 Stein factorization

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:The morphism $g$ is proper with geometrically connected scheme-theoretic fibres, $h$ is a finite morphism, and $g_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Z$. The coherent sheaf $f_{\ast }{\mathcal{O}}_X$ makes the relative spectrum finite over $Y$; the connectedness assertion is the substantive part of the theorem.^[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.