A common theorem is that every projective modules over a PID are free and we know that every projective modules over commutative local rings are free too. Also every finitely generated projective module over a polynomial ring $K[x_1,\ldots, x_n ]$ over a field $K$ is free (Serre Conjecture) and in 1976 proved by Quillen and Suslin. I want to know for $n=2$ how it works.

Any comments and guidances are very welcome.

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    $\begingroup$ The case $n=1$ reduces to the PID case, so nothing to see there, I think? $\endgroup$ – Jyrki Lahtonen May 14 '17 at 8:50
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    $\begingroup$ $K[x,y]$ is not a PID. The ideal $\langle x,y\rangle$ is not principal. $\endgroup$ – Jyrki Lahtonen May 14 '17 at 8:59
  • $\begingroup$ Yes, it is not a PID. So for $n=2$ is not obvious anymore. Is it hard to proof for $n=2$? $\endgroup$ – B.K-Theory May 14 '17 at 9:18
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    $\begingroup$ Not trivial. Seshadri proved this in 1958. @Mohan has a very nice article about this: math.wustl.edu/~kumar/papers/seshadri.pdf $\endgroup$ – user26857 May 14 '17 at 9:38
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    $\begingroup$ @user26857 Thank you for your kind comment. $\endgroup$ – Mohan May 14 '17 at 17:42

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