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The Stacks project

Lemma 40.6.4. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let \tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} 1. Let \mathcal{P} be a property of morphisms of schemes which is \tau -local on the target (Descent, Definition 35.22.1). Assume \{ s : R \to U\} and \{ t : R \to U\} are coverings for the \tau -topology. Let W \subset U be the maximal open subscheme such that s|_{s^{-1}(W)} : s^{-1}(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. In Diagram (40.3.0.1) 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, Lemma 35.22.3 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

[1] The fact that fpqc is missing is not a typo.

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