Generalizing Bondal-Orlov Criteria for Deligne-Mumford Stacks

Abstract

If $F:D^{b}_{coh}(\mathcal{X}) \to T$ is a functor with left and right adjoints from a proper smooth Deligne--Mumford stack with projective coarse moduli space to a triangulated category, there is a Bondal--Orlov criterion determining the full-faithfulness of $F$. We develop techniques that allow for the proof of this criterion absent the assumption that the course moduli space of $\mathcal{X}$ is projective. Furthermore, if the functor $F$ has a dg or infinity category enhancement, the assumption that $F$ has left and right adjoints may also be relaxed.