence we may replace H^{r}(S_{n}^{∼}(r)) with H^{r}(S′_{n}(r)).

We denote by {a_{1}, …, a_{r}}_{ét} (resp. {a_{1}, …, a_{r}}_{syn}) the image of a_{1} ⊗ ⋯ ⊗ a_{r} (a_{i} ∈ M^{′gp}) under the symbol map (3.1.4) (resp. (i^{′∗}M^{′gp})^{⊗r} → H^{r}(S′_{n}(r)) ([Ts2](2.2.1)). Then the local section {a_{1}, …, a_{r}}_{syn} is sent to {a_{1}, …, a_{r}}_{ét} by (E) ([Ts2] Proposition 3.2.4 (2)) and hence to p^{m}d log(\overline{a_{1}}) ∧ ⋯ ∧ d log(\overline{a_{r}}) by the composite of the second sequence. On the other hand, by the explicit description of the syntomic symbol map [Ts2] Lemma 2.4.6, which is still valid for S_{n}(r)′, and Lemma 2.1.10, the image of {a_{1}, …, a_{r}}_{syn} under the composite of the first sequence is also p^{m}d log(\overline{a_{1}}) ∧ ⋯ ∧ d log(\overline{a_{r}}). By [Ts2] Proposition 2.4.1 (1), we see that the symbol map (i^{′∗}M^{′gp})^{⊗r} → H^{r}(S′_{n}(r)) is surjective by induction on n. □

By taking the inductive limit with respect to K′ and ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n}, we obtain the following commutative diagram from Lemma 3.4.4: (3.4.5)

\xymatrix{V^{r}(s) \ar[d] & H^{r}(\overline{Y} ,S_{ℚp} (s)_{(\overline{X},\overline{M})}) \ar[l]_{∼}^{p−s·(3.3.6)} \ar[r] & (\overline{D}^{r})^{N=0,φ=ps} \ar[d]_{_{\overline{pr}0}} \\ H^{r−s}(\overline{Y} ,W ω_{\overline{Y} /\overline{s}, log}^{s})_{ℚp} \ar[r]^{∼} & (\underleftarrow{lim}_{n}H^{r−s}(\overline{Y} ,H^{s}(W_{n}ω_{\overline{Y} /\overline{s}}^{•})))_{ℚp}^{φ=ps} & (P_{0}⊗_{K0} D)^{φ=ps} . \ar[l]}

where s = r or s = d = dim X_{K} and we assume r ≥ d in the latter case. The left vertical (resp. lower right) homomorphism is induced by (3.1.10) (resp. τ_{≤s}W_{n}ω_{Y/s}^{•} → H^{s}(W_{n}ω_{Y/s}^{•})). Note W_{n}ω_{Y/s}^{•} = τ_{≤d}W_{n}ω_{Y/s}^{•} in the case s = d. The lower left one is an isomorphism by Lemma 2.3.6. See (2.1.7) for the relation between φ and V′. To prove the commutativity in the case s = d, we need the remark after Theorem 3.3.7. On the other hand, we have a commutative diagram: (3.4.6)

\xymatrix{(\overline{D}^{r})^{N=0,φ=p^{s}} \ar[dr]_{_{\overline{pr}_{0}}} & (B̂_{st}^{+} ⊗_{R_{E,ℚp}} D^{r})^{N=0,φ=p^{s}} \ar[l]_{∼} \ar[d]^{\overline{f}_{0}⊗pr_{0}} & (B_{st}^{+} ⊗_{K_{0}} D^{r})^{N=0,φ=p^{s}} \ar[l]^{ι⊗s}_{∼} \ar[dl]^{\overline{f}_{0}⊗1} \\  & (P_{0} ⊗_{K_{0}} D^{r})^{φ=p^{s}}}

Note that the first line is (3.3.2). Combining the above two commutative diagrams, we obtain a commutative diagram:

(3.4.7)

V^{r}(s)  ⟶_{(3.3.8)}^{∼} F il^{r}(B_{st}^{+} ⊗_{K_{0}} D^{r})^{N=0,φ=p^{s}}
↓  ↓^{\overline{f}_{0}⊗1}
ℚ_{p} ⊗_{ℤ_{p}} H^{r−s}(\overline{Y} , W ω_{\overline{Y} /\overline{s}, log}^{s}) ⟶_{Prop. 2.3.5}^{∼}  (P_{0} ⊗_{K_{0}} D^{r})^{φ=p^{s}}.

Note that the composite of the lower horizontal map of (3.4.7) with the lower right one of (3.4.5) coincides with the lower left one of (3.4.5). By Lemma 3.4.2, (3.4.3) and (3.4.7), we see that the isomorphisms (3.2.6) and (3.2.7) induced by the comparison isomorphism D_{st}(V^{r}) ≅ D^{r} are induced by (3.1.10).

§4. The semi-stable conjecture in the open case.

§4.1. Statement of the theorem. Recently, G. Yamashita [Y] gave a proof of the semi-stable conjecture via syntomic cohomology for (X, M)/(S, N) satisfying (3.1.2) in the open case, i.e. without assuming s = 0. (Moreover he proved it also for cohomologies with partial compact supports.) In this section, we will give an alternative proof in the special case that the “horizontal divisors at infinity do not have self-intersections” i.e. when (X, M)/(S, N) satisfies the following condition:

(4.1.1) There exist a finite number of divisors D_{i} (i ∈ I) on X such that X_{triv} = X − (Y ∪ (∪_{i∈I}D_{i})) and an étale covering {X_{λ} → X}_{λ∈Λ} satisfying the following condition. For each λ ∈ Λ, there exists an étale morphism over S:

X → Spec((O_{K}[T_{1}, …, T_{u}]/(T_{1} ⋯ T_{u} − π^{e}))[U_{i} (i ∈ I_{λ}), V_{1}, …, V_{t}])

for some integers u ≥ 1, t ≥ 0 and e ≥ 1 such that e|[K : K_{0}] and D_{i} ×_{X} X_{λ} is the inverse image of {U_{i} = 0} for each i ∈ I_{λ}. Here I_{λ} denotes the set of all i ∈ I such that D_{i} ×_{X} X_{λ} ≠ ∅.

Theorem 4.1.2. Assume that (X, M)/(S, N) satisfies the condition (4.1.1). Then, for any integer q ≥ 0, the homomorphism (3.3.9) is an isomorphism preserving the filtrations after taking B_{dR}⊗_{B_{st}}. Hence V^{q} is a semi-stable padic representation and (3.3.9) induces an isomorphism: D_{st}(V^{q}) ≅ D^{q} in \underline{MF}_{K}(φ, N).

We will prove this theorem by removing the divisors D_{i} at infinity one by one and using the Gysin exact sequences.

§4.2 Gysin sequence for crystalline cohomology. Associated to an effective Cartier divisor X′ on a scheme X, one can construct a log structure on X′ as follows: If X′ is defined by a global section f ∈ Γ(X , O_{X}), then f is a non-zero divisor and unique up to multiplication by units. Hence the fine log structure on X associated to the pre-log structure ℕ_{X} → O_{X}; 1 ↦ f is independent on the choice of f up to canonical isomorphisms. In the general case, one obtains a fine log structure by gluing the above log structures étale locally. For a fine log scheme (X , M), we define the fine log structure associated to a Cartier divisor X′ ⊂ X to be the co-product of M and the log structure constructed above. We say that a morphism of fine log scheme f: (X , M) → (Y, N) is syntomic ([Ka3] (2.5)) if it is integral, the underlying morphism of schemes X → Y is flat and locally of finite presentation, and étale locally on X, there exists a (Y, N)-exact closed immersion of (X , M) into a smooth fine log scheme (Z, L) over (Y, N) such that the underlying closed immersion of schemes is transversally regular relative to Y ([EGA IV] Définition (19.2.2). See also Proposition (19.2.4)). If f is syntomic, for any (Y, N)-exact closed immersion of (X , M) into a smooth fine log scheme (Z, L) over (Y, N), the underlying closed immersion is transversally regular relative to S. Syntomic morphisms are stable under base changes and compositions.

Proposition 4.2.1. Let (S, N) be a fine log scheme endowed with a PD-ideal (I, γ) such that p is nilpotent on O_{S}, let S_{0} be the closed subscheme of S defined by a sub PD-ideal of I, and let N_{0} be the inverse image of N. Let (X_{0}, M_{0}) be a syntomic fine log scheme over (S_{0}, N_{0}), let X′_{0} ⊂ X_{0} be a Cartier divisor flat over S_{0}, and let M′_{0} be the inverse image of M_{0} on X′_{0}.

(1) Étale locally on X_{0}, there exist an (S, N)-closed immersion of (X_{0}, M_{0}) into a smooth fine log scheme (Y, L) over (S, N) and a Cartier divisor Y′ ⊂ Y such that X′_{0} is the inverse image of Y′ and Y′ endowed with the inverse image L′ of L is smooth over (S, N).

(2) Suppose that we are given i: (X_{0}, M_{0}) ↪ (Y, L) and Y′ as in (1) globally. Let (D, M_{D}) (resp. (D′, M_{D′})) be the PD-envelope of (X_{0}, M_{0}) in (Y, L) (resp. (X′_{0}, M′_{0}) in (Y′, L′)) compatible with the PD-structure (I, γ). Let J_{D} (resp. J_{D′}) be the PD-ideal of O_{D} (resp. O_{D′}). If Y′ is defined by a global section f ∈ Γ(Y, O_{Y}), then f is a non-zero divisor on O_{D} and we have isomorphisms O_{D}/fO_{D} ≅ O_{D′} and J_{D}^{[r]}/fJ_{D}^{[r]} ≅ J_{D′}^{[r]} (r ≥ 1).

(3) Under the notation of (2), let L^{∘} (resp. M_{0}^{∘}) be the log structure on Y (resp. X_{0}) defined by the log structure L (resp. M_{0}) and the Cartier divisor Y′ (resp. X′_{0}). Then the PD-envelope of (X_{0}, M_{0}^{∘}) in (Y, L^{∘}) compatible with the PD-structure (I, γ) has the same underlying scheme as (D, M_{D}). Furthermore (Y, L^{∘}) is smooth over (S, N), and, for each integer r ≥ 0, we have a canonical exact sequence:

0 → J_{D}^{[r−•]} ⊗_{O_{Y}} Ω_{Y/S}^{•}(log(L/N)) → J_{D}^{[r−•]} ⊗_{O_{Y}} Ω_{Y/S}^{•}(log(L^{∘}/N))

→^{(∗)} (J_{D′}^{[r−1−•]} ⊗_{O_{Y′}} Ω_{Y′/S}^{•}(log(L′/N)))[−1] → 0

such that (∗) sends ω_{1}+d log(g)∧ω_{2} to \overline{ω_{2}} for ω_{1} ∈ J_{D}^{[r−q]}⊗_{O_{Y}} Ω_{Y/S}^{q}(log(L/N)),

ω_{2} ∈ J_{D}^{[r−q]} ⊗_{O_{Y}} Ω_{Y/S}^{q−1}(log(L/N)) and a local equation g = 0 defining Y′ in Y,

where \overline{ω_{2}} denotes the image of ω_{2} in J_{D′}^{[r−q]} ⊗_{O_{Y′}} Ω_{Y′/S}^{q−1}(log(L′/N)).

Proof. (1) Étale locally on X_{0}, there exists an (S, N)-closed immersion i of (X_{0}, M_{0}) into a smooth fine log scheme (Z, M_{Z}) over (S, N), and X′_{0} is defined by the global equation f = 0 for some f ∈ Γ(X_{0}, O_{X_{0}}). Set (Y, L) := (Z, M_{Z}) ×_{Spec(ℤ)} Spec(ℤ[T]) an let Y′ be the closed subscheme of Y defined by T = 0. Then the closed immersion (X_{0}, M_{0}) ↪ (Y, L) defined by i and T ↦ f satisfies the required condition.

(2) Since the question is étale local on X_{0}, we may assume that we have a

factorization (X_{0}, M_{0}) ↪^{j} (Z, M_{Z}) →^{α} (Y, L) such that j is an exact closed immersion and α is étale, and that Y′ is defined by the global equation g = 0 for some g ∈ Γ(Y, O_{Y}). Since (X_{0}, M_{0}) is integral over (S_{0}, N_{0}), we may also assume that (Z, M_{Z}) is integral over (S, N). Let Z′ ⊂ Z be the pull-back of Y′ ⊂ Y and let h be the inverse image of g in Γ(Z, O_{Z}). Since (Z, M_{Z}) and Z′ endowed with the pull-back of M_{Z} are smooth and integral over (S, N), the morphism (Z, M_{Z}) → (S, N) ×_{Spec(ℤ)} Spec(ℤ[T]) defined by T ↦ g is smooth and integral on a neighbourhood of Z′. Especially the underlying morphism of schemes is flat on the neighbourhood. Hence h is a non-zero divisor. Thus, by the construction of PD-envelopes ([Ka2] (5.6)), we may replace (Y, L) with (Z, M_{Z}) and assume that i is an exact closed immersion and (Y, L) is integral over (S, N). Since (X_{0}, M_{0}) is syntomic over (S_{0}, N_{0}), the closed immersion X_{0} ↪ Y_{0} := Y ×_{S} S_{0} is transversally regular relative to S_{0} and hence we may assume that there exists a sequence g_{1}, …, g_{d} ∈ Γ(Y, O_{Y}) whose image in O_{Y_{0}} is transversally O_{Y_{0}}-regular relative to S_{0}, and X_{0} ⊂ Y_{0} is defined by the ideal ∑_{1≤i≤d} g_{i} · O_{Y_{0}}. Since S_{0} ↪ S is a nilimmersion and Y is flat over S, we see that the sequence g_{1}, …, g_{d} is transversally O_{Y}-regular relative to S. (Since Y is locally of finite presentation over S, we are reduced to the case S is noetherian and then to the case S_{0} is defined by an ideal J of O_{S} such that J^{2} = 0). Let X be the closed subscheme of Y defined by the ideal ∑_{1≤i≤d} g_{i} · O_{Y} and set X′ := X ×_{Y} Y′(⊂ Y′). Since the image of the sequence g_{1}, ⋯, g_{d}, g in O_{Y_{0}} is transversally O_{Y_{0}}-regular relative to S_{0} and S_{0} ↪ S is a nilimmersion, the sequence g_{1}, …, g_{d}, g is transversally O_{Y}-regular relative to S and hence the image of g_{1}, …, g_{d} in O_{Y′} is transversally O_{Y′}-regular relative to S. Hence the morphism Y → S[T_{1}, …, T_{d}] (resp. Y′ → S[T_{1}, …, T_{d}]) defined by T_{i} ↦ g_{i} (resp. T_{i} ↦ the image of g_{i} in O_{Y′}) is flat on a neighbourhood of X (resp. X′). (Since Y and Y′ are locally of finite presentation over S, we are easily reduced to the case S is noetherian, where we can use [EGA IV] Chap. 0 Proposition (15.1.21).) Furthermore, since X_{0} = X ×_{S}S_{0} (resp. X′_{0} = X′×_{S}S_{0}), D (resp. D′) is isomorphic to the PD-envelope of X in Y (resp. X′ in Y′). Hence, by [Be-O] 3.2.1, we have D ≅ Y ×_{S[T_{1},... ,T_{d}]} S < T_{1}, …, T_{d} >, D′ ≅ Y′ ×_{S[T_{1},... ,T_{d}]} S < T_{1}, …, T_{d} >, which implies the claim.

