JournalsjemsVol. 22, No. 9pp. 2879–2942

Homotopy finiteness of some DG categories from algebraic geometry

  • Alexander I. Efimov

    Steklov Mathematical Institute of RAS and National University Higher School of Mathematics, Moscow, Russia
Homotopy finiteness of some DG categories from algebraic geometry cover
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Abstract

In this paper, we prove that the bounded derived category Dcohb(Y)D^b_{\mathrm {coh}}(Y) of coherent sheaves on a separated scheme YY 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: Dcohb(Y)D^b_{\mathrm {coh}}(Y) is equivalent to a DG quotient Dcohb(Y~)/T,D^b_{\mathrm {coh}}(\tilde{Y})/T, where Y~\tilde{Y} is some smooth and proper variety, and the subcategory TT 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\mathbb{Z}/2-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of Dcohb(Y~)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 Ak1\mathbb{A}_{\mathrm{k}}^1.

Cite this article

Alexander I. Efimov, Homotopy finiteness of some DG categories from algebraic geometry. J. Eur. Math. Soc. 22 (2020), no. 9, pp. 2879–2942

DOI 10.4171/JEMS/979