Bounded complexes of permutation modules

TL;DR

This paper investigates bounded complexes of permutation modules over fields of characteristic p, establishing conditions under which such complexes are contractible, especially for elementary abelian p-groups and certain finite groups.

Contribution

It introduces collections of permutation modules ensuring contractibility of bounded exact complexes, extending results to broader classes of finite groups.

Findings

Bounded exact complexes with modules from specific collections are contractible.

For certain finite groups, complexes with one-dimensional and projective modules are contractible.

Provides conditions under which complexes of permutation modules are trivial in homology.

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 $C^*$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^*$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^*$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^*$ is contractible.