A generalized Bondal-Orlov full faithfulness criterion for Deligne-Mumford stacks

Let X, Y be smooth projective varieties over C. Let K be a bounded complex of coherent sheaves on X Y and let K Dᵇ₂₎₇ (X) Dᵇ₂₎₇ (Y) be the resulting Fourier-Mukai functor. There is a well-known criterion due to Bondal-Orlov for K to be fully faithful. This criterion was recently extended to smooth Deligne-Mumford stacks with projective coarse moduli schemes by Lim-Polischuk. We extend this to all smooth, proper Deligne-Mumford stacks over arbitrary fields of characteristic 0. Along the way, we establish a number of foundational results for bounded derived categories of proper and tame morphisms of noetherian algebraic stacks (e. g. , coherent duality).