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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.

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 14L30 Group actions on varieties or schemes (quotients)
##### Keywords:
group actions; homology; étale cohomology
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##### References:
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