Paper detail

Central elements of the Jennings basis and certain Morita invariants

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite-dimensional algebra $A$ is a 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 $ZS^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^n(FG)$. In fact, such elements form a basis of $ZS^n(FG)$ for every integer $1 \le n \le p$ if $G$ is powerful. As a corollary we have $\operatorname{Soc}^p(FG) \subseteq Z(FG)$ if $G$ is powerful.