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Singular equivalences and homological conjectures

Zhenxian Chen, Changchang Xi

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.

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 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.
Paper Structure (16 sections, 39 theorems, 69 equations)

This paper contains 16 sections, 39 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact and Frobenius categories
  4. Singularity categories of algebras
  5. Basic facts on modules over quotients of polynomial algebras
  6. Centralizer algebras of matrices
  7. Basics from number theory
  8. New types of equivalence relations of matrices
  9. Singularity categories
  10. Gorenstein projective modules over endomorphism algebras: General theory
  11. Special case: Generators over quotients of polynomial algebras
  12. Singularity categories of centralizer matrix algebras
  13. Singular equivalences
  14. Main result on singular equivalences
  15. Singular equivalences for permutation matrices
  16. ...and 1 more sections

Key Result

Theorem 1.1

[Theorem seocma] Let $A:=S_n(c,R)$ and $B:=S_m(d,R)$ for $c\in M_n(R)$ and $d\in M_m(R)$. Then $(1)$$A$ and $B$ are isomorphic $R$-algebras if and only if $c$ and $d$ are $I$-equivalent matrices. $(2)$ The following statements are equivalent. $\quad (i)$${\mathscr D}_{sg}(A)$ and ${\mathscr D}_{sg}(

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 41 more