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

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

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.)

References

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
Stein, Karl (1956), "Analytische Zerlegungen komplexer Räume", Mathematische Annalen, 132: 63–93, doi:10.1007/BF01343331, ISSN 0025-5831, MR 0083045

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