The Stacks project

Proof. We first prove the second assertion. By Groupoids in Spaces, Lemma 78.6.1 we have to show that $e : S \to G$ is a closed immersion. This follows from Decent Spaces, Lemma 68.14.5.

Next, we prove the first assertion. To do this we may replace $B$ by $S$. By the paragraph above the stabilizer group scheme $G \to U$ is separated. By Groupoids in Spaces, Lemma 78.29.2 the morphism $j = (t, s) : R \to U \times _ S U$ is separated. As $U$ is the spectrum of a field the scheme $U \times _ S U$ is affine (by the construction of fibre products in Schemes, Section 26.17). Hence $R$ is separated, see Morphisms of Spaces, Lemma 67.4.9. $\square$