# On the local structure of stacks

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.

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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

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

Lemma 2.2.3 of http://arxiv.org/pdf/math/9908167v2.pdf

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.

EDIT:

for 1) I should say etale topology.

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.