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


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


Zbl 0632.14016
Full Text: DOI


[1] Bourbaki, N.: Algebre Commutative. Chap. 8 et 9. Masson, Paris (1983). · Zbl 0579.13001
[2] Bloch, S. and Kato, K.: p-adic etale cohomology. Publ. Math. IHES, 63, 107-152 (1986). · Zbl 0613.14017
[3] Fontaine, J.-M. and Messing, W.: p-adic periods and p-adic etale cohomology, MSRI (1986) (preprint). · Zbl 0632.14016
[4] Kato, K.: On p-adic vanishing cycles (Application of ideas of Fontaine-Messing). Adv. Studies in Pure Math., 10, 207-251 (1987). · Zbl 0645.14009
[5] Kato, K. and Saito, S.: Unramified class field theory of arithmetical surfaces. Ann. of Math., 118, 241-275 (1983). JSTOR: · Zbl 0562.14011
[6] Kurihara, M.: Abelian extensions of absolutely unramified local fields with general residue fields (1987) (preprint). · Zbl 0666.12012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.