The singularity category via the stabilization

TL;DR

This paper proves that the singularity category of a ring is equivalent to the stabilization of its stable module category, providing new insights into singular equivalences between different rings using Leavitt rings.

Contribution

It offers a detailed proof of the equivalence between the singularity category and the stabilization of the stable module category, with applications to artinian rings.

Findings

Singularity category is triangle equivalent to stabilization of stable module category

Leavitt rings describe singularity categories of artinian rings

Establishes singular equivalences between rings of different types

Abstract

We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different nature. We use Leavitt rings to describe singularity categories of artinian rings.