Definition 29.43.1. Let $f : X \to S$ be a morphism of schemes.

We say $f$ is

*projective*if $X$ is isomorphic as an $S$-scheme to a closed subscheme of a projective bundle $\mathbf{P}(\mathcal{E})$ for some quasi-coherent, finite type $\mathcal{O}_ S$-module $\mathcal{E}$.We say $f$ is

*H-projective*if there exists an integer $n$ and a closed immersion $X \to \mathbf{P}^ n_ S$ over $S$.We say $f$ is

*locally projective*if there exists an open covering $S = \bigcup U_ i$ such that each $f^{-1}(U_ i) \to U_ i$ is projective.

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