p-ADIC ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES

GO YAMASHITA

Contents

1. Introduction 2 2. The main theorems of p-adic Hodge theory 3 3. The main results 7 References 10

This text is a report of a talk “p-adic étale cohomology and crystalline cohomology for open varieties” in the symposium “Hodge Theory and Algebraic Geometry” (7-11/Oct/2002 at Hokkaido University). It seemed that more algebraic geometers rather than arithmeticians participated in this symposium. Thus, in that talk, the auther began with an introduction to the p-adic Hodge theory in the view of the theory of p-adic representations. However, in this report, we do not treat the theory of p-adic representations, and we treat only a review of the main theorems and the main results.

The aim of the talk was, roughly speaking, “to extend the main theorems of p-adic Hodge theory for open varieties” by the method of Fontaine-Messing-Kato-Tsuji (see [FM],[Ka2], and [Tsu1]). Here, the main theorems of p-adic Hodge theory are: the Hodge-Tate conjecture (C_{HT} for short), the de Rham conjecture (C_{dR}), the crystalline conjecture (C_{crys}), the semi-stabele conjecture (C_{st}), and the potentially semi-stable conjecture (C_{pst}). The theorems C_{dR}, C_{crys}, and C_{st} are called the “comparison theorems”.

The section 1 is an introduction to the p-adic Hodge theory. In the section 2, we review the main theorems of the p-adic Hodge theory. In the section 3, we state the main results. In this report, we only announce the main results.

The auther thanks to Takeshi Saito, Takeshi Tsuji, Seidai Yasuda for helpful discussions. He thanks to Atsushi Shiho, Yukiyoshi Nakkajima for suggesting weight filtrations. Finally, he also thanks to the organizers of the symposium Sanpei Usui, Daisuke Matsushita, Masanori Asakura for giving me an occasion of the talk.

Notations Let K be a complete discrete valuation field of characteristic 0, k the residue field of K, perfect, characteristic p > 0, and O_{K} the valuation ring of K. Denote \overline{K} be the algebraic closure of K, \overline{k} the algebraic closure of k, G_{K} the absolute Galois group of K, and ℂ_{p} the p-adic completion of \overline{K}. (Note that it is an abuse of the notation. If [K : ℚ_{p}] < ∞, it coincide the usual notations.) Let W be the ring of Witt vectors with coefficient in k, and K_{0} the fractional field of W. It is the maximum absolutely unramified (i.e., p is a uniformizer in K_{0}) subfield of K. Let P_{0} be the fractional field of the ring of Witt vectors with coefficient of \overline{k}, and σ the Frobenius endmorphism on W, K_{0}, W (\overline{k}), and P_{0}, indeced by the absolute Frobenius on k, and \overline{k}. The word “log-structure” means Fontaine-Illusie-Kato’s log-structure (see [Ka1]). We do not review the notion of log-structure in this report.

Date: Dec/2002.

1. Introduction

The p-adic Hodge theory is called to be a p-adic analogue of the Hodge theory over ℂ. This means that for p-adic étale cohomologies of varieties over p-adic field, there exist similar decompositions, which is called the Hodge-Tate decomopositon to the Hodge decompositions for singular cohomologies of varieties over ℂ. However, we can say that the p-adic Hodge theory is not just a p-adic analogue of the Hodge decomposition.

The p-adic Hodge theory has no logical relations with the Hodge theory over ℂ. One can learn the p-adic Hodge theory without knowing the Hodge theory over ℂ, however, we begin with comparing with the Hodge theory over ℂ to see the conceptual relation.

The following theorem is classical.

Theorem 1.1 (Hodge decomposition (Kodaira-Hodge)). For compact Kähler manifold X, there exists a canonical isomorphism ：

ℂ ⊗_{ℚ} H_{sing}^{m}(X, ℚ) ≅ H_{dR}^{m}(X/ℂ) ≅ H^{m}(X, O_{X}) ⊕ H^{m−1}(X, Ω_{X/ℂ}^{1}) ⊕ ⋯ ⊕ H^{0}(X, Ω_{X/ℂ}^{m})

(H_{sing}^{m} means singular cohomology.)

