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.

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 \( g: S' \to S \)is a finite morphism and f': 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 \( f'^{-1}(s) i \) s 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' = Specf_* \mathcal{O}_X \(

where Spec is the relative Spec. The construction gives us the natural map \( g: S' \to S \), which is finite since \( \mathcal{O}_X \) is coherent and f is proper. f factors through g and so we get \( f': X \to S' \)., 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

The writing of this article benefited from [1].

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 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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