(3) As in (2), we may assume that i is an exact closed immersion, and (Y, L) is integral over (S, N), and Y′ ⊂ Y is defined by the global equation g = 0 for some g ∈ Γ(Y, O_{Y}). Then (X , M^{∘}) ↪ (Y, L^{∘}) is an exact closed immersion and we obtain the first claim. For the second claim, we may replace Y with a neighbourhood of Y′. Hence, we may assume (Y, L) → (S, N)×_{Spec(ℤ)}Spec(ℤ[T]) defined by T ↦ g is smooth, and there exists a chart P → Γ(S, N), Q → Γ(Y, L), P → Q of (Y, L) → (S, N) such that (Y, L) → (S, N)×_{Spec(ℤ[P])} Spec(ℤ[Q][T]) is étale and the kernel and the torsion part of the cokernel of P^{gp} → Q^{gp} have orders invertible on S. Then Q ⊕ ℕ → Γ(Y, L^{∘}); (0, 1) ↦ g becomes a chart of L^{∘}. Hence (Y, L^{∘}) is smooth over (S, N), and we have

Ω_{Y/S}(log(L^{∘})) ≅ O_{Y} ⊗_{ℤ} P^{gp} ⊕ O_{Y} · d log(g),
Ω_{Y/S}(log(L)) ≅ O_{Y} ⊗_{ℤ} P^{gp} ⊕ O_{Y} · dg,
Ω_{Y′/S′}(log(L′)) ≅ O_{Y′} ⊗_{ℤ} P^{gp}.

Now the claim follows from (2). □

Proposition 4.2.2. Let (S, N , I, γ), (S_{0}, N_{0}), (X_{0}, M_{0}) and X′_{0} be the same as in Proposition 4.2.1. Assume that X_{0} is quasi-compact and separated. Then, there exist an étale hypercovering X_{0}^{•} of X_{0}, a simplicial smooth and integral fine log scheme (Y^{•}, L^{•}) over (S, N) with a Cartier divisor Y^{′ν} ⊂ Y^{ν} for each ν ≥ 0 and an (S, N)-closed immersion of (X_{0}^{•}, M|_{X_{0}^{•}}) into (Y^{•}, L^{•}) such that X_{0}^{ν} (ν ≥ 0) is affine, Y^{′ν} (ν ≥ 0) endowed with the inverse image of L^{ν} is smooth over (S, N), Y^{′0} is defined by the global equation g = 0 for some g ∈ Γ(Y^{0}, O_{Y^{0}}), for any non-decreasing map s: {0, 1, …, ν} → {0, 1, …, μ}, Y′_{μ} is the pull-back of Y′_{ν} by the morphism Y_{μ} → Y_{ν} corresponding to s, and X_{0}^{′ν} := X′_{0} ×_{X_{0}} X_{0}^{ν} (ν ≥ 0) is the pull-back of Y^{′ν} by the closed immersion X_{0}^{ν} ↪ Y^{ν}.

Proof. We will write X for X_{0} to simplify the notation. Since X is quasicompact, by Proposition 4.2.1 (1), there exist an étale covering X^{0} → X with X^{0} affine, an (S, N)-closed immersion of (X^{0}, M|_{X^{0}}) into a fine log scheme (Y^{0}, L^{0}) smooth and integral over (S, N), and a Cartier divisor Y^{′0} ⊂ Y^{0} defined by the global equation g = 0 for some g ∈ Γ(Y^{0}, O_{Y^{0}}) such that X^{′0} ⊂ X^{0} is the pull-back of Y^{′0} by the closed immersion X^{0} ↪ Y^{0} and Y^{′0} endowed with the inverse image of L^{0} is smooth over (S, N). For each ν ≥ 0, we define X^{ν} to be the fiber product of ν +1 copies of X^{0} over X, which is affine, and X^{′ν} to be X′ ×_{X} X^{ν}. We define (Y(ν), L(ν)) (resp. Y′(ν)) to be the fiber product of ν + 1 copies of (Y^{0}, L^{0}) (resp. Y^{′0}) over (S, N) (resp. S), and \overline{Y^{ν}} to be the blowing-up of Y(ν) along Y′(ν). We define Y′′(ν) to be the sum of the pullbacks of Y^{′0} by the ν + 1 projections Y(ν) → Y and Y^{ν} to be the complement of the strict transform of Y′′(ν) ⊂ Y(ν) on \overline{Y^{ν}}. We denote by \overline{L^{ν}} and L^{ν} the inverse images of L(ν) on \overline{Y^{ν}} and Y^{ν} respectively. We define Y^{′ν} to be Y′(ν) ×_{Y(ν)} Y^{ν}, which is a Cartier divisor on Y^{ν}. Let pr_{i}^{ν}: Y(ν) → Y^{1} (0 ≤ i ≤ ν) be the projection to the (i + 1)-th component and set g_{i}^{ν} := (pr_{i}^{ν})^{∗}(g). Then the morphism (Y(ν), L(ν)) → (S, N)[T_{0}, …, T_{ν}] defined by T_{i} ↦ g_{i}^{ν} is smooth and integral on a neighbourhood of Y′(ν) in Y(ν), especially, the underlying morphism of scheme is flat on the neighbourhood. Hence \overline{Y^{ν}} is the pull-back of the blowing-up of S[T_{0}, …, T_{ν}] along T_{0} = T_{1} = ⋯ = T_{ν} = 0. If we choose an integer i_{0} such that 0 ≤ i_{0} ≤ ν, Y^{ν} is the pull-back of S[T_{i_{0}}, U_{i}, U_{i}^{−1}(0 ≤ i ≤ ν, i ≠ i_{0})] → S[T_{0}, …, T_{ν}] where T_{i} = T_{i_{0}}U_{i} (i ≠ i_{0}).This implies that (\overline{Y^{ν}}, \overline{L^{ν}}), (Y^{ν}, L^{ν}) and Y^{′ν} endowed with the inverse images of L^{ν} are smooth and integral over (S, N). We also see that Y^{′ν} ⊂ Y^{ν} is defined by the equation g_{i_{0}}^{ν} = 0 and g_{i_{0}}^{ν} is a non-zero divisor on Y^{ν}. By the universality of blowingup, the closed immersion i(ν): (X^{ν}, M|_{X^{ν}}) ↪ (Y(ν), L(ν)) canonically factors through a closed immersion \overline{i^{ν}}: (X^{ν}, M|_{X^{ν}}) ↪ (\overline{Y^{ν}}, \overline{L^{ν}}). If we denote by h_{i}^{ν} the inverse image of g_{i}^{ν} in O_{X^{ν}}, then, for each i, the closed subscheme X^{′ν} of X^{ν} is defined by h_{i}^{ν} = 0. Hence h_{i}^{ν} = h_{i_{0}}^{ν} · u_{i} for some u_{i} ∈ O_{X^{ν}}^{∗}. This implies that \overline{i^{ν}} factors through (Y^{ν}, L^{ν}), which we denote by i^{ν}. Furthermore we see that X^{′ν} is the pull-back of Y^{′ν}. Let s: {0, 1, ⋯, ν} → {0, 1, …, μ} be a non-decreasing map. By the universality of blowing-up, the composite of (Y^{μ}, L^{μ}) → (Y(μ), L(μ)) with the morphism (Y(μ), L(μ)) → (Y(ν), L(ν)) corresponding to s uniquely factors through (\overline{Y^{ν}}, \overline{L^{ν}}). The inverse images of g_{i}^{ν} (0 ≤ i ≤ ν) in O_{Y^{μ}} are g_{s(i)}^{μ} and coincide up to the multiplication by units. Hence it further factors through (Y^{ν}, L^{ν}) and Y^{′μ} is the pull-back of Y^{′ν}. Thus {(Y^{ν}, L^{ν})}_{ν≥0} become a simplicial fine log scheme and {i^{ν}} are compatible with the simplicial structures. □

Corollary 4.2.3. Let (S, N , I, γ), (S_{0}, N_{0}), (X_{0}, M_{0}) and (X′_{0}, M′_{0}) be as in Proposition 4.2.1 and let M_{0}^{∘} be the fine log structure on X_{0} defined by M_{0} and the Cartier divisor X′_{0} ⊂ X_{0}. Assume that X_{0} is quasi-compact and separated. Let u_{(X_{0},M_{0})/(S,N)} denote the morphism of topoi

((X_{0}, M_{0})/(S, N , I, γ))_{crys}^{∼} ⟶ (X_{0})_{ét}^{∼},

and define u_{(X_{0},M_{0}^{∘})/(S,N)} and u_{(X′_{0},M′_{0})/(S,N)} similarly. Then we have a canonical distinguished triangle:

Ru_{(X_{0},M_{0})/(S,N),∗}J^{[r]} → Ru_{(X_{0},M_{0}^{∘})/(S,N),∗}J^{[r]}

→ Ru_{(X′_{0},M′_{0})/(S,N),∗}J^{[r−1]}[−1] →

for each integer r. Here O denotes the structure sheaf on the relevant crystalline site, J denotes the PD-ideal of O and, for an integer r, J^{[r]} denotes the r-th divided power of J if r ≥ 1 and O if r ≤ 0.

Proof. Choose (X_{0}^{•}, M_{0}|_{X_{0}^{•}}) ↪ (Y^{•}, L^{•}), Y^{′•} ⊂ Y^{•} and g as in Proposition 4.2.2 Then we can apply Proposition 4.2.1 (3) to (X_{0}^{ν}, M_{0}|_{X_{0}^{ν}}) ↪ (Y^{ν}, L^{ν}), X_{0}^{′ν} and Y^{′ν} for each ν ≥ 0. Furthermore, for each non-decreasing map s: {0, 1, …, ν} → {0, 1, …, μ}, since Y^{′μ} is the pull-back of Y^{′ν} by the morphism f_{s}: Y^{μ} → Y^{ν} corresponding to s, the short exact sequences are functorial with respect to f_{s}. Hence, by the cohomological descent ([Ka3](2.18)– (2.21)), we obtain the required distinguished triangles. If we are given another (X̃_{0}^{•}, M_{0}|X̃_{0}^{•}) ↪ (Ỹ^{•}, L̃^{•}), Ỹ^{′•} ⊂ Ỹ^{•} and g̃, we define \overline{Z^{ν}} to be the blowing-up of

Y^{ν} ×_{S} Ỹ^{ν} along Y^{′ν} ×_{S} Ỹ^{′ν} and let Z^{ν} be the complement of the strict transform of Y^{′ν} × Ỹ^{ν} ∪ Y^{ν} × Ỹ^{′ν} on \overline{Z^{ν}}. Let M_{Z^{ν}} be the inverse image of the log structure of (Y^{ν}, L^{ν}) ×_{(S,N)} (Ỹ^{ν}, L̃^{ν}) to Z^{ν}, and let Z^{′ν} ⊂ Z^{ν} be the pull-back of Y^{′ν}×Ỹ^{′ν}. Then, similarly as the proof of Proposition 4.2.2, using g and g̃, we see that {(Z^{ν}, M_{Z^{ν}})}_{ν≥0} naturally become a simplicial fine log scheme, there exists a closed immersion (X_{0}^{ν} ×_{X_{0}} X̃_{0}^{ν}, the inverse image of M_{0}) ↪ (Z^{ν}, M_{Z^{ν}}) inducing a morphism between simplicial fine log schemes, and this closed immersion with Z^{′ν} ⊂ Z^{ν} satisfies the conditions in Proposition 4.2.2. Furthermore, we have natural morphisms (Z^{•}, M_{Z^{•}}) → (Y^{•}, L^{•}) and (Z^{•}, M_{Z^{•}}) → (Ỹ^{•}, L̃^{•}) compatible with the closed immersions, and isomorphisms Z^{′ν} ≅ Y^{′ν} ×_{Y^{ν}} Z^{ν} ≅ Ỹ^{′ν} ×Ỹ^{ν} Z^{ν}. Hence the distinguished triangle is independent of the choice of X_{0}^{•} etc. □

The distinguished triangle in Corollary 4.2.3 is functorial with respect to (X_{0}, M_{0}) → (S_{0}, N_{0}) ↪ (S, N , I, γ) and X′_{0} as follows: Suppose that we are given another (X̃_{0}, M̃_{0}) → (S̃_{0}, Ñ_{0}) ↪ (S̃, Ñ , Ĩ, γ̃) and X̃′_{0}, a morphism α: (X̃_{0}, M̃_{0}) → (X_{0}, M_{0}) and a PD-morphism β: (S̃, Ñ) → (S, N) inducing a morphism β_{0} : (S̃_{0}, Ñ_{0}) → (S_{0}, N_{0}) in a compatible manner in the obvious sense.

