Nilpotence and Duality in the Complete Cohomology of a Module

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, ${\mathcal{E}}_M^{\ast }\mathrm{/}{J}^{\ast }$ is bounded below in degree and $J^{\ast }\mathrm{/}{I}^{\ast }$ 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.