Classifying torsion classes of gentle algebras

TL;DR

This paper refines the combinatorial classification of torsion classes in gentle algebras, extending known results to possibly infinite-dimensional cases and unifying classifications across related algebra types.

Contribution

It provides a refined combinatorial approach to classify torsion classes in gentle algebras, including infinite-dimensional cases, and unifies existing classifications for related algebra classes.

Findings

Refined combinatorial classification of torsion classes.

Extension of classification to infinite-dimensional gentle algebras.

Unification of classifications across gentle and Brauer graph algebras.

Abstract

For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support $\tau$-tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, which gives us a classification of torsion classes in the category of finite length modules over a (possibly infinite-dimensional) gentle algebra. As a consequence, our result also unifies the functorially finite torsion class classification of finite-dimensional gentle algebras with certain classes of special biserial algebras - such as Brauer graph algebras.