Generically Laurent polynomial algebras over a DVR which are not quasi Laurent polynomial algebras

Let (R, pi) 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[T-1, ..., T-n, (a(1)T(1) + b(1))(-1), ..., (a(n)T(n) + b(n))(-1)], where T-1, ..., T-n are algebraically independent over R and a(i) is an element of R \ 0, b(i) is an element of R are such that (a(i), b(i))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 circle times(R) K is a Laurent polynomial algebra in n variables over K and (3) the closed fibre A/pi A congruent to k[X-1, ..., X-r, Y-1, Y-1(-1), ..., Y-s, Y-s(-1)], where r + s = n. Therefore, it is natural to ask: if an R-algebra A satisfies the above three conditions, then is A necessarily quasi LP in n variables? We give examples to show that, in general, this question does not have an affirmative answer if n = 2 and r >= 1. (C) 2013 Elsevier B.V. All rights reserved.