Lemma 78.6.1. Let $B \to S$ be as in Section 78.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. 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 (Descent on Spaces, Definition 73.10.1). 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 $s^{-1}(W) \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (Groupoids in Spaces, Definition 77.18.1).
Proof. The existence and properties of the open $W \subset U$ are described in Descent on Spaces, Lemma 73.10.3. In Diagram (78.3.0.1) let $W_1 \subset R$ be the maximal open subscheme over which the morphism $\text{pr}_1 : R \times _{s, U, t} R \to R$ has property $\mathcal{P}$. It follows from the aforementioned Descent on Spaces, Lemma 73.10.3 and the assumption that $\{ s : R \to U\} $ and $\{ t : R \to U\} $ are coverings for the $\tau $-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired. $\square$
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