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A note on quasi Laurent polynomial algebras in \(n\) variables. (English) Zbl 1303.13020

Consider a domain \(S\). A quasi Laurent polynomial algebra in \(n\) variables over \(S\) is an \(S-\)algebra of the form \(S[T_1,\ldots,T_n,f_1^{-1},\ldots,f_n^{-1}]\) where \(T_1,\ldots,T_n\) are algebraically independent over \(S\) and \(f_i=a_iT_i+b_i\) with \(a_i\in S\setminus (0)\) and \(b_i\in S\) such that \((a_i,b_i)S=S\). In this paper, the authors investigate the following question. Suppose \(A\) is a locally quasi Laurent polynomial algebra in \(n\) variables. Is \(A\) necessarily quasi Laurent polynomial algebra in \(n\) variables over \(S\)? They give an example in two variables showing that the answer is negative in general. However, they prove that if \(S\) is a factorial domain and \(A\) a faithfully flat \(S-\)algebra such that
(1) \(A_P\) is quasi Laurent polynomial in \(n\) variables over \(S_P\) for every height one prime ideal \(P\) of \(S\).
(2) \(A[{1\over \pi}]\) is a Laurent polynomial algebra in \(n\) variables over \(S[{1\over \pi}]\) for some prime element \(\pi\) in \(S\).
Then \(A\) is quasi Laurent polynomial in \(n\) variables over \(S\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14R25 Affine fibrations

References:

[1] 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. · Zbl 1219.14070 · doi:10.1016/j.jpaa.2011.02.013
[2] —-, The structure of a Laurent polynomial fibration in \(n\) variables , J. Algebra 353 (2012), 142-157. · Zbl 1245.14062 · doi:10.1016/j.jalgebra.2011.11.032
[3] P. Russell and A. Sathaye, On finding and cancelling variables in \(k[X,Y,Z]\) , J. Algebra 57 (1979), 151-166. · Zbl 0411.13011 · doi:10.1016/0021-8693(79)90214-X
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