Year: 2026
Type: preprint
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 ${\scriptstyle{\bf LH}}$ and Talelli's operation $Φ$. 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.
Source: arXiv (Cornell University)
Authors: Emmanouil, Ioannis, Talelli, Olympia
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214999260