Serre–Swan theorem

In the mathematical fields of differential geometry, topology and algebraic geometry, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules are like vector bundles".

The three precise formulations of the theorem differ somewhat. The original theorem, as proved 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). Richard Swan in 1962 proved an analytic variant, concerning smooth vector bundles on a smooth manifold (real, complex, or quaternionic). His topological variant is about continuous (real or complex) vector bundles on a compact Hausdorff space.