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## Homotopy finiteness of some DG categories from algebraic geometry

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary resolution of singularities. We believe that homotopy finiteness holds also over perfect fields of finite characteristic. We also prove the analogous result for $\Z/2$-graded DG categories of coherent matrix factorizations on such schemes. In both cases, we represent our DG category as a DG quotient of a smooth and proper DG category by a subcategory generated by one object (we call this a smooth categorical compactification).