New enhancements of derived categories of coherent sheaves and applications

We We introduce new enhancements for the bounded derived category D-b(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 D-b(Coh(X)). Moreover, we characterize properness of Perf(X) and D-b(Coh(X)) geometrically. (C) 2015 Elsevier Inc. All rights reserved.