Lemma 29.43.15. Let g : Y \to S and f : X \to Y be morphisms of schemes. If g \circ f is projective and g is separated, then f is projective.
Proof. Choose a closed immersion X \to \mathbf{P}(\mathcal{E}) where \mathcal{E} is a quasi-coherent, finite type \mathcal{O}_ S-module. Then we get a morphism X \to \mathbf{P}(\mathcal{E}) \times _ S Y. This morphism is a closed immersion because it is the composition
X \to X \times _ S Y \to \mathbf{P}(\mathcal{E}) \times _ S Y
where the first morphism is a closed immersion by Schemes, Lemma 26.21.10 (and the fact that g is separated) and the second as the base change of a closed immersion. Finally, the fibre product \mathbf{P}(\mathcal{E}) \times _ S Y is isomorphic to \mathbf{P}(g^*\mathcal{E}) and pullback preserves quasi-coherent, finite type modules. \square
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