On Artin-Verdier duality for function fields
type: Item
instance of: scholarly article
title: On Artin-Verdier duality for function fields
zbMATH Open document ID: 0585.14014
DOI: 10.1007/BF01163876
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173522
review text: Let X be a smooth, complete curve over a finite field and F a constructible sheaf on the étale site \(U_{et}\) of a nonempty, open subset \(U\subset X\). In this note the author proves that, for all \(i\in {\mathbb{Z}}\), the Yoneda pairing \[ H^ i_ c(U,F)\times Ext_ U^{3- i}(F,{\mathbb{G}}_ m)\to H^ 3_ c(U,{\mathbb{G}}_ m)\overset \sim \rightarrow^{tr}{\mathbb{Q}}\quad /{\mathbb{Z}} \] is a nondegenerate pairing of finite groups, and that the trace map tr (induced by the degree map) is an isomorphism. If \(X=Spec({\mathfrak O}_ K)\) where \({\mathfrak O}_ K\) is the ring of integers of a totally imaginary number field, then the analogous statement is known as the Artin-Verdier duality theorem; cf. \textit{M. Artin}, \textit{J. L. Verdier} [Sémin. on étale cohomologie of number fields, Woods Hole 1964] and \textit{B. Mazur} [Ann. Sci. École Norm. Supér., IV. Sér. 6(1973), 521-556 (1974; Zbl 0282.14004); the author remarks that in the latter paper there is a gap in the argument, which he fills here.] Similar duality theorems (for flat cohomology) were given by \textit{M. Artin} and \textit{J. Milne} [Invent. Math. 35, 111-129 (1976; Zbl 0342.14007)] and \textit{G. Frey} [J. Reine Angew. Math. 301, 46-58 (1978; Zbl 0375.12015)] In recent papers the author extended the above result in two directions: (1) to \({\mathbb{Z}}\)-constructible sheaves [J. Reine Angew. Math. 366, 18-31 (1986)]; and (2) to singular curves [''Duality in the étale cohomology of one-dimensional proper schemes and generalizations'' (preprint)].
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3937316
zbMATH Keywords: etale cohomology
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Yoneda pairing
zbMATH Keywords: Artin-Verdier duality
Wikidata QID: Q56813077
author: Christopher Deninger
MaRDI profile type: Publication
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OpenAlex ID: W2074828496
description: scientific article; zbMATH DE number 3937316