Quillen-Suslin theorem



The most memorable formulation of the Quillen-Suslin theorem states that for a field kk, finitely generated projective modules over a finitary polynomial algebra A=k[x 1,,x r]A = k[x_1, \ldots, x_r] are free.

In Serre’s FAC appears the sentence “It is not known if there exist projective A-modules of finite type which are not free.” This question became known as Serre’s problem or Serre’s conjecture (over repeated objections from Serre). Serre had made partial progress by proving that f.g. projective AA-modules are stably free?, but the question remained unresolved until 1976 when an affirmative solution was produced by Daniel Quillen and independently by Andrei Suslin.

A later simplified proof was given by Leonid Vaserstein; this is recounted in Lang’s Algebra.