The Stacks project

Lemma 29.43.11. Let $f : X \to S$ be a H-quasi-projective morphism. Then $f$ factors as $X \to X' \to S$ where $X \to X'$ is an open immersion and $X' \to S$ is H-projective.

Proof. By definition we can factor $f$ as a quasi-compact immersion $i : X \to \mathbf{P}^ n_ S$ followed by the projection $\mathbf{P}^ n_ S \to S$. By Lemma 29.7.7 there exists a closed subscheme $X' \subset \mathbf{P}^ n_ S$ such that $i$ factors through an open immersion $X \to X'$. The lemma follows. $\square$

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