Cycle Classes for p-Adic Étale Tate Twists

and the Image of p-Adic Regulators

Kanetomo Sato^{1}

Received: December 23, 2011 Revised: February 8, 2013

Communicated by Stephen Lichtenbaum

Abstract. In this paper, we construct Chern class maps and cycle class maps with values in p-adic étale Tate twists [Sa2]. We also relate the p-adic étale Tate twists with the finite part of Bloch-Kato. As an application, we prove that the integral part of p-adic regulator maps has values in the finite part of Galois cohomology under certain assumptions.

2000 Mathematics Subject Classification: Primary 19F27, 14F30; Secondary 14C25

1 Introduction

Let p be a rational prime number. Let A be a Dedekind ring whose fraction field has characteristic zero and which has a residue field of characteristic p. Let X be a regular scheme of pure-dimension which is flat of finite type over S := Spec(A) and which is a smooth or semistable family around its fibers over S of characteristic p. Extending the idea of Schneider [Sc], the author defined in [Sa2] the objects 𝔗_{n}(r)_{X} (r, n ≥ 0) of the derived category of étale ℤ/p^{n}-sheaves on X playing the role of the r-th Tate twist with ℤ/p^{n}-coefficients, which are endowed with a natural product structure with respect to r and both contravariantly and covariantly functorial (i.e., there exist natural pull-back and trace morphisms) for arbitrary separated morphisms of finite type of such schemes. Those pull-back and trace morphisms satisfy the projection formula.

The first aim of this paper is to construct the following Chern class map and cycle class map for m, r ≥ 0:

\xymatrix{K_{m}(X) \ar[dr]^{C_{r,m} (Chern class)} \ar@{.>}[d]_{(missing)} \\ CH^{r}(X, m) \ar[r]^{_{cl{r,m \atop X}}}_{(cycle class)} & H_{ét}^{2r−m}(X, 𝔗_{n}(r)_{X}),}

1supported by JSPS Core-to-Core Program 18005 and Grant-in-Aid for Scientific Research B-23340003

Documenta Mathematica 18 (2013) 177–247

where K_{m}(X) denotes the algebraic K-group [Q] and CH^{r}(X, m) denotes the higher Chow group [B2]. The Chern classes with values in higher Chow groups have not been defined in this arithmetic situation for the lack of a product structure on them, which we do not deal with in this paper. We will prepare the following results to construct the above maps:

(a) The Dold-Thom isomorphism, i.e., the projective bundle formula for p-adicétale Tate twists. See Theorem 4.1 below for details. We note here that the p-adic étale Tate twists do not satisfy homotopy invariance.

(b) For a regular closed immersion i : X_{⋆} ↪ X′_{⋆} of codimension c of simplicial schemes for which the p-adic étale Tate twists are defined, we construct a Gysin morphism

gys_{i} : 𝔗_{n}(r)_{X_{⋆}} ⟶ Ri^{!}𝔗_{n}(r + c)_{X′_{⋆}}[2c] in D((X_{⋆})_{ét}, ℤ/p^{n})

satisfying transitivity and projection formula. See Proposition 5.4 below for details.

(c) We introduce a version of p-adic étale Tate twists with log poles along horizontal normal crossing divisors (see §3 below), and prove a p-adic analogue of the usual homotopy invariance (see Corollary 4.3 below) and a semi-purity property (see Theorem 6.5 below) for this new coefficient.

The existence of C_{r,m} will be verified by (a) and the general framework due to Gillet [Gi1]. The additivity of C_{r,m} for m ≥ 1 will follow from (b) (cf. §5). On the other hand, we will need the results in (c) to construct cl_{X}^{r,m} (cf. §7). This ‘higher’ cycle class map will be a fundamental object to study in ‘higher’ higher classfield theory [Sai]. We will mention a local behavior of cl_{X}^{r,m} in Remark 7.2 below.

The second aim of this paper is to relate the p-adic étale Tate twists with the finite part of Galois cohomology [BK2], using the Fontaine-Jannsen conjecture proved by Hyodo, Kato and Tsuji ([HK], [K4], [Ts1], cf. [Ni2]). We assume here that A is a p-adic integer ring and that X is projective over A with strict semistable reduction. Let K be the fraction field of A and put X_{K} := X ⊗_{A} K = X[p^{−1}]. We define

H^{i}(X, 𝔗_{ℚ_{p}}(r)_{X}) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H_{ét}^{i}(X, 𝔗_{n}(r)_{X}),
H^{i}(X_{K}, ℚ_{p}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H_{ét}^{i}(X_{K}, μ_{p^{n}}^{⊗r}),

where μ_{p^{n}} denotes the étale sheaf of p^{n}-th roots of unity on X_{K}, i.e, the usual Tate twist on X_{K}. We have a natural restriction map H^{i}(X, 𝔗_{ℚ_{p}}(r)_{X}) → H^{i}(X_{K}, ℚ_{p}(r)) and a canonical descending filtration F^{•} on H^{i}(X_{K}, ℚ_{p}(r)) resulting from the Hochschild-Serre spectral sequence for the covering X\overline{K} := X_{K} ⊗_{K} \overline{K} → X_{K} (cf. (9.0.1)). We define a (not necessarily exhaustive) filtration F^{•} on H^{i}(X, 𝔗_{ℚ_{p}}(r)_{X}) as the inverse image of F^{•} on H^{i}(X_{K}, ℚ_{p}(r)), which induces obvious inclusions for m ≥ 0

\xymatrix{gr_{F}^{m}H^{i}(X, 𝔗_{ℚ_{p}}(r)_{X}) \ar@{^{(}->}[r] & gr_{F}^{m}H^{i}(X_{K}, ℚ_{p}(r)) ≃ H^{m}(K, H^{i−m}(X\overline{K}, ℚ_{p}(r))).}

Cycle Class and p-Adic Regulator 179

Here H^{∗}(K, −) denotes the continuous Galois cohomology of the absolute Galois group G_{K} = Gal(\overline{K}/K) defined by Tate [Ta]. We will prove that

gr_{F}^{1} H^{i}(X, 𝔗_{ℚ_{p}}(r)_{X}) ⊂ H_{f}^{1}(K, H^{i−1}(X\overline{K}, ℚ_{p}(r))) (f = finite part)

assuming that p is sufficiently large and that the monodromy-weight conjecture [Mo] holds for the log crystalline cohomology of the reduction of X in degree i − 1 (see Theorem 9.1 below). This result is an extension of the p-adic point conjecture ([Sc], [LS], [Ne1]) to the semistable reduction case and gives an ‘unramified’ version of results of Langer [La] and Nekovář [Ne2] relating the log syntomic cohomology of X with the geometric part H_{g}^{1}(K, H^{i−1}(X\overline{K}, ℚ_{p}(r))).

There is an application of the above results as follows. Let K be a number field and let V be a proper smooth geometrically integral variety over K. Put \overline{V} := V ⊗_{K} \overline{K}. Let i and r be non-negative integers with 2r ≥ i + 1, and let p be a prime number. The étale Chern characters (cf. [So]) induce p-adic regulator maps

reg_{p}^{2r−i−1,r} : K_{2r−i−1}(V)_{𝔬} ⟶ H^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r)))  (2r > i + 1),
reg_{p}^{0,r} : K_{0}(V)_{hom} ⟶ H^{1}(K, H^{2r−1}(\overline{V} , ℚ_{p}(r)))  (2r = i + 1).

Here K_{0}(V)_{hom} denotes the homologically trivial part of K_{0}(V), and K_{m}(V)_{𝔬} denotes the integral part of K_{m}(V) in the sense of Scholl (see §10 below). Motivated by the study of special values of L-functions, Bloch and Kato [BK2] conjecture that the image of reg_{p}^{2r−i−1,r} is contained in the finite part H_{f}^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))) and spans it over ℚ_{p}. In the direction of this conjecture, we will prove the following result, which extends a result of Nekovář [Ne2] Theorem 3.1 on reg_{p}^{0,r} to the case 2r > i + 1 and extends a result of Niziol [Ni1] on the potentially good reduction case to the general case:

Theorem 1.1 (§10) Assume r ≤ p − 2 and the monodromy-weight conjecture for the log crystalline cohomology of degree i of projective strict semistable varieties over 𝔽_{p}. Then we have

Im(reg_{p}^{2r−i−1,r}) ⊂ H_{f}^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))).

Here a projective strict semistable variety over 𝔽_{p} means the reduction of a regular scheme which is projective flat over a p-adic integer ring with strict semistable reduction.

We use the alteration theorem of de Jong [dJ] to prove Theorem 1.1, and the projective strict semistable varieties concerned in the assumption mean those obtained from alterations of scalar extensions of V to the completion of K at places dividing p.

This paper is organized as follows. In §2, we introduce cohomological and homological logarithmic Hodge-Witt sheaves with horizontal log poles on normal crossing varieties over a field of characteristic p > 0. In §3, we define p-adic étale Tate twists with horizontal log poles, and construct a localization sequence using this object (Theorem 3.12). In §4 and §5, we prove the Dold-Thom isomorphisms and define the Chern class maps for p-adic étale Tate twists. The sections 6 and 7 will be devoted to the construction of cycle class maps for p-adic étale Tate twists. In §8 we will introduce Hodge-Witt cohomology and homology of normal crossing varieties and prove that the monodromy-weight conjecture implies a certain invariant cycle theorem. In §9, we establish the comparison between p-adic étale Tate twists and the finite part of Bloch-Kato. We will prove Theorem 1.1 in §10. In the appendix, we will formulate a continuous crystalline cohomology and a continuous syntomic cohomology to prove several technical compatibility results which will have been used in §9.

The author expresses his gratitude to Professors Shuji Saito and Takeshi Tsuji, and Masanori Asakura for valuable comments and discussions on the subjects of this paper.

Notation

For an abelian group M and a positive integer n, _{n}M and M/n denote the kernel and the cokernel of the map M ⟶^{×n} M, respectively. For a field k, \overline{k} denotes a fixed separable closure, and G_{k} denotes the absolute Galois group Gal(\overline{k}/k). For a topological G_{k}-module M, H^{∗}(k, M) denote the continuous Galois cohomology groups H_{cont}^{∗}(G_{k}, M) in the sense of Tate [Ta]. If M is discrete, then H^{∗}(k, M) agree with the étale cohomology groups of Spec(k) with coefficients in the étale sheaf associated with M.

Unless indicated otherwise, all cohomology groups of schemes are taken over the étale topology. For a scheme X, an abelian sheaf ℱ on X_{ét} (or more generally an object in the derived category of abelian sheaves on X_{ét}) and a point x ∈ X, we often write H_{x}^{∗}(X, ℱ) for H_{x}^{∗}(Spec(𝒪_{X,x}), ℱ). For a positive integer m which is invertible on X, we write μ_{m} for the étale sheaf of the m-th roots of unity on X. For a prime number p which is invertible on X and integers m, r ≥ 0, we define

H^{m}(X, ℤ_{p}(r)) := \underleftarrow{lim}_{n≥1} H^{i}(X, μ_{p^{n}}^{⊗r}),
H^{m}(X, ℚ_{p}(r)) := ℚ_{p} ⊗_{ℤ_{p}} H^{m}(X, ℤ_{p}(r)).

For an equi-dimensional scheme X and a non-negative integer q, we write X^{q} for the set of all points on X of codimension q.

For a scheme (or a simplicial scheme) X and an integer n ≥ 2, Shv(X_{ét}, ℤ/n) denotes the category of étale ℤ/n-sheaves on X_{ét}, and D(X_{ét}, ℤ/n) denotes the derived category of complexes of objects of Shv(X_{ét}, ℤ/n).

2 Logarithmic Hodge-Witt sheaves

We first fix the following terminology.

Definition 2.1 (1) A normal crossing varity over a field k is a pure-dimensional scheme which is separated of finite type over k and everywhere étale locally isomorphic to

Spec(k[t_{0}, …, t_{N}]/(t_{0} ⋯ t_{a})) for some 0 ≤ a ≤ N := dim(Y).

(2) We say that a normal crossing variety Y is simple if all irreducible components of Y are smooth over k.

(3) An admissible divisor on a normal crossing varity Y is a reduced effective Cartier divisor D such that the immersion D ↪ Y is everywhere étale locally isomorphic to

Spec(k[t_{0}, …, t_{N}]/(t_{0} ⋯ t_{a}, t_{a+1} ⋯ t_{a+b})) ↪ Spec(k[t_{0}, …, t_{N}]/(t_{0} ⋯ t_{a}))

for some a, b ≥ 0 with a + b ≤ N = dim(Y).

Let p be a prime number, and let k be a field of characteristic p. Let Y be a normal crossing variety over k, and let D be an admissible divisor on Y. Put V := Y − D and let f and g be as follows:

\xymatrix{D \ar@{^{(}->}[r]^{f} & Y & V = Y − D. \ar@{^{(}->}[l]_{g}}

For r ∈ ℤ, we define étale sheaves ν_{(Y,D),n}^{r} and λ_{(Y,D),n}^{r} on Y as follows:

λ_{(Y,D),n}^{r} := Im (dlog : (g_{∗}𝒪_{V}^{×})^{⊗r} → ⨁_{x∈V^{0}} i_{x∗}W_{n} Ω_{x,log}^{r} )
ν_{(Y,D),n}^{r} := Ker (∂ : ⨁_{x∈V^{0}} i_{x∗}W_{n} Ω_{x,log}^{r} → ⨁_{x∈V^{1}} i_{x∗}W_{n} Ω_{x,log}^{r−1} ) .

Here for x ∈ V, i_{x} denotes the composite map x ↪ V ↪ Y, and W_{n} Ω_{x,log}^{q} denotes the étale subsheaf of the logarithmic part of the Hodge-Witt sheaf W_{n} Ω_{x}^{q} (cf. [B1], [I1]), which means the zero sheaf for q < 0. The map ∂ is the sum of boundary maps due to Kato [K1]. By definition, we have

λ_{(Y,D),n}^{r} = ν_{(Y,D),n}^{r} = 0 for r < 0.

When D = ∅, we put

ν_{Y,n}^{r} := ν_{(Y,∅),n}^{r} and λ_{Y,n}^{r} := λ_{(Y,∅),n}^{r},

which are considered in [Sa1]. Although we assumed the perfectness of k in [Sa1], the results in loc. cit. §§2.1–2.4 and §§3.1–3.2 are extended to the case that k is not necessarily perfect by results of Shiho [Sh] Theorems 3.2, 4.1 and the compatibility mentioned in [AS] Remark 7.3 (1). We have ν_{(Y,D),n}^{r} = g_{∗}ν_{V,n}^{r} by the left exactness of g_{∗}, and ν_{D,n}^{r−1} ≃ R^{1}f^{!}ν_{Y,n}^{r} by the purity of ν_{Y,n}^{r} ([Sa1] Theorem 2.4.2). Hence there is a short exact sequence

0 ⟶ ν_{Y,n}^{r} ⟶ ν_{(Y,D),n}^{r} ⟶ f_{∗}ν_{D,n}^{r−1} ⟶ 0.  (2.1.1)

By this fact, we have

λ_{(Y,D),n}^{r} = ν_{(Y,D),n}^{r} = g_{∗}W_{n} Ω_{V,log}^{r}

if Y is smooth (loc. cit. (2.4.9)), which we denote by W_{n} Ω_{(Y,D),log}^{r}. The following proposition will be useful later, where Y is not necessarily smooth:

Proposition 2.2 Assume that Y is simple, and let Y_{1}, Y_{2}, ⋯, Y_{q} be the distinct irreducible components of Y. Then there is an exact sequence on Y_{ét}

0 ⟶ λ_{(Y,D),n}^{r} ⟶^{ř^{0}} ⨁_{|I|=1} W_{n} Ω_{(Y_{I},D_{I}),log}^{r} ⟶^{ř^{1}} ⨁_{|I|=2} W_{n} Ω_{(Y_{I},D_{I}),log}^{r} ⟶^{ř^{2}}

⋯ ⟶^{ř^{q−1}} ⨁_{|I|=q} W_{n} Ω_{(Y_{I},D_{I}),log}^{r} ⟶ 0,

where the notation ‘|I| = t’ means that I runs through all subsets of {1, 2, …, q} consisting of t elements, and for such I = {i_{1}, i_{2}, …, i_{t}} (i_{j}’s are pair-wise distinct), we put

Y_{I} := Y_{i_{1}} ∩ Y_{i_{2}} ∩ ⋯ ∩ Y_{i_{t}} and D_{I} := D ×_{Y} Y_{I}.

The arrow ř^{0} denotes the natural restriction map. For (I, I′) with I = {i_{1}, i_{2}, …, i_{t}} (i_{1} < i_{2} < ⋯ < i_{t}) and |I′| = t + 1, the (I, I′)-factor of ř^{t} is defined as

\begin{cases}0 & (if I ⊄ I′) \\ (−1)^{t−a} · (β_{I′I})^{∗} & (if I′ = I ∪ \{i_{t+1}\} and i_{1} < ⋯ < i_{a} < i_{t+1} < i_{a+1} < ⋯ < i_{t}), \end{cases}

where β_{I′I} denotes the closed immersion Y_{I′} ↪ Y_{I}.

We need the following lemma to prove this proposition:

Lemma 2.3 Assume that (Y, D, V) fits into cartesian squares of schemes

[figure: D; f; ι; □; Y; i; □; V; g; i′; 𝒟; 𝒴; 𝒱; h]

such that 𝒴 is regular, such that the vertical arrows are closed immersions and such that Y, 𝒟 and Y ∪ 𝒟 are simple normal crossing divisors on 𝒴 , where we put 𝒱 := 𝒴 − 𝒟. Then the pull-back map i^{∗}h_{∗}𝒪_{𝒱}^{×} → g_{∗}𝒪_{V}^{×} on Y_{ét} is surjective.

Proof of Lemma 2.3. We use the same notation as in Proposition 2.2. For a Cartier divisor E on a scheme Z, let c_{1}^{Z}(E) ∈ H_{|E|}^{1}(Z, 𝒪_{Z}^{×}) be the localized first Chern class of the invertible sheaf 𝒪_{Z}(E). Since 𝒴 is regular, h_{∗}𝒪_{𝒱}^{×} is generated by 𝒪_{𝒴}^{×} and local uniformizers of the irreducible components {𝒟_{j}}_{j∈J} of 𝒟. Put

D_{j} := Y ×_{𝒴} 𝒟_{j} (j ∈ J),

which is an admissible divisor on Y. Since (g_{∗}𝒪_{V}^{×})/𝒪_{Y}^{×} ≃ R^{1}f^{!}𝒪_{Y}^{×}, it is enough to show that the Gysin map

φ : ⨁_{j∈J} ℤ_{D_{j}} ⟶ R^{1}f^{!}𝒪_{Y}^{×}

sending 1 ∈ ℤ_{D_{j}} to c_{1}^{Y} (D_{j}) is bijective on D_{ét}. By [Sa1] Lemma 3.2.2, there is an exact sequence on Y_{ét}

0 ⟶ 𝒪_{Y}^{×} ⟶^{ř^{0}} ⨁_{|I|=1} 𝒪_{Y_{I}}^{×} ⟶^{ř^{1}} ⨁_{|I|=2} 𝒪_{Y_{I}}^{×} ⟶^{ř^{2}} ⋯ ⟶^{ř^{q−1}} ⨁_{|I|=q} 𝒪_{Y_{I}}^{×} ⟶ 0,  (2.3.1)

where ř’s are defined in the same way as ř’s in Proposition 2.2, and q denotes the number of the distinct irreducible components of Y. Since f^{!}𝒪_{Y_{I}}^{×} = 0 for any nonempty subset I ⊂ {1, 2, …, q}, the exactness of (2.3.1)implies that of the lower row of the following commutative diagram with exact rows:

\xymatrix{0 \ar[r] & ⨁_{j∈J} ℤ_{D_{j}} \ar[r]^{ř^{0}} \ar[d]_{φ} & ⨁_{j∈J} ⨁_{|I|=1} ℤ_{D_{j}∩Y_{I}} \ar[d]_{≀} \ar[r]^{ř^{1}} & ⨁_{j∈J} ⨁_{|I|=2} ℤ_{D_{j}∩Y_{I}} \ar[d]_{≀} \\ 0 \ar[r] & R^{1}f^{!}𝒪_{Y}^{×} \ar[r]^{ř^{0}} & ⨁_{|I|=1} R^{1}f_{1}^{!}𝒪_{Y_{I}}^{×} \ar[r]^{ř^{1}} & ⨁_{|I|=2} R^{1}f_{1}^{!}𝒪_{Y_{I}}^{×} ,}

where D_{j} ∩ Y_{I} is regular for each j ∈ J and I ⊂ {1, 2, …, q} by the assumption that Y ∪ 𝒟 has simple normal crossings on 𝒳 . The middle and the right vertical arrows are defined in the same way as for φ, and bijective by the standard purity for 𝒪^{×} ([Gr1] III §6). Hence φ is bijective as well. □

Proof of Proposition 2.2. Since the problem is étale local on Y, we may assume that (Y, D, V) fits into a diagram as in Lemma 2.3. Then there is an exact sequence

W_{n} Ω_{(𝒴 ,𝒟) ,log}^{r} ⟶ ⨁_{|I|=1} W_{n} Ω_{(Y_{I},D_{I}),log}^{r} ⟶^{ř^{1}} ⋯ ⟶^{ř^{q−1}} ⨁_{|I|=q} W_{n} Ω_{(Y_{I},D_{I}),log}^{r} ⟶ 0

on 𝒴_{ét} by Lemma 2.3 and induction on the number of components of Y which is similar as for [Sa1] Lemma 3.2.2. The assertion follows from this exact sequence. □

3 p-adic étale Tate twists with log poles

In §§3–7, we are mainly concerned with the following setting.

Setting 3.1 Let A be a Dedekind domain whose fraction field has characteristic 0 and which has a maximal ideal of positive characteristic. Put

S := Spec(A).

Let p be a prime number which is not invertible in A. Let X be a regular scheme which is flat of finite type over A and whose fibers over the closed points of S of characteristic p are empty or reduced normal crossing divisors on X. We write Y ⊂ X for the union of those fibers, which may be empty.

Let D ⊂ X be a normal crossing divisor such that D ∪ Y has normal crossings on X (D may be empty). Put U := X − (Y ∪ D) and V := Y − (Y ∩ D), and consider a diagram of immersions

[figure: V; g; X − D; h; U; j; Y; i; □; X]

Let n and r be positive integers. We first state the Bloch-Kato-Hyodo theorem on the structure of the sheaf M_{n}^{r} := i^{∗}R^{r}j_{∗}μ_{p^{n}}^{⊗r}, which will be useful in this paper. We define the étale sheaf 𝒦_{r}^{M} on Y as

𝒦_{r}^{M} := (i^{∗}j_{∗}𝒪_{U}^{×})^{⊗n}/J,

where J denotes the subsheaf of (i^{∗}j_{∗}𝒪_{U}^{×})^{⊗n} generated by local sections of the form a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r} (a_{1}, a_{2}, …, a_{r} ∈ i^{∗}j_{∗}𝒪_{U}^{×}) with a_{s} + a_{t} = 0 or 1 for some 1 ≤ s < t ≤ n. There is a homomorphism of étale sheaves ([BK1] 1.2)

𝒦_{r}^{M} ⟶ M_{n}^{r},  (3.1.1)

which is a geometric version of Tate’s Galois symbol map. For local sections a_{1}, a_{2}, …, a_{r} ∈ i^{∗}j_{∗}𝒪_{U}^{×}, we denote the class of a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{r} in 𝒦_{r}^{M} by {a_{1}, a_{2}, …, a_{r}}, and denote the image of {a_{1}, a_{2}, …, a_{r}} ∈ 𝒦_{r}^{M} under (3.1.1)again by {a_{1}, a_{2}, …, a_{r}}. We define filtrations 𝒰^{•} and 𝒱 ^{•} on M_{n}^{r} as follows.

Definition 3.2 Put 𝔭 := Ker(𝒪_{X} → i_{∗}𝒪_{Y}) and 1 + 𝔭^{q} := Ker(𝒪_{X}^{×} → (𝒪_{X}/𝔭^{q})^{×}) for q ≥ 1.

(1) We define 𝒰^{0}𝒦_{r}^{M} as the full sheaf 𝒦_{n}^{M}. For q ≥ 1, we define 𝒰^{q}𝒦_{r}^{M} ⊂ 𝒦_{r}^{M} as the image of i^{∗}(1 + 𝔭^{q}) ⊗ (i^{∗}j_{∗}𝒪_{U}^{×})^{⊗r−1}.

(2) For q ≥ 0, we define 𝒰^{q}M_{n}^{r} as the image of 𝒰^{q}𝒦_{r}^{M} under the map (3.1.1).

(3) When A is local and its residue field k has characteristic p, we fix a prime element π ∈ A and define 𝒱^{q}M_{n}^{r} ⊂ M_{n}^{r} as the part generated by 𝒰^{q+1}M_{n}^{r} and the image of 𝒰^{q}𝒦_{r−1}^{M} ⊗ ⟨π⟩ under (3.1.1).

Let L_{X^{∘}} be the log structure on X associated with the normal crossing divisor Y ∪ D ([K3]), and let L_{Y^{∘}} be its inverse image log structure onto Y_{ét} (loc. cit. (1.4)). The following theorem is a variant of theorems of Bloch-Kato-Hyodo ([BK1] Theorem 1.4, [Hy1] Theorem 1.6), and the case D = ∅ corresponds to their theorems.

Theorem 3.3 (1) The symbol map (3.1.1)is surjective, i.e., 𝒰^{0}M_{n}^{r} = M_{n}^{r}.

(2) Assume that A is local and that its residue field k has characteristic p. Then there are isomorphisms

M_{n}^{r}/𝒱^{0}M_{n}^{r} ^{∼} ⟶ W_{n} ω_{Y^{∘},log}^{r} ,
𝒱^{0}M_{n}^{r}/𝒰^{1}M_{n}^{r} ^{∼} ⟶ W_{n} ω_{Y^{∘},log}^{r−1} ,

where W_{n} ω_{Y^{∘},log}^{m} denotes the image of the logarithmic differential map

dlog : (L_{Y^{∘}}^{gp})^{⊗m} ⟶ ⨁_{y∈Y^{0}} i_{y∗}W_{n} Ω_{y,log}^{m} ,

and for a point y ∈ Y, i_{y} denotes the natural map y ↪ Y.

(3) Under the same assumption as in (2), let e be the absolute ramification index of A, and let L_{k} be the log structure on Spec(k) associated with the pre-log structure ℕ → k sending 1 ↦ 0. Put e′ := pe/(p − 1). Then for 1 ≤ q < e′, there are isomorphisms

𝒰^{q}M_{1}^{r}/𝒱^{q}M_{1}^{r} ^{∼} ⟶ \begin{cases}ω_{Y^{∘}}^{r−1}/ℬ_{Y^{∘}}^{r−1} & (p |̸ q), \\ ω_{Y^{∘}}^{r−1}/𝒵_{Y^{∘}}^{r−1} & (p | q), \end{cases}
𝒱^{q}M_{1}^{r}/𝒰^{q+1}M_{1}^{r} ^{∼} ⟶ ω_{Y^{∘}}^{r−2}/𝒵_{Y^{∘}}^{r−2} .

Here ω_{Y^{∘}}^{m} denotes the differential module of (Y, L_{Y^{∘}}) over (Spec(k), L_{k}) ([K3] (1.7)), and ℬ_{Y^{∘}}^{m} (resp. 𝒵_{Y^{∘}}^{m}) denotes the image of d : ω_{Y^{∘}}^{m−1} → ω_{Y^{∘}}^{m} (resp. the kernel of d : ω_{Y^{∘}}^{m} → ω_{Y^{∘}}^{m+1}).

(4) Under the same assumption and notation as in (3), we have 𝒰^{q}M_{1}^{r} = 𝒱^{q}M_{1}^{r} = 0 for q ≥ e′.

Proof. Note that the irreducible components of D are semistable families around the fibers of characteristic p by the assumption that Y ∪ D has normal crossings on X. The assertions (1) and (2) are reduced to the case that X is smooth over S and that D = ∅ (i.e., the Bloch-Kato theorem) by Tsuji’s trick in [Ts2] Proof of Theorem 5.1 and a variant of Hyodo’s lemma [Hy1] Lemma 3.5, whose details will be explained in a forthcoming paper [KSS]. The assertion (4) follows from [BK1] Lemma 5.1.

We prove (3). Let π ∈ A be the fixed prime element. Let \overline{π} ∈ L_{Y^{∘}} be the image of π, and let [\overline{π}] ⊂ L_{Y^{∘}} be the subsheaf of monoids generated by \overline{π}. The quotient L_{Y^{∘}}/[\overline{π}] is a subsheaf of monoids of 𝒪_{Y} (with respect to the multiplication of functions) generated by 𝒪_{Y}^{×} and local equations defining D and irreducible components of Y. There is a surjective homomorphism

δ_{m} : 𝒪_{Y} ⊗ \{(L_{Y^{∘}}/[\overline{π}])^{gp}\}^{⊗m} ⟶ ω_{Y^{∘}}^{m}

defined by the local assignment

z ⊗ y_{1} ⊗ ⋯ ⊗ y_{m} ⟼ z · dlog(y_{1}) ∧ ⋯ ∧ dlog(y_{m}),

with z ∈ 𝒪_{Y} and each y_{i} ∈ (L_{Y^{∘}}/[\overline{π}])^{gp}. The kernel of δ_{m} is generated by local sections of the following forms (cf. [Hy1] Lemma 2.2):

(1) z ⊗ y_{1} ⊗ ⋯ ⊗ y_{m} such that y_{s} belongs to 𝒪_{Spec(k)}^{×}|_{Y} for some 1 ≤ s ≤ m.

