39.22 Separation conditions
This really means conditions on the morphism j : R \to U \times _ S U when given a groupoid (U, R, s, t, c) over S. As in the previous section we first formulate the corresponding diagram.
Lemma 39.22.1. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G \to U be the stabilizer group scheme. The commutative diagram
\xymatrix{ R \ar[d]^{\Delta _{R/U \times _ S U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times _{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times _{(U \times _ S 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 39.22.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G \to U be the stabilizer group scheme.
The following are equivalent
j : R \to U \times _ S 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 _ S U is quasi-separated,
G \to U is quasi-separated, and
e : U \to G is quasi-compact.
Proof.
The group scheme G \to U is the base change of R \to U \times _ S U by the diagonal morphism U \to U \times _ S U, see Lemma 39.17.1. Hence if j is separated (resp. quasi-separated), then G \to U is separated (resp. quasi-separated). (See Schemes, Lemma 26.21.12). Thus (a) \Rightarrow (b) in both (1) and (2).
If G \to U is separated (resp. quasi-separated), then the morphism U \to G, as a section of the structure morphism G \to U is a closed immersion (resp. quasi-compact), see Schemes, Lemma 26.21.11. Thus (b) \Rightarrow (a) in both (1) and (2).
By the result of Lemma 39.22.1 (and Schemes, Lemmas 26.18.2 and 26.19.3) we see that if e is a closed immersion (resp. quasi-compact) \Delta _{R/U \times _ S U} is a closed immersion (resp. quasi-compact). Thus (c) \Rightarrow (a) in both (1) and (2).
\square
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