Nilpotence and Duality in the Complete Cohomology of a Module

TL;DR

This paper investigates the structure of complete cohomology rings of modules over finite groups, revealing nilpotent behavior in negative degrees under certain conditions and identifying distinguished ideals with specific boundedness and periodicity properties.

Contribution

It introduces a novel analysis of the complete cohomology ring structure, identifying distinguished ideals and their properties, and establishes nilpotence of negative degree elements for non-projective, non-periodic modules.

Findings

Existence of two distinguished ideals with boundedness and periodicity properties.

Negative degree elements form a nilpotent algebra for certain modules.

The structure of the complete cohomology ring is characterized by these ideals and nilpotence.

Abstract

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. We show that the ring has two distinguished ideals $I^* \subseteq J^* \subseteq \mathcal{E}_M^*$ such that $I^*$ is bounded above in degrees, $\mathcal{E}_M^*/J^*$ is bounded below in degree and $J^*/I^*$ is eventually periodic with terms of bounded dimension. We prove that if $M$ is neither projective nor periodic, then the subring of all elements in negative degrees in $\mathcal{E}_M^*$ is a nilpotent algebra.