Stein factorization

In algebraic geometry, the Stein factorization, introduced by 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.

In algebraic geometry, the Stein factorization, introduced by 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. ==Statement== One version for schemes states the following: Let X be a scheme, S a locally noetherian scheme and f: X \to S a proper morphism. Then one can write f = g \circ f' where g\colon S' \to S is a finite morphism and f'\colon X \to S' is a proper morphism so that 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) is connected for any s \in S' . It follows: Corollary: For any s \in S , the set of connected components of the fiber f^{-1}(s) is in bijection with the set of points in the fiber g^{-1}(s) . S' = \operatorname{Spec}_S f_* \mathcal{O}_X where Spec S is the relative Spec. The construction gives the natural map g\colon S' \to S , which is finite since \mathcal{O}_X is coherent and f is proper. The morphism f factors through g and one gets f'\colon X \to S' , which is proper. By construction, f'_* \mathcal{O}_X = \mathcal{O}_{S'} . One then uses the theorem on formal functions to show that the last equality implies f' has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)

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