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.



Serre's theorem on global generations on stacks
About 1 & 2, I doubt there are sensible results without some hypothesis like the existence an ample family of line bundles that makes the stack actually a scheme, in fact a so-called divisorial scheme.


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 $n$.


It is:
Gross, Philipp: 
Tensor generators on schemes and stacks. 
*Algebr. Geom.* **4** (2017), no. 4, 501–522.


ArXiv version: <https://arxiv.org/abs/1306.5418> 


