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In this paper, we prove that the bounded derived category of coherent sheaves on a separated scheme of finite type over a field k of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: is equivalent to a DG quotient where is some smooth and proper variety, and the subcategory is generated by a single object.
The proof uses categorical resolution of singularities of Kuznetsov and Lunts [KL], and a theorem of Orlov [Or1] stating that the class of geometric smooth and proper DG categories is stable under gluing.
We also prove the analogous result for -graded DG categories of coherent matrix factorizations on such schemes. In this case instead of we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over .
Cite this article
Alexander I. Efimov, Homotopy finiteness of some DG categories from algebraic geometry. J. Eur. Math. Soc. 22 (2020), no. 9, pp. 2879–2942DOI 10.4171/JEMS/979