Locally finite derivations and modular coinvariants

We consider a finite dimensional $\kk G$-module V of a p-group G over a field $\kk$ of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic this yields that the algebra $\kk[V]_G$ of coinvariants is a free module over its subalgebra generated by $\kk G$-module generators of V∗. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order, \cite{SezerCoinv}. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank \cite{SezerShank}.