# Serre's theorem on global generations on stacks

Let $$X$$ be a quasi-projective scheme, the followings are quite useful.

1. Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
2. Every coherent sheaf has trivial higher cohomology groups after tensoring with a suitable line bundle.
3. 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.

As for 3, there is a paper that settles the issue, namely a quasi-compact and quasi-separated algebraic stack has affine stabilizer groups at closed points and satisfies the resolution property if and only if it is the quotient stack of a quasi-affine scheme by an action of $$\mathop{GL}(n)$$ for some $$n$$.