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

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

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 $\Delta _{G/S}$ 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). $\square$

There are also:

• 3 comment(s) on Section 39.6: Properties of group schemes

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).