Generalized Burnside rings and group cohomology
نویسنده
چکیده
We define the cohomological Burnside ring B(G,M) of a finite group G with coefficients in a ZG-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(G,M). The cohomology groups H ∗ X(G,M) are defined in such a way that H∗ X(G, M) ∼= ⊕iH∗(Hi,M) when X is the disjoint union of transitive G-sets G/Hi. If A is an abelian group with trivial action, then B(G,A) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B(G, M) is equal to the crossed Burnside ring B(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(G,M) in terms of twisted group rings when M = k× is the unit group of a commutative ring. 2000 Mathematics Subject Classification. Primary: 19A22, 20J06.
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