Homotopy finiteness of some DG categories from algebraic geometry

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 sta...

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Bibliographic Details
Published in:arXiv.org 2018-12
Main Author: Efimov, Alexander I
Format: Article
Language:English
Subjects:
Categories
Gluing
Sheaves
Singularities
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Summary: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 \(\mathbb{A}_{\mathrm{k}}^1\).
ISSN:2331-8422