Stein factorization
Template:Short description In algebraic geometry, the Stein factorization, introduced by Template:Harvs 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.
Statement
One version for schemes states the following: Template:Harv
Let X be a scheme, S a locally noetherian scheme and <math>f: X \to S</math> a proper morphism. Then one can write
- <math>f = g \circ f'</math>
where <math>g\colon S' \to S</math> is a finite morphism and <math>f'\colon X \to S'</math> is a proper morphism so that <math>f'_* \mathcal{O}_X = \mathcal{O}_{S'}.</math>
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 <math>f'^{-1}(s)</math> is connected for any <math>s \in S'</math>. It follows:
Corollary: For any <math>s \in S</math>, the set of connected components of the fiber <math>f^{-1}(s)</math> is in bijection with the set of points in the fiber <math>g^{-1}(s)</math>.
Proof
Set:
- <math>S' = \operatorname{Spec}_S f_* \mathcal{O}_X</math>
where SpecS is the relative Spec. The construction gives the natural map <math>g\colon S' \to S</math>, which is finite since <math>\mathcal{O}_X</math> is coherent and f is proper. The morphism f factors through g and one gets <math>f'\colon X \to S'</math>, which is proper. By construction, <math>f'_* \mathcal{O}_X = \mathcal{O}_{S'}</math>. One then uses the theorem on formal functions to show that the last equality implies <math>f'</math> has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)