Stein factorization

In algebraic geometry, the Stein factorization states the following: Let X be a scheme, S a locally noetherian scheme and $f:X\to S$ be 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. Furthermore, Failed to parse (syntax error): {\displaystyle f'_*(\mathcal{O}_X)} = {\mathcal{O}_S'}} .

Proof

In EGA, the theorem is deduced from the theorem on formal functions. 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.