\(p\)-adic étale cohomology
This paper gives the complete proofs of the results summarized by \textit{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 \(p^{\dim A}\)).
scientific article; zbMATH DE number 3991665
scientific article