Stable tameness of coordinates in two variables
type: Item
instance of: scholarly article
title: Stable tameness of coordinates in two variables
zbMATH Open document ID: 1409.14099
author: M'hammed El Kahoui
author: Najoua Essamaoui
author: M. Ouali
published in: Journal of Algebra
publication date: 2019-03-11T00:00:00Z
review text: Consider $z$-automorphisms (i.e. sending $z$ to $z$) of $k[x,y,z]$. It has been proved that they are stably tame in the famouse paper ``Stable tameness of two-dimensional polynomial automorphisms over a regular ring'' by Joost Berson, Arno van den Essen, David Wright [\textit{J. Berson} et al., Adv. Math. 230, No. 4--6, 2176--2197 (2012; Zbl 1246.14075)]. In fact, $k$ can be replaced by regular ring, instead of one variable $z$ one can consider any set $z_1,\dots,z_n$. \par A non-commutative version of this result was obtained in the paper [Sel. Math., New Ser. 18, No. 4, 799--802 (2012; Zbl 1268.16025)] by \textit{A. Belov-Kanel} and \textit{J.-T. Yu}. \par The paper under review is devoted to a coordinate analog of the Joost Berson, Arno van den Essen, David Wright result. It was proven that for a given commutative ring $R$ with unity, every coordinate of $R^2$ is stably tame ($R^2=R[x,y]$ by definition as usual). \par The method of proof is based on localization and factorisation technique used in module theory. The paper is clear and nicely written.
reviewed by: A. Ya Belov
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14R25
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 7034178
zbMATH Keywords: residual coordinate
zbMATH Keywords: stable tameness
zbMATH Keywords: tame and wild authomorphisms, $z$-authomorphism
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.01.029
OpenAlex ID: W2914300191
cites work: On the uniqueness of the coefficient ring in a polynomial ring
cites work: Stably tame coordinates
cites work: Two-dimensional stable tameness over Noetherian domains of dimension one
cites work: Stable tameness of two-dimensional polynomial automorphisms over a regular ring
cites work: Tame and Wild Coordinates of 𝐾[𝑧][𝑥,𝑦]
cites work: Totally stably tame variables
cites work: Length 2 variables of \(A[x,y]\) and transfer
cites work: \({\mathcal D}_n(A)\) for a class of polynomial automorphisms and stably tameness
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: On the \(R\)-invariance of \(R[X]\)
cites work: Q5818521
cites work: Vénéreau-type polynomials as potential counterexamples
cites work: Strongly Residual Coordinates over A[x]
cites work: Q4063564
cites work: Stably tame automorphisms
DOI: 10.1016/J.JALGEBRA.2019.01.029
description: scientific article; zbMATH DE number 7034178