Lemma 29.43.10. A projective morphism is quasi-projective.

**Proof.**
Let $f : X \to S$ be a projective morphism. Choose a closed immersion $i : X \to \mathbf{P}(\mathcal{E})$ where $\mathcal{E}$ is a quasi-coherent, finite type $\mathcal{O}_ S$-module. Then $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ is $f$-very ample. Since $f$ is proper (Lemma 29.43.5) it is quasi-compact. Hence Lemma 29.38.2 implies that $\mathcal{L}$ is $f$-ample. Since $f$ is proper it is of finite type. Thus we've checked all the defining properties of quasi-projective holds and we win.
$\square$

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