(pro)Étale cohomology of adic spaces and inverse limit

Question

I am studying Peter Scholze's paper $p$-adic Hodge theory for rigid-analytic varieties and I am confused by the following.

Let $X$ be a finite type scheme over $\mathbb{C}_p$ (proper and smooth if you want) and let $X_{\mathrm{ét}}$ be the usual étale site. Usually, the (algebraic) étale cohomology with coefficients in $\mathbb{Z}_p$ is defined as $$ H^i(X_{\mathrm{ét}}, \mathbb{Z}_p) := \varprojlim_n H^i(X_{\mathrm{ét}}, \mathbb{Z}/p^n), $$ and we cannot move the inverse limit inside. Indeed one can consider the cohomology of the constant sheaf $\mathbb{Z}_p$, but this is not well behaved. Let $X^{\mathrm{ad}}$ the adic space associated to $X$, with étale site $X_{\mathrm{ét}}^{\mathrm{ad}}$. The étale cohomology is defined as above, i.e. as the inverse limit of the cohomology groups of the sheaves $\mathbb{Z}/p^n$.

Now let us consider the pro-étale cohomology. If I understand correctly, the paper, Lemma 3.18 (together with other results) says that cohomology commutes with inverse limit, so

$$ H^i (X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \mathbb{Z}_p) \cong \varprojlim_n H^i(X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \mathbb{Z}/p^n), $$ where this time the group on the left is just the cohomology of the constant sheaf $\mathbb{Z}_p$.

So far so good, but let $\nu \colon X^{\mathrm{ad}}_{\mathrm{pro-ét}} \to X^{\mathrm{ad}}_{\mathrm{ét}}$ be the natural morphism of sites. Corollary 3.17 says that for any abelian sheaf $\mathcal{F}$ on $X^{\mathrm{ad}}_{\mathrm{ét}}$ we have $$ H^i(X^{\mathrm{ad}}_{\mathrm{ét}}, \mathcal{F}) \cong H^i(X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \nu^\ast\mathcal{F}), $$ and if we apply this to $\mathcal{F} = \mathbb{Z}_p$ (the constant sheaf, no inverse system here) we find $$ H^i(X^{\mathrm{ad}}_{\mathrm{ét}}, \mathbb{Z}_p) \cong H^i(X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \nu^\ast\mathbb{Z}_p) = H^i(X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \mathbb{Z}_p)= \\ \varprojlim_n H^i(X^{\mathrm{ad}}_{\mathrm{pro-ét}}, \mathbb{Z}/p^n) \cong \varprojlim_n H^i(X^{\mathrm{ad}}_{\mathrm{ét}}, \mathbb{Z}/p^n). $$ In particular it seems that, for the étale cohomology of adic spaces, there is no need of taking the inverse limit outside the cohomology.

Is this correct/well known or am I missing something?

Thank you!

Answer 1

This problem came up in a reading group yesterday, and the consensus was that the problem is exactly the one that user45878 pointed out in the comments. Thanks to Bogdan Zavyalov for explaining this issue to me and providing the example below. Let's see why these are different with an example:

Consider the scheme $X = \mathrm{Spec}(\mathbf C)$ and its pro-étale site. Note that the category $X_{\mathrm{et}}$ is equivalent to the category of finite sets, with covers surjective maps.

Via this equivalence, we can see that $X_{\mathrm{pro-et}}$ consists of pro-finite sets, i.e. topological spaces $S$ which are isomorphic to $\varprojlim_{i \in I} S_i$ for some finite sets $S_i$ with surjective transition maps. Covers are jointly surjective families of open maps, at least when the index sets $I$ for the limits are countable. (See the erratum to the paper for a subtle set-theoretic issue coming up here; the definition used in more recent papers requires that $\{S^i \rightarrow S\}$ is a covering if for each quasi-compact open in $S$, there is a quasi-compact open in $\sqcup_{i \in I} S^i$ covering it).

