Generalized Burnside rings and group cohomology

Abstract

We define the cohomological Burnside ring B n (G, M) of a finite group G with coefficients in a Z G-module M as the Grothendieck ring of the isomorphism classes of pairs [X, u] where X is a G-set and u is a cohomology class in a cohomology group H X n (G, M). The cohomology groups H X * (G, M) are defined in such a way that H X * (G, M) ≅ ⊕ i H * (H i, M) when X is the disjoint union of transitive G-sets G / H i. If A is an abelian group with trivial action, then B 1 (G, A) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B 0 (G, M) is equal to the crossed Burnside ring B c (G, M). We discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological Burnside rings. We also give an interpretation of B 2 (G, M) in terms of twisted group rings when M = k × is the unit group of a commutative ring. © 2006 Elsevier Inc. All rights reserved.