A note on p-adic étale cohomology
type: Item
instance of: scholarly article
title: A note on p-adic étale cohomology
zbMATH Open document ID: 0647.14006
author: Masato Kurihara
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: 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)], \textit{J.-M. Fontaine} and \textit{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 \(\eta\).
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 4055755
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: syntomic site
zbMATH Keywords: étale cohomology
zbMATH Keywords: de Rham cohomology
MaRDI profile type: Publication
full work available at URL: pjaa.63.275
OpenAlex ID: W1986761987
cites work: Q3701517
cites work: \(p\)-adic étale cohomology
cites work: Q3769025
cites work: Q3788120
cites work: Unramified class field theory of arithmetical surfaces
cites work: Abelian extensions of an absolutely unramified local field with general residue field
DOI: 10.3792/PJAA.63.275
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description: scientific article; zbMATH DE number 4055755