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 $S_{affine, Zar}$, $S_{Zar}$, $S_{affine, {\acute{e}tale}}$, $S_{\acute{e}tale}$, $(\mathit{Sch}/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\mathit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.

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

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

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 $U \mapsto U(k)$ commutes with direct limits..." I think you meant "finite" instead of "direct"

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