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. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)