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% Author Package file for use with AMS-LaTeX 1.2
\controldates{14-MAR-2001,14-MAR-2001,14-MAR-2001,14-MAR-2001}
 
\documentclass{era-l}

\issueinfo{7}{02}{}{2001}
\dateposted{March 16, 2001}
\pagespan{5}{7}
\PII{S 1079-6762(01)00088-9}
%\newcommand{\copyrightyear}{2001}
\copyrightinfo{2001}{American Mathematical Society}

\begin{document}

\title{On Noether's bound for polynomial invariants of a finite group}

\author{John Fogarty}
\address{Department of Mathematics and Statistics,
University of Massachusetts, Amherst, MA 01003-4515}
\email{jfoga9786@aol.com}

\commby{Efim Zelmanov}

\date{October 25, 1999}

\subjclass[2000]{Primary 13A50}

\begin{abstract}
E. Noether's a priori bound, viz., the group order $g$, for
the degrees of generating polynomial invariants of a finite
group, is extended from characteristic 0 to characteristic prime
to $g$.
\end{abstract}

\maketitle

\section{Introduction}

The present paper contains an extension to characteristic $p$---prime 
to the order $g$ of the finite group $G$---of Emmy
Noether's {\em a priori} bound (viz., $g$) for the degrees of
generators for rings of polynomial invariants of $G$ (see \cite{ref3}, p.
275).  Although Noether's original estimate was in characteristic
zero, it turns out that her method, viz., embedding the quotient
variety in a Chow variety, remains valid under the assumption
that $g$! is prime to the characteristic. The question as to
whether the number $g$! can be replaced with $g$ apparently has
remained open since the publication of \cite{ref1} in 1916.

Dave Benson has apprised the author that the same result has been
obtained in work of Fleischmann.  Benson has also accomplished a
compression of the original matrix proof in the ratio of 6:1!

In \cite{ref1}, Noether used the Chow coordinates of the orbits of $G$,
regarded   as $0$-cycles of degree $g$, as generators of the ring of
polynomial invariants.  If the characteristic divides $g$, there are
examples where the Chow coordinates do not generate the ring of
invariants.  We do not use Chow varieties to establish Noether's
bound, and the important question as to whether the canonical map of
the orbit space to the Chow variety parametrizing $0$-cycles of degree
$g$ in affine space is an embedding remains open in the coprime case.
If the characteristic divides the group order, this map is almost
always {\em radiciel}.

Up to the present, almost all the results on invariants of finite
linearly reductive groups can be made to follow from the complete
reducibility of representations together with either general
results on noetherian rings or explicit calculations in
polynomial rings.  The reasoning in the present proof is more
elementary---in a sense, it is a variation on the theme
$(1-1)^{g}=0$ (see (2) below).

\section{Calculations}

Let $A$ be a commutative ring.  Let $G$ be a finite group of
automorphisms of $A$, of order $g$.  Let $\underline{m}$ be a
$G$-stable ideal in $A$.  Let $J$ be the ideal in $A$ generated
by all $G$-invariants in $\underline{m}$.  Then
\begin{equation}
g\underline{m}^{g}\subset J.
\end{equation}
If $g$ is invertible in $A$, then $\underline{m}^{g}\subset J$
and it is from this that the {\em a priori} bound follows. (1)
follows from an explicit syzygy (in the 19th century sense---see
(3) below).  We apply (1) in the case where $k$ is a field in
which $g$ is invertible, $E$ is the space of a representation of
$G$ over $k, \underline{m}$ is the ideal of forms of positive
degree in the polynomial ring $A=k[E]$ and $J$ is the ideal of
nullforms in $k[E]$, i.e., the ideal generated by invariants of
positive degree.  Then $\underline{m}^{g}\subset J$ implies that
$J$ is generated by forms $f_{1},\dots,f_{r}$ of degree $\leq g, $
for $J$ contains all forms of degree $\geq g$ and hence is
generated by a basis of forms of degree $g$ \underline{plus} some
forms of degree $