The Stacks project

Lemma 40.6.5. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be its stabilizer group scheme. Let $\tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Let $\mathcal{P}$ be a property of morphisms 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 subscheme such that $G_ W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids, Definition 39.19.1).

Proof. The existence and properties of the open $W \subset U$ are described in Descent, Lemma 35.22.3. 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 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, Lemma 35.22.3. $\square$

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