Stein factorization

In algebraic geometry, the Stein factorization, introduced by 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:

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 $f'_* \mathcal_X = \mathcal_.$

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 $f'^(s)$ is connected for any $s \in S$. It follows:

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

Proof

Set:
:$S' = \operatorname_S f_* \mathcal_X$
where Spec_''S'' is the relative Spec. The construction gives the natural map $g\colon S' \to S$, which is finite since $\mathcal_X$ 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, $f'_* \mathcal_X = \mathcal_$. 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.)

See also

*Contraction morphism

References

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*{{Citation , last1=Stein , first1=Karl , authorlink=Karl Stein (mathematician) , title=Analytische Zerlegungen komplexer Räume , doi=10.1007/BF01343331 , mr=0083045 , year=1956 , journal=Mathematische Annalen , issn=0025-5831 , volume=132 , pages=63–93

Algebraic geometry