The Stacks project

Lemma 26.23.8. An immersion of schemes is a monomorphism. In particular, any immersion is separated.

Proof. We can see this by checking that the criterion of Lemma 26.23.7 applies. More elegantly perhaps, we can use that Lemmas 26.3.5 and 26.4.6 imply that open and closed immersions are monomorphisms and hence any immersion (which is a composition of such) is a monomorphism. $\square$

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