Stable equivalences and homological dimensions
Xiaogang Li, Changchang Xi
TL;DR
The paper resolves when two centralizer matrix algebras $S_n(c,R)$ and $S_m(d,R)$ are stably equivalent by introducing an $S$-equivalence on matrices, which matches elementary divisors and their residue-structure across blocks. It proves that over perfect fields, stable equivalences are of Morita type and preserve dominant, finitistic, and global dimensions, and that such algebras maintain the same number of non-projective simple modules, thereby validating the Auslander–Reiten conjecture in this context. The authors decompose algebras into non-semisimple blocks, relate stable equivalences to End-algebras of generators over quotients of $R[x]$, and lift these correspondences to the full algebras via Frobenius parts and node-elimination techniques. They also derive concrete corollaries for permutation matrices and for homological dimensions, highlighting the special features of centralizer matrix algebras in stability questions and offering open problems for further study.
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.