Cohen-Macaulay modules of covariants for cyclic $p$-groups

Let $G$ be a a finite group, $k$ a field of characteristic dividing $\mathrm{\mid }G\mathrm{\mid }$ and and $V,W$ $kG$-modules. Broer and Chuai showed that if $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}(V^G)\le 2$ then the module of covariants $k[V,W{]}^G=(k[V]\otimes W{)}^G$ is a Cohen-Macaulay module, hence free over a homogeneous system of parameters for the invariant ring $k[V{]}^G$. In the present article we prove a general result which allows us to determine whether a set of elements of a free $A$-module is a generating set, for any $k$-algebra $A$. We use this result to find generating sets for all modules of covariants $k[V,W{]}^G$ over a homogeneous system of parameters, where $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}(V^G)\le 2$ and $G$ is a cyclic $p$-group.