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Tag 0991

52.12. Points of the pro-étale site

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

Lemma 52.12.1. Let $S$ be a scheme. The pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have enough points.

Proof. The big topos is equivalent to the topos defined by $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$, see Lemma 52.11.15. 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 52.11.23. 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, Proposition 7.38.3. $\square$

Let $S$ be a scheme. Let $\overline{s} : \mathop{\rm Spec}(k) \to S$ be a geometric point. We define a pro-étale neighbourhood of $\overline{s}$ to be a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\ & S } $$ with $U \to S$ weakly étale. In exactly the same manner as in Étale Cohomology, Lemma 50.29.4 one shows that the category of pro-étale neighbourhoods of $\overline{s}$ is cofiltered. Moreover, if $(U, \overline{u})$ is a pro-étale neighbourhood, and if $\{U_i \to U\}$ is a pro-étale covering, then there exists an $i$ and a lift of $\overline{u}$ to a geometric point $\overline{u}_i$ of $U_i$. For $\mathcal{F}$ in $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$ define the stalk of $\mathcal{F}$ at $\overline{s}$ by the formula $$ \mathcal{F}_{\overline{s}} = \mathop{\rm colim}\nolimits_{(U, \overline{u})} \mathcal{F}(U) $$ where the colimit is over all pro-étale neighbourhoods $(U, \overline{u})$ of $\overline{s}$ with $U \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$. A formal argument using the facts above shows the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ defines a point of the topos $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$: it is an exact functor which commutes with arbitrary colimits. In fact, this functor has another description.

Lemma 52.12.2. In the situation above the scheme $\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an object of $X_{pro\text{-}\acute{e}tale}$ and there is a canonical isomorphism $$ \mathcal{F}(\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}} $$ 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 36.53.11. The second statement follows as by Olivier's theorem (More on Algebra, Theorem 15.85.25) the scheme $\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})$ 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 $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$ is of this form, and it is not true that the collection of points associated to geometric points is conservative. Namely, suppose that $S = \mathop{\rm Spec}(k)$ where $k$ is an algebraically closed field. Let $A$ be an abelian group. Consider the sheaf $\mathcal{F}$ on $S_{pro\text{-}\acute{e}tale}$ defined by the rule $$ \mathcal{F}(U) = \frac{\{\text{functions }U \to A\}}{\{\text{locally constant functions}\}} $$ Then $\mathcal{F}(U) = 0$ if $U = S = \mathop{\rm Spec}(k)$ but in general $\mathcal{F}$ is not zero. Namely, $S_{pro\text{-}\acute{e}tale}$ contains quasi-compact objects with infinitely many points. For example, let $E = \mathop{\rm lim}\nolimits E_n$ be an inverse limit of finite sets with surjective transition maps, e.g., $E = \mathop{\rm lim}\nolimits \mathbf{Z}/n\mathbf{Z}$. The scheme $\mathop{\rm Spec}(\mathop{\rm colim}\nolimits \text{Map}(E_n, k))$ is an object of $S_{pro\text{-}\acute{e}tale}$ because $\mathop{\rm colim}\nolimits \text{Map}(E_n, k)$ is weakly étale (even ind-Zariski) over $k$. Thus $\mathcal{F}$ is a nonzero abelian sheaf whose stalk at the unique geometric point of $S$ is zero.

The solution is to use the existence of quasi-compact, weakly contractible objects. First, there are enough quasi-compact, weakly contractible objects by Lemma 52.11.27. Second, if $W \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$ is quasi-compact, weakly contractible, then the functor $$ \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \longrightarrow \textit{Sets},\quad \mathcal{F} \longmapsto \mathcal{F}(W) $$ is an exact functor $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \to \textit{Sets}$ 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 7.17.5), hence commutes with all limits and colimits. Moreover, we can check exactness of a complex of abelian sheaves by evaluation at the quasi-compact, weakly contractible objects of $S_{pro\text{-}\acute{e}tale}$, see Cohomology on Sites, Proposition 21.41.2.

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2673–2800 (see updates for more information).

    \section{Points of the pro-\'etale site}
    \label{section-points}
    
    \noindent
    We first apply Deligne's criterion to show that there are enough points.
    
    \begin{lemma}
    \label{lemma-points-proetale}
    Let $S$ be a scheme. The pro-\'etale sites
    $S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and
    $(\textit{Aff}/S)_\proetale$ have enough points.
    \end{lemma}
    
    \begin{proof}
    The big topos is equivalent to the topos defined by
    $(\textit{Aff}/S)_\proetale$, see
    Lemma \ref{lemma-affine-big-site-proetale}.
    The topos of sheaves on $S_\proetale$ is equivalent to the topos
    associated to $S_{affine, \proetale}$, see
    Lemma \ref{lemma-alternative}.
    The result for the sites $(\textit{Aff}/S)_\proetale$ and
    $S_{affine, \proetale}$ follows immediately from Deligne's result
    Sites, Proposition \ref{sites-proposition-criterion-points}.
    \end{proof}
    
