Finite group actions and étale cohomology
type: Item
instance of: scholarly article
title: Finite group actions and étale cohomology
zbMATH Open document ID: 0838.14013
DOI: 10.1007/BF02698896
author: Jeremy Rickard
published in: Publications Mathématiques
publication date: 1995-05-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104101
review text: Let \(G\) be a finite group, let \(X\) be a quasi-projective variety over an algebraically closed field, and let \(R\) be a finite commutative coefficient ring. The author proves that if \(G\) acts (not necessarily free) on \(X\), then there is a bounded chain complex \(\Lambda_c (X,R)\) of finitely-generated direct summands of permutation \(R[G]\)-modules whose homology is the étale cohomology with compact support of \(X\) with coefficients in \(R\). Moreover, this chain complex is natural up to chain homotopy. (A permutation \(R[G]\)-module is a free \(R\)-module \(M\) on which \(G\) acts in such a way that it fixes setwise an \(R\)-basis of \(M. )\) This generalizes a respective result of \textit{P. Deligne} and \textit{G. Lusztig} [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] for free actions to the case of arbitrary actions. In addition, the author shows how constructions that can be performed on the variety \(X\), taking quotients and fixed points for the action of subgroups of \(G\), correspond to constructions on the chain complex.
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 753833
zbMATH Keywords: group actions
zbMATH Keywords: homology
zbMATH Keywords: étale cohomology
reviewed by: Konrad Drechsler
MaRDI profile type: Publication
cites work: Characters and local structure in G-algebras
cites work: Representations of reductive groups over finite fields
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Derived Equivalences As Derived Functors
cites work: Modular Permutation Representations
cites work: Representations of finite Chevalley groups. A survey
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description: scientific article; zbMATH DE number 753833