We further assume that X̃′_{0} = X′_{0} ×_{X_{0}} X̃_{0}. Then the distinguished triangles in Corollary 4.2.3 for (X_{0}, M_{0})/(S, N , I, γ), X′_{0} and for (X̃_{0}, M̃_{0})/(S̃, Ñ , Ĩ, γ̃), X̃′_{0} are compatible with the morphisms between the each component induced by α and β: Choose (X_{0}^{•}, M_{0}^{•}) ↪ (Y^{•}, L^{•}), Y^{′•} ⊂ Y^{•} for (X_{0}, M_{0})/(S, N), and (X̃_{0}^{•}, M̃_{0}^{•}) ↪ (Ỹ^{•}, L̃^{•}), Ỹ^{′•} ⊂ Ỹ^{•} for (X̃_{0}, M̃_{0})/(S̃, Ñ) as in Proposition 4.2.2. Let \overline{Z^{ν}} be the blowing-up of Ỹ^{ν} ×_{S} Y^{ν} along Ỹ^{′ν} ×_{S} Y^{′ν}, let Z^{ν} be the complement of the strict transform of Ỹ^{′ν} ×_{S} Y^{ν} ∪ Ỹ^{ν} ×_{S} Y^{′ν} on \overline{Z^{ν}}, and let Z^{′ν} be the inverse image of Ỹ^{′ν} ×_{S} Y^{′ν}. Let M_{Z^{ν}} be the inverse image of the log structure on (Ỹ^{ν}, L̃^{ν}) ×_{(S,N)} (Y^{ν}, L^{ν}) to Z^{ν}. Then similarly as in the proof of Corollary 4.2.3, we see that {(Z^{ν}, M_{Z^{ν}})}_{ν≥0} naturally become a simplicial fine log scheme smooth and integral over (S̃, Ñ), there exists a closed immersion of (X̃_{0}^{ν} ×_{X_{0}} X_{0}^{ν}, the inverse image of M̃_{0}) into (Z^{ν}, M_{Z^{ν}}) over (S̃, Ñ) compatible with the simplicial structures, and this closed immersion with Z^{′ν} ⊂ Z^{ν} satisfies the conditions in Proposition 4.2.2. Furthermore we have a natural morphism to the closed immersion (X_{0}^{•}, M_{0}^{•}) ↪ (Y^{•}, L^{•}) (resp. (X̃_{0}^{•}, M̃_{0}^{•}) ↪ (Ỹ^{•}, L̃^{•})) such that Z^{′ν} is the pull-back of Y^{′ν} (resp. Ỹ^{′ν}). This implies the required functoriality.

§4.3. Gysin sequence for syntomic cohomology. Let the notation and the assumption as in §4.1. Assume that I is non-empty and choose one i_{0} ∈ I. We will change the notation as follows: We write M^{∘} for M, and M will denote the log structure defined by the union of the special fiber of X and the divisors D_{i} (i ∈ I, i ≠ i_{0}). We define (X′, M′) to be D_{i_{0}} endowed with the inverse image of M. Then (X, M), (X, M^{∘}) and (X′, M′) satisfies the condition (4.1.1). Note that X′ is a Cartier divisor on X and M^{∘} is the co-product of M and the log structure on X defined by X′ (cf. §4.2). We can construct Gysin sequence for syntomic cohomology as follows. We choose an affine étale covering X^{0} → X, a closed immersion of (X^{0}, M|_{X^{0}}) into a fine log scheme (Z^{0}, M_{Z^{0}}) smooth over Spec(W) endowed with a Cartier divisor Z^{′0} defined by a global equation g = 0 and with a compatible system of liftings of Frobenius {F_{Z_{n}^{0}} : (Z_{n}^{0}, M_{Z_{n}^{0}}) → (Z_{n}^{0}, M_{Z_{n}^{0}})}_{n≥1} such that X′ ×_{X} X^{0} is the pull-back of Z^{′0} and F_{Z_{n}^{0}}^{∗}(g) = g^{p} · (1 + py) for some y ∈ O_{Z_{n}^{0}}. Here the subscript n denotes the reduction mod p^{n}. Such a covering and an embedding exist by a similar argument as the proof of Proposition 4.2.1 (1). Starting from this embedding, we can construct an étale hypercovering X^{•} → X, a closed immersion (X^{•}, M^{•}) ↪ (Z^{•}, M_{Z^{•}}) and a Cartier divisor Z^{′•} ⊂ Z^{•} as in Proposition 4.2.2 endowed with a compatible system of liftings of Frobenius on {(Z_{n}^{•}, M_{Z_{n}^{•}})}_{n≥1}. By taking the PD-envelope of (X_{n}^{•}, M_{n}^{•}) in (Z_{n}^{•}, M_{Z_{n}^{•}}) compatible with the canonical PD-structure on pW_{n} and applying Proposition 4.2.1 (3), we obtain a short exact sequence on the étale site of the simplicial scheme X_{1}^{•} for each r ≥ 0. By using the property F_{Z_{n}^{0}}^{∗}(g) = g^{p} · (1 + py), we see that the short exact sequences are compatible with the Frobenius induced by F_{Z_{n}^{•}} and obtain a short exact sequence:

0→S_{n}^{∼}(r)_{(X•,M•),(Z•,MZ•)}→S_{n}^{∼}(r)_{(X•,M∘•),(Z•,M{∘ \atop Z•})}→S_{n}^{≈}(r−1)_{(X′•,M′•),(Z′•,MZ′•)}[−1]→0.

Here S_{n}^{∼}(s) denotes the syntomic complex defined in [Ts2] §2.1, and S_{n}^{≈}(s) denotes the complex obtained by replacing p^{s} − φ with p^{s+1} − pφ in the definition of S_{n}^{∼}(s). By taking the derived direct image by the morphism of topoi (X_{1}^{•})_{ét}^{∼} → (X_{1})_{ét}^{∼}. We obtain the required distinguished triangle:

(4.3.1)  → S_{n}^{∼}(r)_{(X,M)} → S_{n}^{∼}(r)_{(X,M^{∘})} → S_{n}^{≈}(r − 1)_{(X′,M′)}[−1]

on (X_{1})_{ét}. We can verify the independence and the functoriality similarly as in the crystalline case by taking care of liftings of Frobenius. We define V_{syn,1}^{q}(r) to be the syntomic cohomology H^{q}(\overline{Y} , S_{ℚ_{p}}(r)_{(\overline{X},\overline{M})}) (cf.§3.3) of (\overline{X}, \overline{M}). We define V_{syn,2}^{q}(r) and V_{syn,3}^{q}(r) to be the syntomic cohomology of (\overline{X}, \overline{M^{∘}}) and (\overline{X′}, \overline{M′}) respectively. Then, by taking ℚ ⊗ \underleftarrow{lim}_{n} \underrightarrow{lim}_{K′} H^{∗}((X_{1} ×_{Spec(O_{K})} Spec(O_{K′}))_{ét}, −) of the triangle (4.3.1) for the base changes of (X_{n}, M_{n}), (X_{n}, M_{n}^{∘}) and (X′_{n}, M′_{n}) by O_{K} → O_{K′}, we obtain a complex:

(4.3.2)  ⋯ → V_{syn,1}^{q}(r) → V_{syn,2}^{q}(r) → V_{syn,3}^{q−1}(r − 1) → V_{syn,1}^{q+1}(r) → ⋯

§4.4 Compatibility of Gysin sequences 1. We will prove the compatibility of Gysin sequences with the isomorphisms (3.2.1), (3.3.2) and the homomorphism (3.3.4). We follow the notation in §3.3. We denote by D_{1}^{q} (resp. D_{1}^{q}, resp. \overline{D}_{1}^{q}) for the crystalline cohomology D^{q} (resp. D^{q}, resp. \overline{D}^{q}) for (X, M)/(S, N) defined in §3.2 (resp. §3.3, resp. §3.3). We denote by D_{2}^{q}, D_{2}^{q} and \overline{D}_{2}^{q} for the cohomologies of (X, M^{∘})/(S, N), and D_{3}^{q}, D_{3}^{q} and \overline{D}_{3}^{q} for the cohomologies of (X′, M′)/(S, N). We denote by D_{3}^{q}(−1), D_{3}^{q}(−1) and \overline{D}_{3}^{q}(−1) the same modules as D_{3}^{q}, D_{3}^{q} and \overline{D}_{3}^{q} whose Frobenius endomorphisms φ are replaced with pφ. By taking ℚ ⊗ \underleftarrow{lim}_{n} of the Gysin sequences for the crystalline cohomologies over the bases (W_{n}, W_{n}(L), pW_{n}, γ) and (E_{n}, M_{E_{n}}, \overline{J_{E_{n}}}, \overline{δ}), we obtain an exact sequence:

(4.4.1)  ⋯ → D_{1}^{q} → D_{2}^{q} → D_{3}^{q−1}(−1) → D_{1}^{q+1} → ⋯

and a complex:

(4.4.2)  ⋯ → D_{1}^{q} → D_{2}^{q} → D_{3}^{q−1}(−1) → D_{1}^{q+1} → · · · .

Lemma 4.4.3. Let D_{i} (i = 1, 2) be finitely generated free R_{E,ℚ_{p}}-modules endowed with φ_{E}-semilinear endomorphisms φ_{D_{i}} whose linearizations R_{E,φ} ⊗_{R_{E}} D_{i} → D_{i} are isomorphisms. Let D_{i} be the reduction of D_{i} with respect to R_{E,ℚ_{p}} → K_{0} induced by {i_{E_{n},0}} (§3.3) and let φ_{D_{i}} be the σ-semilinear automorphism of D_{i} induced by φ_{D_{i}}. Suppose that we are given an R_{E,ℚ_{p}}-linear homomorphism f: D_{1} → D_{2} compatible with φ_{D_{i}} and K_{0}-linear sections s_{i}: D_{i} → D_{i} of the canonical surjections p_{i}: D_{i} → D_{i} compatible with φ_{D_{i}} and φ_{D_{i}}. Let g: D_{1} → D_{2} be the K_{0}-linear homomorphism induced by f. Then we have f ∘ s_{1} = s_{2} ∘ g.

Proof. We apply [Ts2] Lemma 4.4.11. Let I_{n} be as in loc. cit. For any a ∈ D_{1}, f(s_{1}(φ_{D_{1}}^{−n}(a))) is a lifting of g(φ_{D_{1}}^{−n}(a)) = φ_{D_{2}}^{−n}(g(a)) ∈ D_{2} in D_{2}. Hence, by loc. cit., s_{2}(g(a)) ≡ φ_{D_{2}}^{n}(f(s_{1}(φ_{D_{1}}^{−n}(a)))) = f(s_{1}(a)) mod I_{n}D_{2}. Since ∩_{n}(I_{n}⊗ ℚ) = 0 and D_{2} is a free R_{E,ℚ_{p}}-module, this implies s_{2}(g(a)) = f(s_{1}(a)). □

By functoriality, the projections D_{i}^{q} → D_{i}^{q} (cf. §3.3) are compatible with (4.4.1) and (4.4.2). On the other hand, by a similar argument as the proof of the functoriality of the Gysin sequences, we also see that the sequences (4.4.1) and (4.4.2) are compatible with the Frobenius endomorphisms. Hence, from Lemma 4.4.3, we obtain the following:

Lemma 4.4.4. The isomorphisms R_{E,ℚ_{p}} ⊗_{K_{0}} D_{i}^{q} ≅ D_{i}^{q} (i = 1, 2, 3) (§3.3) are compatible with the sequences (4.4.1) and (4.4.2).

Note that this and the exactness of (4.4.1) implies that (4.4.2) is also exact. We denote by D_{dR,1}^{q} the de Rham cohomology H^{q}(X_{K}, Ω_{X_{K}}^{•} (log(M_{K}))) endowed with the Hodge filtration (cf. (3.2.1)), which is canonically isomorphic to the projective limit of the crystalline cohomology of (X_{n}, M_{n}) over (S_{n}, N_{n}, pO_{S_{n}}, γ) with respect to n tensored with K over O_{K}. We denote by D_{dR,2}^{q} and D_{dR,3}^{q} the de Rham cohomology of (X_{K}, M_{K}^{∘}) and (X′_{K}, M_{K′}) respectively. Recall that the Hodge spectral sequences for D_{dR,i}^{q} degenerate (cf. [Ts2] Proposition 4.7.9). We denote by D_{dR,3}^{q}(−1) the same K-vector space as D_{dR,3}^{q} whose filtration is defined by F il^{r}(D_{dR,3}^{q}(−1)) = F il^{r−1}D_{dR,3}^{q}. Then, by taking ℚ ⊗ \underleftarrow{lim}_{n} of the Gysin sequence for the crystalline cohomology over the base (S_{n}, N_{n}, pO_{S_{n}}, γ), we obtain an exact sequence of filtered K-vector spaces:

(4.4.5)  ⋯ → D_{dR,1}^{q} → D_{dR,2}^{q} → D_{dR,3}^{q−1}(−1) → D_{dR,1}^{q+1} → ⋯

By functoriality, the projections D_{i}^{q} → D_{dR,i}^{q} induced by {i_{E_{n},π}} (§3.3) are compatible with the exact sequences (4.4.2) and (4.4.5). Hence by combining with Lemma 4.4.4, we obtain the following compatibility:

Lemma 4.4.6. The isomorphisms ρ_{π}: K⊗_{K_{0}}D_{i}^{q} ≅ D_{dR,i}^{q} (3.2.1) are compatible with the exact sequences (4.4.1) and (4.4.5).

By taking ℚ ⊗ \underleftarrow{lim}_{n} \underrightarrow{lim}_{K′} of the Gysin sequence for the base changes of (X_{n}, M_{n}), (X_{n}, M_{n}^{∘}) and (X′_{n}, M′_{n}) by (S′, N′) → (S, N) over (E_{n}, M_{E_{n}}), we obtain a complex:

(4.4.7)  ⋯ → \overline{D}_{1}^{q} → \overline{D}_{2}^{q} → \overline{D}_{3}^{q−1}(−1) → \overline{D}_{1}^{q+1} → · · · .

Here S′ = Spec(O_{K′}) and N′ is the log structure defined by the closed point. By functoriality, the natural homomorphisms D_{i}^{q} → \overline{D}_{i}^{q} are compatible with (4.4.2) and (4.4.7). Hence, by Lemma 4.4.4, we obtain the following compatibility:

Lemma 4.4.8. The isomorphisms B̂_{st}^{+} ⊗_{K_{0}} D_{i}^{q} ≅ \overline{D}_{i}^{q} ([Ts2] Proposition 4.4.6) are compatible with the sequence (4.4.1) and (4.4.7).

