A complete derived invariant and silting theory for graded gentle algebras
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912283397980160 |
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| author | Jin, Haibo Schroll, Sibylle Wang, Zhengfang |
| author_facet | Jin, Haibo Schroll, Sibylle Wang, Zhengfang |
| contents | We confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth graded (not necessarily proper) gentle algebras form a complete derived invariant. Hence, we obtain a complete invariant of triangle equivalences for partially wrapped Fukaya categories of graded surfaces with stops.
A key ingredient of the proof is the full description of homologically smooth graded gentle algebras whose perfect derived categories admit silting objects. We also apply this to classify which graded gentle algebras admit pre-silting objects that are not partial silting. In particular, this allows us to construct a family of counterexamples to the question whether any pre-silting object in the derived category of a finite-dimensional algebra is partial silting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_17474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A complete derived invariant and silting theory for graded gentle algebras Jin, Haibo Schroll, Sibylle Wang, Zhengfang Representation Theory Rings and Algebras Symplectic Geometry 16E35, 16E45, 57M50 We confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth graded (not necessarily proper) gentle algebras form a complete derived invariant. Hence, we obtain a complete invariant of triangle equivalences for partially wrapped Fukaya categories of graded surfaces with stops. A key ingredient of the proof is the full description of homologically smooth graded gentle algebras whose perfect derived categories admit silting objects. We also apply this to classify which graded gentle algebras admit pre-silting objects that are not partial silting. In particular, this allows us to construct a family of counterexamples to the question whether any pre-silting object in the derived category of a finite-dimensional algebra is partial silting. |
| title | A complete derived invariant and silting theory for graded gentle algebras |
| topic | Representation Theory Rings and Algebras Symplectic Geometry 16E35, 16E45, 57M50 |
| url | https://arxiv.org/abs/2303.17474 |