Journal of Commutative Algebra

A note on quasi Laurent polynomial algebras in n variables

A.M. Abhyankar and S.M. Bhatwadekar

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Let $S$ be a principal ideal domain. Recall that a Laurent polynomial algebra over $S$ is an $S$-algebra of the form $S[T_{1},\ldots, T_{n},T_{1}^{-1}, \ldots, T_{n}^{-1}]$. Generalizing this notion, we call an $S$-algebra of the form $S[T_{1},\dots, T_{n},f_{1}^{-1}, \ldots, f_{n}^{-1}]$ a quasi Laurent polynomial algebra in $n$ variables over $S$ if $T_{1},\ldots, T_{n}$ are algebraically independent over $S$ and $f_{i}=a_{i}T_{i}+b_{i}$, where $a_{i} \in S \backslash 0$ and $b_{i} \in S$ are such that $(a_{i}, b_{i})S=S$, for each $i=1, \ldots,n$. It has been shown recently that a locally Laurent polynomial algebra in $n$ variables over $S$ is itself a Laurent polynomial algebra. Now suppose $A$ is a locally quasi Laurent polynomial algebra in $n$ variables over $S$. In this note, we investigate the question: `is $A$ necessarily quasi Laurent polynomial in $n$ variables over $S$?' We first give a sufficient condition for the question to have an affirmative answer. Moreover, when $S$ is semi-local with two maximal ideals and contains the field of rationals $\mathbf{Q}$, we give examples of $S$-algebras which are locally quasi Laurent polynomial in two variables but not quasi Laurent polynomial in two variables.

Article information

J. Commut. Algebra, Volume 6, Number 2 (2014), 127-147.

First available in Project Euclid: 11 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16S34: Group rings [See also 20C05, 20C07], Laurent polynomial rings
Secondary: 13F10: Principal ideal rings 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]

Polynomial algebra Laurent polynomial algebra quasi Laurent polynomial algebra principal ideal domain


Abhyankar, A.M.; Bhatwadekar, S.M. A note on quasi Laurent polynomial algebras in n variables. J. Commut. Algebra 6 (2014), no. 2, 127--147. doi:10.1216/JCA-2014-6-2-127.

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  • S.M. Bhatwadekar and Neena Gupta, On locally quasi $\mathbf{A}^*$ algebras in codimension-one over a Noetherian normal domain, J. Pure Appl. Alg. 215 (2011), 2242–2256.
  • ––––, The structure of a Laurent polynomial fibration in $n$ variables, J. Algebra 353 (2012), 142–157.
  • P. Russell and A. Sathaye, On finding and cancelling variables in $k[X,Y,Z]$, J. Algebra 57 (1979), 151–166.