A note on residual coordinates of polynomial rings
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 also results for the stability of strongly residual coordinates. In particular, if \({\mathbb Q}\subset R\), then every strongly residual coordinate of \(R^{[n]}\) is \(n\)-stable.
scientific article; zbMATH DE number 6976317
scientific article