Periodic cohomology

TL;DR

The paper addresses the problem of understanding cohomological periods of finite groups and their consequences for module theory over group rings. It offers an elementary, direct proof that a finite group with cohomological period $2$ must be cyclic, and uses this to derive freeness results for stably free resolutions: any finitely generated stably free $\\mathbb{Z}[G]$-resolution with $S_1,S_2$ nonfree forces $G$ to be cyclic, implying $S_1$ and $S_2$ are free. Key contributions include Theorem 2.1 (period $2$ implies cyclic), Proposition 3.1 (high-rank stably free modules are free), and Theorem A (freeness of $S_1,S_2$ in finitely generated stably free resolutions over finite groups). The work connects algebraic topology (via universal covers and Wall's $D2$ problem) with module-theoretic consequences, enabling matrix representations of maps and clarifying when stable freeness collapses to freeness in finite-group settings.

Abstract

We offer a direct proof of an elementary result concerning cohomological periods. As a corollary we show that given a finitely generated stably free resolution of Z over a finite group, two of its modules are free.