(2) z ⊗ y_{1} ⊗ ⋯ ⊗ y_{m} such that y_{s} = y_{t} for some 1 ≤ s < t ≤ m.

(3) ∑ _{i=1}^{ℓ} (a_{i} ⊗ a_{i} ⊗ y_{1} ⊗ ⋯ ⊗ y_{m−1}) − ∑ _{j=1}^{ℓ′} (b_{j} ⊗ b_{j} ⊗ y_{1} ⊗ ⋯ ⊗ y_{m−1}) with each a_{i}, b_{j} ∈ L_{Y^{∘}}/[\overline{π}] such that the sums ∑ _{i=1}^{ℓ} a_{i} and ∑ _{j=1}^{ℓ′} b_{j} taken in 𝒪_{Y}

belong to L_{Y^{∘}}/[\overline{π}] and satisfy ∑_{i=1}^{ℓ} a_{i} = ∑_{j=1}^{ℓ′} b_{j}.

Hence the assertion follows from the arguments in loc. cit. p. 551. □

We define the étale subsheaf F M_{n}^{r} ⊂ M_{n}^{r} as the part generated by 𝒰^{1}M_{n}^{r} and the image of (i^{∗}h_{∗}𝒪_{X−D}^{×})^{⊗r}, where h denotes the open immersion X − D ↪ X. By Theorem 3.3 (2), Proposition 2.2 and the same arguments as in [Sa2] §3.4, we obtain the following theorem:

Theorem 3.4 There are short exact sequences of sheaves on Y_{ét}

0 ⟶ F M_{n}^{r} ⟶ M_{n}^{r} ⟶^{σ} ν_{(Y,D∩Y),n}^{r−1} ⟶ 0,
0 ⟶ 𝒰^{1}M_{n}^{r} ⟶ F M_{n}^{r} ⟶^{τ} λ_{(Y,D∩Y),n}^{r} ⟶ 0,

where σ is induced by the boundary map of Galois cohomology groups due to Kato [K1], and τ is given by the local assignment

\{a_{1}, a_{2}, …, a_{r}\} ⟼ dlog(\overline{a_{1}} ⊗ \overline{a_{2}} ⊗ ⋯ ⊗ \overline{a_{r}}).

Here a_{1}, a_{2}, …, a_{r} are local sections of i^{∗}h_{∗}𝒪_{X−D}^{×}, and for a ∈ i^{∗}h_{∗}𝒪_{X−D}^{×}, \overline{a} denotes its residue class in g_{∗}𝒪_{V}^{×}.

Now we define the p-adic étale Tate twists.

Definition 3.5 For n ≥ 1 and r ∈ ℤ, we define a cochain complex C_{n}(r)_{(X,D)}^{•} of sheaves on X_{ét} as follows.

(1) For r = 0, we define C_{n}(0)_{(X,D)}^{•} := ℤ/p^{n}, the constant sheaf ℤ/p^{n} placed in degree 0. For r < 0, we define C_{n}(r)_{(X,D)}^{•} := j_{!}ℋom_{U}(μ_{p^{n}}^{⊗−r}, ℤ/p^{n}), the sheaf j_{!}ℋom_{U}(μ_{p^{n}}^{⊗−r}, ℤ/p^{n}) placed in degree 0.

(2) Assume r ≥ 1, and let ℐ^{•} be the Godement resolution of μ_{p^{n}}^{⊗r} on U_{ét}. We define C_{n}(r)_{(X,D)}^{•} as

j_{∗}ℐ^{0} ⟶ j_{∗}ℐ^{1} ⟶ ⋯

⟶ j_{∗}ℐ^{r−1} ⟶ Ker(d : j_{∗}ℐ^{r} → j_{∗}ℐ^{r+1}) ⟶^{σ_{n}^{r}} i_{∗}ν_{(Y,D∩Y),n}^{r−1}, (3.5.1)

where j_{∗}ℐ^{0} is placed in degree 0 and i_{∗}ν_{(Y,D∩Y),n}^{r−1} is placed in degree r + 1. The last arrow σ_{n}^{r} is defined as the composite map

\xymatrix{σ_{n}^{r} : Ker(d : j_{∗}ℐ_{U}^{r} → j_{∗}ℐ_{U}^{r+1}) \ar@{->>}[r] & R^{r}j_{∗}μ_{p^{n}}^{⊗r} \ar@{->>}[r]^{i_{∗}σ} & i_{∗}ν_{(Y,D∩Y),n}^{r−1},}

and σ denotes the surjective map in Theorem 3.4.

We often write 𝔗_{n}(r)_{(X,D)} for C_{n}(r)_{(X,D)}^{•} regarded as an object of D^{b}(X_{ét}, ℤ/p^{n}). When D = ∅, we often write C_{n}(r)_{X}^{•} and 𝔗_{n}(r)_{X} for C_{n}(r)_{(X,∅)}^{•} and 𝔗_{n}(r)_{(X,∅)}, respectively.

Proposition 3.6 For r ≥ 0, 𝔗_{n}(r)_{(X,D)} is concentrated in [0, r], and there is a distinguished triangle in D^{b}(X_{ét}, ℤ/p^{n})

\xymatrix{i_{∗}ν_{(Y,D∩Y ),n}^{r−1}[−r − 1] \ar[r] & 𝔗_{n}(r)_{(X,D)} \ar[r]^{t} & τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r} \ar[r]^{σ_{n}^{r}[−r]} & i_{∗}ν_{(Y,D∩Y ),n}^{r−1}[−r].}

Proof. The first assertion follows from the surjectivity of σ_{n}^{r} in the definition of C_{n}(r)_{(X,D)}^{•}. The second assertion is straight-forward. □

Remark 3.7 Proposition 3.6 implies that 𝔗_{n}(r)_{X} defined here agrees with that in [Sa2] §4 by a unique isomorphism compatible with the identity map of μ_{p^{n}}^{⊗r} on U (loc. cit. Lemma 4.2.2). Although we assumed that the residue fields of A of characteristic p are perfect in loc. cit., the results in loc. cit. §4 – §7.2 hold true without this assumption by results of Shiho [Sh] Theorems 3.2, 4.1 and the compatibility result in [AS] Remark 7.3 (1).

Proposition 3.8 The complex C_{n}(r)_{(X,D)}^{•} is contravariantly functorial in the pair (X, D). Here a morphism of pairs (X, D) → (X′, D′) means a morphism of schemes f : X → X′ satisfying f(X − D) ⊂ X′ − D′.

Proof. The case r ≤ 0 is clear. As for the case r ≥ 1, it is enough to show that the map

i_{∗}σ : R^{r}j_{∗}μ_{p^{n}}^{⊗r} ⟶ i_{∗}ν_{(Y,D∩Y),n}^{r−1}

is contravariant in (X, D). Let f : (X, D) → (X′, D′) be a morphism of pairs, and consider the following diagram of immersions:

\xymatrix{Y′ \ar@{^{(}->}[r]^{i′} & X′ & U′ := X′ − D′, \ar@{^{(}->}[l]_{j′}}

where Y′ denotes the union of the fibers of X′ → S of characteristic p. By the first exact sequence in Theorem 3.4, σ′ (:= σ for (X′, D′)) is surjective and Ker(i′_{∗}σ′) maps into Ker(i_{∗}σ) under the base-change map

f^{∗} : f^{∗}R^{r}j′_{∗}μ_{p^{n}}^{⊗r} ⟶ R^{r}j_{∗}μ_{p^{n}}^{⊗r}.

Hence this map induces a pull-back map

f^{∗} : f^{∗}i′_{∗}ν_{(Y′,D′∩Y′),n}^{r−1} ⟶ i_{∗}ν_{(Y,D∩Y),n}^{r−1}.  (3.8.1)

These maps are obviously compatible with σ’s and satisfy transitivity. Thus we obtain the proposition. □

Corollary 3.9 (1) Let V be the category whose objects are schemes X as in Setting 3.1 and whose morphisms are morphisms of schemes. Then the complexes C_{n}(r)_{X}^{•} = C_{n}(r)_{(X,∅)}^{•} with X ∈ Ob(V) form a complex C_{n}(r)^{•} of sheaves on the big étale site V_{ét}.

(2) The Godement resolution G_{n}(r)_{(X,D)}^{•} on X_{ét} of C_{n}(r)_{(X,D)}^{•} is contravariantly functorial in (X, D).

Remark 3.10 The object 𝔗_{n}(r)_{(X,D)} is also contravariantly functorial in the pair (X, D). More precisely, for a morphism of pairs f : (X, D) → (X′, D′), there is a natural morphism

f^{∗} : f^{∗}𝔗_{n}(r)_{(X′,D′)} ⟶ 𝔗_{n}(r)_{(X,D)} in D^{b}(X_{ét}, ℤ/p^{n})

by Proposition 3.8, which is, in fact, the unique morphism that extends the pull-back isomorphism of μ_{p^{n}}^{⊗r} for U → U′ (cf. [Sa2] Proposition 4.2.8).

Remark 3.11 Let V and C_{n}(r)^{•} be as in Corollary 3.9, and let 𝔗_{n}(r) be the complex C_{n}(r)^{•} regarded as an object of the derived category D^{b}(V_{ét}, ℤ/p^{n}). Let V′ ⊂ V be the full subcategory consisting of schemes X ∈ Ob(V) with p^{−1} ∈ Γ (X, 𝒪_{X}). The following facts will be useful later in §5:

(1) There exists a unique product structure

𝔗_{n}(q) ⊗^{𝕃} 𝔗_{n}(r) ⟶ 𝔗_{n}(q + r) in D(V_{ét}, ℤ/p^{n})

that extends the isomorphism μ_{p^{n}}^{⊗q}⊗μ_{p^{n}}^{⊗r} ^{∼} ⟶ μ_{p^{n}}^{⊗q+r} on the big étale site (V′)_{ét}, which follows from the same arguments as in [Sa2] Proposition 4.2.6.

(2) There exists a unique isomorphism

𝔾_{m} ⊗^{𝕃} ℤ/p^{n}[−1] ^{∼} ⟶ 𝔗_{n}(1) in D(V_{ét}, ℤ/p^{n})

that extends the canonical isomorphism 𝔾_{m} ⊗^{𝕃} ℤ/p^{n}[−1] ⟶ μ_{p^{n}} on (V′)_{ét}, which follows from the same arguments as in loc. cit. Proposition 4.5.1.

Theorem 3.12 When D is regular, there is a canonical morphism

δ : 𝔗_{n}(r)_{(X,D)} ⟶ α_{∗}𝔗_{n}(r − 1)_{D}[−1] in D^{b}(X_{ét}, ℤ/p^{n})

fitting into a distinguished triangle

α_{∗}𝔗_{n}(r −1)_{D}[−2] ⟶^{α_{∗}} 𝔗_{n}(r)_{X} ⟶^{β^{∗}} 𝔗_{n}(r)_{(X,D)} ⟶^{δ} α_{∗}𝔗_{n}(r −1)_{D}[−1], (3.12.1)

where α denotes the closed immersion D ↪ X, and β denotes the natural morphism of pairs (X, D) → (X, ∅). The arrow α_{∗} denotes the Gysin morphism [Sa2] Theorem 6.1.3.

Proof. The case r < 0 immediately follows from the absolute purity [FG]. To prove the case r ≥ 0, we first construct the morphism δ. Consider a diagram of immersions

[figure: U; j; u; Y; i; □; X; □; X[p^{−1}]; w; E; i′; D; α; D[p^{−1}].; φ; v]

There is a distinguished triangle on X[p^{−1}]_{ét}

v_{∗}μ_{p^{n}}^{⊗r−1}[−2] ⟶^{v_{∗}} μ_{p^{n}}^{⊗r} ⟶^{u^{∗}} Ru_{∗}μ_{p^{n}}^{⊗r} ⟶^{δ_{1}} v_{∗}μ_{p^{n}}^{⊗r−1}[−1],

where δ_{1} is defined as the composite

δ_{1} : Ru_{∗}μ_{p^{n}}^{⊗r} ⟶^{−δ^{loc}} v_{∗}Rv^{!}μ_{p^{n}}^{⊗r}[1] ^{∼} ⟵ v_{∗}μ_{p^{n}}^{⊗r−1}[−1]

and δ^{loc} denotes the connecting morphism of a localization sequence. We used the absolute purity [FG] for the last isomorphism. Applying Rw_{∗}, we get a distinguished triangle on X_{ét}

α_{∗}Rφ_{∗}μ_{p^{n}}^{⊗r−1}[−2] ⟶^{v_{∗}} Rw_{∗}μ_{p^{n}}^{⊗r} ⟶^{u^{∗}} Rj_{∗}μ_{p^{n}}^{⊗r} ⟶^{δ_{2}} α_{∗}Rφ_{∗}μ_{p^{n}}^{⊗r−1}[−1]. (3.12.2)

Consider a diagram

\xymatrix{𝔗_{n}(r)_{(X,D)} \ar[r]^{t} \ar@{.>}[d]_{δ} & τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r} \ar[d]^{δ_{3}:=τ_{≤r}(δ_{2})} \\ α_{∗}𝔗_{n}(r − 1)_{D}[−1] \ar[r] & τ_{≤r}(α_{∗}Rφ_{∗}μ_{p^{n}}^{⊗r−1}[−1]) \ar[r]^{b′} & γ_{∗}ν_{E,n}^{r−2}[−r],}

where the lower row is a part of a distinguished triangle (cf. Proposition 3.6 for (D, ∅)) and γ denotes the composite morphism α ∘ i′ : E ↪ X. Since 𝔗_{n}(r)_{(X,D)} is concentrated in [0, r], we see that the composite b′ ∘ δ_{3} ∘ t is zero by Theorem 3.4 and a simple computation on symbols. On the other hand, we have

Hom_{D^{b}(X_{ét},ℤ/p^{n})}(𝔗_{n}(r)_{(X,D)}, γ_{∗}ν_{E,n}^{r−2}[−r − 1]) = 0,

again by the fact that 𝔗_{n}(r)_{(X,D)} is concentrated in [0, r]. Hence there exists a unique morphism δ fitting into the above diagram (cf. [Sa2] Lemma 2.1.2 (1)), which is the desired morphism. Finally the triangle (3.12.1) is distinguished by (3.12.2) and a commutative diagram with exact rows on X_{ét} (γ = α ∘ i′)

\xymatrix{ & R^{r}w_{∗}μ_{p^{n}}^{⊗r} \ar[r] \ar[d]^{σ_{X}} & R^{r}j_{∗}μ_{p^{n}}^{⊗r} \ar[r] \ar[d]^{σ_{(X,D)}} \ar[r]^{δ_{3}} & α_{∗}R^{r−1}φ_{∗}μ_{p^{n}}^{⊗r−1} \ar[r] \ar[d]^{σ_{D}} & 0 \\ 0 \ar[r] & i_{∗}ν_{Y,n}^{r−1} \ar[r] & i_{∗}ν_{(Y,E),n}^{r−1} \ar[r] & γ_{∗}ν_{E,n}^{r−2} \ar[r] & 0,}

where the surjectivity of δ_{3} in the upper row follows from Theorem 3.3 (1) for (D, ∅), and the exactness of the lower row follows from (2.1.1). □

Remark 3.13 Let ζ_{p} be a primitive p-th root of unity, and let A′ be the normalization of A[ζ_{p}]. Then all the definitions and results for the pair (X, D) in this section are extended to the scalar extension (X ⊗_{A} A′, D ⊗_{A} A′). Indeed, Theorem 3.3 (1) and (2) will be proved in [KSS] for this generalized situation. One can check Theorem 3.3 (3) and (4) for (X ⊗_{A} A′, D ⊗_{A} A′) by the same arguments as for (X, D). See [Sa2] §3.5 for an argument to extend Theorem 3.4.

4 Dold-Thom isomorphism

Let S, p and X be as in Setting 3.1, and let n and r be integers with n ≥ 1. In this section we prove the Dold-Thom isomorphism for p-adic étale Tate twists. Let E be a vector bundle of rank a + 1 on X, and let f : ℙ := ℙ(E) → X be the associated projective bundle, which is a projective smooth morphism of relative dimension a. Let 𝒪(1)_{E} be the tautological invertible sheaf on ℙ, and let ξ ∈ H^{2}(ℙ, 𝔗_{n}(1)_{ℙ}) be the value of the first Chern class c_{1}(𝒪(1)_{E}) ∈ H^{1}(ℙ, 𝒪_{ℙ}^{×}) under the connecting map associated with the Kummer distinguished triangle

𝒪_{ℙ}^{×} ⟶ 𝒪_{ℙ}^{×} ⟶ 𝔗_{n}(1)_{ℙ}[1] ⟶ 𝒪_{ℙ}^{×}[1]

(cf. [Sa2] Proposition 4.5.1). The composite morphisms

\xymatrix{𝔗_{n}(r − q)_{X}[−2q] \ar[r]^{f^{∗}} & Rf_{∗}𝔗_{n}(r − q)_{ℙ}[−2i] \ar[r]^{−∪ξ^{q}} & Rf_{∗}𝔗_{n}(r)_{ℙ}  (0 ≤ q ≤ a)}

induce a canonical morphism

γ_{E} : ⨁_{q=0}^{a} 𝔗_{n}(r − q)_{X}[−2q] ⟶ Rf_{∗}𝔗_{n}(r)_{ℙ} in D^{b}(X_{ét}, ℤ/p^{n}).

Theorem 4.1 (Dold-Thom isomorphism) γ_{E} is an isomorphism.

Proof. γ_{E} is an isomorphism outside of Y by [M] VI Theorem 10.1. The case r < 0 follows from this fact and the proper base-change theorem. To prove the case r ≥ 0, we consider a diagram of schemes

[figure: P; γ; g; □; ℙ; f; Y; i; X; X[p^{−1}].; j]

We have to show that i^{∗}(γ_{E}) is an isomorphism:

i^{∗}(γ_{E}) : ⨁_{q=0}^{a} i^{∗}𝔗_{n}(r − q)_{X}[−2q] ^{∼} ⟶ Rg_{∗}γ^{∗}𝔗_{n}(r)_{ℙ},  (4.1.1)

where we identified i^{∗}Rf_{∗}𝔗_{n}(r)_{ℙ} with Rg_{∗}γ^{∗}𝔗_{n}(r)_{ℙ} by the proper base-change theorem. By a standard norm argument (cf. [Sa2] §10.3) using Bockstein triangles (loc. cit. §4.3), we are reduced to the case that n = 1 and that Γ (X, 𝒪_{X}) contains a primitive p-th root of unity (see Remark 3.13). We need the following lemma:

Lemma 4.2 Let \overline{ξ} ∈ H^{1}(P, λ_{P,1}^{1}) be the image of ξ under the pull-back map

γ^{∗} : H^{2}(ℙ, 𝔗_{1}(1)_{ℙ}) ⟶ H^{1}(P, λ_{P,1}^{1})

(cf. [Sa2] Proposition 4.4.10). Then there are isomorphisms in D^{b}(Y_{ét}, ℤ/p)

⨁_{q=0}^{a} \overline{ξ}^{q} ∪ − : ⨁_{q=0}^{a} λ_{Y,1}^{r−q}[−q] ^{∼} ⟶ Rg_{∗}λ_{P,1}^{r}  (1)

⨁_{q=0}^{a} \overline{ξ}^{q} ∪ − : ⨁_{q=0}^{a} ν_{Y,1}^{r−q}[−q] ^{∼} ⟶ Rg_{∗}ν_{P,1}^{r}  (2)

⨁_{q=0}^{a} \overline{ξ}^{q} ∪ − : ⨁_{q=0}^{a} ω_{Y}^{r−q}[−q] ^{∼} ⟶ Rg_{∗}ω_{P}^{r}  (3)

⨁_{q=0}^{a} \overline{ξ}^{q} ∪ − : ⨁_{q=0}^{a} 𝒵_{Y}^{r−q}[−q] ^{∼} ⟶ Rg_{∗}𝒵_{P}^{r}.  (4)

Here 𝒵_{Y}^{m} (resp. 𝒵_{P}^{m}) denotes the kernel of d : ω_{Y}^{m} → ω_{Y}^{m+1} (resp. d : ω_{P}^{m} → ω_{P}^{m+1}).

Proof of Lemma 4.2. Note that \overline{ξ} agrees with the first Chern class of the tautological invertible sheaf on P = ℙ(i^{∗}E). Since the problems are étale local on Y, we may assume that Y is simple. If Y is smooth, then (1) and (2) are due to Gros [Gr1] I Théorème 2.1.11. The general case is reduced to the smooth case by [Sa1] Proposition 3.2.1, Corollary 2.2.7. As for (3), since we have

Rg_{∗}ω_{P}^{r} ≃ ⨁_{q=0}^{r} Rg_{∗}(Ω_{P/Y}^{q} ⊗_{𝒪_{Y}} ω_{Y}^{r−q}) ≃ ⨁_{q=0}^{r} (Rg_{∗}Ω_{P/Y}^{q}) ⊗_{𝒪_{Y}}^{𝕃} ω_{Y}^{r−q}

by projection formula, the assertion follows from the isomorphisms

\overline{ξ}^{q} ∪ − : 𝒪_{Y} [−q] ^{∼} ⟶ Rg_{∗}Ω_{P/Y}^{q} (0 ≤ q ≤ a).

(4) follows from the same arguments as for [Gr1] I (2.2.3). □

We return to the proof of (4.1.1) for r ≥ 0. The case r = 0 follows from Lemma 4.2 (1) with r = 0. To prove the case r > 1, we use the objects 𝕂(r − q)_{X} ∈ D^{b}(Y_{ét}, ℤ/p) and 𝕂(r)_{ℙ} ∈ D^{b}(P_{ét}, ℤ/p) defined in [Sa2] Lemma 10.4.1, which fit into distinguished triangles

𝕂(r − q)_{X}[−1] → μ′ ⊗^{𝕃} i^{∗}𝔗_{1}(r − q − 1)_{X} → i^{∗}𝔗_{1}(r − q)_{X} → 𝕂(r − q)_{X},
𝕂(r)_{ℙ}[−1] ⟶ g^{∗}μ′ ⊗^{𝕃} γ^{∗}𝔗_{1}(r − 1)_{ℙ} ⟶ γ^{∗}𝔗_{1}(r)_{ℙ} ⟶ 𝕂(r)_{ℙ}.

Here μ′ denotes the constant sheaf i^{∗}j_{∗}μ_{p}(≃ ℤ/p) on Y_{ét} and the central arrows are induced by the product structure of Tate twists. By induction on r ≥ 0, our task is to show that the morphism

⨁_{q=0}^{a} ξ^{q} ∪ − : ⨁_{q=0}^{a} 𝕂(r − q)_{X}[−2q] ⟶ Rg_{∗}𝕂(r)_{ℙ}  (4.2.1)

is an isomorphism, where we have used the pull-back morphisms

f^{∗} : 𝕂(r − q)_{X} ⟶ 𝕂(r − q)_{ℙ} (0 ≤ q ≤ a)

induced by the pull-back morphisms for Tate twists (loc. cit. Proposition 4.2.8, Lemma 2.1.2 (2)). By loc. cit. Lemma 10.4.1 (2), we have

ℋ ^{m}(𝕂(r − q)_{X}) ≃ \begin{cases}μ′ ⊗ ν_{Y}^{r−q−2} & (m = r − q − 1) \\ F M_{1}^{r−q} & (m = r − q) \\ 0 & (otherwise) \end{cases}

and similar facts holds for 𝕂(r)_{P} (see §3 for F M_{n}^{q}). Therefore (4.2.1) is an isomorphism by Lemma 4.2 and Theorems 3.3 (3), (4) and 3.4 with D = ∅ (see also the projection formula in loc. cit. 4.4.10). This completes the proof of Theorem 4.1. □

The following corollary 4.3 follows immediately from Theorems 3.12, 4.1 and the projection formulra ([Sa2] Corollary 7.2.4), which is a p-adic version of homotopy invariance and plays an important role in our construction of cycle class maps (see §7 below).

Corollary 4.3 Let E be a vector bundle on X of ranki a, and put ℙ = ℙ(E ⊕ 1), the projective completion of E. Let ℙ′ be the projective bundle ℙ(E) regarded as the infinite hyperplane, and let f : ℙ → X be the natural projection. Then the composite morphism

𝔗_{n}(r)_{X} ⟶^{f^{∗}} Rf_{∗}𝔗_{n}(r)_{ℙ} ⟶ Rf_{∗}𝔗_{n}(r)_{(ℙ,ℙ′)}

is an isomorphism in D^{b}(X_{ét}, ℤ/p^{n}).

5 Chern class

The main aim of this section is to construct the Chern class map (5.6.1) below. Let S and p be as in Setting 3.1, and let V be the category whose objects are S-schemes satisfying the conditions in Setting 3.1 for X and whose morphisms are morphisms of schemes. Let X_{⋆} be a simplicial object in V, i.e., a contravariant functor

X_{⋆} : Δ^{op} ⟶ V,

where Δ denotes the simplex category. For a morphism γ : [a] → [b] in Δ, we write

γ^{X} : X_{b} ⟶ X_{a} (X_{a} := X_{⋆}([a])) for X_{⋆}(γ), which is a morphism in V. For 0 ≤ i ≤ a, let d^{i} be the coface map in Δ:

d^{i} : [a] ⟶ [a + 1], j ↦ \begin{cases}j & (0 ≤ j < i) \\ j + 1 & (i ≤ j ≤ a). \end{cases}

For integers 0 ≤ i ≤ a, we often write

d_{i} : X_{a+1} → X_{a}

for (d^{i})^{X}. See [Fr] §1 for the definition of the small étale site (X_{⋆})_{ét} on X_{⋆}.

Definition 5.1 (1) For r ∈ ℤ, we define a complex C_{n}(r)_{X_{⋆}}^{•} of sheaves on (X_{⋆})_{ét} by restricting the complex C_{n}(r)^{•} on V_{ét}, cf. Corollary 3.9. We will often write 𝔗_{n}(r)_{X_{⋆}} for the complex C_{n}(r)_{X_{⋆}}^{•} regarded as an object of D^{b}((X_{⋆})_{ét}, ℤ/p^{n}).

(2) We define a canonical morphism

ϱ : 𝔾_{m}[−1] ⟶ 𝔗_{n}(1)_{X_{⋆}} in D^{b}((X_{⋆})_{ét})

by the composite morphism

𝔾_{m}[−1] ⟶ 𝔾_{m} ⊗^{𝕃} ℤ/p^{n}[−1] ^{∼} ⟶ 𝔗_{n}(1)_{X_{⋆}},

where the left arrow denotes the canonical morphism induced by ℤ → ℤ/p^{n} and the right arrow is the restriction of the isomorphism in Remark 3.11 (2).

We next review the following basic notions:

Definition 5.2 (1) A vector bundle over X_{⋆} is a morphism f : E_{⋆} → X_{⋆} of simplicial schemes such that f_{a} : E_{a} → X_{a} is a vector bundle for any a ≥ 0 and such that the commutative diagram

\xymatrix{E_{b} \ar[r]^{f_{b}} \ar[d]_{γ^{E}} & X_{b} \ar[d]^{γ^{X}} \\ E_{a} \ar[r]^{f_{a}} & X_{a}}  (5.2.1)

induces an isomorphism E_{b} ≅ γ^{X∗}E_{a} := E_{a} ×_{X_{a}} X_{b} of vector bundles over X_{b} for any morphism γ : [a] → [b] in Δ (cf. [Gi2] Example 1.1).

(2) A regular closed immersion of simplicial schemes is a morphism i : X_{⋆} → Y_{⋆} such that i_{a} : X_{a} → Y_{a} is a regular closed immersion for any a ≥ 0 and such that the diagram

\xymatrix{X_{b} \ar[r]^{i_{b}} \ar[d]_{γ^{X}} & Y_{b} \ar[d]^{γ^{Y}} \\ X_{a} \ar[r]^{i_{a}} & Y_{a}}  (5.2.2)

is cartesian for any morphism γ : [a] → [b] in Δ. An effective Cartier divisor X_{⋆} on Y_{⋆} is a regular closed immersion X_{⋆} → Y_{⋆} of pure codimension 1.

We now define the first Chern classes of effective Cartier divisors and line bundles.

Definition 5.3 (1) For an effective Cartier divisor D_{⋆} on X_{⋆}, we define the first Chern class c_{1}(D_{⋆}) ∈ H_{D_{⋆}}^{2}(X_{⋆}, 𝔗_{n}(1)) as the value of the first Chern class c_{1}(D_{⋆}) ∈ H_{D_{⋆}}^{1}((X_{⋆})_{Zar}, 𝒪^{×}) under the composite map

H_{D_{⋆}}^{1}((X_{⋆})_{Zar}, 𝒪^{×}) →^{ϵ^{∗}} H_{D_{⋆}}^{1}(X_{⋆}, 𝔾_{m}) →^{ϱ} H_{D_{⋆}}^{2}(X_{⋆}, 𝔗_{n}(1)),

where ϵ : (X_{⋆})_{ét} → (X_{⋆})_{Zar} denotes the continuous map of small sites. The arrow ϱ denotes that in Definition 5.1 (2).

(2) For a line bundle L_{⋆} on X_{⋆}, we define the first Chern class

c_{1}(L_{⋆}) ∈ H^{2}(X_{⋆}, 𝔗_{n}(1))

as the value of the isomorphism class [L_{⋆}] ∈ H^{1}((X_{⋆})_{Zar}, 𝒪^{×}) (cf. [Gi2] Example 1.1) under the composite map

H^{1}((X_{⋆})_{Zar}, 𝒪^{×}) ⟶^{ϵ^{∗}} H^{1}(X_{⋆}, 𝔾_{m}) ⟶^{ϱ} H^{2}(X_{⋆}, 𝔗_{n}(1)).  (5.3.1)

The following proposition plays a key role in the proof of the Whitney sum formula in Proposition 5.5 (3) below.

