# Stein factorization

In algebraic geometry, the Stein factorization, introduced by Template:Harvs for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.

One version for schemes states the following:Template:Harv

Let X be a scheme, S a locally noetherian scheme and $f:X\to S$ a proper morphism. Then one can write

$f=g\circ f'$ The existence of this decomposition itself is not difficult. (see below) But, by Zariski's connectedness theorem, the last part in the above says that the fiber $f'^{-1}(s)$ is connected for any $s\in S$ . It follows:

## Proof

Set:

$S'=$ Spec$f_{*}{\mathcal {O}}_{X}$ where Spec is the relative Spec. The construction gives us the natural map $g:S'\to S$ , which is finite since ${\mathcal {O}}_{X}$ is coherent and f is proper. f factors through g and so we get $f':X\to S'.$ , which is proper. By construction $f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}$ . One then uses the theorem on formal functions to show that the last equality implies $f'$ has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)