Let $M$ be a $ \mathbb{Z}\_{p}[[T]] $-module and $X=Hom(M,\mathbb{Q}\_{p}/\mathbb{Z}\_{p})$ be the dual of $M$. Let $X[p^n]$ denotes the $p^n$-torsion points of $X$. Is $X/X[p^n]$ the dual of $M[p^n]$ $?$ If not, then whose dual is $X/X[p^n]$ in terms of $M$ $?$



Dual of a module
As @S.Carnahan shows. this is not true.


Perhaps the statement you want is that the dual of $M[p^n]$ is $X/p^nX$?


Take the exact sequence


$$0\to M[p^n] \to M \to M \to M/p^nM \to 0,$$


where the middle map is multiplication by $p^n$, and dualize it.


**Edit:** This also shows that $X/X[p^n]$ is the dual of $p^nM$, to answer the second question.


