Lemma 78.6.1. 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 67.4.7 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 65.12.3 (resp. Morphisms of Spaces, Lemma 67.8.4, resp. Spaces, Lemma 65.12.3). \square
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