The Stacks project

Lemma 29.40.8. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. If $g \circ f$ is quasi-projective and $f$ is quasi-compact1, then $f$ is quasi-projective.

Proof. Observe that $f$ is of finite type by Lemma 29.15.8. Thus the lemma follows from Lemma 29.37.10 and the definitions. $\square$

[1] This follows if $g$ is quasi-separated by Schemes, Lemma 26.21.14.

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