Pontryagin dual

Question

Suppose $M$ is a $Z_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero $\Longleftrightarrow$ $M[p]$($p$-torsion of $M$) is finite $?$

Answer 1

Let $\Lambda=\mathbf{Z}_p[[T]]$. The group $M[p]$ is dual to $M^\vee/pM^\vee$. So if $M[p]$ is finite, then $M^\vee/pM^\vee$ is finite, and the form of Nakayama's lemma applicable to profinite modules over profinite rings implies that $M^\vee$ is finitely generated over $\mathbf{Z}_p$, and in particular that its $p$-primary subgroup is finite. Since $\Lambda/p^n\Lambda$ is never a finite $\mathbf{Z}_p$-module for $n\geq 1$, this implies that the $\mu$-invariant of $M^\vee$ is equal to zero.

Conversely, if the $\mu$-invariant is zero, then the structure theory for $\Lambda$-modules plus the assumption that $M^\vee$ is torsion shows that $M^\vee/pM^\vee$ is finite, so $M[p]$ is as well.

Answer 2

Let $X$ be the Pontryagin dual of $M$. Then the dual of $M[p]$ is $X/pX$. Now it is easy to see from the decomposition theorem for modules over the Iwasawa algebra $\Lambda = \mathbb{Z}_p[\![T]\!]$ that $X/pX$ is finite for a finitely generated torsion $\Lambda$-module if and only if $\mu(X)=0$.