Does Zariski's Main Theorem come with a canonical factorization?

Question

Zariski's Main Theorem (EGA IV, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as $X\xrightarrow{g} Z\xrightarrow{h} Y$, where $g$ is an open immersion and $h$ is finite.

Is there a canonical choice for the factorization $f=h\circ g$, at least under some circumstances?

For example, suppose $f$ factors as $X\to U\to Y$, where $X\to U$ is finite étale and $U\to Y$ is a Stein open immersion (i.e. the pushforward of $\mathcal O_U$ is $\mathcal O_Y$). Then I'm pretty sure the Stein factorization $X\to \mathit{Spec}_Y(f_*\mathcal O_X)\to Y$ witnesses Zariksi's Main Theorem (i.e. is an open immersion followed by a finite map).

In general, when does the Stein factorization witness ZMT? In the cases where it fails to witness ZMT (e.g. $X$ finite over an affine open in $Y$), is there some other canonical witness?

Answer 1

I think an initial object exists if you work with integral excellent schemes (maybe integral is not really necessarily, but then require that $X$ be schematically dense in $Z$).

So suppose $X, Y$ are integral and excellent. Consider all possible factorizations $X\to Z_{\alpha} \to Y$ with $Z_{\alpha}$ integral. Then $K(Z_{\alpha})=K(X)$. For any pair $Z_{\alpha}, Z_{\beta}$, the closure $Z_{\gamma}$ of $X$ in $Z_{\alpha}\times_Y Z_{\beta}$ gives a factorization $X\to Z_{\gamma}\to Y$ with $Z_{\gamma}$ dominating $Z_{\alpha}$ and $Z_{\beta}$, finite over $Y$, and $X\to Z_{\gamma}$ is an open immersion (one checks that $X\to Z_{\gamma}$ is an immersion, hence open in some closed subscheme $F$, but $X$ is birational to $Z_{\gamma}$, so $F=Z_{\gamma}$). Thus we can consider the projective limite $Z$ of the $Z_{\alpha}$'s.

By construction $Z$ is affine and integral over $Y$. As $Z_{\alpha}$ is dominated by the normalization $\widetilde{Y}$ of $Y$ in $K(X)$ and $\widetilde{Y}$ is finite over $Y$ by excellent hypothesis, $Z$ is finite over $Y$. It remains to see that the canonical map $X\to Z$ is an open immersion. This property is local over $Y$. So we suppose $Y$ is affine. Cover $X$ by principal affine open subsets $D(h)$'s of some $Z_{\alpha_0}$. Then $D(h) \to D_Z(h)$ is a closed immersion because $D(h)\to D_{Z_{\alpha_0}}(h)$ is, and it is birational, so it is an isomorphism and we are done.

It would interesting to compute explicitely the projective limite in some concrete situations. For exemple, consider a surface $S$, finite over $\mathbb A^2_{\mathbb C}$, with non-normal locus $\Delta$. Let $X$ be an open subset of $S$ with $\Delta\cap X$ non-empty and not equal to $\Delta$. The inclusion $X\to S=Z_{\alpha_0}$ is a factorisation. But what is the $Z$ constructed as above ?

Answer 2

I realized that I completely missed the second part of the question (the example). Note that ZMT implies that $f$ is a quasi-affine morphism. Then $X\to \mathit{Spec}(f_*\mathcal O_X)$ is always an open immersion (see stack project, tag 01SM). So the Stein factorization witness ZMT if and only if $f_*\mathcal O_X$ is finite over $\mathcal O_Y$.

Some comments: one should note that in general, the quasi-coherent algebra $f_*\mathcal O_X$ is not finite over $\mathcal O_Y$ and even worse, the morphism $\mathit{Spec}(f_*\mathcal O_X)\to Y$ may not be of finite type (take $Y$ an algebraic variety and $f$ an open immersion. Then $\mathcal O(X)$ is or not finitely generated is related to Hilbert's 14th problem). Now consider a ZMT factorisation $X\to Z\to Y$. If the complementary of $X$ in $Z$ only consists in points of depth at least 2 (see discussions here), then $f_*\mathcal O_X=h_*\mathcal O_Z$ is finite and we are happy. This happens when $X$ is normal (or with non-normal locus finite over $Y$) and surjective to $Y$ with complementary in $Z$ of codimension at least 2. But I don't have a general criterion.