# Stein factorization

In algebraic geometry, the **Stein factorization**, introduced by Template:Harvs 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:Template:Harv

Let

Xbe a scheme,Sa locally noetherian scheme and a proper morphism. Then one can writewhere 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 .

## Proof

Set:

where **Spec** is the relative **Spec**. The construction gives us the natural map , which is finite since is coherent and *f* is proper. *f* factors through *g* and so we get , 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.)

## References

The writing of this article benefited from [1].

- Template:Hartshorne AG
- Template:EGA
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