Editing Projective module (section)

== Projective modules over a polynomial ring ==
The [[Quillen–Suslin theorem]], which solves Serre's problem, is another [[deep result]]: if ''K'' is a field, or more generally a [[principal ideal domain]], and {{nowrap|1=''R'' = ''K''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>]}} is a [[polynomial ring]] over ''K'', then every projective module over ''R'' is free.
This problem was first raised by Serre with ''K'' a field (and the modules being finitely generated). [[Hyman Bass|Bass]] settled it for non-finitely generated modules,<ref>{{cite journal|title=Big projective modules are free|last=Bass|first=Hyman|journal=[[Illinois Journal of Mathematics]]|volume=7|number=1|year=1963|publisher=Duke University Press|doi=10.1215/ijm/1255637479|at=Corollary 4.5|doi-access=free}}</ref> and [[Dan Quillen|Quillen]] and [[Andrei Suslin|Suslin]] independently and simultaneously treated the case of finitely generated modules.

Since every projective module over a principal ideal domain is free, one might ask this question: if ''R'' is a commutative ring such that every (finitely generated) projective ''R''-module is free, then is every (finitely generated) projective ''R''[''X'']-module free?  The answer is ''no''.  A [[counterexample]] occurs with ''R'' equal to the local ring of the curve {{nowrap|1=''y''<sup>2</sup> = ''x''<sup>3</sup>}} at the origin.  Thus the Quillen–Suslin theorem could never be proved by a simple [[mathematical induction|induction]] on the number of variables.