Homotopy finiteness of some DG categories from algebraic geometry

TL;DR

This paper proves that the derived category of coherent sheaves on certain algebraic schemes is homotopically finitely presented, confirming a conjecture of Kontsevich, and extends the result to DG categories of matrix factorizations.

Contribution

It establishes homotopy finiteness of derived categories for schemes over characteristic zero fields and extends to DG categories of matrix factorizations, using categorical resolutions and gluing techniques.

Findings

Derived category $D^b_{coh}(Y)$ is homotopically finitely presented.

$D^b_{coh}(Y)$ is equivalent to a DG quotient of a smooth proper variety.

Analogous results hold for DG categories of matrix factorizations.

Abstract

In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\tilde{Y})/T,$ 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 \cite{KL}, and a theorem of Orlov \cite{Or} 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 $D^b_{coh}(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\mathbb{A}_{\mathrm{k}}^1$.