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 {{{2}}})
Let X be a scheme, S a locally noetherian scheme and [math]\displaystyle{ f: X \to S }[/math] a proper morphism. Then one can write
- [math]\displaystyle{ f = g \circ f' }[/math]
where [math]\displaystyle{ g\colon S' \to S }[/math] is a finite morphism and [math]\displaystyle{ f'\colon X \to S' }[/math] is a proper morphism so that [math]\displaystyle{ 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]\displaystyle{ f'^{-1}(s) }[/math] is connected for any [math]\displaystyle{ s \in S }[/math]. It follows:
Corollary: For any [math]\displaystyle{ s \in S }[/math], the set of connected components of the fiber [math]\displaystyle{ f^{-1}(s) }[/math] is in bijection with the set of points in the fiber [math]\displaystyle{ g^{-1}(s) }[/math].
Proof
Set:
- [math]\displaystyle{ S' = \operatorname{Spec}_S f_* \mathcal{O}_X }[/math]
where SpecS is the relative Spec. The construction gives the natural map [math]\displaystyle{ g\colon S' \to S }[/math], which is finite since [math]\displaystyle{ \mathcal{O}_X }[/math] is coherent and f is proper. The morphism f factors through g and one gets [math]\displaystyle{ f'\colon X \to S' }[/math], which is proper. By construction, [math]\displaystyle{ f'_* \mathcal{O}_X = \mathcal{O}_{S'} }[/math]. One then uses the theorem on formal functions to show that the last equality implies [math]\displaystyle{ f' }[/math] has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)
See also
References
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9
- 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. http://www.numdam.org/item/PMIHES_1961__11__5_0.
- Stein, Karl (1956), "Analytische Zerlegungen komplexer Räume", Mathematische Annalen 132: 63–93, doi:10.1007/BF01343331, ISSN 0025-5831
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