In the geometric setting it is a statement about the triviality of vector bundles on affine space.
The theorem states that every finitely generated projective module over a polynomial ring is free.
This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending
A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles.
Jean-Pierre Serre, in his 1955 paper 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.
"[2]) The statement does not immediately follow from the proofs given in the topological or holomorphic case.
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.
-bundles on affine space are all trivial, this is not true for G-bundles where G is a general reductive algebraic group.