Q915778
type: Item
instance of: scholarly article
title: Le théorème de Tate-Poitou pour les corps de fonctions des courbes définies sur les corps locaux de dimension N. (The Tate-Poitou theorem for function fields of curves defined over local fields of dimension N)
zbMATH Open document ID: 0703.11064
DOI: 10.1016/0021-8693(89)90300-1
author: Jean-Claude Douai
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let X be a projective, geometrically irreducible, smooth curve over a field k. Assume that k is a local field of dimension N in the sense of Parshin-Kato; i.e., there exists a chain of fields \(k_ 0,k_ 1,...,k_ N=k\), such that \(k_ 0\) is a finite field of characteristic p and, for each i, the field \(k_ i\) is complete with respect to a discrete valuation with residue field equal to \(k_{i-1}\). Assume, moreover, that the field \(k_ 0\) contains the q-th-roots of unity, where q is a given integer prime to p. The author shows in the paper the existence of a duality between the finite groups \(H^{N+2}(X,\mu_ q)\) and \(H^ 1(X,\mu_ q)\), if \(0\leq N\leq 4\). For \(N=0\), the theorem yields results which are already well-known.
reviewed by: Pilar Bayer
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 4152511
zbMATH Keywords: Tate-Poitou theorem
zbMATH Keywords: duality
zbMATH Keywords: finite groups
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90300-1
OpenAlex ID: W1973800546
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label: Le théorème de Tate-Poitou pour les corps de fonctions des courbes définies sur les corps locaux de dimension N. (The Tate-Poitou theorem for function fields of curves defined over local fields of dimension N)
description: scientific article; zbMATH DE number 4152511