    \noindent
    Let $S$ be a scheme. Let $\overline{s} : \Spec(k) \to S$ be a geometric
    point. We define a {\it pro-\'etale neighbourhood} of $\overline{s}$
    to be a commutative diagram
    $$
    \xymatrix{
    \Spec(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\
    & S
    }
    $$
    with $U \to S$ weakly \'etale. In exactly the same manner as in
    \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cofinal-etale}
    one shows that the category of pro-\'etale
    neighbourhoods of $\overline{s}$ is cofiltered. Moreover, if
    $(U, \overline{u})$ is a pro-\'etale neighbourhood, and if $\{U_i \to U\}$
    is a pro-\'etale covering, then there exists an $i$ and a lift of
    $\overline{u}$ to a geometric point $\overline{u}_i$ of $U_i$.
    For $\mathcal{F}$ in $\Sh(S_\proetale)$ define the {\it stalk
    of $\mathcal{F}$ at $\overline{s}$} by the formula
    $$
    \mathcal{F}_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{F}(U)
    $$
    where the colimit is over all pro-\'etale neighbourhoods $(U, \overline{u})$
    of $\overline{s}$ with $U \in \Ob(S_\proetale)$.
    A formal argument using the facts above shows the functor
    $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
    defines a point of the topos $\Sh(S_\proetale)$:
    it is an exact functor which commutes with arbitrary colimits.
    In fact, this functor has another description.
    
    \begin{lemma}
    \label{lemma-classical-point}
    In the situation above the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$
    is an object of $X_\proetale$ and there is a canonical isomorphism
    $$
    \mathcal{F}(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})) =
    \mathcal{F}_{\overline{s}}
    $$
    functorial in $\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    The first statement is clear from the construction of the strict henselization
    as a filtered colimit of \'etale algebras over $S$, or by the characterization
    of weakly \'etale morphisms of
    More on Morphisms, Lemma
    \ref{more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings}.
    The second statement follows as by Olivier's theorem
    (More on Algebra, Theorem \ref{more-algebra-theorem-olivier})
    the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$
    is an initial object of the category of pro-\'etale neighbourhoods
    of $\overline{s}$.
    \end{proof}
    
    \noindent
    Contrary to the situation with the \'etale topos of $S$ it is not true
    that every point of $\Sh(S_\proetale)$ is of this form, and it is not
    true that the collection of points associated to geometric points is
    conservative. Namely, suppose that $S = \Spec(k)$ where $k$ is an
    algebraically closed field. Let $A$ be an abelian group.
    Consider the sheaf $\mathcal{F}$ on $S_\proetale$ defined by the rule
    $$
    \mathcal{F}(U) = \frac{\{\text{functions }U \to A\}}{\{\text{locally constant functions}\}}
    $$
    Then $\mathcal{F}(U) = 0$ if $U = S = \Spec(k)$ but in general $\mathcal{F}$
    is not zero. Namely, $S_\proetale$ contains quasi-compact objects
    with infinitely many
    points. For example, let $E = \lim E_n$ be an inverse limit of
    finite sets with surjective transition maps,
    e.g., $E = \lim \mathbf{Z}/n\mathbf{Z}$.
    The scheme $\Spec(\colim \text{Map}(E_n, k))$
    is an object of $S_\proetale$ because $\colim \text{Map}(E_n, k)$
    is weakly \'etale (even ind-Zariski) over $k$.
    Thus $\mathcal{F}$ is a nonzero abelian sheaf whose stalk at the
    unique geometric point of $S$ is zero.
    
    \medskip\noindent
    The solution is to use the existence of quasi-compact, weakly contractible
    objects. First, there are enough quasi-compact, weakly contractible objects by
    Lemma \ref{lemma-proetale-enough-w-contractible}.
    Second, if $W \in \Ob(S_\proetale)$ is quasi-compact, weakly contractible,
    then the functor
    $$
    \Sh(S_\proetale) \longrightarrow \textit{Sets},\quad
    \mathcal{F} \longmapsto \mathcal{F}(W)
    $$
    is an exact functor $\Sh(S_\proetale) \to \textit{Sets}$ which commutes
    with all limits. The functor
    $$
    \textit{Ab}(S_\proetale) \longrightarrow \textit{Ab},\quad
    \mathcal{F} \longmapsto \mathcal{F}(W)
    $$
    is exact and commutes with direct sums (as $W$ is quasi-compact, see
    Sites, Lemma \ref{sites-lemma-directed-colimits-sections}), hence
    commutes with all limits and colimits. Moreover, we can check exactness of
    a complex of abelian sheaves by evaluation at the
    quasi-compact, weakly contractible objects of $S_\proetale$, see
    Cohomology on Sites, Proposition
    \ref{sites-cohomology-proposition-enough-weakly-contractibles}.

    Comments (7)

    Comment #311 by BB on September 18, 2013 a 10:12 am UTC

    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 = \mathop{\rm lim}\nolimits mu_n, then the LHS is 0, while the RHS is not. It seems these are two different points...

    Comment #312 by Johan (site) on September 18, 2013 a 1:20 pm UTC

    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 February 29, 2016 a 3:05 am UTC

    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 #1889 by Johan (site) on April 1, 2016 a 11:16 pm UTC

    Thanks, fixed here.

    Comment #2445 by Ben Moonen (site) on March 8, 2017 a 2:46 pm UTC

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

    Comment #2446 by Ben Moonen (site) on March 8, 2017 a 2:48 pm UTC

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

    Comment #2488 by Johan (site) on April 13, 2017 a 10:54 pm UTC

    Thanks, fixed here. Hi Ben!

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