The Noether Map I
Mara D Neusel and M"ufit Sezer
mara.d.neusel@ttu.edu mufit.sezer@boun.edu.tr
Abstract:
Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a
finite group G. In this paper we study the image of the associated
Noether map
$
\eta_G^G: F[V(G)]^G \longrightarrow 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
$\overline{Im(\eta_G^G)} =F[V]^G$. This is true without any
restrictions on the group, representation, or ground field. Moreover,
we show that the extension $Im (\eta_G^G) \subseteq F[V]^G$ is a finite
$p$-root extension. Furthermore, we show that the Noether map is
surjective, i.e., its image integrally closed, 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.
Note that this paper together with noether-map-II contain stronger results
than the authors' previous paper Neusel-Sezer/noether.