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.
is a finite morphism and
is a proper morphism so that
The existence of this decomposition itself is not difficult.
But, by Zariski's connectedness theorem, the last part in the above says that the fiber
, the set of connected components of the fiber
is in bijection with the set of points in the fiber
The construction gives the natural map
is coherent and f is proper.
The morphism f factors through g and one gets
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.)