Lemma 29.20.15. Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite.

**Proof.**
A monomorphism is separated by Schemes, Lemma 26.23.3. A monomorphism is injective, hence we get $f$ is quasi-finite at every $x \in X$ for example by Lemma 29.20.6.
$\square$

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