On the other hand, one of the conclusion of p-adic Hodge theory is : There exists the following decomposition, which is called Hodge-Tate decomposition for a p-adic étale cohomology of a variety X, which is proper smooth over K. Former, it was called the Hodge-Tate conjecture, C_{HT} for short.

\begin{matrix}ℂ_{p} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ \\ ℂ_{p} ⊗_{K} H^{m}(X, O_{X}) ⊕ ℂ_{p}(−1) ⊗_{K} H^{m−1}(X, Ω_{X/K}^{1}) ⊕ ⋯ ⊕ ℂ_{p}(−m) ⊗_{K} H^{0}(X, Ω_{X/K}^{m}). \end{matrix}

Moreover, this is compatible with the action of the Galois group G_{K}. ℂ_{p}(−i) means the (−i)-th Tate twist of ℂ_{p}.

In the classical Hodge theory, we relate topological cohomologies, that is, singular cohomologies with analytic cohomologies, that is, de Rham cohomologies (it is only the holomorphic Poincaré lemma, not “Hodge theory”). The singular cohomology has ℚ-structure (or ℤ-structure), and the de Rham cohomology has Hodge filtration, which comes from Hodge decomposition (this is the “Hodge theory”).

For example, we can not distinguish elliptic curves by using only singular cohomologies (they are topologically homeomorphic), and by using only de Rham cohomologies (Fil^{0} = whole space, Fil^{1} = 1-dimensional, Fil^{2} = 0). However, we can recover an elliptic curve by using the both cohomologies:

0 → H_{1}(E, ℤ) → Lie(E) → E(ℂ) → 0,
0 → coLie(E^{∗}) → H_{dR}^{1}(X/ℂ)^{∗} → Lie(E) → 0.

p-ADIC ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES 3

By such a way, we can get deeper information from a comparison isomorphism of two cohomology theories and additional structures of cohomology theories.

The p-adic Hodge theory treats varieties over p-adic field, and it compares topological cohomologies, that is, étale cohomologies and analytic cohomologies, that is, de Rham cohomologies and (log-)crystalline cohomologies. By using p-adic Hodge theory, which relates étale cohomologies with differential forms, we can formulate a conjecture (Tamagawa number conjecture of Bloch-Kato), which precisely predicts special values of Hasse-Weil Lfunctions of varieties (or, motives). That conjecture is not the theme of this report, thus we do not further mention that.

The p-adic Hodge theory compares cohomology theories with additional structures, that is, Galois actions, Hodge filtrations, Frobenius endmorphisms, Monodoromy operators:

1. the Hodge theory over ℂ

• singular cohomology H_{sing}^{m}(X, ℚ) —topological:

ℚ-vector space (+ℤ-structure)

• de Rham cohomology H_{dR}^{m}(X/ℂ) —analytic:

ℂ-vector space +Hodge filtration

2. the p-adic Hodge theory

• étale cohomology H_{ét}^{m}(X\overline{K}, ℚ_{p}) —topological:

ℚ_{p}-vector space +Galois action

• (algebraic) de Rham cohomology H_{dR}^{m}(X_{K}/K) —analytic: K-vector space +Hodge filtration

• (log-)crystalline cohomology K_{0} ⊗_{W} H_{crys}^{m}(Y/W) —analytic:

K_{0}-vector space +Frobenius endmorphism (+ Monodromy operator). For arithmetic geometers, the singular cohomology is called the Betti cohomology. In the proof of the comparison theorems, we use the “syntomic cohomology”. This is a vector space endowed with the Galois action. However, being different from the étale cohomology it is an analytic cohomology defined by differential forms. It is the theoritical heart of the p-adic Hodge theory by the method of Fontaine-Messing-Kato-Tsuji that the syntomic cohomology is isomorphic to the étale cohomology compatible with Galois action.

