Generalized Burnside rings and group cohomology

JOURNAL OF ALGEBRA

卷:310

Article

Hartmann, Robert ; Yalcin, Ergun

关键词: cohomology of groups;    monomial G-sets;    generalized Burnside rings;   

DOI  :  10.1016/j.jalgebra.2006.10.037

来源: Elsevier

【 摘 要 】

We define the cohomological Burnside ring B-n(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(n) (G, M). The cohomology groups H-X* (G, M) are defined in such a way that H-X* (G, M) congruent to circle plus(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(x) is the unit Group of a commutative ring. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

Free