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

Let $G$ be a a finite group, $k$ a field of characteristic dividing $|G|$ and and $V,W$ $kG$-modules. Broer and Chuai showed that if $\mathrm{codim}(V^G) \leq 2$ 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 $\mathrm{codim}(V^G) \leq 2$ and $G$ is a cyclic $p$-group.