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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## Comments (1)

Comment #1806 by Giulia Battiston on