I am asking about the properties of rank over an inverse limit.
Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be the $\pi$-adic completion of $R[[X]]$.
Let $S \subseteq T$ be two $R [[X]]$-modules contained in an $\mathscr{R}$-module $M$. We may write $M = \varprojlim M / \pi^\ell M$ and suppose that we have $T/S = \varprojlim (T_\ell / S_\ell)$ with $T_\ell,S_\ell$ denoting the projection of $T,S$ into $M/\pi^\ell M$ respectively, where additionally each $T_\ell$ and $S_\ell$ are finitely-generated over $R [[X]]$. Note that each transition map is constructively surjective.
Question: If we have that $\mathrm{rank}_{R [[X]]} T_\ell /S_\ell \leq \mathrm{dim}_\mathscr{R} (M/\pi^\ell M)$ for all $\ell \in \mathbb{N}$, does it also follow that $\mathrm{rank}_{R [[X]]} (T / S) \leq \mathrm{dim}_\mathscr{R} M$? I.e. does the inverse limit preserve the rank inequality?
It seems reasonable to me personally that this holds, but I cannot seem to prove it for myself. I have looked everywhere online for a source to confirm/deny this but to no avail. The closest I have come was a paper on the topic of finite-rank free groups (see below), which states it either holds or we have an isomorphism to a universal group - an object I am not familiar with.
Any help would be greatly appreciated!
M
https://arxiv.org/abs/1105.0403
Edit: I have since found the following paper, which in the introduction claims there to be an "obvious result for compact spaces", without stating said result. Here we may assume that compact is equivalent to finite generation (since $R$ is noetherian).