Bounded complexes of permutation modules

Abstract

Let $k$ be a field of characteristic $p>0$. For $G$ an elementary abelian $p$-group, there exist collections of permutation module such that if $C^{\ast }$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^{\ast }$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^{\ast }$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^{\ast }$ is contractible.

7 pages