Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra
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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000490/type/journal_article |
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| Summary: | 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 the 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. Consequently, the mutation graph of support
$\tau $
-tilting modules over a skew-gentle algebra is connected. This generalizes one main result in [49]. |
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| ISSN: | 2050-5094 |