In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism of schemes 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: <blockquote> Let X be a scheme, S a locally noetherian scheme and a proper morphism. Then one can write
where is a finite morphism and is a proper morphism so that </blockquote>
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 is connected for any . It follows:
Corollary: For any , the set of connected components of the fiber is in bijection with the set of points in the fiber .
Set:
where Spec<sub>S</sub> is the relative Spec. The construction gives the natural map , which is finite since is coherent and f is proper. The morphism f factors through g and one gets , which is proper. By construction, . One then uses the theorem on formal functions to show that the last equality implies has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)