Mutation graph of support $\ddot{I}„$-tilting modules over a skew-gentle algebra

Abstract

Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $\hat{I}›={\mathrm{End}}_{\mathcal{D}}R$ 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 $\ddot{I}„$-tilting $\hat{I}›$-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 $\ddot{I}„$-tilting $\hat{I}›$-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 $\ddot{I}„$-tilting modules over a skew-gentle algebra is connected.

45 pages, 22 figures