Proposition 5.4 Let f : X_{⋆} ↪ X′_{⋆} be a regular closed immersion of simplicial objects in V of pure codimension c ≥ 1. Let r be a non-negative integer. Then there exists a Gysin morphism

gys_{f} : 𝔗_{n}(r)_{X_{⋆}} ⟶ Rf^{!}𝔗_{n}(r + c)_{X′_{⋆}}[2c] in D^{+}((X_{⋆})_{ét}, ℤ/p^{n})

satisfying the following three properties:

(a) (Consistency with the first Chern class) If r = 1, then the value of 1 ∈ ℤ/p^{n} = H^{0}(X_{⋆}, 𝔗_{n}(0)) under the Gysin map

gys_{f} : H^{0}(X_{⋆}, 𝔗_{n}(0)) ⟶ H_{X}^{2}(X′_{⋆}, 𝔗_{n}(1))

agrees with the first Chern class c_{1}(X_{⋆}) in Definition 5.3 (1).

(b) (Transitivity) For another regular closed immersion g : X′_{⋆} ↪ X′′_{⋆} of simplicial objects in V of pure codimension c′ ≥ 1, the composite morphism

𝔗_{n}(r)_{X_{⋆}}  ⟶^{gys_{f}}  Rf^{!}𝔗_{n}(r + c)_{X′_{⋆}}[2c]
⟶^{Rf^{!}(gys_{g})} Rf^{!}Rg^{!}𝔗_{n}(r + c + c′)_{X′′_{⋆}}[2c + 2c′]
R(g ∘ f)^{!}𝔗_{n}(r + c + c′)_{X′′_{⋆}}[2(c + c′)]

agrees with gys_{g∘f} .

(c) (Projection formula) The following diagram commutes in D((X_{⋆})_{ét}, ℤ/p^{n}):

\xymatrix{Rf^{!}𝔗_{n}(q)_{X′_{⋆}} ⊗^{𝕃} 𝔗_{n}(r)_{X_{⋆}} \ar[r]^{id⊗gys_{f}} \ar[d]_{π} & Rf^{!}𝔗_{n}(q)_{X′_{⋆}} ⊗^{𝕃} Rf^{!}𝔗_{n}(n + r)_{X′_{⋆}}[2r] \ar[d]^{product} \\ 𝔗_{n}(q + r)_{X_{⋆}} \ar[r]^{_{gys_{f}}} & Rf^{!}𝔗_{n}(q + r + c)_{X′_{⋆}}[2c] ,}

where the left vertical arrow π is the composite morphism

Rf^{!}𝔗_{n}(q)_{X′_{⋆}} ⊗^{𝕃} 𝔗_{n}(r)_{X_{⋆}} ⟶^{f^{∗}⊗id} 𝔗_{n}(q)_{X_{⋆}} ⊗^{𝕃} 𝔗_{n}(r)_{X_{⋆}}
⟶^{product}  𝔗_{n}(q + r)_{X_{⋆}}

and the product structures on {𝔗_{n}(r)_{X_{⋆}}}_{r∈ℤ} and {𝔗_{n}(r)_{X_{⋆}}}_{r∈ℤ} are obtained from that in Remark 3.11.

Proof. Put U_{⋆} := X_{⋆} ⊗ ℤ[p^{−1}] and V_{⋆} := X′_{⋆} ⊗ ℤ[p^{−1}]. Let φ : U_{⋆} ↪ V_{⋆} be the regular closed immersion induced by f. By the absolute purity [FG] and the spectral sequence

E_{1}^{a,b} = H_{U_{a}}^{b}(V_{a}, μ_{p^{n}}^{⊗c}) ⟹ H_{U_{⋆}}^{a+b}(V_{⋆}, μ_{p^{n}}^{⊗c}),

we have

H_{U_{⋆}}^{2c}(V_{⋆}, μ_{p^{n}}^{⊗c}) ≃ Ker(d_{0}^{∗} − d_{1}^{∗} : H_{U_{0}}^{2c}(V_{0}, μ_{p^{n}}^{⊗c}) ⟶ H_{U_{1}}^{2c}(V_{1}, μ_{p^{n}}^{⊗c})).

By the definition of regular closed immersions in Definition 5.2 (2), the cycle class cl_{V_{0}}(U_{0}) ∈ H_{U_{0}}^{2c}(V_{0}, μ_{p^{n}}^{⊗c}) lies in the group on the right hand side, loc. cit. Proposition 1.1.3. We thus define the cycle class

cl_{V_{⋆}}(U_{⋆}) ∈ H_{U_{⋆}}^{2c}(V_{⋆}, μ_{p^{n}}^{⊗c})

as the element corresponding to cl_{V_{0}}(U_{0}). Since φ^{∗}μ_{p^{n},V_{⋆}}^{⊗r} ≃ μ_{p^{n},U_{⋆}}^{⊗r} on (U_{⋆})_{ét} , the cup product with cl_{V_{⋆}}(U_{⋆}) defines a Gysin morphism

gys_{φ} : μ_{p^{n},U_{⋆}}^{⊗r} ≃ φ^{∗}μ_{p^{n},V_{⋆}}^{⊗r} ⟶^{cl_{V⋆} (U_{⋆}) ∪ −} Rφ^{!}μ_{p^{n},V_{⋆}}^{⊗r+c}[2c] in D^{+}((U_{⋆})_{ét}, ℤ/p^{n}),

which satisfies the three properties (a) – (c) listed above (see loc. cit. Proposition 1.2.1 for (b)). We show that there exists a unique morphism

gys_{f} : 𝔗_{n}(r)_{X_{⋆}} ⟶ Rf^{!}𝔗_{n}(r + c)_{X′_{⋆}}[2c] in D^{+}((X_{⋆})_{ét}, ℤ/p^{n})

that extends gys_{φ}. Put

Y_{⋆} := X_{⋆} ⊗ ℤ/pℤ, 𝔏 := 𝔗_{n}(r)_{X_{⋆}} and 𝔐 := Rf^{!}𝔗_{n}(r + c)_{X′_{⋆}}[2c].

and let α and β be as follows:

\xymatrix{U_{⋆} \ar@{^{(}->}[r]^{β} & X_{⋆} & Y_{⋆} . \ar@{^{(}->}[l]_{α}}

Consider the following diagram in D^{+}((X_{⋆})_{ét}, ℤ/p^{n}) whose lower row is distinguished:

\xymatrix{ & 𝔏 \ar[r]^{t} & τ_{≤r}Rβ_{∗}μ_{p^{n}}^{⊗r} \ar[d]_{Rβ_{∗}(gys_{φ})} \\ α_{∗}Rα^{!}𝔐 \ar[r]^{α_{∗}} & 𝔐 \ar[r]^{β^{∗}} & Rβ_{∗}β^{∗}𝔐 \ar[r]^{−δ} & α_{∗}Rα^{!}𝔐[1].}  (5.4.1)

Here the upper horizontal arrow is the canonical morphism (cf. Propositions 3.6 and 3.8), and the lower row is the localization distinguished triangle for 𝔐 (cf. [Sa2] (1.9.2)). We have

τ_{≤r}α_{∗}Rα^{!}𝔐 = 0  (5.4.2)

by the purity in loc. cit. Theorem 4.4.7, which implies that

Hom_{D^{+}((X_{⋆})_{ét},ℤ/p^{n})}(τ_{≤r}Rβ_{∗}μ_{p^{n}}^{⊗r}, α_{∗}Rα^{!}𝔐[1])

= Hom_{Shv((X_{⋆})_{ét},ℤ/p^{n})}(R^{r}β_{∗}μ_{p^{n}}^{⊗r}, α_{∗}R^{r+1}α^{!}𝔐),

where Shv((X_{⋆})_{ét}, ℤ/p^{n}) denotes the category of étale sheaves of ℤ/p^{n}-modules on X_{⋆}. By this fact and the compatibility fact in loc. cit. Theorem 6.1.1, one can easily check that the composite (−δ) ∘ Rβ_{∗}(gys_{φ}) is zero in D^{+}((X_{⋆})_{ét}, ℤ/p^{n}). Therefore we obtain a unique morphism gys_{f} that extends gys_{φ} again by (5.4.2) and by loc. cit. Lemma 2.1.2 (1). The property (a) of gys_{f} is straight-forward, and the property (b) follows from the uniqueness of gys_{f}. The property (c) follows from the same argument as for loc. cit. Corollary 7.2.4. □

Following the method of Grothendieck [G] and Gillet [Gi1], we define Chern classes

c(E_{⋆}) = (c_{i}(E_{⋆}))_{i≥0} ∈ ⨁_{i≥0} H^{2i}(X_{⋆}, 𝔗(i))

of a vector bundle E_{⋆} over X_{⋆} as follows. Let E_{⋆} be of rank a, and let f be the natural projection ℙ(E_{⋆}) → X_{⋆}. Let ξ ∈ H^{2}(ℙ(E_{⋆}), 𝔗_{n}(1)) be the value of the first Chern class of the tautological line bundle on ℙ(E_{⋆}), cf. Definition 5.3 (2). Noting the Dold-Thom isomorphism

⨁_{i=1}^{a} H^{2i}(X_{⋆}, 𝔗_{n}(i)) ≃ H^{2a}(ℙ(E_{⋆}), 𝔗_{n}(a)), (b_{i})_{i=1}^{a} ↦ ∑_{i=1}^{a} f^{∗}(b_{i}) ∪ ξ^{a−i}

(5.4.3)

obtained from Theorem 4.1 and [Gi1] Lemma 2.4, we define

c_{0}(E_{⋆}) := 1 and c_{i}(E_{⋆}) := 0 for i > a,

and define c_{i} = c_{i}(E_{⋆}) for i = 1, 2, …, a by the equation

ξ^{a} + f^{∗}(c_{1}) ∪ ξ^{a−1} + ⋯ + f^{∗}(c_{a−1}) ∪ ξ + f^{∗}(c_{a}) = 0

in H^{2a}(ℙ(E_{⋆}), 𝔗_{n}(a)).

Proposition 5.5 Let X_{⋆} be a simplicial object in V. Then the Chern classes c(E_{⋆}) of vector bundles E_{⋆} over X_{⋆} satisfy the following properties:

(1) (Normalization) If E_{⋆} is a line bundle, then we have c_{0}(E_{⋆}) = 1 and c_{i}(E_{⋆}) = 0 for i > 1, and c_{1}(E_{⋆}) is the value of the first Chern class c_{1}(E_{⋆}) ∈ H^{1}((X_{⋆})_{Zar}, 𝒪^{×}) under the map (5.3.1).

(2) (Functoriality) For a morphism f : X_{⋆} → X′_{⋆} of simplicial objects in V and a vector bundle E_{⋆} on X′_{⋆}, we have

c(f^{∗}E_{⋆}) = f^{∗}c(E_{⋆}),

where f^{∗} on the right hand side denotes the pull-back map obtained from the contravariant functoriality of 𝔗_{n}(r), cf. Remark 3.10.

(3) (Whitney sum) For a short exact sequence 0 → E′_{⋆} → E_{⋆} → E′′_{⋆} → 0 of vector bundles on X_{⋆}, we have

c_{r}(E_{⋆}) = ∑_{s+t=r} c_{s}(E′_{⋆}) ∪ c_{t}(E′′_{⋆}) ∈ H^{2r}(X_{⋆}, 𝔗_{n}(r))

for each r ≥ 0, where the cup product is taken with respect to the product structure of {𝔗_{n}(r)_{X_{⋆}}}_{r≥0} in the derived category of ℤ/p^{n}-sheaves on (X_{⋆})_{ét} , cf. Remark 3.11 (1).

Moreover, the Chern classes c(E_{⋆}) are characterized by these three properties.

Proof. The properties (1) and (2) immediately follow from this definition of Chern classes. The last assertion on the uniqueness follows from the splitting principle of vector bundles, whose details are straight-forward and left to the reader.

We prove the property (3) using the arguments of Grothendieck in [G] p. 144 Theorem 1 (iii), as follows. Let π′ : D′_{⋆} → X_{⋆} and π′′ : D′′_{⋆} → X_{⋆} be the (simplicial) flag schemes of E′_{⋆} and E′′_{⋆}, respectively, and put

D_{⋆} := D′_{⋆} ×_{X_{⋆}} D′′_{⋆} ,

which is identified with the flag scheme of π^{′′∗}E′_{⋆} over D′′_{⋆}. Let f : D_{⋆} → X_{⋆} be the natural projection. Since the pull-back map

f^{∗} : H^{2i}(X_{⋆}, 𝔗_{n}(i)) ⟶ H^{2i}(D_{⋆}, 𝔗_{n}(i)),

is injective by (5.4.3), we may replace (X_{⋆}, E_{⋆}, E′_{⋆}, E′′_{⋆}) with (D_{⋆}, f^{∗}E_{⋆}, f^{∗}E′_{⋆}, f^{∗}E′′_{⋆}) to assume that E_{⋆} has a filtration by subbundles

E_{⋆} = E_{⋆}^{0} ⊃ E_{⋆}^{1} ⊃ ⋯ ⊃ E_{⋆}^{a} = 0 (a := rank(E_{⋆}))

such that the quotient E_{⋆}^{i}/E_{⋆}^{i+1} is a line bundle over X_{⋆} for 0 ≤ i ≤ a − 1 and such that E_{⋆}^{b} = E′_{⋆} for b = rank(E′′_{⋆}). Now let

g : X′_{⋆} := ℙ(E_{⋆}) ⟶ X_{⋆}

be the projective bundle associated with E_{⋆}. Let L_{⋆} be the tautological line bundle over X′ = ℙ(E_{⋆}) and let s : X′_{⋆} → g^{∗}E_{⋆} ⊗ L_{⋆} =: F_{⋆} be the section induced by the canonical inclusion (L_{⋆})^{∨} ↪ g^{∗}E_{⋆}:

s : X′_{⋆} ⟶^{1} 𝔸_{X′_{⋆}}^{1} ⟶ g^{∗}E_{⋆} ⊗ L_{⋆} = F_{⋆} .

Put F_{⋆}^{i} := g^{∗}E_{⋆}^{i} ⊗ L and V_{⋆}^{i} := s^{−1}(F_{⋆}^{i}) for 0 ≤ i ≤ a. Then V_{⋆}^{a} is empty because s does not vanish. Moreover V_{⋆}^{i} is a simplicial object in V and the section

s_{i} : V_{⋆}^{i} ⟶^{s} F_{⋆}^{i}|_{V_{⋆}^{i}} ⟶ (F_{⋆}^{i}/F_{⋆}^{i+1})|_{V_{⋆}^{i}}

meets the zero section transversally for 0 ≤ i ≤ a − 1, cf. [G] p. 147. On the other hand, to prove the Whitney sum formula, it is enough to show

∏_{i=0}^{m+n−1} c_{1}(F_{⋆}^{i}/F_{⋆}^{i+1}) = 0.

We are thus reduced to the following simplicial analogue of loc. cit. p.141 Lemma 2:

Lemma 5.6 Let X_{⋆} be a simplicial object in V. Let E_{⋆} be a vector bundle of rank a over X_{⋆}. Let (E_{⋆}^{i})_{0≤i≤a} be a descending filtration consisting of subbundles on E_{⋆} such that E_{⋆}^{i} has rank a − i. For 1 ≤ i ≤ a, put

ξ_{i} := c_{1}(E_{⋆}^{i−1}/E_{⋆}^{i}) ∈ H^{2}(X_{⋆}, 𝔗_{n}(1)).

Let s : X_{⋆} → E_{⋆} be a section of E_{⋆} → X_{⋆}. For 0 ≤ i ≤ a, put

V_{⋆}^{i} := s^{−1}(E_{⋆}^{i}), L_{⋆}^{i} := (E_{⋆}^{i}/E_{⋆}^{i+1})|_{V_{⋆}^{i}} (restriction of E_{⋆}^{i}/E_{⋆}^{i+1} onto V_{⋆}^{i}),

and let s_{i} : V_{⋆}^{i} → L_{⋆}^{i} be the section induced by s. Assume the following condition:

• V_{⋆}^{i} is a simplicial object in V for 0 ≤ i ≤ a, and s_{i} intersects the zero section transversally for 0 ≤ i ≤ a − 1.

Then we have

cl_{X_{⋆}}(V_{⋆}^{a}) = ∏_{i=1}^{a} ξ_{i} in H^{2a}(X_{⋆}, 𝔗_{n}(a)) ,

where cl_{X_{⋆}}(V_{⋆}^{a}) denotes the value of 1 under the Gysin map

ℤ/p^{n} = H^{0}(V_{⋆}^{a}, 𝔗_{n}(0)) ⟶ H^{2a}(X_{⋆}, 𝔗_{n}(a)) .

One can easily check this lemma by the properties of the Gysin morphisms in Proposition 5.4 and the arguments in loc. cit. p.141 Lemma 2. This completes the proof of Proposition 5.5. □

Now let X be a scheme which belongs to V. Applying the construction of Chern classes to the case X_{⋆} = B_{⋆}GL_{s,X} and E_{⋆} = universal rank s bundle over B_{⋆}GL_{s}/X, we obtain Chern classes

c_{j}(E_{⋆}) ∈ H^{2j}(B_{⋆}GL_{s,X}, 𝔗_{n}(j)) (j ≥ 0), which are called the universal rank s Chern classes. On the other hand, let I_{n}(j)_{X}^{•} be an injective resolution of the complex C_{n}(j)_{X}^{•} on X_{ét} defined in Definition 3.5, and consider the following complex of abelian sheaves on X_{Zar}:

⋯ ⟶ ϵ_{∗}I_{n}(j)_{X}^{2j−2} ⟶^{d^{2j−2}} ϵ_{∗}I_{n}(j)_{X}^{2j−1} ⟶^{d^{2j−1}} Ker(d^{2j} : ϵ_{∗}I_{n}(j)_{X}^{2j} → ϵ_{∗}I_{n}(j)_{X}^{2j+1}).

Here ϵ : X_{ét} → X_{Zar} denotes the continuous map of small sites, and we regarded this sequence as a chain complex with the most right term placed in degree 0. We apply the Dold-Puppe construction [DP] to this complex to obtain a sheaf of simplicial abelian groups, which we denote by K(𝔗_{n}(j), 2j). For a closed subset Z of X and non-negative integers i, r ≥ 0, we define the Chern class map

C_{r,i}^{Z} : K_{i}^{Z}(X) ⟶ H_{Z}^{2r−i}(X, 𝔗_{n}(r))  (5.6.1)

as the following composite map (cf. [Gi1] Definition 2.22):

K_{i}^{Z}(X) ⟶ H_{Z}^{−i}(X_{Zar}, ℤ × ℤ_{∞}B_{⋆}GL(𝒪_{X}))

⟶^{pr} H_{Z}^{−i}(X_{Zar}, ℤ_{∞}B_{⋆}GL(𝒪_{X}))

⟶^{π_{i}(ℤ_{∞}c_{r})} H_{Z}^{−i}(X_{Zar}, ℤ_{∞}K(𝔗_{n}(r), 2r)) ≅ H_{Z}^{2r−i}(X_{ét}, 𝔗_{n}(r)).

Here ℤ_{∞} denotes the Bousfield-Kan completion [BoK], and we have used the universal rank s Chern class c_{r}(E_{⋆}) for a sufficiently large s to define the arrow π_{i}(ℤ_{∞}c_{r}). See [Gi1] Proposition 2.15 for the first arrow and see loc. cit. p. 226 for the last isomorphism. The map C_{r,0}^{X} agrees with c_{r} for X_{⋆} = X (constant simplicial scheme) defined before.

Theorem 5.7 (1) C_{r,i}^{Z} is contravariantly functorial in the pair (X, Z), that is, for a morphism f : X → X′ in V and a closed subset Z′ ⊂ X′ with f^{−1}(Z′) ⊂ Z, there is a commutative diagram

\xymatrix{K_{i}^{Z′}(X′) \ar[r]^{C_{r,i}^{Z′}} \ar[d]_{f^{∗}} & H_{Z′}^{2r−i}(X′, 𝔗_{n}(r)) \ar[d]^{f^{∗}} \\ K_{i}^{Z}(X) \ar[r]^{C_{r,i}^{Z}} & H_{Z}^{2r−i}(X, 𝔗_{n}(r)).}

(2) C_{r,i}^{Z} is additive for i > 0.

(3) The induced Chern character

ch : ⨁_{i≥0} K_{i}(X) ⟶ ∏_{i,r≥0} H^{i}(X, 𝔗_{ℚ_{p}}(r))

with H^{i}(X, 𝔗_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{i}(X, 𝔗_{n}(r)) is a ring homomorphism.

Proof. (1) follows from the functoriality results in Proposition 5.5 (2) and Proposition 3.8. The assertion (2) follows from Proposition 5.5 (3) and [Gi1] Lemma 2.26. See loc. cit. Definition 2.34 and Proposition 2.35 for the assertion (3). □

6 Purity along log poles

This and the next section are devoted to the construction of cycle class maps from higher Chow groups to p-adic étale Tate twists.

Let k be a field of characteristic p > 0, and let Y be a normal crossing variety over k. Let c be a positive integer. Let D be a non-empty admissible divisor on Y, and let Z be a reduced closed subscheme of Y which has codimension ≥ c and contained in D. Let i : Z ↪ Y be the natural closed immersion. In this section, we first prove the following purity result:

Theorem 6.1 (Purity) We have R^{q}i^{!}ν_{(Y,D),n}^{r} = 0 for q ≤ c.

See Theorem 6.5 below for a consequence of this theorem. Note that the assertion for q = c does not follow directly from Gros’ purity [Gr1] II Théorème 3.5.8, even when Y is smooth. The case that Y is smooth has been considered and proved independently by Mieda ([Mi1] Proof of Theorem 2.4, [Mi2]). Although the proof of Theorem 6.1 given below is essentially a variant of Gros’ proof of [Gr1] II Théorème 3.5.8, we include detailed computations for the convenience of the reader.

Proof of Theorem 6.1. Since the problem is étale local, we may assume that Y is simple and that there exists a pair (𝒴 , 𝒟) satisfying the following (1) and (2):

(1) 𝒴 is smooth over k and contains Y as a normal crossing divisor,

(2) 𝒟 is a normal crossing divisor on 𝒴 such that Y ∪ 𝒟 has normal crossings on 𝒴 and such that Y ∩ 𝒟 = D.

Then we have a short exact sequence on 𝒴_{ét}

0 ⟶ W_{n} Ω_{(𝒴 ,𝒟) ,log}^{r+1} ⟶ W_{n} Ω_{(𝒴 ,Y ∪𝒟) ,log}^{r+1} ⟶ ν_{(Y,D),n}^{r} ⟶ 0

(a variant of (2.1.1)), which reduces the assertion to the case that Y is smooth over k and that D has normal crossings on Y. Furthermore, we may assume that Z is regular of pure codimension c by a standard devissage argument. In what follows, we prove

R^{q}i^{!}W_{n} Ω_{(Y,D),log}^{r} = 0 for q ≠ c + 1,  (6.1.1)

assuming that Y is smooth over k (and that Z is regular and contained in D). By the short exact sequences on Y_{ét}

0 ⟶ W_{n−1} Ω_{(Y,D),log}^{r} ⟶ W_{n} Ω_{(Y,D),log}^{r} ⟶ Ω_{(Y,D),log}^{r} ⟶ 0,

0 ⟶ Ω_{(Y,D),log}^{r} ⟶ 𝒵_{Y}^{r}(logD) ⟶^{1−C} Ω_{Y}^{r} (logD) ⟶ 0

(cf. (2.1.1)and [Sa1] Corollary 2.2.5 (2), Lemma 2.4.6), it is enough to show that

R^{q}i^{!}𝒵_{Y}^{r}(logD) = 0 = R^{q}i^{!}Ω_{Y}^{r} (logD) for q ≠ c,  (6.1.2)
1 − C : R^{c}i^{!}𝒵_{Y}^{r}(logD) ⟶ R^{c}i^{!}Ω_{Y}^{r} (logD) is injective.  (6.1.3)

The assertion (6.1.2) follows from the smoothness of Y and the fact that Ω_{Y}^{r} (logD) and 𝒵_{Y}^{r}(logD) are locally free over 𝒪_{Y} and (𝒪_{Y})^{p}, respectively.

We prove (6.1.3) in what follows. Since the problem is étale local on Z, we may assume the following condition:

(⋆) Y is affine, and there exists a regular sequence t_{1}, ⋯, t_{c} ∈ Γ (Y, 𝒪_{Y}) such that the ideal (t_{1}, ⋯, t_{c}) ⊂ 𝒪_{Y} defines Z and such that t_{1}, ⋯, t_{a} (1 ≤ a ≤ c) are uniformizers of the irreducible components of D. Moreover, Ω_{Y}^{1} is free over 𝒪_{Y} and has a basis {dt_{λ}}_{λ∈Λ} which contains dt_{1}, …, dt_{c}.

Let τ be the natural open immersion

\xymatrix{τ : Y [t_{1}^{−1}, …, t_{c}^{−1}] \ar@{^{(}->}[r] & Y.}

For j = 1, …, c, let σ_{j} be the natural open immersion

\xymatrix{σ_{j} : Y [t_{1}^{−1}, …, t_{j−1}^{−1}, t_{j+1}^{−1}, …, t_{c}^{−1}] \ar@{^{(}->}[r] & Y.}

We recall here the following standard fact (cf. [Gr1] II (3.3.6)):

Lemma 6.2 We have isomorphisms of sheaves on Z_{ét}

R^{c}i^{!}Ω_{Y}^{r} (logD) ≃ τ_{∗}τ^{∗}Ω_{Y}^{r} (logD)/ ∑_{j=1}^{c} σ_{j∗}σ_{j}^{∗}Ω_{Y}^{r} (logD)  (6.2.1)

= τ_{∗}τ^{∗}Ω_{Y}^{r} /( ∑_{j=1}^{a} σ_{j∗}σ_{j}^{∗}Ω_{Y}^{r} (logD) + ∑_{j=a+1}^{c} σ_{j∗}σ_{j}^{∗}Ω_{Y}^{r} ),

R^{c}i^{!}𝒵_{Y}^{r}(logD) ≃ τ_{∗}τ^{∗}𝒵_{Y}^{r}(logD)/ ∑_{j=1}^{c} σ_{j∗}σ_{j}^{∗}𝒵_{Y}^{r}(logD)  (6.2.2)
= τ_{∗}τ^{∗}𝒵_{Y}^{r}/( ∑_{j=1}^{a} σ_{j∗}σ_{j}^{∗}𝒵_{Y}^{r}(logD) + ∑_{j=a+1}^{c} σ_{j∗}σ_{j}^{∗}𝒵_{Y}^{r}),

where we regarded the sheaves on the right hand side as sheaves on Z_{ét} naturally.

We define an ascending filtration Fil_{m} (m ≥ 0) on τ_{∗}τ^{∗}Ω_{Y}^{r} as

Fil_{m}(τ_{∗}τ^{∗}Ω_{Y}^{r}) := \{\frac{1}{(t_{1}t_{2} ⋯ t_{c})^{pm}} ω ∈ τ_{∗}τ^{∗}Ω_{Y}^{r} | ω ∈ Ω_{Y}^{r} \}.

Let Fil_{•}(R^{c}i^{!}Ω_{Y}^{r} (logD)) be the induced filtration, and let Fil_{•}(R^{c}i^{!}𝒵_{Y}^{r}(logD)) be its inverse image under the canonical map

R^{c}i^{!}𝒵_{Y}^{r}(logD) ⟶ R^{c}i^{!}Ω_{Y}^{r} (logD))  (6.2.3)

induced by the natural inclusion 𝒵_{Y}^{r}(logD) ↪ Ω_{Y}^{r} (logD)). Note that

R^{c}i^{!}𝒵_{Y}^{r}(logD) = ⋃_{m=0}^{∞} Fil_{m}(R^{c}i^{!}𝒵_{Y}^{r}(logD)).

We prove here the following lemma.

202 K. Sato Lemma 6.3 (1) The map (6.2.3)is injective.

(2) Fil_{m}(R^{c}i^{!}𝒵_{Y}^{r}(logD)) is generated by elements of the form

[\frac{1}{(t_{1}t_{2} ⋯ t_{c})^{pm}} ω] with ω ∈ 𝒵_{Y}^{r},

where [−] denotes the residue class in Fil_{m}(R^{c}i^{!}𝒵_{Y}^{r}(logD)) via (6.2.2).

(3) The kernel of the projection via (6.2.1)

Fil_{m}(τ_{∗}τ^{∗}Ω_{Y}^{r}) ⟶ Fil_{m}(R^{c}i^{!}Ω_{Y}^{r} (logD))

agrees with the subgroup

\{ ∑_{j=1}^{c} \frac{1}{(t_{1} ⋯ t_{j−1}t_{j+1} ⋯ t_{c})^{pm}} ω_{j} | \begin{matrix}ω_{j} ∈ Ω_{Y}^{r} (logD_{j}) & (1 ≤ j ≤ a) \\ ω_{j} ∈ Ω_{Y}^{r} & (a < j ≤ c) \end{matrix} \},

where D_{j} ⊂ Y denotes the regular divisor defined by t_{j} for 1 ≤ j ≤ a. Proof. We prove (1). By the short exact sequence

0 ⟶ 𝒵_{Y}^{r}(logD) ⟶ Ω_{Y}^{r} (logD) ⟶^{d} ℬ_{Y}^{r+1}(logD) ⟶ 0,

we have a long exact sequence on Z_{ét}

⋯ ⟶ R^{c−1}i^{!}ℬ_{Y}^{r+1}(logD) ⟶ R^{c}i^{!}𝒵_{Y}^{r}(logD) ⟶ R^{c}i^{!}Ω_{Y}^{r} (logD) ⟶ · · · . The assertion follows from the smoothness of Y and the fact that ℬ_{Y}^{r+1}(logD) is locally free over (𝒪_{Y})^{p}. The assertion (2) is a consequence of (1), and (3) is straightforward. □

We return to the proof of (6.1.3), and compute the map 1 − C in (6.1.3):

1 − C : R^{c}i^{!}𝒵_{Y}^{r}(logD) ⟶ R^{c}i^{!}Ω_{Y}^{r} (logD)),  (6.3.1)

using the filtration Fil_{•}. Put

gr_{m}(−) := Fil_{m}(−)/Fil_{m−1}(−).

