In mathematics, the Quillen-Suslin theorem is a theorem in abstract algebra about the relationship between free modules and projective modules. Projective modules are modules that are locally free. Not all projective modules are free, but in the mid-1950s, Jean-Pierre Serre found evidence that a limited converse might hold. He asked the question:

Is every projective module over a polynomial ring over a field a free module?

This was open until 1976, when Daniel Quillen and Andrei Suslin independently proved that the answer is yes. Quillen was awarded the Fields Medal in 1978 in part for his proof. Soon after Quillen's and Suslin's proofs, Leonid Vaserstein gave a substantially simpler proof.

Reference

An account of this topic is provided by: Lam, T.Y., Serre's Conjecture, Springer-Verlag, 1978.