Now, we want to consider the constant sheaves $\underline{\mathbf{Z}/\ell^n\mathbf{Z}}$ on $X_{\mathrm{pro-et}}$ and compute their limit. I claim that the sheaf $\widehat{\mathbf{Z}_\ell} := \varprojlim_n \underline{\mathbf{Z}/\ell^n\mathbf{Z}}$ is isomorphic to the sheaf $S \mapsto \mathrm{Hom}_{\mathrm{cont}}(S, \mathbf{Z}_\ell)$, where $\mathbf{Z}_\ell$ has its \emph{$\ell$-adic topology}. On the other hand, we can think of the constant sheaf $\underline{\mathbf{Z}_\ell}$ as the sheaf $S \mapsto \mathrm{Hom}_{\mathrm{loc. cont}}(S, \mathbf{Z}_\ell)$ (note that the topology on $\mathbf{Z}_\ell$ does not effect the notion of locally constant maps to it).

Now, the constant sheaves $\underline{\mathbf{Z}/\ell^n}\mathbf{Z}$ are given by $$S \mapsto \mathrm{Hom}_{\mathrm{loc. const.}}(S, \mathbf{Z}/\ell^n \mathbf{Z}) = \mathrm{Hom}_{\mathrm{cont}}(S, \mathbf{Z}/\ell^n\mathbf{Z})$$ with the groups $\mathbf{Z}/\ell^n\mathbf{Z}$ given the discrete topology. Thus, the inverse limit presheaf is given by $$S \mapsto \varprojlim_n \mathrm{Hom}_{\mathrm{cont.}}(S, \mathbf{Z}/\ell^n \mathbf{Z}) =: \mathrm{Hom}_{\mathrm{cont.}}(S, \varprojlim_n \mathbf{Z}/\ell^n \mathbf{Z}) = \mathrm{Hom}_{\mathrm{cont.}}(S, \mathbf{Z}_\ell)$$

Here, we are using the definition of the inverse limit of topological spaces. Now, it turns out that this is actually already a sheaf: the continuity condition is local in the pro-étale topology since pro-étale covers are open.

Here's a weird thing that can happen: this argument shows that we have an injective map of sheaves $\underline{\mathbf{Z}_\ell} \rightarrow \widehat{\mathbf{Z}_\ell}$. Call its cokernel $\mathscr{F}$. Note that in the terminology of Scholze's paper, we've shown that $X_{\mathrm{pro-et}}$ is equivalent as a site to the site $G-\mathrm{pfsets}$ with $G$ the trivial group. Here, $\widehat{\mathbf{Z}_\ell}$ is the sheaf $\mathscr{F}_M$ associated to the topological $G$-module (i.e. topological space) $\mathbf{Z}_\ell$ given the $\ell$-adic topology and $\underline{\mathbf{Z}_\ell}$ is $\mathscr{F}_M$ with $M = \mathbf{Z}_\ell$ with the discrete topology.

Now, since $G$ acts freely on the profinite set $\{1\}$, corresponding to $\mathrm{Spec}(\mathbf{C})$, Scholze's Proposition 3.7 shows us that the functor $\mathscr{F} \rightarrow \mathscr{F}(\mathrm{Spec} \mathbf{C})$ is exact. Thus, we have an exact sequence: $$ 0 \rightarrow \underline{\mathbf{Z}_\ell(\mathrm{Spec} \mathbf{C})} \rightarrow \widehat{\mathbf{Z}_\ell}(\mathrm{Spec} \mathbf{C})) \rightarrow \mathscr{F}(\mathrm{Spec} \mathbf{C}) \rightarrow 0 $$

But we know that $\underline{\mathbf{Z}_\ell}(\mathrm{Spec}\mathbf{C}) \rightarrow \widehat{\mathbf{Z}_\ell}(\mathrm{Spec}\mathbf{C})$ is an isomorphism (as locally constant functions and continuous functions from the one-point set to $\mathbf{Z}_\ell$ are the same thing), so $\mathscr{F}$ furnishes an example of a non-zero sheaf on the pro-etale site of $\mathrm{Spec}(\mathbf{C})$ which has no sections over the spectrum of any geometric point of $\mathrm{Spec}(\mathbf{C})$.

Note that in the erratum, Scholze removed the discussion of points after Proposition 3.13, but this is surely the sort of phenomenon he had in mind: in the notation of that proposition, the functors $i_x^*$ are jointly conservative (i.e. they detect whether a sheaf is $0$) for $x$ ranging over geometric points, but (unlike in the case of the étale topology) the functors $\Gamma \circ i_x^*$ are not!