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 ΦK DᵇCoh(X) → DᵇCoh(Y) be the resulting Fourier-Mukai functor. There is a well-known criterion due to Bondal-Orlov for ΦK 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).