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Singular equivalences induced by ring extensions

Yongyun Qin

Abstract

Let $B \subseteq A$ be an extension of finite dimensional algebras. We provide a sufficient condition for the existence of triangle equivalences of singularity categories (resp. Gorenstein defect categories) between $A$ and $B$. This result is applied to trivial extensions, Morita rings and triangular matrix algebras to give several reduction methods on singularity categories and Gorenstein defect categories of algebras.

Singular equivalences induced by ring extensions

Abstract

Let be an extension of finite dimensional algebras. We provide a sufficient condition for the existence of triangle equivalences of singularity categories (resp. Gorenstein defect categories) between and . This result is applied to trivial extensions, Morita rings and triangular matrix algebras to give several reduction methods on singularity categories and Gorenstein defect categories of algebras.
Paper Structure (4 sections, 11 theorems, 29 equations)

This paper contains 4 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Definitions and conventions
  3. Singularity categories and ring extensions
  4. Applications and examples

Key Result

Theorem 1.1

Let $A$ and $B$ be two algebras and $B\subseteq A$ be an extension. Assume that $\mathrm{pd} _{B^e}(A/B)<\infty$, $(A/B)^{\otimes _Bp}=0$ for some integer $p$ and $\mathrm{Tor} _i^B(A/B, (A/B)^{\otimes _Bj})=0$ for each $i,j\geq 1$. Then there is a singular equivalence of Morita type with level betw

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 14 more