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Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all $i$ there is an etale finite surjective morphism $\phi_i: Z_i \to \mathcal{U}_i$, where $Z_i$ is a scheme? (Of course it will be true if we remove the condition that the morphisms $\phi_i$ must be finite.)

Upd. Oh, I think by Theorem 6.1 of Laumon, Moret-Bailly "Champs algebriques" it is equivalent to saying that $\mathcal{X}$ admits a Zariski covering by quotient stacks.

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Proposition 5.2 of https://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf has a partial answer to you question. If X is a DM stack whose coarse module space is a scheme, then it is Zariski locally a quotient stack if and only if it is Zariski locally a finite group quotient.

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Yes! A nice reference for this is Lurie's new book Theorem 1.2.5.9. where he began (more or less) with what you said as a definition.

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  • $\begingroup$ I am confused by your reference. As the OP says the statement is well known if you do not require the étale morphisms to be finite. What am I missing? $\endgroup$ – Denis Nardin Sep 7 '16 at 20:51

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