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 f: X \to S a proper morphism. Then one can write

f = g \circ f'

where g: S' \to S is a finite morphism and f': 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).

Proof[edit]

Set:

S' = Specf_* \mathcal{O}_X

where Spec is the relative Spec. The construction gives us the natural map g: S' \to S, which is finite since \mathcal{O}_X is coherent and f is proper. f factors through g and so we get f': 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.)

References[edit]

The writing of this article benefited from [1].