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 a proper morphism. Then one can write

is a finite morphism

The existence of this decomposition itself is not difficult. See below. But, by Zariski's connectedness theorem. It follows:

Corollary.

Proof

Set:

where SpecS is the relative Spec is coherent and f is proper. The morphism f factors through g. One then uses the theorem on formal functions has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)

See also

• Contraction morphism

References