Counterexamples to Conjectures of Wehlau on Noether Numbers

Abstract

Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to several questions and conjectures of Wehlau concerning Noether numbers. For the $2$-group $G=D_8$, we exhibit a submodule $U\subseteq V$, with ${\dim }_{k}U=5$ and ${\dim }_{k}V=6$, such that ${\hat{I}}^2(k[U{]}^G)=6>5={\hat{I}}^2(k[V{]}^G)$, thereby disproving submodule monotonicity. Writing $X=V^{\ast }$ and $Q=U^{\ast }$, the corresponding nonsplit exact sequence $0⟶k⟶X⟶Q⟶0$ also satisfies ${\hat{I}}^2(k[Q{]}^G)=8>6={\hat{I}}^2(k[Q^{\ast }{]}^G)$ and ${\hat{I}}^2(k[Q{]}^G)=8>5={\hat{I}}^2(k[X{]}^G)$. Thus the modular Noether number need not be invariant under duality, and quotient monotonicity also fails. Notably, the basic counterexamples already occur for $2$-groups in defining characteristic. The constructions remain valid over every field of characteristic $2$, and the $D_8$ conclusions propagate by inflation to every finite group admitting $D_8$ as a quotient.