# Journal of the European Mathematical Society

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**Published online first: 2020-06-15**

**DOI: 10.4171/JEMS/979**

Homotopy finiteness of some DG categories from algebraic geometry

Alexander I. Efimov^{[1]}(1) Steklov Mathematical Institute of RAS and National University Higher School of Mathematics, Moscow, Russia

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

*Keywords: *Derived categories, differential graded categories, homotopy finiteness, Verdier localization, resolution of singularities

Efimov Alexander: Homotopy finiteness of some DG categories from algebraic geometry. *J. Eur. Math. Soc.* Electronically published on June 15, 2020. doi: 10.4171/JEMS/979 (to appear in print)