2. The main theorems of p-adic Hodge theory

In this section, we state the main theorems of p-adic Hodge theory: C_{HT}, C_{dR}, C_{crys}, C_{st}, and C_{pst}. Roughly spealing, we can state the main theorems as the following way:

• the Hodge-Tate conjecture (C_{HT}): There exists a Hodge-Tate decomposition on the p-adic étale cohomology.

• the de Rham conjecture (C_{dR}): There exists a comparison isomorphism between the p-adic étale cohomology and the de Rham cohomology.

• the crystalline conjecture (C_{crys}): In the good reduction case, we have stronger result than C_{dR}, that is, there exists a comparison isomorphism between the p-adic étale cohomology and the crystalline cohomology.

• the semi-stable conjecture (C_{st}): In the semi-stable reduction case, we have stronger result than C_{dR}, that is, there exists a comparison isomorphism between the p-adic étale cohomology and the logcrystalline cohomology.

• the potentially semi-stable conjecture (C_{pst}): The p-adic étale cohomology has “only a finite monodromy”.

We will state precisely in the following. In the p-adic Hodge theory, we use Fontaine’s p-adic period rings: B_{dR}, B_{crys}, B_{st} (see

[Fo]). In the Hodge theory over ℂ, we can compare the singular cohomology and de Rham cohomology after tensoring ℂ. On the other hand, in the p-adic Hodge theory, we can compare the étale cohomology and the de Rham cohomoloy after tensoring B_{dR}. We pick up the fundamental properties of them.

1. B_{dR} ： a complete discrete valuation field over K with residue field ℂ_{p}. It contains \overline{K}. (It does not contain ℂ_{p}.) The Galois group G_{K} acts on B_{dR}. It has the filtration by its valuation, and its graded quotient gr^{i}B_{dR} is ℂ_{p}(i). And, ℚ_{p}(1) ⊂ Fil^{1}B_{dR}.

B_{dR}^{G_{K}} = K.

2. B_{crys} ： It is an algebra over K_{0}, and G_{K}-stable subring of B_{dR}. It contains P_{0}.(It does not contain \overline{K}.) K ⊗_{K_{0}} B_{crys} → B_{dR} is also injective. For the filtration, which comes from B_{dR}, we get gr^{i}B_{crys} = ℂ_{p}(i). And, ℚ_{p}(1) ⊂ Fil^{1}B_{dR} ∩ B_{crys}. There exists a σ-semi-linear injective endomorphism φ, which commutes with the action of G_{K}. (Frobenius endmorphism)

B_{crys}^{G_{K}} = K_{0}, Fil^{0}B_{dR} ∩ B_{crys}^{φ=1} = ℚ_{p}.

3. B_{st} ： It is an algebra over K_{0}, and has G_{K}-action. It contains B_{crys}. It contains P_{0}.(It does not contain \overline{K}.) After fixing an uniformizer π of K, we can regard it as a subring of B_{dR}. K ⊗_{K_{0}} B_{st} → B_{dR} is also injective. The Frobenius endmorphism on B_{crys} is extended to B_{st}. The ring B_{st} has a B_{crys}-derivation N : B_{st} → B_{st}, which commutes with the G_{K}-action and satisfies Nφ = pφN.

B_{st}^{G_{K}} = K_{0}, B_{st}^{N=0} = B_{crys}, Fil^{0}B_{dR} ∩ B_{st}^{φ=1,N=0} = ℚ_{p}.

The following theorems were formulated by Tate, Fontaine, Jannsen, proved by Tate, Faltings, Fontaine-Messing, Kato under various assumptions, and proved by Tsuji under no assumptions (1999 [Tsu1]). Later, Faltings and Niziol got alternative proofs (see [Fa],[Ni]).

Theorem 2.1 (the Hodge-Tate conjecture (C_{HT})). Let X_{K} be a proper smooth variety over K. Then, there exists the following canonical isomorphism, which is compatible with the Galois action.

ℂ_{p} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ ⨁_{0≤i≤m} ℂ_{p}(−i) ⊗_{K} H^{m−i}(X_{K}, Ω_{X_{K}/K}^{i}).

