The Noether map II

Abstract:

Let $\rho : G\hookrightarrow \mathrm {GL}(n,\ \mathbb {F})$ be a faithful representation of a finite group $G$. In this paper we proceed with the study of the image of the associated Noether map \[ \eta _G^G: \mathbb {F}[V(G)]^G \rightarrow \mathbb {F}[V]^G. \] In our 2005 paper it has been shown that the Noether map is surjective if $V$ is a projective $\mathbb {F} G$-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for $p$-groups (where $p$ is the characteristic of the ground field $\mathbb {F}$) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of $V$.