The Stacks project

61.18 Points of the pro-étale site

We first apply Deligne's criterion to show that there are enough points.

Let $S$ be a scheme. Let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. We define a pro-étale neighbourhood of $\overline{s}$ to be a commutative diagram

with $U \to S$ weakly étale.

Let $S$ be a scheme and let $\overline{s}$ be a geometric point of $S$. For $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale})$ define the stalk of $\mathcal{F}$ at $\overline{s}$ by the formula

where the colimit is over all pro-étale neighbourhoods $(U, \overline{u})$ of $\overline{s}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$. It follows from the two lemmas above that the functor

defines a point of the site $S_{pro\text{-}\acute{e}tale}$, see Sites, Definition 7.32.2 and Lemma 7.33.1. Hence the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ defines a point of the topos $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale})$, see Sites, Definition 7.32.1 and Lemma 7.32.7. In particular this functor is exact and commutes with arbitrary colimits. In fact, this functor has another description.

Contrary to the situation with the étale topos of $S$ it is not true that every point of $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale})$ is of this form, and it is not true that the collection of points associated to geometric points is conservative. Namely, suppose that $S = \mathop{\mathrm{Spec}}(k)$ where $k$ is an algebraically closed field. Let $A$ be a nonzero abelian group. Consider the sheaf $\mathcal{F}$ on $S_{pro\text{-}\acute{e}tale}$ defined by the

for $U$ affine and by sheafification in general, see Example 61.19.12. Then $\mathcal{F}(U) = 0$ if $U = S = \mathop{\mathrm{Spec}}(k)$ but in general $\mathcal{F}$ is not zero. Namely, $S_{pro\text{-}\acute{e}tale}$ contains affine objects with infinitely many points. For example, let $E = \mathop{\mathrm{lim}}\nolimits E_ n$ be an inverse limit of finite sets with surjective transition maps, e.g., $E = \mathbf{Z}_ p = \mathop{\mathrm{lim}}\nolimits \mathbf{Z}/p^ n\mathbf{Z}$. The scheme $U = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits \text{Map}(E_ n, k))$ is an object of $S_{pro\text{-}\acute{e}tale}$ because $\mathop{\mathrm{colim}}\nolimits \text{Map}(E_ n, k)$ is weakly étale (even ind-Zariski) over $k$. Thus $\mathcal{F}(U)$ is nonzero as there exist maps $E \to A$ which aren't locally constant. Thus $\mathcal{F}$ is a nonzero abelian sheaf whose stalk at the unique geometric point of $S$ is zero. Since we know that $S_{pro\text{-}\acute{e}tale}$ has enough points, we conclude there must be a point of the pro-étale site which does not come from the construction explained above.

The replacement for arguments using points, is to use affine weakly contractible objects. First, there are enough affine weakly contractible objects by Lemma 61.13.4. Second, if $W \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$ is affine weakly contractible, then the functor

is an exact functor $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \to \textit{Sets}$ which commutes with all limits. The functor

is exact and commutes with direct sums (as $W$ is quasi-compact, see Sites, Lemma 7.17.7), hence commutes with all limits and colimits. Moreover, we can check exactness of a complex of abelian sheaves by evaluation at these affine weakly contractible objects of $S_{pro\text{-}\acute{e}tale}$, see Cohomology on Sites, Proposition 21.51.2.

A final remark is that the functor $\mathcal{F} \mapsto \mathcal{F}(W)$ for $W$ affine weakly contractible in general isn't a stalk functor of a point of $S_{pro\text{-}\acute{e}tale}$ because it doesn't preserve coproducts of sheaves of sets if $W$ is disconnected. And in fact, $W$ is disconnected as soon as $W$ has more than $1$ closed point, i.e., when $W$ is not the spectrum of a strictly henselian local ring (which is the special case discussed above).