New enhancements of derived categories of coherent sheaves and applications

We introduce new enhancements for the bounded derived category Db(Coh(X)) of coherent sheaves on a suitable scheme X and for its subcategory Perf(X) of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of Perf(X) to geometric smoothness of X, and for proving homological smoothness of Db(Coh(X)). Moreover, we characterize properness of Perf(X) and Db(Coh(X)) geometrically.