Editing Quillen–Suslin theorem (section)

===Background===
Geometrically, finitely generated projective modules over the ring <math>R[x_1,\dots,x_n]</math> correspond to [[vector bundle]]s over [[affine space]] <math>\mathbb{A}^n_R</math>, where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending <math> M\to \widetilde{M}</math> (Hartshorne II.5, page 110).  [[Affine space]] is topologically [[contractible space|contractible]], so it admits no non-trivial topological vector bundles.  A simple argument using the [[exponential sheaf sequence|exponential exact sequence]] and the [[d-bar Poincaré lemma]] shows that it also admits no non-trivial [[holomorphic vector bundle]]s.

[[Jean-Pierre Serre]], in his 1955 paper [[List of important publications in mathematics#Faisceaux Algébriques Cohérents|''Faisceaux algébriques cohérents'']], remarked that the corresponding question was not known for algebraic vector bundles: "It is not known whether there exist projective ''A''-modules of finite type which are not free."<ref>"On ignore s'il existe des ''A''-modules projectifs de type fini qui ne soient pas libres."  Serre, ''FAC'', p. 243.</ref>  Here <math>A</math> is a [[polynomial ring]] over a [[field (mathematics)|field]], that is, <math>A</math> = <math>k[x_1,\dots,x_n]</math>.

To Serre's dismay, this problem quickly became known as Serre's conjecture.  (Serre wrote, "I objected as often as I could [to the name]."<ref>Lam, p. 1</ref>)  The statement does not immediately follow from the proofs given in the topological or holomorphic case. These cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization.

Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was [[stably free]], meaning that after forming its [[direct sum of modules|direct sum]] with a finitely generated free module, it became free.  The problem remained open until 1976, when [[Daniel Quillen]] and [[Andrei Suslin]] independently proved the result. Quillen was awarded the [[Fields Medal]] in 1978 in part for his proof of the Serre conjecture. [[Leonid Vaseršteĭn]] later gave a simpler and much shorter proof of the theorem, which can be found in [[Serge Lang]]'s ''Algebra''.