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:
f is separated,
\Delta _{X/Y} : X \to X \times _ Y X is universally closed, and
\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 Étale Morphisms, Lemma 41.7.2 implies that it is a closed immersion. Thus (2) \Rightarrow (1) and we win. \square
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