Here, G_{K} acts by g ⊗ g on LHS, by g ⊗ 1 on RHS.

remark . This is an analogue of the Hodge decomopositon. In this isomorphism, the following fact is remarkable: In general, it is very difficult to know the action of Galois group on the étale cohomology. However, afer tensoring ℂ_{p}, the Galois action is very easy:

⨁_{0≤i≤m} ℂ_{p}(−i)^{⊕h^{i,m−i}}

p-ADIC ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES 5

(h^{i,m−i} := dim_{K} H^{m−i}(X, Ω_{X/K}^{i}).)

Theorem 2.2 (the de Rham conjecture (C_{dR})). Let X_{K} be a proper smooth variety over K. Then, there exists the following canonical isomorphism, which is compatible with the Galois action and filtrations.

B_{dR} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K).

Here, G_{K} acts by g ⊗ g on LHS, by g ⊗ 1 on RHS. We endow filtrations by Fil^{i} ⊗ H_{ét}^{m} on LHS, by Fil^{i} = Σ_{i=j+k}Fil^{j} ⊗ Fil^{k} on RHS.

remark . By takin graded quotient, we get C_{dR}⇒C_{HT}.

Theorem 2.3 (the crystalline conjecture (C_{crys})). Let X_{K} be a proper smooth variety over K, X be a proper smooth model of X_{K} over O_{K}. Y be the special fiber of X.

Then, there exists the following canonical isomorphism, which is compatible with the Galois action, and Frobenius endmorphism.

B_{crys} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{crys} ⊗_{W} H_{crys}^{m}(Y/W)

Moreover, after tensoring B_{dR} over B_{crys}, and using the Berthelo-Ogus isomorphism (see [Be]):

K ⊗_{W} H_{crys}^{m}(Y/W) ≅ H_{dR}^{m}(X_{K}/K),

we get an isomorphism:

B_{dR} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K),

which is compatible with filtrations. Here, G_{K} acts by g ⊗ g on LHS, by g ⊗ 1 on RHS, Frobenius endmorphism acts by φ ⊗ φ on LHS, by φ ⊗ 1 on RHS. We endow filtrations by Fil^{i} ⊗ H_{ét}^{m} on LHS, by Fil^{i} = Σ_{i=j+k}Fil^{j} ⊗ Fil^{k} on RHS.

remark . By taking the Galois invariant part of the comparison isomorphism:

B_{crys} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{crys} ⊗_{W} H_{crys}^{m}(Y/W),

we get:

(B_{crys} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}))^{G_{K}} ≅ K_{0} ⊗_{W} H_{crys}^{m}(Y/W).

By taking Fil^{0}(B_{dR} ⊗_{B_{crys}} •) ∩ (•)^{φ=1} of the comparison isomorphism, we get:

H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ Fil^{0}(B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K)) ∩ (B_{crys} ⊗_{W} H_{crys}^{m}(Y/W))^{φ=1}.

We can, that is, recover the crystalline cohomology & de Rham cohomology from the étale cohomology and vice versa with all additional strucuture. (Grothendieck’s mysterious functor.)

Theorem 2.4 (the semi-stable conjecture (C_{st})). Let X_{K} be a proper smooth variety over K, X be a proper semi-stable model of X_{K} over O_{K}. (i.e., X is regular and proper flat over O_{K}, its general fiber is X_{K} and its special fiber is normal crossing divisor.) Let Y be the special fiber of X, and M_{Y} be a natural log-structure on Y.

Then, there exists the following canonical isomorphism, which is compatible with the Galois action, and Frobenius endmorphism, monodromy operator.

