Let me mention that I’m working with ‘classical’ rigid analytic spaces.

Let $f : Arightarrow B$ be an etale map of affinoid algebras over $mathbb{C}_p$.

Since there are possibly various notions of `etale’ maps of affinoids out there let me clarify what I mean. Let $Y = operatorname{Sp}(B)$ and $X = operatorname{Sp}(A)$. For every point $y in Y$ corresponding to the maximal ideal $mathfrak{n} subset B$, if its image in $X$ is denoted $x$ and corresponds to the maximal ideal $mathfrak{m} subset A$, then the induced map on local rings $mathcal{O}_{X,x} rightarrow mathcal{O}_{Y,y}$ is a flat map and $mathfrak{m}mathcal{O}_{Y,y} = mathfrak{n}mathcal{O}_{Y,y}$ (and the residue field extension is separable – but this is automatic over $mathbb{C}_p$). Note that the definition does not include “finite” hypothesis for the map.

Back to the question now:

**Question**: So $f : A rightarrow B$ is an etale map of affinoid algebras over $mathbb{C}_p$ such that the induced map $Y:=operatorname{Sp}(B)rightarrow operatorname{Sp}(A)=:X$, is a set-theoretic bijection. Suppose also that $A$ and $B$ are integral domains. Can I conclude that $Arightarrow B$ is an isomorphism. If in addition, $f :A rightarrow B$ were known to be a *finite* map (i.e. making $B$ a finite $A$-module) then indeed it is fairly straightforward to prove that such a map must be an isomorphism. However, apriori not knowing whether or not the map is finite leads to some difficulty. I wonder if someone could help me arrive at a proof.

**Some random observations of mine that might or might not be useful**:

(1) There is a cool theorem which says basically that etale maps of affinoids can be written locally as a composition of an open immersion and then a *finite* etale map. In our situation, this basically means that we can reduce to the case that $f : A rightarrow B$ has a factorization as $A xrightarrow{f’} B’ rightarrow B$ where $A rightarrow B’$ is a *finite* etale map of affinoid algebras, and $operatorname{Sp}(B) hookrightarrow operatorname{Sp}(B’)$ is an embedding as an affinoid subdomain. We can even assume that $B’$ is an integral domain.

(2) I can prove that $f :A hookrightarrow B$ is a faithfully flat ring map, and hence it must be an inclusion. Also, it follows fairly easily that for every maximal ideal $mathfrak{m}subset A$, that $mathfrak{m}B$ is maximal in $B$. Similarly, $A hookrightarrow B’$ is a finite-etale map (in the usual ring-theoretic sense now) and is injective (being faithfully flat). If I am able to show that $mathfrak{m}B’$ is a maximal ideal in $B’$, I would also be done.

(3) The map $operatorname{Spec}(B) rightarrow operatorname{Spec}(A)$ being faithfully flat, is submersive. Affinoids are Noetherian, Jacobson and so closed points are dense in every closed subset of Specs. Somehow my intuition now seems to be telling me that if on closed points $operatorname{Max}(B)rightarrow operatorname{Max}(A)$ is a bijection then maybe, just maybe, $operatorname{Spec}(B) rightarrow operatorname{Spec}(A)$ is also a bijection. If it were a bijection then it would also be a homeomorphism (being a submersion), which I feel is a step in the right direction, (but again that doesn’t still finish the proof since we want the ring map $Arightarrow B$ to be an isomorphism).

I am fairly convinced that the answer should be that $f$ is an isomorphism under the given hypotheses, but feel free to assume that $A$ is regular at every closed point if that helps.

Thanks in advance.