Let $k$ be a field and we consider the ring $R=k[x,y,x^{-1},y^{-1}]$. Can we classify all finitely generated projective modules over $R$? In particular, are there non-free examples?
I considered the similar question for $k[x,x^{-1}]$. Since $k[x,x^{-1}]$ is a PID, any finitely generated projective module is free.
If we think of the analogue of topology, we know that there are definitely non-free vector bundles over $(\mathbb{C}^{*})^{2}$ since it is not contractible. However the same is true for $\mathbb{C}^{*}$ but there is no non-free finitely generated projective modules over $k[x,x^{-1}]$, which shows that this analogue is not always precise.
Again, my question is : Can we classify all finitely generated projective modules over $R$? In particular, are there non-free examples?
Best Answer
Swan pointed out that all finitely generated projective modules over the Laurent polynomial ring $k[x_{1}^{\pm},\dotsc,x_{n}^{\pm}]$ are free (here $k$ is a field). See [2] in which he says Quillen's proof works with only minor modifications, and see also Lam's discussion in [1], V, ยง4.
[1] Lam, Serre's Problem on Projective Modules, Springer-Verlag, 2006
[2] Swan, Projective modules over Laurent polynomial rings, Transactions of the AMS, Vol 237, 1978