In 1977, Bass, Connell and Wright established that any finitely
generated locally polynomial algebra in n variables over an
integral domain R is
isomorphic to the symmetric algebra of a finitely generated projective
R-module of rank n. In this talk,
we shall present an analogous structure theorem
for any R-algebra which is locally a Laurent polynomial
algebra in n variables.
Next we shall give sufficient conditions
for a faithfully flat R-algebra A
to be a locally Laurent polynomial algebra.
We shall see that over a discrete valuation ring R any
Laurent polynomial fibration is necessarily a Laurent
polynomial algebra. We shall then consider
fibre conditions over more general domains.
If time permits, we shall also mention a few results on
the structure of certain algebras whose generic
fibres are {\mathbb A}^*.
The results have been obtained jointly with S.M. Bhatwadekar.