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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % 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 $