[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 ${\mathbb{A}^n_k}$, for ${k}$ an algebraically closed field. In other words, it was to determine whether a finitely generated projective module over the ring ${k[x_1, \dots, x_n]}$ is necessarily free. The topological analog, whether (topological) vector bundles on ${\mathbb{C}^n}$ are trivial is easy because ${\mathbb{C}^n}$ 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…)