TITLE: Vector bundles on adic spaces
QUESTION [6 upvotes]: Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? In other words, does there exist an analytic adic space $E$ with a morphism to $X$, unique up to isomorphism as an analytic adic space over $X$, such that for every open immersion of analytic adic spaces $S\rightarrow X$ there is a natural isomorphism $\mathcal{E}(S) \cong \mathrm{Hom}_X(S,E)$?
By a theorem of Kedlaya-Liu, we do know that there is a correspondence between $\mathcal{E}$'s and finite projective $A$-modules, which is achieved by mapping $\mathcal{E}$ to the global sections $M=\mathrm{H}^0(X,\mathcal{E})$. So a natural guess for $E$ would be to take $\mathrm{Spa}(B,B^+)$ with $B=\mathrm{Sym}_A(M^\vee)$; however, I do not know what to take for $B^+$. (Is there some sort of $M^+$?)
Thanks!

REPLY [8 votes]: The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ is an adic space. As this is covered by an increasing union of balls $\mathbb B^n_X=\mathrm{Spa}(A\langle T_1,\ldots,T_n\rangle,A^+\langle T_1,\ldots,T_n\rangle)$, it is also equivalent to ask whether $A\langle T_1,\ldots,T_n\rangle$ is sheafy. (Note that your choice of $B$ is not correct, as your $B$ is not a Huber ring. You have to complete $B$ in a certain way -- this is like the difference between $A[T_1,\ldots,T_n]$ and $A\langle T_1,\ldots,T_n\rangle$ -- and then take a union over all possible such choices of completions.)
Now I'm not up to speed about the known counterexamples to sheafyness, but I think there ought to be examples where $A$ is sheafy but $A\langle T\rangle$ is not, so in this sense the answer to your question would be no.
On the other hand, any practical condition guaranteeing sheafyness of $A$ usually also implies sheafyness of $A\langle T\rangle$ -- e.g., strongly noetherian, or sousperfectoid, or ... .
Finally, let me also add again the advertisement that in the setting of Analytic Geometry, it is possible to generalize Huber's theory of adic spaces to incorporate non-sheafy Huber rings (in particular, by allowing the structure sheaf to be a sheaf of animated condensed rings). So in this generalized setting, the answer to the question would also be Yes.