Lemma 79.6.2. Let B \to S be as in Section 79.2. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let G \to U be its stabilizer group algebraic space. Let \tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} . Let \mathcal{P} be a property of morphisms of algebraic spaces which is \tau -local on the target. Assume \{ s : R \to U\} and \{ t : R \to U\} are coverings for the \tau -topology. Let W \subset U be the maximal open subspace such that G_ W \to W has property \mathcal{P}. Then W is R-invariant (see Groupoids in Spaces, Definition 78.18.1).
Proof. The existence and properties of the open W \subset U are described in Descent on Spaces, Lemma 74.10.3. The morphism
G \times _{U, t} R \longrightarrow R \times _{s, U} G, \quad (g, r) \longmapsto (r, r^{-1} \circ g \circ r)
is an isomorphism of algebraic spaces over R (where \circ denotes composition in the groupoid). Hence s^{-1}(W) = t^{-1}(W) by the properties of W proved in the aforementioned Descent on Spaces, Lemma 74.10.3. \square
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