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