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The Stacks project

Lemma 7.38.4. Let \mathcal{C} be a site. Let U be an object of \mathcal{C}. let \{ (p_ i, u_ i)\} _{i\in I} be a family of points of \mathcal{C}. For x \in u_ i(U) let q_{i, x} be the point of \mathcal{C}/U constructed in Lemma 7.35.1. If \{ p_ i\} is a conservative family of points, then \{ q_{i, x}\} _{i \in I, x \in u_ i(U)} is a conservative family of points of \mathcal{C}/U. In particular, if \mathcal{C} has enough points, then so does every localization \mathcal{C}/U.

Proof. We know that j_{U!} induces an equivalence j_{U!} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# , see Lemma 7.25.4. Moreover, we know that (j_{U!}\mathcal{G})_{p_ i} = \coprod _ x \mathcal{G}_{q_{i, x}}, see Lemma 7.35.3. Hence the result follows formally. \square


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