For $L/K$, any totally ramified cyclic extension of degree $p^2$ of local fields which are finite extensions of the field of $p$-adic numbers, we describe the ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-module structure of each fractional ideal of $L$ explicitly in terms of the $4p+1$ indecomposable ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. In particular, if $t_n-[t_n/p]\<p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i\equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-Galois...