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The Stacks project

Lemma 7.38.5. Let \mathcal{C} be a site. Let \{ U_ i\} _{i \in I} be a family of objects of \mathcal{C}. Assume

  1. \coprod h_{U_ i}^\# \to * is a surjective map of sheaves, and

  2. each localization \mathcal{C}/U_ i has enough points.

Then \mathcal{C} has enough points.

Proof. For each i \in I let \{ p_ j\} _{j \in J_ i} be a conservative family of points of \mathcal{C}/U_ i. For j \in J_ i denote q_ j : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) the composition of p_ j with the localization morphism \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}). Then q_ j is a point, see Lemma 7.34.3. We claim that the family of points \{ q_ j\} _{j \in \coprod J_ i} is conservative. Namely, let \mathcal{F} \to \mathcal{G} be a map of sheaves on \mathcal{C} such that \mathcal{F}_{q_ j} \to \mathcal{G}_{q_ j} is an isomorphism for all j \in \coprod J_ i. Let W be an object of \mathcal{C}. By assumption (1) there exists a covering \{ W_ a \to W\} and morphisms W_ a \to U_{i(a)}. Since (\mathcal{F}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{F}_{q_ j} and (\mathcal{G}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{G}_{q_ j} by Lemma 7.34.3 we see that \mathcal{F}|_{U_{i(a)}} \to \mathcal{G}|_{U_{i(a)}} is an isomorphism since the family of points \{ p_ j\} _{j \in J_{i(a)}} is conservative. Hence \mathcal{F}(W_ a) \to \mathcal{G}(W_ a) is bijective for each a. Similarly \mathcal{F}(W_ a \times _ W W_ b) \to \mathcal{G}(W_ a \times _ W W_ b) is bijective for each a, b. By the sheaf condition this shows that \mathcal{F}(W) \to \mathcal{G}(W) is bijective, i.e., \mathcal{F} \to \mathcal{G} is an isomorphism. \square


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