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.
Read more about this topic: Projective Module
Famous quotes containing the word ring:
“Generally, about all perception, we can say that a sense is what has the power of receiving into itself the sensible forms of things without the matter, in the way in which a piece of wax takes on the impress of a signet ring without the iron or gold.”
—Aristotle (384–323 B.C.)