# Zariski's main theorem in the form of Grothendieck, universal properties

Zariski's main theorem in the form of Grothendieck [EGA 4-3, Thm. 8.12.6] asserts that any quasi-finite morphism $a : V^{\circ} → W$ factors as the composition of an open immersion and a finite morphism, say $$V^{\circ} \xrightarrow{\alpha\text{, open immersion}} V \xrightarrow{\beta\text{, finite}} W.$$

I believe that if $V^{\circ}$ is normal, this factorisation satisfies a number of universal properties. Below, I list three properties which I believe hold true at least for varieties defined over $\mathbb C$. All this must be known to experts, but I was not able not find anything in the literature. I would like to ask if anyone could point me to a reference for these matters.

(1) The intermediate variety $V$ can be chosen to be normal. With this choice, the variety $V$ and the factorisation is unique up to unique isomorphism.

(2) If $G$ is any group which acts algebraically on $V°$ and $W$ and if $a$ is equivariant with respect to these actions, then $G$ also acts on $V$ and the morphisms $α$ and $β$ are equivariant with respect to these actions.

(3) If $W° := \mathrm{Image}(a)$ is open in $W$ and $a : V° \to W°$ is Galois with group $G$, then $β: V → W$ is Galois with group $G$.

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I don't think that (3) holds. This would imply that every ${\bf Z}/n$-torsor on $W^\circ$ can be extended to a ${\bf Z}/n$-torsor over $W$. There is an obstruction for this to be possible, which lies in a $H^2$ group. – Damian Rössler Apr 29 '13 at 14:00
I guess that "Galois with group $G$" should not be interpreted as being necessarily étale. Anyway, it seems to me that (2) and (3) follows immediately from (1), personally I would not worry with a reference. Of course, I am not very good with references. – Angelo Apr 29 '13 at 14:22
Laumon and Moret-Bailly, Chapitre 16, particularly Corollaire 16.6.2. Although I agree with Angelo in spirit, it is certainly helpful to readers of an article to have a secondary source that they can refer to as necessary. – Jason Starr Apr 29 '13 at 17:04
Thanks for the comments so far. Angelo is right: I did not intent "Galois" to imply "étale", but did not say so clearly. I think of "Galois" as "quotient map for quotient by a finite group". – Stefan Kebekus Apr 29 '13 at 17:59