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.