Lemma 37.50.1. Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$X \to S$ is projective,

$X \to S$ is H-projective,

$X \to S$ is quasi-projective and proper,

$X \to S$ is H-quasi-projective and proper,

$X \to S$ is proper and $X$ has an ample invertible sheaf,

$X \to S$ is proper and there exists an $f$-ample invertible sheaf,

$X \to S$ is proper and there exists an $f$-very ample invertible sheaf,

there is a quasi-coherent graded $\mathcal{O}_ S$-algebra $\mathcal{A}$ generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ with $\mathcal{A}_1$ a finite type $\mathcal{O}_ S$-module such that $X = \underline{\text{Proj}}_ S(\mathcal{A})$.

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