The Noether map

AUTHORS: Mara D. Neusel (Texas Tech University),
         M\"ufit Sezer   (Bo\u gazici \"Universitesi)

EMAILS:  mara.d.neusel@ttu.edu 
         mufit.sezer@boun.edu.tr

ABSTRACT:
Let $\rho: G\hra GL(n\/,\ \F)$ 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.  Furthermore, we show that the Noether
map is surjective, i.e., its image integrally closed, if $V=\F^n$ is a
projective $\F G$-module.  Moreover, we show that the converse of this
statement is true if $G$ is a $p$-group and $\F$ has characteristic $p$,
or if $\rho$ is a permutation representation. We apply these results and
obtain upper bounds on the Noether number and the Cohen-Macaulay defect
of $\F[V]^G$. We illustrate our results with several examples.