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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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