The Stacks project

[Exposé VI, Appendix by Deligne, Proposition 9.0, SGA4]

Proposition 7.39.3. Let $\mathcal{C}$ be a site. Assume that

  1. finite limits exist in $\mathcal{C}$, and

  2. every covering $\{ U_ i \to U\} _{i \in I}$ has a refinement by a finite covering of $\mathcal{C}$.

Then $\mathcal{C}$ has enough points.

Proof. We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_ p$. See Lemma 7.38.3. Consider the system $(J, \geq , V_ j, g_{jj'})$ with $J = \{ 1\} $, $V_1 = U$, $g_{11} = \text{id}_ U$. Apply Lemma 7.39.2. By the result of that lemma we get a system $(I, \geq , U_ i, f_{ii'})$ refining our system such that $s_ p \not= s'_ p$ and such that moreover for every finite covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$ the map $\coprod _ k u(W_ k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod _ k u(W_ k) \to u(W)$ is surjective for any covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition 7.33.3 (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits). $\square$

Comments (3)

Comment #5087 by Anna on

Is there any published reference for this result?

Comment #5088 by on

Yes, see introduction enclosing section. I will formally add the reference here when I next go through all the comments.

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Your email address will not be published.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 00YQ. Beware of the difference between the letter 'O' and the digit '0'.