Singular equivalences and homological conjectures

TL;DR

This paper investigates the homological properties of centralizer matrix algebras, providing complete descriptions of their singularity categories, verifying key conjectures, and establishing their homological invariants.

Contribution

It offers the first comprehensive analysis of singularity categories and homological conjectures for centralizer matrix algebras, confirming several longstanding conjectures in this context.

Findings

Complete descriptions of singularity categories for centralizer matrix algebras

Verification of the Auslander--Reiten and Cartan determinant conjectures

Proof that all major homological conjectures hold for these algebras

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 centralizer algebras of matrices. 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 complete descriptions of the singularity categories and singular equivalences of centralizer matrix algebras, and verify the Auslander--Reiten (or Gorenstein projective) 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 for centralizer matrix algebras.