Let $f : X \rightarrow Y$ be a smooth proper morphism of noetherian schemes. Consider the Stein factorization $X \xrightarrow{f'} X' \xrightarrow{g} Y$, where $f'$ has connected fibers and $g$ is a finite map. I have the following question
Question : Is it possible to claim something more about the properties of $f'$ and $g$. For example is it possible that either one or both of these maps are smooth/flat? How about in char 0?
My thoughts so far : If I want to prove that $g$ is etale then I must have $f_*\mathcal{O}_X$ locally free. This statement will follow from Grauert's theorem once I could prove that number of connected components of fibers is constant. I am not sure how to proceed from here or even if this is a correct line of reasoning.
I would appreciate some counterexamples if nothing further could be said about the maps $f'$ and $g$.
Let me assume characteristic 0 (I am a bit afraid of positive characteristic). Then indeed, $f'$ is smooth and $g$ is etale. Let us first show the second.
Since $g$ is finite, it is enough to check it is unramified. If $x'$ is a ramification point and $y = g(x')$, then $g^{-1}(y)$ is non-reduced at $x'$, hence ${f'}^{-1}(y)$ is also non-reduced, hence not smooth.
To show that $f'$ is smooth note that smoothness of a morphism is etale-local over the target, and $X'$ is etale-locally isomorphic to $Y$ by etaleness of $g$, so smoothness of $f'$ follows from that of $f$.