A geometric model for the module category of a skew-gentle algebra

Abstract

In this article, we realize skew-gentle algebras as skew-tiling algebras associated to admissible partial triangulations of punctured marked surfaces. Based on this, we establish a bijection between tagged permissible curves and certain indecomposable modules, interpret the Auslander-Reiten translation via the tagged rotation, and show intersection-dimension formulas. As an application, we classify support $\tau$-tilting modules via maximal collections of non-crossing tagged generalized permissible curves.

arXiv commentPresentation was improved, two tables of notations were added, and a correction for the last condition in the definition of tagged permissible curves was made

References in corpus (9)

• Flat surfaces and stability structures
• The representation type of Jacobian algebras
• Cluster categories for marked surfaces: punctured case
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• A geometric model for the derived category of gentle algebras
• Decorated marked surfaces: spherical twists versus braid twists
• Coloured quivers for rigid objects and partial triangulations: The unpunctured case
• Non-kissing and non-crossing complexes for locally gentle algebras
• On auto-equivalences and complete derived invariants of gentle algebras