Lemma 29.43.6 (0B5N)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.43: Projective morphisms
Lemma 29.43.6 (cite)

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Lemma 29.43.6. Let $f : X \to S$ be a proper morphism of schemes. If there exists an $f$ -ample invertible sheaf on $X$ , then $f$ is locally projective.

Proof. If there exists an $f$ -ample invertible sheaf, then we can locally on $S$ find an immersion $i : X \to \mathbf{P}^ n_ S$ , see Lemma 29.39.4. Since $X \to S$ is proper the morphism $i$ is a closed immersion, see Lemma 29.41.7. $\square$

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Comment #1806 by Giulia Battiston on January 25, 2016 at 12:29

There is a small typo in the proof, "sheavf" instead of "sheaf".

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