Calabi-Yau completions for roots of dualizing dg bimodules
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910764240994304 |
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| author | Hanihara, Norihiro |
| author_facet | Hanihara, Norihiro |
| contents | Roots of shifted Serre functors appear naturally in representation theory and algebraic geometry. We give an analogue of Keller's Calabi-Yau completion for roots of shifted inverse dualizing bimodules over dg categories. Given a positive integer $a$, we introduce the notion of the $a$-th root pair on smooth dg categories and define its Calabi-Yau completion. We prove that the Calabi-Yau completion has the Calabi-Yau property when the $a$-th root pair has certain invariance under an action of the cyclic group of order $a$, and observe that it is only twisted Calabi-Yau in general. Next, we establish a bijection between Adams graded Calabi-Yau dg categories of Gorenstein parameter $a$ and $a$-th root pairs on a dg category with the cyclic invariance. Applying this bijection, we prove that a certain operation on dg categories, called the $a$-Segre product, allows us to reproduce Calabi-Yau dg categories. Furthermore, we discuss the cluster category of these Calabi-Yau completions, and prove that it is a $\mathbb{Z}/a\mathbb{Z}$-quotient of the usual cluster category, which thereby establishes the $a$-th root versions of cluster categories. In the appendix, we give a generalization of Beilinson's theorem on tilting bundles on projective spaces to the setting of Adams graded dg categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Calabi-Yau completions for roots of dualizing dg bimodules Hanihara, Norihiro Representation Theory Commutative Algebra Algebraic Geometry Rings and Algebras Roots of shifted Serre functors appear naturally in representation theory and algebraic geometry. We give an analogue of Keller's Calabi-Yau completion for roots of shifted inverse dualizing bimodules over dg categories. Given a positive integer $a$, we introduce the notion of the $a$-th root pair on smooth dg categories and define its Calabi-Yau completion. We prove that the Calabi-Yau completion has the Calabi-Yau property when the $a$-th root pair has certain invariance under an action of the cyclic group of order $a$, and observe that it is only twisted Calabi-Yau in general. Next, we establish a bijection between Adams graded Calabi-Yau dg categories of Gorenstein parameter $a$ and $a$-th root pairs on a dg category with the cyclic invariance. Applying this bijection, we prove that a certain operation on dg categories, called the $a$-Segre product, allows us to reproduce Calabi-Yau dg categories. Furthermore, we discuss the cluster category of these Calabi-Yau completions, and prove that it is a $\mathbb{Z}/a\mathbb{Z}$-quotient of the usual cluster category, which thereby establishes the $a$-th root versions of cluster categories. In the appendix, we give a generalization of Beilinson's theorem on tilting bundles on projective spaces to the setting of Adams graded dg categories. |
| title | Calabi-Yau completions for roots of dualizing dg bimodules |
| topic | Representation Theory Commutative Algebra Algebraic Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2412.18753 |