Since we have

C(\frac{1}{(t_{1}t_{2} ⋯ t_{c})^{pm}} ω) = \frac{1}{(t_{1}t_{2} ⋯ t_{c})^{m}} C(ω) for ω ∈ 𝒵_{Y}^{r},

the Cartier operator preserves Fil_{•} (by Lemma 6.3 (2)) and induces a map

gr_{m}(C) : gr_{m}(R^{c}i^{!}𝒵_{Y}^{r}(logD)) ⟶ gr_{m}(R^{c}i^{!}Ω_{Y}^{r} (logD)),

Documenta Mathematica 18 (2013) 177–247

which is the zero map for m ≥ 2. Hence we have

the kernel of (6.3.1) ⊂ Fil_{1}(R^{c}i^{!}𝒵_{Y}^{r}(logD)).

Let D_{j} ⊂ Y be as we defined in Lemma 6.3 (3) for 1 ≤ j ≤ a. Fix an arbitrary ω ∈ 𝒵_{Y}^{r}, and assume that

x := [\frac{1}{(t_{1}t_{2} ⋯ t_{c})^{p}} ω] ∈ Fil_{1}(R^{c}i^{!}𝒵_{Y}^{r}(logD))

belongs to the kernel of (6.3.1). To show (6.1.3), we have to prove that

ω = ∑_{j=1}^{c} t_{j}^{p} η_{j} for some \begin{cases}η_{j} ∈ Ω_{Y}^{r} (logD_{j}) & (1 ≤ j ≤ a) \\ η_{j} ∈ Ω_{Y}^{r} & (a < j ≤ c) \end{cases}  (6.3.2)

in Ω_{Y}^{r} (logD), which is equivalent to that x = 0 in R^{c}i^{!}𝒵_{Y}^{r}(logD) by Lemma 6.3 (1) and (6.2.1). Since x = C(x) in Fil_{1}(R^{c}i^{!}Ω_{Y}^{r} (logD)) by assumption, we have

\frac{1}{(t_{1}t_{2} ⋯ t_{c})^{p}} ω − \frac{1}{t_{1}t_{2} ⋯ t_{c}} C(ω) = ∑_{j=1}^{c} \frac{1}{(t_{1}t_{2} ⋯ t_{j−1}t_{j+1} ⋯ t_{c})^{p}} α_{j}

in τ_{∗}τ^{∗}Ω_{Y}^{r} for some α_{j} ∈ Ω_{Y}^{r} (logD_{j}) (1 ≤ j ≤ a) and α_{j} ∈ Ω_{Y}^{r} (a < j ≤ c) by Lemma 6.3 (3). This implies

ω = (t_{1}t_{2} ⋯ t_{c})^{p−1}C(ω) + ∑_{j=1}^{c} t_{j}^{p} α_{j} in Ω_{Y}^{r} ,  (6.3.3)

and our task is to prove that

(t_{1}t_{2} ⋯ t_{c})^{p−1}C(ω) = ∑_{j=1}^{c} t_{j}^{p} ζ_{j} for some \begin{cases}ζ_{j} ∈ Ω_{Y}^{r} (logD_{j}) & (1 ≤ j ≤ a) \\ ζ_{j} ∈ Ω_{Y}^{r} & (a < j ≤ c). \end{cases}

(6.3.4)

Take a basis {dt_{λ}}_{λ∈Λ} of Ω_{Y}^{1} over 𝒪_{Y} which contain dt_{1}, …, dt_{c} (see the condition (⋆)). Fix an ordering on Λ and let J be the set of all r-tuples λ = (λ_{1}, λ_{2}, …, λ_{r}) of elements of Λ satisfying

λ_{1} < λ_{2} < ⋯ < λ_{r}.

For λ = (λ_{1}, λ_{2}, …, λ_{r}) ∈ J, put

dt_{λ} := dt_{λ_{1}} ∧ dt_{λ_{2}} ∧ ⋯ ∧ dt_{λ_{r}} ∈ Ω_{Y}^{r} .

Using the basis {dt_{λ}}_{λ∈J} of Ω_{Y}^{r} over 𝒪_{Y}, we decompose C(ω) as

C(ω) = ϵ + δ  (ϵ, δ ∈ Ω_{Y}^{r}).  (6.3.5)

Here ϵ is an 𝒪_{Y} -linear combination of dt_{λ}’s which contain dt_{j} for some 1 ≤ j ≤ a, and δ is an 𝒪_{Y} -linear combination of dt_{λ}’s which contain none of dt_{1}, …, dt_{a}. We have

(t_{1}t_{2} ⋯ t_{c})^{p−1}ϵ = ∑_{j=1}^{a} t_{j}^{p} β_{j} for some β_{j} ∈ Ω_{Y}^{r} (logD_{j}) (1 ≤ j ≤ a), (6.3.6)

which implies (6.3.4)if δ = 0. Otherwise, we proceed as follows. Put

y := (t_{2}t_{3} ⋯ t_{c})^{p−1}δ ∈ Ω_{Y}^{r} .

Since dω = 0, the equalities (6.3.3)and (6.3.6)imply

t_{1}^{p−2} dt_{1} ∧ y = t_{1}^{p−1} dy + ∑_{j=1}^{c} t_{j}^{p} β′_{j} for some \begin{cases}β′_{j} ∈ Ω_{Y}^{r} (logD_{j}) & (1 ≤ j ≤ a) \\ β′_{j} ∈ Ω_{Y}^{r} & (a < j ≤ c). \end{cases}

By this equality and the assumption on δ, one can easily check that

t_{1}^{p−2} dt_{1} ∧ y = ∑_{j=1}^{c} t_{j}^{p} β′′_{j} for some \begin{cases}β′′_{1} ∈ Ω_{Y}^{r} (logD_{1}) \\ β′′_{j} ∈ Ω_{Y}^{r} & (2 ≤ j ≤ c). \end{cases} (6.3.7)

To proceed with the proof of (6.3.4), we need the following lemma:

Lemma 6.4 Let f and g be the following 𝒪_{Y} -linear maps, respectively:

f : Ω_{Y}^{r} ⟶ Ω_{Y}^{r+1}, z ↦ dt_{1} ∧ z,
g : Ω_{Y}^{r} (logD_{1}) ⟶ Ω_{Y}^{r+1}(logD_{1}), z ↦ \frac{dt_{1}}{t_{1}} ∧ z.

Then the following holds:

(1) For m ≥ 1, we have

Im(f) ∩ (t_{1}^{m} Ω_{Y}^{r+1}(logD_{1}) + ∑_{j=2}^{c} t_{j}^{m} Ω_{Y}^{r+1}) = t_{1}^{m} Im(g) + ∑_{j=2}^{c} t_{j}^{m} Im(f).

(2) For m, n ≥ 0, we have

t_{1}^{n} Ω_{Y}^{r+1} ∩ ( ∑_{j=2}^{c} t_{j}^{m} Im(f)) = ∑_{j=2}^{c} t_{1}^{n} t_{j}^{m} Im(f).

(3) For m ≥ 0, the sequence

\xymatrix{t_{1}^{m} Ω_{Y}^{r−1} \ar[r]^{f} & t_{1}^{m} Ω_{Y}^{r} \ar[r]^{f} & t_{1}^{m} Ω_{Y}^{r+1}}

is exact.

Proof. These assertions follow from linear algebra over 𝒪_{Y}. The details are straightforward and left to the reader. □

By (6.3.7)and Lemma 6.4 (1) for m = p, we have

t_{1}^{p−2} dt_{1} ∧ y = t_{1}^{p−1} dt_{1} ∧ γ_{1} + ∑_{j=2}^{c} t_{j}^{p} dt_{1} ∧ γ_{j}  (6.4.1)

for some γ_{1} ∈ Ω_{Y}^{r} (logD_{1}) and some γ_{j} ∈ Ω_{Y}^{r} (2 ≤ j ≤ c). If p = 2, this equality implies

t_{1}^{p−2} dt_{1} ∧ y = t_{1}^{p−2} dt_{1} ∧ (t_{1}γ_{1} + ∑_{j=2}^{c} t_{j}^{p} γ_{j}).  (6.4.2)

If p ≥ 3, we obtain (6.4.2) from (6.4.1) by replacing γ_{2}, …, γ_{c} suitably in Ω_{Y}^{r} (2 ≤ j ≤ c), where we used Lemma 6.4 (2) for (m, n) = (p, p − 2). Noting that t_{1}γ_{1} and t_{j}^{p} γ_{j} (2 ≤ j ≤ c) belong to Ω_{Y}^{r}, we have

t_{1}^{p−2} y = t_{1}^{p−2}dt_{1} ∧ ϑ + t_{1}^{p−1}γ_{1} + ∑_{j=2}^{c} t_{j}^{p} (t_{1}^{p−2}γ_{j}) for some ϑ ∈ Ω_{Y}^{r−1}

by (6.4.2)and Lemma 6.4 (3) for m = p − 2. Thus we have

(t_{1}t_{2} ⋯ t_{c})^{p−1}δ = t_{1}^{p−1} y = t_{1}^{p}(\frac{dt_{1}}{t_{1}} ∧ ϑ + γ_{1}) + ∑_{j=2}^{c} t_{j}^{p} (t_{1}^{p−1}γ_{j}),

which implies (6.3.4) (see also (6.3.5), (6.3.6)). This completes the proof of (6.3.2), (6.1.3)and Theorem 6.1. □

Theorem 6.5 Let S, p and X be as in Setting 3.1, and let c be a positive integer. Let D be a non-empty normal crossing divisor on X, and let Z be a closed subset of X which has codimension ≥ c. Put U := X − D. Then we have

H_{Z}^{q}(X, 𝔗_{n}(r)_{(X,D)}) ≃ \begin{cases}0 & (q < r + c) \\ H_{Z∩U}^{r+c}(U, 𝔗_{n}(r)_{U}) & (q = r + c). \end{cases}  (6.5.1)

In particular, when Z has pure codimension r on X, we have

H_{Z}^{q}(X, 𝔗_{n}(r)_{(X,D)}) ≃ \begin{cases}0 & (q < 2r) \\ ℤ/p^{n}[Z^{0} ∩ U] & (q = 2r), \end{cases}  (6.5.2)

where ℤ/p^{n}[Z^{0} ∩ U] means the free ℤ/p^{n}-module generated by the set Z^{0} ∩ U.

Proof. Admitting (6.5.1), we obtain (6.5.2)from the purity of 𝔗_{n}(r)_{U} ([Sa2] Theorem 4.4.7, Corollary 4.4.9) and the absolute purity of μ_{p^{n}}^{⊗r} on U[p^{−1}] ([FG]). To show (6.5.1), we divide the problem into the following 4 cases:

(1) Z ⊂ D ∩ Y (2) Z ⊂ D and Z ⊄ Y

(3) Z is arbitrary and q < r + c (4) Z is arbitrary and q = r + c.

The case (1) follows from Theorems 3.3, 6.1 (see also Remark 3.13) and the same arguments as in [Sa2] Theorem 4.4.7. The case (2) follows from the case (1) and the same arguments as in loc. cit. Corollary 4.4.9. The case (3) also follows from similar arguments as for the previous cases. To prove the case (4), write Z as the union of closed subsets

Z = Z_{1} ∪ Z_{2},

where Z_{1} has pure codimension c on X, Z_{2} has codimension ≥ c + 1 on X and we suppose that no irreducible components of Z_{2} are contained in Z_{1}. Since Z_{1} ∩ Z_{2} has codimension ≥ c + 2 on X, the assertion in the cases (1)–(3) and a Mayer-Vietoris long exact sequence

⋯ ⟶ H_{Z_{1}∩Z_{2}}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ ⨁_{j=1,2} H_{Z_{j}}^{r+c}(X, 𝔗_{n}(r)_{(X,D)})

⟶ H_{Z}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ H_{Z_{1}∩Z_{2}}^{r+c+1}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ ⋯

imply isomorphisms

H_{Z}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ≃ ⨁_{j=1,2} H_{Z_{j}}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ≃ H_{Z_{1}}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}).

Similarly, we have H_{Z∩U}^{r+c}(U, 𝔗_{n}(r)_{U}) ≃ H_{Z_{1}∩U}^{r+c}(U, 𝔗_{n}(r)_{U}) by the purity of 𝔗_{n}(r)_{U} ([Sa2] Corollary 4.4.9). Hence we may assume that Z has pure codimension c on X. Moreover we may assume that no irreducible components of Z are contained in D by the cases (1)+(2) and a similar devissage argument. Then noting that Z ∩ D has codimension ≥ c + 1 on X, we obtain the case (4) from the cases (1)+(2) and a long exact sequence

⋯ ⟶ H_{Z∩D}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ H_{Z}^{r+c}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ H_{Z∩U}^{r+c}(U, 𝔗_{n}(r)_{U})

⟶ H_{Z∩D}^{r+c+1}(X, 𝔗_{n}(r)_{(X,D)}) ⟶ · · · .

This completes the proof of Theorem 6.5. □

Definition 6.6 Let X and D be as in Theorem 6.5, and let C be a cycle on U := X − D of codimension r. Let W ⊂ U be the support of C and let \overline{W} be its closure in X. Then we define the cycle class

cl_{X}(C) ∈ H2r ; \overline{W}(X, 𝔗_{n}(r)_{(X,D)})

as the inverse image of cl_{U}(C) ([Sa2] 5.1.2) under the isomorphism in Theorem 6.5

H2r ; \overline{W}(X, 𝔗_{n}(r)_{(X,D)}) ^{∼} ⟶ H_{W}^{2r}(U, 𝔗_{n}(r)_{U}).

7 Cycle class map

Let S, p and X be as in Setting 3.1. In this section, we construct a cycle class morphism

cl_{X} : ℤ(r)_{X}^{ét} ⊗^{𝕃} ℤ/p^{n} ⟶ 𝔗_{n}(r)_{X} in D(X_{ét}, ℤ/p^{n}),  (7.0.1)

following the method of Bloch [B3] §4 (cf. [GL] §3, see also Remark 7.3 below). Here ℤ(r)_{X}^{ét} denotes the étale sheafification on X of the presheaf of cochain complexes

U ⟼ z^{r}(U, ⋆)[−2r],

and z^{r}(U, ⋆) denotes Bloch’s cycle complex

⋯ ⟶ z^{r}(U, q) ⟶^{d_{q}} z^{r}(U, q − 1) ⟶^{d_{q−1}} ⋯ ⟶^{d_{1}} z^{r}(U, 0).
(degree −q)  (degree −q + 1)  (degree 0)

We review the definition of this complex briefly, which will be useful later. Let Δ^{⋆} be the standard cosimplicial scheme over Spec(ℤ):

Δ^{q} := Spec(ℤ[t_{0}, t_{1}, …, t_{q}]/(t_{0} + t_{1} + ⋯ + t_{q} = 1)).

A face of Δ^{q} (of codimension a ≥ 1) is a closed subscheme defined by the equation

t_{i_{1}} = t_{i_{2}} = ⋯ = t_{i_{a}} = 0 for some 0 ≤ i_{1} < i_{2} < ⋯ < i_{a} ≤ q.

Now z^{r}(U, q) is defined as the free abelian group generated by the set of all integral closed subschemes on U × Δ^{q} of codimension r which meet all faces of U × Δ^{q} properly. Here a face of U ×Δ^{q} means the product of U and a face of Δ^{q}. Noting that the faces of codimension 1 are effective Cartier divisors, we define the differential map d_{q} as the alternating sum of pull-back maps along the faces of codimension 1, which defines the above complex z^{r}(U, ⋆).

We fix a projective completion \overline{Δ^{q}} of Δ^{q} as follows:

\overline{Δ^{q}} := Proj (ℤ[T_{0}, T_{1}, …, T_{q}, T_{∞}]/(T_{0} + T_{1} + ⋯ + T_{q} = T_{∞})).

Let D^{q} ⊂ \overline{Δ^{q}} be the hyperplane at infinity:

D^{q} : T_{∞} = 0.

The following proposition plays a key role in our construction of the morphism (7.0.1).

Proposition 7.1 Let q and r be integers with q, r ≥ 0, and let U be étale of finite type over X. Let Σ^{r,q} be the set of all closed subsets on U × Δ^{q} of pure codimension r which meet the faces of U × Δ^{q} properly. For W ∈ Σ^{r,q}, let \overline{W} be the closure of W in U × \overline{Δ^{q}}. Then:

(1) There is an isomorphism

cl^{r,q} : z^{r}(U, q) ⊗ ℤ/p^{n} ^{∼} ⟶ \underrightarrow{lim}_{W ∈Σ^{r,q}} H2r ; \overline{W} (U × \overline{Δ^{q}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})})

sending a cycle C ∈ z^{r}(U, q) to the cycle class cl_{U×\overline{Δ^{q}}}(C) (see Definition 6.6).

(2) For W ∈ Σ^{r,q}, the natural morphism

τ_{≤2r} RΓ\overline{W} (U × \overline{Δ^{q}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})})

⟶ H2r ; \overline{W} (U × \overline{Δ^{q}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})}) [−2r]

is an isomorphism in the derived category of ℤ/p^{n}-modules.

(3) Let C be a cycle which belongs to z^{r}(U, q), and let W be the support of C (note that W belongs to Σ^{r,q}). Let \overline{i} : U × \overline{Δ^{q−1}} ↪ U × \overline{Δ^{q}} be the closure of a face map i : U × Δ^{q−1} ↪ U × Δ^{q}. Then the pull-back map

\overline{i}^{∗} : H2r ; \overline{W} (U × \overline{Δ^{q}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})})

⟶ H_{\overline{i}^{−1}(\overline{W})}^{2r} (U × \overline{Δ^{q−1}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q−1}},U×D^{q−1})})

sends the cycle class cl_{U×\overline{Δ^{q}}}(C) to cl_{U×\overline{Δ^{q−1}}}(i^{∗}C), where i^{∗}C denotes the pull-back of the cycle C along i.

Proof. (1) and (2) follow from Theorem 6.5 and the definition of z^{r}(U, q). We show (3). By Theorem 6.5, it is enough to show that the pull-back map

i^{∗} : H_{W}^{2r} (U × Δ^{q}, 𝔗_{n}(r)_{U×Δ^{q}}) ⟶ H_{i^{−1}(W)}^{2r} (U × Δ^{q−1}, 𝔗_{n}(r)_{U×Δ^{q−1}})

sends the cycle class cl_{U×Δ^{q}}(C) to cl_{U×Δ^{q−1}}(i^{∗}C). Put

𝒰 := U × Δ^{q}, 𝒟 := i(U × Δ^{q−1}) ⊂ 𝒰 and 𝒱 := 𝒰 − 𝒟

and let t ∈ Γ (𝒰, 𝒪_{𝒰} ) be a defining equation of 𝒟. Noting that i^{−1}(W) = W ∩ 𝒟, consider the following diagram:

\xymatrix{H_{W}^{2r}(𝒰, 𝔗_{n}(r)_{𝒰} ) \ar[d]_{α} \ar[r]^{i^{∗}} & H_{W ∩𝒟}^{2r} (𝒟, 𝔗_{n}(r)_{𝒟} ) \ar[d]^{i_{∗}}|{≀} \\ H_{W ∩𝒱}^{2r+1} (𝒱, 𝔗_{n}(r + 1)_{𝒱} ) \ar[r]^{−δ} \ar[d] & H_{W ∩𝒟}^{2r+2} (𝒰, 𝔗_{n}(r + 1)_{𝒰} ) \ar[d]|{≀} \\ ⨁_{y∈W^{0}} H_{y}^{2r+1}(𝒱, 𝔗_{n}(r + 1)_{𝒱} ) \ar[r]^{−δ} & ⨁_{x∈(W ∩𝒟)^{0}} H_{x}^{2r+2}(𝒰, 𝔗_{n}(r + 1)_{𝒰} ) \\ ⨁_{y∈W^{0}} κ(y)^{×}/p^{n} \ar[u]^{g}|{≀} \ar[r]^{div} & ⨁_{x∈(W ∩𝒟)^{0}} ℤ/p^{n}. \ar[u]_{g′}|{≀}}

Here we defined α by sending ω ∈ H_{W}^{2r}(𝒰, 𝔗_{n}(r)_{𝒰} ) to {t|_{𝒱} } ∪ ω|_{𝒱} , where {t|_{𝒱} } denotes the class of t|_{𝒱} ∈ Γ (𝒱, 𝒪_{𝒱}^{×}) in H^{1}(𝒱, 𝔗_{n}(1)_{𝒱} ). The arrows i_{∗}, g and g′ are Gysin isomorphisms (cf. [Sa2] Theorem 4.4.7). The arrows δ are boundary maps of localization long exact sequences, and the central square commutes obviously. The arrow div denotes the divisor map, and the bottom square commutes by the compatibility in loc. cit. Theorem 6.1.1 and [JSS] Theorem 3.1.1 (see also Remark 3.7 of this paper). The top square commutes by the following equalities (ω ∈ H_{W}^{2r}(𝒰, 𝔗_{n}(r)_{𝒰} )):

i_{∗} ∘ i^{∗}(ω) = cl_{𝒰} (𝒟) ∪ ω = −δ({t|_{𝒱} }) ∪ ω = −δ({t|_{𝒱} } ∪ ω|_{𝒱} ) = −δ ∘ α(ω),

where the first (resp. second) equality follows from the projection formula in [Sa2] Corollary 7.2.4 (resp. the same compatibility as mentioned before). Hence the assertion follows from the transitivity of Gysin maps for W ∩ 𝒟 ↪ 𝒟 ↪ 𝒰 (loc. cit. Corollary 6.3.3) and the fact that the divisor map div sends t|_{y} ∈ κ(y)^{×}/p^{n} to the cycle i^{∗}[y], where [y] means the cycle on 𝒰 given by the closure of y. This completes the proof. □

We construct the morphism (7.0.1). For U as in the proposition, there is a diagram of cochain complexes concerning • (see Corollary 3.9 (2) for G_{n}(r)^{•}):

z^{r}(U, q) ⊗ ℤ/p^{n}[−2r] ⟶^{cl^{r,q}} \underrightarrow{lim}_{W ∈Σ^{r,q}} H2r ; \overline{W} (U × \overline{Δ^{q}}, 𝔗_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})}) [−2r]

⟵^{α^{r,q}} \underrightarrow{lim}_{W ∈Σ^{r,q}} τ_{≤2r} Γ\overline{W} (U × \overline{Δ^{q}}, G_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})}^{•})

⟶^{β^{r,q}} Γ (U × \overline{Δ^{q}}, G_{n}(r)_{(U×\overline{Δ^{q}},U×D^{q})}^{•}) .

Here α^{r,q} and β^{r,q} are natural maps of complexes, which are obviously contravariant for the face maps U × Δ^{q−1} ↪ U × Δ^{q}. The map cl^{r,q} is bijective and contravariant for these face maps by Proposition 7.1 (1) and (3). Hence we get homomorphisms of double complexes concerning (⋆, •)

z^{r}(U, ⋆) ⊗ ℤ/p^{n}[−2r] ⟵^{(cl^{r,⋆})^{−1}∘α^{r,⋆}} \underrightarrow{lim}_{W ∈Σ^{r,⋆}} τ_{≤2r} Γ\overline{W} (U × \overline{Δ^{⋆}}, G_{n}(r)_{(U×\overline{Δ^{⋆}},U×D^{⋆})}^{•})

⟶^{β^{r,⋆}} Γ (U × \overline{Δ^{⋆}}, G_{n}(r)_{(U×\overline{Δ^{⋆}},U×D^{⋆})}^{•})

⟵ Γ (U, G_{n}(r)_{U}^{•}),

where the differentials in the ⋆-direction are alternating sums of pull-back maps along the faces of codimension 1, and the last arrow is the inclusion to the factor of ⋆ = 0. The first and the last arrows are quasi-isomorphisms on the associated total complexes by Proposition 7.1 (2) and Corollary 4.3, respectively. We thus obtain the desired morphism (7.0.1)in D(X_{ét}, ℤ/p^{n}) by sheafifying the diagram of total complexes.

Remark 7.2 The Rost-Voevodsky theorem ([Vo1], [Vo2], [We], cf. [BK1] §3) and the Suslin-Voevodsky theorem [SV] (cf. [GL]) imply that the morphism (7.0.1)induces isomorphisms (cf. [Sa2] Conjecture 1.4.1)

τ_{≤r} (ℤ(r)_{X}^{ét} ⊗^{𝕃} ℤ/p^{n}) ^{∼} ⟶ 𝔗_{n}(r)_{X}  in D^{b}(X_{ét}, ℤ/p^{n}),
τ_{≤r} (ℤ(r)_{X}^{Zar} ⊗^{𝕃} ℤ/p^{n}) ^{∼} ⟶ τ_{≤r}Rε_{∗}𝔗_{n}(r)_{X}  in D^{b}(X_{Zar}, ℤ/p^{n}),

which one can easily check by similar arguments as in [SS] §A.2. Here ℤ(r)_{X}^{Zar} denotes the complex of Zariski sheaves on X defined as U ↦ z^{r}(U, ⋆)[−2r], and ε denotes the continuous map of sites X_{ét} → X_{Zar}.

Remark 7.3 If X is smooth over S, then ℤ(r)_{X}^{Zar} is concentrated in degrees ≤ r by a result of Geisser [Ge] Corollary 4.4. Hence his arguments in loc. cit. §6, Proof of Theorem 1.3 gives an alternative construction of (7.0.1)in this case.

8 de Rham-Witt cohomology and homology

In this section we define two kinds of de Rham-Witt complexes for normal crossing varieties playing the roles of cohomology and homology, and relate them with the modified de Rham-Witt complex of Hyodo [Hy2]. The main results of this section are Theorems 8.8 and 8.12 below.

Let k be a field of characteristic p > 0, and let Y be a normal crossing variety over k. Let U ⊂ Y be a dense open subset which is smooth over k, and let σ : U ↪ Y be the natural open immersion.

Definition 8.1 We define the complex (W_{n} Λ_{Y}^{•} , d) of étale sheaves on Y as the differential graded subalgebra of σ_{∗}W_{n} Ω_{U}^{•} generated by

W_{n}𝒪_{Y} and λ_{Y,n}^{1}(⊂ σ_{∗}W_{n} Ω_{U}^{1}),

which we call the cohomological de Rham-Witt complex of Y. It is easy to see that W_{n} Λ_{Y}^{•} does not depend on the choice of U. For n = 1, W_{1} Λ_{Y}^{•} is the same as the complex Λ_{Y}^{•} defined in [Sa1] §3.3.

By the relations

\begin{cases}V (aR(x)) = V (a)x ∈ W_{n+1} Ω_{U}^{r} & (a ∈ W_{n}𝒪_{U}, x ∈ W_{n+1} Ω_{U,log}^{r}) \\ F (ax) = F (a)R(x) ∈ W_{n} Ω_{U}^{r} & (a ∈ W_{n+1}𝒪_{U}, x ∈ W_{n+1} Ω_{U,log}^{r}) \\ R(ax) = R(a)R(x) ∈ W_{n} Ω_{U}^{r} & (a ∈ W_{n+1}𝒪_{U}, x ∈ W_{n+1} Ω_{U,log}^{r}) \end{cases} (8.1.1)

(cf. [I1] I Théorème 2.17, Proposition 2.18), the operators V, F, R on W_{n} Ω_{U}^{r} induce operators

V : W_{n} Λ_{Y}^{r} ⟶ W_{n+1} Λ_{Y}^{r} ,
F : W_{n+1} Λ_{Y}^{r} ⟶ W_{n} Λ_{Y}^{r} ,
R : W_{n+1} Λ_{Y}^{r} ⟶ W_{n} Λ_{Y}^{r} ,

which satisfy relations

F V = V F = p, F dV = d, dF = pF d, V d = pdV,  (8.1.2)
RV = V R,  RF = F R,  Rd = dR.  (8.1.3)

The local structure of W_{n} Λ_{Y}^{r} can be written in terms of the usual Hodge-Witt sheaves of the strata of Y.

Proposition 8.2 (1) Assume that Y is simple, and let Y_{1}, Y_{2}, ⋯, Y_{q} be the irreducible components of Y. Then there is an exact sequence on Y_{ét}

0 ⟶ W_{n} Λ_{Y}^{r} ⟶^{ř^{0}} ⨁_{|I|=1} W_{n} Ω_{Y_{I}}^{r} ⟶^{ř^{1}} ⨁_{|I|=2} W_{n} Ω_{Y_{I}}^{r} ⟶^{ř^{2}}

⋯ ⟶^{ř^{q−1}} ⨁_{|I|=q} W_{n} Ω_{Y_{I}}^{r} ⟶ 0,

where the notation is the same as in Proposition 2.2.

(2) Assume that Y is embedded into a smooth k-variety 𝒴 as a normal crossing divisor. Let i : Y ↪ 𝒴 be the closed immersion. Then there is a short exact sequence on 𝒴_{ét}

0 ⟶ W_{n} Ω_{𝒴}^{r} (−logY) ⟶ W_{n} Ω_{𝒴}^{r} ⟶^{i^{∗}} i_{∗}W_{n} Λ_{Y}^{r} ⟶ 0.

