$\tau$-tilting finite gentle algebras are representation-finite

TL;DR

This paper proves that gentle algebras over a field are $ au$-tilting finite if and only if they are representation-finite, establishing a precise equivalence between these properties using combinatorial methods.

Contribution

It establishes a complete characterization of $ au$-tilting finiteness for gentle algebras, linking it directly to their representation-finiteness.

Findings

Gentle algebras are $ au$-tilting finite if and only if they are representation-finite.

The proof uses the brick-$ au$-tilting correspondence and combinatorial analysis.

Provides a clear criterion for $ au$-tilting finiteness in gentle algebras.

Abstract

We show that a gentle algebra over a field is $\tau$-tilting finite if and only if it is representation-finite. The proof relies on the "brick-$\tau$-tilting correspondence" of Demonet-Iyama-Jasso and on a combinatorial analysis.