Stein factorization explained

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.

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 writef = 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' _{*} l{O} _{X} = l{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

Set:

S' = \operatorname{Spec} _{S} f _{*} l{O} _{X} where Spec_S is the relative Spec. The construction gives the natural map g\colon S' \to S , which is finite since l{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' _{*} l{O} _{X} = l{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.)

See also

• Contraction morphism

References