$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a uniformizer $\pi$. If $Y=\Sp(B)\subset X$ is a connected affinoid subdomain, the conditions above imply that the rings of power-bounded elements $A^{\circ}, B^{\circ}$ are noetherian affine formal models of $A$ and $B$ respectively. Furthermore, we have an inclusion $A^{\circ}\rightarrow B^{\circ}$, and we can denote its quotient by $C=B^{\circ}/A^{\circ}$. This quotient will, in general, not be flat as a $\mathcal{R}$-module as it will have $\pi$-torsion. For example, if $X=\mathbb{B}^{1}_{\vert \frac{1}{\pi}\vert}$ is the disk of radius $\vert \frac{1}{\pi}\vert$ and $Y=Sp(K\langle t\rangle)$ is the unit disc, then $t$ is power bounded on $Y$ and not on $X$. However, $\pi t$ is power-bounded in the ring of rigid functions of both spaces. My question is whether $C$ has bounded $\pi$-torsion. That is if there is a natural $n$ such that every element killed by a power of $\pi$ is already killed by $\pi^{n}$. I am not sure if this is the case, as a counterexample to this would be finding a family of functions $f_{n}$ on $\mathbb{B}^{1}_{\vert \frac{1}{\pi}\vert}$
such that they are power-bounded on $\mathbb{B}^{1}_{1}$ and such that there is a point $x\in \mathbb{B}^{1}_{\vert \frac{1}{\pi}\vert}$ such that $\vert f_{n}(x)\vert \geq \vert \frac{1}{\pi^{n}}\vert$, which seems like a natural thing to happen. I would also be interested in knowing if there are any conditions on $Y$ that would imply this kind of behaviour.
Bounded Torsion of Quotients of Affine Formal Models
ac.commutative-algebraag.algebraic-geometryarithmetic-geometryrigid-analytic-geometry
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Best Answer
As it turns out, the quotient $C$ does not necessarily have bounded $\pi$-torsion. For example, let $X=Sp(K\langle t\rangle)$ and $Y=Sp(K\langle \frac{t}{\pi}\rangle)$ be the disc of radius $\pi$. Then all the elements of the form $\frac{t^{n}}{\pi^{n}}$ are power bounded in $K\langle \frac{t}{\pi}\rangle$. Furthermore, their equivalence classes in $C$ are $\pi^{n}$-torsion but not $\pi^{n-1}$ torsion for each $n$. Thus, in general, the answer seems to be no. I would, however, still be interested in knowing any conditions on an admissible affinoid subdomain that would imply this kind of behavior.