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1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?

2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack $[X/G]$ with a non necessarily finite group $G$? For example, Geigle and Lenzing give such a presentation for stacky projective lines here.

3) What about Artin stacks?

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Please define what you mean by "locally" in (1). For example, the Lemma 2.2.3 reference in Jacob Bell's answer (for suitable separated DM stacks) is weaker than "Zariski-locally" but stronger (in general) than "etale-locally". – user30379 Feb 4 '13 at 4:05
@pranavk: I deliberately did not specify, because I am interested in any kind of results. – Nullstellensatz Feb 4 '13 at 16:41

I am not an expert in the algebraic category, however, I know that 2) is an open problem in the differentiable category; it is not known if every smooth orbifold is a global quotient stack. It is true in the differentiable category when you ask for the orbifold to be effective ("reduced" in algebrogeometric lingo). I have good (topos-theoretic) reasons to suspect that if $\mathscr{X}$ is a Deligne-Mumford stack such that for each scheme $T,$ the subgroupoid of $\mathscr{X}\left(T\right)$ on those maps $T \to \mathscr{X}$ which are etale is equivalent to a set, then $\mathscr{X}$ is a global quotient. (This condition is basically means that the isotropy groups act faithfully). I would be very surprised if 3) were true.

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the one I know without having to look in the literature is 1)

Lemma 2.2.3 of

I think 2) is true as well (maybe you need to add the adjective tameness appropriately?) and for 3) there should be a result of Kresch saying that your stack can be stratified by quotient stacks. But I'd have to look this stuff up.


for 1) I should say etale topology.

for 2) I was thinking about this result (Theorem 4.4 and Proposition 5.1) by Kresch

for 3), the result I was misremembering was Proposition 3.5 of, and the first paragraph of the proof. (it's for stacks with affine stabilisers)


There is a paper by Edidin-Hassett-Kresch-Vistoli where the investigate when an Artin stack is a quotient stack. It turns out that this is closely related to the pushforward of the structure sheaf of a smooth atlas to admit a surjection from a vector bundle. Here is the review by Vezzosi.

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