The Stacks project

Lemma 29.43.13. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of schemes. Then

  1. $f$ is projective if and only if $f$ is quasi-projective and proper, and

  2. $f$ is H-projective if and only if $f$ is H-quasi-projective and proper.

Proof. If $f$ is projective, then $f$ is quasi-projective by Lemma 29.43.10 and proper by Lemma 29.43.5. Conversely, if $X \to S$ is quasi-projective and proper, then we can choose an open immersion $X \to X'$ with $X' \to S$ projective by Lemma 29.43.12. Since $X \to S$ is proper, we see that $X$ is closed in $X'$ (Lemma 29.41.7), i.e., $X \to X'$ is a (open and) closed immersion. Since $X'$ is isomorphic to a closed subscheme of a projective bundle over $S$ (Definition 29.43.1) we see that the same thing is true for $X$, i.e., $X \to S$ is a projective morphism. This proves (1). The proof of (2) is the same, except it uses Lemmas 29.43.3 and 29.43.11. $\square$

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