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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