Stein factorization

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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.

One version for schemes states the following:(EGA, III.4.3.1)

Let X be a scheme, S a locally noetherian scheme and a proper morphism. Then one can write

where is a finite morphism and is a proper morphism so that .

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[edit]

Set:

where SpecS is the relative Spec. The construction gives the natural map , which is finite since is coherent and f is proper. The morphism f factors through g and one gets , 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.)

References[edit]