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