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
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