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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.
Efimov A.
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).
Keywords: resolution of singularitiesdifferential graded categorieshomotopy finitenessVerdier localizationderived categories
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