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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?

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    $\begingroup$ @QiaochuYuan Quillien-Suslin theorem works for polynomial rings but $k[x,y.x^{-1},y^{-1}]$ is not a polynomial ring. $\endgroup$ Commented Aug 8, 2022 at 15:03
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    $\begingroup$ Ah, my apologies, I had a reduction argument in mind that doesn't actually work. $\endgroup$ Commented Aug 8, 2022 at 15:28
  • $\begingroup$ Thank you for improving the post. $\endgroup$
    – rschwieb
    Commented Aug 9, 2022 at 15:45
  • $\begingroup$ 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?! $\endgroup$
    – rschwieb
    Commented Aug 9, 2022 at 16:07
  • $\begingroup$ @rschwieb Sorry I should add "finitely generated". $\endgroup$ Commented Aug 9, 2022 at 16:22

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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.

[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

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