A note on partial coordinate system in a polynomial ring

TL;DR

This paper extends a result on partial coordinate systems in polynomial rings, showing that the property persists under more general conditions involving residual variables and arbitrary elements.

Contribution

It generalizes previous theorems by incorporating the theory of residual variables to broader cases where the element is not necessarily a non-zerodivisor.

Findings

Extension of Berson, Bikker, and van den Essen's result to arbitrary elements.

Application of residual variables theory to polynomial rings.

Broader conditions under which partial coordinate systems are preserved.

Abstract

J. Berson, J. W. Bikker and A. van den Essen proved that for a non-zerodivisor $a$ in a commutative ring $R$ containing $Q$ if the polynomials $f_1,\dots,f_{n-1}$ in $R[X_1,\dots,X_n]$ form a partial coordinate system over the rings $R_a$ and $\dfrac{R}{aR}$ then $f_1,\dots,f_{n-1}$ form a partial coordinate system over the ring $R$. In this note we show that the theory of residual variables of Bhatwadekar-Dutta and its recent extension by Das-Dutta, extends their result to the case when $a$ is an arbitrary element of $A$.

Equations

$$\frac{R}{aR}[X_1,\dots ,X_n]=\frac{R}{aR}[\overline{f_1},\dots ,\overline{f_{n-1}},\overline{g}]$$

$$R_a[X_1,\dots ,X_n]=R_a[f_1,\dots ,f_{n-1},h].$$

$$X_i=G_i+a{H}_i$$

$$X_i=\sum \frac{e_{i,m_1,\dots ,m_{n-1},m}}{a^{k_i}}{f}_1^{m_1}{f}_2^{m_2}\dots f_{n-1}^{m_{n-1}}{h}^m$$

Full text

Keywords. Polynomial algebra, Coordinate, Residual coordinate.

AMS Subject classifications (2010). Primary: 13B25; Secondary: 14R25.

1 Introduction

We will assume all rings to be commutative containing unity. Throughout this note we will use the notation $R[X_{1},\dots,X_{n}]$ to mean a polynomial algebra in $n$ variables over $R$. Sometimes we will denote this ring by $R^{[n]}$.

Let $A:=R[X_{1},\dots,X_{n}]$. We recall that $m$ polynomials $f_{1},\dots,f_{m}$ ($m\leq{n}$) in $A$ are said to form a partial coordinate system (of colength $n-m$) in $A$ if $A=R[f_{1},\dots,f_{m}]^{[n-m]}$. If $m=n$ then we will say $f_{1},\dots,f_{n}$ form a coordinate system in $A$. For an arbitrary $a\in{R}$, $f_{1},\dots,f_{m}$ in $A$ are said to form an $a$-strongly partial residual coordinate system (of colength $n-m$) in $A$ if the images of $f_{1},\dots,f_{m}$ form a partial coordinate system (of colength $n-m$) in $\dfrac{A}{aA}$ and also in $A_{a}$. We will say $f_{1},\dots,f_{m}$ form a partial residual coordinate system (of colength $n-m$) in $A$ if, for each prime ideal $\mathpzc{p}$ of $R$ we have $k(\mathpzc{p})\otimes_{R}{A}=(k(\mathpzc{p})\otimes_{R}{R[f_{1},\dots,f_{m}]})^{[n-m]}$, where $k(\mathpzc{p}):=\dfrac{R_{\mathpzc{p}}}{\mathpzc{p}R_{\mathpzc{p}}}$ is the residue field of $R$ at $\mathpzc{p}$.

The following result has been proved by J. Berson, J. W. Bikker and A. van den Essen in [1].

Theorem 1.1**.**

Let $R$ be a ring containing $\mathbb{Q}$, $a\in{R}$ a non-zerodivisor and $A=R[X_{1},\dots,X_{n}]$. If $n-1$ polynomials $f_{1},\dots,f_{n-1}$ in $A$ form an $a$-strongly partial residual coordinate system in $A$, then $f_{1},\dots,f_{n-1}$ form a partial coordinate system in $A$.

