Projective Modules Over A Polynomial Ring
The Quillen–Suslin theorem, which solves Serre's problem is another deep result; it states that if K is a field, or more generally a principal ideal domain, and R = K 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). Bass settled it for non-finitely generated modules and Quillen and Suslin independently and simultaneously treated the case of finitely generated modules.
Since every projective module over a principal ideal domain is free, one might conjecture that following is true: if R is a commutative ring such that every (finitely generated) projective R-module is free then every (finitely generated) projective R-module is free. This is false. A counterexample occurs with R equal to the local ring of the curve y2 = x3 at the origin. So Serre's problem can not be proved by a simple induction on the number of variables.
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