Modular Covariants of Cyclic Groups of Order p

Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$. We conjecture that a similar set of covariants generates $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$ when $W$ has arbitrary dimension.