A note on residual coordinates of polynomial rings
type: Item
instance of: scholarly article
title: A note on residual coordinates of polynomial rings
zbMATH Open document ID: 1403.13013
DOI: 10.1216/JCA-2018-10-3-317
author: M'hammed El Kahoui
author: M. Ouali
published in: Journal of Commutative Algebra
publication date: 2018-11-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1541754161
review text: Let \(R^{[n]}=R[x_1,\ldots,x_n]\) be the polynomial algebra in \(n\) variables over the commutative ring \(R\). A polynomial \(p\in R^{[n]}\) is a coordinate if \(R^{[n]}=R[p,q_2,\ldots,q_n]\) for some polynomials \(q_2,\ldots,q_n\). The polynomial \(p\) is a stable coordinate if it is a coordinate in \(R^{[n+m]}=R[x_1,\ldots,x_n,y_1,\ldots,y_m]\) for some \(m\). It is a residual coordinate if it is a coordinate in the factor ring \(R/{\mathfrak p}^{[n]}\) when the ideal \(\mathfrak p\) of \(R\) runs on the spectrum of \(R\). By results in [\textit{S. M. Bhatwadekar} and \textit{A. K. Dutta}, Commun. Algebra 21, No. 2, 635--645 (1993; Zbl 0778.13016)] and [\textit{A. van den Essen} and \textit{P. van Rossum}, Trans. Am. Math. Soc. 356, No. 5, 1691--1703 (2004; Zbl 1040.13006)] when \({\mathbb Q}\subset R\) and \(n=2\) the three notions -- coordinate, stable coordinate, and residual coordinate -- are equivalent. In the general case by \textit{M. Hochster} [Proc. Am. Math. Soc. 34, 81--82 (1972; Zbl 0233.13012)] the residual coordinates of \(R^{[n]}\) are not obliged to be coordinates when \(n\geq 3\). The first main result of the paper under review shows that when \(K\) is an algebraically closed field of characteristic 0 residual coordinates of \(K^{[n]}\), \(n\geq 3\), are 1-stable, i.e., become coordinates of \(K^{[n+1]}\). Recently it was shown in [\textit{P. Das} and \textit{A. K. Dutta}, J. Pure Appl. Algebra 218, No. 10, 1792--1799 (2014; Zbl 1291.14090)] that for any Noetherian domain \(R\) residual coordinates of \(R^{[n]}\), \(n\geq 3\), are stable coordinates. The second main result is that when the ring \(R\) contains \(\mathbb Q\) and is Noetherian of Krull dimension \(d\), then every residual coordinate of \(R^{[n]}\) is \((2^d-1)n\)-stable. Let \(c\in R\) be not a zero divisor. The polynomial \(p\in R^{[n]}\) is a \(c\)-strongly residual coordinate if it is a coordinate over the localization \(R_c\) and over the quotient \(R/cR\). The paper contains al…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 6976317
zbMATH Keywords: residual coordinate
zbMATH Keywords: stable coordinate
MaRDI profile type: Publication
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OpenAlex ID: W2899631888
Wikidata QID: Q128963226
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description: scientific article; zbMATH DE number 6976317