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

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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?

  • $\begingroup$ Coherent sheaf always means locally is finitely generated as module. And proper implies that direct image preserves coherence... $\endgroup$ – xarles Sep 30 '18 at 18:04
  • $\begingroup$ @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? $\endgroup$ – KarlPeter Sep 30 '18 at 19:15
  • $\begingroup$ The proofs I know are long and "by devissage", so not really illuminating. Try stacks.math.columbia.edu/tag/02O3 $\endgroup$ – xarles Sep 30 '18 at 19:24

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