CENTRAL ELEMENTS OF THE JENNINGS BASIS AND CERTAIN MORITA INVARIANTS (Cohomology theory of finite groups and related topics)

Abstract

From Morita theoretic viewpoint, computing Morita invariants is important. We proved that the intersection of the center and the nth socle ZS^{n}(A) :=Z(A)cap mathrm{S}mathrm{o}mathrm{c}^{n}(A) of a finite dimensional algebra A is Morita invariant; This is a generalization of important Morita invariants, the center Z(A) and the Reynolds ideal ZS^{1}(A). As an example, we also studied ZS^{n}(FP) for the group algebra FP of a finite p-group P over a field F of positive characteristic p. Such an algebra has a basis along the radical filtration, known as the Jennings basis. We show sufficient conditions under which an element of the Jennings basis is central and a lower bound for the dimension of ZS^{n}(FP) for every positive integer n. Equalities hold for 1 leq n leq p if P is powerful. As a corollary we have Somathrm{c}^{p} (FP) subseteq Z(FP) if P is powerful. This is a report of a talk based on [Sakurai, arXiv:1701.03799v2].