Cohomology of Burnside Rings

Abstract

Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \subset G$ let $\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete description of the $A(G)$-modules $\textrm{Ext}^l_{A(G)}(\mathbb{Z}_H, \mathbb{Z}_J)$ and $\textrm{Tor}^{A(G)}_l(\mathbb{Z}_H, \mathbb{Z}_J)$ for any $H, J \subset G$ and $l \geq 0$. We show that if the order of $G$ is not square-free then there exist $H, J \subset G$ such that $\textrm{Ext}^l_{A(G)}(\mathbb{Z}_H, \mathbb{Z}_J)$ and $\textrm{Tor}^{A(G)}_l(\mathbb{Z}_H, \mathbb{Z}_J)$ have unbounded rank as finite groups.