\(p\)-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


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


Zbl 0584.14009
Full Text: DOI Numdam EuDML


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