Lemma 59.30.1. Let $S$ be a scheme. All of the following sites have enough points $S_{affine, Zar}$, $S_{Zar}$, $S_{affine, {\acute{e}tale}}$, $S_{\acute{e}tale}$, $(\mathit{Sch}/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\mathit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.
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.
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 $S_{affine, Zar}$ follows from Deligne's result. The result for $S_{\acute{e}tale}$ either follows from (the proof of) Theorem 59.29.10 or from Topologies, Lemma 34.4.12 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.
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 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.
\[ B = \mathop{\mathrm{colim}}\nolimits _{(U, x)} \mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits _{(\mathop{\mathrm{Spec}}(C), x_ C)} C \]
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