Let $p$ be a prime and let $\{A\_n\}\_{n > 0}$ be an inverse limit of abelian groups such that $A\_n$ is $p^n$-torsion with $A\_n/p^{n - 1} \cong A\_{n - 1}$ (these isomorphisms are part of the data). Let $A = \varprojlim A\_n$ be the inverse limit, which surjects onto each $A\_n$. Is it necessarily true that $A/p^n \cong A\_n$?


This should be elementary but I can neither prove it nor find a likely to exist counterexample. It is equivalent to ask whether $lim^1$ vanishes for the system $\{ A\_n[p^m]\}\_{n > 0}$ for each fixed $m$.



Inverse limit of $p^n$-torsion abelian groups
See [Tag 09B8](https://stacks.math.columbia.edu/tag/09B8). Here are some more characters.


