Lemma 39.6.1. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Then $G \to S$ is separated (resp. quasi-separated) if and only if the identity morphism $e : S \to G$ is a closed immersion (resp. quasi-compact).

**Proof.**
We recall that by Schemes, Lemma 26.21.11 we have that $e$ is an immersion which is a closed immersion (resp. quasi-compact) if $G \to S$ is separated (resp. quasi-separated). For the converse, consider the diagram

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 $\Delta _{G/S}$ is a base change of $e$. Hence if $e$ is a closed immersion (resp. quasi-compact) so is $\Delta _{G/S}$, see Schemes, Lemma 26.18.2 (resp. Schemes, Lemma 26.19.3). $\square$

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