= Stein factorization
{c}
{wiki}
For a <proper morphism> $f:X\to Y$ with $Y$ a <locally Noetherian scheme>, Stein factorization separates its connected-fiber part from its finite part:
$$
X\xrightarrow{g}Z\xrightarrow{h}Y,\qquad Z=\operatorname{Spec}_Y(f_*\mathcal O_X).
$$
The <morphism> $g$ is proper with geometrically connected <scheme-theoretic fibres>, $h$ is a <finite morphism>, and $g_*\mathcal O_X=\mathcal O_Z$. The <coherent sheaf> $f_*\mathcal O_X$ makes the relative spectrum finite over $Y$; the connectedness assertion is the substantive part of the theorem.<The Stacks project>[Theorem 37.53.4, tag 03H0]{ref}
For a <finite morphism>, $Z=X$ and $g$ is the identity. For $X=Y\times\mathbb P^1$ with the projection to $Y$, the intermediate <scheme> is $Y$. Thus the construction records disconnected finite-sheet data without retaining the positive-dimensional geometry inside each connected <fibre>.
If $X$ is normal, so is $Z$. For a surjective <proper morphism> from a <smooth algebraic surface> to a <smooth algebraic curve> over an <algebraically closed field>, $Z$ is therefore a <smooth algebraic curve>. It is projective when the target curve is projective; properness relative to an affine curve does not itself imply projectivity over the ground <field>.
Back to article page
