The chain complex of an acyclic simplicial \(G\)-complex (Q1818858)

For a finite group \(G\), the reduced chain complex of an acyclic finite-dimensional simplicial \(G\)-complex is a split acyclic complex of \(\mathbb{Z} G\)-modules. The main tool of this result is the notion of a permutation module (i.e. having a stable basis). A key theorem says that some complexes of permutation modules are split acyclic iff all of their Brauer quotients are acyclic.