The Stacks project

59.30 Points in other topologies

In this section we briefly discuss the existence of points for some sites other than the étale site of a scheme. We refer to Sites, Section 7.38 and Topologies, Section 34.2 ff for the terminology used in this section. All of the geometric sites have enough points.

Lemma 59.30.1. Let $S$ be a scheme. All of the following sites have enough points $S_{Zar}$, $S_{\acute{e}tale}$, $(\mathit{Sch}/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\mathit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.

Proof. For each of the big sites the associated topos is equivalent to the topos defined by the site $(\textit{Aff}/S)_\tau $, see Topologies, Lemmas 34.3.10, 34.4.11, 34.5.9, 34.6.9, and 34.7.11. The result for the sites $(\textit{Aff}/S)_\tau $ follows immediately from Deligne's result Sites, Lemma 7.39.4.

The result for $S_{Zar}$ is clear. The result for $S_{\acute{e}tale}$ either follows from (the proof of) Theorem 59.29.10 or from Lemma 59.21.2 and Deligne's result applied to $S_{affine, {\acute{e}tale}}$. $\square$

The lemma above guarantees the existence of points, but it doesn't tell us what these points look like. We can explicitly construct some points as follows. Suppose $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ is a geometric point with $k$ algebraically closed. Consider the functor

\[ u : (\mathit{Sch}/S)_{fppf} \longrightarrow \textit{Sets}, \quad u(U) = U(k) = \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(k), U). \]

Note that $U \mapsto U(k)$ commutes with finite limits as $S(k) = \{ \overline{s}\} $ and $(U_1 \times _ U U_2)(k) = U_1(k) \times _{U(k)} U_2(k)$. Moreover, if $\{ U_ i \to U\} $ is an fppf covering, then $\coprod U_ i(k) \to U(k)$ is surjective. By Sites, Proposition 7.33.3 we see that $u$ defines a point $p$ of $(\mathit{Sch}/S)_{fppf}$ with stalks

\[ \mathcal{F}_ p = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{F}(U) \]

where the colimit is over pairs $U \to S$, $x \in U(k)$ as usual. But... this category has an initial object, namely $(\mathop{\mathrm{Spec}}(k), \text{id})$, hence we see that

\[ \mathcal{F}_ p = \mathcal{F}(\mathop{\mathrm{Spec}}(k)) \]

which isn't terribly interesting! In fact, in general these points won't form a conservative family of points. A more interesting type of point is described in the following remark.

reference

Remark 59.30.2. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $(p, u)$ be a point of the site $(\textit{Aff}/S)_{fppf}$, see Sites, Sections 7.32 and 7.33. Let $B = \mathcal{O}_ p$ be the stalk of the structure sheaf at the point $p$. Recall that

\[ B = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits _{(\mathop{\mathrm{Spec}}(C), x_ C)} C \]

where $x_ C \in u(\mathop{\mathrm{Spec}}(C))$. It can happen that $\mathop{\mathrm{Spec}}(B)$ is an object of $(\textit{Aff}/S)_{fppf}$ and that there is an element $x_ B \in u(\mathop{\mathrm{Spec}}(B))$ mapping to the compatible system $x_ C$. In this case the system of neighbourhoods has an initial object and it follows that $\mathcal{F}_ p = \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ for any sheaf $\mathcal{F}$ on $(\textit{Aff}/S)_{fppf}$. It is straightforward to see that if $\mathcal{F} \mapsto \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ defines a point of $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{fppf})$, then $B$ has to be a local $A$-algebra such that for every faithfully flat, finitely presented ring map $B \to B'$ there is a section $B' \to B$. Conversely, for any such $A$-algebra $B$ the functor $\mathcal{F} \mapsto \mathcal{F}(\mathop{\mathrm{Spec}}(B))$ is the stalk functor of a point. Details omitted. It is not clear what a general point of the site $(\textit{Aff}/S)_{fppf}$ looks like.


Comments (2)

Comment #1708 by Yogesh More on

minor comment: In the line "Note that commutes with direct limits..." I think you meant "finite" instead of "direct"


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 06VW. Beware of the difference between the letter 'O' and the digit '0'.