The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911178756718592 |
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| author | Jasso, Gustavo Muro, Fernando |
| author_facet | Jasso, Gustavo Muro, Fernando |
| contents | Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\mathbb{Z}$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. For $d=1$ this correspondence specialises to previous work of the second-named author on algebraic triangulated categories of finite type. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures Jasso, Gustavo Muro, Fernando Representation Theory Primary: 18G80 Secondary: 18N40 Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\mathbb{Z}$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. For $d=1$ this correspondence specialises to previous work of the second-named author on algebraic triangulated categories of finite type. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule. |
| title | The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures |
| topic | Representation Theory Primary: 18G80 Secondary: 18N40 |
| url | https://arxiv.org/abs/2509.22625 |