Stein factorization

In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism of schemes 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 → S {\displaystyle f:X\to S} a proper morphism. Then one can write

f = g ∘ f ′ {\displaystyle f=g\circ f'}

where g : S ′ → S {\displaystyle g\colon S'\to S} is a finite morphism and f ′ : X → S ′ {\displaystyle f'\colon X\to S'} is a proper morphism so that f ∗ ′ O X = O S ′ . {\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 f ′ − 1 ( s ) {\displaystyle f'^{-1}(s)} is connected for any s ∈ S ′ {\displaystyle s\in S'} . It follows:

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

Set:

S ′ = Spec S ⁡ f ∗ O X {\displaystyle S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}

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

• Contraction morphism

• 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