Stein factorization

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. The version for schemes states the following:(EGA, 4.3.1)

Let X be a scheme, S a locally noetherian scheme and $f:X\to S$ a proper morphism. Then one can write

$f=g\circ f'$

where $g:S'\to S$ is a finite morphism and $f':X\to S'$ is a proper morphism so that $f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}$.

By Zariski's connectedness theorem, the last part in the theorem 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 $f^{-1}(s)$ is in bijection with the set of points in the fiber $g^{-1}(s)$.

Sketch of proof

In EGA, the theorem is deduced from the theorem on formal functions. Indeed, we set:

$S'=$ Spec$f_{*}{\mathcal {O}}_{X}$

where Spec is the relative Spec. The construction gives us: $X\to S'\to S$. We then shows this has the required properties using the theorem on formal functions, or rather its corollary.

The red book has an alternative non-cohomological proof.

References

The writing of this article benefited from [1].

• 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, MR0083045