Note that this lemma and the exactness of (4.4.1) imply the exactness of (4.4.7). By construction, it is clear that the distinguished triangle (4.3.1) is compatible with the distinguished triangle of Corollary 4.2.3 for (X_{n}, M_{n}), (X′_{n}, M′_{n}), … over (W_{n}, pW_{n}, γ). Hence, by the functoriality of the Gysin sequence for crystalline cohomology, we obtain:

Lemma 4.4.9. The natural homomorphisms V_{syn,i}^{q} → \overline{D}_{i}^{q} (3.3.4) are compatible with the sequences (4.3.2) and (4.4.7).

§4.5. Compatibility of Gysin sequences 2. To prove Theorem 4.1.2, we also need to verify the compatibility of the Gysin sequence (4.3.2) of the syntomic cohomology with that of the étale cohomology. For simplicity, we omit the log structures from the notation of log schemes; we simply write X, X^{∘} and X′ for the log schemes (X, M), (X, M^{∘}) and (X′, M′) appearing in §4.4. As in [O], we denote by \underline{X}, \underline{S},... the underlying schemes of

log schemes X, S,... (We do not adopt the notation X^{∘}, S^{∘},... in [Na] Notation (1.1.2) and [I2]1.2 to avoid the confusion with the notation X^{∘}. ) Let X_{triv} (resp. (X^{∘})_{triv}, resp. X′_{triv}) be the maximal open subschemes of X (resp. X^{∘}, resp. X′) on which the log structure is trivial. We have (X^{∘})_{triv} = X_{triv}\X′_{triv}. We denote by V_{1}^{q}, V_{2}^{q} and V_{3}^{q} the q-th étale cohomology of (X_{triv})\overline{K}, ((X^{∘})_{triv})\overline{K} and (X′_{triv})\overline{K} with coefficients ℚ_{p} respectively. Then we have the Gysin exact sequence:

(4.5.1)  ⋯ → V_{1}^{q} → V_{2}^{q} → V_{3}^{q−1}(−1) → V_{1}^{q+1} → ⋯

Proposition 4.5.2. For any integer r ≥ 0, the homomorphisms V_{syn,i}^{q}(r) → V_{i}^{q}(r) defined by p^{−r} · (3.3.6) are compatible with the sequences (4.5.1) and (4.3.2).

Let i and i′ denote the closed immersions \underline{Y} → \underline{X} and \underline{Y′} → \underline{X′} and let j, j^{∘} and j′ denote the open immersions X_{triv} → X, (X^{∘})_{triv} → X and X′_{triv} → X′ respectively. Proposition 4.5.2 follows from the following local version:

Proposition 4.5.3. For any integer r ≥ 0, the following diagram is commutative:

⟶  S_{n}^{∼}(r)_{X}  ⟶  S_{n}^{∼}(r)_{X^{∘}}  ⟶  S_{n}^{≈}(r − 1)_{X′}[−1]
↓^{(3.3.5)}  ↓^{(3.3.5)}  ↓
⟶ i^{∗}Rj_{∗}ℤ/p^{n}ℤ(r)′ ⟶ i^{∗}Rj_{∗}^{∘}ℤ/p^{n}ℤ(r)′ ⟶ i^{′∗}Rj′_{∗}ℤ/pℤ(r)′(−1)[−1].

Here the right vertical homomorphism is the composite of (4.5.4)

S_{n}^{≈}(r − 1)_{X′} → S_{n}^{∼}(r − 1)_{X′} ⟶^{(3.3.5)} i^{′∗}Rj′_{∗}ℤ/p^{n}(r − 1)′ → i^{′∗}Rj′_{∗}ℤ/p^{n}(r)′(−1),

where the first map is defined by the multiplication by p on J_{D′}^{[r−1−•]} ⊗ Ω^{•} and the identity map on O_{D′} ⊗ Ω^{•}.

We will prove Proposition 4.5.3 in §4.8 after some preliminaries in §4.6 and §4.7. We will prove it along the following lines. By using the Gysin sequence (4.3.1) and explicit descriptions of i^{∗}Rj_{∗}ℤ/p^{n}ℤ(r)′ and i^{∗}Rj_{∗}^{∘}ℤ/p^{n}ℤ(r)′ as complexes in terms of Godment resolutions, we construct a map

(4.5.5)  α: S_{n}^{≈}(r − 1)_{X′} → i^{′∗}Rj′_{∗}ℤ/p^{n}ℤ(r)′(−1)

such that the diagram in Proposition 4.5.3 with (4.5.4) replaced by α is commutative. The main difficulty to compare α with (4.5.4) comes from the fact that the resolution \overline{S}_{n}(r) of ℤ/p^{n}ℤ(r)′ relating ℤ/p^{n}ℤ(r)′ with S_{n}^{∼}(r) (cf. [Ts2] §3.1) does not behave well with respect to the closed immersion X′ ↪ X. We overcome this problem by replacing X′ → X ← X^{∘} with X′ →^{id} X′ ← X^{′∘}, where X^{′∘} is the scheme \underline{X′} endowed with the inverse image of M_{X^{∘}}. Although X^{′∘} is not log smooth, we still have a Gysin sequence for S_{n}^{∼}(r) (4.7.11) and we can construct a map

(4.5.6)  β: S_{n}^{≈}(r − 1)_{X′} → i^{′∗}Rj′_{∗}ℤ/p^{n}ℤ(r)′(−1)

in the same way as α, which is easily seen to be equal to α above. (For the étale side, we need to use the Kummer étale sites of fs log schemes ([Na]).) For X′ →^{id} X′ ← X^{′∘}, we also have a Gysin sequence for \overline{S}_{n}(r) (4.7.10), which allows us to compare the morphism β with (4.5.4).

§4.6. Preliminaries on log fundamental groups. We will summarize some basic facts on log fundamental groups ([I2]§4) which we use in the proof of the compatibility of Gysin sequences for syntomic and étale cohomologies. We leave the most of their proofs to the readers. We continue to omit the log structures in the notation of log schemes. A logarithmic point s is Spec(k) for a separably closed field k with a saturated log structure such that the multiplication by n is bijective on M_{s}/k^{∗} for any positive integer n prime to the characteristic of k. A log geometric point of an fs log scheme S is a morphism s → S for a log geometric point s ([Na] Definition (2.5), [I2] Definition 4.1).

Lemma 4.6.1. Let T → S be a Kummer étale morphism ([Na] Definition (2.1.2), [I2] 1.6) and let s̃ → S be a log geometric point. If the image of s̃ in S is contained in the image of T in S, then there exists a lifting s̃ → T.

Using the fact that a Kummer étale closed immersion is an open immersion, we can prove the following lemma in the same way as [SGA1] 5.3, 5.4.

Lemma 4.6.2. Let T → S be a Kummer étale separated morphism of fs log schemes, let U be an fs log scheme over S and let φ: ũ → U be a log geometric point of U. Then the map Hom_{S}(U, T) → Hom_{S}(ũ, T); f ↦ f ∘ φ is injective.

Let S be a locally noetherian fs log scheme and let s̃ → S be a log geometric point. Then the category Kcov(S) of Kummer étale covers of S ([I2] 3.1) with the fiber functor Fs̃: Kcov(S) → (Sets) satisfies the axioms (G1) to (G6) of [SGA1] V§4. We define the log fundamental group of S at s̃ to be Aut(Fs̃) ([I2]4.6). By Lemma 4.6.2, we see that the fiber functor Fs̃(−) is canonically identified with Hom_{S}(s̃, −). We consider an equi-characteristic connected normal scheme X with an fs log structure such that there exists a global chart α: ℕ → Γ(X, M_{X}) whose composite with Γ(X, M_{X}) → Γ(X, O_{X}) sends ℕ\{0} to 0. Let x = Spec(k) be the generic point of X with the inverse image log structure. Let k^{sep} be a separable closure of k and let \overline{x} be Spec(k^{sep}) endowed with the inverse image of M_{x}. Choose a chart α as above and define a log geometric point x̃ of X to be Spec(k^{sep}) with the log structure associated to ℕ̃ → k; a ≠ 0 ↦ 0. Here ℕ̃ = ∪_{n∈ℕ,n∈k^{∗}} \frac{1}{n}ℕ. We define the morphism x̃ → \overline{x} → x → X by the natural inclusion ℕ → ℕ̃ and the chart α. x̃ is independent of the choice of the chart α up to non-canonical isomorphisms over \overline{x}. For n ∈ ℕ invertible in k, let t_{n} denote the image of \frac{1}{n} ∈ ℕ̃ in Γ(x̃, Mx̃). Then the homomorphism Aut(x̃/\overline{x}) → ℤ̂′(1) := \underleftarrow{lim}_{n∈ℕ,n∈k^{∗}} μ_{n}(k^{sep}) defined by σ ↦ (σ(t_{n})t_{n}^{−1})_{n≥1} is an isomorphism. Any automorphism of x̃ over x induces an automorphism of \overline{x} over x and thus we obtain a surjective homomorphism Aut(x̃/x) → Aut(\overline{x}/x) = Gal(k^{sep}/k)^{∘} with kernel Aut(x̃/\overline{x}) ≅ ℤ̂′(1). Here (−)^{∘} denotes the the opposite group. We define k^{ur} to be the union of all finite extensions k′ of k contained in k^{sep} such that the normalizations of the underlying scheme of X in k′ are unramified. We define x^{ur} and x̃^{ur} similarly as \overline{x} and x̃ using k^{ur} instead of k^{sep}. We have a canonical surjective homomorphism Aut(x̃/x) → Aut(x̃^{ur}/x) inducing an isomorphism ℤ̂′(1) ≅ Aut(x̃/\overline{x}) →^{∼} Aut(x̃^{ur}/x^{ur}). We also have a natural surjection Aut(x̃^{ur}/x) → Aut(x^{ur}/x) ≅ Gal(k^{ur}/k)^{∘}. The fiber functor Fx̃: Kcov(X) → (Sets) is explicitly pro-represented as follows. For each finite extension k′ of k contained in k^{ur}, let X_{k′} be a strict étale cover of X whose function field is k′, and for a positive integer n invertible on X, we define X_{k′,n} to be the Kummer étale cover X_{k′}×_{Spec(ℤ[ℕ])}Spec(ℤ[\frac{1}{n}ℕ]) of X. Then, the inclusions k′ ↪ k^{sep} and \frac{1}{n}ℕ ↪ ℕ̃ define a morphism x̃ → X_{k′,n}. If k′/k is Galois and μ_{n}(k^{sep}) ⊂ k′, then X_{k′,n}/X is Galois i.e. Aut(X_{k′,n}/X) acts transitively on Fx̃(X_{k′,n}) = Hom_{X}(x̃, X_{k′,n}). We assert that Fx̃ is pro-represented by \{X_{k′,n}\} i.e. \underrightarrow{lim}_{k′,n} Hom_{X}(X_{k′,n}, Y) → Hom_{X}(x̃, Y) is an isomorphism for any Y ∈ Kcov(X). The injectivity follows from Lemma 4.6.2. For the surjectivity, by Lemma 4.6.1 and Lemma 4.6.3 below, we may replace Y by X_{k′,n} with k′/k Galois and μ_{n}(k^{sep}) ⊂ k′. In this case Hom_{X}(X_{k′,n}, X_{k′,n}) → Hom_{X}(x̃, X_{k′,n}) is surjective.

Lemma 4.6.3. For any Kummer étale cover Y → X, there exists n ∈ ℕ invertible on X and a strict étale cover X′ → X such that Y ×_{X} X′_{n} → X′_{n} is trivial. Here X′_{n} = X_{n} ×_{Spec(ℤ[ℕ])} Spec(ℤ[\frac{1}{n}ℕ]).

The automorphism group Aut(x̃/x) naturally acts on the fiber functor Fx̃: Kcov(X) → (Sets) and we obtain a homomorphism Aut(x̃/x)^{∘} → π_{1}(X, x̃).

Proposition 4.6.4. The above homomorphism factors through an isomorphism Aut(x̃^{ur}/x)^{∘} →^{≅} π_{1}(X, x̃).

