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A note on p-adic étale cohomology. (English) Zbl 0647.14006

Let X be a projective smooth scheme over a complete discrete valuation ring of mixed characteristics (0,p). Let \(i: X_ s\hookrightarrow X\) and \(j: X_{\eta}\to X\) be the canonical morphisms of the special and generic fibre as usual. In Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)], J.-M. Fontaine and W. Messing defined the syntomic site \(X_{syn}\) and a sheaf \(S^ r_ n\) on \(X_{syn}\) in order to link the étale cohomology to the de Rham cohomology. The aim of this paper is a local study of the p- adic étale vanishing cycles \(i^*Rj_*{\mathbb{Z}}/p^ n(r)\). The main result is that there is a distinguished triangle \(i^*R\pi_*S^ r_ n\to \tau_{\leq r}i^*Rj_*{\mathbb{Z}}/p^ n(r)\to W_ n\Omega^{r- 1}_{Y,\log}[-r]\) where \(\pi: X_{syn}\to X_{et}\) is the canonical morphism and \(W_ n\Omega^{r-1}_{Y,\log}\) the logarithmic Hodge-Witt sheaf. Some consequences are given, among them the statement that every abelian étale covering of \(X_{\eta}\) comes from some abelian étale covering of \(X_ s\) and an abelian extension of \(\eta\).
Reviewer: H.Lange

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
14F40 de Rham cohomology and algebraic geometry

Citations:

Zbl 0632.14016
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References:

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[3] Fontaine, J.-M. and Messing, W.: p-adic periods and p-adic etale cohomology, MSRI (1986) (preprint). · Zbl 0632.14016
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