Proof. (1) follows from the contravariant functoriality of Hodge-Witt sheaves of smooth varieties and the same arguments as for [Sa1] Proposition 3.2.1. The assertion (2) follows from (1) and [Mo] Lemma 3.15.1. □

We next introduce homological Hodge-Witt sheaves.

Definition 8.3 For r ≥ 0, we define the étale sheaf W_{n} Ξ_{Y}^{r} on Y as the W_{n}𝒪_{Y} - submodule of σ_{∗}W_{n} Ω_{U}^{r} generated by ν_{Y,n}^{r}(⊂ σ_{∗}W_{n} Ω_{U}^{r}). Similarly as for W_{n} Λ_{Y}^{r}, the sheaf W_{n} Ξ_{Y}^{r} does not depend on the choice of U. For n = 1, W_{1} Ξ_{Y}^{r} agrees with the sheaf Ξ_{Y}^{r} defined in [Sa1] §2.5.

Since λ_{Y,n}^{r} ⊂ ν_{Y,n}^{r} by definition, we have W_{n} Λ_{Y}^{r} ⊂ W_{n} Ξ_{Y}^{r}, which agree with W_{n} Ω_{Y}^{r} when Y is smooth (cf. [Sa1] Remark 3.1.4). The following local description will be useful later.

Proposition 8.4 (1) Under the same setting and notation as in Proposition 8.2 (1), there is a canonical ascending filtration F_{a} (a ≥ 0) on W_{n} Ξ_{Y}^{r} satisfying

F_{0}(W_{n} Ξ_{Y}^{r}) = 0 and gr_{a}^{F} W_{n} Ξ_{Y}^{r} ≃ ⨁_{|I|=a} W_{n} Ω_{Y_{I}}^{r−a+1},

where the isomorphism for a ≥ 2 depends on the fixed ordering of the irreducible components of Y.

(2) Under the same setting and notation as in Proposition 8.2 (2), there is a short exact sequence on 𝒴_{ét}

0 ⟶ W_{n} Ω_{𝒴}^{r+1} ⟶ W_{n} Ω_{𝒴}^{r+1}(logY) ⟶^{ϱ} i_{∗}W_{n} Ξ_{Y}^{r} ⟶ 0,

where ϱ is induced by the Poincaré residue mapping

ϱ_{0} : W_{n} Ω_{𝒴}^{r+1}(logY) ⟶ (iσ)_{∗}W_{n} Ω_{U}^{r}.

Proof. We first recall the following fact due to Mokrane [Mo] Proposition 1.4.5. Under the setting of (2), let F_{a}W_{n} Ω_{𝒴}^{m}(logY) (a ≥ 0) be the image of the product

W_{n} Ω_{𝒴}^{a} (logY) × W_{n} Ω_{𝒴}^{m−a} ⟶ W_{n} Ω_{𝒴}^{m}(logY).

Put F_{−1}W_{n} Ω_{𝒴}^{m}(logY) := 0. Then there are isomorphisms

gr_{a}^{F} W_{n} Ω_{𝒴}^{m}(logY) ≃ \begin{cases}W_{n} Ω_{𝒴}^{m} & (a = 0) \\ ⨁_{|I|=a} W_{n} Ω_{Y_{I}}^{m−a} & (a ≥ 1), \end{cases}  (8.4.1)

which are induced by Poincaré residue mappings for a ≥ 1 and depend on the fixed ordering of the irreducible components of Y for a ≥ 2. We prove (2). Because the complement Σ := Y − U has codimension ≥ 2 on 𝒴 and the sheaves W_{n} Ω_{𝒴}^{r+1} and W_{n} Ω_{𝒴}^{r+1}(logY) are finitely successive extensions of locally free 𝒪_{𝒴} -modules of finite rank ([I1] I Corollaire 3.9), the sequence

0 ⟶ W_{n} Ω_{𝒴}^{r+1} ⟶ W_{n} Ω_{𝒴}^{r+1}(logY) ⟶^{ϱ_{0}} (iσ)_{∗}W_{n} Ω_{U}^{r}

is exact by (8.4.1)for 𝒴 −Σ. Hence it is enough to show that Im(ϱ_{0}) = i_{∗}W_{n} Ξ_{Y}^{r}. Let F_{a}(ν_{Y,n}^{r}) (a ≥ 0) be the filtration on ν_{Y,n}^{r} in [Sa1] Proposition 2.2.1, which is defined under the assumption that Y is simple and satisfies

F_{0}(ν_{Y,n}^{r}) = 0 and gr_{a}^{F} ν_{Y,n}^{r} ≃ ⨁_{|I|=a} W_{n} Ω_{Y_{I},log}^{r−a+1} (non-canonically).

Hence comparing this local description of ν_{Y,n}^{r} with (8.4.1) for m = r + 1, we see that Im(ϱ_{0}) = i_{∗}W_{n} Ξ_{Y}^{r}. As for (1), we define the desired filtration F_{a}(W_{n} Ξ_{Y}^{r}) as the W_{n}𝒪_{Y} -submodule generated by F_{a}(ν_{Y,n}^{r}). This filtration satisfies the required properties by (2) and again by (8.4.1). □

Since the residue mapping ϱ_{0} in Proposition 8.4 (2) commutes with the operators d, V, F, R (cf. (8.1.1)), these operators on W_{n} Ω_{U}^{•} induce operators

d : W_{n} Ξ_{Y}^{r} ⟶ W_{n} Ξ_{Y}^{r+1},
V : W_{n} Ξ_{Y}^{r} ⟶ W_{n+1} Ξ_{Y}^{r} ,
F : W_{n+1} Ξ_{Y}^{r} ⟶ W_{n} Ξ_{Y}^{r} ,
R : W_{n+1} Ξ_{Y}^{r} ⟶ W_{n} Ξ_{Y}^{r}

by Proposition 8.4 (2) (without the assumptions in Proposition 8.4 (1) or (2)), which satisfy the relations listed in (8.1.2) and (8.1.3). We call the resulting complex (W_{n} Ξ_{Y}^{•} , d) the homological de Rham-Witt complex of Y.

Remark 8.5 There are natural injective homomorphisms of complexes

⨁_{r=0}^{dim(Y)} λ_{Y,n}^{r}[−r] ↪ W_{n} Λ_{Y}^{•} ,  ⨁_{r=0}^{dim(Y)} ν_{Y,n}^{r}[−r] ↪ W_{n} Ξ_{Y}^{•} ,

where the differentials on the complexes on the left hand side are defined as zero. Indeed the differentials on W_{n} Ω_{U}^{•} are zero on W_{n} Ω_{U,log}^{•}.

The following Proposition 8.6 and Theorem 8.8 explain a fundamental relationship between W_{n} Λ_{Y}^{•} and W_{n} Ξ_{Y}^{•}.

Proposition 8.6 The complex W_{n} Ξ_{Y}^{•} is a W_{n} Λ_{Y}^{•} -module, that is, there is a natural W_{n}𝒪_{Y} -bilinear pairing

W_{n} Λ_{Y}^{r} × W_{n} Ξ_{Y}^{s} ⟶ W_{n} Ξ_{Y}^{r+s}  (r, s ≥ 0)  (8.6.1)

satisfying the Leibniz rule

d(x · y) = (dx) · y + (−1)^{r}x · dy  (x ∈ W_{n} Λ_{Y}^{r}, y ∈ W_{n} Ξ_{Y}^{s}).

Moreover this pairing is compatible with R and satisfies relations

\begin{matrix}F (x · y) = F (x) · F (y) \\ x · V (y) = V (F (x) · y) \\ V (x) · y = V (x · F (y)) \end{matrix}  (x ∈ W_{n} Λ_{Y}^{r}, y ∈ W_{n} Ξ_{Y}^{s}).

Proof. The pairing (8.6.1) is induced by the product of W_{n} Ω_{U}^{•} and the biadditive pairing

λ_{Y,n}^{r} × ν_{Y,n}^{s} ⟶ ν_{Y,n}^{r+s}

defined in [Sa1] Definition 3.1.1. The properties of the pairing (8.6.1)follow from the corresponding properties of the product of W_{n} Ω_{U}^{•} ([I1] I Théorème 2.17, Proposition 2.18). □

Remark 8.7 When k is perfect and Y is the special fiber of a regular semistable family X over a discrete valuation ring with residue field k, we have

W_{n} Λ_{Y}^{•} ⊂ W_{n} ω_{Y}^{•} ⊂ W_{n} Ξ_{Y}^{•}

by [Sa1] Proposition 4.2.1. Here W_{n} ω_{Y}^{•} denotes the modified de Rham-Witt complex associated with X [Hy2]. The complex W_{n} Ξ_{Y}^{•} does not in general have a product structure unless Y is smooth.

Theorem 8.8 Assume that k is perfect and that Y is proper over k. Put b := dim(Y) and W_{n} := W_{n}(k). Then:

(1) There is a canonical trace map tr_{n} : ℍ^{2b}(Y, W_{n} Ξ_{Y}^{•}) → W_{n}, which is bijective if Y is geometrically connected over k.

(2) The pairing

\xymatrix{ℍ^{i}(Y, W_{n} Λ_{Y}^{•}) × ℍ^{2b−i}(Y, W_{n} Ξ_{Y}^{•}) \ar[r]^{(8.6.1)} & ℍ^{2b}(Y, W_{n} Ξ_{Y}^{•}) \ar[r]^{^{tr^{n}}} & W_{n}}

is a non-degenerate pairing of finitely generated W_{n}-modules for any i ≥ 0.

Proof. (1) Let f_{n} be the canonical morphism W_{n}Y → W_{n}, where W_{n}Y means the scheme consisting of the topological space Y and the structure sheaf W_{n}𝒪_{Y}. Since ℍ^{2b}(Y, W_{n} Ξ_{Y}^{•}) ≃ H^{b}(Y, W_{n} Ξ_{Y}^{b}), our task is to show that

W_{n} Ξ_{Y}^{b} ^{∼} ⟶ f_{n}^{!}W_{n} in D_{qc}^{+}(W_{n}Y, W_{n}𝒪_{Y}),  (8.8.1)

where the subscript qc means the triangulated subcategory consisting of complexes with quasi-coherent cohomology sheaves and f_{n}^{!} means the twisted inverse image functor of Hartshorne [Ha]. If Y is smooth, (8.8.1) is a theorem of Ekedahl [E]. In the general case, (8.8.1) is reduced to the smooth case by Proposition 8.4 (2) and the same arguments as for [Sa1] Proposition 2.5.9.

(2) The case n = 1 follows from [Sa1] Corollary 2.5.11, Proposition 3.3.5. The case n ≥ 2 follows from a standard induction argument on n and the following lemma: □

Lemma 8.9 (1) There are injective homomorphisms

\underline{p} : W_{n−1} Λ_{Y}^{r} ↪ W_{n} Λ_{Y}^{r} and \underline{p} : W_{n−1} Ξ_{Y}^{r} ↪ W_{n} Ξ_{Y}^{r}

induced by the multiplication by p on W_{n} Λ_{Y}^{r} and W_{n} Ξ_{Y}^{r}, respectively.

(2) The following natural projections of complexes are quasi-isomorphisms:

W_{n} Λ_{Y}^{•} /\underline{p}(W_{n−1} Λ_{Y}^{•}) ⟶ Λ_{Y}^{•} and W_{n} Ξ_{Y}^{•} /\underline{p}(W_{n−1} Ξ_{Y}^{•}) ⟶ Ξ_{Y}^{•} .

Proof. When Y is smooth, the assertions are due to Illusie [I1] I Proposition 3.4 and Corollaire 3.15. In the general case, (1) follows from that for the dense open subset U ⊂ Y we fixed before. To prove (2), we may assume that Y is simple. Then the assertions follow from those for the strata of Y and Propositions 8.2 (1) and 8.4 (1). □

In the rest of this section, we work under the following setting. Let A be a discrete valuation ring with perfect residue field k, and let X be a regular semistable family over A. Put Y := X ⊗_{A} k. Recall that

W_{n} Λ_{Y}^{•} ⊂ W_{n} ω_{Y}^{•} ⊂ W_{n} Ξ_{Y}^{•}

by Remark 8.7. We define a Frobenius endomorphism φ on these complexes by p^{m}F on degree m ≥ 0. There is a short exact sequence of complexes with Frobenius action

0 ⟶ W_{n} ω_{Y}^{•} (−1)[−1] ⟶^{\frac{dt}{t} ∧} W_{n} ω̃_{Y}^{•} ⟶ W_{n} ω_{Y}^{•} ⟶ 0  (8.9.1)

by [Hy2] (1.4.3), where the complex W_{n} ω_{Y}^{•} (−1) means the complex W_{n} ω_{Y}^{•} with Frobenius endomorphism p · φ. The monodromy operator

N : W_{n} ω_{Y}^{•} ⟶ W_{n} ω_{Y}^{•} (−1)

is defined as the connecting morphism associated with this sequence.

Proposition 8.10 There is a short exact sequence of complexes with Frobenius action

0 ⟶ W_{n} Λ_{Y}^{•} ⟶ W_{n} ω̃_{Y}^{•} ⟶ W_{n} Ξ_{Y}^{•} (−1)[−1] ⟶ 0  (8.10.1)

fitting into a commutative diagram of complexes

\xymatrix{ & W_{n} ω_{Y}^{•} (−1)[−1] \ar[d]_{\frac{dt}{t} ∧} \ar@{^{(}->}[dr] \\ W_{n} Λ_{Y}^{•} \ar[r] \ar@{^{(}->}[dr] & W_{n} ω̃_{Y}^{•} \ar[r] \ar[d] & W_{n} Ξ_{Y}^{•} (−1)[−1] \\  & W_{n} ω_{Y}^{•} .}

Consequently, we obtain a complex of W_{n}(k)-modules with Frobenius action for i ≥ 0

ℍ^{i}(Y, W_{n} Λ_{Y}^{•}) ⟶ ℍ^{i}(Y, W_{n} ω_{Y}^{•})

⟶^{N} ℍ^{i}(Y, W_{n} ω_{Y}^{•})(−1) ⟶ ℍ^{i}(Y, W_{n} Ξ_{Y}^{•})(−1). (8.10.2)

Proof. Let σ : U ↪ Y be as before and define a homomorphism ∂ : W_{n} ω̃_{Y}^{r} → σ_{∗}W_{n} Ω_{U}^{r−1} as the composite

∂ : W_{n} ω̃_{Y}^{r} ⟶ σ_{∗}σ^{∗}W_{n} ω̃_{Y}^{r} ^{∼} ⟶ σ_{∗}(W_{n} Ω_{U}^{r} ⊕ W_{n} Ω_{U}^{r−1}) ⟶^{pr_{2}} σ_{∗}W_{n} Ω_{U}^{r−1},

where the second arrow is obtained from the fact that the sequence (8.9.1)splits on U. It is easy to see that ∂ satisfies

d∂ = ∂d and ∂F = pF ∂,

which induces a map of complexes ∂^{•} : W_{n} ω̃_{Y}^{•} ⟶ σ_{∗}W_{n} Ω_{U}^{•}(−1)[−1]. To show that ∂^{•} induces the exact sequence (8.10.1), we may assume the assumption in Proposition 8.2 (2). Then we have an isomorphism of complexes

W_{n} ω̃_{Y}^{•} ≃ W_{n} Ω_{𝒴}^{•} (logY)/W_{n} Ω_{𝒴}^{•} (−logY) ([Hy2] p. 247 Lemma)

and the assertion follows from Propositions 8.2 (2) and 8.4 (2). The commutativity of the diagram in the proposition is straight-forward and left to the reader. □

We prove that the monodromy-weight conjecture for log crystalline cohomology implies an invariant cycle ‘theorem’ (cf. [I2] 2.4.5), which will be useful in §§9–10 below. Let A be a discrete valuation ring with finite residue field k, and X be a regular scheme which is projective flat over A with strict semistable reduction. Put K_{0} := Frac(W (k)) and Y := X ⊗_{A} k. For integer i ≥ 0, we define

C^{i} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} ℍ^{i}(Y, W_{n} Λ_{Y}^{•}),  (8.10.3)
D^{i} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} ℍ^{i}(Y, W_{n} ω_{Y}^{•}),  (8.10.4)
E^{i} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} ℍ^{i}(Y, W_{n} Ξ_{Y}^{•}),  (8.10.5)

which are finite-dimensional over K_{0} and F -isocrystals over Spec(k). The content of the monodromy-weight conjecture due to Mokrane is the following:

Conjecture 8.11 ([Mo] Conjecture 3.27) The filtration on D^{i} induced by the weight spectral sequence (loc. cit. 3.25, cf. [Na] (2.0.9;p)) agrees with the monodromy filtration induced by the monodromy operator N : D^{i} → D^{i}(−1) ([Mo] 3.26).

Theorem 8.12 Let i ≥ 0 be an integer, and assume Conjecture 8.11 for D^{i}. Then the following sequence of F -isocrystals over Spec(k) induced by (8.10.2)is exact:

\xymatrix{C^{i} \ar[r] & D^{i} \ar[r]^{N} & D^{i}(−1) \ar[r] & E^{i}(−1).}

Proof. Suppose that #(k) = p^{a}. Then φ^{a} is K_{0}-linear and we use the notion of weights concerning φ^{a}. Because the groups

ℍ^{i}(Y, W_{n} Λ_{Y}^{•}), ℍ^{i}(Y, W_{n} ω_{Y}^{•}), ℍ^{i}(Y, W_{n} Ξ_{Y}^{•}), ℍ^{i}(Y, W_{n} ω̃_{Y}^{•})

are finite for any n ≥ 1, there is a diagram of F -isocrystals with exact rows by (8.9.1) and Proposition 8.10

[figure: D̃^{i}; D^{i}; N; D^{i}(−1); D̃^{i+1}; C^{i}; _{D̃_{i}}; ∂; E^{i−1}(−1); C^{i+1}; _{D̃_{i+1}}; ∂; E^{i}(−1),]

where we put

D̃^{i} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} ℍ^{i}(Y, W_{n} ω̃_{Y}^{•}).

By the assumption on D^{i}, Ker(N) has weights ≤ i and Coker(N) has weights ≥ i+2. Hence for the reason of weights, it is enough to show the following lemma, where we do not need Conjecture 8.11:

Lemma 8.13 For integers i ≥ 0, C^{i} has weights ≤ i and E^{i} has weights ≥ i.

Proof of Lemma 8.13. When Y is smooth, the lemma is a consequence of the Katz-Messing theorem [KM]. In the general case, C^{i} has weights ≤ i by the spectral sequence of F -isocrystals obtained from Proposition 8.2 (1)

E_{1}^{s,t} = ⨁_{|I|=s+1} D_{I}^{t} ⟹ C^{s+t},

where we put

D_{I}^{t} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} ℍ^{t}(Y_{I}, W_{n} Ω_{Y_{I}}^{•}).

Moreover E^{i} has weights ≥ i by the duality result in Theorem 8.8. Thus we obtain Lemma 8.13 and Theorem 8.12. □

Remark 8.14 One can show Lemma 8.13 for E^{i} alternatively by the Katz-Messing theorem and the spectral sequence obtained from Proposition 8.4 (1)

E_{1}^{s,t} = ⨁_{|I|=1−s} D_{I}^{2s+t}(s) ⟹ E^{s+t}.

9 Comparison with the finite part of Bloch-Kato

In this section we relate the cohomology of p-adic étale Tate twists with the finite part of Bloch-Kato [BK2]. Let p be a prime number, and let K be a p-adic local field, i.e., a finite extension of ℚ_{p}. Let O_{K} be the integer ring of K, and let k be the residue field of O_{K}. In this section we consider Tate twists only in the sense of G_{K}-ℤ_{p}-modules (and not in the sense of F -crystals). For a topological G_{K}-ℤ_{p}-module M and an integer r ∈ ℤ, we define

M(r) := \begin{cases}M ⊗_{ℤ_{p}} ℤ_{p}(1)^{⊗r} & (if r ≥ 0), \\ Hom_{ℤ_{p}}(ℤ_{p}(1)^{⊗(−r)}, M) & (if r < 0), \end{cases}

where ℤ_{p}(1) denotes the topological G_{K}-module \underleftarrow{lim}_{n≥1} μ_{p^{n}}(\overline{K}).

Let X be a regular scheme which is projective flat over O_{K} with strict semistable reduction, and put X_{K} := X ⊗_{O_{K}} K and Y := 𝒳 ⊗_{O_{K}} k. Let j : X_{K} ↪ X be the natural open immersion, and let i and r be non-negative integers. Put 𝒱 ^{i} := H^{i}(X\overline{K}, ℚ_{p}) (see also Notation) and

H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{i+1}(X, 𝔗_{n}(r)_{X}),

which are finite-dimensional over ℚ_{p}. Let F^{•} be the filtration on H^{i+1}(X_{K}, ℚ_{p}(r)) resulting from the Hochschild-Serre spectral sequence

E_{2}^{a,b} = H^{a}(K, 𝒱 ^{b}(r)) ⟹ H^{a+b}(X_{K}, ℚ_{p}(r)).  (9.0.1)

Let F^{q}H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) be the inverse image of F^{q}H^{i+1}(X_{K}, ℚ_{p}(r)) under the canonical map j^{∗} : H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) → H^{i+1}(X_{K}, ℚ_{p}(r)). We have

Φ^{i,r} := gr_{F}^{1} H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) ↪ gr_{F}^{1} H^{i+1}(X_{K}, ℚ_{p}(r)) = H^{1}(K, 𝒱 ^{i}(r))

by definition. We relate Φ^{i,r} with the finite part of Bloch-Kato [BK2]:

H_{f}^{1}(K, 𝒱 ^{i}(r)) := Ker(ι : H^{1}(K, 𝒱 ^{i}(r)) → H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys})).

where B_{crys} denotes the period ring of crystalline representations defined by Fontaine [Fo1], and ι is induced by the natural inclusion ℚ_{p}(r) ↪ B_{crys}. The main result of this section is as follows

Theorem 9.1 Assume 0 ≤ r ≤ p − 2 and Conjecture 8.11 for D^{i} := D^{i}(Y). Then we have

Φ^{i,r} ⊂ H_{f}^{1}(K, 𝒱 ^{i}(r)).

Further if r ≥ dim(X_{K}) − p + 3, then we have Φ^{i,r} = H_{f}^{1}(K, 𝒱 ^{i}(r)).

Remark 9.2 When X is smooth over O_{K}, Theorem 9.1 is a part of the p-adic point conjecture raised by Schneider [Sc] and proved by Langer-Saito [LS] and Nekovář [Ne1].

Put

H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{i+1}(X, τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r})

and let F^{q}H^{i+1}(X, τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r}) be the inverse image of F^{q}H^{i+1}(X_{K}, ℚ_{p}(r)) under the map

j^{∗} : H^{i+1}(X, τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r}) → H^{i+1}(X_{K}, ℚ_{p}(r)).

We first prepare the following key commutative diagram to prove Theorem 9.1, assuming 0 ≤ r ≤ p − 2:

[figure: H^{1}(K, 𝒱 ^{i}(r)); ι; F^{1}H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)); σ′; α; (A); H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}); D^{i}/ND^{i}; δ; ϵ; (B); E^{i},] (9.2.1)

where E^{i} := E^{i}(Y) is as in (8.10.5), and D^{i}/ND^{i} denotes the cokernel of the monodromy operator N : D^{i} → D^{i}. The top arrow is induced by j^{∗} and an edge homomorphism of the spectral sequence (9.0.1). The arrow ϵ is induced by the complex (8.10.2), and the arrow σ′ is induced by the composite map

H^{i+1}(X, τ_{≤r}Rj_{∗}μ_{p^{n}}^{⊗r}) ⟶^{σ} H^{i+1−r}(Y, ν_{Y,n}^{r−1}) ⟶ ℍ^{i}(Y, W_{n} Ξ_{Y}^{•}) (n ≥ 1).

See Theorem 3.4 for σ and see Remark 8.5 for the last map. The constructions of the arrows α and δ and the commutativity of the square (A) deeply rely on the p-adic Hodge theory. The commutativity of the triangle (B) is rather straight-forward once we define α (see Proposition 9.10 below). The proof of Theorem 9.1 will be given after Proposition 9.10.

Remark 9.3 In his paper [La], Langer considered the diagram (A) assuming that O_{K} is absolutely unramified (loc. cit. Proposition 2.9). We remove this assumption and the assumption (∗) stated in loc. cit. p. 191, using a continuous version of crystalline cohomology (see Appendix A below). For this purpose, we include a detailed construction of α and a proof of the commutativity of (A).

Construction of δ. We first construct the map δ. There is an exact sequence of topological G_{K}-ℚ_{p}-modules

0 ⟶ 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys} ⟶ 𝒱 ^{i} ⊗_{ℚ_{p}} B_{st} ⟶^{1⊗N} 𝒱 ^{i} ⊗_{ℚ_{p}} B_{st} ⟶ 0,  (9.3.1)

where N denotes the monodromy operator on B_{st}. Taking continuous Galois cohomology groups, we obtain a long exact sequence

0 ⟶ (𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys})^{G_{K}} ⟶ (𝒱 ^{i} ⊗_{ℚ_{p}} B_{st})^{G_{K}} ⟶^{1⊗N} (𝒱 ^{i} ⊗_{ℚ_{p}} B_{st})^{G_{K}}
⟶ H^{1}(G_{K}, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}) ⟶ · · · .

By the C_{st}-conjecture proved by Hyodo, Kato and Tsuji ([HK], [K4], [Ts1], cf. [Ni2]), the last arrow of the top row is identified with the map N : D^{i} → D^{i} (see (9.3.3) below for the construction of C_{st}-isomorphisms). We thus define the desired map δ as the connecting map of this sequence.

C_{st}-isomorphism. Before constructing α, we recall the construction of C_{st}- isomorphisms, which will be useful later ([K4], [Ts1] §4.10). See [K3] for the general framework of log structures and log schemes. Put

W := W (k) and K_{0} := Frac(W) = W [p^{−1}].

For a log scheme (Z, M_{Z}) and an integer n ≥ 1, we define

(Z_{n}, M_{Z_{n}}) := (Z, M_{Z}) ×_{Spec(ℤ)} Spec(ℤ/p^{n}),

where we regarded Spec(ℤ) and Spec(ℤ/p^{n}) as log schemes by endowing them with the trivial log structures and the fiber product is taken in the category of log schemes. Let M be the log structure on X associated with the normal crossing divisor Y. Let \overline{O_{K}} be the integral closure of O_{K} in \overline{K}, and let \overline{M} be the log structure on \overline{X} := X ⊗_{O_{K}} \overline{O_{K}} define by base-change (see (A.0.2) below). Put \overline{Y} := Y ⊗_{k} \overline{k}. For n ≥ 1 and s ≥ 0, let 𝒮_{n} (s)_{(\overline{X},\overline{M})} ∈ D^{+}(\overline{Y}_{ét}, ℤ/p^{n}) be Tsuji’s version of syntomic complex of (\overline{X}, \overline{M}) (§A.5). Let \overline{j} : X\overline{K} ↪ \overline{X} be the natural open immersion. For m, s ≥ 0, we define the syntomic cohomology of (\overline{X}, \overline{M}) as

H_{syn}^{m}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(s)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{m}(\overline{Y} , 𝒮_{n} (s)_{(\overline{X},\overline{M})}).

The following isomorphism due to Tsuji ([Ts1] Theorem 3.3.4 (2)) plays a crucial role:

H_{syn}^{m}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(s)) ≃ H^{m}(\overline{X}, τ_{≤s}R\overline{j}_{∗}ℚ_{p}(s)).  (9.3.2)

The right hand side is isomorphic to 𝒱 ^{m}(s) = H^{m}(X\overline{K}, ℚ_{p}(s)) if s ≥ dim(X_{K}) or s ≥ m. We now introduce the crystalline cohomology of (\overline{X}, \overline{M}) over W:

H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H_{crys}^{m}((\overline{X}_{n}, \overline{M}_{n})/W_{n}),

where H_{crys}^{m}((\overline{X}_{n}, \overline{M}_{n})/W_{n}) denotes the crystal cohomology of (\overline{X}_{n}, \overline{M}_{n}) over W_{n} (see §A.1 below). We define a G_{K}-equivariant homomorphism

g^{m} : 𝒱 ^{m} ⟶ D^{m} ⊗_{K_{0}} B_{st}

as the composite map (d := dim(X_{K}))

𝒱 ^{m} ^{∼} ⟶^{(1)} H_{syn}^{m}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(d)) ⊗_{ℚ_{p}} ℚ_{p}(−d)  (9.3.3)

⟶^{(2)} H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}} ⊗_{ℚ_{p}} ℚ_{p}(−d)

∼ ⟶^{(3)} (D^{m} ⊗_{K_{0}} B_{st}^{+})^{N=0} ⊗_{ℚ_{p}} ℚ_{p}(−d)

⟶^{(4)} D^{m} ⊗_{K_{0}} B_{st} ,

where (1) is obtained from the isomorphism (9.3.2) with s = d, the arrow (2) is the p^{d}-times of the canonical map induced by (A.5.2) below, and (3) is the crystalline interpretation of (D^{m} ⊗_{K_{0}} B_{st}^{+})^{N=0} due to Kato ([K4] §4, cf. Proposition A.3.1 (3) below). The map (4) is induced by the natural inclusion ℚ_{p}(−d) ↪ B_{st}. The B_{st}- linear extension of g^{m} is an isomorphism by [Ts2] Theorem 4.10.2:

𝒱 ^{m} ⊗_{ℚ_{p}} B_{st} ^{∼} ⟶ D^{m} ⊗_{K_{0}} B_{st} ,  (9.3.4)

which we call the C_{st}-isomorphism. Now we have a commutative diagram of G_{K}- modules for m, s ≥ 0

[figure: H_{syn}^{m}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(s)); (1); (9.3.2); H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}}; (2); ∼; (D^{m} ⊗_{K_{0}} B_{st}^{+})^{N=0}; 𝒱 ^{m}(s); ℚ_{p}(s)↪B_{st}; 𝒱 ^{m} ⊗_{ℚ_{p}} B_{st}; (9.3.4); ∼; D^{m} ⊗_{K_{0}} B_{st},]

(9.3.5)

where the arrow (1) is the p^{r}-times of the canonical map (A.5.2), and (2) is the crystalline interpretation of (D^{m} ⊗_{K_{0}} B_{st}^{+})^{N=0} mentioned in (9.3.3). This diagram commutes by the construction of (9.3.4) and the compatibility facts in [Ts1] Corollaries 4.8.8 and 4.9.2.

