Lemma 77.29.2. Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be the stabilizer group algebraic space.

1. The following are equivalent

1. $j : R \to U \times _ B U$ is separated,

2. $G \to U$ is separated, and

3. $e : U \to G$ is a closed immersion.

2. The following are equivalent

1. $j : R \to U \times _ B U$ is locally separated,

2. $G \to U$ is locally separated, and

3. $e : U \to G$ is an immersion.

3. The following are equivalent

1. $j : R \to U \times _ B U$ is quasi-separated,

2. $G \to U$ is quasi-separated, and

3. $e : U \to G$ is quasi-compact.

Proof. The group algebraic space $G \to U$ is the base change of $R \to U \times _ B U$ by the diagonal morphism $U \to U \times _ B U$, see Lemma 77.16.1. Hence if $j$ is separated (resp. locally separated, resp. quasi-separated), then $G \to U$ is separated (resp. locally separated, resp. quasi-separated). See Morphisms of Spaces, Lemma 66.4.4. Thus (a) $\Rightarrow$ (b) in (1), (2), and (3).

Conversely, if $G \to U$ is separated (resp. locally separated, resp. quasi-separated), then the morphism $e : U \to G$, as a section of the structure morphism $G \to U$ is a closed immersion (resp. an immersion, resp. quasi-compact), see Morphisms of Spaces, Lemma 66.4.7. Thus (b) $\Rightarrow$ (c) in (1), (2), and (3).

If $e$ is a closed immersion (resp. an immersion, resp. quasi-compact) then by the result of Lemma 77.29.1 (and Spaces, Lemma 64.12.3, and Morphisms of Spaces, Lemma 66.8.4) we see that $\Delta _{R/U \times _ B U}$ is a closed immersion (resp. an immersion, resp. quasi-compact). Thus (c) $\Rightarrow$ (a) in (1), (2), and (3). $\square$

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