Let $X$ be a quasi-projective scheme, the followings are quite useful.
- Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
- Every coherent sheaf has trivial higher cohomology groups after tensoring with a suitable line bundle.
- Every coherent sheaf is a quotient of a finite rank locally free sheaf.
Question Are there any analogous discussions for Deligne- Mumford/Artin Stacks. My primary interests would be on the stacks of principal bundles/Higgs bundles on a curve. Thank you in advance.