Lemma 40.10.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $U$ is the spectrum of a field, then $R$ is a separated scheme.
Proof. By Groupoids, Lemma 39.7.3 the stabilizer group scheme $G \to U$ is separated. By Groupoids, Lemma 39.22.2 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 a separated scheme, see Schemes, Lemma 26.21.12. $\square$
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