The Stacks project

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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