77.29 Separation conditions

This really means conditions on the morphism $j : R \to U \times _ B U$ when given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$. As in the previous section we first formulate the corresponding diagram.

Lemma 77.29.1. 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. The commutative diagram

$\xymatrix{ R \ar[d]^{\Delta _{R/U \times _ B U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times _{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times _{(U \times _ B U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times _{s, U} G \ar[r] & G }$

the two left horizontal arrows are isomorphisms and the right square is a fibre product square.

Proof. Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry. $\square$

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). $\square$

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).