Jon Carlson: 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. Let M be a finitely generated kG-module. We consider the complete cohomology ring ℰ̂_M^∗=Σ_n∈ℤ Ex̂t_kGⁿ(M, M). We show that the ring has two distinguished ideals I^∗ ⊆ J^∗ ⊆ ℰ̂_M^∗ such that I^∗ is bounded above in degrees, ℰ̂_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 ℰ̂_M^∗ is a nilpotent algebra.

This video is part of the University of Georgia‘s Algebra seminar.