The Stacks project

Lemma 67.40.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is separated,

  2. $\Delta _{X/Y} : X \to X \times _ Y X$ is universally closed, and

  3. $\Delta _{X/Y} : X \to X \times _ Y X$ is proper.

Proof. The implication (1) $\Rightarrow $ (3) follows from Lemma 67.40.5. We will use Spaces, Lemma 65.5.8 without further mention in the rest of the proof. Recall that $\Delta _{X/Y}$ is a representable monomorphism which is locally of finite type, see Lemma 67.4.1. Since proper $\Rightarrow $ universally closed for morphisms of schemes we conclude that (3) implies (2). If $\Delta _{X/Y}$ is universally closed then √Čtale Morphisms, Lemma 41.7.2 implies that it is a closed immersion. Thus (2) $\Rightarrow $ (1) and we win. $\square$

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