A note on quasi Laurent polynomial algebras in n variables

Abstract

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.