Stable equivalences and homological dimensions

TL;DR

This paper characterizes stable equivalences between centralizer matrix algebras over arbitrary fields using a new matrix relation, showing they preserve important homological dimensions and support the Alperin--Auslander conjecture.

Contribution

It provides a complete characterization of stable equivalences between centralizer matrix algebras and demonstrates their preservation of key homological properties.

Findings

Stable equivalences are characterized by a new matrix relation.

Stable equivalences induce stable Morita equivalences.

Homological dimensions are preserved under these equivalences.

Abstract

As is known, every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of \textbf{two} matrices. So it is fundamental to study first the centralizer algebra of a single matrix, called a centralizer matrix algebra. In this article, stable equivalences between centralizer matrix algebras over arbitrary fields are completely characterized in terms of a new type of equivalence relation on matrices. Moreover, stable equivalences of centralizer matrix algebras over any fields induce stable equivalences of Morita type, thus preserve dominant, finitistic and global dimensions. Our methods also show that the Alperin--Auslander/Auslander--Reiten conjecture holds true for stable equivalences between an arbitrary algebra and a centralizer matrix algebra over a common field.