Mutation graph of support $\tau$-tilting modules over a skew-gentle algebra

TL;DR

This paper explores the mutation graph of support τ-tilting modules over skew-gentle algebras, establishing its connectivity and linking it to geometric flips in punctured surface cluster categories.

Contribution

It introduces a mutation framework for maximal rigid objects in 2-Calabi-Yau categories and connects it to support τ-tilting module mutations, proving the graph's connectedness for skew-gentle algebras.

Findings

Mutation graph of support τ-tilting modules is connected.

Mutation graph is isomorphic to flip graph of tagged arcs.

Supports geometric interpretation via punctured surface cluster categories.

Abstract

Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $\Lambda=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with mutation of support $\tau$-tilting $\Lambda$-modules. In the case that $\mathcal{D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $\tau$-tilting $\Lambda$-modules is isomorphic to the graph of flips of certain collections of tagged arcs on the surface, which is moreover proved to be connected. As a direct consequence, the mutation graph of support $\tau$-tilting modules over a skew-gentle algebra is connected.