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In algebraic geometry, the Stein factorization, introduced by 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:
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 is connected for any . It follows: Corollary: For any , the set of connected components of the fiber is in bijection with the set of points in the fiber . == Proof == Set: : Spec where Spec is the relative Spec. The construction gives us the natural map , which is finite since is coherent and ''f'' is proper. ''f'' factors through ''g'' and so we get , which is proper. By construction . One then uses the theorem on formal functions to show that the last equality implies has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stein factorization」の詳細全文を読む スポンサード リンク
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