Nilpotence and Duality in the Complete Cohomology of a Module

Abstract

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring ${\mathcal{E}}_M^{\ast }={\sum }_{n\in \mathbb{Z}}{\widehat{Ext}}_{kG}^n(M,M)$. We show that the ring has two distinguished ideals $I^{\ast }\subseteq J^{\ast }\subseteq {\mathcal{E}}_M^{\ast }$ such that $I^{\ast }$ is bounded above in degrees, ${\mathcal{E}}_M^{\ast }/J^{\ast }$ is bounded below in degree and $J^{\ast }/I^{\ast }$ 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 ${\mathcal{E}}_M^{\ast }$ is a nilpotent algebra.

15 pages, The Version of Record of this article is published in Beiträge zur Algebra und Geometrie and is available on line at https://doi.org/10.1007/s13366-021-00595-y

References in corpus (1)

• Hopf algebra structures and tensor products for group algebras