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In algebraic geometry, the '''Stein factorization''', introduced by {{harvs|txt|authorlink=Karl Stein (mathematician)|first=Karl |last=Stein|year=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.
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One version for schemes states the following:{{harv|EGA|loc=4.3.1}}
<blockquote>
Let ''X'' be a [[scheme]], ''S'' a locally noetherian scheme and <math>f: X \to S</math> a [[proper morphism]]. Then one can write
:<math>f = g \circ f'</math>
where <math>g: S' \to S</math> is a [[finite morphism]] and <math>f': X \to S'</math> is a proper morphism so that <math>f'_* \mathcal{O}_X = \mathcal{O}_{S'}</math>.
</blockquote>
 
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>f'^{-1}(s)</math> is connected for any <math>s \in S</math>. It follows:
 
'''Corollary''': For any <math>s \in S</math>, the set of connected components of the fiber <math>f^{-1}(s)</math> is in bijection with the set of points in the fiber <math>g^{-1}(s)</math>.
 
== Proof ==
Set:
:<math>S' = </math> '''Spec'''<math>f_* \mathcal{O}_X</math>
where '''Spec''' is the [[relative Spec|relative '''Spec''']]. The construction gives us the natural map <math>g: S' \to S</math>, which is finite since <math>\mathcal{O}_X</math> is coherent and ''f'' is proper. ''f'' factors through ''g'' and so we get <math>f': X \to S'.</math>, which is proper. By construction <math>f'_* \mathcal{O}_X = \mathcal{O}_{S'}</math>. One then uses the [[theorem on formal functions]] to show that the last equality implies <math>f'</math> has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)
 
== References ==
{{reflist}}
The writing of this article benefited from [http://amathew.wordpress.com/2010/12/23/the-baby-version-of-zariskis-main-theorem/].
*{{Hartshorne AG}}
*{{EGA|book=III-1}}
*{{Citation | last1=Stein | first1=Karl | title=Analytische Zerlegungen komplexer Räume | doi=10.1007/BF01343331 | id={{MR|0083045}} | year=1956 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=132 | pages=63–93}}
 
 
 
[[Category:Algebraic geometry]]
 
 
{{geometry-stub}}

Latest revision as of 09:08, 9 April 2014

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