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| Natura: | Preprint |
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2013
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| Accesso online: | https://arxiv.org/abs/1308.0135 |
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| _version_ | 1866915141692424192 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\tilde{Y})/T,$ where $\tilde{Y}$ is some smooth and proper variety, and the subcategory $T$ is generated by a single object.
The proof uses categorical resolution of singularities of Kuznetsov and Lunts \cite{KL}, and a theorem of Orlov \cite{Or} stating that the class of geometric smooth and proper DG categories is stable under gluing.
We also prove the analogous result for $\mathbb{Z}/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D^b_{coh}(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\mathbb{A}_{\mathrm{k}}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1308_0135 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Homotopy finiteness of some DG categories from algebraic geometry Efimov, Alexander I. Algebraic Geometry Category Theory Rings and Algebras In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\tilde{Y})/T,$ where $\tilde{Y}$ is some smooth and proper variety, and the subcategory $T$ is generated by a single object. The proof uses categorical resolution of singularities of Kuznetsov and Lunts \cite{KL}, and a theorem of Orlov \cite{Or} stating that the class of geometric smooth and proper DG categories is stable under gluing. We also prove the analogous result for $\mathbb{Z}/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D^b_{coh}(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\mathbb{A}_{\mathrm{k}}^1$. |
| title | Homotopy finiteness of some DG categories from algebraic geometry |
| topic | Algebraic Geometry Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/1308.0135 |