B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{W} H_{log-crys}^{m}((Y, M_{Y})/(W, O^{×}))

Moreover, after tensoring B_{dR} over B_{st}, and using the Hyodo-Kato isomorphism (see [H-Ka]) (it depens on the choice of the uniformizer pi of K):

K ⊗_{W} H_{log-crys}^{m}((Y, M_{Y})/(W, O^{×})) ≅ H_{dR}^{m}(X_{K}/K)

we get an isomorphism:

B_{dR} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K)

which is compatible with filtrations. Here, G_{K} acts by g ⊗ g on LHS, by g ⊗ 1 on RHS, Frobenius endmorphism acts by φ ⊗ φ on LHS, by φ ⊗ 1 on RHS, monodromy operator acts by N ⊗ 1 on LHS, by N ⊗ 1 + 1 ⊗ N on RHS. We endow filtrations by Fil^{i} ⊗ H_{ét}^{m} on LHS, by Fil^{i} = Σ_{i=j+k}Fil^{j} ⊗ Fil^{k} on RHS.

remark . By taking the Galois invariant part of the comparison isomorphism:

B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{W} H_{log-crys}^{m}((Y, M_{Y})/(W, O^{×}))

we get:

(B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}))^{G_{K}} ≅ K_{0} ⊗_{W} H_{log-crys}^{m}((Y, M_{Y})/(W, O^{×}))

By taking Fil^{0}(B_{dR} ⊗_{B_{st}} •) ∩ (•)^{φ=1,N=0} of the comparison isomorphism, we get:

H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ Fil^{0}(B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K)) ∩ (B_{st} ⊗_{W} H_{log-crys}^{m}((Y, M_{Y})/(W, O^{×})))^{φ=1,N=0}

We can, that is, recover the log-crystalline cohomology & de Rham cohomology from the étale cohomology and vice versa with all additional strucuture. (Grothendieck’s mysterious functor.)

remark . From B_{st}^{N=0} = B_{crys}, we get C_{st}⇒C_{crys}.

remark . By using de Jong’s alteration(see [dJ]), we get C_{st}⇒C_{dR}. We need a slight argument to showing that it is compatible not only with the action of Gal(\overline{K}/L) for a suitable finite extention L of K, but also with the aciton of G_{K}. (see [Tsu4])

In the following theorem, we do not review the definition of the potentially semi-stable representation.

Theorem 2.5 (the potentially semi-stable conjecture (C_{pst})). Let X_{K} be a proper variety over K. Then, the p-adic étale cohomology H_{ét}^{m}(X\overline{K}, ℚ_{p}) is a potentially semi-stable representation of G_{K}.

remark . By using de Jong’s alteration （see [dJ]） and truncated simplicial schemes, we get C_{st}⇒C_{pst}. (see [Tsu3])

The logical dependence is the following:

C_{pst} ⇐ C_{st} ⇒ C_{crys}, C_{st} ⇒ C_{dR} ⇒ C_{HT}.

C_{st} ⇒ C_{crys} and C_{dR} ⇒ C_{HT} are trivial. For C_{st} ⇒ C_{dR}, we use de Jong’s alteration. For C_{st} ⇒ C_{pst}, we use de Jong’s alteration and truncated simplicial scheme. i.e., C_{st} is the deepest theorem.

p-ADIC ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES 7

3. The main results

In this section, we state the main results without proof (see [Y]). In this report, we do not mention weight filtrations and functoriarities.

We call C_{HT}(resp. C_{dR}, C_{crys}, C_{st}, C_{pst}) in the previous section proper smooth C_{HT}(resp. proper smooth C_{dR}, proper C_{crys}, proper C_{st}, proper C_{pst}). Roughly speaking, we remove conditions of the main theorems in the following way.

1. proper smooth C_{HT}⇝ open non-smooth C_{HT} X_{K} is separated of finite type over K. Or, “open” smooth C_{HT} X_{K} can be compactified into a proper smooth variety over K, such that its complement is a normal crossing divisor.

2. proper smooth C_{dR}⇝ open non-smooth C_{dR} X_{K} is separated of finite type over K. Or, “open” smooth C_{dR} X_{K} can be compactified into a proper smooth variety over K, such that its complement is a normal crossing divisor.

