[Apologies in the delay in posts on the Segal paper — there are a couple of things I’m confused on that are preventing me from proceeding.]

A classical problem (posed by Serre) was to determine whether there were any nontrivial algebraic vector bundles over affine space , for an algebraically closed field. In other words, it was to determine whether a finitely generated projective module over the ring is necessarily free. The topological analog, whether (topological) vector bundles on are trivial is easy because is contractible. The algebraic case is harder.

The problem was solved affirmatively by Quillen and Suslin. In this post, I would like to describe an elementary proof, due to Vaserstein, of the Quillen-Suslin theorem. (more…)