Generically Laurent polynomial algebras over a D.V.R. which are not quasi Laurent polynomial algebras

Abstract

Let (R,π) be a D.V.R. with quotient field K and residue field k. We call an R-algebra A to be quasi Laurent polynomial (abbreviated as quasi LP) in n variables over R if A=R[T1,…,Tn,(a1T1+b1)−1,…,(anTn+bn)−1], where T1,…,Tn are algebraically independent over R and ai∈R∖0, bi∈R are such that (ai,bi)R=R, for i=1,…,n. If an R-algebra A is quasi LP in n variables, then (1) A is a finitely generated, faithfully flat R-algebra, (2) the generic fibre A⊗RK is a Laurent polynomial algebra in n variables over K and (3) the closed fibre A/πA≅k[X1,…,Xr,Y1,Y1−1,…,Ys,Ys−1], where r+s=n. Therefore, it is natural to ask: if an R-algebra A satisfies the above three conditions, then isAnecessarily quasi LP innvariables? We give examples to show that, in general, this question does not have an affirmative answer if n=2 and r≥1.