Stein factorization

In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) 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.

Statement[edit]

One version for schemes states the following:(EGA, III.4.3.1)

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

$f=g\circ f'$

where $g\colon S'\to S$ is a finite morphism and $f'\colon X\to S'$ is a proper morphism so that $f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.$

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:

Corollary: For any $s\in S$ , the set of connected components of the fiber $f^{-1}(s)$ is in bijection with the set of points in the fiber $g^{-1}(s)$ .

Proof[edit]

Set:

S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}

where Spec_S is the relative Spec. The construction gives the natural map $g\colon S'\to S$ , which is finite since ${\mathcal {O}}_{X}$ is coherent and f is proper. The morphism f factors through g and one gets $f'\colon X\to S'$ , 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.)