They conjectured the following ([1, Conjecture 4.4]):

Conjecture. Theorem 1.1 also holds even when $a$ is a zerodivisor in $R$.

In this note we observe that an affirmative solution to the above conjecture can be deduced from the following formulation of a result of Das-Dutta ([3, Corollary 3.19]).

Theorem 1.2**.**

Let $R$ be a Noetherian ring containing $\mathbb{Q}$, $A=R^{[n]}$ and $f_{1},\dots,f_{n-1}\in{A}$. Then the following are equivalent:

1. (i)

$f_{1},\dots,f_{n-1}$* form a partial coordinate system in $A$.* 2. (ii)

$f_{1},\dots,f_{n-1}$* form a partial residual coordinate system in $A$.*

Remark 1.3**.**

We note the following observations on the above result

(i) Although the result is stated in the paper [3] for Noetherian domains containing $\mathbb{Q}$, the proof is an application of Theorem 3.13 and Theorem 2.4 in [3] which hold over any Noetherain ring containing $\mathbb{Q}$ (not necessarily a domain).

(ii) The case $n=2$ was previously proved by Bhatwadekar-Dutta ([2, Theorem 3.1]). **

2 Proof of the conjecture

Theorem 2.1**.**

Let $R$ be a ring containing $\mathbb{Q}$, $a\in{R}$ be arbitrary and $A=R[X_{1},\dots,X_{n}]$. If $n-1$ polynomials $f_{1},\dots,f_{n-1}$ in $A$ form an $a$-strongly partial residual coordinate system of colength $1$ in $A$, then $f_{1},\dots,f_{n-1}$ form a partial coordinate system in $A$.

Proof.

Since $f_{1},\dots,f_{n-1}$ form an $a$-strongly partial residual coordinate system in $A$, there exist $g,h\in{A}$ such that

[TABLE]

and

[TABLE]

Hence we have,

[TABLE]

for some $G_{i}\in{R[f_{1},\dots,f_{n-1},g]}$ and $H_{i}\in{A}$ $(1\leq{i}\leq{n})$;

and

[TABLE]

where $e_{i,m_{1},\dots,m_{n-1},m}\in{R}$, for all ${i,m_{j},m}$ $(1\leq{i}\leq{n},1\leq{j}\leq{n-1},{m_{j},m,k_{i}}\geq{0})$.

Now, let $S$ be the $\mathbb{Q}$-subalgebra of $R$ generated by the subset of $R$ consisting of $a$; all coefficients of $G_{i}$ (for all $i$) as polynomials in $f_{1},\dots,f_{n-1},g$; coefficients of $H_{i}$ (for all $i$); coefficients of $g,h,f_{i}$ (for all $i$) and $e_{i,m_{1},\dots,m_{n-1},m}$ (for all $i$ and for all ${m,m_{j}})$. Then $S$ is a Noetherian $\mathbb{Q}$-algebra. Let $\mathpzc{p}$ be an arbitrary prime ideal of $S$ and $B:=S[X_{1},\dots,X_{n}]$. If $a\in{\mathpzc{p}}$ then from (1), we have $k(\mathpzc{p})\otimes_{S}{B}=(k(\mathpzc{p})\otimes_{S}{S[f_{1},\dots,f_{n-1}]})^{[1]}$. If $a\notin{\mathpzc{p}}$ then from (2), we have $k(\mathpzc{p})\otimes_{S}{B}=(k(\mathpzc{p})\otimes_{S}{S[f_{1},\dots,f_{n-1}]})^{[1]}$. So, $f_{1},\dots,f_{n-1}$ form a partial residual coordinate system (of colength $1$) in $B$. Since $S$ is Noetherian, hence by Theorem 1.2, $f_{1},\dots,f_{n-1}$ form a partial coordinate system in $B$ and hence in $A$. ∎

Acknowledgements. The author acknowledges Council of Scientific and Industrial Research (CSIR) for their research grant.