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. $\square$

Comment #1806 by Giulia Battiston on

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

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).