Singularity categories, Schur Functors and Triangular Matrix Rings

TL;DR

This paper investigates Schur functors that preserve singularity categories and applies these to analyze the singularity categories of triangular matrix rings, providing new insights into non-Gorenstein rings and concrete algebra examples.

Contribution

It introduces new Schur functors preserving singularity categories and applies them to describe singularity categories of triangular matrix rings and certain non-Gorenstein rings.

Findings

Schur functors that preserve singularity categories are identified.

Singularity categories of specific non-Gorenstein rings are characterized.

Examples of finite-dimensional algebras sharing the same singularity category are provided.

Abstract

We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed.