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Derived equivalences, new matrix equivalences, and homological conjectures

Xiaogang Li, Changchang Xi

TL;DR

The paper funds a program to understand finite-dimensional algebras via centralizers of matrices, introducing three matrix-equivalence notions $M$-equivalence, $D$-equivalence, and $AD$-equivalence. It proves that Morita, derived, and almost $\nu$-stable derived equivalences between centralizer matrix algebras correspond exactly to these matrix equivalences, reducing derived-equivalence problems to invariant data from elementary divisors and power indices. The results yield that the finitistic dimension conjecture and the Nakayama conjecture hold for centralizer matrix algebras, and they illuminate how derived equivalences interact with permutation matrices, including implications for $p$-regular and $p$-singular parts. The work also provides a rich set of consequences and questions, including representation-finiteness criteria, reduction to matrix-level invariants, and field-dependent behavior for permutation centralizers, offering a bridge between linear algebra and deep homological questions in representation theory.

Abstract

Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of \textbf{two} matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost $ν$-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for $p$-regular parts and for $p$-singular parts. As an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras.

Derived equivalences, new matrix equivalences, and homological conjectures

TL;DR

The paper funds a program to understand finite-dimensional algebras via centralizers of matrices, introducing three matrix-equivalence notions -equivalence, -equivalence, and -equivalence. It proves that Morita, derived, and almost -stable derived equivalences between centralizer matrix algebras correspond exactly to these matrix equivalences, reducing derived-equivalence problems to invariant data from elementary divisors and power indices. The results yield that the finitistic dimension conjecture and the Nakayama conjecture hold for centralizer matrix algebras, and they illuminate how derived equivalences interact with permutation matrices, including implications for -regular and -singular parts. The work also provides a rich set of consequences and questions, including representation-finiteness criteria, reduction to matrix-level invariants, and field-dependent behavior for permutation centralizers, offering a bridge between linear algebra and deep homological questions in representation theory.

Abstract

Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of \textbf{two} matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost -stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for -regular parts and for -singular parts. As an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras.

Paper Structure

This paper contains 14 sections, 30 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notation
  4. Basic facts on derived equivalences of algebras
  5. Modules over quotients of polynomial algebras
  6. New equivalence relations of matrices
  7. Definitions of matrix equivalences
  8. Representation-finite centralizer matrix algebras
  9. Derived equivalences and homological conjectures
  10. Characterizations of Morita and derived equivalences: Proof of Theorem \ref{['main1']}
  11. Homological conjectures: Proof of Theorem \ref{['fdc']}
  12. Derived equivalences imply Morita equivalences: Proof of Corollary \ref{['derp']}
  13. Derived equivalences for permutation matrices: Proof of Corollary \ref{['cor1.5']}
  14. Examples and further questions

Key Result

Theorem 1.1

Let $R$ be a field, $c\in M_n(R)$ and $d\in M_m(R)$. Then the centralizer matrix algebras of $c$ and of $d$ are Morita equivalent (respectively, derived equivalent, or almost $\nu$-stable derived equivalent) if and only if the matrices $c$ and $d$ are $M$-equivalent (respectively, $D$-equivalent, o

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4: Proposition \ref{['regular-singular']}
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 32 more