# Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids

Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ and the non-archimedean analytic space $X^{an} = \operatorname{sp} \mathcal{A}$. The former has an etale theory from the 1960's, and the latter from V. Berkovich's '93 IHES paper. In particular for an abelian etale sheaf $\mathcal{L}$ I can analytify the sheaf to get a sheaf on the etale topology of $X^{an}$, which I'll call $\mathcal{L}^{an}$.

Are there any comparison results between $H^i(X_{\acute{e}t},\mathcal{L})$ and $H^i(X^{an}_{\acute{e}t},\mathcal{L}^{an})$?

The comparison theorems in that paper seem to work for algebras of finite type over $K$, or of finite type over an affinoid base. It doesn't seem directly possible to set it up so that the derived pushforward $R^q\varphi_*$ maps to $\operatorname{sp} K$.

I am principally interested in smooth affinoids over a discretely valued field, and $\mathcal{L}$ a locally constant sheaf of finite abelian groups whose orders are prime to the characteristic of the residue field.

I have a very convoluted argument in mind using Berkovich's most recent pre-print but I imagine there is an easier way.

EDIT: I suppose I should probably indicate that I've put some thought into this: You can take a closed immersion of $X^{an}$ into some ball (e.g. higher dimensional analogues of $E(0,r) \times D(0,s)$). This closed immersion is algebraically of finite type, and so we can use the comparison theorem to compare the two push-forwards. This plus the Leray spectral sequence would reduce the problem to showing the analogous result for constructible sheaves on balls. However there doesn't really seem to be any tools that I can obviously use to attack this.

Not sure if this is still relevant, but the desired comparison in the case of "principal interest" to you can be proved in two different ways.

On one hand, you can deduce it from some recent results of Achinger, which imply that both cohomologies in question are unchanged if you replace the etale sites by the finite etale sites. Since the finite etale sites of $X$ and $X^{an}$ are canonically equivalent, this gives what you want. This works for any affinoid $\mathcal{A}$ over a complete discretely valued $K$ and any finite locally constant sheaf of abelian groups $\mathcal{L}$ on $X_{et}$.

Alternatively, and now for $K$ any complete nonarchimedean field, you can make a devissage to the case where $\mathcal{L}$ is a constant sheaf of finite abelian groups, which then is handled by Corollary 3.2.3 in Huber's book. To do this, pick a finite etale Galois cover $f:X'\to X$ with Galois group $G$ such that $f^{\ast}\mathcal{L}$ is constant, and use the Hochschild-Serre spectral sequences $H^i(G,H^j_{et}(X',f^{\ast}\mathcal{L})) \Rightarrow H^{i+j}_{et}(X,\mathcal{L})$ and $H^i(G,H^j_{et}(X'^{an},f^{an,\ast}\mathcal{L}^{an})) \Rightarrow H^{i+j}_{et}(X^{an},\mathcal{L}^{an})$. There is a canonical morphism from the first spec. seq. to the second, and it's an isomorphism on all terms of the $E_2$-page (by Huber's Corollary 3.2.3 plus some nonsense about compatibilities), so it gives an isomorphism on the abutments.

• Yes, thank you! I had skimmed the introduction of his paper, but I hadn't connected it to this question. – Joe Berner Aug 8 '17 at 18:47

Yes, there is such a comparison theorem. It works not only for finitely generated $K$-algebras, but more generally for finitely generated $A$-schemes for $A$ an affinoid algebra. This is done by Berkovich in the paper you mentioned (IHES 93), for coefficients in a ring whose torsion is prime to the residue characteristic.

He has extended those results later in a paper published in Israel Journal of Maths. The torsion (of the coefficients) is now allowed to be prime to the characteristic of the ground field. But I do not remember whether this holds for finitely generated schemes over an arbitrary affinoid algebra, or only over a field.

• I just checked. Unfortunately, this is only proved (in a relative form) for schemes of finite type over a field. – ACL Jun 12 '16 at 21:21
• The comparison theorem in that paper applies for finite type morphisms of finite type $A$ schemes, but I want to attack $\operatorname{spec} A$ itself. As a concrete example, I don't see how I can apply these results to compare the classical etale cohomology $H^i(\operatorname{spec}K\{rT\}, \mathbb{Z}/l)$ with the etale cohomology of the Berkovich disk $H^i(\operatorname{sp} K\{rT\},\mathbb{Z}/l)$. The latter is zero for $i>0$ by some of the results in the '93 paper, but I don't know that the scheme's cohomology also vanishes. – Joe Berner Jun 13 '16 at 0:08
• Oh yes sorry, you are right, now I see the point. To my knowledge the answer is not known. For instance, Berkovich does not know whether the étale cohomology of an $n$-dimensional affinoid space over an algebraically closed field vanishes in rank $>n$ (he thinks that it is true, but cannot prove it). I remember that this paper "Mieda, Yoichi Variants of formal nearby cycles. J. Inst. Math. Jussieu 13 (2014), no. 4, 701–752" provides a definition of Berkovich nearby cycles that uses the scheme Spec A. Maybe you could find some ideas there that could be applied to your question? – Antoine Ducros Jun 14 '16 at 6:59
• I've decided to mark this as answered. The Mieda article doesn't have the result, but it seems to have an analogous result for the cohomology of $X_{\overline{K}}$ phrased in terms of adic spaces. I'm not very familiar with them, but I believe in the scenario I care about (hausdorff and paracompact) the etale theories should be same. Mieda uses a special case of a result from Huber's book, and the more general statement in the book (hopefully) implies what I want. It's 3.5.13 in the book. – Joe Berner Jun 18 '16 at 16:08
• Simply be careful that the compactly supported étale cohomology of Huber is not the same than that of Berkovich on spaces that have boundary (eg. the unit disc). But if I'm not wrong the usual étale cohomologies of both theories coincide. – Antoine Ducros Jun 18 '16 at 20:00