Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has geometrically closed fibers. I'm interested in an applicaton of Stein factorisation on this meanwhile solved former question.
recall this rigidity lemma from Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online available)
Lemma. Let $X$, $Y$ and $Z$ be algebraic varieties over a field $k$. Suppose that $X$ is complete. If $f : X × Y \rightarrow Z$ is a morphism with the property that, for some $y \in Y (k)$, the fibre $X \times_k \{y\}$ is mapped to a point $z \in Z(k)$ then $f$ factors through the projection $pr_Y : X \times_k Y → Y$.
The proof starts with reduction to $k=\bar{k}$. In the linked thread I asked why it can reduced to the case $k=\bar{k}$? A very nice argument is given by Piotr Achinger, futhermore an elementary application of descent theory would also provide an answer.
The concern of this question is if it also possible to justify this reduction to $k=\bar{k}$ by applying Stein factorisation in a sophisticated way. first of all, following diagram with canonical vertical maps commute
$$ \require{AMScd} \begin{CD} X \times Y \times \bar{k} @>{g'} >> Specf_*(\mathcal{O}_{X \times Y} \otimes_k \bar{k}) @>{h'} >> Z \times \bar{k} \\ @VVV @VVV @VVV \\ X \times Y @>{g}>>Specf_*\mathcal{O}_{X \times Y} @>{h}>> Spec(R); \end{CD} $$
In we substitute $f$ by $g$ and resp. $f'$ by $g'$ we can futhermore assume that $f$ and $f'$ have connected fibers. Can we from here go ahead? The goal is to show that for every $y \in Y$ the product $X \times \{y\}$ is mapped to a point under assumption that we already know that for $f'$. Does this suffice to show that $f$ factorize over $pr_Y$? My idea was, if $Z$ has an ample bundle $\mathcal{L}$ then it's restrictions to closed image $f(X \times \{y\})$ stays ample and then consider the pullback $f^*\mathcal{L}$. If it is not ample, then $f(X \times \{y\})$ was a point. All restrictions and pullbacks can parallel aplied to $\mathcal{L} \otimes_k \bar{k}$ and using the diagram compared. this was my rough idea. does it make any sense or is Stein factorisation a wrong tool to attack this problem?
