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Finite group actions and étale cohomology. (English) Zbl 0838.14013
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 P. Deligne and 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.

14F20 Étale and other Grothendieck topologies and (co)homologies
14L30 Group actions on varieties or schemes (quotients)
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