# Zariski's Main Theorem for stacks

Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$.

Assume there is a quasifinite, representable, surjective and smooth morphism $f:X\rightarrow\ Y$.

Is there a canonical way to write $f$ as $X\rightarrow Z \rightarrow Y$ where $X\rightarrow Z$ is an open immersion and $Z\rightarrow Y$ is finite representable? That is to describe $Z$ in terms of $X,Y$.

I'm not sure $Spec(C)$, where $C$ is the integral closure of $\mathcal O_Y$ in $f_{*} \mathcal O_X$ works as in Laumon-Moret Bailey, or does it? Also, I'm not assuming $Y$ is normal so the Stein factorization might not work.

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Lack of properness of $f$ makes Stein factorization irrelevant, not lack of normality. And properness of $Y$ is too strong, as it blocks localizing on $Y$; relax properness to "finite type" (or less). The quasi-coherent $C$ is a limit of coherent $O_Y$-submodules (as on any DM stack), hence of coherent $O_Y$-subalgebras $C_i$ via integrality. To show $X\rightarrow{\rm{Spec}}(C_i)$ an open immersion for large $i$, it suffices to work etale-locally on $Y$, so you can assume $Y$ is a scheme (dropping properness!), hence $X$ too. By EGA IV$_3$, 8.12.3 and ZMT, QED. (Isn't this all in L-MB?) – user36938 Jul 24 '13 at 12:00