Failure of Tate acyclicity for integral structure sheaves Let $(A,A^+)$ be a sheafy Tate-Huber pair, and let $X=\operatorname{Spa}(A,A^+)$.  It is well-known that $H^i(X,\mathcal{O}_X)=0$ for $i>0$.  I assume it is generally not true that $H^i(X,\mathcal{O}_X^+)=0$ for $i>0$, but I don't think that I have seen an explicit counterexample.  Is there a simple example of a nonzero cohomology class in some $H^i(X,\mathcal{O}_X^+)$ for $i>0$?
 A: Below is an example of a formal scheme $\mathfrak{X}$ such that its rigid analytic generic fiber $X$ is affinoid, while its special fiber contains a complete elliptic curve $E$.  Because $H^1(E,\mathcal{O}_E) \ne 0$, $H^1(X,\mathcal{O}_X^+) \ne 0$ as well.
Let $p$ be an odd prime, and let $X$ be the rigid analytic elliptic curve over $\mathbb{Q}_p$ defined by $$y^2 = x^3-x, \quad |x| \le |p^{-2}|\,.$$
It has a formal model over $\mathfrak{X}$ over $\mathbb{Z}_p$ that is covered by two affines
$$\mathfrak{U}=\operatorname{Spf}\mathbb{Z}_p\left<x,y\right>/(y^2-x^3+x)$$
$$\mathfrak{V}=\operatorname{Spf}\mathbb{Z}_p\left<u,v,w\right>/(v-u^3+uv^2,uw-p)$$
where $u=x/y$, $v=1/y$, $w=py/x$.  These correspond to rational subsets $U$ and $V$, defined by the inequalities $|x| \le 1$ and $|y| \ge 1$, respectively.
One can check that the above rings are normal using Serre's criterion.  They are also flat over $\mathbb{Z}_p$.  By Theorem 7.4.1 in de Jong's paper Crystalline Diedonné module theory via formal and rigid geometry, the above rings can be identified with $\mathcal{O}_X^+(U)$ and $\mathcal{O}_X^+(V)$.
Now consider $y/x = xu - v \in \mathcal{O}_X^+(U \cap V)$.  I claim that $y/x \notin \mathcal{O}_X^+(U) + \mathcal{O}_X^+(V)$.  Indeed, consider the elliptic curve defined by $p=0$ on $\mathfrak{U}$ and $w=0$ on $\mathfrak{V}$.  On this curve, $y/x$ has simple poles at the points $x=y=0$ and $u=0$, while elements of $\mathcal{O}_X^+(U) + \mathcal{O}_X^+(V)$ have either no pole or a pole of order $\ge 2$ at these points.  Hence $y/x$ determines a nontrivial element of $H^1(X,\mathcal{O}_X^+)$.
Since $H^1(X,\mathcal{O}_X)=0$, the above cocycle must be $p$-power torsion.  In fact, it is annihilated by $p$, since $py/x=w$.
A: Here is another example, taken from remark 9.3.4 in Bhargav Bhatt's lecture notes.
Let $A^+=\mathbb{F}_p[t,y,z]/(y^2-z^3)$, with the $t$-adic topology, let $A=A^+[1/t]$, and let $X=\operatorname{Spa}(A,A^+)$.  To construct a nontrivial class in $H^1(X,\mathcal{O}_X^+)$, we will use the exact sequence
$$0 \to H^0(X,\mathcal{O}_X^+)/t^n \to H^0(X,\mathcal{O}_X^+/t^n) \to H^1(X,\mathcal{O}_X^+)[t^n] \to 0.$$
For any $n$, one can form a section of $H^0(X,\mathcal{O}_X^+/t^n)$ that restricts to $\frac{y}{z}$ on the locus $z \ne 0$ and to $0$ on the locus $|z| \le |t|^{2n}$.  One can check that this section does not come from $H^0(X,\mathcal{O}_X^+) = \widehat{A^+}$, so its image in $H^1(X,\mathcal{O}_X^+)[t^n]$ must be nonzero.
