[II, Definition 5.3.1, EGA] and [page 103, H]

Definition 29.40.1. Let $f : X \to S$ be a morphism of schemes.

1. We say $f$ is quasi-projective if $f$ is of finite type and there exists an $f$-relatively ample invertible $\mathcal{O}_ X$-module.

2. We say $f$ is H-quasi-projective if there exists a quasi-compact immersion $X \to \mathbf{P}^ n_ S$ over $S$ for some $n$.1

3. We say $f$ is locally quasi-projective if there exists an open covering $S = \bigcup V_ j$ such that each $f^{-1}(V_ j) \to V_ j$ is quasi-projective.

[1] This is not exactly the same as the definition in Hartshorne. Namely, the definition in Hartshorne (8th corrected printing, 1997) is that $f$ should be the composition of an open immersion followed by a H-projective morphism (see Definition 29.43.1), which does not imply $f$ is quasi-compact. See Lemma 29.43.11 for the implication in the other direction.

Comment #1162 by prefaisceau on

There is an extra "if $f$" in item (2).

Comment #2734 by Matt Stevenson on

Part (1) of the definition is EGA II Def 5.3.1.

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