Finite group actions and étale cohomology

Abstract

If a finite group G acts on a quasi-projective variety X, then H*c(X,Z/n), the etale cohomology with compact support of X with coefficients inZ/n, has aZ/n[G]-module structure. It is well known that there is a finer invariant, an object RΓc(X,Z/n) of the derived category ofZ/n[G]-modules, whose cohomology is H*c(X,Z/n). We show that there is a finer invariant still, a bounded complex Λc(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓc(X,Z/n) in the derived category. This complex has many properties analogous to those of the simplicial chain complex of a simplicial complex with a group action. There are similar results forl-adic cohomology.