Lemma 79.6.1. 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 \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 74.10.1). 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 78.18.1).
Proof. The existence and properties of the open W \subset U are described in Descent on Spaces, Lemma 74.10.3. In Diagram (79.3.0.1) 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 74.10.3 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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