# Bounded torsion of quotients of affine formal models

$$\DeclareMathOperator\Sp{Sp}$$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.

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.