Remark 9.4 We modified the C_{st}-isomorphisms in [Ts1] by constant multiplications to make them consistent with the canonical map from syntomic cohomology of Kato to (continuous) crystalline cohomology (see the map (1) of (9.5.1) and the equality (A.5.4) below). Under this modification, the element t_{syn} in [Ts1] p. 378 is replaced with p^{−1}t_{syn}, and Corollary 4.8.8 of loc. cit. remains true.

Construction of α. We start the construction of α assuming 0 ≤ r ≤ p − 2, which will be complete in Corollary 9.8 below. For n ≥ 1, let 𝒮_{n}(r)_{(X,M)} ∈ D^{+}(Y_{ét}, ℤ/p^{n}) be the syntomic complex of Kato (see §A.5 below). Since 0 ≤ r ≤ p − 2, there is an isomorphism

(τ_{≤r}Rj_{∗}μ_{p^{n}})|_{Y} ≃ 𝒮_{n}(r)_{(X,M)} in D^{+}(Y_{ét}, ℤ/p^{n})  (9.4.1)

for each n ≥ 1 ([Ts2] Theorem 5.1), and H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)) is isomorphic to the syntomic cohomology

H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{i+1}(Y, 𝒮_{n}(r)_{(X,M)})

by the proper base-change theorem. Therefore we consider H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r)) instead of H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)), which is the major reason we assume r ≤ p − 2. We first prove:

Lemma 9.5 The kernel of the composite map

ξ : H_{syn}^{m}((X, M), 𝒮_{ℚ_{p}}(r)) ⟶ H_{crys}^{m}((X, M)/W)_{ℚ_{p}} ⟶ H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}}

agrees with that of the composite map

η : H_{syn}^{m}((X, M), 𝒮_{ℚ_{p}}(r)) ≃^{(9.4.1)} H^{m}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r))

⟶ 𝒱 ^{m}(r) = H^{m}(X\overline{K}, ℚ_{p}(r)).

Proof. The map η factor through (9.3.2)with s = r, and the map ξ factors through the map (1) in (9.3.5)with s = r by (A.5.4)below. Hence the diagram

\xymatrix{H_{syn}^{m}((X, M), 𝒮_{ℚ_{p}}(r)) \ar[r]^{ξ} \ar[d]_{η} & H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}} \ar@{^{(}->}[d] \\ 𝒱 ^{m}(r) \ar@{^{(}->}[r] & D^{m} ⊗_{ℚ_{p}} B_{st}}

commutes by (9.3.5), which implies the assertion. □

Let H_{cont-cr}^{m}((X, M)/W) be the continuous crystalline cohomology of (X, M) over W (see §A.2 below). Put

H_{syn}^{m}((X, M), 𝒮_{ℚ_{p}}(r))^{0} := Ker(η : H_{syn}^{m}((X, M), 𝒮_{ℚ_{p}}(r)) ⟶ 𝒱 ^{m}(r)),
H_{cont-cr}^{m}((X, M)/W)_{ℚ_{p}} := ℚ_{p} ⊗_{ℤ_{p}} H_{cont-cr}^{m}((X, M)/W),
H_{cont-cr}^{m}((X, M)/W)_{ℚ_{p}}^{0} := Ker(H_{cont-cr}^{m}((X, M)/W)_{ℚ_{p}} → H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}}).

There are canonical maps

H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} ⟶^{(1)} H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0}  (9.5.1)

⟶^{(2)} H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}})

∼ ⟶^{(3)} H^{1}(K, (D^{i} ⊗_{K_{0}} B_{st}^{+})^{N=0})

⟶^{(4)} H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}),

where (1) is the canonical map obtained from Lemma 9.5 and Proposition A.6.1 (2) below, the arrow (3) is the isomorphism (2) in (9.3.5) for m = i, and (4) is obtained from the C_{st}-isomorphism (9.3.4) and (9.3.1). The most crucial map (2) is obtained from an edge homomorphism of the spectral sequence in Theorem A.4.2 (see also Corollary A.4.5 below). To proceed with the construction of α, we need to introduce some notation.

Definition 9.6 Put S := Spec(O_{K}), and fix a prime π of O_{K}. Let M_{S} be the log structure on S associated with the closed point {π = 0}. Put V := Spec(W [t]), and let M_{V} be the log structure on V associated with the divisor {t = 0}. Let

ι : (S, M_{S}) ⟶ (V, M_{V})

be the exact closed immersion over W sending t ↦ π. For each n ≥ 1, let (ℰ_{n}, M_{ℰ_{n}}) be the PD-envelope of the exact closed immersion ι_{n} : (V_{n}, M_{V_{n}}) ↪ (S_{n}, M_{S_{n}}) compatible with the canonical PD-structure on pW_{n}. We define

H_{crys}^{m}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H_{crys}^{m}((X_{n}, M_{n})/(ℰ_{n}, M_{ℰ_{n}})),

which is an R_{ℰ} := \underleftarrow{lim}_{n≥1} Γ (ℰ_{n}, 𝒪_{ℰ_{n}})-module endowed with a φ_{ℰ} -semilinear endomorphism φ and a K_{0}-linear endomorphism N satisfying Nφ = pφN. Here φ_{ℰ} is a certain canonical Frobenius operator on R_{ℰ} . See [Ts1] p. 253 for φ_{ℰ} , loc. cit. §4.3 for N and φ, and see loc. cit. Remark 4.3.9 for N.

The following lemma plays a key role in our construction of α, where we do not need to assume 0 ≤ r ≤ p − 2 (compare with the assumption (∗) in [La] p. 191).

Lemma 9.7 (1) There exists a long exact sequence

⋯ ⟶ H_{cont-cr}^{i}((X, M)/W)_{ℚ_{p}} ⟶ H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}}

⟶^{N} H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} ⟶^{∂} H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}} ⟶ ⋯

and the image of ∂ agrees with H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0}.

(2) There is a commutative diagram whose lower row is exact

\xymatrix{ & H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} \ar@{->>}[r]^{∂} \ar[d] & H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0} \ar[d] \\ D^{i} \ar[r]^{N} & D^{i} \ar[r]^{(−1)^{i+1}δ} & H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}).}

Here the right vertical arrow is the composite of (2) – (4) in (9.5.1), and the left vertical arrow is the specialization maps with t ↦ 0 (cf. [Ts1] (4.4.4)).

Proof. (1) follows from Proposition A.2.4 and Corollaries A.3.2 (1) and A.3.4 below. As for (2), the exactness of the lower row has been explained in the construction of δ. See Corollary A.4.5 below for the commutativity of the square. □

Corollary 9.8 The composite map (9.5.1)factors as follows:

H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} ⟶ D^{i}/ND^{i} ↪^{δ} H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}).

We define α as the first arrow of this decomposition.

Cycle Class and p-Adic Regulator 223

Commutativity of the diagram (9.2.1). We first prove that the square (A) in (9.2.1)is commutative. By the construction of α and δ, it is enough to show Theorem 9.9 The following square is commutative:

[figure: H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0}; (9.5.1); ∼; F^{1}H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)); H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}); H^{1}(K, 𝒱 ^{i}(r)),; B_{crys}↩ℚ_{p}(r)]  (9.9.1)

where the right vertical arrow is the top arrow in (9.2.1). Proof. Put d := dim(X_{K}). We define a G_{K}-homomorphism

β^{i,r} : 𝒱 ^{i}(r) ⟶ \begin{cases}H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d) & (if r < d) \\ H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}} & (if r ≥ d) \end{cases}

as the following composite map for r ≥ d:

\xymatrix{β^{i,r} : 𝒱 ^{i}(r) & H_{syn}^{i}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(r)) \ar[l]_{(9.3.2)}^{∼} \ar[r]^{(9.3.5)(1)} & H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}},}

and as the following map for r < d:

\xymatrix{β^{i,r} : 𝒱 ^{i}(r) ≃ 𝒱 ^{i}(d) ⊗_{ℚ_{p}} ℚ_{p}(r − d) \ar[r]^{β^{i,d}⊗id} & H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d).}

We next define a homomorphism

γ^{i,r} : H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0}

⟶ \begin{cases}H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d)) & (if r < d) \\ H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}) & (if r ≥ d) \end{cases}

as the following composite map, if r < d:

H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} ⟶^{(i)} H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}})

≃ H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d) ⊗_{ℚ_{p}} ℚ_{p}(d − r))

⟶^{(ii)} H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d)),

where (i) is the composite of (1) and (2) in (9.5.1), and (ii) is induced by the inclusion ℚ_{p}(d − r) ↪ H_{crys}^{0}((\overline{X}, \overline{M})/W)_{ℚ_{p}} (cf. (A.6.5) below) and the cup product of crystalline cohomology. If r ≥ d, then we define γ^{i,r} as the composite of (1) and (2) in (9.5.1).

Documenta Mathematica 18 (2013) 177–247

We prove the commutativity of (9.9.1) assuming r < d. In this case, the following triangle commutes by Theorem A.6.7 below:

\xymatrix{H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} \ar[d] \ar@{-->}[dr]^{γ^{i,r}} \\ H^{1}(K, 𝒱 ^{i}(r)) \ar[r]^{β^{i,r}} & H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d)),} (9.9.2)

where the left vertical arrow is the composite of the top horizontal and the right vertical arrows of (9.9.1). Moreover the following square commutes by the construction of the C_{st}-isomorphism (9.3.4)(compare with (9.3.5)):

\xymatrix{𝒱 ^{i}(r) \ar[d]_{ℚ_{p}(r)↪B_{crys}} \ar[r]^{β^{i,r}} & H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d) \ar[d] \\ 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys} \ar[r]^{(9.3.1), (9.3.4)}_{∼} & (D^{i} ⊗_{ℚ_{p}} B_{st})^{N=0},}  (9.9.3)

where the right vertical arrow is given by the isomorphism (2) of (9.3.5) and the inclusion ℚ_{p}(r −d) ↪ B_{crys}. The commutativity of (9.9.1)follows from these commutative diagrams and the fact that the composite map (9.5.1)agrees with the composite of γ^{i,r} and the map

H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d)) ⟶ H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys})

induced by the right vertical arrow and the bottom isomorphism in (9.9.3). When r ≥ d, the following triangle commutes by Corollary A.6.3 below:

\xymatrix{H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} \ar[d] \ar@{-->}[dr]^{γ^{i,r}} \\ H^{1}(K, 𝒱 ^{i}(r)) \ar[r]^{β^{i,r}} & H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}),}

where the left vertical arrow is defined in the same way as for that of (9.9.2). The commutativity of (9.9.1) follows from this diagram and the diagram (9.3.5). This completes the proof of Theorem 9.9. □

Proposition 9.10 The triangle (B) in (9.2.1)is commutative.

Proof. There is a commutative diagram by the definition of ∂ in Proposition A.2.4

\xymatrix{H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} \ar[r]^{t↦0} \ar[d]_{∂} & D^{i} \ar[d]^{\frac{dt}{t} ∧} \\ H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}} \ar[r]^{t↦0} & D̃^{i+1},}

where D̃^{i+1} = D̃^{i+1}(Y) is as in the proof of Theorem 8.12, the left (resp. right) vertical arrow is as in Lemma 9.7 (1) (resp. given by the product with dlog(t), cf. (8.9.1)), and the horizontal arrows are the specialization maps by t ↦ 0. Therefore in view of the constructions of α and σ′, it is enough to check that the following diagram of canonical morphisms commutes in D^{+}(Y_{ét}, ℤ/p^{n}) for each n ≥ 0:

\xymatrix{𝒮_{n}(r)_{(X,M)} \ar[r] \ar[d]_{σ} & 𝕁_{(X_{n},M_{n})/W_{n}}^{[r]} \ar[r]^{s} & W_{n} ω̃_{Y}^{•≥r} \ar[r] \ar[d]_{(8.10.1)} & W_{n} ω̃_{Y}^{•} \ar[d]^{(8.10.1)} \\ ν_{Y,n}^{r−1}[−r] \ar[rr]^{Remark 8.5} &  & W_{n} Ξ_{Y}^{•≥r−1}[−1] \ar[r] & W_{n} Ξ_{Y}^{•} [−1],}

where 𝕁_{(X_{n},M_{n})/W_{n}}^{[r]} is as in §A.5 below, and s denotes the specialization map by t ↦ 0. The right square commutes obviously in the category of complexes of sheaves. As for the left square, we have only to show the commutativity on the diagram of the r-th cohomology sheaves because W_{n} Ξ_{Y}^{•≥r−1}[−1] is concentrated in degrees ≥ r and 𝒮_{n}(r)_{(X,M)} is concentrated in degrees ≤ r by (9.4.1). Since we have

ℋ ^{r}(𝒮_{n}(r)_{(X,M)}) ≃^{(9.4.1)} (R^{r}j_{∗}μ_{p^{n}}^{⊗r})|_{Y}

and the sheaf on the right hand side is generated by symbols (cf. the case D = ∅ of Theorem 3.3 (1)), one can check this commutativity of the diagram of sheaves by a straight-forward computation on symbols using the compatibility in [Ts1] Proposition 3.2.4 (2). □

Proof of Theorem 9.1. We have the following commutative diagram by (9.2.1):

[figure: F^{1}H^{i+1}(X, 𝔗_{ℚ_{p}}(r)); F^{1}H^{i+1}(X, τ_{≤r}Rj_{∗}ℚ_{p}(r)); σ′; α; (B); (A); E^{i}; H^{1}(K, 𝒱 ^{i}(r)); ι; D^{i}/ND^{i}; δ; ϵ; H^{1}(K, 𝒱 ^{i} ⊗_{ℚ_{p}} B_{crys}),]

where the left triangle commutes obviously. The kernel of ι is H_{f}^{1}(K, 𝒱 ^{i}(r)) by definition. The top row is a complex by the definitions of σ′ and 𝔗_{n}(r)_{X}. The map ϵ is injective by Theorem 8.12 and Conjecture 8.11 for D^{i}. We obtain the first assertion

Φ^{i,r} ⊂ H_{f}^{1}(K, 𝒱 ^{i}(r))  (9.10.1)

by a simple diagram chase on this diagram. To obtain the second assertion, we repeat the same arguments for Φ^{2d−i,d−r+1}, where we put d := dim(X_{K}). The map

ϵ : D^{2d−i}/ND^{2d−i} ⟶ E^{2d−i}

is injective by Conjecture 8.11 for D^{i}, Theorems 8.12 and 8.8 and the Poincaré duality between D^{i} and D^{2d−i} [Hy2]. Hence we have

Φ^{2d−i,d−r+1} ⊂ H_{f}^{1}(K, 𝒱 ^{2d−i}(d − r + 1)).  (9.10.2)

The results (9.10.1) and (9.10.2) imply Φ^{i,r} = H_{f}^{1}(K, 𝒱 ^{i}(r)), because Φ^{i,r} and Φ^{2d−i,d−r+1} (resp. H_{f}^{1}(K, 𝒱 ^{i}(r)) and H_{f}^{1}(K, 𝒱 ^{2d−i}(d − r + 1))) are the exact annihilators of each other under the non-degenerate pairing of the Tate duality

H^{1}(K, 𝒱 ^{i}(r)) × H^{1}(K, 𝒱 ^{2d−i}(d − r + 1)) ⟶ ℚ_{p}

by [Sa1] Corollary 10.6.1 (resp. [BK2] Proposition 3.8). □

10 Image of p-adic regulators

In this section, we prove Theorem 1.1. We first review the definitions of the finite part of Galois cohomology [BK2] and the integral part of algebraic K-groups [Sch] over number fields.

Let K be a number field and let 𝒱 be a topological G_{K}-ℚ_{p}-module which is finitedimensional over ℚ_{p}. Let 𝔬 be the integer ring of K. For a place v of K, let K_{v} (resp. 𝔬_{v}) be the completion of K (resp. 𝔬) at v. The finite part H_{f}^{1}(K, 𝒱 ) ⊂ H^{1}(K, 𝒱 ) is defined as

H_{f}^{1}(K, 𝒱 ) := Ker(H^{1}(K, 𝒱 ) ⟶ ∏_{v: finite} \frac{H^{1}(K_{v}, 𝒱 )}{H_{f}^{1}(K_{v}, 𝒱 )} ),

where v runs through all finite places of K, and H_{f}^{1}(K_{v}, 𝒱 ) is defined as follows:

H_{f}^{1}(K_{v}, 𝒱 ) := \begin{cases}Ker(H^{1}(K_{v}, 𝒱 ) → H^{1}(I_{v}, 𝒱 )) & (if v|̸p), \\ Ker(H^{1}(K_{v}, 𝒱 ) → H^{1}(K_{v}, 𝒱 ⊗_{ℚ_{p}} B_{crys})) & (if v|p), \end{cases}

where I_{v} denotes the inertia subgroup of G_{K_{v}}. We next review the definition of the integral part of algebraic K-groups. Let V be a proper smooth variety over the number field K. First fix a finite place v of K. By de Jong’s alteration theorem [dJ], there is a proper generically finite morphism

π : V′ ⟶ V_{v} := V ⊗_{K} K_{v}

such that V′ has a projective regular model X′ with strict semistable reduction over the integer ring of some finite extension of K_{v}. The integral part K_{m}(V_{v})_{𝔬_{v}} ⊂ K_{m}(V_{v}) ⊗ ℚ is the kernel of the composite map

K_{m}(V_{v}) ⊗ ℚ ⟶^{π^{∗}} K_{m}(V′) ⊗ ℚ ⟶ \frac{K_{m}(V′) ⊗ ℚ}{Image of K_{m}(X′) ⊗ ℚ},

which is in fact independent of X′ ([Sch] §1). The integral part K_{m}(V)_{𝔬} is defined as

K_{m}(V)_{𝔬} := Ker(K_{m}(V) ⊗ ℚ ⟶ ∏_{v: finite} \frac{K_{m}(V_{v}) ⊗ ℚ}{K_{m}(V_{v})_{𝔬_{v}}}),

Documenta Mathematica 18 (2013) 177–247

where v runs through all finite places of K. If V admits a regular model which is proper flat over the integer ring of K, then K_{m}(V)_{𝔬} agrees with the image of the Kgroup of the model, i.e., the integral part considered by Beilinson. By these definitions of H_{f}^{1}(K, 𝒱 ) and K_{2r−i−1}(V)_{𝔬}, Theorem 1.1 is immediately reduced to Theorem 10.1 below, which is an analogue of Theorem 1.1 over local fields.

Let ℓ and p be prime numbers. We change the setting slightly, and let K be an ℓ-adic local field, i.e., a finite extension of ℚ_{ℓ}. Let O_{K} be the integer ring of K, and let V be a proper smooth variety over K. Let i and r be integers with 2r ≥ i + 1 ≥ 1. There is a p-adic regulator map obtained from étale Chern character

reg_{p} : K_{2r−i−1}(V)_{O_{K}, hom} ⟶ H^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))),

where \overline{V} denotes V ⊗_{K} \overline{K}, and K_{2r−i−1}(V)_{O_{K}, hom} denotes the subspace of K_{2r−i−1}(V)_{O_{K}} consisting of all elements which vanish in H^{i+1}(\overline{V} , ℚ_{p}(r)) under the Chern character.

Theorem 10.1 If ℓ = p, assume that r ≤ p − 2 and that Conjecture 8.11 holds for projective strict semistable varieties over 𝔽_{p} in degree i. Then Im(reg_{p}) is contained in H_{f}^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))).

Remark 10.2 When ℓ = p, we need Conjecture 8.11 for the reduction of an alteration of V. When ℓ ≠ p, we do not need the monodromy-weight conjecture, but use Deligne’s proof of the Weil conjecture [D] to show that reg_{p} is zero.

Proof of Theorem 10.1. We first reduce the problem to the case that V has a regular model which is projective flat over O_{K} with strict semistable reduction. By de Jong’s alteration theorem [dJ], there exists a proper generically finite morphism π : V′ → V such that V′ has a projective regular model with strict semistable reduction over the integer ring O_{L} of L := Γ (V′, 𝒪_{V′}). Then there is a commutative diagram

\xymatrix{K_{2r−i−1}(V)_{O_{K}, hom} \ar[r]^{^{reg^{p}}} \ar[d]_{π^{∗}} & H^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))) \ar[r] \ar[d]_{π^{∗}} & \frac{H^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r)))}{H_{f}^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r)))} \ar@{^{(}->}[d]_{π^{∗}} \\ K_{2r−i−1}(V′)_{O_{L}, hom} \ar[r]^{_{reg_{p}}} & H^{1}(L, H^{i}(\overline{V′}, ℚ_{p}(r))) \ar[r] & \frac{H^{1}(L, H^{i}(\overline{V′}, ℚ_{p}(r)))}{H_{f}^{1}(L, H^{i}(\overline{V′}, ℚ_{p}(r)))}}

where \overline{V′} denotes V′ ⊗_{L} \overline{K}, and the right (and the middle) vertical arrows are split injective by a standard argument using a corestriction map of Galois cohomology and a trace map of étale cohomology. By this diagram and the definition of K_{∗}(V)_{O_{K}}, Theorem 10.1 for V is reduced to that for V′. Thus we may assume that V has a projective regular model X with strict semistable reduction over O_{K}. Then the case ℓ ≠ p follows from [Ne1] II Theorem 2.2 (cf. [D]). We prove the case ℓ = p. Let Y be the reduction of X. Put

H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{n≥1} H^{i+1}(X, 𝔗_{n}(r)),

and let K_{2r−i−1}(X)_{hom} (resp. H^{i+1}(X, 𝔗_{ℚ_{p}}(r))^{0}) be the kernel of the composite map

K_{2r−i−1}(X) ⟶ K_{2r−i−1}(V) ⟶^{ch} H^{i+1}(V, ℚ_{p}(r)) → H^{i+1}(\overline{V} , ℚ_{p}(r))

(resp. H^{i+1}(X, 𝔗_{ℚ_{p}}(r)) ⟶ H^{i+1}(V, ℚ_{p}(r)) → H^{i+1}(\overline{V} , ℚ_{p}(r)) ).

There is a commutative diagram by Corollary 5.7

\xymatrix{K_{2r−i−1}(X)_{hom} \ar@{->>}[r] \ar[d]_{ch} & K_{2r−i−1}(V)_{O_{K}, hom} \ar[d]^{_{reg_{p}}} \\ H^{i+1}(X, 𝔗_{ℚ_{p}}(r))^{0} \ar[r] & H^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))).}

The image of the bottom arrow is contained in H_{f}^{1}(K, H^{i}(\overline{V} , ℚ_{p}(r))) by Theorem 9.1 and Conjecture 8.11 for D^{i} = D^{i}(Y), which implies Theorem 10.1. This completes the proof of Theorems 10.1 and 1.1. □

A Continuous crystalline cohomology

In this appendix, we formulate continuous versions of crystalline and syntomic cohomology of log schemes, combining the methods of Jannsen, Kato and Tsuji ([J], [K2], [K3], [Ts1], [Ts2]). The results of this appendix have been used in §9 of this paper. See [K3] for the general framework of log structures and log schemes.

Let p be a prime number, and let K be a complete discrete valuation field of characteristic 0 whose residue field k is a perfect field of characteristic p. Let O_{K} be the integer ring of K. Put W := W (k), W_{n} := W_{n}(k) (n ≥ 1) and K_{0} := Frac(W). Let X be a regular scheme which is projective flat over O_{K} with semistable reduction, and put

X_{K} := X ⊗_{O_{K}} K, Y := X ⊗_{O_{K}} k
X\overline{K} :=X_{K} ⊗_{K} \overline{K}, \overline{Y} := Y ⊗_{k} \overline{k} and \overline{X} := X ⊗_{O_{K}} \overline{O_{K}},

where \overline{O_{K}} denotes the integral closure of O_{K} in \overline{K}. Let M be the log structure on X associated with the normal crossing divisor Y. We endow \overline{X} with a log structure \overline{M} as follows. For a finite field extension L/K, put S_{L} := Spec(O_{L}), and let M_{S_{L}} be the log structure on S_{L} associated with its closed point. We denote (S_{K}, M_{S_{K}}) simply by (S, M_{S}), and define (X_{O_{L}}, M_{O_{L}}) by base-change in the category of log schemes

(X_{O_{L}}, M_{O_{L}}) := (X, M) ×_{(S,M_{S})} (S_{L}, M_{S_{L}}) .  (A.0.1)

We then define the log structure \overline{M} on \overline{X} as that associated with the pre-log structure

\underrightarrow{lim}_{K⊂L⊂\overline{K}} M_{O_{L}}|\overline{X},  (A.0.2)

where L runs through all finite field extensions of K contained in \overline{K}, M_{O_{L}}|\overline{X} denotes the topological inverse image of M_{O_{L}} onto \overline{X}, and the inductive limit is taken in the category of étale sheaves of monoids on \overline{X}.

For a log scheme (Z, M_{Z}) and an integer n ≥ 1, we define

(Z_{n}, M_{Z_{n}}) := (Z, M_{Z}) ×_{Spec(ℤ)} Spec(ℤ/p^{n}),

where we regarded Spec(ℤ) and Spec(ℤ/p^{n}) as log schemes by endowing them with the trivial log structures and the fiber product is taken in the category of log schemes.

We denote by Mod(ℤ/p^{•}) the category of the projective systems {F_{n}}_{n≥1} of ℤ_{p}- modules such that F_{n} is a ℤ/p^{n}-module for each n ≥ 1. For a pro-finite group G, Mod(G-ℤ/p^{•}) denotes the category of the projective systems {F_{n}}_{n≥1} of discrete G-modules such that F_{n} is a ℤ/p^{n}-module.

For a scheme T, Shv(T_{ét}, ℤ/p^{•}) denotes the category of the projective systems {ℱ_{n}}_{n≥1} of étale sheaves on T such that ℱ_{n} is a ℤ/p^{n}-sheaf for each n ≥ 1. For a pro-finite group G acting on T, Shv(T_{ét}, G-ℤ/p^{•}) denotes the category of the projective systems {ℱ_{n}}_{n≥1} of étale G-sheaves on T such that ℱ_{n} is a ℤ/p^{n}-sheaf for each n ≥ 1. We write D(T_{ét}, ℤ/p^{•}) (resp. D(T_{ét}, G-ℤ/p^{•})) for the derived category of Shv(T_{ét}, ℤ/p^{•}) (resp. Shv(T_{ét}, G-ℤ/p^{•})).

For an additive category 𝒞, we write ℚ ⊗ 𝒞 for the ℚ-tensor category of 𝒞, i.e., the category whose objects are the same as 𝒞 and such that for objects A, B ∈ 𝒞, the group of morphisms Hom_{ℚ⊗𝒞} (A, B) is given by ℚ ⊗ Hom_{𝒞} (A, B). We often write ℚ ⊗ A for A ∈ 𝒞 regarded as an object of ℚ ⊗ 𝒞 to avoid confusions.

A.1 Crystalline complexes

We construct the following objects:

𝔼_{(X_{•},M_{•})/W_{•}}, 𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )} ∈ D^{+}(Y_{ét}, ℤ/p^{•})
𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}, 𝔼_{(\overline{X}_{•},\overline{M}_{•})/(ℰ_{•},M_{ℰ•} )} ∈ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}),

where W_{n} means Spec(W_{n}) endowed with the trivial log structure for each n ≥ 1. See Definition 9.6 for (ℰ_{n}, M_{ℰ_{n}}). To construct 𝔼_{(X_{•},M_{•})/W_{•}}, we fix an étale hypercovering (X^{⋆}, M^{⋆}) → (X, M) and a closed immersion (X^{⋆}, M^{⋆}) ↪ (Z^{⋆}, M_{Z^{⋆}}) of simplicial fine log schemes over W such that (Z^{i}, M_{Z^{i}}) is smooth over W for each i ∈ ℕ (cf. [HK] (2.18)). Put

Y^{⋆} := X^{⋆} ⊗_{O_{K}} k,

which is an étale hypercovering of Y. For n ≥ 1, let (𝒟_{n}^{⋆}, M_{𝒟_{n}^{⋆}} ) be the PD-envelope of (X_{n}^{⋆}, M_{n}^{⋆}) ↪ (Z_{n}^{⋆}, M_{Z_{n}^{⋆}}) with respect to the canonical PD-structure on (p) ⊂ W_{n} ([K3] Definition 5.4), and we define a complex 𝔼_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}} of sheaves on Y_{ét}^{⋆} as

𝒪_{𝒟_{n}^{⋆}} →^{d} 𝒪_{𝒟_{n}^{⋆}} ⊗_{𝒪_{Zn}^{⋆}} ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{1} →^{d} ⋯ →^{d} 𝒪_{𝒟_{n}^{⋆}} ⊗_{𝒪_{Z^{⋆}_{n}}} ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{q} →^{d} ⋯,

where the first term is placed in degree 0 and ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{q} denotes the q-th differential module of (Z_{n}^{⋆}, M_{Z_{n}^{⋆}}) over W_{n} cf. [K3] (1.7). See loc. cit. Theorem 6.2 for d.

