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 ( 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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