In this post I want to outline an argument that proves “most” algebraic stacks are generically “global” quotient stacks. I don’t have the time to add this to the stacks project now, but I hope to return to it in the not too distant future.

To fix ideas suppose that X is a Noetherian, reduced, irreducible algebraic stack whose geometric generic stabilizer is affine. Then I would like to show there exists a dense open substack U ⊂ X such that U ≅ [W/GL_n] for some Noetherian scheme W endowed with action of GL_n. The proof consists in repeatedly replacing X by dense open substacks each of which has some additional property:

- We may assume that X is a gerbe, i.e., that there exists an algebraic space Y and a morphism X —> Y such that X is a gerbe over Y. This follows from Proposition Tag 06RC.
- We may assume Y is an affine Noetherian integral scheme. This holds because X —> Y is surjective, flat, and locally of finite presentation, so Y is reduced, irreducible, and locally Noetherian by descent. Thus we get what we want by replacing Y be a nonempty affine open.
- We may assume there exists a surjective finite locally free morphism Z —> Y such that there exists a morphism s : Z —> X over Y. Namely, pick a finite type point of the generic fibre of X —> Y and do a limit argument.
- We may assume the projections R = Z ×_X Z —> Z are affine. Namely, the geometric generic fibres of R —> Z ×_Y Z are torsors under the geometric generic stabilizer which we assumed to be affine. A limit argument does the rest (note that we may shrink Z and Z ×_Y Z by shrinking Y).
- We may assume the projections s, t : R —> Z are free, i.e., s_*O_R and t_*O_R are free O_Z-modules. This follows from generic freeness.
- General principle. Suppose that (U, R, s, t, c) is a groupoid scheme with U, R affine and s, t free and of finite presentation. Consider the morphism p : U —> [U/R]. Then p_*O_U is a filtered colimit of finite free modules V_i on the algebraic stack [U/R]. This follows from a well known trick with basis elements.
- General principle, continued. For sufficiently large i the stabilizer groups of [U/R] act faithfully on the fibres of the vector bundle V_i.
- General principle, continued. [U/R] ≅ [W/GL_n] for some algebraic space W and integer n. Namely W is the quotient by R of the frame bundle of the vector bundle V_i.
- We conclude that X = [W/GL_n] for some Noetherian, reduced irreducible algebraic space W.
- The set of points where W is not a scheme is GL_n-invariant and not dense, hence we may assume W is a scheme by shrinking. (I think this works — there should be something easier you can do here, but I don’t see it right now.)

Note that we can’t assume that W is affine (a counter example is X = [Spec(k)/B] where B is the Borel subgroup of SL_{2, k} and k is a field). But with a bit more work it should be possible to get W quasi-affine as in the paper by Totaro (which talks about the harder question of when the entire stack X is of the form [W/GL_n] and relates it to the resolution property).