Proof. Since Fx̃ is pro-represented by {X_{k′,n}} and x̃ → X_{k′,n} factors through x̃^{ur}, we see that the action of Aut(x̃/x) on Fx̃ factors through Aut(x̃^{ur}/x). For m ∈ ℕ invertible on X, we denote by t_{m} the image of (\frac{1}{m}, 1) by the (non-canonical) isomorphism ℕ̃ ⊕ (k^{ur})^{∗} ≅ M_{x^{ur}}. We have a bijection as sets Aut(x̃^{ur}/x) → Gal(k^{ur}/k) × ℤ̂′(1)(k^{ur}) sending σ to the pair of σ^{∗}: k^{ur} → k^{ur} and (σ^{∗}(t_{m})t_{m}^{−1})_{m}. On the other hand, we have Aut(Fx̃)^{∘} ≅ \underleftarrow{lim}_{k′,n} Aut(X_{k′,n}), where (k′, n) ranges over all finite Galois extensions k′ of k contained in k^{ur} and n ∈ ℕ invertible on X such that μ_{n}(k^{ur}) ⊂ k′. For such (k′, n), we have a bijection Aut(X_{k′,n}) → Gal(k′/k) × μ_{n}(k′) sending τ to the pair of τ^{∗}: k′ → k′ and τ^{∗}(t′_{n})(t′_{n})^{−1}, where t′_{n} denotes the image of \frac{1}{n} by the chart \frac{1}{n}ℕ → Γ(X_{k′,n}, M_{X_{k′,n}}). Hence Aut(x̃^{ur}/x)^{∘} ≅ π_{1}(X, x̃). □

Next we consider an equi-characteristic connected regular scheme Z with the fs log structure associated to a regular divisor defined by the equation t = 0 for some t ∈ Γ(Z, O_{Z}), and assume that X is the divisor with the inverse image of M_{Z}. Set Z_{triv} = Z\X. Then the functor Kcov(Z) → Etcov(Z_{triv}); W ↦ W ×_{Z} Z_{triv} induces an equivalence of categories from Kcov(Z) to the subcategory consisting of étale covers of Z_{triv} tamely ramified along X ([I2] Theorem 7.6). Let z = Spec(K) be the generic point of Z, choose a separable closure K^{sep} of K and set \overline{z} := Spec(K^{sep}). Let K^{ur} be the union of all finite extensions K′ of K contained in K^{sep} such that the normalizations of Z_{triv} in K′ are unramified and tamely ramified along X. Set z^{ur} := Spec(K^{ur}). We will give a way to construct a path from \overline{z} to x̃. For a finite extension K′ of K contained in K^{ur}, we denote by Z_{K′} a Kummer étale cover of Z whose function field is K′. Then we have a natural morphism \overline{z} → Z_{K′} and the fiber functor F\overline{z}: Kcov(Z) → (Sets) is pro-represented by \{Z_{K′}\}. By Lemma 4.5.1, \underleftarrow{lim}_{K′} Hom_{Z}(x̃, Z_{K′}) is non-empty. An element φ = \{φ_{K′}\}_{K′} of this set defines a path from \overline{z} to x̃; it induces a map

Hom_{Z}(\overline{z}, W) ←^{∼} \underrightarrow{lim}_{K′⊂K^{ur}} Hom_{Z}(Z_{K′}, W) ⟶^{φ∘−} Hom_{Z}(x̃, W)

for W ∈ Kcov(Z). For any σ ∈ Aut(x̃^{ur}/x), there exists a unique automorphism σ_{K′} of Z_{K′} such that σ_{K′} ∘ φ_{K′} = φ_{K′} ∘ σ for each finite Galois extension K′ of K contained in K^{ur}, and {σ_{K′}} defines an automorphism τ ∈ Aut(z^{ur}/z). The homomorphism Aut(x̃^{ur}/x)^{∘} ≅ π_{1}(X, x̃) → π_{1}(Z, \overline{z}) ≅ Aut(z^{ur}/z)^{∘} induced by the above path sends σ to τ. For another φ′, there exists a unique τ ∈ Aut(z^{ur}/z) such that φ′ = σ ∘ φ.

§4.7. The complexes \overline{S}_{n}(r) and S_{n}^{∼}(r). We keep the notation of §4.5. Working étale locally on \underline{X}, we assume that \underline{X} is affine and we are given a W -closed immersion of X into a fine log scheme Z smooth over W endowed with a Cartier divisor \underline{Z′} ⊂ \underline{Z} defined by a global equation g = 0 (g ∈ Γ(Z, O_{Z})) and with a compatible system of liftings of Frobenius \{F_{Z_{n}}: Z_{n} → Z_{n}\}_{n≥1} such that \underline{X′} is the pull-back of \underline{Z′} and \underline{Z′} endowed with the pull-back of M_{Z} is smooth over W. We denote by Z^{∘} the scheme \underline{Z} endowed with the log structure defined by M_{Z} and the Cartier divisor \underline{Z′} (cf. the beginning of §4.2), and by Z′ (resp. Z^{′∘}) the scheme \underline{Z′} with the the inverse image of M_{Z} (resp. M_{Z^{∘}}). Note that Z^{′∘} is not smooth over W. We further assume that there exists t_{1}, …, t_{d} ∈ Γ(Z, M_{Z}) and t ∈ Γ(Z, O_{Z}) such that \underline{Z′} is defined by t = 0 in \underline{Z}, \{d log(t_{i})(1 ≤ i ≤ d), dt\} (resp. \{d log(t_{i})(1 ≤ i ≤ d), d log(t)}, resp. {d log(t_{i})}) form a basis of Ω_{Z/W} (resp. Ω_{Z^{∘}/W}, resp. Ω_{Z′/W}), and F_{Z_{n}}^{∗}(t_{i}) = t_{i}^{p}, F_{Z_{n}}^{∗}(t) = t^{p} for each n ≥ 1. Choose and fix such t_{i} and t. We have closed immersions X^{∘} ↪ Z^{∘}, X′ ↪ Z′ and X^{′∘} ↪ Z^{′∘}. Recall that X^{′∘} is \underline{X′} with the inverse image of M_{X^{∘}}. Let U = Spec(A) → X be a strict étale morphism and set U^{∘} := X^{∘} ×_{X} U, U′ := X′ ×_{X} U and U^{′∘} := X^{′∘} ×_{X} U. By replacing U with a suitable affine open covering, we assume that U, U^{∘} and U′ satisfy the condition [Ts2] (1.5.2). (See [Ts2] Lemma 1.3.3). We may further assume that U^{′∘} also satisfies the equivalent conditions in [Ts2] Lemma 1.3.2 and Γ(U′, M_{U′})/Γ(U′, O_{U′}^{∗}) → Γ(U′, M_{U′}/O_{U′}^{∗}) is an isomorphism. (See the proof of [Ts2] Lemma 1.3.3 for the latter.) Let A′ be the coordinate ring of U′. As in [Ts2] 1.4, let A^{h} be the henselization of A with respect to the ideal pA. Let U^{h} be Spec(A^{h}) with the inverse image of M_{U} and set U^{h∘} := X^{∘} ×_{X} U^{h}, U^{′h} := X′ ×_{X} U^{h} and U^{′h∘} := X^{′∘} ×_{X} U^{h}. The coordinate ring of \underline{U^{′h}} = \underline{U^{′h∘}} is the henselization of A′ with respect to pA′, which we denote by A^{′h}. Let U_{triv}^{h}, (U^{h∘})_{triv} and U_{triv}^{′h} denote the maximal open subschemes of U^{h}, U^{h∘} and U^{′h} respectively on which the log structures are trivial and let A_{triv}^{h}, (A^{h∘})_{triv} and A_{triv}^{′h} denote their coordinate rings. Finally we define (X_{triv})^{∘} (resp. (X′_{triv})^{∘}) to be X_{triv} (resp. X′_{triv}) endowed with the inverse image of M_{X^{∘}}, and define (U_{triv}^{h})^{∘} and (U_{triv}^{′h})^{∘} similarly. Note that the log structure of (X_{triv})^{∘} is the one defined by the divisor X′_{triv} ↪ X_{triv}. Now we have the following commutative diagrams:

(4.7.1)

((X^{∘})_{triv})_{ét} → ((X_{triv})^{∘})_{Két} →^{ε} (X_{triv})_{ét}
↑  ↑
((X′_{triv})^{∘})_{Két} →^{ε′} (X′_{triv})_{ét}

(4.7.2)

((U^{h∘})_{triv})_{ét} → ((U_{triv}^{h})^{∘})_{Két} →^{ε_{U}} (U_{triv}^{h})_{ét}
↑  ↑
((U_{triv}^{′h})^{∘})_{Két} →^{ε′_{U}} (U_{triv}^{′h})_{ét}

Here Két denotes the Kummer étale site ([Na]). Note ét = Két for schemes with trivial log structures. We have natural morphisms from (4.7.2) to (4.7.1). We will construct a resolution \overline{S}_{n}(r) of ℤ/p^{n}ℤ(r)′ on each site in the diagram (4.7.2) in a compatible manner. For (U^{h∘})_{triv}, (U_{triv}^{h})^{∘} and U_{triv}^{′h}, we can directly apply [Ts2]§3.1, but for the other two, we need some modifications.

Let η := Spec(K) be the generic point of U_{triv}^{h}. Choose an algebraic closure \overline{K} of K and set \overline{η} := Spec(\overline{K}). We define K^{ur} (resp. K^{∘ur}) to be the union of all finite extensions L of K contained in \overline{K} such that the normalizations of U_{triv}^{h} (resp. (U^{h∘})_{triv}) in L are unramified. We define η′, K′, \overline{K′}, \overline{η′} and K^{′ur} similarly using U_{triv}^{′h}. We set

G_{U} := Gal(K^{ur}/K) ≅ π_{1}(U_{triv}^{h}, \overline{η}),
G_{U}^{∘} := Gal(K^{∘ur}/K) ≅ π_{1}((U^{h∘})_{triv}, \overline{η}) ≅ π_{1}((U_{triv}^{h})^{∘}, \overline{η}),
G′_{U} := Gal(K^{′ur}/K′) ≅ π_{1}(U_{triv}^{′h}, \overline{η′}).

See [I2]Theorem 7.6 for the last isomorphism in the second line. We define η^{′∘} to be Spec(K′) with the inverse image of M_{(U_{triv}^{′h})^{∘}} and define η̃^{′∘} and η̃^{′∘ur} similarly as x̃ and x̃^{ur} in §4.6 using \overline{K′} and K^{′ur}. We set

G_{U}^{′∘} := Aut(η̃^{′∘ur}/η^{′∘})^{∘} ≅ π_{1}((U_{triv}^{′h})^{∘}, η̃^{′∘}).

See Proposition 4.6.4 for the second isomorphism. For a finite extension L of K contained in K^{∘ur}, denote by V_{L} a Kummer étale cover of (U_{triv}^{h})^{∘} whose function field is L. We choose and fix a compatible system {f_{L}: η̃^{′∘} → V_{L}}_{L⊂K^{∘ur}}, which gives a path from η̃^{′∘} → (U_{triv}^{′h})^{∘} → (U_{triv}^{h})^{∘} to \overline{η} → (U_{triv}^{h})^{∘} (§4.6). It also induces a compatible system \{\underline{f_{L}}: \overline{η′} → \underline{V_{L}}\}_{L⊂K^{ur}}, which gives a path from \overline{η′} to \overline{η}. These paths induce homomorphisms G_{U}^{′∘} → G_{U}^{∘} and G′_{U} → G_{U} which are compatible with the natural homomorphisms G_{U}^{∘} → G_{U} and G_{U}^{′∘} → G′_{U}. We define \overline{A^{h}} (resp. \overline{A_{triv}^{h}}) to be the normalization of A^{h} (resp. A_{triv}^{h}) in K^{ur}. Similarly, we define \overline{A^{h∘}} and \overline{(A^{h∘})_{triv}} (resp. \overline{A^{′h}} and \overline{A_{triv}^{′h}}) using A^{h}, (A^{h∘})_{triv} and K^{∘ur} (resp. A^{′h}, A_{triv}^{′h} and K^{′ur}). By applying [Ts2]§1.4 and §1.5 to U, U^{∘}, U′ and \overline{A^{h}}, \overline{A^{h∘}}, \overline{A^{′h}}, we obtain a commutative diagram:

U^{′h} ← \overline{U′} ↪ \overline{D′}
↓ ↓ ↓
U^{h} ← \overline{U} ↪ \overline{D}
↑ ↑ ↑
U^{h∘} ← \overline{U^{∘}} ↪ \overline{D^{∘}}

compatible with the actions of G_{U}^{∗} on \overline{U^{∗}} and \overline{D^{∗}} and with the liftings of Frobenius on \overline{D^{∗}} (∗ = ∅, ′, ∘). The upper vertical maps are induced by the path from \overline{η′} to \overline{η} chosen above. Since Z^{∘} and Z′ satisfy the condition [Ts2] (2.1.1), we can construct resolutions \overline{S}_{n}(r)_{U^{∘},Z^{∘}} and \overline{S}_{n}(r)_{U′,Z′} of ℤ/p^{n}ℤ(r)′ on ((U^{h∘})_{triv})_{ét} (or ((U_{triv}^{h})^{∘})_{Két}) and on (U_{triv}^{′h})_{ét} as in [Ts2] §3.1. Although Z does not satisfy [Ts2] (2.1.1), the construction in [Ts2] §3.1 still works as follows.

Lemma 4.7.3. (1) The absolute Frobenius of \overline{A^{h}}/p\overline{A^{h}} is surjective.

(2) The homomorphism F il^{1}A_{crys}(\overline{A^{h}}) → ˆ \overline{A^{h}} defined by x ↦ p^{−1}φ(x) mod F il^{1} is surjective.

Proof. (1) For any a ∈ \overline{A^{h}}, there exists u ∈ (\overline{A^{h}})^{∗} such that 1 + p^{1/2}a = u^{p} and hence p^{1/2}a ≡ (u − 1)^{p} mod p\overline{A^{h}}. Set v = (u − 1) · p^{−1/2p}. Then v^{p} ∈ \overline{A^{h}} and hence v ∈ \overline{A^{h}}. Thus we obtain a ≡ v^{p} mod p^{1/2}\overline{A^{h}}. Set b = (a − v^{p}) · p^{−1/2}. Then, by the same argument, there exists w ∈ \overline{A^{h}} such that b ≡ w^{p} mod p^{1/2}\overline{A^{h}}. Now we have a ≡ (v + p^{1/2p}w)^{p} mod p\overline{A^{h}}. (2) By (1), the

homomorphism A_{crys}(\overline{A^{h}}) → ˆ \overline{A^{h}} is surjective (cf. [Ts2] Lemma A1.1). Hence the claim follows from the surjectivity of 1−p^{−1}φ: F il^{1}A_{crys}(\overline{A^{h}}) → A_{crys}(\overline{A^{h}}) (cf. [Ts2] Theorem A3.26 and Proposition A3.33). □

As in [Ts2] §3.1, let \overline{E}_{n} be the PD-envelope of \overline{U}_{n} ↪ \overline{D}_{n} ×_{W_{n}} Z_{n} compatible with the PD-structure on J_{\overline{D}_{n}} + pO_{\overline{D}_{n}}. For each t_{i} ∈ Γ(Z, M_{Z}), choose and fix a lifting a_{i} ∈ Γ(\overline{D}, M\overline{D}) of the image of t_{i} in Γ(\overline{U}, M\overline{U}) such that F ∗ ; \overline{D}(a_{i}) = a_{i}^{p} ([Ts2] Lemma 3.1.5). Let u_{i} ∈ Γ(\overline{E}_{n}, 1 + J_{\overline{E}_{n}}) be the unique element such that t_{i} = a_{i} · u_{i} in Γ(\overline{E}_{n}, M_{\overline{E}_{n}}). For t ∈ Γ(Z, O_{Z}), we choose a lifting a ∈

Γ(\overline{D}, O\overline{D}) = A_{crys}(\overline{A^{h}}) of the image of t in Γ(\overline{U}, O\overline{U}) = ˆ \overline{A^{h}}. By Lemma 4.7.3 (2), we may assume that a^{p}−φ(p) ∈ pF il^{1}A_{crys}(\overline{A^{h}}). Similarly as [Ts2] Lemma 3.1.4, we see that there exists a PD-isomorphism over O_{\overline{D}_{n}}:

O_{\overline{D}_{n}}⟨V_{1}, …, V_{d}, V ⟩ →^{∼} O_{\overline{E}_{n}}; V_{i} ↦ u_{i} − 1, V ↦ t − a

We define J̃_{\overline{D}_{n}}^{[r]′} and J̃_{\overline{E}_{n}}^{[r]′} as in [Ts2] §3.1.