Regarding this complex as a complex of projective systems (with respect to n ≥ 1) of sheaves on Y_{ét}^{⋆}, we obtain a complex 𝔼_{(X_{•}^{⋆},M_{•}^{⋆})/W_{•}} of objects of Shv(Y_{ét}^{⋆}, ℤ/p^{•}). We then define

𝔼_{(X_{•},M_{•})/W_{•}} := Rθ_{∗}(𝔼_{(X_{•}^{⋆},M_{•}^{⋆})/W_{•}}) ∈ D^{+}(Y_{ét}, ℤ/p^{•}),

where θ : Shv(Y_{ét}^{⋆}, ℤ/p^{•}) → Shv(Y_{ét}, ℤ/p^{•}) denotes the natural morphism of topoi. The resulting object 𝔼_{(X_{•},M_{•})/W_{•}} is independent of the choice of the pair ((X^{⋆}, M^{⋆}), (Z^{⋆}, M_{Z^{⋆}})) by a standard argument (cf. [K2] p. 212).

To construct 𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )}, we put V := Spec(W [t]), and define M_{V} as the log structure on V associated with the divisor {t = 0}. We regard (X, M) as a log scheme over (V, M_{V}) by the composite map

(X, M) → (S, M_{S}) → (V, M_{V}),

where the last map is given by T ↦ π, the prime element of O_{K} we fixed in Definition 9.6 to define (ℰ_{n}, M_{ℰ_{n}}). We then apply the same construction as for 𝔼_{(X_{•},M_{•})/W_{•}} to the morphism (X, M) → (V, M_{V}), i.e, fix an étale hypercovering (X^{⋆}, M^{⋆}) → (X, M) and a closed immersion i^{⋆} : (X^{⋆}, M^{⋆}) ↪ (Z^{⋆}, M_{Z^{⋆}}) of simplicial log schemes over (V, M_{V}) such that (Z^{i}, M_{Z^{i}}) is smooth over W for each i ∈ ℕ. We define 𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )} by replacing ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{q} with ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/(V_{n},M_{Vn})}^{q} in the above definition of 𝔼_{(X_{•},M_{•})/W_{•}}.

We construct 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}} as follows. Fix an étale hypercovering (X^{⋆}, M^{⋆}) → (X, M) and for each finite extension L/K contained in \overline{K}, fix a closed immersion (X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) ↪ (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}) ((X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) := (X^{⋆}, M^{⋆}) ×_{(S,M_{S})} (S_{L}, M_{S_{L}})) of simplicial fine log schemes over W such that (Z_{L}^{i}, M_{Z_{L}^{i}}) is smooth over W for each i ∈ ℕ and L/K, and such that for finite extensions L′/L there are morphisms τ_{L′/L} : (Z_{L′}^{⋆}, M_{Z_{L′}^{⋆}}) → (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}) satisfying transitivity. For a finite extension L/K, let k_{L} be the residue field of L, and put

Y_{L}^{⋆} := X^{⋆} ⊗_{O_{K}} k_{L} and \overline{Y}^{⋆} := X^{⋆} ⊗_{O_{K}} \overline{k},

which are étale hypercoverings of Y_{L} := Y ⊗_{k} k_{L} and \overline{Y}, respectively. We define a complex 𝔼_{(X_{O_{L},•}^{⋆},M_{O_{L},•}^{⋆})/W_{•}} on Y_{L}^{⋆} applying the same construction as for 𝔼_{(X_{•}^{⋆},M_{•}^{⋆})/W_{•}} to the embedding (X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) ↪ (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}), whose inverse image onto \overline{Y}^{⋆} yields an inductive system of complexes of objects of Shv(\overline{Y} _{ét}^{⋆}, G_{K}-ℤ/p^{•}) with respect to finite extensions L/K. We then define

𝔼_{(\overline{X}_{•}^{⋆},\overline{M}_{•}^{⋆})/W_{•}} := \underrightarrow{lim}_{K⊂L⊂\overline{K}} 𝔼_{(X_{O_{L},•}^{⋆},M_{O_{L},•}^{⋆})/W_{•}} |_{\overline{Y}_{⋆}},
𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}} := R\overline{θ}_{∗}(𝔼_{(\overline{X}_{•}^{⋆},\overline{M}_{•}^{⋆})/W_{•}}) ∈ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}),

where \overline{θ} : Shv(\overline{Y} _{ét}^{⋆}, G_{K}-ℤ/p^{•}) → Shv(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}) denotes the natural morphism of topoi. We construct 𝔼_{(\overline{X}_{•},\overline{M}_{•})/(ℰ_{•},M_{ℰ•} )} applying similar arguments to the complexes 𝔼_{(X_{O_{L},•}^{⋆},M_{O_{L},•}^{⋆})/(ℰ_{•},M_{ℰ•} )} |_{\overline{Y}_{⋆}}.

We define the following objects of Mod(ℤ/p^{•}):

H_{crys}^{i}((X_{•}, M_{•})/W_{•}) := R^{i}Γ (𝔼_{(X_{•},M_{•})/W_{•}}),  (A.1.1)
H_{crys}^{i}((X_{•}, M_{•})/(ℰ_{•}, M_{ℰ_{•}})) := R^{i}Γ (𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )}),  (A.1.2)
H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/W_{•}) := R^{i}\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}})  (A.1.3)
H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/(ℰ_{•}, M_{ℰ_{•}})) := R^{i}\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/(ℰ_{•},M_{ℰ•} )}),  (A.1.4)

where Γ and \overline{Γ} denote the following left exact functors, respectively:

Γ := Γ (Y, −) : Shv(Y_{ét}, ℤ/p^{•}) → Mod(ℤ/p^{•}),  (A.1.5)
\overline{Γ} := Γ (\overline{Y} , −) : Shv(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}) → Mod(G_{K}-ℤ/p^{•}).  (A.1.6)

We define ‘naive’ crystalline cohomology groups

H_{crys}^{i}((X, M)/W), H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} )),
H_{crys}^{i}((\overline{X}, \overline{M})/W)  and  H_{crys}^{i}((\overline{X}, \overline{M})/(ℰ , M_{ℰ} ))

as the projective limit of (A.1.1)–(A.1.4), respectively.

Remark A.1.6 Let π_{n} : Shv(Y_{ét}, ℤ/p^{•}) → Shv(Y_{ét}, ℤ/p^{n}) be the natural functor sending {ℱ_{m}}_{m≥1} to ℱ_{n}. Since π_{n} is exact, it extends to a triangulated functor

π_{n} : D^{+}(Y_{ét}, ℤ/p^{•}) ⟶ D^{+}(Y_{ét}, ℤ/p^{n}),

which sends 𝔼_{(X_{•},M_{•})/W_{•}} ↦ 𝔼_{(X_{n},M_{n})/W_{n}}, an object computing the crystalline cohomology of (X_{n}, M_{n})/W_{n}, because π_{n} is compatible with the gluing functor Rθ_{∗}. Moreover by [J] Proposition 1.1 (b), we have

H_{crys}^{i}((X_{•}, M_{•})/W_{•}) ≃ {H_{crys}^{i}((X_{n}, M_{n})/W_{n})}_{n≥1}.

We have similar facts for (A.1.2), (A.1.3)and (A.1.4)as well.

A.2 Continuous crystalline cohomology

We define continuous crystalline cohomology groups as follows:

H_{cont-cr}^{i}((X, M)/W) := R^{i}( \underleftarrow{lim} ∘Γ )(𝔼_{(X_{•},M_{•})/W_{•}}),  (A.2.1)
H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} )) := R^{i}( \underleftarrow{lim} ∘Γ )(𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )}),  (A.2.2)

where Γ is as in (A.1.5). Because Γ has an exact left adjoint, it preserves injectives and there exists a spectral sequence

E_{2}^{a,b} = R^{a} \underleftarrow{lim} H_{crys}^{b}((X_{•}, M_{•})/W_{•}) ⟹ H_{cont-cr}^{a+b}((X, M)/W).

Because R^{a} \underleftarrow{lim} = 0 for a ≥ 2, this spectral sequence breaks up into short exact sequences

0 ⟶ R^{1} \underleftarrow{lim} H_{crys}^{i−1}((X_{•}, M_{•})/W_{•}) ⟶ H_{cont-cr}^{i}((X, M)/W)

⟶ H_{crys}^{i}((X, M)/W) ⟶ 0. (A.2.3)

We have similar exact sequences for (A.2.2)by the same arguments.

Proposition A.2.4 There exists a distinguished triangle

𝔼_{(X_{•},M_{•})/W_{•}} ⟶^{can} 𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )} ⟶^{ν} 𝔼_{(X_{•},M_{•})/(ℰ_{•},M_{ℰ•} )} ⟶^{\frac{dt}{t} ∧} 𝔼_{(X_{•},M_{•})/W_{n}}[1]

in D^{+}(Y_{ét}, ℤ/p^{•}). Consequently we obtain a long exact sequence

⋯ ⟶ H_{cont-cr}^{i}((X, M)/W) ⟶ H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} ))

⟶^{ν} H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} )) ⟶^{∂} H_{cont-cr}^{i+1}((X, M)/W) ⟶ · · · .

Proof. The assertion follows from the same arguments as in the proof of [K4] Lemma 4.2 with F = 𝒪^{crys}. □

There is a commutative diagram (see Definition 9.6 for N)

\xymatrix{H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} )) \ar[r]^{ν} \ar[d] & H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} )) \ar[d] \\ H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} )) \ar[r]^{N} & H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} )) .}

We will show later that the vertical arrows are bijective in Corollary A.3.2 (1) below.

A.3 Comparison of projective systems of crystalline cohomology

We recall here the following comparison facts on projective systems of crystalline cohomology groups, which will be useful later. Note that for A_{•} ∈ Mod(ℤ/p^{•}), both \underleftarrow{lim} A_{•} and R^{1} \underleftarrow{lim} A_{•} have finite exponents if ℚ ⊗ (A_{•}) ≃ 0 in ℚ ⊗ Mod(ℤ/p^{•}).

Proposition A.3.1 Let i be a non-negative integer, and put D_{n}^{i} := ℍ^{i}(Y, W_{n} ω_{Y}^{•}). Let P_{n} be the ring defined in [Ts1] §1.6, and put R_{ℰ_{n}} := Γ (ℰ_{n}, 𝒪_{ℰ_{n}}).

(1) There is an isomorphism in ℚ ⊗ Mod(ℤ/p^{•})

ℚ ⊗ (H_{crys}^{i}((X_{•}, M_{•})/(ℰ_{•}, M_{ℰ_{•}}))) ≃ ℚ ⊗ {R_{ℰ_{n}} ⊗_{W_{n}} D_{n}^{i}}_{n≥1}

and R^{1} \underleftarrow{lim} H_{crys}^{i}((X_{•}, M_{•})/(ℰ_{•}, M_{ℰ_{•}})) has a finite exponent.

(2) There is an isomorphism in ℚ ⊗ Mod(G_{K}-ℤ/p^{•})

ℚ ⊗ (H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/(ℰ_{•}, M_{ℰ_{•}}))) ≃ ℚ ⊗ \{P_{n} ⊗_{W_{n}} D_{n}^{i}\}_{n≥1}

and R^{1} \underleftarrow{lim} H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/(ℰ_{•}, M_{ℰ_{•}})) has a finite exponent.

(3) There is an isomorphism in ℚ ⊗ Mod(G_{K}-ℤ/p^{•})

ℚ ⊗ (H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/W_{•})) ≃ ℚ ⊗ \{(P_{n} ⊗_{W_{n}} D_{n}^{i})^{N=0}\}_{n≥1}

and R^{1} \underleftarrow{lim} H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/W_{•}) has a finite exponent. Here N acts on P_{n} ⊗_{W_{n}} D_{n}^{i} by N_{P_{n}} ⊗ 1 + 1 ⊗ N and N_{P_{n}} denotes the monodromy operator on P_{n}, cf. [Ts1] p. 253.

Proof. (1) The first assertion follows from [HK] Lemma 5.2 (see also (∗) in the proof of [Ts1] Proposition 4.4.6). To show the second assertion, we check that the projective system {R_{ℰ_{n}} ⊗_{W_{n}}D_{n}^{i}}_{n≥1} satisfies the Mittag-Leffler condition. Indeed, D_{n}^{i} is finitely generated over W_{n}, and the projection R_{ℰ_{n+1}} → R_{ℰ_{n}} is surjective by the presentation

R_{ℰ_{n}} = W [t, t^{eν}/ν! (ν ≥ 1)] ⊗_{W} W_{n} (e := [K : K_{0}])

obtained from the definition of (ℰ_{n}, M_{ℰ_{n}}) (see the proof of [Ts1] Proposition 4.4.6).

(2) The first assertion follows from that of (1) and [Ts1] Proposition 4.5.4. The second assertion follows from the fact that {P_{n} ⊗_{W_{n}} D_{n}^{i}}_{n≥1} satisfies the Mittag-Leffler condition. Indeed, the natural projection P_{n+1} → P_{n} is surjective (loc. cit. Lemma 1.6.7) and D_{n}^{i} is finitely generated over W_{n}.

(3) See the proof of loc. cit. Proposition 4.5.6 for the first assertion.. We show the second assertion. Note that P_{n} is flat over W_{n}, because R_{ℰ_{n}} is flat over W_{n} and P_{n} is flat over R_{ℰ_{n}} by the above presentation of R_{ℰ_{n}} and loc. cit. Proposition 4.1.5. Let {D_{n}}_{n≥1} be a projective system of W -modules such that D_{n} is finitely generated over W_{n} for each n, and let N : {D_{n}}_{n≥1} → {D_{n}}_{n≥1} be a nilpotent W -endomorphism. Our task is to show that

R^{1} \underleftarrow{lim} \{(P_{n} ⊗_{W_{n}} D_{n})^{N=0}\}_{n≥1} = 0.

Consider a short exact sequence of projective systems

0 ⟶ {(D_{n})^{N=0}}_{n≥1} ⟶ {D_{n}}_{n≥1} ⟶ {N(D_{n})}_{n≥1} ⟶ 0.

Note that (D_{n})^{N=0} and N(D_{n}) are finitely generated over W_{n} for each n. Let b be the minimal integer for which N^{b} = 0 on {D_{n}}_{n≥1}. By [K4] Lemma 4.3 and the flatness of P_{n} over W_{n}, we have a short exact sequence for each n ≥ 1

0 → (P_{n} ⊗_{W_{n}} (D_{n})^{N=0})^{N=0} → (P_{n} ⊗_{W_{n}} D_{n})^{N=0} → (P_{n} ⊗_{W_{n}} N(D_{n}))^{N=0} → 0,

which yields a short exact sequence of projective systems with respect to n ≥ 1. Since N^{b−1} = 0 on {N(D_{n})}_{n≥1}, we may assume b = 1 by induction on b ≥ 1. Now let B_{n} be as in [Ts1] §1.6 and let A_{crys} be as in loc. cit. §1.1. Then we have isomorphisms

(P_{n} ⊗_{W_{n}} D_{n})^{N=0} ≃^{(1)} (P_{n})^{N=0} ⊗_{W_{n}} D_{n} ≃^{(2)} B_{n} ⊗_{W_{n}} D_{n} =^{(3)} (A_{crys}/p^{n}) ⊗_{W_{n}} D_{n},

where (1) follows from the assumption b = 1 and the flatness of P_{n} over W_{n}. The isomorphism (2) (resp. (3)) follows from loc. cit. Corollary 1.6.6 (resp. the definition of B_{n} in loc. cit. §1.6). Thus R^{1} \underleftarrow{lim} \{(P_{n} ⊗_{W_{n}} D_{n})^{N=0}\}_{n≥1} is zero, and we obtain the assertion. □

Corollary A.3.2 (1) For i ≥ 0, we have

H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} ≃ H_{cont-cr}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} .

(2) The torsion subgroups of H_{crys}^{i}((\overline{X}, \overline{M})/(ℰ , M_{ℰ} )) and H_{crys}^{i}((\overline{X}, \overline{M})/W) have finite exponents for any i ≥ 0.

Proof. (1) follows from Proposition A.3.1 (1) and the remark after (A.2.3). Since \underleftarrow{lim}_{n≥1} P_{n} is p-torsion free, the assertion (2) follows from the isomorphisms in Proposition A.3.1 (3). □

Remark A.3.3 If K is absolutely unramified, then we have

H_{crys}^{i}((X_{n}, M_{n})/W_{n}) ≃ ℍ^{i}(Y, W_{n} ω̃_{Y}^{•}),

which is finitely generated over W_{n} by Theorem 8.8 (2) and (8.10.1) (see also [Hy2] (1.4.3), (2.4.2)). Consequently, the projective system H_{crys}^{i}((X_{•}, M_{•})/W_{•}) satisfies the Mittag-Leffler condition and we obtain a long exact sequence

⋯ ⟶ H_{crys}^{i}((X, M)/W)_{ℚ_{p}} ⟶ H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}}

⟶^{N} H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} ⟶ H_{crys}^{i+1}((X, M)/W)_{ℚ_{p}} ⟶ ⋯,

which removes the assumption (∗) in [La] p. 191. On the other hand, the author does not know if H_{crys}^{i}((X_{•}, M_{•})/W_{•}) satisfies the Mittag-Leffler condition even up to torsion, when K is not absolutely unramified.

The following corollary has been used in the proof of Lemma 9.7 (1):

Corollary A.3.4 In the following commutative diagram of canonical maps, the arrows (3) and (4) are injective:

\xymatrix{H_{cont-cr}^{i}((X, M)/W)_{ℚ_{p}} \ar[r]^{(1)} \ar[d]_{(2)} & H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} \ar[d]^{(3)} \\ H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}} \ar[r]^{(4)} & H_{crys}^{i}((\overline{X}, \overline{M})/(ℰ , M_{ℰ} ))_{ℚ_{p}} .}

In particular, the kernel of (1) agrees with that of (2).

Proof. The injectivity of (3) follows from Proposition A.3.1 (1) and (2) and the injectivity of the natural maps R_{ℰ_{n}} → P_{n} for n ≥ 1 ([Ts1] Proposition 4.1.5). The injectivity of (4) follows from Proposition A.3.1 (2) and (3). □

A.4 continuous-Galois crystalline cohomology

We define the continuous-Galois crystalline cohomology as follows:

H_{cG-cr}^{i}((X, M)/W) := R^{i}( \underleftarrow{lim} Γ_{Gal}\overline{Γ} )(𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}),

where \overline{Γ} is as in (A.1.6), and Γ_{Gal} denotes the functor taking G_{K}-invariant subgroups:

Γ_{Gal} := Γ (G_{K}, −) : Mod(G_{K}-ℤ/p^{•}) ⟶ Mod(ℤ/p^{•}).

There is a natural map

H_{cont-cr}^{i}((X, M)/W) ⟶ H_{cG-cr}^{i}((X, M)/W)  (A.4.1)

by definition.

Theorem A.4.2 There exists a Hochschild-Serre spectral sequence

E_{2}^{a,b} = H^{a}(K, H_{crys}^{b}((\overline{X}, \overline{M})/W)_{ℚ_{p}}) ⟹ H_{cG-cr}^{a+b}((X, M)/W)_{ℚ_{p}}.

Proof. Because \overline{Γ} has an exact left adjoint functor, it preserves injectives and there exists a spectral sequence

E_{2}^{a,b} = H^{a}(K, H_{crys}^{b}((\overline{X}_{•}, \overline{M}_{•})/W_{•})) ⟹ H_{cG-cr}^{a+b}((X, M)/W),

where H^{∗}(K, {F_{n}}_{n≥1}) for a projective system {F_{n}}_{n≥1} of discrete G_{K}-modules denotes the continuous Galois cohomology of G_{K} in the sense of Jannsen [J] §2. We show that the canonical map

α^{a,b} : H^{a}(K, H_{crys}^{b}((\overline{X}, \overline{M})/W)_{ℚ_{p}}) ⟶ H^{a}(K, H_{crys}^{b}((\overline{X}_{•}, \overline{M}_{•})/W_{•}))_{ℚ_{p}}

(see the proof of loc. cit. Theorem 2.2) is bijective for a, b ≥ 0. Put

L_{•}^{b} := \{P_{n} ⊗_{W_{n}} D_{n}^{b}\}_{n≥1} ,  L^{b} := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim} L_{•}^{b} = B_{st}^{+} ⊗_{K_{0}} D^{b} ,
T_{•}^{b} := H_{crys}^{b}((\overline{X}_{•}, \overline{M}_{•})/W_{•}) and T^{b} := H_{crys}^{b}((\overline{X}, \overline{M})/W)_{ℚ_{p}} .

Since the torsion part of \underleftarrow{lim} L_{•}^{b} has a finite exponent, the canonical map

β^{a,b} : H^{a}(K, L^{b}) ⟶ H^{a}(K, L_{•}^{b})_{ℚ_{p}}  (A.4.3)

is bijective by loc. cit. Theorem 5.15 (c). We have a short exact sequence of objects in ℚ ⊗ Mod(G_{K}-ℤ/p^{•})

0 ⟶ ℚ ⊗ (T_{•}^{b}) ⟶ ℚ ⊗ (L_{•}^{b}) ⟶^{N} ℚ ⊗ (L_{•}^{b}) ⟶ 0

and a short exact sequence of topological G_{K}-modules

0 ⟶ T^{b} ⟶ L^{b} ⟶^{N} L^{b} ⟶ 0  (A.4.4)

by Proposition A.3.1 (3) and [K4] Lemma 4.3. These exact sequences yield a commutative diagram with exact rows

⋯ ⟶ H^{a−1}(K, L^{b}) ⟶ H^{a}(K, T^{b}) ⟶ H^{a}(K, L^{b}) ⟶^{N} H^{a}(K, L^{b}) ⟶ ⋯
β^{a−1,b}↓  _{α^{a,b}}↓  _{β^{a,b}}↓  _{β^{a,b}}↓
⋯ ⟶ H^{a−1}(K, L_{•}^{b})_{ℚ_{p}} ⟶ H^{a}(K, T_{•}^{b})_{ℚ_{p}} ⟶ H^{a}(K, L_{•}^{b})_{ℚ_{p}} ⟶^{N} H^{a}(K, L_{•}^{b})_{ℚ_{p}} ⟶ ⋯,

where the arrows β^{a−1,b} and β^{a,b} are bijective by (A.4.3). Hence the assertion follows from the five lemma. □

Corollary A.4.5 Put

H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0} := Ker(H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}} → H_{crys}^{i+1}((\overline{X}, \overline{M})/W)_{ℚ_{p}}).

Then the canonical homomorphism

H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0} ⟶ H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}),

obtained from (A.4.1) and an edge homomorphism of the spectral sequence in Theorem A.4.2, fits into a commutative diagram

\xymatrix{H_{crys}^{i}((X, M)/(ℰ , M_{ℰ} ))_{ℚ_{p}} \ar@{->>}[r]^{∂} \ar[d] & H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0} \ar[d] \\ (B_{st}^{+} ⊗_{K_{0}} D^{i})^{G_{K}} \ar[r]^{(−1)^{i+1}δ} & H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}),}

where ∂ is the connecting map induced by that in Proposition A.2.4 (see Corollary A.3.4 for the existence and the surjectivity), and δ denotes the connecting map of continuous Galois cohomology associated with (A.4.4) with b = i. The left vertical arrow is obtained from Proposition A.3.1 (2).

Proof. The assertion follows from Proposition A.3.1 (2) and a simple computation on boundary maps arising from the distinguished triangle

R\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}) ⟶^{can} R\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/(ℰ_{•},M_{ℰ•} )})

⟶^{ν} R\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/(ℰ_{•},M_{ℰ•} )}) ⟶^{\frac{dt}{t} ∧} R\overline{Γ} (𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}})[1]

in D^{+}(Mod(G_{K}-ℤ/p^{•})) (a variant of Proposition A.2.4). The sign (−1)^{i+1} in the diagram arises from the difference of the orientations between this distinguished triangle and (A.4.4), and the fact that the construction of connecting morphisms associated with short exact sequences of complexes commutes with the shift functor [i] up to the sign (−1)^{i}. The details are straight-forward and left to the reader. □

A.5 Syntomic complexes

We construct the following objects for r ≥ 0:

𝒮_{•} (r)_{(X,M)} ∈ D^{+}(Y_{ét}, ℤ/p^{•}) and 𝒮_{•} (r)_{(\overline{X},\overline{M})} ∈ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}),

and the following objects for r with 0 ≤ r ≤ p − 1

𝒮_{•}(r)_{(X,M)} ∈ D^{+}(Y_{ét}, ℤ/p^{•}) and 𝒮_{•}(r)_{(\overline{X},\overline{M})} ∈ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}).

Definition A.5.1 Let (T, M_{T}) be a log scheme over ℤ_{p}. A Frobenius endomorphism φ : (T, M_{T}) → (T, M_{T}) is a morphism over ℤ_{p} such that φ ⊗ ℤ/p : (T_{1}, M_{T_{1}}) → (T_{1}, M_{T_{1}}) is the absolute Frobenius endomorphism in the sense of [K3] Definition 4.7.

To construct the objects 𝒮_{•} (r)_{(X,M)} and 𝒮_{•}(r)_{(X,M)}, we fix an étale hypercovering (X^{⋆}, M^{⋆}) → (X, M) and a closed immersion (X^{⋆}, M^{⋆}) ↪ (Z^{⋆}, M_{Z^{⋆}}) of simplicial fine log schemes over W such that (Z^{i}, M_{Z^{i}}) is smooth over W and has a Frobenius endomorphism for each i ∈ ℕ. Let n ≥ 1 be an integer, and let (𝒟_{n}^{⋆}, M_{𝒟_{n}^{⋆}} ) be the PD-envelope of (X_{n}^{⋆}, M_{n}^{⋆}) in (Z_{n}^{⋆}, M_{Z_{n}^{⋆}}) with respect to the canonical PD-structure on (p) ⊂ W_{n}. For i ≥ 1, let 𝒥^{[i]} ⊂ 𝒪_{𝒟_{n}^{⋆}} be the i-th divided power of the ideal 𝒥 = Ker(𝒪_{𝒟_{n}^{⋆}} → 𝒪_{X_{n}^{⋆}}). For i ≤ 0, put 𝒥^{[i]} := 𝒪_{𝒟_{n}^{⋆}} . Let 𝕁_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}}^{[r]} be the complex of sheaves on Y_{ét}^{⋆}

𝒥^{[r]} →^{d} 𝒥^{[r−1]}⊗_{𝒪_{Z^{⋆}_{n}}} ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{1} →^{d} ⋯ →^{d} 𝒥^{[r−q]}⊗_{𝒪_{Z^{⋆}_{n}}} ω_{(Z_{n}^{⋆},M_{Z^{⋆}_{n}})/W_{n}}^{q} →^{d} ⋯,

where 𝒥^{[r]} is placed in degree 0. See [Ts2] Corollary 1.10 for d. The complex 𝔼_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}} we considered in §A.1 agrees with 𝕁_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}}^{[0]}. We define a complex 𝒮_{n} (r)_{(X^{⋆},M^{⋆})} on Y_{ét}^{⋆} as the mapping fiber of the homomorphism

p^{r} − φ_{n}^{∗} : 𝕁_{(X_{n}^{⋆},M_{n}^{⋆})}^{[r]} → 𝔼_{(X_{n}^{⋆},M_{n}^{⋆})}

For 0 ≤ r ≤ p − 1, the Frobenius endomorphism on (Z_{n+r}^{⋆}, M_{Z_{n+r}^{⋆}}) induces a homomorphism of complexes

f_{r} := \overline{p^{−r} · φ_{n+r}^{∗}} : 𝕁_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}}^{[r]} ⟶ 𝔼_{(X_{n}^{⋆},M_{n}^{⋆})/W_{n}}

(cf. [Ts2] p. 540). We define a complex 𝒮_{n}(r)_{(X^{⋆},M^{⋆})} on Y_{ét}^{⋆} as the mapping fiber of the homomorphism

1 − f_{r} : 𝕁_{(X_{n}^{⋆},M_{n}^{⋆})}^{[r]} → 𝔼_{(X_{n}^{⋆},M_{n}^{⋆})}

Regarding 𝒮_{n} (r)_{(X^{⋆},M^{⋆})} and 𝒮_{n}(r)_{(X^{⋆},M^{⋆})} as complexes of projective systems (on n ≥ 1) of sheaves on Y_{ét}^{⋆}, we define

𝒮_{•} (r)_{(X,M)} := Rθ_{∗}(𝒮_{•} (r)_{(X^{⋆},M^{⋆})})  (r ≥ 0),
𝒮_{•}(r)_{(X,M)} := Rθ_{∗}(𝒮_{•}(r)_{(X^{⋆},M^{⋆})})  (0 ≤ r ≤ p − 1),

where θ : Shv(Y_{ét}^{⋆}, ℤ/p^{•}) → Shv(Y_{ét}, ℤ/p^{•}) denotes the natural morphism of topoi. The resulting objects are independent of the pair ((X^{⋆}, M^{⋆}), (Z^{⋆}, M_{Z^{⋆}})) (cf. [K2] p. 212).

We construct 𝒮_{•} (r)_{(\overline{X},\overline{M})} for r ≥ 0 and 𝒮_{•}(r)_{(\overline{X},\overline{M})} for 0 ≤ r ≤ p − 1 as follows. Fix an étale hypercovering (X^{⋆}, M^{⋆}) → (X, M) and for each finite extension L/K contained in \overline{K}, fix a closed immersion (X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) ↪ (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}) ((X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) := (X^{⋆}, M^{⋆}) ×_{(S,M_{S})} (S_{L}, M_{S_{L}})) of simplicial fine log schemes over W such that (Z_{L}^{i}, M_{Z_{L}^{i}}) is smooth over W for each i ∈ ℕ and L/K, such that Z_{L}^{i} has a Frobenius endomorphism for each i ∈ ℕ and L/K, and such that for finite extensions L′/L there are morphisms τ_{L′/L} : (Z_{L′}^{⋆}, M_{Z_{L′}^{⋆}}) → (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}) which satisfy transitivity and compatibility with Frobenius morphisms. For a finite extension L/K, we define Y_{L}^{⋆} in a similar way as in §A.1. We define complexes 𝒮_{•} (r)_{(X_{O_{L}} ,M_{O_{L}})} and 𝒮_{•}(r)_{(X_{O_{L}} ,M_{O_{L}})} on Y_{L}^{⋆} applying the same constructions as for the complexes 𝒮_{•} (r)_{(X,M)} and 𝒮_{•}(r)_{(X,M)}, respectively, to the embedding (X_{O_{L}}^{⋆}, M_{O_{L}}^{⋆}) ↪ (Z_{L}^{⋆}, M_{Z_{L}^{⋆}}), whose inverse images onto \overline{Y}^{⋆} := X^{⋆} ⊗_{O_{K}} \overline{k} yield inductive systems of complexes of objects of Shv(\overline{Y} _{ét}^{⋆}, G_{K}-ℤ/p^{•}) with respect to finite extensions L/K. We define

𝒮_{•} (r)_{(\overline{X}_{•}^{⋆},\overline{M}_{•}^{⋆})} := \underrightarrow{lim}_{K⊂L⊂\overline{K}} 𝒮_{•} (r)_{(X_{O_{L},•}^{⋆},M_{O_{L},•}^{⋆})}|_{\overline{Y}_{⋆}},
𝒮_{•} (r)_{(\overline{X}_{•},\overline{M}_{•})} := R\overline{θ}_{∗}(𝒮_{•} (r)_{(\overline{X}_{•}^{⋆},\overline{M}_{•}^{⋆})}) ∈ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}),

where \overline{θ} : Shv(\overline{Y} _{ét}^{⋆}, G_{K}-ℤ/p^{•}) → Shv(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}) denotes the natural morphism of topoi. We construct 𝒮_{•}(r)_{(\overline{X}_{•},\overline{M}_{•})} from 𝒮_{•}(r)_{(X_{O_{L},•}^{⋆},M_{O_{L},•}^{⋆})}’s in a similar way. When 0 ≤ r ≤ p − 1, there is a canonical projection

ψ_{•}^{r} : 𝒮_{•} (r)_{(\overline{X},\overline{M})} ⟶ 𝒮_{•}(r)_{(\overline{X},\overline{M})}

induced by the identity map of 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}. There are canonical morphisms

c_{•}^{r} : 𝒮_{•} (r)_{(\overline{X},\overline{M})} ⟶ 𝕁_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}^{[r]} ⟶^{1} 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}} (r ≥ 0)  (A.5.2)
c_{•}^{r} : 𝒮_{•}(r)_{(\overline{X},\overline{M})} ⟶ 𝕁_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}^{[r]} ⟶^{1} 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}} (0 ≤ r ≤ p − 1), (A.5.3)

which satisfy

p^{r} · c_{•}^{r} = c_{•}^{r} ∘ ψ_{•}^{r}  (when 0 ≤ r ≤ p − 1).  (A.5.4)

For r ≥ 0, we define

ℤ/p^{•}(r)′ := (p^{a}a!)^{−1}ℤ_{p}(r) ⊗ ℤ/p^{•} ∈ Shv(X\overline{K}, G_{K}-ℤ/p^{•}),

where a denotes the maximal integer ≤ r/(p − 1). We have ℤ/p^{•}(r)′ = μ_{p^{•}}^{⊗r} canonically when r ≤ p − 2. Similarly for r < 0, we define

ℤ/p^{•}(r)′ := p^{a}a! ℤ_{p}(r) ⊗ ℤ/p^{•} ∈ Shv(X\overline{K}, G_{K}-ℤ/p^{•})  (A.5.5)

with a the maximal integer with a ≤ −r/(p−1), which will be used in (A.6.4)below. Let \overline{j} : X\overline{K} ↪ \overline{X} and \overline{ι} : \overline{Y} ↪ \overline{X} be the natural immersions. Let R\overline{j}_{∗} and \overline{ι}^{∗} be the following functors, respectively:

R\overline{j}_{∗} : D^{+}((X\overline{K})_{ét}, G_{K}-ℤ/p^{•}) ⟶ D^{+}(\overline{X}_{ét}, G_{K}-ℤ/p^{•}),
\overline{ι}^{∗} : D^{+}(\overline{X}_{ét}, G_{K}-ℤ/p^{•}) ⟶ D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}).

Theorem A.5.6 ([Ts1] §3.1, [K4] Theorem 5.4) For r ≥ 0, there exists a canonical morphism in D^{+}(\overline{Y}_{ét}, G_{K}-ℤ/p^{•}) compatible with product structures

η_{•}^{r} : 𝒮_{•} (r)_{(\overline{X},\overline{M})} ⟶ \overline{ι}^{∗}R\overline{j}_{∗}ℤ/p^{•}(r)′.

If r ≤ p − 2, then η_{•}^{r} factors through an isomorphism

𝒮_{•}(r)_{(\overline{X},\overline{M})} ^{∼} ⟶ τ_{≤r} \overline{ι}^{∗}R\overline{j}_{∗}μ_{p^{•}}^{⊗r}.  (A.5.7)

Proof. We define η_{•}^{r} applying the arguments in [Ts1] §3.1 in the category of ℤ/p^{•}- sheaves. If r ≤ p − 2, then we have ℤ/p^{•}(r)′ = μ_{p^{•}}^{⊗r}, and η_{•}^{r} factors through a morphism \overline{η}_{•}^{r} : 𝒮_{•}(r)_{(\overline{X},\overline{M})} → \overline{ι}^{∗}R\overline{j}_{∗}μ_{p^{•}}^{⊗r} by the construction of η_{•}^{r} (loc. cit. (3.1.11)). The morphism \overline{η}_{•}^{r} induces an isomorphism as claimed, because 𝒮_{•}(r)_{(\overline{X},\overline{M})} is concentrated in [0, r] and \overline{η}_{•}^{r} induces isomorphisms on the q-th cohomology objects with 0 ≤ q ≤ r by [K4] Theorem 5.4. □

We define

H_{syn}^{i}((X, M), 𝒮_{•}(r)) := R^{i}Γ (𝒮_{•}(r)_{(X,M)}),
H_{syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) := \underleftarrow{lim} H_{syn}^{i}((X, M), 𝒮_{•}(r)),
H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} H_{syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)),
H_{syn}^{i}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim} R^{i}\overline{Γ} (𝒮_{•}(r)_{(\overline{X},\overline{M})}).

where Γ and \overline{Γ} are as in (A.1.5)and (A.1.6), respectively.

A.6 continuous(-Galois) syntomic cohomology

We assume r ≤ p − 2 in what follows. For i ≥ 0, we define the continuous syntomic cohomology as follows:

H_{cont-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) := R^{i}( \underleftarrow{lim} Γ )(𝒮_{•}(r)_{(X,M)}).

Similarly, we define the continuous-Galois syntomic cohomology as follows:

H_{cG-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) := R^{i}( \underleftarrow{lim} Γ_{Gal}\overline{Γ} )(𝒮_{•}(r)_{(\overline{X},\overline{M})}).

We put

H_{cont-syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) := ℚ_{p} ⊗_{ℤ_{p}} H_{cont-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)).

Proposition A.6.1 Let i ≥ 0 be an integer.

(1) Let η : H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) → H^{i}(X\overline{K}, ℚ_{p}(r)) be as in Lemma 9.5. Then the kernel of the composite map

H_{cont-syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) ⟶ H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) ⟶^{η} H^{i}(X\overline{K}, ℚ_{p}(r))

agrees with that of the composite map

H_{cont-syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) → H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) → H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}},

which we denote by H_{cont-syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r))^{0}, in what follows.

(2) If K is a p-adic local field (i.e., k is finite), then we have

H_{cont-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) ^{∼} ⟶ H_{syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)).

In particular, we have the following canonical map in this case:

H_{syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) ⟶ H_{cont-cr}^{i}((X, M)/W).

(3) If r ≥ d := dim(X_{K}), then we have

H_{cG-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) ^{∼} ⟶ H_{cont}^{i}(X_{K}, ℤ_{p}(r)),

where H_{cont}^{∗}(X_{K}, ℤ_{p}(r)) denotes the continuous étale cohomology [J].

Proof. (1) The assertion immediately follows from Lemma 9.5.

(2) There is a short exact sequence analogous to (A.2.3)

0 ⟶ R^{1} \underleftarrow{lim} H_{syn}^{i−1}((X, M), 𝒮_{•}(r)) ⟶ H_{cont-syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r))
⟶ H_{syn}^{i}((X, M), 𝒮_{ℤ_{p}}(r)) ⟶ 0.

Since X is proper over k and k is finite by assumption, H_{syn}^{i−1}((X, M), 𝒮_{n}(r)) is finite for any i, n ≥ 1 by [Ts1] Proposition 2.4.1. See also the remark in loc. cit., p. 263. The assertion follows from these facts.

(3) Since r ≥ d by assumption, (A.5.7) implies 𝒮_{•}(r)_{(\overline{X},\overline{M})} ≃ \overline{ι}^{∗}R\overline{j}_{∗}μ_{p^{•}}^{⊗r}. Hence the assertion follows from the isomorphisms in D^{+}(Mod(G_{K}-ℤ/p^{•}))

RΓ (X\overline{K}, μ_{p^{•}}^{⊗r}) = RΓ (\overline{X}, R\overline{j}_{∗}μ_{p^{•}}^{⊗r}) ^{∼} ⟶ RΓ (\overline{Y} , \overline{ι}^{∗}R\overline{j}_{∗}μ_{p^{•}}^{⊗r}),  (A.6.2)

where the last isomorphism is a consequence of the proper base-change theorem for the usual étale cohomology. □

For i ≥ 0, put

H^{i}(X_{K}, ℚ_{p}(r))^{0} := Ker(H^{i}(X_{K}, ℚ_{p}(r)) → H^{i}(X\overline{K}, ℚ_{p}(r))),
H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r))^{0} := Ker(η : H_{syn}^{i}((X, M), 𝒮_{ℚ_{p}}(r)) → H^{i}(X\overline{K}, ℚ_{p}(r))).

We have H^{i}(X_{K}, ℚ_{p}(r)) = H_{cont}^{i}(X_{K}, ℚ_{p}(r)) when K is a p-adic local field. The following corollary is a consequence of Proposition A.6.1 (1), (2) and the covariant functoriality of Hochschild-Serre spectral sequences in coefficients (see also the diagram in the proof of Theorem A.6.7 below).

Corollary A.6.3 Assume that K is a p-adic local field, and let e be the composite map

e : H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} → H^{i+1}(X_{K}, ℚ_{p}(r))^{0} → H^{1}(K, H^{i}(X\overline{K}, ℚ_{p}(r))),

where the last map is an edge homomorphism of the Hochschild-Serre spectral sequence (9.0.1). If r ≥ d, then e fits into a commutative diagram

\xymatrix{H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} \ar[r] \ar[d]_{e} & H_{cont-cr}^{i+1}((X, M)/W)_{ℚ_{p}}^{0} \ar[d] \\ H^{1}(K, H^{i}(X\overline{K}, ℚ_{p}(r))) \ar[r]^{β^{i,r}} & H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}),}

where the top arrow is obtained from Proposition A.6.1 (1) and (2), the right vertical arrow is the map in Corollary A.4.5 and β^{i,r} is as in the proof of Theorem 9.9.

We next construct a commutative diagram in ℚ ⊗ D^{+}(Mod(G_{K}-ℤ/p^{•})) assuming r < d

\xymatrix{ℚ ⊗ RΓ (\overline{Y} , 𝒮_{•}(r)_{(\overline{X},\overline{M})}) \ar[d]_{f^{r}} \ar@{-->}[dr]^{h^{r}} \\ ℚ ⊗ RΓ (X\overline{K}, μ_{p^{•}}^{⊗r}) \ar[r]^{g^{r}} & ℚ ⊗ (RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}) ⊗ ℤ/p^{•}(r − d)′),}

(A.6.4)

which is a key ingredient of the commutative diagram (A) of §9 for the case r < d. See (A.5.5)for ℤ/p^{•}(r − d)′. We define h^{r} as the composite morphism

ℚ ⊗ RΓ (\overline{Y} , 𝒮_{•}(r)_{(\overline{X},\overline{M})})

⟶^{(A.5.3)} ℚ ⊗ RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}})

∼ ⟶ ℚ ⊗ \{RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}) ⊗ ℤ/p^{•}(d − r)′ ⊗ ℤ/p^{•}(r − d)′\}

⟶ ℚ ⊗ \{RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}) ⊗ ℤ/p^{•}(r − d)′\} ,

where the last arrow is induced by the p^{d−r}-times of the composite map

ℚ ⊗ ℤ/p^{•}(d − r)′ ≃^{(1)} ℚ ⊗ H_{syn}^{0}((\overline{X}_{•}, \overline{M}_{•}), 𝒮_{•} (d − r))  (A.6.5)

↪^{(A.5.2)} ℚ ⊗ H_{crys}^{0}((\overline{X}_{•}, \overline{M}_{•})/W_{•})

(see (A.6.6)below for the isomorphism (1)) and the product of crystalline complexes. We define f^{r} as the morphism induced by (A.5.7) and the isomorphisms in (A.6.2). To define g^{r}, we need an isomorphism

f̃^{d} : ℚ ⊗ RΓ (\overline{Y} , 𝒮_{•} (d)_{(\overline{X},\overline{M})}) ≃ ℚ ⊗ RΓ (X\overline{K}, ℤ/p^{•}(d)′)  (A.6.6)

induced by η_{•}^{d} in Theorem A.5.6 (cf. [Ts1] Theorem 3.3.2 (1)). We define

h̃^{d} : ℚ ⊗ RΓ (\overline{Y} , 𝒮_{•} (d)_{(\overline{X},\overline{M})}) ⟶ ℚ ⊗ RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}})

in the same way as h^{r} (using (A.5.2) instead of (A.5.3)) and define g^{d} := p^{d} · (h̃^{d} ∘ (f̃^{d})^{−1}). Finally we define g^{r} as the composite of the natural map

RΓ (X\overline{K}, μ_{p^{•}}^{⊗r}) ⟶ RΓ (X\overline{K}, ℤ/p^{•}(d)′) ⊗ ℤ/p^{•}(r − d)′

and g^{d} ⊗ id. The above diagram is commutative by the definition of g^{r} and the compatibility of η_{•}^{r} with products (cf. [Ts1] §3.1). Now we prove

Theorem A.6.7 Assume that K is a p-adic local field (i.e., k is finite). Then the diagram (9.9.2)commutes for r < d.

Proof. We first note the isomorphisms

H_{cont-syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r)) ^{∼} ⟶ H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r)),
H_{cont}^{i+1}(X_{K}, ℚ_{p}(r)) ^{∼} ⟶ H^{i+1}(X_{K}, ℚ_{p}(r))

by the assumption that k is finite (cf. Proposition A.6.1 (2)). For integers m ≥ 0 and s < 0, put

H_{cG-cr}^{m}((X, M)/W; s) := R^{m}( \underleftarrow{lim} Γ_{Gal})\{RΓ (\overline{Y} , 𝔼_{(\overline{X}_{•},\overline{M}_{•})/W_{•}}) ⊗ ℤ/p^{•}(s)′\},
H_{cG-cr}^{m}((X, M)/W; s)_{ℚ_{p}}^{0} := Ker(H_{cG-cr}^{m}((X, M)/W; s)_{ℚ_{p}} → H_{crys}^{m}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(s)).

By (A.6.4), there is a commutative diagram

\xymatrix{H_{syn}^{i+1}((\overline{X}, \overline{M}), 𝒮_{ℚ_{p}}(r)) \ar[d]_{f^{r}} \ar@{-->}[dr]^{h^{r}} \\ H^{i+1}(X\overline{K}, ℚ_{p}(r))_{ℚ_{p}} \ar[r]^{g^{r}} & H_{crys}^{i+1}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d),}

where the bottom arrow is the same as β^{i+1,r}. By this diagram we obtain the arrows (g^{r})^{0} and (h^{r})^{0} in the following commutative diagram:

\xymatrix{H_{syn}^{i+1}((X, M), 𝒮_{ℚ_{p}}(r))^{0} \ar[d]_{(f^{r})^{0}} \ar@{-->}[dr]^{(h^{r})^{0}} \\ H_{cont}^{i+1}(X, ℚ_{p}(r))^{0} \ar[r]^{(g^{r})^{0}} \ar[d]_{edge} & H_{cG-cr}^{i+1}((X, M)/W; r − d)_{ℚ_{p}}^{0} \ar[d]^{edge} \\ H^{1}(K, H^{i}(X\overline{K}, μ_{p^{•}}^{⊗r}))_{ℚ_{p}} \ar[r]^{g^{r}} & H^{1}(K, H_{crys}^{i}((\overline{X}_{•}, \overline{M}_{•})/W_{•}) ⊗ ℤ/p^{•}(r − d))_{ℚ_{p}} \\ H^{1}(K, H^{i}(X\overline{K}, ℚ_{p}(r))) \ar[u]|{≀}^{c} \ar[r]^{β^{i,r}} & H^{1}(K, H_{crys}^{i}((\overline{X}, \overline{M})/W)_{ℚ_{p}}(r − d)), \ar[u]|{≀}_{c′}}

where the top triangle is induced by (A.6.4) and the central square commutes by the functoriality of Hochschild-Serre spectral sequences. The arrows c and c′ are canonical maps, and the bottom square commutes by the definitions of β^{i,r} and g^{r}. The map c is bijective by [J] Theorem 5.15 (c) and the finiteness of H^{i}(X\overline{K}, μ_{p^{n}}^{⊗r}) for n ≥ 1. The bijectivity of c′ is obtained from a similar argument as for the proof of Theorem A.4.2. Moreover it is easy to see that the composite of the left column agrees with the left vertical arrow of (9.9.2), and that the composite of h^{r} and the right column agrees with γ^{i,r} in (9.9.2). The commutativity in question follows from these facts. □

References

[AS] Asakura, M., Sato, K.: Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance (preliminary version). preprint 2013,

http://front.math.ucdavis.edu/1301.5829

[B1] Bloch, S.: Algebraic K-theory and crystalline cohomology. Inst. Hautes Études Sci. Publ. Math. 47, 187–268 (1977)

[B2] Bloch, S.: Algebraic cycles and higher K-theory. Adv. Math. 61, 267–304 (1986)

[B3] Bloch, S.: Algebraic cycles and the Beilinson conjectures. In: The Lefschetz Centennial Conference, (Contemp. Math. 58), Part I, pp. 65–79, Providence, Amer. Math. Soc., 1986

[BK1] Bloch, S., Kato, K.: p-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. 63, 107–152 (1986)

[BK2] Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. In: Cartier, P., Illusie, L., Katz, N. M., Laumon, G., Manin, Yu. I., Ribet, K. A. (eds.) The Grothendieck Festscherift I, (Progr. Math. 86), pp. 333–400, Boston, Birkhäuser, 1990

[BoK] Bousfield, A. K., Kan, D. M.: Homotopy limits, Completions and Localizations. Lecture Notes in Math. 304, Berlin, Springer, 1973

[dJ] de Jong, A. J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)

[D] Deligne, P.: La conjecture de Weil II. Inst. Hautes Études Sci. Publ. Math. 52, 313–428 (1981)

[DP] Dold, A., Puppe, D.: Homologie nicht-additiver Functoren Anwendungen. Ann. Inst. Fourier 11, 201–312 (1961)

[E] Ekedahl, T.: On the multiplicative properties of the de Rham-Witt complex I. Ark. Mat. 22, 185–238 (1984)

[Fo1] Fontaine, J.-M.: Sur certain types de représentations p-adiques du groupe de Galois d’un corps locaux; construction d’un anneau de Barsotti-Tate. Ann. of Math. 115, 529–577 (1982)

[Fo2] Fontaine, J.-M.: Représentations p-adiques semi-stables. In: Périodes padiques. Séminaire de Bures, 1988 (Astérisque 223), pp. 113–184, Marseille, Soc. Math. France, 1994

[Fr] Friedlander, E. M.: Etale Homotopy of Simplicial Schemes. Ann. of Math. Studies 104, Providence, Princeton Univ. Press, 1982 [FG] Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). In:

Usui, S., Green, M., Illusie, L., Kato, K., Looijenga, E., Mukai, S., Saito, S. (eds.) Algebraic Geometry, Azumino, 2001, (Adv. Stud. in Pure Math. 36), pp. 153–184, Tokyo, Math. Soc. Japan, 2002

[Fu] Fulton, W.: Intersection Theory, 2nd ed. Ergeb. Math. Grenzgeb. (3) 2, Berlin, Springer, 1988

[Ge] Geisser, T.: Motivic cohomology over Dedekind rings. Math. Z. 248, 773–794 (2004)

[GL] Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530, 55–103 (2001)

[Gi1] Gillet, H.: Riemann-Roch theorems for higher algebraic K-theory. Adv. Math. 40, 203–289 (1981)

[Gi2] Gillet, H.: Universal cycle class. Compositio Math. 49, 3–49 (1983)

[Gr1] Gros, M.: Classes de Chern et classes des cycles en cohomologie logarithmique. Mém. Soc. Math. Fr. (2) 21, 1–87 (1985)

[Gr2] Gros, M.: Régulators syntomiques et valeurs de fonction L p-adiques I, avec un appendice par Kurihara, M. Invent. Math. 99, 293-320 (1990)

[GrS] Gros, M., Suwa, N.: La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique. Duke Math. J. 57, 615-628 (1988)

[G] Grothendieck, A.: Théorie des classes de Chern. Bull. Soc. Math. France 86, 137–154 (1958)

[Ha] Hartshorne, R.: Residues and Duality. (Lecture Notes in Math. 20), Berlin, Springer, 1966

[Hy1] Hyodo, O.: A note on p-adic etale cohomology in the semi-stable reduction case. Invent. Math. 91, 543–557 (1988)

[Hy2] Hyodo, O.: On the de Rham-Witt complex attached to a semi-stable family. Compositio Math. 78, 241–260 (1991)

[HK] Hyodo, O., Kato, K.: Semi-stable reduction and crystalline cohomology with logarithmic poles. In: Périodes p-adiques. Séminaire de Bures, 1988 (Astérisque 223), pp. 221–268, Marseille, Soc. Math. France 1994

[I1] Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. École Norm. Sup. (4) 12, 501–661 (1979)

[I2] Illusie, L.: Autour du théorème de monodromie locale. In: Périodes padiques. Séminaire de Bures, 1988 (Astérisque 223), pp. 9–57, Marseille, Soc. Math. France, 1994 [J] Jannsen, U.: Continuous étale cohomology. Math. Ann. 280, 207–245 (1987)

[JSS] Jannsen, U., Saito, S., Sato, K.: Étale duality for constructible sheaves on arithmetic schemes. to appear in J. Reine Angew. Math.

[K1] Kato, K.: A Hasse principle for two-dimensional global fields. (with an appendix by Colliot-Thélène, J.-L.), J. Reine Angew. Math. 366, 142–183 (1986)

[K2] Kato, K.: On p-adic vanishing cycles (Application of ideas of Fontaine-Messing). In: Algebraic Geometry, Sendai 1985, (Adv. Stud. in Pure Math. 10), pp. 207–251, Tokyo, Kinokuniya, 1987

[K3] Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Igusa, J. (ed.) Algebraic Analysis, Geometry and Number Theory, pp. 191–224, Baltimore, The Johns Hopkins Univ. Press, 1988

[K4] Kato, K.: Semi-stable reduction and p-adic étale cohomology. In: Périodes p-adiques, Séminaire de Bures, 1988, (Astérisuque 223), pp. 269–293, Marseille, Soc. Math. France, 1994

[KSS] Kato, K., Saito, S., Sato, K.: p-adic vanishing cycles and p-adic étale Tate twists on generalized semistable families. in preparation

[KM] Katz, N., Messing, W.: Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math. 23, 93–97 (1974)

[La] Langer, A.: Local points of motives in semistable reduction. Compositio Math. 116, 189–217 (1999)

[LS] Langer A., Saito, S.: Torsion zero-cycles on the self-product of a modular elliptic curve. Duke Math. J. 85, 315–357 (1996)

[Mi1] Mieda, Y: Cycle class, Lefschetz trace formula and integrality for p-adic cohomology. In: Asada, M., Nakamura, H. and Takahashi, H. (eds.) Algebraic Number Theory and Related Topics 2007 (RIMS Kôkyûroku Bessatsu B12), pp. 57–66, Kyoto, RIMS, Kyoto Univ. 2009

[Mi2] Mieda, Y: Integrality of traces for p-adic cohomology. in preparation

[M] Milne, J. S.: Étale Cohomology. (Princeton Math. Ser. 33), Princeton, Princeton Univ. Press, 1980

[Mo] Mokrane, A.: La suite spectrale des poids en cohomologie de Hyodo-Kato. Duke Math. J. 72, 301–337 (1993)

[Na] Nakkajima, Y.: Signs in weight spectral sequences, monodromy-weight conjectures, log Hodge symmetry and degenerations of surfaces. Rend. Sem. Mat. Univ. Padova 116, 71–185 (2006)

[Ne1] Nekovář, J.: Syntomic cohomology and p-adic regulators. preprint, 1997

[Ne2] Nekovář, J.: p-adic Abel-Jacobi maps and p-adic height pairings. In: Gordon, B. B., Lewis, J. D., Müller-Stach, S., Saito, S., Yui, N. (eds.) The Arithmetic and Geometry of Algebraic Cycles, Banff, 1998, (CRM Proc. Lecture Notes 24), pp. 367–379, Providence, Amer. Math. Soc., 2000

[Ni1] Niziol, W.: On the image of p-adic regulators. Invent. Math. 127, 375–400 (1997)

[Ni2] Niziol, W.: Semistable conjecture via K-theory. Duke Math. J. 141, 151–178 (2008)

[Q] Quillen, D.: Higher algebraic K-theory I. In: Bass, H. (ed.) Algebraic K-theory I. (Lecture Notes in Math. 341), pp. 85–147 Berlin, Springer, 1973

[Sai] Saito, S.: Cohomological Hasse principle and motivic cohomology of arithmetic schemes. In: Bhatia, R., Pal, A., Rangarajan, G., Srinivas, V., Vanninathan, M. (eds.) Proc. Internat. Congr. Math. Hyderabad, 2010, pp. 558–585, Hindustan Book Agency, New Delhi, 2011

[SS] Saito, S., Sato, K.: A p-adic regulator map and finiteness results for arithmetic schemes. Documenta Math. Andrei A. Suslin’s Sixtieth Birthday, 525–594 (2010)

[Sa1] Sato, K.: Logarithmic Hodge-Witt sheaves on normal crossing varieties. Math. Z. 257, 707–743 (2007)

[Sa2] Sato, K.: p-adic étale Tate twists and arithmetic duality (with an appendix by Hagihara, K.). Ann. Sci. École Norm. Sup. (4) 40, 519–588 (2007)

[Sc] Schneider, P.: p-adic point of motives. In: Jannsen, U. (ed.) Motives, (Proc. Symp. Pure Math. 55-II), pp. 225–249, Providence, Amer. Math. Soc., 1994

[Sch] Scholl, A. J.: Integral elements in K-theory and products of modular curves. In: Gordon, B. B., Lewis, J. D., Müller-Stach, S., Saito, S., Yui, N. (eds.) The arithmetic and geometry of algebraic cycles, Banff, 1998, (NATO Sci. Ser. C Math. Phys. Sci., 548), pp. 467–489, Dordrecht, Kluwer, 2000.

[Sh] Shiho, A.: On logarithmic Hodge-Witt cohomology of regular schemes. J. Math. Sci. Univ. Tokyo 14, 567–635 (2007)

[So] Soulé, C.: Operations on étale K-theory. Applications. In: Dennis, R. K. (ed.) Algebraic K-theory, Oberwolfach, 1980, Part I. (Lecture Notes in Math. 966), pp. 271–303, Berlin, Springer, 1982

[SV] Suslin, A. A., Voevodsky, V.: Bloch-Kato conjecture and motivic cohomology with finite coefficients. In: Gordon, B. B., Lewis, J. D., Müller-Stach, S., Saito, S., Yui, N. (eds.) The Arithmetic and Geometry of Algebraic Cycles, Banff, 1998, (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 548), pp. 117–189, Dordrecht, Kluwer, 2000 [Ta] Tate, J.: Relations between K_{2} and Galois cohomology. Invent. Math. 36,

257–274 (1976)

[Ts1] Tsuji, T.: p-adic étale cohomology and crystalline cohomology in the semistable reduction case. Invent. Math. 137, 233–411 (1999)

[Ts2] Tsuji, T.: On p-adic nearby cycles of log smooth families. Bull. Soc. Math. France 128, 529–575 (2000)

[Vo1] Voevodsky, V.: Motivic cohomology with ℤ/2-coefficients. Inst. Hautes Études Sci. Publ. Math. 98, 59–104 (2003)

[Vo2] Voevodsky, V.: On motivic cohomology with ℤ/ℓ-coefficients. Ann. of Math. 174 401–438 (2011)

[We] Weibel, C.: The norm residue isomorphism theorem. J. Topol. 2, 346–372 (2009)

K. Sato

Department of Mathematics Chuo University 1-13-27 Kasuga, Bunkyo-ku 112-8552 Tokyo, Japan kanetomo sato@hotmail.com 248

Documenta Mathematica 18 (2013)