The Stacks project

Lemma 29.43.14. Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective, then $g \circ f$ is projective.

Proof. By Lemmas 29.43.10 and 29.43.5 we see that $f$ and $g$ are quasi-projective and proper. By Lemmas 29.41.4 and 29.40.3 we see that $g \circ f$ is proper and quasi-projective. Thus $g \circ f$ is projective by Lemma 29.43.13. $\square$

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