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I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".

First of all, to define the etale site of an adic space, we only consider adic spaces that are locally of the form $Spa(A,A^+)$ where $A$ has one of the following properties
i) $\hat{A}$ is discrete
ii) $A$ is strongly Noetherian Tate
iii) $\hat{A}$ is has a Noetherian ring of definition over which it is finitely generated.

Then the etale site of $X$ is defined in the same way as with schemes (namely by considering surjective families of etale morphisms). Why do we restrict ourselves only to adic spaces of the above form? Could we not, for example, consider stable adic spaces instead? Let me explain that. I give the definitions, as I haven't seen those being widely used. Those are taken from Wedhorn's notes, found here https://arxiv.org/pdf/1910.05934.pdf.

Definition 1 A Huber ring $A$ is called stably sheafy if every $\hat{A}$-algebra topologically of finite type is sheafy, i.e given such an algebra $B$ and any subring of integral elements $B^+$, $(B,B^+)$ is sheafy.

Definition 2 A stable adic space $X$ is an adic space that is covered by affinoid adic spaces $Spa(A,A^+)$ where $A$ is stably sheafy.

Now consider only adic spaces that are stable. This includes the adic spaces that Huber considers in his book. Then the category of adic spaces that are etale over $X$ (a stable adic space) is well defined. Moreover if we define $X_{et}$ in the same way as above, everything seems to work well (for example fiber products exist under the same assumptions as in Huber's book). So what is it (if there is something) that restricts us to work only in the case Huber treats in his book? Perhaps there are properties that we want the etale site to have and fail in this more general setting. Let me suggest one, that I cannot see if it is true or not.

Question 1 Let $f:X\rightarrow Y$ be an etale morphism of affinoid stable adic spaces. Is $\mathcal{O}_Y(Y)\rightarrow\mathcal{O}_X(X)$ flat?

If this is true, I expect the proof to be hard. The analogous statement ,in the usual setting, is lemma 1.7.6 in Huber's book. In any case, regardless whether we can work in this more general setting or not, I would like to see an answer related to question 1.

I haven't thought much of the latter and I haven't checked every detail to make sure everything works fine (but I hope it does). Any answer is very much appreciated.

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  • $\begingroup$ Let me add as a comment that even in a more general setting, the known results would not change, as etale adic spaces over $X$ (where $X$ is as in Huber's book) remain the same. So even if we can do the above, an important question is whether we care about the etale cohomology of stable adic spaces that are not of the form that Huber considers. $\endgroup$
    – RumDiary
    Dec 27 '20 at 19:43
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Great question!

The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat obscure for some time, so he probably didn't want to further scare people by allowing the most general non-noetherian case! Also, the detailed analysis of sheafyness that happened in recent years wasn't known then. In particular, it was not known that there are interesting large classes of stably sheafy spaces outside of the cases considered by Huber.

On the other hand, it turns out that for the definition of the étale site, sheafyness is not really important. See, for instance, Chapter 9 (Definition 9.1.2) of Kedlaya-Liu 1, https://arxiv.org/abs/1301.0792, for a definition of the étale site for general preadic spaces (that is allowing non-sheafy Huber pairs). In www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf (see especially Section 15), I also discuss étale sites of preadic spaces (in the analytic case). Mostly all basic properties of the étale site, especially pertaining to étale cohomology, extend to this more general setting. One important property is that étale maps of (qcqs) adic spaces admit "noetherian approximation", so one can often formally reduce to the cases considered by Huber.

Re your comment: There has recently been a large interest in studying the étale site of adic spaces not falling into the settings considered by Huber; the prototypical case is that of perfectoid spaces.

Incidentally, in the case of perfectoid spaces, we very much expect the answer to your question to be negative: Already the map $$ \mathbb C_p\langle T^{1/p^\infty}\rangle\to \mathbb C_p\langle (\tfrac Tp)^{1/p^\infty}\rangle $$ corresponding to an inclusion of perfectoid discs is probably not flat (although, I'm embarassed to say, we still don't know this??).

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