Nilpotence and duality in the complete cohomology of a module

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