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?
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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.