Group action on polynomial and power series rings
type: Item
instance of: scholarly article
title: Group action on polynomial and power series rings
zbMATH Open document ID: 1030.13002
author: Peter Symonds
published in: Pacific Journal of Mathematics
publication date: 2000-12-03T00:00:00Z
review text: Let \(S\) be a finitely generated graded integral domain over a field \(k\), with finite dimensional homogeneous components, such as a polynomial ring. Let \(G\) be a finite group of grading preserving automorphisms of \(S\). The paper concerns the structure of \(S\) as a \(kG\)-module. The main result is that \(S\) contains a free \(kG\)-module summand, which is large in an appropriate sense, made precise in terms of the Hilbert-Serre theory relating the growth of the dimension of the homogeneous components of \(S\). Versions of this theorem (with restrictions on \(k\) or \(S\)) were proved by \textit{R. Howe} [J. Algebra 122, 384-379 (1989; Zbl 0693.20005)] and by \textit{R. M. Bryant} [J. Algebra 154, 416-436 (1993; Zbl 0828.20002)]. A similar result holds for filtered algebras, such as power series rings.
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 1537849
zbMATH Keywords: graded integral domain
zbMATH Keywords: polynomial ring
zbMATH Keywords: automorphisms
zbMATH Keywords: projective module
zbMATH Keywords: Hilbert polynomial
zbMATH Keywords: ring of invariants
zbMATH Keywords: power series rings
reviewed by: Matyas Domokos
MaRDI profile type: Publication
full work available at URL: pjm.2000.195.225
OpenAlex ID: W1979446484
DOI: 10.2140/PJM.2000.195.225
description: scientific article; zbMATH DE number 1537849