Lemma 4.7.4. (cf. [Ts2] Lemma 3.1.6). For each r, we have

J̃_{\overline{E}_{n}}^{[r]′} = ⨁_{\underline{m}∈ℕ^{d+1}} J̃_{\overline{D}_{n}}^{[r]′} ∏_{1≤i≤d}(u_{i} − 1)^{[m_{i}]} · (t − a)^{[m]},

where \underline{m} = (m_{1}, …, m_{d}, m).

Proof. Since we can apply the same argument as the proof of [Ts2] Lemma 3.1.6 to the ring R_{n} := O_{\overline{D}_{n}}⟨u_{1} − 1, …, u_{d} − 1⟩, it suffices to show the following: For x = ∑_{m∈ℕ} x_{m}(t − a)^{[m]} ∈ O_{\overline{E}_{n}} = R_{n}⟨t − a⟩, we have φ(x) ∈ p^{r}O_{\overline{E}_{n}} if and only if φ(x_{m}) ∈ p^{max{r−m,0}}R_{n} for all m ∈ ℕ. The sufficiency follows from

φ(t − a) = t^{p} − φ(a) = (t − a + a)^{p} − φ(a)

= p\{(p − 1)!(t − a)^{[p]} + ∑_{ν=1}^{p−1} \frac{1}{p}\binom{p}{ν}(t − a)^{p−ν}a^{ν}\} + a^{p} − φ(p) ∈ pJ_{\overline{E}_{n}}.

Assume x ≠ 0 and φ(x) ∈ p^{r}R_{n}⟨t − a⟩, and let M be the largest integer such that x_{M} ≠ 0. Then the coefficient of (t − a)^{[Mp]} in φ(x) is φ(x_{M})p^{M}c for some c ∈ ℤ_{p}^{∗} ∩ ℚ. Hence p^{M}φ(x_{M}) ∈ p^{r}R_{n}, which implies φ(x_{M}) ∈ p^{max{r−M,0}}R_{n}. By the sufficiency, we can subtract x_{M}(t−a)^{[M]}, repeat the argument and show φ(x_{m}) ∈ p^{max{r−m,0}}R_{n} for all m. □

By Lemma 4.7.4, we can construct a complex J_{\overline{E}_{n}}^{[r]′} ⊗_{O_{Zn}} Ω_{Z_{n}/W_{n}}^{•} and show

that it gives a resolution of J_{\overline{D}_{n}}^{[r]′} as in [Ts2] Lemma 3.1.7. (Since d(t − a)^{[m]} = (t − a)^{[m−1]}dt, it is enough to use dt instead of u_{i}d log(t_{i}) for the indeterminate t − a. ) Thus we obtain a resolution \overline{S}_{n}(r)_{U,Z} of ℤ/p^{n}ℤ(r)′ on (U_{triv}^{h})_{ét} by the same method as [Ts2] §3.1. We have natural maps from the pull-backs of \overline{S}_{n}(r)_{U,Z} to \overline{S}_{n}(r)_{U′,Z′} and \overline{S}_{n}(r)_{U^{∘},Z^{∘}}. We will need the following lemma to construct the map α (4.5.5).

Lemma 4.7.5. The natural map \overline{S}_{n}(r)_{U,Z} → ε_{U∗}\overline{S}_{n}(r)_{U^{∘},Z^{∘}} is injective. (See (4.7.2) for ε_{U}.)

Let \overline{E^{∘}}_{n} be the PD-envelope of \overline{U^{∘}}_{n} ↪ \overline{D^{∘}}_{n} × Z_{n}^{∘}. Choose a lifting a′ ∈ Γ(\overline{D^{∘}}, M\overline{D^{∘}}) of the image of t in Γ(\overline{U^{∘}}, M\overline{U^{∘}}) such that F ∗ ; \overline{D^{∘}}(a′) = (a′)^{p} ([Ts2] Lemma 3.1.5) and let u ∈ Γ(\overline{E^{∘}}_{n}, 1 + J_{\overline{E^{∘}}_{n}}) be the unique element such that a′ · u = t in Γ(\overline{E^{∘}}_{n}, M_{\overline{E^{∘}}_{n}}). We denote the image of u_{i} ∈ Γ(\overline{E}_{n}, 1 + J_{\overline{E}_{n}}) (defined above) in Γ(\overline{E^{∘}}_{n}, 1 + J_{\overline{E^{∘}}_{n}}) by the same letter u_{i}. Then we have an isomorphism:

O_{\overline{E^{∘}}_{n}} ≅ O_{\overline{D^{∘}}_{n}}⟨u_{1} − 1, …, u_{d} − 1, u − 1⟩.

Proof Lemma 4.7.5. It suffices to prove that the natural map J_{\overline{E}_{n}}^{[r]′} → J_{\overline{E^{∘}}_{n}}^{[r]′}

and the multiplication by t on J_{\overline{E^{∘}}_{n}}^{[r]′} are injective. We first prove the claim for r = 0. Since the map O_{\overline{E}_{n}} → O_{\overline{E^{∘}}_{n}} factors through O_{\overline{D^{∘}}_{n}}⟨u_{i} − 1, t − a⟩ ≅ O_{\overline{D^{∘}}_{n}}⟨u_{i} − 1, t − a′⟩ and t = a′u, it is enough to prove that O_{\overline{D}_{n}} → O_{\overline{D^{∘}}_{n}} and the multiplication by a′ on O_{\overline{D^{∘}}_{n}} are injective. We are easily reduced to the case n = 1. Define R\overline{A^{h}} and R\overline{A^{h∘}} as in [Ts2] §1.1 and let z be a generator of the

kernel of W (R\overline{A^{h}}) → ˆ \overline{A^{h}} ([Ts2] Corollary A2.2). Then, by [Ts2] Lemma A2.11, we see that Γ(\overline{D}_{1}, O_{\overline{D}_{1}}) = A_{crys}(\overline{A^{h}})/p (resp. Γ(\overline{D^{∘}}_{1}, O_{\overline{D^{∘}}_{1}}) = A_{crys}(\overline{A^{h∘}})/p) is a free R\overline{A^{h}}/z^{p} (resp. R\overline{A^{h∘}}/z^{p})-module with a base \{z^{[pn]}|n ≥ 0\}. Especially, the filtration is separated and each graded quotient is isomorphic to R\overline{A^{h}}/z ≅ \overline{A^{h}}/p (resp. R\overline{A^{h∘}}/z ≅ \overline{A^{h∘}}/p) (Lemma 4.7.3 (1) and [Ts2] Lemma A2.1). Hence the claim follows from the injectivity of the natural map \overline{A^{h}}/p → \overline{A^{h∘}}/p and the

multiplication by t on \overline{A^{h∘}}/p. Next we consider the case r ≥ 1. Let x ∈ { \atop J_{\overline{E}_{n+r}}^{[r]′}}

and assume that its image in { \atop J_{\overline{E^{∘}}_{n+r}}^{[r]′}} is contained in p^{n}{ \atop J_{\overline{E^{∘}}_{n+r}}^{[r]′}}. Then it is

contained in p^{n}J_{\overline{E^{∘}}_{n+r}}^{[r]} and hence x ∈ p^{n}J_{\overline{E}_{n+r}}^{[r]} by the case r = 0. Choose y ∈ J_{\overline{E}_{n+r}}^{[r]} such that x = p^{n}y and let y^{∘} be the image of y in J_{\overline{E^{∘}}_{n+r}}^{[r]}. Then

p^{n}y^{∘} ∈ p^{n}{ \atop J_{\overline{E^{∘}}_{n+r}}^{[r]′}}, which implies y^{∘} ∈ { \atop J_{\overline{E^{∘}}_{n+r}}^{[r]′}} i.e. φ(y^{∘}) ∈ p^{r}O_{\overline{E^{∘}}_{n+r}}. By the

case r = 0, this implies φ(y) ∈ p^{r}O_{\overline{E}_{n+r}} and hence y ∈ { \atop J_{\overline{E}_{n+r}}^{[r]′}}. We can prove the second assertion similarly. Note φ(a′) = (a′)^{p}. □

Now it remains to construct a resolution \overline{S}_{n}(r)_{U^{′∘},Z^{′∘}} of ℤ/p^{n}ℤ(r)′ on ((U′_{triv})^{∘})_{Két}. To do this we need to construct \overline{U′}^{∘} ↪ \overline{D′}^{∘} for the non-smooth U^{′∘} modifying the construction in [Ts2] §1.4, §1.5. The underlying scheme of \overline{D′}^{∘} is the same as \overline{D′} i.e. Spec(A_{crys}(\overline{A^{′h}})). For the log structure, we use the fiber product Q^{′∘} of the diagram of monoids:

\underleftarrow{lim}(M_{η^{′∘ur}} ←^{f} M_{η^{′∘ur}} ←^{f} M_{η^{′∘ur}} ←^{f} ⋯) → M_{η^{′∘ur}} ← Γ(U^{′∘}, M_{U^{′∘}}),

where f(x) = x^{p} and the left map is the projection to the first component. Choose a chart ℕ̃ → M_{η^{′∘ur}} compatible with the chart ℕ → Γ(U^{′∘}, M_{U^{′∘}}) send-

ing 1 to the image of t (cf. the definition of x̃^{ur} in §4.6). The chart induces an iso-

morphism ℕ̃ ⊕ (K^{′ur})^{∗} ≅ M_{η^{′∘ur}}. By Γ(U′, M_{U′}/O_{U′}^{∗}) ⊕ ℕ ≅ Γ(U^{′∘}, M_{U^{′∘}}/O_{U^{′∘}}^{∗})

and the assumption Γ(U′, M_{U′})/Γ(U′, O_{U′}^{∗}) ≅ Γ(U′, M_{U′}/O_{U′}^{∗}), we see Γ(U′, M_{U′})/Γ(U′, O_{U′}^{∗}) ⊕ ℕ ≅ Γ(U^{′∘}, M_{U^{′∘}})/Γ(U^{′∘}, O_{U^{′∘}}^{∗}) and hence Γ(U^{′∘}, M_{U^{′∘}}) is generated by the images of t and Γ(U′, M_{U′}). This implies that the image of Γ(U^{′∘}, M_{U^{′∘}}) in M_{η^{′∘ur}} ≅ ℕ̃ ⊕ (K^{′ur})^{∗} is contained in ℕ ⊕ ((A_{triv}^{′h})^{∗} ∩ A^{′h}). Hence Q^{′∘} coincides with the fiber product of

\underleftarrow{lim}_{f}(ℕ̃ ⊕ ((\overline{A_{triv}^{′h}})^{∗} ∩ \overline{A^{′h}})) → ℕ̃ ⊕ ((\overline{A_{triv}^{′h}})^{∗} ∩ \overline{A^{′h}}) ← Γ(U^{′∘}, M_{U^{′∘}})

and the morphism Q^{′∘} → \underleftarrow{lim}_{f} M_{η^{′∘ur}} → \underleftarrow{lim}_{f} K^{′ur} factors through \underleftarrow{lim}_{f} \overline{A^{′h}}.

We define the log structure of \overline{D′}^{∘} to be the one associated to

Q^{′∘} → \underleftarrow{lim} \overline{A^{h}} → R\overline{A^{h}} →^{[ ]} W (R\overline{A^{′h}}) ⊂ A_{crys}(\overline{A^{′h}}).

Using the natural action of G_{U}^{′∘} on Q^{′∘} and the multiplication by p on Q^{′∘}, we can define the action of G_{U}^{′∘} and the lifting of Frobenius on \overline{D′}^{∘}.

