TITLE: Relation between projective modules over $R$ and $R[T]$
QUESTION [8 upvotes]: Let $R$ be a commutative ring and $R[U]$ the polynomial ring in one variable. What is the relation between projective modules over $R$ and projective modules over $R[U]$? Is every projective module over $R[U]$ of the form $P[U]$ for a projective $R$-module $P$? If not what are the obstructions?
Edit: I realised that this question was to general for what I was actually looking for. Since the question in the current form seems to be interesting on its own, I refrained from editing it and opened a new question instead.

REPLY [3 votes]: If $R$ is a left regular ring, then the canonical map $K_0(R) \to K_0(R[t])$ is an isomorphism. This result is due to Grothendieck at least when $R$ is commutative. The general case can be found in the paper "The Whitehead group of a polynomial extension" (Bass, Heller, Swan) or in Rosenberg's book on Algebraic K-Theory.
Of course, this does not imply that every f.g. projective $R[t]$-module has the form $P[t]$ for some f.g. projective $R$-module (but we cannot expect that!); but this turns out to be true "up to exact sequences".