On residual coordinates and stable coordinates of \(R^{[3]}\)
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 whether it is a coordinate. More generaly if \(R\) is a polynomial ring over a field of characteristic \(0\), then the question whether residual coordinates or stable coordinates of \(R^{[3]}\) are coordinates is open. It was proved by Freudenburg that the Vénéreau polynomial in \(\mathbb{C}[x]^{[3]}\) is a \(1\)-stable coordinate. The results of the current paper show that residual coordinates of \(\mathbb{C}[x]^{[3]}\) are stable coordinates, but they fell short of showing that they are \(1\)-stable coordinates (say). Other results in the reviewed paper are: If \(x,p\) is a coordinate system of \(R=\mathbb{C}^{[2]}\) and \(A\) is an \(\mathbb{A}^2\)-fibration of \(R\), then \(A\) is an \(\mathbb{A}^3\) fibration over \(\mathbb{C}[x]\) and \(p\) is a residual coordinate of \(A\). Also, assuming that \(\mathbb{A}^3\)-fibrations over \(\mathbb{C}^{[1]}\) are trivial, the following properties are equivalent: (1) Every \(\mathbb{A}^2\)-fibration over \(\mathbb{C}^{[2]}\) is trivial. (2) Every residual coordinate of \(\mathbb{C}[x]^{[3]}\) is a coordinate of \(\mathbb{C}[x]^{[3]}\).
scientific article; zbMATH DE number 6134587
scientific article