On the combinatorics of gentle algebras

Abstract

For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,\tau Y$). We use this to describe support $\tau$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\tau$-tilting modules and functorially finite torsion classes. We give an explicit basis of Ext${}^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

25 pages; v2 new final section on uniqueness of kisses between exchangeable strings added; additional small changes

References in corpus (1)

• A reduction theorem for $\tau$-rigid modules

Cited by in corpus (8)

• Non-kissing complexes and tau-tilting for gentle algebras
• Celebrating Loday's Associahedron
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• Laminations of punctured surfaces as $\tau$-regular irreducible components
• Morphisms and extensions between bricks over preprojective algebras of type A
• On $\tau$-tilting modules over trivial extensions of gentle tree algebras
• Jordan recoverability of some subcategories of modules over gentle algebras
• On homomorphisms and generically $\tau$-regular components for skewed-gentle algebras