3. proper C_{crys}⇝ “open” C_{crys} X can be compactified into a proper smooth variety over O_{K}, such that its complement is a horizontal normal crossing divisor, which is also normal crossing to the special fiber.

4. proper C_{st}⇝ “open” C_{st} X can be compactified into a proper semi-stable family over O_{K}, such that its complement is a horizontal normal crossing divisor, which is also normal crossing to the special fiber.

5. proper C_{pst}⇝ open non-smooth C_{pst} X_{K} is separated of finite type over K.

In the above, the word open means arbitrary open, on the other hand, the word “open” means “proper minus normal crossing divisor”.

We consider cohomologies with proper support H_{c}^{m} and cohomologies without proper support H^{m}. Moreover, we can consider “partially proper support cohomologies” in “open” smooth cases: If we decompose the normal crossing divisor D into D = D^{1} ∪ D^{2}, “partially proper support cohomologies” are cohomologies with support only on D^{i} (i = 1, 2). We denote them H_{i}^{m} (i = 1, 2), that is,

H_{ét,1}^{m}((X \ D)\overline{K}, ℚ_{p}) := H_{ét}^{m}(X\overline{K}, Rj_{2∗}j_{1!}ℚ_{p}),
H_{ét,1}^{m}((X \ D)\overline{K}, ℚ_{p}) := H_{ét}^{m}(X\overline{K}, Rk_{1∗}k_{2!}ℚ_{p}).

(Here, j_{1} : (X \ D)\overline{K} ↪ (X \ D^{2})\overline{K}, j_{2} : (X \ D^{2})\overline{K} ↪ X\overline{K}, k_{1} : (X \ D^{1})\overline{K} ↪ X\overline{K}, and k_{2} : (X \ D)\overline{K} ↪ (X \ D^{1})\overline{K}.)

H_{dR,i}^{m}((X \ D)_{K}/K) := H^{m}(X_{K}, I(D^{i})Ω_{X_{K}/K}(logD)).

H_{log-crys,i}^{m}(Y \ C) := K_{0} ⊗_{W} H_{log-crys,i}^{m}((Y, M_{Y})/(W, O^{×}), K(C^{i})O_{(Y,M_{Y} )/(W,O^{×})}). Here, Y (resp. C, C^{i}) are the special fiber of X(resp. D, D^{i}), and I(D^{i})(resp. K(D^{i})) are the ideal sheaf of O_{X}(resp. O_{(Y,M_{Y} )/(W,O^{×})}) defined by D^{i}(resp. C^{i}) (see [Tsu2]). They are called the “minus log”.

When we consider algebraic correspondences on open varieties, we need to consider partially proper support cohomologies. Thus, in a sense, when we consider not only a comparison between varieties but also a comparison of Hom, we have to consider partially proper support cohomologies. In this way, it is impotant to show comparison isomorphisms for partially proper support cohomologies.

We state the main result.

First, we prove a extended version of Hyodo-Kato isomorphism:

Proposition 3.1. Let X be a proper semi-stable model over O_{K}, D be a horizontal normal crossing divisor of X, which is also normal crossing to the special fiber. We decompose D into D = D^{1} ∪D^{2}. Put Y (resp. C) to be the special fiber of X(resp. D). Fix a uniformizer pi of K. Then, we have the following isomorphism:

K ⊗_{K_{0}} H_{log-crys,i}^{m}(Y \ C) ≅ H_{dR,i}^{m}((X \ D)_{K}/K).

Thus, the pair

(H_{log-crys,i}^{m}(Y \ C), H_{dR,i}^{m}((X \ D)_{K}/K))

has a filtered (φ, N)-module structure.

