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 $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. 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.