# Can we classify all finitely generated projective modules over $k[x,y,x^{-1},y^{-1}]$?

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?

• @QiaochuYuan Quillien-Suslin theorem works for polynomial rings but $k[x,y.x^{-1},y^{-1}]$ is not a polynomial ring. Commented Aug 8, 2022 at 15:03
• Ah, my apologies, I had a reduction argument in mind that doesn't actually work. Commented Aug 8, 2022 at 15:28
• Thank you for improving the post. Commented Aug 9, 2022 at 15:45
• Wow, I had never heard that over a Noetherian domain, nonfinitely generated projective modules are free. Amazing! Somehow the finitely generated projectives have a different flavor?! Commented Aug 9, 2022 at 16:07
• @rschwieb Sorry I should add "finitely generated". Commented Aug 9, 2022 at 16:22

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, §4.