Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929684932984832 |
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| author | Karmazyn, Joseph Lepri, Emma Wemyss, Michael |
| author_facet | Karmazyn, Joseph Lepri, Emma Wemyss, Michael |
| contents | To any periodic module over any algebra, this paper introduces an associated trivial extension DG-algebra T. After first passing to a strictly unital $A_\infty$-minimal model, it then constructs a particular $A_\infty$-algebra N, called the unitally positive $A_\infty$-algebra, which roughly speaking describes the identity in degree zero and all the positive cohomology. The object N is fundamental, and can be constructed for any DG-category satisfying very mild assumptions.
The main application is to birational geometry. When applied to contraction algebras, the construction gives a simple and direct proof of the Donovan-Wemyss conjecture, namely that smooth irreducible 3-fold flops are classified by their contraction algebras, and thus by noncommutative data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications Karmazyn, Joseph Lepri, Emma Wemyss, Michael Algebraic Geometry Representation Theory To any periodic module over any algebra, this paper introduces an associated trivial extension DG-algebra T. After first passing to a strictly unital $A_\infty$-minimal model, it then constructs a particular $A_\infty$-algebra N, called the unitally positive $A_\infty$-algebra, which roughly speaking describes the identity in degree zero and all the positive cohomology. The object N is fundamental, and can be constructed for any DG-category satisfying very mild assumptions. The main application is to birational geometry. When applied to contraction algebras, the construction gives a simple and direct proof of the Donovan-Wemyss conjecture, namely that smooth irreducible 3-fold flops are classified by their contraction algebras, and thus by noncommutative data. |
| title | Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2411.17359 |