A characterisation of Morita algebras in terms of covers

TL;DR

This paper characterizes Morita algebras using the concept of covers, relating them to self-injective algebras and faithful projective-injective modules through the Schur functor.

Contribution

It provides a new characterization of Morita algebras in terms of covers and establishes a converse relationship involving self-injective algebras and projective-injective modules.

Findings

Morita algebras are characterized as covers of self-injective algebras.

The Schur functor's full faithfulness on projective modules is key.

A converse is shown: covers imply Morita and self-injective properties.

Abstract

A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ 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 $\operatorname{End}_A(P)^{op}$ is self-injective whenever $(A, P)$ is a cover of $\operatorname{End}_A(P)^{op}$ for a faithful projective-injective module $P$.