Stable equivalences and homological dimensions

Abstract: As is known, every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of 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 equivalence relation on matrices. Moreover, stable equivalences of centralizer matrix algebras over perfect fields induce stable equivalences of Morita type, thus preserve dominant, finitistic and global dimensions. Our methods also show that algebras stably equivalent to a centralizer matrix algebra over a field have the same number of non-isomorphic, non-projective simple modules. Thus the Auslander--Reiten conjecture on stable equivalences holds true for centralizer matrix algebras.