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$$p$$-adic étale cohomology. (English) Zbl 0613.14017
This paper gives the complete proofs of the results summarized by S. Bloch in Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 13-26 (1983; Zbl 0584.14009). Let $$K$$ be a field, complete with respect to a discrete valuation, of characteristic 0 and residue characteristic $$p>0$$. Let $$V$$ be a complete smooth variety over $$K$$ and $$X$$ a proper smooth model of $$V$$ over the valuation ring $$\Lambda$$ of $$K$$. The central object of the paper are the étale cohomology groups $$H^ q(\bar V,{\mathbb{Q}}_ p)$$, where $$\bar V$$ is the extension of $$V$$ to the algebraic closure $$\bar K$$ of $$K$$. They (more precisely: certain subquotients of them) are compared as Gal$$(\bar K/K)$$-modules to suitable de Rham-Witt cohomology groups and crystalline cohomology groups. As a corollary the authors prove that $$H^ q(\bar V,{\mathbb{Q}}_ p)$$ has a Hodge-Tate decomposition under the assumption that the reduction of $$\bar X$$ be “ordinary”, a notion which for an abelian variety $$A$$ coincides with the usual one (the group of $$p$$-torsion points has order $$p^{\dim A}$$).
Reviewer: F.Herrlich

MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry 14G20 Local ground fields in algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
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References:
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