## 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

Zbl 0632.14016
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### References:

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