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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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000490/type/journal_article |
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| author | Ping He Yu Zhou Bin Zhu |
| author_facet | Ping He Yu Zhou Bin Zhu |
| author_sort | Ping He |
| collection | DOAJ |
| description | 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]. |
| format | Article |
| id | doaj-art-9a05ce9de05e4a64af72f8d30963b28b |
| institution | DOAJ |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-9a05ce9de05e4a64af72f8d30963b28b2025-08-20T03:16:39ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.49Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebraPing He0https://orcid.org/0009-0007-6470-9413Yu Zhou1https://orcid.org/0000-0002-8260-7791Bin Zhu2https://orcid.org/0009-0004-0607-3726Beijing Institute of Mathematical Sciences and Applications, No. 544, Hefangkou Village Huaibei Town, Huairou District, Beijing, 101408, China; E-mail:School of Mathematical Sciences, Beijing Normal University, No.19, Xinjiekouwai Street, Haidian District, Beijing, 100875, ChinaDepartment of Mathematical Sciences, Tsinghua University, No. 1, Qinghuayuan Street, Haidian District, Beijing, 100084, China; E-mail: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].https://www.cambridge.org/core/product/identifier/S2050509425000490/type/journal_article16G2005E1005C10 |
| spellingShingle | Ping He Yu Zhou Bin Zhu Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra Forum of Mathematics, Sigma 16G20 05E10 05C10 |
| title | Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra |
| title_full | Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra |
| title_fullStr | Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra |
| title_full_unstemmed | Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra |
| title_short | Mutation graph of support $\tau $ -tilting modules over a skew-gentle algebra |
| title_sort | mutation graph of support tau tilting modules over a skew gentle algebra |
| topic | 16G20 05E10 05C10 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000490/type/journal_article |
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