Swan's theorem

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Swan's theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".

[edit] Differential geometry

Suppose M is a compact C^∞-manifold, and a smooth vector bundle V is given on M. The space of smooth sections of V is then a module over C^∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated and projective over C^∞(M).

Even more: every finitely generated projective module over C^∞(M) arises in this way from some smooth vector bundle on M, in essentially only one way. More precisely: the category of smooth vector bundles on M is equivalent to the category of finitely generated projective modules over C^∞(M).

[edit] Topology

Suppose X is a compact Hausdorff space, and C(X) is the ring of continuous real-valued functions on X. Analogous to the result above, the category of real vector bundles on X is equivalent to the category of finitely generated projective modules over C(X). The same result holds if one replaces "real-valued" by "complex-valued" and "real vector bundle" by "complex vector bundle", but it does not hold if one replace the field by a totally disconnected field like the rational numbers.