Calabi-Yau structures on derived and singularity categories of symmetric orders
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912865600929792 |
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| author | Hanihara, Norihiro Liu, Junyang |
| author_facet | Hanihara, Norihiro Liu, Junyang |
| contents | We construct left and right Calabi-Yau structures on derived respectively singularity categories of symmetric orders $Λ$ over commutative Gorenstein rings $R$. For this, we first construct Calabi-Yau structures over $R$ by lifting Amiot's construction of Calabi-Yau structures on Verdier quotients to the dg level. Then we prove base change properties relating Calabi-Yau structures over $R$ to those over the base field $k$. As a result, we prove the existence of a right Calabi-Yau structure on the dg singularity category associated with $Λ$ which is a cyclic lift of the weak Calabi-Yau structure constructed by the first-named author and Iyama. We also show the existence of a left Calabi-Yau structure on the dg bounded derived category of $Λ$. This is a non-commutative generalization of a result by Brav and Dyckerhoff. By combining the existence of the right Calabi-Yau structure on the dg singularity category with a structure theorem by Keller and the second-named author, we deduce that under suitable hypotheses, the singularity category associated with $Λ$ is triangle equivalent to a generalized cluster category in the sense of Amiot. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Calabi-Yau structures on derived and singularity categories of symmetric orders Hanihara, Norihiro Liu, Junyang Representation Theory Algebraic Geometry Category Theory K-Theory and Homology 18G80, 18G35, 13C14 We construct left and right Calabi-Yau structures on derived respectively singularity categories of symmetric orders $Λ$ over commutative Gorenstein rings $R$. For this, we first construct Calabi-Yau structures over $R$ by lifting Amiot's construction of Calabi-Yau structures on Verdier quotients to the dg level. Then we prove base change properties relating Calabi-Yau structures over $R$ to those over the base field $k$. As a result, we prove the existence of a right Calabi-Yau structure on the dg singularity category associated with $Λ$ which is a cyclic lift of the weak Calabi-Yau structure constructed by the first-named author and Iyama. We also show the existence of a left Calabi-Yau structure on the dg bounded derived category of $Λ$. This is a non-commutative generalization of a result by Brav and Dyckerhoff. By combining the existence of the right Calabi-Yau structure on the dg singularity category with a structure theorem by Keller and the second-named author, we deduce that under suitable hypotheses, the singularity category associated with $Λ$ is triangle equivalent to a generalized cluster category in the sense of Amiot. |
| title | Calabi-Yau structures on derived and singularity categories of symmetric orders |
| topic | Representation Theory Algebraic Geometry Category Theory K-Theory and Homology 18G80, 18G35, 13C14 |
| url | https://arxiv.org/abs/2512.03836 |