Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?

Question

Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \hookrightarrow \Bbb C}|\sigma(\cdot)|$.

My very naïve and ignorant understanding is that one of the "selling points" of considering $X$ is that it incorporates both Archimedean and non-Archimedean information. Hence I would expect that the "correct" theory of étale cohomology for $X$ reflects this. There's already a variant of étale cohomology that is "sensitive to Archimedean information", namely the cohomology of the Artin–Verdier site as defined here. Therefore, I would expect étale cohomology of $X$ (whatever it is precisely) to be isomorphic to the cohomology of the Artin–Verdier site. However it, from looking at some sources on étale morphisms in the setting of global Berkovich geometry, it seems that there's a GAGA theorem of the sort that claims that étale cohomology of $X$ is just the étale cohomology of $\operatorname{Spec}(\mathcal O_K)$, which ignores ramification at infinity.

This has me confused. I guess on a more basic level, my question is related to the question: Does an extension of number fields that is ramified only at infinite places induce a ramified or an unramified morphism on analytic spectra?

Answer 1

$\DeclareMathOperator\Spec{Spec}$Another way to see it is that, by allowing ramification at infinite primes, we see Archimedean phenomena in the cohomology. The Artin–Verdier site is a way to get rid of this phenomena. That is, the étale cohomology of $\Spec \mathcal{O}_K$ does incorporate Archimedean information, whereas the cohomology of the Artin–Verdier site does not. For instance, the usual étale cohomology $H^n(\Spec \mathcal{O}_K, \mathbb{Z}/2\mathbb{Z})$ is non-trivial in arbitrarily high even degrees $n$ when $K$ has real embeddings, which reflects the Tate cohomology of $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$.

The Artin–Verdier site is one way to make Artin–Verdier duality work in the presence of real embeddings, simply by disregarding the Archimedean phenomena. But there is another way to make Artin–Verdier duality work in the presence of real embeddings, without changing the étale cohomology, namely by using flat cohomology with compact support as the dual cohomology. The equivalent of the paper you linked to is this preprint.

Artin–Verdier duality using flat cohomology with compact support first appeared in Milne’s Arithmetic Duality Theorems, but contains some flaws. The construction of the correct cohomology complex is due to Alexander Schmidt (see this paper) and a corrected version of Artin–Verdier duality was written down by Demarche–Harari here.

The answer to your last question can be found in Proposition 8.8 of this paper, which says that étaleness can be checked at the level of schemes in this case.

I hope that this answers your question to some extent.