The singularity category via the stabilization
Xiao-Wu Chen
TL;DR
The paper proves that the singularity category $D_{sg}(R)$ is triangle equivalent to the stabilization of the stable module category, providing a unified framework to compare singularities across diverse algebras. It develops the stabilization for looped and left triangulated categories, establishes a universal property for functors out of stabilized categories, and then uses this to realize $D_{sg}(R)$ as the stabilization of $R$-module data. The authors apply the result to derive singular equivalences, including artinian and monomial/quiver-related cases, and connect these equivalences to Leavitt rings, giving explicit graded module descriptions of singularity categories. This framework yields concrete tools for transferring homological singularities and yields new descriptions of $D_{sg}$ for artinian rings via Leavitt ring techniques, with broader implications for noncommutative geometry and representation theory.
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.