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## Homotopy finiteness of some DG categories from algebraic geometry

Journal of the European Mathematical Society. 2020. Vol. 22. No. 9. P. 2879-2942.

In this paper, we prove that the bounded derived category D-coh(b) (Y) of coherent sheaves on a separated scheme Y 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: D-coh(b) (Y) is equivalent to a DG quotient D-coh(b) ((Y) over tilde)/T, where (Y) over tilde 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 [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 Z/2-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of D-coh(b) ((Y) over tilde) we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over A(k)(1).