Editing Serre–Swan theorem (section)

{{short description|Relates the geometric vector bundles to algebraic projective modules}}
In the [[mathematics|mathematical]] fields of [[topology]] and [[K-theory]], the '''Serre–Swan theorem''', also called '''Swan's theorem''', relates the geometric notion of [[vector bundle]]s to the algebraic concept of [[projective module]]s and gives rise to a common intuition throughout [[mathematics]]: "projective modules over [[commutative ring]]s are like vector bundles on [[compact space]]s".

The two precise formulations of the theorems differ somewhat. The original theorem, as stated by [[Jean-Pierre Serre]] in 1955, is more algebraic in nature, and concerns vector bundles on an [[algebraic variety]] over an [[algebraically closed field]] (of any [[characteristic (algebra)|characteristic]]). The complementary variant stated by [[Richard Swan]] in 1962 is more analytic, and concerns ([[real number|real]], [[complex number|complex]], or [[quaternions|quaternionic]]) vector bundles on a [[smooth manifold]] or [[Hausdorff space]].