[II Proposition 5.1.6, EGA]

Lemma 29.37.6. Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-affine if and only if $\mathcal{O}_ X$ is $f$-relatively ample.

Proof. Follows from Properties, Lemma 28.27.1 and the definitions. $\square$

Comment #2708 by Matt Stevenson on

This is closely related to EGA II Prop 5.1.6 (it is the equivalence of (a) and (c'), but in EGA the morphism f is assumed to be quasi-compact).

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