I know that finitely generated projective modules over $k[x_1,...,x_n]$ are free does this result hold for any projective module?

Does there exist a projective module over $k[x_1,...,x_n]$ which is not free?

  • 5
    $\begingroup$ I think this is a corollary of the Theorem 4.3 from this paper of Bass. $\endgroup$
    – user26857
    Apr 22, 2017 at 16:22
  • 3
    $\begingroup$ @spohreis Bass' theorem shows that all projective modules over $k[x_1,...,x_n]$ are free. What do you want more then this? $\endgroup$
    – user26857
    Oct 11, 2017 at 18:40
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    $\begingroup$ Please add this as an answer. $\endgroup$
    – user23505
    Oct 11, 2017 at 20:27

1 Answer 1


A CW answer to remove this from the unanswered list: All projective modules are free; this is the main result of this paper by Hyman Bass.

  • 2
    $\begingroup$ According to the paper, shouldn't the answer to the question in the title be 'no'?, i.e. non-f.g. projectives over polynomials rings are free? $\endgroup$
    – Pedro
    Jul 17, 2018 at 1:01
  • $\begingroup$ Edited, thanks! $\endgroup$ Jul 17, 2018 at 2:19

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