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This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism of (analytic) adic spaces (suppose affinoid) is étale if and only if the induced morphism between the first components of the Huber pairs defining the spaces is algebraically étale, that one can even prove it in purely algebraic setting getting again the result.

The question is the following. Consider an étale morphism $f:X\rightarrow Y$ of locally of finite type Tate adic spaces over $(\mathbb{Q}_{p},\mathbb{Z}_{p})$. Let $u\in Y(K, K^{+})$ be a point of $Y$ with values in an affinoid pair $(K,K^{+})$, where $K$ is a nonarchimedean extension of $\mathbb{Q}_{p}$, and $K^{+}$ is a valuation ring of $K$. Suppose that this point has a lifting to a $(K,K^{+})$-point of $X$. Then the same is true on a small enough neighborhood of $u$, i.e. there exists a neighborhood of $u$, say $U$, such that every $(K,K^{+})$-valued point of $U$ has a lift to a $(K,K^{+})$-point of $X$.

My idea is the following. First, the whole problem is local both in the target and in the domain, hence we can assume that $X=\text{Spa}(B,B^{+})$, and that $Y=\text{Spa}(A,A^{+})$ for $A,B$ complete affinoid. Moreover, since the map is étale, by a result of Huber, we can assume that $B=A\langle X_{1},\ldots ,X_{n}\rangle/(f_{1},\ldots ,f_{n})$, with $f_{i}\in A[X_{1},\ldots ,X_{n}]$ for all $i$ such that $\text{det}\left(\frac{\partial f_{i}}{\partial X_{j}}\right)_{i,j=1,\ldots ,n}$ is a unit in $B$. Now, we have a morphism $u:(A,A^{+})\rightarrow (K,K^{+})$, which is the $(K,K^{+})$-point of $Y$, lifting to $x:(B,B^{+})\rightarrow (K,K^{+})$. But now I was not able to go on, how to construct the right neighborhood of $u$? Thanks for any suggestion!

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  • $\begingroup$ math.stackexchange.com/q/1829116/77622 $\endgroup$
    – Simone
    Jun 16, 2016 at 22:27
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    $\begingroup$ In the setting of classical affinoid algebras that you are considering (you may as well take $X$ and $Y$ to be affinoid), etaleness is equivalent to the notion by that name in classical rigid-analytic geometry, and it is very far from etaleness in the ring/scheme-theoretic sense (e.g., consider an open immersion of affinoids, even just an inclusion of discs). So one cannot analyze this using just the theory of etale maps of schemes (though some of the ideas do adapt, working with rigid-analytic relative 1-forms and rigid-analytic flatness, etc.). $\endgroup$
    – nfdc23
    Jun 16, 2016 at 23:14
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    $\begingroup$ Have you worked out the analogue for classical rigid-analytic spaces (i.e., the case when $K$ is finite over the non-archimedean ground field)? If not, do that first! Hint: the key is to prove that the local ring on a rigid-analytic space is henselian (the proof of which rests on the analytic fact that a point isolated in its fiber admits an open neighborhood that is finite over an open neighborhood of its image; the details require some serious input about henselian local rings); even for complex-analytic spaces this is a serious theorem (see Houzel's Seminaire Cartan lectures). $\endgroup$
    – nfdc23
    Jun 17, 2016 at 14:23
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    $\begingroup$ The algebraic version of what you're looking for is Lemma 15.7.13 of stacks.math.columbia.edu/tag/07LW. $\endgroup$
    – js21
    Jun 17, 2016 at 16:15

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