A note on quasi Laurent polynomial algebras in \(n\) variables (Q405377)

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 thatNEWLINENEWLINE (1) \(A_P\) is quasi Laurent polynomial in \(n\) variables over \(S_P\) for every height one prime ideal \(P\) of \(S\).NEWLINENEWLINE (2) \(A[{1\over \pi}]\) is a Laurent polynomial algebra in \(n\) variables over \(S[{1\over \pi}]\) for some prime element \(\pi\) in \(S\).NEWLINENEWLINE Then \(A\) is quasi Laurent polynomial in \(n\) variables over \(S\).