The Stacks project

Lemma 29.38.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. If $\mathcal{L}$ is relatively very ample on $X/S$ then $f$ is separated.

Proof. Being separated is local on the base (see Schemes, Section 26.21). An immersion is separated (see Schemes, Lemma 26.23.8). Hence the lemma follows since locally $X$ has an immersion into the homogeneous spectrum of a graded ring which is separated, see Constructions, Lemma 27.8.8. $\square$

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