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)
where is a finite morphism and is a proper morphism so that .
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 is connected for any . It follows:
Corollary: For any , the set of connected components of the fiber is in bijection with the set of points in the fiber .
where SpecS is the relative Spec. The construction gives the natural map , which is finite since is coherent and f is proper. The morphism f factors through g and one gets , which is proper. By construction, . 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.)
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR0463157
- Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
- Stein, Karl (1956), "Analytische Zerlegungen komplexer Räume", Mathematische Annalen, 132: 63–93, doi:10.1007/BF01343331, ISSN 0025-5831, MR 0083045