  Let $L/K$ be a Galois extension of local fields of characteristic $0$ with
Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers
in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a
$G$-module $F_L^n$ which as a set is the $n$-th power of the maximal ideal of
the ring of integers in $L$. We give explicit necessary and sufficient
conditions under which $F_L^n$ is a cohomologically trivial $G$-module. This
has applications to elliptic curves over local fields and to ray class groups
of number fields.
