Quillen-Suslin Theorem (Paperback)

High Quality Content by WIKIPEDIA articles! The Quillen-Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings. It states that every finitely generated projective module over a polynomial ring is free. Geometrically, finitely generated projective modules correspond to vector bundles over affine space, and free modules to trivial vector bundles. Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the d-bar Poincare lemma shows that it also admits no non-trivial holomorphic vector bundles.