Title: Generalized Burnside rings and group cohomology

Authors: Robert Hartmann & Ergun Yalcin

Contact info:

Robert Hartmann
University of Leicester,
Department of Mathematics
University Road, Leicester LE1 7RH, UK

Erg\" un Yal\c c\i n
Bilkent University,
Department of Mathematics,
Ankara, 06800, Turkey

Paper Status: Submitted.

Abstract: We define the cohomological Burnside ring
$B^n (G, M)$ of a finite group $G$ with coefficients in a $\ZZ
G$-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 $H^n _X (G, M)$. The cohomology groups $H^*_X
(G, M)$ are defined in such a way that $H^* _X (G, M)\cong \oplus _i
H^* (H _i , M)$ 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 $M=k^\times$ is the unit group of a commutative ring.