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

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.

arXiv comment7 pages. v2: updated references. v3: Fixed proof of Proposition 3.3

References in corpus (4)

• Flat surfaces and stability structures
• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Quivers with potentials associated to triangulated surfaces
• A $\tau$-Tilting Approach to Dissections of Polygons

Cited by in corpus (11)

• Silting objects
• On $\tau$-tilting finiteness of the Schur algebra
• Report on the finiteness of silting objects
• On $\tau$-tilting finite simply connected algebras
• Schurian-finiteness of blocks of type $A$ Hecke algebras
• Pairwise Compatibility for 2-Simple Minded Collections
• Schemes of modules over gentle algebras and laminations of surfaces
• On $\tau$-tilting finiteness of symmetric algebras of polynomial growth
• $\tau$-tilting finiteness of two-point algebras II
• On $\tau$-tilting modules over trivial extensions of gentle tree algebras
• $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras