Singular equivalences and homological conjectures

Abstract

The fact that each finite-dimensional algebra over a field is isomorphic to the centralizer of two matrices, has suggested to investigate representation theoretical problems of finite-dimensional algebras through the centralizer algebras of matrices. Therefore the first natural question is to study the problems for the centralizer algebra of one matrix, called a centralizer matrix algebra. In this paper we give an elementary and explicit approach to the singularity categories and singular equivalences of centralizer matrix algebras, and verify the Auslander--Reiten and Cartan determinant conjectures for centralizer matrix algebras. Consequently, all historical homological conjectures (the finitistic dimension, Wakamatsu tilting, tilting (projective) complement, strong Nakayama, generalized Nakayama and Nakayama conjectures) are true for centralizer matrix algebras over fields. Moreover, we prove some homological invariants of singular equivalences of centralizer matrix algebras.