Stable equivalences and homological dimensions

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.