On the $τ$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917177601294336 |
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| author | Chang, Wen Jin, Haibo Schroll, Sibylle Wang, Qi |
| author_facet | Chang, Wen Jin, Haibo Schroll, Sibylle Wang, Qi |
| contents | This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $τ$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $τ$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras Chang, Wen Jin, Haibo Schroll, Sibylle Wang, Qi Representation Theory Rings and Algebras This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $τ$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models. |
| title | On the $τ$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2512.24316 |