The noether map i

Abstract

Let æ : G GL(n, F) be a faithful representation of a finite group G. In this paper we study the image of the associated Noether map J G G : F[V(G)]G → F [V]G. It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure Im (JG G = F [V]G. This is true without any restrictions on the group, representation, or ground field. Moreover, we show that the extension Im(J G G) ⊆ F [V]G is a finite p-root extension if the characteristic of the ground field is p. Furthermore, we show that the Noether map is surjective, if V = Fn is a projective FG-module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of FVG and the Cohen-Macaulay defect of FV G. We illustrate our results with several examples. © de Gruyter 2009.