Serre–Swan theorem

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In the mathematical field of topology, the Serre–Swan theorem is a result relating vector bundles on a compact Hausdorff space to projective modules.

Let X be a compact Hausdorff space and Vec(X) be the category of complex vector bundles over X, and let ProjMod(C(X)) be the category of finitely generated projective modules over the C*-algebra C(X). There is a functor Γ : Vec(X)→ProjMod(C(X)) which sends each complex vector bundle E over X to the C(X)-module Γ(X,E) of sections. The Serre–Swan theorem asserts that the functor Γ is an equivalence of categories.

[edit] References

• Karoubi, Max (1978), K-theory: An introduction, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, ISBN 978-0387080901 

This article incorporates material from Serre-Swan theorem on PlanetMath, which is licensed under the GFDL.