# Stein Factorization

I have a question about an argument in Hartshorne's "Algebraic Geometry" (see p 280); here the excerpt:

Here we introduce the Stein factorization of projective morphism (btw.: I think that by Chow's lemma we can replace the condition projective by proper: see https://en.wikipedia.org/wiki/Stein_factorization)

So let us from here consider that $$f:X \to Y$$ is proper.

My problem is that I don't understand how the argument that the induced morphism $$g: Spec(f_*(\mathcal{O}_X) \to Y$$ is finite works.

My considerations: Firsly, of corse the problem is local so let consider the affine subset $$Spec(R) := V \subset Y$$. Since $$f$$ is proper by definition of properness $$\mathcal{\Gamma}(V, f_*\mathcal{O}_X)= \mathcal{\Gamma}(f^{-1}(V), \mathcal{O}_X)$$ is a finitely generated $$R =\mathcal{\Gamma}(V, \mathcal{O}_Y)$$-algebra(!), NOT finitely generated as $$R$$-module! Therefore locally we have $$\mathcal{\Gamma}(V, f_*\mathcal{O}_X) \cong R[X_1,..., X_n]/ (I)$$ for appropriate $$n$$.

This states indeed that $$f_*\mathcal{O}_X$$ is a coherent $$\mathcal{O}_Y$$ -algebra!

And here occurs the problem: How we conclude from here that $$g$$ is finite. To show this we need that $$f_*\mathcal{O}_X$$ is a coherent $$\mathcal{O}_Y$$ -module, right?

I guess that there exist some finiteness theorem for coherent $$\mathcal{O}_Y$$-sheaves. But here occures again the cruical point: A coherent $$\mathcal{O}_Y$$-sheave has the coherent-property as $$\mathcal{O}_Y$$ -module, not algebra.

So how the argument above really works? What theorem is used here?

• Coherent sheaf always means locally is finitely generated as module. And proper implies that direct image preserves coherence... – xarles Sep 30 '18 at 18:04
• @xarles: Do you have a nice reference for the statement that proper morphisms preserve coherence. I only know it for affine morphisms. Or - if that not too awkwardly - could you give a sketch of the proof? – KarlPeter Sep 30 '18 at 19:15
• The proofs I know are long and "by devissage", so not really illuminating. Try stacks.math.columbia.edu/tag/02O3 – xarles Sep 30 '18 at 19:24