On residual coordinates and stable coordinates of \(R^{[3]}\)
type: Item
instance of: scholarly article
title: On residual coordinates and stable coordinates of \(R^{[3]}\)
zbMATH Open document ID: 1262.14073
published in: Archiv der Mathematik
publication date: 2013-02-05T00:00:00Z
review text: Let \(R\) be a commutative ring with a unit and \(A=R^{[n]}\) the polynomial \(R\)-algebra in \(n\)-variables. A polynomial \(p\in A\) is called a residual coordinate if \(\forall\,\rho\in {\text{ Spec}}(R)\), \(\pi_{\rho}(p)\) is a coordinate of \(K(\rho)\otimes_{R}A\cong_{K(\rho)}K(\rho)^{[n]}\). Here \(K(\rho)\) is the residue field of \(\rho\) and \(\pi_{\rho}\,:\,A\rightarrow K(\rho)\otimes_{R}A\) is the canonical homomorphism. The polynomial \(p\) is called a stable coordinate if it is a coordinate of \(R^{[n+m]}\) for some \(m\geq 0\). An \(R\)-algebra \(A\) is called an \(\mathbb{A}^n\)-fibration over \(R\) if: (1) \(A\) is finitely generated as an \(R\)-algebra. (2) \(A\) is flat as an \(R\)-module. (3) For every \(\rho\in {\text{Spec}}(R)\), \(K(\rho)\otimes_{R}A\cong_{K(\rho)}K(\rho)^{[n]}\). An \(\mathbb{A}^n\)-fibration \(A\) over \(R\) is called trivial if \(A\cong_{R}R^{n}\). Obviously the concepts of residual coordinate and stable coordinate can also be defined in affine fibrations (and not just in polynomial rings). In this paper the relations among the three notions: residual coordinate, stable coordinate and a coordinate in \(\mathbb{A}^3\)-fibrations are investigated. It is shown that in \((\mathbb{C}^{[n]})^{[3]}\) (\(n\geq 0\)) residual coordinates and stable coordinates are the same objects and so (the case \(n=0\)) in \(\mathbb{C}^{[3]}\) a polynomial is a coordinate if and only if it is a stable coordinate. More generaly, if \(R=\mathbb{C}^{[n]}\) (\(n\geq 0\)) and \(A\) is an \(\mathbb{A}^3\)-fibration over \(R\) and \(p\in A\) then the following three properties are shown to be equivalent: (1) \(p\) is a residual coordinate of \(A\). (2) \(p\) is a stable coordinate of \(A\). (3) The inclusion homomorphism of \(R[\rho]\) into \(A\) makes \(A\) an \(\mathbb{A}^2\)-fibration. The intriguing polynomial \(y+x(xz+y(yu+z^2))\) of Vénéreau is a residual coordinate and a stable coordinate of \(\mathbb{C}[x]^{[3]}\). However, it is an open question…
Mathematics Subject Classification ID: 14R25
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 6134587
zbMATH Keywords: affine fibration
zbMATH Keywords: residual coordinate
zbMATH Keywords: stable coordinate
author: M'hammed El Kahoui
reviewed by: Ronen Peretz
MaRDI profile type: Publication
full work available at URL: s00013-012-0469-9
OpenAlex ID: W2166258754
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DOI: 10.1007/S00013-012-0469-9
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description: scientific article; zbMATH DE number 6134587