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$$p$$-torsion étale cohomology and crystalline cohomology in semi-stable reduction. (Cohomologie étale de $$p$$-torsion et cohomologie cristalline en réduction semi-stable.) (French) Zbl 0961.14010
Let $$K$$ be a $$p$$-adic field (e.g. a finite extension of $$\mathbb{Q}_p)$$ and $$X_K$$ be a proper and smooth scheme over $$\text{Spec}(K)$$. There are two $$p$$-adic cohomologies on $$X_K$$: the étale $$p$$-adic cohomology $$H^*_{ \text{ét}} (X_{\overline K}, \mathbb{Q}_p)$$ and the de Rham cohomology $$H^*_{ \text{dR}} (X_K)$$. Grothendieck sensed the existence of a mysterious functor between these two invariants. Using the rings of Fontaine $$B_{\text{cris}}$$, $$B_{\text{st}}$$ and $$B_{\text{dR}}$$, comparison theorems were conjectured by Fontaine and shown by Fontaine and Messing, Faltings, Kato and Tsuji.
We have $$H^*_{\text{ét}}(X_{\overline K},\mathbb{Q}_p)= \mathbb{Q}_p \otimes_{\mathbb{Z}_p} \varprojlim H^*((X_{\overline K})_{\text{ét}},\mathbb{Z}/p^n \mathbb{Z})$$. The author studies the $$p$$-torsion étale cohomology $$H^*((X_{\overline K})_{\text{ét}}, \mathbb{Z}/p^n \mathbb{Z})$$ where $$K$$ is the field of quotients of a Witt vector ring $$W$$ with coefficients in a perfect field of characteristic $$p>0$$. J.-M. Fontaine and W. Messing [in: Current trends in arithmetical algebraic geometry, Proc. Summer. Res. Conf., Arcata 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)] showed that if $$X_K$$ has good reduction for $$i\in \{0,\dots, p-2\}$$ there is a comparison theorem between $$H^i((X_{\overline K})_{\text{ét}}, \mathbb{Z}/p^n \mathbb{Z})$$ and $$H^i_{\text{dR}}(X/p^n)$$. They used the construction of crystalline representations of J.-M. Fontaine and G. Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15, 547-608 (1982; Zbl 0579.14037)] and the computation of the vanishing $$p$$-adic cycles by Kato. So $$H^i_{\text{ét}}(X_{K_0}, \mathbb{Q}_p)$$ is crystalline for $$i\in\{0,\dots,p-2\}$$.
Fontaine and Laffaille constructed abelian categories $$\underline {MF}^{f, i}_{\text{tor}}$$ of filtered $$W$$-modules of finite length with a Frobenius for $$0\leq i\leq p-2$$ and a functor $$V_{\text{cris}}$$ to the category of $$\mathbb{Z}_p$$-modules of finite length with an action of $$\text{Gal} (\overline K/K)$$. If $$X$$ is a proper and smooth model of $$X_K$$ then $$H^i_{\text{dR}}$$ is an object of $$\underline {MF}^{f,i}_{\text{tor}}$$ for $$0\leq i\leq p-2$$. If $$V$$ is $$\mathbb{Z}_p$$-module and $$V^\wedge= \operatorname{Hom}_{\mathbb{Z}_p} (V,\mathbb{Q}_p/ \mathbb{Z}_p)$$ then there is an isomorphism of Galois modules from $$V_{\text{cris}}(H^i_{\text{dR}}(X/p^n))$$ to $$H^i((X_{\overline K})_{\text{ét}}, \mathbb{Z}/p^n \mathbb{Z})^\wedge$$. So $$H^i_{ \text{ét}} (X_{\overline K_0}, \mathbb{Q}_p)$$ is crystalline for $$0\leq i\leq p-2$$.
In this paper, the author generalizes these results to the case where $$X_K$$ has a semi-stable reduction over $$W$$. The proof uses the generalization of the theory of Fontaine and Lafaille given by C. Breuil [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 3, 281-327 (1998; Zbl 0907.14006)] and his computation on the log-syntomic site [Bull. Soc. Math. Fr. 124, No. 4, 587-647 (1996; Zbl 0865.19004)], a generalization of the Deligne-Illusie isomorphism and the computation by Kato, Hyodo and Tsuji of the vanishing $$p$$-adic cycles in the semi-stable reduction case.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14G20 Local ground fields in algebraic geometry
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##### References:
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