78.29 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 78.29.1. Let $B \to S$ be as in Section 78.3. 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 78.29.2. Let $B \to S$ be as in Section 78.3. 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 following are equivalent
$j : R \to U \times _ B U$ is separated,
$G \to U$ is separated, and
$e : U \to G$ is a closed immersion.
The following are equivalent
$j : R \to U \times _ B U$ is locally separated,
$G \to U$ is locally separated, and
$e : U \to G$ is an immersion.
The following are equivalent
$j : R \to U \times _ B U$ is quasi-separated,
$G \to U$ is quasi-separated, and
$e : U \to G$ 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 78.16.1. 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 67.4.4. 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 67.4.7. 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 78.29.1 (and Spaces, Lemma 65.12.3, and Morphisms of Spaces, Lemma 67.8.4) 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).
$\square$
Comments (0)