# A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a coarse moduli space.

Question:Is it true that $X$ is also a global quotient stack?

• Two questions on your question: 1) the normalization of a DM-stack is well-defined because normalization is an étale-local property, right ? is it well-defined for Artin stacks ? 2) What are the reasons for your proper/projective assumptions ? Do you have easy counterexamples without them ? – ACL Mar 28 '14 at 8:25
• @ACL: Regarding the need for "proper", one silly observation is that one does, at least, need "quasi-compact". Otherwise the disjoint union of $B\mu_n$ over all integers $n$ is a counterexample. But this counterexample doesn't address, say, "proper" versus "quasi-compact and locally finitely presented over the base". – Jason Starr Mar 28 '14 at 11:59
• @ACL: Also, I think the counterexamples of Edidin, Hassett, Kresch and Vistoli show why one should assume that $X$ is separated. Still, I don't see why one should prefer "proper" to "separated, quasi-compact and locally finitely presented" for this question. – Jason Starr Mar 28 '14 at 13:37
• @Jason Starr: You are right. People believe that a stack with the conditions you mentioned above should be a quotient stack. In my situation I can also assume these extra conditions. – matthew Mar 28 '14 at 14:03
• @ACL: in fact normalization commutes with smooth localization, hence is well-defined for Artin stacks. This is proved in Laumon and Moret-Bailly's book. – Matthieu Romagny Mar 28 '14 at 21:06