Generically a quotient

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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).
  5. 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.
  6. 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.
  7. General principle, continued. For sufficiently large i the stabilizer groups of [U/R] act faithfully on the fibres of the vector bundle V_i.
  8. 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.
  9. We conclude that X = [W/GL_n] for some Noetherian, reduced irreducible algebraic space W.
  10. 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).

4 thoughts on “Generically a quotient

      • That’s an important question: are Noetherian algebraic stacks locally global quotients; we could also ask for etale local or smooth local. I don’t know the answer. (If X is separated DM and has a coarse moduli scheme, then the answer is yes etale locally, I think.) Maybe the answer is no in general (can’t remember if I’ve ever seen a counter example).

  1. Proposition 3.5.9 in Cycle Groups for Artin Stacks might be relevant:

    Let Y be a stack. Then Y admits a stratification by global quotient stacks if and only if for every geometric point $x: \mathrm{Spec} \to Y$ , the stabilizer group $\mathrm{Isom}(x, x)$ is affine.

    The results is slightly different (all geometric points instead of a geometric generic point and there is no assumption on irreducibility–reducibility is hidden in the definition of stratification by global quotient stacks, noetherian in the convention of stack as finite type over a field.) so that it achieves a necessary and sufficient condition.

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