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
where
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
Corollary: For any
Proof
Set:
where SpecS is the relative Spec. The construction gives the natural map