We define \overline{U′}^{∘} to be Spec(ˆ\overline{A^{′h}}) with the log structure associated to

Γ(U^{′∘}, M_{U^{′∘}}) → ˆ \overline{A^{′h}}. We have a natural action of G_{U}^{′∘} (through G′_{U}) on \overline{U′}^{∘} and G_{U}^{′∘}-equivariant morphism \overline{U′}^{∘} → \overline{D′}^{∘} induced by the surjection

A_{crys}(\overline{A^{′h}}) → ˆ \overline{A^{′h}} (cf. Lemma 4.7.3) and Q^{′∘} → Γ(U^{′∘}, M_{U^{′∘}}). We assert that \overline{U′}^{∘} and \overline{D′}^{∘} are fs log schemes and the morphism \overline{U′}^{∘} → \overline{D′}^{∘} is an exact closed immersion. To prove this, we choose a chart P → Γ(U^{′∘}, M_{U^{′∘}}) such that P → Γ(U^{′∘}, M_{U^{′∘}})/Γ(U^{′∘}, O_{U^{′∘}}^{∗}) is an isomorphism ([Ts2] Lemma 1.3.2). Then the composite P → Γ(U^{′∘}, M_{U^{′∘}}) → ℕ̃ ⊕ ((\overline{A_{triv}^{′h}})^{∗} ∩ \overline{A^{′h}}) can be lifted to the projective limit \underleftarrow{lim}_{f}(ℕ̃ ⊕ ((\overline{A_{triv}^{′h}})^{∗} ∩ \overline{A^{′h}})). Hence P → Γ(U^{′∘}, M_{U^{′∘}}) can be lifted to P → Q^{′∘}. On the other hand, since the image of Γ(U^{′∘}, O_{U^{′∘}}^{∗}) in ℕ̃ ⊕ (\overline{A_{triv}^{′h}}^{∗} ∩ \overline{A^{′h}}) is contained in \overline{A^{′h}}^{∗}, the inverse image G of {1} under Q^{′∘} → Γ(U^{′∘}, M_{U^{′∘}})/Γ(U^{′∘}, O_{U^{′∘}}^{∗}) is a group and we have Q^{′∘}/G ≅ Γ(U^{′∘}, M_{U^{′∘}})/Γ(U^{′∘}, O_{U^{′∘}}^{∗}) ≅ P. Hence, by [Ts2] Lemma 1.3.1, P → Γ(U^{′∘}, M_{U^{′∘}}) → Γ(\overline{U′}^{∘}, M\overline{U′}^{∘}) and P → Q^{′∘} → Γ(\overline{D′}^{∘}, M\overline{D′}^{∘}) are charts, and \overline{U′}^{∘} → \overline{D′}^{∘} is an exact closed immersion. Next we compare \overline{U′}^{∘} → \overline{D′}^{∘} with \overline{U′} → \overline{D′} and \overline{U^{∘}} → \overline{D^{∘}}. The fiber product Q′ of \underleftarrow{lim}_{f} \overline{A^{′h}} → \overline{A^{′h}} ← Γ(U′, M_{U′}) used in the definition of the log structure of \overline{D′}

is the same as the fiber product of the diagram with \overline{A^{′h}} replaced by \overline{A_{triv}^{′h}}^{∗}∩\overline{A^{′h}}. Hence, there exists a natural map Q′ → Q^{′∘} compatible with the actions of G′_{U} and G_{U}^{′∘}. Using this and the natural map Γ(U′, M_{U′}) → Γ(U^{′∘}, M_{U^{′∘}}), we obtain a commutative diagram:

(4.7.6)  \begin{aligned}U^{′h∘} & ← & \overline{U′}^{∘} & ↪ & \overline{D′}^{∘} \\ ↓ & & ↓ & & ↓ \\ U^{′h} & ← & \overline{U′} & ↪ & \overline{D′} \end{aligned}

compatible with the actions of G′_{U}, G_{U}^{′∘} and the liftings of Frobenius. Similarly, the fiber product Q^{∘} of \underleftarrow{lim}_{f} \overline{A^{h∘}} → \overline{A^{h∘}} ← Γ(U^{∘}, M_{U^{∘}}) used in the construction of the log structure of \overline{D^{∘}} is the same as the the fiber product of the diagram with \overline{A^{h∘}} replaced with \overline{(A^{h∘})_{triv}}^{∗} ∩ \overline{A^{h∘}}. On the other hand, we have \overline{(A^{h∘})_{triv}}^{∗}∩\overline{A^{h∘}} ⊂ \underrightarrow{lim}_{L⊂K^{∘ur}} Γ(V_{L}, M_{L}), where V_{L} is as in the construction of a path from η̃^{′∘} to \overline{η}. The fixed system of morphisms \{f_{L}: η̃^{′∘} → V_{L}\} induces a morphism Q^{∘} → Q^{′∘} compatible with the actions of G_{U}^{∘} and G_{U}^{′∘}. Using this and the natural map Γ(U^{∘}, M_{U^{∘}}) → Γ(U^{′∘}, M_{U^{′∘}}), we obtain a commutative diagram:

(4.7.7)  \begin{aligned}U^{h∘} & ← & \overline{U^{∘}} & ↪ & \overline{D^{∘}} \\ ↑ & & ↑ & & ↑ \\ U^{′h∘} & ← & \overline{U′}^{∘} & ↪ & \overline{D′}^{∘} \end{aligned}

compatible with the actions of G_{U}^{′∘}, G_{U}^{∘} and the liftings of Frobenius. Now we are ready to construct \overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}. Let \overline{E^{′∘}}_{n} be the PD-envelope of \overline{U′}_{n}^{∘} ↪ \overline{D′}_{n}^{∘} × Z_{n}^{∘}, which is endowed with an action of G_{U}^{′∘} and a lifting of Frobenius by a natural way. The diagram (4.7.7) and id: Z^{∘} → Z^{∘} induce a PD-morphism \overline{E^{′∘}}_{n} → \overline{E^{∘}}_{n} compatible with the actions of G_{U}^{∘} and G_{U}^{′∘} and with the liftings of Frobenius. If we denote by the images of u_{i}, u ∈ Γ(\overline{E^{∘}}_{n}, O_{\overline{E^{∘}}_{n}}^{∗}) in Γ(\overline{E^{′∘}}_{n}, O_{\overline{E^{′∘}}_{n}}^{∗}) by the same symbols, we have (cf. [Ts2] Lemma 3.1.4)

O_{\overline{E^{′∘}}_{n}} ≅ O\overline{D′}_{n}^{∘}⟨u_{i} − 1, u − 1⟩.

Hence, if we define { \atop J_{\overline{E^{′∘}}_{n}}^{[r]′}} similarly as in [Ts2] §3.1, [Ts2] Lemma 3.1.6 still

holds. Similarly as in [Ts2] Lemma 3.1.7, we obtain a resolution J_{\overline{D′}_{n}^{∘}}^{[r]′} →

J_{\overline{E^{′∘}}_{n}}^{[r−•]′} ⊗ Ω_{Z_{n}^{∘}/W_{n}}^{•}. Thus, as in [Ts2] §3.1, we obtain the required resolution

\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}} of ℤ/p^{n}ℤ(r)′ on ((U_{triv}^{′h})^{∘})_{Két} by taking the global section of the mapping fiber of

1 − φ_{r}: J_{\overline{E^{′∘}}_{n}}^{[r−•]′} ⊗ Ω_{Z_{n}^{∘}/W_{n}}^{•} → O_{\overline{E^{′∘}}_{n}} ⊗ Ω_{Z_{n}^{∘}/W_{n}}^{•}.

The morphism \overline{E^{′∘}}_{n} → \overline{E^{∘}}_{n} induces a map from the pull-back of \overline{S}_{n}(r)_{U^{∘},Z^{∘}} to \overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}. The comparison with \overline{S}_{n}(r)_{U′,Z′} is non-trivial. Consider the morphism \overline{D′}^{∘} × Z^{∘} → Spec(ℤ[ℕ ⊕ ℕ]) defined by sending (1, 0), (0, 1) ∈ ℕ ⊕ ℕ to the images of a′ ∈ Γ(\overline{D^{∘}}, M\overline{D^{∘}}) and t ∈ Γ(Z^{∘}, M_{Z^{∘}}) in Γ(\overline{D′}^{∘} × Z^{∘}, M). Define the log étale morphism Spec(ℤ[ℕ ⊕ ℤ]) → Spec(ℤ[ℕ ⊕ ℕ]) by ℕ ⊕ ℕ → ℕ ⊕ ℤ; (m, n) ↦ (m, m − n), and consider the following cartesian diagrams:

(\overline{D′}^{∘} × Z^{′∘})^{∼} → (\overline{D′}^{∘} × Z^{∘})^{∼} → Spec(ℤ[ℕ ⊕ ℤ])
↓ ↓ ↓
\overline{D′}^{∘} × Z^{′∘} → \overline{D′}^{∘} × Z^{∘} → Spec(ℤ[ℕ ⊕ ℕ]).

Then, the closed immersion \overline{U′}^{∘} ↪ \overline{D′}^{∘} × Z^{′∘} naturally factors through (\overline{D′}^{∘} × Z^{′∘})^{∼} because the images of a′ and t coincide in Γ(\overline{U′}^{∘}, M). On the other hand, the vanishing of a′ in Γ(\overline{D′}^{∘} × Z^{∘}, O) implies that (\overline{D′}^{∘} × Z^{′∘})^{∼} → (\overline{D′}^{∘} × Z^{∘})^{∼} is an isomorphism. Hence \overline{E^{′∘}}_{n} is isomorphic to the PD-envelope of \overline{U′}_{n}^{∘} ↪ \overline{D′}_{n}^{∘} × Z_{n}^{′∘}, and the diagram (4.7.6) and Z^{′∘} → Z′ induce a PD-morphism \overline{E^{′∘}}_{n} → \overline{E′}_{n}. Here \overline{E′}_{n} denotes the PD-envelope of \overline{U′}_{n} ↪ \overline{D′}_{n} × Z′_{n} used in the construction of \overline{S}_{n}(r)_{U′,Z′}. One can verify the compatibility with the actions of G′_{U}, G_{U}^{′∘}, with the liftings of Frobenius and with the connections. Thus we obtain a canonical map \overline{S}_{n}(r)_{U′,Z′} → ε′_{U∗}\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}.

Lemma 4.7.8. The map \overline{S}_{n}(r)_{U′,Z′} → ε′_{U∗}\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}} is injective.

Proof. This follows from O_{\overline{E′}_{n}} ≅ O_{\overline{D′}_{n}}⟨u_{i} − 1⟩ and O_{\overline{E^{′∘}}_{n}} ≅ O_{\overline{D′}_{n}}⟨u_{i} − 1, u − 1⟩. □

Next we discuss on Gysin sequence for \overline{S}_{n}(r) on (U′_{triv})_{ét}. Recall that Z^{′∘} is not smooth over W.

Lemma 4.7.9. (1) The natural map Ω_{Z^{∘}/W} ⊗_{O_{Z^{∘}}} O_{Z^{′∘}} → Ω_{Z^{′∘}/W} is an isomorphism.

(2) Let F be an O_{Z′}(= O_{Z^{′∘}})-module with an integrable connection ∇ : F → F ⊗ Ω_{Z′/W}. Then the composite

F → F ⊗_{O_{Z′}} Ω_{Z′/W} → F ⊗_{O_{Z′}} Ω_{Z^{′∘}/W} ≅ F ⊗_{O_{Z}} Ω_{Z^{∘}/W} is an integrable connection on F as an O_{Z^{∘}}-module. Furthermore, the natural maps F ⊗Ω_{Z′}^{q} → F ⊗Ω_{Z^{′∘}}^{q} ≅ F ⊗Ω_{Z^{∘}}^{q} induce a morphism between the de Rham complexes: F ⊗ Ω_{Z′}^{•} → F ⊗ Ω_{Z^{∘}}^{•}.

Proof. Straightforward. □

By Lemma 4.7.9 above, we can replace Ω_{Z′}^{q} with Ω_{Z^{′∘}}^{q} in \overline{S}_{n}(r)_{U′,Z′}, and obtain a complex on (U_{triv}^{′h})_{ét}, which we denote by \overline{S}_{n}(r)_{U′,Z′}^{∘}. We can construct a short exact sequence:

(4.7.10)  0 ⟶ \overline{S}_{n}(r)_{U′,Z′} ⟶ \overline{S}_{n}(r)_{U′,Z′}^{∘} ⟶ \overline{S}_{n}(r − 1)_{U′,Z′}[−1] ⟶ 0

in an obvious way. On the other hand, the PD-morphism \overline{E^{′∘}}_{n} → \overline{E′}_{n} induces a map \overline{S}_{n}(r)_{U′,Z′}^{∘} → ε′_{U∗}\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}} compatible with the map in Lemma 4.7.8. Finally, we discuss on the complex S_{n}^{∼}(r). As in [Ts2] §2.1, using the PDenvelopes of X ↪ Z, X^{∘} ↪ Z^{∘} and X′ ↪ Z′, we can define complexes S_{n}^{∼}(r)_{X,Z}, S_{n}^{∼}(r)_{X^{∘},Z^{∘}} and S_{n}^{∼}(r)_{X′,Z′} on (\underline{X_{1}})_{ét}, (\underline{X_{1}^{∘}})_{ét} = (\underline{X_{1}})_{ét} and (\underline{X′_{1}})_{ét}. We have natural maps from the first complex to the latter two. We also have natural maps from the sections over \underline{U_{1}}, \underline{U_{1}^{∘}} = \underline{U_{1}} and \underline{U′_{1}} to the global sections of \overline{S}_{n}(r)_{U,Z}, \overline{S}_{n}(r)_{U^{∘},Z^{∘}} and \overline{S}_{n}(r)_{U′,Z′} (cf. [Ts2] (3.1.9) and (2.1.2)). For X^{′∘} ↪ Z^{′∘}, we define the complex S_{n}^{∼}(r)_{X^{′∘},Z^{′∘}} on (\underline{X_{1}^{′∘}})_{ét} = (\underline{X′_{1}})_{ét} to be the one obtained by replacing Ω_{Z′}^{q} with Ω_{Z^{′∘}}^{q} in S_{n}^{∼}(r)_{X′,Z′}, using Lemma 4.7.9. We can construct a short exact sequence:

(4.7.11) 0 ⟶ S_{n}^{∼}(r)_{X′,Z′} ⟶ S_{n}^{∼}(r)_{X^{′∘},Z^{′∘}} ⟶ S_{n}^{≈}(r − 1)_{X′,Z′}[−1] ⟶ 0.

Here S_{n}^{≈}(r − 1)_{X′,Z′} is the complex obtained from S_{n}^{∼}(r − 1)_{X′,Z′} by replacing p^{r−1} − φ with p^{r} − pφ. We have a natural map from Γ(\underline{U′_{1}}, S_{n}^{∼}(r)_{X^{′∘},Z^{′∘}}) to the global section of \overline{S}_{n}(r)_{U′,Z′}^{∘}, and (4.7.11) is compatible with (4.7.10). Noting that the PD-envelope of U^{′∘} ↪ Z^{′∘} is isomorphic to the PD-envelope of U′ ↪ Z′, one can also construct a natural map from the pull-back of S_{n}^{∼}(r)_{X^{∘},Z^{∘}} to S_{n}^{∼}(r)_{X^{′∘},Z^{′∘}} and (4.7.11) is compatible with the short exact sequence (cf. (4.3.1)):

(4.7.12) 0 ⟶ S_{n}^{∼}(r)_{X,Z} ⟶ S_{n}^{∼}(r)_{X^{∘},Z^{∘}} ⟶ S_{n}^{≈}(r − 1)_{X′,Z′}[−1] ⟶ 0.

