Bondal-Orlov Fully Faithfulness Criterion for Deligne-Mumford Stacks

TL;DR

This paper extends the Bondal-Orlov fully faithfulness criterion from smooth projective varieties to smooth proper Deligne-Mumford stacks with projective coarse moduli spaces, providing a broader applicability in derived category theory.

Contribution

The authors generalize the Bondal-Orlov criterion to Deligne-Mumford stacks, enabling the assessment of functor full faithfulness in this more general setting.

Findings

Extended the criterion to DM stacks with projective coarse moduli space

Provided conditions under which an exact functor is fully faithful

Simplified the verification process for functor faithfulness in derived categories

Abstract

Suppose $F\colon \mathcal{D}(X)\to \mathcal{T}$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety $X$ to a triangulated category $\mathcal{T}$. If $F$ possesses left and right adjoints, then the Bondal-Orlov criterion gives a simple way of determining if $F$ is fully faithful. We prove a natural extension to the case when $X$ is a smooth and proper DM stack with projective coarse moduli space.