Locally finite derivations and modular coinvariants

TL;DR

This paper studies the structure of coinvariant rings for finite p-group modules, providing new descriptions of Hilbert ideals and showing conditions under which these rings are free modules or complete intersections.

Contribution

It introduces a generating set for the Hilbert Ideal and characterizes the algebra of coinvariants as a free module over a subalgebra in cyclic cases, extending known results.

Findings

The algebra of coinvariants is a free module over a subalgebra generated by module generators in cyclic groups.

The Hilbert Ideal is a complete intersection for the Klein 4-group under certain conditions.

Provides explicit descriptions of Hilbert ideals for specific p-groups.

Abstract

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}.