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 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 $f:X\to S$ a proper morphism. Then one can write

$f=g\circ f^{\prime }$

where $g:S^{\prime }\to S$ is a finite morphism and $f^{\prime }:X\to S^{\prime }$ is a proper morphism so that $f{\text{'}}_{*}{𝒪}_X={𝒪}_{S^{\prime }}.$

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 $f{\text{'}}^{-1}(s)$ is connected for any $s\in S^{\prime }$. 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

Set:

$S^{\prime }={\mathrm{Spec}}_S{f}_{*}{𝒪}_X$

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

See also

• Contraction morphism

References

• Template:Hartshorne AG
• Template:EGA
• Template:Citation