**Neena Gupta**

July 19, 2012

**On locally Laurent polynomial algebras**:
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 $\A^*$.
The results have been obtained jointly with S.M. Bhatwadekar.