The commuting derivations conjecture
type: Item
instance of: scholarly article
title: The commuting derivations conjecture
zbMATH Open document ID: 1052.13012
DOI: 10.1016/S0022-4049(02)00198-6
author: Stefan Maubach
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-09T00:00:00Z
review text: The main result of the paper (theorem 3.6) is an application of Kaliman's theorem [see e.g. \textit{S. Kaliman}, Pac. J. Math. 203, No. 1, 161--190 (2002; Zbl 1060.14085)] to pairs of commuting locally nilpotent derivations of \({\mathbb C}[x,y,z]\), or equivalently, to algebraic actions of the vector group \(({\mathbb C}^2,+)\) on \({\mathbb C}^3\). The quotient morphism for such an action is of the form \(\pi :{\mathbb C}^3\to {\mathbb C}^1\), which may be viewed as a regular function on \({\mathbb C}^3\). The author shows that the general fiber of \(\pi\) is a plane. In this case, Kaliman's theorem implies that \(\pi\) is a coordinate function on \({\mathbb C}^3\). Reviewer's remark: There exist algebraic actions of \({\mathbb C}^2\) on \({\mathbb C}^3\) which are not linear in any coordinate system. However, the results of this paper show that any such action has rank at most 2. From this, it is possible to classify all \({\mathbb C}^2\)-actions on \({\mathbb C}^3\), though the paper under review does not follow this course: see \textit{D. Daigle} and \textit{G. Freudenburg}, J. Algebra 204, No.2, 353-371 (1998; Zbl 0956.13007).
reviewed by: Gene Freudenburg
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 1895979
zbMATH Keywords: quotient morphism for vector group actions
zbMATH Keywords: commuting locally nilpotent derivations
Wikidata QID: Q123246387
MaRDI profile type: Publication
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full work available at URL: s0022-4049(02)00198-6
OpenAlex ID: W2048452447
description: scientific article; zbMATH DE number 1895979