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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06UL. Beware of the difference between the letter 'O' and the digit '0'.