The noether map i

TL;DR: In this paper, it was shown that the integral closure of the Noether map characterizes the ring of invariants in the sense that its integral closure is closed without any restrictions on the group, representation, or ground field.

Abstract: 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 . It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure . This is true without any restrictions on the group, representation, or ground field. Moreover, we show that the extension 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 = 𝔽 n is a projective 𝔽G-module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of 𝔽[V] G and the Cohen-Macaulay defect of 𝔽[V] G . We illustrate our results with several examples.