The main result is the following:

Theorem 3.2 (“open” C_{st}). Let X be a proper semi-stable model over O_{K}, D be a horizontal normal crossing divisor of X, which is also normal crossing to the special fiber. We decompose D into D = D^{1} ∪ D^{2}. Put Y (resp. C) to be the special fiber of X(resp. D). Then, for i = 1, 2, we have the following canonical B_{st}-linear isomorphism:

B_{st} ⊗_{ℚ_{p}} H_{ét,i}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{K_{0}} H_{log-crys,i}^{m}(Y \ C)

Here, that is compatible the additional structures equipped by the following table:

B_{st} ⊗_{ℚ_{p}}H_{ét,i}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅  B_{st} ⊗_{K_{0}}H_{log-crys,i}^{m}(Y \ C)
Gal  g ⊗g  g ⊗1
Frob  φ ⊗1  φ ⊗φ
Monodromy N ⊗1  N ⊗ 1 +1 ⊗ N
\begin{matrix}Fil & i & after \\ B_{dR}⊗_{B_{st}} \end{matrix} \} Fil^{i} ⊗H_{ét}^{m}  ∑_{i=j+k}Fil^{j} ⊗Fil^{k}

Moreover, this is compatible with product structures.

In particular, if D^{1} = ϕ, then we get

B_{st} ⊗_{ℚ_{p}} H_{ét}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{K_{0}} H_{log-crys}^{m}(Y \ C),
B_{st} ⊗_{ℚ_{p}} H_{ét,c}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{K_{0}} H_{log-crys,c}^{m}(Y \ C).

remark . A proof for cohomologies with proper support (H_{c}) in the case of D^{2} = ∅ and D is simple normal crossing was given by T. Tsuji in personal conversations. That proof asserts there exist a comparison isomorphism of H_{c}’s. Taking dual, we get the comparison isomorphism of H’s, but we can not verify that the isomorphism is the one which has constructed in [Tsu2], because the proof neglects product structures. Later, he also gave an alternative proof for cohomologies without support (H) in the case of D^{2} = ∅ and D is simple normal crossing, by removing smooth divisors one by one (see [Tsu5]). That proof asserts there exist a comparison isomorphism of H’s. Taking dual, we get

p-ADIC ÉTALE COHOMOLOGY AND CRYSTALLINE COHOMOLOGY FOR OPEN VARIETIES 9

the comparison isomorphism of H_{c}’s, but we can not verify that the isomorphism is the one which has constructed in the above personal conversations, because the proof neglects product structures.

Anyway, we want to construct comparison maps of H and H_{c} (more generally, H_{1} and H_{2}), which is compatible with product structures, and to show the comparison maps are isomorphism.

From this “open”C_{st}, by the similar argument of

C_{pst} ⇐ C_{st} ⇒ C_{crys}, C_{st} ⇒ C_{dR} ⇒ C_{HT}

in the previous section, we can extend C_{HT}, C_{dR}, C_{crys}, and C_{pst}.

The “open”C_{crys}is immediately deduced from the “open”C_{st}.

Theorem 3.3 (“open”C_{crys}). Let X be a proper smooth model over O_{K}, D be a horizontal normal crossing divisor of X, which is also normal crossing to the special fiber. We decompose D into D = D^{1} ∪ D^{2}. Put Y (resp. C) to be the special fiber of X(resp. D). Then, for i = 1, 2, we have the following canonical B_{st}-linear isomorphism, which is compatible with the Galois actions, the Frobenius endmorphisms, the filtrations after tensoring B_{dR} over B_{crys}:

B_{st} ⊗_{ℚ_{p}} H_{ét,i}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅ B_{st} ⊗_{K_{0}} H_{log-crys,i}^{m}(Y \ C)

By de Jong’s alteration and truncated simplicial scheme argument (see [Tsu3]), we can deduce the open non-smooth C_{dR}from the “open”C_{st}. Here, in the case of open non-smooth, we use the de Rham cohomology of (Deligne-)Hartshorne. (see [Ha1][Ha2])

Theorem 3.4 (open non-smooth C_{dR}). Let X_{K} be a separated variety of finite type over K. Then, we have the following canonical isomorphism, which is compatible with the Galois actions and filtrations:

B_{dR} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR}^{m}(X_{K}/K)
B_{dR} ⊗_{ℚ_{p}} H_{ét,c}^{m}(X\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR,c}^{m}(X_{K}/K).

