Locally finite derivations and modular coinvariants

We consider a finite-dimensional $\mathbb {K}G$-module V of a p-group G over a field $\mathbb {K}$ of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic, this yields that the algebra $\mathbb {K}[V]_G$ of coinvariants is a free module over its subalgebra generated by $\mathbb {K}G$-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 [M. Sezer, Decomposing modular coinvariants, J. Algebra423 (2015), 87–92]. 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 Shank [M. Sezer and R. J. Shank, Rings of invariants for modular representations of the Klein four group, Trans. Amer. Math. Soc. 368 (2016), 5655–5673].