On cohomologically Trivially modules over finite $p$-groups

TL;DR

This paper proves that finitely generated cohomologically trivial modules over group rings of finite p-groups decompose into a direct sum of trivial and free modules, advancing understanding of their structure.

Contribution

It introduces a decomposition theorem for finitely generated cohomologically trivial modules over group rings of finite p-groups, revealing their internal structure.

Findings

Finitely generated cohomologically trivial modules split into trivial and free parts.

Established results on generators and relators of these modules.

Provided structural insights into modules over p-group group rings.

Abstract

We show that every finitely generated cohomologically trivial module over $RG$, where $G$ is a finite $p$-group and $R$ is a $p$-adic ring, splits as the direct sum of a finite cohomologically trivial $RG$-module and a free $RG$-module. Along the way, we also establish other results concerning generators and relators of such modules.