Homotopy finiteness of some DG categories from algebraic geometry
Alexander I. EfimovSteklov Mathematical Institute of RAS and National University Higher School of Mathematics, Moscow, Russia
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