Homotopy finiteness of the derived categories of coherent sheaves

I will briefly recall the notion of homotopy finiteness of enhanced triangulated categories. Then it will be explained how to show that derived categories of coherent sheaves on any separated scheme of finite type over $C$ are homotopically finite, as well as categories of matrix factorizations.
The proof uses Kuznetsov-Lunts construction of categorical resolution of singularities, which in turn uses Hironaka's theorem.