Lemma 77.6.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then $G \to B$ is separated (resp. quasi-separated, resp. locally separated) if and only if the identity morphism $e : B \to G$ is a closed immersion (resp. quasi-compact, resp. an immersion).

Proof. We recall that by Morphisms of Spaces, Lemma 66.4.7 we have that $e$ is a closed immersion (resp. quasi-compact, resp. an immersion) if $G \to B$ is separated (resp. quasi-separated, resp. locally separated). For the converse, consider the diagram

$\xymatrix{ G \ar[r]_-{\Delta _{G/B}} \ar[d] & G \times _ B G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\ B \ar[r]^ e & G }$

It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that $\Delta _{G/B}$ is a base change of $e$. Hence if $e$ is a closed immersion (resp. quasi-compact, resp. an immersion) so is $\Delta _{G/B}$, see Spaces, Lemma 64.12.3 (resp. Morphisms of Spaces, Lemma 66.8.4, resp. Spaces, Lemma 64.12.3). $\square$

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