§4.8. Proof of Proposition 4.5.3. We keep the notation and the assumption in §4.7. We first construct the morphisms α (4.5.5) and β (4.5.6). Choose sufficiently large algebraically closed fields Ω of characteristic 0 and Ω′ of characteristic p. Let S be the set of all isomorphic classes of fs monoids P such that P^{∗} = {1}. For each isomorphic class c ∈ S, choose a representative P_{c} of c and define the log geometric point Ω_{c} to be Spec(Ω) with M_{Ω_{c}} = Ω ⊕ ∪_{n∈ℕ,n≠0} \frac{1}{n}P_{c} and Ω′_{c} to be Spec(Ω′) with M_{Ω′_{c}} = Ω′ ⊕ ∪_{n∈ℕ,p∤n} \frac{1}{n}P_{c}. In the following, we denote by C^{∗} the Godement resolution with respect to all log geometric points whose sources are Ω_{c} or Ω′_{c} for some c ∈ S. Note that such log geometric points form a set. To simplify the notation, we write Θ for the operation C^{∗}i_{∗}i^{∗}j_{∗}C^{∗} and Θ′ for C^{∗}i′_{∗}i^{′∗}j′_{∗}C^{∗}. Denote by i_{U} and i′_{U} the closed immersions \underline{U} ⊗k = \underline{U^{h}} ⊗k → \underline{U^{h}} and \underline{U′}⊗k = \underline{U^{′h}}⊗k → \underline{U^{′h}}, and by j_{U}, j_{U}^{∘}, j′_{U} the open immersions U_{triv}^{h} → \underline{U^{h}}, (U^{h∘})_{triv} → \underline{U^{h}} and U_{triv}^{′h} → \underline{U^{′h}}. Similarly, as above, we denote by Θ_{U} and Θ′_{U} the operations C^{∗}i_{U∗}i_{U}^{∗}j_{U∗}C^{∗} and C^{∗}i′_{U∗}i_{U}^{′∗}j′_{U∗}C^{∗}. Since the derived direct images of ℤ/p^{n}ℤ(r)′ by the left morphisms in the first lines of the diagrams (4.7.1) and (4.7.2) are again ℤ/p^{n}ℤ(r)′ ([I2] Theorem 7.4), we see that the left and middle vertical morphisms in the diagram in Proposition 4.5.3 are induced by sheafifying the following morphisms of presheaves on \underline{X}_{ét}:

Γ(\underline{U}, Θ(Λ)) →^{q.i.} Γ(\underline{U^{h}}, Θ_{U}(Λ)) →^{q.i.} Γ(\underline{U^{h}}, Θ_{U}(\overline{S}_{n}(r)_{U,Z})) ← Γ(\underline{U}, i_{∗}S_{n}^{∼}(r)_{X,Z})

Γ(\underline{U}, Θ(ε_{∗}C^{∗}(Λ))) →^{q.i} Γ(\underline{U^{h}}, Θ_{U}(ε_{U∗}C^{∗}(Λ))) →^{q.i.} Γ(\underline{U^{h}}, Θ_{U}(ε_{U∗}C^{∗}(\overline{S}_{n}(r)_{U^{∘},Z^{∘}}))) ← Γ(\underline{U}, i_{∗}S_{n}^{∼}(r)_{X^{∘},Z^{∘}})

Here Λ = ℤ/p^{n}ℤ(r)′ and q.i. means a quasi-isomorphism. See (4.7.1) and (4.7.2) for ε and ε_{U}. Let K^{•}, K_{U}^{•} and L_{U}^{•} be the cokernels of the injective homomorphisms Λ_{X_{triv}} → ε_{∗}C^{∗}(Λ_{(X_{triv})^{∘}}), Λ_{U_{triv}^{h}} → ε_{U∗}C^{∗}(Λ_{(U_{triv}^{h})^{∘}}), and \overline{S}_{n}(r)_{U,Z} → ε_{U∗}C^{∗}(\overline{S}_{n}(r)_{U^{∘},Z^{∘}}) (Lemma 4.7.5). We have a natural injective homomorphism from the first line to the second one. Taking its quotient and using (4.7.12), we obtain

(4.8.1)  Γ(\underline{U},Θ(K^{•}))→^{q.i.}Γ(\underline{U^{h}},Θ_{U} (K_{U}^{•}))→^{q.i.}Γ(\underline{U^{h}},Θ_{U} (L_{U}^{•}))←Γ(\underline{U},i_{∗}S_{n}^{≈}(r)_{X′,Z′} [−1])

We have quasi-isomorphisms K^{•} ← τ_{≤1}K^{•} → H^{1}(K^{•})[−1] ≅ Λ(−1)_{X′_{triv}}[−1] ([I2]Theorem 7.4). Hence varying U and sheafifying, we obtain the required morphism α (4.5.5). We can apply the same argument to ε′: ((X′_{triv})^{∘})_{Két} → (X′_{triv})_{ét}, ε′_{U}: ((U_{triv}^{′h})^{∘})_{Két} → (U_{triv}^{′h})_{ét} and the resolutions Λ_{U_{triv}^{′h}} → \overline{S}_{n}(r)_{U′,Z′} and Λ_{(U_{triv}^{′h})^{∘}} → \overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}. We define K^{′•}, K_{U}^{′•} and L_{U}^{′•} to be the cokernels of the injective homomorphisms Λ_{X′_{triv}} → ε′_{∗}C^{∗}(Λ_{(X′_{triv})^{∘}}), Λ_{U_{triv}^{′h}} → ε′_{U∗}C^{∗}(Λ_{(U_{triv}^{′h})^{∘}}) and \overline{S}_{n}(r)_{U′,Z′} → ε′_{U∗}C^{∗}(\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}) (Lemma 4.7.8). Then using (4.7.11), we obtain

(4.8.2)  Γ(\underline{U′},Θ′(K^{′•}))→^{q.i.}Γ(\underline{U^{′h}},Θ′_{U} (K_{U}^{′•}))→^{q.i.}Γ(\underline{U^{′h}},Θ′_{U} (L_{U}^{′•}))←Γ(\underline{U′},i′_{∗}S_{n}^{≈}(r)_{X′,Z′} [−1])

By [I2]Theorem 7.4 and [Na] Theorem (5.1), we have quasi-isomorphisms K^{′•} ← τ_{≤1}K^{′•} → H^{1}(K^{′•})[−1] ≅ Λ(−1)_{X′_{triv}}[−1]. Hence, varying U and sheafifying, we obtain the required β. We have a natural map from (4.8.1) to (4.8.2) and hence the two maps α and β coincide. Let us compare β with the map (4.5.4). By (4.7.10), we have a morphism \overline{S}_{n}(r − 1)_{U′,Z′}[−1] → L_{U}^{′•} and the last map of (4.8.2) factors through Γ(\underline{U^{′h}}, Θ′_{U}(\overline{S}_{n}(r)_{U′,Z′}[−1])). We have the following commutative diagram of complexes on (U_{triv}^{′h})_{ét}:

K_{U}^{′•}  ←^{q.i.}  τ_{≤1}K_{U}^{′•}  →^{q.i.} ℤ/p^{n}(r)′(−1)[−1]
q.i. ↓  q.i. ↓  ∥
L_{U}^{′•}  ←^{q.i.}  τ_{≤1}L_{U}^{′•}  →^{q.i.} ℤ/p^{n}(r)′(−1)[−1]
↑  ↑  (∗) ↑
\overline{S}_{n}(r − 1)_{U′,Z′}[−1] ←^{q.i.} τ_{≤1}(\overline{S}_{n}(r − 1)_{U′,Z′}[−1]) ≅ ℤ/p^{n}(r − 1)′[−1]

Here the morphism (∗) is the composite (4.8.3)

ℤ/p^{n}(r − 1)′ ≅ H^{0}(\overline{S}_{n}(r − 1)_{U′,Z′}) → H^{1}(L_{U}^{′•}) ←^{∼} H^{1}(K_{U}^{′•}) ≅ ℤ/p^{n}(r)′(−1)

Hence to prove the coincidence of β and (4.5.4), it suffices to prove the following:

Proposition 4.8.4. The map (4.8.3) is the natural map.

Proof. By the definition of K_{U}^{′•} and L_{U}^{′•}, the map (4.8.3) coincides with the composite of

ℤ/p^{n}(r − 1)′ ≅ H^{0}(\overline{S}_{n}(r − 1)_{U′,Z′}) ←^{≅} H^{1}(\overline{S}_{n}(r)_{U′,Z′}^{∘}) → R^{1}ε′_{U∗}\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}}

←^{≅} R^{1}ε′_{U∗}ℤ/p^{n}(r)′ ≅^{(∗∗)} ℤ/p^{n}(r)′(−1)

where the second isomorphism is defined by (4.7.10). Note that all sheaves appearing above are ind locally constant. Let I be the kernel of the surjection G_{U}^{′∘} → G′_{U}, which is canonically isomorphic to ℤ̂(1). Then we have a natural isomorphism H^{1}(I, ℤ/p^{n}(r)′) ≅ Hom(ℤ̂(1), ℤ/p^{n}(r)′) ≅ ℤ/p^{n}(r)′(−1), and it is compatible with the isomorphism (∗∗) above. Hence we may replace R^{1}ε′_{U∗}(−) with H^{1}(I, −) regarding locally constant sheaves on ((U_{triv}^{′h})^{∘})_{Két} and (U_{triv}^{′h})_{ét} as G_{U}^{′∘} and G′_{U}-modules. Let α ∈ ℤ/p^{n}(r − 1)′ ≅ H^{0}(\overline{S}_{n}(r − 1)_{U′,Z′}). Since F_{Z_{n}}^{∗}(t) = t^{p}, we see that the image of α in H^{1}(\overline{S}_{n}(r)_{U′,Z′}^{∘}) is the class of (α·d log(t), 0). Choose an isomorphism M_{η^{′∘ur}} ≅ ℕ̃⊕(K^{′ur})^{∗} as in the definition

of \overline{D′}^{∘} in §4.7. Then the pair (t, \{(1/p^{n}, 1)\}_{n∈ℕ}) defines an element of Q^{′∘} and we denote its image under Q^{′∘} → Γ(\overline{D′}_{n}^{∘}, M) → Γ(\overline{E^{′∘}}_{n}, M) by [\underline{t}]. Since the images of t and [\underline{t}] in Γ(\overline{U′}_{n}^{∘}, M) coincide, there exists a unique u ∈ Γ(\overline{E^{′∘}}_{n}, 1 + J_{\overline{E^{′∘}}_{n}}) such that u · t = [\underline{t}] in Γ(\overline{E^{′∘}}_{n}, M). We have φ(u) = u^{p} and d log(u) = −d log(t). Hence (α · d log(t), 0) ∈ (\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}})^{1} is the image of −α · log(u) ∈ (\overline{S}_{n}(r)_{U^{′∘},Z^{′∘}})^{0} by the differential map. Hence the image of (α · d log(t), 0) in H^{1}(I, ℤ/p^{n}ℤ(r)′) is given by the cocycle σ ↦ −(σ(−α log(u))−(−α log(u))) = α · log(σ([\underline{t}])[\underline{t}]^{−1}). This completes the proof because I ≅ ℤ̂(1) → ℤ/p^{n}(1)′ ⊂ Γ(\overline{E^{′∘}}_{n}, J_{\overline{E^{′∘}}_{n}}) is given by σ ↦ log(σ([\underline{t}])[\underline{t}]^{−1}). □

In the case that X does not have a global embedding into Z as in the beginning of §4.7, we choose a strict étale covering X^{0} → X, and X^{0} ↪ Z^{0}, \underline{Z^{′0}} ⊂ \underline{Z^{0}}, and \{F_{Z_{n}^{0}} : Z_{n}^{0} → Z_{n}^{0}\} satisfying the conditions in the beginning of §4.7. Such a covering and an embedding exist by a similar argument as the proof of Proposition 4.2.1 (1). From this embedding, we can construct X^{•} ↪ Z^{•} and \underline{Z^{′•}} ⊂ \underline{Z^{•}} as in Proposition 4.2.2 endowed with \{F_{Z_{n}^{•}}\}. We can verify that X^{ν} ↪ Z^{ν}, \underline{Z^{′ν}} ⊂ \underline{Z^{ν}} and \{F_{Z_{n}^{ν}}\} satisfy the conditions in the beginning of §4.7 for each ν ∈ ℕ. By applying the above argument to each level, we can construct α and β on (\underline{X^{•}})_{ét}, which coincide with each other, and show that β coincides with (4.5.4) on (\underline{X^{•}})_{ét}. Note that our construction does not depend on {t_{1}, …, t_{d}, t} chosen in the beginning of §4.7. By taking Rθ_{∗} for the morphism of topoi θ: (\underline{X^{•}})_{ét}^{∼} → (\underline{X})_{ét}^{∼}, we obtain Proposition 4.5.3 for a general X.

§4.9. Proof of Theorem 4.1.2. We will prove Theorem 4.1.2 by the induction on the number of elements of I. In the case that I is empty, the theorem is nothing but Theorem 3.2.2. Assume that I is non-empty, choose i_{0} ∈ I and we define (X, M), (X, M^{∘}) and (X′, M′) as in the beginning of §4.3. As the induction hypothesis, we assume that Theorem 4.1.2 is true for (X, M) and (X′, M′). By Lemma 4.4.8, Lemma 4.4.9 and Proposition 4.5.2, for an integer r ≥ 2 dim(X_{K}), the comparison maps B_{st} ⊗_{ℚ_{p}} V_{i}^{q}(r) → B_{st} ⊗_{K_{0}} D_{i}^{q}(r) (i = 1, 2) and B_{st} ⊗_{ℚ_{p}} V_{3}^{q}(r−1) → B_{st} ⊗_{K_{0}} D_{3}^{q}(r−1) are compatible with the Gysin exact sequences (4.5.1) and (4.4.1). Since the comparison maps are isomorphisms for (X, M) and (X′, M′) for every q by the induction hypothesis, we see that the comparison map for (X, M^{∘}) is also an isomorphism for every q. Furthermore, by Lemma 4.4.6, the comparison maps above tensored with B_{dR} over B_{st} send F il^{i} to F il^{i} and are compatible with the Gysin exact sequences (4.5.1) and (4.4.5). By the induction hypothesis, the comparison maps tensored with B_{dR} are filtered isomorphisms for (X, M) and (X′, M′). Hence by five lemma, it also holds for (X, M^{∘}). Thus we see that Theorem 4.1.2 is true for (X^{∘}, M^{∘}).

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Takeshi Tsuji

Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro Tokyo 153-8914 Japan t-tsuji@ms.u-tokyo.ac.jp