On cohomologically Trivially modules over finite $p$-groups

TL;DR

We address the problem of understanding cohomologically trivial modules over the group ring $RG$ for finite $p$-groups. The authors introduce invariants $d_R(A)$ and $r_R(A)$, show that $r_R(A)=d_R(A_G)$ and relate presentations to the CT property, and establish CT criteria linking cohomology vanishing to freeness over $\bar K G$. The main result proves that every finitely generated CT $RG$-module $A$ decomposes as $A= T\oplus (A/T)$ with $T$ the torsion part and $A/T$ free over $RG$, reducing the argument to the cyclic $p$-group case; this clarifies the structural building blocks for CT modules and informs questions about automorphisms and Schmid-type conjectures for finite $p$-groups. The work connects group cohomology, module presentations, and $p$-adic and zeta-function perspectives to yield a precise decomposition theorem with potential implications for counterexamples and related cohomological phenomena.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['Number of relations for the ZpG-module A']}
• Lemma 3.1
• proof