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

We say $f$ is

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

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