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

${\displaystyle f=g\circ f'}$

where ${\displaystyle g\colon S'\to S}$ is a finite morphism and ${\displaystyle f'\colon X\to S'}$ is a proper morphism so that ${\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.}$

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 ${\displaystyle f'^{-1}(s)}$ is connected for any ${\displaystyle s\in S'}$. It follows:

Corollary: For any ${\displaystyle s\in S}$, the set of connected components of the fiber ${\displaystyle f^{-1}(s)}$ is in bijection with the set of points in the fiber ${\displaystyle g^{-1}(s)}$.

Proof

Set:

${\displaystyle S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}$

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

See also

• Contraction morphism

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