Singularity categories of gentle algebras

TL;DR

This paper characterizes the singularity categories of gentle algebras, showing they decompose into products of cluster categories, and relates them to stable categories of selfinjective gentle algebras, with applications to surface triangulations.

Contribution

It provides a complete description of the singularity category of any gentle algebra, linking it to cluster categories and surface triangulations, which was previously unknown.

Findings

Singularity category of gentle algebra is a product of cluster categories of type A1.

Equivalence with the stable module category of a selfinjective gentle algebra.

Number of factors corresponds to the count of inner triangles in surface triangulation.

Abstract

We determine the singularity category of an arbitrary finite dimensional gentle algebra $\Lambda$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $\Lambda$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.