A characterisation of Morita algebras in terms of covers

Abstract

A pair $(A,P)$ is called a cover of ${\mathrm{End}}_A(P{)}^{op}$ if the Schur functor ${\mathrm{Hom}}_A(P,-)$ is fully faithful on the full subcategory of projective $A$-modules, for a given projective $A$-module $P$. By definition, Morita algebras are the covers of self-injective algebras and then $P$ is a faithful projective-injective module. Conversely, we show that $A$ is a Morita algebra and ${\mathrm{End}}_A(P{)}^{op}$ is self-injective whenever $(A,P)$ is a cover of ${\mathrm{End}}_A(P{)}^{op}$ for a faithful projective-injective module $P$.

8 pages, An example was added and several typos were corrected