Homotopy finiteness of some DG categories from algebraic geometry

TL;DR

The paper proves that for any separated finite-type scheme $Y$ over a characteristic zero field, the DG enhancement of $D^b_{coh}(Y)$ is homotopically finitely presented, validating Kontsevich’s conjecture. It strengthens this by showing $D^b_{coh}(Y)$ is Morita-equivalent to a DG quotient $D^b_{coh}( ilde{Y})/T$ with $ ilde{Y}$ smooth and proper and $T$ generated by one object, using Kuznetsov–Lunts categorical resolutions and Orlov’s stability of smooth/proper DG categories under gluing. The authors also extend these results to $oldsymbol{Z}/2$-graded DG categories of coherent matrix factorizations, where a gluing of a finite set of MF-categories on smooth, proper models yields a similar hfp property. A central methodological theme is the construction of smooth categorical compactifications via Auslander-type constructions and categorical blow-ups, enabling explicit localizations with kernels generated by single objects. These tools provide a principled path to universal enhancements and compactifications in noncommutative geometric contexts with both sheaf and matrix-factorization data.

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 $\mathbb{A}_{\mathrm{k}}^1$.

Theorems & Definitions (183)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Definition 1.7
• Theorem 1.8
• Conjecture 1.9
• Theorem 1.10