The Stacks project

Lemma 29.37.3. Let $f : X \to S$ be a morphism of schemes. If there exists an $f$-ample invertible sheaf, then $f$ is separated.

Proof. Being separated is local on the base (see Schemes, Lemma 26.21.7 for example; it also follows easily from the definition). Hence we may assume $S$ is affine and $X$ has an ample invertible sheaf. In this case the result follows from Properties, Lemma 28.26.8. $\square$

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