Complexes of modules over hierarchical groups

Abstract

Let $\overline{\mathfrak{F}}$ be the group class obtained from the class $\mathfrak{F}$ of finite groups by iterated applications of Kropholler's operation $\mathbf{LH}$ and Talelli's operation $\Phi$. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of $\overline{\mathfrak{F}}$-groups with coefficients in a commutative ring. If $G$ is an $\overline{\mathfrak{F}}$-group, we show that any acyclic complex of projective (respectively, flat) $\mathbb{Q}G$-modules is contractible (respectively, pure acyclic). If $G$ is torsion-free, the same conclusions hold for acyclic complexes of projective or flat $\mathbb{Z}G$-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.