Homotopy finiteness of some DG categories from algebraic geometry

Abstract

In this paper, we prove that the bounded derived category $D_{\mathrm{coh}}^b(Y)$ 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: $D_{\mathrm{coh}}^b(Y)$ is equivalent to a DG quotient $D_{\mathrm{coh}}^b(\tilde{Y})/T,$ 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 [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 $\mathbb{Z}/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D_{\mathrm{coh}}^b(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over ${\mathbb{A}}_{\mathrm{k}}^1$.