Corner replacement for Morita contexts

TL;DR

This paper explores how Morita equivalences can be adapted when replacing corner subrings, providing a canonical method and examining the ascent of these equivalences with applications across various algebraic structures.

Contribution

It introduces a canonical approach to replace corner subrings with Morita equivalent ones and studies the ascent of Morita equivalences in this context.

Findings

A canonical method for corner replacement preserving Morita equivalence

Extensions of results to trivial extensions and tensor rings

Applications to diverse algebraic structures like orbifolds and functor categories

Abstract

We consider how Morita equivalences are compatible with the notion of a corner subring. Namely, we outline a canonical way to replace a corner subring of a given ring with one which is Morita equivalent, and look at how such an equivalence ascends. We use the language of Morita contexts, and then specify these more general results. We give applications to trivial extensions of finite-dimensional algebras, tensor rings of pro-species, semilinear clannish algebras arising from orbifolds, and functor categories.