In the case of “open” smooth, we can consider partially proper support cohomologies by de Jong’s alteration and diagonal class argument (see [Tsu4]).

Theorem 3.5 (“open” C_{dR}). Let X_{K} be a proper smooth variety over K, and D_{K} be a normal crossing divisor of X_{K}. We decompose D into D_{K} = D_{K}^{1} ∪ D_{K}^{2}. Then, for i = 1, 2, we have the following canonical isomorphism, which is compatible with the Galois actions and filtrations:

B_{dR} ⊗_{ℚ_{p}} H_{ét,i}^{m}((X \ D)\overline{K}, ℚ_{p}) ≅ B_{dR} ⊗_{K} H_{dR,i}^{m}((X \ D)_{K}/K)

By taking graded quotient, we can deduce the open non-smooth C_{HT} from the open non-smooth C_{dR}. However, the Hodge-Tate decomposition of the open non-smooth C_{HT} is a formal decomposition, and it relates cohomologies of the sheaf of differential forms only in the “open” smooth case.

Theorem 3.6 (open non-smooth C_{HT}). Let X_{K} be a separated variety of finite type over K. Then, we have the following canonical isomorphism, which is compatible with the Galois actions:

ℂ_{p} ⊗_{ℚ_{p}} H_{ét}^{m}(X\overline{K}, ℚ_{p}) ≅ ⨁_{−∞≪i≪∞} ℂ_{p}(−i) ⊗_{K} gr^{i}H_{dR}^{m}(X_{K}/K)

ℂ_{p} ⊗_{ℚ_{p}} H_{ét,c}^{m}(X\overline{K}, ℚ_{p}) ≅ ⨁_{−∞≪i≪∞} ℂ_{p}(−i) ⊗_{K} gr^{i}H_{dR,c}^{m}(X_{K}/K).

Theorem 3.7 (“open” C_{HT}). Let X_{K} be a proper smooth variety over K. and D_{K} be a normal crossing divisor of X_{K}. We decompose D into D_{K} = D_{K}^{1} ∪ D_{K}^{2}. Then, for i = 1, 2, we have the following canonical isomorphism, which is compatible with the Galois actions:

ℂ_{p} ⊗_{ℚ_{p}} H_{ét,i}^{m}(X\overline{K}, ℚ_{p}) ≅ ⨁_{0≤j≤m} ℂ_{p}(−j) ⊗_{K} H^{m−j}(X_{K}, I(D^{i})Ω_{X_{K}/K}^{j}(logD)).

By de Jong’s alteration and truncated simplicial scheme argument (see [Tsu3]), we can deduce the open non-smooth C_{pst} from the “open”C_{st}:

Theorem 3.8 (open non-smooth C_{pst}). Let X_{K} be a separated variety of finite type over K. Then, the p-adic étale cohomologies H_{ét}^{m}(X\overline{K}, ℚ_{p}), H_{ét,c}^{m}(X\overline{K}, ℚ_{p}) are potentially semistable representations.

Finally, we mention with a few words about the proof of the main result (“open” C_{st}). In the method of Fontaine-Messing-Kato-Tsuji, we use the intermediate cohomology “syntomic cohomology” (see [FM][Ka2][Tsu1]). In the open case, we find difficulties in making product structures. To make product structures, we consider “bettari-log” schemes. (By the Japanese word “bettari”, we image that the log-structure is spread on the whole scheme.) However, log-crystalline cohomologies for these “bettari-log” schemes are in general infinite dimensional. Thus, we overcome difficulties by finding a modified crystalline sheaf, whose log-crystalline cohomology is finite dimensional. We construct a spectral sequence relating (étale, log-crystalline, and syntomic) cohomologies of the open variety with (étale, log-crystalline, and syntomic) cohomologies of the “bettari-log” schemes, and a spectral sequence relating (étale, log-crystalline, and syntomic) cohomologies of the “bettari-log” schemes with (étale, log-crystalline, and syntomic) cohomologies of logsmooth schemes. By using these ingredients, we finish the proof.

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