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61.18 Points of the pro-étale site

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

Lemma 61.18.1. Let $S$ be a scheme. The pro-étale sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have enough points.

Proof. The big pro-étale topos of $S$ is equivalent to the topos defined by $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$, see Lemma 61.12.11. The topos of sheaves on $S_{pro\text{-}\acute{e}tale}$ is equivalent to the topos associated to $S_{affine, {pro\text{-}\acute{e}tale}}$, see Lemma 61.12.20. The result for the sites $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and $S_{affine, {pro\text{-}\acute{e}tale}}$ follows immediately from Deligne's result Sites, Lemma 7.39.4. The case $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is handled because it is equal to $(\mathit{Sch}/\mathop{\mathrm{Spec}}(\mathbf{Z}))_{pro\text{-}\acute{e}tale}$. $\square$

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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\ & S } \]

with $U \to S$ weakly étale.

Lemma 61.18.2. Let $S$ be a scheme and let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. The category of pro-étale neighbourhoods of $\overline{s}$ is cofiltered.

Proof. The proof is identitical to the proof of Étale Cohomology, Lemma 59.29.4 but using the corresponding facts about weakly étale morphisms proven in More on Morphisms, Lemmas 37.64.5, 37.64.6, and 37.64.13. $\square$

Lemma 61.18.3. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Let $\mathcal{U} = \{ \varphi _ i : S_ i \to S\} _{i\in I}$ be a pro-étale covering. Then there exist $i \in I$ and geometric point $\overline{s}_ i$ of $S_ i$ mapping to $\overline{s}$.

Proof. Immediate from the fact that $\coprod \varphi _ i$ is surjective and that residue field extensions induced by weakly étale morphisms are separable algebraic (see for example More on Morphisms, Lemma 37.64.11. $\square$

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

\[ \mathcal{F}_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{F}(U) \]

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

\[ S_{pro\text{-}\acute{e}tale}\textit{Sets},\quad U \longmapsto \{ \overline{u}\text{ geometric point of }U\text{ mapping to }\overline{s}\} \]

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.

Lemma 61.18.4. In the situation above the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an object of $X_{pro\text{-}\acute{e}tale}$ and there is a canonical isomorphism

\[ \mathcal{F}(\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}} \]

functorial in $\mathcal{F}$.

Proof. The first statement is clear from the construction of the strict henselization as a filtered colimit of étale algebras over $S$, or by the characterization of weakly étale morphisms of More on Morphisms, Lemma 37.64.11. The second statement follows as by Olivier's theorem (More on Algebra, Theorem 15.104.24) the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an initial object of the category of pro-étale neighbourhoods of $\overline{s}$. $\square$

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

\[ \mathcal{F}(U) = \frac{\{ \text{functions }U \to A\} }{\{ \text{locally constant functions}\} } \]

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

\[ \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \longrightarrow \textit{Sets},\quad \mathcal{F} \longmapsto \mathcal{F}(W) \]

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

\[ \textit{Ab}(S_{pro\text{-}\acute{e}tale}) \longrightarrow \textit{Ab},\quad \mathcal{F} \longmapsto \mathcal{F}(W) \]

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).

Comments (7)

Comment #311 by BB on

With the formula immediately preceding Lemma 45.12.2 defining the stalk F_s, I do not see why F_s = F(O_{S,s}^{sh}): the former is a colimit over all etale neighbourhoods, not all weakly etale or pro-etale neighbourhoods, so the colimit need not be realized by one of its terms. In fact, unless I'm mistaken, if S = Q and F = \lim mu_n, then the LHS is 0, while the RHS is not. It seems these are two different points...

Comment #312 by on

This is a stupid typo. We should be taking the colimit over the proetale neighbourhoods of the geometric point. Otherwise it won't be a point of the proetale site. Fixed here. Thank you very much!

Comment #1850 by Aravind Asok on

Silly comment: in Lemma 51.12.1, in the sentence beginning In exactly the same manner..." maybe instead of referencingthe chapter on etale cohomology" you want to add a reference to Tag 03PQ (Lemma 49.29.4).

Comment #2445 by on

Typo: in the sentence following the proof of Lemma 12.2: '...associated to geometric points'

Comment #2446 by on

In the example after Lemma 12.2, the point seems to be that S_{pro-etale} contains quasi-compact objects with infinitely many points.

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