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

中文翻译：

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代数几何中某些 DG 范畴的同伦有限性

在本文中，我们证明了在特征为零的域 $\mathrm{k}$ 上有限类型的分离方案 $Y$ 上相干滑轮的有界派生范畴 $D^b_{coh}(Y)$ 是同调的有限地呈现。这证实了康采维奇的猜想。我们实际上证明了一个更强的陈述：$D^b_{coh}(Y)$ 等价于 DG 商 $D^b_{coh}(\tilde{Y})/T,$ where $\tilde{Y}$是一些平滑和适当的品种，子类别 $T$ 是由单个对象生成的。证明使用 Kuznetsov 和 Lunts \cite{KL} 的奇点的分类分辨率，以及 Orlov \cite{Or} 的定理，表明几何光滑和适当的 DG 类别在胶合下是稳定的。我们还证明了此类方案上相干矩阵分解的 $\mathbb{Z}/2$-graded DG 类别的类似结果。